- Email: [email protected]
Low-energy hadron physics from effective chiral Lagrangians with vector mesons
LOW-ENERGY HADRON PHYSICS FROM EFFECTIVE CHIRAL LAGRANGIANS WITH VECTOR MESONS
Ulf-G. MEISSNER Center for Theoretical Physics, Laboratory for Nuclear Science and Department of Physics. Massachusetts Institute of Technology, Cambridge, MA 02139, U.S.A.
I
NORTH-HOLLAND
-
AMSTERDAM
PHYSICS REPORTS (Review Section of Physics Letters) 161, Nos. 5 & 6 (1988) 213—362. North-Holland, Amsterdam
LOW-ENERGY HADRON PHYSICS FROM EFFECTIVE CHIRAL LAGRANGIANS WITH VECTOR MESONS Ulf-G. MEISSNER Center for Theoretical Physics. Laboratory for Nuclear Science and Department of Physics. Massachusetts Institute of Technology, Cambridge, MA 02139, U.S.A. Received October 1987
Contents: 1. Introduction and motivation 2. The massive Yang—Mills approach 2.1. A minimal chiral Lagrangian with spin-I mesons 2.2. Inclusion of non-minimal terms for the ~rpA 1-system 2.3. A subtracted Lagrangian with Vector Meson Dominance 2.4. A Lagrangian of pseudoscalar and vector mesons 3. The hidden symmetry approach 3.1. The p-meson as a dynamical gauge boson 3.2. The w-meson as a dynamical gauge boson 3.3. The non-Abelian anomaly in the framework of the hidden symmetry approach 3.4. Extensions of the hidden symmetry (axial-vector mesons) 4. Equivalence proofs 4.1. The Stiickelberg construction 4.2. Generalized proof in the presence of external gauge bosons 4.3. Gauge symmetric constraints to eliminate the A 5. Connection to QCD 5.1. QCD in the large N~limit and microscopic derivation of the pion decay constant 5.2. Vector mesons and QCD 5.3. Bosonization of the Nambu—Jona-Lasinio model 6. Baryons from the effective Lagrangian
215 220 220 231 235 239 242 243 247 250 255 257 257 260 264 266 267 272 279 284
6.1. Nucleons and ~-isobarsin the irpw-system 6.2. The electromagnetic structure of the nucleon 6.3. Axial properties of the nucleon 6.4. Meson—nucleon form factors and the nucleon— nucleon interaction 6.5. The baryon spectrum 6.6. Complete vector meson dominance? 6.7. Results from the ~rpA1w-system 6.8. NN-annihilation and the H-dibaryon 7. Further developments 7.1. The QCD trace anomaly and scalar particles 7.2. Nuclear binding and scalar particles 7.3. Comments on the derivative expansion 7.4. Remarks on vector mesons coupled to two-phase models 8. Summary and outlook Appendices A. Non-linear realization, PCAC and VMD B. p-mesons in the Skyrme model C. Calculation of the moment of inertia and the electroweak currents in the ~rpw-system D. Some remarks on the SU(6) approach References Note added in proof
285 294 302 306 314 317 318 324 327 328 331 333 337 339 341 346 348 354 355 362
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U. -G. Meijiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
215
Abstract: We review the use of effective chiral Lagrangians with vector mesons in the description of low-energy hadron physics. The massive Yang—Mills approach (vector mesons as heavy gauge particles of the Yang—Mills type) and the hidden symmetry scheme (vector mesons as dynamical gauge bosons) used to enlarge the field space beyond the Goldstone pions are discussed in detail, with particular emphasis on the gauged Wess—Zumino action and Vector Meson Dominance. It is demonstrated how all parameters can be fixed in the meson sector, and in particular, that there are only a few. The equivalence of these two seemingly different approaches is proven. The QCD generating functional is investigated and its connection to effective theories of spin-0 and spin-i mesons is shown. Baryons arise as topological solitons, with all parameters fixed in the meson sector. We present in great detail the phenomenological aspects of nucleons such as their static properties, electromagnetic, axial and strong form factors. The close connection to boson-exchange is elucidated, and the application to NN-physics is stressed. We furthermore discuss the inclusion of scalar particles and their relevance to OCD as well as the problem of nuclear binding. Further developments are sketched.
1. Introduction and motivation Low-energy hadron physics down to length scales of 0.5 fm has been rather successfully described in terms of mesons and baryons in the sixties [1.1, 1.2, 1.3]. Hadrons were used as elementary fields in effective local theories, namely chiral effective Lagrangians. With the advent of Quantum Chromodynamics, QCD, most of the notions developed in that era like e.g. Vector Meson Dominance, boson exchange, lost their importance or even disappeared completely. The interest in effective meson theories has been revived in the early eighties, when it was shown [1.4, 1.5] that QCD in the limit of large number of colors, N~,reduces to a nonlinear, effective theory of infinitely many weakly interacting mesons. Baryons emerge as topological soliton solutions of the mesonic Lagrangian, so their properties are intimately tied to the parameters of the meson sector. To allow for quantitative predictions, one obviously has to truncate the number of meson fields. Since we are interested in hadron physics at energies below 1—2 GeV, a reasonable approximation is to restrict ourselves to the low mass mesons. The power of the effective Lagrangian approach lies in the fact that it embodies the symmetries of the QCD Lagrangian and the structured vacuum. To first order, one simply works on the field space of pseudoscalar Goldstone bosons, JPC = 0~, related to the spontaneous breakdown of chiral symmetry, SU(Nf) ®SU(Nf) to diagonal SU(Nf)V. Here, Nf is the number of flavors, we will mostly restrict ourselves to the non-strange sector of u- and d-quarks, SU(2). The pertinent Goldstone bosons are the pions, with their interactions at low energy described by the non-linear if-model [1.6, 1.7, 1.8] leading to the successful predictions of current algebra. As first pointed out by Skyrme [1.9, 1.10] this theory allows for stable soliton solutions when one adds a higher order term in derivatives of the pion field. These solitons resemble very much nucleons, i.e. they carry a conserved topological charge, they interact strongly and they have a rich quantum sector (like the nucleon excited states). Witten [1.11] demonstrated the link of this model to QCD by investigating the underlying symmetries and connecting the anomalies of QCD to the Wess—Zumino term [1.12], which renewed the interest in effective chiral models (for reviews on the Skyrme model, which operates with pions only, see ref. [1.131). It soon became clear that a more accurate description of baryon properties at energy scales below 1—2 GeV could only be achieved if one enlarges the mesonic field space to incorporate other mesons, especially the vector mesons p, w and A1. A detailed study of these effective theories including .
. .
,
pions and vector mesons will be the main topic of this review. We will show how to make contact to concepts like Vector Meson Dominance and current algebra, therefore combining physics of the sixties with physics of the eighties. To make our motivation stronger, let us focus on some properties and facts concerning the vector mesons. The existence of the vector mesons became first obvious in the study of the electromagnetic structure of the nucleon. The w-meson, a resonant three-pion correlation, was introduced by Nambu
216
U. -G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
[1.141to explain the apparent discrepancy in the proton and the neutron charge radii. The dispersion theoretical analysis of the proton charge form factor lead Frazer and Fulco [1.15] to the assumption of a resonant two-pion correlation, the p-meson. In analogy to electromagnetism, Sakurai [1.16]conjectured that the conservation of isospin and hypercharge could be explained through the coupling of hadronic vector particles via the Yang—Mills mechanism [1.17]. These two seemingly different aspects of electromagnetic and hadronic interaction were unified in the framework of the Vector Meson Dominance model (VMD) [1.1], according to which the exchange of vector mesons dominates the electromagnetic and final state interactions of hadrons [1.18, 1.19]. In a more field theoretic language, the VMD hypothesis leads to the current-field identity [1.20] ~
p~(x)- ~
w~(x)-
~-
~(x)
(1.1)
for flavor SU(3). In the language of QCD, eq. (1.1) can be interpreted as follows: A photon materializes in a q~pair propagating as a vector meson, which then couples to the quarks inside the target hadron. All the complexities of QCD, namely gluon self-interactions and confinement, are incorporated in the physical vector meson intermediate state. An important test concerning the applicability of VMD is to compare the coupling constants g~,g,~, obtained from direct (leptonic) reactions like e~e—~V (q2 = m~)with indirect determinations via photoreactions like -yN-—~VN, ~yN—~-yN,...with q2 0 and E~ 2—20 GeV. From the leptonic decays one finds g~= 5.67, g(~,= 15.18 and g 4 = 11.92, whereas photoreactions give g~= 5.53, g(,, = 15.81—19.38 and g4 = 17.06—23.12. For comparison, broken SU(3) with ideal mixing gives g~= (1I3)g(,~= (\/~I3)g4. The VMD model, furthermore, allows to parametrize the form factors of baryons and mesons in terms of vector meson poles [1.21], as a pertinent example us consider theofelectromagnetic form factor of the pion. In the 2 <0, with q theletfour-momentum the incoming photon), VMD predicts space-like region (q F,~(q2)= m~/(m~q2), (1.2) .
. .
—
—
whereas in the time-like region (q2 > 0) the form factor is determined by the p-meson resonance at 770 MeV; for q2 > 4m~one also has to take into account the finite width of the decay p—~‘rri~via m F,~(q2)= 2 2 2 m~—q —im~f(q)
(1.3)
with F~(m~) = 153 MeV. As shown in fig. 1.1, the VMD model describes the data at momentum transfer q2~~ 1 (GeV/c)2 within 10—20% accuracy. Of particular interest is the slope of the form factor F~,(q2) at q2 = 0 which gives the pion charge radius. From eq. (1.2), we find (r2)~2= \/‘~Im~ = 0.63 fm close to the experimental value of Kr2)~/2= (0.66 ±0.01) fm [1.22]. This means that although the pion is an elementary, point-like particle in the underlying effective theory, it is seen by the photon as an extended object through the coupling to the p-meson. VMD also determines the hadronic and radiative decays of vector mesons. Durso [1.23] has recently re-examined these decays and shown that with the inclusion of small symmetry-breaking terms, the VMD assumption indeed works quite well for the radiative decays of vector and pseudoscalar mesons (cf. table 1.1). Matching VMD and soft-pion predictions for given scattering amplitudes implies a set of consistency relations of which the most famous one is the KSFR relation [1.24] a = m~Ig2f = 2 (aexp = 2.13) from S-wave ‘rrN-scattering. In
U.-G. MeiJ3ner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
217
b_____________________________________ PION FORM FACTOR IN THE TIME LIKE REGION
a
—
I
______
I
I
2)
FORM FACTOR F~(q ~ CERN NA7
PION
6
~ORSAY ~ NA7
~ CEA ‘7
//\ \~
—4.
-
p17701
05
SPACELIKE
I
-0.5
-0.4
I
—0.3
-0.2
TIMELIKE
-
0
0.1
0.2
~~°~-°/ I
4m,,
I
-0.1
2
0 4m~,0.2
0.3
q2 [Gev2]
-
I
I
0.4 0.6 qz[GevZ]
0.8
Fig. 1.1. The pion charge form factor F~(q2)in the space-like (a) and the time-like (b) region. In the time-like region, the form factor is dominated by the p-meson. The solid line gives the prediction of the VMD model, which is close to the data for momentum transfer q ~ 1 GeV/c.
Table 1.1 Radiative meson decays. VMD gives the prediction of a Lagrangian incorporating Vector Meson Dominance with small SU(3) breaking terms [1.23] in comparison to the empirical data. The first three decays were used to fix the parameters. Width (key) Decay
VMD
Experiment
w—~ry
888.0 6.1 63.0 5.4 55.0 12.5 119.0 0.24 149.0 68.0 71.0
861.0±56.0 5.9±2.1 71.0±8.0 3.0±2.0 55.0±15.0 8.1 ±8.8 87.0±20.0
4—*~iry
p—~r~-y
11’—+ory
K*o~_~.KOl
—
118.0±8.1 57.0±5.0 51.0±10.0
general, we can say that VMD works within 20% for the SU(2) sector (p, o) and somewhat poorer for SU(3). It therefore seems vital to include this concept in any effective meson Lagrangian which is supposed to describe low-energy hadron physics. Another example showing the relevance of mesons is the description of the nucleon—nucleon interaction in semi-phenomenological boson-exchange-models [1.25, 1.26]. Following the work of Yukawa [1.27] which describes the long-range interaction between two nucleons through one-pion exchange (cf. fig. 1.2), the inclusion of higher mass mesons and resonant multi-pion states has led to a very successful description of low-energy NN scattering data. The p-meson shows up at intermediate
218
U.-G. MeiJiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons IS
NUCLEON-NUCLEON POTENTIAL
150
~.1OO
50
~
-50
-100 I
0
I
1.0
I
I
2.0
r[fm] Fig. 1.2. The nucleon—nucleon interaction in the ‘S
0 channel. For large separations, the one-pion exchange dominates. In the intermediate range, correlated two-pion exchange gives the attraction between two nucleons. At smaller distances, the potential is very repulsive, with a large part of the repulsion coming from the exchange of w-mesons.
distances, especially in the isovector tensor potential, where it cancels part of the pion tensor potential. A good part of the short-range repulsion between nucleons can be understood in terms of w-meson exchange. The intermediate-range attraction in the central potential is due to correlated two-pion exchange in the scalar—isoscalar channel [1.281as indicated in fig. 1.2. The notion of boson exchange will appear naturally in the framework of the effective Lagrangian to be constructed here. From these considerations it seems that low-energy hadron physics can be described quite satisfactorily in terms of mesons, when one incorporates the underlying symmetries and anomalies of QCD, not quarks and gluons as dictated by QCD. Further indications of this are the success of the shell model and meson exchange currents. The main assumption of the very successful nuclear shell model is that nucleons move as independent particles inside the nucleus. In terms of quarks and gluons, one expects quarks to be confined within volumes of —4 fm in radius [1.29]. On the other hand, the average distance between nucleons in nuclei is d 1.8 fm, which would imply that the quark cores would overlap frequently. How can an independent-particle picture as confirmed by experiments through mapping 3He out single particle emerge? The electro-disintegration of the deuteron d(e, probe e’)pn and the magnetic formdistributions factor are the best examples of meson exchange currents, which momentum transfer up to l~I 1 GeVIc [1.30]. Again, one would expect that the meson picture breaks down at these length scales and explicit quark degrees of freedom are needed, but again this is not the case. So meson theory works down to length scales considerably smaller than A~D 1 fm, which is not expected in view of the large electromagnetic radii of the hadrons, e.g. r~roion 0.85 fm and r 10~ 0.7 fm. What we are going to demonstrate is that the relevant degrees of freedom which govern the strong interactions of hadrons at energies up to —1 GeV are indeed mesons, with their interactions dictated by chiral symmetry and the anomaly structure of QCD at low energies, not quarks and gluons. Does all this mean that there are no quarks? This certainly would be the wrong conclusion. What it really means is that there is a beautiful correspondence principle between the meson and the quark P—
U. -G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
219
language, with its theoretical justification given by the so-called “Cheshire Cat” principle (CCP) [1.31]. This principle states that a theory of quarks and gluons chirally coupled to mesons is equivalent to a chiral meson theory. * Having said all this, the reader might wonder whether inclusion of more meson fields beyond the pions in the effective Lagrangian does not proliferate the numbers of tuneable parameters, therefore limiting the predictive power of this approach. This is indeed not the case. In section 2 of this article, we demonstrate how one can make use of linear representations of chiral symmetry and introduce the vector mesons as heavy gauge particles. We will show how the Wess—Zumino term links a variety of 2y, w—* 3’rr,... and therefore predicts most of the coupling hadronic and radiative decays like ~ constants in terms of the gauge couplings g related to the gauge group we are considering (e.g. SU(2)L®SU(2)R®U(l)v for p, A 1 and w). In the non-anomalous sector, one has to go beyond the minimal form of the effective Lagrangian as dictated by chiral symmetry to cope with problems like the irA1-mixing and the A1 parameters. To circumvent the latter problems, one can also use a non-linear realization of chiral symmetry a la Weinberg [1.32], this approach is nowadays called the hidden symmetry scheme. The vector mesons arise as dynamical gauge bosons of a hidden symmetry of the non-linear if-model, with the gauge principle again limiting the number of meson parameters3)flavor quite drastically. We discuss in detail how the pand w-mesons fit in this scheme and extensions to SU( as well as axial-vector mesons. In section 4, we prove the equivalence of both of these schemes (massive Yang—Mills and hidden symmetry). This equivalence proof puts further constraints on the seemingly arbitrariness how to introduce vector mesons in the effective theory. Section 5 deals with the question of how these chiral Lagrangians are related to the generating functional of QCD. For that, we investigate the QCD partition function assuming spontaneous breakdown of chiral symmetry and show that indeed vector meson Lagrangians of the form discussed in sections 2 and 3 can emerge. Section 6 gives a detailed analysis of the baryon phenomenology as predicted by these models. We show how to construct soliton solutions, which have the hedgehog structure exhibited in fig. 1.3, quantize them as fermions and give results for static properties, electromagnetic and axial form factors, strong form factors, the baryon spectrum and nucleon—antinucleon annihilation. In section 7, we comment on
THE DEFENSIVE
HEDGEHOG
(1) Al
Fig. 1.3. The defensive hedgehog in the presence of vector and scalar mesons. The classical soliton solution to the effective chiral theories discussed in this review can be visualized with the familiar hedgehog pion field (radial in (iso)space) with a core of o-mesons and p-, A 1-mesons (projected along the i direction). S denotes any scalar meson or glueball (see section 7). *
This is a very simplified version of the CCP.
220
U. -G. MeiJlner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
further developments, in particular, the inclusion of scalar particles. The motivation for including scalars is twofold: first, scalar glueballs are the Goldstone modes of broken scale invariance and, second, the nuclear binding discussed before can be described by the exchange of scalar mesons. Comments on the derivative expansion and chiral two-phase models are also made. The appendices contain some technicalities and some more speculative ideas, as well as developments related but not in direct line with the ones described before. For the sake of completeness, let me mention that I have only included material which was available for use by the end of May 1987.
2. The massive Yang—Mills approach In this section, we will demonstrate how one can construct a chiral Lagrangian of spin-0 and spin-i particles, with the latter treated as massive Yang—Mills bosons* of chiral U(Nf)L 0 U(Nf)R symmetry, eventually broken down to chiral SU(Nf) (with N~denoting the number of flavors). These ideas originated twenty years ago in the description of mesons coupled to point-like nucleons, and were most strongly advocated by Sakurai [2.1] and Schwinger [2.2]. For a review on most of the older ideas and phenomenology we refer the reader to ref. [2.3]. With the advent of Quantum Chromodynamics (QCD), these ideas were mainly forgotten, but by now it has become clear that QCD at low energies can most economically be described by effective Lagrangians which embody the relevant symmetries of QCD (especially chiral symmetry). Following the work of Schechter and collaborators [2.4—2.7]we will show how the anomaly structure of low-energy QCD at low energies can be described in terms of the gauged Wess—Zumino action. * * This amounts to the construction of a Lagrangian of pseudoscalars and (axial) vector mesons. In the following subsection, modifications to this minimal ansatz will be discussed, due to the difficulties coming from the ~rA1-mixingwhich induces a strong momentum dependence of certain vertices. Then, following the work of Brihaye, Pak and Rossi [2.9—2.11],we show how one can implement the concept of Vector Meson Dominance (VMD) in such effective Lagrangians. A more general framework on the basis of differential geometry [2.12, 2.13] will be discussed in Appendix A (for the non-anomalous sector). Finally, we will show how one can eliminate the axial-meson A1, triggered by the fact that the A1 is not a sharp resonance (F(A1)Im(A1) 1/3) and that its parameters (mass, decay width, etc.) are not as uniquely established as the ones of the vector mesons p and w. Furthermore, the A1 plays only a minor role in nuclear physics, e.g. its contribution to the NN interaction is almost completely wiped out by the pir-continuum [2.14]. 2.1. A minimal chiral Lagrangian with spin-i mesons Here, let us start from QCD at low energies. The chiral model we will construct can be considered as based on an order-parameter multiplet Mab transforming like the quark-field configuration ~iRbqLa~ M condenses in the chiral phase, i.e. (Mab) ~ 0. It is most convenient to make a polar decomposition M = U H where H is a unitary matrix including the usual nonet of pseudoscalar mesons 4 (4 = * Literally, the term “massive Yang—Mills boson” does not make sense, what is meant is that symmetry-breaking mass terms have to be added by hand. ** Since the review by Zahed and Brown [2.8] covers this topic in an excellent fashion, we will be bnef on this point.
U. -G. MeiJiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
U=exp(2i4/\/~f1,)=exp(2i4ilF.~)
221
(2.1)
with F,~= = 132 MeV the weak decay Notice that U transforms in a so-called 3)L 0pion SU(3)R suchconstant. that under a transformation by unitary matrices non-linear representation of SU( (A, B), U becomes A U B’. The chiral Lagrangian for pseudoscalar interactions at low energies is nothing but the non-linear if-model [2.3] =
Tr(9~U c~~Ut) = ~ Tr(c9.~U ~Ut).
~
(2.2)
One can furthermore add a chiral-symmetry breaking term proportional to ~ ?IZaUaa, where the are the current quark masses. With that, the octet of pseudoscalar mesons (‘it, K, 11) will be described satisfactorily. The mass of the ‘q’ can be understood by introducing a phenomenological pseudoscalar glueball field to describe the U( 1) A anomaly [2.15—2.18].We will not go into the details of this point here. Similarly, the QCD trace anomaly can be modeled by the introduction of a scalar glueball field [2.19, 2.20], as we will discuss in section 7. In this section, we are particularly interested in the Wess—Zumino term [2.21] which describes the non-Abelian anomaly structure of QCD [2.8, 2.22]. Let us first discuss the non-anomalous part ‘~, of our Lagrangian. For that, we follow Sakurai [2.1], and introduce left- and right-handed spin-i fields A L~ and A R~ respectively, which are related to the vector and axial-vector mesons V~,and A, 2 by (V,2 = v; ~i’~h,A,2 = A~ra/V~) AL,2=~(V,2+A,2),
AR,2=~(V,2—A,2).
(2.3)
The couplings of these mesons to matter are introduced as gauge couplings, which is the most economical way to minimize the number of appearing parameters. This amounts to gauging the non-linear if-model [2.2]
~
Tr(D,~UDbLUt) DU= ô.,~,U—igA,2~U+igUA,2~ ~J,gauged
=
(2.4)
with D,2 the covariant derivative. g is the gauge coupling constant, for the normalizations used here, we have g = \/~f~, where f~,,is related to the p—÷‘rr’rr decay via 2
~
(m~_4m~)3/2
~
(2.5)
which gives (m~= 770 MeV, 1 = 153 MeV, m,~= 139.6 MeV) f~/4ir = 2.95,
*
f~
=
6.08.
We will specialize to the non-strange sector of u- and d-quarks later on.
(2.5a)
222
U. -G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
In the derivation of eq. (2.5), a minimal momentum dependence of the pint-vector was assumed, as we will see in what follows. The difference to a gauge theory stems from the necessity of introducing vector and axial-vector meson mass terms. Therefore, the pertinent Lagrangian involving spin-i and spin-0 mesons takes the form: Tr(D,2 U D~Ut)+ ~
~
=
+
Tr[M(U + U~—2)]
—
~ Tr(F,2VLF~-I- F~RF~)
m~(A,2LA~+ A,2RA~)— i~Tr(D~UDOUtF~+ D~LUtD0UF~).
(2.6)
The last term in eq. (2.6) is a so-called non-minimal term since it is of higher order in derivatives than the other terms in eq. (2.6). Notice that at this level the vector and the axial-vector mesons have the same “bare” mass m0. The corresponding field strength tensors are given by F~ = ~
— ~AL~R
~ig[A~”~, Ar].
(2.7)
The mass matrix M appearing in eq. (2.6) can be written as M
~(m~ + ~m~)i
=
—
(2/1/~)(m~ — m~)A8
(2.8)
in terms of the pion and kaon masses m,~and m~,respectively. The quadratic piece of the Lagrangian (2.6) is given by [2.23] 2— ~2)
=
~ —
~
m
Tr(ô,2~— ~F,,gA~) ~Tr(F~F’~” + FA
14~ Fb0~A)+
~
Tr(V~+ A~)
(2.9)
in terms of the vector and axial-vector fields. Obviously, (2.9) is non-diagonal in oI,,,4 and A,2, i.e. the pseudoscalar and axial fields mix. To get rid of the A~~4-vertex, we introduce physical pseudoscalar and axial-vector fields 4~iand A via ~
2m0
FZ’F~, (2.10)
2F~/4m~) = (1 + g2F~J4m~)~. (1 — g This is the conventional diagonalization scheme. Since also terms of the type 3 -
=
1~[V,2, cb] appear, one
can generalize the definition of the axial field A,2 by substituting o~by V,~= + i(g/2)[4, V,2]. We will come back to this point in subsection 2.3. Now, the quadratic part ~2) reads ~2)
=
~ Tr(ô,2~e9~4) ~~
+
—
2)m~ ~ Tr(V~)+ im~Tr(A~)— with physical masses given by +
=
m~IZ2,
th~= m~/Z2,
(2.11)
~ (4~IZ
m,~.= m~= m~,
m~= m~,IZ2.
(2.12)
U. -G. Meiflner, Low-energy hadron physics from effective chiral Lagranglans with vector mesons
Of particular interest is the mass difference of the A1- and the p-meson, 2~2 m 2 m~= ~-~--~ (—~)
223
(2.13)
—
which arises due to a partial Higgs mechanism. If the vector meson mass term would not have been present in (2.6), the ordinary pion would be completely eaten up by the A 1 this underlines the crucial manner in which the present Lagrangian differs from a true gauge theory. Notice that (2.13) has originally been derived by2 Weinberg using sum rule techniques [2.24]. F~=the 132KSFR MeV, = 0.46, i.e. m~ = 1135 MeV~ ~ Notice With that gZ2= =8.60, ~ gives m0 = 770 MeV, we find Z relation [2.25]. The part of the Lagrangian of third order in the meson fields and A,2 yields the VJxf and AV4 couplings —
=
~AV4
~ Tr(V~4 9,2~)+ Tr{ V,L[A,2,
=
+
Tr[(ô,2VV
~]}+
—
c9~V,2)ô,24ô~,4],
[(1—8)Tr{(d~V~ ô~V~)[A,2,~]} —
Z2 Tr{(9~A~ — dVA,2)[V,2,
8 = 1 — Z2
—
(2.14)
i9~]}],
2Z4~g/(1— Z2)
where one integration by parts has been performed. From (2.14) we can now read off the pint and pA 1i’i transition amplitudes as follows (from now on we omit the tilde on the physical fields and constants): ““
2
1
2
my
2
my
m m~-~m~]’
~ 2
x
~
@
~)
1/2
2
(i+2g~m~m~)m~~ 2
1/2
~ Defining the p—* 2’rr and A1
(2.15)
~
—~
pit transition matrix elements via
(it.np)nsg~~p,2•(4~X (2.16) + hA~p,,r
~
qA~eA,
q~,
where the r’s are the relevant polarization vectors of the p and the A1, we can read off the coupling constant g~~1-7~, gA1P~and hA~p,~ as ~
=
(gI\h)(~+ ~
224
U. -G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
=
m2—m2\
gm
/
~
I~1 2g~ 2m~ —
.11),
(2.17)
=
Notice that for ~ = 0, the pint-coupling is decreased by a factor 3/4 and that there is no d-wave contribution to the amplitude A 1 —~pit. Since we want to have a minimal momentum dependence of ~ so that F(p—* 2’rr) Unfortunately, is given by (2.7)this andchoice g = \/~g~~ = ~a too small we choose 2g~= ito recover the eq. 7~fT. of ~gives A (2.4) with g = ~ 1-width, with m~= 1275 MeV* we find from
~A1~T = 41Tm~ [~ ~ + ~ (~ gA1P~T+ ~I q~2hA)] (2.18) that 120 MeV, much smaller than the experimental value FAP,, 350 MeV. We will return to this point in the following subsection, for the moment let us assume that we work with ~so that the p ‘itt decay width is given in its minimal form (2.4). This is consistent with the universality of the pint and pNN-couplings [2.1]. Let us now discuss the anomalous sector, i.e. the vector meson processes connected to the gauged Wess—Zumino term. For that, let us briefly remind you of the Wess—Zumino term, for more details see ref. [2.8]. In terms of the differential one-forms —~
L
=
(~9,2U)Ut~
=
(dU)Ut (2.19)
R= UtdU= UtLU where L and R transform as nonets under left and right U(3)-flavor groups, respectively, the WZ action can be written as [2.22] 5]
=
Cf Tr[R5]
(2.20)
F~~(U) = Cf Tr[L with ô M5 = M4, i.e. the boundary of the five-dimensional manifold M5 is ordinary Minkowski space M4. The constant C has been determined by Witten [2.22] by calculating the anomalous process it°—* 2y, it follows to be C = —iN 2, with Nc (=3) the number of colors. The physics of F~becomes most 0/2401T expansion L = (2i/F,~)d4 +..., then transparent in the weak-field T~~= ~
(2.21)
where we have used Stokes’ theorem and Poincaré’s lemma (d2 = 0). Physically, F~ accounts for processes like KK—*3’ir and removes the spurious symmetries of the non-linear if-model (2.2). (Unfortunately, the reaction KK—3 3’it is not known experimentally.) Stated in another way, the WZ action contains all processes involving the four-dimensional Levi—Civita tensor e~’3.If we now think ~f *
The experimental situation concerning the A,-parameters is discussed in the following subsection.
U.-G. Melliner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
225
introducing the spin-i particles, it becomes clear that all abnormal parity processes (processes which do not conserve intrinsic parity*) will be described by the gauged WZ-term. Some typical examples are the decays o—* 3i’i and K*_~*Kinit, for which experimental information exists. To clarify the gauging of the WZ action, we will first consider electromagnetism, i.e. the variation ~U=ie[Q, U]
(2.22)
where Q is the SU(3) charge operator changes by
Q
=
diag( ~,
—
~‘,
—
~ Under the local variation (2.22), F~
4 — R4)] aF~~(U) = 5Cif dr Tr[Q(L _5Cif d{dr Tr[Q(L3
=
+
R3)]}
_5Cif dr Tr{Q[L3 + R3]}
=
(2.23)
making use of the fact that even powers of L and R are exact forms, i.e. 0=dL—L2=dL3—L4=dR+R2=dR3+R4. Furthermore, we have used Stokes’ theorem and the fact that M4 = 9M5. Let us now introduce a gauge field A,2 and the corresponding one-form A = A,2 ~ with the transformation property (e = 1) = dr. We consider F~’~(U, A)
F~~(U) + 5CiJ A Tr[Q(L3
=
+
R3)]
(2.24)
The variation of F~’~(U, A) is ~F~1~(U,A)
=
5Cif A Tr[Q6(L3
=
iOCf d~A Tr[Q2(L2 — R2) +
+
R3)]
Q d(Ut)Q dU].
(2.25)
The quantity inside the trace is a closed form, thus it can be written as an exact form L2—R2=d(L+R), (2.26)
Q d(Ut)Q dU *
=
d[aQUtQ dU — bQ d(Ut)QU],
The intrinsic parity of a particle is defined as follows: it is
pseudotensor, e.g.
is, ~y, p
+1
if a particle transforms as a true tensor of that rank, and —1 if it transforms as a
and A 1 have intrinsic parity —1, +1, +1 and —1.
226
U. -G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
where a, b are constants satisfying a ~F~1~(U,A)
=
+
b = 1. Integration by parts allows us to rewrite ~FW(U, A) as
iOCf dr dA Tr[Q2(L + R)
+
aQUtQ dU
—
bQ d(Ut)QU].
(2.27)
We end up with a gauge invariant expression when we consider F~21(U,A)
=
F11~(U,A) — 1OC
A dA Tr[Q2(L
+
R)
+
aQUtQ dU
—
bQ d(Ut)QU].
(2.28)
SM5
This is the desired gauging of the WZ-term. Introducing r = a — b, and putting pieces together, we find 1~21(U,A)= F~~(U)+5Ci J ATr[Q(L3
—
1OC
f
+
R3)]
SM5
A dATr[Q2(L
+
R)
+
~QUtQ dU - ~QUQ d(Ut)]
SM5
—
5Ci J dA dA Tr[QUtQU].
(2.29)
SM5
Notice that the last term proportional to r is gauge-invariant by itself, it therefore does not affect the gauge invariance of p(2)~ But since dA dA = ~d~xE’~’~F,2pFap is not parity invariant, we must choose r = a b = 0, i.e. a = b = 1/2. Let us now discuss the gauging of non-Abelian subgroups of U(3)L®U(3)R. We will again use the trial and error method advocated by Witten [2.22] and demonstrated above. * There will always appear a gauge invariant term which is multiplied by an arbitrary coefficient in addition to the terms whose variation reproduces the anomalies. This additional term should be dropped if one wants to represent the anomalies in an irreducible way. For an arbitrary subgroup of U(3)L 0 U(3)R, the relevant group parameters are —
rLR
= ~
iELR/k/2
with A*/2 the generators of U(3). In terms of left- and right-handed gauge fields, AL and AR, we have the following transformation properties: U—*exp{irL}U exp{—ieR} ML
*
=dEL
~[rL,
AL],
~AR
=drR +~[ER, AR].
A more direct way of integrating the WZ anomaly equation can be found in refs. [2.26, 2.27]; see also section 3.3.
(2.30)
U. -G. Meifiner, Low-energy hadron physics from effective chiral Lagrangian., with vector mesons
227
The trial and error gauging procedure outlined before gives the following result F~~(U, AL, AR)
3 F~~(U) + 5Cif Tr[ALL
—
+
ARR3]
5CJ Tr[(dALAL + AL dAL)L +~dA~AR + AR dAR)R]
+5Cf Tr[dALdUARUt—dARd(Ute)ALU]
+
5Cf Tr[ARU~LUR2 - ALUARUtL2]
+
2.5Cf Tr[(ALL)2
—
(ARR)2]
+
5Cif Tr[A~L + A~R]
+ 5Cif Tr[(dARAR + AR dAR)UtALU
+
5CiJ Tr[ALUARU5ALL
+
—
(dALAL + AL dAL)UARUt]
ARUtALUARR]
+ 5Cf Tr[A~UtALU - A~LUARU+ ~(UARU~L)2]
-
5Crf Tr[FLUFRUt] (2.31)
with FLR = dALR — iA~R.The last term in (2.31) with the arbitrary parameter r disappears if the gauge group is such that one can define a parity operation A~(x)*-+ AR,2(—x), U(x) ~-* U1(—x). All terms in (2.31) are parity invariant apart from the last one, thus r = 0 should be taken. Of course, the variation of FWZ(U, AL, AR) under a gauge transformation is nothing but the Wess—Zumino anomaly equation ~
AL, AR)= _iOCiJTr{EL[(dAL)2_ ~idA~]—(L~R)}.
(2.32)
The WZ action is invariant under global U(3)L 0 U(3)R. It also contains the gauge coupling g via rescaling the fields ALR gA~~ and its vector meson content can be expressed by using (2.2). Now, let us assume that the electromagnetic interactions at low energies are dominated by the neutral vector mesons [2.1, 2.28], i.e. —*
~‘em
=
(\ e/g)A~[m~p~°~ + (1/3)m~,w,~ — (V~/3)m~,~]
for the nonet ansatz p°—— (i/V~)(u~ — dd), w
-~(1 /V~)(ut~ + dd),
(2.33) ~ -~s.f, and that other hadrons
228
U. -G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
interact with the electromagnetic field A,2 through their couplings to the vector mesons (vector meson dominance). In this scheme, the decay i’i°—*2-ycan be calculated from (2.31) via ‘rt°—*wp°-—*2-y.The it°p°w-vertex is given as a piece of F~~(U, AL, AR). Defining the appropriate Lagrangian density as gVV
=
with
p0123 =
4~~
Tr(~~V~9aV~)
(2.34)
+1, the relevant coupling constant gvv4 follows to be (2.35)
2/i6it2F~)(1 g2F~/6m~). —
~vv~ = (3g This leads to 22
8~ag~~
0
F(i’i —*yy)=--~jwith a
=
4~ m,,
(2.36)
1/137. Comparing with the current algebra result
F(it°—*2y)=
(2.37)
a2
gives 2/16ir2F~. (2.38) ~vv~ = 3g Now gvv 4 in (2.38) is roughly 1.6 larger than gvv4 as predicted from (2.35), i.e. F(’rt°—.*2-y) as calculated from (2.36) comes out almost a factor three too small. This result is not surprising since the WZ action (2.31) was constructed in a left—right symmetric way, i.e. it reproduces the left—right symmetric form of the WZ anomaly equation (2.32). As first pointed out by Bardeen [2.29], the anomaly sits entirely in the axial-vector current, and the vector current is conserved. By a suitable addition of counterterms, one can always shift the anomaly in one given sector [2.30]. In the form (2.31), both the vector and axial-vector currents are anomalous. Let us therefore consider the following counterterm
=
5Ci
J
Tr[(dARAR
+ AR dAR)AL
—
(L~R)]+ 5Cf Tr[A~AL A~AR+ —
(2.39)
M
The variation of the Bardeen subtracted WZ action becomes ~(F-I~) 4— ~(F~A2+ AF~A+A2F~)]}, =
—3OCif Tr{(EL— ER)[4FV
+
—
~A
(2.40)
U. -G. Melfiner, Low-energy hadron physics from effective chiral Lagranglans with vector mesons
229
where F~= dV
~i(V2 + A2),
+
FA
=
dA + ~i(AV + VA)
(2.41)
with A and V defined in (2.2). For vector gauge transformations (EL = ER) the anomaly in Bardeen’s form vanishes. Of particular interest is the fact that the Bardeen counterterm I~(2.39) can be obtained from the gauged Wess—Zumino action (2.31) via I~=FwjU=~, AL, AR).
(2.42)
Therefore, the effective action which reproduces the Bardeen form of the anomaly is F~~(U, AL, AR)= F~~(U, AL, AR)
—
F~~(l, AL, AR)
(2.43)
so that F~vanishes when U= ~fi[2.21]. Notice that the action (2.43) breaks the global U(3)LOU(3)R chiral symmetry down to U(3)~0 U(i)A. This is most obvious from (2.42), the special choice U = .1 clearly breaks chiral symmetry. Nevertheless, this breaking does not affact any of the usual currentalgebra theorems (which involve amplitudes independent of ~ )~ * Before discussing the meson phenomenology arising from (2.43), let us give the Bardeen subtracted WZ-term for the group SU(2)L®SU(2)R®U(l)v, incorporating the p-, A 1- and w-mesons most relevant to the non-strange sector. It reads [2.31] using Stokes’ theorem -
Tr[LpLaLp]
482 ~
EW~
—
~pTr[ARaLp ~ALaR~ +ARaU~AL$U~ARaALp],
(2.44)
with A~R=~(P,2+w,2±a,2), W,2
=lgw,211
,
P~
a
lgp,2r
a
,
.
bb
a,~=lga,2r
L,2=Ut~U and R,2=Uc9~,Ut. Integrating by parts and throwing away a total divergence, we find iN =
~2
iN EtLuaIJW,L
Tr[LpL aLp]
+ ~T
x Tr[pjR~ — L~)+ a~(R~ + L~)+ p~a~ ~(p~ —
—
aa)Ut(pp
—
a~)U].
(2.45)
Let us now consider some typical anomalous meson decays. First, notice that the counterterm I~exactly 3”If~,F”= I from the anomalous sector, but this is not a strong interaction process. *
It breaks the prediction eF
U. -G. Mcifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
230
cancels the second term in the VV4-coupling constant (2.35) and hence we recover the current-algebra theorem for i’i°—* 2’y. Therefore, we will now use the effective action (2.43) to calculate other processes and obtain some predictions which indeed agree with the data. The most well-known abnormal parity hadronic process is the decay o—* 3’rr. Actually, it was motivated by the success of the Gell-Mann— Sharp—Wagner model [2.32] of pole dominance, i.e. w—* p’ir—* 3i’t, to which later on a possible contact term was included [2.33], which contributes very little to the width F(w—* 3’ir). In our framework, both these terms arise naturally, with their (relative) coupling strengths completely determined by the gauge coupling g. For the special choice of ~ to have the minimal form of the V~4vertex, we have 2
a/S
~VV4
=
=
gvv4r,20~pTr[a V 8 V 4}, (ig/2) Tr[V~
=
—
J~
=
22
3g /l6it F,,,
~L~]
~hE~pap Tr[V~ 8’çf 2i~2F~, t g
gvv4
a~a~4],
(2.46)
2F~]~ 3 [g2F~]2 43 L~’~i 32 L m~] [g
The first two vertices are needed for calculating the pole diagrams w—* ‘itp—* ‘it’rt’tt, the third one determines the direct (contact) o—*3’it interaction. For w,2(p)—~’rt~(q1) + ‘rt°(q°) + ‘tt(q) we can calculate the amplitude as M=ir~~Pgogg;F F=3h+
3~3 F,,
81T
The experimental width T(u—* ~t~’ii°i’t)
192~ where E~and E gives
2
(2.47)
q)2]F2,
(2.48)
a2
(p
— q
)
+
m~
follows to be
~ JdE~fdE
are the energies of the
F(u—*’ir~’it°’it’)=7.6MeV
~ a+.0,-
[(q)2(q~)2—(q~
~
and
‘it,
respectively. Evaluating the integral in (2.48)
(2.49)
for m,, = 140 MeV, F,, = 132 MeV, m~= 768 MeV, ma = 782 MeV, and g = 8.6. Experimentally, we have F(~—*3’tr) = (8.9 ±0.3) MeV, i.e. the prediction (2.49) is 17% too low. Notice that roughly 10% of the width F(w—* 3’rt) comes from the contact term ~ and that the contact term decreases the width. For comparison, the current algebra result for ‘ir°—*2-y (7.1 keV) is about 11% lower than the experimental value (7.9 keV). Furthermore, the left—right symmetric form of the WZ action would lead to F(w—* 3ii) = 2.6 MeV, due to the too small VV~-couplingconstant. Again, the importance of the Bardeen counterterm becomes clear. Let us now study a radiative process, the simplest one being the decay of a vector meson into a pseudoscalar and a photon, like e.g. w—~‘rt°-y.In our model, this process goes via the op’rt vertex, i.e. o—* p°’rr°and the p°couples to the photon via (2.33). Straightforward
U.-G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
231
Table 2.1 Radiative vector-meson decays from the Bardeen-subtracted Wess—Zumino action. Symmetry-breaking corrections FKI F_ F,/F,, = 1.2 are incorporated, the w—4 mixing is parametrized by = 0.076 as determined from the 4—* pis decay. The experimental data are taken from ref. [2.34]. Width (keV) Process
Prediction 80
Experiment 63.0±4
p°—*is°’y
80 29 117 800 38 77 5
63.0±4 51.0±5 75.0±35 789.0±92 725.0±14 93.1±25 3.2±2.6
7 68 1 5
8.4±27 67.7±9
K**~_~K*’t
4—011-y
—
6.5 ±1.9
evaluation of the pertinent Feynman diagram gives ~
641T
3=0.8OMeV F,, qj
(2.50)
where q,, = 380 MeV is the pion momentum in the w-rest-frame, in good agreement with the experimental value F(w—s~it°’~y) = (0.86 ±0.05) MeV. Other allowed SU(3) radiative decays like ~ — 41—s. ‘q-y,. . can also be calculated from the action (2.43), the results in comparison to the empirical data are shown in table 2.1. * The overall agreement is rather satisfactory. To discuss the hadronic and radiative decays of axial-vector mesons, we have to come back to the problem of the A 1-parameters as discussed following eq. (2.18). To satisfactorily describe p and A1 parameters, we have to add other non-minimal terms to the non-anomalous part of the Lagrangian. This will be done in the following subsection. .
2.2. Inclusion of non-minimal terms for the irpA 1-system As we have seen, the non-anomalous Lagrangian (2.6) could only describe the p-width properly at the expense of the A1 —s. pit decay. Therefore, it is necessary to add at least one more non-minimal term to cure this problem. Before doing so, let us remind you about the rather unsettled question of the mass and width of the A1-meson. Fifteen years ago, we had m~= 1070 MeV, F~= (30 ± 130) MeV and = (128 ±5) MeV, so the mass ratio mA1/mP \/~as predicted by Weinberg [2.24] was nicely fulfilled and the Lagrangian (2.6) with the free parameter ~ could explain these data. However, the 1986 edition of the Particle Data Tables [2.34] gives 1A
mA=(l275±28)MeV, *
For further details, see ref.
[2.61.
1
=(316±45)MeV,
[=(153±2)MeV,
U.-G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
232
i.e. mA/me = 1.66 V~, and the Lagrangian (2.6) can no longer accommodate the width of the p and the A1 at the same time. Recent data from ‘i-decays report [2.35, 2.36] m~= (1056 ±35) MeV,
TA
=
(476i~i~) MeV,
1A
mA
=(1097±i3)MeV,
1
=(378±40)MeV.
This confirms the large width, but brings the mass again down to the Weinberg ratio. To accommodate for the large A1-width, we follow the work of ref. [2.6] and add another non-minimal term to the Lagrangian ~ (2.6) t]. y Tr[F~UF~U The new term is clearly gauge invariant, y is a free parameter. The quadratic part of =
(2.51)
.~ +
=
~(i
—
—
y) Tr[(ô,2V~
— 8017,2)2]
—
~(i
+
‘y) Tr[(ô,2A,,
—
8
~(2)
reads
2] 0A,2)
~Tr[8~4 ~
+
—
gF,, 8/J~A,2+ m~A~ + ~g2F2,,A~ +...],
(2.52)
where 17,2 = A L,2 + A R,2 and A = A L,2 — A R,2 are the vector- and axial-vector fields. The Lagrangian (2.52) is diagonalized via the definitions V~= (1—
~=2~
)‘~2v
A,2 = (1 + )‘~2A +
F F,,=-~,
2m
0 2F21’’~2 [ g Z=[i_-~----~j
(2.53)
where the tilde denotes physical quantities. The vector and axial-vector meson masses follow to be: m~=-j-~-~--~~
(2.54)
For convenience, we will introduce a renormalized coupling constant ~ via —
g=g/(1—y) so that gV,2 =1
=
—
1/2
(2.55)
,
gV,2. With that, we can rewrite Z2 in terms of the physical quantities as
4m~
(2.56)
= (1)[m]2
i+y
m 5,
2 = 1/2 [2.251, while the second Notice be thatthetheWeinberg first equality corresponds KSFR forgeneral, Z would formula [2.24] for toZ2the = 1/2 andrelation y = 0. In these two relations do not imply each other. (For a somewhat different approach to this problem insisting on the KSFR relation
U. -G. Meijiner, Low-energy hadron physics from effective chiral Lagranglans with vector mesons
233
and the Weinberg formula, see ref. [2.37].) From (2.51) we can read off the V4141 interaction as: =
6=1
~ Tr[V~,4 8,241] 2
—
Z
2[ —
+
~ Tr[(8,2V0 2m~
—
80V,2)o,2~8~4l], (2.57)
Z4
1 1_Z2]~~
where a non-minimal piece proportional to 6 Tr[V,2
041,2~0] appears. The piti’i width derived from (2.57)
follows to be 3~2 F(p—s.it’it)j-~-— m~ [i_~] 1 Iqj
2
(2.58)
with q,, the pion momentum in the p rest-frame. The other vertex of interest is the AV41-vertex, it is given by ~AV~
1/2 g,, 11y1 Tr{V,2[A,2,~]} 4Z2 Li+~i
i~2fr,,
+
+
—
4m~,Z2(1
6
—
ii—,,] ~Lir~] Tr{(ô,2V 0 1/2
2 i~+yi” 4m~ Li] Tr{(ô,2A,,
~
2
4-
2
7 F,, [1~2j
—
—
80V’,2)[A,2’ 8,4~]}
ÔPA,2)[V,2,
~
(2.59)
11/2
Tr{(8~A,~ —
,2)(8,2VP —
Notice that (2.57) and (2.59) will be modified if one adds SU(3) symmetry-breaking pieces. For our present discussion, however, we simply assume that this breaking is adequately described by the physical masses of the various mesons. For further applications, let us write down the the transition amplitude A—.*V + 41, e.g. for the decay A~,2(p)—s.p~(k) + ‘rr°(q) T,2V(p, q) = V~[T~(p,q) 6,2~+ TD(p, q) ~ 11
1
~ pq—
L 1—yJ21
T 5(p, q)= —T ~ [m~+(1—6)kq]— 2 (2 (1 [—4y T~(p,q)=7—T----~(i—6) Li~2]
-‘1/2
4~2
11/2
1
F-,, p~k,
(2.60)
1 ~
where we have exhibited only the 5- and D-wave amplitudes which contribute to the A1 —s. pit decay width as follows F(A1—s.pit)2
q,,I2 ITmA
12T2+1(EP ~
(2.61)
234
U.-G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
We are now able to simultaneously fit the mass of the A1 as well as the p and A1 decay widths, since we have three parameters g, ~and y at our disposal. * For mA = 1275 MeV and TA = 315 MeV, one finds two possible solutions (i) y = 0.25, ~ = 0.34 and g = 10.1; and (ii) y = —0.26, ~ = 0.065, g = 7.0. But since the mass and the width of the A1 are not conclusively determined, we prefer to study the parameter dependence of m~,1 and TA. It is most convenient to insist on a minimal momentum dependence of the p’trit coupling by setting 8 = 0, i.e. then (~t~t~p) = g~lT,,p,2 (4 X 19,20) has the standard form, and 1 depends only on g. This is shown in fig. 2.1 by the dashed line, the cross denotes the experimental width F(p—s. 2it) = (153 ±3) MeV, i.e. g = ~ = x 6.08 = 8.6. The g-dependence of T~is rather moderate. In fig. 2.1, we also show the width F(A1 —s. pit) as a function of g and the A1 mass (we use F(A1 —s. 3’it) as an upper bound on T(A1 —s. pit)). For m~= 1275 MeV, we find = 315 MeV for g 7.6 and g 9.6, close to the values obtained above. Therefore, setting 8 = 0 does not change the essential features of the model (for a somewhat different opinion see ref. [2.37]). To decide on which value of g we should take, we calculate the D-wave and S-wave contribution to the A1 —s. pit decay width. For g = 9.6, we find D/S 0.23 whereas for g = 7.6, D/S 0, in agreement with the strong D-wave suppression indicated in T decay experiments [2.38—2.40].As a check on the parameters and on the vector meson dominance assumption, [2.33], we can calculate the A1 —~‘it -y decay from A~—s. ‘it + p°via F(Ait)2~
T(Ait~p°)~730keV
~
(2.62)
for TA = 300 MeV and g = 7.6. This is consistent with the recent measurement [2.71] T(A~—s. ‘tt~-y)=
PARAMETERS I
OF THE A1
I
I
I
MESON
—
I
I
400
I
-
1230
1275
‘~‘3OO
-
1150
-~ ~2OO
...—
--+--
IO5O
-
100
-
I
6
-
— - — - -
I
7
-
I
I
8
9
I
9 ‘/~‘
10
II
l’pirir
Fig. 2.1. The width of the decays A, —* pit and p—o isis for different values of the A,-mass (covering the experimental uncertainties) as a function of the pins-coupling constant g = v’~f0_,,.The cross denotes the empirical value of F(p—* 2ir) = (153 ±3) MeV. *
Clearly, one wants to get g close to its empirical value so that the predictions of the anomalous decays discussed before are only slightly
changed.
U.-G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
235
(640 ±246) keV. One can now also calculate the anomalous decays of axial-vector mesons, with the AA41 Lagrangian given by ~AA~
=
15~2F,,(i+Y)
,2va/3
Tr[ô,2A
8aAp41].
0
(2.63)
This allows e.g. to predict T(D—s. p’rri’i) -~3 MeV, somewhat lower than the experimental bound T(D—s.pinrr) < T(D—s.4i’i) (14.4 ±3.9) MeV [2.34]. We will not elaborate on this point but rather refer the reader to ref. [2.6]. What is important is the fact that one again has to use Bardeen’s form of the anomaly, in which case decays like D—s. pp, D—s. ww, A1 —s. pw, . are suppressed, i.e. the equivalent radiative decays D—s.p°’y,D—s.w-y, A~—s.p°-y vanish. Finally, let us point out that the lower value of g = 7.6 favored by axial-vector meson decays as compared to g = 8.6 fitted from the p—~2-it decay lowers the V—s.41-y amplitudes by 10% and the one for w—s.3it by approximately 20%. Altogether, with the choice of the gauge coupling g close to its non-anomalous value g = ~ and the two non-minimal terms proportional to ‘y and ~, one can describe a huge amount of mesonic decays — anomalous as well as non-anomalous decays of vector and axial-vector mesons. * Golterman and Han Dass [2.45, 2.46] have constructed a similar effective Lagrangian starting from the QCD generating functional. They use a slightly different extra non-minimal term to account for the p- and A1-properties simultaneously. Since in the present framework Vector Meson Dominance was put in through (2.32), we will now demonstrate how VMD can be enforced on the level of the effective action by adding suitable counterterms. .
.
2.3. A subtracted Lagrangian with Vector Meson Dominance Following the work of Brihaye, Pak and Rossi [2.9—2.11]we will develop a systematic functional subtraction formalism to implement Vector Meson Dominance (VMD) unambiguously. The idea is to add local counterterms in a way so that the low-energy predictions involving pseudoscalar and electromagnetic processes only are not affected by the vector meson contributions without any constraints on the new parameters. This is nothing but implementing Zumino’s ideas [2.47] of VMD on a gauged chiral Lagrangian. For that, let us start with the non-linear if-model (2.2) and gauge it electromagnetically — this can be done without any ambiguity since the photon is indeed a gauge boson. We will restrict ourselves to the two-flavor case, the generalization to SU(N) can be found in ref. [2.10]. The electromagnetically gauged non-linear if-model reads /~1 /
./_~
~‘
0,em
A
-~
N — -~
[ ~
,2
\
]_~iEL_~_i I
~
~
2
,2
JE,2*a/3
.~‘
2
r~
(~_~)E,20if/3B~0Ba Tr[r3(L~ + Ru)],
v
a
/3
(2.64)
where B,2 is the photon field, B,2,, = a,2B,, — d,,B,2 the Abelian field strength tensor, and D,2 U = 8,2 U — ieB,2[Q, U] the pertinent covariant derivative. The charge matrix Q is given by Q = ~( ~ + r3). The Lagrangian (2.64) incorporates all low-energy ‘try interactions in accordance with current algebra. * For a different approach relating anomalous decays to Lagrangian, see refs. [2.42—2.44].
I’~~(U)
via VMD-assumptions without including spin-i particle in the effective
236
U. -G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
For later purposes, we will recast (2.64) into the following form: 2+ ~
~0,em = ~
[5~’t
(D,2~)
~](c
~
x D,2~)2+
~
N 0e —
~
[2~—sin2~] ~
L
]~(D~ox ~
E~0,~B”
(2.65)
.
using the weak-field expansion 3abçob.
U=i+irir/f,,=i+ir~, and D,2q~=o(pa+eBa We will now introduce p-, w- and A 2)R ®U(1)~gauge particles as we did 1-mesons SU(2)L ®SU( already in section 2.1. The only difference to theas Lagrangian (7.6) for the non-anomalous sector (with = 0) and (2.44) for the anomalous sector is an enlarged definition of the physical A 1-field, i.e. we use A,2 = A~,hYS+ (a/f,,g)D,2 ITPhYS
(2.66)
with a a parameter to be fixed later. Previously, we have used A,2 = + (a /f,,g) 8,2 ~ (where the tilde denotes the physical quantities), this alternative diagonalization procedure is suggested by the appearance of terms like D,2 ‘t~— gfA,2 in the non-anomalous part of the effective Lagrangian and will prove useful in the computation of meson processes, e.g. -rtp-scattering reduces to a sum of three diagrams by using (2.66) instead of five by using the standard procedure. The extra two diagrams are due to the exchange of A1-particles. Of course, the scattering amplitude turns out to be the same. With the definition, = 17,2 ±
V,2=—~i(T•V,2+w,2),
(2.67)
A,2=—~iT•A,2,
D,2U=8,2U+gA~U-gUA, the diagonalized effective action T = F(i~,2,V,2, A,2,
1
2
2
W,2)
reads
2
1 1 (D,21r)2 + (f,,g) T=~j 2a ~ (V,2+w,2) +
—
2(i)
JA~
E,2Oap J
(~)
ô,2w
2)it 0[
—
J[~xD,2~-
71- (1—
3a
+
(1~2a)~(AaxD~~)—
~
/~
~(Da1T
~xA,2] x D~i~)
~a (8a1~p— 8pVa +gV~XV~~gA~
XA~)]
(2.68)
U. -G. Meifiner, Low-energy hadron physics from effective chiral Lagranglans with vector mesons
237
in terms of the physical fields. We now have to implement VMD on the naive effective action (2.68), which gives a consistency criterion for the gauge procedure employed. Let us now assume that at low-energy VMD holds exactly [2.11i.e. every process described by the iry-Lagrangian (2.64) should be reproduced with the same amplitude by the corresponding processes involving vector mesons. As before, we assume that all electromagnetic couplings can be obtained from vector-meson couplings through the insertion of the extra term (2.33) in the Lagrangian. In the two-flavor case and using the parameter a, this term reads ~VMD
=
+ C(B~), —f2,,(eg/a)(V~+ ~w,2)B,2
(2.69)
which allows for an unambiguous identification of V,2 since it coupled to the external electromagnetic source. In what follows, we will set V,2 = and first discuss the vector case (A,2 = 0). As will become clear, we will have to add in a systematic way local counterterms to remove the spurious vertices introduced by the vector meson propagators. For that, consider an effective action I~(ço,B,2) that reproduces all low-energy theorems. The aim is now to find an action T(ço, 17,2) such that VMD in the functional integral formalism holds, i.e. e”~’B~) =
f
[dV,2]exp[iT(~, 17,2)
+
~im2(V,2— B,2)2],
(2.70)
where we have removed all internal indices and coupling constants. Expanding around 17,2 = B,2 we find the following defining equation for T(p, 17,2): * ~
[m~g~ 0+ ~2r]-1
~
(2.71)
which can be easily solved. Expanding the solution in powers of m-2 we find T(~,V,2)=T0(~,V,2)+~ [~(~,V,2)]+C(m4). 2m aB,2
(2.72)
The 2g,2second term is nothing but a subtraction of the contact current—current term generated by the m 0-part of the vector meson propagator. To see how this subtraction procedure works in the non-anomalous sector, let us discuss the ‘wit-scattering amplitude. From current algebra, it follows to be = (—116f ~,)(1T X 9 ir)2. From the action (2.68), we get 2. (2.73) 12a (irx8~ir) To recover the current algebra result, we have to add a counterterm as suggested by (2.72), i.e. = (—a/3f~,)(1TX 19 ~.)2 In a similar fashion, one can now systematically determine by comparison all subtracted effective vertices for the itp-theory in order to reproduce the low-energy theorems. In other words, we are looking for an action T(~c,P,2~w) such that the electromagnetic vertices, generated by T(~C,P,2~(LI) + TvMD, reproduce those already contained in the electromagnetic Lagrangian (2.64). *
We assume that the action is at most quadratic in the external vector sources.
238
U.-G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
For the non-anomalous sector, the procedure is straightforward. Concerning the anomalous sector, it is important to realize that to lowest order in momentum only one anomalous vertex can contribute, i.e. no w-propagation is allowed. With that, one can generalize (2.71) by expanding the anomalous action around P,2 = B,2 and ((5,2 = (1 /3)B,2. With that, one finally arrives at [2.91 F = T’ + ~
.
Tna(~,P,2~w)=~J(o~)2
f,, f . 2j p [i—asin T*n(~,P,2~‘~e*)
~7
2
‘P]
2
2
sin 2 ‘P ‘P
1
+
2
f,,g 1I 2aj
2
2
+gp~X Pp) g~.(8P0— 8~P~
E,2vapJo,2wv{
sin2’P 3
‘P
.
1 ~C.(D~cc)
2
(2.74)
2’P(1—asln ‘P)
Notice that in (2.74) the transverse and longitudinal parts of the vector fields are separated. The combination F” + F” + FVMD is a unique low-energy Lagrangian of pions, p- and t~o-mesonswhich reproduces all low-energy theorems incorporated in Trn” (2.64). It can therefore be used as an extension of the effective theories discussed in sections 2.1 and 2.2, after one reinstates the axial-vector meson. Since there is no external source to which the 1-meson can couple, the construction of the effective Lagrangian is not as unambiguous as before. One way to resolve this problem is to introduce a source to which the A,2 couples and the demand that the interaction between the axial sources J~,indeed gives the anomalous part of the Lagrangian (2.68) before field diagonalization. For the non-anomalous sector, one simply wants to recover the original “naive” chirally symmetric Lagrangian with the SU(2)L 0 SU(2)R 0 U(1)~ softly broken by chiral symmetric degenerate mass term. This leads to F na =
f~ 1 [p~(D,2~~gA,2)]
i 2~1— a~ .1
+
~
2
+
2~Isin2’P I 2
f
‘~
L
‘P
[~X(D,2~—gA,2)]2 2 2 .
1—a sin ‘P
.
—
a(sln
‘P)Ic°2
egJ(p°~+ ~w,2)B,2+gJA,2J~,
(2.75)
which after the diagonalization procedure (2.66) leads to an effective mass term for the pions and the axial-vector mesons =
—am~2+ 2a(1— a) A~.
(2.76)
If one assumes that the pion mass is entirely generated through this mechanism, one needs a = 1/2 and recovers the KSFR relation [2.25]. The anomalous part of the action follows to be
U. -G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
2
F””(ç, P,2~W,2,A,2)(NcIi6it2)E,2~~/3Jo,2W0[g~(28~p/3 +gp~x p4)— ~
+ sin~ (C~x C~)— sin2’P ~ .(~2 x C~)+ 2 sin2 ‘P 2’P ‘P ‘P 1
C,2
— —
2 —
C,2
—
239
x D~~)/’P ~.
(C~x
(2.77)
(1—a(sin2’P)/’P2)D,2~—agA,2sin2Q 1—a sin2 ‘P — a(sin2 ‘P)/’P2 (1 —
a sin2 (p)gA,2 — (a(sin2 1 — a sin2 ‘P — a(sin2
‘P)1~’P2) D,2~
‘P)/’P2
Now (F~fl+ F11”)~~~ 0 gives a complete itpA 1w-Lagrangian at low energies (E ~ mA1) after performing the3’7F~ field =diagona1i~ation.It produces all purely pionic and theorems, e.g.it f~,,where F3~denotes the -y—-s. 3ir amplitude andelectromagnetic F~the one for low-energy ir°—s.2~y.*Furthermore, eF gives the non-Abelian anomaly in Bardeen’s form. Finally, notice that in the limit a —*0, the action (2.77) goes smoothly to the anomalous part of the action (2.68) (before diagonalization) since in this limit the vector mesons are identified with the external sources. Furthermore, all the counterterms needed to eliminate the spurious local vertices generated by vector-meson propagation are built in by construction. Notice that this procedure does not impose any constraints on the fundamental parameters of the model. 2.4. A Lagrangian of pseudoscalar and vector mesons As we have seen before, the inclusion of the axial-vector mesons like e.g. the A 1 poses several problems due to uncertainties with A1 parameters. The A1 also plays no important role in the NN-interaction [2.14]; we will therefore now modify the model presented in section 2.1 to eliminate the A1 by a non-linear realization of the vector particle (the p), following the ideas of Weinberg [2.49]and the work of Kaymakcalan and Schechter [2.7]. The starting point is the following non-anomalous Lagrangian: ~01~2’
~= -~
t], ~F~Tr[D,2UD~U = ~ Tr[F~ 0F~’+ F0~~PR]+ 7 Tf[FLUFtLORUt] —
=
m~Tr[A~A~L+ ~R~~R] + B Tr[A~UA~RUt],
using the standard definitions U=exp{2i41!F,,}, *
F,,=\/~f.,,.,
This relation is not fulfilled by the Lagrangian constructed in section 2.2 as first pointed out by Rudaz [2.48].
(2.78)
240
U.-G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
D,2U= ~U—igA~U+igUA~, F,2,, = 8,2A,,
80A,2 —ig[A,2, A,,] (for Land R). 3)L 0 U(3)L gauge transformations, the local invariance is broken Clearly, ..~ + wayisby invariant U( in a minimal ~ Theunder generalized mass in ~2 with the constant B will turn out to be of great importance. We want to eliminate the A 1, i.e. find a (primed) gauge in which —
~
A,2
=
0,
(A~)’= (A~)’= P~
(2.79)
.
For that, let us consider the U(3)®U(3) gauge transformations U—s. ULUU;
ULR E
U(3)LR,
A~—s.ULA~U~ + (i/g)U~8,2U~,
(2.80)
A~—s.URA~U + (i/g)UR 8,2U, 1/2
—1/2
with UL = U112~U”2, and UR = U , i.e. the ~lOflS as gauge parameters: U= U = U112p,2U112 + (i/g)U~2ô,2U~2, =
(2.81a) (2.81b)
U~2p,2U112+ (i/g)U112 8,2U112.
(2.81c)
We have U’ = ~l,which means that the pseudoscalar fields will only appear in the non-gauge invariant terms. Equations (2.81b, c) can now be solved and give the following chiral-invariant constraint: A~= UA~Ut+ (ilg)U e9,2Ut.
(2.82)
We can now eliminate the axial field by injecting (2.81) into the Lagrangian (2.78). From (2.82) it is clear that D,2 U vanishes, i.e. one gets no contributions from any term involving D,2 U. Using the fact that the p-meson transforms under the diagonal subgroup U(3)v of U(3)L 0 U(3)R [2.49] we find that ~ simply becomes a Yang—Mills type term for the p-field Sf~=(y_1)Tr[F,2,,(p)F~*(p)],
F,2 0(p)=ô,2p,,—80p,2 —ig[p,,,p0].
With y = 3 / 4, we obtain the conventional normalization. The =
~
(2.83)
term reads
112~U~2 + 8,2 U”2 U112)] (B + 2m~)Tr[p,2p~] + (ilg)(B + 2m~)Tr[p,2 . (19,2 U + 2(m~Ig2)Tr[c9,2 U112 19~U~2] (B/g2) Tr[U112 8,2 U”2 . U”2 2] —
.
(2.84)
19~Ull
now contains all terms of the pseudoscalars and the vector meson mass term, with the vector meson
241
U. -G. Meijiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
mass given by m~,=4m~+2B.
(2.85)
The p4)4) interaction from (2.84) follows to be 1-
2
Tr[p~4) t9~~4)1, ~
(2.86) 3(g~~)2I12~rm~, where the coupling constant ~ is related to the p-meson width via F(p—~21T)= q i.e. ~ = 8.6. Notice that differs from the original gauge coupling g, the special choice 2g = ~ gives the KSFR relation g~F~= 2m~,[2.25].It is important that we have taken the term proportional to B in ~2’ otherwise we would have ended up with the result = m~IF~5.85, i.e. F(p) 70 MeV, an old problem noticed in particular in ref. [2.3]. By adding an ordinary pion mass term to account for PCAC, one can recover the low-energy i’rrr-scattering results. For S-wave scattering in the I = 0 and I = 2 channels we indeed find the well-known result a 0 = 7m~!161TF~, a2 = —m~!8irF~ [2.50,2.511. Let us now turn to the anomalous sector. In principle, one can start from the Bardeen subtracted Wess—Zumino action (2.43), which breaks chiral U(3)L®U(3)R down to U(2)v®U(1)A. Notice that one can think of the Bardeen subtracted WZ term as a remnant of a chirally invariant action involving another set of pseudoscalars 4)’ which are not excited at low energies, i.e. =
~
2
F~~(U, AL, AR)
—
=
m~/gF~,
E~~(U’, AL, AR)
(2.87)
has the same low-energy predictions F~ = F~~(U, AL, AR) F~~(U = AL, AR). Let us first see how the Bardeen form of the WZ action with the axial fields eliminated via (2.81b, c) works. For that, consider again the process w—+3’rr. Using a weak field expansion for U, we have —
~,
AA~—A~=(2IgF~)a~4)+..., (2.88)
~ so that the relevant pieces of the anomalous Lagrangian read =
~GSW + ~contact
=
~
Tr[ô~w”s9~Xp~4)] + ihE~~~ Tr[w~od~4)d’~4)].
(2.89)
The GSW-term coefficient follows to be 2g~F~ ~vv~ = 3mI4ir and the coefficient of the contact term is given by* h
=
(2.90)
—m~Ii2g~~F~.
* Notice that the coefficient h is related to the dimensionless parameter $ introduced by Adkins and Nappi [2.52]via $ whereas in ref. [2.52]it is fitted to the nucleon and i’-mass, yielding $ = 15.6.
(2.91) =
—3 ir2F~h/V~= 8
242
U.-G. Meif3ner, Low-energy hadron physics from effective chiral Lagrangians with vector me~ons
The computation of the w—~3ir amplitude proceeds as before (cf. eq. (2.47)), with the coefficients given by (2.90) and (2.91). Here, the contact term adds to the GSW term (in contrast to the situation before) and is about four times larger in magnitude. With that, we find that F(w—~3’rr)comes out 16% above the experimental value, whereas it was 15% below pexP(~~ 3’rr) for the model with explicit A1-mesons. Proceeding along this direction means that all anomalous couplings are predicted in terms of g~~• If one is willing to give up this ability to predict the anomalous couplings and insists on chiral invariance without introducing a new set of pseudoscalars 4)’ via (2.87), one can go back to the WZ term (2.31) and eliminate the axial mesons via (2.81). Doing this, we end up with three parity-and-chiral-invariant terms for the anomalous action, i.e. Fan(U,
~)
=
T~~(U) + c, JTr[iALL~1 + c2f Tr[dALLAL —ALL dAL+ ALLALLI
+ c3
J
Tr[—2iA~L + gA~LALL],
(2.92)
2p~ h/2 + (i/g) U”2 ~ U-”2 Obviously, where AL hasthree to becoefficients written in terms the p-field,c A~= U” we have now to be of determined. 2 is related to gvv4, and c1 and c3 might be adjusted to give the contact term h. The problem of fixing these coefficients is recently under investigation [2.531. The action (2.92) together with (2.83) and (2.84) might be used to construct a fairly simple effective Lagrangian of pseudoscalar and vector mesons. In light of the uncertainties related to the A,parameters, this seems to be a useful exercise. Furthermore, the precise relation to the hidden symmetry approach discussed in the next section should be worked out, especially concerning the anomalous sector (for the non-anomalous sector, cf. section 4.3).
3. The hidden symmetry approach Any non-linear sigma model based on the manifold GIH is gauge equivalent to another model with GgIObaI®HIOCaI symmetry [3.11. The gauge fields of the hidden local symmetry are generated by quantum effects and poles of the gauge fields are developed [3.2]. Starting from this observation, Bando et al. [3.3] have proposed that the p-meson is a dynamical gauge boson of a hidden local symmetry in the non-linear chiral Lagrangian 2)L®SU(2)R describing the islow-energy effective two-flavor dynamics. The broken spontaneously to theQCD diagonal subgroup global chiral symmetry G = SU( H = SU(2)~.In the first subsection, we will explore the consequences of this idea and derive low-energy theorems. Furthermore, the relation to the conventional Skyrme model is pointed out. The next two subsections are devoted to extensions of Htocai so as to introduce the o-meson and determine its coupling from the Wess—Zumino action. The inclusion of the w in the two-flavor sector can be achieved by enlarging Hiocai to SU(2) 0 U(1), as explored by Meil3ner et al. [3.4]. Bando, Kugo and Yamawaki [3.5] proposed the more general extension to U(3), therefore also including the 4)-meson. We will derive and discuss the pertinent low-energy theorems according to these enlarged hidden symmetries. Following Fujiwara et al. [3.6], we will examine the Wess—Zumino action in the presence of vector
U. -G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
243
2-y, w—~3’rr, ~rr—~ 3y). Finally, in section 3.4, mesons and calculate various anomalous processes (‘~i°—~ we discuss the inclusion of axial-vector mesons and external gauge bosons (y, W~,Z) in the hidden symmetry approach following the work of Bando et al. [3.5, 3.6, 3.7]. 3.1. The p-meson as a dynamical gauge boson Our starting point is the non-linear cr-model SU(2)L®SU(2)R/SU(2)v with its Lagrangian given by =
defined
on
the
coset
(f~i4)Tr(~9~U o~U~).
space
G1H =
(3.1)
U is the standard representation of the Nambu—Goldstone pions U(x) = exp{iir(x)if~},with IT = ITTa and ~a are the usual SU(2) generators. f~ is the pion decay constant. Obviously, the Lagrangian (3.1) is invariant under global chiral symmetry U(x)—~g~ U(x) g~
g~~E[SU(2)~R]g,oba,.
,
(3.2)
The hidden SU(2)~symmetry of (3.1) becomes apparent if one rewrites the SU(2)-valued field U(x) in terms of SU(2)-valued new variables ~LR(x) such that [3.3]
U(x) = ~~(x)~ 4~(x)
(3.3)
Connected to the SU(2)~-symmetryis a gauge field V~,
v~=~v:T”i2.
(3.4)
The new fields ~LR(x) and V,~(x)transform under ~L,R(x)~/~@)
~
[SU(2)L®SU(2)R]global®[SU(2)v]Iocat
as follows (3.5a)
,
V~(x)—~. (jig) h(x) ô.~h~(x) + h(x) V,A(x)h~(x) ,
(3.5b)
where h(x) parametrizes [SU(2)~]iocai. Insisting on the G®H symmetry and parity, one can construct two linear independent terms of second order in derivatives of = —(f~i4) Tr[(ô,~ igV~j~~ ~ + (~ igV~)~. —
—
(3.6) =
-(f~l4)Tr[9~L~~
-
~R
.
~+]2
(3.7)
using the covariant derivative 8.,~ igV~.g is a gauge coupling to be specified later. It is now easy to show that any combination of (3.6) and (3.7) —
(3.8)
244
U. -G. Me,f3ner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
where a is an arbitrary parameter, is equivalent to the non-linear cr-model (3.1). For that, we work in the unitary gauge =
~R
=
exp{+iITi2fTj
(3.9)
to eliminate the unphysical scalar degrees of freedom. By introducing independent left- and righthanded fields ~LR we have actually three pions plus three unphysical scalar modes (see section 3.4). We find ~
(3.10) (3.11)
using ~ + ~
~+
~.
~
—
~.
So
~A
reduces to the original Lagrangian
whereas ~
~,
vanishes
identically when one uses the equation of motion for V,~ gV
=
_iTr{~ ~
+
ô~
.
~)}~—
~
(~X
(3.12)
~
using the weak field expansion of (3.9). Since vanishes identically, it can be multiplied by an arbitrary parameter a as done before. So far, the vector field V~ (x) is a purely auxiliary field without any dynamics. The second line in (3.12) is already suggestive of V~being the p-meson. We will therefore assume that V,~has its own dynamics and that a kinetic term for it is generated by the QCD-dynamics or quantum effects at the composite level [3.1, 3.8]. For a detailed discussion on this point we refer the reader to the forthcoming review by Bando, Kugo and Yamawaki [3.9]. Adding by hand a kinetic term, we have ~,
~=
~A +
~
—
(3.13)
~
F~is the corresponding non-Abelian field strength tensor of the vector meson (3.14)
~
The physics of the Lagrangian (3.13) becomes transparent when one makes the weak-field expansion ~(x) = 1 + (ii2f~)i-.17(x) ~ . Then the Lagrangian (3.3) becomes* Tr[~U d~U~] —
(~
+ p~+ ~ g2f~p~
g)p~ (~X .
~)
+ U(174).
(3.15)
Here, we have identified the vector particle V°~ with the p-meson. From (3.15), we can read of the p-meson mass and the pinT coupling constant as 2 22 m~=agf~,,
*
Notice that the Lagrangian (3.15) is precisely the irp-Lagrangian already given by Weinberg [3.10]solely based on chiral symmetry arguments.
U. -G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
~
~ag.
=
245
(3.17)
Equation (3.16) has the typical form of a Higgs mechanism. Indeed, the unphysical scalar modes in ~L,R= exp(i4)/2fTT) exp(±iiri2f,,)are absorbed into the p-mesons. As already mentioned, the parameter a is not specified by these symmetry considerations. However, for a = 2, we find the KSFR relation [3.11] m~=2g2f~
(3.18)
and universality (3.19) The pinir coupling constant is determined by the p—> mm decay width. We have g2
m2 (3.20)
42I~*2~Tjqj3
For m~= 770 MeV, q,,, = 358 MeV (~ q,~ is the momentum of the pions in the decay rest frame) and = 153 MeV this gives ~ = 6.11. With f,~ = 93 MeV, we see that the KSFR relation is fulfilled within 5%. Equation (3.19) only gives universality if the p-coupling to matter fields ~‘~A is saturated by the minimal term ~ ~ with g~AA = g. This agrees with the dominance of p-exchange in ‘~~“Ascattering as discussed in section 1. At this stage, we can make contact to the Skyrme model [3.13].In the limit m~—~ ~ with g held fixed, the vector particle decouples. This leads to the constraint (pt, = p Ta12) —
gp~, (1!2i)(ô~L~~ —
+
ô~L~:R
~j~)=0
(3.21)
and reduces the field strength tensor ~ F~~(PF~(~(~L~
-
4g
[(~L
~L
+~R.~R))
R~R)’(LL~R~R)],
(3.22)
so that the kinetic term for the vector boson becomes the Skyrme term:
=
+
~
Tr[~UU~,~~UU~]2
(3.23)
with e = g the Skyrme parameter [3.12]. This result has been derived by Igarashi et al. [3.13], Abud et
246
U. -G. Meif3ner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
al. [3.14] and Balog and Vecsernyes [3.5].* Furthermore, it agrees with the original result of Iketani [3.16], e = g, because in the presence of the A, meson the pirir vertex receives a strong momentum dependence [cf. sections 2.2 and 3.4]. The main virtue of the hidden symmetry approach is the unique way of introducing electromagnetic interactions as we will demonstrate now. The electromagnetic field coupled to the charge operators 2)L®SU(2)R]gtoba, isospins and hypercharge, = J(L+R) + Yi2, with 1~L. R) are andcompletely Y being independent the [SU( respectively. These generators of the hidden SU(2)~to which the p-meson couples. Let us for the moment neglect the weak interactions and concentrate on the electromagnetic fields (the generalization of coupling to the electroweak standard model will be discussed later on). Coupling the fields ~LR to the electromagnetic U(l)Q gauge field B,~means to redefine the covariant derivative as (Y = 0):
=
~
v:)~LR+ieO~LRB~
—ig ~
(3.24)
~,
with e 0 being the coupling constant of U(l)Q. The Lagrangian (3.15) reads 5~=~f~Tr[9~U 9’~U~] — e0m
1
e
~
+ a
2
(~) (~g)v~
V~B~ + ~ + (1 — aI2)e0V,~ (~ X gP
-
—
.
(m
m~B~ +
(3.25)
X a~17)
ti9~L1T)~ +
with Ba,, = 9~B~ — ~ For a = 2, this Lagrangian coincides with Sakurai’s [3.17] y—p mixing and leads to vector meson dominance. To see that, let us diagonalize the mass matrix of V~and B~.We find 2 + e~)f~, m~± = ag2f~, (3.26) m~= 0, m~o= a(g where the mass eigenstates are +
e 0V~.
A~=
2
2
~
gV~— e0B~ 2
‘
e0
\/g
(3.27)
2
+
e0
The electromagnetic charge is given by ge0 e=
328 e,~.
~
The p-dominance model of the pion form factor emerges naturally for a
*
=
2 since (cf. section 1):
g~,~=e(1—ai2)=0,
(3.29)
~
(3.30~)
=
m~Ig,
Actually, Balog and Vecsernyes find e = 2s~-= g + ~i, where ~lstems entirely from the non-anomalous part of the Wess—Zumino action not
considered here.
U.-G. Me,Jjner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
where the -yp-vertex is defined via two a-independent relations [3.18] ~ 1 m~ 2f~g —
‘
=
247
—eg~~p~A~. Together with (3.18) and (3.19), one can derive
~ 1 m~ g
(3.31)
—
which combined lead to the a-independent generalized KSFR relation (3.32)
~
Experimentally, ~ follows from the width F(p°—~e~e) =6.62keV to be ~ =0.12GeV2 (if one assumes a minimal photon coupling) and 2g~~f~ = 0.11 GeV2 (for ~ = 6.11 and f,~ = 93MeV). Notice that the relation (3.32) follows solely from symmetry considerations and can be considered as a “low-energy theorem” of the hidden symmetry approach [3.5]. In summary we can conclude that if the p-meson is indeed a dynamical gauge boson of the hidden [SU(2)~]tocaisymmetry of the non-linear cr-model (3.1), the following phenomenology emerges naturally for a = 2: (i) the KSFR relation m~= 2g2f~ (ii) universality of the p-coupling to pions and other hadrons g = ~ = g~HH; (iii) p-dominance of (isovector) photon couplings to pions as seen in the pion charge form factors; and (iv) p-exchange dominance in pion—matter scattering. 3.2. The w-meson as a dynamical gauge boson From the low-energy nuclear phenomenology discussed in section 3.1, the relevance of a strongly coupled isoscalar vector meson is well-established in addition to the fact that the w-meson joins the p-meson as a member of the vector meson octet. It is therefore necessary to incorporate the w-meson and its dynamics in addition to the p-meson. This is achieved by either extending the hidden symmetry to [SU(2)v®U(1)]tocai [3.4] or to [U(3)~], 0~~,[3.18]. In the latter case, [U(3)LOU(3)R]global breaking effects have to be taken into account. For the discussion of the two-flavor sector, it is most convenient to introduce the w-meson as a [U(1)]iocatgauge particle thus enlarging the hidden symmetry to SU(2)~®U(1) as discussed by Meil3ner, Kaiser, Wirzba and Weise [3.4]. Starting again with the Lagrangian (3.8), with h(x) E SU(2)~®U(1),the gauge covariant derivative now reads =
~ —i(gi2)(rp~+
Whl)ns
~
(3.33)
—iV~ ,
where we have chosen equal gauge coupling constants for SU(2) and U(1). Adding again a kinetic term for the gauge bosons V,~ 2)Tr[F~~F~”], (3.34) —(1/2g with ~
=
—
ô,,V~—i[V~,Vj
we find a generalized KSFR relation using again the weak-field expansion ~ = 1 m~= m~,= 2g2f~
(3.35)
,
+
(ii2fTT)’r 17, (3.36)
248
U. -G. Meij3ner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
for a = 2. The almost mass degeneracy of the p and the w (m~= 769 MeV, m~= 783 MeV) justifies the choice of equal gauge couplings for SU(2)~and U(1). From the covariant derivative ~ (3.33) it is obvious that the u and p cannot couple in the non-anomalous sector. The w-couplings are indeed dictated by the Wess—Zumino term to be discussed in section 3.3. Bando, Kugo and Yamawaki [3.5] have proposed an alternative scheme to incorporate the by exploring the full vector meson nonet (p, w, 4), K*, K*). This amounts to working in the nonet space GIH= [U(3)LOU(3)R]iU(3)v. ~LR(x) are now U(3) matrix-valued fields, and in the unitary gauge, we have a nonet of pseudoscalar mesons (m, K, K*, ~ .~f)
U(x)
=
~(x)
~(x)
=
exp{iIr~~t*If}
(3.37)
,
where ta (a = 1,. . 9) are the U(3)-generators and f,~ = fk = f. The non-Abelian gauge field ~ (x) = ~tai2 contains the vector mesons (p, to, K*, K*, 4)). The transformation properties of these fields under [U(3)L ®U(3)R]gIobal 0 [U(3)xj],oca, are . ,
~L.R(x) e’~R
~L,R(x)~e
(3.38)
,
~~(h) ~i,(x) = ~h(x) + ig[h(x), ~~x)],
(3.39)
h(x) is the group parameter of the hidden local U(3)~symmetry. The pertinent covariant derivatives in the presence of global U(3)LOU(3)R gauge fields are =
(a~— ig1~)~L~+ i~LRA~.
(3.40)
If we are only interested in coupling the electromagnetic field B~,we have AL,.
AR,~= e0B,.~Q
1
where gauge
Q
(341)
,
diag(+ ~). The Lagrangian is given by (again, a kinetic term for the hidden local field has been added by hand), =
~,
— ~-,
—
2 ~[F~VR + F~L]. a~v Tr[F~] For the external gauge field being the electromagnetic field B,.~, we find after gauge-fixing exp{iiri2f} and expanding to second order in pseudoscalar fields:
(3.42)
—
~A +
~=
~
~
~
=
=
bag,
(3.43)
ag2f~,
(3.44)
=
=
—
=
~(a 2)e —
0.
(3.45)
Constructing mass eigenstates as before (cf. (3.26) to (3.28)), we find
g~1~=2agf~Tr(~t”Q).
(3.46)
U. -G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
249
This leads to the generalized a-independent relations [3.5] (cf. eq. (3.31))
m~ m~
m,~,
(3.47)
g
which are experimentally confirmed. Using the leptonic decay width F( p e + e ) = 6.9 keV, F(w—s.e~e)=0.66keVand F(4)—*e~e)=1.31keVand the fact that F(V~+e~e)ctcm~g~,, we find that (3.43) holds within 10%. As it stands, this extension of the hidden symmetry is not satisfactory since it predicts m~= m0) = m4. One therefore has to add [U(3)L®U(3)R]gtoba, breaking terms which do not affect the hidden symmetry and therefore leave (3.47) intact. This can be achieved in the following way —~
~
(3.48)
~
(3.49)
with ~ given by eq. (3.37) and the breaking matrices CAy are taken to be of U(2)L 0 U(2)R invariant form [3.5]: CAN
=
diag(0, 0, cAy)
(3.50)
with CAy being constants. Obviously, the additional terms field via IT=V1+EAITV1+
~
lead to a renormalization of the pion
(3.51)
CA.
Performing a weak-field expansion ~ = 1 2
2f~= 1 m +Kcv m~= m~= ag
=
—
(ii2f)t~”,
one obtains the following relations
2 (1+m, c~)2’ 1~
f~=f~IV1+ CA
(3.52) (3.53)
and the relation (3.47) remains valid. Equation (3.48) can be cast into the following form m mK~
m m~
m m(,~
(3.54)
which holds within 2% (and justifies the ansatz (3.48), (3.49) which is by no means unique. For a general discussion on SU(3)-breaking terms, see the review by Gasiorowicz and Geffen [3.19] and references therein.) If one assumes CA = cs,, one gets [2.5] fK’f~= mKImP
—
1.15,
(3.55)
in good agreement with experimental data (fK = 1.22fT~= 113 MeV [3.20]). Again, to-couplings to pions are not given by (3.42) but through the Wess—Zumino term to be discussed in what follows.
250
U.-G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
3.3. The non-Abelian anomaly in the framework of the hidden symmetry approach The non-Abelian anomaly gives important restrictions on the amplitudes of processes which are of abnormal intrinsic parity* and include vector mesons. Particularly, the intrinsic odd parity processes to—~3m, w—~iT°-yand the anomalous low-energy theorems for in°—~2-y, -y—~ 3m are entirely dictated by 2-y and -y—~3m, and one the Wess—Zumino term. Vector meson dominance relates the to-decays to m°—+ can therefore test whether complete vector meson dominance holds. Complete vector dominance means that there are no contact terms like -y’rr’rr or -yiririT. Let us now briefly discuss the Wess—Zumino anomaly equation and the Wess—Zumino effective action [3.6]. For the time being, we work on the manifold G1H = U(3)L®U(3)RiU(3)v and will reduce it to the to-coupling later on. U(x) as defined in (3.37) transforms under chiral U(3)LOU(3)R as U(x)—~e’~’-U(x)~ Gauging these symmetries amounts to introducing left- and right-handed external gauge fields A,.1LR = ALR(tai2) which transform as =
+i[EL R’
~6LR
ALR~].
(3.56)
As pointed out first by Wess and Zumino [3.21], the effective action F must satisfy the same anomalous Ward identities as does the underlying fundamental theory, QCD. The Wess—Zumino (WZ) anomaly equation reads ~F[U, AL, AR] = G[AL, AR] =
NCJ
-
Tr[EL{(dAL)2
-
~
dA~}
-
(dAR)2
-
~ dA~}]
(3.57)
with N~the number of colors, M 4 ordinary Minkowski space—time, and we use the following differential one-forms ALR
ALR,.L df 1 dx~=(dU)U’ R = U1 dU = U’LU. L =(o,.1U)U As we already discussed in section 2.1, the Wess—Zumino action ~ =
(3.58)
,
FWZ[U, AL, AR]
=
C
f M5
Tr[L5]
+
SCi
f
Tr[ALL3
+
is given by [3.22, 3.23, 3.24]
ARR3I
M4
—scf Tr[(dALAL+ALdAL)L+(dARAR+ARdAR)R] + SCJ Tr[dAL dUARU’
*
—
dAR dU’ALU]
The intrinsic parity of a particle is defined to be even if its parity equals (_)S~*.
U.-G. MeiJiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
+ 5CJ Tr[ARU~’ALUR2
-
ALUARU1L2I
+
~cf
+
5CiJ Tr[(dARAR
+
5CiJ Tr[ALUARU’ALL + ARU’ALUARR1
J
+ 5C
[(ALL)2
—
251
(ARR)2] + 5CiJ Tr[A~L + A~R]
+
AR dAR)U’ALU
Tr[A~U’ALU
-
—
(dALAL + AL dAL)UARU~’}
A~UARU1+ ~(UARU~’AL)21
M4
=
f
[c
Tr[L5]]
(359)
gauged
M5
with C = —iN~/24O1T2and M5 is a five-dimensional manifold whose boundary is Minkowski space M4. In the presence of vector mesons, the WZ anomaly equation reads ~F(~L, ~R’ ~ AL, AR)
=
—lOCi
J
Tr[eL{(dAL)2
—
~idA~}
—
(L~R)]
(3.60)
so it has the same form as (3.57), but the gauge transformation ~ also contains hidden local symmetry transformations such that + ~R(ER)
= ~L(EL)
~
+
8(h).
(3.61)
The general solutions to eq. (3.60) are given by special solution of the inhomogeneous equation plus general solutions of the homogeneous equation ~F = 0, i.e. anomaly-free terms. Obviously, we can choose the WZ action (3.59) as4 aand special solution. The anomaly-free termsblocks. can beForwritten as are made of gauge-covariant building that, we four-dimensional integrals over M define R
D~LR ~L,R
F~= di~ igi~, —
d~LR ~L,R FLR
=
~L,R
—
igi~+ i~LRALR~LR,
FLR~~L,R.
(3.62)
There are six independent forms that conserve parity but violate intrinsic parity: ~‘
3R-R3L],
~
1=Tr[L
2—I~2)],
~t
3=iTr[F,,(L 2- 1’RLI, ~ =iTr[FLR
~6
4=iTr[F,,(LI~—I~L)], =iTr[FLLR- FRRL].
(3.63)
252
U.-G. MeiJ3ner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
Further possible terms can be reduced to these six ones using identities like DDL = i[L, F~]and so on. Therefore, the part of the effective action responsible for intrinsic parity violation processes is given by
f
F=Fwz[~~R,AL,AR]+~ c~.,
(3.64)
4
M
where c 1 are arbitrary constants to be determined later, which are essentially fixed by experimental data. Let us first check the consistency of (3.64) with the low-energy theorems given ir° 2-y and -y-—~3’rr. These processes are determined solely by the anomaly, and do not depend on other parameters (like the c’s) of the theory. For that, let us examine [3.61the term ~ in the case of electromagnetism M=eB (3.65) AL=AR=eB,~Qdx and using the expansion d~LR~ ~L,R = +(iIf,T) dir The relevant terms in ~6 are —~
±~“.
=
ief~i’{2gTr[i~dB diT]
+
2g Tr[dB
i~ dir]
—
4e Tr[B dB d7T]} + 4ef1 Tr[B(dir)3] + (3.66)
Obviously, the third term Tr[B dB dir] directly contributes to ir°—~2-y and the fourth term to y—~ir ir ‘rr. They are, however, cancelled by the first two terms involving ‘rr -y ~ vertices. It is a straightforward exercise to show that one can substitute the vector mesons i~3~by [3.6] —
—
g~—~eB+—~—[irô ir]. 2f~
(3.67)
‘~
This replacement is only exact under the following conditions: (i) the square of the four-momentum of the i~3i~ propagator is zero; (ii) the -y and the ir are on the zero-mass shell; and (iii) that -y has physical polarization. With (3.67), we can replace the first two terms of ~6 by Tr{i~,dB}~dir—p
Tr[B dB dir] +
Tr{[ir, dir], dB} dir.
(3.68)
Obviously, the terms in (3.68) exactly cancel the third and fourth term in (3.67), i.e. ~6 does neither contribute to ‘rr0—~2-y nor to -y--~3’rr. In a similar fashion, one can easily show that none of the ~ contribute to these processes. Now we can address the question of vector meson dominance (VMD). Although the Lagrangians ~ do not contribute to the anomalous processes like ‘rr° 2-y, -y--* 3’rr,... they indeed determine the rate of direct to vector meson mediated diagrams as shown in fig. 3.1. There is a particular choice of the parameters c 1 for which exact VMD is realized for the process in question. For that, we introduce the following abbreviations 21f,,) Tr[A dA dir + dA A dir], (yiririr) (4eIf~)Tr[A(dir)3], (yyir) (—2ie (~~yir)+(—2ieg/f,,) Tr[i~dA dir ±dA ~ dir], (i~irnir) (4gIf~)Tr[i~(dir)3], (3.69) —~
(t~ir)
(—2ig2If,,) Tr[i~ di~dir + di~ti~dir].
,
U. -G. Meijiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
253
There terms show up in the action (3.64) as follows: FFwz+~cJ~ 4
M
3(yyir)
=
+
4(yiririr) + c 6(yyir) + (—c3
—
c4(i~J1~iT)+(—c1 + c2
—
c5)(i~yir)~ + (c4
+ c6)(y7riTlT)+(c1
—
—
(3.70)
c6)(i~yir)+
c2 + c4)(~iririr)
where we have scaled out the factors (5C)’ for convenience. Exact vector meson dominance for the processes follows by choosing c1,. c6 such that . .
1= ~
—
15Cf
,
3)]~
(~ + ~) + 5CJ
(a~1+ /3~2)
(3.71)
5C[3(~ir)—2(~ir
where the parameters a and /3 can be determined by higher order processes like pp—p rr’ir For a /3 = 1, we have exact vector dominance, e.g. ir°—~p° + w°—~-y + y. But there is still a contact term —(lYir3), with its strength entirely determined by the anomaly. The amplitude of the decay (O—~3ir, F~3’~, can be evaluated from the Feynman diagrams in fig. 3.2 to give —
— —
ig ~ 22
4irf~
cci L.i a+,O,-
f
1 (p—q
—
)
a2
2J
—m~
3g 23’ 8irf,~
where g ~ is the i9~gauge coupling constant and the last term comes from the contact co—~3ir vertex. The first part of (3.72) obviously describes the well-known Gell-Mann—Sharp—Wagner process (GSW) [3.25], which we already discussed in section 2.1. This gives for the width E(w—~3ir) = 6.1 GeV, approximately 30% below the experimental value of 8.6 MeV. This result is not in contradiction to the one presented in section 2.1, because there the contact term was approximately a factor 5 smaller in
VECTOR MESON DOMINANCE :
7T~.-2y, y.~31T
iT---
W..-3iT
~7T
~q•)
iT
V
V
iT———
_7r(q)
—
_77~
--
~ir(q) ~iTo(q~)
~ iT
(a)
DECAY: KINEMATICS
(b)
Fig. 3.1. Direct (contact term) contributions versus vector-meson mediated contributions to the anomalous processes (a) ir°—* 2y and (b) y—* 3ir. The wavy lines denote photons (~y),the dashed lines pions (IT) and the double lines vector mesons (p or
(a)
+
w(p)~.__7T0(q0)
(b)
Fig. 3.2. Kinematics for the decay w—~3ir with (a) the Gell-Mann— Sharp—Wagner process and (b) the direct contribution.
254
U.-G. Meijiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
magnitude due to the Bardeen subtraction. Fujiwara et a!. [3.6] therefore concluded to give up complete VMD and allow for a direct (yir1r1T)~term.*The coefficients a, /3 in (3.71) are modified such that there is no more contact term for the u—~3’rr decay. (This means c1 = 7.5C, c2 = —7.5C, = —15 C. This Lagrangian will be referred to as the “PVMD Lagrangian.” The complete VMD (CVMD) Lagrangian (3.71) is realized for c1 = —c2 5.5C and c4 = —15 C.) This is essentially the old GSW model, which leads to the following results
F(ir°—~2-y)=7.6eV,
[(7.9±0.1)eV],
F(~—+‘rr°-y)= 0.84 MeV F(o—~3ir) = 9.1 MeV,
[(0.86±0.05) MeV]
(3.73)
,
[(8.9 ±0.3) MeV],
where the numbers in brackets give the experimental values. Notice that F(’rr°—+ 2-y) is identical to the current algebra result [3.22] but it stems entirely from the anomaly. Finally, we want to discuss some simplifications of the WZ action F~ to be used in soliton calculations. Meifiner et al. [3.4] have proposed to use the following Lagrangian of pions, p and u mesons:
~minimaI
=
Tr[~U ~U~]
Tr[(~~)~+
—
(~~+)~]2
—
Tr[F~F~]+
~
(3.74)
based on the hidden SU(2)~®U(1) symmetry discussed in section 3.2. This is called the “minimal model” because the homogeneous part of the WZ term is chosen in a specific way, namely 3I~ =
1OCJ Tr[(L
The special choice of
-
~3L) (L1~LJ~)].
(3.75)
-
c
1 = —c2 = 2, c3 = c4 c5 = c6 = 0, ensures vector meson dominance in the isoscalar channel. As can be read off from (3.70), the action (3.75) has only (i~innr) terms, i.e., all (yiTinT) terms are cancelled and therefore photons do not couple to pions (to leading orders). Furthermore, expressing the p-meson field as p~= _iTr[ra(ô~~~ + ~ ~)]/2g, the action (3.75) exactly reduces to a Lagrangian of the form .
(3.76) which represents w-meson couplings to the conserved baryon current B~’,given by 2)E~Tr{U~ô~U U~U U~8~U}. (3.77) B~=(1/24ir The o-coupling is exactly the one used by Adkins and Nappi [3.28]in their model of the w-stabilized skyrmion. Furthermore, we find g,~= (N~I2)gin agreement with the results of Kaymakcalan et a!. [3.23] and MeiBner and Zahed [3.29].The restriction to the two terms in (3.75) with the special choice of their coefficients is vital if one wants to discuss VMD based on meson currents coupled to conserved currents. It constitutes the simplest way of coupling the o to the ‘rrp Lagrangian (3.15) consistent with *
Notice that VMD any way breaks down at the two-photon level as pointed out by Brihaye et al. 13.251.
U. -G. Meif3ner, Low.energy hadron physics from effective chiral Lagrangians with vector mesons
255
the WZ conditions. Obviously, the decay width w—~3ir is largely underestimated by this ansatz. Therefore, Meil3ner, Kaiser and Weise [3.30] have also constructed a more realistic WZ action which violates exact VMD, but gives the proper w—~3ir decay width (see section 4.3). The pertinent Lagrangian derived from this action is given by ~~plete
=
(~
4w~v
g)w~~+
e
Tr{ip~(U~ 8~U+ ~UU~) + ~ PaU~P~U}. (3.78)
This ansatz will be referred to as the “complete model”. Notice that (3.78) violates the current algebra prediction of -y--~3ir, so higher order terms should be added for consistency. As we will see later on, these higher order terms will be of no big relevance for the properties of the soliton solutions emerging in this model. 3.4. Extensions of the hidden symmetry (axial-vector mesons) The inclusion of axial-vector mesons in the hidden symmetry scheme can be done either by enlarging the local gauge symmetry to [SU(2)L®SU(2)R]gjobal[3.31]or [U(3)L®U(3)R] local [3.5, 3.7]. We follow here the second approach as outlined by Bando, Kugo and Yamawaki [3.5].For details, we refer the reader to the extensive review in ref. [3.7]. The hidden local symmetry can be formally extended to [U(3)L® U(3)R]local by introducing new variables ~M such that U(x) = ~(x)~M(x) ~R(x). ~L’ ~R’ ~M transform under [U(3)L® (3)R]gIobal ® [U(3)L® U(3)R]local as: ~L,R(x)~1zL,R(x)
~L,R(x) gL,R,
with hLR(x) E [U(3)LR]locaI. ~l~L(x)
=
~M(x)~h1L(x)
~M(x)
h~(x)
(3.79)
The pertinent covariant derivatives read
(9~ igV~(x)+ ig A~(x))~L(x), —
~i~R(x)— (~ —igV~(x)—igA~(x)) ~R(x), =
ô~M(x) ig(V,jx) —
—
(3.80)
A~(x))~M(x)+ ig ~M(x)(V~(x)+ A~(x)).
A,
3)L]lOcal and V~(x) formthe leftand right-handed nonet gauge A~(x)are axial vector mesons including the A fields invariant under [U( 1. Insisting on parity, one can construct four invariant terms out of ~L,R,Mand ~L,R,Mof second order in derivatives:
2(x) and [U(3)R]local.
~?=
—a(f~J4)Tr[~~L. 5~ —
~
b(f~I4)Tr[~~L.
+ —
4M ~L~R
~M ~
~±~+]2
~RM_~.~M~]2,
—
c(f~I4)Tr[~.~M .
(3.81)
a, b, c, d being arbitrary parameters. In the unitary gauge ~L = = = exp{iir/2fIT} and ~M 1, one recovers the usual non-linear Lagrangian of pseudoscalar, vector and axial-vector mesons. Further2ir due to the A,~ô~ir-vertex.Adding more, onekinetic has to terms redefine pion field ir’ = [bcl(b by hand andthegauging withasexternal fields+ c) ~ + d]” (-y, W~,Z°),one finds that for a = 2 all
256
U.-G. Meif3ner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
vertices (Vext’Wff) vanish, i.e. g~,~ = 0. That means that p-dominance of the pion form factor is preserved. To satisfy Weinberg’s mass relation m~= 2m~and Weinberg’s sum rule g~1= g~[3.32], one has to choose a=b=c=2,
d=0.
(3.82)
Recent experiments support these relations [3.33, 3.34]. m~ 1056 ±35 MeV~\/~m~ and g~Ig~ 1.1 from T-decays. Unfortunately, the redefinition of the pseudoscalar and axial-vector fields to get rid of the mixing terms proportional to (A 9 /Lir) induce a strong momentum dependence of gPIT~T~Defining r=b/(b+c), we find 2), g~~(0,0,0)=(aI2)g, (3.83) ~ r which modifies the successful “low-energy theorem” (3.32) —
~
=
(1
—
r2)I2f~
(3.84)
and the KSFR relation m~= 2f~g~~(1r2)2.
(3.85)
—
Furthermore, the decay width [‘(A 1 p’rr) comes out to be roughly half of the experimental value. Even more disastrous, for any choice of the parameters the A1-yii-vertex vanishes [3.7], in sharp contrast to experiment: [‘(A1 -yir) = (640 ±246) keV [3.35]. Bando et al. therefore proposed to add higher order terms, restricting to the lowest number of derivatives. For that, use the differential one-forms (cf. eq. (3.62)) —~
—~
L, i~—~ —i1, —iI~
(1/i) D~LR~
and construct the following six invariants
]1~ —i D~M~
R’
(3.86)
(1.: L~dx~and so on) =
2)Tr[I.:,J.:~L~ + I~I~~R~] ~t a1(—iIg + a 2) Tr[~MR~ ~ + ~Lv~LV~MR~] 2(—i/g + a 2) Tr[L MRVM’~ + R MLV~MR] + h.c. 3(i/2g + a 2) Tr[M~M~L~ + ~MM~M~4MRI 4(—i14g +a 2 ) Tr[L~A~L~R~ M~MR~I + h.c. 5(i14g +a 2 ) Tr[~k~ ~IcI~L~ ~L ~MRfl + h.c. (3.87) 6(—i14g The contributions to the various vertices pir’rr, A 1 p’rr,... can be worked out for all these terms. If one cancels the strong momentum dependence of (3.83) and insists on p-dominance for the A1 yir decay, one finds that independently of the choice of a, (i = 1~. . , 6) satisfying these conditions, the —
—
g~lT~~
—*
.
U.-G. Meiflner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
257
A1 pir and A1-yir amplitudes are uniquely given; their widths turn out to be F(A1—~.pir)~360MeV, F(A1—~-yir)=*300keV,
(3.88)
which is consistent with data (see the discussion on the A1 properties in section 2.2). The most obvious choice to fix the parameters a1,. , a6 further is to demand that the new terms do not induce momentum dependence of the p’rrir vertex in the absence of the A1-meson. This can be achieved by setting a1 = a2 = a3 = 0, —a4 = a5 = a6 = 1. So we end up with the following Lagrangian: . .
~[=~
~{(a = b = c = 2, d
0)— ~(a1
=
a2
=
a3
= 0,
—a4 = a5
=
a6 =
1)
(3.89)
which reproduces all the phenomenology of pA1-system. Let us end this section by summarizing the phenomenology arising from the Lagrangian (3.89) [3.7]: vector meson dominance of the pion electromagnetic form factor, universality of the p-meson 2f~, coupling, the KSFR relation m~ = 2g Weinberg’s mass relation mA 1= — — — —
Weinberg s sum rule g~1= F(A1—~pir)~360MeV and F(A1—~yir)=300keV. Notice that if one wants to treat the A1-meson as a dynamical gauge boson, simultaneous generation of both the kinetic term and the higher derivative terms ~t is necessary to be consistent with the effective low-energy theory without A1 the strong momentum dependence of the p-coupling comes from the A1 kinetic term and is exactly cancelled by the higher derivative terms. One again is tempted to conclude that low-energy hadron physics can be more effectively described in terms of pions, p- and ü-mesons alone. — —
—
4. Equivalence proofs In this section, we will demonstrate that the models based on gauging rigid chiral symmetry (cf. section 2) are similar in nature with the models based on concepts of the hidden local gauge symmetry as discussed in section 3. This gauge equivalence can be most easily demonstrated by making use of the so-called Stückelberg transformation [4.1], as outlined in section 4.1. Then, we will generalize the proof incorporating external gauge bosons in the hidden symmetry scheme as emphasized by Yamawaki [4.2]. These proofs are fairly general, we will therefore explicitly make contact between the massive Yang—Mills ansatz without A1 and the irp-Lagrangian based on the [SU(2)~]iocaihidden symmetry as outlined in section 3.1. In historical perspective, these proofs relate the ideas of Schwinger [4.3], Sakurai [4.4], Weinberg [4.5] and many others of how to implement spin-i particles in effective Lagrangians. 4.1. The Stuckelberg construction In the massive Yang—Mills approach, the vector mesons are introduced as chiral gauge this multiplets by 2)R 0 U(i)v]gIobal (or [U(3)L0 U(3)R]globaI). To relate approach explicitly gauging [SU(2)L® SU(
258
U. -G. MeiJ3ner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
to the hidden symmetry scheme where the spin-i mesons emerge as gauge bosons of a hidden local symmetry, we will make use of the Stückelberg construction [4.1, 4.6] following the work of MeiBner and Zahed [4.7]. First, let us outline the main idea of the proof. For that, consider the following massive Yang—Mills Lagrangian =
Tr[F~
~
+
F~]
—
[A~
~
+
A~],
(4.1)
with ALR~being left- and right-handed gauge fields and F~LRtheir respective field strength tensors. In the unitary gauge (valid for both L and R) gL,~(A~~R + ~)g~
=
=
gL,R
~
(4.2)
we can rewrite the Lagrangian (4.1) as follows ~
Tr[F~
+
F~]
—
~
Tr[F~
+
F~]
—
+2 (g~V~~g~)2] Tr[(g~V~~g~)
Tr[(g~V~~g~)2 + (g~V~~g~)2].
(4.3)
In other words, the massive Yang—Mills model (4.1) is gauge equivalent to a massless Yang—Mills model gauge coupled to non-linear if-models [4.8]. The Goldstone modes in (4.3) compensate for the longitudinal modes in (4.1). This is, of course, the well-known Higgs mechanism. Notice that if the unitary gauge is identified with the pion field, (4.3) suggests a relationship between f,,, m and g of the form 2
22
m =gf
22
agf,,
(4.4)
where a is a parameter to be determined by proper diagonalization of the pion field. For a = 2, this is just the KSFR relation. So, this sometimes mysterious relationship finds a natural explanation in the Stuckelberg construction. Now, let us consider the massive Yang—Mills model (2.6) without the w (for convenience), i.e. Tr[~U with ~
U] + ~
Tr[FL + F~]
—
~
Tr[A~ + A~]
(4.5)
the covariant derivative in the adjoint representation,
U—~U+A~UUA~.
(4.5a)
In the unitary gauge,
u
=
~
U~R,
ALR~=
~L.R VL,R~~L,R
(4.6)
U. -G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
259
we use the Stückelberg construction to recast (4.5) in the following form Tr[~U ~U]
Tr[F~~+ FL]
+ ~
Tr[(~ VL~L) +
-
(~VRRfl.
(4.7)
Notice that the Lagrangian (4.7) enjoys both global and local gauge invariance, GO H = [SU(2)L0 SU(2)R]global 0 [SU(2)L0
SU(2)R]local,
(4.8)
as defined by the following transformations: ~L,R(x)
=
hLR(x) ~L.R(x) g~.
(4.9)
Since we have two gauge groups, we can choose two gauge conditions. If we choose to work in gauge where U—fl,
(4.10)
~L~R~~’’
and use the KSFR relation, then we have Tr[(A~
-
AR~)2]+ ~
Tr[F~
+
FL]
-fW~VL~L)2+ (~R VR~R)~.
(4.11)
We should stress that (4.11) is a simple rewriting of (4.5), which has no dynamical content aside from assuming the KSFR relation. Let us now compare with the Lagrangian of Bando, Kugo and Yamawaki [4.9]. It reads (setting ~M = 1): 2’ =
+
+
+
2’d
+
=
-a(f~/4)Tr[(VL~L~+
=
—b(f~!4)Tr[(VL~L~VR~R~R)],
=
—c(f~/4)Tr[(AL~
=
-d(f~I4)Tr[(VL~L~
= (i
!4g2) Tr[F~
+
—
(4.12)
AR 2], 1j VR~R~R-
(AL~ -
AR))2],
FL].
It is obvious that (4.12) reduces to (4.11) for the following choices of parameters a=b=2,
c=—2,
d=0
(4.13)
because j~,,and the pion-field ir(x) are rescaled by \/~due to the irA 1~mixing.*Thus, the massive Yang—Mills approach is entirely analogous in nature to the approach based on a hidden local symmetry. *
This is most obvious if one works in more general gauges where u =
~
~
260
U, -G. MeiJ3ner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
Both approaches follow naturally from the Stückelberg construction. The above arguments show that the choice (4.5) for is just a particular case of (4.12). 4.2. Generalized proof in the presence of external gauge bosons In section 4.1 we made no use of the rather unique way of coupling electroweak interactions in the hidden symmetry scheme. Following Yamawaki [4.2, 4.10] we will now present a unified description of pions, and A1-mesons coupled to electroweak interactions and show its relation to the massive Yang—Mills approach. Furthermore, we can give a natural explanation why the arbitrary parameter a indeed should be a = 2 if one reduces to the Hiocai Lagrangian discussed in section 3.1. For that, let us start with theThe generalized model to which is invariant GglobaI 0 ~ where 2)L®SU(2)R. generalization arbitrary groupsunder is straightforward and can we be choose G = SU( found in ref. [4.11]. Gglobal is assumed to be fully gauged with external gauge fields L~(x)= ~
R~(x)= ~
T!2,
(4.14)
TI2,
where ~ is the (external) gauge coupling constant. For example, L~(x)and R~(x)may contain -y, and Z bosons (standard electroweak interactions). The dynamical variables are GgIobaI valued matrix fields ~L(x), ~R(x) and U(x), which are introduced as U(x)
=
~~(x) U(x) ~(x)
(4.15)
with U(x) = exp{i’r. ir/f~}.The transformation properties of U, section 3.4):
~L’ ~R
and U are given by (U
~M’
cf.
U(x)—~U’(x) = gjx) U(x) g~(x), +
~L.R(x)—~ ~L.R(x)
U(x)-~U’(x) with
gL,R
=
hLR(x) ~L,R(x) g~R(x),
=
hL(x)
(4.16)
U(x) hR(x),
E Gglobal and hLR(x)
E
Giocat. Accordingly, the covariant derivatives are defined as
D~L(x) d~L(x) iL~,(x)~L(x)+ i~L(x)L~jx), -
D~R(x)
D~U(x)
YR(X) —iR~(x)~R(x)+ c9,~U(x)
—
iL,~(x)U(x)
+
i~R(x)R~jx),
iU(x) R,jx),
where the gauge bosons of the generalized hidden local symmetry given by L,,~(x)gLr*I2,
(4.17)
R,~(x)=gRTaI2.
Giocai
with coupling constant g are
(4.18)
L,~(x)and R~(x)transform under Ggiobai as L~(x)~L~(x)= ig~(x)~g~(x)
+
g~(x)~(x)
g~(x), (4.19)
R~(x)-+R~(x)= ig~(x)~g(x)
+
g~(x)R~(x)g~(x).
U. -G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
261
It is now easy to write down the most general Lagrangian with the smallest number of derivatives which is invariant under gauged Ggiobai 0 G10~10 parity* 2’
=
a02’~+ bo2’A
+
cQ2’M
+
dQ2’IT
—
-~-~
Tr[L~ + R~]
—
4~2 Tr[L~~ +
R~~]
(4.20)
with =
-(f~I4)Tr[D~L~~
+
U D~R.~U~]
(4.20a)
—(f~/4)Tr[D~L.~ UDR.~U’~’] 2 = —(f~I4) Tr[D~U~U’~] = (f~/4)Tr[D~L~ UD~R ~U” —D,,~U U’~’]2.
2’A=
(4.20b) (4.20c)
—
(4.20d)
—
a 0, b0, c0 and d0 are arbitrary parameters. The kinetic terms of section 3). The gauge field strengths are defined via ~
—i[F~,F~],
~
.
and
are put in by hand (cf.
(4.21)
The idea is now to rewrite the Lagrangian (4.20) in two different bases, the so-called “hidden basis” and “external basis”, respectively. For that, define L~ns~L(X)L~jX)~ x)—iô~L(x)~ ~~(x), (4.22) R~
~R (x) R~ (x)
~~(x)
—
~ ~R(x)~~~(x),
and L~(x)~(x) L~(x)~L(x)+i~(x)
ô~L(x),
(4.23) R~(x) ~(x) R~(x)~R(x)
+
i~(x) ~R(x).
The pertinent covariant derivatives are D~U(x)
~ U(x)
—
iL~(x) U(x)
D,~U(x)
~
U(x)
—
iL~(x)U(x) + iU(x) R~(x),
+
i U(x) R~ (x), (4.24)
D~U(x)nsc~U(x) iL~jx)U(x) + iU(x) R1~(x). —
With that, the Lagrangian (4.20) can be rewritten as 214)Tr[(L —L~)+U(R,, —R~)U~’]2 2’v(f = (f~,I4)Tr[(L~ L~)+ U(R,~ R~)U”]2, —
*
—
Keep in mind that we are interested in strong interaction physics.
(4.25a)
262
U. -G. Meif3ner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
=
(f~I4)Tr[(L~
—
L)
=
(f~I4)Tr[(L~
—
L~) U(R~ R~)U~]2,
=
(f~I4)Tr[D~UD’~U~] = (f~I4)Tr[D~UD’~U~],
(4.25c)
(f~I4)Tr[D~UD~U”]= (f~I4)Tr[D~UD~U~].
(4.25d)
=
—
U(R
—
—
(4.25b)
—
The first basis (with the tilde) is the “hidden basis.” In this case, everything transforms under Giocat. In the “external basis” (without tilde), however, everything transforms under the gauged Ggiobal. Notice that 2’~,in the external basis is nothing but the gauged non-linear if-model discussed in section 2. Now, let us consider the hidden basis in the gauge U(x) = ~, so that Giocai is reduced to its subgroup Hiocai [SU(2)~] 10~51. The transformation properties of ~L(x) and 4R(x) are the same as in eq. (4.16) with the only difference that now h(x) = hjx) = hR(x) E Hiocai. Introducing vector and axial-vector fields V~(x)and A,,jx), respectively,
A~(x)=
V~(x)= ~[R~(x) + L~(x)], and similarly for V~jx)and
A~(x),one can
(4.22): V~jx)—~. V~(x)= ih(x) ~h~(x)
+
-
~[R~(x)
-
L~(x)]
(4.26)
read off their transformation properties from (4.19) and
h(x) V~(x)h~(x), (4.27)
-
A~(x)—~ A~(x)= h(x) A~(x)h ~(x), and similarly for V~jx)and A~(x).In this gauge, one can rewrite part of the Lagrangian (4.20) as + 2 [bb0~0
bo2’A + co2’M
+
d~
=
(b0
+
c0)f~Tr[A~
—
+ d
b0(b0 + c0)~A~]
0]f~ Tr[A~]. 0
0
(4.28)
Because of the ‘rr—A,~mixing, 4Tr[A~]=Tr[D~UD~U~]
(4.29)
and one has to normalize the pion kinetic term via simultaneous rescaling of 1T(x) and f,~,i.e. ~
\/~ir(x)~ir(x),
Z— b0+c0
+
d0.
(4.30)
Then, eq. (4.20) is given by ~ +
Tr[D~UD~U~] ~ —
Tr[V~+
A~] ~ —
Tr[V~ + A~]
(4.31)
U.-G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
with a
a0IZ, b
263
b0/Z and c = c0/Z. The vector and axial-vector field strength tensors are
~
~E{V~,V~}, (4.32)
~
A~E{A,~,A~}.
We have seen in section 3, the phenomenology of the p and A1 mesons dictates the choice of parameters: a = b = c = 2, and d = 0. In terms of the “bare” parameters this reads: a0 = b0 = c0 x, d0 = 0. The simplest choice is x = 1, in which case the Lagrangian (4.20) is given by 2) Tr[I~~+ R~] (1 /4~2) Tr[L~~+ R~]. (4.33) 2’ = + + (1 /4g Let us now consider the Lagrangian (4.33) in the external basis. In this case, Giocai is completely gauge fixed and no longer exists. Using (4.25), we have =
—
—
2’=f~Tr[V~ —V~]2+f~Tr[A~ A~]2+ (f~I4)Tr[D~UD~U~] —
—(iI2g2)Tr[V~ + A~~]—(1I2j2)Tr[V~~ +A~~].
(4.34)
If one identifies the p and A
1 mesons with V~and A~instead of and A~,respectively, the Lagrangian (4.34) is exactly the massive Yang—Mills Lagrangian discussed in section 2 except that it contains external gauge fields V~,A~ (-y, W, Z). The equivalence of both approaches becomes most transparent if one switches off the electroweak interactions as shown in fig. 4.1. Stremmitzer [4.12] has derived a Lagrangian like (4.34) with A, = V,~= A,~= 0. The Lagrangian (4.34) generalizes the massive Yang—Mills approach, with its mass term being gauged-G invariant (Stuckelberg mass term [4.1]). Finally, let us come back to the Lagrangian (4.33), from which the phenomenologically successful
hidden (-7r,1o,
A,
“local
,
“global
Switch off external gauge bosons E Gglobal
1’
_________________________ ‘~YM(iT, p. A,
Zhidden1~,P,I~t1 ,..) ‘‘local
,~
—
m
Gglobol
Fig. 4.1. Gauge equivalence between the hidden symmetry and the massive Yang—Mills approach. The hidden symmetry Lagrangian ~~‘hidd, contains dynamical gauge bosons (vector and axial-vector mesons) as well as external gauge bosons (here the standard electroweak bosons -y, Z°, W~).Switching off the external gauge bosons, ~hdden becomes gauge equivalent to .~yM(P’o, A ).
264
U. -G. Meiflner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
Ggiobal
A
0 Hiocai Lagrangian 2’
+
=
a2’v +
2’gauge
with a
can be derived. For this, assume that the
=2
1-meson mass is much larger than the p-meson mass, i.e., let us freeze out the A1 degrees of freedom. In that case, one can substitute A,,, in (4.33) by its equation of motion*
A,,,
=
[bI(b
+
c)]A,,,
=
,
(4.35)
and one arrives at the irp-Lagrangian originally proposed by Bando et al. [4.13] with a = 2+ (f214)(D U D~U~) -
= 2f~
Tr[V~]
~
Tr(V~ V~
2:
-
4~2TrW~~ + A~V]. (4.36)
A similar mechanism to eliminate the A1 degrees of freedom will be studied in the next section with the inclusion of the WZ term. 4.3. Gauge symmetric constraints to eliminate the
1
Here we want to demonstrate that the massive Yang—Mills Lagrangian derived by Kaymakcalan and Schechter [4.14] (cf. section 2.5) is indeed equivalent to the non-linear Lagrangian of Bando et a!. [4.13]. We will further discuss the WZ term in Bardeen’s form (cf. section 2.2). For that, let us start with the gauged non-linear if-model in terms of pions, p and A1 mesons (4.37)
2=21+22,
22
=
~ Tr[~,,U 2lJ’~U~] ~ Tr[F~,,.~+ F~~] + y Tr[FL,,,,~UF~U~1,
(4.37a)
=
m~Tr[A~+ A~]+ B Tr[A,,,LUA~U~],
(4.37b)
—
with the usual definitions of the covariant derivative U = U igA~U + igUA, with the left- and right-handed generators ~ = ~Ta(pa(±)aa)given in terms of the p and A1 mesons, p~ = p~r*I2 and A~= aT*12 [4.14, 4.15]. The field strength tensors F~R are given by F~’~’~ = ~A~R ÔAL.R ig[A~’~,Ar]. Of particular interest are the third term of 2’~and the second term of 22, these are higher-order chiral invariant terms (non-minimal terms). The coupling constant y and the generalized mass parameters B will be fixed in what follows. Let us now eliminate the A1 mesons from the Lagrangian 1. This can be achieved by imposing the chiral gauge symmetric constraint —
—
U=0.
—
(4.38)
This constraint has the virtue that it preserved all symmetries of the effective Lagrangian. It is equivalent to the Weinberg trick [4.16]to eliminate the A1. The pions are taken as gauge parameters of chiral symmetry, i.e., one chooses g~= VU and g = VU. In that case, we can rewrite A~ and *
This is the same procedure used in section 3.lto eliminate the p-meson from the non-linear if-model (cf. section 3.1).
U. -G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
265
U=exp{ir~ir/f,,}as A~=
VUp~VU~ + (ilg)VU ~
A~= VU~p~\[U + (iIg)VU~~[U,
(4.39)
U=VUIVU. Under global SU(2)L ®SU(2)R (parameterized by ~LR) we find the following transformation properties
VU-~VU’= g~VUK=KVUg~, (4.40)
p,,,,—*p,’,, =Kp,,,K’~—(i/g)9,,,KK’~ with K = K(\~U,g~,g~)an auxiliary matrix as defined by Coleman, Wess and Zumino [4.17]. Under diagonal isospin SU(2)v* one has the conventional transformations
VU-+ VU’ = g~VUg~, (4.41) = gvp~g~ (iIg)~g~g~. This shows that the p-meson is still a gauge field of diagonal isospin, whereas the A -~
—
1-meson is given in terms of ‘rr and p as a~ flebcdpcird Enforcing (4.38) on the Lagrangian (4.37), we observe that the non-linear if-model part of 2’~vanishes and only the gauge kinetic term survives: =
(y
—
1) Tr[F,~,,(p)F~(p)],
(4.42)
where the field strength tensor F,,,,, ( p) contains only the p-mesons: F,,,,, ( p) = ô,,, ô,, p,,, ig[ p,~,p,,]. Under the constraint (4.38), the non-minimal part of 2’~contributes in the same way as the canonical term ~ To achieve conventional normalization, one has to set y = 1/2. In a similar way, the generalized mass term 22 transforms into 2’s, —
=
(B +
~
+
2mg) Tr[p~,,] + (ilg)(B + 2m~)Tr[p’~9,,,VU VU~+ .
Tr[a~VU o~VU~] -
~
Tr[VU~
.
—
o1~VU~ VU)]
~VUVU~o~VU].
(4.43)
Complete equivalence to the hidden symmetry Lagrangian derived by Bando et al. [4.13] =
m~Tr[p~,,] + iag~~~f~ Tr[p”(o.,,,, +
*
(a+ 1)Tr[~.~~
—
SU(2)~may be parametrized by group elements g~,.
~.
~
+
~
4:+
.
(a— 1)Tr[~~
.
~
(4.44)
266
U. -G. MetJ3ner. Low-energy hadron physics from effective chiral Lagrangians with vector mesons
for a = 2 follows by fixing the parameters B, m0 and g as m~=~m~, ~
B=~m~
(4.45)
and identifying 4 with VU = exp{i~i~I2f~}. Obviously, the KSFR relation holds (mp = Let us now turn to the anomalous sector. Imposing the constraint (4.38) on the gauged Wess— Zumino action F,~from (cf. section 2.2, eq. (2.45)), with the (U-field introduced via U(1)~-gauging,we find (without Bardeen-subtraction) ~
p~,w~]= (~g)w~B~ + ~
(4.46)
This is nothing but the general results (eq. (2.45)) with A1 = 0. This can be easily understood. The Lagrangians with A1~equal to or different from zero are related by a chiral gauge transformation. A gauge variation of the WZ term gives the anomaly, and for SU(2) ® SU(2) the anomaly vanishes. Evidently, the procedures (4.38) or (4.39) do not influence the anomalous processes involving pseudoscalar and vector mesons. * So for a given WZ action in terms of p, A1 and o, we have a unique way to eliminate the A1 degrees of freedom.
5. Connection to QCD In this section, we will make contact between the effective Lagrangians considered in sections 2 and 3 and the fundamental underlying theory, QCD. First, we will demonstrate how a microscopic derivation of chiral dynamics from QCD can be carried out in the low-energy and large-Ne domain of QCD, following the work of Karchev and Slavnov [5.1]. The main assumption going into this work is that chiral symmetry is spontaneously broken, with the appearance of massless (pseudoscalar) Goldstone bosons. This enables us to give an approximate calculation of the first coefficient appearing in the chiral expansion, namely the pion decay constant. Furthermore, the anomalous Wess—Zumino action can be derived. In a similar fashion, following the work of Ball [5.2],we will demonstrate that with the same assumptions one can indeed “derive” the hidden symmetry Lagrangian of Bando et al. [5.3]. Finally, we will show how to convert the generalized Nambu—Jona-Lasinio (NJL) model [5.4] into a massive Yang—Mills theory including vector mesons. There, we argue that the Nambu—Jona-Lasinio model is an intermediate model between QCD and purely mesonic Lagrangians. The NJL model can be motivated from lattice QCD in the strong coupling (g—* cis) and l/d-expansion [5.5, 5.6, 5.7], where d is the number of space—time dimensions. In this limit, lattice QCD reduces to a nearest-neighbor, i.e. four-fermion, interaction reminiscent of the NJL model. We will follow the work presented in refs. [5.8, 5.9, 5.10]. For earlier studies on these topics, namely the relation of the generating QCD functional with passive gluons and effective chiral theories (like e.g. the Skyrme model), we refer the reader to the review by Zahed and Brown [5.11]. *
See also the discussion given in section 2.5.
U.-G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
267
5.1. QCD in the large N~limit and microscopic derivation of the pion decay constant As first pointed out by ‘t Hooft [5.12]and Witten [5.13] QCD reduces to a theory of weakly interacting mesons for the gauge group SU(N~),with N~—~and g2N~* fixed. For that, let us rewrite the QCD-Lagrangian with explicit color-factors —(N~/4g2)F~F~”~ +i~fr(/—54’)~,
~=
(5.1)
with A,~= A T~,when Ta are the generators of SU(N~).The Lagrangian (5.1) exhibits a U(Nf) X U(N~)chiral symmetry. ~fr denotes the quark fields in the fundamental representation. If ta denote the generators of the U(Nf) group, one can introduce composite operators (“mesons”) and consider the generating functional
f
Z[i~] = Z~ [dA][d~][d~] exp{iJ [~(x) + ~ 75t~0(x)]},
(5.2)
where gauge-fixing and ghost-term have been suppressed. Defining chiral quark field via tp~p)~, ~ch= ~(~P+P)
(5.3)
~ch=(~
with ~RL = ~(1 ±‘y5) and L~(x)= exp{2i~”(x)t’~}, we can use the Faddeev—Popov trick to impose a color-singlet subsidiary condition on the quark fields this is necessary for constructing an effective theory of colorless objects. With that and the change of variables (5.3) the generating functional Z[~] takes the form —
f
Z = Z~’ [dA][d~][d~][dQ]~Jl~, x expf
if [~(x)
+
~)
i~(x)y~L~PR ~(x)
+
(~ch)-l 5ta(~ch)-l
(5.4)
a()]}
7
with L,,
(1~’o.,~11 and we have used the exponential representation of the ô-function 1 = (5.3), it contains the information about the Wess—Zumino term, i.e. the anomalies, as we will demonstrate later on. Let us focus for a moment on the source term in (5.4). To leading order in i~= 2~~ta, and using 12(x) = 1 + ~r(x),we see that it behaves like (— i /2) ~!i { IT, t’~} + ~ so that terms of higher order in ~a are 1 / N~suppressed. Therefore, we integrate out all color-fields in (5.4) and obtain ~
i/i)
=
J ~(~ry5t~i)[d~2].**~I is the Jacobian of the transformation
Z[i7]
f
[dfi] expfi
J
~eff(IT,
~~)}
(5.5)
with an effective Lagrangian ~eff’ which depends only on the meson fields. In the low-energy limit and to leading order in ~ we can calculate ~ff(Ir, ~). Before doing so, let us briefly return to the * **
Here, g denotes the running coupling constant of QCD (not to be confused with the gauge coupling g = ~
which was used before).
For the reader who is not familiar with these manipulations, we refer to the book by Faddeev and Slavnov 15.11].
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Jacobian IJ~of the transformation (5.3). As we will outline in detail in section 5.3, it contains the anomaly structure of QCD and is given by In jJ~= ~
J
~
+
E~
TrJ F~(x)F~(x)IT°(x).
(5.6)
is the Wess—Zumino (WZ) term, which is topologically quantized:
~wz
—
NC
f
dTJ
E~
Tr[L5(r)
L~(T) L~(r)L~(T)Lff(T)],
(5.7)
5 = M4 ® [0, 1] to define the fields and currents via [5.11, 5.13] where we use the normalized manifold M L~(x,T)= 12’(x, T) ô~Q(x,T), Q(x, T)=exp{iTIr(x)} (5.8) L 5(x, r) = 12’(x, T) o~f2(x,r). The second term in (5.6) is obviously the Abelian anomaly connected to IT°—~2y. Here, ~r°corresponds to the generator t°= Let us now return to the representation (5.5) of the generating functional. To arrive at this form, one first has to integrate over all gluon fields. Keep in mind that we want to integrate out all colored degrees of freedom. The gluons interact with matter via ~ and ir°F F (where Fap = ~ is the dual gluon field strength tensor), therefore, the integration over the gluon fields will contribute to the generating functional via 1y~/i,IT0]} . exp{iS0[~/iT’ The important observation to make is that the functional S
(5.9)
0 depends only on color-singlet, bilocal fields of the form ifr~(x)~fr~(x),where a is a color-index and i, j denotes flavor. This can be easily understood if one uses the double-line notation of ‘t Hooft [5.12] and Witten [5.13]. The gluons and ghost field propagators can be represented by two differently directed lines, as shown in fig. 5.1. The index lines are continuous or form closed lines, so that only factors N~or color-singlet combinations ~l’a (x) ~l~a ( Y) can appear. It therefore is convenient to introduce bilocal variables B”(x, y) [5.15, 5.16] and rewrite (5.6) as exp{iS0) =
f
[dB] 6(N~B”(x, y)
-
~(x) ~(y)) exp{iV0(B,
IT)).
(5.10)
Injecting this into (5.4) and using the exponential parametrization of the 6-function in (5.10) with a A’. IJ,]
I
I
I
Fig. 5.1. Double line representation for the gluon propagator in the adjoint representation. The solid lines denote quarks and anti-quarks.
U. -G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
269
Lagrange-multiplier ~iU gives
f
Z[~] = Z~ [dQ][d~][d~] xexp{i[V(B,
iT) +
~~~(L)+ N~Tr(~tB) —N~Trln~’]},
(5.11)
where J2~’is the fermion operator D = ~ + LPR ~ to be discussed in section 5.3 [5.17] and we have omitted any ghost-field contributions, since the integral over the ghost fields ~, c does not contribute to leading order in N~[5.1]. ~a is the Lagrangian multiplier related to the 6-function imposing the Our main interest now is to study the functional color-singlet subsidiary condition F~= ~/Jy5t~/i. V(B, IT°). For that, notice that we can divide it into two parts V(B, ir)
V(B, 0)
=
+
V(B, IT°),
(5.12)
where V(B, IT°)includes at least one source IT°.According to the large Ne-counting rules, the behavior of V(ir°)is entirely determined by planar gluon diagrams and does not depend on B* i.e. V(B,
ITo)
=
V~(IT°) + C(N~’).
(5.13)
In a similar fashion, we can expand the part V(B, 0) as V(B, 0)
=
N~[V~(B)+
V~(B)+
~f~)].
(5.14)
It is now straightforward to expand the action in (5.11) to leading order in N~for integrals over the fields B, ill, ç Near the stationary point, the action is extremized by (the subscript “st” denotes quantities at the stationary point) iB~(x y) —
=
[(is — (5.15) a_
a_
—
~iU~~(x—y)— ~B ) 40 — IT — —0, where we have neglected the ghost-fields. It is now crucial to assume that flavor U(Nf) ® U(Nf) is spontaneously broken down to the diagonal subgroup U(Nf). In that case, the stationary solution (5.15) acquires the particularly simple structure B~=511B~~, ~
(5.16)
The assumption of spontaneous breaking of chiral symmetry means that the Fourier transform F[..~U] = ~1(p) tends to a non-vanishing constant as the momentum goes to zero. This is equivalent to the statement that the quarks acquire a dynamical mass, mdYfl cC [d~((p)/dp]~ 0 [5.18, 5.19], which breaks chiral symmetry. We are now at the point where we can calculate the effective action for the 1. *
Remember that quark loops are suppressed by a factor N~
270
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fields IT’2(x). At low energies, the terms quadratic in the current L,1 are dominant, therefore we will now calculate this contribution. For that, we expand the exponent in (5.11) to second order in the fields .A1 and B (these are the leading terms as N~tends to infinity) around the stationary point. It has the form ~,
~=
~~L)
+
V
0(ir0) + NC Tr{—~LPR~PLPR + + ~‘ttcM 5+~i 5~LPR_~Wcoy5+.AIB+ ~BKB}, ~iLPRcPJ/1+~iLPR’P40y 40y
—
(5.17)
where the coefficient functions in the expansions of Tr ln P~ and 17 0(B) are denoted by ‘P and K, respectively. It is now straightforward to integrate over the fields B, so that oneequation obtains a quadratic 1 = — K’. At large N~,G satisfies a Bethe—Salpeter form for the field At, G G=K(1+G) (5.18) which is graphically represented in fig. 5.2, where to each fermion line corresponds to the quark propagator B 55(x y). The poles of G(x, x’; y, y’) determine fermion—antifermion bound states, one of them being massless due to Goldstone’s theorem [5.20]. This massless boson is obviously a pseudoscalar particle (cf. discussion after eq. (5.4)). We will assume that no other massless states exist in the spectrum. Integrating now over the fields .~1t,the Lagrange function ~ (eq. (5.17)) takes the form —
~[=$~~~(L)+V0(ir°)+
~
5~~Y5çP +
—LP~’P~y5ço +i40y5~GLP~}+...
(5.19)
xTr{—jLPR~GLPR+ çDY
with
~1G=
~
F,alT’ 1 (x
+
cPGq~.For convenience, let us introduce the following abbreviations
—
y)
=
—
y)
=
F~(x— yu)
Tr{ySt~PGy5tb}, . a 5b Tr{1y,1PRt ~‘~GY t =
}
(5.20)
,
~
Using (5.18), it is easy to realize that the two-point functions F’~”(X— y) are represented by a sum of 1,we have diagrams as shown in fig. 5.3. From there it is obvious that to leading order in N~ N~F2l)(x— y) = i( T ~/i(x)y5t” ~/J(x)~/i(y)y5tb ~i(y)) (5.21) and similar expressions for F” and F~. The Fourier transforms of the functions F exhibit the
+ Fig. 5.2. Bethe—Salpeter equation for the quadratic form G for large N, determining the field Al defined in (5.11).
÷
+...
Fig. 5.3. Iterative solution of the Bethe—Salpeter equations to determine the Green’s functions F~”(x— y).
U. -G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
271
Goldstone pole at low momenta F[F
ab
ab
ab
2
]=F (p)=—C6 Ip +C(p),
F[F~] =
rab()
(5.22)
+2C(p), ~iCf6~p,1/p
=
F[F~] = F~(p)= ~ 6~(_g,1P+
+ C(p),
where f and C are constants to be determined. Finally, we have to integrate over the fields 40a,
which
gives the promised effective Lagrangian: 2~~(L) + V(ir°)+ ~NCL~F~L — ~NCL~F~CF FbL~. ~ Using the expansion (5.22), this can be cast into the familiar form ~eff
(5.23)
=
5~=_~f~Tr{L,1L~} + L~L,1V 0(IT0)
—
~ 48 ir
f
dr ~
Tr{L5(r) L,1(r) Lv(T) L~fr)Lff(T)},
(5.24)
0
with the pion decay constant f~, given by 2. (5.25) f~= (N~/4)f Some remarks concerning the derivation of (5.24) are in order. Although we have restricted ourselves to low momenta, the Goldstone poles could give rise to non-singular coefficients in any order in L,1 * As we have shown, in the leading order (second order in L,1) these poles are explicitly cancelled. This cancellation is intimately related to our assumption of imposing color-singlet subsidiary conditions on the collective variables ~/iy,1t’2~fr. In ref. [5.1], arguments are given that these cancellations hold in higher orders, too. For that, it is vital that the constant C as defined in (5.22) is non-vanishing. Finally, let us comment on the functional V~,(ir°). Obviously, since ir° is related to t°= ~\/~7N,it contains information about the ~‘-meson. To leading order in N~1,we can evaluate V 0(ir0). Assuming that it has a non-vanishing vacuum expectation value, it generates a mass for the i~’which is given by .
m~
-~
1
f~,256ir
41
(T[Tr(P(x) F(x)) Tr(F(0) F(0))])
(5.26)
.
in agreement with Witten’s result [5.21]. Since f~ N~and ([Tr(FF) Tr(FF)]~ N~,we see that N~’.Therefore, the restoration of the U( 1)-symmetry does not contradict the assumption that to 1,U(N~)®U(N~) is broken. leading order in N~ In principle, one could continue the present analysis to calculate the coefficients of the higher order terms in L,1. Unfortunately, for that purpose it is vital to know the pole as well as the regular terms in -~
-~
*
Only terms of even power in La are allowed because of time-reversal invariance.
U. -G. Meijiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
272
the Bethe—Salpeter amplitude to determine the function V0(B). This would require a complete summation of all planar gluon diagrams. This can only be done if one makes further model assumptions. We will therefore leave this microscopic picture of deriving effective chiral Lagrangians and present in the next two sections how the generating functional of QCD can be linked directly to an effective meson-theory including vector (and axial-vector) mesons. 5.2. Vector mesons and QCD Here, we will show how one can reduce QCD at low energies to an effective theory which incorporates vector mesons, the hidden gauge symmetry discussed in section 3 and vector meson dominance. The assumptions going into these manipulations are similar to the ones made in section 6.1, namely that QCD at low energies corresponds to a theory of weakly interacting, color singlet and local meson fields (and glueballs), with the U(N~)® U(N1) chiral symmetry spontaneously broken down to SU(Nf) ®U(1). With these assumptions, one can “derive” the successful phenomenology of the spin-i mesons, with their masses and decay rates determined by the scalar/pseudoscalar sector. We will follow the recent work of Ball [5.2]. Let us consider the QCD generating functional in Euclidean space—time, ZE
=
f
[d~][d~][dG~] exp{_
J
~(/
+
+
th)~+ ~
+~
Tr(F~)}.
(5.27)
are quark-fields which interact with N~— 1 gluon fields G,1 = G T’~,where T~’are the generators of SU(NC). Again, we have omitted the gauge fixing and ghost field contributions. A~LW is an external electroweak gauge field which couples to flavor, we will later on restrict A~to be the photon field. th is flavor-diagonal bare quark mass matrix, for ,fI = 0 we have an exact U(N1) ® U(N1) chiral symmetry. Our aim is now to express the generating functional in terms of meson fields, since these are the relevant degrees of freedom at low energies. Mesons are quark—antiquark composites, therefore, we should integrate out the gluon fields. This can be achieved formally by summing up all one gluon irreducible Green’s functions, so that ~s
~,
~ x
x1
(5.28)
x*
where we have introduced the covariant derivative D,1 = + A~W.j (x) ~/J(x)y,1 Ta qi(x) is the chiral singlet local current to which the gluon field G couples. The idea is now to make statements about QCD at low energies without knowing the exact structure of OGI Green’s functions x~).For that, introduce bilocal fields B which at low energies describe the mesons, using ~,
1=f [dB][dA~]exp{If Tr ~1t(y,x) (B(y, x)
-
~(y) ~(x))},
(5.29)
with At(x, y) an auxiliary (Lagrange-multiplier) field. Injecting (5.29) into (5.28), we find ZE
=
J
[d~][d~][d~][dB] exp{_
J
~+
i$i)~+JfJ~(B -
~)
-
F[B]},
(5.30)
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273
with
X
Tr[’y,~T~B(x1, x2)
Tan B(x~,x
12...
1)]
y~T’
(5.31)
.
Since B(x — y) are non-propagating fields, we can easily integrate them out in terms of a Legendre transform on F. We end up with ZE
=
f
[d~][d~][d~] exp{-
[1
~(~+ m)~+
Jf
~(x) J~(x,y) ~(y)
+
G[J~]]}.
(5.32)
G[.At] sums up all quark self-energy insertions via the bilocal field .At(x, y) since .itt[B] generates one-meson irreducible Green’s functions,
Z[~1t] = e~
=
f
[dB] exp{—[F[B]
—
Tr(~ttB)]}.
(5.33)
Up to now, we have only performed formal manipulations on ZE.* We will have to make some further assumptions on color confinement and locality to show that dynamics of the color singlet part of the fields ~~1t is indeed that of the mesons at low energy. For that, consider integrating out the high energy modes of QCD down to a scale A. (A might be estimated to be A 4irf,~ 1 GeV [5.22].) This new action with the cut-off scale A must allow us to describe low-energy scattering processes 1 1.and(xtherefore has the following features. First, glue must be confined, i.e., the Green’s functions F, 1 .. x~) must be color singlets for x~— x,j > A’. If one furthermore assumes that possible glueballs have masses larger than A, the ~ are entirely determined by the gluon-condensate and have no poles. This means that we can factorize F~in the following way: -~
.‘~
.
F~ .ff;(x1
x0)
(5.34) 1 and 611 ~n (i = a, ,u) are symmetrical products where the ca’s are constants on should scales be Jx~confined, x,,j > Ai.e. B(x, y) and iU(x, y) must be strongly localized of 6-functions. Second, quarks on the scale A’ since we do not want to have free quarks at energies ~A.Therefore, we can expand .At and B in the form . .
. . .
=
6a1
an6
. ..
,1Acfl , —
A4 A1(x, y)
=
z
M(z) f(t) + At,,(z) çf’(t)
+
+
4t,,~. . . ,1(z) ç 1
~ f~”~(t), (5.35)
B(x, y)
=
B(z) g(t) + B,1(z) 1,1 g’(t) +
.
+
B~,.. ,1(Z) t,11 .
t,1
g~(t),
where we have defined z = (x + y)A/2 and t = (x — y)A/2. f(t) and g(t) have to vanish rapidly as ~. Furthermore, if we expand ~/i around z, i.e. ~i(x) = ~i(z) + (c/A) ô,1~l(z+ *
t)~~0+...,
Note that we have not taken care about the trace and the U(1) anomalies.
(5.36)
U. -G. MeiJ3ner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
274
we can separate each term in the action (5.30) as
ff fJ Jf
~(x) ~t(x, y)
y)
~(
=f
B(x, y) ~t(x, y)
A4
=
J
J
~(z) ~(z)~(z) f(t)
f
B(z) ~1(z) f(t) g(t) +...,
y,1 B(x, y) Y,1 B(y, x)
=f
Y~B(z) Y,1 B(z)
(5.37)
f
g2(t) +...,
and similarly for all other terms. From (5.37) it is obvious that integrating over momenta larger than A means integrating over f and g, rather than itt(z) and B(z). Therefore, one gets resonances at the stationary (semi-classical) points of f and g. f(t) can be interpreted as the meson wavefunction, for A —~1 GeV mesons have a size r~—- A~-~0.2 fm. We further have to assume that dissipation of a meson into a quark—antiquark pair above threshold* is suppressed since localization of .At(x, y) does not exclude this possibility. Third, since .At and B contain also color-octet mesons [5.23], we have to assume that these states acquire masses larger than A, and thus decouple from the low-energy dynamics. This means that chiral symmetry remains unbroken in the octet channel. With these assumptions, we can rewrite the action (5.29) in terms of the color-singlet fields i1t and B as follows
JJ
~(x) ~t(x, y)
y)
~(
=
J
a~ ~(z)At(z) ~(z)
JJB(x,y)At(x, y)=~ofB(z)~i1(x)+.~,
ff
2~(y, x) Y,1 B(x,
~)
=
~~2f
(~
(5.38)
B(z))2
2y,1 B(x, y) F~
with a~,f3~and ~s, (i = 0, 1, 2,
. . .)
unknown constants. Obviously the local fields .
1t1, Jtt,1, At~, correspond to generations of mesons of increasingly complicated tensor structure and, presumably, increasing masses. Expanding them in spinor space gives for .~U (5.39)
~
5q, ~y,1qand qy57,1q, i.e. scalar, where the different parts correspond to the composite operators cjq, qy pseudoscalar, vector, and axial-vector mesons, respectively. Similarly, decompositions can be written down for dtt,1, .At etc. With this, the gluonic potential G[~iU] takes the form G[~]=fJ{_Eo+[2Tr(M_N,1N,1)+~.] +
*
[~r Tr((!~IM)2— N,1N,1IcIM + ~Ic1N,,I~1N,1 + 8(N,1N,1)2)
This threshold is around 0.6 GeV, i.e. twice the constituent quark mass.
+ . . .1± O(A16/A2)).
(5.40)
U. -G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
275
E0 is the vacuum energy density and r = (3/8)(~sI~8), where /h8G[At] is the isconstant to the colorand octet 4~(y,1B)4. locally related chiral invariant, it pieces stemming from the part of the action —~F~ contains no derivatives in ~ [5.24]. Obviously, we have here an expansion of G[AtI in powers of Al2, which makes sense since the weakly interacting mesons At cannot have strong gluonic self-couplings, otherwise meson bound states (exotics) would form, and these are not observed in the low-energy spectrum. The suppression of purely gluonic interactions, mediated by Zweig suppressed decays, is consistent with Regge phenomenology, i.e. the exchange of hadrons. Notice that all these statements are nothing but a reformulation of Wittens large N~rules [5.13] as it should be. Before addressing the properties of mesons from the effective action, some remarks concerning the pseudoscalar singlet in G[A1], i.e. including the U(i) anomalies, are in order. For that, U(N~)®U(N~) has to be broken to U(1) ® SU(Nf) ®SU(Nf). This can be achieved by introducing a 0 + chiral singlet field proportional to with non-vanishing expectation value [5.25, 5.26]. In a similar fashion, the QCD trace anomaly can be reproduced by including a scalar 0~-fieldproportional to ~ We will consider here only a minimal version where the e-terms, the U(1) and the trace anomalies are neglected. In that case, G[A1] depends only on one parameter ~t for each generation of mesons. Therefore, we can get quantitative predictions without detailed knowledge of the non-linear gluon dynamics. Let us now consider the first generation of mesons .A1 = M + 7,1N,1 (cf. eq. (5.39)), i.e. the scalar, pseudoscalar, vector and axial-vector mesons. The generating functional ZE (5.32) reads* ZE =
f +
[dM]A[dN1A[d~]A[d~]A expf
J [~+
th)~+ ~
+~
~+...]}
~ ô,1~(M+~)ô,1~+4Tr(2MM N,1N,1)+(eterms)+7
(5.41)
where the subscript A on the measures denotes that we have to integrate only over fields which vary more slowly than A’ (low-energy approximation). ir~is a chiral pseudoscalar singlet and the related unknown coefficient. g is a fermion—meson coupling constant. It is interesting to study the zeroth order approximation to (5.41) for which g = = = 0. In that case, the integration over N and M becomes trivial and we end up with an extended Nambu—Jona-Lasinio model [5.4, 5.8, 5.9, 5.10] ~
Z~=
f
[d~]~[d~]~ exp{_
J [~(~ ~
4
-
~)2
-
~5~~)2]]}.
(5.42)
2,chiral symmetry alone would Notice that the* * interaction hasback onlytoone —2/j~ allow for two. This can beterm traced the overall fact thatconstant the above four-fermion interaction is a Fierz rearrangement of ~ ~fr)2 i.e., that QCD is a vector-like theory. It is important to stress that the more complete model (5.41) shares most of the qualitative features of the NJL model, e.g. the meson kinetic terms are entirely due to the quark kinetic term ~Ii/~/i. If ~2 ~ A2, the quarks will not appear as asymptotic states, since they couple strongly to mesons. Integrating out the quarks, we get ZE
* **
=
f
[dMJA[dNIA exp{N~ln detA (4~ + th + M + ~Y)— ~—
i is defined by xy,, = with the exception of the bare quark mass matrix. We will come back to this point in the following section 5.3.
f
Tr( ~ 1dM
—
N,1N,1)
+
. . .
}.
(5.43)
U.-G. MeiJ3ner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
276
The baryons will emerge as chiral solitons of this action. To deduce the physics contained in (5.43), let us reinstate the fermions for a short while and try to find the vacuum. By Lorentz invariance, only M can acquire a v.e.v., so that we can set N 0. For that let us rewrite M in terms of a “chiral phase” u = exp{iy5r. ii/2f), i.e. =
M=ü.~ut,
(5.44)
where u is unitary (u~u= 1) and 2~Hermitian
(.~ = ).
With th
=
0 and
A~,W=
0, the generating
functional takes the form of the chiral quark loop model [5.30, 5.31, 5.321, i.e. =
f
[du]A[d~] 4[d~]A[d~]Aexp{_
J
u~Iut)~ + (p2/4) Tr(2~/2)}}.
+
[~(/
(5.45)
Even more conveniently, we use Simic’s constituent gauge [5.33] to rewrite (5.45) in terms of constituent quarks T~/i, j=~iü, (5.46) q=u which are nothing but current quarks dressed by a pion cloud as shown in fig. 5.4. If J~denotes the Jacobian of the inverse of the transformation (5.46), this leads to ZE
=
f
exp{-f
~
[~(/
+
üt/u)q + ~q + (p2/4) Tr(2Z/2)I}.
(5.47)
The mechanism of chiral symmetry breaking is the same as in the NJL model; for that, let us derive the effective potential of the scalar field. We define = ~ + &~= oi + &~, the effective potential V(u2) follows using standard procedures: .~‘‘
V(u2)
[u4(ln(~-~)
= —~-~
+
~
0)c1A2O~2]+ —
~—
Tr(~—),
(5.48)
c0 and c1 being constants of order unity. They parametrize the dynamics in the transition region from 2= (~q) the high-energy to the low-energy modes. Minimizing the effective action (5.47), we find o~ which is non-zero as long as u2(c 0, c1, p~) 0. Expanding the fermion determinant in (5.43) to second order in derivatives of (/o-) and introducing U Ciu~(with U incorporating the isovector Goldstone modes), we find ~(2)
=
~ Tr[a,1U ~~U] +
~
Tr[~,1(a2) ~
(5.49)
= /
mq
m~
Fig. 5.4. Constituent quarks as current quarks dressed by a pion cloud.
U.-G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
277
where f is given by
f
“~C
2[1(A)
c~ 3]
+
(5.50)
—
and has to be identified with mq the ispion constant. Frommass. the effective potential can three read 2mq, where the decay constituent quark The model action(5.48) (5.43)wehas off m~ = 2oparameters (~, c 3, one 0, c1),~aqq if one chooses to fitf,1 = = 93MeV, mq so = 350 (~q~ = be —(220 MeV) as a finds rn, = 700 MeV, = 4.3 and F(o—t. 2iT) 1.5 GeV, thatMeV theand r cannot observed resonance. If one restores the external photon field AW, the induced Wess—Zumino term describes properly the abnormal parity processes ir°—*2-y and -y----* 3ir. For non-zero current quark masses th = 0, one recovers the usual chiral perturbation theory results, though m~= i80 MeV ~(m~+ rnd) 7.5 MeV induces the usual problems in the SU(3) sector. Let us now come back to the vector meson content of At by restoring N, the spin-i field. For that, we decompose N,, N,,v + y5N~in vector and axial-vector parts. First, we have to relate the physical spin-i mesons S 1,, to N,,. For SU(2), we can conveniently write them as 5A,, iw +iTp,, +iy5D,, +iy5TA S1,, = V,, + y 1,, (5.51) ~‘-
The 5y,,q. p and the A1 are most naturally described by constituent quark composites of the form t~T*y,,qand Using qi-”y
~/i(/~+M+fr1)çb=~(/+I
+u~u+u~/u+
üt~u)q,
(5.52)
one defines S 1,,
u~N,,u+ u~~,,u + utA~~u.
This implies N,,’uS1,,u t +u9,,u t —A,,ew =u~,,u
(5.53)
.
(5.54)
Note that N,, and S1,, are related non-linearly, so they will not give the same results for multiple pion scattering at tree level. It is important to stress that with these definitions the action possesses a hidden U(Nf) ® U(Nf) chiral gauge symmetry [5.3]. If we parametrize the external chiral symmetry by group elements g and the internal (hidden) symmetry by group elements h, the various fields in the action transforms as follows: q—*q’=hq, t, -*~‘=h.h~,
i/J—~~~J’=g’/i,
M—*M’=~Mg u—* u’ = guht,
A~—3(A~)’ = gA~gt + ggt,
(5.55)
S 1,, —* Si,,
= hS1,,ht
+
h c9h~,
N—* N’
=
gNgt.
Obviously, the vector and axial-vector mesons are gauge bosons of the hidden symmetry. Furthermore, it is interesting to note that in terms of the hidden gauge, the relevant fermions are the constituent
U. -G. Meiflner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
278
quarks — therefore, this gauge has been called “constituent gauge” by Simic [5.24, 5.33]. With 5A,,, the generating functional (5.43) takes the form = V,, + y ZE =
f
[duIA[d~}A[dV]
tthu + ~ + 4[dA]AJ~exp~N~ ln det(~+ü
~
~
-
(Si,,
-
-
u~(ô,,+ A~)u)2+
.. .]}.
(5.56)
Notice that the pion field enters the functional through the Jacobian J~,and that the pion kinetic term is generated by the gluon potential. Performing a derivative expansion of the quark determinant, we find for th = 0 (after chiral symmetry breaking)
—
2
—
~—
Tr[D,,U D~U]— ~Tr(V~~ + A~)— ~2 Tr[V,, —~—j-Tr[y~A,, ~—~-~-— (ut D,,u — 12t D,,i~)] ,
—
~(u~D,,u + ~ D,,~)]2
—
(5.57)
where we have ase-,
2,
ai
f, is
(5.58)
g2~_~
~
the physical pion decay constant. We have f2 >f~,so that in the presence of p, A
1-mesons the cut-off A will be larger to accommodate these particles. As before (sections 2.1, 2.3), we have to redefine the ii and A1-fields to get mass eigenstates. This amounts to A,,—* A,, — (a — 1)19,, IT/afT. Integrating out the A1, which should be a good approximation below 1 GeV, by using its equation of motion and ignoring the kinetic term of the A1, (5.57) gives (to second order in derivatives): Tr[D,,
=
U
D~U]
Tr[V~j
~
—
—
~
f~Tr[2V~
2, —
(5.59)
(ut D,,u + â~D,,z2)]
which is the hidden symmetry Lagrangian of Bando et al. [5.3]where V,, is identified with the p-meson. The only bound on the parameter a we have up to now is a> 1 since af2 = ~2(a — 1)_i. If we reinstate the A 1-degrees of freedom, we can now read off couplings, masses, etc. from the Lagrangian (5.57). Before doing so, let us mention that all parameters (is, c0, c1) are already fit from the ‘rr—cr sector, chiral symmetry alone would allow for six independent parameters if one sticks to terms with two derivatives. We find the following predictions: 2 m~ = 6(a
1)o- 2 , mA2 = 6au 2 , ma2 = a 4o~2 2=0)=~ag, g~,,.,,(q2=—m~)= ~a(1—i/a2)g,
—
g~,,,,(q g~ = (1 a/2)ge, —
2
2
gaqq
—
~
g
2
~
=
2 ,
g.,,qq
em~!g, i — 6
g
2
(5.60)
U.-G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
279
Fixing fIT = 93 MeV, rnq = 300 MeV, and varying a between 1.8 and 2.2 (which is the typical accuracy of soft-pion predictions, e.g. for the KSFR relation) we find m~=(750±80)MeV, mA =(1060±50)MeV, rn~=(975±50)MeV, (cjq)=—(175±5MeV)3,
(5.61)
where we have used (cjq ~ = u~t2 Obviously, the value of (~q)comes out too small, to improve on it without changing the other results, one can either include spin-dependent terms in the meson wavefunction* or include small s-terms defined in the gluon potential (5.40). For e 1/25, one gets (tjq) = (—225 MeV)2, m~= (780 ±150) MeV and mA = (1270 ±100) MeV. A will be reduced by almost a factor of two (A -=0.6 GeV), which seems a bit too small. The other interesting features exhibited in (5.60) are (i) approximate VMD of the -y’Trlr vertex, ~ = (0.0 ±0.1)ge, and (ii) approximate universality in the p-channel: g = (5.6 ±0.3), g~~(q2 = 0) = (5.6 ±0.8) and ~ = —m~) = (4.2 ± 0.9). The KSFR relation is fulfilled for a = 2. Because of the strong momentum dependence of the pirir-coupling, the width F(p—~2iT)comes out too small (—80MeV); similar statements hold for F(p—~eke) and F(A 2 = 0) = 0 is the same bad prediction of any chiral Lagrangian 1 pir).** gA(q with A 1-mesons and second derivatives only. The width of the decay o~—*2ir is largely suppressed, we find F(ff—~2ir) (200 ±150) MeV. This is consistent if one identifies the if with the observed S*(975) or ~(980) resonances. * * * It is worth to stress some more general properties of the action (5.56) which are not dependent on the particular choice of the parameters. In (5.56), photons couple to vector mesons as a normal parity combination of ‘rr and A1. Therefore, all abnormal parity decays involving photons will be given entirely by the vector meson mediated graphs, i.e. there are no contact terms. This justifies the Gell-Mann— Sharp—Wagner model [5.34]. For normal parity processes, we have only approximate VMD, direct photon—2n-pion vertices might be small but not necessarily zero. So we have found an argument through the manipulations which lead to (5.56) which links the concept of VMD directly to the QCD-generating functional. Therefore, Sakurai’s original concept can be embedded quite naturally in the framework of QCD. Finally, let us mention that one might (approximately) calculate the parameters j~t,c~and c1 in the “dilute instanton vacuum” model of QCD [5.35].It remains to be seen whether this more constraining action (5.56) gives a more accurate description of baryons and their properties than the effective actions derived from chiral symmetry as discussed in sections 2 and 3. —*
-=
5.3. Bosonization of the Narnbu—Jona-Lasinio model In the introduction to this section, we had argued that lattice QCD in the strong coupling and 1 / d-expansion resembles a four-fermion contact interaction like the Nambu—Jona-Lasinio (NJL) model. Furthermore, the NJL model is supposed to describe the spontaneous breaking of chiral symmetry in the same way as it happens in QCD. It therefore seems natural that an extended NJL model in terms of quark-fields gives some intermediate step from the full QCD-Lagrangian to the purely mesonic theories considered here. Following Ebert and Reinhardt [5.10, 5.36], we will briefly *
This would make the p smaller in size than the
ir
by a factor ~
which is not very comforting.
These are exactly the problems encountered already in sections 2 and 3 when incorporating the A1-meson. Keep in mind that the relevance of these mesons to low-energy hadron physics is not undoubted and this identification is not entirely convincing. **
U.-G. Meij3ner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
280
outline how a massive Yang—Mills theory arises via bosonization of the generalized NJL Lagrangian. For that, consider the following effective quark Lagrangian invariant under color SU(NC), ~=q(i/_tho)q+2G1[(q~ q)2+(~iy~~ q)2] —
2G 2[(~y~ ~
q) +
~
~
q)],
(5.62)
where we have suppressed color indices on the quark-fields q, cj. The generators the flavor U(2) are 2, of notice that would we T0 = ~, ‘r,(i = i, 2, 3). G1 and G2 are coupling constants of dimension (mass) have started from a vector-like theory and Fierz-transformed it, we could have set Gl = 2G 2. The ansatz (5.62) has therefore one degree of freedom more than the underlying fundamental theory, QCD. The bare quark masses th = diag(th~,thd) explicitly break flavor U(2)®U(2). To bosonize the Lagrangian (5.62), let us introduce the following color-singlet collective variables [5.37] 3
2 q—~S=~ t=0 S1 ~,2
T
~ q—*P= ~ P 2 3
5
~iy
2
5T
(5.63)
~y~y ~ q—*A~=—i~A’
~ q—*V~’=—i~V~’ ~,
-~
in the generating functional Z[q, ~j
=
f
[dq][d~]exp{-i[S(q,
~)
+
~q +
~]}
corresponding to (5.62). With that, and the 6-function representation (5.29) the action becomes bilinear in quark and anti-quark fields, so that these fields can be easily integrated out. One finds
f
t) = ~ ~
A, M, M
—
~
Tr[(M
J
—
—
ph 0)]
Tr[V~ + A~]— iN~Tr[ln(i~)],
where M is a complex field defined via external fields M, M~, V and A, i.e.
iJ~iy,,D~= i[y,,ô~
th t(M 0)
M = S
+
iP.
(ia’)
is the Dirac operator in the presence of the
5)M + ~(i +
y,,V~+ y,,A~]— [~(1 + y
(5.64)
—
y 5)Mt]
(5.65)
exhibiting clearly the underlying chiral symmetry. Notice that (5.62) can be written as 2 + P2] —2G 2 + A2]. +2G1[S 2[V The last term in (5.64) obviously stems from the quark determinant ~
~kin
[det(i p)]N~
=
exp{N~Tr[In(i,D’)j}
.
(5.66)
U. -G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
281
In terms of meson fields, we can rewrite V~and A,~as ~
A~=—~i(rA1~+D~)
(5.67)
where (w~,p~) and (D,2, A denote the isosinglet and isotriplet vector- and axial-vector fields, respectively. D,~represents the D-meson (nowadays called f1) at mD = 1285 MeV. To make the particle content of the complex field M more clear, let us work in a polar decomposition such that M=~IU,
(5.68)
where 2~(x)is the field of the scalar (if) particle and the chiral field U(x) is parametrized by the Goldstone bosons i~(x),*U(x) = exp{i~(x)If}with v~(x)= ~(x) r. f is the bare pion decay constant (up to renormalization factors). Notice that the effective Lagrangian (5.64) looks already quite similar to a massive Yang—Mills theory, with the kinetic terms of the gauge bosons hidden in the fermion determinant. We will come back to this point later on. Let us first study the effective theory (5.64) for large numbers of colors, N~—~ In this case, the meson functional Seff = exp{— fx “~eff} is dominated by the stationary phase at ~.
U=11,
A~1=V~=0,
~0=m
(5.69)
with m being the constituent quark mass (cf. fig. 5.4), which is determined by the gap-equation it
J
‘
m = th0 + i8G1N~
d~k 4
(2i~)
_______
2
k —m
(5.70)
2
Obviously, one has to cut off the momentum integral at some scale A since we are dealing with a non-renormalizable four-fermion interaction. A is related to the quark condensate value (~q~ and should therefore be of the order of the scale of chiral symmetry breaking [5.18, 5.22]. We now return to the quark determinant appearing in (5.64). Its phase is not invariant under chiral transformations 5} (5.71) i~ø—~ (li p~Q, ~11= exp{—ia ry and contains the gauged Wess—Zumino action [5.38]. We will not go into details on this point here but rather refer the reader to the extensive reviews refs. [5.11]and [5.39].The modulus of det(i)D’) can be calculated from [det(i,D’)t(i~’)]”2using proper-time regularization [5.40], {1n(~D’t~ø)} =
ln~det(i,Ø’)~ =
—
~J
~ Tr[exp(—p’t~’s)].
(5.72)
i/A2
Performing a heat-kernel expansion [5.17, 5.40] we find lnIdet(i~)I~ =
—~
J
16~r x
Tr{F(0, m2IA2)[S ~S
+ ~(F~FV~~ + F~VFA~~V) ~4] —
*
we
assume to be in the chirally broken phase.
+
+
~P
~1YLP
2exp((—m2IA2)A2)}
+...
(5.73)
282
U.-G. Meij3ner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
where the dots stand for higher order terms in U [5.10, 5.36]. The pertinent covariant derivatives appearing in (5.73) are D~S=o.~S+[V~, S]+i{A~,P}, (5.74)
D~P=o.~P+[V~,P]+i{A~,S}. The respective vector and axial-vector field strength tensors are
~
=
~
~
=
~
[Vp, VP] + [Ak, Ar],
+
—
(5.75)
~
—
+ [Vp, A~]+ [Ag, Vu].
F(a, x) is the incomplete F-function, F(a, x)
.f°dt e_tta_l. As
promised, the kinetic terms for the (axial) vector particles appear in the quark determinant (and do not have to be put in by hand). Let us now discuss the consequences of the effective Lagrangian (5.64) for the meson sector. For that we will expand U(x) in powers of IT(x), i.e., U(x) = 1 + uT/f and bring all kinetic terms in standard form by proper renormalization. Furthermore, the pion- and A1-fields have to be redefined to get rid of =
the 8.~i~ A~vertex. Writing the scalar field as 1~(x)= m + o-(x), we get .
2V~,
~
=
z”2
~l/2,
V~1= \~Z~’
and can be read off the
if,
p and
t~omasses
=
[J~ r(o, m2)]1’2
(5.76)
from the bilinear part of the effective Lagrangian as
m~= g~/4G 2, g~= (5.77) 2 m~2= rui m~= m~ + 4m 0mIf~G1 2the bare pion mass and decay constant.* To diagonalize the ‘n~A with m~band fb = 2mZ~’ 1-mixing, we introduce physical pion (~) and physical A1-fields (A1) via 2Z1~
fL2fb
~
=
~
—~
+
m
2—Z” ‘
~
(5.78) 22
Z
=
2 1/2
mPImA
(1 g~f.TIm~) with f,~ = 93 MeV the physical pion decay constant. f,~ and j satisfy the Goldberger—Treiman relation on the quark level. The masses of the physical pions, A1- and D-mesons are 2D = m~+ 6m2, m~ m~b(m~ m~1= m 1/m~). (5.79) 1
* fb
=
—
is different from f due to the field redefinitions as it becomes clear from eq. (5.78).
U. -G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
283
Notice that the A1 and the D are degenerate in mass, as well as the p and the to. Furthermore, the p—A1 mass splitting is entirely due to partial Higgs mechanism as already discussed in section 2.1 in the framework of massive Yang—Mills Lagrangians. In the case of vanishing current quark masses th0, we find m~= 0 in agreement with Goldstone’s theorem. All phenomenological relations like KSFR or the Weinberg mass that ratio are now determined by the value of the constituent quark mass. For that, let us set 2 = am~so 6m m~= (1 With Z
(1
+
+
a)m~,
a)”2
=
(1
m~= (6/a)m2. —
g~f~/m~)”2we
(5.80) obtain
[(1 + a)/a]g~f~.
=
(5.81)
To arrive at (5.81), we have used the universality of p-couplings inherent in the model [5.10]. For a = 1, we recover Weinberg’s ratio m~= m~ 1!V~ and the KSFR relation m~= ~ With m~= 770 MeV, the constituent quark mass follows to be m = 314 MeV, which is just m~10~0~/3. If one insists on the higher mass of the A1, i.e. m~1= 1275 MeV, one can choose another relation between m~and m. = 2m (a = 3/2) e.g. gives 2 f2)112 = 737 MeV (5.82) \/~m~ = 1217 MeV, m~= (~g on the expense of the KSFR relation. Nevertheless, the Weinberg ratio and the KSFR relation are incorporated in the Lagrangian (5.64), with this precise form depending on the value of the constituent quark mass. Let us mention that the whole scheme has been extended to broken flavor SU(3) ® SU(3) [5.10]. In that case, one has five parameters th~= th~,th~,G 1, G2 and A which can be fixed by demanding ,j,, = 140 MeV, JTT = 93 MeV, m~= 770 MeV, g~/4ir = 3 and m~= m~I2= 510 MeV. Some pertinent results are mA1
=
th~=2MeV, ,h~=82MeV, 2/4ii~= 0.6, G 2/41T = 2.5* A = 1.4 GeV, G1A 2A fK/fm.=1~1(1.2±0.1)~ gfIg~=1.3(2.0±0.2) mU
m~J 315MeV,
(5.83)
The current quark masses come out on the small side, from QCD sum rules one expects e.g. th~= (110 ±10) MeV [5.41]. The meson mass spectrum is summarized in table 5.1. It shows a fair overall agreement. The quark condensates do not come out too well, we find (uu)~3 = —340 MeV and (~~s)I(iiu) =1.4 to be compared with (uü)~=—250MeVand ((~s~/(üu))sR=0.8[5.41]. Finally, let us mention that treatment of the quark determinant outlined here gives a unique way of fixing the so-called non-minimal (normal parity) contributions to the gauged Wess—Zumino action [5.23, 5.36, 5.42]. To summarize, we can say that an “intermediate model” like the extended NJL Lagrangian (5.62) gives rise to massive Yang—Mills Lagrangians with all parameters fixed in the meson sector, *
Notice that G,
2G 2, i.e. the NJL model (5.62) does not stem from Fierz-transforming a vector-like interaction when it explores its
phenomenological consequences.
284
U. -G. MeiJ3ner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons Table 5.1 Meson mass spectrum from the bosonized Nambu—JonaLasinio model. The two values for the A 1-mass give the experimental uncertainty already discussed in sections 2 and 3. The E-meson is nowadays called “f1(1420)” and the Q1meson runs under ‘~K1(1280)”.
[MeV~ m~ [MeV] mK. [MeV] mA [MeV] m~[MeV] m0 [MeV~ ~
ref. [5.36]
experiment
600 970 9(X) 1090 1582 1332
494 1020 892 1060/1275 1422 1270
therefore adding further credit to the use of chiral meson Lagrangians. This tightens the link between QCD and the effective theories discussed before.
6. Baryons from the effective Lagrangian In the previous sections, we have only examined the meson sector related to the effective Lagrangian constructed by symmetry principles, e.g., chiral symmetry, with pseudoscalar and vector mesons. This is actually “old stuff” and has been explored in great detail already 20 years ago (cf. references given in section 1). The novel feature, triggered by the developments in the conventional Skyrme model, is that baryons can emerge as topological solitons from the underlying non-linear meson theory in the spirit of Skyrme’s and Witten’s ideas. Furthermore, the properties of the topological solitons are entirely fixed by the mesonic data like the meson coupling constants and meson masses, i.e., all baryon properties will emerge as parameter-free predictions. This gives very stringent tests on the quality of the models considered and is in sharp contrast to other approximate approaches to treat low-energy QCD like e.g. the potential models [6.1]. Furthermore, starting from a relativistic meson theory, the models with pseudoscalar and vector mesons offer rich flexibility in calculating a variety of static and dynamical baryon properties let us only mention a few: static properties, electromagnetic and axial form factors, strong form factors, excited baryon states, qualitative features of the nucleon—nucleon interaction and of nucleon—antinucleon annihilation. In what follows, we will demonstrate in some detail how a model with pions, p- and w-mesons works in comparison with experimental data, following the work of Meifiner, Kaiser and Weise [6.2]. Then calculations including the A1 -meson will be discussed, followed by a brief review concerning the situation on NN and dibaryon physics. Before going into details, let us make a short historical digression. Actually, the first models to include vector-mesons did not account for p, w, A1 but just for o- or p-mesons coupled to the skyrmions. It was the work of Adkins and Nappi [6.3] which triggered interest in vector meson stabilization. Since this model is discussed in detail in the review of Zahed and Brown [6.4], we will refer the reader to this reference. The B = 2 solutions in this model have been investigated by Kiabucar [6.5]. The link of the p-meson coupling to the Skyrme parameter e [6.6] has been first made by Iketani, who showed that the Skyrme stabilizing term stems from the exchange of infinitely heavy p-mesons as discussed in section 3.1. Following that work, Adkins [6.7] proposed a model coupling the p-meson to the non-linear if-model based solely on symmetry grounds. A variant of this model has been studied by —
U. -G. Mei/iner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
285
Meil3ner [6.8, 6.9]. Since these simplified models are not in the center of the work presented here, we will give only a brief outline in appendix B. For other attempts to include the p-meson, see the work by Igarashi et al. [6.10, 6.11]. The first full-scale attempt to investigate the ‘rrpwA1-system has been carried out by MeiBner and Zahed [6.12, 6.13], who showed that a more realistic Lagrangian indeed gives an improved description of the static nucleon properties. Similar results have been obtained by Chemtob [6.14]. After that, a flurry of groups has studied various effective Lagrangians looking mainly at static nucleon properties. These results will be discussed in section 6.7. The most detailed discussion on baryon properties has been done in the ‘rrpü-system which we are going to present now. Nucleons and L1-isobars in the irpw-system
6.1.
The first step in constructing nucleons, z~-isobars,.. . from the effective Lagrangian under consideration is to find the classical solution, the hedgehog. This state has neither good spin nor isospin, but the grand-spin K = I + S is a good quantum number. To be specific*, let us discuss a Lagrangian with pions, p- and w-mesons following the work of Meil3ner, Kaiser, Wirzba and Weise [6.2, 6.15] based on the hidden symmetry scheme outlined in section 4. p- and w-mesons are introduced as dynamical gauge bosons of the hidden [SU(2)~ ® U(l)~]10~~1 symmetry. The Lagrangian reads ~
2 ) Tr[F,~~F~’] + ~ + (1/2)m~f~ Tr[U 1]. (6.1) (112g U(x) = exp{ir. IT(x)/f,j embodies the isovector pion field, and ~L = = = ~ in the unitary gauge. f,~is the weak pion constant which we will take on its empirical value, JTT = 93 MeV.** ~ is the contribution from the Wess—Zumino (WZ) term to be discussed later. is the gauge covariant derivative and F,LP the pertinent field strength tensor —
—
(6.2) ~
=
~
—
o~V~ —i[V~,V~].
We have chosen the same gauge coupling constant g for SU(2)~and U(1)~which leads to a mass degeneracy of the w- and p-meson: 2
2
2
22
22
m~=m~=m~=af,~g =2f~g with a = 2 giving g
=
~
and the KSFR relation (6.3). Setting my
(6.3) =
770 MeV, we get
g=5.8545
(6.4)
in good agreement with the value ~ = 6.11 determined from the width of the decay p—~2ir. The discrepancy is of the order of 5% and just shows how good the KSFR relation holds. For the following *
Keep in mind that although we are using a specific Lagrangian, these considerations hold in general.
**
16.17].
Recent measurements give values for f,, between (92.16 ±1.01) MeV [6.16J.We use here the more standard value quoted by Nagels et al.
286
U.-G. MeiJiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
discussion, we will disregard the small mass difference m~ 13 MeV. The contribution from the WZ action ~ is given by (after making use of Stokes’ theorem) —
~
/7.j
\
Ic
I
j.*eaf3
e
+
L’41T
+
ô~U~U~U~U
+ c(~)~vaPw~pTr{iT.pa(U+.
~U
+
+
ô~U}
o~U~ U~)+ ~ r.pU±r.pU}
(6.5)
The first part of ~ gives the o-coupling to the topologically conserved baryon current B ~ as first discussed by Adkins and Nappi [6.3], with the oNN coupling given via universality as g=~=
=
g~,= (N~/2)g—9.
(6.5a)
For c = 0, we talk of the “minimal model” as discussed in section 3.3. For c = 1, the wpir-interaction is taken into account, the WZ action for that case is equivalent to the rigid gauged WZ action and elimination of the A1 via the constraint U 0. We will refer to this model as the “complete model”; it embodies the p-dominance in the o—~3~decay amplitude. The last term in (6.1) is the conventional pion mass term, we take m,~= 139 MeV. This avoids the problem of infinite quantities such as the isovector charge radius in the exact chiral limit m,~= 0 [6.18]. The Lagrangian (6.1) (and any similar effective Lagrangian possibly also including A1-mesons) describes correlated mesons as well as solitons. The latter 3r Bones are finite energy-configurations with non-vanishing winding number (baryon number) B = f d 0(r). Their stability is established by the repulsive character of the vector mesons at short distances. To investigate the baryon number B = 1 sector, we will specialize to hedgehog skyrmions. This amounts to working in a space of maximal symmetry and brings the coupled equations of motion for the meson fields into a tractable form. Keep in mind that one could instead start with deformed meson fields and allowing for more degrees of freedom. Here, additional calculations are in order, but we feel that the guidance from the conventional skyrmion where the hedgehog indeed turns out to be the minimum energy-configuration [6.19] should be taken seriously. So, for the pion field, we choose the usual hedgehog configuration U(r)
=
exp{ir IF(r)}
,
~(r) = exp{ir. i~F(r)/2},
(6.6)
which connects the intrinsic symmetry space (isospin-space) with r-space in a radially symmetric way. F(r) is the so-called chiral chiral field.number, The baryon number density reduces to B0(r) = 2)(F’/r2) sin2F(r). Toangle ensureorunit baryon we have to choose (—1/21T F(0)rr,
F(c~)0.
(6.7)
In the minimal model, the w-meson couples only to the topologically conserved baryon current B’~(x). Since on the classical level only the time component of the latter is non-vanishing, the space components w’ of the W-meson field have no sources, so that =
w(r)6~°
(6.8)
as first pointed out by Adkins and Nappi [6.3]. These arguments also hold in the case of the complete model, because the additional term in the Wess—Zumino action (6.5) can be cast into the form of a total
U. -G. Meljiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
287
derivative which does not change baryon number. Equation (6.8) can also be inferred from the facts that w0 is a scalar and transforms into itself under parity x—~—x. We will refer to the function w(r) as the w-meson profile. The negative parity p-meson has non-vanishing space components only. For that, let us rewrite the ~ term (the second term) in (6.1) as
f~Tr[J~—V~J2,
(6.9)
where J~= (1I2i)(~UU~ + ô.,~U~U)/det(1 + U). With the hedgehog ansatz (6.6) for the chiral angle F(r), this leads to J~(U)=0,
J~(U)=r,
1~F.,(1—cosF(r))/r.
(6.10)
It is therefore natural to take the Wu—Yang—’t Hooft—Polyakov [6.20] ansatz for V,~(x), i.e. the p-meson p,~(x): Pza(T)
=
EI/~Fk G(r)/gr
poa(r)
;
=0,
(6.11)
where the SU(2) gauge coupling g has been scaled out for convenience. In (6.11) a = 1, 2, 3 are isospin indices, whereas i denotes the three space components of the vector fields. We will refer to the radial function G(r) as the p-meson profile. Using the hedgehog ansatz (6.6), (6.8) and (6.11), we can derive the static energy functional from the Lagrangian (6.1). This static energy has to be identified with the hedgehog or skyrmion mass MH: M11
=
E[F(r), G(r), w(r)]
=
—
f
3r~(F,G, w).
(6.12)
d
Straightforward algebra gives [6.2, 6.15]
E[F, G, w] =
41T
J
dr{f~[ir2FI2 +
+ m~,f~r2(1cos F) —
+
sin2F] + 2f~[(G +1— cos F)2] —
f?-~i + G2(G+2)2 + g 2gr
r2[f~g2w2 +
—~--~
wF’sin~F+
—~-~-~
cw’G(G+2)sin2F}
l6lT
(6.13) with the vector meson mass terms appearing via the KSFR relations and x’ = dxldr. The coupled equations of motion for F(r), G(r), and w(r) follow after functional minimization of (6.13). These are exactly the Euler—Lagrange equations due to the hedgehog symmetry. They are given by: F” =
—
+
F’
+
2
4 [4(G
8irf,~ r
+
1) sin F
—
sin 2F]
+
m~sin F
~[—2sin2F + cG(G +2) cos 2F],
(6.14a)
U.-G. MeiJJner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
288
G” =2g2f~(G+ 1
—
cos F) +
4 G(G
+
1)(G
+
2) +
w’(G
C ~2
+
1) sin 2F,
(6.14b)
3~
w”——w’+2f~g2w—
2F1sin2F +
c2
[G(G+2)F’cos2F+
G’(G+1)sin2F].
(6.14c)
Notice that for the minimal model (c = 0), the equations of motion for G and w decouple. In that case, (6.14c) agrees with the w-equation of motion derived by Adkins and Nappi [6.3], and the p-equation of motion agrees with the one given by Igarashi et al. [6.10]. To ensure singularity-free solutions and finiteness of the energy, the following boundary conditions have to be imposed on G(r) and w(r): G(0)2,
G(oo)0,
w’(O)O,
w(oo)0.
(6.15)
For a baryon number B = n solution, we have G(0) = —[1 (—1)~],and F(0) derive the asymptotic behavior of the profiles, for small r one has (B = 1) 3) F(r)—~IT + c1r + ~(r G(r)—~—2+c 2+C(r4) as r—’O, w(r)—~c 22r+ U(r4) —
=
nrr.
We can easily
(6.16)
3 + c4r
where c 1, c2, c5 and c4 are constants. For large r, all meson fields fall off exponentially, i.e.
as r—*
2m~~7r2
f~ G(r) F(r)—~e_3m~/rl e_em~ (i + w(r)
(6.17)
~.
Notice that for the vector-meson profiles the coupling to the 2ii- and 3’rr-continuum (for p and to, respectively) takes over as r becomes much larger than — m~,with ~ the Compton wavelength of the p, to-meson. The set of coupled equations (6.14a, b, c) together with (6.7) and (6.15) constitutes a boundary value problem of ordinary differential equations that can be solved using standard procedures. Using m,~= 138 MeV, f~ = 93 MeV, g = 5.8545 (i.e. m~= = 770 MeV), the numerical solutions to the variational problem are shown in fig. 6.1 (complete model, c = 1) and fig. 6.2 (minimal model, c = 0, in comparison to the results of ref. [6.10]). Because of the additional WplT-correlations in the complete model, the profiles F(r) and G(r) extend further out in space than in the minimal model. For the W-profiles the situation is reversed. Comparing the results of ref. [6.10],* where no w-meson is present, we see that due to the additional w-repulsion the chiral field F(r) and the p-meson profile G(r) are much less concentrated than in their case. In table 6.1, we give the skyrmion mass MH and the (hedgehog) soliton radius TH, with rH g~’enby (r~) =
*
r~= 4ir
J
r4 B
2F’ sin2F dr. 0(r) dr =
—
r
Compare also ref. [6.11]for a critical discussion on the work presented in ref. [6.10].
(6.18)
289
U. -G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
COMPLETE MODEL: MESON PROFILES
F(r)
-~
MINIMAL MODEL
.
ii-
AND p
PROFILES
3Kr)
I
____
O~5~~I.5
r[fm]
r [fm]
Fig. 6.1. Meson profiles for the complete model: the chiral angle F(r), and the vector meson profiles — G(r) and w(r) for g = 5.85, = 93MeV and m~= 138 MeV. The scale for the w includes a factor
Fig. 6.2. The chiral angle F(r) and the p-meson profile — G(r) for the minimal model in comparison with the results of ref. [6.10] for f = 93MeV, g = 5.85 and m~= 0.
f~.
We will refer to r 11 as the baryonic root mean square (rms) radius, it measures the extension of the baryonic charge, i.e. the baryon number distribution, of the soliton. Notice that the inclusion of the w-meson has two main effects [6.2, 6.15]: first, the skyrmion mass increases by —40% as compared to the model without (0 [6.10, 6.11]; secondly, the soliton radius increases by almost a factor of 2. As we will demonstrate in what follows, a small baryonic charge radius rH 0.5 fm is not inconsistent with typical nucleonic charge radii 0.8 fm, the difference is due to vector meson propagation. In table 6.2, we show the various contributions to the skyrmion mass as defined in (6.13). As in the case of the co-stabilized skyrmion [6.3, 6.4], we can derive a virial theorem relating the energy from the co-meson —
-=
Table 6.1 Properties of the skyrmion resulting from the lagrangian (6.1) with ir, p- and w-mesons. For comparison, the results of the model of ref. [6.10]including pions and p-mesons are also given (see ref. [6.11]for further details). The parameters used are m = 138 MeV, f = 93MeV, and g = 5.8545. MH is the static soliton mass, and rH the baryonic rms radius. PVMD denotes the model discussed by Kunz, Masak and Reitz [6.23]which incorporates only partial vector meson dominance, and is very similar to the complete model (c = 1) (cf. section 3.3).
MH[MeV] TH [fm]
Table 6.2 Various contributions to the hedgehog mass MR for the minimal (c = 0) and the complete (c = 1) model as defined in (6.13). The virial theorem 2E~ = —E~~ holds within ~ 4 X ~ Minimal model Complete model (c=0) ~c1) ________________________________________________________ E, [MeV] 773.2 751.3 E~[MeV] 42.6 39.0
c= 0
c= I
c= 0 w(r) = 0
PVMD
~
~
[MeV]
1474.4 0.50
1465.3 0.48
1057.0 0.27
1367.0 0.51
E~~[MeV] I2EJEWZ~— 1
MH [MeV]
33.0 511.9 3.9 X 10~6 1474.4
49.3 513.4 3.9 x 10_6 1465.3
290
U. -G. Meifiner, Low-energy hadron physics from effective chiral Lagranglans with vector mesons
and the WZ term via ~
=
(6.19)
—2E~,
where E~sums up the co kinetic and co mass term. Equation (6.19) follows from the fact that the energy-functional (6.13) is bilinear in w(r) and w ‘(r). Performing a partial integration and eliminating the term proportional to r2w’2, one can use the equation of motion (614c) to show that E,,~+ ~ = ~ This virial theorem serves as an excellent check on the numerical calculations, for the actual solutions given before we find that 2E()/Ewz~ 1 ~ 4 x 10_6. The hedgehog configuration discussed so far does contain a mixture of the nucleon, the isobar, - . , i.e. —
~-
-
(6.20) with I denoting isospin and J angular momentum. U) is the coherent state peaked around the classical hedgehog configuration (U~U(x)~U) = U(x) and H is the parity operator. Adkins, Nappi and Witten [6.6] proposed a semiclassical scheme to discriminate between the N and the i~at the quantum level, by quantizing the spinning modes of the skyrmion. The idea consists of spinning adiabatically the hedgehog in isospace, generating a classical angular momentum, and then quantizing it using standard canonical rules. For the chiral field, this procedure amounts to the substitution U(r, t) = A(t) U(r) A~(t)
(6.21)
in the Lagrangian, and quantization of A. A(t) is a space constant, time-dependent SU(2) matrix. (Keep in mind that we restrict ourselves to the flavor SU(2)-sector.) Evidently, the space components of the co-field and the time-component of the p-field, w’ and p°,will be excited by this procedure. Whereas the Euler—Lagrange equations have sources only for w° and p’ on the classical level (hedgehog), this is no longer the case for a time-dependent U-field as given by (6.21). Defining A~dA!dt = A~A i~K
(6.22)
we have to leading order in the rotational frequency w —— K~*and consistent with the intrinsic parity of the vector mesons the following most general ansätze for the vector-meson excitations: T
p°(r,t) = [2/g] A(t) T [K ~ 1(r)+ FK F~2(r)]A~(t) , (6.23)
w(r,
t)
=
[cP(r)/r]K x F= 12(r) K X F.
~1(r), ~2(r)and b(r) are radial functions to be specified later. The classical p-field transforms as r~p’(r, t)
=
A(t)
T
p’(r) A~(t).
(6.23a)
Before proceeding, we should point out that this quantization procedure is by no means complete, since *
This is in accordance with the large Ne-expansion [6.6].
U. -G. Meifiner, Low-energy hadron physics from effective chiral Lagranglans with vector mesons
291
both translational and vibrational quantum effects are ignored, but it is justified in the framework of the large Ne-expansion. Injecting (6.22) to (6.24) into the Lagrangian density, we find the following time-dependent Lagrange function
=f
L(t)
d3x~=—MH + ATr(AA~)
(6.24)
with the moment of inertia A given by A=
4IT
f
~
A[F, G, w;
~2’
~]dr.
(6.25)
Straightforward algebra (cf. appendix C) gives the integrand A as follows: A[F, G, w; =
~
~2’
~]
~ f~r2[sin2F + —
8 sin4( F) 8 sin2( ~F) ~ + 3 ~ + 2~~2 + + 2~P2Ir2+ m2P2] + (1I3g2)[r2(3~2+ ~ +
~
+
4G2(~+
+
—‘~-~
41T
~
~1~2
—
~
—
~2 +
—
1) + 2~(G2+ 2G
F’CP(2sin2F— C~
c 8IT
1— C~2)+
—~--~
+
2)]
sin 2F(G
~‘
G~1
—
—
~)
(6.26)
with m = \/~gf~. Minimizing the moment of inertia gives the coupled Euler—Lagrange equations for the vector meson excitations ~ 1(r), ~2(r), and cP(r): 2sin2(~) =
—
+
1) —4f~g (1/r2)[G2(~ 2~ 1 1)— 2(G + 1)~2]+ m 1, —
C
3222
~‘
sin 2F~(G
—
2f~sin2(~) =
+
-
—
~
+4 [G2(~1
~F’ + ~ 3g3 2 ~‘(G 32irr
8ITr
+
=
m~+ 4g
+
m2tP
—
42
F’(2 sin2F
—
+
[2F’ cos2F(G
—
~(~) 0, =
~(0)
=
+
+
2G
+
2)~2]
3(G
1) sin2F,
(6.27b)
G~1
—
c~2)
~) + (G’
for the minimal (C = 0) and the complete model finiteness of the moment of inertia are =
2 1)
~2
-
C~ 1—
+ C ~2
(6.27a)
~2(co)=
0,
G’~1 G~
—
(C
—
=
~(0)
—
~)sin2F],
(6.27c)
1). The boundary conditions following from
=
~~c)
=
0.
(6.28)
292
U. -G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
By virtue of the small r-behavior of the vector meson excitations +
=
0(r3)
+
~ 2(r)34r+0(r) 2 + 0(r3) = E5r
asr—*0,
(6.29)
cP(r)
and the small r-behavior of the classical fields (6.16), one can deduce the following constraint: 2~~(0) + ~2(0) = 2
(6.30)
-
This can be used to check the numerics. The asymptotic behavior of the vector meson excitations for large r is given by ~ 2~/y2 1(r), ~2(r)~e ~(r) —— e3~/y3 as r—~
(6.31)
with y m~r.As first pointed out by MeiBner and Kaiser [6.21], the rather lengthy expression of A (6.26) can be simplified by performing a partial integration on the terms proportional to r2~2,r2~2, r2~~ and ~ and using the equations of motion (6.27) to eliminate all terms bilinear in ~ ~2’ ‘P, and derivatives. This procedure leaves the linear terms multiplied by a factor 1/2, and all other terms remain unchanged. The so-called reduced form of the moment of inertia ‘tred reads: ‘iced
=
~f~,r2[sin2F+ 8 sin4(F/2)] + ~f~r2[—4sin2(F/2)~ 2F
3g G~(2 ~2 2~~) + 4ir F’~sin =A~+A~+A~+ A~+A~, +
~
—
—
+ C
~ 16IT
1] (~‘Gsin2F) (6.32)
so that the contributions to the moment of inertia as defined in (6.32) are given by A,~= 4IT j~° A,~dr This reduced form of the moment of inertia serves as a further check on the numerical accuracy. To project onto states of good spin and isospin we now have to perform a collective quantization of the adiabatic rotation. This procedure relates the spin- and isospin-operators (J and I, respectively) to the rotational velocity K = ~iTr(TA~A)via —
J= —iATr(A~A~-), 1= +iATr(AA~).
(6.33)
The Hamiltonian corresponding to (6.24) reads: H = M~+ J2/2A
=
MH + I2/2A.
Evidently, the nucleon mass MN, the z~-massMA and the N~Xmass-splitting MA—MN follow as
(6.34)
293
U. -G. Meiflner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
MN=MH+3/8A,
(6.35a)
MA=MH+15/8A,
(6.35b)
MA—MN=3/2A.
(6.35c)
In fig. 6.3, we show the vector meson excitations ~1(r), ~2(r)and (2(r) = P(r)/r for the minimal model. The vector meson excitations for the complete model can be found in ref. [6.2]: In table 6.3, we give the moment of inertia, the various contributions to it as defined in (6.32), the nucleon mass MN, and the nucleon—z~mass-splitting MA—MN. Due to the too large skyrmion mass MN, the masses come out too high. Quantum fluctuations tend to decrease the mass substantially. The same statement applies to the N~mass-splitting [6.22]. We will come back to this point later on. It is 4IT interesting ~f to observe that almost all of the moment of inertia is given by the pionic contribution A,~= A~ 1~° dr. Kunz, Masak and Reitz [6.23] have recently investigated the PVMD Lagrangian of pions, p- and co-mesons discussed in section 3.3 (after eq. (3.27)). The WZ term is chosen such as to give the proper co—+3-rr decay width. Their results are very similar to the ones obtained in ref. [6.2], they find M11 = 1367, MN = 1456 and MA—MN = 358 MeV. These authors perform also a stability analysis and show that the (0-meson indeed stabilizes the soliton (for N~> 1).
MINIMAL MODEL: VECTOR MESON EXCITATIONS I I I
1.0• Table 6.3 .3I-,
Various moment of ine~ia defined (6.32) and the contributions total momenttoofthe inertia A. The nucleonas mass MN inand the nucleon—s mass-splitting are also given. Furthermore, the ratio of
U)
_________________________________________ o I-,
~ -0
~rl
—0.5
moment of inertia as calculated with (6.26) and (6.32) is given as a good check on the numerics. Deviations from 1 are ~ 10’. Minimal model Complete model c=0 c1 A,, [fm] 0.7569 0.6923
A,,~ A~ MN A~,, A~ A
—
-0.2
-
1.0
III
0.5
1.0
1.5
[f ml Fig. 6.3. The p-meson excitations ~1(r)and ~2(r)and the a-meson excitation ~1(r)= ‘P(r)Ir for the minimal model. Notice the different scale for all fields.
A/Ad
fm] [fin] [MeV] fm] fm] [fml MN [MeV]
—0.2519 0.1632 1564.2 0.1608 0.0000 0.8241 359.2 0.999992
—0.2258 0.1506 1574.6 0.0758 —0.0161 0.6768 437.3 0.999998
294 6.2.
U. -G. Meiflner, Low-energy hadron physics from effective chiral Lagranglans with vector mesons
The electromagnetic structure of the nucleon
As we already discussed in sections 3.2 and 3.3, electromagnetism can be uniquely coupled to the Lagrangian (6.1). On the meson level, this leads to the characteristic current-field identities [6.22] 2
jisoscaiar(X)
~ovector(X)
=
=
—(m~/3g)w (x)
(6.36a)
,
—(m~/g)p~3)(x)+...,
(6.36b)
where the dots in (6.36b) stand for higher order corrections due to the non-Abelian character of the p-meson, i.e., for strong fields higher order terms are induced which modify the simple structure current p-field. J~(x)evidently is the electromagnetic current. Equation (6.36) embodies Vector Meson Dominance (VMD) in the isoscalar channel and asymptotic VMD in the isovector channel, leading to the successful description of the pion form factor as described in section 1. Before discussing how this scheme works in the soliton (baryon) sector, let us briefly remind you of the basic definitions of electromagnetic currents and nucleon form factors. For a spin- ~ particle, the most general form of the matrix elements of the electromagnetic current operator is: (Nf(P’)~J~m(0)~Ni(P)) = ~(pI)[F1(q2)y~
+
~
u~q~]u(p),
(6.37)
where p, p’ are ingoing and outgoing four-momenta, and q = p’ p2) isand thethe four-momentum transfer Dirac form factor F of 2) the virtual photon as shown in fig. 6.4a. The Pauli form factor F1(q q are real functions of the Lorentz invariant q2 = q~q~’. Time reversal invariance, parity invariance2( and current conservation have been used to obtain the decomposition (6.37). A convenient coordinate system to work with is the Breit frame defined by p = (E, —q/2), p’ = (E, q/2) with E = \[M~ + q2/4 the nucleon energy. In the Breit frame, one has q = (0, q), i.e. no energy is transferred by the photon. Furthermore, the electric and magnetic parts of the electromagnetic current J~= (p, j) separate —
~
=
G~(q2)x~x 1,
(6.38a)
x
(6.38b)
(Nf(q/2)~J’(0)~N~(—q/2)) = GM(~) ~
PHOTON-NUCLEON
SCATTERING: KINEMATICS Nf(p~)
Nq(E,~/2)
N~(ps)
N~(E~/2)
(b)
(a)
2) and F,( q2). (b) In the Breit-frame, the incoming photon transfers no energy. Fig. 6.4. (a) Kinematics of the electromagnetic form factors F1 ( q
U. -G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
295
for where X~,Xt are two-component Pauli (iso)spinors and we have dropped the sign in G~~(—q2) convenience. GE and GM are the charge and the magnetic form factor, respectively, which are related to F 1 and F2 via: 2)= ~(q2)
+
F
~
G~(q
2),
G~(q2)= ~(q2)
2(q
+
F
2).
(6.39)
2(q
For static charge and current distributions the form factors GE and GM in the Breit frame can be directly related to the Fourier transform of the corresponding charge and current densities. * This identification is not possible in any other frame. The nucleon isoscalar (5) and isovector (V) form factors are related to the proton and neutron ones by G~’~ =Gs +GV ~640 EM
E,M
—
EM ~
and are normalized to the respective charges and magnetic moments: G~(0)=1,
G~(0)=0,
(6.41)
G~(0)=~~=2.79,G~(0)=jt~=—1.91. Let us now return to the soliton model with pions, p- and co-mesons and compute the pertinent isoscalar and isovector currents. The starting point is the classical solution, the hedgehog. The isoscalar current is associated with the isoscalar symmetry, which involves the co-field as the U(1)~gauge field. Performing an isoscalar gauge transformation exp{—ia(x)/2}, the co-meson transforms as ~.o,~(x)—~ a) 1~(x)= w~(x) (1 /g) ~a(x),
(6.42)
—
and we can read off the isoscalar currents from the terms proportional to o~a(x) (Gell-Mann—Levy method [6.24]). Because of (6.42) one would expect a contribution B~(x)to the isoscalar current from the w~B~-term in the Lagrangian (6.1), but this contribution is exactly cancelled by the ungauged Wess—Zumino term, which gives a contribution B ~‘(x).This holds for the minimal as well as for the complete model. Consequently, only the symmetry-breaking co mass term contributes to the current, we have —
JP~s(x)=
J~0(x)= —(m~/g)to~’(x).
(6.43)
The current-field identity (6.43) reflects the underlying Vector Meson Dominance of the models discussed so far. Similarly, the isovector current is the symmetry current of the isospin transformations hESU(2)~ 1
~‘rp *
1 ~L+ + —*h~rp h —(i/g)ô.~~h~h
t~
-
(6.44)
This procedure is justified from the large Ne-expansion, it follows after collective quantization of the center-of-mass motion as shown by
Braaten, Tse and Willcox [6.23].
296
U.-G. Metfiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
The resulting current is (cf. appendix C) J~’(x)
JP..V(x) =
—if~Tr{r[e
.
—i(1
rw~~Tr{~[U~
—
c)
~
ô~.~
~_
—
~(U~
U~) —i~g(~r.p~ +
. 8~U— ~
U~U~ U
—
~
UU +
.
UU
~]}.
(6.45)
Notice that the last term in (6.45) is only present in the minimal model due to the non-gauge invariance of its Wess—Zumino term. The electromagnetic currents follow after projection of the symmetry currents (6.43) and (6.45) on the charge operator Q = ~ + r3), as given by the quark charges of the underlying fundamental theory. One has Js1~s(x)= —(m~,/3g)w~’(x), J~’(x)= J~)(x),
(6.46)
where S and V stand for isoscalars and isovectors, respectively. The explicit form of the currents follows after inserting the meson fields of the rotated hedgehog and using the quantization rules =
—2iA Tr[A~AT],
~ = 2iA Tr[AAT],
a.jr~=
—
~Tr[Ar,A~j.
(6.47)
Straightforward algebra outlined in appendix C gives* 1cm(X) Os
=
—(mU)/3g) w(r), 2
(6.48)
J~(x)= ~ +
J~(x)=
{
{~
[4sin~(~)+(1+2cosF(r)) (1— c) ~ 2 ~(r) F’(r) sin2F(r)},
~1(r)+~2(r)]
8IT r ~
+
f~[2 sin4(~) (1
—
c)
17 —~—~
-
G(r) cos F(r)]
w(r) F’(r) sin2F(r)
41T
~oXr 2
~
)
-
r
All information about the electromagnetic structure of the nucleon is contained in the currents (6.48). Taking Fourier transforms one can immediately read off the nucleon electromagnetic form factors: G~(q2)=
—
~
4IT
J
r2 j 0(qr) w(r) dr,
(6.49a)
The w- and A-dependent terms of the isovector currents are not present in the complete model, in this case the WZ term is invariant under
isospin gauge transformations.
297
U.-G. Metfiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
iflU)
G~(q2)=~J
~2ITf(~)
j1(qr)~(r)dr,
(6.49b)
io(qr)[~~ ~
‘pFIsin2F]dr, (6.49c)
V
2
G~(q)= ~
IT
r
-
M~j ~—)j1(qr)
x [2f~(2 sin4(~) G cos F) —
+
(1
c)
—
42
wF’ sin2F] dr,
(6.49d)
with j0(qr) and j1(qr) the usual spherical Bessel functions. Note that the isoscalar form factors GSEM embody complete Vector Meson Dominance: they are entirely given in terms of the co-meson content of the soliton. In the case of the isovector form factors, VMD holds only asymptotically, i.e. at large distances r from the center of the soliton. In this region, F(r) is small, and the form factors are determined by the p-meson fields ~1(r), 42(r) and G(r) (for r ~ 1 fm). Before actually computing the form factors (6.49), let us examine whether the electric charges are properly normalized. This requires S 2 V 2 1 G~(q =0)=G~(q =0)=~.
(6.50)
For the isoscalar electric form factor this means 2 w(r)dr= —~g/m~.
(6.51)
4irf r
This can be easily proven by using the equation of motion for the co-field: [—V2+ m2j w(r) =
-~
~y
4IT
r
sin2F
—
3g 2 2 c [G(G l6irr dr -~-
+
2) sin 2F],
(6.52)
so that the total derivative appearing on the right-hand 2 = 0: side of (6.52) is only present in the complete model. The Fourier transform of (6.52) yields at q 4ITm~f r2
co(r)dr=
~ 4n-f
~--~
sin2Fdr= _(~f)B.
(6.53)
Observe that the correct normalization is obtained by virtue of the topological quantization of the baryon number (here B = 1). In the construction of the isovector charge density, the moment of inertia A enters. Comparing the integrand of (6.49c) at q2 =0 with A[F, G, ~ ~ ~ (6.26), it is not immediately obvious that the normalization G~(0)= 1/2 will indeed be satisfied. To prove this, we use
298
U. -G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
a trick proposed in ref. [6.21]. We separate a constant from the p-meson excitation ~1(r),i.e. we define ~(r) = ~1(r) 1, and rewrite the A-functional (6.26) in terms of m ~2’ and ‘p: —
A[F, G, w; r~, 2[—sin2F+ 2~(l+ 2cos F) =
+ 32 +
2(~+ 1)~2+
+ 3]—
+
~
~f~r +
~
[3r2~’2+ r2~2+ 4G2~(~ + ~2)
+
—~-~
+
2(G2
c
—~--~
2G
+
~2
+ m2’p2]
2)~]
‘pF’[2sin2F— c~ 2 c’q —
—
c]
—
4Ir
The Euler—Lagrange equations for ~f(0)=0,
+
2
i~are
8IT
~‘sin2F(1
+ i~+
G’~j).
(6.54)
subject to the boundary conditions
~(x~)=—1,
(6.55)
whereas the ones for 42(r) and ~(r) can be read off from (6.27). To make contact with the isovector charge functional, we again bring the moment of inertia in its reduced form by performing a partial integration and using the equations of motion for n, ~2 and ‘p (cf. (6.26) to (6.32)). This leaves us with
Ared[F,
G, ~
m ~2’
2[3
—
~f~r
‘p]
sin2F
+
~(1 + 2 cos F)
+ ~2] +
(1— c)
—~-~
4i~
‘pF’ sin2F,
(6.55a)
where we have thrown out a total derivative in the complete model. Rewriting (6.49c) in terms of Th and ‘p, it follows immediately that the isovector charge is indeed properly normalized, i.e., G~(0)= 1/2. Let us now discuss the electromagnetic properties of the nucleon as given by the form factors (6.49). We begin by a discussion of the isoscalar charge distribution which reflects the co-meson content of the nucleon. It is particularly interesting to compare the normalized isoscalar charge density Ps(r)
O,S
=
2Jem(X)
2
=
—(2mU)/3g)w(r),
(6.56)
4ITfpS(r)r2drl to the baryon number distribution B 0(r) which is the source of the co-field. In fig. 6.5, we show ps(r) and
B0(r) for the minimal model. Note that the isoscalar photon couples to the co-meson, and not directly to B0(r), due to the underlying VMD. Hence, the2which isoscalar chargethe radius as seen ofbybaryon the photon is measures distribution number considerably larger than the radius rH = (r~)V inside the soliton. To illustrate this, let us consider the isoscalar charge form factor (6.39a) G~(q2)=
-
J
r~j 2+q2 0(qr) a)(r) dr = rn
J
r~j 0(qr) B0(r) dr,
(6.57)
U. -G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
299
ISOSCt,LAR CHARGE DISTRIBUTION
r[fm] Fig.
6.5.
Isoscalar charge distributions for the minimal model. B
0(r) is the normalized baryon charge density, whereas the photon couples to the
isoscalar electric charge density as given by the s-meson field.
2 m~)w(r) = [(~ g) B where we have used (V 0(r) + total derivative] and the convolution theorem of Fourier transforms. From (6.51), we can immediately derive a relation between the isoscalar charge 2 and r~,it reads radius (r~)~ GW=0) ~ (6.58) —
Hence with (r~)1/2 0.5 fm (cf. table 6.1), one obtains (r~)1/2 0.8 fm, in good agreement with the empirical value (r~)~2 =0.79fm. The significant difference between (r~)~2 and ~r~)U2 is due to the propagating co-meson which connects the isoscalar photon with the baryon number distribution. Note that in the limit of “frozen” vector meson degrees of freedom (mU)—~,gU)/mU) = constant) this difference would disappear. This is in fact the case in Skyrme-type models [6.6]. It is obvious, however, that important physics at length scales \/~/m,, 3 0.5 fm is lost in this limit. Let us now discuss the static nucleon properties and the proton/ neutron charge distributions. The latter ones are determined from the isoscalar and isovector charge distribution given in (6.48). The resulting proton and neutron charge distributions are shown in fig. 6.6 for the complete model. The static electromagnetic nucleon properties are summarized in table 6.4. The proton and neutron magnetic moments are somewhat too large in the minimal model and reasonably well-reproduced in the complete model. In both models, the isoscalar magnetic moment comes out close to the experimental value ~ = 0.44 (p~= 0.40 for c = 0 and ~ = 0.47 for c = 1). For the minimal model, the isovector magnetic moment comes out too large (p~= 2.97 for c = 0), the empirical value is ~ = 2.35. The improvement in the complete model (c = 1) is due to the cop-i-r interaction contributing mainly to the isovector magnetic moment. The electromagnetic charge radii are in good agreement with the data for both models, with the exception of the neutron charge radius, which is somewhat2 overestimated since it = 0. The electric and involves a very delicate cancellation of the isoscalar and isovector currents at q magnetic form factors of the proton and the neutron are displayed in figs. 6.7—6.10. In the minimal —
-~
300
U.-G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons NUCLEON CHARGE DISTRIBUTIONS
PROTON
015
NEUTRON
O~”I~O
[III,
r[fm]
“2.0
r[f ml 2
Fig. 6.6. The proton (left) and neutron (right) electric charge distribution for the complete model. Notice that the factor 41Tr
has been included.
Table 6.4 Electromagnetic properties of the proton (p) and the neutron (n). The electric and magnetic charge radii as well as the magnetic moments are given. For comparison, we show the results of the conventional Skyrme model [6.6]. Complete model
0.92 —0.22 0.84 0.85 3.36 —2.57 1.31
0.97 —0.25 0.94 0.94 2.77 —1.84 1.51
(r~)~2 [fin] (r~), [fin2] (r~)~’2 [fm] (r~,)~2 [fm] ~&,, [nm] is,, [nm]
Minimal model
I)L~//Lj
Conventional skyrmion 0.88
—0.31 0.79 0.82
1.97 —1.24 1.59
Experiment 0.86
±0.01
—0.119±0.004 0.86±0.06 0.88 ±0.07 2.79 —1.91 1.46
model, the q2-behavior of all form factors is well-reproduced. * The complete model tends to produce charge and magnetic moment distributions which are slightly too large in size, but keep in mind that the normalization of the magnetic form factors comes out much better in the complete than in the minimal model. For comparison, we also display in figs. 6.7—6.10 the empirical dipole fits [6.27] G~(q2)= (1
—
q2/M~2,
G~(q2)= ~~(1 n
—
2
q2IM~)2, 2
(6.59)
2—2
G~(q ) = ~t~(1 q /MD) G~(q2)= 0.20:2 (i L13:2yl~Gfl(q2)~ —
-
*
We show a comparison with data up to q2 = 0.6 (GeVIc)2, these statements hold true up to q2
hold beyond the applicability of the model (cf. section 7.2).
1 (GeV/c)~Any further extrapolation would
301
U.-G. Mei/3ner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
2)
PROTON CHARGE FORM FACTOR G~(q I
I
I
I
I
NEUTRON CHARGE FORM FACTOR G~(q2) I
1.0
0.08 0.06
0.5 ~
I
I
J7~~TT) dipole
0.02 0.04
I
0
I
0.1
0.2
0.3
0.4
—IJ.02
I
0.1
0.5
I
0.2
1q
21
~2[GevZ]
0.3 [GeV2]
0.4
Fig. 6.7. The proton electric form factor. The solid lines gives the result of the minimal model, the long-dashed line is the result of the complete model. For comparison, the empirical dipole fit (e.g. (6.59)) is also shown. The data are taken from ref. [6.85].
Fig. 6.8. The neutron electric form factor. The solid line gives the result of the minimal model, the long-dashed line refers to the complete model. The data are taken from ref. 16.851.
PROTON MAGNETIC FORM FACTOR G~(q2)/G~(0)
NEUTRON MAGNETIC FORM FACTOR G~(q2)/G~,(0)
I
I
I
I
I
I
I
I
1.0 1.0
0.5
— S..
S.
dipole
dipole I
O
0.1
0.2
I
0.3
0.4
0.5
1q21[Gev2] Fig. 6.9. The proton magnetIc form factor. The solid line is the result of the minimal model, the long-dashed line the one of the complete model. The data are taken from ref. [6.85].
0
0.1
~
I
I
I
I
0.2
0.3
0.4
0.5
q2J[Gev2] Fig. 6.10. The neutron magnetic form factor, notations as in fig. 6.9. The data are from ref. [6.85].
302
U. -G. MeiJiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
with M~= 0.71 GeV2. Keep in mind that in the calculation of the electromagnetic form factors we have neglected any recoil effects. To leading order in N~,this approximation is justified [6.251,but it deserves further studies. This problem is presently under investigation [6.28, 6.29]. We should stress, however, that the results exhibited in this section are extremely encouraging if one keeps in mind (i) the simplicity of the model and (ii) that we did not allow for any readjustment of the parameters g and f~, as it is commonly done in Skyrme moaels [6.3, 6.6, 6.7, 6.14]. 6.3. Axial properties of the nucleon In addition to the electromagnetic form factors, the axial form factors give further information about the structure of the nucleon. Particularly, the axial current is only partially conserved (PCAC [6.301) and the axial charge is not equal to one. These facts have to be explored in the context of the chiral models we are discussing. Before going into details, we shall remind you about the definitions for the axial current. Let N(p)) be a nucleon state with four-momentum p, then the most general matrix-element of the axial current operator which conserves parity and time reversal invariance is (N(p’)~A~(0)IN(p))= ~(pt)[G~(q2) y~+ ~
G~(q2)q~+ 2MN GT(q2) ~q~]y5
~ u(p’)
(6.60) with q2 (p p’)2, and MN the nucleon mass. G~(q2)is the axial form factor, its value at q2 = 0 gives the axial charge gA of the nucleon which can be measured in /3-decay. Experimentally, one has —
g~= (1.25 ±0.01). G~(q2)can be measured in neutrino reactions (p + v~—~n + ~i~) [6.31, 6.32, 6.33, 6.34] and through pion electroproduction of protons [6.35, 6.36, 6.37]. G~(q2)is the so-called induced pseudoscalar form factor, which is experimentally not well-known [6.171and will not be considered in what follows.* G~(q2)is the so-called induced pseudotensor form factor, it is only non-vanishing if one allows for second class currents, i.e., currents which have opposite G-parity to the ones defined by the operators y y~and q~y~.For calculating these form factors, it is most convenient to work in the Breit frame. One finds ~ =
~
G~(q2)~T
+
(GA(q2) =
—
~
2MN GT(q2)
Gp(q2))ff~]~
x~-’q
~-
x~’
(6.61a)
x~,
(6.61b)
so that the induced pseudotensor form factor related to_possible second class currents is given entirely by the time-component of the axial current. E = VM~N+ q214 is the nucleon energy, ~T = 4 is the transversal and ~L = 4 the longitudinal component of the spin operator. Let us first discuss the question of second class currents related to the time-component ~ of the —
*
Keep in mind that we could easily calculate G~(q2)in the models presented.
U. -G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
303
axial current. For the hedgehog ansatz, the most general form of the time-component A0’~is given by =
A 0(r) Tr{TaAT (K .
X
r)A~}
(6.62)
with A0(r) a model-dependent radial function. A(t) and K, the rotational velocity of the soliton, are defined in (6.21) and (6.22), respectively. The trace expression in (6.62) now has to be quantized in terms of spin- and isospin-operators. A naive application of the quantization rules (6.47) would give after inverting the order of A and K: Tr{TaATA+ (K .
X
r)}
—
2abbcd
if
rd =
—
(6.63)
a
However, K is a differential operator in the coordinates parametrizing the rotation matrix A (t) [6.4]: A=a0+ira,
~a~1; (6.64)
ô K=a.-~-——a0-~--—aX1 1/
Therefore, K and A do not commute, one obtains [K, A~]= —(1 /4A)r. A~,
(6.65)
so that the commutator contributes to the spin—isospin matrix element Tr{raAT[(_
A~x
~
=
—
~ Tr[raAi~ rA~]—~ +
,
a
(6.66)
2) 0 for all q2. Hence, chiral Therefore, the spin—isospin matrix feature element that of (6.62) vanishes, GT( q soliton models have the desirable they are free ofi.e.second-class currents. This is indeed consistent with the data [6.17]. Let us now discuss the space-component of the axial current. First, we consider it on the classical level, i.e. for the hedgehog. Its most general form follows to be =
A 1(r) ota
+
A2(r) ~
(6.67)
where i and a are space and isospin indices, respectively (i, a 1,2, 3). A1(r) and A2(r) are radial functions to be determined later. Performing the adiabatic rotations, we find AI~a(r)= [A1(r) ~l,b + A2(r)I’F”] x ~ Tr [ArA+r~2].
(6.68)
Collective quantization of the isospin rotation and using (6.47), we can read off the axial form factor 2) from the Fourier transform of (6.68): GA(q G~(q2)= [r2 j 0(qr) A1(r) + j1(qr) A2(r)] dr. (6.69)
-(~)~fJ
(~)
304
U. -G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
It is now straightforward to determine the axial vector charge g~
gA =
G~(q2= 0)
=
—
~
J
r2[A 1(r)
~A2(r)]dr
+
(6.70)
2: and the nucleon axial radius t~r~)’~ =
6 dG~(q2)
-
=
4M
-
~
J
r~[A 1(r) + ~A2(r)1dr.
(6.71)
Furthermore, the PCAC relation I9.~A~ = m~f~T,where ~ is the interpolating pion-field, can be written
9A~”~ =V~A’~
(6.72)
with Da(r) =
—
(~r~+ A~(r)+ ~ A2(r))u.
~
~ns Ta
D(r) ~ F.
(6.73)
Now, we have to determine the radial functions A1 2(r). This can be done in the following way. The axial current is the symmetry current of axial rotations of the U- and 4LR-fields (6.6), namely 2)A, ~
(6.74)
sESU(
~
U—~sUs. Using the Gell-Mann—Lévy method [6.26], this symmetry current can be read off from the term proportional to s s Fixing the gauge ~ = = ~ at the end, we obtain (cf. appendix C). ~.
2~
A 1 (r)
=
f~[
A 2(r) = —A1(r)
F (G
+
1)
—
sin2F]
—
~-
(sin~F cG cos2F), —
(6.75)
+f~F’(r),
where we have performed one partial integration on the Bp-term. * It does not influence the results presented in this section. In table 6.5 we summarize the axial properties of the nucleon, in comparison with the experimental data and the results of the conventional Skyrme model [6.6, 6.29]. The axial vector coupling constant g~comes out a bit too small in both models, but the value g~= 0.99 for the complete model is clearly improved as compared to the conventional Skyrme model. In the complete model, the wpir-interaction mocks up A1-degrees of freedom (cf. section 4.3); therefore, simulating some kind of axial-vector meson dominance as discussed in refs. [6.38] and [6.39]. The axial radius comes out close to the empirical value for both models. In fig. 6.11 we show the normalized *
This seemingly arbitrariness finds its natural explanation in the discussion of the strong NN~form factor (section 6.4).
U.-G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
305
Table 6.5 Nucleon axial properties. For comparison, the results of the conventional Skyrme model are also shown [6.6]. Minimal model
Complete model
Conventional Skyrme model
Experiment
0.99 0.62
0.65 0.35
1.25 ±0.01 0.68 ±0.03
0.88 (ri) ~ [fm] 0.64
2)/G~(O)
NUCLEON AXIAL FORM FACTOR G~(q 1.0
\
0.5
\~.
0
0.5
~,,/diPoIe
1.0
1.5
I~2I[Gev2] Fig. 6.11. The normalized axial form factor GA ( q2 ) Ig~for the minimal model (solid curve) and the complete model (long-dashed curve). Data are from refs. [6.31—6.37].
axial form factor GA ( q2) Ig~in comparison with data and the empirical dipole fit of ref. takes into account data up to q2 3(GeVIc)2 and is given by g~(q~ = 1.23(1
—
MA)
MA
=
1.05~~ GeV.
[6.341, which
(6.76)
The agreement of both models up to momentum transfer q2 ~ 1 (GeV/c)2 is very good, due to the op’rr-interaction the complete model gives a somewhat better description of the data. Let us finish this section by discussing the PCAC relation (6.72, 6.73). With A 1(r) and A2(r) given by
306
U. -G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
(6.75), we find D(r) =
—
3 m~sin F + c 8ir
~—
—~-~
Since the pion-field is given by ir
—
2F ~4 r G(2 cos
—
(G
+
1) cos 2F).
(6.77)
—(f
1,13) sin F, we see that the PCAC relation is exact in the minimal
model, i.e. ~
=f,m~,ir.
(6.78)
In the case of the complete model, there is a small deviation from the exact PCAC result due to the up’Tr-term in the WZ action (cf. section 3.4, eq. (3.78)). We will come back to this point in the following section. 6.4. Meson—nucleon form factors and the nucleon—nucleon interaction In One-Boson-Exchange models (OBE), the meson—nucleon interaction is cut down at large momentum transfer by the use of ad hoc form factors, commonly of the monopole-type = 2) m~~MNN G~~~(q
A -q
(6.79)
where M = {ir, p, o} is the meson under consideration and A a cut-off, typically of the order of 1 GeV. q is the four-momentum transfer of the meson as shown in fig. 6.12. ~MNN is the pertinent meson—nucleon coupling constant defined by G(q2 = —m~)= gMNN, i.e., the coupling constant is defined at the respective meson pole. For more details on this point we refer the reader to the extensive review by Machleidt, Holinde and Elster [6.40]. These strong form factors can be calculated in our models as we will show in what follows. We can make further contact with the OBE models keeping in mind that
STRONG MESON NUCLEON FORM FACTORS
N(p~)
Fig. 6.12. Kinematics of the strong meson—nucleon form factors
G~~~(q2), where
M stands for i~,p, w,.
..
U. -G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
307
meson exchange in lowest order can indeed be discussed at the classical level, as a pertinent example let us quote Yukawa’s original papers [6.41] on the pion exchange. Let us now describe how the strong meson—nucleon form factors emerge. Denote by ~M any meson field ii, p, w, A1,. , its static equation of motion can always be cast into the form .
2
.
m~)~M(r)
= JM(r) (6.80) (V with m~the pertinent meson mass. The source functions are highly non-linear functions of all the mesons under consideration; their Fourier transforms correspond to the meson—nucleon form factors. First, we will study the pion field surrounding a static nucleon. For that, we have to define the pion-field. Naively, one would identify it with the field ~n~(x)that enters in the unitary matrix U(x) = exp{ira ~()/f} but this representation involves ‘7rrr-coupling to arbitrary orders. Therefore, it is more convenient to work in the quaternion representation —
U(x)
=
(1 If.,,)[o~(x) + ir ~c(x)]
(6.81)
and identify the physical pion-field with ~a(X) This agrees with the way the pion appears in the linear if-model [6.26]. There the pion—nucleon sector is represented by the familiar Lagrangian of “oldfashioned” meson theory =
~rN
~(o~)2
+
~m~p2 —g
/iiy
5rVi~p+ 4n(t/i, ~fr),
(6.82)
where ~i, ~1iare Dirac spinors for the nucleon. Using the hedgehog ansatz, rotating the hedgehog state and performing a collective quantization, we can read off ~a(T) in terms of the chiral angle F(r) as follows: =
~ Tr{AT.
FA+Ta}
using (N!Tr(AT. FA+ra)IN) = reads 2 m~)4~(r)= J., (V 1(r).
sin F(r) = ~ (NIo
—
—
~
~ sin F(r)
FT°~N~.The
(6.83)
corresponding static field equation for ~c(r)
—
(6.84) fig. 6.12 with the pion as the Now, let meson. us derive strong ITN form factor ofGN~,~(q incoming Thethepertinent matrix element the pionic current in the Breit frame reads* 2).Consider
(N
~2) ~ 1(q/2)~J~(0)~N1(—q/2)) = G~~~(q
(6.85)
where MN is the nucleon mass and ~ are Pauli (iso)spinors. The left-hand side of (6.85) is nothing but the Fourier transform of the pion current as defined in (6.84) G
~
2\
*
a
~.
NN~TI~~J~ O~ N
qr
—
=
~j3
lq-r
—(q
f
2re+ m~) d3r ~l~r 1a
~a(r).
Defining G,,NN in this way, we avoid the problem of the vanishing linear coupling of the pion-fiuctuations to the hedgehog.
(6.86)
308
U. -G. Meifiner, Low-energy hadron physics from effective chiral Lagrangiaris with vector mesons
Furthermore, we can reduce (6.86) to a one-dimensional integral by defining the radial source function J~,(r)as J~(r)=
i,..
FT~
J1,(r)
(6.87)
which follows naturally from the hedgehog structure of the pion field. With that, the form factor GNN~(q) reads 8~rMN 1 2 = j r j 1(qr) J~(r)dr. (6.88) Using the equation of motion (6.14a), we can determine the source function J~,(r),it follows to be J~(r)= ~ f(F’)~sin F
+
m~(1 cos F) sin F + 2 sin F [1 2(G —
2F
+
8irf~ 2 r cos F[2 sin
—
—
+
1
—
cos F)
cos
cG(G +2) cos 2F1}.
F] (6.89)
J 1,(r) is shown for the minimal (c = 0) as well as for the complete model (c = 1) in fig. 6.13 (in arbitrary units) [6.42, 6.431. It is peaked around 0.5 fm away from the nucleon center, consistent with the baryon number charge radius Kr~)1/2 0.5 fm. It is interesting to extract the leading behavior of J1,(r) as r becomes large; using the expansion (6.17), we find 3(r) + /3 F(r) G(r), r-+ (6.90) J~(r) a F where a and /3 are constants. These terms correspond to 3’rr and ‘Tip-contributions, in agreement with ~,
—
PION SOURCE FUNCTION J7r(r)
r[fm] Fig. 6.13. Profile of the pion source function J~(r),eq. (6.89). The overall factor (f~I3)is scaled out.
U.-G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
309
the dispersion theoretical approach to the NN’rr-vertex function [6.44]. The ITNN-form factor in the spacelike region is shown in fig. 6.14 for the minimal and the complete model. For small momentum transfer q2, it follows closely a monopole with a cut-off A = 0.86 GeV, quite consistent with OBE models [6.4]. For larger momentum transfer q2 ~ 10 fm2, G~N~(q2) falls off somewhat faster than a monopole, consistent with perturbative QCD [6.45] and bag model [6.461calculations. In the timelike region, GNN~,(q2)defines the nucleon—pion coupling constant gNN,~, i.e. (6.91) We find ~NN~ = 15.1 for the minimal model and ~NN~ = 14.5 for the complete model. Experimentally, one has ~NN~ = (13.5 ±0.1). For the minimal model, this ~NN~ agrees with the one as determined from the Goldberger—Treiman relation g~If~ = ~
(6.91a)
whereas for the complete model ~ = 1.15, i.e., the Goldberger—Treiman relation is violated by 15% due to the Wess—Zumino term. Experimentally, the GTR holds within 7%. Finally, let us mention that the nucleon—pion form factor has a branch point at q = —9m~which represents the starting of the cut corresponding to the 3ir-continuum, i.e. J,~(r) exp{—3m~r}1r3. Notice that the GTR (6.91) is nothing but the q2—*0 limes of the PCAC relation Ô,~A~’ =f~m~r in —~
gl,NN
(q2) -glrNN
(q2 r-m~,)G~NN(q2)
G~NN(q2) IN THE SPACE-LIKE REGION (MINIMAL & COMPLETE MODEL)
GNN~.(~2) (Complete)
0
I
I
I
I
I 5
I
I
I
I
I 10
I
I
I
I
15
I
I
I
I
I
20
I
I
I
I
25
q2 [fm_2]
Fig. 6.14. The pion—nucleon form factor G~~~(q2) for the minimal and the complete model. For comparison, a monopole form factor with .4 = 0.86 GeV is shown.
310
(1. -G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
terms of the axial, the induced pseudoscalar and the strong irNN-form factor M~[G~(q2)
—
Gp(q2)]
~
=
~rn~2 G~~~(q2).
(6.91b)
For this relation to be satisfied, it is crucial to perform a partial integration on the w~B~-contribution of the axial current as discussed after eq. (6.75). This partial integration does not affect the Goldberger—Treiman relation at = 0 but is justified by the generalized PCAC relation (6.91b). For the o- and p-meson, the starting field equations are (V2
—
m~)w’~(r)= J~(r)
(V2
,
—
m~)p~(r)= J~(r)
,
(6.92)
where the prime on J~(r)denotes that the p-meson self-couplings are included in the p-meson source function for calculating the pNN-form factors. Before actually computing the (radial) source functions J~(r)and J~(r),let us consider the most general matrix elements of .J~(r),Jjr) between nucleon states, they read (N(pJ~(0)~N(p)~ =
u(pI)[F~P)(q2)
~
+
i F~(q~)if qP]r u(p), N
(N(p’)~J~(0)~N(p)) = ~(p~)[F~(q2) y~+
(6.93)
~
which define the vector and tensor coupling form factor F 2) and F 2), respectively. Empirically, in 1( q 2(q the case of the p-meson the tensor coupling —F 2 dominates the coupling, towhereas for the 2) at q2 = 0vector is proportional the anomalous o-meson the situation is reversed. The form factor F2(q magnetic coupling of the particle under consideration. We have F~(0)= 14.1, but F~0~(0) = —0.12 [6.40]. The actual evaluation of the form factors F~’~(q2)is performed in the Breit frame with q~q~ = —q2. For that, we need the radial source functions J(r) and J~(r).These are defined via (in the Breit frame) Jo.a(r)
=
J~°(r)a
,
J(r)
=
J,~(r)cr x
rTa,
(6.94) J~(r)= J~,(r)~
,
J~(r)= J~(r)o~x
~.
The p- and o-electric form factors G~’°~(q2) follow as Fourier transforms of the time components of the sources ~ the magnetic form factors are given by Fourier transforming the spatial components of the sources. The form factors F 2) and F 2) are related to G~(q)and G~(q2)via 1(q 2(q G~’~( q2) F~( q2) + [q2I4(M~)2IF~( q2) G~(q2) = F~°’~(q2) + F~’°~(q2). The radial source functions defined in eq. (6.94) are given by
(6.95)
U.-G. Meifiner, Low-energy hadron physicsfrom effective chiral Lagrangians with vector mesons
r2 J~(r)
~
~G2(2—2~
{~g2f2r2sin2(~) +
1 ~)+ c
[~‘ (G + 1)sin2F+2F’~1},
—
2 J~(r)=
{2g2f~r(1 cos F)
~
—
+
311
[G2(G
+
3)]
+
c ~2
w’r(G +1) sin 2F},
r 3 r2 J~(r)= r2
J1()
3 F’ sin2F
c
~T—
—~
~
+c
+
(6.96) [F’G(G + 2) cos 2F + G’(G
—~
rF’(2sin2F- ~
r[2F’(G
—
-
+
1) sin 2F],
c~2)
G~ 1
—
~)
cos 2F + sin 2F(G’
—
G’~1 G~ —
—
2)which is related to the dominant contribution of the In fig. 6.15 we show the pNN-form factor F~(q matrix element (6.93) for the p-meson current. Figure 6.16 shows the wNN-form factor F~(q2)which dominates the w-exchange. Notice the branch at q2 = —9m~when the source function enters the 3’ir continuum.* For smaller momentum transfer (q2 ~ 0.5 (GeVIc)2) both form factors follow closely a monopole with A 1 GeV, for larger momentum transfer they fall off somewhat faster. The results for F~(q2)and F~(q2)are similar. This is in rough agreement with phenomenological OBE potential fits [6.40]. In table 6.6 we present the normalizations F~(0) and F~(0)together with the ratio i~ = F~(0)IF~(0) which determines the tensor to vector coupling. For comparison, the recent results of the Bonn OBE potential [6.40] are also shown. The strength of the tensor coupling comes out remarkably well. However, the model does not account for the empirical fact that KPIKV, where Ky is the isovector anomalous magnetic moment (K~= F~(0)IF~’(0)), is 1.6—1.8. From the magnetic moments given in —
STRONG wNN FORM FACTOR F
STRONG pNN FORM FACTOR Ft
_
~2
~IIIT
[GeV2/c2]
Fig. 6.15. The pNN form factor F~(q2)in comparison with a monopole with cut-off A 1GeV.
*
A similar branch point exists for F~(q2)(fig. 6.15) at q2 = —9m~,.
2) 1w(~
(~2)
-9m~.O
0.5
1.0 1.5 ~2[GeV2/c2]
2.0
Fig. 6.16. The oNN form factor F~’(q2)in comparison with a monopole (A = 1GeV). At q2 = —9m~,,the source function for F~’(q2) develops a cut.
U. -G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
312
Table 6.6 Normalizations of the vector (F 1) and tensor (F,) couplings of the p-mesons to the nucleon. = F2(0) /F~(0)and the empirical values are taken from ref. [6.40]. Complete model (c=1)
Empirical values
2.7 14.4 5.3
3.2 14.2 4.4
2.3 14.1 6.1
F~(0) F~(0)
Minimal model (c=0)
Kp
table 6.4 we find that KpIKy = 1.1 for the minimal model and Kp/KV = 1.2 for the complete model. (There exists up to date no serious model calculation explaining this effect, although a suggestion has been made in ref. [6.45] in the framework of a chiral bag model.) For the o-meson, we find F~(0)= 8.78 for both models to be compared with F~°~ 11.5 [6.40]. The oNN coupling constant can be read off from the u electric current, J~(r)= [~~)NN B°(r)+ total derivative] as [6.46, 6.131 ——
~
(6.97)
which is roughly half of the empirical value g~,~~/41T = 10—12 [6.47, 6.48]. We will come back to this point later. We are now at the point to make some more qualitative statements about the nucleon—nucleon interaction. For that, consider two nucleons at positions r1 and r2 at a sufficiently large distance r = r~ r21 so that their baryon number distributions do not overlap and the mutual polarizations of the two nucleons can be neglected. In this case, the nucleon—nucleon (NN) potential mediated by the meson fields M can be written as —
V(rI,r2)=~Jd~rJM(rI,r)~M(r2,r)
(6.98)
in terms of the source function of nucleon 1 and the field generated by nucleon 2. Within this classical framework, we can discuss certain aspects of the NN interaction. First, from the discussion of the ‘irNN form factor it is obvious that the basic features of the one-pion exchange at distances r ~ 1 fm are well-reproduced. Second, the dominant contribution to the p-meson exchange, the tensor potential proportional to (o~x q) (u2 x q), comes out remarkably well as shown in table 6.6. Let us therefore consider the isovector tensor NN-interaction. It is known to be dominated by ii- and p-exchange which contribute with opposite signs [6.49]. In our approach, the tensor interaction in momentum space follows to be 2) S V~(q)= V~~(q 12(q) r1 T2. (6.99) 2) is given by S12(q) = 3o1 . ~ is the tensor operator, and V~~(q —
2 2 V~~(q )
—
~1
L
2
G~~~(q (2MN)2 ) q2
2
q +
2
m~
—
f2\
~ (2MN)
2
)
q q2 +
m~
(6.100)
U. -G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
313
is nothing but the p-meson magnetic form factor as defined in where the pNN-form factor G~N~(q2) (6.95). V~~(q2) is shown in fig. 6.17 for the minimal model. It is plotted such that one can see first the influence of the pion—nucleon form factor as compared to one-pion-exchange with pointlike nucleon sources, and then the role of p-exchange in cutting down the tensor force as q2 increases. For comparison, some data from NN scattering amplitudes are shown [6.50]. The model reproduces nicely the destructive interference between IT- and p-exchange and accounts well for the data [6.421.The results for the complete model are similar. As discussed in section 1, the w-meson accounts to large extent for the repulsion of two nucleons in the central channel. For large enough separation of the two nucleons, i.e. no substantial overlap, the NN central potential due to the w-meson is given by* 2
V~(r)=
~11N~
f
—m,,Ix—yI
d3x d3y B 0(~x rj) e —
B0(~y r2~)
—
(6.101)
—
for the minimal model using J~(r)= g,aNN B°(r). This potential gives the solid line in fig. 6.18. At distances much shorter than 1 fm, one cannot use (6.101) any more due to the strong overlap of the baryon number distributions and the deformation effects from the mutual polarizations. These shape
NN SOVECTOR TENSOR INTERACTION 1,’V 1(OPE) I
0.3 ,2
I.)
,
~i 3fm3J
11.1 l~I
IGeV 0.2
/
I
/
/I
._—--_..~ ~
1.5
-
TI
only 1.0
-
0.5;
I~~I [fm11 Fig. 6.17. The isovector tensor potential (6.100) as a function of momentum transfer qI. The short-dashed curve gives the one-pion exchange with point-like nucleons. The long-dashed curve shows the influence of the ~irNNform factor G~~~(q2) with the coupling constant readjusted to the empirical one. The solid curve is obtained with the complete expression (6.100) including p-exchange. The empirically deduced data of energy E= 140 MeV are taken from ref. [6.50] (dots). The vertical scale includes the empirical nucleon mass. *
NN CENTRAL POTENTIAL
-
-
I
r [fm] Fig. 6.18. The repulsive central NN potential generated by the wmeson field. The dashed part indicates that the product-ansatz used for separations R ~ 1 fm is only a rough approximation to the true potential l’~fr).
For the complete model there are small corrections from the total derivative term.
314
U.-G. Metfiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
EFFECTIVE wNN COUPLING CONSTANT
1~1
~eff
__ N
=
N
__
N
N
N
N
Fig. 6.19. Important contributions to the effective wNN coupling constant used in OBE-potentials. These effects (intermediate z~’s)are not included in the classical framework [eq. (6.98)] considered here.
deformations have been indeed observed in extensive computer simulations of the skyrmion—skyrmion interaction [6.51, 6.52, 6.53]. We can nevertheless get a rough estimate of the repulsion in the NN-system by investigating the baryon number B = 2 solution of the coupled equations (6.14a, b, c) with the pertinent boundary conditions F(0) = 21T and G(0) = 0. Comparing the mass of this object with twice the mass of the B = 1 solution gives =
MH(B
=
2)— 2M
11(B
=
1)
=
1.25 GeV.
(6.102)
t~Mis almost entirely given by the mass difference from the Wess—Zumino term, which generates the dominant B~B~-coupling of the baryon current at short distances. ~M is therefore a qualitative measure of the short distance repulsion generated by the *o-field. It turns out to be too small by a factor of two if one compares the volume integral of this interaction as shown in fig. 6.18 (dashed line) with the corresponding one of the Bonn OBE potential. The missing repulsion in the central NN-potential is clearly related to the too small wNN coupling constant (cf. eq. (6.97)). It has to be seen in the same context as the missing intermediate range attraction in these models. Both effects are associated with two-pion exchange and go beyond the classical framework used here [6.44]. As an example, let us mention that g~°~ gets roughly half its strength from diagrams like the one shown in fig. 6.19, and these are clearly not treated in the framework of classical meson theory. Nevertheless, we can say that some important qualitative features of the NN-interaction are described properly by the rather simple models developed before. 6.5. The baryon spectrum In this section, we want to demonstrate how the inclusion of vector mesons improves the baryon spectrum in comparison to calculation in the framework of the conventional Skyrme model which have been extensively carried out at SLAC [6.54, 6.55] and at Siegen [6.56, 6.57]. For a more complete set of references we refer the reader to the reviews [6.58, 6.59]. We will not outline the whole calculational scheme but rather give some specific results obtained by Weigel, Schwesinger and Hayashi [6.60, 6.61] who worked out the baryon spectrum in the minimal model (c = 0). The procedure to calculate baryon resonances is the same as in the conventional Skyrme model.
U. -G. MeiJlner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
315
First, one constructs the hedgehog solution and the scattering of pions, p- and w-mesons off the soliton is described by small amplitude fluctuations around the static solution. This amounts to the ansätze U(r, t) = exp{ir. [FF(r) w(r, t) = (w0(r) p,L(r,
t)
=
+
(p0(r,
O(r, t)]}
+
W0(r, t), W(r, t))
t),
(6.103)
,
p1(r) +p~(r,t)).
We therefore have to consider three pionic, four w-meson and twelve p-meson excitations (0, W~,p~, respectively). Injecting the ansätze (6.103) into the Lagrangian (6.1) and expanding to second order in the small oscillations one obtains coupled equations of motion upon variation. Due to the hedgehog structure of the static solution, i.e., its invariance under combined space and isospin-rotations, one ends up with eight transverse and seven longitudinal coupled differential equations for the small oscillations. Here we have made use of current conservation which eliminates four vector meson degrees of freedom. From this, we can now calculate the scattering matrix S in the soliton-fixed frame, with all input parameters given from the meson sector
f,~=93MeV,
m~= m~=770MeV,
g6.12.*
(6.104)
A technical subtlety involves treating ~= \/iJ, only one of the two possible squares leads to a regular solution at r = 0. A way to circumvent this problem is to add a small extra mass term for the p-meson, = Sm~p~ p~with ~ 10MeV [6.60]. Notice that this problem does not occur in the calculation of the static properties discussed before. To compare to the empirical data, a further step has to be taken. For that, the physical S-matrix elements SL2T2J have to be calculated as linear superposition of the soliton-fixed ones via —
SL2T2J = ~ G(K, T, J, L) SKLL
(6.105)
where G(K, T, J, L) is some geometrical coefficient, L denotes the angular momentum of the pion, J the angular momentum of the ‘uN system and T the total isospin. To arrive at (6.105), we have assumed an adiabatic rotation which neglects the dynamical effects of the rotating skyrmion. This is a severe assumption and deserves further study. The physical S-matrix may be parametrized by phase shifts 6 and absorption coefficients (inelasticities) ~j via T1L,2T,2J
SL2T2J =
exp{2i6L 2T2J}
.
(6.106)
In fig. 6.20, we show some pertinent results of the phase-shift analysis [6.60] in comparison to the Skyrme model results. For the higher partial waves, like e.g. D13 and F15, there is a very good agreement between the predictions of the model and the data. Notice furthermore that inclusion of the vector-mesons solves the problem of the rising phase shifts in the Skyrme model, i.e. the high-energy behavior of the phase shifts and inelasticities is apparently improved. However, the old problems with the missing resonances in the S11 ** and the P11 channel remain, for the latter one has some *
The authors of ref. [6.60]prefer to work with g = ~ and not g as determined by the KSFR relation. By including a scalar meson one finds some structure in this channel (cf. section 7.1)).
* *
316
U. -G. Meijiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
PION-NUCLEON
:~1
SCATTERING
~:1TTJ
:::1i111~~ ~
~
~
-
F
15
05
1
15
ENERGY
0.5
1
1.5
EGeVJ
Fig. 6.20. Phase-shifts ~ (in degrees) and inelasticities (‘i) for some partial waves. The dashed lines represent the results for the minimal model [6.61], dashed—dotted lines correspond to the results of the conventional Skyrme model, and the solid lines are the data taken from ref. [6.86].
improvement at higher energies. * Clearly, these problems are related to the improper treatment of the zero modes, in fact, the rotational and translational zero modes of the soliton influence the L = 0, 1, 2 phase shifts. Furthermore, let us make some remarks on the opening of other channels. Apart from the ‘urN—*irN channels, there is also a strong flux into channels like ‘rrN—*Nw, Np, ~‘ur,~w and ~p. This can be deduced from the large deviations of the inelasticities from unity at higher energies. Below the p-meson threshold, only the &rr-channel is open and is strongly excited before other channels open. The large variations of i~at higher energies are due to the effect of channel openings at threshold. Overall, we can summarize by saying that the minimal model works rather well in describing the excitation spectrum of the nucleon. Finally, let us mention that Mattis [6.63] has given model-independent predictions for vector-meson scattering off skyrmions which are in fair agreement with the available data. *
Notice that the Roper shows up in the helicity amplitudes of pion-electroproduction [6.62].
U. -G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
317
6.6. Complete vector meson dominance? In the minimal and complete model discussed so far, vector meson dominance was only asymptotically realized in the isovector channel, although the pion form factor is nicely described by p-meson exchange alone. One might therefore wonder how important the deviations from the naive currentfield-identify 1 J~~~’—(m2Ig)p’~
(6.107)
in the baryon sector are. For that we will consider a model which incorporates exact VMD, i.e. it has no pionic corrections to the isoscalar/isovector currents. This model is constructed according to Sakurai’s ideas of vector-mesons as gauge fields of the isoscalar and isospin symmetries [6.24]. Following the work of Meil3ner and Kaiser [6.21, 6.39, 6.64], such a model is based on the U(2)~symmetry of the non-linear if-model. It is essentially equivalent to the complete model discussed before, apart from the fact that it does not contain non-minimal terms like Tr[A~UA~Ut]and Tr[F~~UF~~RUt]. Notice that we are now considering a massive Yang—Mills Lagrangian with pions, p- and w-mesons. The U(2)~Lagrangian is given by Tr[D~U D~Ut] ~(p~ —
+
w~)+
xTr{ir.pa(ô~U Ut + Ut ~U)+
~
(p~+ w~)+
~-
(~
g)w~B~’ +
paUtTpsLT~ ~m~f ~Tr(U— 1)
(6.108)
with D~U = U (igI2)[T ~ U] the pertinent covariant derivative. We will not go through all details in constructing nucleon observables as done before, but rather outline the differences to the models presented so far. First, due to the simpler structure in the non-anomalous sector, the relevant boundary condition for the p-meson profile at small r is G(0) = 0, not G(0) = —2 as before. Therefore, the p-meson profile is much less pronounced than in the previous cases. Due to the omission of the non-minimal terms* which are repulsive, the hedgehog mass is lowered to MH = 1004 MeV for our standard set of parameters f.,, = 93 MeV, g = 5.85 and a = 2. The hedgehog radius decreases to rH = 0.43 fm. Projecting onto nucleon states and minimizing the moment of inertia functional, we find again that the p-excitation profiles ~ 1(r) and 42(r) are much less pronounced than in the complete model. This has immediate consequences for the moment of inertia, it turns out to be too small: A = 0.358 fm, i.e. M~ MN = 827 MeV. Furthermore, the isovector currents are entirely given by the p-meson in this model, we have for the isovector form factors [6.64]** —
—
2)= ~ G~(q 2)= G~(q
* **
2~fr2[3~ 1(r)+ ~2(r)]j0(qr) dr,
-
MNJ G(r)
(~)
(6.108a)
j
1(qr) dr.
(6.108b)
This modifies the PCAC relation via vp-corrections. The isoscalar form factors are given in (6.49a, b), the w- and clLprofiles are not very different from the ones in the complete model.
318
U. -G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
The electric charge radii are not too much affected since the decreased integrand in (6.108a) is balanced by the factor A However, for the isovector magnetic moment we find a much too small value, namely = 0.68 to be compared to the empirical value of ~ = 2.35. Furthermore, g~ = 0.64 is as small as in the conventional Skyrme model [6.66], i.e. it is half of the empirical value. From these observations we conclude that exact vector meson dominance is not realized in the baryon sector of low-energy hadron physics. Chemtob [6.65] has investigated a similar massive Yang—Mills model with ‘ii-, p- and ü-mesons with VMD imposed via the order-by-order construction of local counterterms a Ia Brihaye, Pak and Rossi [6.66]. For the parameters f,~and ~ on their empirical values, he finds M1~= 983 MeV,
~.
MN = 719 MeV, ~ = 0.80 and g~= 0.43, very similar to the results of Meil3ner and Kaiser. Only by changing the parameters to f~ = 76 MeV and g 9.10 does the Nz~-splittinggo down to 297 MeV and j~ increase to 1.77. In both cases, ,u~lies in the range 0.3 ~ ~ 0.4, in agreement with the results of ref. [6.64]. Similar observations have been made in ref. [6.23]. These authors discuss the hidden symmetry Lagrangian with pions, p- and ‘.o-mesons. The Wess—Zumino term is chosen to ensure exact VMD on the one-photon-level as proposed by Fujiwara et al. [6.66]. The contribution of the WZ term to the static soliton energy is given by (cf. eq. (3.71)) —
-~
12r dri1~ fgN~ 22 16ir r
4irj
~ VMD =~
2F ~ (wF’G(G + 2) +2 sin F(G’w + Gw’) + w’(2 sin F sin 2F)1}. (6.109) x [wF’ sin The equations of motion for the meson fields change accordingly, the boundary conditions are the same as in (6.15). The results are similar to the ones discussed before; the hedgehog mass decreases to 1143 MeV, the hedgehog radius comes out to be rH = 0.36 fm. * Therefore, the axial-vector charge gA is also very small (g~= 0.47) since it is proportional to the extension of the pion-source function. * * The authors of ref. [6.23] have also calculated the moment of inertia of the spinning soliton, they find A = 0.411, i.e. M~ MN = 705 MeV. These findings are rather consistent with the ones of Meil3ner and Kaiser [6.21, 6.39, 6.64]. Furthermore, recent investigations of the nucleon electromagnetic form factors in a QCD-based phenomenological model (meson theory of small momentum transfer and perturbative QCD for high momentum transfer) suggests that half of the isovector photons coupling to the proton are given by multipion terms [6.67]. In the context of chiral bag models, the authors of ref. [6.45] also propose sizable deviations from exact VMD via photon—quark couplings. All this indicates that although exact VMD works well in the meson sector, the non-linearities inherent in the solitonic nature of the baryons exclude such a simple picture in the baryon sector. We will come back to this point when we discuss the static properties of nucleons in the ‘Trp(0A 1 -system. Again, we will encounter the situation that insisting on exact (complete) VMD will lead to a too small moment of inertia and isovector magnetic moment. —
—
—
6.7. Results from the ITpA1 w-system Up to now, we have only considered Lagrangians involving the pion and the low-lying vector mesons We will now extend our analysis to systems including the axial-vector meson A1. Due to the
p and *
~.
The same input parameters (f,, = 93MeV, g = 5.85) are used. Keep in mind that g~measures the i~NNcoupling strength through the Goldberger—Treiman relation.
**
U. -G. Meifiner, Low-energy hadron physics from effective chira! Lagrangians with vector mesons
319
difficulties related to the A1 parameters and ITA1-mixing as discussed in sections 2 and 3, there exist various calculations using different Lagrangians [6.12, 6.13, 6.14, 6.68, 6.69, 6.70] reaching slightly different conclusions. In refs. [6.68]and [6.70],the Lagrangian includes in addition a scalar particle (the e-meson), the discussion of this work is therefore delegated to section 7. Historically, Meil3ner and Zahed [6.12, 6.13] were the first to demonstrate that the inclusion of spin-i mesons with all parameters fixed in the meson sector indeed improves the static baryon properties as anticipated by Witten [6.71]. We will give a brief outline of their work and only comment on the other calculations. Following the work at Syracuse [6.46], the authors of refs. [6.12, 6.13] consider the following massive Yang—Mills Lagrangian with the p-, A1-, w-mesons as SU(2)L 0 SU(2)R 0 U(1)~gauge particles: =
+ ~wz’
~o +
~
—L~)} t} —
Tr{a~}+
~
—
Tr{p~Ua~Ut}
Tr{a~[U,a~]Ut}+ ~ f~m~ Tr{U
—
8g
~
Tr{a~Up~U
Tr{p~[U, p~]Ut}
+
=
~
Tr{w~w~ + p~p~+ a~~a~}~ Tr{p~+ 4g —
—
IC
—
—i
**‘a$
r
+
+
Ut
—
2}
(6.110)
a~},
mn ir 1L w~1rl~p~a~,sf -‘- ;-;—--1- ~ IT
x Tr{p~,(R~ L~)+ aa(Rp —
+
L~)+ Paas
—
~(Pa aa)Ut(p~+ a~)U}, —
where ~ is the contribution from the gauged non-linear if-model, .9?~ gives the minimally broken SU(2)~0 SU(2)R 0 U( 1 )~ Lagrangian for the spin-i mesons (p, w, A 1) and ~ is the Wess—Zumino term in the Bardeen-subtracted form (cf. section 2.1). L,~ = Ut ô.,~U left-handed current in the 1 [o-(x) + ir• i~(x)]defined to incorporateis the the isovector pion field. The pSugawara form, with U(x) = f and A 2)L®SU(2)R doublet, the o-meson is associated with the U(1)~gauge 1-mesons formthean following SU( symmetry. We use conventions: co~= ijw~ p~= igp~T”, and a~= iga~Ta. For convenience, the gauge couplings j of the U(i)~and g of the SU(2)L 0 SU(2)R are chosen to be the same, j = g. The pertinent field strength tensors are ~,
~
—
p~’
9~p1’—,~9PpU+
a~L=ô~La*_9*a/L
+
~
~
p*]
+
~
av]+ ~
ar],
(6.llOa)
p*].
As discussed already in section 2, the Lagrangian (6.1) needs proper diagonalization due to the ‘urA 1-vertex —Tr{a~~IT}. The decoupling of the pion and the A1 is achieved by defining the physical fields ~ and c via 2Im2)d,~*, ~=a~—if,,.(g
~=Zir,
(6.111)
320
U. -G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
where f,~is the physical pion decay constant given by* =
where m
Z
=
(1
—
g212/m2)l/2
(6.112)
,
m~= m~is the vector meson mass. As a consequence, the bare A1 mass m gets shifted to = m/Z = \/~m.In what follows we will omit the ““ when writing physical ii- and A1-fields. We will neglect the momentum dependence of the pIrIT-coupling and fix g = ~ through the width of the decay p°—s~ IT IT. This requires g = 6.05. Actually, in ref. [6.701this momentum dependence has been taken into account, we will discuss this point later on. The choice of parameters g = 6, f,, = 93 MeV and m = 780 MeV leads to m~= 1116 MeV close to the Weinberg ratio. Notice that the KSFR relation does not follow automatically in this scheme. Let us now construct solitons (baryons) from the Lagrangian (6.110). For that, we make again the time-independent hedgehog ansätze, the most general form for spin-i fields reads (cf. section 6.1) =
w°igw(r),
p=igR(r)(FXr), (6.113)
a=ig[a1(r) r+a2(r) F’r.F]. Notice that the a involves two radial functions a1(r) and a2(r), just as the p°-componentin eq. (6.23). Obviously, w = p°= a°= 0 at the classical level. For U(r), we choose the usual hedgehog ansatz. Injecting the ansatze (6.113) into the Lagrangian (6.110), the ground-state energy of a hedgehog skyrmion reads E = MH
=
E~+ E.ITPA
+
E,~A+ E~A+ E~
+
~
(6.114)
where the E’s are functionals of F(r), R(r), w(r) and a12(r) given by 2 dr [F12
J
+
2 sin2F + 2m~(i cos F)], —
E~= 2ITf~ r
J
E~~A = 4ITf~ r2 dr[g2Ra sin 2F
+
g2 sin2F(R2
—
a2) +
2gR
~
~
2
2
r
E~A= 4’rrm J r dr [R *
2
~ —
2
-~-
2
+
~
2
(3a +
/3
2
+
Notice that there appears a sign error in refs. [6.12, 6.13] In this equatIon.
2a13)
,
(6.115)
U. -G. Meif3ner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
321
~
+2g ~(2a2+~2+2ap)+2g(a+$)(R~a_Ra1) + ~ [(R2+
2)2 +
2(a2
~-~-~--~
—
+
R2)(a + ~)21}
~
~
+ 2a
gaR
—
~
(a2
—
R2) sin2F + ga
1R cos 2F)],
in which a and f3 are defined via
/
j~. sinF
iTT
\21
sinF
.
a=a1—g—-2---—, f3=a2—gy--~)[(S1flF) The pertinent Euler—Lagrange equations can be obtained by functional minimization of the energyfunctional (6.115). The relevant boundary conditions to ensure unit baryon numbers and finiteness of the energy are F(0) = ~T, R(0) = 0, w’(O) = 0 and a’(O) = 0. The equation of motion for /3(r) is only of first order and can be used to eliminate one degree of freedom from the system of coupled equations. The vector-meson profiles of the p-, w- and the physical A1-meson are shown in fig. 6.21, for the parameters f~ = 93 MeV, m = 780 MeV, g = 6.0 and m~= 138 MeV. The hedgehog mass comes out to be M~= 847 MeV. Since the w-meson couples entirely through the Wess—Zumino, its coupling constant —----~-—
MESON
PROFILES
IN THE lrpA1w
-
SYSTEM
r[f ml Fig. 6.21. Vector- and axial-vector meson profiles in the irpüA1-system, from ref. 16.131, for g = g= 6. Curve 1 represents the w-profile w(r), curve 2 the p-meson profile R(r), curve 3 the scalar part of the physical A1-meson A~(r)and curve 4 the tensor part of the physical A1, A~(r).The scale on the left-hand side stands for the R(r), whereas the scale on the right-hand side corresponds to w(r), A~(r),and A~(r).
322
U. -G. Meif3ner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
can be read off from (U
+
(6.116)
m2)w~= —(gN~I2)b~ ,
where b,~is the conserved isoscalar current,
b~=
—
Tr[LpLaLpI
~
—
32ir
~
(6.117)
From (6.116) we conclude that the w-hedgehog coupling is g’~~fl/47r = N~g2/16ir2= 6.45, a bit too low compared to the empirical value of the w-nucleon coupling, g~/4i~= 10—12 [6.47, 6.48]. Thus, we do expect sizeable contributions from quantum effects. Instead, one can use a U(1)~-coupling~ different from the SU(2) one, and fit it to the empirical value. For j= 7.47, i.e. g~fl~I47r = 10, we find M 11 920 MeV. To project onto states of good spin and isospin, we again perform an adiabatic rotation of the soliton andmost quantize theansãtze spinning To fields leading rotational velocity tA)the general for modes. the vector are order (using in thethe notation of refs. [6.12,2Ka 6.131, —i Tr(r’~A cf. eqs. (6.23, 6.24)): w~(r,t)
=
ig w(r),
(OA(r, t)
=
ig WA(r) (P x K),
p~(r,t)
=
ig ~A(r) Ar~KA1
pA(r, t)
=
A p(r) At
+
ig ~A(r) K~PAr PAt, (6.118)
0
t
.
aA(r, t) = ig ~A(r)
Ar (r x K) A
a 0(r, t)
=
A a(r) A~
where WA(r), ,iiA(r), ~A(r) and CPA(T) are smooth functions of r to be determined. Notice that the isorotation of the skyrmion excites all odd-parity components of the vector fields. For the calculation of the moment of inertia A[WA, /~A’~A’ ~A] we will assume that jz~= = = 0. This approximation is motivated by the predominance of the w-correlations in the static analysis.* In this case, the moment of inertia depends only on the w-excitation WA(r) and the classical background fields F(r), R(r) and w(r). The equation of motion for the profile function WA(r) reads 2WA 2 F’sin2F WA+2+mwA2F ~ 2w~ (Fg\IF’ 2 1F’F 2R ~ 2 +(—J1--—sinF+t--——————R’IsinF .
.
\r
—2F’Rcos2F+ *
~
~
For a full scale calculation involving
~
~A
r
+
and
RF’
~A
/
+
R’F)}
~O, see ref. [6.70].
(6.119)
U. -G. Meifiner, Low-energy hadron physicsfrom effective chiral Lagrangians with vector mesons
323
with F = N~g(4ir2and F = —2f~,gF/m.Solving (6.119) subject to the boundary conditions WA(0) = WA(cc) 0, we find A = 1.178 fm, i.e. M~ MN = 251 MeV, a bit lower than the empirical value. For a larger U(1)~-coupling(j = 7.47) we have M~ MN 306 MeV. The axial-vector coupling constant g~ follows from the axial current A~(x,t) via —
—
hnjdxeT~.1(N~Aa(xt)~N~ = ~
(6.120)
and is given by gA
=
{~(~‘ + ~F)_gf2(3acos2F+
—
~
+
—1-~--
+
4aR’
~
+
~
sin2F + ~2F)
+
—~
a sin2F+ (3 +2Rsin2F) w’(a sin2F + R cos2F)
—4a’R + 8gaR2 + 4g13R2}.
(6.121)
In a similar fashion, the angle averaged isoscalar and isovector currents follow to be
J f
d12 V~= ~
Kaf2f~sin2F +
dfl~V~q x =
(~f)ff~ sin2F
WA
+
F’ sin2F}
(gf~)r(Rsin2F
Tr[AtT~AT~],
+
a sin
+2R2 +2/32+2grR(R2 +4a2 +2a/3 +
r2w’(a sin2F
—
~p2)~
~2
F’sin2F
R sin 2F)}e~1qk Tr[r~Atr~4].
(6.122)
In Table 6.7, we summarize the nucleon properties in comparison to the results of Chemtob [6.14] and Lacombe et al. [6.701. Chemtob has explored the VMD-Lagrangian constructed in ref. [6.65], with additional soft-pion corrections which reduce the pion decay constant to f ~ 87.4 MeV in the spirit of Gasser and Leutwyler [6.72]. Chemtob [6.14] chooses a universal gauge coupling g = 5.7 fixed from the p 2’ir decay. For the moment of inertia, only the contributions from the classical fields are taken into account and the electromagnetic properties are calculated via current-field identities. This leads to the problems in the isovector channels already discussed in the previous section. Lacombe et al. [6.70]have taken a similar Lagrangian as it is used in refs. [6.12, 6.13], with the exception of an extra term in the non-anomalous sector to correct for the momentum dependence of the pirir-coupling. This term reads —~
~extra = (~m2/8)Tr[(AL~U
—
UAR~)(UtAL~ A~Ut)1 —
(6.123)
with A~L= (1I\h)(p” ±a’~)(before diagonalization of the ‘rrA 2)is a coefficient of 1-mixing). (z~m dimension (mass)2, typically ~m2 = 0.3m~.The gauge coupling g is fixed to the p—~iry decay width, and the U(1) gauge coupling j to the value g~,~~/41r = 12.45 from the NN-interaction. The results in table 6.7 indicate indeed a substantial improvement over the conventional Skryme model, especially for the
324
U. -G. Meif3ner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons Table 6.7 Static nucleon properties in the ~rpA
1w-system. The parameters for the calculation are (i) ref. [6.13]: 93MeV, g = 6,j=7.47, m,, = 138 MeV, mA = 1116MeV; (ii) ref. [6.14]:f,, = 87.3 MeV, g = 5.68, 0.3m~).For for gthe conventional 1=11.4, m,, = 140MeV, mA = 2=1060MeV; (iii) comparison, ref. [6.70]:the f~ =results 93MeV, = 5.52, 1=8.34,Skyrme m~= 139 MeV, mA = 1082 model MeV [6.74]with (~m f~ = 54MeV and e, = g,,,,, = 4.84 are also shown.
f,,
=
MN [MeV] M~— MN 1MeV] gA
/.4~
[nm]
Pv (r2)~
[nm]
0 [fm] 2 1/2 (r2)~ )EJ=1 [fm] (r 2)~, 0[fm] ~r 1 [fm]
ref. [6.13]
ref. [6.14]
ref. [6.70]
ref. [6.74]
993.0 306.0 1.09 0.39 1.98
1019.0 336.0 1.50
1360.0 292.0 0.92 0.21 2.07
939.0 293.0 0.65 0.37 1.61
0.61
0.68
0.92
0.95
—
0.66
0.67 1.05 0.87 0.99
0.82 —
1.16 —
1.04
—
1.04
—
axial-vector coupling constant g~.There is still some discrepancy concerning the hedgehog mass in these models, this point remains to be solved. The least we can say is that the soliton picture indeed works when one puts in all mesons relevant at nuclear length scales. Finally, let us mention the work of Baacke, Pottinger and Golterman [6.69]. These authors worked out the hedgehog properties of the ‘rrpA1w-Lagrangian proposed by Golterman and Hari-Dass [6.73], which embodies exact VMD. They find a small hedgehog mass MH 0.86—1.09 GeV depending on the value of ~ and rH (0.4 ±0.05)fm. They estimate the moment of inertia from the purely pionic term and find it to be too small, of the order of 0.3 fm. This does not come as a surprise when one recalls the discussion in section 6.6. Of particular interest in their analysis is the stability analysis of the energy-functional. The condition that E should be a positive-definite quadratic form of the fields F, R and a12 is indeed matched for their Lagrangian. Actually, in ref. [6.69]an additional term of third order in derivatives as proposed in ref. [6.73] is omitted. Although the situation in the ‘rrpA1w-system is not at all unique, we believe that the more realistic the meson theory, the better the baryon properties come out as was first pointed out in refs. [6.12, 6.13]. =-
—
6.8. NN-annihilation and the H-dibaryon Up to now, we have only considered baryon number B = 1 and B = 2 configurations, exploiting the static properties of low-lying baryons and their electroweak structure. Another important issue in low-energy hadron physics is the understanding of NN-annihilation, mainly triggered through the experiments at LEAR. A large fraction of NN-annihilation is dominated by four and five pion decays, as shown in fig. 6.22 indicating the short-range character of the annihilation process. Furthermore, most of the multi-pion channels are vector-meson dominated, and therefore a solitonic description of the (anti)nucleon with vector mesons seems to give the most straightforward approach to describe the existing data. As a pertinent example, let us remind you that preliminary ASTERIX analysis [6.75] indicates that the p’rr-channel accounts for 60—80% of the p~—~ ‘rr~iriT° reaction._Here, we follow the work of Zahed and Brown [6.761 who set up a framework of calculating NN-annihilation in the presence of vector mesons as a highly non-perturbative process. Their idea is to look at superimposed
U. -G. MeiJiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
325
Ni~—.~nir AT LEAR ENERGIES
50
~4O-
—
30
-
20
-
CD
z
= U
NUMBER OF PIONS Fig. 6.22. Histogram for p~into n pions at LEAR energies [6.75].
NN-configurations at rest decaying into multi-meson states, relying on the validity of the effective mesonic description at short distances. Alternatively, one could think of calculating the annihilation of a nucleon and an anti-nucleon by numerical integration of the equations of motion and project out multi-pion final states. In light of the problems encountered in full-scale calculations of the NNinteraction [6.51,6.52, 6.33] this program does not seem feasible at the present time. To circumvent the problem of looking for skyrmion—anti-skyrmion configurations as a classical solution of the Euler— Lagrange equations, we will start from a product-ansatz of the form UHH(d, /3)
=
U(r
—
d/2) B Ut(r + d12) Bt
(6.124)
where the soliton (U) and the anti-soliton (Ut) are separated by a distance d, and B = B(/3)* is a constant isospin-rotation. Obviously, (6.124) follows from a hedgehog product-ansatz after G-parity transformation. The existence of such saddle-point solutions in the vacuum sector with MHfi = 2MH has been first discussed by Bagger, Goldstein and Soldate [6.77], these configurations are clearly unstable against decay into pions and vector mesons. For the moment, let us concentrate on a simplified model with pions and to-mesons only, the w-stabilized skyrmion of Adkins and Nappi [6.3]. The generalization to the gauged non-linear if-model including p- and A 1-mesons is straightforward. The Lagrangian to be discussed reads ~Z~= ~f~Tr[L~]— ~w~w~”’—~
,
(6.125) The relevant
Tr[LpLaLp]. 2) where B~is the topological baryon current, B’~= (E~”~”~I241T w-couplings to the HH-configuration can be deduced from (6.125) by expanding around the ‘~‘
*
In the most attractive tensor channel, we can write B(ii) = ir
=
ir 5.
326
U.-G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
classical configuration ~
=
U1~r~f3 U i~r~/3 using
4 (6.126) &~, U~(4) e~U11~ e1 where is the classical co-field and a fluctuation. Similarly, pionic fluctuations are parametrized in terms of 4 r~.~rI2f. With that, we can now deduce the pertinent co, u-couplings to the HH configuration. The w—HH coupling is identically zero since the baryon density vanishes locally for configurations of the form (6.124). For the most general isospin-rotation B = + ir /3 we find for the winding number W,
+
=
W= Tr(L~)= 6if3 2 Tr(Li- /3) = 0 (6.127) 0d employing differential one-forms (L = L~df) and Poincaré’s lemma. The co—HH—ur-coupling follows .
in a similar way to be ~o,aflH
=
~(g~/24u2)
Tr[L~(~)L//,(4) L~(4)]
=
~(ig~I8ir2f~)r ~
~
~ Tr[~ u(L~+ R~)].
(6.128)
Assuming plane waves for the IT- and co-mesons, which means that the HH-configuration does not distort considerably the meson decay products, we can write the HH—> cour transition amplitude M(HH—~uw)as M(ÜH~uw)= (w(q) ua(p)~Jdti~H~(t)~H~,
(6.129)
where the HH-configuration is assumed to be at rest, the pion carries momentum p = (Eq,, q) and the co-meson q = (E 02, + q). Clearly, this process contributes to the 4’rr-channel since the co decays further into three pions. HH) is the coherent soliton state around the classical background, —
HH~= cb11(U) 1~ 1(U) RJ~j0),
(6.130)
is the soliton operator and R~(U)the rotation operator in the Hilbert space of functionals of U. This construction follows from a generalization of the Mandeistam operator [6.78]; for further details we refer the reader to ref. [6.76]. Using proper plane-wave normalizations, the transition matrixelement (6.129) can be written as 4~(U)
M(HH~uco)=
—
32u~f ~(E~ + E~ E~~) ~ —
Pa Jd3x(0~Tr[Ta(Ly
+
R~)]~H), (6.131)
where r~(q) is the polarization vector of the vector particle. The evaluation of the matrix element ‘(01 Tr[u-~’(L~+ R~)]lHH~ is straightforward in the semiclassical approximation, we find
(01 Tr[ra(L~ + RY)]IHH~ Tr{Ta[L7(Uu~) + R(U~~)I}+ =
3 FTr(ra[r. ~ r./~2r/~}
=
—2Wa~{cosF sin
(6.132)
U.-G. MeiJiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
327
which is a total divergence. Therefore, the matrix element M(HH—s~cou) vanishes, since there is no contribution from the surface at infinity (w(r) exp{—3m~r}).Thus, the decay rate HH—4 wu vanishes in the semiclassical limit, which is intuitively clear. The co-couplings are dictated by the Wess—Zumino term, and when H and H overlap strongly, the WZ-term vanishes globally (untwisting of the baryon number). From this argument one is tempted to conclude that this kind of suppression will hold true for most co-related processes. Since the argument is of purely geometrical nature, it holds also for the NN-system. The present data seem to support this suppression, although the experiments on p~—*wit° are somewhat controversial,* one can say that pj3—~co’rr° contributes less than 1% to the total cross section. Notice that for channels including the p-meson (NN—~pu, p3ur,. .) this geometrical argument does not hold since the p couples through the isovector current and the WZ term. This seems to be confirmed by the data [6.79] which show an enhancement of the p-channels. Let us now give some coupling involving p-mesons. They can be read off from the Lagrangian (6.110) in a similar fashion as was done for HHco’rr. Two pertinent examples are -~
.
2
~-
ç
I~
fIT
~
~i
‘~~a f
2.!ir
~HHPIT=1g
+R )—g —~-[p~[U,p ]U]
4Tr[P~(L
NC
—
a~Tr[(RL)
~Ut~U1
(6.133)
64 u ~fiHtop1r
=
—i
/4V**~
Tr[i~a(Rp
—
L~)
—
aUt~pU].
A full-scale investigation of 3ir-, 4ir- and 5’rr-channels is recently under investigation at Stony Brook [6.80] using realistic wave functions to evaluate the final matrix elements. Finally, let us finish this section with a short comment on dibaryons. Kunz and Masak [6.81] have shown that the H-dibaryon predicted by Jaffe [6.82] corresponds to the lowest-lying SO(3) solution [6.83, 6.84] of the SU(3)~hidden gauge theory. They pointed out that a more realistic vector meson Lagrangian will indeed improve on the uncertainties in calculation of the H-properties as encountered in the conventional Skyrme model. Without symmetry-breaking terms and using a Skyrme term to stabilize the soliton instead of the co-meson, they find the mass of the H to be 2.14GeVsM(H) 2.57 GeV depending on the strength of the Skyrme term. The rms radius of the H is rather small and lies between 0.5 and 0.6 fm. It remains to be seen how much a more realistic calculation including co-mesons and SU(3)-breaking terms will narrow down the H parameters.
7. Further developments In this section, we will discuss some further aspects we have omitted so far, like e.g. the inclusion of scalar particles, the introduction of vector mesons by means of the derivative expansion and vector mesons coupled to chiral two-phase models. Although the scalar mesons 5, ~, ~,. . are experimentally elusive, and the scalar glueballs even more so, their incorporation in an extrapolation of the effective action up to 1 GeV may be relevant. Furthermore, the scalar gluonic components are intimately related to two broken symmetries of QCD, namely the U(l)A anomaly** and the trace anomaly. For example, one can introduce an effective order parameter field which takes into account the effects of the trace .
*
Recent measurements give (0.9 ±0.45)% or (0.17 ±0.06)% for the contribution of wii° to the total cross section [6.75]. In what follows, we will not discuss the U(l)A anomaly in detail (cf. section 5.1).
**
328
LI. -G. MeiJ3ner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
anomaly (broken scale invariance) via H(x) = /3(g~) Tr(G~~)/g~ and couple it to the meson field of the effective theory. This idea has been initiated and explored by Schechter and collaborators [7.1, 7.2, 7.3], mainly in the context of the conventional Skyrme model. We will focus here on the work presented in ref. [7.4], where a more realistic Lagrangian including u-, p- and co-mesons is modified to account for the trace anomaly and its implications are discussed. Another, though more phenomenological approach to include scalars, comes from the fact that the intermediate range attraction in the nucleon—nucleon interaction is described by the exchange of a scalar particle. In the dispersion-theoretical language this particle is nothing but a 2’rr-resonance in the S = 0, T = .0 channel [7.5]. For that reason, Lacombe, Loiseau, Vinh Mau and Cottingham [7.6, 7.7] have constructed and studied an effective Lagrangian with vector and axial-vector mesons plus a scalar particle. Alternatively, one can calculate soft-pion corrections to the non-linear if-model which also give rise to an intermediate range attraction [7.8, 7.9, 7.10]. We will outline the pertinent results of the Lagrangian proposed in refs. [7.6, 7.7] omitting any calculational detail. After that, we will briefly summarize the alternative idea of putting in vector mesons by means of the derivative expansion [7.11, 7.12]. In this framework, vector mesons are identified with higher order terms in derivatives of the pion field, like e.g. the Skyrme term can be related to the exchange of an infinitely heavy p-meson (cf. section 3.1). Our main focus will be to demonstrate the limitations of this approach as compared to the one including vector mesons as dynamical degrees of freedom. Finally, we will give some comments on vector mesons coupled to the chiral two-phase models [7.13—7.15],a field which is only at its very beginnings. The more speculative ideas of the SU(6) approach can be found in appendix D [7.16—7.18]. 7.1. The QCD trace anomaly and scalar particles Schechter and collaborators [7.1, 7.2, 7.3] have pointed out the importance of another property of QCD at low energy that we have not considered so far (apart from the remarks made in section 5.2), namely the anomalous behavior of QCD under scale transformations. The trace of the QCD energy-momentum tensor, i.e. the divergence of the dilatation current, is non-vanishing. Performing a scale-transformation on LQCD, one finds [7.19, 7.20] p.
p. =
s
Op.
2
V
Tr(Gp.,~)+ [1 + y(g~)]
=
,~
g~
i=1
m~q1q,,
(7.1)
where J~denotes the dilatation current, O’~”the canonical energy-momentum tensor, g~the running coupling constant and /3( g,~)the QCD /3-function. For non-vanishing current quark masses th,(i = u, d, s,.. .), one has an additional contribution proportional to the associated anomalous dimension. For the SU(2) sector we are considering, we will neglect the second term in (7.1). Let us start to consider an effective theory for QCD without matter. Following ref. [7.1], we assume that there exists an order parameter field H(x) such that (7.2)
O~H(x)
which is suggestive of a scalar glueball. To second order in derivatives of H, the Lagrangian ~ uniquely given by 32 (9p.H)2 =
~aH
—
~H ln[H1A4],
is
(7.3)
U. -G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
329
where a is a dimensionless parameter and A is the QCD scale. Obviously, the potential term in (7.3) has a minimum at (H~= A41e, where e is the Euler number, yielding a negative vacuum energy density of ‘(H~i/4. A bubble of higher-energy density would increase the total energy and hence be unstable. We will see later on how solitonic matter can stabilize such a bubble. Expanding H around its v.e.v. gives a fluctuation field for a scalar glueball with mass m~= (H)”214a. For the discussion to follow, we will rewrite the Lagrangian ~ in terms of an exponential parametrization [7.4], H(x) = A4 exp{u(x)
—
1} ,
(7.4)
and redefine the constants so that
(7.5)
(1I2)F~e2~p.u)2 (F~I16)m~ e4~(4u 1),
=
—
—
where mG is the glueball mass and f~is related to v.e.v. of the gluon condensate via* ((g~/u)Gp.~G~’)”2=17Jm~I2. (7.6) Experimentally, we have 0.1 GeV~A~0.5GeV[7.21], and ((g~Iu)Gp..~G~’) ~~*s(0.34GeV)4 [7.22]. The glueball mass is not known exactly, it should be higher than 1 GeV. Therefore, our strategy [7.4] is to adjust m~and I~in a way such that mG 1—2 GeV and ((g~/ir)Gp.~G~)(0.20—0.40 GeV)4. Since the Lagrangian (7.5) already satisfies the trace anomaly, the Lagrangian of pions and vector mesons has to be made scale invariant.** The chiral field U(x) can be chosen to scale with dimension zero, so all terms with four derivatives on the pion field are evidently scale-invariant (like e.g. the Skyrme term p. U, Ut d~U]2). The vector-meson fields scale with dimension one. To make any term in the effective Lagrangian under consideration scale-invariant, we have to multiply it by exp{(4 d where d is the number of derivatives and n~the number of vector-meson fields of that particular term. This modifies the Lagrangian (6.1) (we consider again the irpco-models discussed in section 6.1) in the following way ‘—‘
—
—
~ —
‘j(p~+ w~)+ ~ ~
Tr[U
—
11
+
+
+
(F~I16)m~.
(7.7)
We have not specified the Wess—Zumino term, it is scale invariant and does not couple to the glueball field. Furthermore, we have added the term (—F~m~I16) to measure the soliton energy with respect to the vacuum. Notice also that (7.7) obeys the large-Ne counting rules: f~,,F~and A scale as whereas m~,m~= m = \/~gj~~ and the glueball-meson coupling scale as ~1). It is now straightforward to derive the energy-functional E[~r, pp., cop., a] and the equations of motion using the hedgehog ansatze (6.6, 6.8, 6.11) and o(r) = o~(r).Here, we only display the equation for the glueball field cr(r) 2
0
—U
—
—-ci
+mGcTe
2o’
~
~
1
,2
+—~-[~F +
n
~
r
r
2
‘*
1
2
+-~(G+1—~cosF) r
[1—cosF].
m2 w2+e2’~
(7.8)
The equations of motion for the meson fields are the same as in eqs. (6.14) with the substitutions m m e°,m,~ m~e°,and fIT fT, e°~ The boundary conditions on the meson fields remain unchanged —~
* **
—+
—~
The relation of the parameters 1 and m
2aI’I~and 4m~= A212’/ë. in (7.5) to theas parameters a and A in (7.3) are L~= 16A A scale transformation on a meson-field0 cI~ is defined = —(d + x,j’P,,, where d is the scale dimension.
330
U. -G. MeiJlner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
(6.16, 6.17), the pertinent boundary conditions for ci(r) are ci’(O) = 0 and u(oc) = 0. In the conventional Skyrme model, solutions with exp[o-(0)] = 0 are allowed [7.2], this is not the case here, i.e. the expectation value of the gluon condensate (H~will always be finite and non-vanishing. A typical solution for glueball masses 1 and 1.5 GeV and F,~= 0.15, 0.1 GeV, respectively, i.e. in both cases 4, is shown in fig. 7.1. The square of the glueball field ~/i2(r)= exp{2ci(r)} = (0.27 0eV) shows a pronounced depletion around r 0.5 fm, in this region the baryon density B°(r) is sharply peaked. The static nucleon properties are only mildly affected by the presence of the glueball field as shown in table 7.1. The glueball-profile resembles very much a bag model with finite inside energydensity, in the spirit of Shifman, Vainstein and Zakharov [7.23]. We can read off the bag constant B as the difference of the outside and the inside energy density via B = (F~m~/16){e4~°~ [4ci(0)
—
1] + 1)
(7.9)
.
GLUEBALL PROFILE AND BARYON DENSITY
-
r[f m] Fig. 7.1. The square of the glueball field ~li2(r) = exp{2o-(r)} and the baryon charge density B°(r)for g = 5.85, f,, = 93MeV and m,, = 139 MeV. The glueball parameters are: mG = 1.5 GeV, T~= 0.1 GeV (solid lines) and m 0 = 1 GeV, I~= 0.15 GeV (dashed lines). Notice that with increasing glueball mass the depletion of the energy density inside the bag gets more pronounced (for fixed A).
Table 7.1 Static nucleon properties in the presence of a scalar glueball field and a scalar quarkonium (columns (3) and (4)) for the complete model (eqs. (6.1, 6.5)). For comparison, the results of the complete model without any scalar are shown in (5). The glueball4); mass (2):is I’~ m6 =0.1GeV = 1.5 GeV ((GG) in all cases. = (0.27 The GeV)4); other (3) 1 = 0.1are GeV, 0 = 39°; (4): f~=0.04 ((GG) = (0.17 GeV)’), m~=0.92GeV, parameters (1): m, [ == 0.72GeV, 0.15 GeV ((GG) = (0.34GeV) 0 = 44°.m, is the mass of the scalar quarkomum and 0 the glueball—quarkonium mixing angle.
M~
[Me’s’]
(1) 1419.0
(2) 1353.0
(3) 1343.0
(4) 1054.0
(5) 1465.0
1537.0 473.0 1.00 0.96 —0.23 2.68 —1.73
1487.0 536.0 1.04 0.90 —0.21 2.53
1472.0 516.0 1.05 0.91 —0.21 2.59 —1.54
1220.0 664.0 1.24 0.59 —0.04 2.23 —1.20
1575.0 437.0 0.97 0.99 —0.25 2.77 —1.84
M
[MeV]
5 — gA
MN [MeV]
2[fm]
(r~)~ [fm2] (r~/~~ /L~ ~
[nm] [nmj
—1.55
U. -G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
331
We have B = (105 MeV)4 for mG = 1.5 0eV and I~= 0.1 0eV. Varying the parameters mG and I~in the ranges given above (0.7 5 mG 3 GeV, 0.05 c ~ 0.25 0eV), we find that in all cases bags form with typical confinement sizes* r~ 0.4—0.6 fm. These bags have a finite inside energy density, with the bag constant of the order B = (100—130 MeV)4. This is in agreement with typical bag model fits [7.24]. An unpleasant feature of this simple approach is that the width F(H—÷2’rr) is too large. This can be cured by adding a scalar quarkonium state as proposed in ref. [7.3]. Recent work done at Syracuse [7.25] shows that the addition of a scalar quarkonium state with a mass of 1—1.5 0eV lowers the width of the decay H—* 2u to some hundred MeV, depending on the mixing angle between the scalar glueball and the scalar quarkonium. The static nucleon properties are again not drastically affected. In table 7.1, we show results for MG = 1.5 GeV, m~= 0.72 and 0.92 0eV, i7~= 0.1 and 0.04 0eV, and a glueball— quarkonium mixing angle of 39° and 44°, respectively. For the very low value of ~ i.e. ((g~/ = (0.17 0eV)4, the hedgehog mass is lowered at the expense of the other static properties, especially gA comes out much too small. This mechanism of bag formation is particularly interesting since it provides a natural bridge between the purely mesonic theories considered so far and the chiral two-phase models to be discussed in section 7.4. Furthermore, it might put the ad hoc assumption of confining bag walls on firmer grounds. Finally, let us point out that one can include the U(l)~-anoma1y of QCD proportional to Gp.,,G~” in a similar fashion by an order parameter field Q(x) = (g~Nf/16u2)Tr(Gp.~G~), with Gp.~= ~=‘~Gap the dual gluon field strength tensor. -~
7.2. Nuclear binding and scalar particles The discussion of the nuclear binding in the Skyrme model [7.16], i.e. intermediate-range attraction between two nucleons, led the authors of refs. [7.6, 7.7] to construct a chiral effective Lagrangian including p-, A 1-, co-mesons and a chiral singlet-meson c, which might be the S(980) or the E(l300) which is responsible for the enhancement around 1 0eV in the irir-scattering S-wave phase shifts. We will be brief in presenting their work, the Lagrangian proposed in refs. [7.6, 7.7] reads ~~IrpA1
~ 2 8l’vrv lrLrp.OL
— —
CD
+
~ E’2
~p.vRJ
1
-i-
~
2’-rr.42
lrVlp.L
A2
f Tr[Dp. U D~Ut]+ (~m2I8)Tr[(ALp.U
1
2
~4(c9p.co~c9,,wp.) 2— ..~E
1
—
UA Rp.)(LTAL
—
A~Ut)],
122
+2m 4,wp.,
~
~(o,ie)
-~wz=
g,
2 4wp.B’ x Tr[ALaLp —
—
ARaRP +
ig(UA~~UtA~~ ALaARII)], —
(7.10)
with the p- and A 1-mesons incorporated in the left- and right-handed gauge fields AL Rp. = AL Rp. The Lagrangian (7.10) is nothing but the gauged non-linear if-model considered by MeiBner and Zahed *
The confinement size is defined by the region in which the baryon density is peaked.
332
U. -G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
[7.27] with the addition of the r-meson and the non-minimal term proportional to z~m2.g is the gauge coupling of the SU(2)R®SU(2)L and g,~, the one of U(1)~.(Actually, g,,, is related to gaNN, the omega-nucleon coupling constant.) The parametersf and ~m2 can be related to the meson data via [7.7] m~= m~+ ~m2 + 8g2f,
f=f~m~I4(m~ + ~m2).
(7.11)
In terms of the physical ‘rr-, p- and A 1-fields the p—~2rr decay-width follows to be 2
23/2
2
g m8 f 4mIT~ I ~m F(p—s2’rr)-—--——~1————1--~ 1+ 2 2 L’+IT \ m~/ L m~+ L~m
222
gf~m~1 2
(7.12)
221.
(m8 + z~m) i
Adding electromagnetism via minimal coupling, one can easily determine the electroweak currents of the Lagrangian (7.10). For details we refer the reader to ref. [7.7]. Fixing the SU(2) gauge coupling g to give the empirical widths F(p—~iry) = (70 ±10) keV, and the U(1) coupling g~to F(co—3 iry) = (850 ±70) keV, one finds g = 3~9*and g,1, = 9.3. Actually, g1~will later on be fixed to be close to the2 coNNnow coupling constant 12) to allow a better of the This static gives properties. parameter ~m= can be used to fit(~~NN the p-meson decayforwidth via fit(7.12). .~m2=The 0.3m~and m~ 1082 MeV. m 0 is set to be 800 MeV,** the coupling constant ô,~could be estimated from irrr S-wave scattering. The authors of refs. [7.6, 7.7] prefer to leave it as a free parameter, but it is important to notice that t5,~ is bounded because the term ~ r Tr[ Lp. L ~] is attractive and tends to destabilize the soliton. In table 7.2 we summarize the pertinent results of the Lagrangian (7.10) which follow using standard procedures. The overall agreement with the data is rather satisfactory, as we already discussed in section 6.7. It is worth to stress that the attraction gained by including the e-meson also tends to decrease the axial-vector coupling constant g~[7.8, 7.9]. Most important, however, is the fact that this Lagrangian indeed gives attraction between two nucleons in the central channel when the product ansatz is used [7.6]. It furthermore shows again, as already pointed out in ref. [7.27], that a more realistic effective Lagrangian indeed gives better baryon properties than the conventional Skyrme model. Table 7.2 Static nucleon properties for the upA1wr-Lagrangian of refs. 2 and [7.6,7.7]. 6,(MeV)The aremeson given masses by (1) (3.9, are rn,14.3, = 769 0.3MeV, rn,2, m~= 782.6 24.8), (2) {3.5, and m, 14.3, = 800 0.73MeV. m~,2 The 1.8), parameters (3) (3.9, g, 12.45, g~,Am 0.3 rn,2, 0), (4) (3.5, 12.45, 0.73 m2 0, 0} and (5) {6.1, 13.65, 0, 27.3}, respectively. The A1-mass follows from eq. (7.11). (1) MH MN
[MeV] [MeV] — MN [MeVI p.,, mm] p. [nm] gA 2)~, [fm] (r (r2)~$~,,. [fm] 0 f [MeV] mA1 [MeV]
*
(2)
(3)
(4)
(5)
1265.0 1339.0 293.0 2.43 —1.99 0.93 0.64
865.0 938.0 293.0 2.33 —1.70 0.75 0.71
1287.0 1360.0 292.0 2.27 —1.86 0.92 0.61
865.0 938.0 294.0 2.22 0.74 0.68
866.0 939.0 295.0 2.33 —1.63 0.62 0.66
0.94 93.0 1082.0
0.98 73.0 1083.0
0.92 93.0 1082.0
0.98 73.0 1082.0
0.83 71.2 1275.0
—1.60
Experiment —
939.0 293.0 2.79 —1.91 1.25 0.72 0.81 93.0 1060/ 1275
Notice that this agrees with the value g = 5.52 previously given since the authors of ref. [7.7]scale out a factor of V~from the IT-coupling. This is a somewhat unjustified procedure since there is no scalar meson at 800 MeV.
**
U. -G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector rnesons
333
7.3. Comments on the derivative expansion Aitchison and collaborators [7.11, 7.28, 7.29, 7.30] and Truong and collaborators [7.12, 7.31, 7.32] have investigated the derivative expansion in the chiral field of the chiral action by integrating out the heavy meson fields. This amounts to identifying particular terms in the derivative expansion with the exchange of very heavy p-, to-, . mesons and fix the respective parameters from irir-scattering data. Let us first discuss the work presented in refs. [7.12, 7.31, 7.32]. The starting Lagrangian in terms of the chiral field U = exp{ir~~~/f} is .
.9~=c2~~2 + ~ =
+
.
~
+
c6.~’~ +
+
cSb.~b,
Tr(dp. U c9~Ut), 2,
,2)
=Tr[9p.U~ Ut, i1~U~ Ut] = [Tr(cp.U i3p.Ut)l2,
~(1)
= Tr(Bp.B’~)
~(1)
(7.13)
~b=Tr(U1), where ~ stems from the non-linear if-model, ~~1) is the Skyrme term reminiscent of the exchange of a very heavy p-meson [7.33]. ~2) is the so-called non-Skyrme term* which accounts for attraction (if c~2~ >0) and therefore is related to the “if”-meson and ~‘6 is the heavy mass limit of the cop. BtL~coupling proposed by Adkins and Nappi [7.34, 7.35] with Bp. the topological baryon current. ~ is the symmetry breaking pion mass term. The coefficients c~and c~2~ can obviously be estimated from uu-scattering, indeed Pham and Truong [7.32] find from sum rules in the 5, P-wave ITu~channels** yIe2
=
lie2
=
~
(7.14a)
2f~im~,
(7.14b)
making use of the fact that c 2 = f~I4and m~= 26m~somewhat below the physical p-mass because of unitarity corrections. For these values, we find e = 5.39, close to the pirrr-coupling constant g~ITIT= 6.1. The S-wave phase shifts below 1 0eV are not that well-known; with m~= (22—32)m~one can estimate y = 0.28—0.34. For y >0 the term 2~2) is attractive, it tends to destabilize the soliton, therefore, one finds solitonic solutions only for y
= ~0
+
+
4
~2
+ ~
=
* **
32e
+ £~+ +~
8e
-
m4,
+
~
There exist three different fourth-order terms, two of them can be lumped together in the Skyrme term ~ Notice that in ref. [7.32] a factor 2 is missing in (7.14b).
(7.15)
334
U. -G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
For /3 = 17 and ~ = 0.21, the mass of the hedgehog comes out to be 1751 MeV [7.31],which is too high. Performing the usual collective quantization, one finds MN = 1771 MeV and MA MN = 86 MeV, i.e. too large masses for N and ~ and a too low splitting. From these numbers one can conclude that the derivative expansion, i.e. freezing out propagator effects of the heavy mesons, does not give a satisfactory description of the nucleon as a topological soliton. Similar observations concerning this failure of the derivative expansion have been made in refs. [7.36—7.38].For demonstrating that, let us focus on the electromagnetic properties derived from a Lagrangian of the form (7.14) using minimal electromagnetic coupling. The electric and magnetic nucleon form factors derived from such a purely pionic Lagrangian are given by —
G~(q2)= 4u
S 2 GM(q )=4u
J
r2 j 0(qr) 1(r) dr, NI j r 2/T~ ~—,j j1(qr)I(r)dr,
—,~--
0
=
(7.16)
q
V 2 4irf2. GE(q)=1~j r j0(qr)V(r)dr,
V
8u
2
G~(q )2_~_MNj
f(r\. ~,,—)J1(qr)V(r)dr.
1(r) and V(r) are the isoscalar and isovector radial distributions, respectively, subject to the normalizations 4uJ1(r)dr=~,
~Jr2V(r)dr=A.
(7.17)
1(r) = B°(r)12 is model-independent with B°(r) the baryon number density and V(r) is a modeldependent radial function. Obviously, two of the form factors appearing in (7.16) can be expressed in terms of the two other ones. This leads to the model-independent relations 2)= _2(~)
—~-~
G~(q2),
(7.18a)
G~(q G~(q2)=
(~
+q2
G~(q2).
(7.18b)
Equation (7.18a) has also been derived by Nyman and Riska [7.39]. Neither one of these relations agrees well with the existing data, they are even inconsistent with the emporical dipole fits. Furthermore, in the limit q2 = 0 they imply the following relations between electromagnetic radii and magnetic moments ~P+~fl 9MN(MAMNX’(rE)P+’(rE)fl1,
(7.19a)
U. -G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons =
—
2
MNI(MA 5
2 —
—
(rE)fl
~
=
MN), A
335
(7.19b) N
MN
2
2
[(rM)P
—
(7.19c)
(rM)fl].
Equation (7.19a) is fulfilled within 10%, (7.19b) within 32% and (7.19c) within 52%. From these considerations, we conclude that a description of electromagnetic form factors based on the derivative expansion is necessarily limited in accuracy. This has been indeed demonstrated in refs. [7.37, 7.38], using a Lagrangian (7.13) with the following VMD-inspired parameters: c
(1) =
1 1 32g~~~32g
c
=—
(2)
=0 (7.20)
= 8fIT
CSb =
‘
~
c2
=
~
2and the SU(3) relation g~,= (N~gI2)to arrive We usevalue the generalized 2f~gnucleon properties for this model (“modified” at the for c~’1.InKSFR-relation table 7.3, we m~ give= m~ the =static skyrmion) in comparison with the complete model (cf. section 6.1) and the conventional skyrmion. The nucleon electromagnetic and axial form factors are exhibited in figs. 7.2a, 7.2b, 7.2c, 7.2d and 7.2e. The limited accuracy of the derivative expansion is indeed demonstrated and the importance of propagator effects, i.e., explicit vector meson degrees of freedom, becomes obvious. Aitchison and collaborators [7.11, 7.28, 7.29, 7.30], have derived a purely pionic Lagrangian like
Table 7.3 Static nucleon properties. MH is the hedgehog mass, and r~,its radius. MN denotes the nucleon mass, M~— MN the N—A mass-splitting. The charge radii and magnetic moments of the proton and the neutron are also given, as well as the axial vector coupling constant g~and the axial radius rA. “Complete model” denotes the vector meson Lagrangian (6.1, 6.5), the results for the VMD-inspired modified Skyrme model with fourth and sixth order term (eq. (7.20)) are given in the column “Modified skyrmion”. “Conventional skyrmion” gives the results of Adkins, Nappi and Witten [7.39].The asterisks (°)denote input quantities. All meson masses are taken on their empirical values. Quantity MH rH
[MeV]
[fml
MN [MeVI M 5 — MN2 [fm] [MeV] (r~)~ (r~) 2 (r~.,)~2 0 [fm] (r~)~’2 [fm] [nm] [nm]
It~,,Iis~l gA
(r~)”2
[fm] [MeV]
Complete model
Modified skyrmion
Conventional skyrmion
1465.0 0.48 1574.0
1650.0 0.60 1700.0
865.0 0.59 939.0°
437.0 0.97 —0.25 0.94 0.94 2.77
206.0 0.80 —0.27 0.73 0.75
4.55
293.0* 0.88 —0.31 0.79 0.82 1.97
293.0 0.86±0.01 —0.119 ±0.004 0.86±0.06 0.88±0.07 2.79
—1.84 1.51 0.99 0.60 93.0*
—3.82 119 1.27 0.48 93.0°
—1.24 1.59 0.65 0.35 54.0
—1.91 1.46 1.25 ±0.01 0.68±0.02 93.3 ±0.3
Experiment — —
939.0
)
336
U. -G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
a
b
~
0.6
::
::
G~(~2)
006
O’2
G~(~2
I .
/
.
~2 [GeV2/c2]
~2
c________________
G~(~2) 4.0
[GeV2/c2]
d________________
G~(~2)
-4.0
\
\ -3.0
:: ~z [GeV2/c2]
~a [GeVa/c2]
e GA (~2)
1.0
N
——
—
DIPOLE FIT A
4’5fm’
~2
[Gev2/czj
Fig. 7.2. (a) The proton charge form factor for the complete model (solid line) and the modified Skyrme model (dashed—dotted line). The dashed line gives the empirical dipole fit. (b) The neutron charge form factor (same notations as in (a)). (c) The proton magnetic form factor (same notations as in (a)). (d) The neutron magnetic form factor (same notations as in (a)). (e) The axial form factor (same notations as in (a)).
337
U. -G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector rnesons
(7.13) by starting from an effective meson Lagrangian ~
=
~(~‘
ci, p, co, A
1) and integrating out the heavy meson fields ci, p, A1 and co. This amounts to substituting the heavy mesons by their of 2 =equations m~I2f~, in motion ~SmIao = ~Sm/~W = 0. The result for the parameter of the Skyrme term is e agreement with (7.14b), and is rather model-independent, where the model-dependence comes from the choice of the non-anomalous ITpA 1-Lagrangian. The result for the non-Skyrme parameter depends strongly on the ITA1-mixing and the particular choice of the irpA1-Lagrangian, which might be related to the fact that the ci does not show up as a clear resonance in the meson spectrum. They also investigate the most general sixth-order term, not restricting to the particular choice Tr(Bp. B ~‘) Using approximate chiral profiles like F(r) = IT exp(—rIR), where R measures the extension of the soliton, they conclude that the derivative expansion does not provide a useful approximation for discussing stability of the soliton at sizes relevant to nuclear physics. For further details on this, we refer the reader to refs. [7.11, 7.30] and the recent review by Aitchison [7.40]. ...
~
‘—
7.4. Remarks on vector mesons coupled to two-phase models
Up to this point, we have made no reference to the quark degrees of freedom inherent in the QCD Lagrangian. The effective meson theories treated so far are just opposite to the bag model [7.24]which pictures the baryon as an assemblage of quarks confined by fiat in a bubble where the perturbative QCD is assumed to hold (cf. section 7.1). One might argue that to incorporate also perturbative QCD it seems necessary to construct models which interpolate between these two extremes.* Here, we will demonstrate on a specific example (the to-meson coupled to the chiral bag) the physics one expects from these models and the actual difficulties one encounters. First, let us elaborate a..bit on the expectations. In (1 + 1)-dimensions, it has been proven that bag boundary conditions arise as bosonization conditions on the quark fields [7.42, 7.43, 7.44, 7.45], i.e. that there is a complete equivalence between a purely bosonic or two-phase (mesons chirally coupled to quarks at the boundary of the bag) description, the so-called “Cheshire Cat principle”. This exact bosonization is assumed to hold also in (3 + 1)-dimensions, and can be cast into the statement that all physical observables are indeed independent of the bag radius R. Stated in another way, this means that there exists a smooth limit from a purely mesonic model (like e.g. a skyrmion with vector mesons) to a chiral two-phase model, with free quarks inside a cavity of radius R and mesons (IT, p, to,. .) in the outside. The coupling of the mesons to the quarks can be direct (like e.g. surface couplings) or indirect (through currents which leak out of the bag, like e.g. the baryon number current). Unfortunately, these models have to overcome some major obstacles not present in (1 + 1)-dimensions, namely the chiral Casimir effects [7.46—7.51],i.e. in general infinite contributions of the vacuum to any observable C. At the present stage, no completely convincing renormalization program to cure these divergences exists, and therefore the calculations performed so far [7.13, 7.14, 7.15] can only be considered as a first step towards a chiral bag with pions and vector mesons. To be more specific, let us briefly outline the two-phase model with co-meson stabilization considered by Klabucar and Brown [7.13]. The soliton phase of their model is a non-linear if-model whose topological baryon current is coupled to a vector meson field cop., as proposed in ref. [7.34]. (The inclusion of possible surface couplings has been explored in ref. [7.14].) On the bag boundary S at r = R the soliton is coupled to the quarks inside in a chirally symmetric way. The pertinent action reads .
*
For a review of the history and further references on the chiral bag model, we refer the reader to the work by Rho and Brown [7.41].
338
U. -G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
~ +
~
Tr(U
—
1)] 0(V)
~U5q s(S) + [+~ic~/q
—
—
B] 0(V),
(7.21)
where U = exp{ir lrIfIT}, U5 = exp{i’r. IT1/sIfIT}, s(S) the surface delta-function, 0 the step function and V(V) gives the volume inside (outside) the bag. The constant B gives the strength of the bag volume energy. Using hedgehog ansãtze, one can write down the equations of motion for the quark-fields q, cj~in V, the meson fields IT and to in V and the continuity equation for the flux of axial current. The latter reads
+
f~F’=
—
c~y5~ Pie. Pq .
(r = R).
(7.22)
Now, for any non-vanishing chiral profile F(R) at the surface the flux of axial current in (7.22) will have a vacuum contribution which influences the solutions F(r) and w(r) outside the bag. For any observable C, one has to calculate the vacuum asymmetry =
~ lim ~0~[q~(x,s), Cq(x, 0)]~0~
(7.23)
+
which can be done analytically in some cases (C = baryon number) or numerically. Apart from the baryon number [7.51] all other relevant observables (energy, flux of axial current,. . .) develop infinities proportional to Ins, s’, and so on. Vepstas, Goldhaber and Jackson [7.49] have given a numerical regularization procedure based on the evaluation of the pertinent mode sums,* which renders the energy, the flux of axial current,. . . , finite. Kiabucar and Brown [7.13]use this procedure to take the Casimir effects into account. Their energy-functional reads 25
)+f~m~(1_cosF)+~°
F’
E=4lrJdrr2[t~ (F~2+ —~w’ _~m~co~]+~°+~RB+E~, where
~
=
(7.24)
E
0R is the eigenfrequency of the hedgehog and E~the Casimir 4, fIT = 93 MeV, mIT lowest-lying = 139 MeV, m,,, = 782.4quark MeV, state and g~I4IT= 10 they find energy. With B = (145 MeV) the energy of the hedghog as displayed in fig. 7.3. The total energy is rather insensitive to the bag radius R, nicely confirming the ideas of the “Cheshire Cat” principle. A similar observation has been made in ref. [7.14], in a model with an to surface-coupling and a similar procedure to render the Casimir energies finite. (The parameters used in ref. [7.14] are fIT = 93MeV, g, 4, the 1, = 7.04, = (160 MeV) surface coupling parameter does not enter the energy-functional.) Again, let usBpoint out that this R-independence of the energy certainly depends strongly on the regularization procedure to extract the finite part of the energy. The isoscalar rms radius (r~)~ shows an approximate R-independence. The axial coupling gA is not insensitive to the choice of the bag radius R, the authors of ref. [7.13]therefore conclude that it might be most economical to choose a certain bag radius. This statement seems to be a *
The non-uniqueness of this procedure is discussed in ref. [7.48].
U. -G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons TWO-PHASE MODEL WITH
w
339
MESONS
.5
R[fm] Fig. 7.3. Total energy E,
0, of the chiral phase-model with o-meson stabilization. E,01 gives the contribution from the soliton (pions and w-mesons), ~ the contribution from the valence quarks, ~ the vacuum contribution (evaluated according to ref. [7.49])and E~0the volume energy By. E,01, gives the result of ref. [7.14],where the w-meson is coupled to the surface of the bag.
bit premature in the light of the mesonic models discussed before only when one includes all mesons relevant at nuclear length scales, a satisfactory description of the static nucleon properties emerges. Notice also that a similar study of the p-meson coupled to the chiral bag via the hidden symmetry approach [7.15] does not give an energy-insensitivity to the bag radius. It remains to be seen how a more realistic two-phase model with p- and to-mesons coupled to it will indeed exhibit R-independence of the physical observables. More work in this direction is certainly needed. —
8. Summary and outlook We will be brief in summarizing the results discussed in the previous sections. We have shown that an effective chiral Lagrangian of pions and vector mesons can give a rather satisfactory description of the baryon properties, with all parameters fixed in the meson sector. The soliton scenario of baryons offers a flexible approach to investigate various aspects of hadron structure, in particular for the low-lying non-strange hadrons (N, ~,. . Let us emphasize that here the question of giving the nucleon a certain size (“bag radius”) does not arise, indeed depending upon the probe, the apparent nucleon radius differs. In agreement with experiment, we find that relectromagnetic> raxial > rbaryon chargeS The baryon charge radius cannot be measured directly but can be estimated from models of NN-annihilation. Our finding of r~ 0.5 fm is consistent with some models of the NN-annihilation [8.1]. No other hadron model considered so far can reproduce these facts in such a simple manner, and is easily applicable to different situations like the nucleon properties, the NN-interaction and nucleon—antinucleon physics. We would like to stress again that concepts like (asymptotic) Vector Meson Dominance and Boson-exchange are embedded naturally in the framework of the effective theories discussed, and these concepts are known to be important for low-energy hadron physics. It is worth to stress that the number of parameters does not proliferate when one includes the vector mesons. Indeed, for the rrpco-system we are left with only three, namely the pion decay constant, the
340
U. -G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
pion mass and the (universal) gauge coupling. The latter is even constrained by the pion charge radius to be g~ITIT 2IT, close to its empirical value. For the strange particles, the situation is not that well understood. Various approaches to describe the physics of strange baryons have been investigated in the conventional Skyrme model [8.2, 8.3, 8.4, 8.5, 8.6]. Treating the symmetry-breaking terms, which have to be included in SU(3)f, via perturbation theory leads to a drastic reduction in fIT if one wants to get an overall fit to the baryon octet [8.4]. One can improve on this if one includes the symmetry-breaking terms in the full diagonalization procedure, still the static properties of the low-lying hadrons are not too well described [8.6]. Therefore, Callan and Klebanov [8.5] have proposed an alternative scheme to treat strangeness in this kind of effective models. In their framework, hyperons are described as bound states of kaons and an SU(2)-soliton made out of pions only. To leading order in N~,this approach works quite well, but for subleading Ne-orders it sometimes gives wrong predictions, e.g. the ~ comes out heavier than the ~:,*. It therefore seems natural to extend this approach to the SU(2) sector including also vector mesons, i.e., binding kaons to the solitons discussed here. As has been shown by Rho and collaborators [8.7], a minimal extension of the conventional Skyrme model including the to- and the ~-meson indeed gives the proper sign for the v—I splitting for values of the toNN coupling constant close to its empirical value. This is somewhat surprising in light of the work of Adkins and Nappi [8.7], where a good fit to the Nz~-system is only obtained for a large gm~It seems obvious that the inclusion of the p-meson (and the A 1) is necessary before one can make a final statement. Certainly, chiral symmetry is broken much more strongly in the strange sector than in SU(2) and one therefore faces all problems already encountered in the sixties, when soft-K predictions on the basis of chiral perturbation theory were worked out and showed much larger discrepancies to data than equivalent soft-pion theorems. As was already pointed out in the Introduction, Vector Meson Dominance is also not as good in the SU(3) sector. A description of the hyperons which is of similar quality than the one of the nucleon presented here certainly has to account for deviations from VMD, inducing even more complexity in the strange sector. Of more speculative nature is the assumption that the Cheshire Cat principle breaks down in the hyperon sector, based on the observation that in chiral two-phase models, some observables like the mass or magnetic moments are not radius-independent [8.10]. At the present stage, the SU(3) soliton picture and the SU(3) chiral bag are far from being understood well enough to draw a decisive conclusion on this point. Finally, let us point out that the soliton picture of the baryons we have presented here is non-perturbative, and therefore goes beyond any perturbative QCD calculation. The connection of the soliton picture to well-known concepts of conventional nuclear physics and hadron phenomenology is obvious and shows how one can naturally incorporate phenomenological and empirical constraints in a serious model of the baryon structure.
Acknowledgements I am particularly grateful to Norbert Kaiser for a critical reading of the manuscript and to Véronique Bernard for pertinent criticism. The work presented here could not have been done without the help and support of Gerry Brown, Wolfram Weise, Andreas Wirzba and Ismail Zahed. Fruitful discussions with Marc Chemtob, Vincent Pasquier, M. Praszalowicz, R. Rajaraman, S.0. Rajeev, Mannque Rho, Joe Schechter and Koichi Yamawaki are also acknowledged.
U. -G. MeiJJner, Low-energy hadron physics from effective chiral Lagranglans with vector mesons
341
This work is supported in part by funds provided by the U.S. Department of Energy (D.O.E.) under contract number DE-ACO2-76ER03069.
Appendix A. Non-linear realization, PCAC and VMD Here, we want to show one can construct the most general Lagrangian (for the non-anomalous sector) which incorporates the PCAC and Zumino’s form of VMD [2.47]. We will make extensive use of the notations developed by Callan, Coleman, Wess and Zumino [2.12, 2.49} and follow the work of Giler, Kosinski, Rembielinski, Maslanka and Szymanowski [2.131.As already pointed out by Weinberg [2.50] the most general framework for spontaneously broken chiral symmetry (PCAC) is provided by non-linear realizations of chiral symmetry. For that, consider a compact group G of the dynamical symmetry and H C G the group to which it is broken down (the algebraic symmetry). The following fields will now be considered. First, there are the preferred fields 4 which parametrize the coset space GIH, transforming via gL~=L4,,
h=h(cb,g)EH,
L~EGIH,
cb’=çb’(cb,g).
(A.1)
Furthermore, there are vector fields transforming as gauge fields, actually, there are two types, one connected to the global symmetry g E G parametrized by the generators T~= Tg: —
(A.2)
V,~=V~(Tg)k,
and vector fields connected to the coset space (generators T
=
—
Ta),
(A.3) It is convenient to introduce the following Cartan forms 1(~ + V~+ A~)L~ = W~(Tg)k+ ~(T~)~ ~ + L~ L~1~L~ = W~(Tg)k+ ~(Tc)a w~+
(A.4)
which transform under the local (tilde) and the global (no tilde) group, respectively. Any other field ~li transforms as =
D(h(g,
4))
çb-
(A.5)
where D is a linear representation of H. The pertinent covariant derivatives are defined via =
+
(A.6)
with Th the generators of H in the representation D. The most general Lagrangian invariant under the
342
1]. -G. McifIner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
local transtormation has the form
~
=
~ocal
F,1V(~),D~v~ D~, ~),
F,1JX) =
D~X,1+ [X,1,Xj,
—
ñ,1x= CX
(A.7)
Xj.
+[~,
Similarly, the most general Lagrangian invariant under global transformations reads F,1jw). D,17J~,D,1~I,~i)
~(Tha’
“~globat =
(A.8)
F,1v(X) = + [X,1,Xv]. 2)R and H = SU(2)~diagonal. The preferred fields defined by Let us now specialize to G = SU(2)L ® SU( L~ 1~ (~,~) E G!H transform as —
g1~= ~‘h,
~
=
(A.9)
and the gauge fields via ~
=g(~R)Ag(L
R)
+g(L~)ôg(LR),
(A.10)
where A,1 = A,1 T for the left- and right-handed fields. Vector and axial vector fields can be introduced by V,1 = ~(A~ + A~)and A,12,= 0), ~(A~ the basic representation of (g~~ the generators TR =A~).In (0, ir!2). Using the defining equations (A.4), we can rewrite Tlft~ are given TL = (iT! in terms of ~, A,1, and V,1 as the Cartan forms by ‘~ and .
—
~~
= ~
w~=
+ ~
(All) =
+
~[~(V~
+
A,1)~+ ~(V,1 A,1)~]. —
If we choose to work in a parametrization U = =
—
UV,1Ut + A,1 + UA,1Ut
—
with U transforming as U’ = g~Ug~, we have
~2
U (A.12)
=
+
UV~U~ + A~ UA,1U~+ U —
or using the u--model parametrization U = o-
(
iT
____
~=-~9~r--
i+u-)’
+ iT~~7,*
iT W,1[\\
we have
(lTX9,11T
1+~r (A.13)
i~
-
W,1W,1
(
_______
2~t7V,17T,1
1+u
To construct the most general Lagrangian invariant under the local SU(2)~ and global For the moment, we Set f~ 1. *
=
U.-G. Meiflner, Low-energy hadron physics from effective chiral Lagrangians with vector inesons
343
(SU(2)L ® SU(2)R) transformations, one has to use (A.7) and (A.8). Before doing so, let us see which constraints VMD according to Kroll, Lee and Zumino [2.28] imposes. The electromagnetic Lagrangian is supposed to take the form =
+
)+
(eIg)A~
~m~(p~3~)2 ~(F~)2,
(A.14)
-
where A~mdenotes the photon field, F~the (Abelian) electromagnetic field strength tensor, p~the neutral component of the p-field and other vector fields are denoted by the dots. Obviously, ~~ 3(p~) is m =0, the Lagrangian invariant underweU(l) p~invariant p~+g~’ set A (A.14) (which will gauge call ~)transformation is globally chiral and ô,1A. takes Iftheweform (A.15) with ~‘invariant under SU(2)~.Obviously, the p-mass term can only be invariant under global SU(2)~. Because ~(A.l5) should also be globally SU(2)®SU(2) invariant, the problem is to find the proper choice of the physical p-field to reconcile these conditions. Now, the most general ansatz up to fourth order in the Cartan forms i~, w, ~, and ~ reads =
‘~oca1 + ~g1obal
=
a Tr[D,1
77~ —
+
+ d~Tr[~,1, ~v]2
b Tr[F~v(~)]+ c Tr{F,1V(~)+ Tr[~2]2
+~
—
~
~v]}2
d Tr[~2] ,
(A.l6)
2] +..., 0 Tr[(~ )2 + (~ )2] + a1 Tr[~ + (a2 a1) Tr{?) where we have written down the terms which survive after imposing VMD in ~Iobal and in “~ocaIa term containing second order derivatives in the fields has been omitted. Let us now see which restrictions VMD imposes on the parameters appearing in (A.16). Working out the U(1) gauge transformation acting on ~gIobaI’ we find to lowest order in derivatives: ~g1obal
=
a
—
=
with a
—
(m)~Tr[(~
-
)2
—
-
(
-
)2] + (~)2a
—
Tr[(~
-
)2
-
2~
(A.17)
a,g~Im~. The corresponding p-field reads
P,1 r=g~1[V,1
—
(A.18) V,1 =
1[~t(~
W,1
~
+fl,1)~+~(W,1
W,1 ~~15L
m~)~1~
where V,1 is the vector part of the linear multiplet (V,1, A,1). For SU(2)L®SU(2)R, we can express V,1 in terms of ~T, O~,P,1 and A,1 as ~
(A.l9)
Up to now, we have only identified the physical fields for the pseudoscalar Goldstone modes and the
344
U.-G. MeiJ3ner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
vector fields via VMD. For the physical axial-mesons ~ there is no such principle. We only know that they should transform according to the adjoint representation of the diagonal subgroup of the chiral group, and the quadratic part of the Lagrangian should be diagonal in the physical fields. To fulfill these requirements, we use the appropriately shifted axial part of the linear multiplet, i.e. =
+f~S(o-) ~
~
—
g~p
yr),
X
(A.19a)
where we have reinstated the pion decay constant f~.S(o) is a function of o~= (1 v.2)1/2 For S(1) = d[(a 1)(m~/g~)2 dl the non-diagonal terms are eliminated, this is the only condition on S(u). Usually, we set S(cr) = S(l) = constant, but using a more general S(u) does not influence the on-shell S-matrix elements; indeed, a specific choice of S(u) allows for a smooth off-shell extension (see below). Normalizing the kinetic terms gives the following relations: —
—
—
2a=g~2,
m d=(~)
2(b+c)=g~2,
2
m +(a_1)(~)
gA
2
g~
)
/m g~\2 S(l)=_1_(a—1)(\ ‘) , m~g~
(A.20)
~ [1(1)(m~)2p] with /3 = a 2. For VMD to hold, we have to choose /3 = 0. Therefore, the Lagrangian in terms 2(m~Ig~) of the normalized physical fields reads after imposing VMD (0 = = = ~2 = ~VMD
=
2g~Tr[D,1 li~ Dii] —
+
cTr{F,1v(~)+~
+
d~Tr[~2j2
-
+
[~—~
+
(m)2
—
C]
Tr[F~~(~)]
d~Tr[~,1,~V}2
Tr[~2]
-
(m)2
(A.21)
Tr[ii2]
m 2 m -2(a-1)(--~)Tr[ii~]-(--~)Tr[~-w]2, where the Cartan forms ij, ~, w and ~ are given in terms of the physical fields ~, p, Ay~ (we will omit the superscript (phys) on the axial field) as: 11 /
~r (1T.o~~.~T)\1
‘t),1TfIT+f 2
2+)1~
n,1T{[l+a(1u)][f(l+~s(u))(o,1~gPP~)gAffA~
+
1 +a(i
+ ~
((1
~ A,1 )]
~) + g~f~
+ ~ S(~))(~~,1
}
U. -G. MeiJlner, Low-energy hadron physics from effective chiral Lagranglans with vector mesons
w =T•I
345
(A.22)
/1
w,1T{[l+a(1~)][gP(u(1+a)s(u)(lu))p,1 (+a) ~xA (1+a(1—ff2))
f.TT
f~r+1) with T = iT!2. The generalization to SU(N) 1 ® SU(N)R is straightforward but tedious. If we now want to restrict ourselves to the lowest possible numbers in derivatives (quadratic), the quartic terms in (A.21) can be uniquely eliminated giving the following constraints: 2, S(o-) = E[1 , ~ = 1 (mAg~Ig~m~) (A.23) 2 2 a=0, d (+) =0, d (—) +c(1—2cg~)=0, l—e 2 =2cg~, —
—
so that we recover the Weinberg relation f~= m~Ig~[1 (m~gAIg~mA)2]. The Lagrangian reads —
~TMD
Tr[D,1~v
=
-
D~i 7,1] +
Tr{F,1~(~) + (1-
gA
2] - (m)2
-
(m)2
Tr[(~ ii)2 -
+
Tr[~
(~ w)2 -
~2)[~,
-
~2j
~]}2
(A.24)
where the KSFR relation follows for ~ —1. Of particular importance to the general Lagrangian (A .21) is the question of universality. From (A.21) one can read off the relation between the gauge coupling g~ and the pirn decay constant ~ as l+(a2—1)Z2 l—(a —l)2Z2—/3
Z
‘
so that the universality condition g~ ~ —
/3
= 2a(l
—
a)Z2.
mggA g~m~
(A.25)
reads (A.26)
This means that even if VMD is fulfilled (/3 = 0), for universality (a = 0 or a = 1) to hold, one needs further restrictions. Now, a = 1 is excluded by the KSFR relation. So even if we restrict ourselves to lowest order in derivatives and have implemented VMD, we have a one-parameter family of equivalent Lagrangians, so that further conditions like KSFR and universality have to be imposed to make it unique. The most general Lagrangian fulfilling VMD, KSFR and universality is therefore given by (A.21) for a = 0, (m~!g~)2 = 2(m~Ig~)2. The remaining freedom can be used to eliminate the terms of fourth order in derivatives to lead to (A.24) with e = —1. What remains to be done in this framework is to work out the anomalous sector and construct baryons from it. At this state, we guess that the strict
346
U. -G. Meijiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
implementation of VMD will give rise to problems in the baryon sector (cf. the discussion in section 6.6).
Appendix B. p-mesons in the Skyrme model In this appendix, we want to show how the p-meson can be included in the Skyrme model in a simple, chirally invariant way following the work of Adkins [6.71and MeiBner [6.8, 6.9]. For that, consider the following Lagrangian:
~
=
Tr[c~ U 9~U+]+ ~ Tr[ô~U U~,~
+
~
—
~Tr[R~R~]+ ~
Tr(U
—
1)
U~]2—(1—
/3)
~ Tr[B,1B~]
Tr[R~R~]+ a Tr[R,1~ô~U+ U
The pion field is incorporated in the SU(2)-valued matrix U(x) decay constant. The p-field is a 2 x 2 four-vector R,1
=
(B.1)
= exp{iT. lTIf,j ,
with f~ the (weak) pion
(B.2)
+ ~T
where and p~are real. R,1~is the Abelian field strength tensor, R,1V = tTl,1Rv For /3 = 1, we have the model of Adkins [6.7], where stable solitons arise due to the fourth-order Skyrme term proportional to e~2.For /3 = 0, the stabilization is due to a very heavy w-meson as proposed in ref. [6.9], indeed the stabilizing term ‘—~Tr[B,1, B~]is nothing but the w-meson stabilization discussed by Adkins and Nappi [6.3]. ~ is the baryon current, —
=
U~3
~
0U~ U~a~U] and the constant 2
2
can be related to the ioNN-coupling constant g~,via
2
e6 = gjm,~
(B.3)
.
Empirically, g~is found to be g~/4~r10—12. The pion mass term explicitly breaks chiral invariance, so does the p-meson mass term. The last term in eq. (B .1) gives the pirr~-interaction,the coupling constant a can be determined from the decay width ~,-*2,T to be a 0.0444. The whole Lagrangian (B.1) is invariant under chiral transformations U LUR ~, R ,1 ~ LR ~ ~, * as well as under P, C and T. The number of degrees of freedom of the p-field has to be reduced to ensure unit isospin, one therefore imposes the chirally invariant constraint —~
Tr[R~U]=0. *
Except for the mass terms, of course.
(B.4)
U. -G. Meij3ner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
347
To calculate the properties of the static hedgehog soliton, one does the usual hedgehog ansätze for the meson fields, i.e. U(r) = exp{iT~PF(r)} and R’(r) = ~Ta6~~7 R(r), subject to the following boundary conditions: F(0) = ir, F(cc) = 0 to have baryon number 1, and R(0) = R(~)= 0 to have finiteness of the energy. The hedgehog properties can be read off from table B.l, Adkins chose to use the meson data (e
4 ~ and ~ as parameter to fit the N and s-mass, whereas MeiBner preferred not to vary the mesonic input. The static nucleon properties follow after introduction of time-dependent collective coordinates and quantizing them. Through that procedure, the time components of the p-field ~1(r)and ~2(r) (cf. eq. (6.23)) are excited. It is interesting to note that the moment of inertia A as defined by —
3x~[U(r, t), R~(r,t)] = —M~+ ATr(A~A) J
(B.5)
d
is almost entirely given by the parts from the classical fields U(r) and Ri(r),* the parts involving ~
12(r) contribute approximately 1% to the nucleon mass [6.8]. Some static properties of the nucleon are given in table B.l. Similar to the Skyrme model calculations of Adkins, Nappi and Witten [6.6] fixingf,~and e4 ~ to get the proper N, ~ masses results in too smallf,, and g,~.On the other hand, leavingf~and g~,!4iron their empirical values gives too high masses, a reasonable Na-splitting and an improved gA~ In general, the predictions of this model are a few percent better than the ones of the conventional Skyrme model. For further details we refer the reader to refs. [6.7, 6.9]. In ref. [6.8], the baryon number B = 2 solution for /3 = 1 was constructed, it turns out to be quite similar to the Skyrme model B = 2 solution. The main difference between these two solutions is that the two-body repulsion between two strongly overlapping solitons is significantly weakened in the model with explicit p-mesons due to the strong attractive p’rr-interaction. For /3 = 0, the situation is rather different. For the parameters f,~ = 93MeV and g~j4ir= 10, one finds a repulsion of 1.9 GeV, due to the strongly repulsive character of the (0-meson at short distances. This again points to the importance of building models which incorporate all mesons relevant at nuclear length scales. Finally, let us point out that the p-meson introduced here is not the same as a massive Yang—Mills particle (for m~—~ ~) or gauge boson of a hidden local symmetry, it is a constraint field connected to pions by the minimal interaction term in (B.l). Table B.1 Some hedgehog and static nucleon properties in models with pions and p-mesons alone. (*) denotes input quantities. g,, is the oNN coupling constant, and M0 the energy of the static soliton solution, the hedgehog. The results are taken from the work of Adkins [6.7]and Meillner [6.9]. ref. [6.7]
f~
*
[MeV]
ref. [6.9]
52.4
g~I4sT
—
e4 M,, [MeV] M,., [MeV} M1 — M~ 2 [fm] lMeV] (r~)” gA
4.65 865.0 939.0* 293.0* 0.70 0.65
93.0* 6.45*
93.0* 10.0*
Experiment
93.0* 12.0*
75.0* 10.0*
—
—
—
—
1219.0 1290.0 285.0 0.50 0.85
1390.0 1442.0 206.0 0.55 1.12
1468.0 1503.0 180.0 0.58 1.23
1012.0 1069.0 230.0 0.49 0.89
93.0 10—12 6.11 —
939.0 — 293.0 1.25
It is mostly the pionic part from the non-linear r-model which gives the moment of inertia (cf. table I of ref. 16.81).
348
U. -G. MeiJ3ner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
Appendix C. Calculation of the moment of inertia and the electroweak currents in the Trpw-system Here, we want to give some explicit intermediate steps in the evaluation of the moment of inertia in the ~rpto-system (eq. (6.26)) and the electromagnetic and axial currents (6.48) and (6.75). First, let us consider the moment of inertia A. For that, we make the time-dependent ansãtze for U(r, t) (6.11), p°(r,t) and ca(r, t) (6.23), and evaluate all terms which are quadratic in the rotational velocity of the soliton, i.e. L(t)_jd3r~[~, TP,1, with A =
j~° A[F, G,
4ir
A(t)
=
~
;
‘l dr.
~
(Cl)
—MH +2AK2
W,1]
K is treated as a time-independent K =
0,
exp{iT Kt}
uniform rotation (C.2)
with iT . K = A~A.To order K2, the following terms contribute A
3rTr{(a 1= ~
J
d
2}, 0~~
+
.
(C.3a)
~—igT.p~)
A 2=— ~-Jd3rw.w,
(C.3b) 3r (curl w)2
A3 =
—
J
d~rW,JW,J
=
— ~
J
(C.3c)
d
A 4=~Jd3rp0j.p0j,
(C.3d)
As=_~Jd3ro.B,
(C.3e) tr PkU},
A6 =
~2
A 7
=
~
Id3r ~um0k~
Trf iT
J
Tr{iT pk(ô~U~ Ut + Ut. d~U)+
3r Eujko
~
po(~kU
Ut
+
Ut ~U)
+
(C.3f)
T ~p0U
r
.PkUT .poU}.
(C.3g)
d
We have omitted the contribution from the non-linear u--model which can be found e.g. in ref. [6.6]. Because of the hedgehog structure, all three-dimensional integrals can be reduced to one-dimensional ones by use of the following angular averages
J
dQ (K.
~)2 =
~K2,
~—
J
dQ (K x ~)= ~K2,
(C.4)
where S2 is the unit-sphere in two dimensions. Our conventions are = diag(1, —1, —1, —1) and = — = 1. The following useful identities will be employed quite frequently
U.-G. Meifiner, Low-energy hadron physics from effective chiral Lagranglans with vector mesons
UT.VUt=(cosF+irsinF)rV(cosF—iT.PsinF) = cos(2F)r V— sin(2F)’r~(r x V) + 2 sin2FT PV P, {~kU,Ut}+
2i~r.~(F’
+i~k
—
349
(C.5)
sin(2F)
where V is a vector. With that, let us evaluate the contributions A 1,.
~=VU=exp{i’r~PFI2}
.
. ,
A7. For A1, we use
and find 2(FI2))]A~.
~ ô0~—igr•p~= A[2i’r~K(l —cosF— ~1)+2iT With (C.4) and multiplying by r2 leaves us with ~
—
A~=2f~[(2sin2(~)-
)2
-
~
(2sin2(~) - ~i)( 2sin2(ç)
=
~f~r2[8sin4(ç)
-
PK~(—42—2sin
8sin2(~1+ 3~~
+
+ ~2) +
~].
~ (2sin2(~) +
(C.6a)
Similarly, (C.3b) gives
=f (— ~-~) ~-(K x
A~
“) =
—
in
(C.6b)
~2
and for (C.3c) we find
~ =f (— ~)curl[ ~
(K x r)]r2
=f(_ ~)[~-~+
(~-~
—
~~~.)pK.p]2r2
—
1
(~~2+ ~2)
(C.6c)
The contribution from p-kinetic term (C.3d) is somewhat more lengthy; we will first evaluate r p0~. One finds a a a T.p0,=r(ô0p,—d,p0+g~ =
abc
b
p0p1)
(2Irg)A[r Ki~’U+ r rK’V + r’K PW + ‘r. PK~PP’X]A~,
with U=—r~, W G— ~2
—
Vr=G~1_G—~, G(~1+ ~2)’ X —r~+2~2+ G~2.
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U. -G. MeiJ3ner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
The contribution to the moment of inertia functional reads 112 r Tr[Tp0~r.p0,1 2
k
~,=
~j
S
[(U + V
=
+
W + Z)2 + 2U2
+ 2V2 +
2W2]
(C.6d)
~
The w,1B~~coupling from the Wess—Zumino term is much easier to evaluate,
=
=
—(3g/2)
J ~ (K x ~)(—flsin2F (—2r
X
K) (C.6e)
S2
=
(g/2ir2)~F’sin2F.
A 6 and A7 are contributions from the ‘rrpw-interaction in the Wess—Zumino term; we start with the evaluation of A6. For that, notice that = (curl w)’~and
t + Ut•
~k~’)}
Tr{iTpO(akU U =
(~,+ ~
—
2)(F’
—
sin~F))PkK. P
K~1sin(2F)
—
tT PkU} Tr{i’r.p0U =2 cos(2F) (K x P)~—2 sin(2F)
—
~2
sin(2F)
~kK
P,
(C.7a)
(~x (ek x P)),
~
(C.7b)
where ek is a unit vector in k-direction. Putting pieces together and performing the angular integration, we end up with =
—
—~--~
[~P’~sin(2F) + 2cPF’(~ 1+
8~r
42)1
—
—~--~
l6~r
G~1X. sin(2F).
(C.6f)
For A7, similar algebra using t Tr{iT
Pk(ôO~
U
+
U~~ 0U)} =
2
sin(2F)(ek x P) . (P x K)
leads to = +
—~--~
8ir
GP’
sin(2F)
—
—~---~
l6ir
G~1P’ sin(2F),
where the last term can be calculated from the last term in (C.6f) via the substitutions
(C.6g) qkO __
—
EIOk
U.-G. MeiJlner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
351
F—* —F and U—* Ut. Putting all the pieces [(C.6a) to (C.6f)] together, we arrive at the moment of inertia functional A as given in eq. (6.26) for the complete model (c = 1). For the minimal model, we just have to drop the contributions Let us now turn to the calculation of the electromagnetic currents. As outlined in section 6.2, they are the isoscalar and isovector symmetry currents obtained by the Gell-Mann—Lévy method. For that, consider the transformation Tp,1+w,1—*h(x)[T~p,1+w,1}h~(x)—(1Ig)t~h(x)h~(x), h(x) = exp{— ~i[a0(x)11+ a(x) with U—* hUh~and
~—*
~
h(x) E SU(2),
T]} ,
(C.8)
The variation of the Lagrangian reads
=~[h~h~,hUh~,h(rp,1 + w,1)h —(i/g) d,1hh~] 2). =(ô~a0(x))J~0(x)+(d,1a(x))J~1(x)+U(a~,a
(C.9)
Let us consider the isoscalar current. For that we have to discuss the ungauged WZ action F~ = (—iN~/240ir2) Tr(dUUt)5. U(x) is extended to be an element of SU(3) and the isoscalar generator = diag(l, 1) is substituted by A 8 = diag(l, 1, —2). Only at the end of the calculation can we restrict U to be an element of SU(2). This gives 2/g)w~, (C.lO) J~0= ~B’~ ~B’~ (m fM5
—
—
where the first term obviously comes from the ungauged WZ action, the second from the coupling and third from the (0-meson mass term. Therefore, the isoscalar current is entirely dictated by symmetry-breaking (0 in mass term. In the case of the p-meson, things are more complicated due to multi-pion contributions to the isovector current. We will just discuss this current for the complete model. In terms of the ‘rr- and p-fields it is given by J~=~ = —
~Ut~ ttu t9 i(g/2)(~~.p~~± ~T.ptL~+)I}
if~Tr{T . ~
. d’1~ —
~
—
+
~ovLU.Ut
To get the nucleon current, we first have to project on the charge operator Q
(C.ll) =
~(~ + ri),
which gives
J~= ~J~ 0+J~3~.
(C.12)
In the isoscalar sector, (C.10) immediately gives m ~em =
—
~em~(~’’~) 2 ‘P(r) S,i
—
m
~
m2
~P(r)
(C.13)
352
U. -G. Meiffner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
where we have made use of the quantization rule K = ~!4A. For the calculation of the isovector current, it is important to notice that K is a differential operator in the space defining A = a0 + ia T, i.e. AT,
[K,A]=
[K,A~]=—
(C.14)
~
Therefore, two relevant identities to be used in the evaluation of J~1are Tr[Ar
KA+ra1~_÷ T~’I2A,
Tr[AT. rK~PATa]5.
Ta/6A
(C.15)
The actual evaluation of (C.ll) via the time-dependent ansätze (6.21, 6.23) gives (for the time component J~1) ~ + —(i!2g)(~~
=
0
0+
)]
—2if~A{iT•K~+~iT.K~—2iT.K t+ ~iT•K hjUtiT.KU_ ~UiT•KU eiT~(Kg, + PK P~ 2)~~iT~(K~1+ PK —
—
—
=2f~,A{rK(2cosF—2— ~cos2F+ ~ —2~cosF) 2(F!2) sin2F 4~ sin2(F/2) 2~2cos F) + T PK P(4 sin ~ sin2(F!2)} —
—
—
so that the nucleon current by use of the quantization rules outlined before follows to be =
~
sin~(F/2)+ (1
+
2 cos F)~ 1+
~21.
(C.16)
Similar algebra for the space-components of the isovector current leads to 2jG cos F —2 sin4(F/2)] ~ X r J~ =
—
(C.17)
~f
Notice that the isovector as well as the isoscalar currents are given by =
(I°(r),i(r) ~ x P),
(C.18)
where I°(r)is the charge-density, and i(r) the pertinent current density. From the form (C.18) it is obvious that the static nucleon currents are conserved, i.e. = a 01°(r)—V~[i(r) ~ X =—(ifxf).Pi’(r)+i(r)ifVX(Vr)O.
(C.19)
U.-G. MeiJlner, Low-energy hadron physics from effective chiral Lagranglans with vector mesons
353
Finally, let us derive the axial-current radial function A12(r) (6.75). The axial current is the symmetry current of axial rotations (C.20)
s(x)=exp{—~r.a5(x)}.
Obviously, under s(x) U transforms into sUs and r + into2s. TTherefore, + (0,1. It important to notice weishave to calculate the that there exists no transformation ~—* ~ s) such that ~2 s~ axial-current via the Gell-Mann—Lévy method with the Lagrangian given in terms of ~L and ~ and fix the gauge at the end. ~L and ~R transform under axial rotations as —~
~
(C.2l)
~
so that U = 4~
—~
s~ ~
=
sUs. We can read off ~
from the term proportional to ô,1s s~=
i.e.
&~E=~t[sUs, &S,
~
W,1~ TP,1] =
(~a 5(x))A~(x).
It follows
to
(C.22)
be t.
=
~if~Tr{T.[—U
19~~_ 9’~U U~+2~ t
g
+
~FaVa4Wva
.
f95~+2~9~~• ~
—ig~~.p~ +ig~~.p~~+]}
Ut8~U+ ~ ig(UT.p
—
Tr{T. [9~U. U
t + UtT.p 4U
4U)]}
(C.23)
where a partial integration in the B ~ has been performed. To see that, let us discuss this term in some more detail. For convenience, we will use differential one-forms and define (3.58) df, R = Ut dU, L = dUUt 2 and dL = L2. Under an axial rotation (C.20) R transforms into with dR = —R R~s+.ds+s+Rs+s+Uts+.dsUs W
(0,1
and similarly for L. The 2
W,1B d4x =
—~--~
l6lT
WA
Tr(R3).
(C.26) we find
~‘,
wATr[dss~ +R+ Uts+ .ds(J]3 WA
(C.25)
B -term in this language reads
Let us now perform an axial rotation on
=
(C.24)
Tr[R3
—
T
da
2+ 5(L
so that the axial variation reads
R2)] +
0(a2),
354
U. -G. MeiJfner, Low-energy hadron physics from effective chiral Lagrangians with vector ,nesons
=
—
w A Tr[r . da5(L2 + R2)]
32ir =
=
—~-~-~
—~-~--~-~
dw
—~-~--~-~
A Tr[r
32 ~
. da5(L
w A
Tr[T . da
5(dR
32ir
—
dL)]
-
R)],
—
(C.27)
where we have performed a partial integration and thrown away the surface term proportional to d(w A Tr[T da(R L)]). Using dw = A dx’~, we can convert (C.27) back into the more conventional form —
=
(~
w~Tr[r. (o~U Ut
3ig
),1PO~
—
Ut. ô~U)].
(C.28)
Injecting the time-dependent ansätze (6.21), (6.23) and performing the adiabatic rotation of the hedgehog as outlined in section 6.3, the radial functions A1(r) and A2(r) as given by (6.75) follow immediately.
Appendix D. Some remarks on the SU(6) approach Based on the success of the non-relativistic SU(6) symmetry, which is only exact in the static limit, Caldi and Pagels [7.16, 7.17] have developed a model which treats pseudoscalar and vector mesons on a similar footing. Let us consider chiral SU(3)L 0 SU(3)R. Pseudoscalars transform like (3, 3) ~ (3, 3), i.e. in terms of quark fields ~a~i~y5(AaI2)q whereas vector mesons transform as (1,8)~(8,1), i.e. -~ qy,1(A12)q. To reconcile with the static SU(6), 3)L®SU(3)R. let us assumeIn that pseudothis vectors case, theand vector field scalars belong to a (3, 3)~(3,3) representation of SU( operators have to be described by an antisymmetric tensor, e.g. ~ = ~o,1~(A’~/2)q.The phenomenological vector-mesons follow by projection via pa~ =
~
(D.1)
where Z~2is a normalization constant. The relation (D.1) is called PCTC (Partial Conservation of Tensor-Current) in analogy to PCAC. Automatically, we have current conservation ~v = 0, i.e. we have three independent components for the spin-i fields. The chiral partners to the vector-field can be constructed from the dual of ~ ~,avapT~’~ and therefore projection gives the B(1235)*, jPC = 1 + axial meson as the chiral partner of the p. One also has a conserved current for the axial meson, I’PBa 9v(,~Ta4Lv)= 0. Going now to a linear representation, we can group the odd-parity 36-plet and the even-parity 36-plet of static SU(6) in terms of the meson fields (ir’1, p~) and (~.** B~)via ~
-
MAB = {a.(jo.a
+
~
+
(u. e), 1(ip° +
Ba)}(Aa)ABI~~S/~
(D.2)
with A = ai; i = 1,2,3; a = 1,2,3 and a = 1,2,. . , 8. Imagine now that chiral SU(6)L ®SU(6)R is broken down by SU(6) spin-flavor’ in this case the whole 36~p1et** ~f pseudoscalar and vector becomes massless. Of course, there are no Goldstone bosons of spin 1, Caldi and Pagels [7.16] argue that the .
*
This meson is nowadays labeled “b,’.
**
Let us for the time-being forget the U,f1)-anomaly so that we need not bother to distinguish between SU(6) and U(6).
U. -G. Mei/3ner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
355
vectors acquire mass through relativistic effects. Insisting on Lorentz-invariance, it is obvious that the kinetic energy terms give rise to spin-dependent mass terms (Lorentz-boosts affect spin but not U-spin), one indeed gets the following mass formula 2 2 2 2 D3 —
m,~= mB
—
m~
which is fulfilled within a few percent when one identifies the scalar particle u- with the ~(980) = a0(980). The vector mesons are therefore called “dormant Goldstone bosons”. An important consequence of putting the vector mesons in a (3,3)~(3,3) representation is that VMD is a consequence of spontaneous symmetry breaking. Indeed, since the pion couples to the axial current via (oi, ~—~f,~,the p-meson couples to the vector current in the same way. Consequently, we have (0~A(0)Irrb(k))=ik,1f~,8ab, (D.4) =
is,18m~/y~
which leads to
2, (D.5) (m~/f~)(Z~/Z,,)~ where the pion normalization can be determined from f,~= \/2Z~I3(othe SU(6) Z,~= Z~, 2 3/2 one obtains the empirical width F(p —* e + e 0).In ). Equation (D. limit, 5) is similar to while for (Z.,T/ Z~)” the KSFR relation. This approach also leads to universality ~ = ~ for more details, cf. ref. [7.16]. Notice, however, that the interpretation of VMD departs from the usual one since here it applies to (~r.~~(A**I2)q) rather than to the vector current ~y,1(Aa/2)q. One can now write down a generalized Skyrme model [7.17] =
tr(8~U~~Ut)±
m~f~Tr[u U. ifUt]
+
(D.6)
with U(x) = exp{ii1~f(x)Ij,~},M = iA’~1T’21 + (u~E)A”p°,with the Wess—Zumino term determined from the fact that 1T 5(SU(6)) -~ Z in the standard fashion. Obviously, baryons fall in multiplets of SU(6). To allow for stable solitons, one has to add a stabilizing higher order term. In ref. [7.18], a coupling to quarks was used to find solitons of this approach, where (o-mesons are included as components of Unfortunately, the quantization of these tensor modes can only be performed for the non-interacting theory. With two input parameters (f,, = 93MeV, mq 370 MeV) we find MH 1.4 GeV and the (ONN-coupling constant comes to be g~,~~/41T = 10.9, close to its empirical value. Obviously, all SU(6)-predictions like ~ = —3 / 2 come out for free. It is worth stressing that in this approach the spherical shape is not an exact solution to the equations of motion, but rather the ground state is deformed and the N~-splittingoccurs through shape deformations. At the present stage, it remains to be seen whether this approach offers a realistic alternative to the ones presented in sections 2 and 3. ~
References [1.1] J.J. Sakurai, Currents and Mesons (University of Chicago Press, Chicago, 1969). [1.2] V. de Alfaro et al., Currents in Hadron Physics (North-Holland, Amsterdam, 1973). [1.3] M. Gourdin, Phys. Rep. 11(1979) 29. [1.4] G. ‘t Hooft, NucI. Phys. B 72 (1975) 461; B 75 (1974) 461. [1.5] E. witten, Nuci. Phys. B 160 (1979) 57.
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U.-G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
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Ulf-G. MEISSNER Center for Theoretical Physics, Laboratory for Nuclear Science and Department of Physics. Massachusetts Institute of Technology, Cambridge, MA 02139, U.S.A.
I
NORTH-HOLLAND
-
AMSTERDAM
PHYSICS REPORTS (Review Section of Physics Letters) 161, Nos. 5 & 6 (1988) 213—362. North-Holland, Amsterdam
LOW-ENERGY HADRON PHYSICS FROM EFFECTIVE CHIRAL LAGRANGIANS WITH VECTOR MESONS Ulf-G. MEISSNER Center for Theoretical Physics. Laboratory for Nuclear Science and Department of Physics. Massachusetts Institute of Technology, Cambridge, MA 02139, U.S.A. Received October 1987
Contents: 1. Introduction and motivation 2. The massive Yang—Mills approach 2.1. A minimal chiral Lagrangian with spin-I mesons 2.2. Inclusion of non-minimal terms for the ~rpA 1-system 2.3. A subtracted Lagrangian with Vector Meson Dominance 2.4. A Lagrangian of pseudoscalar and vector mesons 3. The hidden symmetry approach 3.1. The p-meson as a dynamical gauge boson 3.2. The w-meson as a dynamical gauge boson 3.3. The non-Abelian anomaly in the framework of the hidden symmetry approach 3.4. Extensions of the hidden symmetry (axial-vector mesons) 4. Equivalence proofs 4.1. The Stiickelberg construction 4.2. Generalized proof in the presence of external gauge bosons 4.3. Gauge symmetric constraints to eliminate the A 5. Connection to QCD 5.1. QCD in the large N~limit and microscopic derivation of the pion decay constant 5.2. Vector mesons and QCD 5.3. Bosonization of the Nambu—Jona-Lasinio model 6. Baryons from the effective Lagrangian
215 220 220 231 235 239 242 243 247 250 255 257 257 260 264 266 267 272 279 284
6.1. Nucleons and ~-isobarsin the irpw-system 6.2. The electromagnetic structure of the nucleon 6.3. Axial properties of the nucleon 6.4. Meson—nucleon form factors and the nucleon— nucleon interaction 6.5. The baryon spectrum 6.6. Complete vector meson dominance? 6.7. Results from the ~rpA1w-system 6.8. NN-annihilation and the H-dibaryon 7. Further developments 7.1. The QCD trace anomaly and scalar particles 7.2. Nuclear binding and scalar particles 7.3. Comments on the derivative expansion 7.4. Remarks on vector mesons coupled to two-phase models 8. Summary and outlook Appendices A. Non-linear realization, PCAC and VMD B. p-mesons in the Skyrme model C. Calculation of the moment of inertia and the electroweak currents in the ~rpw-system D. Some remarks on the SU(6) approach References Note added in proof
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Abstract: We review the use of effective chiral Lagrangians with vector mesons in the description of low-energy hadron physics. The massive Yang—Mills approach (vector mesons as heavy gauge particles of the Yang—Mills type) and the hidden symmetry scheme (vector mesons as dynamical gauge bosons) used to enlarge the field space beyond the Goldstone pions are discussed in detail, with particular emphasis on the gauged Wess—Zumino action and Vector Meson Dominance. It is demonstrated how all parameters can be fixed in the meson sector, and in particular, that there are only a few. The equivalence of these two seemingly different approaches is proven. The QCD generating functional is investigated and its connection to effective theories of spin-0 and spin-i mesons is shown. Baryons arise as topological solitons, with all parameters fixed in the meson sector. We present in great detail the phenomenological aspects of nucleons such as their static properties, electromagnetic, axial and strong form factors. The close connection to boson-exchange is elucidated, and the application to NN-physics is stressed. We furthermore discuss the inclusion of scalar particles and their relevance to OCD as well as the problem of nuclear binding. Further developments are sketched.
1. Introduction and motivation Low-energy hadron physics down to length scales of 0.5 fm has been rather successfully described in terms of mesons and baryons in the sixties [1.1, 1.2, 1.3]. Hadrons were used as elementary fields in effective local theories, namely chiral effective Lagrangians. With the advent of Quantum Chromodynamics, QCD, most of the notions developed in that era like e.g. Vector Meson Dominance, boson exchange, lost their importance or even disappeared completely. The interest in effective meson theories has been revived in the early eighties, when it was shown [1.4, 1.5] that QCD in the limit of large number of colors, N~,reduces to a nonlinear, effective theory of infinitely many weakly interacting mesons. Baryons emerge as topological soliton solutions of the mesonic Lagrangian, so their properties are intimately tied to the parameters of the meson sector. To allow for quantitative predictions, one obviously has to truncate the number of meson fields. Since we are interested in hadron physics at energies below 1—2 GeV, a reasonable approximation is to restrict ourselves to the low mass mesons. The power of the effective Lagrangian approach lies in the fact that it embodies the symmetries of the QCD Lagrangian and the structured vacuum. To first order, one simply works on the field space of pseudoscalar Goldstone bosons, JPC = 0~, related to the spontaneous breakdown of chiral symmetry, SU(Nf) ®SU(Nf) to diagonal SU(Nf)V. Here, Nf is the number of flavors, we will mostly restrict ourselves to the non-strange sector of u- and d-quarks, SU(2). The pertinent Goldstone bosons are the pions, with their interactions at low energy described by the non-linear if-model [1.6, 1.7, 1.8] leading to the successful predictions of current algebra. As first pointed out by Skyrme [1.9, 1.10] this theory allows for stable soliton solutions when one adds a higher order term in derivatives of the pion field. These solitons resemble very much nucleons, i.e. they carry a conserved topological charge, they interact strongly and they have a rich quantum sector (like the nucleon excited states). Witten [1.11] demonstrated the link of this model to QCD by investigating the underlying symmetries and connecting the anomalies of QCD to the Wess—Zumino term [1.12], which renewed the interest in effective chiral models (for reviews on the Skyrme model, which operates with pions only, see ref. [1.131). It soon became clear that a more accurate description of baryon properties at energy scales below 1—2 GeV could only be achieved if one enlarges the mesonic field space to incorporate other mesons, especially the vector mesons p, w and A1. A detailed study of these effective theories including .
. .
,
pions and vector mesons will be the main topic of this review. We will show how to make contact to concepts like Vector Meson Dominance and current algebra, therefore combining physics of the sixties with physics of the eighties. To make our motivation stronger, let us focus on some properties and facts concerning the vector mesons. The existence of the vector mesons became first obvious in the study of the electromagnetic structure of the nucleon. The w-meson, a resonant three-pion correlation, was introduced by Nambu
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[1.141to explain the apparent discrepancy in the proton and the neutron charge radii. The dispersion theoretical analysis of the proton charge form factor lead Frazer and Fulco [1.15] to the assumption of a resonant two-pion correlation, the p-meson. In analogy to electromagnetism, Sakurai [1.16]conjectured that the conservation of isospin and hypercharge could be explained through the coupling of hadronic vector particles via the Yang—Mills mechanism [1.17]. These two seemingly different aspects of electromagnetic and hadronic interaction were unified in the framework of the Vector Meson Dominance model (VMD) [1.1], according to which the exchange of vector mesons dominates the electromagnetic and final state interactions of hadrons [1.18, 1.19]. In a more field theoretic language, the VMD hypothesis leads to the current-field identity [1.20] ~
p~(x)- ~
w~(x)-
~-
~(x)
(1.1)
for flavor SU(3). In the language of QCD, eq. (1.1) can be interpreted as follows: A photon materializes in a q~pair propagating as a vector meson, which then couples to the quarks inside the target hadron. All the complexities of QCD, namely gluon self-interactions and confinement, are incorporated in the physical vector meson intermediate state. An important test concerning the applicability of VMD is to compare the coupling constants g~,g,~, obtained from direct (leptonic) reactions like e~e—~V (q2 = m~)with indirect determinations via photoreactions like -yN-—~VN, ~yN—~-yN,...with q2 0 and E~ 2—20 GeV. From the leptonic decays one finds g~= 5.67, g(~,= 15.18 and g 4 = 11.92, whereas photoreactions give g~= 5.53, g(,, = 15.81—19.38 and g4 = 17.06—23.12. For comparison, broken SU(3) with ideal mixing gives g~= (1I3)g(,~= (\/~I3)g4. The VMD model, furthermore, allows to parametrize the form factors of baryons and mesons in terms of vector meson poles [1.21], as a pertinent example us consider theofelectromagnetic form factor of the pion. In the 2 <0, with q theletfour-momentum the incoming photon), VMD predicts space-like region (q F,~(q2)= m~/(m~q2), (1.2) .
. .
—
—
whereas in the time-like region (q2 > 0) the form factor is determined by the p-meson resonance at 770 MeV; for q2 > 4m~one also has to take into account the finite width of the decay p—~‘rri~via m F,~(q2)= 2 2 2 m~—q —im~f(q)
(1.3)
with F~(m~) = 153 MeV. As shown in fig. 1.1, the VMD model describes the data at momentum transfer q2~~ 1 (GeV/c)2 within 10—20% accuracy. Of particular interest is the slope of the form factor F~,(q2) at q2 = 0 which gives the pion charge radius. From eq. (1.2), we find (r2)~2= \/‘~Im~ = 0.63 fm close to the experimental value of Kr2)~/2= (0.66 ±0.01) fm [1.22]. This means that although the pion is an elementary, point-like particle in the underlying effective theory, it is seen by the photon as an extended object through the coupling to the p-meson. VMD also determines the hadronic and radiative decays of vector mesons. Durso [1.23] has recently re-examined these decays and shown that with the inclusion of small symmetry-breaking terms, the VMD assumption indeed works quite well for the radiative decays of vector and pseudoscalar mesons (cf. table 1.1). Matching VMD and soft-pion predictions for given scattering amplitudes implies a set of consistency relations of which the most famous one is the KSFR relation [1.24] a = m~Ig2f = 2 (aexp = 2.13) from S-wave ‘rrN-scattering. In
U.-G. MeiJ3ner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
217
b_____________________________________ PION FORM FACTOR IN THE TIME LIKE REGION
a
—
I
______
I
I
2)
FORM FACTOR F~(q ~ CERN NA7
PION
6
~ORSAY ~ NA7
~ CEA ‘7
//\ \~
—4.
-
p17701
05
SPACELIKE
I
-0.5
-0.4
I
—0.3
-0.2
TIMELIKE
-
0
0.1
0.2
~~°~-°/ I
4m,,
I
-0.1
2
0 4m~,0.2
0.3
q2 [Gev2]
-
I
I
0.4 0.6 qz[GevZ]
0.8
Fig. 1.1. The pion charge form factor F~(q2)in the space-like (a) and the time-like (b) region. In the time-like region, the form factor is dominated by the p-meson. The solid line gives the prediction of the VMD model, which is close to the data for momentum transfer q ~ 1 GeV/c.
Table 1.1 Radiative meson decays. VMD gives the prediction of a Lagrangian incorporating Vector Meson Dominance with small SU(3) breaking terms [1.23] in comparison to the empirical data. The first three decays were used to fix the parameters. Width (key) Decay
VMD
Experiment
w—~ry
888.0 6.1 63.0 5.4 55.0 12.5 119.0 0.24 149.0 68.0 71.0
861.0±56.0 5.9±2.1 71.0±8.0 3.0±2.0 55.0±15.0 8.1 ±8.8 87.0±20.0
4—*~iry
p—~r~-y
11’—+ory
K*o~_~.KOl
—
118.0±8.1 57.0±5.0 51.0±10.0
general, we can say that VMD works within 20% for the SU(2) sector (p, o) and somewhat poorer for SU(3). It therefore seems vital to include this concept in any effective meson Lagrangian which is supposed to describe low-energy hadron physics. Another example showing the relevance of mesons is the description of the nucleon—nucleon interaction in semi-phenomenological boson-exchange-models [1.25, 1.26]. Following the work of Yukawa [1.27] which describes the long-range interaction between two nucleons through one-pion exchange (cf. fig. 1.2), the inclusion of higher mass mesons and resonant multi-pion states has led to a very successful description of low-energy NN scattering data. The p-meson shows up at intermediate
218
U.-G. MeiJiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons IS
NUCLEON-NUCLEON POTENTIAL
150
~.1OO
50
~
-50
-100 I
0
I
1.0
I
I
2.0
r[fm] Fig. 1.2. The nucleon—nucleon interaction in the ‘S
0 channel. For large separations, the one-pion exchange dominates. In the intermediate range, correlated two-pion exchange gives the attraction between two nucleons. At smaller distances, the potential is very repulsive, with a large part of the repulsion coming from the exchange of w-mesons.
distances, especially in the isovector tensor potential, where it cancels part of the pion tensor potential. A good part of the short-range repulsion between nucleons can be understood in terms of w-meson exchange. The intermediate-range attraction in the central potential is due to correlated two-pion exchange in the scalar—isoscalar channel [1.281as indicated in fig. 1.2. The notion of boson exchange will appear naturally in the framework of the effective Lagrangian to be constructed here. From these considerations it seems that low-energy hadron physics can be described quite satisfactorily in terms of mesons, when one incorporates the underlying symmetries and anomalies of QCD, not quarks and gluons as dictated by QCD. Further indications of this are the success of the shell model and meson exchange currents. The main assumption of the very successful nuclear shell model is that nucleons move as independent particles inside the nucleus. In terms of quarks and gluons, one expects quarks to be confined within volumes of —4 fm in radius [1.29]. On the other hand, the average distance between nucleons in nuclei is d 1.8 fm, which would imply that the quark cores would overlap frequently. How can an independent-particle picture as confirmed by experiments through mapping 3He out single particle emerge? The electro-disintegration of the deuteron d(e, probe e’)pn and the magnetic formdistributions factor are the best examples of meson exchange currents, which momentum transfer up to l~I 1 GeVIc [1.30]. Again, one would expect that the meson picture breaks down at these length scales and explicit quark degrees of freedom are needed, but again this is not the case. So meson theory works down to length scales considerably smaller than A~D 1 fm, which is not expected in view of the large electromagnetic radii of the hadrons, e.g. r~roion 0.85 fm and r 10~ 0.7 fm. What we are going to demonstrate is that the relevant degrees of freedom which govern the strong interactions of hadrons at energies up to —1 GeV are indeed mesons, with their interactions dictated by chiral symmetry and the anomaly structure of QCD at low energies, not quarks and gluons. Does all this mean that there are no quarks? This certainly would be the wrong conclusion. What it really means is that there is a beautiful correspondence principle between the meson and the quark P—
U. -G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
219
language, with its theoretical justification given by the so-called “Cheshire Cat” principle (CCP) [1.31]. This principle states that a theory of quarks and gluons chirally coupled to mesons is equivalent to a chiral meson theory. * Having said all this, the reader might wonder whether inclusion of more meson fields beyond the pions in the effective Lagrangian does not proliferate the numbers of tuneable parameters, therefore limiting the predictive power of this approach. This is indeed not the case. In section 2 of this article, we demonstrate how one can make use of linear representations of chiral symmetry and introduce the vector mesons as heavy gauge particles. We will show how the Wess—Zumino term links a variety of 2y, w—* 3’rr,... and therefore predicts most of the coupling hadronic and radiative decays like ~ constants in terms of the gauge couplings g related to the gauge group we are considering (e.g. SU(2)L®SU(2)R®U(l)v for p, A 1 and w). In the non-anomalous sector, one has to go beyond the minimal form of the effective Lagrangian as dictated by chiral symmetry to cope with problems like the irA1-mixing and the A1 parameters. To circumvent the latter problems, one can also use a non-linear realization of chiral symmetry a la Weinberg [1.32], this approach is nowadays called the hidden symmetry scheme. The vector mesons arise as dynamical gauge bosons of a hidden symmetry of the non-linear if-model, with the gauge principle again limiting the number of meson parameters3)flavor quite drastically. We discuss in detail how the pand w-mesons fit in this scheme and extensions to SU( as well as axial-vector mesons. In section 4, we prove the equivalence of both of these schemes (massive Yang—Mills and hidden symmetry). This equivalence proof puts further constraints on the seemingly arbitrariness how to introduce vector mesons in the effective theory. Section 5 deals with the question of how these chiral Lagrangians are related to the generating functional of QCD. For that, we investigate the QCD partition function assuming spontaneous breakdown of chiral symmetry and show that indeed vector meson Lagrangians of the form discussed in sections 2 and 3 can emerge. Section 6 gives a detailed analysis of the baryon phenomenology as predicted by these models. We show how to construct soliton solutions, which have the hedgehog structure exhibited in fig. 1.3, quantize them as fermions and give results for static properties, electromagnetic and axial form factors, strong form factors, the baryon spectrum and nucleon—antinucleon annihilation. In section 7, we comment on
THE DEFENSIVE
HEDGEHOG
(1) Al
Fig. 1.3. The defensive hedgehog in the presence of vector and scalar mesons. The classical soliton solution to the effective chiral theories discussed in this review can be visualized with the familiar hedgehog pion field (radial in (iso)space) with a core of o-mesons and p-, A 1-mesons (projected along the i direction). S denotes any scalar meson or glueball (see section 7). *
This is a very simplified version of the CCP.
220
U. -G. MeiJlner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
further developments, in particular, the inclusion of scalar particles. The motivation for including scalars is twofold: first, scalar glueballs are the Goldstone modes of broken scale invariance and, second, the nuclear binding discussed before can be described by the exchange of scalar mesons. Comments on the derivative expansion and chiral two-phase models are also made. The appendices contain some technicalities and some more speculative ideas, as well as developments related but not in direct line with the ones described before. For the sake of completeness, let me mention that I have only included material which was available for use by the end of May 1987.
2. The massive Yang—Mills approach In this section, we will demonstrate how one can construct a chiral Lagrangian of spin-0 and spin-i particles, with the latter treated as massive Yang—Mills bosons* of chiral U(Nf)L 0 U(Nf)R symmetry, eventually broken down to chiral SU(Nf) (with N~denoting the number of flavors). These ideas originated twenty years ago in the description of mesons coupled to point-like nucleons, and were most strongly advocated by Sakurai [2.1] and Schwinger [2.2]. For a review on most of the older ideas and phenomenology we refer the reader to ref. [2.3]. With the advent of Quantum Chromodynamics (QCD), these ideas were mainly forgotten, but by now it has become clear that QCD at low energies can most economically be described by effective Lagrangians which embody the relevant symmetries of QCD (especially chiral symmetry). Following the work of Schechter and collaborators [2.4—2.7]we will show how the anomaly structure of low-energy QCD at low energies can be described in terms of the gauged Wess—Zumino action. * * This amounts to the construction of a Lagrangian of pseudoscalars and (axial) vector mesons. In the following subsection, modifications to this minimal ansatz will be discussed, due to the difficulties coming from the ~rA1-mixingwhich induces a strong momentum dependence of certain vertices. Then, following the work of Brihaye, Pak and Rossi [2.9—2.11],we show how one can implement the concept of Vector Meson Dominance (VMD) in such effective Lagrangians. A more general framework on the basis of differential geometry [2.12, 2.13] will be discussed in Appendix A (for the non-anomalous sector). Finally, we will show how one can eliminate the axial-meson A1, triggered by the fact that the A1 is not a sharp resonance (F(A1)Im(A1) 1/3) and that its parameters (mass, decay width, etc.) are not as uniquely established as the ones of the vector mesons p and w. Furthermore, the A1 plays only a minor role in nuclear physics, e.g. its contribution to the NN interaction is almost completely wiped out by the pir-continuum [2.14]. 2.1. A minimal chiral Lagrangian with spin-i mesons Here, let us start from QCD at low energies. The chiral model we will construct can be considered as based on an order-parameter multiplet Mab transforming like the quark-field configuration ~iRbqLa~ M condenses in the chiral phase, i.e. (Mab) ~ 0. It is most convenient to make a polar decomposition M = U H where H is a unitary matrix including the usual nonet of pseudoscalar mesons 4 (4 = * Literally, the term “massive Yang—Mills boson” does not make sense, what is meant is that symmetry-breaking mass terms have to be added by hand. ** Since the review by Zahed and Brown [2.8] covers this topic in an excellent fashion, we will be bnef on this point.
U. -G. MeiJiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
U=exp(2i4/\/~f1,)=exp(2i4ilF.~)
221
(2.1)
with F,~= = 132 MeV the weak decay Notice that U transforms in a so-called 3)L 0pion SU(3)R suchconstant. that under a transformation by unitary matrices non-linear representation of SU( (A, B), U becomes A U B’. The chiral Lagrangian for pseudoscalar interactions at low energies is nothing but the non-linear if-model [2.3] =
Tr(9~U c~~Ut) = ~ Tr(c9.~U ~Ut).
~
(2.2)
One can furthermore add a chiral-symmetry breaking term proportional to ~ ?IZaUaa, where the are the current quark masses. With that, the octet of pseudoscalar mesons (‘it, K, 11) will be described satisfactorily. The mass of the ‘q’ can be understood by introducing a phenomenological pseudoscalar glueball field to describe the U( 1) A anomaly [2.15—2.18].We will not go into the details of this point here. Similarly, the QCD trace anomaly can be modeled by the introduction of a scalar glueball field [2.19, 2.20], as we will discuss in section 7. In this section, we are particularly interested in the Wess—Zumino term [2.21] which describes the non-Abelian anomaly structure of QCD [2.8, 2.22]. Let us first discuss the non-anomalous part ‘~, of our Lagrangian. For that, we follow Sakurai [2.1], and introduce left- and right-handed spin-i fields A L~ and A R~ respectively, which are related to the vector and axial-vector mesons V~,and A, 2 by (V,2 = v; ~i’~h,A,2 = A~ra/V~) AL,2=~(V,2+A,2),
AR,2=~(V,2—A,2).
(2.3)
The couplings of these mesons to matter are introduced as gauge couplings, which is the most economical way to minimize the number of appearing parameters. This amounts to gauging the non-linear if-model [2.2]
~
Tr(D,~UDbLUt) DU= ô.,~,U—igA,2~U+igUA,2~ ~J,gauged
=
(2.4)
with D,2 the covariant derivative. g is the gauge coupling constant, for the normalizations used here, we have g = \/~f~, where f~,,is related to the p—÷‘rr’rr decay via 2
~
(m~_4m~)3/2
~
(2.5)
which gives (m~= 770 MeV, 1 = 153 MeV, m,~= 139.6 MeV) f~/4ir = 2.95,
*
f~
=
6.08.
We will specialize to the non-strange sector of u- and d-quarks later on.
(2.5a)
222
U. -G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
In the derivation of eq. (2.5), a minimal momentum dependence of the pint-vector was assumed, as we will see in what follows. The difference to a gauge theory stems from the necessity of introducing vector and axial-vector meson mass terms. Therefore, the pertinent Lagrangian involving spin-i and spin-0 mesons takes the form: Tr(D,2 U D~Ut)+ ~
~
=
+
Tr[M(U + U~—2)]
—
~ Tr(F,2VLF~-I- F~RF~)
m~(A,2LA~+ A,2RA~)— i~Tr(D~UDOUtF~+ D~LUtD0UF~).
(2.6)
The last term in eq. (2.6) is a so-called non-minimal term since it is of higher order in derivatives than the other terms in eq. (2.6). Notice that at this level the vector and the axial-vector mesons have the same “bare” mass m0. The corresponding field strength tensors are given by F~ = ~
— ~AL~R
~ig[A~”~, Ar].
(2.7)
The mass matrix M appearing in eq. (2.6) can be written as M
~(m~ + ~m~)i
=
—
(2/1/~)(m~ — m~)A8
(2.8)
in terms of the pion and kaon masses m,~and m~,respectively. The quadratic piece of the Lagrangian (2.6) is given by [2.23] 2— ~2)
=
~ —
~
m
Tr(ô,2~— ~F,,gA~) ~Tr(F~F’~” + FA
14~ Fb0~A)+
~
Tr(V~+ A~)
(2.9)
in terms of the vector and axial-vector fields. Obviously, (2.9) is non-diagonal in oI,,,4 and A,2, i.e. the pseudoscalar and axial fields mix. To get rid of the A~~4-vertex, we introduce physical pseudoscalar and axial-vector fields 4~iand A via ~
2m0
FZ’F~, (2.10)
2F~/4m~) = (1 + g2F~J4m~)~. (1 — g This is the conventional diagonalization scheme. Since also terms of the type 3 -
=
1~[V,2, cb] appear, one
can generalize the definition of the axial field A,2 by substituting o~by V,~= + i(g/2)[4, V,2]. We will come back to this point in subsection 2.3. Now, the quadratic part ~2) reads ~2)
=
~ Tr(ô,2~e9~4) ~~
+
—
2)m~ ~ Tr(V~)+ im~Tr(A~)— with physical masses given by +
=
m~IZ2,
th~= m~/Z2,
(2.11)
~ (4~IZ
m,~.= m~= m~,
m~= m~,IZ2.
(2.12)
U. -G. Meiflner, Low-energy hadron physics from effective chiral Lagranglans with vector mesons
Of particular interest is the mass difference of the A1- and the p-meson, 2~2 m 2 m~= ~-~--~ (—~)
223
(2.13)
—
which arises due to a partial Higgs mechanism. If the vector meson mass term would not have been present in (2.6), the ordinary pion would be completely eaten up by the A 1 this underlines the crucial manner in which the present Lagrangian differs from a true gauge theory. Notice that (2.13) has originally been derived by2 Weinberg using sum rule techniques [2.24]. F~=the 132KSFR MeV, = 0.46, i.e. m~ = 1135 MeV~ ~ Notice With that gZ2= =8.60, ~ gives m0 = 770 MeV, we find Z relation [2.25]. The part of the Lagrangian of third order in the meson fields and A,2 yields the VJxf and AV4 couplings —
=
~AV4
~ Tr(V~4 9,2~)+ Tr{ V,L[A,2,
=
+
Tr[(ô,2VV
~]}+
—
c9~V,2)ô,24ô~,4],
[(1—8)Tr{(d~V~ ô~V~)[A,2,~]} —
Z2 Tr{(9~A~ — dVA,2)[V,2,
8 = 1 — Z2
—
(2.14)
i9~]}],
2Z4~g/(1— Z2)
where one integration by parts has been performed. From (2.14) we can now read off the pint and pA 1i’i transition amplitudes as follows (from now on we omit the tilde on the physical fields and constants): ““
2
1
2
my
2
my
m m~-~m~]’
~ 2
x
~
@
~)
1/2
2
(i+2g~m~m~)m~~ 2
1/2
~ Defining the p—* 2’rr and A1
(2.15)
~
—~
pit transition matrix elements via
(it.np)nsg~~p,2•(4~X (2.16) + hA~p,,r
~
qA~eA,
q~,
where the r’s are the relevant polarization vectors of the p and the A1, we can read off the coupling constant g~~1-7~, gA1P~and hA~p,~ as ~
=
(gI\h)(~+ ~
224
U. -G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
=
m2—m2\
gm
/
~
I~1 2g~ 2m~ —
.11),
(2.17)
=
Notice that for ~ = 0, the pint-coupling is decreased by a factor 3/4 and that there is no d-wave contribution to the amplitude A 1 —~pit. Since we want to have a minimal momentum dependence of ~ so that F(p—* 2’rr) Unfortunately, is given by (2.7)this andchoice g = \/~g~~ = ~a too small we choose 2g~= ito recover the eq. 7~fT. of ~gives A (2.4) with g = ~ 1-width, with m~= 1275 MeV* we find from
~A1~T = 41Tm~ [~ ~ + ~ (~ gA1P~T+ ~I q~2hA)] (2.18) that 120 MeV, much smaller than the experimental value FAP,, 350 MeV. We will return to this point in the following subsection, for the moment let us assume that we work with ~so that the p ‘itt decay width is given in its minimal form (2.4). This is consistent with the universality of the pint and pNN-couplings [2.1]. Let us now discuss the anomalous sector, i.e. the vector meson processes connected to the gauged Wess—Zumino term. For that, let us briefly remind you of the Wess—Zumino term, for more details see ref. [2.8]. In terms of the differential one-forms —~
L
=
(~9,2U)Ut~
=
(dU)Ut (2.19)
R= UtdU= UtLU where L and R transform as nonets under left and right U(3)-flavor groups, respectively, the WZ action can be written as [2.22] 5]
=
Cf Tr[R5]
(2.20)
F~~(U) = Cf Tr[L with ô M5 = M4, i.e. the boundary of the five-dimensional manifold M5 is ordinary Minkowski space M4. The constant C has been determined by Witten [2.22] by calculating the anomalous process it°—* 2y, it follows to be C = —iN 2, with Nc (=3) the number of colors. The physics of F~becomes most 0/2401T expansion L = (2i/F,~)d4 +..., then transparent in the weak-field T~~= ~
(2.21)
where we have used Stokes’ theorem and Poincaré’s lemma (d2 = 0). Physically, F~ accounts for processes like KK—*3’ir and removes the spurious symmetries of the non-linear if-model (2.2). (Unfortunately, the reaction KK—3 3’it is not known experimentally.) Stated in another way, the WZ action contains all processes involving the four-dimensional Levi—Civita tensor e~’3.If we now think ~f *
The experimental situation concerning the A,-parameters is discussed in the following subsection.
U.-G. Melliner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
225
introducing the spin-i particles, it becomes clear that all abnormal parity processes (processes which do not conserve intrinsic parity*) will be described by the gauged WZ-term. Some typical examples are the decays o—* 3i’i and K*_~*Kinit, for which experimental information exists. To clarify the gauging of the WZ action, we will first consider electromagnetism, i.e. the variation ~U=ie[Q, U]
(2.22)
where Q is the SU(3) charge operator changes by
Q
=
diag( ~,
—
~‘,
—
~ Under the local variation (2.22), F~
4 — R4)] aF~~(U) = 5Cif dr Tr[Q(L _5Cif d{dr Tr[Q(L3
=
+
R3)]}
_5Cif dr Tr{Q[L3 + R3]}
=
(2.23)
making use of the fact that even powers of L and R are exact forms, i.e. 0=dL—L2=dL3—L4=dR+R2=dR3+R4. Furthermore, we have used Stokes’ theorem and the fact that M4 = 9M5. Let us now introduce a gauge field A,2 and the corresponding one-form A = A,2 ~ with the transformation property (e = 1) = dr. We consider F~’~(U, A)
F~~(U) + 5CiJ A Tr[Q(L3
=
+
R3)]
(2.24)
The variation of F~’~(U, A) is ~F~1~(U,A)
=
5Cif A Tr[Q6(L3
=
iOCf d~A Tr[Q2(L2 — R2) +
+
R3)]
Q d(Ut)Q dU].
(2.25)
The quantity inside the trace is a closed form, thus it can be written as an exact form L2—R2=d(L+R), (2.26)
Q d(Ut)Q dU *
=
d[aQUtQ dU — bQ d(Ut)QU],
The intrinsic parity of a particle is defined as follows: it is
pseudotensor, e.g.
is, ~y, p
+1
if a particle transforms as a true tensor of that rank, and —1 if it transforms as a
and A 1 have intrinsic parity —1, +1, +1 and —1.
226
U. -G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
where a, b are constants satisfying a ~F~1~(U,A)
=
+
b = 1. Integration by parts allows us to rewrite ~FW(U, A) as
iOCf dr dA Tr[Q2(L + R)
+
aQUtQ dU
—
bQ d(Ut)QU].
(2.27)
We end up with a gauge invariant expression when we consider F~21(U,A)
=
F11~(U,A) — 1OC
A dA Tr[Q2(L
+
R)
+
aQUtQ dU
—
bQ d(Ut)QU].
(2.28)
SM5
This is the desired gauging of the WZ-term. Introducing r = a — b, and putting pieces together, we find 1~21(U,A)= F~~(U)+5Ci J ATr[Q(L3
—
1OC
f
+
R3)]
SM5
A dATr[Q2(L
+
R)
+
~QUtQ dU - ~QUQ d(Ut)]
SM5
—
5Ci J dA dA Tr[QUtQU].
(2.29)
SM5
Notice that the last term proportional to r is gauge-invariant by itself, it therefore does not affect the gauge invariance of p(2)~ But since dA dA = ~d~xE’~’~F,2pFap is not parity invariant, we must choose r = a b = 0, i.e. a = b = 1/2. Let us now discuss the gauging of non-Abelian subgroups of U(3)L®U(3)R. We will again use the trial and error method advocated by Witten [2.22] and demonstrated above. * There will always appear a gauge invariant term which is multiplied by an arbitrary coefficient in addition to the terms whose variation reproduces the anomalies. This additional term should be dropped if one wants to represent the anomalies in an irreducible way. For an arbitrary subgroup of U(3)L 0 U(3)R, the relevant group parameters are —
rLR
= ~
iELR/k/2
with A*/2 the generators of U(3). In terms of left- and right-handed gauge fields, AL and AR, we have the following transformation properties: U—*exp{irL}U exp{—ieR} ML
*
=dEL
~[rL,
AL],
~AR
=drR +~[ER, AR].
A more direct way of integrating the WZ anomaly equation can be found in refs. [2.26, 2.27]; see also section 3.3.
(2.30)
U. -G. Meifiner, Low-energy hadron physics from effective chiral Lagrangian., with vector mesons
227
The trial and error gauging procedure outlined before gives the following result F~~(U, AL, AR)
3 F~~(U) + 5Cif Tr[ALL
—
+
ARR3]
5CJ Tr[(dALAL + AL dAL)L +~dA~AR + AR dAR)R]
+5Cf Tr[dALdUARUt—dARd(Ute)ALU]
+
5Cf Tr[ARU~LUR2 - ALUARUtL2]
+
2.5Cf Tr[(ALL)2
—
(ARR)2]
+
5Cif Tr[A~L + A~R]
+ 5Cif Tr[(dARAR + AR dAR)UtALU
+
5CiJ Tr[ALUARU5ALL
+
—
(dALAL + AL dAL)UARUt]
ARUtALUARR]
+ 5Cf Tr[A~UtALU - A~LUARU+ ~(UARU~L)2]
-
5Crf Tr[FLUFRUt] (2.31)
with FLR = dALR — iA~R.The last term in (2.31) with the arbitrary parameter r disappears if the gauge group is such that one can define a parity operation A~(x)*-+ AR,2(—x), U(x) ~-* U1(—x). All terms in (2.31) are parity invariant apart from the last one, thus r = 0 should be taken. Of course, the variation of FWZ(U, AL, AR) under a gauge transformation is nothing but the Wess—Zumino anomaly equation ~
AL, AR)= _iOCiJTr{EL[(dAL)2_ ~idA~]—(L~R)}.
(2.32)
The WZ action is invariant under global U(3)L 0 U(3)R. It also contains the gauge coupling g via rescaling the fields ALR gA~~ and its vector meson content can be expressed by using (2.2). Now, let us assume that the electromagnetic interactions at low energies are dominated by the neutral vector mesons [2.1, 2.28], i.e. —*
~‘em
=
(\ e/g)A~[m~p~°~ + (1/3)m~,w,~ — (V~/3)m~,~]
for the nonet ansatz p°—— (i/V~)(u~ — dd), w
-~(1 /V~)(ut~ + dd),
(2.33) ~ -~s.f, and that other hadrons
228
U. -G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
interact with the electromagnetic field A,2 through their couplings to the vector mesons (vector meson dominance). In this scheme, the decay i’i°—*2-ycan be calculated from (2.31) via ‘rt°—*wp°-—*2-y.The it°p°w-vertex is given as a piece of F~~(U, AL, AR). Defining the appropriate Lagrangian density as gVV
=
with
p0123 =
4~~
Tr(~~V~9aV~)
(2.34)
+1, the relevant coupling constant gvv4 follows to be (2.35)
2/i6it2F~)(1 g2F~/6m~). —
~vv~ = (3g This leads to 22
8~ag~~
0
F(i’i —*yy)=--~jwith a
=
4~ m,,
(2.36)
1/137. Comparing with the current algebra result
F(it°—*2y)=
(2.37)
a2
gives 2/16ir2F~. (2.38) ~vv~ = 3g Now gvv 4 in (2.38) is roughly 1.6 larger than gvv4 as predicted from (2.35), i.e. F(’rt°—.*2-y) as calculated from (2.36) comes out almost a factor three too small. This result is not surprising since the WZ action (2.31) was constructed in a left—right symmetric way, i.e. it reproduces the left—right symmetric form of the WZ anomaly equation (2.32). As first pointed out by Bardeen [2.29], the anomaly sits entirely in the axial-vector current, and the vector current is conserved. By a suitable addition of counterterms, one can always shift the anomaly in one given sector [2.30]. In the form (2.31), both the vector and axial-vector currents are anomalous. Let us therefore consider the following counterterm
=
5Ci
J
Tr[(dARAR
+ AR dAR)AL
—
(L~R)]+ 5Cf Tr[A~AL A~AR+ —
(2.39)
M
The variation of the Bardeen subtracted WZ action becomes ~(F-I~) 4— ~(F~A2+ AF~A+A2F~)]}, =
—3OCif Tr{(EL— ER)[4FV
+
—
~A
(2.40)
U. -G. Melfiner, Low-energy hadron physics from effective chiral Lagranglans with vector mesons
229
where F~= dV
~i(V2 + A2),
+
FA
=
dA + ~i(AV + VA)
(2.41)
with A and V defined in (2.2). For vector gauge transformations (EL = ER) the anomaly in Bardeen’s form vanishes. Of particular interest is the fact that the Bardeen counterterm I~(2.39) can be obtained from the gauged Wess—Zumino action (2.31) via I~=FwjU=~, AL, AR).
(2.42)
Therefore, the effective action which reproduces the Bardeen form of the anomaly is F~~(U, AL, AR)= F~~(U, AL, AR)
—
F~~(l, AL, AR)
(2.43)
so that F~vanishes when U= ~fi[2.21]. Notice that the action (2.43) breaks the global U(3)LOU(3)R chiral symmetry down to U(3)~0 U(i)A. This is most obvious from (2.42), the special choice U = .1 clearly breaks chiral symmetry. Nevertheless, this breaking does not affact any of the usual currentalgebra theorems (which involve amplitudes independent of ~ )~ * Before discussing the meson phenomenology arising from (2.43), let us give the Bardeen subtracted WZ-term for the group SU(2)L®SU(2)R®U(l)v, incorporating the p-, A 1- and w-mesons most relevant to the non-strange sector. It reads [2.31] using Stokes’ theorem -
Tr[LpLaLp]
482 ~
EW~
—
~pTr[ARaLp ~ALaR~ +ARaU~AL$U~ARaALp],
(2.44)
with A~R=~(P,2+w,2±a,2), W,2
=lgw,211
,
P~
a
lgp,2r
a
,
.
bb
a,~=lga,2r
L,2=Ut~U and R,2=Uc9~,Ut. Integrating by parts and throwing away a total divergence, we find iN =
~2
iN EtLuaIJW,L
Tr[LpL aLp]
+ ~T
x Tr[pjR~ — L~)+ a~(R~ + L~)+ p~a~ ~(p~ —
—
aa)Ut(pp
—
a~)U].
(2.45)
Let us now consider some typical anomalous meson decays. First, notice that the counterterm I~exactly 3”If~,F”= I from the anomalous sector, but this is not a strong interaction process. *
It breaks the prediction eF
U. -G. Mcifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
230
cancels the second term in the VV4-coupling constant (2.35) and hence we recover the current-algebra theorem for i’i°—* 2’y. Therefore, we will now use the effective action (2.43) to calculate other processes and obtain some predictions which indeed agree with the data. The most well-known abnormal parity hadronic process is the decay o—* 3’rr. Actually, it was motivated by the success of the Gell-Mann— Sharp—Wagner model [2.32] of pole dominance, i.e. w—* p’ir—* 3i’t, to which later on a possible contact term was included [2.33], which contributes very little to the width F(w—* 3’ir). In our framework, both these terms arise naturally, with their (relative) coupling strengths completely determined by the gauge coupling g. For the special choice of ~ to have the minimal form of the V~4vertex, we have 2
a/S
~VV4
=
=
gvv4r,20~pTr[a V 8 V 4}, (ig/2) Tr[V~
=
—
J~
=
22
3g /l6it F,,,
~L~]
~hE~pap Tr[V~ 8’çf 2i~2F~, t g
gvv4
a~a~4],
(2.46)
2F~]~ 3 [g2F~]2 43 L~’~i 32 L m~] [g
The first two vertices are needed for calculating the pole diagrams w—* ‘itp—* ‘it’rt’tt, the third one determines the direct (contact) o—*3’it interaction. For w,2(p)—~’rt~(q1) + ‘rt°(q°) + ‘tt(q) we can calculate the amplitude as M=ir~~Pgogg;F F=3h+
3~3 F,,
81T
The experimental width T(u—* ~t~’ii°i’t)
192~ where E~and E gives
2
(2.47)
q)2]F2,
(2.48)
a2
(p
— q
)
+
m~
follows to be
~ JdE~fdE
are the energies of the
F(u—*’ir~’it°’it’)=7.6MeV
~ a+.0,-
[(q)2(q~)2—(q~
~
and
‘it,
respectively. Evaluating the integral in (2.48)
(2.49)
for m,, = 140 MeV, F,, = 132 MeV, m~= 768 MeV, ma = 782 MeV, and g = 8.6. Experimentally, we have F(~—*3’tr) = (8.9 ±0.3) MeV, i.e. the prediction (2.49) is 17% too low. Notice that roughly 10% of the width F(w—* 3’rt) comes from the contact term ~ and that the contact term decreases the width. For comparison, the current algebra result for ‘ir°—*2-y (7.1 keV) is about 11% lower than the experimental value (7.9 keV). Furthermore, the left—right symmetric form of the WZ action would lead to F(w—* 3ii) = 2.6 MeV, due to the too small VV~-couplingconstant. Again, the importance of the Bardeen counterterm becomes clear. Let us now study a radiative process, the simplest one being the decay of a vector meson into a pseudoscalar and a photon, like e.g. w—~‘rt°-y.In our model, this process goes via the op’rt vertex, i.e. o—* p°’rr°and the p°couples to the photon via (2.33). Straightforward
U.-G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
231
Table 2.1 Radiative vector-meson decays from the Bardeen-subtracted Wess—Zumino action. Symmetry-breaking corrections FKI F_ F,/F,, = 1.2 are incorporated, the w—4 mixing is parametrized by = 0.076 as determined from the 4—* pis decay. The experimental data are taken from ref. [2.34]. Width (keV) Process
Prediction 80
Experiment 63.0±4
p°—*is°’y
80 29 117 800 38 77 5
63.0±4 51.0±5 75.0±35 789.0±92 725.0±14 93.1±25 3.2±2.6
7 68 1 5
8.4±27 67.7±9
K**~_~K*’t
4—011-y
—
6.5 ±1.9
evaluation of the pertinent Feynman diagram gives ~
641T
3=0.8OMeV F,, qj
(2.50)
where q,, = 380 MeV is the pion momentum in the w-rest-frame, in good agreement with the experimental value F(w—s~it°’~y) = (0.86 ±0.05) MeV. Other allowed SU(3) radiative decays like ~ — 41—s. ‘q-y,. . can also be calculated from the action (2.43), the results in comparison to the empirical data are shown in table 2.1. * The overall agreement is rather satisfactory. To discuss the hadronic and radiative decays of axial-vector mesons, we have to come back to the problem of the A 1-parameters as discussed following eq. (2.18). To satisfactorily describe p and A1 parameters, we have to add other non-minimal terms to the non-anomalous part of the Lagrangian. This will be done in the following subsection. .
2.2. Inclusion of non-minimal terms for the irpA 1-system As we have seen, the non-anomalous Lagrangian (2.6) could only describe the p-width properly at the expense of the A1 —s. pit decay. Therefore, it is necessary to add at least one more non-minimal term to cure this problem. Before doing so, let us remind you about the rather unsettled question of the mass and width of the A1-meson. Fifteen years ago, we had m~= 1070 MeV, F~= (30 ± 130) MeV and = (128 ±5) MeV, so the mass ratio mA1/mP \/~as predicted by Weinberg [2.24] was nicely fulfilled and the Lagrangian (2.6) with the free parameter ~ could explain these data. However, the 1986 edition of the Particle Data Tables [2.34] gives 1A
mA=(l275±28)MeV, *
For further details, see ref.
[2.61.
1
=(316±45)MeV,
[=(153±2)MeV,
U.-G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
232
i.e. mA/me = 1.66 V~, and the Lagrangian (2.6) can no longer accommodate the width of the p and the A1 at the same time. Recent data from ‘i-decays report [2.35, 2.36] m~= (1056 ±35) MeV,
TA
=
(476i~i~) MeV,
1A
mA
=(1097±i3)MeV,
1
=(378±40)MeV.
This confirms the large width, but brings the mass again down to the Weinberg ratio. To accommodate for the large A1-width, we follow the work of ref. [2.6] and add another non-minimal term to the Lagrangian ~ (2.6) t]. y Tr[F~UF~U The new term is clearly gauge invariant, y is a free parameter. The quadratic part of =
(2.51)
.~ +
=
~(i
—
—
y) Tr[(ô,2V~
— 8017,2)2]
—
~(i
+
‘y) Tr[(ô,2A,,
—
8
~(2)
reads
2] 0A,2)
~Tr[8~4 ~
+
—
gF,, 8/J~A,2+ m~A~ + ~g2F2,,A~ +...],
(2.52)
where 17,2 = A L,2 + A R,2 and A = A L,2 — A R,2 are the vector- and axial-vector fields. The Lagrangian (2.52) is diagonalized via the definitions V~= (1—
~=2~
)‘~2v
A,2 = (1 + )‘~2A +
F F,,=-~,
2m
0 2F21’’~2 [ g Z=[i_-~----~j
(2.53)
where the tilde denotes physical quantities. The vector and axial-vector meson masses follow to be: m~=-j-~-~--~~
(2.54)
For convenience, we will introduce a renormalized coupling constant ~ via —
g=g/(1—y) so that gV,2 =1
=
—
1/2
(2.55)
,
gV,2. With that, we can rewrite Z2 in terms of the physical quantities as
4m~
(2.56)
= (1)[m]2
i+y
m 5,
2 = 1/2 [2.251, while the second Notice be thatthetheWeinberg first equality corresponds KSFR forgeneral, Z would formula [2.24] for toZ2the = 1/2 andrelation y = 0. In these two relations do not imply each other. (For a somewhat different approach to this problem insisting on the KSFR relation
U. -G. Meijiner, Low-energy hadron physics from effective chiral Lagranglans with vector mesons
233
and the Weinberg formula, see ref. [2.37].) From (2.51) we can read off the V4141 interaction as: =
6=1
~ Tr[V~,4 8,241] 2
—
Z
2[ —
+
~ Tr[(8,2V0 2m~
—
80V,2)o,2~8~4l], (2.57)
Z4
1 1_Z2]~~
where a non-minimal piece proportional to 6 Tr[V,2
041,2~0] appears. The piti’i width derived from (2.57)
follows to be 3~2 F(p—s.it’it)j-~-— m~ [i_~] 1 Iqj
2
(2.58)
with q,, the pion momentum in the p rest-frame. The other vertex of interest is the AV41-vertex, it is given by ~AV~
1/2 g,, 11y1 Tr{V,2[A,2,~]} 4Z2 Li+~i
i~2fr,,
+
+
—
4m~,Z2(1
6
—
ii—,,] ~Lir~] Tr{(ô,2V 0 1/2
2 i~+yi” 4m~ Li] Tr{(ô,2A,,
~
2
4-
2
7 F,, [1~2j
—
—
80V’,2)[A,2’ 8,4~]}
ÔPA,2)[V,2,
~
(2.59)
11/2
Tr{(8~A,~ —
,2)(8,2VP —
Notice that (2.57) and (2.59) will be modified if one adds SU(3) symmetry-breaking pieces. For our present discussion, however, we simply assume that this breaking is adequately described by the physical masses of the various mesons. For further applications, let us write down the the transition amplitude A—.*V + 41, e.g. for the decay A~,2(p)—s.p~(k) + ‘rr°(q) T,2V(p, q) = V~[T~(p,q) 6,2~+ TD(p, q) ~ 11
1
~ pq—
L 1—yJ21
T 5(p, q)= —T ~ [m~+(1—6)kq]— 2 (2 (1 [—4y T~(p,q)=7—T----~(i—6) Li~2]
-‘1/2
4~2
11/2
1
F-,, p~k,
(2.60)
1 ~
where we have exhibited only the 5- and D-wave amplitudes which contribute to the A1 —s. pit decay width as follows F(A1—s.pit)2
q,,I2 ITmA
12T2+1(EP ~
(2.61)
234
U.-G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
We are now able to simultaneously fit the mass of the A1 as well as the p and A1 decay widths, since we have three parameters g, ~and y at our disposal. * For mA = 1275 MeV and TA = 315 MeV, one finds two possible solutions (i) y = 0.25, ~ = 0.34 and g = 10.1; and (ii) y = —0.26, ~ = 0.065, g = 7.0. But since the mass and the width of the A1 are not conclusively determined, we prefer to study the parameter dependence of m~,1 and TA. It is most convenient to insist on a minimal momentum dependence of the p’trit coupling by setting 8 = 0, i.e. then (~t~t~p) = g~lT,,p,2 (4 X 19,20) has the standard form, and 1 depends only on g. This is shown in fig. 2.1 by the dashed line, the cross denotes the experimental width F(p—s. 2it) = (153 ±3) MeV, i.e. g = ~ = x 6.08 = 8.6. The g-dependence of T~is rather moderate. In fig. 2.1, we also show the width F(A1 —s. pit) as a function of g and the A1 mass (we use F(A1 —s. 3’it) as an upper bound on T(A1 —s. pit)). For m~= 1275 MeV, we find = 315 MeV for g 7.6 and g 9.6, close to the values obtained above. Therefore, setting 8 = 0 does not change the essential features of the model (for a somewhat different opinion see ref. [2.37]). To decide on which value of g we should take, we calculate the D-wave and S-wave contribution to the A1 —s. pit decay width. For g = 9.6, we find D/S 0.23 whereas for g = 7.6, D/S 0, in agreement with the strong D-wave suppression indicated in T decay experiments [2.38—2.40].As a check on the parameters and on the vector meson dominance assumption, [2.33], we can calculate the A1 —~‘it -y decay from A~—s. ‘it + p°via F(Ait)2~
T(Ait~p°)~730keV
~
(2.62)
for TA = 300 MeV and g = 7.6. This is consistent with the recent measurement [2.71] T(A~—s. ‘tt~-y)=
PARAMETERS I
OF THE A1
I
I
I
MESON
—
I
I
400
I
-
1230
1275
‘~‘3OO
-
1150
-~ ~2OO
...—
--+--
IO5O
-
100
-
I
6
-
— - — - -
I
7
-
I
I
8
9
I
9 ‘/~‘
10
II
l’pirir
Fig. 2.1. The width of the decays A, —* pit and p—o isis for different values of the A,-mass (covering the experimental uncertainties) as a function of the pins-coupling constant g = v’~f0_,,.The cross denotes the empirical value of F(p—* 2ir) = (153 ±3) MeV. *
Clearly, one wants to get g close to its empirical value so that the predictions of the anomalous decays discussed before are only slightly
changed.
U.-G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
235
(640 ±246) keV. One can now also calculate the anomalous decays of axial-vector mesons, with the AA41 Lagrangian given by ~AA~
=
15~2F,,(i+Y)
,2va/3
Tr[ô,2A
8aAp41].
0
(2.63)
This allows e.g. to predict T(D—s. p’rri’i) -~3 MeV, somewhat lower than the experimental bound T(D—s.pinrr) < T(D—s.4i’i) (14.4 ±3.9) MeV [2.34]. We will not elaborate on this point but rather refer the reader to ref. [2.6]. What is important is the fact that one again has to use Bardeen’s form of the anomaly, in which case decays like D—s. pp, D—s. ww, A1 —s. pw, . are suppressed, i.e. the equivalent radiative decays D—s.p°’y,D—s.w-y, A~—s.p°-y vanish. Finally, let us point out that the lower value of g = 7.6 favored by axial-vector meson decays as compared to g = 8.6 fitted from the p—~2-it decay lowers the V—s.41-y amplitudes by 10% and the one for w—s.3it by approximately 20%. Altogether, with the choice of the gauge coupling g close to its non-anomalous value g = ~ and the two non-minimal terms proportional to ‘y and ~, one can describe a huge amount of mesonic decays — anomalous as well as non-anomalous decays of vector and axial-vector mesons. * Golterman and Han Dass [2.45, 2.46] have constructed a similar effective Lagrangian starting from the QCD generating functional. They use a slightly different extra non-minimal term to account for the p- and A1-properties simultaneously. Since in the present framework Vector Meson Dominance was put in through (2.32), we will now demonstrate how VMD can be enforced on the level of the effective action by adding suitable counterterms. .
.
2.3. A subtracted Lagrangian with Vector Meson Dominance Following the work of Brihaye, Pak and Rossi [2.9—2.11]we will develop a systematic functional subtraction formalism to implement Vector Meson Dominance (VMD) unambiguously. The idea is to add local counterterms in a way so that the low-energy predictions involving pseudoscalar and electromagnetic processes only are not affected by the vector meson contributions without any constraints on the new parameters. This is nothing but implementing Zumino’s ideas [2.47] of VMD on a gauged chiral Lagrangian. For that, let us start with the non-linear if-model (2.2) and gauge it electromagnetically — this can be done without any ambiguity since the photon is indeed a gauge boson. We will restrict ourselves to the two-flavor case, the generalization to SU(N) can be found in ref. [2.10]. The electromagnetically gauged non-linear if-model reads /~1 /
./_~
~‘
0,em
A
-~
N — -~
[ ~
,2
\
]_~iEL_~_i I
~
~
2
,2
JE,2*a/3
.~‘
2
r~
(~_~)E,20if/3B~0Ba Tr[r3(L~ + Ru)],
v
a
/3
(2.64)
where B,2 is the photon field, B,2,, = a,2B,, — d,,B,2 the Abelian field strength tensor, and D,2 U = 8,2 U — ieB,2[Q, U] the pertinent covariant derivative. The charge matrix Q is given by Q = ~( ~ + r3). The Lagrangian (2.64) incorporates all low-energy ‘try interactions in accordance with current algebra. * For a different approach relating anomalous decays to Lagrangian, see refs. [2.42—2.44].
I’~~(U)
via VMD-assumptions without including spin-i particle in the effective
236
U. -G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
For later purposes, we will recast (2.64) into the following form: 2+ ~
~0,em = ~
[5~’t
(D,2~)
~](c
~
x D,2~)2+
~
N 0e —
~
[2~—sin2~] ~
L
]~(D~ox ~
E~0,~B”
(2.65)
.
using the weak-field expansion 3abçob.
U=i+irir/f,,=i+ir~, and D,2q~=o(pa+eBa We will now introduce p-, w- and A 2)R ®U(1)~gauge particles as we did 1-mesons SU(2)L ®SU( already in section 2.1. The only difference to theas Lagrangian (7.6) for the non-anomalous sector (with = 0) and (2.44) for the anomalous sector is an enlarged definition of the physical A 1-field, i.e. we use A,2 = A~,hYS+ (a/f,,g)D,2 ITPhYS
(2.66)
with a a parameter to be fixed later. Previously, we have used A,2 = + (a /f,,g) 8,2 ~ (where the tilde denotes the physical quantities), this alternative diagonalization procedure is suggested by the appearance of terms like D,2 ‘t~— gfA,2 in the non-anomalous part of the effective Lagrangian and will prove useful in the computation of meson processes, e.g. -rtp-scattering reduces to a sum of three diagrams by using (2.66) instead of five by using the standard procedure. The extra two diagrams are due to the exchange of A1-particles. Of course, the scattering amplitude turns out to be the same. With the definition, = 17,2 ±
V,2=—~i(T•V,2+w,2),
(2.67)
A,2=—~iT•A,2,
D,2U=8,2U+gA~U-gUA, the diagonalized effective action T = F(i~,2,V,2, A,2,
1
2
2
W,2)
reads
2
1 1 (D,21r)2 + (f,,g) T=~j 2a ~ (V,2+w,2) +
—
2(i)
JA~
E,2Oap J
(~)
ô,2w
2)it 0[
—
J[~xD,2~-
71- (1—
3a
+
(1~2a)~(AaxD~~)—
~
/~
~(Da1T
~xA,2] x D~i~)
~a (8a1~p— 8pVa +gV~XV~~gA~
XA~)]
(2.68)
U. -G. Meifiner, Low-energy hadron physics from effective chiral Lagranglans with vector mesons
237
in terms of the physical fields. We now have to implement VMD on the naive effective action (2.68), which gives a consistency criterion for the gauge procedure employed. Let us now assume that at low-energy VMD holds exactly [2.11i.e. every process described by the iry-Lagrangian (2.64) should be reproduced with the same amplitude by the corresponding processes involving vector mesons. As before, we assume that all electromagnetic couplings can be obtained from vector-meson couplings through the insertion of the extra term (2.33) in the Lagrangian. In the two-flavor case and using the parameter a, this term reads ~VMD
=
+ C(B~), —f2,,(eg/a)(V~+ ~w,2)B,2
(2.69)
which allows for an unambiguous identification of V,2 since it coupled to the external electromagnetic source. In what follows, we will set V,2 = and first discuss the vector case (A,2 = 0). As will become clear, we will have to add in a systematic way local counterterms to remove the spurious vertices introduced by the vector meson propagators. For that, consider an effective action I~(ço,B,2) that reproduces all low-energy theorems. The aim is now to find an action T(ço, 17,2) such that VMD in the functional integral formalism holds, i.e. e”~’B~) =
f
[dV,2]exp[iT(~, 17,2)
+
~im2(V,2— B,2)2],
(2.70)
where we have removed all internal indices and coupling constants. Expanding around 17,2 = B,2 we find the following defining equation for T(p, 17,2): * ~
[m~g~ 0+ ~2r]-1
~
(2.71)
which can be easily solved. Expanding the solution in powers of m-2 we find T(~,V,2)=T0(~,V,2)+~ [~(~,V,2)]+C(m4). 2m aB,2
(2.72)
The 2g,2second term is nothing but a subtraction of the contact current—current term generated by the m 0-part of the vector meson propagator. To see how this subtraction procedure works in the non-anomalous sector, let us discuss the ‘wit-scattering amplitude. From current algebra, it follows to be = (—116f ~,)(1T X 9 ir)2. From the action (2.68), we get 2. (2.73) 12a (irx8~ir) To recover the current algebra result, we have to add a counterterm as suggested by (2.72), i.e. = (—a/3f~,)(1TX 19 ~.)2 In a similar fashion, one can now systematically determine by comparison all subtracted effective vertices for the itp-theory in order to reproduce the low-energy theorems. In other words, we are looking for an action T(~c,P,2~w) such that the electromagnetic vertices, generated by T(~C,P,2~(LI) + TvMD, reproduce those already contained in the electromagnetic Lagrangian (2.64). *
We assume that the action is at most quadratic in the external vector sources.
238
U.-G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
For the non-anomalous sector, the procedure is straightforward. Concerning the anomalous sector, it is important to realize that to lowest order in momentum only one anomalous vertex can contribute, i.e. no w-propagation is allowed. With that, one can generalize (2.71) by expanding the anomalous action around P,2 = B,2 and ((5,2 = (1 /3)B,2. With that, one finally arrives at [2.91 F = T’ + ~
.
Tna(~,P,2~w)=~J(o~)2
f,, f . 2j p [i—asin T*n(~,P,2~‘~e*)
~7
2
‘P]
2
2
sin 2 ‘P ‘P
1
+
2
f,,g 1I 2aj
2
2
+gp~X Pp) g~.(8P0— 8~P~
E,2vapJo,2wv{
sin2’P 3
‘P
.
1 ~C.(D~cc)
2
(2.74)
2’P(1—asln ‘P)
Notice that in (2.74) the transverse and longitudinal parts of the vector fields are separated. The combination F” + F” + FVMD is a unique low-energy Lagrangian of pions, p- and t~o-mesonswhich reproduces all low-energy theorems incorporated in Trn” (2.64). It can therefore be used as an extension of the effective theories discussed in sections 2.1 and 2.2, after one reinstates the axial-vector meson. Since there is no external source to which the 1-meson can couple, the construction of the effective Lagrangian is not as unambiguous as before. One way to resolve this problem is to introduce a source to which the A,2 couples and the demand that the interaction between the axial sources J~,indeed gives the anomalous part of the Lagrangian (2.68) before field diagonalization. For the non-anomalous sector, one simply wants to recover the original “naive” chirally symmetric Lagrangian with the SU(2)L 0 SU(2)R 0 U(1)~ softly broken by chiral symmetric degenerate mass term. This leads to F na =
f~ 1 [p~(D,2~~gA,2)]
i 2~1— a~ .1
+
~
2
+
2~Isin2’P I 2
f
‘~
L
‘P
[~X(D,2~—gA,2)]2 2 2 .
1—a sin ‘P
.
—
a(sln
‘P)Ic°2
egJ(p°~+ ~w,2)B,2+gJA,2J~,
(2.75)
which after the diagonalization procedure (2.66) leads to an effective mass term for the pions and the axial-vector mesons =
—am~2+ 2a(1— a) A~.
(2.76)
If one assumes that the pion mass is entirely generated through this mechanism, one needs a = 1/2 and recovers the KSFR relation [2.25]. The anomalous part of the action follows to be
U. -G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
2
F””(ç, P,2~W,2,A,2)(NcIi6it2)E,2~~/3Jo,2W0[g~(28~p/3 +gp~x p4)— ~
+ sin~ (C~x C~)— sin2’P ~ .(~2 x C~)+ 2 sin2 ‘P 2’P ‘P ‘P 1
C,2
— —
2 —
C,2
—
239
x D~~)/’P ~.
(C~x
(2.77)
(1—a(sin2’P)/’P2)D,2~—agA,2sin2Q 1—a sin2 ‘P — a(sin2 ‘P)/’P2 (1 —
a sin2 (p)gA,2 — (a(sin2 1 — a sin2 ‘P — a(sin2
‘P)1~’P2) D,2~
‘P)/’P2
Now (F~fl+ F11”)~~~ 0 gives a complete itpA 1w-Lagrangian at low energies (E ~ mA1) after performing the3’7F~ field =diagona1i~ation.It produces all purely pionic and theorems, e.g.it f~,,where F3~denotes the -y—-s. 3ir amplitude andelectromagnetic F~the one for low-energy ir°—s.2~y.*Furthermore, eF gives the non-Abelian anomaly in Bardeen’s form. Finally, notice that in the limit a —*0, the action (2.77) goes smoothly to the anomalous part of the action (2.68) (before diagonalization) since in this limit the vector mesons are identified with the external sources. Furthermore, all the counterterms needed to eliminate the spurious local vertices generated by vector-meson propagation are built in by construction. Notice that this procedure does not impose any constraints on the fundamental parameters of the model. 2.4. A Lagrangian of pseudoscalar and vector mesons As we have seen before, the inclusion of the axial-vector mesons like e.g. the A 1 poses several problems due to uncertainties with A1 parameters. The A1 also plays no important role in the NN-interaction [2.14]; we will therefore now modify the model presented in section 2.1 to eliminate the A1 by a non-linear realization of the vector particle (the p), following the ideas of Weinberg [2.49]and the work of Kaymakcalan and Schechter [2.7]. The starting point is the following non-anomalous Lagrangian: ~01~2’
~= -~
t], ~F~Tr[D,2UD~U = ~ Tr[F~ 0F~’+ F0~~PR]+ 7 Tf[FLUFtLORUt] —
=
m~Tr[A~A~L+ ~R~~R] + B Tr[A~UA~RUt],
using the standard definitions U=exp{2i41!F,,}, *
F,,=\/~f.,,.,
This relation is not fulfilled by the Lagrangian constructed in section 2.2 as first pointed out by Rudaz [2.48].
(2.78)
240
U.-G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
D,2U= ~U—igA~U+igUA~, F,2,, = 8,2A,,
80A,2 —ig[A,2, A,,] (for Land R). 3)L 0 U(3)L gauge transformations, the local invariance is broken Clearly, ..~ + wayisby invariant U( in a minimal ~ Theunder generalized mass in ~2 with the constant B will turn out to be of great importance. We want to eliminate the A 1, i.e. find a (primed) gauge in which —
~
A,2
=
0,
(A~)’= (A~)’= P~
(2.79)
.
For that, let us consider the U(3)®U(3) gauge transformations U—s. ULUU;
ULR E
U(3)LR,
A~—s.ULA~U~ + (i/g)U~8,2U~,
(2.80)
A~—s.URA~U + (i/g)UR 8,2U, 1/2
—1/2
with UL = U112~U”2, and UR = U , i.e. the ~lOflS as gauge parameters: U= U = U112p,2U112 + (i/g)U~2ô,2U~2, =
(2.81a) (2.81b)
U~2p,2U112+ (i/g)U112 8,2U112.
(2.81c)
We have U’ = ~l,which means that the pseudoscalar fields will only appear in the non-gauge invariant terms. Equations (2.81b, c) can now be solved and give the following chiral-invariant constraint: A~= UA~Ut+ (ilg)U e9,2Ut.
(2.82)
We can now eliminate the axial field by injecting (2.81) into the Lagrangian (2.78). From (2.82) it is clear that D,2 U vanishes, i.e. one gets no contributions from any term involving D,2 U. Using the fact that the p-meson transforms under the diagonal subgroup U(3)v of U(3)L 0 U(3)R [2.49] we find that ~ simply becomes a Yang—Mills type term for the p-field Sf~=(y_1)Tr[F,2,,(p)F~*(p)],
F,2 0(p)=ô,2p,,—80p,2 —ig[p,,,p0].
With y = 3 / 4, we obtain the conventional normalization. The =
~
(2.83)
term reads
112~U~2 + 8,2 U”2 U112)] (B + 2m~)Tr[p,2p~] + (ilg)(B + 2m~)Tr[p,2 . (19,2 U + 2(m~Ig2)Tr[c9,2 U112 19~U~2] (B/g2) Tr[U112 8,2 U”2 . U”2 2] —
.
(2.84)
19~Ull
now contains all terms of the pseudoscalars and the vector meson mass term, with the vector meson
241
U. -G. Meijiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
mass given by m~,=4m~+2B.
(2.85)
The p4)4) interaction from (2.84) follows to be 1-
2
Tr[p~4) t9~~4)1, ~
(2.86) 3(g~~)2I12~rm~, where the coupling constant ~ is related to the p-meson width via F(p—~21T)= q i.e. ~ = 8.6. Notice that differs from the original gauge coupling g, the special choice 2g = ~ gives the KSFR relation g~F~= 2m~,[2.25].It is important that we have taken the term proportional to B in ~2’ otherwise we would have ended up with the result = m~IF~5.85, i.e. F(p) 70 MeV, an old problem noticed in particular in ref. [2.3]. By adding an ordinary pion mass term to account for PCAC, one can recover the low-energy i’rrr-scattering results. For S-wave scattering in the I = 0 and I = 2 channels we indeed find the well-known result a 0 = 7m~!161TF~, a2 = —m~!8irF~ [2.50,2.511. Let us now turn to the anomalous sector. In principle, one can start from the Bardeen subtracted Wess—Zumino action (2.43), which breaks chiral U(3)L®U(3)R down to U(2)v®U(1)A. Notice that one can think of the Bardeen subtracted WZ term as a remnant of a chirally invariant action involving another set of pseudoscalars 4)’ which are not excited at low energies, i.e. =
~
2
F~~(U, AL, AR)
—
=
m~/gF~,
E~~(U’, AL, AR)
(2.87)
has the same low-energy predictions F~ = F~~(U, AL, AR) F~~(U = AL, AR). Let us first see how the Bardeen form of the WZ action with the axial fields eliminated via (2.81b, c) works. For that, consider again the process w—+3’rr. Using a weak field expansion for U, we have —
~,
AA~—A~=(2IgF~)a~4)+..., (2.88)
~ so that the relevant pieces of the anomalous Lagrangian read =
~GSW + ~contact
=
~
Tr[ô~w”s9~Xp~4)] + ihE~~~ Tr[w~od~4)d’~4)].
(2.89)
The GSW-term coefficient follows to be 2g~F~ ~vv~ = 3mI4ir and the coefficient of the contact term is given by* h
=
(2.90)
—m~Ii2g~~F~.
* Notice that the coefficient h is related to the dimensionless parameter $ introduced by Adkins and Nappi [2.52]via $ whereas in ref. [2.52]it is fitted to the nucleon and i’-mass, yielding $ = 15.6.
(2.91) =
—3 ir2F~h/V~= 8
242
U.-G. Meif3ner, Low-energy hadron physics from effective chiral Lagrangians with vector me~ons
The computation of the w—~3ir amplitude proceeds as before (cf. eq. (2.47)), with the coefficients given by (2.90) and (2.91). Here, the contact term adds to the GSW term (in contrast to the situation before) and is about four times larger in magnitude. With that, we find that F(w—~3’rr)comes out 16% above the experimental value, whereas it was 15% below pexP(~~ 3’rr) for the model with explicit A1-mesons. Proceeding along this direction means that all anomalous couplings are predicted in terms of g~~• If one is willing to give up this ability to predict the anomalous couplings and insists on chiral invariance without introducing a new set of pseudoscalars 4)’ via (2.87), one can go back to the WZ term (2.31) and eliminate the axial mesons via (2.81). Doing this, we end up with three parity-and-chiral-invariant terms for the anomalous action, i.e. Fan(U,
~)
=
T~~(U) + c, JTr[iALL~1 + c2f Tr[dALLAL —ALL dAL+ ALLALLI
+ c3
J
Tr[—2iA~L + gA~LALL],
(2.92)
2p~ h/2 + (i/g) U”2 ~ U-”2 Obviously, where AL hasthree to becoefficients written in terms the p-field,c A~= U” we have now to be of determined. 2 is related to gvv4, and c1 and c3 might be adjusted to give the contact term h. The problem of fixing these coefficients is recently under investigation [2.531. The action (2.92) together with (2.83) and (2.84) might be used to construct a fairly simple effective Lagrangian of pseudoscalar and vector mesons. In light of the uncertainties related to the A,parameters, this seems to be a useful exercise. Furthermore, the precise relation to the hidden symmetry approach discussed in the next section should be worked out, especially concerning the anomalous sector (for the non-anomalous sector, cf. section 4.3).
3. The hidden symmetry approach Any non-linear sigma model based on the manifold GIH is gauge equivalent to another model with GgIObaI®HIOCaI symmetry [3.11. The gauge fields of the hidden local symmetry are generated by quantum effects and poles of the gauge fields are developed [3.2]. Starting from this observation, Bando et al. [3.3] have proposed that the p-meson is a dynamical gauge boson of a hidden local symmetry in the non-linear chiral Lagrangian 2)L®SU(2)R describing the islow-energy effective two-flavor dynamics. The broken spontaneously to theQCD diagonal subgroup global chiral symmetry G = SU( H = SU(2)~.In the first subsection, we will explore the consequences of this idea and derive low-energy theorems. Furthermore, the relation to the conventional Skyrme model is pointed out. The next two subsections are devoted to extensions of Htocai so as to introduce the o-meson and determine its coupling from the Wess—Zumino action. The inclusion of the w in the two-flavor sector can be achieved by enlarging Hiocai to SU(2) 0 U(1), as explored by Meil3ner et al. [3.4]. Bando, Kugo and Yamawaki [3.5] proposed the more general extension to U(3), therefore also including the 4)-meson. We will derive and discuss the pertinent low-energy theorems according to these enlarged hidden symmetries. Following Fujiwara et al. [3.6], we will examine the Wess—Zumino action in the presence of vector
U. -G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
243
2-y, w—~3’rr, ~rr—~ 3y). Finally, in section 3.4, mesons and calculate various anomalous processes (‘~i°—~ we discuss the inclusion of axial-vector mesons and external gauge bosons (y, W~,Z) in the hidden symmetry approach following the work of Bando et al. [3.5, 3.6, 3.7]. 3.1. The p-meson as a dynamical gauge boson Our starting point is the non-linear cr-model SU(2)L®SU(2)R/SU(2)v with its Lagrangian given by =
defined
on
the
coset
(f~i4)Tr(~9~U o~U~).
space
G1H =
(3.1)
U is the standard representation of the Nambu—Goldstone pions U(x) = exp{iir(x)if~},with IT = ITTa and ~a are the usual SU(2) generators. f~ is the pion decay constant. Obviously, the Lagrangian (3.1) is invariant under global chiral symmetry U(x)—~g~ U(x) g~
g~~E[SU(2)~R]g,oba,.
,
(3.2)
The hidden SU(2)~symmetry of (3.1) becomes apparent if one rewrites the SU(2)-valued field U(x) in terms of SU(2)-valued new variables ~LR(x) such that [3.3]
U(x) = ~~(x)~ 4~(x)
(3.3)
Connected to the SU(2)~-symmetryis a gauge field V~,
v~=~v:T”i2.
(3.4)
The new fields ~LR(x) and V,~(x)transform under ~L,R(x)~/~@)
~
[SU(2)L®SU(2)R]global®[SU(2)v]Iocat
as follows (3.5a)
,
V~(x)—~. (jig) h(x) ô.~h~(x) + h(x) V,A(x)h~(x) ,
(3.5b)
where h(x) parametrizes [SU(2)~]iocai. Insisting on the G®H symmetry and parity, one can construct two linear independent terms of second order in derivatives of = —(f~i4) Tr[(ô,~ igV~j~~ ~ + (~ igV~)~. —
—
(3.6) =
-(f~l4)Tr[9~L~~
-
~R
.
~+]2
(3.7)
using the covariant derivative 8.,~ igV~.g is a gauge coupling to be specified later. It is now easy to show that any combination of (3.6) and (3.7) —
(3.8)
244
U. -G. Me,f3ner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
where a is an arbitrary parameter, is equivalent to the non-linear cr-model (3.1). For that, we work in the unitary gauge =
~R
=
exp{+iITi2fTj
(3.9)
to eliminate the unphysical scalar degrees of freedom. By introducing independent left- and righthanded fields ~LR we have actually three pions plus three unphysical scalar modes (see section 3.4). We find ~
(3.10) (3.11)
using ~ + ~
~+
~.
~
—
~.
So
~A
reduces to the original Lagrangian
whereas ~
~,
vanishes
identically when one uses the equation of motion for V,~ gV
=
_iTr{~ ~
+
ô~
.
~)}~—
~
(~X
(3.12)
~
using the weak field expansion of (3.9). Since vanishes identically, it can be multiplied by an arbitrary parameter a as done before. So far, the vector field V~ (x) is a purely auxiliary field without any dynamics. The second line in (3.12) is already suggestive of V~being the p-meson. We will therefore assume that V,~has its own dynamics and that a kinetic term for it is generated by the QCD-dynamics or quantum effects at the composite level [3.1, 3.8]. For a detailed discussion on this point we refer the reader to the forthcoming review by Bando, Kugo and Yamawaki [3.9]. Adding by hand a kinetic term, we have ~,
~=
~A +
~
—
(3.13)
~
F~is the corresponding non-Abelian field strength tensor of the vector meson (3.14)
~
The physics of the Lagrangian (3.13) becomes transparent when one makes the weak-field expansion ~(x) = 1 + (ii2f~)i-.17(x) ~ . Then the Lagrangian (3.3) becomes* Tr[~U d~U~] —
(~
+ p~+ ~ g2f~p~
g)p~ (~X .
~)
+ U(174).
(3.15)
Here, we have identified the vector particle V°~ with the p-meson. From (3.15), we can read of the p-meson mass and the pinT coupling constant as 2 22 m~=agf~,,
*
Notice that the Lagrangian (3.15) is precisely the irp-Lagrangian already given by Weinberg [3.10]solely based on chiral symmetry arguments.
U. -G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
~
~ag.
=
245
(3.17)
Equation (3.16) has the typical form of a Higgs mechanism. Indeed, the unphysical scalar modes in ~L,R= exp(i4)/2fTT) exp(±iiri2f,,)are absorbed into the p-mesons. As already mentioned, the parameter a is not specified by these symmetry considerations. However, for a = 2, we find the KSFR relation [3.11] m~=2g2f~
(3.18)
and universality (3.19) The pinir coupling constant is determined by the p—> mm decay width. We have g2
m2 (3.20)
42I~*2~Tjqj3
For m~= 770 MeV, q,,, = 358 MeV (~ q,~ is the momentum of the pions in the decay rest frame) and = 153 MeV this gives ~ = 6.11. With f,~ = 93 MeV, we see that the KSFR relation is fulfilled within 5%. Equation (3.19) only gives universality if the p-coupling to matter fields ~‘~A is saturated by the minimal term ~ ~ with g~AA = g. This agrees with the dominance of p-exchange in ‘~~“Ascattering as discussed in section 1. At this stage, we can make contact to the Skyrme model [3.13].In the limit m~—~ ~ with g held fixed, the vector particle decouples. This leads to the constraint (pt, = p Ta12) —
gp~, (1!2i)(ô~L~~ —
+
ô~L~:R
~j~)=0
(3.21)
and reduces the field strength tensor ~ F~~(PF~(~(~L~
-
4g
[(~L
~L
+~R.~R))
R~R)’(LL~R~R)],
(3.22)
so that the kinetic term for the vector boson becomes the Skyrme term:
=
+
~
Tr[~UU~,~~UU~]2
(3.23)
with e = g the Skyrme parameter [3.12]. This result has been derived by Igarashi et al. [3.13], Abud et
246
U. -G. Meif3ner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
al. [3.14] and Balog and Vecsernyes [3.5].* Furthermore, it agrees with the original result of Iketani [3.16], e = g, because in the presence of the A, meson the pirir vertex receives a strong momentum dependence [cf. sections 2.2 and 3.4]. The main virtue of the hidden symmetry approach is the unique way of introducing electromagnetic interactions as we will demonstrate now. The electromagnetic field coupled to the charge operators 2)L®SU(2)R]gtoba, isospins and hypercharge, = J(L+R) + Yi2, with 1~L. R) are andcompletely Y being independent the [SU( respectively. These generators of the hidden SU(2)~to which the p-meson couples. Let us for the moment neglect the weak interactions and concentrate on the electromagnetic fields (the generalization of coupling to the electroweak standard model will be discussed later on). Coupling the fields ~LR to the electromagnetic U(l)Q gauge field B,~means to redefine the covariant derivative as (Y = 0):
=
~
v:)~LR+ieO~LRB~
—ig ~
(3.24)
~,
with e 0 being the coupling constant of U(l)Q. The Lagrangian (3.15) reads 5~=~f~Tr[9~U 9’~U~] — e0m
1
e
~
+ a
2
(~) (~g)v~
V~B~ + ~ + (1 — aI2)e0V,~ (~ X gP
-
—
.
(m
m~B~ +
(3.25)
X a~17)
ti9~L1T)~ +
with Ba,, = 9~B~ — ~ For a = 2, this Lagrangian coincides with Sakurai’s [3.17] y—p mixing and leads to vector meson dominance. To see that, let us diagonalize the mass matrix of V~and B~.We find 2 + e~)f~, m~± = ag2f~, (3.26) m~= 0, m~o= a(g where the mass eigenstates are +
e 0V~.
A~=
2
2
~
gV~— e0B~ 2
‘
e0
\/g
(3.27)
2
+
e0
The electromagnetic charge is given by ge0 e=
328 e,~.
~
The p-dominance model of the pion form factor emerges naturally for a
*
=
2 since (cf. section 1):
g~,~=e(1—ai2)=0,
(3.29)
~
(3.30~)
=
m~Ig,
Actually, Balog and Vecsernyes find e = 2s~-= g + ~i, where ~lstems entirely from the non-anomalous part of the Wess—Zumino action not
considered here.
U.-G. Me,Jjner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
where the -yp-vertex is defined via two a-independent relations [3.18] ~ 1 m~ 2f~g —
‘
=
247
—eg~~p~A~. Together with (3.18) and (3.19), one can derive
~ 1 m~ g
(3.31)
—
which combined lead to the a-independent generalized KSFR relation (3.32)
~
Experimentally, ~ follows from the width F(p°—~e~e) =6.62keV to be ~ =0.12GeV2 (if one assumes a minimal photon coupling) and 2g~~f~ = 0.11 GeV2 (for ~ = 6.11 and f,~ = 93MeV). Notice that the relation (3.32) follows solely from symmetry considerations and can be considered as a “low-energy theorem” of the hidden symmetry approach [3.5]. In summary we can conclude that if the p-meson is indeed a dynamical gauge boson of the hidden [SU(2)~]tocaisymmetry of the non-linear cr-model (3.1), the following phenomenology emerges naturally for a = 2: (i) the KSFR relation m~= 2g2f~ (ii) universality of the p-coupling to pions and other hadrons g = ~ = g~HH; (iii) p-dominance of (isovector) photon couplings to pions as seen in the pion charge form factors; and (iv) p-exchange dominance in pion—matter scattering. 3.2. The w-meson as a dynamical gauge boson From the low-energy nuclear phenomenology discussed in section 3.1, the relevance of a strongly coupled isoscalar vector meson is well-established in addition to the fact that the w-meson joins the p-meson as a member of the vector meson octet. It is therefore necessary to incorporate the w-meson and its dynamics in addition to the p-meson. This is achieved by either extending the hidden symmetry to [SU(2)v®U(1)]tocai [3.4] or to [U(3)~], 0~~,[3.18]. In the latter case, [U(3)LOU(3)R]global breaking effects have to be taken into account. For the discussion of the two-flavor sector, it is most convenient to introduce the w-meson as a [U(1)]iocatgauge particle thus enlarging the hidden symmetry to SU(2)~®U(1) as discussed by Meil3ner, Kaiser, Wirzba and Weise [3.4]. Starting again with the Lagrangian (3.8), with h(x) E SU(2)~®U(1),the gauge covariant derivative now reads =
~ —i(gi2)(rp~+
Whl)ns
~
(3.33)
—iV~ ,
where we have chosen equal gauge coupling constants for SU(2) and U(1). Adding again a kinetic term for the gauge bosons V,~ 2)Tr[F~~F~”], (3.34) —(1/2g with ~
=
—
ô,,V~—i[V~,Vj
we find a generalized KSFR relation using again the weak-field expansion ~ = 1 m~= m~,= 2g2f~
(3.35)
,
+
(ii2fTT)’r 17, (3.36)
248
U. -G. Meij3ner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
for a = 2. The almost mass degeneracy of the p and the w (m~= 769 MeV, m~= 783 MeV) justifies the choice of equal gauge couplings for SU(2)~and U(1). From the covariant derivative ~ (3.33) it is obvious that the u and p cannot couple in the non-anomalous sector. The w-couplings are indeed dictated by the Wess—Zumino term to be discussed in section 3.3. Bando, Kugo and Yamawaki [3.5] have proposed an alternative scheme to incorporate the by exploring the full vector meson nonet (p, w, 4), K*, K*). This amounts to working in the nonet space GIH= [U(3)LOU(3)R]iU(3)v. ~LR(x) are now U(3) matrix-valued fields, and in the unitary gauge, we have a nonet of pseudoscalar mesons (m, K, K*, ~ .~f)
U(x)
=
~(x)
~(x)
=
exp{iIr~~t*If}
(3.37)
,
where ta (a = 1,. . 9) are the U(3)-generators and f,~ = fk = f. The non-Abelian gauge field ~ (x) = ~tai2 contains the vector mesons (p, to, K*, K*, 4)). The transformation properties of these fields under [U(3)L ®U(3)R]gIobal 0 [U(3)xj],oca, are . ,
~L.R(x) e’~R
~L,R(x)~e
(3.38)
,
~~(h) ~i,(x) = ~h(x) + ig[h(x), ~~x)],
(3.39)
h(x) is the group parameter of the hidden local U(3)~symmetry. The pertinent covariant derivatives in the presence of global U(3)LOU(3)R gauge fields are =
(a~— ig1~)~L~+ i~LRA~.
(3.40)
If we are only interested in coupling the electromagnetic field B~,we have AL,.
AR,~= e0B,.~Q
1
where gauge
Q
(341)
,
diag(+ ~). The Lagrangian is given by (again, a kinetic term for the hidden local field has been added by hand), =
~,
— ~-,
—
2 ~[F~VR + F~L]. a~v Tr[F~] For the external gauge field being the electromagnetic field B,.~, we find after gauge-fixing exp{iiri2f} and expanding to second order in pseudoscalar fields:
(3.42)
—
~A +
~=
~
~
~
=
=
bag,
(3.43)
ag2f~,
(3.44)
=
=
—
=
~(a 2)e —
0.
(3.45)
Constructing mass eigenstates as before (cf. (3.26) to (3.28)), we find
g~1~=2agf~Tr(~t”Q).
(3.46)
U. -G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
249
This leads to the generalized a-independent relations [3.5] (cf. eq. (3.31))
m~ m~
m,~,
(3.47)
g
which are experimentally confirmed. Using the leptonic decay width F( p e + e ) = 6.9 keV, F(w—s.e~e)=0.66keVand F(4)—*e~e)=1.31keVand the fact that F(V~+e~e)ctcm~g~,, we find that (3.43) holds within 10%. As it stands, this extension of the hidden symmetry is not satisfactory since it predicts m~= m0) = m4. One therefore has to add [U(3)L®U(3)R]gtoba, breaking terms which do not affect the hidden symmetry and therefore leave (3.47) intact. This can be achieved in the following way —~
~
(3.48)
~
(3.49)
with ~ given by eq. (3.37) and the breaking matrices CAy are taken to be of U(2)L 0 U(2)R invariant form [3.5]: CAN
=
diag(0, 0, cAy)
(3.50)
with CAy being constants. Obviously, the additional terms field via IT=V1+EAITV1+
~
lead to a renormalization of the pion
(3.51)
CA.
Performing a weak-field expansion ~ = 1 2
2f~= 1 m +Kcv m~= m~= ag
=
—
(ii2f)t~”,
one obtains the following relations
2 (1+m, c~)2’ 1~
f~=f~IV1+ CA
(3.52) (3.53)
and the relation (3.47) remains valid. Equation (3.48) can be cast into the following form m mK~
m m~
m m(,~
(3.54)
which holds within 2% (and justifies the ansatz (3.48), (3.49) which is by no means unique. For a general discussion on SU(3)-breaking terms, see the review by Gasiorowicz and Geffen [3.19] and references therein.) If one assumes CA = cs,, one gets [2.5] fK’f~= mKImP
—
1.15,
(3.55)
in good agreement with experimental data (fK = 1.22fT~= 113 MeV [3.20]). Again, to-couplings to pions are not given by (3.42) but through the Wess—Zumino term to be discussed in what follows.
250
U.-G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
3.3. The non-Abelian anomaly in the framework of the hidden symmetry approach The non-Abelian anomaly gives important restrictions on the amplitudes of processes which are of abnormal intrinsic parity* and include vector mesons. Particularly, the intrinsic odd parity processes to—~3m, w—~iT°-yand the anomalous low-energy theorems for in°—~2-y, -y—~ 3m are entirely dictated by 2-y and -y—~3m, and one the Wess—Zumino term. Vector meson dominance relates the to-decays to m°—+ can therefore test whether complete vector meson dominance holds. Complete vector dominance means that there are no contact terms like -y’rr’rr or -yiririT. Let us now briefly discuss the Wess—Zumino anomaly equation and the Wess—Zumino effective action [3.6]. For the time being, we work on the manifold G1H = U(3)L®U(3)RiU(3)v and will reduce it to the to-coupling later on. U(x) as defined in (3.37) transforms under chiral U(3)LOU(3)R as U(x)—~e’~’-U(x)~ Gauging these symmetries amounts to introducing left- and right-handed external gauge fields A,.1LR = ALR(tai2) which transform as =
+i[EL R’
~6LR
ALR~].
(3.56)
As pointed out first by Wess and Zumino [3.21], the effective action F must satisfy the same anomalous Ward identities as does the underlying fundamental theory, QCD. The Wess—Zumino (WZ) anomaly equation reads ~F[U, AL, AR] = G[AL, AR] =
NCJ
-
Tr[EL{(dAL)2
-
~
dA~}
-
(dAR)2
-
~ dA~}]
(3.57)
with N~the number of colors, M 4 ordinary Minkowski space—time, and we use the following differential one-forms ALR
ALR,.L df 1 dx~=(dU)U’ R = U1 dU = U’LU. L =(o,.1U)U As we already discussed in section 2.1, the Wess—Zumino action ~ =
(3.58)
,
FWZ[U, AL, AR]
=
C
f M5
Tr[L5]
+
SCi
f
Tr[ALL3
+
is given by [3.22, 3.23, 3.24]
ARR3I
M4
—scf Tr[(dALAL+ALdAL)L+(dARAR+ARdAR)R] + SCJ Tr[dAL dUARU’
*
—
dAR dU’ALU]
The intrinsic parity of a particle is defined to be even if its parity equals (_)S~*.
U.-G. MeiJiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
+ 5CJ Tr[ARU~’ALUR2
-
ALUARU1L2I
+
~cf
+
5CiJ Tr[(dARAR
+
5CiJ Tr[ALUARU’ALL + ARU’ALUARR1
J
+ 5C
[(ALL)2
—
251
(ARR)2] + 5CiJ Tr[A~L + A~R]
+
AR dAR)U’ALU
Tr[A~U’ALU
-
—
(dALAL + AL dAL)UARU~’}
A~UARU1+ ~(UARU~’AL)21
M4
=
f
[c
Tr[L5]]
(359)
gauged
M5
with C = —iN~/24O1T2and M5 is a five-dimensional manifold whose boundary is Minkowski space M4. In the presence of vector mesons, the WZ anomaly equation reads ~F(~L, ~R’ ~ AL, AR)
=
—lOCi
J
Tr[eL{(dAL)2
—
~idA~}
—
(L~R)]
(3.60)
so it has the same form as (3.57), but the gauge transformation ~ also contains hidden local symmetry transformations such that + ~R(ER)
= ~L(EL)
~
+
8(h).
(3.61)
The general solutions to eq. (3.60) are given by special solution of the inhomogeneous equation plus general solutions of the homogeneous equation ~F = 0, i.e. anomaly-free terms. Obviously, we can choose the WZ action (3.59) as4 aand special solution. The anomaly-free termsblocks. can beForwritten as are made of gauge-covariant building that, we four-dimensional integrals over M define R
D~LR ~L,R
F~= di~ igi~, —
d~LR ~L,R FLR
=
~L,R
—
igi~+ i~LRALR~LR,
FLR~~L,R.
(3.62)
There are six independent forms that conserve parity but violate intrinsic parity: ~‘
3R-R3L],
~
1=Tr[L
2—I~2)],
~t
3=iTr[F,,(L 2- 1’RLI, ~ =iTr[FLR
~6
4=iTr[F,,(LI~—I~L)], =iTr[FLLR- FRRL].
(3.63)
252
U.-G. MeiJ3ner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
Further possible terms can be reduced to these six ones using identities like DDL = i[L, F~]and so on. Therefore, the part of the effective action responsible for intrinsic parity violation processes is given by
f
F=Fwz[~~R,AL,AR]+~ c~.,
(3.64)
4
M
where c 1 are arbitrary constants to be determined later, which are essentially fixed by experimental data. Let us first check the consistency of (3.64) with the low-energy theorems given ir° 2-y and -y-—~3’rr. These processes are determined solely by the anomaly, and do not depend on other parameters (like the c’s) of the theory. For that, let us examine [3.61the term ~ in the case of electromagnetism M=eB (3.65) AL=AR=eB,~Qdx and using the expansion d~LR~ ~L,R = +(iIf,T) dir The relevant terms in ~6 are —~
±~“.
=
ief~i’{2gTr[i~dB diT]
+
2g Tr[dB
i~ dir]
—
4e Tr[B dB d7T]} + 4ef1 Tr[B(dir)3] + (3.66)
Obviously, the third term Tr[B dB dir] directly contributes to ir°—~2-y and the fourth term to y—~ir ir ‘rr. They are, however, cancelled by the first two terms involving ‘rr -y ~ vertices. It is a straightforward exercise to show that one can substitute the vector mesons i~3~by [3.6] —
—
g~—~eB+—~—[irô ir]. 2f~
(3.67)
‘~
This replacement is only exact under the following conditions: (i) the square of the four-momentum of the i~3i~ propagator is zero; (ii) the -y and the ir are on the zero-mass shell; and (iii) that -y has physical polarization. With (3.67), we can replace the first two terms of ~6 by Tr{i~,dB}~dir—p
Tr[B dB dir] +
Tr{[ir, dir], dB} dir.
(3.68)
Obviously, the terms in (3.68) exactly cancel the third and fourth term in (3.67), i.e. ~6 does neither contribute to ‘rr0—~2-y nor to -y--~3’rr. In a similar fashion, one can easily show that none of the ~ contribute to these processes. Now we can address the question of vector meson dominance (VMD). Although the Lagrangians ~ do not contribute to the anomalous processes like ‘rr° 2-y, -y--* 3’rr,... they indeed determine the rate of direct to vector meson mediated diagrams as shown in fig. 3.1. There is a particular choice of the parameters c 1 for which exact VMD is realized for the process in question. For that, we introduce the following abbreviations 21f,,) Tr[A dA dir + dA A dir], (yiririr) (4eIf~)Tr[A(dir)3], (yyir) (—2ie (~~yir)+(—2ieg/f,,) Tr[i~dA dir ±dA ~ dir], (i~irnir) (4gIf~)Tr[i~(dir)3], (3.69) —~
(t~ir)
(—2ig2If,,) Tr[i~ di~dir + di~ti~dir].
,
U. -G. Meijiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
253
There terms show up in the action (3.64) as follows: FFwz+~cJ~ 4
M
3(yyir)
=
+
4(yiririr) + c 6(yyir) + (—c3
—
c4(i~J1~iT)+(—c1 + c2
—
c5)(i~yir)~ + (c4
+ c6)(y7riTlT)+(c1
—
—
(3.70)
c6)(i~yir)+
c2 + c4)(~iririr)
where we have scaled out the factors (5C)’ for convenience. Exact vector meson dominance for the processes follows by choosing c1,. c6 such that . .
1= ~
—
15Cf
,
3)]~
(~ + ~) + 5CJ
(a~1+ /3~2)
(3.71)
5C[3(~ir)—2(~ir
where the parameters a and /3 can be determined by higher order processes like pp—p rr’ir For a /3 = 1, we have exact vector dominance, e.g. ir°—~p° + w°—~-y + y. But there is still a contact term —(lYir3), with its strength entirely determined by the anomaly. The amplitude of the decay (O—~3ir, F~3’~, can be evaluated from the Feynman diagrams in fig. 3.2 to give —
— —
ig ~ 22
4irf~
cci L.i a+,O,-
f
1 (p—q
—
)
a2
2J
—m~
3g 23’ 8irf,~
where g ~ is the i9~gauge coupling constant and the last term comes from the contact co—~3ir vertex. The first part of (3.72) obviously describes the well-known Gell-Mann—Sharp—Wagner process (GSW) [3.25], which we already discussed in section 2.1. This gives for the width E(w—~3ir) = 6.1 GeV, approximately 30% below the experimental value of 8.6 MeV. This result is not in contradiction to the one presented in section 2.1, because there the contact term was approximately a factor 5 smaller in
VECTOR MESON DOMINANCE :
7T~.-2y, y.~31T
iT---
W..-3iT
~7T
~q•)
iT
V
V
iT———
_7r(q)
—
_77~
--
~ir(q) ~iTo(q~)
~ iT
(a)
DECAY: KINEMATICS
(b)
Fig. 3.1. Direct (contact term) contributions versus vector-meson mediated contributions to the anomalous processes (a) ir°—* 2y and (b) y—* 3ir. The wavy lines denote photons (~y),the dashed lines pions (IT) and the double lines vector mesons (p or
(a)
+
w(p)~.__7T0(q0)
(b)
Fig. 3.2. Kinematics for the decay w—~3ir with (a) the Gell-Mann— Sharp—Wagner process and (b) the direct contribution.
254
U.-G. Meijiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
magnitude due to the Bardeen subtraction. Fujiwara et a!. [3.6] therefore concluded to give up complete VMD and allow for a direct (yir1r1T)~term.*The coefficients a, /3 in (3.71) are modified such that there is no more contact term for the u—~3’rr decay. (This means c1 = 7.5C, c2 = —7.5C, = —15 C. This Lagrangian will be referred to as the “PVMD Lagrangian.” The complete VMD (CVMD) Lagrangian (3.71) is realized for c1 = —c2 5.5C and c4 = —15 C.) This is essentially the old GSW model, which leads to the following results
F(ir°—~2-y)=7.6eV,
[(7.9±0.1)eV],
F(~—+‘rr°-y)= 0.84 MeV F(o—~3ir) = 9.1 MeV,
[(0.86±0.05) MeV]
(3.73)
,
[(8.9 ±0.3) MeV],
where the numbers in brackets give the experimental values. Notice that F(’rr°—+ 2-y) is identical to the current algebra result [3.22] but it stems entirely from the anomaly. Finally, we want to discuss some simplifications of the WZ action F~ to be used in soliton calculations. Meifiner et al. [3.4] have proposed to use the following Lagrangian of pions, p and u mesons:
~minimaI
=
Tr[~U ~U~]
Tr[(~~)~+
—
(~~+)~]2
—
Tr[F~F~]+
~
(3.74)
based on the hidden SU(2)~®U(1) symmetry discussed in section 3.2. This is called the “minimal model” because the homogeneous part of the WZ term is chosen in a specific way, namely 3I~ =
1OCJ Tr[(L
The special choice of
-
~3L) (L1~LJ~)].
(3.75)
-
c
1 = —c2 = 2, c3 = c4 c5 = c6 = 0, ensures vector meson dominance in the isoscalar channel. As can be read off from (3.70), the action (3.75) has only (i~innr) terms, i.e., all (yiTinT) terms are cancelled and therefore photons do not couple to pions (to leading orders). Furthermore, expressing the p-meson field as p~= _iTr[ra(ô~~~ + ~ ~)]/2g, the action (3.75) exactly reduces to a Lagrangian of the form .
(3.76) which represents w-meson couplings to the conserved baryon current B~’,given by 2)E~Tr{U~ô~U U~U U~8~U}. (3.77) B~=(1/24ir The o-coupling is exactly the one used by Adkins and Nappi [3.28]in their model of the w-stabilized skyrmion. Furthermore, we find g,~= (N~I2)gin agreement with the results of Kaymakcalan et a!. [3.23] and MeiBner and Zahed [3.29].The restriction to the two terms in (3.75) with the special choice of their coefficients is vital if one wants to discuss VMD based on meson currents coupled to conserved currents. It constitutes the simplest way of coupling the o to the ‘rrp Lagrangian (3.15) consistent with *
Notice that VMD any way breaks down at the two-photon level as pointed out by Brihaye et al. 13.251.
U. -G. Meif3ner, Low.energy hadron physics from effective chiral Lagrangians with vector mesons
255
the WZ conditions. Obviously, the decay width w—~3ir is largely underestimated by this ansatz. Therefore, Meil3ner, Kaiser and Weise [3.30] have also constructed a more realistic WZ action which violates exact VMD, but gives the proper w—~3ir decay width (see section 4.3). The pertinent Lagrangian derived from this action is given by ~~plete
=
(~
4w~v
g)w~~+
e
Tr{ip~(U~ 8~U+ ~UU~) + ~ PaU~P~U}. (3.78)
This ansatz will be referred to as the “complete model”. Notice that (3.78) violates the current algebra prediction of -y--~3ir, so higher order terms should be added for consistency. As we will see later on, these higher order terms will be of no big relevance for the properties of the soliton solutions emerging in this model. 3.4. Extensions of the hidden symmetry (axial-vector mesons) The inclusion of axial-vector mesons in the hidden symmetry scheme can be done either by enlarging the local gauge symmetry to [SU(2)L®SU(2)R]gjobal[3.31]or [U(3)L®U(3)R] local [3.5, 3.7]. We follow here the second approach as outlined by Bando, Kugo and Yamawaki [3.5].For details, we refer the reader to the extensive review in ref. [3.7]. The hidden local symmetry can be formally extended to [U(3)L® U(3)R]local by introducing new variables ~M such that U(x) = ~(x)~M(x) ~R(x). ~L’ ~R’ ~M transform under [U(3)L® (3)R]gIobal ® [U(3)L® U(3)R]local as: ~L,R(x)~1zL,R(x)
~L,R(x) gL,R,
with hLR(x) E [U(3)LR]locaI. ~l~L(x)
=
~M(x)~h1L(x)
~M(x)
h~(x)
(3.79)
The pertinent covariant derivatives read
(9~ igV~(x)+ ig A~(x))~L(x), —
~i~R(x)— (~ —igV~(x)—igA~(x)) ~R(x), =
ô~M(x) ig(V,jx) —
—
(3.80)
A~(x))~M(x)+ ig ~M(x)(V~(x)+ A~(x)).
A,
3)L]lOcal and V~(x) formthe leftand right-handed nonet gauge A~(x)are axial vector mesons including the A fields invariant under [U( 1. Insisting on parity, one can construct four invariant terms out of ~L,R,Mand ~L,R,Mof second order in derivatives:
2(x) and [U(3)R]local.
~?=
—a(f~J4)Tr[~~L. 5~ —
~
b(f~I4)Tr[~~L.
+ —
4M ~L~R
~M ~
~±~+]2
~RM_~.~M~]2,
—
c(f~I4)Tr[~.~M .
(3.81)
a, b, c, d being arbitrary parameters. In the unitary gauge ~L = = = exp{iir/2fIT} and ~M 1, one recovers the usual non-linear Lagrangian of pseudoscalar, vector and axial-vector mesons. Further2ir due to the A,~ô~ir-vertex.Adding more, onekinetic has to terms redefine pion field ir’ = [bcl(b by hand andthegauging withasexternal fields+ c) ~ + d]” (-y, W~,Z°),one finds that for a = 2 all
256
U.-G. Meif3ner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
vertices (Vext’Wff) vanish, i.e. g~,~ = 0. That means that p-dominance of the pion form factor is preserved. To satisfy Weinberg’s mass relation m~= 2m~and Weinberg’s sum rule g~1= g~[3.32], one has to choose a=b=c=2,
d=0.
(3.82)
Recent experiments support these relations [3.33, 3.34]. m~ 1056 ±35 MeV~\/~m~ and g~Ig~ 1.1 from T-decays. Unfortunately, the redefinition of the pseudoscalar and axial-vector fields to get rid of the mixing terms proportional to (A 9 /Lir) induce a strong momentum dependence of gPIT~T~Defining r=b/(b+c), we find 2), g~~(0,0,0)=(aI2)g, (3.83) ~ r which modifies the successful “low-energy theorem” (3.32) —
~
=
(1
—
r2)I2f~
(3.84)
and the KSFR relation m~= 2f~g~~(1r2)2.
(3.85)
—
Furthermore, the decay width [‘(A 1 p’rr) comes out to be roughly half of the experimental value. Even more disastrous, for any choice of the parameters the A1-yii-vertex vanishes [3.7], in sharp contrast to experiment: [‘(A1 -yir) = (640 ±246) keV [3.35]. Bando et al. therefore proposed to add higher order terms, restricting to the lowest number of derivatives. For that, use the differential one-forms (cf. eq. (3.62)) —~
—~
L, i~—~ —i1, —iI~
(1/i) D~LR~
and construct the following six invariants
]1~ —i D~M~
R’
(3.86)
(1.: L~dx~and so on) =
2)Tr[I.:,J.:~L~ + I~I~~R~] ~t a1(—iIg + a 2) Tr[~MR~ ~ + ~Lv~LV~MR~] 2(—i/g + a 2) Tr[L MRVM’~ + R MLV~MR] + h.c. 3(i/2g + a 2) Tr[M~M~L~ + ~MM~M~4MRI 4(—i14g +a 2 ) Tr[L~A~L~R~ M~MR~I + h.c. 5(i14g +a 2 ) Tr[~k~ ~IcI~L~ ~L ~MRfl + h.c. (3.87) 6(—i14g The contributions to the various vertices pir’rr, A 1 p’rr,... can be worked out for all these terms. If one cancels the strong momentum dependence of (3.83) and insists on p-dominance for the A1 yir decay, one finds that independently of the choice of a, (i = 1~. . , 6) satisfying these conditions, the —
—
g~lT~~
—*
.
U.-G. Meiflner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
257
A1 pir and A1-yir amplitudes are uniquely given; their widths turn out to be F(A1—~.pir)~360MeV, F(A1—~-yir)=*300keV,
(3.88)
which is consistent with data (see the discussion on the A1 properties in section 2.2). The most obvious choice to fix the parameters a1,. , a6 further is to demand that the new terms do not induce momentum dependence of the p’rrir vertex in the absence of the A1-meson. This can be achieved by setting a1 = a2 = a3 = 0, —a4 = a5 = a6 = 1. So we end up with the following Lagrangian: . .
~[=~
~{(a = b = c = 2, d
0)— ~(a1
=
a2
=
a3
= 0,
—a4 = a5
=
a6 =
1)
(3.89)
which reproduces all the phenomenology of pA1-system. Let us end this section by summarizing the phenomenology arising from the Lagrangian (3.89) [3.7]: vector meson dominance of the pion electromagnetic form factor, universality of the p-meson 2f~, coupling, the KSFR relation m~ = 2g Weinberg’s mass relation mA 1= — — — —
Weinberg s sum rule g~1= F(A1—~pir)~360MeV and F(A1—~yir)=300keV. Notice that if one wants to treat the A1-meson as a dynamical gauge boson, simultaneous generation of both the kinetic term and the higher derivative terms ~t is necessary to be consistent with the effective low-energy theory without A1 the strong momentum dependence of the p-coupling comes from the A1 kinetic term and is exactly cancelled by the higher derivative terms. One again is tempted to conclude that low-energy hadron physics can be more effectively described in terms of pions, p- and ü-mesons alone. — —
—
4. Equivalence proofs In this section, we will demonstrate that the models based on gauging rigid chiral symmetry (cf. section 2) are similar in nature with the models based on concepts of the hidden local gauge symmetry as discussed in section 3. This gauge equivalence can be most easily demonstrated by making use of the so-called Stückelberg transformation [4.1], as outlined in section 4.1. Then, we will generalize the proof incorporating external gauge bosons in the hidden symmetry scheme as emphasized by Yamawaki [4.2]. These proofs are fairly general, we will therefore explicitly make contact between the massive Yang—Mills ansatz without A1 and the irp-Lagrangian based on the [SU(2)~]iocaihidden symmetry as outlined in section 3.1. In historical perspective, these proofs relate the ideas of Schwinger [4.3], Sakurai [4.4], Weinberg [4.5] and many others of how to implement spin-i particles in effective Lagrangians. 4.1. The Stuckelberg construction In the massive Yang—Mills approach, the vector mesons are introduced as chiral gauge this multiplets by 2)R 0 U(i)v]gIobal (or [U(3)L0 U(3)R]globaI). To relate approach explicitly gauging [SU(2)L® SU(
258
U. -G. MeiJ3ner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
to the hidden symmetry scheme where the spin-i mesons emerge as gauge bosons of a hidden local symmetry, we will make use of the Stückelberg construction [4.1, 4.6] following the work of MeiBner and Zahed [4.7]. First, let us outline the main idea of the proof. For that, consider the following massive Yang—Mills Lagrangian =
Tr[F~
~
+
F~]
—
[A~
~
+
A~],
(4.1)
with ALR~being left- and right-handed gauge fields and F~LRtheir respective field strength tensors. In the unitary gauge (valid for both L and R) gL,~(A~~R + ~)g~
=
=
gL,R
~
(4.2)
we can rewrite the Lagrangian (4.1) as follows ~
Tr[F~
+
F~]
—
~
Tr[F~
+
F~]
—
+2 (g~V~~g~)2] Tr[(g~V~~g~)
Tr[(g~V~~g~)2 + (g~V~~g~)2].
(4.3)
In other words, the massive Yang—Mills model (4.1) is gauge equivalent to a massless Yang—Mills model gauge coupled to non-linear if-models [4.8]. The Goldstone modes in (4.3) compensate for the longitudinal modes in (4.1). This is, of course, the well-known Higgs mechanism. Notice that if the unitary gauge is identified with the pion field, (4.3) suggests a relationship between f,,, m and g of the form 2
22
m =gf
22
agf,,
(4.4)
where a is a parameter to be determined by proper diagonalization of the pion field. For a = 2, this is just the KSFR relation. So, this sometimes mysterious relationship finds a natural explanation in the Stuckelberg construction. Now, let us consider the massive Yang—Mills model (2.6) without the w (for convenience), i.e. Tr[~U with ~
U] + ~
Tr[FL + F~]
—
~
Tr[A~ + A~]
(4.5)
the covariant derivative in the adjoint representation,
U—~U+A~UUA~.
(4.5a)
In the unitary gauge,
u
=
~
U~R,
ALR~=
~L.R VL,R~~L,R
(4.6)
U. -G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
259
we use the Stückelberg construction to recast (4.5) in the following form Tr[~U ~U]
Tr[F~~+ FL]
+ ~
Tr[(~ VL~L) +
-
(~VRRfl.
(4.7)
Notice that the Lagrangian (4.7) enjoys both global and local gauge invariance, GO H = [SU(2)L0 SU(2)R]global 0 [SU(2)L0
SU(2)R]local,
(4.8)
as defined by the following transformations: ~L,R(x)
=
hLR(x) ~L.R(x) g~.
(4.9)
Since we have two gauge groups, we can choose two gauge conditions. If we choose to work in gauge where U—fl,
(4.10)
~L~R~~’’
and use the KSFR relation, then we have Tr[(A~
-
AR~)2]+ ~
Tr[F~
+
FL]
-fW~VL~L)2+ (~R VR~R)~.
(4.11)
We should stress that (4.11) is a simple rewriting of (4.5), which has no dynamical content aside from assuming the KSFR relation. Let us now compare with the Lagrangian of Bando, Kugo and Yamawaki [4.9]. It reads (setting ~M = 1): 2’ =
+
+
+
2’d
+
=
-a(f~/4)Tr[(VL~L~+
=
—b(f~!4)Tr[(VL~L~VR~R~R)],
=
—c(f~/4)Tr[(AL~
=
-d(f~I4)Tr[(VL~L~
= (i
!4g2) Tr[F~
+
—
(4.12)
AR 2], 1j VR~R~R-
(AL~ -
AR))2],
FL].
It is obvious that (4.12) reduces to (4.11) for the following choices of parameters a=b=2,
c=—2,
d=0
(4.13)
because j~,,and the pion-field ir(x) are rescaled by \/~due to the irA 1~mixing.*Thus, the massive Yang—Mills approach is entirely analogous in nature to the approach based on a hidden local symmetry. *
This is most obvious if one works in more general gauges where u =
~
~
260
U, -G. MeiJ3ner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
Both approaches follow naturally from the Stückelberg construction. The above arguments show that the choice (4.5) for is just a particular case of (4.12). 4.2. Generalized proof in the presence of external gauge bosons In section 4.1 we made no use of the rather unique way of coupling electroweak interactions in the hidden symmetry scheme. Following Yamawaki [4.2, 4.10] we will now present a unified description of pions, and A1-mesons coupled to electroweak interactions and show its relation to the massive Yang—Mills approach. Furthermore, we can give a natural explanation why the arbitrary parameter a indeed should be a = 2 if one reduces to the Hiocai Lagrangian discussed in section 3.1. For that, let us start with theThe generalized model to which is invariant GglobaI 0 ~ where 2)L®SU(2)R. generalization arbitrary groupsunder is straightforward and can we be choose G = SU( found in ref. [4.11]. Gglobal is assumed to be fully gauged with external gauge fields L~(x)= ~
R~(x)= ~
T!2,
(4.14)
TI2,
where ~ is the (external) gauge coupling constant. For example, L~(x)and R~(x)may contain -y, and Z bosons (standard electroweak interactions). The dynamical variables are GgIobaI valued matrix fields ~L(x), ~R(x) and U(x), which are introduced as U(x)
=
~~(x) U(x) ~(x)
(4.15)
with U(x) = exp{i’r. ir/f~}.The transformation properties of U, section 3.4):
~L’ ~R
and U are given by (U
~M’
cf.
U(x)—~U’(x) = gjx) U(x) g~(x), +
~L.R(x)—~ ~L.R(x)
U(x)-~U’(x) with
gL,R
=
hLR(x) ~L,R(x) g~R(x),
=
hL(x)
(4.16)
U(x) hR(x),
E Gglobal and hLR(x)
E
Giocat. Accordingly, the covariant derivatives are defined as
D~L(x) d~L(x) iL~,(x)~L(x)+ i~L(x)L~jx), -
D~R(x)
D~U(x)
YR(X) —iR~(x)~R(x)+ c9,~U(x)
—
iL,~(x)U(x)
+
i~R(x)R~jx),
iU(x) R,jx),
where the gauge bosons of the generalized hidden local symmetry given by L,,~(x)gLr*I2,
(4.17)
R,~(x)=gRTaI2.
Giocai
with coupling constant g are
(4.18)
L,~(x)and R~(x)transform under Ggiobai as L~(x)~L~(x)= ig~(x)~g~(x)
+
g~(x)~(x)
g~(x), (4.19)
R~(x)-+R~(x)= ig~(x)~g(x)
+
g~(x)R~(x)g~(x).
U. -G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
261
It is now easy to write down the most general Lagrangian with the smallest number of derivatives which is invariant under gauged Ggiobai 0 G10~10 parity* 2’
=
a02’~+ bo2’A
+
cQ2’M
+
dQ2’IT
—
-~-~
Tr[L~ + R~]
—
4~2 Tr[L~~ +
R~~]
(4.20)
with =
-(f~I4)Tr[D~L~~
+
U D~R.~U~]
(4.20a)
—(f~/4)Tr[D~L.~ UDR.~U’~’] 2 = —(f~I4) Tr[D~U~U’~] = (f~/4)Tr[D~L~ UD~R ~U” —D,,~U U’~’]2.
2’A=
(4.20b) (4.20c)
—
(4.20d)
—
a 0, b0, c0 and d0 are arbitrary parameters. The kinetic terms of section 3). The gauge field strengths are defined via ~
—i[F~,F~],
~
.
and
are put in by hand (cf.
(4.21)
The idea is now to rewrite the Lagrangian (4.20) in two different bases, the so-called “hidden basis” and “external basis”, respectively. For that, define L~ns~L(X)L~jX)~ x)—iô~L(x)~ ~~(x), (4.22) R~
~R (x) R~ (x)
~~(x)
—
~ ~R(x)~~~(x),
and L~(x)~(x) L~(x)~L(x)+i~(x)
ô~L(x),
(4.23) R~(x) ~(x) R~(x)~R(x)
+
i~(x) ~R(x).
The pertinent covariant derivatives are D~U(x)
~ U(x)
—
iL~(x) U(x)
D,~U(x)
~
U(x)
—
iL~(x)U(x) + iU(x) R~(x),
+
i U(x) R~ (x), (4.24)
D~U(x)nsc~U(x) iL~jx)U(x) + iU(x) R1~(x). —
With that, the Lagrangian (4.20) can be rewritten as 214)Tr[(L —L~)+U(R,, —R~)U~’]2 2’v(f = (f~,I4)Tr[(L~ L~)+ U(R,~ R~)U”]2, —
*
—
Keep in mind that we are interested in strong interaction physics.
(4.25a)
262
U. -G. Meif3ner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
=
(f~I4)Tr[(L~
—
L)
=
(f~I4)Tr[(L~
—
L~) U(R~ R~)U~]2,
=
(f~I4)Tr[D~UD’~U~] = (f~I4)Tr[D~UD’~U~],
(4.25c)
(f~I4)Tr[D~UD~U”]= (f~I4)Tr[D~UD~U~].
(4.25d)
=
—
U(R
—
—
(4.25b)
—
The first basis (with the tilde) is the “hidden basis.” In this case, everything transforms under Giocat. In the “external basis” (without tilde), however, everything transforms under the gauged Ggiobal. Notice that 2’~,in the external basis is nothing but the gauged non-linear if-model discussed in section 2. Now, let us consider the hidden basis in the gauge U(x) = ~, so that Giocai is reduced to its subgroup Hiocai [SU(2)~] 10~51. The transformation properties of ~L(x) and 4R(x) are the same as in eq. (4.16) with the only difference that now h(x) = hjx) = hR(x) E Hiocai. Introducing vector and axial-vector fields V~(x)and A,,jx), respectively,
A~(x)=
V~(x)= ~[R~(x) + L~(x)], and similarly for V~jx)and
A~(x),one can
(4.22): V~jx)—~. V~(x)= ih(x) ~h~(x)
+
-
~[R~(x)
-
L~(x)]
(4.26)
read off their transformation properties from (4.19) and
h(x) V~(x)h~(x), (4.27)
-
A~(x)—~ A~(x)= h(x) A~(x)h ~(x), and similarly for V~jx)and A~(x).In this gauge, one can rewrite part of the Lagrangian (4.20) as + 2 [bb0~0
bo2’A + co2’M
+
d~
=
(b0
+
c0)f~Tr[A~
—
+ d
b0(b0 + c0)~A~]
0]f~ Tr[A~]. 0
0
(4.28)
Because of the ‘rr—A,~mixing, 4Tr[A~]=Tr[D~UD~U~]
(4.29)
and one has to normalize the pion kinetic term via simultaneous rescaling of 1T(x) and f,~,i.e. ~
\/~ir(x)~ir(x),
Z— b0+c0
+
d0.
(4.30)
Then, eq. (4.20) is given by ~ +
Tr[D~UD~U~] ~ —
Tr[V~+
A~] ~ —
Tr[V~ + A~]
(4.31)
U.-G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
with a
a0IZ, b
263
b0/Z and c = c0/Z. The vector and axial-vector field strength tensors are
~
~E{V~,V~}, (4.32)
~
A~E{A,~,A~}.
We have seen in section 3, the phenomenology of the p and A1 mesons dictates the choice of parameters: a = b = c = 2, and d = 0. In terms of the “bare” parameters this reads: a0 = b0 = c0 x, d0 = 0. The simplest choice is x = 1, in which case the Lagrangian (4.20) is given by 2) Tr[I~~+ R~] (1 /4~2) Tr[L~~+ R~]. (4.33) 2’ = + + (1 /4g Let us now consider the Lagrangian (4.33) in the external basis. In this case, Giocai is completely gauge fixed and no longer exists. Using (4.25), we have =
—
—
2’=f~Tr[V~ —V~]2+f~Tr[A~ A~]2+ (f~I4)Tr[D~UD~U~] —
—(iI2g2)Tr[V~ + A~~]—(1I2j2)Tr[V~~ +A~~].
(4.34)
If one identifies the p and A
1 mesons with V~and A~instead of and A~,respectively, the Lagrangian (4.34) is exactly the massive Yang—Mills Lagrangian discussed in section 2 except that it contains external gauge fields V~,A~ (-y, W, Z). The equivalence of both approaches becomes most transparent if one switches off the electroweak interactions as shown in fig. 4.1. Stremmitzer [4.12] has derived a Lagrangian like (4.34) with A, = V,~= A,~= 0. The Lagrangian (4.34) generalizes the massive Yang—Mills approach, with its mass term being gauged-G invariant (Stuckelberg mass term [4.1]). Finally, let us come back to the Lagrangian (4.33), from which the phenomenologically successful
hidden (-7r,1o,
A,
“local
,
“global
Switch off external gauge bosons E Gglobal
1’
_________________________ ‘~YM(iT, p. A,
Zhidden1~,P,I~t1 ,..) ‘‘local
,~
—
m
Gglobol
Fig. 4.1. Gauge equivalence between the hidden symmetry and the massive Yang—Mills approach. The hidden symmetry Lagrangian ~~‘hidd, contains dynamical gauge bosons (vector and axial-vector mesons) as well as external gauge bosons (here the standard electroweak bosons -y, Z°, W~).Switching off the external gauge bosons, ~hdden becomes gauge equivalent to .~yM(P’o, A ).
264
U. -G. Meiflner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
Ggiobal
A
0 Hiocai Lagrangian 2’
+
=
a2’v +
2’gauge
with a
can be derived. For this, assume that the
=2
1-meson mass is much larger than the p-meson mass, i.e., let us freeze out the A1 degrees of freedom. In that case, one can substitute A,,, in (4.33) by its equation of motion*
A,,,
=
[bI(b
+
c)]A,,,
=
,
(4.35)
and one arrives at the irp-Lagrangian originally proposed by Bando et al. [4.13] with a = 2+ (f214)(D U D~U~) -
= 2f~
Tr[V~]
~
Tr(V~ V~
2:
-
4~2TrW~~ + A~V]. (4.36)
A similar mechanism to eliminate the A1 degrees of freedom will be studied in the next section with the inclusion of the WZ term. 4.3. Gauge symmetric constraints to eliminate the
1
Here we want to demonstrate that the massive Yang—Mills Lagrangian derived by Kaymakcalan and Schechter [4.14] (cf. section 2.5) is indeed equivalent to the non-linear Lagrangian of Bando et a!. [4.13]. We will further discuss the WZ term in Bardeen’s form (cf. section 2.2). For that, let us start with the gauged non-linear if-model in terms of pions, p and A1 mesons (4.37)
2=21+22,
22
=
~ Tr[~,,U 2lJ’~U~] ~ Tr[F~,,.~+ F~~] + y Tr[FL,,,,~UF~U~1,
(4.37a)
=
m~Tr[A~+ A~]+ B Tr[A,,,LUA~U~],
(4.37b)
—
with the usual definitions of the covariant derivative U = U igA~U + igUA, with the left- and right-handed generators ~ = ~Ta(pa(±)aa)given in terms of the p and A1 mesons, p~ = p~r*I2 and A~= aT*12 [4.14, 4.15]. The field strength tensors F~R are given by F~’~’~ = ~A~R ÔAL.R ig[A~’~,Ar]. Of particular interest are the third term of 2’~and the second term of 22, these are higher-order chiral invariant terms (non-minimal terms). The coupling constant y and the generalized mass parameters B will be fixed in what follows. Let us now eliminate the A1 mesons from the Lagrangian 1. This can be achieved by imposing the chiral gauge symmetric constraint —
—
U=0.
—
(4.38)
This constraint has the virtue that it preserved all symmetries of the effective Lagrangian. It is equivalent to the Weinberg trick [4.16]to eliminate the A1. The pions are taken as gauge parameters of chiral symmetry, i.e., one chooses g~= VU and g = VU. In that case, we can rewrite A~ and *
This is the same procedure used in section 3.lto eliminate the p-meson from the non-linear if-model (cf. section 3.1).
U. -G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
265
U=exp{ir~ir/f,,}as A~=
VUp~VU~ + (ilg)VU ~
A~= VU~p~\[U + (iIg)VU~~[U,
(4.39)
U=VUIVU. Under global SU(2)L ®SU(2)R (parameterized by ~LR) we find the following transformation properties
VU-~VU’= g~VUK=KVUg~, (4.40)
p,,,,—*p,’,, =Kp,,,K’~—(i/g)9,,,KK’~ with K = K(\~U,g~,g~)an auxiliary matrix as defined by Coleman, Wess and Zumino [4.17]. Under diagonal isospin SU(2)v* one has the conventional transformations
VU-+ VU’ = g~VUg~, (4.41) = gvp~g~ (iIg)~g~g~. This shows that the p-meson is still a gauge field of diagonal isospin, whereas the A -~
—
1-meson is given in terms of ‘rr and p as a~ flebcdpcird Enforcing (4.38) on the Lagrangian (4.37), we observe that the non-linear if-model part of 2’~vanishes and only the gauge kinetic term survives: =
(y
—
1) Tr[F,~,,(p)F~(p)],
(4.42)
where the field strength tensor F,,,,, ( p) contains only the p-mesons: F,,,,, ( p) = ô,,, ô,, p,,, ig[ p,~,p,,]. Under the constraint (4.38), the non-minimal part of 2’~contributes in the same way as the canonical term ~ To achieve conventional normalization, one has to set y = 1/2. In a similar way, the generalized mass term 22 transforms into 2’s, —
=
(B +
~
+
2mg) Tr[p~,,] + (ilg)(B + 2m~)Tr[p’~9,,,VU VU~+ .
Tr[a~VU o~VU~] -
~
Tr[VU~
.
—
o1~VU~ VU)]
~VUVU~o~VU].
(4.43)
Complete equivalence to the hidden symmetry Lagrangian derived by Bando et al. [4.13] =
m~Tr[p~,,] + iag~~~f~ Tr[p”(o.,,,, +
*
(a+ 1)Tr[~.~~
—
SU(2)~may be parametrized by group elements g~,.
~.
~
+
~
4:+
.
(a— 1)Tr[~~
.
~
(4.44)
266
U. -G. MetJ3ner. Low-energy hadron physics from effective chiral Lagrangians with vector mesons
for a = 2 follows by fixing the parameters B, m0 and g as m~=~m~, ~
B=~m~
(4.45)
and identifying 4 with VU = exp{i~i~I2f~}. Obviously, the KSFR relation holds (mp = Let us now turn to the anomalous sector. Imposing the constraint (4.38) on the gauged Wess— Zumino action F,~from (cf. section 2.2, eq. (2.45)), with the (U-field introduced via U(1)~-gauging,we find (without Bardeen-subtraction) ~
p~,w~]= (~g)w~B~ + ~
(4.46)
This is nothing but the general results (eq. (2.45)) with A1 = 0. This can be easily understood. The Lagrangians with A1~equal to or different from zero are related by a chiral gauge transformation. A gauge variation of the WZ term gives the anomaly, and for SU(2) ® SU(2) the anomaly vanishes. Evidently, the procedures (4.38) or (4.39) do not influence the anomalous processes involving pseudoscalar and vector mesons. * So for a given WZ action in terms of p, A1 and o, we have a unique way to eliminate the A1 degrees of freedom.
5. Connection to QCD In this section, we will make contact between the effective Lagrangians considered in sections 2 and 3 and the fundamental underlying theory, QCD. First, we will demonstrate how a microscopic derivation of chiral dynamics from QCD can be carried out in the low-energy and large-Ne domain of QCD, following the work of Karchev and Slavnov [5.1]. The main assumption going into this work is that chiral symmetry is spontaneously broken, with the appearance of massless (pseudoscalar) Goldstone bosons. This enables us to give an approximate calculation of the first coefficient appearing in the chiral expansion, namely the pion decay constant. Furthermore, the anomalous Wess—Zumino action can be derived. In a similar fashion, following the work of Ball [5.2],we will demonstrate that with the same assumptions one can indeed “derive” the hidden symmetry Lagrangian of Bando et al. [5.3]. Finally, we will show how to convert the generalized Nambu—Jona-Lasinio (NJL) model [5.4] into a massive Yang—Mills theory including vector mesons. There, we argue that the Nambu—Jona-Lasinio model is an intermediate model between QCD and purely mesonic Lagrangians. The NJL model can be motivated from lattice QCD in the strong coupling (g—* cis) and l/d-expansion [5.5, 5.6, 5.7], where d is the number of space—time dimensions. In this limit, lattice QCD reduces to a nearest-neighbor, i.e. four-fermion, interaction reminiscent of the NJL model. We will follow the work presented in refs. [5.8, 5.9, 5.10]. For earlier studies on these topics, namely the relation of the generating QCD functional with passive gluons and effective chiral theories (like e.g. the Skyrme model), we refer the reader to the review by Zahed and Brown [5.11]. *
See also the discussion given in section 2.5.
U.-G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
267
5.1. QCD in the large N~limit and microscopic derivation of the pion decay constant As first pointed out by ‘t Hooft [5.12]and Witten [5.13] QCD reduces to a theory of weakly interacting mesons for the gauge group SU(N~),with N~—~and g2N~* fixed. For that, let us rewrite the QCD-Lagrangian with explicit color-factors —(N~/4g2)F~F~”~ +i~fr(/—54’)~,
~=
(5.1)
with A,~= A T~,when Ta are the generators of SU(N~).The Lagrangian (5.1) exhibits a U(Nf) X U(N~)chiral symmetry. ~fr denotes the quark fields in the fundamental representation. If ta denote the generators of the U(Nf) group, one can introduce composite operators (“mesons”) and consider the generating functional
f
Z[i~] = Z~ [dA][d~][d~] exp{iJ [~(x) + ~ 75t~0(x)]},
(5.2)
where gauge-fixing and ghost-term have been suppressed. Defining chiral quark field via tp~p)~, ~ch= ~(~P+P)
(5.3)
~ch=(~
with ~RL = ~(1 ±‘y5) and L~(x)= exp{2i~”(x)t’~}, we can use the Faddeev—Popov trick to impose a color-singlet subsidiary condition on the quark fields this is necessary for constructing an effective theory of colorless objects. With that and the change of variables (5.3) the generating functional Z[~] takes the form —
f
Z = Z~’ [dA][d~][d~][dQ]~Jl~, x expf
if [~(x)
+
~)
i~(x)y~L~PR ~(x)
+
(~ch)-l 5ta(~ch)-l
(5.4)
a()]}
7
with L,,
(1~’o.,~11 and we have used the exponential representation of the ô-function 1 = (5.3), it contains the information about the Wess—Zumino term, i.e. the anomalies, as we will demonstrate later on. Let us focus for a moment on the source term in (5.4). To leading order in i~= 2~~ta, and using 12(x) = 1 + ~r(x),we see that it behaves like (— i /2) ~!i { IT, t’~} + ~ so that terms of higher order in ~a are 1 / N~suppressed. Therefore, we integrate out all color-fields in (5.4) and obtain ~
i/i)
=
J ~(~ry5t~i)[d~2].**~I is the Jacobian of the transformation
Z[i7]
f
[dfi] expfi
J
~eff(IT,
~~)}
(5.5)
with an effective Lagrangian ~eff’ which depends only on the meson fields. In the low-energy limit and to leading order in ~ we can calculate ~ff(Ir, ~). Before doing so, let us briefly return to the * **
Here, g denotes the running coupling constant of QCD (not to be confused with the gauge coupling g = ~
which was used before).
For the reader who is not familiar with these manipulations, we refer to the book by Faddeev and Slavnov 15.11].
268
U.-G. MeiJ3ner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
Jacobian IJ~of the transformation (5.3). As we will outline in detail in section 5.3, it contains the anomaly structure of QCD and is given by In jJ~= ~
J
~
+
E~
TrJ F~(x)F~(x)IT°(x).
(5.6)
is the Wess—Zumino (WZ) term, which is topologically quantized:
~wz
—
NC
f
dTJ
E~
Tr[L5(r)
L~(T) L~(r)L~(T)Lff(T)],
(5.7)
5 = M4 ® [0, 1] to define the fields and currents via [5.11, 5.13] where we use the normalized manifold M L~(x,T)= 12’(x, T) ô~Q(x,T), Q(x, T)=exp{iTIr(x)} (5.8) L 5(x, r) = 12’(x, T) o~f2(x,r). The second term in (5.6) is obviously the Abelian anomaly connected to IT°—~2y. Here, ~r°corresponds to the generator t°= Let us now return to the representation (5.5) of the generating functional. To arrive at this form, one first has to integrate over all gluon fields. Keep in mind that we want to integrate out all colored degrees of freedom. The gluons interact with matter via ~ and ir°F F (where Fap = ~ is the dual gluon field strength tensor), therefore, the integration over the gluon fields will contribute to the generating functional via 1y~/i,IT0]} . exp{iS0[~/iT’ The important observation to make is that the functional S
(5.9)
0 depends only on color-singlet, bilocal fields of the form ifr~(x)~fr~(x),where a is a color-index and i, j denotes flavor. This can be easily understood if one uses the double-line notation of ‘t Hooft [5.12] and Witten [5.13]. The gluons and ghost field propagators can be represented by two differently directed lines, as shown in fig. 5.1. The index lines are continuous or form closed lines, so that only factors N~or color-singlet combinations ~l’a (x) ~l~a ( Y) can appear. It therefore is convenient to introduce bilocal variables B”(x, y) [5.15, 5.16] and rewrite (5.6) as exp{iS0) =
f
[dB] 6(N~B”(x, y)
-
~(x) ~(y)) exp{iV0(B,
IT)).
(5.10)
Injecting this into (5.4) and using the exponential parametrization of the 6-function in (5.10) with a A’. IJ,]
I
I
I
Fig. 5.1. Double line representation for the gluon propagator in the adjoint representation. The solid lines denote quarks and anti-quarks.
U. -G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
269
Lagrange-multiplier ~iU gives
f
Z[~] = Z~ [dQ][d~][d~] xexp{i[V(B,
iT) +
~~~(L)+ N~Tr(~tB) —N~Trln~’]},
(5.11)
where J2~’is the fermion operator D = ~ + LPR ~ to be discussed in section 5.3 [5.17] and we have omitted any ghost-field contributions, since the integral over the ghost fields ~, c does not contribute to leading order in N~[5.1]. ~a is the Lagrangian multiplier related to the 6-function imposing the Our main interest now is to study the functional color-singlet subsidiary condition F~= ~/Jy5t~/i. V(B, IT°). For that, notice that we can divide it into two parts V(B, ir)
V(B, 0)
=
+
V(B, IT°),
(5.12)
where V(B, IT°)includes at least one source IT°.According to the large Ne-counting rules, the behavior of V(ir°)is entirely determined by planar gluon diagrams and does not depend on B* i.e. V(B,
ITo)
=
V~(IT°) + C(N~’).
(5.13)
In a similar fashion, we can expand the part V(B, 0) as V(B, 0)
=
N~[V~(B)+
V~(B)+
~f~)].
(5.14)
It is now straightforward to expand the action in (5.11) to leading order in N~for integrals over the fields B, ill, ç Near the stationary point, the action is extremized by (the subscript “st” denotes quantities at the stationary point) iB~(x y) —
=
[(is — (5.15) a_
a_
—
~iU~~(x—y)— ~B ) 40 — IT — —0, where we have neglected the ghost-fields. It is now crucial to assume that flavor U(Nf) ® U(Nf) is spontaneously broken down to the diagonal subgroup U(Nf). In that case, the stationary solution (5.15) acquires the particularly simple structure B~=511B~~, ~
(5.16)
The assumption of spontaneous breaking of chiral symmetry means that the Fourier transform F[..~U] = ~1(p) tends to a non-vanishing constant as the momentum goes to zero. This is equivalent to the statement that the quarks acquire a dynamical mass, mdYfl cC [d~((p)/dp]~ 0 [5.18, 5.19], which breaks chiral symmetry. We are now at the point where we can calculate the effective action for the 1. *
Remember that quark loops are suppressed by a factor N~
270
U. -G. Meif3ner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
fields IT’2(x). At low energies, the terms quadratic in the current L,1 are dominant, therefore we will now calculate this contribution. For that, we expand the exponent in (5.11) to second order in the fields .A1 and B (these are the leading terms as N~tends to infinity) around the stationary point. It has the form ~,
~=
~~L)
+
V
0(ir0) + NC Tr{—~LPR~PLPR + + ~‘ttcM 5+~i 5~LPR_~Wcoy5+.AIB+ ~BKB}, ~iLPRcPJ/1+~iLPR’P40y 40y
—
(5.17)
where the coefficient functions in the expansions of Tr ln P~ and 17 0(B) are denoted by ‘P and K, respectively. It is now straightforward to integrate over the fields B, so that oneequation obtains a quadratic 1 = — K’. At large N~,G satisfies a Bethe—Salpeter form for the field At, G G=K(1+G) (5.18) which is graphically represented in fig. 5.2, where to each fermion line corresponds to the quark propagator B 55(x y). The poles of G(x, x’; y, y’) determine fermion—antifermion bound states, one of them being massless due to Goldstone’s theorem [5.20]. This massless boson is obviously a pseudoscalar particle (cf. discussion after eq. (5.4)). We will assume that no other massless states exist in the spectrum. Integrating now over the fields .~1t,the Lagrange function ~ (eq. (5.17)) takes the form —
~[=$~~~(L)+V0(ir°)+
~
5~~Y5çP +
—LP~’P~y5ço +i40y5~GLP~}+...
(5.19)
xTr{—jLPR~GLPR+ çDY
with
~1G=
~
F,alT’ 1 (x
+
cPGq~.For convenience, let us introduce the following abbreviations
—
y)
=
—
y)
=
F~(x— yu)
Tr{ySt~PGy5tb}, . a 5b Tr{1y,1PRt ~‘~GY t =
}
(5.20)
,
~
Using (5.18), it is easy to realize that the two-point functions F’~”(X— y) are represented by a sum of 1,we have diagrams as shown in fig. 5.3. From there it is obvious that to leading order in N~ N~F2l)(x— y) = i( T ~/i(x)y5t” ~/J(x)~/i(y)y5tb ~i(y)) (5.21) and similar expressions for F” and F~. The Fourier transforms of the functions F exhibit the
+ Fig. 5.2. Bethe—Salpeter equation for the quadratic form G for large N, determining the field Al defined in (5.11).
÷
+...
Fig. 5.3. Iterative solution of the Bethe—Salpeter equations to determine the Green’s functions F~”(x— y).
U. -G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
271
Goldstone pole at low momenta F[F
ab
ab
ab
2
]=F (p)=—C6 Ip +C(p),
F[F~] =
rab()
(5.22)
+2C(p), ~iCf6~p,1/p
=
F[F~] = F~(p)= ~ 6~(_g,1P+
+ C(p),
where f and C are constants to be determined. Finally, we have to integrate over the fields 40a,
which
gives the promised effective Lagrangian: 2~~(L) + V(ir°)+ ~NCL~F~L — ~NCL~F~CF FbL~. ~ Using the expansion (5.22), this can be cast into the familiar form ~eff
(5.23)
=
5~=_~f~Tr{L,1L~} + L~L,1V 0(IT0)
—
~ 48 ir
f
dr ~
Tr{L5(r) L,1(r) Lv(T) L~fr)Lff(T)},
(5.24)
0
with the pion decay constant f~, given by 2. (5.25) f~= (N~/4)f Some remarks concerning the derivation of (5.24) are in order. Although we have restricted ourselves to low momenta, the Goldstone poles could give rise to non-singular coefficients in any order in L,1 * As we have shown, in the leading order (second order in L,1) these poles are explicitly cancelled. This cancellation is intimately related to our assumption of imposing color-singlet subsidiary conditions on the collective variables ~/iy,1t’2~fr. In ref. [5.1], arguments are given that these cancellations hold in higher orders, too. For that, it is vital that the constant C as defined in (5.22) is non-vanishing. Finally, let us comment on the functional V~,(ir°). Obviously, since ir° is related to t°= ~\/~7N,it contains information about the ~‘-meson. To leading order in N~1,we can evaluate V 0(ir0). Assuming that it has a non-vanishing vacuum expectation value, it generates a mass for the i~’which is given by .
m~
-~
1
f~,256ir
41
(T[Tr(P(x) F(x)) Tr(F(0) F(0))])
(5.26)
.
in agreement with Witten’s result [5.21]. Since f~ N~and ([Tr(FF) Tr(FF)]~ N~,we see that N~’.Therefore, the restoration of the U( 1)-symmetry does not contradict the assumption that to 1,U(N~)®U(N~) is broken. leading order in N~ In principle, one could continue the present analysis to calculate the coefficients of the higher order terms in L,1. Unfortunately, for that purpose it is vital to know the pole as well as the regular terms in -~
-~
*
Only terms of even power in La are allowed because of time-reversal invariance.
U. -G. Meijiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
272
the Bethe—Salpeter amplitude to determine the function V0(B). This would require a complete summation of all planar gluon diagrams. This can only be done if one makes further model assumptions. We will therefore leave this microscopic picture of deriving effective chiral Lagrangians and present in the next two sections how the generating functional of QCD can be linked directly to an effective meson-theory including vector (and axial-vector) mesons. 5.2. Vector mesons and QCD Here, we will show how one can reduce QCD at low energies to an effective theory which incorporates vector mesons, the hidden gauge symmetry discussed in section 3 and vector meson dominance. The assumptions going into these manipulations are similar to the ones made in section 6.1, namely that QCD at low energies corresponds to a theory of weakly interacting, color singlet and local meson fields (and glueballs), with the U(N~)® U(N1) chiral symmetry spontaneously broken down to SU(Nf) ®U(1). With these assumptions, one can “derive” the successful phenomenology of the spin-i mesons, with their masses and decay rates determined by the scalar/pseudoscalar sector. We will follow the recent work of Ball [5.2]. Let us consider the QCD generating functional in Euclidean space—time, ZE
=
f
[d~][d~][dG~] exp{_
J
~(/
+
+
th)~+ ~
+~
Tr(F~)}.
(5.27)
are quark-fields which interact with N~— 1 gluon fields G,1 = G T’~,where T~’are the generators of SU(NC). Again, we have omitted the gauge fixing and ghost field contributions. A~LW is an external electroweak gauge field which couples to flavor, we will later on restrict A~to be the photon field. th is flavor-diagonal bare quark mass matrix, for ,fI = 0 we have an exact U(N1) ® U(N1) chiral symmetry. Our aim is now to express the generating functional in terms of meson fields, since these are the relevant degrees of freedom at low energies. Mesons are quark—antiquark composites, therefore, we should integrate out the gluon fields. This can be achieved formally by summing up all one gluon irreducible Green’s functions, so that ~s
~,
~ x
x1
(5.28)
x*
where we have introduced the covariant derivative D,1 = + A~W.j (x) ~/J(x)y,1 Ta qi(x) is the chiral singlet local current to which the gluon field G couples. The idea is now to make statements about QCD at low energies without knowing the exact structure of OGI Green’s functions x~).For that, introduce bilocal fields B which at low energies describe the mesons, using ~,
1=f [dB][dA~]exp{If Tr ~1t(y,x) (B(y, x)
-
~(y) ~(x))},
(5.29)
with At(x, y) an auxiliary (Lagrange-multiplier) field. Injecting (5.29) into (5.28), we find ZE
=
J
[d~][d~][d~][dB] exp{_
J
~+
i$i)~+JfJ~(B -
~)
-
F[B]},
(5.30)
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with
X
Tr[’y,~T~B(x1, x2)
Tan B(x~,x
12...
1)]
y~T’
(5.31)
.
Since B(x — y) are non-propagating fields, we can easily integrate them out in terms of a Legendre transform on F. We end up with ZE
=
f
[d~][d~][d~] exp{-
[1
~(~+ m)~+
Jf
~(x) J~(x,y) ~(y)
+
G[J~]]}.
(5.32)
G[.At] sums up all quark self-energy insertions via the bilocal field .At(x, y) since .itt[B] generates one-meson irreducible Green’s functions,
Z[~1t] = e~
=
f
[dB] exp{—[F[B]
—
Tr(~ttB)]}.
(5.33)
Up to now, we have only performed formal manipulations on ZE.* We will have to make some further assumptions on color confinement and locality to show that dynamics of the color singlet part of the fields ~~1t is indeed that of the mesons at low energy. For that, consider integrating out the high energy modes of QCD down to a scale A. (A might be estimated to be A 4irf,~ 1 GeV [5.22].) This new action with the cut-off scale A must allow us to describe low-energy scattering processes 1 1.and(xtherefore has the following features. First, glue must be confined, i.e., the Green’s functions F, 1 .. x~) must be color singlets for x~— x,j > A’. If one furthermore assumes that possible glueballs have masses larger than A, the ~ are entirely determined by the gluon-condensate and have no poles. This means that we can factorize F~in the following way: -~
.‘~
.
F~ .ff;(x1
x0)
(5.34) 1 and 611 ~n (i = a, ,u) are symmetrical products where the ca’s are constants on should scales be Jx~confined, x,,j > Ai.e. B(x, y) and iU(x, y) must be strongly localized of 6-functions. Second, quarks on the scale A’ since we do not want to have free quarks at energies ~A.Therefore, we can expand .At and B in the form . .
. . .
=
6a1
an6
. ..
,1Acfl , —
A4 A1(x, y)
=
z
M(z) f(t) + At,,(z) çf’(t)
+
+
4t,,~. . . ,1(z) ç 1
~ f~”~(t), (5.35)
B(x, y)
=
B(z) g(t) + B,1(z) 1,1 g’(t) +
.
+
B~,.. ,1(Z) t,11 .
t,1
g~(t),
where we have defined z = (x + y)A/2 and t = (x — y)A/2. f(t) and g(t) have to vanish rapidly as ~. Furthermore, if we expand ~/i around z, i.e. ~i(x) = ~i(z) + (c/A) ô,1~l(z+ *
t)~~0+...,
Note that we have not taken care about the trace and the U(1) anomalies.
(5.36)
U. -G. MeiJ3ner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
274
we can separate each term in the action (5.30) as
ff fJ Jf
~(x) ~t(x, y)
y)
~(
=f
B(x, y) ~t(x, y)
A4
=
J
J
~(z) ~(z)~(z) f(t)
f
B(z) ~1(z) f(t) g(t) +...,
y,1 B(x, y) Y,1 B(y, x)
=f
Y~B(z) Y,1 B(z)
(5.37)
f
g2(t) +...,
and similarly for all other terms. From (5.37) it is obvious that integrating over momenta larger than A means integrating over f and g, rather than itt(z) and B(z). Therefore, one gets resonances at the stationary (semi-classical) points of f and g. f(t) can be interpreted as the meson wavefunction, for A —~1 GeV mesons have a size r~—- A~-~0.2 fm. We further have to assume that dissipation of a meson into a quark—antiquark pair above threshold* is suppressed since localization of .At(x, y) does not exclude this possibility. Third, since .At and B contain also color-octet mesons [5.23], we have to assume that these states acquire masses larger than A, and thus decouple from the low-energy dynamics. This means that chiral symmetry remains unbroken in the octet channel. With these assumptions, we can rewrite the action (5.29) in terms of the color-singlet fields i1t and B as follows
JJ
~(x) ~t(x, y)
y)
~(
=
J
a~ ~(z)At(z) ~(z)
JJB(x,y)At(x, y)=~ofB(z)~i1(x)+.~,
ff
2~(y, x) Y,1 B(x,
~)
=
~~2f
(~
(5.38)
B(z))2
2y,1 B(x, y) F~
with a~,f3~and ~s, (i = 0, 1, 2,
. . .)
unknown constants. Obviously the local fields .
1t1, Jtt,1, At~, correspond to generations of mesons of increasingly complicated tensor structure and, presumably, increasing masses. Expanding them in spinor space gives for .~U (5.39)
~
5q, ~y,1qand qy57,1q, i.e. scalar, where the different parts correspond to the composite operators cjq, qy pseudoscalar, vector, and axial-vector mesons, respectively. Similarly, decompositions can be written down for dtt,1, .At etc. With this, the gluonic potential G[~iU] takes the form G[~]=fJ{_Eo+[2Tr(M_N,1N,1)+~.] +
*
[~r Tr((!~IM)2— N,1N,1IcIM + ~Ic1N,,I~1N,1 + 8(N,1N,1)2)
This threshold is around 0.6 GeV, i.e. twice the constituent quark mass.
+ . . .1± O(A16/A2)).
(5.40)
U. -G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
275
E0 is the vacuum energy density and r = (3/8)(~sI~8), where /h8G[At] is the isconstant to the colorand octet 4~(y,1B)4. locally related chiral invariant, it pieces stemming from the part of the action —~F~ contains no derivatives in ~ [5.24]. Obviously, we have here an expansion of G[AtI in powers of Al2, which makes sense since the weakly interacting mesons At cannot have strong gluonic self-couplings, otherwise meson bound states (exotics) would form, and these are not observed in the low-energy spectrum. The suppression of purely gluonic interactions, mediated by Zweig suppressed decays, is consistent with Regge phenomenology, i.e. the exchange of hadrons. Notice that all these statements are nothing but a reformulation of Wittens large N~rules [5.13] as it should be. Before addressing the properties of mesons from the effective action, some remarks concerning the pseudoscalar singlet in G[A1], i.e. including the U(i) anomalies, are in order. For that, U(N~)®U(N~) has to be broken to U(1) ® SU(Nf) ®SU(Nf). This can be achieved by introducing a 0 + chiral singlet field proportional to with non-vanishing expectation value [5.25, 5.26]. In a similar fashion, the QCD trace anomaly can be reproduced by including a scalar 0~-fieldproportional to ~ We will consider here only a minimal version where the e-terms, the U(1) and the trace anomalies are neglected. In that case, G[A1] depends only on one parameter ~t for each generation of mesons. Therefore, we can get quantitative predictions without detailed knowledge of the non-linear gluon dynamics. Let us now consider the first generation of mesons .A1 = M + 7,1N,1 (cf. eq. (5.39)), i.e. the scalar, pseudoscalar, vector and axial-vector mesons. The generating functional ZE (5.32) reads* ZE =
f +
[dM]A[dN1A[d~]A[d~]A expf
J [~+
th)~+ ~
+~
~+...]}
~ ô,1~(M+~)ô,1~+4Tr(2MM N,1N,1)+(eterms)+7
(5.41)
where the subscript A on the measures denotes that we have to integrate only over fields which vary more slowly than A’ (low-energy approximation). ir~is a chiral pseudoscalar singlet and the related unknown coefficient. g is a fermion—meson coupling constant. It is interesting to study the zeroth order approximation to (5.41) for which g = = = 0. In that case, the integration over N and M becomes trivial and we end up with an extended Nambu—Jona-Lasinio model [5.4, 5.8, 5.9, 5.10] ~
Z~=
f
[d~]~[d~]~ exp{_
J [~(~ ~
4
-
~)2
-
~5~~)2]]}.
(5.42)
2,chiral symmetry alone would Notice that the* * interaction hasback onlytoone —2/j~ allow for two. This can beterm traced the overall fact thatconstant the above four-fermion interaction is a Fierz rearrangement of ~ ~fr)2 i.e., that QCD is a vector-like theory. It is important to stress that the more complete model (5.41) shares most of the qualitative features of the NJL model, e.g. the meson kinetic terms are entirely due to the quark kinetic term ~Ii/~/i. If ~2 ~ A2, the quarks will not appear as asymptotic states, since they couple strongly to mesons. Integrating out the quarks, we get ZE
* **
=
f
[dMJA[dNIA exp{N~ln detA (4~ + th + M + ~Y)— ~—
i is defined by xy,, = with the exception of the bare quark mass matrix. We will come back to this point in the following section 5.3.
f
Tr( ~ 1dM
—
N,1N,1)
+
. . .
}.
(5.43)
U.-G. MeiJ3ner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
276
The baryons will emerge as chiral solitons of this action. To deduce the physics contained in (5.43), let us reinstate the fermions for a short while and try to find the vacuum. By Lorentz invariance, only M can acquire a v.e.v., so that we can set N 0. For that let us rewrite M in terms of a “chiral phase” u = exp{iy5r. ii/2f), i.e. =
M=ü.~ut,
(5.44)
where u is unitary (u~u= 1) and 2~Hermitian
(.~ = ).
With th
=
0 and
A~,W=
0, the generating
functional takes the form of the chiral quark loop model [5.30, 5.31, 5.321, i.e. =
f
[du]A[d~] 4[d~]A[d~]Aexp{_
J
u~Iut)~ + (p2/4) Tr(2~/2)}}.
+
[~(/
(5.45)
Even more conveniently, we use Simic’s constituent gauge [5.33] to rewrite (5.45) in terms of constituent quarks T~/i, j=~iü, (5.46) q=u which are nothing but current quarks dressed by a pion cloud as shown in fig. 5.4. If J~denotes the Jacobian of the inverse of the transformation (5.46), this leads to ZE
=
f
exp{-f
~
[~(/
+
üt/u)q + ~q + (p2/4) Tr(2Z/2)I}.
(5.47)
The mechanism of chiral symmetry breaking is the same as in the NJL model; for that, let us derive the effective potential of the scalar field. We define = ~ + &~= oi + &~, the effective potential V(u2) follows using standard procedures: .~‘‘
V(u2)
[u4(ln(~-~)
= —~-~
+
~
0)c1A2O~2]+ —
~—
Tr(~—),
(5.48)
c0 and c1 being constants of order unity. They parametrize the dynamics in the transition region from 2= (~q) the high-energy to the low-energy modes. Minimizing the effective action (5.47), we find o~ which is non-zero as long as u2(c 0, c1, p~) 0. Expanding the fermion determinant in (5.43) to second order in derivatives of (/o-) and introducing U Ciu~(with U incorporating the isovector Goldstone modes), we find ~(2)
=
~ Tr[a,1U ~~U] +
~
Tr[~,1(a2) ~
(5.49)
= /
mq
m~
Fig. 5.4. Constituent quarks as current quarks dressed by a pion cloud.
U.-G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
277
where f is given by
f
“~C
2[1(A)
c~ 3]
+
(5.50)
—
and has to be identified with mq the ispion constant. Frommass. the effective potential can three read 2mq, where the decay constituent quark The model action(5.48) (5.43)wehas off m~ = 2oparameters (~, c 3, one 0, c1),~aqq if one chooses to fitf,1 = = 93MeV, mq so = 350 (~q~ = be —(220 MeV) as a finds rn, = 700 MeV, = 4.3 and F(o—t. 2iT) 1.5 GeV, thatMeV theand r cannot observed resonance. If one restores the external photon field AW, the induced Wess—Zumino term describes properly the abnormal parity processes ir°—*2-y and -y----* 3ir. For non-zero current quark masses th = 0, one recovers the usual chiral perturbation theory results, though m~= i80 MeV ~(m~+ rnd) 7.5 MeV induces the usual problems in the SU(3) sector. Let us now come back to the vector meson content of At by restoring N, the spin-i field. For that, we decompose N,, N,,v + y5N~in vector and axial-vector parts. First, we have to relate the physical spin-i mesons S 1,, to N,,. For SU(2), we can conveniently write them as 5A,, iw +iTp,, +iy5D,, +iy5TA S1,, = V,, + y 1,, (5.51) ~‘-
The 5y,,q. p and the A1 are most naturally described by constituent quark composites of the form t~T*y,,qand Using qi-”y
~/i(/~+M+fr1)çb=~(/+I
+u~u+u~/u+
üt~u)q,
(5.52)
one defines S 1,,
u~N,,u+ u~~,,u + utA~~u.
This implies N,,’uS1,,u t +u9,,u t —A,,ew =u~,,u
(5.53)
.
(5.54)
Note that N,, and S1,, are related non-linearly, so they will not give the same results for multiple pion scattering at tree level. It is important to stress that with these definitions the action possesses a hidden U(Nf) ® U(Nf) chiral gauge symmetry [5.3]. If we parametrize the external chiral symmetry by group elements g and the internal (hidden) symmetry by group elements h, the various fields in the action transforms as follows: q—*q’=hq, t, -*~‘=h.h~,
i/J—~~~J’=g’/i,
M—*M’=~Mg u—* u’ = guht,
A~—3(A~)’ = gA~gt + ggt,
(5.55)
S 1,, —* Si,,
= hS1,,ht
+
h c9h~,
N—* N’
=
gNgt.
Obviously, the vector and axial-vector mesons are gauge bosons of the hidden symmetry. Furthermore, it is interesting to note that in terms of the hidden gauge, the relevant fermions are the constituent
U. -G. Meiflner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
278
quarks — therefore, this gauge has been called “constituent gauge” by Simic [5.24, 5.33]. With 5A,,, the generating functional (5.43) takes the form = V,, + y ZE =
f
[duIA[d~}A[dV]
tthu + ~ + 4[dA]AJ~exp~N~ ln det(~+ü
~
~
-
(Si,,
-
-
u~(ô,,+ A~)u)2+
.. .]}.
(5.56)
Notice that the pion field enters the functional through the Jacobian J~,and that the pion kinetic term is generated by the gluon potential. Performing a derivative expansion of the quark determinant, we find for th = 0 (after chiral symmetry breaking)
—
2
—
~—
Tr[D,,U D~U]— ~Tr(V~~ + A~)— ~2 Tr[V,, —~—j-Tr[y~A,, ~—~-~-— (ut D,,u — 12t D,,i~)] ,
—
~(u~D,,u + ~ D,,~)]2
—
(5.57)
where we have ase-,
2,
ai
f, is
(5.58)
g2~_~
~
the physical pion decay constant. We have f2 >f~,so that in the presence of p, A
1-mesons the cut-off A will be larger to accommodate these particles. As before (sections 2.1, 2.3), we have to redefine the ii and A1-fields to get mass eigenstates. This amounts to A,,—* A,, — (a — 1)19,, IT/afT. Integrating out the A1, which should be a good approximation below 1 GeV, by using its equation of motion and ignoring the kinetic term of the A1, (5.57) gives (to second order in derivatives): Tr[D,,
=
U
D~U]
Tr[V~j
~
—
—
~
f~Tr[2V~
2, —
(5.59)
(ut D,,u + â~D,,z2)]
which is the hidden symmetry Lagrangian of Bando et al. [5.3]where V,, is identified with the p-meson. The only bound on the parameter a we have up to now is a> 1 since af2 = ~2(a — 1)_i. If we reinstate the A 1-degrees of freedom, we can now read off couplings, masses, etc. from the Lagrangian (5.57). Before doing so, let us mention that all parameters (is, c0, c1) are already fit from the ‘rr—cr sector, chiral symmetry alone would allow for six independent parameters if one sticks to terms with two derivatives. We find the following predictions: 2 m~ = 6(a
1)o- 2 , mA2 = 6au 2 , ma2 = a 4o~2 2=0)=~ag, g~,,.,,(q2=—m~)= ~a(1—i/a2)g,
—
g~,,,,(q g~ = (1 a/2)ge, —
2
2
gaqq
—
~
g
2
~
=
2 ,
g.,,qq
em~!g, i — 6
g
2
(5.60)
U.-G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
279
Fixing fIT = 93 MeV, rnq = 300 MeV, and varying a between 1.8 and 2.2 (which is the typical accuracy of soft-pion predictions, e.g. for the KSFR relation) we find m~=(750±80)MeV, mA =(1060±50)MeV, rn~=(975±50)MeV, (cjq)=—(175±5MeV)3,
(5.61)
where we have used (cjq ~ = u~t2 Obviously, the value of (~q)comes out too small, to improve on it without changing the other results, one can either include spin-dependent terms in the meson wavefunction* or include small s-terms defined in the gluon potential (5.40). For e 1/25, one gets (tjq) = (—225 MeV)2, m~= (780 ±150) MeV and mA = (1270 ±100) MeV. A will be reduced by almost a factor of two (A -=0.6 GeV), which seems a bit too small. The other interesting features exhibited in (5.60) are (i) approximate VMD of the -y’Trlr vertex, ~ = (0.0 ±0.1)ge, and (ii) approximate universality in the p-channel: g = (5.6 ±0.3), g~~(q2 = 0) = (5.6 ±0.8) and ~ = —m~) = (4.2 ± 0.9). The KSFR relation is fulfilled for a = 2. Because of the strong momentum dependence of the pirir-coupling, the width F(p—~2iT)comes out too small (—80MeV); similar statements hold for F(p—~eke) and F(A 2 = 0) = 0 is the same bad prediction of any chiral Lagrangian 1 pir).** gA(q with A 1-mesons and second derivatives only. The width of the decay o~—*2ir is largely suppressed, we find F(ff—~2ir) (200 ±150) MeV. This is consistent if one identifies the if with the observed S*(975) or ~(980) resonances. * * * It is worth to stress some more general properties of the action (5.56) which are not dependent on the particular choice of the parameters. In (5.56), photons couple to vector mesons as a normal parity combination of ‘rr and A1. Therefore, all abnormal parity decays involving photons will be given entirely by the vector meson mediated graphs, i.e. there are no contact terms. This justifies the Gell-Mann— Sharp—Wagner model [5.34]. For normal parity processes, we have only approximate VMD, direct photon—2n-pion vertices might be small but not necessarily zero. So we have found an argument through the manipulations which lead to (5.56) which links the concept of VMD directly to the QCD-generating functional. Therefore, Sakurai’s original concept can be embedded quite naturally in the framework of QCD. Finally, let us mention that one might (approximately) calculate the parameters j~t,c~and c1 in the “dilute instanton vacuum” model of QCD [5.35].It remains to be seen whether this more constraining action (5.56) gives a more accurate description of baryons and their properties than the effective actions derived from chiral symmetry as discussed in sections 2 and 3. —*
-=
5.3. Bosonization of the Narnbu—Jona-Lasinio model In the introduction to this section, we had argued that lattice QCD in the strong coupling and 1 / d-expansion resembles a four-fermion contact interaction like the Nambu—Jona-Lasinio (NJL) model. Furthermore, the NJL model is supposed to describe the spontaneous breaking of chiral symmetry in the same way as it happens in QCD. It therefore seems natural that an extended NJL model in terms of quark-fields gives some intermediate step from the full QCD-Lagrangian to the purely mesonic theories considered here. Following Ebert and Reinhardt [5.10, 5.36], we will briefly *
This would make the p smaller in size than the
ir
by a factor ~
which is not very comforting.
These are exactly the problems encountered already in sections 2 and 3 when incorporating the A1-meson. Keep in mind that the relevance of these mesons to low-energy hadron physics is not undoubted and this identification is not entirely convincing. **
U.-G. Meij3ner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
280
outline how a massive Yang—Mills theory arises via bosonization of the generalized NJL Lagrangian. For that, consider the following effective quark Lagrangian invariant under color SU(NC), ~=q(i/_tho)q+2G1[(q~ q)2+(~iy~~ q)2] —
2G 2[(~y~ ~
q) +
~
~
q)],
(5.62)
where we have suppressed color indices on the quark-fields q, cj. The generators the flavor U(2) are 2, of notice that would we T0 = ~, ‘r,(i = i, 2, 3). G1 and G2 are coupling constants of dimension (mass) have started from a vector-like theory and Fierz-transformed it, we could have set Gl = 2G 2. The ansatz (5.62) has therefore one degree of freedom more than the underlying fundamental theory, QCD. The bare quark masses th = diag(th~,thd) explicitly break flavor U(2)®U(2). To bosonize the Lagrangian (5.62), let us introduce the following color-singlet collective variables [5.37] 3
2 q—~S=~ t=0 S1 ~,2
T
~ q—*P= ~ P 2 3
5
~iy
2
5T
(5.63)
~y~y ~ q—*A~=—i~A’
~ q—*V~’=—i~V~’ ~,
-~
in the generating functional Z[q, ~j
=
f
[dq][d~]exp{-i[S(q,
~)
+
~q +
~]}
corresponding to (5.62). With that, and the 6-function representation (5.29) the action becomes bilinear in quark and anti-quark fields, so that these fields can be easily integrated out. One finds
f
t) = ~ ~
A, M, M
—
~
Tr[(M
J
—
—
ph 0)]
Tr[V~ + A~]— iN~Tr[ln(i~)],
where M is a complex field defined via external fields M, M~, V and A, i.e.
iJ~iy,,D~= i[y,,ô~
th t(M 0)
M = S
+
iP.
(ia’)
is the Dirac operator in the presence of the
5)M + ~(i +
y,,V~+ y,,A~]— [~(1 + y
(5.64)
—
y 5)Mt]
(5.65)
exhibiting clearly the underlying chiral symmetry. Notice that (5.62) can be written as 2 + P2] —2G 2 + A2]. +2G1[S 2[V The last term in (5.64) obviously stems from the quark determinant ~
~kin
[det(i p)]N~
=
exp{N~Tr[In(i,D’)j}
.
(5.66)
U. -G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
281
In terms of meson fields, we can rewrite V~and A,~as ~
A~=—~i(rA1~+D~)
(5.67)
where (w~,p~) and (D,2, A denote the isosinglet and isotriplet vector- and axial-vector fields, respectively. D,~represents the D-meson (nowadays called f1) at mD = 1285 MeV. To make the particle content of the complex field M more clear, let us work in a polar decomposition such that M=~IU,
(5.68)
where 2~(x)is the field of the scalar (if) particle and the chiral field U(x) is parametrized by the Goldstone bosons i~(x),*U(x) = exp{i~(x)If}with v~(x)= ~(x) r. f is the bare pion decay constant (up to renormalization factors). Notice that the effective Lagrangian (5.64) looks already quite similar to a massive Yang—Mills theory, with the kinetic terms of the gauge bosons hidden in the fermion determinant. We will come back to this point later on. Let us first study the effective theory (5.64) for large numbers of colors, N~—~ In this case, the meson functional Seff = exp{— fx “~eff} is dominated by the stationary phase at ~.
U=11,
A~1=V~=0,
~0=m
(5.69)
with m being the constituent quark mass (cf. fig. 5.4), which is determined by the gap-equation it
J
‘
m = th0 + i8G1N~
d~k 4
(2i~)
_______
2
k —m
(5.70)
2
Obviously, one has to cut off the momentum integral at some scale A since we are dealing with a non-renormalizable four-fermion interaction. A is related to the quark condensate value (~q~ and should therefore be of the order of the scale of chiral symmetry breaking [5.18, 5.22]. We now return to the quark determinant appearing in (5.64). Its phase is not invariant under chiral transformations 5} (5.71) i~ø—~ (li p~Q, ~11= exp{—ia ry and contains the gauged Wess—Zumino action [5.38]. We will not go into details on this point here but rather refer the reader to the extensive reviews refs. [5.11]and [5.39].The modulus of det(i)D’) can be calculated from [det(i,D’)t(i~’)]”2using proper-time regularization [5.40], {1n(~D’t~ø)} =
ln~det(i,Ø’)~ =
—
~J
~ Tr[exp(—p’t~’s)].
(5.72)
i/A2
Performing a heat-kernel expansion [5.17, 5.40] we find lnIdet(i~)I~ =
—~
J
16~r x
Tr{F(0, m2IA2)[S ~S
+ ~(F~FV~~ + F~VFA~~V) ~4] —
*
we
assume to be in the chirally broken phase.
+
+
~P
~1YLP
2exp((—m2IA2)A2)}
+...
(5.73)
282
U.-G. Meij3ner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
where the dots stand for higher order terms in U [5.10, 5.36]. The pertinent covariant derivatives appearing in (5.73) are D~S=o.~S+[V~, S]+i{A~,P}, (5.74)
D~P=o.~P+[V~,P]+i{A~,S}. The respective vector and axial-vector field strength tensors are
~
=
~
~
=
~
[Vp, VP] + [Ak, Ar],
+
—
(5.75)
~
—
+ [Vp, A~]+ [Ag, Vu].
F(a, x) is the incomplete F-function, F(a, x)
.f°dt e_tta_l. As
promised, the kinetic terms for the (axial) vector particles appear in the quark determinant (and do not have to be put in by hand). Let us now discuss the consequences of the effective Lagrangian (5.64) for the meson sector. For that we will expand U(x) in powers of IT(x), i.e., U(x) = 1 + uT/f and bring all kinetic terms in standard form by proper renormalization. Furthermore, the pion- and A1-fields have to be redefined to get rid of =
the 8.~i~ A~vertex. Writing the scalar field as 1~(x)= m + o-(x), we get .
2V~,
~
=
z”2
~l/2,
V~1= \~Z~’
and can be read off the
if,
p and
t~omasses
=
[J~ r(o, m2)]1’2
(5.76)
from the bilinear part of the effective Lagrangian as
m~= g~/4G 2, g~= (5.77) 2 m~2= rui m~= m~ + 4m 0mIf~G1 2the bare pion mass and decay constant.* To diagonalize the ‘n~A with m~band fb = 2mZ~’ 1-mixing, we introduce physical pion (~) and physical A1-fields (A1) via 2Z1~
fL2fb
~
=
~
—~
+
m
2—Z” ‘
~
(5.78) 22
Z
=
2 1/2
mPImA
(1 g~f.TIm~) with f,~ = 93 MeV the physical pion decay constant. f,~ and j satisfy the Goldberger—Treiman relation on the quark level. The masses of the physical pions, A1- and D-mesons are 2D = m~+ 6m2, m~ m~b(m~ m~1= m 1/m~). (5.79) 1
* fb
=
—
is different from f due to the field redefinitions as it becomes clear from eq. (5.78).
U. -G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
283
Notice that the A1 and the D are degenerate in mass, as well as the p and the to. Furthermore, the p—A1 mass splitting is entirely due to partial Higgs mechanism as already discussed in section 2.1 in the framework of massive Yang—Mills Lagrangians. In the case of vanishing current quark masses th0, we find m~= 0 in agreement with Goldstone’s theorem. All phenomenological relations like KSFR or the Weinberg mass that ratio are now determined by the value of the constituent quark mass. For that, let us set 2 = am~so 6m m~= (1 With Z
(1
+
+
a)m~,
a)”2
=
(1
m~= (6/a)m2. —
g~f~/m~)”2we
(5.80) obtain
[(1 + a)/a]g~f~.
=
(5.81)
To arrive at (5.81), we have used the universality of p-couplings inherent in the model [5.10]. For a = 1, we recover Weinberg’s ratio m~= m~ 1!V~ and the KSFR relation m~= ~ With m~= 770 MeV, the constituent quark mass follows to be m = 314 MeV, which is just m~10~0~/3. If one insists on the higher mass of the A1, i.e. m~1= 1275 MeV, one can choose another relation between m~and m. = 2m (a = 3/2) e.g. gives 2 f2)112 = 737 MeV (5.82) \/~m~ = 1217 MeV, m~= (~g on the expense of the KSFR relation. Nevertheless, the Weinberg ratio and the KSFR relation are incorporated in the Lagrangian (5.64), with this precise form depending on the value of the constituent quark mass. Let us mention that the whole scheme has been extended to broken flavor SU(3) ® SU(3) [5.10]. In that case, one has five parameters th~= th~,th~,G 1, G2 and A which can be fixed by demanding ,j,, = 140 MeV, JTT = 93 MeV, m~= 770 MeV, g~/4ir = 3 and m~= m~I2= 510 MeV. Some pertinent results are mA1
=
th~=2MeV, ,h~=82MeV, 2/4ii~= 0.6, G 2/41T = 2.5* A = 1.4 GeV, G1A 2A fK/fm.=1~1(1.2±0.1)~ gfIg~=1.3(2.0±0.2) mU
m~J 315MeV,
(5.83)
The current quark masses come out on the small side, from QCD sum rules one expects e.g. th~= (110 ±10) MeV [5.41]. The meson mass spectrum is summarized in table 5.1. It shows a fair overall agreement. The quark condensates do not come out too well, we find (uu)~3 = —340 MeV and (~~s)I(iiu) =1.4 to be compared with (uü)~=—250MeVand ((~s~/(üu))sR=0.8[5.41]. Finally, let us mention that treatment of the quark determinant outlined here gives a unique way of fixing the so-called non-minimal (normal parity) contributions to the gauged Wess—Zumino action [5.23, 5.36, 5.42]. To summarize, we can say that an “intermediate model” like the extended NJL Lagrangian (5.62) gives rise to massive Yang—Mills Lagrangians with all parameters fixed in the meson sector, *
Notice that G,
2G 2, i.e. the NJL model (5.62) does not stem from Fierz-transforming a vector-like interaction when it explores its
phenomenological consequences.
284
U. -G. MeiJ3ner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons Table 5.1 Meson mass spectrum from the bosonized Nambu—JonaLasinio model. The two values for the A 1-mass give the experimental uncertainty already discussed in sections 2 and 3. The E-meson is nowadays called “f1(1420)” and the Q1meson runs under ‘~K1(1280)”.
[MeV~ m~ [MeV] mK. [MeV] mA [MeV] m~[MeV] m0 [MeV~ ~
ref. [5.36]
experiment
600 970 9(X) 1090 1582 1332
494 1020 892 1060/1275 1422 1270
therefore adding further credit to the use of chiral meson Lagrangians. This tightens the link between QCD and the effective theories discussed before.
6. Baryons from the effective Lagrangian In the previous sections, we have only examined the meson sector related to the effective Lagrangian constructed by symmetry principles, e.g., chiral symmetry, with pseudoscalar and vector mesons. This is actually “old stuff” and has been explored in great detail already 20 years ago (cf. references given in section 1). The novel feature, triggered by the developments in the conventional Skyrme model, is that baryons can emerge as topological solitons from the underlying non-linear meson theory in the spirit of Skyrme’s and Witten’s ideas. Furthermore, the properties of the topological solitons are entirely fixed by the mesonic data like the meson coupling constants and meson masses, i.e., all baryon properties will emerge as parameter-free predictions. This gives very stringent tests on the quality of the models considered and is in sharp contrast to other approximate approaches to treat low-energy QCD like e.g. the potential models [6.1]. Furthermore, starting from a relativistic meson theory, the models with pseudoscalar and vector mesons offer rich flexibility in calculating a variety of static and dynamical baryon properties let us only mention a few: static properties, electromagnetic and axial form factors, strong form factors, excited baryon states, qualitative features of the nucleon—nucleon interaction and of nucleon—antinucleon annihilation. In what follows, we will demonstrate in some detail how a model with pions, p- and w-mesons works in comparison with experimental data, following the work of Meifiner, Kaiser and Weise [6.2]. Then calculations including the A1 -meson will be discussed, followed by a brief review concerning the situation on NN and dibaryon physics. Before going into details, let us make a short historical digression. Actually, the first models to include vector-mesons did not account for p, w, A1 but just for o- or p-mesons coupled to the skyrmions. It was the work of Adkins and Nappi [6.3] which triggered interest in vector meson stabilization. Since this model is discussed in detail in the review of Zahed and Brown [6.4], we will refer the reader to this reference. The B = 2 solutions in this model have been investigated by Kiabucar [6.5]. The link of the p-meson coupling to the Skyrme parameter e [6.6] has been first made by Iketani, who showed that the Skyrme stabilizing term stems from the exchange of infinitely heavy p-mesons as discussed in section 3.1. Following that work, Adkins [6.7] proposed a model coupling the p-meson to the non-linear if-model based solely on symmetry grounds. A variant of this model has been studied by —
U. -G. Mei/iner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
285
Meil3ner [6.8, 6.9]. Since these simplified models are not in the center of the work presented here, we will give only a brief outline in appendix B. For other attempts to include the p-meson, see the work by Igarashi et al. [6.10, 6.11]. The first full-scale attempt to investigate the ‘rrpwA1-system has been carried out by MeiBner and Zahed [6.12, 6.13], who showed that a more realistic Lagrangian indeed gives an improved description of the static nucleon properties. Similar results have been obtained by Chemtob [6.14]. After that, a flurry of groups has studied various effective Lagrangians looking mainly at static nucleon properties. These results will be discussed in section 6.7. The most detailed discussion on baryon properties has been done in the ‘rrpü-system which we are going to present now. Nucleons and L1-isobars in the irpw-system
6.1.
The first step in constructing nucleons, z~-isobars,.. . from the effective Lagrangian under consideration is to find the classical solution, the hedgehog. This state has neither good spin nor isospin, but the grand-spin K = I + S is a good quantum number. To be specific*, let us discuss a Lagrangian with pions, p- and w-mesons following the work of Meil3ner, Kaiser, Wirzba and Weise [6.2, 6.15] based on the hidden symmetry scheme outlined in section 4. p- and w-mesons are introduced as dynamical gauge bosons of the hidden [SU(2)~ ® U(l)~]10~~1 symmetry. The Lagrangian reads ~
2 ) Tr[F,~~F~’] + ~ + (1/2)m~f~ Tr[U 1]. (6.1) (112g U(x) = exp{ir. IT(x)/f,j embodies the isovector pion field, and ~L = = = ~ in the unitary gauge. f,~is the weak pion constant which we will take on its empirical value, JTT = 93 MeV.** ~ is the contribution from the Wess—Zumino (WZ) term to be discussed later. is the gauge covariant derivative and F,LP the pertinent field strength tensor —
—
(6.2) ~
=
~
—
o~V~ —i[V~,V~].
We have chosen the same gauge coupling constant g for SU(2)~and U(1)~which leads to a mass degeneracy of the w- and p-meson: 2
2
2
22
22
m~=m~=m~=af,~g =2f~g with a = 2 giving g
=
~
and the KSFR relation (6.3). Setting my
(6.3) =
770 MeV, we get
g=5.8545
(6.4)
in good agreement with the value ~ = 6.11 determined from the width of the decay p—~2ir. The discrepancy is of the order of 5% and just shows how good the KSFR relation holds. For the following *
Keep in mind that although we are using a specific Lagrangian, these considerations hold in general.
**
16.17].
Recent measurements give values for f,, between (92.16 ±1.01) MeV [6.16J.We use here the more standard value quoted by Nagels et al.
286
U.-G. MeiJiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
discussion, we will disregard the small mass difference m~ 13 MeV. The contribution from the WZ action ~ is given by (after making use of Stokes’ theorem) —
~
/7.j
\
Ic
I
j.*eaf3
e
+
L’41T
+
ô~U~U~U~U
+ c(~)~vaPw~pTr{iT.pa(U+.
~U
+
+
ô~U}
o~U~ U~)+ ~ r.pU±r.pU}
(6.5)
The first part of ~ gives the o-coupling to the topologically conserved baryon current B ~ as first discussed by Adkins and Nappi [6.3], with the oNN coupling given via universality as g=~=
=
g~,= (N~/2)g—9.
(6.5a)
For c = 0, we talk of the “minimal model” as discussed in section 3.3. For c = 1, the wpir-interaction is taken into account, the WZ action for that case is equivalent to the rigid gauged WZ action and elimination of the A1 via the constraint U 0. We will refer to this model as the “complete model”; it embodies the p-dominance in the o—~3~decay amplitude. The last term in (6.1) is the conventional pion mass term, we take m,~= 139 MeV. This avoids the problem of infinite quantities such as the isovector charge radius in the exact chiral limit m,~= 0 [6.18]. The Lagrangian (6.1) (and any similar effective Lagrangian possibly also including A1-mesons) describes correlated mesons as well as solitons. The latter 3r Bones are finite energy-configurations with non-vanishing winding number (baryon number) B = f d 0(r). Their stability is established by the repulsive character of the vector mesons at short distances. To investigate the baryon number B = 1 sector, we will specialize to hedgehog skyrmions. This amounts to working in a space of maximal symmetry and brings the coupled equations of motion for the meson fields into a tractable form. Keep in mind that one could instead start with deformed meson fields and allowing for more degrees of freedom. Here, additional calculations are in order, but we feel that the guidance from the conventional skyrmion where the hedgehog indeed turns out to be the minimum energy-configuration [6.19] should be taken seriously. So, for the pion field, we choose the usual hedgehog configuration U(r)
=
exp{ir IF(r)}
,
~(r) = exp{ir. i~F(r)/2},
(6.6)
which connects the intrinsic symmetry space (isospin-space) with r-space in a radially symmetric way. F(r) is the so-called chiral chiral field.number, The baryon number density reduces to B0(r) = 2)(F’/r2) sin2F(r). Toangle ensureorunit baryon we have to choose (—1/21T F(0)rr,
F(c~)0.
(6.7)
In the minimal model, the w-meson couples only to the topologically conserved baryon current B’~(x). Since on the classical level only the time component of the latter is non-vanishing, the space components w’ of the W-meson field have no sources, so that =
w(r)6~°
(6.8)
as first pointed out by Adkins and Nappi [6.3]. These arguments also hold in the case of the complete model, because the additional term in the Wess—Zumino action (6.5) can be cast into the form of a total
U. -G. Meljiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
287
derivative which does not change baryon number. Equation (6.8) can also be inferred from the facts that w0 is a scalar and transforms into itself under parity x—~—x. We will refer to the function w(r) as the w-meson profile. The negative parity p-meson has non-vanishing space components only. For that, let us rewrite the ~ term (the second term) in (6.1) as
f~Tr[J~—V~J2,
(6.9)
where J~= (1I2i)(~UU~ + ô.,~U~U)/det(1 + U). With the hedgehog ansatz (6.6) for the chiral angle F(r), this leads to J~(U)=0,
J~(U)=r,
1~F.,(1—cosF(r))/r.
(6.10)
It is therefore natural to take the Wu—Yang—’t Hooft—Polyakov [6.20] ansatz for V,~(x), i.e. the p-meson p,~(x): Pza(T)
=
EI/~Fk G(r)/gr
poa(r)
;
=0,
(6.11)
where the SU(2) gauge coupling g has been scaled out for convenience. In (6.11) a = 1, 2, 3 are isospin indices, whereas i denotes the three space components of the vector fields. We will refer to the radial function G(r) as the p-meson profile. Using the hedgehog ansatz (6.6), (6.8) and (6.11), we can derive the static energy functional from the Lagrangian (6.1). This static energy has to be identified with the hedgehog or skyrmion mass MH: M11
=
E[F(r), G(r), w(r)]
=
—
f
3r~(F,G, w).
(6.12)
d
Straightforward algebra gives [6.2, 6.15]
E[F, G, w] =
41T
J
dr{f~[ir2FI2 +
+ m~,f~r2(1cos F) —
+
sin2F] + 2f~[(G +1— cos F)2] —
f?-~i + G2(G+2)2 + g 2gr
r2[f~g2w2 +
—~--~
wF’sin~F+
—~-~-~
cw’G(G+2)sin2F}
l6lT
(6.13) with the vector meson mass terms appearing via the KSFR relations and x’ = dxldr. The coupled equations of motion for F(r), G(r), and w(r) follow after functional minimization of (6.13). These are exactly the Euler—Lagrange equations due to the hedgehog symmetry. They are given by: F” =
—
+
F’
+
2
4 [4(G
8irf,~ r
+
1) sin F
—
sin 2F]
+
m~sin F
~[—2sin2F + cG(G +2) cos 2F],
(6.14a)
U.-G. MeiJJner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
288
G” =2g2f~(G+ 1
—
cos F) +
4 G(G
+
1)(G
+
2) +
w’(G
C ~2
+
1) sin 2F,
(6.14b)
3~
w”——w’+2f~g2w—
2F1sin2F +
c2
[G(G+2)F’cos2F+
G’(G+1)sin2F].
(6.14c)
Notice that for the minimal model (c = 0), the equations of motion for G and w decouple. In that case, (6.14c) agrees with the w-equation of motion derived by Adkins and Nappi [6.3], and the p-equation of motion agrees with the one given by Igarashi et al. [6.10]. To ensure singularity-free solutions and finiteness of the energy, the following boundary conditions have to be imposed on G(r) and w(r): G(0)2,
G(oo)0,
w’(O)O,
w(oo)0.
(6.15)
For a baryon number B = n solution, we have G(0) = —[1 (—1)~],and F(0) derive the asymptotic behavior of the profiles, for small r one has (B = 1) 3) F(r)—~IT + c1r + ~(r G(r)—~—2+c 2+C(r4) as r—’O, w(r)—~c 22r+ U(r4) —
=
nrr.
We can easily
(6.16)
3 + c4r
where c 1, c2, c5 and c4 are constants. For large r, all meson fields fall off exponentially, i.e.
as r—*
2m~~7r2
f~ G(r) F(r)—~e_3m~/rl e_em~ (i + w(r)
(6.17)
~.
Notice that for the vector-meson profiles the coupling to the 2ii- and 3’rr-continuum (for p and to, respectively) takes over as r becomes much larger than — m~,with ~ the Compton wavelength of the p, to-meson. The set of coupled equations (6.14a, b, c) together with (6.7) and (6.15) constitutes a boundary value problem of ordinary differential equations that can be solved using standard procedures. Using m,~= 138 MeV, f~ = 93 MeV, g = 5.8545 (i.e. m~= = 770 MeV), the numerical solutions to the variational problem are shown in fig. 6.1 (complete model, c = 1) and fig. 6.2 (minimal model, c = 0, in comparison to the results of ref. [6.10]). Because of the additional WplT-correlations in the complete model, the profiles F(r) and G(r) extend further out in space than in the minimal model. For the W-profiles the situation is reversed. Comparing the results of ref. [6.10],* where no w-meson is present, we see that due to the additional w-repulsion the chiral field F(r) and the p-meson profile G(r) are much less concentrated than in their case. In table 6.1, we give the skyrmion mass MH and the (hedgehog) soliton radius TH, with rH g~’enby (r~) =
*
r~= 4ir
J
r4 B
2F’ sin2F dr. 0(r) dr =
—
r
Compare also ref. [6.11]for a critical discussion on the work presented in ref. [6.10].
(6.18)
289
U. -G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
COMPLETE MODEL: MESON PROFILES
F(r)
-~
MINIMAL MODEL
.
ii-
AND p
PROFILES
3Kr)
I
____
O~5~~I.5
r[fm]
r [fm]
Fig. 6.1. Meson profiles for the complete model: the chiral angle F(r), and the vector meson profiles — G(r) and w(r) for g = 5.85, = 93MeV and m~= 138 MeV. The scale for the w includes a factor
Fig. 6.2. The chiral angle F(r) and the p-meson profile — G(r) for the minimal model in comparison with the results of ref. [6.10] for f = 93MeV, g = 5.85 and m~= 0.
f~.
We will refer to r 11 as the baryonic root mean square (rms) radius, it measures the extension of the baryonic charge, i.e. the baryon number distribution, of the soliton. Notice that the inclusion of the w-meson has two main effects [6.2, 6.15]: first, the skyrmion mass increases by —40% as compared to the model without (0 [6.10, 6.11]; secondly, the soliton radius increases by almost a factor of 2. As we will demonstrate in what follows, a small baryonic charge radius rH 0.5 fm is not inconsistent with typical nucleonic charge radii 0.8 fm, the difference is due to vector meson propagation. In table 6.2, we show the various contributions to the skyrmion mass as defined in (6.13). As in the case of the co-stabilized skyrmion [6.3, 6.4], we can derive a virial theorem relating the energy from the co-meson —
-=
Table 6.1 Properties of the skyrmion resulting from the lagrangian (6.1) with ir, p- and w-mesons. For comparison, the results of the model of ref. [6.10]including pions and p-mesons are also given (see ref. [6.11]for further details). The parameters used are m = 138 MeV, f = 93MeV, and g = 5.8545. MH is the static soliton mass, and rH the baryonic rms radius. PVMD denotes the model discussed by Kunz, Masak and Reitz [6.23]which incorporates only partial vector meson dominance, and is very similar to the complete model (c = 1) (cf. section 3.3).
MH[MeV] TH [fm]
Table 6.2 Various contributions to the hedgehog mass MR for the minimal (c = 0) and the complete (c = 1) model as defined in (6.13). The virial theorem 2E~ = —E~~ holds within ~ 4 X ~ Minimal model Complete model (c=0) ~c1) ________________________________________________________ E, [MeV] 773.2 751.3 E~[MeV] 42.6 39.0
c= 0
c= I
c= 0 w(r) = 0
PVMD
~
~
[MeV]
1474.4 0.50
1465.3 0.48
1057.0 0.27
1367.0 0.51
E~~[MeV] I2EJEWZ~— 1
MH [MeV]
33.0 511.9 3.9 X 10~6 1474.4
49.3 513.4 3.9 x 10_6 1465.3
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U. -G. Meifiner, Low-energy hadron physics from effective chiral Lagranglans with vector mesons
and the WZ term via ~
=
(6.19)
—2E~,
where E~sums up the co kinetic and co mass term. Equation (6.19) follows from the fact that the energy-functional (6.13) is bilinear in w(r) and w ‘(r). Performing a partial integration and eliminating the term proportional to r2w’2, one can use the equation of motion (614c) to show that E,,~+ ~ = ~ This virial theorem serves as an excellent check on the numerical calculations, for the actual solutions given before we find that 2E()/Ewz~ 1 ~ 4 x 10_6. The hedgehog configuration discussed so far does contain a mixture of the nucleon, the isobar, - . , i.e. —
~-
-
(6.20) with I denoting isospin and J angular momentum. U) is the coherent state peaked around the classical hedgehog configuration (U~U(x)~U) = U(x) and H is the parity operator. Adkins, Nappi and Witten [6.6] proposed a semiclassical scheme to discriminate between the N and the i~at the quantum level, by quantizing the spinning modes of the skyrmion. The idea consists of spinning adiabatically the hedgehog in isospace, generating a classical angular momentum, and then quantizing it using standard canonical rules. For the chiral field, this procedure amounts to the substitution U(r, t) = A(t) U(r) A~(t)
(6.21)
in the Lagrangian, and quantization of A. A(t) is a space constant, time-dependent SU(2) matrix. (Keep in mind that we restrict ourselves to the flavor SU(2)-sector.) Evidently, the space components of the co-field and the time-component of the p-field, w’ and p°,will be excited by this procedure. Whereas the Euler—Lagrange equations have sources only for w° and p’ on the classical level (hedgehog), this is no longer the case for a time-dependent U-field as given by (6.21). Defining A~dA!dt = A~A i~K
(6.22)
we have to leading order in the rotational frequency w —— K~*and consistent with the intrinsic parity of the vector mesons the following most general ansätze for the vector-meson excitations: T
p°(r,t) = [2/g] A(t) T [K ~ 1(r)+ FK F~2(r)]A~(t) , (6.23)
w(r,
t)
=
[cP(r)/r]K x F= 12(r) K X F.
~1(r), ~2(r)and b(r) are radial functions to be specified later. The classical p-field transforms as r~p’(r, t)
=
A(t)
T
p’(r) A~(t).
(6.23a)
Before proceeding, we should point out that this quantization procedure is by no means complete, since *
This is in accordance with the large Ne-expansion [6.6].
U. -G. Meifiner, Low-energy hadron physics from effective chiral Lagranglans with vector mesons
291
both translational and vibrational quantum effects are ignored, but it is justified in the framework of the large Ne-expansion. Injecting (6.22) to (6.24) into the Lagrangian density, we find the following time-dependent Lagrange function
=f
L(t)
d3x~=—MH + ATr(AA~)
(6.24)
with the moment of inertia A given by A=
4IT
f
~
A[F, G, w;
~2’
~]dr.
(6.25)
Straightforward algebra (cf. appendix C) gives the integrand A as follows: A[F, G, w; =
~
~2’
~]
~ f~r2[sin2F + —
8 sin4( F) 8 sin2( ~F) ~ + 3 ~ + 2~~2 + + 2~P2Ir2+ m2P2] + (1I3g2)[r2(3~2+ ~ +
~
+
4G2(~+
+
—‘~-~
41T
~
~1~2
—
~
—
~2 +
—
1) + 2~(G2+ 2G
F’CP(2sin2F— C~
c 8IT
1— C~2)+
—~--~
+
2)]
sin 2F(G
~‘
G~1
—
—
~)
(6.26)
with m = \/~gf~. Minimizing the moment of inertia gives the coupled Euler—Lagrange equations for the vector meson excitations ~ 1(r), ~2(r), and cP(r): 2sin2(~) =
—
+
1) —4f~g (1/r2)[G2(~ 2~ 1 1)— 2(G + 1)~2]+ m 1, —
C
3222
~‘
sin 2F~(G
—
2f~sin2(~) =
+
-
—
~
+4 [G2(~1
~F’ + ~ 3g3 2 ~‘(G 32irr
8ITr
+
=
m~+ 4g
+
m2tP
—
42
F’(2 sin2F
—
+
[2F’ cos2F(G
—
~(~) 0, =
~(0)
=
+
+
2G
+
2)~2]
3(G
1) sin2F,
(6.27b)
G~1
—
c~2)
~) + (G’
for the minimal (C = 0) and the complete model finiteness of the moment of inertia are =
2 1)
~2
-
C~ 1—
+ C ~2
(6.27a)
~2(co)=
0,
G’~1 G~
—
(C
—
=
~(0)
—
~)sin2F],
(6.27c)
1). The boundary conditions following from
=
~~c)
=
0.
(6.28)
292
U. -G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
By virtue of the small r-behavior of the vector meson excitations +
=
0(r3)
+
~ 2(r)34r+0(r) 2 + 0(r3) = E5r
asr—*0,
(6.29)
cP(r)
and the small r-behavior of the classical fields (6.16), one can deduce the following constraint: 2~~(0) + ~2(0) = 2
(6.30)
-
This can be used to check the numerics. The asymptotic behavior of the vector meson excitations for large r is given by ~ 2~/y2 1(r), ~2(r)~e ~(r) —— e3~/y3 as r—~
(6.31)
with y m~r.As first pointed out by MeiBner and Kaiser [6.21], the rather lengthy expression of A (6.26) can be simplified by performing a partial integration on the terms proportional to r2~2,r2~2, r2~~ and ~ and using the equations of motion (6.27) to eliminate all terms bilinear in ~ ~2’ ‘P, and derivatives. This procedure leaves the linear terms multiplied by a factor 1/2, and all other terms remain unchanged. The so-called reduced form of the moment of inertia ‘tred reads: ‘iced
=
~f~,r2[sin2F+ 8 sin4(F/2)] + ~f~r2[—4sin2(F/2)~ 2F
3g G~(2 ~2 2~~) + 4ir F’~sin =A~+A~+A~+ A~+A~, +
~
—
—
+ C
~ 16IT
1] (~‘Gsin2F) (6.32)
so that the contributions to the moment of inertia as defined in (6.32) are given by A,~= 4IT j~° A,~dr This reduced form of the moment of inertia serves as a further check on the numerical accuracy. To project onto states of good spin and isospin we now have to perform a collective quantization of the adiabatic rotation. This procedure relates the spin- and isospin-operators (J and I, respectively) to the rotational velocity K = ~iTr(TA~A)via —
J= —iATr(A~A~-), 1= +iATr(AA~).
(6.33)
The Hamiltonian corresponding to (6.24) reads: H = M~+ J2/2A
=
MH + I2/2A.
Evidently, the nucleon mass MN, the z~-massMA and the N~Xmass-splitting MA—MN follow as
(6.34)
293
U. -G. Meiflner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
MN=MH+3/8A,
(6.35a)
MA=MH+15/8A,
(6.35b)
MA—MN=3/2A.
(6.35c)
In fig. 6.3, we show the vector meson excitations ~1(r), ~2(r)and (2(r) = P(r)/r for the minimal model. The vector meson excitations for the complete model can be found in ref. [6.2]: In table 6.3, we give the moment of inertia, the various contributions to it as defined in (6.32), the nucleon mass MN, and the nucleon—z~mass-splitting MA—MN. Due to the too large skyrmion mass MN, the masses come out too high. Quantum fluctuations tend to decrease the mass substantially. The same statement applies to the N~mass-splitting [6.22]. We will come back to this point later on. It is 4IT interesting ~f to observe that almost all of the moment of inertia is given by the pionic contribution A,~= A~ 1~° dr. Kunz, Masak and Reitz [6.23] have recently investigated the PVMD Lagrangian of pions, p- and co-mesons discussed in section 3.3 (after eq. (3.27)). The WZ term is chosen such as to give the proper co—+3-rr decay width. Their results are very similar to the ones obtained in ref. [6.2], they find M11 = 1367, MN = 1456 and MA—MN = 358 MeV. These authors perform also a stability analysis and show that the (0-meson indeed stabilizes the soliton (for N~> 1).
MINIMAL MODEL: VECTOR MESON EXCITATIONS I I I
1.0• Table 6.3 .3I-,
Various moment of ine~ia defined (6.32) and the contributions total momenttoofthe inertia A. The nucleonas mass MN inand the nucleon—s mass-splitting are also given. Furthermore, the ratio of
U)
_________________________________________ o I-,
~ -0
~rl
—0.5
moment of inertia as calculated with (6.26) and (6.32) is given as a good check on the numerics. Deviations from 1 are ~ 10’. Minimal model Complete model c=0 c1 A,, [fm] 0.7569 0.6923
A,,~ A~ MN A~,, A~ A
—
-0.2
-
1.0
III
0.5
1.0
1.5
[f ml Fig. 6.3. The p-meson excitations ~1(r)and ~2(r)and the a-meson excitation ~1(r)= ‘P(r)Ir for the minimal model. Notice the different scale for all fields.
A/Ad
fm] [fin] [MeV] fm] fm] [fml MN [MeV]
—0.2519 0.1632 1564.2 0.1608 0.0000 0.8241 359.2 0.999992
—0.2258 0.1506 1574.6 0.0758 —0.0161 0.6768 437.3 0.999998
294 6.2.
U. -G. Meiflner, Low-energy hadron physics from effective chiral Lagranglans with vector mesons
The electromagnetic structure of the nucleon
As we already discussed in sections 3.2 and 3.3, electromagnetism can be uniquely coupled to the Lagrangian (6.1). On the meson level, this leads to the characteristic current-field identities [6.22] 2
jisoscaiar(X)
~ovector(X)
=
=
—(m~/3g)w (x)
(6.36a)
,
—(m~/g)p~3)(x)+...,
(6.36b)
where the dots in (6.36b) stand for higher order corrections due to the non-Abelian character of the p-meson, i.e., for strong fields higher order terms are induced which modify the simple structure current p-field. J~(x)evidently is the electromagnetic current. Equation (6.36) embodies Vector Meson Dominance (VMD) in the isoscalar channel and asymptotic VMD in the isovector channel, leading to the successful description of the pion form factor as described in section 1. Before discussing how this scheme works in the soliton (baryon) sector, let us briefly remind you of the basic definitions of electromagnetic currents and nucleon form factors. For a spin- ~ particle, the most general form of the matrix elements of the electromagnetic current operator is: (Nf(P’)~J~m(0)~Ni(P)) = ~(pI)[F1(q2)y~
+
~
u~q~]u(p),
(6.37)
where p, p’ are ingoing and outgoing four-momenta, and q = p’ p2) isand thethe four-momentum transfer Dirac form factor F of 2) the virtual photon as shown in fig. 6.4a. The Pauli form factor F1(q q are real functions of the Lorentz invariant q2 = q~q~’. Time reversal invariance, parity invariance2( and current conservation have been used to obtain the decomposition (6.37). A convenient coordinate system to work with is the Breit frame defined by p = (E, —q/2), p’ = (E, q/2) with E = \[M~ + q2/4 the nucleon energy. In the Breit frame, one has q = (0, q), i.e. no energy is transferred by the photon. Furthermore, the electric and magnetic parts of the electromagnetic current J~= (p, j) separate —
~
=
G~(q2)x~x 1,
(6.38a)
x
(6.38b)
(Nf(q/2)~J’(0)~N~(—q/2)) = GM(~) ~
PHOTON-NUCLEON
SCATTERING: KINEMATICS Nf(p~)
Nq(E,~/2)
N~(ps)
N~(E~/2)
(b)
(a)
2) and F,( q2). (b) In the Breit-frame, the incoming photon transfers no energy. Fig. 6.4. (a) Kinematics of the electromagnetic form factors F1 ( q
U. -G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
295
for where X~,Xt are two-component Pauli (iso)spinors and we have dropped the sign in G~~(—q2) convenience. GE and GM are the charge and the magnetic form factor, respectively, which are related to F 1 and F2 via: 2)= ~(q2)
+
F
~
G~(q
2),
G~(q2)= ~(q2)
2(q
+
F
2).
(6.39)
2(q
For static charge and current distributions the form factors GE and GM in the Breit frame can be directly related to the Fourier transform of the corresponding charge and current densities. * This identification is not possible in any other frame. The nucleon isoscalar (5) and isovector (V) form factors are related to the proton and neutron ones by G~’~ =Gs +GV ~640 EM
E,M
—
EM ~
and are normalized to the respective charges and magnetic moments: G~(0)=1,
G~(0)=0,
(6.41)
G~(0)=~~=2.79,G~(0)=jt~=—1.91. Let us now return to the soliton model with pions, p- and co-mesons and compute the pertinent isoscalar and isovector currents. The starting point is the classical solution, the hedgehog. The isoscalar current is associated with the isoscalar symmetry, which involves the co-field as the U(1)~gauge field. Performing an isoscalar gauge transformation exp{—ia(x)/2}, the co-meson transforms as ~.o,~(x)—~ a) 1~(x)= w~(x) (1 /g) ~a(x),
(6.42)
—
and we can read off the isoscalar currents from the terms proportional to o~a(x) (Gell-Mann—Levy method [6.24]). Because of (6.42) one would expect a contribution B~(x)to the isoscalar current from the w~B~-term in the Lagrangian (6.1), but this contribution is exactly cancelled by the ungauged Wess—Zumino term, which gives a contribution B ~‘(x).This holds for the minimal as well as for the complete model. Consequently, only the symmetry-breaking co mass term contributes to the current, we have —
JP~s(x)=
J~0(x)= —(m~/g)to~’(x).
(6.43)
The current-field identity (6.43) reflects the underlying Vector Meson Dominance of the models discussed so far. Similarly, the isovector current is the symmetry current of the isospin transformations hESU(2)~ 1
~‘rp *
1 ~L+ + —*h~rp h —(i/g)ô.~~h~h
t~
-
(6.44)
This procedure is justified from the large Ne-expansion, it follows after collective quantization of the center-of-mass motion as shown by
Braaten, Tse and Willcox [6.23].
296
U.-G. Metfiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
The resulting current is (cf. appendix C) J~’(x)
JP..V(x) =
—if~Tr{r[e
.
—i(1
rw~~Tr{~[U~
—
c)
~
ô~.~
~_
—
~(U~
U~) —i~g(~r.p~ +
. 8~U— ~
U~U~ U
—
~
UU +
.
UU
~]}.
(6.45)
Notice that the last term in (6.45) is only present in the minimal model due to the non-gauge invariance of its Wess—Zumino term. The electromagnetic currents follow after projection of the symmetry currents (6.43) and (6.45) on the charge operator Q = ~ + r3), as given by the quark charges of the underlying fundamental theory. One has Js1~s(x)= —(m~,/3g)w~’(x), J~’(x)= J~)(x),
(6.46)
where S and V stand for isoscalars and isovectors, respectively. The explicit form of the currents follows after inserting the meson fields of the rotated hedgehog and using the quantization rules =
—2iA Tr[A~AT],
~ = 2iA Tr[AAT],
a.jr~=
—
~Tr[Ar,A~j.
(6.47)
Straightforward algebra outlined in appendix C gives* 1cm(X) Os
=
—(mU)/3g) w(r), 2
(6.48)
J~(x)= ~ +
J~(x)=
{
{~
[4sin~(~)+(1+2cosF(r)) (1— c) ~ 2 ~(r) F’(r) sin2F(r)},
~1(r)+~2(r)]
8IT r ~
+
f~[2 sin4(~) (1
—
c)
17 —~—~
-
G(r) cos F(r)]
w(r) F’(r) sin2F(r)
41T
~oXr 2
~
)
-
r
All information about the electromagnetic structure of the nucleon is contained in the currents (6.48). Taking Fourier transforms one can immediately read off the nucleon electromagnetic form factors: G~(q2)=
—
~
4IT
J
r2 j 0(qr) w(r) dr,
(6.49a)
The w- and A-dependent terms of the isovector currents are not present in the complete model, in this case the WZ term is invariant under
isospin gauge transformations.
297
U.-G. Metfiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
iflU)
G~(q2)=~J
~2ITf(~)
j1(qr)~(r)dr,
(6.49b)
io(qr)[~~ ~
‘pFIsin2F]dr, (6.49c)
V
2
G~(q)= ~
IT
r
-
M~j ~—)j1(qr)
x [2f~(2 sin4(~) G cos F) —
+
(1
c)
—
42
wF’ sin2F] dr,
(6.49d)
with j0(qr) and j1(qr) the usual spherical Bessel functions. Note that the isoscalar form factors GSEM embody complete Vector Meson Dominance: they are entirely given in terms of the co-meson content of the soliton. In the case of the isovector form factors, VMD holds only asymptotically, i.e. at large distances r from the center of the soliton. In this region, F(r) is small, and the form factors are determined by the p-meson fields ~1(r), 42(r) and G(r) (for r ~ 1 fm). Before actually computing the form factors (6.49), let us examine whether the electric charges are properly normalized. This requires S 2 V 2 1 G~(q =0)=G~(q =0)=~.
(6.50)
For the isoscalar electric form factor this means 2 w(r)dr= —~g/m~.
(6.51)
4irf r
This can be easily proven by using the equation of motion for the co-field: [—V2+ m2j w(r) =
-~
~y
4IT
r
sin2F
—
3g 2 2 c [G(G l6irr dr -~-
+
2) sin 2F],
(6.52)
so that the total derivative appearing on the right-hand 2 = 0: side of (6.52) is only present in the complete model. The Fourier transform of (6.52) yields at q 4ITm~f r2
co(r)dr=
~ 4n-f
~--~
sin2Fdr= _(~f)B.
(6.53)
Observe that the correct normalization is obtained by virtue of the topological quantization of the baryon number (here B = 1). In the construction of the isovector charge density, the moment of inertia A enters. Comparing the integrand of (6.49c) at q2 =0 with A[F, G, ~ ~ ~ (6.26), it is not immediately obvious that the normalization G~(0)= 1/2 will indeed be satisfied. To prove this, we use
298
U. -G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
a trick proposed in ref. [6.21]. We separate a constant from the p-meson excitation ~1(r),i.e. we define ~(r) = ~1(r) 1, and rewrite the A-functional (6.26) in terms of m ~2’ and ‘p: —
A[F, G, w; r~, 2[—sin2F+ 2~(l+ 2cos F) =
+ 32 +
2(~+ 1)~2+
+ 3]—
+
~
~f~r +
~
[3r2~’2+ r2~2+ 4G2~(~ + ~2)
+
—~-~
+
2(G2
c
—~--~
2G
+
~2
+ m2’p2]
2)~]
‘pF’[2sin2F— c~ 2 c’q —
—
c]
—
4Ir
The Euler—Lagrange equations for ~f(0)=0,
+
2
i~are
8IT
~‘sin2F(1
+ i~+
G’~j).
(6.54)
subject to the boundary conditions
~(x~)=—1,
(6.55)
whereas the ones for 42(r) and ~(r) can be read off from (6.27). To make contact with the isovector charge functional, we again bring the moment of inertia in its reduced form by performing a partial integration and using the equations of motion for n, ~2 and ‘p (cf. (6.26) to (6.32)). This leaves us with
Ared[F,
G, ~
m ~2’
2[3
—
~f~r
‘p]
sin2F
+
~(1 + 2 cos F)
+ ~2] +
(1— c)
—~-~
4i~
‘pF’ sin2F,
(6.55a)
where we have thrown out a total derivative in the complete model. Rewriting (6.49c) in terms of Th and ‘p, it follows immediately that the isovector charge is indeed properly normalized, i.e., G~(0)= 1/2. Let us now discuss the electromagnetic properties of the nucleon as given by the form factors (6.49). We begin by a discussion of the isoscalar charge distribution which reflects the co-meson content of the nucleon. It is particularly interesting to compare the normalized isoscalar charge density Ps(r)
O,S
=
2Jem(X)
2
=
—(2mU)/3g)w(r),
(6.56)
4ITfpS(r)r2drl to the baryon number distribution B 0(r) which is the source of the co-field. In fig. 6.5, we show ps(r) and
B0(r) for the minimal model. Note that the isoscalar photon couples to the co-meson, and not directly to B0(r), due to the underlying VMD. Hence, the2which isoscalar chargethe radius as seen ofbybaryon the photon is measures distribution number considerably larger than the radius rH = (r~)V inside the soliton. To illustrate this, let us consider the isoscalar charge form factor (6.39a) G~(q2)=
-
J
r~j 2+q2 0(qr) a)(r) dr = rn
J
r~j 0(qr) B0(r) dr,
(6.57)
U. -G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
299
ISOSCt,LAR CHARGE DISTRIBUTION
r[fm] Fig.
6.5.
Isoscalar charge distributions for the minimal model. B
0(r) is the normalized baryon charge density, whereas the photon couples to the
isoscalar electric charge density as given by the s-meson field.
2 m~)w(r) = [(~ g) B where we have used (V 0(r) + total derivative] and the convolution theorem of Fourier transforms. From (6.51), we can immediately derive a relation between the isoscalar charge 2 and r~,it reads radius (r~)~ GW=0) ~ (6.58) —
Hence with (r~)1/2 0.5 fm (cf. table 6.1), one obtains (r~)1/2 0.8 fm, in good agreement with the empirical value (r~)~2 =0.79fm. The significant difference between (r~)~2 and ~r~)U2 is due to the propagating co-meson which connects the isoscalar photon with the baryon number distribution. Note that in the limit of “frozen” vector meson degrees of freedom (mU)—~,gU)/mU) = constant) this difference would disappear. This is in fact the case in Skyrme-type models [6.6]. It is obvious, however, that important physics at length scales \/~/m,, 3 0.5 fm is lost in this limit. Let us now discuss the static nucleon properties and the proton/ neutron charge distributions. The latter ones are determined from the isoscalar and isovector charge distribution given in (6.48). The resulting proton and neutron charge distributions are shown in fig. 6.6 for the complete model. The static electromagnetic nucleon properties are summarized in table 6.4. The proton and neutron magnetic moments are somewhat too large in the minimal model and reasonably well-reproduced in the complete model. In both models, the isoscalar magnetic moment comes out close to the experimental value ~ = 0.44 (p~= 0.40 for c = 0 and ~ = 0.47 for c = 1). For the minimal model, the isovector magnetic moment comes out too large (p~= 2.97 for c = 0), the empirical value is ~ = 2.35. The improvement in the complete model (c = 1) is due to the cop-i-r interaction contributing mainly to the isovector magnetic moment. The electromagnetic charge radii are in good agreement with the data for both models, with the exception of the neutron charge radius, which is somewhat2 overestimated since it = 0. The electric and involves a very delicate cancellation of the isoscalar and isovector currents at q magnetic form factors of the proton and the neutron are displayed in figs. 6.7—6.10. In the minimal —
-~
300
U.-G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons NUCLEON CHARGE DISTRIBUTIONS
PROTON
015
NEUTRON
O~”I~O
[III,
r[fm]
“2.0
r[f ml 2
Fig. 6.6. The proton (left) and neutron (right) electric charge distribution for the complete model. Notice that the factor 41Tr
has been included.
Table 6.4 Electromagnetic properties of the proton (p) and the neutron (n). The electric and magnetic charge radii as well as the magnetic moments are given. For comparison, we show the results of the conventional Skyrme model [6.6]. Complete model
0.92 —0.22 0.84 0.85 3.36 —2.57 1.31
0.97 —0.25 0.94 0.94 2.77 —1.84 1.51
(r~)~2 [fin] (r~), [fin2] (r~)~’2 [fm] (r~,)~2 [fm] ~&,, [nm] is,, [nm]
Minimal model
I)L~//Lj
Conventional skyrmion 0.88
—0.31 0.79 0.82
1.97 —1.24 1.59
Experiment 0.86
±0.01
—0.119±0.004 0.86±0.06 0.88 ±0.07 2.79 —1.91 1.46
model, the q2-behavior of all form factors is well-reproduced. * The complete model tends to produce charge and magnetic moment distributions which are slightly too large in size, but keep in mind that the normalization of the magnetic form factors comes out much better in the complete than in the minimal model. For comparison, we also display in figs. 6.7—6.10 the empirical dipole fits [6.27] G~(q2)= (1
—
q2/M~2,
G~(q2)= ~~(1 n
—
2
q2IM~)2, 2
(6.59)
2—2
G~(q ) = ~t~(1 q /MD) G~(q2)= 0.20:2 (i L13:2yl~Gfl(q2)~ —
-
*
We show a comparison with data up to q2 = 0.6 (GeVIc)2, these statements hold true up to q2
hold beyond the applicability of the model (cf. section 7.2).
1 (GeV/c)~Any further extrapolation would
301
U.-G. Mei/3ner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
2)
PROTON CHARGE FORM FACTOR G~(q I
I
I
I
I
NEUTRON CHARGE FORM FACTOR G~(q2) I
1.0
0.08 0.06
0.5 ~
I
I
J7~~TT) dipole
0.02 0.04
I
0
I
0.1
0.2
0.3
0.4
—IJ.02
I
0.1
0.5
I
0.2
1q
21
~2[GevZ]
0.3 [GeV2]
0.4
Fig. 6.7. The proton electric form factor. The solid lines gives the result of the minimal model, the long-dashed line is the result of the complete model. For comparison, the empirical dipole fit (e.g. (6.59)) is also shown. The data are taken from ref. [6.85].
Fig. 6.8. The neutron electric form factor. The solid line gives the result of the minimal model, the long-dashed line refers to the complete model. The data are taken from ref. 16.851.
PROTON MAGNETIC FORM FACTOR G~(q2)/G~(0)
NEUTRON MAGNETIC FORM FACTOR G~(q2)/G~,(0)
I
I
I
I
I
I
I
I
1.0 1.0
0.5
— S..
S.
dipole
dipole I
O
0.1
0.2
I
0.3
0.4
0.5
1q21[Gev2] Fig. 6.9. The proton magnetIc form factor. The solid line is the result of the minimal model, the long-dashed line the one of the complete model. The data are taken from ref. [6.85].
0
0.1
~
I
I
I
I
0.2
0.3
0.4
0.5
q2J[Gev2] Fig. 6.10. The neutron magnetic form factor, notations as in fig. 6.9. The data are from ref. [6.85].
302
U. -G. MeiJiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
with M~= 0.71 GeV2. Keep in mind that in the calculation of the electromagnetic form factors we have neglected any recoil effects. To leading order in N~,this approximation is justified [6.251,but it deserves further studies. This problem is presently under investigation [6.28, 6.29]. We should stress, however, that the results exhibited in this section are extremely encouraging if one keeps in mind (i) the simplicity of the model and (ii) that we did not allow for any readjustment of the parameters g and f~, as it is commonly done in Skyrme moaels [6.3, 6.6, 6.7, 6.14]. 6.3. Axial properties of the nucleon In addition to the electromagnetic form factors, the axial form factors give further information about the structure of the nucleon. Particularly, the axial current is only partially conserved (PCAC [6.301) and the axial charge is not equal to one. These facts have to be explored in the context of the chiral models we are discussing. Before going into details, we shall remind you about the definitions for the axial current. Let N(p)) be a nucleon state with four-momentum p, then the most general matrix-element of the axial current operator which conserves parity and time reversal invariance is (N(p’)~A~(0)IN(p))= ~(pt)[G~(q2) y~+ ~
G~(q2)q~+ 2MN GT(q2) ~q~]y5
~ u(p’)
(6.60) with q2 (p p’)2, and MN the nucleon mass. G~(q2)is the axial form factor, its value at q2 = 0 gives the axial charge gA of the nucleon which can be measured in /3-decay. Experimentally, one has —
g~= (1.25 ±0.01). G~(q2)can be measured in neutrino reactions (p + v~—~n + ~i~) [6.31, 6.32, 6.33, 6.34] and through pion electroproduction of protons [6.35, 6.36, 6.37]. G~(q2)is the so-called induced pseudoscalar form factor, which is experimentally not well-known [6.171and will not be considered in what follows.* G~(q2)is the so-called induced pseudotensor form factor, it is only non-vanishing if one allows for second class currents, i.e., currents which have opposite G-parity to the ones defined by the operators y y~and q~y~.For calculating these form factors, it is most convenient to work in the Breit frame. One finds ~ =
~
G~(q2)~T
+
(GA(q2) =
—
~
2MN GT(q2)
Gp(q2))ff~]~
x~-’q
~-
x~’
(6.61a)
x~,
(6.61b)
so that the induced pseudotensor form factor related to_possible second class currents is given entirely by the time-component of the axial current. E = VM~N+ q214 is the nucleon energy, ~T = 4 is the transversal and ~L = 4 the longitudinal component of the spin operator. Let us first discuss the question of second class currents related to the time-component ~ of the —
*
Keep in mind that we could easily calculate G~(q2)in the models presented.
U. -G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
303
axial current. For the hedgehog ansatz, the most general form of the time-component A0’~is given by =
A 0(r) Tr{TaAT (K .
X
r)A~}
(6.62)
with A0(r) a model-dependent radial function. A(t) and K, the rotational velocity of the soliton, are defined in (6.21) and (6.22), respectively. The trace expression in (6.62) now has to be quantized in terms of spin- and isospin-operators. A naive application of the quantization rules (6.47) would give after inverting the order of A and K: Tr{TaATA+ (K .
X
r)}
—
2abbcd
if
rd =
—
(6.63)
a
However, K is a differential operator in the coordinates parametrizing the rotation matrix A (t) [6.4]: A=a0+ira,
~a~1; (6.64)
ô K=a.-~-——a0-~--—aX1 1/
Therefore, K and A do not commute, one obtains [K, A~]= —(1 /4A)r. A~,
(6.65)
so that the commutator contributes to the spin—isospin matrix element Tr{raAT[(_
A~x
~
=
—
~ Tr[raAi~ rA~]—~ +
,
a
(6.66)
2) 0 for all q2. Hence, chiral Therefore, the spin—isospin matrix feature element that of (6.62) vanishes, GT( q soliton models have the desirable they are free ofi.e.second-class currents. This is indeed consistent with the data [6.17]. Let us now discuss the space-component of the axial current. First, we consider it on the classical level, i.e. for the hedgehog. Its most general form follows to be =
A 1(r) ota
+
A2(r) ~
(6.67)
where i and a are space and isospin indices, respectively (i, a 1,2, 3). A1(r) and A2(r) are radial functions to be determined later. Performing the adiabatic rotations, we find AI~a(r)= [A1(r) ~l,b + A2(r)I’F”] x ~ Tr [ArA+r~2].
(6.68)
Collective quantization of the isospin rotation and using (6.47), we can read off the axial form factor 2) from the Fourier transform of (6.68): GA(q G~(q2)= [r2 j 0(qr) A1(r) + j1(qr) A2(r)] dr. (6.69)
-(~)~fJ
(~)
304
U. -G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
It is now straightforward to determine the axial vector charge g~
gA =
G~(q2= 0)
=
—
~
J
r2[A 1(r)
~A2(r)]dr
+
(6.70)
2: and the nucleon axial radius t~r~)’~ =
6 dG~(q2)
-
=
4M
-
~
J
r~[A 1(r) + ~A2(r)1dr.
(6.71)
Furthermore, the PCAC relation I9.~A~ = m~f~T,where ~ is the interpolating pion-field, can be written
9A~”~ =V~A’~
(6.72)
with Da(r) =
—
(~r~+ A~(r)+ ~ A2(r))u.
~
~ns Ta
D(r) ~ F.
(6.73)
Now, we have to determine the radial functions A1 2(r). This can be done in the following way. The axial current is the symmetry current of axial rotations of the U- and 4LR-fields (6.6), namely 2)A, ~
(6.74)
sESU(
~
U—~sUs. Using the Gell-Mann—Lévy method [6.26], this symmetry current can be read off from the term proportional to s s Fixing the gauge ~ = = ~ at the end, we obtain (cf. appendix C). ~.
2~
A 1 (r)
=
f~[
A 2(r) = —A1(r)
F (G
+
1)
—
sin2F]
—
~-
(sin~F cG cos2F), —
(6.75)
+f~F’(r),
where we have performed one partial integration on the Bp-term. * It does not influence the results presented in this section. In table 6.5 we summarize the axial properties of the nucleon, in comparison with the experimental data and the results of the conventional Skyrme model [6.6, 6.29]. The axial vector coupling constant g~comes out a bit too small in both models, but the value g~= 0.99 for the complete model is clearly improved as compared to the conventional Skyrme model. In the complete model, the wpir-interaction mocks up A1-degrees of freedom (cf. section 4.3); therefore, simulating some kind of axial-vector meson dominance as discussed in refs. [6.38] and [6.39]. The axial radius comes out close to the empirical value for both models. In fig. 6.11 we show the normalized *
This seemingly arbitrariness finds its natural explanation in the discussion of the strong NN~form factor (section 6.4).
U.-G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
305
Table 6.5 Nucleon axial properties. For comparison, the results of the conventional Skyrme model are also shown [6.6]. Minimal model
Complete model
Conventional Skyrme model
Experiment
0.99 0.62
0.65 0.35
1.25 ±0.01 0.68 ±0.03
0.88 (ri) ~ [fm] 0.64
2)/G~(O)
NUCLEON AXIAL FORM FACTOR G~(q 1.0
\
0.5
\~.
0
0.5
~,,/diPoIe
1.0
1.5
I~2I[Gev2] Fig. 6.11. The normalized axial form factor GA ( q2 ) Ig~for the minimal model (solid curve) and the complete model (long-dashed curve). Data are from refs. [6.31—6.37].
axial form factor GA ( q2) Ig~in comparison with data and the empirical dipole fit of ref. takes into account data up to q2 3(GeVIc)2 and is given by g~(q~ = 1.23(1
—
MA)
MA
=
1.05~~ GeV.
[6.341, which
(6.76)
The agreement of both models up to momentum transfer q2 ~ 1 (GeV/c)2 is very good, due to the op’rr-interaction the complete model gives a somewhat better description of the data. Let us finish this section by discussing the PCAC relation (6.72, 6.73). With A 1(r) and A2(r) given by
306
U. -G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
(6.75), we find D(r) =
—
3 m~sin F + c 8ir
~—
—~-~
Since the pion-field is given by ir
—
2F ~4 r G(2 cos
—
(G
+
1) cos 2F).
(6.77)
—(f
1,13) sin F, we see that the PCAC relation is exact in the minimal
model, i.e. ~
=f,m~,ir.
(6.78)
In the case of the complete model, there is a small deviation from the exact PCAC result due to the up’Tr-term in the WZ action (cf. section 3.4, eq. (3.78)). We will come back to this point in the following section. 6.4. Meson—nucleon form factors and the nucleon—nucleon interaction In One-Boson-Exchange models (OBE), the meson—nucleon interaction is cut down at large momentum transfer by the use of ad hoc form factors, commonly of the monopole-type = 2) m~~MNN G~~~(q
A -q
(6.79)
where M = {ir, p, o} is the meson under consideration and A a cut-off, typically of the order of 1 GeV. q is the four-momentum transfer of the meson as shown in fig. 6.12. ~MNN is the pertinent meson—nucleon coupling constant defined by G(q2 = —m~)= gMNN, i.e., the coupling constant is defined at the respective meson pole. For more details on this point we refer the reader to the extensive review by Machleidt, Holinde and Elster [6.40]. These strong form factors can be calculated in our models as we will show in what follows. We can make further contact with the OBE models keeping in mind that
STRONG MESON NUCLEON FORM FACTORS
N(p~)
Fig. 6.12. Kinematics of the strong meson—nucleon form factors
G~~~(q2), where
M stands for i~,p, w,.
..
U. -G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
307
meson exchange in lowest order can indeed be discussed at the classical level, as a pertinent example let us quote Yukawa’s original papers [6.41] on the pion exchange. Let us now describe how the strong meson—nucleon form factors emerge. Denote by ~M any meson field ii, p, w, A1,. , its static equation of motion can always be cast into the form .
2
.
m~)~M(r)
= JM(r) (6.80) (V with m~the pertinent meson mass. The source functions are highly non-linear functions of all the mesons under consideration; their Fourier transforms correspond to the meson—nucleon form factors. First, we will study the pion field surrounding a static nucleon. For that, we have to define the pion-field. Naively, one would identify it with the field ~n~(x)that enters in the unitary matrix U(x) = exp{ira ~()/f} but this representation involves ‘7rrr-coupling to arbitrary orders. Therefore, it is more convenient to work in the quaternion representation —
U(x)
=
(1 If.,,)[o~(x) + ir ~c(x)]
(6.81)
and identify the physical pion-field with ~a(X) This agrees with the way the pion appears in the linear if-model [6.26]. There the pion—nucleon sector is represented by the familiar Lagrangian of “oldfashioned” meson theory =
~rN
~(o~)2
+
~m~p2 —g
/iiy
5rVi~p+ 4n(t/i, ~fr),
(6.82)
where ~i, ~1iare Dirac spinors for the nucleon. Using the hedgehog ansatz, rotating the hedgehog state and performing a collective quantization, we can read off ~a(T) in terms of the chiral angle F(r) as follows: =
~ Tr{AT.
FA+Ta}
using (N!Tr(AT. FA+ra)IN) = reads 2 m~)4~(r)= J., (V 1(r).
sin F(r) = ~ (NIo
—
—
~
~ sin F(r)
FT°~N~.The
(6.83)
corresponding static field equation for ~c(r)
—
(6.84) fig. 6.12 with the pion as the Now, let meson. us derive strong ITN form factor ofGN~,~(q incoming Thethepertinent matrix element the pionic current in the Breit frame reads* 2).Consider
(N
~2) ~ 1(q/2)~J~(0)~N1(—q/2)) = G~~~(q
(6.85)
where MN is the nucleon mass and ~ are Pauli (iso)spinors. The left-hand side of (6.85) is nothing but the Fourier transform of the pion current as defined in (6.84) G
~
2\
*
a
~.
NN~TI~~J~ O~ N
qr
—
=
~j3
lq-r
—(q
f
2re+ m~) d3r ~l~r 1a
~a(r).
Defining G,,NN in this way, we avoid the problem of the vanishing linear coupling of the pion-fiuctuations to the hedgehog.
(6.86)
308
U. -G. Meifiner, Low-energy hadron physics from effective chiral Lagrangiaris with vector mesons
Furthermore, we can reduce (6.86) to a one-dimensional integral by defining the radial source function J~,(r)as J~(r)=
i,..
FT~
J1,(r)
(6.87)
which follows naturally from the hedgehog structure of the pion field. With that, the form factor GNN~(q) reads 8~rMN 1 2 = j r j 1(qr) J~(r)dr. (6.88) Using the equation of motion (6.14a), we can determine the source function J~,(r),it follows to be J~(r)= ~ f(F’)~sin F
+
m~(1 cos F) sin F + 2 sin F [1 2(G —
2F
+
8irf~ 2 r cos F[2 sin
—
—
+
1
—
cos F)
cos
cG(G +2) cos 2F1}.
F] (6.89)
J 1,(r) is shown for the minimal (c = 0) as well as for the complete model (c = 1) in fig. 6.13 (in arbitrary units) [6.42, 6.431. It is peaked around 0.5 fm away from the nucleon center, consistent with the baryon number charge radius Kr~)1/2 0.5 fm. It is interesting to extract the leading behavior of J1,(r) as r becomes large; using the expansion (6.17), we find 3(r) + /3 F(r) G(r), r-+ (6.90) J~(r) a F where a and /3 are constants. These terms correspond to 3’rr and ‘Tip-contributions, in agreement with ~,
—
PION SOURCE FUNCTION J7r(r)
r[fm] Fig. 6.13. Profile of the pion source function J~(r),eq. (6.89). The overall factor (f~I3)is scaled out.
U.-G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
309
the dispersion theoretical approach to the NN’rr-vertex function [6.44]. The ITNN-form factor in the spacelike region is shown in fig. 6.14 for the minimal and the complete model. For small momentum transfer q2, it follows closely a monopole with a cut-off A = 0.86 GeV, quite consistent with OBE models [6.4]. For larger momentum transfer q2 ~ 10 fm2, G~N~(q2) falls off somewhat faster than a monopole, consistent with perturbative QCD [6.45] and bag model [6.461calculations. In the timelike region, GNN~,(q2)defines the nucleon—pion coupling constant gNN,~, i.e. (6.91) We find ~NN~ = 15.1 for the minimal model and ~NN~ = 14.5 for the complete model. Experimentally, one has ~NN~ = (13.5 ±0.1). For the minimal model, this ~NN~ agrees with the one as determined from the Goldberger—Treiman relation g~If~ = ~
(6.91a)
whereas for the complete model ~ = 1.15, i.e., the Goldberger—Treiman relation is violated by 15% due to the Wess—Zumino term. Experimentally, the GTR holds within 7%. Finally, let us mention that the nucleon—pion form factor has a branch point at q = —9m~which represents the starting of the cut corresponding to the 3ir-continuum, i.e. J,~(r) exp{—3m~r}1r3. Notice that the GTR (6.91) is nothing but the q2—*0 limes of the PCAC relation Ô,~A~’ =f~m~r in —~
gl,NN
(q2) -glrNN
(q2 r-m~,)G~NN(q2)
G~NN(q2) IN THE SPACE-LIKE REGION (MINIMAL & COMPLETE MODEL)
GNN~.(~2) (Complete)
0
I
I
I
I
I 5
I
I
I
I
I 10
I
I
I
I
15
I
I
I
I
I
20
I
I
I
I
25
q2 [fm_2]
Fig. 6.14. The pion—nucleon form factor G~~~(q2) for the minimal and the complete model. For comparison, a monopole form factor with .4 = 0.86 GeV is shown.
310
(1. -G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
terms of the axial, the induced pseudoscalar and the strong irNN-form factor M~[G~(q2)
—
Gp(q2)]
~
=
~rn~2 G~~~(q2).
(6.91b)
For this relation to be satisfied, it is crucial to perform a partial integration on the w~B~-contribution of the axial current as discussed after eq. (6.75). This partial integration does not affect the Goldberger—Treiman relation at = 0 but is justified by the generalized PCAC relation (6.91b). For the o- and p-meson, the starting field equations are (V2
—
m~)w’~(r)= J~(r)
(V2
,
—
m~)p~(r)= J~(r)
,
(6.92)
where the prime on J~(r)denotes that the p-meson self-couplings are included in the p-meson source function for calculating the pNN-form factors. Before actually computing the (radial) source functions J~(r)and J~(r),let us consider the most general matrix elements of .J~(r),Jjr) between nucleon states, they read (N(pJ~(0)~N(p)~ =
u(pI)[F~P)(q2)
~
+
i F~(q~)if qP]r u(p), N
(N(p’)~J~(0)~N(p)) = ~(p~)[F~(q2) y~+
(6.93)
~
which define the vector and tensor coupling form factor F 2) and F 2), respectively. Empirically, in 1( q 2(q the case of the p-meson the tensor coupling —F 2 dominates the coupling, towhereas for the 2) at q2 = 0vector is proportional the anomalous o-meson the situation is reversed. The form factor F2(q magnetic coupling of the particle under consideration. We have F~(0)= 14.1, but F~0~(0) = —0.12 [6.40]. The actual evaluation of the form factors F~’~(q2)is performed in the Breit frame with q~q~ = —q2. For that, we need the radial source functions J(r) and J~(r).These are defined via (in the Breit frame) Jo.a(r)
=
J~°(r)a
,
J(r)
=
J,~(r)cr x
rTa,
(6.94) J~(r)= J~,(r)~
,
J~(r)= J~(r)o~x
~.
The p- and o-electric form factors G~’°~(q2) follow as Fourier transforms of the time components of the sources ~ the magnetic form factors are given by Fourier transforming the spatial components of the sources. The form factors F 2) and F 2) are related to G~(q)and G~(q2)via 1(q 2(q G~’~( q2) F~( q2) + [q2I4(M~)2IF~( q2) G~(q2) = F~°’~(q2) + F~’°~(q2). The radial source functions defined in eq. (6.94) are given by
(6.95)
U.-G. Meifiner, Low-energy hadron physicsfrom effective chiral Lagrangians with vector mesons
r2 J~(r)
~
~G2(2—2~
{~g2f2r2sin2(~) +
1 ~)+ c
[~‘ (G + 1)sin2F+2F’~1},
—
2 J~(r)=
{2g2f~r(1 cos F)
~
—
+
311
[G2(G
+
3)]
+
c ~2
w’r(G +1) sin 2F},
r 3 r2 J~(r)= r2
J1()
3 F’ sin2F
c
~T—
—~
~
+c
+
(6.96) [F’G(G + 2) cos 2F + G’(G
—~
rF’(2sin2F- ~
r[2F’(G
—
-
+
1) sin 2F],
c~2)
G~ 1
—
~)
cos 2F + sin 2F(G’
—
G’~1 G~ —
—
2)which is related to the dominant contribution of the In fig. 6.15 we show the pNN-form factor F~(q matrix element (6.93) for the p-meson current. Figure 6.16 shows the wNN-form factor F~(q2)which dominates the w-exchange. Notice the branch at q2 = —9m~when the source function enters the 3’ir continuum.* For smaller momentum transfer (q2 ~ 0.5 (GeVIc)2) both form factors follow closely a monopole with A 1 GeV, for larger momentum transfer they fall off somewhat faster. The results for F~(q2)and F~(q2)are similar. This is in rough agreement with phenomenological OBE potential fits [6.40]. In table 6.6 we present the normalizations F~(0) and F~(0)together with the ratio i~ = F~(0)IF~(0) which determines the tensor to vector coupling. For comparison, the recent results of the Bonn OBE potential [6.40] are also shown. The strength of the tensor coupling comes out remarkably well. However, the model does not account for the empirical fact that KPIKV, where Ky is the isovector anomalous magnetic moment (K~= F~(0)IF~’(0)), is 1.6—1.8. From the magnetic moments given in —
STRONG wNN FORM FACTOR F
STRONG pNN FORM FACTOR Ft
_
~2
~IIIT
[GeV2/c2]
Fig. 6.15. The pNN form factor F~(q2)in comparison with a monopole with cut-off A 1GeV.
*
A similar branch point exists for F~(q2)(fig. 6.15) at q2 = —9m~,.
2) 1w(~
(~2)
-9m~.O
0.5
1.0 1.5 ~2[GeV2/c2]
2.0
Fig. 6.16. The oNN form factor F~’(q2)in comparison with a monopole (A = 1GeV). At q2 = —9m~,,the source function for F~’(q2) develops a cut.
U. -G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
312
Table 6.6 Normalizations of the vector (F 1) and tensor (F,) couplings of the p-mesons to the nucleon. = F2(0) /F~(0)and the empirical values are taken from ref. [6.40]. Complete model (c=1)
Empirical values
2.7 14.4 5.3
3.2 14.2 4.4
2.3 14.1 6.1
F~(0) F~(0)
Minimal model (c=0)
Kp
table 6.4 we find that KpIKy = 1.1 for the minimal model and Kp/KV = 1.2 for the complete model. (There exists up to date no serious model calculation explaining this effect, although a suggestion has been made in ref. [6.45] in the framework of a chiral bag model.) For the o-meson, we find F~(0)= 8.78 for both models to be compared with F~°~ 11.5 [6.40]. The oNN coupling constant can be read off from the u electric current, J~(r)= [~~)NN B°(r)+ total derivative] as [6.46, 6.131 ——
~
(6.97)
which is roughly half of the empirical value g~,~~/41T = 10—12 [6.47, 6.48]. We will come back to this point later. We are now at the point to make some more qualitative statements about the nucleon—nucleon interaction. For that, consider two nucleons at positions r1 and r2 at a sufficiently large distance r = r~ r21 so that their baryon number distributions do not overlap and the mutual polarizations of the two nucleons can be neglected. In this case, the nucleon—nucleon (NN) potential mediated by the meson fields M can be written as —
V(rI,r2)=~Jd~rJM(rI,r)~M(r2,r)
(6.98)
in terms of the source function of nucleon 1 and the field generated by nucleon 2. Within this classical framework, we can discuss certain aspects of the NN interaction. First, from the discussion of the ‘irNN form factor it is obvious that the basic features of the one-pion exchange at distances r ~ 1 fm are well-reproduced. Second, the dominant contribution to the p-meson exchange, the tensor potential proportional to (o~x q) (u2 x q), comes out remarkably well as shown in table 6.6. Let us therefore consider the isovector tensor NN-interaction. It is known to be dominated by ii- and p-exchange which contribute with opposite signs [6.49]. In our approach, the tensor interaction in momentum space follows to be 2) S V~(q)= V~~(q 12(q) r1 T2. (6.99) 2) is given by S12(q) = 3o1 . ~ is the tensor operator, and V~~(q —
2 2 V~~(q )
—
~1
L
2
G~~~(q (2MN)2 ) q2
2
q +
2
m~
—
f2\
~ (2MN)
2
)
q q2 +
m~
(6.100)
U. -G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
313
is nothing but the p-meson magnetic form factor as defined in where the pNN-form factor G~N~(q2) (6.95). V~~(q2) is shown in fig. 6.17 for the minimal model. It is plotted such that one can see first the influence of the pion—nucleon form factor as compared to one-pion-exchange with pointlike nucleon sources, and then the role of p-exchange in cutting down the tensor force as q2 increases. For comparison, some data from NN scattering amplitudes are shown [6.50]. The model reproduces nicely the destructive interference between IT- and p-exchange and accounts well for the data [6.421.The results for the complete model are similar. As discussed in section 1, the w-meson accounts to large extent for the repulsion of two nucleons in the central channel. For large enough separation of the two nucleons, i.e. no substantial overlap, the NN central potential due to the w-meson is given by* 2
V~(r)=
~11N~
f
—m,,Ix—yI
d3x d3y B 0(~x rj) e —
B0(~y r2~)
—
(6.101)
—
for the minimal model using J~(r)= g,aNN B°(r). This potential gives the solid line in fig. 6.18. At distances much shorter than 1 fm, one cannot use (6.101) any more due to the strong overlap of the baryon number distributions and the deformation effects from the mutual polarizations. These shape
NN SOVECTOR TENSOR INTERACTION 1,’V 1(OPE) I
0.3 ,2
I.)
,
~i 3fm3J
11.1 l~I
IGeV 0.2
/
I
/
/I
._—--_..~ ~
1.5
-
TI
only 1.0
-
0.5;
I~~I [fm11 Fig. 6.17. The isovector tensor potential (6.100) as a function of momentum transfer qI. The short-dashed curve gives the one-pion exchange with point-like nucleons. The long-dashed curve shows the influence of the ~irNNform factor G~~~(q2) with the coupling constant readjusted to the empirical one. The solid curve is obtained with the complete expression (6.100) including p-exchange. The empirically deduced data of energy E= 140 MeV are taken from ref. [6.50] (dots). The vertical scale includes the empirical nucleon mass. *
NN CENTRAL POTENTIAL
-
-
I
r [fm] Fig. 6.18. The repulsive central NN potential generated by the wmeson field. The dashed part indicates that the product-ansatz used for separations R ~ 1 fm is only a rough approximation to the true potential l’~fr).
For the complete model there are small corrections from the total derivative term.
314
U.-G. Metfiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
EFFECTIVE wNN COUPLING CONSTANT
1~1
~eff
__ N
=
N
__
N
N
N
N
Fig. 6.19. Important contributions to the effective wNN coupling constant used in OBE-potentials. These effects (intermediate z~’s)are not included in the classical framework [eq. (6.98)] considered here.
deformations have been indeed observed in extensive computer simulations of the skyrmion—skyrmion interaction [6.51, 6.52, 6.53]. We can nevertheless get a rough estimate of the repulsion in the NN-system by investigating the baryon number B = 2 solution of the coupled equations (6.14a, b, c) with the pertinent boundary conditions F(0) = 21T and G(0) = 0. Comparing the mass of this object with twice the mass of the B = 1 solution gives =
MH(B
=
2)— 2M
11(B
=
1)
=
1.25 GeV.
(6.102)
t~Mis almost entirely given by the mass difference from the Wess—Zumino term, which generates the dominant B~B~-coupling of the baryon current at short distances. ~M is therefore a qualitative measure of the short distance repulsion generated by the *o-field. It turns out to be too small by a factor of two if one compares the volume integral of this interaction as shown in fig. 6.18 (dashed line) with the corresponding one of the Bonn OBE potential. The missing repulsion in the central NN-potential is clearly related to the too small wNN coupling constant (cf. eq. (6.97)). It has to be seen in the same context as the missing intermediate range attraction in these models. Both effects are associated with two-pion exchange and go beyond the classical framework used here [6.44]. As an example, let us mention that g~°~ gets roughly half its strength from diagrams like the one shown in fig. 6.19, and these are clearly not treated in the framework of classical meson theory. Nevertheless, we can say that some important qualitative features of the NN-interaction are described properly by the rather simple models developed before. 6.5. The baryon spectrum In this section, we want to demonstrate how the inclusion of vector mesons improves the baryon spectrum in comparison to calculation in the framework of the conventional Skyrme model which have been extensively carried out at SLAC [6.54, 6.55] and at Siegen [6.56, 6.57]. For a more complete set of references we refer the reader to the reviews [6.58, 6.59]. We will not outline the whole calculational scheme but rather give some specific results obtained by Weigel, Schwesinger and Hayashi [6.60, 6.61] who worked out the baryon spectrum in the minimal model (c = 0). The procedure to calculate baryon resonances is the same as in the conventional Skyrme model.
U. -G. MeiJlner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
315
First, one constructs the hedgehog solution and the scattering of pions, p- and w-mesons off the soliton is described by small amplitude fluctuations around the static solution. This amounts to the ansätze U(r, t) = exp{ir. [FF(r) w(r, t) = (w0(r) p,L(r,
t)
=
+
(p0(r,
O(r, t)]}
+
W0(r, t), W(r, t))
t),
(6.103)
,
p1(r) +p~(r,t)).
We therefore have to consider three pionic, four w-meson and twelve p-meson excitations (0, W~,p~, respectively). Injecting the ansätze (6.103) into the Lagrangian (6.1) and expanding to second order in the small oscillations one obtains coupled equations of motion upon variation. Due to the hedgehog structure of the static solution, i.e., its invariance under combined space and isospin-rotations, one ends up with eight transverse and seven longitudinal coupled differential equations for the small oscillations. Here we have made use of current conservation which eliminates four vector meson degrees of freedom. From this, we can now calculate the scattering matrix S in the soliton-fixed frame, with all input parameters given from the meson sector
f,~=93MeV,
m~= m~=770MeV,
g6.12.*
(6.104)
A technical subtlety involves treating ~= \/iJ, only one of the two possible squares leads to a regular solution at r = 0. A way to circumvent this problem is to add a small extra mass term for the p-meson, = Sm~p~ p~with ~ 10MeV [6.60]. Notice that this problem does not occur in the calculation of the static properties discussed before. To compare to the empirical data, a further step has to be taken. For that, the physical S-matrix elements SL2T2J have to be calculated as linear superposition of the soliton-fixed ones via —
SL2T2J = ~ G(K, T, J, L) SKLL
(6.105)
where G(K, T, J, L) is some geometrical coefficient, L denotes the angular momentum of the pion, J the angular momentum of the ‘uN system and T the total isospin. To arrive at (6.105), we have assumed an adiabatic rotation which neglects the dynamical effects of the rotating skyrmion. This is a severe assumption and deserves further study. The physical S-matrix may be parametrized by phase shifts 6 and absorption coefficients (inelasticities) ~j via T1L,2T,2J
SL2T2J =
exp{2i6L 2T2J}
.
(6.106)
In fig. 6.20, we show some pertinent results of the phase-shift analysis [6.60] in comparison to the Skyrme model results. For the higher partial waves, like e.g. D13 and F15, there is a very good agreement between the predictions of the model and the data. Notice furthermore that inclusion of the vector-mesons solves the problem of the rising phase shifts in the Skyrme model, i.e. the high-energy behavior of the phase shifts and inelasticities is apparently improved. However, the old problems with the missing resonances in the S11 ** and the P11 channel remain, for the latter one has some *
The authors of ref. [6.60]prefer to work with g = ~ and not g as determined by the KSFR relation. By including a scalar meson one finds some structure in this channel (cf. section 7.1)).
* *
316
U. -G. Meijiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
PION-NUCLEON
:~1
SCATTERING
~:1TTJ
:::1i111~~ ~
~
~
-
F
15
05
1
15
ENERGY
0.5
1
1.5
EGeVJ
Fig. 6.20. Phase-shifts ~ (in degrees) and inelasticities (‘i) for some partial waves. The dashed lines represent the results for the minimal model [6.61], dashed—dotted lines correspond to the results of the conventional Skyrme model, and the solid lines are the data taken from ref. [6.86].
improvement at higher energies. * Clearly, these problems are related to the improper treatment of the zero modes, in fact, the rotational and translational zero modes of the soliton influence the L = 0, 1, 2 phase shifts. Furthermore, let us make some remarks on the opening of other channels. Apart from the ‘urN—*irN channels, there is also a strong flux into channels like ‘rrN—*Nw, Np, ~‘ur,~w and ~p. This can be deduced from the large deviations of the inelasticities from unity at higher energies. Below the p-meson threshold, only the &rr-channel is open and is strongly excited before other channels open. The large variations of i~at higher energies are due to the effect of channel openings at threshold. Overall, we can summarize by saying that the minimal model works rather well in describing the excitation spectrum of the nucleon. Finally, let us mention that Mattis [6.63] has given model-independent predictions for vector-meson scattering off skyrmions which are in fair agreement with the available data. *
Notice that the Roper shows up in the helicity amplitudes of pion-electroproduction [6.62].
U. -G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
317
6.6. Complete vector meson dominance? In the minimal and complete model discussed so far, vector meson dominance was only asymptotically realized in the isovector channel, although the pion form factor is nicely described by p-meson exchange alone. One might therefore wonder how important the deviations from the naive currentfield-identify 1 J~~~’—(m2Ig)p’~
(6.107)
in the baryon sector are. For that we will consider a model which incorporates exact VMD, i.e. it has no pionic corrections to the isoscalar/isovector currents. This model is constructed according to Sakurai’s ideas of vector-mesons as gauge fields of the isoscalar and isospin symmetries [6.24]. Following the work of Meil3ner and Kaiser [6.21, 6.39, 6.64], such a model is based on the U(2)~symmetry of the non-linear if-model. It is essentially equivalent to the complete model discussed before, apart from the fact that it does not contain non-minimal terms like Tr[A~UA~Ut]and Tr[F~~UF~~RUt]. Notice that we are now considering a massive Yang—Mills Lagrangian with pions, p- and w-mesons. The U(2)~Lagrangian is given by Tr[D~U D~Ut] ~(p~ —
+
w~)+
xTr{ir.pa(ô~U Ut + Ut ~U)+
~
(p~+ w~)+
~-
(~
g)w~B~’ +
paUtTpsLT~ ~m~f ~Tr(U— 1)
(6.108)
with D~U = U (igI2)[T ~ U] the pertinent covariant derivative. We will not go through all details in constructing nucleon observables as done before, but rather outline the differences to the models presented so far. First, due to the simpler structure in the non-anomalous sector, the relevant boundary condition for the p-meson profile at small r is G(0) = 0, not G(0) = —2 as before. Therefore, the p-meson profile is much less pronounced than in the previous cases. Due to the omission of the non-minimal terms* which are repulsive, the hedgehog mass is lowered to MH = 1004 MeV for our standard set of parameters f.,, = 93 MeV, g = 5.85 and a = 2. The hedgehog radius decreases to rH = 0.43 fm. Projecting onto nucleon states and minimizing the moment of inertia functional, we find again that the p-excitation profiles ~ 1(r) and 42(r) are much less pronounced than in the complete model. This has immediate consequences for the moment of inertia, it turns out to be too small: A = 0.358 fm, i.e. M~ MN = 827 MeV. Furthermore, the isovector currents are entirely given by the p-meson in this model, we have for the isovector form factors [6.64]** —
—
2)= ~ G~(q 2)= G~(q
* **
2~fr2[3~ 1(r)+ ~2(r)]j0(qr) dr,
-
MNJ G(r)
(~)
(6.108a)
j
1(qr) dr.
(6.108b)
This modifies the PCAC relation via vp-corrections. The isoscalar form factors are given in (6.49a, b), the w- and clLprofiles are not very different from the ones in the complete model.
318
U. -G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
The electric charge radii are not too much affected since the decreased integrand in (6.108a) is balanced by the factor A However, for the isovector magnetic moment we find a much too small value, namely = 0.68 to be compared to the empirical value of ~ = 2.35. Furthermore, g~ = 0.64 is as small as in the conventional Skyrme model [6.66], i.e. it is half of the empirical value. From these observations we conclude that exact vector meson dominance is not realized in the baryon sector of low-energy hadron physics. Chemtob [6.65] has investigated a similar massive Yang—Mills model with ‘ii-, p- and ü-mesons with VMD imposed via the order-by-order construction of local counterterms a Ia Brihaye, Pak and Rossi [6.66]. For the parameters f,~and ~ on their empirical values, he finds M1~= 983 MeV,
~.
MN = 719 MeV, ~ = 0.80 and g~= 0.43, very similar to the results of Meil3ner and Kaiser. Only by changing the parameters to f~ = 76 MeV and g 9.10 does the Nz~-splittinggo down to 297 MeV and j~ increase to 1.77. In both cases, ,u~lies in the range 0.3 ~ ~ 0.4, in agreement with the results of ref. [6.64]. Similar observations have been made in ref. [6.23]. These authors discuss the hidden symmetry Lagrangian with pions, p- and ‘.o-mesons. The Wess—Zumino term is chosen to ensure exact VMD on the one-photon-level as proposed by Fujiwara et al. [6.66]. The contribution of the WZ term to the static soliton energy is given by (cf. eq. (3.71)) —
-~
12r dri1~ fgN~ 22 16ir r
4irj
~ VMD =~
2F ~ (wF’G(G + 2) +2 sin F(G’w + Gw’) + w’(2 sin F sin 2F)1}. (6.109) x [wF’ sin The equations of motion for the meson fields change accordingly, the boundary conditions are the same as in (6.15). The results are similar to the ones discussed before; the hedgehog mass decreases to 1143 MeV, the hedgehog radius comes out to be rH = 0.36 fm. * Therefore, the axial-vector charge gA is also very small (g~= 0.47) since it is proportional to the extension of the pion-source function. * * The authors of ref. [6.23] have also calculated the moment of inertia of the spinning soliton, they find A = 0.411, i.e. M~ MN = 705 MeV. These findings are rather consistent with the ones of Meil3ner and Kaiser [6.21, 6.39, 6.64]. Furthermore, recent investigations of the nucleon electromagnetic form factors in a QCD-based phenomenological model (meson theory of small momentum transfer and perturbative QCD for high momentum transfer) suggests that half of the isovector photons coupling to the proton are given by multipion terms [6.67]. In the context of chiral bag models, the authors of ref. [6.45] also propose sizable deviations from exact VMD via photon—quark couplings. All this indicates that although exact VMD works well in the meson sector, the non-linearities inherent in the solitonic nature of the baryons exclude such a simple picture in the baryon sector. We will come back to this point when we discuss the static properties of nucleons in the ‘Trp(0A 1 -system. Again, we will encounter the situation that insisting on exact (complete) VMD will lead to a too small moment of inertia and isovector magnetic moment. —
—
—
6.7. Results from the ITpA1 w-system Up to now, we have only considered Lagrangians involving the pion and the low-lying vector mesons We will now extend our analysis to systems including the axial-vector meson A1. Due to the
p and *
~.
The same input parameters (f,, = 93MeV, g = 5.85) are used. Keep in mind that g~measures the i~NNcoupling strength through the Goldberger—Treiman relation.
**
U. -G. Meifiner, Low-energy hadron physics from effective chira! Lagrangians with vector mesons
319
difficulties related to the A1 parameters and ITA1-mixing as discussed in sections 2 and 3, there exist various calculations using different Lagrangians [6.12, 6.13, 6.14, 6.68, 6.69, 6.70] reaching slightly different conclusions. In refs. [6.68]and [6.70],the Lagrangian includes in addition a scalar particle (the e-meson), the discussion of this work is therefore delegated to section 7. Historically, Meil3ner and Zahed [6.12, 6.13] were the first to demonstrate that the inclusion of spin-i mesons with all parameters fixed in the meson sector indeed improves the static baryon properties as anticipated by Witten [6.71]. We will give a brief outline of their work and only comment on the other calculations. Following the work at Syracuse [6.46], the authors of refs. [6.12, 6.13] consider the following massive Yang—Mills Lagrangian with the p-, A1-, w-mesons as SU(2)L 0 SU(2)R 0 U(1)~gauge particles: =
+ ~wz’
~o +
~
—L~)} t} —
Tr{a~}+
~
—
Tr{p~Ua~Ut}
Tr{a~[U,a~]Ut}+ ~ f~m~ Tr{U
—
8g
~
Tr{a~Up~U
Tr{p~[U, p~]Ut}
+
=
~
Tr{w~w~ + p~p~+ a~~a~}~ Tr{p~+ 4g —
—
IC
—
—i
**‘a$
r
+
+
Ut
—
2}
(6.110)
a~},
mn ir 1L w~1rl~p~a~,sf -‘- ;-;—--1- ~ IT
x Tr{p~,(R~ L~)+ aa(Rp —
+
L~)+ Paas
—
~(Pa aa)Ut(p~+ a~)U}, —
where ~ is the contribution from the gauged non-linear if-model, .9?~ gives the minimally broken SU(2)~0 SU(2)R 0 U( 1 )~ Lagrangian for the spin-i mesons (p, w, A 1) and ~ is the Wess—Zumino term in the Bardeen-subtracted form (cf. section 2.1). L,~ = Ut ô.,~U left-handed current in the 1 [o-(x) + ir• i~(x)]defined to incorporateis the the isovector pion field. The pSugawara form, with U(x) = f and A 2)L®SU(2)R doublet, the o-meson is associated with the U(1)~gauge 1-mesons formthean following SU( symmetry. We use conventions: co~= ijw~ p~= igp~T”, and a~= iga~Ta. For convenience, the gauge couplings j of the U(i)~and g of the SU(2)L 0 SU(2)R are chosen to be the same, j = g. The pertinent field strength tensors are ~,
~
—
p~’
9~p1’—,~9PpU+
a~L=ô~La*_9*a/L
+
~
~
p*]
+
~
av]+ ~
ar],
(6.llOa)
p*].
As discussed already in section 2, the Lagrangian (6.1) needs proper diagonalization due to the ‘urA 1-vertex —Tr{a~~IT}. The decoupling of the pion and the A1 is achieved by defining the physical fields ~ and c via 2Im2)d,~*, ~=a~—if,,.(g
~=Zir,
(6.111)
320
U. -G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
where f,~is the physical pion decay constant given by* =
where m
Z
=
(1
—
g212/m2)l/2
(6.112)
,
m~= m~is the vector meson mass. As a consequence, the bare A1 mass m gets shifted to = m/Z = \/~m.In what follows we will omit the ““ when writing physical ii- and A1-fields. We will neglect the momentum dependence of the pIrIT-coupling and fix g = ~ through the width of the decay p°—s~ IT IT. This requires g = 6.05. Actually, in ref. [6.701this momentum dependence has been taken into account, we will discuss this point later on. The choice of parameters g = 6, f,, = 93 MeV and m = 780 MeV leads to m~= 1116 MeV close to the Weinberg ratio. Notice that the KSFR relation does not follow automatically in this scheme. Let us now construct solitons (baryons) from the Lagrangian (6.110). For that, we make again the time-independent hedgehog ansätze, the most general form for spin-i fields reads (cf. section 6.1) =
w°igw(r),
p=igR(r)(FXr), (6.113)
a=ig[a1(r) r+a2(r) F’r.F]. Notice that the a involves two radial functions a1(r) and a2(r), just as the p°-componentin eq. (6.23). Obviously, w = p°= a°= 0 at the classical level. For U(r), we choose the usual hedgehog ansatz. Injecting the ansatze (6.113) into the Lagrangian (6.110), the ground-state energy of a hedgehog skyrmion reads E = MH
=
E~+ E.ITPA
+
E,~A+ E~A+ E~
+
~
(6.114)
where the E’s are functionals of F(r), R(r), w(r) and a12(r) given by 2 dr [F12
J
+
2 sin2F + 2m~(i cos F)], —
E~= 2ITf~ r
J
E~~A = 4ITf~ r2 dr[g2Ra sin 2F
+
g2 sin2F(R2
—
a2) +
2gR
~
~
2
2
r
E~A= 4’rrm J r dr [R *
2
~ —
2
-~-
2
+
~
2
(3a +
/3
2
+
Notice that there appears a sign error in refs. [6.12, 6.13] In this equatIon.
2a13)
,
(6.115)
U. -G. Meif3ner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
321
~
+2g ~(2a2+~2+2ap)+2g(a+$)(R~a_Ra1) + ~ [(R2+
2)2 +
2(a2
~-~-~--~
—
+
R2)(a + ~)21}
~
~
+ 2a
gaR
—
~
(a2
—
R2) sin2F + ga
1R cos 2F)],
in which a and f3 are defined via
/
j~. sinF
iTT
\21
sinF
.
a=a1—g—-2---—, f3=a2—gy--~)[(S1flF) The pertinent Euler—Lagrange equations can be obtained by functional minimization of the energyfunctional (6.115). The relevant boundary conditions to ensure unit baryon numbers and finiteness of the energy are F(0) = ~T, R(0) = 0, w’(O) = 0 and a’(O) = 0. The equation of motion for /3(r) is only of first order and can be used to eliminate one degree of freedom from the system of coupled equations. The vector-meson profiles of the p-, w- and the physical A1-meson are shown in fig. 6.21, for the parameters f~ = 93 MeV, m = 780 MeV, g = 6.0 and m~= 138 MeV. The hedgehog mass comes out to be M~= 847 MeV. Since the w-meson couples entirely through the Wess—Zumino, its coupling constant —----~-—
MESON
PROFILES
IN THE lrpA1w
-
SYSTEM
r[f ml Fig. 6.21. Vector- and axial-vector meson profiles in the irpüA1-system, from ref. 16.131, for g = g= 6. Curve 1 represents the w-profile w(r), curve 2 the p-meson profile R(r), curve 3 the scalar part of the physical A1-meson A~(r)and curve 4 the tensor part of the physical A1, A~(r).The scale on the left-hand side stands for the R(r), whereas the scale on the right-hand side corresponds to w(r), A~(r),and A~(r).
322
U. -G. Meif3ner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
can be read off from (U
+
(6.116)
m2)w~= —(gN~I2)b~ ,
where b,~is the conserved isoscalar current,
b~=
—
Tr[LpLaLpI
~
—
32ir
~
(6.117)
From (6.116) we conclude that the w-hedgehog coupling is g’~~fl/47r = N~g2/16ir2= 6.45, a bit too low compared to the empirical value of the w-nucleon coupling, g~/4i~= 10—12 [6.47, 6.48]. Thus, we do expect sizeable contributions from quantum effects. Instead, one can use a U(1)~-coupling~ different from the SU(2) one, and fit it to the empirical value. For j= 7.47, i.e. g~fl~I47r = 10, we find M 11 920 MeV. To project onto states of good spin and isospin, we again perform an adiabatic rotation of the soliton andmost quantize theansãtze spinning To fields leading rotational velocity tA)the general for modes. the vector are order (using in thethe notation of refs. [6.12,2Ka 6.131, —i Tr(r’~A cf. eqs. (6.23, 6.24)): w~(r,t)
=
ig w(r),
(OA(r, t)
=
ig WA(r) (P x K),
p~(r,t)
=
ig ~A(r) Ar~KA1
pA(r, t)
=
A p(r) At
+
ig ~A(r) K~PAr PAt, (6.118)
0
t
.
aA(r, t) = ig ~A(r)
Ar (r x K) A
a 0(r, t)
=
A a(r) A~
where WA(r), ,iiA(r), ~A(r) and CPA(T) are smooth functions of r to be determined. Notice that the isorotation of the skyrmion excites all odd-parity components of the vector fields. For the calculation of the moment of inertia A[WA, /~A’~A’ ~A] we will assume that jz~= = = 0. This approximation is motivated by the predominance of the w-correlations in the static analysis.* In this case, the moment of inertia depends only on the w-excitation WA(r) and the classical background fields F(r), R(r) and w(r). The equation of motion for the profile function WA(r) reads 2WA 2 F’sin2F WA+2+mwA2F ~ 2w~ (Fg\IF’ 2 1F’F 2R ~ 2 +(—J1--—sinF+t--——————R’IsinF .
.
\r
—2F’Rcos2F+ *
~
~
For a full scale calculation involving
~
~A
r
+
and
RF’
~A
/
+
R’F)}
~O, see ref. [6.70].
(6.119)
U. -G. Meifiner, Low-energy hadron physicsfrom effective chiral Lagrangians with vector mesons
323
with F = N~g(4ir2and F = —2f~,gF/m.Solving (6.119) subject to the boundary conditions WA(0) = WA(cc) 0, we find A = 1.178 fm, i.e. M~ MN = 251 MeV, a bit lower than the empirical value. For a larger U(1)~-coupling(j = 7.47) we have M~ MN 306 MeV. The axial-vector coupling constant g~ follows from the axial current A~(x,t) via —
—
hnjdxeT~.1(N~Aa(xt)~N~ = ~
(6.120)
and is given by gA
=
{~(~‘ + ~F)_gf2(3acos2F+
—
~
+
—1-~--
+
4aR’
~
+
~
sin2F + ~2F)
+
—~
a sin2F+ (3 +2Rsin2F) w’(a sin2F + R cos2F)
—4a’R + 8gaR2 + 4g13R2}.
(6.121)
In a similar fashion, the angle averaged isoscalar and isovector currents follow to be
J f
d12 V~= ~
Kaf2f~sin2F +
dfl~V~q x =
(~f)ff~ sin2F
WA
+
F’ sin2F}
(gf~)r(Rsin2F
Tr[AtT~AT~],
+
a sin
+2R2 +2/32+2grR(R2 +4a2 +2a/3 +
r2w’(a sin2F
—
~p2)~
~2
F’sin2F
R sin 2F)}e~1qk Tr[r~Atr~4].
(6.122)
In Table 6.7, we summarize the nucleon properties in comparison to the results of Chemtob [6.14] and Lacombe et al. [6.701. Chemtob has explored the VMD-Lagrangian constructed in ref. [6.65], with additional soft-pion corrections which reduce the pion decay constant to f ~ 87.4 MeV in the spirit of Gasser and Leutwyler [6.72]. Chemtob [6.14] chooses a universal gauge coupling g = 5.7 fixed from the p 2’ir decay. For the moment of inertia, only the contributions from the classical fields are taken into account and the electromagnetic properties are calculated via current-field identities. This leads to the problems in the isovector channels already discussed in the previous section. Lacombe et al. [6.70]have taken a similar Lagrangian as it is used in refs. [6.12, 6.13], with the exception of an extra term in the non-anomalous sector to correct for the momentum dependence of the pirir-coupling. This term reads —~
~extra = (~m2/8)Tr[(AL~U
—
UAR~)(UtAL~ A~Ut)1 —
(6.123)
with A~L= (1I\h)(p” ±a’~)(before diagonalization of the ‘rrA 2)is a coefficient of 1-mixing). (z~m dimension (mass)2, typically ~m2 = 0.3m~.The gauge coupling g is fixed to the p—~iry decay width, and the U(1) gauge coupling j to the value g~,~~/41r = 12.45 from the NN-interaction. The results in table 6.7 indicate indeed a substantial improvement over the conventional Skryme model, especially for the
324
U. -G. Meif3ner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons Table 6.7 Static nucleon properties in the ~rpA
1w-system. The parameters for the calculation are (i) ref. [6.13]: 93MeV, g = 6,j=7.47, m,, = 138 MeV, mA = 1116MeV; (ii) ref. [6.14]:f,, = 87.3 MeV, g = 5.68, 0.3m~).For for gthe conventional 1=11.4, m,, = 140MeV, mA = 2=1060MeV; (iii) comparison, ref. [6.70]:the f~ =results 93MeV, = 5.52, 1=8.34,Skyrme m~= 139 MeV, mA = 1082 model MeV [6.74]with (~m f~ = 54MeV and e, = g,,,,, = 4.84 are also shown.
f,,
=
MN [MeV] M~— MN 1MeV] gA
/.4~
[nm]
Pv (r2)~
[nm]
0 [fm] 2 1/2 (r2)~ )EJ=1 [fm] (r 2)~, 0[fm] ~r 1 [fm]
ref. [6.13]
ref. [6.14]
ref. [6.70]
ref. [6.74]
993.0 306.0 1.09 0.39 1.98
1019.0 336.0 1.50
1360.0 292.0 0.92 0.21 2.07
939.0 293.0 0.65 0.37 1.61
0.61
0.68
0.92
0.95
—
0.66
0.67 1.05 0.87 0.99
0.82 —
1.16 —
1.04
—
1.04
—
axial-vector coupling constant g~.There is still some discrepancy concerning the hedgehog mass in these models, this point remains to be solved. The least we can say is that the soliton picture indeed works when one puts in all mesons relevant at nuclear length scales. Finally, let us mention the work of Baacke, Pottinger and Golterman [6.69]. These authors worked out the hedgehog properties of the ‘rrpA1w-Lagrangian proposed by Golterman and Hari-Dass [6.73], which embodies exact VMD. They find a small hedgehog mass MH 0.86—1.09 GeV depending on the value of ~ and rH (0.4 ±0.05)fm. They estimate the moment of inertia from the purely pionic term and find it to be too small, of the order of 0.3 fm. This does not come as a surprise when one recalls the discussion in section 6.6. Of particular interest in their analysis is the stability analysis of the energy-functional. The condition that E should be a positive-definite quadratic form of the fields F, R and a12 is indeed matched for their Lagrangian. Actually, in ref. [6.69]an additional term of third order in derivatives as proposed in ref. [6.73] is omitted. Although the situation in the ‘rrpA1w-system is not at all unique, we believe that the more realistic the meson theory, the better the baryon properties come out as was first pointed out in refs. [6.12, 6.13]. =-
—
6.8. NN-annihilation and the H-dibaryon Up to now, we have only considered baryon number B = 1 and B = 2 configurations, exploiting the static properties of low-lying baryons and their electroweak structure. Another important issue in low-energy hadron physics is the understanding of NN-annihilation, mainly triggered through the experiments at LEAR. A large fraction of NN-annihilation is dominated by four and five pion decays, as shown in fig. 6.22 indicating the short-range character of the annihilation process. Furthermore, most of the multi-pion channels are vector-meson dominated, and therefore a solitonic description of the (anti)nucleon with vector mesons seems to give the most straightforward approach to describe the existing data. As a pertinent example, let us remind you that preliminary ASTERIX analysis [6.75] indicates that the p’rr-channel accounts for 60—80% of the p~—~ ‘rr~iriT° reaction._Here, we follow the work of Zahed and Brown [6.761 who set up a framework of calculating NN-annihilation in the presence of vector mesons as a highly non-perturbative process. Their idea is to look at superimposed
U. -G. MeiJiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
325
Ni~—.~nir AT LEAR ENERGIES
50
~4O-
—
30
-
20
-
CD
z
= U
NUMBER OF PIONS Fig. 6.22. Histogram for p~into n pions at LEAR energies [6.75].
NN-configurations at rest decaying into multi-meson states, relying on the validity of the effective mesonic description at short distances. Alternatively, one could think of calculating the annihilation of a nucleon and an anti-nucleon by numerical integration of the equations of motion and project out multi-pion final states. In light of the problems encountered in full-scale calculations of the NNinteraction [6.51,6.52, 6.33] this program does not seem feasible at the present time. To circumvent the problem of looking for skyrmion—anti-skyrmion configurations as a classical solution of the Euler— Lagrange equations, we will start from a product-ansatz of the form UHH(d, /3)
=
U(r
—
d/2) B Ut(r + d12) Bt
(6.124)
where the soliton (U) and the anti-soliton (Ut) are separated by a distance d, and B = B(/3)* is a constant isospin-rotation. Obviously, (6.124) follows from a hedgehog product-ansatz after G-parity transformation. The existence of such saddle-point solutions in the vacuum sector with MHfi = 2MH has been first discussed by Bagger, Goldstein and Soldate [6.77], these configurations are clearly unstable against decay into pions and vector mesons. For the moment, let us concentrate on a simplified model with pions and to-mesons only, the w-stabilized skyrmion of Adkins and Nappi [6.3]. The generalization to the gauged non-linear if-model including p- and A 1-mesons is straightforward. The Lagrangian to be discussed reads ~Z~= ~f~Tr[L~]— ~w~w~”’—~
,
(6.125) The relevant
Tr[LpLaLp]. 2) where B~is the topological baryon current, B’~= (E~”~”~I241T w-couplings to the HH-configuration can be deduced from (6.125) by expanding around the ‘~‘
*
In the most attractive tensor channel, we can write B(ii) = ir
=
ir 5.
326
U.-G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
classical configuration ~
=
U1~r~f3 U i~r~/3 using
4 (6.126) &~, U~(4) e~U11~ e1 where is the classical co-field and a fluctuation. Similarly, pionic fluctuations are parametrized in terms of 4 r~.~rI2f. With that, we can now deduce the pertinent co, u-couplings to the HH configuration. The w—HH coupling is identically zero since the baryon density vanishes locally for configurations of the form (6.124). For the most general isospin-rotation B = + ir /3 we find for the winding number W,
+
=
W= Tr(L~)= 6if3 2 Tr(Li- /3) = 0 (6.127) 0d employing differential one-forms (L = L~df) and Poincaré’s lemma. The co—HH—ur-coupling follows .
in a similar way to be ~o,aflH
=
~(g~/24u2)
Tr[L~(~)L//,(4) L~(4)]
=
~(ig~I8ir2f~)r ~
~
~ Tr[~ u(L~+ R~)].
(6.128)
Assuming plane waves for the IT- and co-mesons, which means that the HH-configuration does not distort considerably the meson decay products, we can write the HH—> cour transition amplitude M(HH—~uw)as M(ÜH~uw)= (w(q) ua(p)~Jdti~H~(t)~H~,
(6.129)
where the HH-configuration is assumed to be at rest, the pion carries momentum p = (Eq,, q) and the co-meson q = (E 02, + q). Clearly, this process contributes to the 4’rr-channel since the co decays further into three pions. HH) is the coherent soliton state around the classical background, —
HH~= cb11(U) 1~ 1(U) RJ~j0),
(6.130)
is the soliton operator and R~(U)the rotation operator in the Hilbert space of functionals of U. This construction follows from a generalization of the Mandeistam operator [6.78]; for further details we refer the reader to ref. [6.76]. Using proper plane-wave normalizations, the transition matrixelement (6.129) can be written as 4~(U)
M(HH~uco)=
—
32u~f ~(E~ + E~ E~~) ~ —
Pa Jd3x(0~Tr[Ta(Ly
+
R~)]~H), (6.131)
where r~(q) is the polarization vector of the vector particle. The evaluation of the matrix element ‘(01 Tr[u-~’(L~+ R~)]lHH~ is straightforward in the semiclassical approximation, we find
(01 Tr[ra(L~ + RY)]IHH~ Tr{Ta[L7(Uu~) + R(U~~)I}+ =
3 FTr(ra[r. ~ r./~2r/~}
=
—2Wa~{cosF sin
(6.132)
U.-G. MeiJiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
327
which is a total divergence. Therefore, the matrix element M(HH—s~cou) vanishes, since there is no contribution from the surface at infinity (w(r) exp{—3m~r}).Thus, the decay rate HH—4 wu vanishes in the semiclassical limit, which is intuitively clear. The co-couplings are dictated by the Wess—Zumino term, and when H and H overlap strongly, the WZ-term vanishes globally (untwisting of the baryon number). From this argument one is tempted to conclude that this kind of suppression will hold true for most co-related processes. Since the argument is of purely geometrical nature, it holds also for the NN-system. The present data seem to support this suppression, although the experiments on p~—*wit° are somewhat controversial,* one can say that pj3—~co’rr° contributes less than 1% to the total cross section. Notice that for channels including the p-meson (NN—~pu, p3ur,. .) this geometrical argument does not hold since the p couples through the isovector current and the WZ term. This seems to be confirmed by the data [6.79] which show an enhancement of the p-channels. Let us now give some coupling involving p-mesons. They can be read off from the Lagrangian (6.110) in a similar fashion as was done for HHco’rr. Two pertinent examples are -~
.
2
~-
ç
I~
fIT
~
~i
‘~~a f
2.!ir
~HHPIT=1g
+R )—g —~-[p~[U,p ]U]
4Tr[P~(L
NC
—
a~Tr[(RL)
~Ut~U1
(6.133)
64 u ~fiHtop1r
=
—i
/4V**~
Tr[i~a(Rp
—
L~)
—
aUt~pU].
A full-scale investigation of 3ir-, 4ir- and 5’rr-channels is recently under investigation at Stony Brook [6.80] using realistic wave functions to evaluate the final matrix elements. Finally, let us finish this section with a short comment on dibaryons. Kunz and Masak [6.81] have shown that the H-dibaryon predicted by Jaffe [6.82] corresponds to the lowest-lying SO(3) solution [6.83, 6.84] of the SU(3)~hidden gauge theory. They pointed out that a more realistic vector meson Lagrangian will indeed improve on the uncertainties in calculation of the H-properties as encountered in the conventional Skyrme model. Without symmetry-breaking terms and using a Skyrme term to stabilize the soliton instead of the co-meson, they find the mass of the H to be 2.14GeVsM(H) 2.57 GeV depending on the strength of the Skyrme term. The rms radius of the H is rather small and lies between 0.5 and 0.6 fm. It remains to be seen how much a more realistic calculation including co-mesons and SU(3)-breaking terms will narrow down the H parameters.
7. Further developments In this section, we will discuss some further aspects we have omitted so far, like e.g. the inclusion of scalar particles, the introduction of vector mesons by means of the derivative expansion and vector mesons coupled to chiral two-phase models. Although the scalar mesons 5, ~, ~,. . are experimentally elusive, and the scalar glueballs even more so, their incorporation in an extrapolation of the effective action up to 1 GeV may be relevant. Furthermore, the scalar gluonic components are intimately related to two broken symmetries of QCD, namely the U(l)A anomaly** and the trace anomaly. For example, one can introduce an effective order parameter field which takes into account the effects of the trace .
*
Recent measurements give (0.9 ±0.45)% or (0.17 ±0.06)% for the contribution of wii° to the total cross section [6.75]. In what follows, we will not discuss the U(l)A anomaly in detail (cf. section 5.1).
**
328
LI. -G. MeiJ3ner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
anomaly (broken scale invariance) via H(x) = /3(g~) Tr(G~~)/g~ and couple it to the meson field of the effective theory. This idea has been initiated and explored by Schechter and collaborators [7.1, 7.2, 7.3], mainly in the context of the conventional Skyrme model. We will focus here on the work presented in ref. [7.4], where a more realistic Lagrangian including u-, p- and co-mesons is modified to account for the trace anomaly and its implications are discussed. Another, though more phenomenological approach to include scalars, comes from the fact that the intermediate range attraction in the nucleon—nucleon interaction is described by the exchange of a scalar particle. In the dispersion-theoretical language this particle is nothing but a 2’rr-resonance in the S = 0, T = .0 channel [7.5]. For that reason, Lacombe, Loiseau, Vinh Mau and Cottingham [7.6, 7.7] have constructed and studied an effective Lagrangian with vector and axial-vector mesons plus a scalar particle. Alternatively, one can calculate soft-pion corrections to the non-linear if-model which also give rise to an intermediate range attraction [7.8, 7.9, 7.10]. We will outline the pertinent results of the Lagrangian proposed in refs. [7.6, 7.7] omitting any calculational detail. After that, we will briefly summarize the alternative idea of putting in vector mesons by means of the derivative expansion [7.11, 7.12]. In this framework, vector mesons are identified with higher order terms in derivatives of the pion field, like e.g. the Skyrme term can be related to the exchange of an infinitely heavy p-meson (cf. section 3.1). Our main focus will be to demonstrate the limitations of this approach as compared to the one including vector mesons as dynamical degrees of freedom. Finally, we will give some comments on vector mesons coupled to the chiral two-phase models [7.13—7.15],a field which is only at its very beginnings. The more speculative ideas of the SU(6) approach can be found in appendix D [7.16—7.18]. 7.1. The QCD trace anomaly and scalar particles Schechter and collaborators [7.1, 7.2, 7.3] have pointed out the importance of another property of QCD at low energy that we have not considered so far (apart from the remarks made in section 5.2), namely the anomalous behavior of QCD under scale transformations. The trace of the QCD energy-momentum tensor, i.e. the divergence of the dilatation current, is non-vanishing. Performing a scale-transformation on LQCD, one finds [7.19, 7.20] p.
p. =
s
Op.
2
V
Tr(Gp.,~)+ [1 + y(g~)]
=
,~
g~
i=1
m~q1q,,
(7.1)
where J~denotes the dilatation current, O’~”the canonical energy-momentum tensor, g~the running coupling constant and /3( g,~)the QCD /3-function. For non-vanishing current quark masses th,(i = u, d, s,.. .), one has an additional contribution proportional to the associated anomalous dimension. For the SU(2) sector we are considering, we will neglect the second term in (7.1). Let us start to consider an effective theory for QCD without matter. Following ref. [7.1], we assume that there exists an order parameter field H(x) such that (7.2)
O~H(x)
which is suggestive of a scalar glueball. To second order in derivatives of H, the Lagrangian ~ uniquely given by 32 (9p.H)2 =
~aH
—
~H ln[H1A4],
is
(7.3)
U. -G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
329
where a is a dimensionless parameter and A is the QCD scale. Obviously, the potential term in (7.3) has a minimum at (H~= A41e, where e is the Euler number, yielding a negative vacuum energy density of ‘(H~i/4. A bubble of higher-energy density would increase the total energy and hence be unstable. We will see later on how solitonic matter can stabilize such a bubble. Expanding H around its v.e.v. gives a fluctuation field for a scalar glueball with mass m~= (H)”214a. For the discussion to follow, we will rewrite the Lagrangian ~ in terms of an exponential parametrization [7.4], H(x) = A4 exp{u(x)
—
1} ,
(7.4)
and redefine the constants so that
(7.5)
(1I2)F~e2~p.u)2 (F~I16)m~ e4~(4u 1),
=
—
—
where mG is the glueball mass and f~is related to v.e.v. of the gluon condensate via* ((g~/u)Gp.~G~’)”2=17Jm~I2. (7.6) Experimentally, we have 0.1 GeV~A~0.5GeV[7.21], and ((g~Iu)Gp..~G~’) ~~*s(0.34GeV)4 [7.22]. The glueball mass is not known exactly, it should be higher than 1 GeV. Therefore, our strategy [7.4] is to adjust m~and I~in a way such that mG 1—2 GeV and ((g~/ir)Gp.~G~)(0.20—0.40 GeV)4. Since the Lagrangian (7.5) already satisfies the trace anomaly, the Lagrangian of pions and vector mesons has to be made scale invariant.** The chiral field U(x) can be chosen to scale with dimension zero, so all terms with four derivatives on the pion field are evidently scale-invariant (like e.g. the Skyrme term p. U, Ut d~U]2). The vector-meson fields scale with dimension one. To make any term in the effective Lagrangian under consideration scale-invariant, we have to multiply it by exp{(4 d where d is the number of derivatives and n~the number of vector-meson fields of that particular term. This modifies the Lagrangian (6.1) (we consider again the irpco-models discussed in section 6.1) in the following way ‘—‘
—
—
~ —
‘j(p~+ w~)+ ~ ~
Tr[U
—
11
+
+
+
(F~I16)m~.
(7.7)
We have not specified the Wess—Zumino term, it is scale invariant and does not couple to the glueball field. Furthermore, we have added the term (—F~m~I16) to measure the soliton energy with respect to the vacuum. Notice also that (7.7) obeys the large-Ne counting rules: f~,,F~and A scale as whereas m~,m~= m = \/~gj~~ and the glueball-meson coupling scale as ~1). It is now straightforward to derive the energy-functional E[~r, pp., cop., a] and the equations of motion using the hedgehog ansatze (6.6, 6.8, 6.11) and o(r) = o~(r).Here, we only display the equation for the glueball field cr(r) 2
0
—U
—
—-ci
+mGcTe
2o’
~
~
1
,2
+—~-[~F +
n
~
r
r
2
‘*
1
2
+-~(G+1—~cosF) r
[1—cosF].
m2 w2+e2’~
(7.8)
The equations of motion for the meson fields are the same as in eqs. (6.14) with the substitutions m m e°,m,~ m~e°,and fIT fT, e°~ The boundary conditions on the meson fields remain unchanged —~
* **
—+
—~
The relation of the parameters 1 and m
2aI’I~and 4m~= A212’/ë. in (7.5) to theas parameters a and A in (7.3) are L~= 16A A scale transformation on a meson-field0 cI~ is defined = —(d + x,j’P,,, where d is the scale dimension.
330
U. -G. MeiJlner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
(6.16, 6.17), the pertinent boundary conditions for ci(r) are ci’(O) = 0 and u(oc) = 0. In the conventional Skyrme model, solutions with exp[o-(0)] = 0 are allowed [7.2], this is not the case here, i.e. the expectation value of the gluon condensate (H~will always be finite and non-vanishing. A typical solution for glueball masses 1 and 1.5 GeV and F,~= 0.15, 0.1 GeV, respectively, i.e. in both cases 4, is shown in fig. 7.1. The square of the glueball field ~/i2(r)= exp{2ci(r)} = (0.27 0eV) shows a pronounced depletion around r 0.5 fm, in this region the baryon density B°(r) is sharply peaked. The static nucleon properties are only mildly affected by the presence of the glueball field as shown in table 7.1. The glueball-profile resembles very much a bag model with finite inside energydensity, in the spirit of Shifman, Vainstein and Zakharov [7.23]. We can read off the bag constant B as the difference of the outside and the inside energy density via B = (F~m~/16){e4~°~ [4ci(0)
—
1] + 1)
(7.9)
.
GLUEBALL PROFILE AND BARYON DENSITY
-
r[f m] Fig. 7.1. The square of the glueball field ~li2(r) = exp{2o-(r)} and the baryon charge density B°(r)for g = 5.85, f,, = 93MeV and m,, = 139 MeV. The glueball parameters are: mG = 1.5 GeV, T~= 0.1 GeV (solid lines) and m 0 = 1 GeV, I~= 0.15 GeV (dashed lines). Notice that with increasing glueball mass the depletion of the energy density inside the bag gets more pronounced (for fixed A).
Table 7.1 Static nucleon properties in the presence of a scalar glueball field and a scalar quarkonium (columns (3) and (4)) for the complete model (eqs. (6.1, 6.5)). For comparison, the results of the complete model without any scalar are shown in (5). The glueball4); mass (2):is I’~ m6 =0.1GeV = 1.5 GeV ((GG) in all cases. = (0.27 The GeV)4); other (3) 1 = 0.1are GeV, 0 = 39°; (4): f~=0.04 ((GG) = (0.17 GeV)’), m~=0.92GeV, parameters (1): m, [ == 0.72GeV, 0.15 GeV ((GG) = (0.34GeV) 0 = 44°.m, is the mass of the scalar quarkomum and 0 the glueball—quarkonium mixing angle.
M~
[Me’s’]
(1) 1419.0
(2) 1353.0
(3) 1343.0
(4) 1054.0
(5) 1465.0
1537.0 473.0 1.00 0.96 —0.23 2.68 —1.73
1487.0 536.0 1.04 0.90 —0.21 2.53
1472.0 516.0 1.05 0.91 —0.21 2.59 —1.54
1220.0 664.0 1.24 0.59 —0.04 2.23 —1.20
1575.0 437.0 0.97 0.99 —0.25 2.77 —1.84
M
[MeV]
5 — gA
MN [MeV]
2[fm]
(r~)~ [fm2] (r~/~~ /L~ ~
[nm] [nmj
—1.55
U. -G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
331
We have B = (105 MeV)4 for mG = 1.5 0eV and I~= 0.1 0eV. Varying the parameters mG and I~in the ranges given above (0.7 5 mG 3 GeV, 0.05 c ~ 0.25 0eV), we find that in all cases bags form with typical confinement sizes* r~ 0.4—0.6 fm. These bags have a finite inside energy density, with the bag constant of the order B = (100—130 MeV)4. This is in agreement with typical bag model fits [7.24]. An unpleasant feature of this simple approach is that the width F(H—÷2’rr) is too large. This can be cured by adding a scalar quarkonium state as proposed in ref. [7.3]. Recent work done at Syracuse [7.25] shows that the addition of a scalar quarkonium state with a mass of 1—1.5 0eV lowers the width of the decay H—* 2u to some hundred MeV, depending on the mixing angle between the scalar glueball and the scalar quarkonium. The static nucleon properties are again not drastically affected. In table 7.1, we show results for MG = 1.5 GeV, m~= 0.72 and 0.92 0eV, i7~= 0.1 and 0.04 0eV, and a glueball— quarkonium mixing angle of 39° and 44°, respectively. For the very low value of ~ i.e. ((g~/ = (0.17 0eV)4, the hedgehog mass is lowered at the expense of the other static properties, especially gA comes out much too small. This mechanism of bag formation is particularly interesting since it provides a natural bridge between the purely mesonic theories considered so far and the chiral two-phase models to be discussed in section 7.4. Furthermore, it might put the ad hoc assumption of confining bag walls on firmer grounds. Finally, let us point out that one can include the U(l)~-anoma1y of QCD proportional to Gp.,,G~” in a similar fashion by an order parameter field Q(x) = (g~Nf/16u2)Tr(Gp.~G~), with Gp.~= ~=‘~Gap the dual gluon field strength tensor. -~
7.2. Nuclear binding and scalar particles The discussion of the nuclear binding in the Skyrme model [7.16], i.e. intermediate-range attraction between two nucleons, led the authors of refs. [7.6, 7.7] to construct a chiral effective Lagrangian including p-, A 1-, co-mesons and a chiral singlet-meson c, which might be the S(980) or the E(l300) which is responsible for the enhancement around 1 0eV in the irir-scattering S-wave phase shifts. We will be brief in presenting their work, the Lagrangian proposed in refs. [7.6, 7.7] reads ~~IrpA1
~ 2 8l’vrv lrLrp.OL
— —
CD
+
~ E’2
~p.vRJ
1
-i-
~
2’-rr.42
lrVlp.L
A2
f Tr[Dp. U D~Ut]+ (~m2I8)Tr[(ALp.U
1
2
~4(c9p.co~c9,,wp.) 2— ..~E
1
—
UA Rp.)(LTAL
—
A~Ut)],
122
+2m 4,wp.,
~
~(o,ie)
-~wz=
g,
2 4wp.B’ x Tr[ALaLp —
—
ARaRP +
ig(UA~~UtA~~ ALaARII)], —
(7.10)
with the p- and A 1-mesons incorporated in the left- and right-handed gauge fields AL Rp. = AL Rp. The Lagrangian (7.10) is nothing but the gauged non-linear if-model considered by MeiBner and Zahed *
The confinement size is defined by the region in which the baryon density is peaked.
332
U. -G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
[7.27] with the addition of the r-meson and the non-minimal term proportional to z~m2.g is the gauge coupling of the SU(2)R®SU(2)L and g,~, the one of U(1)~.(Actually, g,,, is related to gaNN, the omega-nucleon coupling constant.) The parametersf and ~m2 can be related to the meson data via [7.7] m~= m~+ ~m2 + 8g2f,
f=f~m~I4(m~ + ~m2).
(7.11)
In terms of the physical ‘rr-, p- and A 1-fields the p—~2rr decay-width follows to be 2
23/2
2
g m8 f 4mIT~ I ~m F(p—s2’rr)-—--——~1————1--~ 1+ 2 2 L’+IT \ m~/ L m~+ L~m
222
gf~m~1 2
(7.12)
221.
(m8 + z~m) i
Adding electromagnetism via minimal coupling, one can easily determine the electroweak currents of the Lagrangian (7.10). For details we refer the reader to ref. [7.7]. Fixing the SU(2) gauge coupling g to give the empirical widths F(p—~iry) = (70 ±10) keV, and the U(1) coupling g~to F(co—3 iry) = (850 ±70) keV, one finds g = 3~9*and g,1, = 9.3. Actually, g1~will later on be fixed to be close to the2 coNNnow coupling constant 12) to allow a better of the This static gives properties. parameter ~m= can be used to fit(~~NN the p-meson decayforwidth via fit(7.12). .~m2=The 0.3m~and m~ 1082 MeV. m 0 is set to be 800 MeV,** the coupling constant ô,~could be estimated from irrr S-wave scattering. The authors of refs. [7.6, 7.7] prefer to leave it as a free parameter, but it is important to notice that t5,~ is bounded because the term ~ r Tr[ Lp. L ~] is attractive and tends to destabilize the soliton. In table 7.2 we summarize the pertinent results of the Lagrangian (7.10) which follow using standard procedures. The overall agreement with the data is rather satisfactory, as we already discussed in section 6.7. It is worth to stress that the attraction gained by including the e-meson also tends to decrease the axial-vector coupling constant g~[7.8, 7.9]. Most important, however, is the fact that this Lagrangian indeed gives attraction between two nucleons in the central channel when the product ansatz is used [7.6]. It furthermore shows again, as already pointed out in ref. [7.27], that a more realistic effective Lagrangian indeed gives better baryon properties than the conventional Skyrme model. Table 7.2 Static nucleon properties for the upA1wr-Lagrangian of refs. 2 and [7.6,7.7]. 6,(MeV)The aremeson given masses by (1) (3.9, are rn,14.3, = 769 0.3MeV, rn,2, m~= 782.6 24.8), (2) {3.5, and m, 14.3, = 800 0.73MeV. m~,2 The 1.8), parameters (3) (3.9, g, 12.45, g~,Am 0.3 rn,2, 0), (4) (3.5, 12.45, 0.73 m2 0, 0} and (5) {6.1, 13.65, 0, 27.3}, respectively. The A1-mass follows from eq. (7.11). (1) MH MN
[MeV] [MeV] — MN [MeVI p.,, mm] p. [nm] gA 2)~, [fm] (r (r2)~$~,,. [fm] 0 f [MeV] mA1 [MeV]
*
(2)
(3)
(4)
(5)
1265.0 1339.0 293.0 2.43 —1.99 0.93 0.64
865.0 938.0 293.0 2.33 —1.70 0.75 0.71
1287.0 1360.0 292.0 2.27 —1.86 0.92 0.61
865.0 938.0 294.0 2.22 0.74 0.68
866.0 939.0 295.0 2.33 —1.63 0.62 0.66
0.94 93.0 1082.0
0.98 73.0 1083.0
0.92 93.0 1082.0
0.98 73.0 1082.0
0.83 71.2 1275.0
—1.60
Experiment —
939.0 293.0 2.79 —1.91 1.25 0.72 0.81 93.0 1060/ 1275
Notice that this agrees with the value g = 5.52 previously given since the authors of ref. [7.7]scale out a factor of V~from the IT-coupling. This is a somewhat unjustified procedure since there is no scalar meson at 800 MeV.
**
U. -G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector rnesons
333
7.3. Comments on the derivative expansion Aitchison and collaborators [7.11, 7.28, 7.29, 7.30] and Truong and collaborators [7.12, 7.31, 7.32] have investigated the derivative expansion in the chiral field of the chiral action by integrating out the heavy meson fields. This amounts to identifying particular terms in the derivative expansion with the exchange of very heavy p-, to-, . mesons and fix the respective parameters from irir-scattering data. Let us first discuss the work presented in refs. [7.12, 7.31, 7.32]. The starting Lagrangian in terms of the chiral field U = exp{ir~~~/f} is .
.9~=c2~~2 + ~ =
+
.
~
+
c6.~’~ +
+
cSb.~b,
Tr(dp. U c9~Ut), 2,
,2)
=Tr[9p.U~ Ut, i1~U~ Ut] = [Tr(cp.U i3p.Ut)l2,
~(1)
= Tr(Bp.B’~)
~(1)
(7.13)
~b=Tr(U1), where ~ stems from the non-linear if-model, ~~1) is the Skyrme term reminiscent of the exchange of a very heavy p-meson [7.33]. ~2) is the so-called non-Skyrme term* which accounts for attraction (if c~2~ >0) and therefore is related to the “if”-meson and ~‘6 is the heavy mass limit of the cop. BtL~coupling proposed by Adkins and Nappi [7.34, 7.35] with Bp. the topological baryon current. ~ is the symmetry breaking pion mass term. The coefficients c~and c~2~ can obviously be estimated from uu-scattering, indeed Pham and Truong [7.32] find from sum rules in the 5, P-wave ITu~channels** yIe2
=
lie2
=
~
(7.14a)
2f~im~,
(7.14b)
making use of the fact that c 2 = f~I4and m~= 26m~somewhat below the physical p-mass because of unitarity corrections. For these values, we find e = 5.39, close to the pirrr-coupling constant g~ITIT= 6.1. The S-wave phase shifts below 1 0eV are not that well-known; with m~= (22—32)m~one can estimate y = 0.28—0.34. For y >0 the term 2~2) is attractive, it tends to destabilize the soliton, therefore, one finds solitonic solutions only for y
= ~0
+
+
4
~2
+ ~
=
* **
32e
+ £~+ +~
8e
-
m4,
+
~
There exist three different fourth-order terms, two of them can be lumped together in the Skyrme term ~ Notice that in ref. [7.32] a factor 2 is missing in (7.14b).
(7.15)
334
U. -G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
For /3 = 17 and ~ = 0.21, the mass of the hedgehog comes out to be 1751 MeV [7.31],which is too high. Performing the usual collective quantization, one finds MN = 1771 MeV and MA MN = 86 MeV, i.e. too large masses for N and ~ and a too low splitting. From these numbers one can conclude that the derivative expansion, i.e. freezing out propagator effects of the heavy mesons, does not give a satisfactory description of the nucleon as a topological soliton. Similar observations concerning this failure of the derivative expansion have been made in refs. [7.36—7.38].For demonstrating that, let us focus on the electromagnetic properties derived from a Lagrangian of the form (7.14) using minimal electromagnetic coupling. The electric and magnetic nucleon form factors derived from such a purely pionic Lagrangian are given by —
G~(q2)= 4u
S 2 GM(q )=4u
J
r2 j 0(qr) 1(r) dr, NI j r 2/T~ ~—,j j1(qr)I(r)dr,
—,~--
0
=
(7.16)
q
V 2 4irf2. GE(q)=1~j r j0(qr)V(r)dr,
V
8u
2
G~(q )2_~_MNj
f(r\. ~,,—)J1(qr)V(r)dr.
1(r) and V(r) are the isoscalar and isovector radial distributions, respectively, subject to the normalizations 4uJ1(r)dr=~,
~Jr2V(r)dr=A.
(7.17)
1(r) = B°(r)12 is model-independent with B°(r) the baryon number density and V(r) is a modeldependent radial function. Obviously, two of the form factors appearing in (7.16) can be expressed in terms of the two other ones. This leads to the model-independent relations 2)= _2(~)
—~-~
G~(q2),
(7.18a)
G~(q G~(q2)=
(~
+q2
G~(q2).
(7.18b)
Equation (7.18a) has also been derived by Nyman and Riska [7.39]. Neither one of these relations agrees well with the existing data, they are even inconsistent with the emporical dipole fits. Furthermore, in the limit q2 = 0 they imply the following relations between electromagnetic radii and magnetic moments ~P+~fl 9MN(MAMNX’(rE)P+’(rE)fl1,
(7.19a)
U. -G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons =
—
2
MNI(MA 5
2 —
—
(rE)fl
~
=
MN), A
335
(7.19b) N
MN
2
2
[(rM)P
—
(7.19c)
(rM)fl].
Equation (7.19a) is fulfilled within 10%, (7.19b) within 32% and (7.19c) within 52%. From these considerations, we conclude that a description of electromagnetic form factors based on the derivative expansion is necessarily limited in accuracy. This has been indeed demonstrated in refs. [7.37, 7.38], using a Lagrangian (7.13) with the following VMD-inspired parameters: c
(1) =
1 1 32g~~~32g
c
=—
(2)
=0 (7.20)
= 8fIT
CSb =
‘
~
c2
=
~
2and the SU(3) relation g~,= (N~gI2)to arrive We usevalue the generalized 2f~gnucleon properties for this model (“modified” at the for c~’1.InKSFR-relation table 7.3, we m~ give= m~ the =static skyrmion) in comparison with the complete model (cf. section 6.1) and the conventional skyrmion. The nucleon electromagnetic and axial form factors are exhibited in figs. 7.2a, 7.2b, 7.2c, 7.2d and 7.2e. The limited accuracy of the derivative expansion is indeed demonstrated and the importance of propagator effects, i.e., explicit vector meson degrees of freedom, becomes obvious. Aitchison and collaborators [7.11, 7.28, 7.29, 7.30], have derived a purely pionic Lagrangian like
Table 7.3 Static nucleon properties. MH is the hedgehog mass, and r~,its radius. MN denotes the nucleon mass, M~— MN the N—A mass-splitting. The charge radii and magnetic moments of the proton and the neutron are also given, as well as the axial vector coupling constant g~and the axial radius rA. “Complete model” denotes the vector meson Lagrangian (6.1, 6.5), the results for the VMD-inspired modified Skyrme model with fourth and sixth order term (eq. (7.20)) are given in the column “Modified skyrmion”. “Conventional skyrmion” gives the results of Adkins, Nappi and Witten [7.39].The asterisks (°)denote input quantities. All meson masses are taken on their empirical values. Quantity MH rH
[MeV]
[fml
MN [MeVI M 5 — MN2 [fm] [MeV] (r~)~ (r~) 2 (r~.,)~2 0 [fm] (r~)~’2 [fm] [nm] [nm]
It~,,Iis~l gA
(r~)”2
[fm] [MeV]
Complete model
Modified skyrmion
Conventional skyrmion
1465.0 0.48 1574.0
1650.0 0.60 1700.0
865.0 0.59 939.0°
437.0 0.97 —0.25 0.94 0.94 2.77
206.0 0.80 —0.27 0.73 0.75
4.55
293.0* 0.88 —0.31 0.79 0.82 1.97
293.0 0.86±0.01 —0.119 ±0.004 0.86±0.06 0.88±0.07 2.79
—1.84 1.51 0.99 0.60 93.0*
—3.82 119 1.27 0.48 93.0°
—1.24 1.59 0.65 0.35 54.0
—1.91 1.46 1.25 ±0.01 0.68±0.02 93.3 ±0.3
Experiment — —
939.0
)
336
U. -G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
a
b
~
0.6
::
::
G~(~2)
006
O’2
G~(~2
I .
/
.
~2 [GeV2/c2]
~2
c________________
G~(~2) 4.0
[GeV2/c2]
d________________
G~(~2)
-4.0
\
\ -3.0
:: ~z [GeV2/c2]
~a [GeVa/c2]
e GA (~2)
1.0
N
——
—
DIPOLE FIT A
4’5fm’
~2
[Gev2/czj
Fig. 7.2. (a) The proton charge form factor for the complete model (solid line) and the modified Skyrme model (dashed—dotted line). The dashed line gives the empirical dipole fit. (b) The neutron charge form factor (same notations as in (a)). (c) The proton magnetic form factor (same notations as in (a)). (d) The neutron magnetic form factor (same notations as in (a)). (e) The axial form factor (same notations as in (a)).
337
U. -G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector rnesons
(7.13) by starting from an effective meson Lagrangian ~
=
~(~‘
ci, p, co, A
1) and integrating out the heavy meson fields ci, p, A1 and co. This amounts to substituting the heavy mesons by their of 2 =equations m~I2f~, in motion ~SmIao = ~Sm/~W = 0. The result for the parameter of the Skyrme term is e agreement with (7.14b), and is rather model-independent, where the model-dependence comes from the choice of the non-anomalous ITpA 1-Lagrangian. The result for the non-Skyrme parameter depends strongly on the ITA1-mixing and the particular choice of the irpA1-Lagrangian, which might be related to the fact that the ci does not show up as a clear resonance in the meson spectrum. They also investigate the most general sixth-order term, not restricting to the particular choice Tr(Bp. B ~‘) Using approximate chiral profiles like F(r) = IT exp(—rIR), where R measures the extension of the soliton, they conclude that the derivative expansion does not provide a useful approximation for discussing stability of the soliton at sizes relevant to nuclear physics. For further details on this, we refer the reader to refs. [7.11, 7.30] and the recent review by Aitchison [7.40]. ...
~
‘—
7.4. Remarks on vector mesons coupled to two-phase models
Up to this point, we have made no reference to the quark degrees of freedom inherent in the QCD Lagrangian. The effective meson theories treated so far are just opposite to the bag model [7.24]which pictures the baryon as an assemblage of quarks confined by fiat in a bubble where the perturbative QCD is assumed to hold (cf. section 7.1). One might argue that to incorporate also perturbative QCD it seems necessary to construct models which interpolate between these two extremes.* Here, we will demonstrate on a specific example (the to-meson coupled to the chiral bag) the physics one expects from these models and the actual difficulties one encounters. First, let us elaborate a..bit on the expectations. In (1 + 1)-dimensions, it has been proven that bag boundary conditions arise as bosonization conditions on the quark fields [7.42, 7.43, 7.44, 7.45], i.e. that there is a complete equivalence between a purely bosonic or two-phase (mesons chirally coupled to quarks at the boundary of the bag) description, the so-called “Cheshire Cat principle”. This exact bosonization is assumed to hold also in (3 + 1)-dimensions, and can be cast into the statement that all physical observables are indeed independent of the bag radius R. Stated in another way, this means that there exists a smooth limit from a purely mesonic model (like e.g. a skyrmion with vector mesons) to a chiral two-phase model, with free quarks inside a cavity of radius R and mesons (IT, p, to,. .) in the outside. The coupling of the mesons to the quarks can be direct (like e.g. surface couplings) or indirect (through currents which leak out of the bag, like e.g. the baryon number current). Unfortunately, these models have to overcome some major obstacles not present in (1 + 1)-dimensions, namely the chiral Casimir effects [7.46—7.51],i.e. in general infinite contributions of the vacuum to any observable C. At the present stage, no completely convincing renormalization program to cure these divergences exists, and therefore the calculations performed so far [7.13, 7.14, 7.15] can only be considered as a first step towards a chiral bag with pions and vector mesons. To be more specific, let us briefly outline the two-phase model with co-meson stabilization considered by Klabucar and Brown [7.13]. The soliton phase of their model is a non-linear if-model whose topological baryon current is coupled to a vector meson field cop., as proposed in ref. [7.34]. (The inclusion of possible surface couplings has been explored in ref. [7.14].) On the bag boundary S at r = R the soliton is coupled to the quarks inside in a chirally symmetric way. The pertinent action reads .
*
For a review of the history and further references on the chiral bag model, we refer the reader to the work by Rho and Brown [7.41].
338
U. -G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
~ +
~
Tr(U
—
1)] 0(V)
~U5q s(S) + [+~ic~/q
—
—
B] 0(V),
(7.21)
where U = exp{ir lrIfIT}, U5 = exp{i’r. IT1/sIfIT}, s(S) the surface delta-function, 0 the step function and V(V) gives the volume inside (outside) the bag. The constant B gives the strength of the bag volume energy. Using hedgehog ansãtze, one can write down the equations of motion for the quark-fields q, cj~in V, the meson fields IT and to in V and the continuity equation for the flux of axial current. The latter reads
+
f~F’=
—
c~y5~ Pie. Pq .
(r = R).
(7.22)
Now, for any non-vanishing chiral profile F(R) at the surface the flux of axial current in (7.22) will have a vacuum contribution which influences the solutions F(r) and w(r) outside the bag. For any observable C, one has to calculate the vacuum asymmetry =
~ lim ~0~[q~(x,s), Cq(x, 0)]~0~
(7.23)
+
which can be done analytically in some cases (C = baryon number) or numerically. Apart from the baryon number [7.51] all other relevant observables (energy, flux of axial current,. . .) develop infinities proportional to Ins, s’, and so on. Vepstas, Goldhaber and Jackson [7.49] have given a numerical regularization procedure based on the evaluation of the pertinent mode sums,* which renders the energy, the flux of axial current,. . . , finite. Kiabucar and Brown [7.13]use this procedure to take the Casimir effects into account. Their energy-functional reads 25
)+f~m~(1_cosF)+~°
F’
E=4lrJdrr2[t~ (F~2+ —~w’ _~m~co~]+~°+~RB+E~, where
~
=
(7.24)
E
0R is the eigenfrequency of the hedgehog and E~the Casimir 4, fIT = 93 MeV, mIT lowest-lying = 139 MeV, m,,, = 782.4quark MeV, state and g~I4IT= 10 they find energy. With B = (145 MeV) the energy of the hedghog as displayed in fig. 7.3. The total energy is rather insensitive to the bag radius R, nicely confirming the ideas of the “Cheshire Cat” principle. A similar observation has been made in ref. [7.14], in a model with an to surface-coupling and a similar procedure to render the Casimir energies finite. (The parameters used in ref. [7.14] are fIT = 93MeV, g, 4, the 1, = 7.04, = (160 MeV) surface coupling parameter does not enter the energy-functional.) Again, let usBpoint out that this R-independence of the energy certainly depends strongly on the regularization procedure to extract the finite part of the energy. The isoscalar rms radius (r~)~ shows an approximate R-independence. The axial coupling gA is not insensitive to the choice of the bag radius R, the authors of ref. [7.13]therefore conclude that it might be most economical to choose a certain bag radius. This statement seems to be a *
The non-uniqueness of this procedure is discussed in ref. [7.48].
U. -G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons TWO-PHASE MODEL WITH
w
339
MESONS
.5
R[fm] Fig. 7.3. Total energy E,
0, of the chiral phase-model with o-meson stabilization. E,01 gives the contribution from the soliton (pions and w-mesons), ~ the contribution from the valence quarks, ~ the vacuum contribution (evaluated according to ref. [7.49])and E~0the volume energy By. E,01, gives the result of ref. [7.14],where the w-meson is coupled to the surface of the bag.
bit premature in the light of the mesonic models discussed before only when one includes all mesons relevant at nuclear length scales, a satisfactory description of the static nucleon properties emerges. Notice also that a similar study of the p-meson coupled to the chiral bag via the hidden symmetry approach [7.15] does not give an energy-insensitivity to the bag radius. It remains to be seen how a more realistic two-phase model with p- and to-mesons coupled to it will indeed exhibit R-independence of the physical observables. More work in this direction is certainly needed. —
8. Summary and outlook We will be brief in summarizing the results discussed in the previous sections. We have shown that an effective chiral Lagrangian of pions and vector mesons can give a rather satisfactory description of the baryon properties, with all parameters fixed in the meson sector. The soliton scenario of baryons offers a flexible approach to investigate various aspects of hadron structure, in particular for the low-lying non-strange hadrons (N, ~,. . Let us emphasize that here the question of giving the nucleon a certain size (“bag radius”) does not arise, indeed depending upon the probe, the apparent nucleon radius differs. In agreement with experiment, we find that relectromagnetic> raxial > rbaryon chargeS The baryon charge radius cannot be measured directly but can be estimated from models of NN-annihilation. Our finding of r~ 0.5 fm is consistent with some models of the NN-annihilation [8.1]. No other hadron model considered so far can reproduce these facts in such a simple manner, and is easily applicable to different situations like the nucleon properties, the NN-interaction and nucleon—antinucleon physics. We would like to stress again that concepts like (asymptotic) Vector Meson Dominance and Boson-exchange are embedded naturally in the framework of the effective theories discussed, and these concepts are known to be important for low-energy hadron physics. It is worth to stress that the number of parameters does not proliferate when one includes the vector mesons. Indeed, for the rrpco-system we are left with only three, namely the pion decay constant, the
340
U. -G. Meifiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
pion mass and the (universal) gauge coupling. The latter is even constrained by the pion charge radius to be g~ITIT 2IT, close to its empirical value. For the strange particles, the situation is not that well understood. Various approaches to describe the physics of strange baryons have been investigated in the conventional Skyrme model [8.2, 8.3, 8.4, 8.5, 8.6]. Treating the symmetry-breaking terms, which have to be included in SU(3)f, via perturbation theory leads to a drastic reduction in fIT if one wants to get an overall fit to the baryon octet [8.4]. One can improve on this if one includes the symmetry-breaking terms in the full diagonalization procedure, still the static properties of the low-lying hadrons are not too well described [8.6]. Therefore, Callan and Klebanov [8.5] have proposed an alternative scheme to treat strangeness in this kind of effective models. In their framework, hyperons are described as bound states of kaons and an SU(2)-soliton made out of pions only. To leading order in N~,this approach works quite well, but for subleading Ne-orders it sometimes gives wrong predictions, e.g. the ~ comes out heavier than the ~:,*. It therefore seems natural to extend this approach to the SU(2) sector including also vector mesons, i.e., binding kaons to the solitons discussed here. As has been shown by Rho and collaborators [8.7], a minimal extension of the conventional Skyrme model including the to- and the ~-meson indeed gives the proper sign for the v—I splitting for values of the toNN coupling constant close to its empirical value. This is somewhat surprising in light of the work of Adkins and Nappi [8.7], where a good fit to the Nz~-system is only obtained for a large gm~It seems obvious that the inclusion of the p-meson (and the A 1) is necessary before one can make a final statement. Certainly, chiral symmetry is broken much more strongly in the strange sector than in SU(2) and one therefore faces all problems already encountered in the sixties, when soft-K predictions on the basis of chiral perturbation theory were worked out and showed much larger discrepancies to data than equivalent soft-pion theorems. As was already pointed out in the Introduction, Vector Meson Dominance is also not as good in the SU(3) sector. A description of the hyperons which is of similar quality than the one of the nucleon presented here certainly has to account for deviations from VMD, inducing even more complexity in the strange sector. Of more speculative nature is the assumption that the Cheshire Cat principle breaks down in the hyperon sector, based on the observation that in chiral two-phase models, some observables like the mass or magnetic moments are not radius-independent [8.10]. At the present stage, the SU(3) soliton picture and the SU(3) chiral bag are far from being understood well enough to draw a decisive conclusion on this point. Finally, let us point out that the soliton picture of the baryons we have presented here is non-perturbative, and therefore goes beyond any perturbative QCD calculation. The connection of the soliton picture to well-known concepts of conventional nuclear physics and hadron phenomenology is obvious and shows how one can naturally incorporate phenomenological and empirical constraints in a serious model of the baryon structure.
Acknowledgements I am particularly grateful to Norbert Kaiser for a critical reading of the manuscript and to Véronique Bernard for pertinent criticism. The work presented here could not have been done without the help and support of Gerry Brown, Wolfram Weise, Andreas Wirzba and Ismail Zahed. Fruitful discussions with Marc Chemtob, Vincent Pasquier, M. Praszalowicz, R. Rajaraman, S.0. Rajeev, Mannque Rho, Joe Schechter and Koichi Yamawaki are also acknowledged.
U. -G. MeiJJner, Low-energy hadron physics from effective chiral Lagranglans with vector mesons
341
This work is supported in part by funds provided by the U.S. Department of Energy (D.O.E.) under contract number DE-ACO2-76ER03069.
Appendix A. Non-linear realization, PCAC and VMD Here, we want to show one can construct the most general Lagrangian (for the non-anomalous sector) which incorporates the PCAC and Zumino’s form of VMD [2.47]. We will make extensive use of the notations developed by Callan, Coleman, Wess and Zumino [2.12, 2.49} and follow the work of Giler, Kosinski, Rembielinski, Maslanka and Szymanowski [2.131.As already pointed out by Weinberg [2.50] the most general framework for spontaneously broken chiral symmetry (PCAC) is provided by non-linear realizations of chiral symmetry. For that, consider a compact group G of the dynamical symmetry and H C G the group to which it is broken down (the algebraic symmetry). The following fields will now be considered. First, there are the preferred fields 4 which parametrize the coset space GIH, transforming via gL~=L4,,
h=h(cb,g)EH,
L~EGIH,
cb’=çb’(cb,g).
(A.1)
Furthermore, there are vector fields transforming as gauge fields, actually, there are two types, one connected to the global symmetry g E G parametrized by the generators T~= Tg: —
(A.2)
V,~=V~(Tg)k,
and vector fields connected to the coset space (generators T
=
—
Ta),
(A.3) It is convenient to introduce the following Cartan forms 1(~ + V~+ A~)L~ = W~(Tg)k+ ~(T~)~ ~ + L~ L~1~L~ = W~(Tg)k+ ~(Tc)a w~+
(A.4)
which transform under the local (tilde) and the global (no tilde) group, respectively. Any other field ~li transforms as =
D(h(g,
4))
çb-
(A.5)
where D is a linear representation of H. The pertinent covariant derivatives are defined via =
+
(A.6)
with Th the generators of H in the representation D. The most general Lagrangian invariant under the
342
1]. -G. McifIner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
local transtormation has the form
~
=
~ocal
F,1V(~),D~v~ D~, ~),
F,1JX) =
D~X,1+ [X,1,Xj,
—
ñ,1x= CX
(A.7)
Xj.
+[~,
Similarly, the most general Lagrangian invariant under global transformations reads F,1jw). D,17J~,D,1~I,~i)
~(Tha’
“~globat =
(A.8)
F,1v(X) = + [X,1,Xv]. 2)R and H = SU(2)~diagonal. The preferred fields defined by Let us now specialize to G = SU(2)L ® SU( L~ 1~ (~,~) E G!H transform as —
g1~= ~‘h,
~
=
(A.9)
and the gauge fields via ~
=g(~R)Ag(L
R)
+g(L~)ôg(LR),
(A.10)
where A,1 = A,1 T for the left- and right-handed fields. Vector and axial vector fields can be introduced by V,1 = ~(A~ + A~)and A,12,= 0), ~(A~ the basic representation of (g~~ the generators TR =A~).In (0, ir!2). Using the defining equations (A.4), we can rewrite Tlft~ are given TL = (iT! in terms of ~, A,1, and V,1 as the Cartan forms by ‘~ and .
—
~~
= ~
w~=
+ ~
(All) =
+
~[~(V~
+
A,1)~+ ~(V,1 A,1)~]. —
If we choose to work in a parametrization U = =
—
UV,1Ut + A,1 + UA,1Ut
—
with U transforming as U’ = g~Ug~, we have
~2
U (A.12)
=
+
UV~U~ + A~ UA,1U~+ U —
or using the u--model parametrization U = o-
(
iT
____
~=-~9~r--
i+u-)’
+ iT~~7,*
iT W,1[\\
we have
(lTX9,11T
1+~r (A.13)
i~
-
W,1W,1
(
_______
2~t7V,17T,1
1+u
To construct the most general Lagrangian invariant under the local SU(2)~ and global For the moment, we Set f~ 1. *
=
U.-G. Meiflner, Low-energy hadron physics from effective chiral Lagrangians with vector inesons
343
(SU(2)L ® SU(2)R) transformations, one has to use (A.7) and (A.8). Before doing so, let us see which constraints VMD according to Kroll, Lee and Zumino [2.28] imposes. The electromagnetic Lagrangian is supposed to take the form =
+
)+
(eIg)A~
~m~(p~3~)2 ~(F~)2,
(A.14)
-
where A~mdenotes the photon field, F~the (Abelian) electromagnetic field strength tensor, p~the neutral component of the p-field and other vector fields are denoted by the dots. Obviously, ~~ 3(p~) is m =0, the Lagrangian invariant underweU(l) p~invariant p~+g~’ set A (A.14) (which will gauge call ~)transformation is globally chiral and ô,1A. takes Iftheweform (A.15) with ~‘invariant under SU(2)~.Obviously, the p-mass term can only be invariant under global SU(2)~. Because ~(A.l5) should also be globally SU(2)®SU(2) invariant, the problem is to find the proper choice of the physical p-field to reconcile these conditions. Now, the most general ansatz up to fourth order in the Cartan forms i~, w, ~, and ~ reads =
‘~oca1 + ~g1obal
=
a Tr[D,1
77~ —
+
+ d~Tr[~,1, ~v]2
b Tr[F~v(~)]+ c Tr{F,1V(~)+ Tr[~2]2
+~
—
~
~v]}2
d Tr[~2] ,
(A.l6)
2] +..., 0 Tr[(~ )2 + (~ )2] + a1 Tr[~ + (a2 a1) Tr{?) where we have written down the terms which survive after imposing VMD in ~Iobal and in “~ocaIa term containing second order derivatives in the fields has been omitted. Let us now see which restrictions VMD imposes on the parameters appearing in (A.16). Working out the U(1) gauge transformation acting on ~gIobaI’ we find to lowest order in derivatives: ~g1obal
=
a
—
=
with a
—
(m)~Tr[(~
-
)2
—
-
(
-
)2] + (~)2a
—
Tr[(~
-
)2
-
2~
(A.17)
a,g~Im~. The corresponding p-field reads
P,1 r=g~1[V,1
—
(A.18) V,1 =
1[~t(~
W,1
~
+fl,1)~+~(W,1
W,1 ~~15L
m~)~1~
where V,1 is the vector part of the linear multiplet (V,1, A,1). For SU(2)L®SU(2)R, we can express V,1 in terms of ~T, O~,P,1 and A,1 as ~
(A.l9)
Up to now, we have only identified the physical fields for the pseudoscalar Goldstone modes and the
344
U.-G. MeiJ3ner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
vector fields via VMD. For the physical axial-mesons ~ there is no such principle. We only know that they should transform according to the adjoint representation of the diagonal subgroup of the chiral group, and the quadratic part of the Lagrangian should be diagonal in the physical fields. To fulfill these requirements, we use the appropriately shifted axial part of the linear multiplet, i.e. =
+f~S(o-) ~
~
—
g~p
yr),
X
(A.19a)
where we have reinstated the pion decay constant f~.S(o) is a function of o~= (1 v.2)1/2 For S(1) = d[(a 1)(m~/g~)2 dl the non-diagonal terms are eliminated, this is the only condition on S(u). Usually, we set S(cr) = S(l) = constant, but using a more general S(u) does not influence the on-shell S-matrix elements; indeed, a specific choice of S(u) allows for a smooth off-shell extension (see below). Normalizing the kinetic terms gives the following relations: —
—
—
2a=g~2,
m d=(~)
2(b+c)=g~2,
2
m +(a_1)(~)
gA
2
g~
)
/m g~\2 S(l)=_1_(a—1)(\ ‘) , m~g~
(A.20)
~ [1(1)(m~)2p] with /3 = a 2. For VMD to hold, we have to choose /3 = 0. Therefore, the Lagrangian in terms 2(m~Ig~) of the normalized physical fields reads after imposing VMD (0 = = = ~2 = ~VMD
=
2g~Tr[D,1 li~ Dii] —
+
cTr{F,1v(~)+~
+
d~Tr[~2j2
-
+
[~—~
+
(m)2
—
C]
Tr[F~~(~)]
d~Tr[~,1,~V}2
Tr[~2]
-
(m)2
(A.21)
Tr[ii2]
m 2 m -2(a-1)(--~)Tr[ii~]-(--~)Tr[~-w]2, where the Cartan forms ij, ~, w and ~ are given in terms of the physical fields ~, p, Ay~ (we will omit the superscript (phys) on the axial field) as: 11 /
~r (1T.o~~.~T)\1
‘t),1TfIT+f 2
2+)1~
n,1T{[l+a(1u)][f(l+~s(u))(o,1~gPP~)gAffA~
+
1 +a(i
+ ~
((1
~ A,1 )]
~) + g~f~
+ ~ S(~))(~~,1
}
U. -G. MeiJlner, Low-energy hadron physics from effective chiral Lagranglans with vector mesons
w =T•I
345
(A.22)
/1
w,1T{[l+a(1~)][gP(u(1+a)s(u)(lu))p,1 (+a) ~xA (1+a(1—ff2))
f.TT
f~r+1) with T = iT!2. The generalization to SU(N) 1 ® SU(N)R is straightforward but tedious. If we now want to restrict ourselves to the lowest possible numbers in derivatives (quadratic), the quartic terms in (A.21) can be uniquely eliminated giving the following constraints: 2, S(o-) = E[1 , ~ = 1 (mAg~Ig~m~) (A.23) 2 2 a=0, d (+) =0, d (—) +c(1—2cg~)=0, l—e 2 =2cg~, —
—
so that we recover the Weinberg relation f~= m~Ig~[1 (m~gAIg~mA)2]. The Lagrangian reads —
~TMD
Tr[D,1~v
=
-
D~i 7,1] +
Tr{F,1~(~) + (1-
gA
2] - (m)2
-
(m)2
Tr[(~ ii)2 -
+
Tr[~
(~ w)2 -
~2)[~,
-
~2j
~]}2
(A.24)
where the KSFR relation follows for ~ —1. Of particular importance to the general Lagrangian (A .21) is the question of universality. From (A.21) one can read off the relation between the gauge coupling g~ and the pirn decay constant ~ as l+(a2—1)Z2 l—(a —l)2Z2—/3
Z
‘
so that the universality condition g~ ~ —
/3
= 2a(l
—
a)Z2.
mggA g~m~
(A.25)
reads (A.26)
This means that even if VMD is fulfilled (/3 = 0), for universality (a = 0 or a = 1) to hold, one needs further restrictions. Now, a = 1 is excluded by the KSFR relation. So even if we restrict ourselves to lowest order in derivatives and have implemented VMD, we have a one-parameter family of equivalent Lagrangians, so that further conditions like KSFR and universality have to be imposed to make it unique. The most general Lagrangian fulfilling VMD, KSFR and universality is therefore given by (A.21) for a = 0, (m~!g~)2 = 2(m~Ig~)2. The remaining freedom can be used to eliminate the terms of fourth order in derivatives to lead to (A.24) with e = —1. What remains to be done in this framework is to work out the anomalous sector and construct baryons from it. At this state, we guess that the strict
346
U. -G. Meijiner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
implementation of VMD will give rise to problems in the baryon sector (cf. the discussion in section 6.6).
Appendix B. p-mesons in the Skyrme model In this appendix, we want to show how the p-meson can be included in the Skyrme model in a simple, chirally invariant way following the work of Adkins [6.71and MeiBner [6.8, 6.9]. For that, consider the following Lagrangian:
~
=
Tr[c~ U 9~U+]+ ~ Tr[ô~U U~,~
+
~
—
~Tr[R~R~]+ ~
Tr(U
—
1)
U~]2—(1—
/3)
~ Tr[B,1B~]
Tr[R~R~]+ a Tr[R,1~ô~U+ U
The pion field is incorporated in the SU(2)-valued matrix U(x) decay constant. The p-field is a 2 x 2 four-vector R,1
=
(B.1)
= exp{iT. lTIf,j ,
with f~ the (weak) pion
(B.2)
+ ~T
where and p~are real. R,1~is the Abelian field strength tensor, R,1V = tTl,1Rv For /3 = 1, we have the model of Adkins [6.7], where stable solitons arise due to the fourth-order Skyrme term proportional to e~2.For /3 = 0, the stabilization is due to a very heavy w-meson as proposed in ref. [6.9], indeed the stabilizing term ‘—~Tr[B,1, B~]is nothing but the w-meson stabilization discussed by Adkins and Nappi [6.3]. ~ is the baryon current, —
=
U~3
~
0U~ U~a~U] and the constant 2
2
can be related to the ioNN-coupling constant g~,via
2
e6 = gjm,~
(B.3)
.
Empirically, g~is found to be g~/4~r10—12. The pion mass term explicitly breaks chiral invariance, so does the p-meson mass term. The last term in eq. (B .1) gives the pirr~-interaction,the coupling constant a can be determined from the decay width ~,-*2,T to be a 0.0444. The whole Lagrangian (B.1) is invariant under chiral transformations U LUR ~, R ,1 ~ LR ~ ~, * as well as under P, C and T. The number of degrees of freedom of the p-field has to be reduced to ensure unit isospin, one therefore imposes the chirally invariant constraint —~
Tr[R~U]=0. *
Except for the mass terms, of course.
(B.4)
U. -G. Meij3ner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
347
To calculate the properties of the static hedgehog soliton, one does the usual hedgehog ansätze for the meson fields, i.e. U(r) = exp{iT~PF(r)} and R’(r) = ~Ta6~~7 R(r), subject to the following boundary conditions: F(0) = ir, F(cc) = 0 to have baryon number 1, and R(0) = R(~)= 0 to have finiteness of the energy. The hedgehog properties can be read off from table B.l, Adkins chose to use the meson data (e
4 ~ and ~ as parameter to fit the N and s-mass, whereas MeiBner preferred not to vary the mesonic input. The static nucleon properties follow after introduction of time-dependent collective coordinates and quantizing them. Through that procedure, the time components of the p-field ~1(r)and ~2(r) (cf. eq. (6.23)) are excited. It is interesting to note that the moment of inertia A as defined by —
3x~[U(r, t), R~(r,t)] = —M~+ ATr(A~A) J
(B.5)
d
is almost entirely given by the parts from the classical fields U(r) and Ri(r),* the parts involving ~
12(r) contribute approximately 1% to the nucleon mass [6.8]. Some static properties of the nucleon are given in table B.l. Similar to the Skyrme model calculations of Adkins, Nappi and Witten [6.6] fixingf,~and e4 ~ to get the proper N, ~ masses results in too smallf,, and g,~.On the other hand, leavingf~and g~,!4iron their empirical values gives too high masses, a reasonable Na-splitting and an improved gA~ In general, the predictions of this model are a few percent better than the ones of the conventional Skyrme model. For further details we refer the reader to refs. [6.7, 6.9]. In ref. [6.8], the baryon number B = 2 solution for /3 = 1 was constructed, it turns out to be quite similar to the Skyrme model B = 2 solution. The main difference between these two solutions is that the two-body repulsion between two strongly overlapping solitons is significantly weakened in the model with explicit p-mesons due to the strong attractive p’rr-interaction. For /3 = 0, the situation is rather different. For the parameters f,~ = 93MeV and g~j4ir= 10, one finds a repulsion of 1.9 GeV, due to the strongly repulsive character of the (0-meson at short distances. This again points to the importance of building models which incorporate all mesons relevant at nuclear length scales. Finally, let us point out that the p-meson introduced here is not the same as a massive Yang—Mills particle (for m~—~ ~) or gauge boson of a hidden local symmetry, it is a constraint field connected to pions by the minimal interaction term in (B.l). Table B.1 Some hedgehog and static nucleon properties in models with pions and p-mesons alone. (*) denotes input quantities. g,, is the oNN coupling constant, and M0 the energy of the static soliton solution, the hedgehog. The results are taken from the work of Adkins [6.7]and Meillner [6.9]. ref. [6.7]
f~
*
[MeV]
ref. [6.9]
52.4
g~I4sT
—
e4 M,, [MeV] M,., [MeV} M1 — M~ 2 [fm] lMeV] (r~)” gA
4.65 865.0 939.0* 293.0* 0.70 0.65
93.0* 6.45*
93.0* 10.0*
Experiment
93.0* 12.0*
75.0* 10.0*
—
—
—
—
1219.0 1290.0 285.0 0.50 0.85
1390.0 1442.0 206.0 0.55 1.12
1468.0 1503.0 180.0 0.58 1.23
1012.0 1069.0 230.0 0.49 0.89
93.0 10—12 6.11 —
939.0 — 293.0 1.25
It is mostly the pionic part from the non-linear r-model which gives the moment of inertia (cf. table I of ref. 16.81).
348
U. -G. MeiJ3ner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
Appendix C. Calculation of the moment of inertia and the electroweak currents in the Trpw-system Here, we want to give some explicit intermediate steps in the evaluation of the moment of inertia in the ~rpto-system (eq. (6.26)) and the electromagnetic and axial currents (6.48) and (6.75). First, let us consider the moment of inertia A. For that, we make the time-dependent ansãtze for U(r, t) (6.11), p°(r,t) and ca(r, t) (6.23), and evaluate all terms which are quadratic in the rotational velocity of the soliton, i.e. L(t)_jd3r~[~, TP,1, with A =
j~° A[F, G,
4ir
A(t)
=
~
;
‘l dr.
~
(Cl)
—MH +2AK2
W,1]
K is treated as a time-independent K =
0,
exp{iT Kt}
uniform rotation (C.2)
with iT . K = A~A.To order K2, the following terms contribute A
3rTr{(a 1= ~
J
d
2}, 0~~
+
.
(C.3a)
~—igT.p~)
A 2=— ~-Jd3rw.w,
(C.3b) 3r (curl w)2
A3 =
—
J
d~rW,JW,J
=
— ~
J
(C.3c)
d
A 4=~Jd3rp0j.p0j,
(C.3d)
As=_~Jd3ro.B,
(C.3e) tr PkU},
A6 =
~2
A 7
=
~
Id3r ~um0k~
Trf iT
J
Tr{iT pk(ô~U~ Ut + Ut. d~U)+
3r Eujko
~
po(~kU
Ut
+
Ut ~U)
+
(C.3f)
T ~p0U
r
.PkUT .poU}.
(C.3g)
d
We have omitted the contribution from the non-linear u--model which can be found e.g. in ref. [6.6]. Because of the hedgehog structure, all three-dimensional integrals can be reduced to one-dimensional ones by use of the following angular averages
J
dQ (K.
~)2 =
~K2,
~—
J
dQ (K x ~)= ~K2,
(C.4)
where S2 is the unit-sphere in two dimensions. Our conventions are = diag(1, —1, —1, —1) and = — = 1. The following useful identities will be employed quite frequently
U.-G. Meifiner, Low-energy hadron physics from effective chiral Lagranglans with vector mesons
UT.VUt=(cosF+irsinF)rV(cosF—iT.PsinF) = cos(2F)r V— sin(2F)’r~(r x V) + 2 sin2FT PV P, {~kU,Ut}+
2i~r.~(F’
+i~k
—
349
(C.5)
sin(2F)
where V is a vector. With that, let us evaluate the contributions A 1,.
~=VU=exp{i’r~PFI2}
.
. ,
A7. For A1, we use
and find 2(FI2))]A~.
~ ô0~—igr•p~= A[2i’r~K(l —cosF— ~1)+2iT With (C.4) and multiplying by r2 leaves us with ~
—
A~=2f~[(2sin2(~)-
)2
-
~
(2sin2(~) - ~i)( 2sin2(ç)
=
~f~r2[8sin4(ç)
-
PK~(—42—2sin
8sin2(~1+ 3~~
+
+ ~2) +
~].
~ (2sin2(~) +
(C.6a)
Similarly, (C.3b) gives
=f (— ~-~) ~-(K x
A~
“) =
—
in
(C.6b)
~2
and for (C.3c) we find
~ =f (— ~)curl[ ~
(K x r)]r2
=f(_ ~)[~-~+
(~-~
—
~~~.)pK.p]2r2
—
1
(~~2+ ~2)
(C.6c)
The contribution from p-kinetic term (C.3d) is somewhat more lengthy; we will first evaluate r p0~. One finds a a a T.p0,=r(ô0p,—d,p0+g~ =
abc
b
p0p1)
(2Irg)A[r Ki~’U+ r rK’V + r’K PW + ‘r. PK~PP’X]A~,
with U=—r~, W G— ~2
—
Vr=G~1_G—~, G(~1+ ~2)’ X —r~+2~2+ G~2.
350
U. -G. MeiJ3ner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
The contribution to the moment of inertia functional reads 112 r Tr[Tp0~r.p0,1 2
k
~,=
~j
S
[(U + V
=
+
W + Z)2 + 2U2
+ 2V2 +
2W2]
(C.6d)
~
The w,1B~~coupling from the Wess—Zumino term is much easier to evaluate,
=
=
—(3g/2)
J ~ (K x ~)(—flsin2F (—2r
X
K) (C.6e)
S2
=
(g/2ir2)~F’sin2F.
A 6 and A7 are contributions from the ‘rrpw-interaction in the Wess—Zumino term; we start with the evaluation of A6. For that, notice that = (curl w)’~and
t + Ut•
~k~’)}
Tr{iTpO(akU U =
(~,+ ~
—
2)(F’
—
sin~F))PkK. P
K~1sin(2F)
—
tT PkU} Tr{i’r.p0U =2 cos(2F) (K x P)~—2 sin(2F)
—
~2
sin(2F)
~kK
P,
(C.7a)
(~x (ek x P)),
~
(C.7b)
where ek is a unit vector in k-direction. Putting pieces together and performing the angular integration, we end up with =
—
—~--~
[~P’~sin(2F) + 2cPF’(~ 1+
8~r
42)1
—
—~--~
l6~r
G~1X. sin(2F).
(C.6f)
For A7, similar algebra using t Tr{iT
Pk(ôO~
U
+
U~~ 0U)} =
2
sin(2F)(ek x P) . (P x K)
leads to = +
—~--~
8ir
GP’
sin(2F)
—
—~---~
l6ir
G~1P’ sin(2F),
where the last term can be calculated from the last term in (C.6f) via the substitutions
(C.6g) qkO __
—
EIOk
U.-G. MeiJlner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
351
F—* —F and U—* Ut. Putting all the pieces [(C.6a) to (C.6f)] together, we arrive at the moment of inertia functional A as given in eq. (6.26) for the complete model (c = 1). For the minimal model, we just have to drop the contributions Let us now turn to the calculation of the electromagnetic currents. As outlined in section 6.2, they are the isoscalar and isovector symmetry currents obtained by the Gell-Mann—Lévy method. For that, consider the transformation Tp,1+w,1—*h(x)[T~p,1+w,1}h~(x)—(1Ig)t~h(x)h~(x), h(x) = exp{— ~i[a0(x)11+ a(x) with U—* hUh~and
~—*
~
h(x) E SU(2),
T]} ,
(C.8)
The variation of the Lagrangian reads
=~[h~h~,hUh~,h(rp,1 + w,1)h —(i/g) d,1hh~] 2). =(ô~a0(x))J~0(x)+(d,1a(x))J~1(x)+U(a~,a
(C.9)
Let us consider the isoscalar current. For that we have to discuss the ungauged WZ action F~ = (—iN~/240ir2) Tr(dUUt)5. U(x) is extended to be an element of SU(3) and the isoscalar generator = diag(l, 1) is substituted by A 8 = diag(l, 1, —2). Only at the end of the calculation can we restrict U to be an element of SU(2). This gives 2/g)w~, (C.lO) J~0= ~B’~ ~B’~ (m fM5
—
—
where the first term obviously comes from the ungauged WZ action, the second from the coupling and third from the (0-meson mass term. Therefore, the isoscalar current is entirely dictated by symmetry-breaking (0 in mass term. In the case of the p-meson, things are more complicated due to multi-pion contributions to the isovector current. We will just discuss this current for the complete model. In terms of the ‘rr- and p-fields it is given by J~=~ = —
~Ut~ ttu t9 i(g/2)(~~.p~~± ~T.ptL~+)I}
if~Tr{T . ~
. d’1~ —
~
—
+
~ovLU.Ut
To get the nucleon current, we first have to project on the charge operator Q
(C.ll) =
~(~ + ri),
which gives
J~= ~J~ 0+J~3~.
(C.12)
In the isoscalar sector, (C.10) immediately gives m ~em =
—
~em~(~’’~) 2 ‘P(r) S,i
—
m
~
m2
~P(r)
(C.13)
352
U. -G. Meiffner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
where we have made use of the quantization rule K = ~!4A. For the calculation of the isovector current, it is important to notice that K is a differential operator in the space defining A = a0 + ia T, i.e. AT,
[K,A]=
[K,A~]=—
(C.14)
~
Therefore, two relevant identities to be used in the evaluation of J~1are Tr[Ar
KA+ra1~_÷ T~’I2A,
Tr[AT. rK~PATa]5.
Ta/6A
(C.15)
The actual evaluation of (C.ll) via the time-dependent ansätze (6.21, 6.23) gives (for the time component J~1) ~ + —(i!2g)(~~
=
0
0+
)]
—2if~A{iT•K~+~iT.K~—2iT.K t+ ~iT•K hjUtiT.KU_ ~UiT•KU eiT~(Kg, + PK P~ 2)~~iT~(K~1+ PK —
—
—
=2f~,A{rK(2cosF—2— ~cos2F+ ~ —2~cosF) 2(F!2) sin2F 4~ sin2(F/2) 2~2cos F) + T PK P(4 sin ~ sin2(F!2)} —
—
—
so that the nucleon current by use of the quantization rules outlined before follows to be =
~
sin~(F/2)+ (1
+
2 cos F)~ 1+
~21.
(C.16)
Similar algebra for the space-components of the isovector current leads to 2jG cos F —2 sin4(F/2)] ~ X r J~ =
—
(C.17)
~f
Notice that the isovector as well as the isoscalar currents are given by =
(I°(r),i(r) ~ x P),
(C.18)
where I°(r)is the charge-density, and i(r) the pertinent current density. From the form (C.18) it is obvious that the static nucleon currents are conserved, i.e. = a 01°(r)—V~[i(r) ~ X =—(ifxf).Pi’(r)+i(r)ifVX(Vr)O.
(C.19)
U.-G. MeiJlner, Low-energy hadron physics from effective chiral Lagranglans with vector mesons
353
Finally, let us derive the axial-current radial function A12(r) (6.75). The axial current is the symmetry current of axial rotations (C.20)
s(x)=exp{—~r.a5(x)}.
Obviously, under s(x) U transforms into sUs and r + into2s. TTherefore, + (0,1. It important to notice weishave to calculate the that there exists no transformation ~—* ~ s) such that ~2 s~ axial-current via the Gell-Mann—Lévy method with the Lagrangian given in terms of ~L and ~ and fix the gauge at the end. ~L and ~R transform under axial rotations as —~
~
(C.2l)
~
so that U = 4~
—~
s~ ~
=
sUs. We can read off ~
from the term proportional to ô,1s s~=
i.e.
&~E=~t[sUs, &S,
~
W,1~ TP,1] =
(~a 5(x))A~(x).
It follows
to
(C.22)
be t.
=
~if~Tr{T.[—U
19~~_ 9’~U U~+2~ t
g
+
~FaVa4Wva
.
f95~+2~9~~• ~
—ig~~.p~ +ig~~.p~~+]}
Ut8~U+ ~ ig(UT.p
—
Tr{T. [9~U. U
t + UtT.p 4U
4U)]}
(C.23)
where a partial integration in the B ~ has been performed. To see that, let us discuss this term in some more detail. For convenience, we will use differential one-forms and define (3.58) df, R = Ut dU, L = dUUt 2 and dL = L2. Under an axial rotation (C.20) R transforms into with dR = —R R~s+.ds+s+Rs+s+Uts+.dsUs W
(0,1
and similarly for L. The 2
W,1B d4x =
—~--~
l6lT
WA
Tr(R3).
(C.26) we find
~‘,
wATr[dss~ +R+ Uts+ .ds(J]3 WA
(C.25)
B -term in this language reads
Let us now perform an axial rotation on
=
(C.24)
Tr[R3
—
T
da
2+ 5(L
so that the axial variation reads
R2)] +
0(a2),
354
U. -G. MeiJfner, Low-energy hadron physics from effective chiral Lagrangians with vector ,nesons
=
—
w A Tr[r . da5(L2 + R2)]
32ir =
=
—~-~-~
—~-~--~-~
dw
—~-~--~-~
A Tr[r
32 ~
. da5(L
w A
Tr[T . da
5(dR
32ir
—
dL)]
-
R)],
—
(C.27)
where we have performed a partial integration and thrown away the surface term proportional to d(w A Tr[T da(R L)]). Using dw = A dx’~, we can convert (C.27) back into the more conventional form —
=
(~
w~Tr[r. (o~U Ut
3ig
),1PO~
—
Ut. ô~U)].
(C.28)
Injecting the time-dependent ansätze (6.21), (6.23) and performing the adiabatic rotation of the hedgehog as outlined in section 6.3, the radial functions A1(r) and A2(r) as given by (6.75) follow immediately.
Appendix D. Some remarks on the SU(6) approach Based on the success of the non-relativistic SU(6) symmetry, which is only exact in the static limit, Caldi and Pagels [7.16, 7.17] have developed a model which treats pseudoscalar and vector mesons on a similar footing. Let us consider chiral SU(3)L 0 SU(3)R. Pseudoscalars transform like (3, 3) ~ (3, 3), i.e. in terms of quark fields ~a~i~y5(AaI2)q whereas vector mesons transform as (1,8)~(8,1), i.e. -~ qy,1(A12)q. To reconcile with the static SU(6), 3)L®SU(3)R. let us assumeIn that pseudothis vectors case, theand vector field scalars belong to a (3, 3)~(3,3) representation of SU( operators have to be described by an antisymmetric tensor, e.g. ~ = ~o,1~(A’~/2)q.The phenomenological vector-mesons follow by projection via pa~ =
~
(D.1)
where Z~2is a normalization constant. The relation (D.1) is called PCTC (Partial Conservation of Tensor-Current) in analogy to PCAC. Automatically, we have current conservation ~v = 0, i.e. we have three independent components for the spin-i fields. The chiral partners to the vector-field can be constructed from the dual of ~ ~,avapT~’~ and therefore projection gives the B(1235)*, jPC = 1 + axial meson as the chiral partner of the p. One also has a conserved current for the axial meson, I’PBa 9v(,~Ta4Lv)= 0. Going now to a linear representation, we can group the odd-parity 36-plet and the even-parity 36-plet of static SU(6) in terms of the meson fields (ir’1, p~) and (~.** B~)via ~
-
MAB = {a.(jo.a
+
~
+
(u. e), 1(ip° +
Ba)}(Aa)ABI~~S/~
(D.2)
with A = ai; i = 1,2,3; a = 1,2,3 and a = 1,2,. . , 8. Imagine now that chiral SU(6)L ®SU(6)R is broken down by SU(6) spin-flavor’ in this case the whole 36~p1et** ~f pseudoscalar and vector becomes massless. Of course, there are no Goldstone bosons of spin 1, Caldi and Pagels [7.16] argue that the .
*
This meson is nowadays labeled “b,’.
**
Let us for the time-being forget the U,f1)-anomaly so that we need not bother to distinguish between SU(6) and U(6).
U. -G. Mei/3ner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons
355
vectors acquire mass through relativistic effects. Insisting on Lorentz-invariance, it is obvious that the kinetic energy terms give rise to spin-dependent mass terms (Lorentz-boosts affect spin but not U-spin), one indeed gets the following mass formula 2 2 2 2 D3 —
m,~= mB
—
m~
which is fulfilled within a few percent when one identifies the scalar particle u- with the ~(980) = a0(980). The vector mesons are therefore called “dormant Goldstone bosons”. An important consequence of putting the vector mesons in a (3,3)~(3,3) representation is that VMD is a consequence of spontaneous symmetry breaking. Indeed, since the pion couples to the axial current via (oi, ~—~f,~,the p-meson couples to the vector current in the same way. Consequently, we have (0~A(0)Irrb(k))=ik,1f~,8ab, (D.4) =
is,18m~/y~
which leads to
2, (D.5) (m~/f~)(Z~/Z,,)~ where the pion normalization can be determined from f,~= \/2Z~I3(othe SU(6) Z,~= Z~, 2 3/2 one obtains the empirical width F(p —* e + e 0).In ). Equation (D. limit, 5) is similar to while for (Z.,T/ Z~)” the KSFR relation. This approach also leads to universality ~ = ~ for more details, cf. ref. [7.16]. Notice, however, that the interpretation of VMD departs from the usual one since here it applies to (~r.~~(A**I2)q) rather than to the vector current ~y,1(Aa/2)q. One can now write down a generalized Skyrme model [7.17] =
tr(8~U~~Ut)±
m~f~Tr[u U. ifUt]
+
(D.6)
with U(x) = exp{ii1~f(x)Ij,~},M = iA’~1T’21 + (u~E)A”p°,with the Wess—Zumino term determined from the fact that 1T 5(SU(6)) -~ Z in the standard fashion. Obviously, baryons fall in multiplets of SU(6). To allow for stable solitons, one has to add a stabilizing higher order term. In ref. [7.18], a coupling to quarks was used to find solitons of this approach, where (o-mesons are included as components of Unfortunately, the quantization of these tensor modes can only be performed for the non-interacting theory. With two input parameters (f,, = 93MeV, mq 370 MeV) we find MH 1.4 GeV and the (ONN-coupling constant comes to be g~,~~/41T = 10.9, close to its empirical value. Obviously, all SU(6)-predictions like ~ = —3 / 2 come out for free. It is worth stressing that in this approach the spherical shape is not an exact solution to the equations of motion, but rather the ground state is deformed and the N~-splittingoccurs through shape deformations. At the present stage, it remains to be seen whether this approach offers a realistic alternative to the ones presented in sections 2 and 3. ~
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356
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