- Email: [email protected]
The weak pion-nucleon vertex revisited
Nuclear Physics A489 (1988) 671-682 North-Holland, Amsterdam
THE WEAK
PION-NUCLEON Norbert
VERTEX
REVISITED*
KAISER
Institute of Theoretical Physics, University of Regensburg, D-8400 Regensburg, FRG** Ulf-G.
MEISSNER
Center for Theoretical Physics, Laboratory for Nuclear Science and Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 USA Received 19 February 1988 (Revised 14 July 1988) Abstract: We evaluate the parity-violating weak pion-nucleon vertex in the framework of a chiral soliton model including pions and the vector mesons p and w. The weak TN coupling constant G, is enhanced by a factor of 13.3 for the standard electro-weak model as compared to the Cabbibo model, with its absolute value given by G,, = 2.7 1O-8. We compare our results to available data and other theoretical determinations. In particular, we predict the y-asymmetry in ‘sF to be IP,(‘*F)I = 1.2 . 10e4. We also discuss calculational differences to standard quark model estimates.
1. Introduction The aim of this paper is to calculate the weak pion-nucleon vertex G, in the framework of a chiral soliton model. This vertex plays a fundamental role in the description of parity-violating contributions to the nucleon-nucleon interaction 1*2) and parity-violating effects in nuclei itself (longitudinal asymmetry measurements, decays of polarized levels, circular polarizations in electromagnetic decays) 3,4). The large uncertainties in the determination of the weak meson-nucleon couplings as obtained from parity-violating nucleon-nucleon potentials which stem from the uncertainties induced by nucleon structure in these calculations ‘,4,5) do call for new theoretical insight. This is possible if one can calculate the matrix elements of the isospin-breaking
piece of the strangeness-conserving
weak current
x current
hamil-
tonian. This matrix element gives rise to a parity-violating TN-vertex, which is the dominant source of nuclear parity violation. Such a calculation can be done in an extended chiral soliton model “) and will be presented here. In sect. 2, we will briefly review how one can go from the standard model of electro-weak interactions ‘) to an effective current x current hamiltonian of the Fermi type. Apart from the Cabbibo part, it becomes obvious that only a non-zero Weinberg angle can lead to isospin-breaking contributions. In sect. 3, we recall the main l This work is supported in part by funds provided by the U.S. Department of Energy (D.O.E.) contract #DE-AC02-76ER03069, and BMFT grant MEP 0234 REA. l * Present address: Niels Bohr Institute, Blegdamsvej 17, DK-2100 Kebenhavn 0, Denmark.
0375-9474/88/$03.50 @ Elsevier Science Publishers (North-Holland Physics Publishing Division)
B.V.
under
672
N. Kaiser, U.-G. Meissner / Weak TN oertex
ingredients of the extended Skyrme model incorporating the vector mesons p and w. Baryons arise as topological solitons in such a scenario as motivated by the large N, expansion of QCD (where N, is the number of colors) ‘). The lagrangian involves pions, p and o mesons and has essentially three parameters (f,, g,,, and m,) which can be fixed in the meson sector. It leads to a rather satisfactory description of the electromagnetic and axial properties of nucleons as well as the strong meson-nucleon vertices (for further details see ref. “)). The pertinent electromagnetic and axial currents derived from the model lagrangian are summarized in appendix A. Sect. 4 gives the calculational details of how to obtain the weak rrN amplitude. For the calculation of the Cabbibo contribution, one needs a determination of the neutron-proton mass difference. This problem is discussed in some detail in appendix B. in the final section, we compare our results to previous calculations and the available experimental data. We aiso comment on the weak current x current matrix element in the framework of the conventional Skyrme model 9), which has been calculated in the case of a vanishing Weinberg angle by Trampetic I”).
2. Effective current-current
hamiltonian
Here, we recall the steps how to construct an effective current x current hamiltonian for the Glashow-Salam-Weinberg theory. The quantum numbers of the nucleon according to the generators of weak isospin and hypercharge are A$. (I = i, Y = l), pR (I =O, Y = 2) and nR (1 = 0, Y = 0), where PJ, p, n denote the nucleons, the proton and the neutron, respectively. We work in a basis of left- and right-handed fields with NL,R= j( 1 r y.JN The coupling of the electro-weak gauge bosons (W,, BP) to the nucleon reads L??int = $giVLY”(COS
@,( 71
Wh + 72 W:) + 73 W3,)NL
+g’ftP,y’“p,4S~Ly~~L+~RY~PPR)BCL
;
(2.1)
with 0, = 13.3” the Cabbibo mixing angle. After the usual rotation in the W3 - B plane onto the mass-eigenstates (x2”) eq. (2.1) transforms into
cos 28w _ +,g.Z,,y&-pLYPPL-.-----
(
w
1 cos 6w
ALy9iL-
2 sin* 8w cos 8
w
FRYPR
>
where 0w is the Weinberg angle (sin2 Ow= 0.23) and ~~abbibothe contribution the Cabbibo model:
3
(2.2) from
N. Kaiser, U.-G. Meissner / Weak TN vertex
Eq. (2.2) can be easily
cast into
an effective
current
x
673
current
weak
interaction
hamiltonian
-2sin2
ew(N,y’“73NL)(N~~(I+73)N)
+sin40,(N~~(1-!-r3)N)(N~P(l+~3)N)}.
(2.4)
With Gr= g*/(4&hri&) = 1.16 * 1O-5 GeV-* the Fermi constant and B,(cos2 8,= 0.947) the Cabbibo angle. Obviously, the first line in (2.4) comprises the Cabbibo model. One can already read off the following facts from eq. (2.4): The first two terms are isospin symmetric and can therefore only contribute via the neutron-proton mass difference, as it was first emphasized by Schiilke ‘l). The third and fourth terms in eq. (2.4) are not isospin-symmetric and will give an enhancement of the parityviolating TN vertex to be discussed in sect. 4. 3. Chiral soliton model To evaluate
the TN vertex, we obviously need a model for the matrix elements (PIH%P) and (nlH%). S UCh a model is the extended Skyrme model incorporating the vector mesons p and w based on the hidden symmetry approach of Bando, Kugo and Yamawaki 12). Since this model and its phenomenological implications have been recently reviewed by one of the authors 6), we will be brief here. The effective meson lagrangian of pions and p and w mesons as hidden SU(2) x U(1) gauge bosons reads
-;(p,,p’““+
wP,,wP”) +;N,gw,B+
Ncg*
+ cF 64?r2
pvm’p
wW’
xTr{iT.p,(apUUt+UtapU)+~gT.p,Ut~.ppU} +$z~f~Tr(U-1).
(3.1)
The SU(2)-valued chiral field U(x) = exp {il. m(x)/fm} incorporates the pseudoscalar-isotriplet pion-field a(x). fT = 93 MeV is the weak pion decay constant. The auxiliary variable t(x) is related to U(x) via t(x) = m in the unitary gauge. One of the main dynamical predictions of the model is the generalized KFSR-relation rn: = rnt = m2 = 2f;g2, which fixes g = 5.85 for m = 770 MeV. g can also be related term generates the to the decay constant g,,,, we have g = g,,,. The Wess-Zumino anomalous meson couplings. The o meson couples to the topological baryon-current BP = (EP’& /24~r*) Tr (d,UU’ . a,UUt * d,UUi) and through a wpr-contact term (for c = 1 in eq. (3.1)). Furthermore, p,+, = aPpv -a,,~~ -t-go, x py and wfiy = a,w, - a,w, are the usual field-strength tensors of the p and the w meson, respectively. N, = 3 is the number of colors, and m, = 139 MeV the empirical pion mass.
674
IV. Kaiser, U.-G. Meismer
/ Weak TN vertex
The nucleon can be constructed from the effective lagrangian (3.1) in two steps. First, one calculates the classical soliton which is degenerate in spin and isospin, using hedgehog arm&e for the meson fields. To project onto states of good spin and isospin, one then performs an adiabatic rotation and quantizes the spinning modes. The time-dependent meson fields read U(r,t)=A(t)exp{i~*
S(r)}A*(t),
w”(r, t) = o(r), w’(r,t)=mKX;,
(3.2)
r
where K = --ai Tr (A’&) is the angular frequency of the rotation and A(t) a time-dependent, SU(2)-valued matrix. The ciassical meson profiles are obtained by functional minimization of the energy-functional (for A = 1, K = 0) and subject to the boundary conditions F(0) = 71;w’(0) = 0, G(0) = -2, F(co) = G(co) = @(co) = 0. The time-dependent Lagrange-density reads d3r~=-~[F,G,o]+OEF,G,w;5,,5,,~]Tr(AAt).
(3.3)
Functional minimization of the moment of inertia O[ * * *] gives the vector meson excitations &, & and 4. For further details, we refer the reader to ref. “). In what follows, we need the electromagnetic and axial currents associated to the lagrangian (3.1). These follow by the usuai ~oether-construction and are summarized in appendix A. Let us point out here that the model defined by (3.1) gives an overall satisfactory description of nucleon static properties, the electromagnetic and axial form factors as well as the strong meson-nucleon vertices.
4. Weak pion-n~~l~~ vertex
The parity-violating weak amplitude A(n + pr-) is related to the current X current hamiltonian (2.4) via the current commutator I’), AP”(n+ pn-) = limo- i(pC]HP,“(dS
= O)]n>
(4.1)
N, Kaiser, U.-G. Meissner / Weak TN vertex
675
for the strangeness-conserving piece and H C is the parity-conserving part of the weak hamiltonian. In the limit of exact SU(2) symmetry, the amplitude AP” vanishes since then (pJH&jp)=(n(NP,“ln). It can be directIy related to the parity-violating weak TN-vertex* =iG,$(n
2%
x T),N
(4.2)
from which we can immediately read off APY = G,.Jxh. The matrix elements (pjHK[p) and (n(H$,CJn)follow from eq. (2.4). Decomposing the different parts into isoscalar, isovector and axial currents, the energy-shift induced by H,“Ais given by (NIHz(N)
= AE = AEo+&-3AE1 ,
AEo = -J$GF
d3r (( V,P’
-sin* 2ewV; &AE1
= Z&G,
VP’+4
+ ApAp)& 1 + 2
COS' 8,)
sin4 e,f,rfi),
sin2 ew cos 2ew
J
d3rIpV3&.
(4.3)
We have used the obvious identifications ~~~~)~~~(~)~~N~~)
9
V,(r)-+
N(r)y,z
N(r),
A,(r) +
fiWws; N(p)
(4.4)
for the isoscalar, isovector and axial current, respectively. forms for these currents as given in appendix A, we find
J i( co
AL?, = 2afiGF
0
+$(1+2cos’
x 8i(r)‘-$y [ + 4 sin4
AE, = W%GF
dPr2
2A,(r*)+A,(r)‘-T
so(r)” 1
Injecting the expiicit
f(l+2cos*
8,)
8,-sin22ew) (3ih(r)*+2~~(r)h(~)
+k(rY)
1
ew sin* ew cos
2ew
drg &4~Mr) JO3
-~{r~~~~(r)~~~~(r))l.
(4.5)
0
* In contrast to earlier conventions, we do not use f, for the weak nN-coupling G,, to avoid any confusion with the weak pion decay constant.
constant
but rather
676
N. Kaiser, U.-G. Meissner f Weak ?rN vertex
At this point, we can already deduce some results for the conventional Skyrme model, which operates with pions only. In this model, there exist relations between the isoscalar and isovector current densities, namely p(r) = 2rs(r) , p,(r) = -pz(r)=2ri(r),
(4.6)
so that the conventional Skyrme model predicts a vanishing contribution for the neutral current part of the n + p7r- amplitude, i.e. one is simply left with the pure Cabbibo-model cont~bution, which is of course too small (cf. sect. 5). Again, the effective lagrangian including vector mesons proves to be more realistic in that it shows a clear enhancement of the GSW value for G,, as compared to the Cabbibomodel prediction. Finally, we have to evaluate the weak nN coupling constant G, for the Cabbibo model. Factorizing the current x current matrix element, it is given as: GC,= iGr 2 cos2 8&r-IA,lO)(pl V+‘lln) = ~6 cos2 3~G~f~{~~ - IS&,)
(4.7)
where the matrix element (pi V’,ln> is of course proportional to the neutron-proton mass difference and (T-~AJO)defines the weak-pion decay constant fY. The evaluation of the neutron-proton mass difference in the chiral lagrangian is relegated to appendix B. Here, some comments are in order. In the SU(2),f version of the theory we are considering here, eq. (4.7) is the only way of defining Gz in the Cabbibo model. This is, of course, rather different from current SU(3), estimates based on triangular sum rules 4,19).T o h ave access to such relations, we would have to consider an SU(3), effective lagrangian. However, the SU(3) Skyrme model has never given a satisfying phenomenology even on the level of baryon masses “). Therefore, we cannot take into account the terms proportional to sin’ 8, which give a larger value of Gz in the Cabbibo model 1,4,5*13). This obvious shortcoming of our approach can only be overcome in a complete study of the effective SU(3) chiral soliton model which we have not attempted here. 5. Results and discussion
Here, we will give the values for G,, as predicted by our model lagrangian (2.1). We will consider two cases, namely c = 0 in the Wess-Zumino action (“minimal model”) and c = 1 (“complete model”). Furthermore, since the absolute values of G, are not very well known 3,23), it is also useful to study the enhancement of G, in the GSW model due to the neutral currents as compared to the Cabbibo model. We find with fw = 93 MeV, GF = 1.16 x 10e5GeV2, cos2 0, = 0.947 and Ow= 28.5” for the “minimal model” (c = 0) A& = 0.30 keV , AE, = 1.37 eV,
(5.11
N. Kaiser, U.-G. Meissner / Weak TN vertex
and for the “complete
modef”
677
(c = 1) AE, = 0.20 keV , AE, = 2.33 eV .
(5.2)
This translates into the weak pion-nucleon coupling constant Gzsw via eq. (4.1) to lo-’ (c=O) and GFsw= 2.51 . IO-* (c = 1). For the Cabbibo model, G zsw=1.47+ we find (cf. appendix B and eq. (4.7)) G’, = 1.88 . lo-‘, so that for c=O (5.3)
for c= 1. For comparison,
Gari and Reid found
G,Gsw/ G”, = 5 [ref. r3)] whereas
Golowich and Holstein “) quote Gzsw /G”, = 24 for the central the values in ref. 4, are based mainly on theoretical estimates. upper bound on G, is known to be “) ]G,,] < 1.43 * lo-‘.
Donoghue,
values. Note that An experimental
(5.4)
Our results for the GSW model are consistent with this. However, they are rather small, e.g. DDH in ref. “) give G, = 4.56 - 10m7. Haxton 14) has recently re-examined the nuclear structure issues related to nuclear parity non-conservation. He argues that the best measure of G, comes from soft-pion relations between axial-charge in @-decay (18Ne + lsF) and parity mixing in “F. The result for the y-asymmetry “F is ]P,(‘*F)] = (2.0+0.5)
- lo-’
(5.5)
= 1.08 . 10-6.
(5.6)
with & =
&NN
G,/m
Here, &NN = 13.5 is the strong pion-nucleon coupling constant and we have used the central value of DDH ‘) for G,. Experimentally “) PY(lEF) is known to be [Pyp(r8F)(
=(1.7*5.8)
. 1O-4
(5.7)
so that in a conservative estimate by matching lower and upper limits, one finds & 5 0.5d,DDH. For the central values of (5.6) and (5.7), however, we have G, = o.o59G,Db”. For the “complete model” (c = l}, we find &= = 6.43 - lo-* (with the Cabbibo part included) and would predict using (5.5) /P,(‘8F)j
= 1.2. 10-4.
(5.8)
This is well within the experimental errors and close to the central experimental value (eq. (5.7)). However, let us point out again that the values for G, we find are
N. Kaiser, U.-G, Meissner / Weak TN vertex
678
on the small end of the experimentally allowed ones, but they are within experimental and theoretical uncertainties. For comparison, Adelberger and Haxton in ref. ‘*) make a best fit to some of the available data, which are the IongitudinaI asymmet~ A,_ in fit-p, fi+d, and $+ a and the parity mixing matrix elements in “F, 19F and “Ne. They consider these as the most reliable input. They find 6?, = 5 . lo-‘, approximately 8 times larger than the value we predict for the “complete model.” However, if they exclude the *‘Ne calculation, which is very questionable, ref. 22) concludes, that 6,, = 0 is the correct result. The **F data set a very solid constraint on & of 6, < 3.4 - 10m7. Our results confirm the statement that a definitive measurement of G,, is very much needed. Another topic to be discussed is non-appearance of renormalization group enhancement factors which play an important role in quark-model calculations of the weak coupling constants 4,5,2’).In a fundamental gauge theory, these effects are there and they are important. In the case of the effective lagrangian considered here, no such factors arise since the theory does not have any scale dependence. SimilarIy, any quark-mass factor arising from factorization does not appear. Quarks are integrated out in the effective meson theory, so all knowledge about quark effects is subsumed in the meson properties. It should be stressed again that the calculation of baryon observables such as the weak TN-vertex considered here is very different from the point of view of a fundamental theory of quarks and vector bosons or an effective meson theory. The latter offers a simplified framework of calculating these observables as we have shown in here. The chiral soliton model makes a unique prediction of the parity-violating weak rrN amplitude which will or will not survive further experiments, which are very much needed. Of course, in light of the simplifications which arise in our model, one is entitled to doubt the validity of the results. We feel, however, that a thorough investigation of all possible baryonic observables in the framework of chiral soliton models should be pursued to explore its strengths and iimitations. We would like to thank Manfred Gari for stimulating discussions and helpful advice. We are grateful to Ulrich Vogl for providing us with independent checks on the numerics and helpful discussions. We also thank the referee for some very usefu1 remarks. Appendix A CURRENTS
IN THE
SOLITON
MODEL
OF THE
NUCLEON
Here, we want to give the Noether currents and current densities which go into the calculations of the matrix elements (N(H$IN). For the model under consideration, the isoscalar (I,) and isovector (V,) electromagnetic currents as well as the
N. Kaiser, U.-G. Meissner / Weak xN vertex
axial current
(A,)
read (for hedgehog
anstitze
of the meson
679
fields):
IO(r) = s(r), I(r) = p(r)K
X i,
Vo3a(r) = pi(r) Tr (AT * KA+T”)+p,(r)
Tr (AT + i%- GA’?),
V’(r) = i(r) Tr (ATX FA+r") , A’*“(r)
= so(r) Tr (AT - (K x ;)A+?)
A”(r) =$A,(r)
Tr (ArA+?)+$A,(r)iTr
where a = 1, 2, 3 is an isospin-index, s(r), p(r). . . are radial functions
, (AT - ;A+F),
(A.11
K the rotational frequency of the soliton and given below. Applying the usual quantization
rules (see ref. 16)), e.g. K = a/40 and angular integrations, e.g. (K x S)‘-+ fK2, the various components in eq. (4.3) can easily be evaluated. For the lagrangian (3.1), the radial functions in (A.l) follow to be for c = 0 (minimal model) and c = 1 (complete model): 2
s(r)=--mw, 3g
pr(r) = -jz(4sin4
($F)+2
cos Fe [r)-(1
-c)
$!Q 5 F’sin2
F,
~2(r)=-f2(-4sin4(fP)+4sin2(~F)5,+25,)+(1-c)~~F’sin2F, fi i(r) = --(2
sin4 (;F) - G cos F) - (1 - c) 5
r
a,(r)
=fi
sin 2F -Ft2sinF.(l--5,) (
>
‘;” F' sin' F,
-&$(sin’F-cGcos’F),
A,(r)=G(2sinF(G+I)-fsin2F)-$$(sin2F-cGcos’F),
h(r) = -Al(r)+A3(r), h(r) =f;F’ , with f, = constant. it is easy a-model
(A.21
93 MeV the weak pion decay constant, and g = 5.85 the gauge coupling We use the generalized KFSR relation m2 = rnt = rni = 2f ig’. As a check, to convince oneself that the leading terms in f; reduce to the non-linear currents for G = cos F - 1 = t2 = -5, and c,$= w = 0.
N. Kaiser, U.-G. Meissner f Weak TN vertex
680
Appendix B THE NEUTRON-PROTON
MASS DIFFERENCE
In the evaluation of the Cabbibo-model matrix-element (4.7) we encountered the neutro-proton mass difference from the factor (pJ V~jn), where VE is the vector current. In general, the np mass difference AM z M, - jV$ = AM”” + Aj’@StrOng)
(B.1) has two contributions, the electromagnetic one (AM’Y’) and the strong one (A~‘strong’). Both can be evaluated in the model used here, but for a difficulty discussed below we will use the QCD value AM(Strong)= 2.02 MeV for the strong (isospin-breaking) part. The electromagnetic part can be easily evaluated, and gives a result very close to the (phenomenological) improved Cottingham formula discussed by Gasser and Leutwyler 15). The electromagnetic cont~bution to the np mass difference is nothing but the pn difference of the electromagnetic self-energy of the nucleon, i.e. 2 A,+?”
d3r d3r’ 4fl,i_ r,, (NIJ~“,(r)J”“-“.(r’)lN)
= ?m
2
with J;“‘ the electromagnetic A we have
current and e2f47r = a = (137.03))‘. From appendix J;“(r)=
After Fourier transformation,
03.2)
I,(r)+
Vi(r).
(B.3)
this gives 03.4)
with J:“‘(q)
= s(q2) + dl(q2) Tr (AT. ICA?T~) + d2(q2) Tr [A(T * @fl* t$-f~
J”.“‘(q) = p(q*)K
x $+
i(g*)
v K)A+T~],
Tr (ATX $A*T~) .
The qz-dependent functions s(q’), p(q’>, . . . follow as Bessel transforms radial functions S(Y), p(r), . . . , given in appendix A. We find
tfW
of the
s(q’) = GS,(q’) > &(a21 = -2@G:(rt’), d,(q’) = -~Y/‘%G&‘)
,
dq2) = $?G?,(q’), N
i(q2) = -
$
G’&(q’): N
(B-6)
N. Kaiser,
with Gz,&(q2)
the electric/magnetic
ref. 16). GG(q*) is an electric needed
U.-G. Meissner / Weak TN vertex
in the further
isoscalar/isovector
quadrupole
discussion.
form factors
NA-transition
Injecting
681
form factor
calculated which
in
is not
(B.6) into (B.4), we find
-&
G;(q2)G:h2)
3
G”,(q2)G&(q2).
(B-7) Numerical
evaluation
of (B.7) gives AM(Y) =
-0.71
MeV (c = 0)
-0.70
MeV (c = 1) ,
(B.8)
which is astonishingly close to the “empirical” value AM’Y’ = -0.79 MeV derived from the generalized Cottingham formula 15). Notice that in the conventional Skyrme model, one has AM’Y’ = -1.08 MeV [ref. “)I*, somewhat too big. The calculation of the strong contribution, AM(Sfrong), could be done along the following lines. One introduces a term in the meson lagrangian which accounts for the r” - GT*mass difference A.&, =&f~(mf,~-m~*)
Tr[(&-3&3)-1]
(B.9)
with .$ = exp (ir . nn/2fw}. This term is obviously isospin-breaking i.e. it should give a contribution to the neutron-proton mass difference. Unfortunately, we encounter here a limitation of the semi-classical quantization procedure. As long as one operates with hedgehog states, the mass shift derived from (B.9), i.e. drr2
AEib=2&$r&-m2,0)
x r-2 sin* (#‘)-i
sin2 ($F) Tr (ATA+T~ . ATA’T~)]
= -0.9 MeV
(B.lO)
is the same for the proton and the neutron. This follows from the fact that the operator Tr (ATA+T~ . ATA+T~) when quantized is nothing but an isospin-independent c-number, Tr (ATA’Q-~ . ATA+T~) + -2. This problem can certainly be resolved within
the framework
of a more elaborate
quantization
procedure
as e.g. advocated
by Zahed and collaborators 18). We therefore take the strong part of the np mass difference from experiment. This procedure can to a certain extent be justified if one imagines the skyrmion as a chiral two-phase model in the limit of vanishing bag radius (for a similar discussion cf. refs. 6,17,23)). l Notice that these authors the A (1232) mass.
use the reduced
pion decay constant
f, = 64.5 MeV to fit the nucleon
and
682
N. Kaiser, U.-G. Meissner / Weak TN vertex
References 1) M. Gari, Phys. Reports 6C (1973) 317 2) D. Tadic, Rep. Progr. Phys. 43 (1980) 67; E. Fischbach and D. Tadic, Phys. Reports 6C (1973) 123 3) B.R. Holstein, Lectures delivered at Princeton University, report UMHEP-248 (1985) (unpublished) 4) J.F. Donoghue, E. Golowich and B.R. Holstein, Phys. Reports 131 (1986) 319 5) B. Desplanques, J.F. Donoghue and B.R. Holstein, Ann. of Phys. 124 (1980) 449 6) U.-G. Meissner, Phys. Reports 161 (1988) 213 7) S. Weinberg, Phys. Rev. Lett. 19 (1967) 1264; 27 (1971) 1686; A. Salam, in Elementary particle theory: Relativistic groups and analyticity, ed. N. Svartholm (Almquist and Wiksell, Stockholm, 1968) 8) G. ‘tHooft, Nucl. Phys. B72 (1974) 461; B75 (1974) 461; E. Witten, Nucl. Phys. B160 (1979) 57 9) I. Zahed and G.E. Brown, Phys. Reports 142 (1986) 1; U.-G. Meissner and I. Zahed, Adv. in Nucl. Phys. 17 (1986) 143 10) J. Trampetic, Phys. Lett. B144 (1984) 250 11) L. Schiilke, Nucl. Phys. B40 (1972) 386 12) M. Bando, T. Kugo and K. Yamawaki, Phys. Reports 164 (1988) 217 13) M. Gari and J.H. Reid, Phys. Lett. B53 (1974) 237. We are grateful to M. Gari for pointing out to us an error in eq. (10) of this reference 14) W.C. Haxton, invited talk presented at the workshop on “Parity violation in hadronic systems”, TRIUMF, Canada, 1987 15) J. Gasser and H. Leutwyler, Phys. Reports 87C (1982) 77 16) U.G. Meissner, N. Kaiser and W. Weise, Nucl. Phys. A466 (1987) 685 17) M. Durgut and N. Pak, Phys. Lett. B159 (1985) 357; erratum, Phys. Lett. B162 (1985) 405 18) I. Zahed and S.H. Lee, Stony Brook report (1987), unpublished 19) E. Fischbach, Phys. Rev. 170 (1968) 1398 20) M. Praszalowicz, Phys. Lett. B158 (1985) 214 21) J. Donoghue, Phys. Rev. D13 (1976) 2064; D15 (1977) 184 22) E.G. Adelberger and W.C. Haxton, Ann. Rev. Nucl. Part. Sci. 35 (1985) 501 23) M. Durgut, N.K. Pak and T. Yilmaz, Phys. Rev. D36 (1987) 3443
THE WEAK
PION-NUCLEON Norbert
VERTEX
REVISITED*
KAISER
Institute of Theoretical Physics, University of Regensburg, D-8400 Regensburg, FRG** Ulf-G.
MEISSNER
Center for Theoretical Physics, Laboratory for Nuclear Science and Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 USA Received 19 February 1988 (Revised 14 July 1988) Abstract: We evaluate the parity-violating weak pion-nucleon vertex in the framework of a chiral soliton model including pions and the vector mesons p and w. The weak TN coupling constant G, is enhanced by a factor of 13.3 for the standard electro-weak model as compared to the Cabbibo model, with its absolute value given by G,, = 2.7 1O-8. We compare our results to available data and other theoretical determinations. In particular, we predict the y-asymmetry in ‘sF to be IP,(‘*F)I = 1.2 . 10e4. We also discuss calculational differences to standard quark model estimates.
1. Introduction The aim of this paper is to calculate the weak pion-nucleon vertex G, in the framework of a chiral soliton model. This vertex plays a fundamental role in the description of parity-violating contributions to the nucleon-nucleon interaction 1*2) and parity-violating effects in nuclei itself (longitudinal asymmetry measurements, decays of polarized levels, circular polarizations in electromagnetic decays) 3,4). The large uncertainties in the determination of the weak meson-nucleon couplings as obtained from parity-violating nucleon-nucleon potentials which stem from the uncertainties induced by nucleon structure in these calculations ‘,4,5) do call for new theoretical insight. This is possible if one can calculate the matrix elements of the isospin-breaking
piece of the strangeness-conserving
weak current
x current
hamil-
tonian. This matrix element gives rise to a parity-violating TN-vertex, which is the dominant source of nuclear parity violation. Such a calculation can be done in an extended chiral soliton model “) and will be presented here. In sect. 2, we will briefly review how one can go from the standard model of electro-weak interactions ‘) to an effective current x current hamiltonian of the Fermi type. Apart from the Cabbibo part, it becomes obvious that only a non-zero Weinberg angle can lead to isospin-breaking contributions. In sect. 3, we recall the main l This work is supported in part by funds provided by the U.S. Department of Energy (D.O.E.) contract #DE-AC02-76ER03069, and BMFT grant MEP 0234 REA. l * Present address: Niels Bohr Institute, Blegdamsvej 17, DK-2100 Kebenhavn 0, Denmark.
0375-9474/88/$03.50 @ Elsevier Science Publishers (North-Holland Physics Publishing Division)
B.V.
under
672
N. Kaiser, U.-G. Meissner / Weak TN oertex
ingredients of the extended Skyrme model incorporating the vector mesons p and w. Baryons arise as topological solitons in such a scenario as motivated by the large N, expansion of QCD (where N, is the number of colors) ‘). The lagrangian involves pions, p and o mesons and has essentially three parameters (f,, g,,, and m,) which can be fixed in the meson sector. It leads to a rather satisfactory description of the electromagnetic and axial properties of nucleons as well as the strong meson-nucleon vertices (for further details see ref. “)). The pertinent electromagnetic and axial currents derived from the model lagrangian are summarized in appendix A. Sect. 4 gives the calculational details of how to obtain the weak rrN amplitude. For the calculation of the Cabbibo contribution, one needs a determination of the neutron-proton mass difference. This problem is discussed in some detail in appendix B. in the final section, we compare our results to previous calculations and the available experimental data. We aiso comment on the weak current x current matrix element in the framework of the conventional Skyrme model 9), which has been calculated in the case of a vanishing Weinberg angle by Trampetic I”).
2. Effective current-current
hamiltonian
Here, we recall the steps how to construct an effective current x current hamiltonian for the Glashow-Salam-Weinberg theory. The quantum numbers of the nucleon according to the generators of weak isospin and hypercharge are A$. (I = i, Y = l), pR (I =O, Y = 2) and nR (1 = 0, Y = 0), where PJ, p, n denote the nucleons, the proton and the neutron, respectively. We work in a basis of left- and right-handed fields with NL,R= j( 1 r y.JN The coupling of the electro-weak gauge bosons (W,, BP) to the nucleon reads L??int = $giVLY”(COS
@,( 71
Wh + 72 W:) + 73 W3,)NL
+g’ftP,y’“p,4S~Ly~~L+~RY~PPR)BCL
;
(2.1)
with 0, = 13.3” the Cabbibo mixing angle. After the usual rotation in the W3 - B plane onto the mass-eigenstates (x2”) eq. (2.1) transforms into
cos 28w _ +,g.Z,,y&-pLYPPL-.-----
(
w
1 cos 6w
ALy9iL-
2 sin* 8w cos 8
w
FRYPR
>
where 0w is the Weinberg angle (sin2 Ow= 0.23) and ~~abbibothe contribution the Cabbibo model:
3
(2.2) from
N. Kaiser, U.-G. Meissner / Weak TN vertex
Eq. (2.2) can be easily
cast into
an effective
current
x
673
current
weak
interaction
hamiltonian
-2sin2
ew(N,y’“73NL)(N~~(I+73)N)
+sin40,(N~~(1-!-r3)N)(N~P(l+~3)N)}.
(2.4)
With Gr= g*/(4&hri&) = 1.16 * 1O-5 GeV-* the Fermi constant and B,(cos2 8,= 0.947) the Cabbibo angle. Obviously, the first line in (2.4) comprises the Cabbibo model. One can already read off the following facts from eq. (2.4): The first two terms are isospin symmetric and can therefore only contribute via the neutron-proton mass difference, as it was first emphasized by Schiilke ‘l). The third and fourth terms in eq. (2.4) are not isospin-symmetric and will give an enhancement of the parityviolating TN vertex to be discussed in sect. 4. 3. Chiral soliton model To evaluate
the TN vertex, we obviously need a model for the matrix elements (PIH%P) and (nlH%). S UCh a model is the extended Skyrme model incorporating the vector mesons p and w based on the hidden symmetry approach of Bando, Kugo and Yamawaki 12). Since this model and its phenomenological implications have been recently reviewed by one of the authors 6), we will be brief here. The effective meson lagrangian of pions and p and w mesons as hidden SU(2) x U(1) gauge bosons reads
-;(p,,p’““+
wP,,wP”) +;N,gw,B+
Ncg*
+ cF 64?r2
pvm’p
wW’
xTr{iT.p,(apUUt+UtapU)+~gT.p,Ut~.ppU} +$z~f~Tr(U-1).
(3.1)
The SU(2)-valued chiral field U(x) = exp {il. m(x)/fm} incorporates the pseudoscalar-isotriplet pion-field a(x). fT = 93 MeV is the weak pion decay constant. The auxiliary variable t(x) is related to U(x) via t(x) = m in the unitary gauge. One of the main dynamical predictions of the model is the generalized KFSR-relation rn: = rnt = m2 = 2f;g2, which fixes g = 5.85 for m = 770 MeV. g can also be related term generates the to the decay constant g,,,, we have g = g,,,. The Wess-Zumino anomalous meson couplings. The o meson couples to the topological baryon-current BP = (EP’& /24~r*) Tr (d,UU’ . a,UUt * d,UUi) and through a wpr-contact term (for c = 1 in eq. (3.1)). Furthermore, p,+, = aPpv -a,,~~ -t-go, x py and wfiy = a,w, - a,w, are the usual field-strength tensors of the p and the w meson, respectively. N, = 3 is the number of colors, and m, = 139 MeV the empirical pion mass.
674
IV. Kaiser, U.-G. Meismer
/ Weak TN vertex
The nucleon can be constructed from the effective lagrangian (3.1) in two steps. First, one calculates the classical soliton which is degenerate in spin and isospin, using hedgehog arm&e for the meson fields. To project onto states of good spin and isospin, one then performs an adiabatic rotation and quantizes the spinning modes. The time-dependent meson fields read U(r,t)=A(t)exp{i~*
S(r)}A*(t),
w”(r, t) = o(r), w’(r,t)=mKX;,
(3.2)
r
where K = --ai Tr (A’&) is the angular frequency of the rotation and A(t) a time-dependent, SU(2)-valued matrix. The ciassical meson profiles are obtained by functional minimization of the energy-functional (for A = 1, K = 0) and subject to the boundary conditions F(0) = 71;w’(0) = 0, G(0) = -2, F(co) = G(co) = @(co) = 0. The time-dependent Lagrange-density reads d3r~=-~[F,G,o]+OEF,G,w;5,,5,,~]Tr(AAt).
(3.3)
Functional minimization of the moment of inertia O[ * * *] gives the vector meson excitations &, & and 4. For further details, we refer the reader to ref. “). In what follows, we need the electromagnetic and axial currents associated to the lagrangian (3.1). These follow by the usuai ~oether-construction and are summarized in appendix A. Let us point out here that the model defined by (3.1) gives an overall satisfactory description of nucleon static properties, the electromagnetic and axial form factors as well as the strong meson-nucleon vertices.
4. Weak pion-n~~l~~ vertex
The parity-violating weak amplitude A(n + pr-) is related to the current X current hamiltonian (2.4) via the current commutator I’), AP”(n+ pn-) = limo- i(pC]HP,“(dS
= O)]n>
(4.1)
N, Kaiser, U.-G. Meissner / Weak TN vertex
675
for the strangeness-conserving piece and H C is the parity-conserving part of the weak hamiltonian. In the limit of exact SU(2) symmetry, the amplitude AP” vanishes since then (pJH&jp)=(n(NP,“ln). It can be directIy related to the parity-violating weak TN-vertex* =iG,$(n
2%
x T),N
(4.2)
from which we can immediately read off APY = G,.Jxh. The matrix elements (pjHK[p) and (n(H$,CJn)follow from eq. (2.4). Decomposing the different parts into isoscalar, isovector and axial currents, the energy-shift induced by H,“Ais given by (NIHz(N)
= AE = AEo+&-3AE1 ,
AEo = -J$GF
d3r (( V,P’
-sin* 2ewV; &AE1
= Z&G,
VP’+4
+ ApAp)& 1 + 2
COS' 8,)
sin4 e,f,rfi),
sin2 ew cos 2ew
J
d3rIpV3&.
(4.3)
We have used the obvious identifications ~~~~)~~~(~)~~N~~)
9
V,(r)-+
N(r)y,z
N(r),
A,(r) +
fiWws; N(p)
(4.4)
for the isoscalar, isovector and axial current, respectively. forms for these currents as given in appendix A, we find
J i( co
AL?, = 2afiGF
0
+$(1+2cos’
x 8i(r)‘-$y [ + 4 sin4
AE, = W%GF
dPr2
2A,(r*)+A,(r)‘-T
so(r)” 1
Injecting the expiicit
f(l+2cos*
8,)
8,-sin22ew) (3ih(r)*+2~~(r)h(~)
+k(rY)
1
ew sin* ew cos
2ew
drg &4~Mr) JO3
-~{r~~~~(r)~~~~(r))l.
(4.5)
0
* In contrast to earlier conventions, we do not use f, for the weak nN-coupling G,, to avoid any confusion with the weak pion decay constant.
constant
but rather
676
N. Kaiser, U.-G. Meissner f Weak ?rN vertex
At this point, we can already deduce some results for the conventional Skyrme model, which operates with pions only. In this model, there exist relations between the isoscalar and isovector current densities, namely p(r) = 2rs(r) , p,(r) = -pz(r)=2ri(r),
(4.6)
so that the conventional Skyrme model predicts a vanishing contribution for the neutral current part of the n + p7r- amplitude, i.e. one is simply left with the pure Cabbibo-model cont~bution, which is of course too small (cf. sect. 5). Again, the effective lagrangian including vector mesons proves to be more realistic in that it shows a clear enhancement of the GSW value for G,, as compared to the Cabbibomodel prediction. Finally, we have to evaluate the weak nN coupling constant G, for the Cabbibo model. Factorizing the current x current matrix element, it is given as: GC,= iGr 2 cos2 8&r-IA,lO)(pl V+‘lln) = ~6 cos2 3~G~f~{~~ - IS&,)
(4.7)
where the matrix element (pi V’,ln> is of course proportional to the neutron-proton mass difference and (T-~AJO)defines the weak-pion decay constant fY. The evaluation of the neutron-proton mass difference in the chiral lagrangian is relegated to appendix B. Here, some comments are in order. In the SU(2),f version of the theory we are considering here, eq. (4.7) is the only way of defining Gz in the Cabbibo model. This is, of course, rather different from current SU(3), estimates based on triangular sum rules 4,19).T o h ave access to such relations, we would have to consider an SU(3), effective lagrangian. However, the SU(3) Skyrme model has never given a satisfying phenomenology even on the level of baryon masses “). Therefore, we cannot take into account the terms proportional to sin’ 8, which give a larger value of Gz in the Cabbibo model 1,4,5*13). This obvious shortcoming of our approach can only be overcome in a complete study of the effective SU(3) chiral soliton model which we have not attempted here. 5. Results and discussion
Here, we will give the values for G,, as predicted by our model lagrangian (2.1). We will consider two cases, namely c = 0 in the Wess-Zumino action (“minimal model”) and c = 1 (“complete model”). Furthermore, since the absolute values of G, are not very well known 3,23), it is also useful to study the enhancement of G, in the GSW model due to the neutral currents as compared to the Cabbibo model. We find with fw = 93 MeV, GF = 1.16 x 10e5GeV2, cos2 0, = 0.947 and Ow= 28.5” for the “minimal model” (c = 0) A& = 0.30 keV , AE, = 1.37 eV,
(5.11
N. Kaiser, U.-G. Meissner / Weak TN vertex
and for the “complete
modef”
677
(c = 1) AE, = 0.20 keV , AE, = 2.33 eV .
(5.2)
This translates into the weak pion-nucleon coupling constant Gzsw via eq. (4.1) to lo-’ (c=O) and GFsw= 2.51 . IO-* (c = 1). For the Cabbibo model, G zsw=1.47+ we find (cf. appendix B and eq. (4.7)) G’, = 1.88 . lo-‘, so that for c=O (5.3)
for c= 1. For comparison,
Gari and Reid found
G,Gsw/ G”, = 5 [ref. r3)] whereas
Golowich and Holstein “) quote Gzsw /G”, = 24 for the central the values in ref. 4, are based mainly on theoretical estimates. upper bound on G, is known to be “) ]G,,] < 1.43 * lo-‘.
Donoghue,
values. Note that An experimental
(5.4)
Our results for the GSW model are consistent with this. However, they are rather small, e.g. DDH in ref. “) give G, = 4.56 - 10m7. Haxton 14) has recently re-examined the nuclear structure issues related to nuclear parity non-conservation. He argues that the best measure of G, comes from soft-pion relations between axial-charge in @-decay (18Ne + lsF) and parity mixing in “F. The result for the y-asymmetry “F is ]P,(‘*F)] = (2.0+0.5)
- lo-’
(5.5)
= 1.08 . 10-6.
(5.6)
with & =
&NN
G,/m
Here, &NN = 13.5 is the strong pion-nucleon coupling constant and we have used the central value of DDH ‘) for G,. Experimentally “) PY(lEF) is known to be [Pyp(r8F)(
=(1.7*5.8)
. 1O-4
(5.7)
so that in a conservative estimate by matching lower and upper limits, one finds & 5 0.5d,DDH. For the central values of (5.6) and (5.7), however, we have G, = o.o59G,Db”. For the “complete model” (c = l}, we find &= = 6.43 - lo-* (with the Cabbibo part included) and would predict using (5.5) /P,(‘8F)j
= 1.2. 10-4.
(5.8)
This is well within the experimental errors and close to the central experimental value (eq. (5.7)). However, let us point out again that the values for G, we find are
N. Kaiser, U.-G, Meissner / Weak TN vertex
678
on the small end of the experimentally allowed ones, but they are within experimental and theoretical uncertainties. For comparison, Adelberger and Haxton in ref. ‘*) make a best fit to some of the available data, which are the IongitudinaI asymmet~ A,_ in fit-p, fi+d, and $+ a and the parity mixing matrix elements in “F, 19F and “Ne. They consider these as the most reliable input. They find 6?, = 5 . lo-‘, approximately 8 times larger than the value we predict for the “complete model.” However, if they exclude the *‘Ne calculation, which is very questionable, ref. 22) concludes, that 6,, = 0 is the correct result. The **F data set a very solid constraint on & of 6, < 3.4 - 10m7. Our results confirm the statement that a definitive measurement of G,, is very much needed. Another topic to be discussed is non-appearance of renormalization group enhancement factors which play an important role in quark-model calculations of the weak coupling constants 4,5,2’).In a fundamental gauge theory, these effects are there and they are important. In the case of the effective lagrangian considered here, no such factors arise since the theory does not have any scale dependence. SimilarIy, any quark-mass factor arising from factorization does not appear. Quarks are integrated out in the effective meson theory, so all knowledge about quark effects is subsumed in the meson properties. It should be stressed again that the calculation of baryon observables such as the weak TN-vertex considered here is very different from the point of view of a fundamental theory of quarks and vector bosons or an effective meson theory. The latter offers a simplified framework of calculating these observables as we have shown in here. The chiral soliton model makes a unique prediction of the parity-violating weak rrN amplitude which will or will not survive further experiments, which are very much needed. Of course, in light of the simplifications which arise in our model, one is entitled to doubt the validity of the results. We feel, however, that a thorough investigation of all possible baryonic observables in the framework of chiral soliton models should be pursued to explore its strengths and iimitations. We would like to thank Manfred Gari for stimulating discussions and helpful advice. We are grateful to Ulrich Vogl for providing us with independent checks on the numerics and helpful discussions. We also thank the referee for some very usefu1 remarks. Appendix A CURRENTS
IN THE
SOLITON
MODEL
OF THE
NUCLEON
Here, we want to give the Noether currents and current densities which go into the calculations of the matrix elements (N(H$IN). For the model under consideration, the isoscalar (I,) and isovector (V,) electromagnetic currents as well as the
N. Kaiser, U.-G. Meissner / Weak xN vertex
axial current
(A,)
read (for hedgehog
anstitze
of the meson
679
fields):
IO(r) = s(r), I(r) = p(r)K
X i,
Vo3a(r) = pi(r) Tr (AT * KA+T”)+p,(r)
Tr (AT + i%- GA’?),
V’(r) = i(r) Tr (ATX FA+r") , A’*“(r)
= so(r) Tr (AT - (K x ;)A+?)
A”(r) =$A,(r)
Tr (ArA+?)+$A,(r)iTr
where a = 1, 2, 3 is an isospin-index, s(r), p(r). . . are radial functions
, (AT - ;A+F),
(A.11
K the rotational frequency of the soliton and given below. Applying the usual quantization
rules (see ref. 16)), e.g. K = a/40 and angular integrations, e.g. (K x S)‘-+ fK2, the various components in eq. (4.3) can easily be evaluated. For the lagrangian (3.1), the radial functions in (A.l) follow to be for c = 0 (minimal model) and c = 1 (complete model): 2
s(r)=--mw, 3g
pr(r) = -jz(4sin4
($F)+2
cos Fe [r)-(1
-c)
$!Q 5 F’sin2
F,
~2(r)=-f2(-4sin4(fP)+4sin2(~F)5,+25,)+(1-c)~~F’sin2F, fi i(r) = --(2
sin4 (;F) - G cos F) - (1 - c) 5
r
a,(r)
=fi
sin 2F -Ft2sinF.(l--5,) (
>
‘;” F' sin' F,
-&$(sin’F-cGcos’F),
A,(r)=G(2sinF(G+I)-fsin2F)-$$(sin2F-cGcos’F),
h(r) = -Al(r)+A3(r), h(r) =f;F’ , with f, = constant. it is easy a-model
(A.21
93 MeV the weak pion decay constant, and g = 5.85 the gauge coupling We use the generalized KFSR relation m2 = rnt = rni = 2f ig’. As a check, to convince oneself that the leading terms in f; reduce to the non-linear currents for G = cos F - 1 = t2 = -5, and c,$= w = 0.
N. Kaiser, U.-G. Meissner f Weak TN vertex
680
Appendix B THE NEUTRON-PROTON
MASS DIFFERENCE
In the evaluation of the Cabbibo-model matrix-element (4.7) we encountered the neutro-proton mass difference from the factor (pJ V~jn), where VE is the vector current. In general, the np mass difference AM z M, - jV$ = AM”” + Aj’@StrOng)
(B.1) has two contributions, the electromagnetic one (AM’Y’) and the strong one (A~‘strong’). Both can be evaluated in the model used here, but for a difficulty discussed below we will use the QCD value AM(Strong)= 2.02 MeV for the strong (isospin-breaking) part. The electromagnetic part can be easily evaluated, and gives a result very close to the (phenomenological) improved Cottingham formula discussed by Gasser and Leutwyler 15). The electromagnetic cont~bution to the np mass difference is nothing but the pn difference of the electromagnetic self-energy of the nucleon, i.e. 2 A,+?”
d3r d3r’ 4fl,i_ r,, (NIJ~“,(r)J”“-“.(r’)lN)
= ?m
2
with J;“‘ the electromagnetic A we have
current and e2f47r = a = (137.03))‘. From appendix J;“(r)=
After Fourier transformation,
03.2)
I,(r)+
Vi(r).
(B.3)
this gives 03.4)
with J:“‘(q)
= s(q2) + dl(q2) Tr (AT. ICA?T~) + d2(q2) Tr [A(T * @fl* t$-f~
J”.“‘(q) = p(q*)K
x $+
i(g*)
v K)A+T~],
Tr (ATX $A*T~) .
The qz-dependent functions s(q’), p(q’>, . . . follow as Bessel transforms radial functions S(Y), p(r), . . . , given in appendix A. We find
tfW
of the
s(q’) = GS,(q’) > &(a21 = -2@G:(rt’), d,(q’) = -~Y/‘%G&‘)
,
dq2) = $?G?,(q’), N
i(q2) = -
$
G’&(q’): N
(B-6)
N. Kaiser,
with Gz,&(q2)
the electric/magnetic
ref. 16). GG(q*) is an electric needed
U.-G. Meissner / Weak TN vertex
in the further
isoscalar/isovector
quadrupole
discussion.
form factors
NA-transition
Injecting
681
form factor
calculated which
in
is not
(B.6) into (B.4), we find
-&
G;(q2)G:h2)
3
G”,(q2)G&(q2).
(B-7) Numerical
evaluation
of (B.7) gives AM(Y) =
-0.71
MeV (c = 0)
-0.70
MeV (c = 1) ,
(B.8)
which is astonishingly close to the “empirical” value AM’Y’ = -0.79 MeV derived from the generalized Cottingham formula 15). Notice that in the conventional Skyrme model, one has AM’Y’ = -1.08 MeV [ref. “)I*, somewhat too big. The calculation of the strong contribution, AM(Sfrong), could be done along the following lines. One introduces a term in the meson lagrangian which accounts for the r” - GT*mass difference A.&, =&f~(mf,~-m~*)
Tr[(&-3&3)-1]
(B.9)
with .$ = exp (ir . nn/2fw}. This term is obviously isospin-breaking i.e. it should give a contribution to the neutron-proton mass difference. Unfortunately, we encounter here a limitation of the semi-classical quantization procedure. As long as one operates with hedgehog states, the mass shift derived from (B.9), i.e. drr2
AEib=2&$r&-m2,0)
x r-2 sin* (#‘)-i
sin2 ($F) Tr (ATA+T~ . ATA’T~)]
= -0.9 MeV
(B.lO)
is the same for the proton and the neutron. This follows from the fact that the operator Tr (ATA+T~ . ATA+T~) when quantized is nothing but an isospin-independent c-number, Tr (ATA’Q-~ . ATA+T~) + -2. This problem can certainly be resolved within
the framework
of a more elaborate
quantization
procedure
as e.g. advocated
by Zahed and collaborators 18). We therefore take the strong part of the np mass difference from experiment. This procedure can to a certain extent be justified if one imagines the skyrmion as a chiral two-phase model in the limit of vanishing bag radius (for a similar discussion cf. refs. 6,17,23)). l Notice that these authors the A (1232) mass.
use the reduced
pion decay constant
f, = 64.5 MeV to fit the nucleon
and
682
N. Kaiser, U.-G. Meissner / Weak TN vertex
References 1) M. Gari, Phys. Reports 6C (1973) 317 2) D. Tadic, Rep. Progr. Phys. 43 (1980) 67; E. Fischbach and D. Tadic, Phys. Reports 6C (1973) 123 3) B.R. Holstein, Lectures delivered at Princeton University, report UMHEP-248 (1985) (unpublished) 4) J.F. Donoghue, E. Golowich and B.R. Holstein, Phys. Reports 131 (1986) 319 5) B. Desplanques, J.F. Donoghue and B.R. Holstein, Ann. of Phys. 124 (1980) 449 6) U.-G. Meissner, Phys. Reports 161 (1988) 213 7) S. Weinberg, Phys. Rev. Lett. 19 (1967) 1264; 27 (1971) 1686; A. Salam, in Elementary particle theory: Relativistic groups and analyticity, ed. N. Svartholm (Almquist and Wiksell, Stockholm, 1968) 8) G. ‘tHooft, Nucl. Phys. B72 (1974) 461; B75 (1974) 461; E. Witten, Nucl. Phys. B160 (1979) 57 9) I. Zahed and G.E. Brown, Phys. Reports 142 (1986) 1; U.-G. Meissner and I. Zahed, Adv. in Nucl. Phys. 17 (1986) 143 10) J. Trampetic, Phys. Lett. B144 (1984) 250 11) L. Schiilke, Nucl. Phys. B40 (1972) 386 12) M. Bando, T. Kugo and K. Yamawaki, Phys. Reports 164 (1988) 217 13) M. Gari and J.H. Reid, Phys. Lett. B53 (1974) 237. We are grateful to M. Gari for pointing out to us an error in eq. (10) of this reference 14) W.C. Haxton, invited talk presented at the workshop on “Parity violation in hadronic systems”, TRIUMF, Canada, 1987 15) J. Gasser and H. Leutwyler, Phys. Reports 87C (1982) 77 16) U.G. Meissner, N. Kaiser and W. Weise, Nucl. Phys. A466 (1987) 685 17) M. Durgut and N. Pak, Phys. Lett. B159 (1985) 357; erratum, Phys. Lett. B162 (1985) 405 18) I. Zahed and S.H. Lee, Stony Brook report (1987), unpublished 19) E. Fischbach, Phys. Rev. 170 (1968) 1398 20) M. Praszalowicz, Phys. Lett. B158 (1985) 214 21) J. Donoghue, Phys. Rev. D13 (1976) 2064; D15 (1977) 184 22) E.G. Adelberger and W.C. Haxton, Ann. Rev. Nucl. Part. Sci. 35 (1985) 501 23) M. Durgut, N.K. Pak and T. Yilmaz, Phys. Rev. D36 (1987) 3443