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Meson exchange currents in nuclear weak and electromagnetic interactions
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be reprocked
~o~f~-~5i~~
P~~~ki~~ CO., Amsterdam
by photoprint or microfilm without written permission from the publisher
MESON EXCHANGE CURRENTS IN NUCLEAR WEAK AND ELECTROMAGNETIC INTERACTIONS M. CHBMTOB t
Received 6 October
1970
Atrstract: Meson theory is appIied to the study of the exchange magnetic moment and GamowTeller operators. We consider contributions due to the exchange of one pion and of vector mesons and give their detailed expressions in terms of a general classification of the operators. The one-pion-exchange process is separated into Born and non-Born parts corresponding respectively to the Born and non-Born parts in the process of weak or photoproduction of a pion by the nucleon. The results of low-energy theorems for this process enable us to reach a modeiindependent description of the non-Born parts. Corrections to the soft-pion limit are studied by means of simple dynamical models: a phenomeno~ogi~a~ Lagrangian for the weak current and the Chew-Low model for the e-m. current. The exchange operators due to the exchaage, of P- and w-mesons are evaluated in the vector dominance model. The applications are concerned with the exchange effect in the isovector and isoscalar magnetic moments of 3He and “H and in the Gamow-Teller matrix element for 3H /?-decay. The results are obtained with the dominant Sstate with a Gaussian radial function. We also discuss their dependence on short-range correlations described in terms of&&row factors, The exchange corrections are found to be too smslf for the current wave functions of the trinucleons. We discuss briefiY other agencies that may account for the remaining discrepancies.
1. Introduction In a previous paper ‘1, we made use of the low-energy theorem based otl the PCAC hypothesis to obtain the complete onepion-exchange (OPE) two-body contribution to the &mow-Teller matrix element in p-decay and applied it to the process 3H -+ ‘He f e- + &. We have suggested the approach as a successful means of tackling the difficult problem of meson exchange effects in nuclei, In this article, we would like to expand that theme to both weak and electromagnetic interactions by systematically discussing the low-energy theorems, defining as clearty as possible what we mean by the meson exchange corrections, pointing out which pieces of information are model-independent, and emphasizing the basic differences in the meson exchange currents of weak and electromagnetic processes. The problem of exchange currents in the weak and electromagnetic processes is an old one ‘>. However, it still remains unclarified because of the difliculties in t On leave from SPT, CEN Sactay, France.
2
M. CHEMTOB
AND M. RHO
defining interactions in a composite system starting from the field-theoretic description. We do not pretend to resolve these difficulties. The solution to the problem would involve beyond
a fully relativistic
equation
for many-body
systems, which is at present
our capability.
Our approach in obtaining the equivalent two-body current (we neglect many-body currents) relies on the conventional S-matrix method which has the advantage of being both simple and consistent. This has been used before. A renewal of interest in it stems from two sources: one, a recent development in deriving the two-nucleon potential, notably the one-boson-exchange (OBE) models 3), and the other, the success of the low-energy theorems associated with the PCAC and current algebra in describing the X-N interaction “). Within the conventional S-matrix framework, the former tells us how to split the totality of the relevant Feynman graphs, and the latter allows us to treat the OPE term correctly to all orders in the strong interaction coupling constant. In our studies, we encounter both the soft-current and soft-pion theorems. The former is applicable to the Gamow-Teller matrix element where the four-momentum transfer k, is small. The soft-pion theorem supplements the former when we deal with the OPE graph. For the magnetic moment,. however, it is more convenient to take the soft-pion limit, as will become clear in sects. 2 and 3. Let us point out here that the soft-pion theorem has proven to be an extremely powerful technique in understanding both the nuclear force and n-nucleus interaction, thus being applicable to the virtual as well as the real pions. The pion exchanged in nuclei is not really soft; the pion four-momentum qn does not vanish. But the extrapolation to the relevant value of q2 has been studied, and shows that the effect is not significant, at least for the OPE term “). From these considerations emerges an important and hitherto unappreciated fact that the exchange current in the weak process is quite different from that in the electromagnetic process, that is, the sources of the corrections are different. In treating the exchanges of meeons other than a single pion, the situation is not simple. The exchange of two soft pions may be treated, but it cannot represent the true situation. Because of the tensor nature, one or two of the pions can carry considerable momentum. There are also the practical difficulties in the choice of graphs. So we shall not consider the multi-pion exchanges here. We include however the vector meson (p and 0) exchanges in an approximate way with the hope that they simulate, at least partially, the effect of multi-pion exchanges. We apply our theory to the bound three-nucleon systems. Aside from the deuteron, the wave functions of these nuclei are the best known in nuclear physics. There have been innumerable studies, both theoretical and experimental, on the ground state properties of these nuclei: the Coulomb entrgy difference, the electromagnetic and weak properties such as electric and magnetic form factors, n-d radiative capture, photo- and electro-disintegration, P-decay and muon capture. A lucid discussion on the status of the three-body problem can be found in the review article of Delves and
MESON EXCHANGE
CURRENTS
3
Phillips “). Despite such a long history and considerable efforts, the understanding of the three-nucleon system is far short of the extent to which one knows of the deuteron. The reason is, of course, the extra degrees of freedom an additional nucleon brings in. A reliable calculation of meson exchange corrections in triton b-decay, and in the magnetic moments of 3H and 3He would be valuable since, once we know the corrections, a set of linear equations describing the P-decay and the magnetic moments can be used to draw conclusions about the structure of the bound three-nucleon systems. Our calculation is limited in that only the dominant S-state component of the trinucleon wave functions is used. This in only a first step towards more accurate calculations using a more realistic wave function. Furthermore, since our exchange current operators (except for the isoscalar magnetic moment) should be reliable, an extended application to other nuclei would be just an exercise in nuclear structure calculations. In view of the length of the paper, we summarize the essential points here. The twobody meson exchange operator is divided into a one-pion exchange (OPE) term and a heavy-meson exchange (HME) term. We apply the low-energy theorems to the OPE term. Sect. 2 deals with this question, We give the theorems relevant to the Gamow-Teller transition, and the isovector and isoscalar magnetic moments. The principal conclusions are that (i) for the Gamow-Teller operator, the theorem specifies the exchange current in terms of two off-shell nN amplitudes, available from Adler’s works, (ii) for the isovector magnetic moment, it leads to a reliable expression given in terms of generalized Born graphs, and (iii) for the isoscalar magnetic moment, the theorem has no predictive power and hence one needs to go beyond it. Sect. 3 is devoted to the classifi~tion of two-body meson exchange operators based on general invariance considerations, Explicit expressions are given for the OPE operators when certain assumptions are made. We also indicate how the operators are extracted from Feynman graphs. In sect. 4, the one-boson-exchange contributions are explicitIy cal&lated. For both the p-decay and the magnetic moments, models are used to get the HME contribution. For the magnetic moments, the vertex correction coming from N* is calculated by means of the Chew-Low model. The shortcomings in our treatment of the HME graphs are emphasized, and should be kept in mind in assessing the results of sect. 5. Sect. 5 deals with the details of ex~riment~ situations and theoretical calculations of the Gamow-Teller matrix element and the isovector and isoscalar magnetic moments in the three-nucleon system. We use for simplicity the dominant S-component of the wave functions. For this reason, the exchange Gamow-Teller matrix element is small as the Nz3 cannot contribute. The importance of the D-state is pointed out. For the isovector magnetic moment where the Nj, is expected to play a minor role, our calculated value should be reliable, but there still remains some
4
M. CHEMTOB AND M. RHO
sizeable discrepancy. Although the exchange isoscalar moment fills up the existing discrepancy, this may be fortuitous because of its model dependence. Appendix A deals with the problem of defining the meson exchange current from a field theory, and appendix B deals with the model dependence in the nuclear wave function. 2. Low-energy theorems The study of the two-body exchange currents amounts to the description of the process given in fig. 1. We follow the usual procedure used in the nuclear force problem, and split the contributions into a one-pion exchange (OPE) graph and a
Fig. 1. Diagram representing the transition amplitude of a two-nucleon system interacting with the axial or vector currents (denoted by the wavy line).
b)
0) Fig. 2. One-pion exchange (a) and one-boson-exchange
(b) contributions
to the exchange currents.
heavy-meson exchange (HME) graph as given by figs. 2a and 2b. As was pointed out by Blin-Stoyle and Tint ‘) for the triton B-decay and by Ohtsubo et al. ‘) for the axial current in general, a model-independent way would be to look at the two-body exchange contribution by means of the PCAC hypothesis applied directly to the two-body processes N, +N, -+ N,+N,+y, N,+N,
-+ N,+N,+e-+7,,
(2.1)
which may be given in the low-energy limit (that is, when the current momentum k approaches zero) by a non-radiative amplitude T(p,, p4; pz, pl) describing the twoparticle reaction in the absence of a current. We shall not consider this approach here.
MESON EXCHANGE
5
CURRENTS
In this section, we discuss the low-energy theorems relevant to the OPE graph of fig. 2a, although by now these are folk-lores in particle physics. Our philosophy is that tbis is the dominant contribution in most of the exchange current contributions and therefore should be treated as accurately as possible. The HME terms are treated as corrections, and hence are more model dependent. Let us consider the process given by fig. 2a. The matrix element for such a process is 1
---6(Pl+Pz+~--P’,-P;)
(243
x W(dN(p;)lJilN(~d)
dz(q2) q2+mz
R
(N(P;)IJW(P,))+(~
*
9,
(2.2)
where superscripts n and j are the isospin indices for pions and current, dz(q2) is the pion propagator form factor, and Ji is the pion source current. The nNN vertex d,(q2)(NlJ.lN> will be taken to be known from dispersion-theoretic analysis (see subsect. 3.2 for more details). Therefore we need only to know the left vertex which is essentially a matrix element for the production of a pion by a current Ji, which is either J:“* or Jy. In allowed /&transition we are only interested in the soft-current limit (k 3 0). In all cases, however, the pion is assumed to be soft, since it carries a mass (- q2) which is small in the process where a nucleon makes transitions with little momentum transfer, as in the case of B-decay or magnetic moment. On the average, q2 x m,2. Thus the well-known soft-pion theorem is ideally suited to this kind of process occurring in nuclei, in contradistinction to elementary particle physics where the soft-pion limit is only a mathematical limit. We shall argue later that q2 = 0 is a reliable limit, the extrapolation to q2 w ma modifying the results unappreciably.
Fig. 3. Pion-production
amplitude from a single nucleon by the axial or vector currents.
2.1. DEFINITIONS
We shall follow the notations + of Adler “) with minor modifications. The process given by fig. 3 which represents the reaction
J&9 + NP,) + n”(q)+ N(d),
(2.3)
7 We use the natural units 6 = c = 1, e2/4n = &, and the Pauli metric, where (I . b = a$, = a. b =a& = a * b--a&. The Dirac matrices are all Hermitian and satisfy {yp,yy} = 26,,. We also use I+ = YIYZYJY.+ and upY = (1/2i)(y,,y, -IV&
6
M. CHEMTOB AND M. RHO
is described by the set of kinematic invariants k2, q2, v = -(p,
+p;)
* k/2M,
I+, = q * k/2M,
in which M is the nucleon mass. We denote the axial current current by Ji’“’ = Jy + Ji, where Jp and JI are the isovector nents, respectively. When we write simply Jj, it means both otherwise specified. The transition amplitude for the process terms of Mij E i(q2+mz)
s
where T is the usual time-ordering MA by the decompositions
d4xe
by J,“’ and the e.m. and isoscalar compoJy and Ji.“. unless (2.3) is described in
-iq’“
operator. Let us specify the isospin structure of
MA”’ = a’t’A:+‘+J$)A:-), A M :‘“’ E j,&j+MS,”
(2.4)
= a~if)V:+)+a~T)r;,(-)+alO)v,(o),
(2.6)
where (2.7)
and the space-spin structure by
A:” =
vj*“) =
&4iyv, VB)
k2,q2)~Mj4hW~,j,
$)/*‘“‘(V, v,, k2,q2)~(P;)o,(%)fJ(P,j.
(2.8)
Ineq. (2.8), A!*)(. . .j and V/*“)(. . .) are amplitudes which depend on the invariants eq. (2.4), and O,(A,) and O,(VJ are covariants given in table 1. The aim of any theory is to determine the amplitudes Ai*) and V/*,‘). We shall often use the matrix elements of the currents between nucleon states: =
iu(p;)[gA(Q2)y,Y,-ihA(Q2)Q,YS13zju(pl),(2.9a)
= qp;j x(r~[3~~(Q2)f3~~F~(Q2)1-~A4Q1[3F~(Q2)+~~F~(Q2)l}~(~~), (2.9b) where Q = pi -pl,
and the form factors at Q2 = 0 have the values gA(O) = 1.23)0.01, F;(O) = F;(O) = 1, F;(O) = 2% = _ 0.12, 2M
(2.10)
MESON EXCHANGE 2.2. AXIAL
CURRENTS
7
CURRENT
In this section, we discuss in a concise manner what we mean by the low-energy theorem relevant to the /?-decay. In the limit as k + 0 which is valid in b-decay where both energy and three-momentum transfers are small, the Adler-Dothan theorem *#‘) provides an expression for i@’ valid up to order k. The derivation is given in the papers of Adler and Dothan 9), and Adler s); so we shall simply give the results. Let us separate IV; into a “Born term’, (Mf)B and a non-Born term Mf, M; = (Mf)B+m;.
(2.11)
Then the Born term is defined by (M?)B = f(P;)[igpr,Ys
S&r + k)ig.4(k2)3rjYAYs + ig,(k2)~zjYnysSF(~1-q)igrZnYS]U(P1),
Fig. 4. Nucleon pole diagrams in the amplitude for pion production current.
(2.12)
from the nucleon by the axial
where S,(P) = (iy++M)-‘. This corresponds to fig. 4. Rearranging the non-Born term a little from the original expression of refs. **‘) we may write a;
= ~~(p;){~~i”r(~2>~~+~(,~‘[-2P(~2)P~-2~~(q2)M~~])~(~~),
where a($)
= [a~-‘(q”>-~~~-‘cq”)]Y~ve~~2~~,
PCq2) = -[A:-)(q2)+~(;)(qZ)]y’vB’k2’0,
(2.13)
y(q2) = &+)(q2)ly,ve=k2=0. Thanks to the PCAC on which the low-energy theorem is based: (2.14) c(, j?, and y can be given in terms of the invariant nN scattering amplitudes AnN, BnN : U(p;)WN’*h(pl)
= - il(p;)[AnN’*’
- iy * kBxN(*)]u(p,),
8
M. CHEMTOB AND M. RHO
(2.15)
The a and fi are related at q2 = 0 by the Adler-Weisberger relation “) - ~
2M2
[/?(O)+a(O)]
= 1 -gi’.
(2.16)
%(%A
It is thus advantageous to work at q2 = 0, in which case only two off-shell nN amplitudes completely specify Mf since the Born term eq. (2.12) is given unambiguously. One may, however, ask at this point how CI,j3 and y extrapolate from their values at q2 = 0 to q2 z ml. Such an extrapolation necessarily involves models, since it deals with momenta which cannot be reached directly by experiments. It has been shown in particular by Brown and Green that numerically the different models yield similar results up to q2 z 10 rnz [ref. 5)]. Thus, from a practical point of view, the model dependence is not an obstacle. We have applied a method similar to the one used by Brown and Green for threebody force, and found that the extrapolation in q2 changes insignificantly from the values at q2 = 0. More specifically, using the same arguments and procedures [i.e. saturation of the dispersion integral by (3,3) resonance, zero-width approximation, etc.], we found that a(q2) = o(q2)a(0), (2.17)
B(s2) = %12)8(0)9 y(q2) = +12)Y(0)P with a(q2)z
(l+-$$)g9
b(q2)=
(l+&)gy
c(q2) z (l+ &)
$
where the magnitude of each of &, ;1,, Iz, is greater than unity. It is thus consistent with
a(q2) x b(f.p) E c(q2)x
Ag,
MESON EXCHANGE
CURRENTS
9
for q2 < M2. If the pionic form factor of nucleon K(q’) is assumed to extrapolate to q2 = rnz as smoothly as it does from q2 = -m: to q2 = 0, then we would expect that a(m2)
x b(mi)w c(mz) z 0.95,
which gives a 5 y0 variation from q2 = 0 value. We shall neglect this variation from now on. There is a variant of the low-energy theorems that can be obtained by considering directly the soft-pion limit (q + 0). In this limit, the only surviving amplitudes are 2’;’ and Ai-’ for which the PCAC places the constraints
&--)
= - ;?A
[F:(k’)-g,g,(k2)+2MF:(k2)].
One can also take advantage of the relations (2.15) valid at k + 0, arbitrary q, and the above relations valid at q + 0, arbitrary k, describing the same quantity i@, to obtain the combined constraints which would specify Is;if up to 0 (qk, q2, k’). The results are given in Adler-Dothan ‘). The advantage of this method is that it allows ;I\-’ and A:+) to be specified separately in addition A:-‘, AL-‘. 2.3. ELECTROMAGNETIC
CURRENT
Unlike the axial case where the k -+ 0 limit is appropriate, we shall instead take the soft-pion limit for the electromagnetic current. The reason is that the k + 0 limit is not quite appropriate for the magnetic moment, and furthermore we want to rely on the experiments of the threshold production of pions for which the soft-pion theorem is known to be reliable for charged pions. A shortest derivation of the theorem is through the modified PCAC relation in the presence of electromagnetic field Al [ref. ‘“)I a,Jt(*)T
ieA,Q*)
= LX#J~*),
A
=
J;’
JAW) A
=
J;‘3,
JA'*'
c =
(2.18)
T iJt2,
@f&h
gm
= f,%t,
where (+, 0) stand for the charge states of the pion. For charged pions, the photo-
10
production
M. CHEMTOB
AND M. RHO
amplitude is
where aAis the photon polarization four-vector, the r.h.s. means that a pion-pole term
and the bar in the second term on
has been separated out. Similar equations can be written for the neutral pion. Now imposing the gauge condition k,Mi’m’ = 0, one finds that A4Fm*is given to order (m,/M) by the B orn terms of fig. 5.
Fig. 5. Born diagrams representing the amplitude for pion production by the vector current. The separation into the seagull diagram (a), the nucleon-pole diagrams (b) and (c) and the pion-pole diagram (d) provides, when appropriate expressions for the vertices are used, a book-keeping of the terms describing the full amplitude in the soft-pion limit. These modified vertices also follow from the chiral Lagrangians.
The first term in eq. (2.19) yields the seagull term fig. (5a) and a piece of fig. 5d. (Note that the seagull term is equivalent to the Born term with NN intermediate state. It contributes only to the charged pion production.) One can see this by examining the matrix element of Jf in eq. (2.9a) and using the Goldberger-Treiman relation to express hA ill terms of gA and letting Q2 = 0 in the form factors. The figs. 5b, 5c and a part of fig. 5d come from the second term of eq. (2.19). Note that we are to take the full electromagnetic vertex as defined in eq. (2.9b) and F,(q’) -+ F,(O) = 1 to ensure the gauge condition. It is important to note that to the order (m,/M), no non-Born terms contribute, in contradistinction to the axial current where non-trivial contributions come from the non-Born terms. 2.3.1. Thresholdproduction of pions. In order to have an idea how good eq. (2.19) (and a similar equation for the neutral pion) described by lig. 5 can be for the isovector V(‘) and isoscalar V(O) amplitudes for our purpose, it is instructive to look at the experjments on the photoproduction of pions at threshold. We shall assume that the
MESON EXCHANGE
CURRENTS
11
features observed at q2 = -mz (physical pions) can give us some information about the q2 z rni point. It is now well established that fig. 5 describes very reliably the charged pion (n’) production, whereas it appears to be insufficient for no production where the main terms figs. 5a and 5d vanish. Also theoretically, the corrections to the soft-pion theorems are found to be very small for X* , while important for no [ref. I’)]. Now to translate these observations in terms of the isovector and isoscalar amplitudes, we note from eqs. (2.6) and (2.7) that M>“jyp + z’n) =
$ pp+ Vn’O’],
Mk”‘(yn + z-p) =
-
$
[Vp-V,‘“‘],
(2.20)
$[v,‘+‘+ VjO’], Mpm’(yn + non) = p,‘+‘- v,1’O’-J.
Mtm.(yp --f “Op) = Or inversely v,‘-) = $
[M’,‘“‘(yp -+ n+n)-M;m.(yn
-+ n-p)],
vi+’ = [M”;“QJp + ~“p)+M~“~(yn + n’p)], T/(O) E. = $
[Mtm.(yp -+ n+n)+Mtm$yn
(2.21)
3 n-p)],
where M2.m*(yp + z+n) stands for the amplitude for y +p + z+ + n, etc. Substituting into eq. (2.21) the experimental values for z* production and the theoretical estimate of no production by Furlan et al. ‘l), we have found that
(2.22) w 7x 10-2.
Thus, in the isovector part we may safely ignore Vi+‘. The amplitude V,$-’ being the sum of numerically reliable quantities is thus correctly given by the low-energy theorem. The same would hold for the amplitudes we are interested in if the extrapolation from q2 = -mi to q2 z rnz is as smooth as is assumed. On the other hand, Vi” so obtained would not be good, since it is a difference between two large numbers, and any further corrections to it can become significant. 3. Two-body exchange operators 1.1. GENERAL
CLASSIFICATION
Tn this section, we discuss how the non-relativistic two-body operators for the netic moment (MM) and the Gamow-Teller (GT) P-decay transitions are
12
M. CHEMTOB
AND M. RHO
extracted, given the appropriate meson theories (next section) to calculate the Feynman graphs. Let us first classify the functional forms for these operators based on general invarjance requirements. We assume that GT and MM exchange operators have the same transforma~on properties as the corresponding one-body operators. They are both axial vectors The GT operator is an isovector, whereas the MM has an isovector as well as an isoscalar component. For their construction we have at our disposal position coordinates of the two nucleons x1, x2 their conjugate momenta pl, pz, the Pauli spin matrices bi , rr2 and isospin matrices r(l), r(2). For the isospin parts of the operators, al1 possible two-body terms formed with z( I ) and r(2) must reduce to three isoscalars:
and three isovectors: Cz(i)x
2(2)1j~
0.2)
llzj(1)4Tj(2)j~
where the MM operators involve the isovector componentj = 3 and the GT operatorsj = (1)&i(2). For the space-spin parts it is convenient to work with the relative and cm. coordinates: r = x1-x2,
R = +(x1 +x2).
Operators depending on R (i.e. translationally non-invariant terms) or on the momentum operators pr, pz (non-local terms) are, in principle, allowed but we shall not consider them in our classification. The space-spin parts must then be axial vectors formed with ul, b2 and r. They must be of the following seven forms (9 z r/lrl): (01 x @A (@I _+%I, (a, x (r2) * 69, (#I 5%)
(3.3)
+M,
That no new terms can occur upon introducing the isospin and spin exchange operators Pf and PC2 can be seen by using the foIlowing identities among the isospin matrices: Z
[z(l) X @>]PIz Cl -
= i[+)-+)],
Mhw1~;2 = 41) *42)--m42),
(3.4)
and Iikewise for the spin matrices. Because of the antisymmetry of nuclear wave functions Pfl, Pyz Pi2 = - 1, the Majorana operator (Prz) would therefore bring
MESON EXCHANGE
13
CURRENTS
nothing new. Thus the operators (3.3) give the full space-spin dependence within multiplicative scalar functions of the relative coordinate. The complete operators are formed by combining these with the isospin parts (3.1) and (3.2) in all possible ways under the restriction that they are fully symmetric under the interchange of all particle coordinates. We thus arrive at the general classifications of the isovector and isoscalar components:
where we have used the definitions: f;,z = +4-(~1 * f)(u2 Tp
=
[(u
1 0
x P) +(a,
u 2 ) * w-+(u
x P)(@2 1 0
u 2 )] 9
* q-j, 0
=
i-,
x.
(3.6a)
The operators T12 are the D-wave parts of (ui 0 u2)3 * 3, obtained by subtracting out their average over orientations of P. The functions gr, g,r, h,, hi, etc., are arbitrary scalar functions of 1~1= r, which must all be real by virtue of the transformation properties of the operators under time reversal [see appendix 3 of ref. ‘)I. The constant G is defined by e/2M (MM) G= (3.7) ( -gA (GT). We note that of the 20 terms entering our classification, only 12 of them would be allowed in the phenomenological theory of Osborne and Foldy I’). The additional 8 terms are those involving the functions hi,,, , hi,, , and also those involving functions with subscript III, namely j,ir, haI, etc. They had to be rejected in the phenomenological theory of Osborne-Foldy as a consequence of the assumptions used in its construction and the restrictions imposed on the two-body operators by time reversal 13). To be more specific, their assumption that the exchange operators are proportional to the NN potential imposes reality conditions on the radial functions, which for the above-mentioned terms are incompatible with conditions imposed by time reversal. For example, a term such as
transforms properly under time reversal only if we allow G(r) to become pure imaginary. On general grounds there is, however, no reason to discard these terms, which in fact will appear in our explicit meson-theory descriptions. The classification of operators non-invariant under translation is a rather fastidious task which we do not wish to undertake here. We simply write down some typical
14
M. CHEMTOB
AND M. RHO
terms we shall encounter in the calculations: H’ = GI[(a,+a,).iip-(bl~bZ)(P.~j]~~+[(~1-a,).~3-((a,--a,)(p.~)]F,, + C(@l=P)(q x i()+(q
x i&Q2 * P)]&
+ i(P x ii)&),
(3.8)
where FI;*-, F,, are real scalar functions of r which may depend on spin and isospin operators, and have appropriate symmetries so that H’ is fully symmetric under interchange of two coordinates. As for non-local operators, there exists a phenomenological classification of them by Sachs and Austern r4 ). In our calculation where the static approximation is made for nucleons, non-local operators are met only infrequently, so we shall not go into them; they do not contribute in our application to the three-nucleon systems. 3.2. DEFINITION
OF EXCHANGE
OPERATORS
FROM
FEYNMAN
DIAGRAMS
The most convenient way of looking at the meson exchanges is to divide them into an OPE term and a multipion exchange contribution which is to be simulated partly by heavy meson exchanges (HME). We have given the corresponding Feynman graphs in fig. 2. They represent two-nucleon scattering from initial momenta pi, pz to final momenta pi, pi while undergoing an interaction with the external field (electromagnetic or weak), The relevant currents are the e.m. or weak currents. The deduction of exchange operators from the Feynman amplitudes associated with those graphs will now be discussed. Let us first consider the OPE graph, fig. 2a. The matrix element associated with it is denoted by (p;p;lJjlplp2). If the vertex corresponding to the pion production by a current JA is written as (~“(q)N(p~)l~~iN(~~)) where the superscript n stands for the isospin index for pion, then using the conventional form of the zNN vertex
= igrKmfq%(P;)~z ~414,
(3.9)
we may write
J; =
(2~)3s(P,+P2$.k-P;-P;) x
= (2n)-36(p,+P2+k-P;-P;)
dff(q2){N(~~)l~~l~(~~)> +(I * 2) <~“(~)N(P:)IJ~IN(P~)> q2 lr
(3.10)
in which (1 Z$ 2) stands for the term obtained by interchange of nucleons 1 and 2 in initial and final states. The 1: can be considered as a matrix element in momentum space of the two-body current operator for two-nucleon systems. Eq. (3.10) represents an amplitude expressed in renormalized perturbation theory, thus containing a pionic form factor of nucleon K,,, and the pionic propagator form factor d,. These quantities are normalized at the pion pole q2 = -mz. In principle, the off-shell behaviour in these form factors can be very important. However, if we want to pursue the calculation in the line adopted in nuclear force theory, we need to be careful. The reason is that when obtaining two-nucleon potentials by using phenon-
MESON EXCHANGE
CURRENTS
15
enological coupling constants and masses for pion and heavier mesons, the form factor effect is assumed to be automatically taken care of. This fact was duly recognized by Brown and Green “) in their treatment of three-body force. Since we are going to use correlated nuclear wave functions, and put in also the heavy meson exchanges, it is deemed necessary and consistent to put K(q2) s KnNN(q2)d,(q2) on the mass shell K( - mz) = 1. This argument does not apply to the weak or photoproduction vertex (nNI.@N), whose off-shell-dependence cannot be ignored. In our case, we work with the amplitude at the soft-pion limit q2 = 0 which is sufficiently reliable for small q2. As it stands, the representation (3.10) is not suitable for nuclear physics applications. The reason is that conventional description of nuclear structure is usually given in terms of non-relativistic, single-time wave functions depending on the space and intrinsic coordinates of nucleons. The transformations required for such a descrption are standard ones. Matrix elements with respect to Dirac spinors must, firstly, be expressed in terms of Pauli spinors and, secondly, transformed to position space. In the non-relativistic reduction of Dirac to Pauli spinors, the static limit is taken. This is achieved by expressing matrix elements in a particular representation for the Dirac matrices and retaining leading terms in the limit where the nucleon mass M --f co. The procedure reproduces the lowest-order terms in a Foldy-Wouthuysen transformation. The representation in configuration space is then obtained by doing a Fourier transform on each nucleon momentum
where Ji(xl, x2) is the diagonal part of the matrix element. It represents the Fourier transform of the current density of the two-nucleon system as the momentum k of the external field is held fixed. Any momentum-dependent term would come from a derivative contained in Jj(x, , x2) operating on the delta functions. In table 1, non-relativistic expressions in the static limit are given for each covariant O,(A) and O,(V). The nNN vertex is, on the other hand,
and in the static limit q2 w q2. As an illustation, let us consider the axial current. To make an unambiguous separation of local and non-local terms, one should be careful in performing the integrals in eq. (3.11). Of the four integrals over nucleon momenta, one, say pi, is used to eliminate the delta function expressing momentum conservation. The remaining integrals are then expressed in terms of the independent momenta pl, *(pi + PZ) and
M. CHEMTOB
16
AND
M. RHO
TABLE 1 Explicit forms of covariants
O(A,)
{a, bjA z aA b
0,
4 A2 A3
where
NR limits
-(@X4) 2iP
ih
ici
-(Y - W’,
expressions,
rZ= 1,2,3,4
2iPA
A5
A8
* kbA, P z +(p,+p;),
LTllrqr
-MY,%
4
. k-u
Covariants
A4
A6
and O( V,) and their non-relativistic
-4~ - kk, ik, -(Y * k)h
v,
v3 v,
-$w(k-g)[k(q*kj-qk2]
v5
-i[(uxk)xk]
v,
(pi -p2)
E q. The argument of the exponent is written as
The integrals over the first two momenta introduce the delta functions displayed in eq. (3.11) so that any dependence on them is equivalent to the terms Pl -+
3(P;+Pd
$1
+
B2+4t
where hr and F2 are momentum operators acting on the ket wave functions. Putting
MESON EXCHANGE
17
CURRENTS
eq. (3.10) into (3.11), we can write
-iq.r e x -9,~(P;)~Y,~“~(P,)+(l q’+rn:
~5 2)) ,
(3.12)
where s = 1,. . ., 8, and a$) = a[ z,, Zj]* as defined before. After expressing all the dependences on pion momentum by a derivative, i.e. qr + ii?,, I = 1,2, 3, the integration over q brings in the familiar Yukawa function
s
-iq.r dq -%---
= 23czmnY,(q),
(3.13)
q’+m;f
Fig. 6. Representation of the pion-production amplitude by the axial or vector currents in terms of pole diagrams. Graphs (a) and (b) are for the intermediate nucleon isobars (s-channel singularities) and graph (c) is for the vector meson intermediate states (t-channel singularites).
Fig. 7. One-pion-exchange diagrams representing the vertex corrections due to vector meson intermediate states. Due to its odd G-parity, the o-meson (J PG = 1 - -) contributes only to second-class currents.
where Y,,(X) = eWX/x, with X, = m,r. If the invariant amplitudes AL” have no dependence on q, the integrand in eq. (3.12) depends through O,(A,) and ys on the pion momentum, at most quadratically. As we have seen in the previous section, the amplitudes A,(*I have that feature in low-energy limits. But suppose we consider explicitly the vertex correction due to p (fig. 6c), giving rise to the two-body operator of fig. 7. In the limit k + 0, the amplitude relevant to this contains the factor 1 q2’
(3.14)
if the Feynman graph is evaluated in the usual way. In such a case, the integration is
18
M. CHEMTOB
AND M. RHO
MESON EXCHANGE
19
CURRENTS
not eq. (3.13) but
s 4
e (q2 +
-iq.r
m,2)(q2 + m;) =
m4i?^) %2 2_1 m,
[Y&J-
(t)
y0 (?%)I
. (3.15)
1
If, furthermore, pions can propagate with either positive or negative frequencies alone as is the case in some of the Born terms (see the next subsection), then eq. (3.12) does not hold, and radial functions different from the Yukawa types [eq. (3.13) or eq. (3.15)] appear. Now given si, what are the relevant operators for the magnetic moment and the P-decay? To answer this, we recall that only the space part of the current four-vector is needed. The Gamow-Teller operator is simply the monopole part of the axial current, and the magnetic operator is the dipole part of the current, i.e. JAj]k,lJ M = -+iV,x
Je.m.lk=O
for
B-decay
for
MM.
(3.16)
We conclude this section by giving the expressions of the radial dependences of the operators in the case where the amplitudes A$*), VJ*“) are independent of the invariants v, vn and q’. In table 2, we give the functions g, h, j for the GT operators, and in table 3 the functions g, h, j, k, Z, m for the MM operators. We note that this condition is met for the NB parts of the amplitudes found in the application of the low-energy theorems. It does not, however, apply to the Born parts which have singular terms in the soft-pion or soft-current limits. For this and other reasons to which we will return in the next section, the Born parts must be treated separately. Note that for the GT operator where k + 0, A,, . . ., A8 do not contribute, and for the MM operator, V, and V, vanish since k2 = 0 for real photon. Furthermore, in the non-relativistic (NR) limit, VI and V, are negligible compared to V3 and V,, since the matrix elements of O,( V,) and O,(V,) are proportional to k, or M-l. These are the reasons why table 2 contains only A,, s = 1, . . .4 and table 3 only v,, s = 3, 4. We also note that at the limit v = vn = k2 = q2 = 0 which comes in the low-energy theorem, some amplitudes vanish due to crossing symmetry, and these are indicated therein. 4. One-boson exchanges In this chapter we calculate the two-body exchange operators due to OPE and heavy-meson exchange (HME) contributions. For the OPE term, we use the lowenergy theorem information, and supplement the calculation by using other models: phenomenological Lagrangian model for the GT case and the Chew-Low model for the MM case. For the HME, we consider the p- and o-exchange graphs.
M. CHEMTOB
20 4.1.
ONE-PION
EXCHANGE
AND M. RHO
CONTRIBUTION
For the benefit of those readers who have not quite appreciated the power and relevance of low-energy theorems, we summarize here some important points to bear in mind. For both the e.m. and axial currents, we have divided the pion-production amplitudes (and therefore the exchange current) into a well-defined Born term, and the rest into the non-Born term. Then the low-energy theorems tell us essentially the following: (i) for the isovector e.m. current, the NB term is negligible; (ii) for the isoscalar e.m. current, the Born term is small, while the NB term is undetermined; and (iii) for the axial current, the NB term is given in terms of off-shell nN scattering amplitudes, not necessarily small. These results rest on the applicability of the soft-pion theorems, and the question is then: to what extent can the exchanged pions in the OPE process be considered as soft? In practice, there are limitations in eq. (3.12) on the range of integration over the pion momentum. The long-range parts of the interactions are only sensitive to small values of 4. The longer the range, the smaller the q involved. From energymomentum conservation at the rcNN vertex, we also have: q = pi -p2. Thus for uncorrelated nucleons near the Fermi surface, q0 = p;,, -pzO GZ0 and q2 roughly in the range between ma and 4rni with the peak at ml. This is not far from the softpion limit, and thus renders the PCAC applicable. Once this is established, the power of the theorems is that they give us non-trivial relations exact to all orders in strong interaction coupling constants connecting the required amplitudes to either calculable quantities or measurable quantities. 4.1 .l. Non-Born contribution. We reverse the order and treat the non-Born part first. The reason is that the NB part permits the full pion propagator and has no danger of double-counting which the Born term does. Thus the discussion of sect. 3 is directly applicable. In following this part of the section, tables 2 and 3 are of practical importance to us, since once the amplitudes are calculated explicitly, the associated GT and MM operators can be read off from the tables. The assumption that the amplitudes are independent of the invariants applies to most of the situations discussed below, as the relevant point corresponds to v = vn = k2 = q2 = 0. (i) Axial current. Let us consider the NB parts (denoted by a bar) of the pion production matrix element given in eq. (2.16) in the limit k -+ 0. They are regular, so we have m; - (n”(q)N(p;jlJ:“IN(pl))NB
= ~(p;){[A!-‘+A:-)]a~7)ol(A2)
+A:+‘~~~)O,(A,)+[A~-)-~A(,-)JU!~)O~(A~)}U(P~)+ = fi(d){
-B(42)@qA2)
+Y(42)Q01(A3)
O(k)
-2cr(4’)al7’o,(A,)}u(p,)+
O(k)
*
(4.1)
This equation defines CL,j? and y for arbitrary q 2. Note that a and B are related by
MESON EXCHANGE
21
CURRENTS
the Adler-Weisberger sum rule at q2 = 0, as can be seen from eq. (3.19). The softpion limit is a good approximation for the case we are considering; hence we need only a(O), j(O) and y(O). The corrections for the case q2 w rnz can be safely neglected. From eq. (3.5) we have Hy’ = S’(Cr(l) Xf(2)IjCCcl X@2h+ G2X)GI +Cz(1)-2(2>lj[(bl-~2)(h,+h~PI2)+T~(;’(h,,+h~~P~2)1 +[~:z(1)+z(2)ljC(u1+a2)j,+T,(:)j,,l+HrL),
(4.2)
where Sn = - &40Y2(4
91 = 3MO)Y&), 4 = h = -aw-o~~,)~ h; = - HdO>
h, = h = -3wv2(4,
+ B&01xd4A
where 5 =
k E
z&
Y,(x)= There is no translationally
(4.3)
hi, = - tC40) + B(O)1Y2C4,
Y,(x)
=
e:,
(l+;+;)Yo(.)r
non-invariant
CNj)NL= -[~(1)x2(2)lj2[a(o)+Bio)l
term, but there is a non-local term: ;[(@I 7.
‘P)P2+(a2’
p)Pll (l+
:) yO(xJ* x (4.4)
The PCAC implies that a(0)
=
g_!k (o) CB”N(-% 3 I
p(o) _
gA
““1:,I’ 1 gso)
10’
(4.5)
where the subscript 0 means that v = vn = k2 = q2 = 0. These values have been evaluated by Adler “) by extrapolating from on-shell nN scattering amplitudes. They are given in table 4, row 1, denoted for short as “Adler”. The errors involved in these numbers are hard to assess, but we would guess that they are of the same magnitudes as the errors usually quoted in the test of the Adler consistency condition and the Alder-Weisberger sum rule. We shall now turn to an alternative calculation of the amplitudes which may be considered as some sort of book-keeping of the global predictions of the low-energy theorems. The low-energy theorems can be viewed as sum rules derived from current algebra by insertion of a complete set of intermediate states in the equal-time com-
Total
2.62 0.38 1.51
“) Amplitudes are given in units of
PI
P
OprJ e9
2.13 5.2 1.23
rnse2.
0.509
0.578 -0.0693
N4.3
%.I xSl
0.28K(O)
I
4
0.512
0.442
0.0699
0,47K(O)
2.519
2.31 0.139 0.0699
3MC(O)
- 1.297
--1.123
-0.174
-0.83KiOI
1.158
0.562
0.578 -- 0.087 0.0693
0,7OK(O)
-1.021
-0.442
-0.578 -0.0699 0.0693
-0.75X(O)
between Lagrangian and PCAC methods for calculating the amplitudes Ai*) and
Adler
jes
PCAC
Comparisons
TABLE a(O),
p(O),
y(0)
0.0910
0.0970
0 -0.0180 0.0120
-0.0144
“)
MESON EXCHANGE
23
CURRENTS
mutators “). Cheng and Kim 16) exploited this idea in relation to the AdlerWeisberger sum rule. We shall apply here a similar idea and approximate the pionproduction process by pole diagrams with intermediate states corresponding to nucleon isobars (N*) and vector mesons (V). These are illustrated by the diagrams given in figs. 6a to 6c. Nucleon isobars: We describe the intermediate resonances in particle representation. Effective couplings are used for the vertices nNN* and ANN*, where A is a shorthand notation for the axial current. Coupling constants for the former vertices are determined from available experimental widths and for the latter by using one-pion-pole dominance in the axial form factors. The isobars with angular momentum and parity Jp = (n++)*, n > 0, are described by Rarita-Schwinger spinors z+, . . . ,JP). For n = 0, the usual Dirac spinors are used. We give the rcNN* vertices in terms of the matrix elements of the Lagrangian density ’ “):
<~~)N(P’)I~IN*(P)) [(JP)pp
= gnNN8i%,, =
(n++)*,
’ ( 05
. . . c&+fi(~‘)
n
=
ulrl
. ...,(P),
)
1,2, . . .],
(44
(j;5) U(P), WI)N(P’GW*(P)>= gnNWfi(P’) (4.7)
NJPh*= &'I.
Introduction of isospin is straightforward: let T be the isospin of the isobar, s’ its projection along isospin z-axis and let cc (spherical component) and s specify the isospin projections of rc and N, respectively. Inserting the Clebsch-Gordan coefficient (~$~s~Ts’) in the r.h.s. of eqs. (4.6) and (4.7) takes care of the isospin dependence by making them isoscalars, as they should be. The ANN* couplings involve, in general, a number of invariant operators. In the limit k --t 0, relevant to our problem only one invariant operator survives. We write the vertices in terms of the matrix elements of the axial current:
WP’)IJ?IN*(PD = +d in- ‘aA,,, qlrz. . . qPn fi(~‘) i:5 Q~,,~. . JP) + O(k), (
[J’ = (n+$)‘,
>
n = 1,2,. . .],
(N(P’P?IN*(P))
= tid~(p’)y, [J’
= 4’1.
(4.8)
(‘;) u(p)+ O(k), (4.9)
The requirement of time-reversal invariance implies that all the coupling constants introduced so far must be real. The inclusion of isospin dependence is done in a similar way to that for the nNN* vertices.
M. CHEMTOB AND M. RHO
24
The PCAC hypothesis places constraints on the coupling constants gz that we shall use in the following. They are connected to the coupling constants gnNN, through the relation ’ “)
m,2<+~)Nb’)l~lN*(p>>.
(4.10)
g,(O) 4’ + mi
These Goldberger-Treiman-type
relations may be written in general form
2MgA
92
(4.11)
(gnNN,[JP= (n++)*, n = ‘9 2Y *I ’
f
nNN’
=
gnNN*
[J’
M*fM
*
(4.12)
= +*I.
In particular, they fix relative signs of the coupling constants which would otherwise be difficult to know. We have now the necessary ingredients for calculating the diagrams given in figs. 6a and 6b. We restrict to the lowest-excited nucleon resonances Nz+,(1236, 120), N:+,(1400, 120), and NT- 1(1525,40), where the numbers in parentheses specify masses and partial widths (in MeV) for N* --+N + n decay modes, and the subscripts stand for (2Jp, 2T). Corresponding to the Feynman propagator for spin-4 particles we use the following expression for the spin-3 propagator: (iy * Q+M)-‘,
*Q+Mh
(&,+Y,Y,)/(~
where terms of order Q/M which vanish in the static limit and unknown off-massshell terms have been dropped. The results will be expressed in terms of the contribution to the amplitudes A$*‘. Terms of order q2/M2 appear in the calculations but will not be written down: NT+,:
a(0) = -/3(O) = *y(o) = ;
N:+I: $-’
= A$+’
a(0) =
M
=
M*+M
;$y$, $3
=
2&NN*g: M*2-M2
(4.13)
’
gnNN* d
M(M*-M)
’
P(O) = -y(O) = =p$.
(4.14)
MESON EXCHANGE NT_,.
;I;-’
=
_-)&+’
=
_
CURRENTS
25
2 gnNN*g: ~M*+M'
a(0) = -g(O)
= -*y(o)
= - ;
g=&--.
(4.15)
The numerical values are given in table 4 where the low-energy theorem values are also given for comparison. In the last column of the table, we consider a particular combination of the amplitudes given by [2A~-‘-Z~-‘-3A5+‘] within a factor. This is precisely the combination which determines the GT matrix element of triton betadecay if one restricts to the S-state. Note that the contribution from the N:+, is exactly cancelled. Vector mesons: A strict polology treatment of the vertex correction given by fig. 6c presents delicate points because of the vector-meson propagators (q’+m:)-’ and their non-trivial dependence on the pion mass (-4’). Adler “) has given a general treatment of these vector meson exchange amplitudes by a method implementing the vector meson dominance (VMD) hypothesis with the current algebra. It includes the effects of all vector mesons with quantum numbers JPG = l-+. We shall not enter into the details of his method, but shall simply enumerate some definitions and quote his results. They are strictly valid in the soft-pion limit q + 0, but we shall also retain the q2 dependence originating from the vector meson propagator. The VNN vertex is taken to be
iii(p;)[yJ, -o,,(k-q),f233zeU(Pl),
(4.16)
where fi and f2 are form factors, functions of (k-q)2. The p-propagator has the form [&, + (k - q)l(k - qX/rn$]/ [(k - q)2 + m:]. The matrix element of the axial current between vector meson and pion states is found to be (up to order k) (4.17j
where unknown off-shell terms are neglected’. The factor F, is the quantity used in the field-current identity “) to relate the current Ji.“‘. to the vector-meson fields VA, namely: e.m.
JA
=
F, V,
E
f-6V, .
(4.18)
2Yv
t The result (17) is related through the KSFR relation to the result obtained by the application of the PCAC hypothesis to the V,A vertex. The PCAC constraint specifies the coupling constant gvnn in terms off, 3 -\/2[MgJg,(O)] (pion decay constant) and g vRll (the coupling constant for V -+ nn decay) by the relation gvnn = &!gv,,f,. The KSFR relation gives on the other hand g& = (mdf,)’ which leads to gvnA = [g,(O)/g&f](n?:/2yv).
26
The amplitudes,
M. CHEMTOB
AND
M. RHO
as k + 0 are Ap(q2)
-
g,(O)+
mf)
2M2gA q2 + rnp”
(4.19)
where we have used the hypothesis of p-dominance of the isovector e.m. form factor Fz 2(k2) =f1,2FJ(k2+$). If one neglects q2 as compared to rn;, one obtains Ai-,
_
gr(Oh
Ai-’
-2M29,’
= -
Mc&) (If&/), A
or
a(0) =
do) -(l+Kv),
2M2g,
(4.20)
which are almost identical to the results of the low-energy theorems except for an additional term in 2i-j +. The numerical values corresponding to this assumption are given in table 4, fifth row. If the model used to get eq. (4.19) were correct, then the q2 dependence would just reflect the off-shell extrapolation of the amplitudes Ai-’ and A$--‘. However, even though the q2 -+ 0 limit is in agreement with the PCAC prediction, it is not clear whether the q2 dependence in eq. (4.19) is correct. For example, the hard-pion theories give quite different behaviour, with a disagreement + with the PCAC results in the q2 = 0 limit. As already noted, when transformation made, keeping the factor m~(q2+m~)’ in fig. 7 amounts function e-mn’/m,r by
into coordinate space is to replacing the Yukawa
(4.21) Note that this is dominated by the p-meson term at small r( < 1 fm), and for r >> 1 fm by the pion term: it is negative for r 2 1 fm, and changes sign to positive for r 2 1 fm. This is quite different from what one would expect from taking eq. (4.20). For a sensible nuclear wave function, eq. (4.21) gives expectation values with signs opposite to the PCAC results, and is very sensitive to its inner parts. Thus if the extrapolation from q2 = 0 to q2 # 0 is indeed smooth, then we must conclude that a calculation based on eq. (4.21) 1s . a suspect. It might be appropriate at this point to note that this sort of ambiguities also appears in treating K(q2) extrapolation in coordinate space. 7 Note that the PCAC
value
for Ai-)
= -(gr(0)/MZg,)(l
+~v)+g~g,JM’.
MESON EXCHANGE
CURRENTS
27
(ii) Vector current. Unlike the axial case, the limit k + 0 cannot be taken directly in the case of magnetic moment, since the latter may depend on terms linear in k in the matrix elements of the vector current [see eq. (3.16)]. Regardless of what k takes, we can choose to work with the soft-pion limit (q + 0), in which case, as we have seen in sect. 2, only the generalized Born terms survive. This does not mean that the non-Born terms cannot be given by the soft-pion technique. According to Adler and Gilman la), the terms determined in this way are jp+)
1
~(0)
1
_
g@h
--Tgjr’ _
-x
gr(O)Ks
(4.22)
while the other amplitudes are not determined, since they are associated with the invariants on(Vs) linear in q. But these contributions are already of order MV2 and, in accordance with the general results of table 1, the exchange MM (expressed in units of nuclear magneton) corresponding to these terms is of order Me3. Thus we conclude that the NB terms are small. The question remains, however, as to what happens when q2 moves away from zero. In sect. 2 we have given an argument that for the axial current, an extrapolation to some finite value q2 << M2 introduces only a small modification; but what about the e.m. current? There is a systematic way of resolving this question as discussed by Furlan et al. ‘l), and extensively used by Ericson 19) in photoproduction of pion on nuclei. We shall not do such analysis for the reason that even extrapolated to 4 2%m 2, the NB terms remain small for the isovector e.m. current, and hence an elaborate theory is not warranted. Therefore we shall use models to evaluate them. This is perhaps dangerous for the isoscalar moment for the reason that the NB terms are not small corrections any more. Our analysis for this, therefore, may not be as meaningful as for the isovector case. In the same way as for the axial vector current, we shall assume that the NB terms can be described by the N* graphs, and the vector meson (p, w) graphs. Nucleon isobars: Unlike the p-decay case, coupling constants are not readily available for the phenomenological Lagrangian approach. We shall instead use the celebrated dispersion theoretical method of Chew, Goldberger, Low andNambu2’) (CGLN) for treating the pion photoproduction amplitudes. We take the static limit for the nucleons and the Ml approximation. The latter amounts to neglecting all partial waves other than the magnetic dipole coming from the nucleon magnetic moment and is expected to take into account the contributions from low-lying even-parity nN resonances with (2Jp, 2T) 5 (3+, 3). The procedure goes as follows. We use the expressions connecting the invariant amplitudes V, to the c.m. amplitudes $J. These are given in appendix 1 of Adler’s
28
M. CHEMTOB
AND M. RHO
paper “). We then take the static limit for the nucleon M --, co, and find
(4.23)
v, = - 1 2% Iqllkl
where Ikl and 141are the c-m. momenta of the current and the pion, respectively, and o = W-M, where W is the total energy. From partial wave decomposition of the amplitudes fy, keeping only the magnetic dipole M,j, j = _t, 3, we get J%
3(k*q)M,+
/“j 5?1:-3M,,,
f;
= 2M,,-tM,,,
(4.24)
$1 x 0.
The amplitudes Mzj of course depend upon o. It is a well-known result of the CGLN theory that the solution for M,j is
(4.25)
where h,j is related to the partial wave amplitude for elastic n-N scattering fIj by (4.26) Here, Sj is the p-wave phase shift for j = $ or 3. To avoid double-counting, the Born contributions to Mrj must be subtracted out. However, the same eq. (4.25) which connects the full M,j and hi, also connects their Born parts. Thus the NB part of M,j is connected to the NB part of hj by the same equation. To state the results for the functions 9 and h of eq. (3,5), it is convenient to introduce the following combinations of p-wave scattering amplitudes
MO) = (~11-~1,-~,1+~,,)l,=,, la,(O) = #,I
+2h,3+2h,,
(4.27)
+4h33)lw=0,
where the subscripts on h stand for (2J, 27’). We evaluate these quantities at w = 0
MESON EXCHANGE
CURRENTS
29
since as v, vg -+ 0, o = [2M(v -vg) + A4213- M -+ 0. Explicitly we get h,(O) =
s
2_‘mdp
7T 0 wp2pz
[sin2 6,,(p) + sin2 6, ,(p)] z O.O74,/m~, (4.28)
dp [sin2 6,,(p) ++ sin’ 6, ,(p)] h,(O) = 2 m __ 71s 0 o;p2
w O.O658/m~,
where the numerical values are obtained by putting in the known values for 6,, and the Roper 21) parametrization for 6,, . In terms of A,,, 2T, the desired amplitudes are + V’f’ = 0, $1 =
Jq+’ = vi-’ = 35[h,,+h,3-2h,,]l,,o)
35h, - h~~-hs~+hdo=o,
(4.29)
j7(+) = 35~h11+2h,3+2h3,+4h331lo=o, 4 (E
-2h,,)l,,, = 0, Vi:+’ = Vi-’ = 0; so we need the combinations Since (h,,+h,, h, and h2 only. Now from table 3, the following contributions to the MM operator arz obtained:
91 = %3(r,-y,)h,(0)Yo(x,)m,3,
grr = - f Hy,-
h, = _it = -$ 3(r,-y,)4h2(0)Yo(x,)m;l h
y&(O) Y2(4m~
2
= .h = - f Hr, - ~nPUO)Y2k)m~.
Note that we have not considered N* contributions to the isoscalar amplitudes F’(O), since the dispersion integral cannot contribute much because of the factor &P+y.) and of the fact that only (3, 3) resonance (Roper) is allowed. Vector mesons
(p and CD):The CGLN theory based on fixed momentum-transfer
dispersion relations overlooks a piece of the pion photoproduction amplitude associated with the t-channel singularities. It is known that the neglected part is well approximated by a resonating Jp = 1- wave in the t-channel, provided the Ml approximation is made and the dispersion integrals are restricted to the (3, 3) resonance region. We shall study these effects in an approach similar to the one developed in the axial case. We assume them to be described by the vector meson exchange processes given by fig, 6(c). Both p and o are allowed intermediate states and contribute to the matrix elements of the isoscalar and isovector currents, respectively. The V7ryvertices, 7 Note that P, is proportional to l/w but that, since the associated invariant 0~(Vi) is itself proportional to w (see table l), the contribution On( VI) V1 to the matrix element of the current is regular in the limit 6.1+ 0 as it should be. In view of this fact, there may be a contribution to the MM operator associated with P,. However, the Ml approximation gives exactly P1 = 0.
30
M. CHEMTOB
AND
M. RHO
where V is p or o, are taken to be
<~“(q)lJ;lp;(k’)) = - i F
elw,,, k: qc,a”“,
(4.31) (d’(q)JJj;jlcop(k’))
= -i
gz m,
Earn,,k: q,,@,
where .Q,,~~is the fully antisymmetric tensor (&I234 = + 1). These are the only possible gauge-invariant couplings if gradient terms are ignored. In a recent dispersion theory analysis of the nucleon magnetic moments, made by Abarbanel, Callan and Sharp 22), the p and o pole contributions were included and vertices of the form (4.31) used. This study , as well as the analysis of the inelastic form factors of the nucleon by Walecka and Zucker 23), give us some clue to the signs of the coupling constants. The VNN vertices are written as in eq. (4.16) for the axial vector current W(P;)IJ$NP,))
= @,NNz(Pi)zj
(N(p;)lJ;IN($%))
=
hNN
[
Yp-
2%cjtv(Pi-Pl)v1
u(Pl)9
1
qp;)
(4.32)
44.
Note that gpNN
2fl(o)9 Kv=2Mf&,
=
flP)
wheref, and fiare the form factors used before [see eq. (4.16)]. From radiative widths of the vector mesons r(p --f ny) = 0.5 MeV and r(~ + ~7) = 1.3 MeV, =
gPnY
The one-pole approximation
0.448,
gmny= 0.758.
(4.33)
to the e.m. form factors of nucleons gives gVNN
=
YV,
where yv is the coupling constant in the direct V-y coupling [see eq. (4.18)J. The coupling constants yv can be ontained from leptonic decays of the vector mesons or storage ring experiments or photoreactions on hadrons. There is some disagreement among various experiments. We shall use the values deduced from leptonic decays 24), g = 0.52kO.03, 4n
d = 3.7040.7, 4n
which give gpNN
= 2.56,
gmNN
=
6.82.
(4.34)
MESON EXCHANGE CURRENTS
31
With the vertices eqs. (4.31) and (4.32), the V-amplitudes Y(*“) are easily computed: v;~)
_ --
2
v(+) 1
~(cbJ2pNNl.m,)
t+m,2 qo,
=
_ -
2 G7wnyf2aNN/mm) t+mt,
’ vi+)
%bJ2PNNlmfJ
t+m,2
=
2h7any.f2~NN/mJ p-p)
’
vy’ = 2(9,,, s,dm,) t+m,2
’
(4.35)
t+m2, V;+) _ - %dh.JmJ t+mi
’
’
where we use the abbreviations t = (q-k)‘, f2pNN= gpNN(+/2&f), f2mNN= gwNN (tcJ2M). At the symmetry point and to the lowest order in M-‘, we have j7i.o’ = 2g,NNg,n, 3 mP
vi+’
=
@oNN&ony
(4.36)
’ mS
sign
The gVNN gv.y > 0 is favoured in various analyses 22*23). Referring to table 3, the exchange MM operators coming from the vector mesons are
(4.37)
3 hlI
=
j,,
=
35,
J!!F ( m,
y2(x,),
)
where CV
=
-
h’NN
gVny
d12n.
Let us emphasize that the results (4.37) are simple since q2 is taken to be zero. However, if we were to keep the q2 dependence, we would have a situation analogous to that discussed near eq. (4.21). If one keeps the q2 dependence, then the following modification should be made to eq. (4.37): yo,
2kJ
+
m&
[Yo,2hJ-
(5)3yo,24
*
(4.38j
4.1.2. Born contributions. In principle, the Born contributions to the amplitudes are uniquely defined, as it is through the definition of the Born terms as in figs. 4 and 5 that the non-Born terms are given their precise meaning. However, in the spirit of deriving an equivalent operator which is to be sandwiched between nuclear wave functions, there is a problem of double-counting, which is a unique feature of a many-body system. In as far as the nuclear wave function already contains some parts of meson exchanges through the N-N potential, not all the graphs of figs. 4 and 5 can be ascribed to the additional meson exchange effects we are talking about. This can be best illustrated in terms of the time-dependent picture. The pieces already included in the wave function are given in fig. 8, in which emission and reabsorption
M. CHEMTOB AND M. RHO
32
of pions is completed before or after the interaction of a nucleon with the external field. Subtracting these graphs from the Born terms, one obtains the legitimate nucleon Born terms of fig. 9. One notes that in figs. 9a and 9b a neutron /?-decays (for example) to a proton before the pion gets reabsorbed by the spectator nucleon. Figs. 9c and 9d correspond to NH pair intermediate states. Here of course, the pion can propagate with both positive and negative frequencies and has the Feynman propagator. Fig. 9e contributes only to the isovector e.m. current.
Fig. 8. Time-ordered Feynman diagrams representing mechanisms where the interaction with the current occurs after emission of the pion and its absorption. Together with the diagrams where the current interacts before the pion is exchanged, they represent effects already included in the wave functions.
____ 4I ____ 41 d 1 Tc
a)
cl
b)
/i ____ ____
lT
d)
IT
TL
e)
Fig. 9. The parts of the Born contributions to the OPE process that remain after excluding the unwanted parts given in fig. 8. The time-ordered diagrams (a) and (b) give the nucleon-recoil exchange current, and (c) and (d) the pair-excitation exchange current. The Feynman diagram (e) gives the pionic exchange current.
Because the pion is present in the intermediate states in figs. 9a and 9b, an introduction of such terms as corrections would necessitate a normalization correction also. We do not know at this moment how to treat this satisfactorily, the trouble being that our formulation is not a perturbation theory. (It is important to realize that despite the terminology “Born term”, our Born terms are not perturbative
MESON EXCHANGE
33
CURRENTS
objects.) In perturbation theory, one may be able to formulate a systematic theory to take into account terms such as those given by figs. 9a and 9b and corresponding normalization corrections. These questions are discussed in appendix A from a general point of view. The practical aspects pertinent to our application will be left to the next section. We divide the Born terms into three groups. Referring to fig. 9, we shall call the current associated with fig. 9e a “pionic current”, figs. 9c and 9d a “pair excitation current”, and figs. 9a and 9b a “nucleon recoil current”. Both the axial and e.m. currents will be discussed simultaneously. Most of the arguments having been presented already 25), the discussion will be kept as brief as possible. (i) Pionic current. All the time-ordered diagrams are allowed here so that the Feynman rules are to be used in getting the matrix element of the exchange current in momentum space. There cannot be any contributions to the Gamow-Teller operator because of the Lorentz invariance, nor to the isoscalar magnetic moment operator because of the G-parity invariance. The isovector matrix element in momentum space is J;' =
-” 6(p,+p2+k--pi-p;)
1
-
(P1 -Pi)‘+
(2n)3
1
[~(P;)r5Z,U(Pl>liEj*“[~~P~)~5
x(P,-PP;+P;-P2), b2
-A)’
m,2
+
zn”(P2)lm (403g)
d
After some standard calculations, the two-body MM operator associated with pionic current &?T2 is found to be [in units of nuclear magneton (e/2M)] “&f2 = -+NH) -M 2 z
x ,wd~I
- Vl)(fl2
* V2)k
x
mJ(41~
(4.40)
where
Carrying out the differentiation indicated by VI and V, in eq. (4.40), we find among other terms the well-known Sachs’ space-exchange moment: JZ Sachs
=
&f”l”“[‘(‘)
x +)1&
x R)[(%
. a2)YOk)+S12
y2(xz)l,
(4.41) s12 = 3(c: . P)(rQ * 3)-(61
* u2),
which can be identified with the fourth term in eq. (3.8); i.e.,
In addition to &?sa&, we have other operators which can be written in terms of
34
M. CHEMTOB AND M. RHO
the classifications (3.8) and (3.5), Sr = - 5 ~L%2--x,)Y&,)9 x 4m4~yI(&J~
Fn, = -fAN[:(l) Y,(x) z (l+x)Y,(x)
911= - ~~A~(1 x
+x.)Yo(xx), (4.43)
= (l+x)eG.
(ii) Pair excitation current. The relevant diagrams include all possible timeorderings for the lrNN vertex, but the current is allowed to create or annihilate a nucleon-antinucleon pair only. Thus in writing down the contributions to the exchange current, we can still make use of the Feynman rules except that in the nucleon propagator (iy - Q + M)-’ only the part which propagates with negative frequency must be kept. This part is the second term in
(jy . Q+M)-’
=
_
2+
Q+M _
(74EQ-i7Q
3’4~;;.
i&
Qo-EQ+
4’“).
0
Q-=
This term contributes to both the axial and vector current. The contribution isovector e.m. vertex leads to the large seagull term which is just fig. 5a +, (ff”(q)N(p;)IJ~IN(p,)),*,
+ - $
snjkrk 01,
1 = 1,2,3,
to the
(4.44)
whereas there is no contribution to the isoscalar vertex to the order l/M. On the other hand, due to the fact that the axial vector current of the nucleons has an odd Dirac operator, we have to go to one more power of M-‘, with the result
We shall keep this, since terms of that order also occur in the NB parts. It should be clear to everyone at this point that the exchange currents must be basically different between the B-decay and the MM. The notion which is still invoked, that one can determine one matrix element from another independently of exchange corrections, is clearIy erroneous.
The pair contributions
expressed in terms of our classification are:
axial current 91 = 2hf = 5 ~f&Y,(x.), (4.46) gn = 2hp, = -3i,r1
= - ZfzW
Yz(xJ.
t The equivalence between the seagull term (sometimes called Kroll-Ruderman term) and the pair term in the NR limit can be checked explicitly by noting that the negative frequency propagator becomes - [(/I-l)+@ * Q/iU]/4M + -(/?-1)/4M to the lowest order in M-l.
35
MESON EXCHANGE CURRENTS
vector current
Comparing with eq. (4.43), we see that the sum of R-dependent pionic and pair currents vanishes, i.e. F;;I + Firs” = 0.
terms from the
(iii) Nucleon recoil current. The graphs to be considered are the time-ordered ones given by fig. 8. These contribute in the adiabatic limit, since the nucleon recoil is due to the emission or absorption of the exchanged virtual pion. The matrix element of the exchange current can be calculated in the non-covariant perturbation theory, which we do not reproduce here. We simply quote the results: axial current h, = j, = -+g,
= - 3$,&N
I(( ) Kk)1, & o x, -
-
(4.48)
vector current m,
=2jr=2hr=
-$g,=
m,, = 21,~= 2hn = -2gu
- $L, 2
[K&d-
yq
x
3 (4.49)
2
= - --fn~~ K2(4 7r
where K,(x,) are the Bessel functions of the second kind.
4.2. HEAVY MESON EXCHANGE
CONTRIBUTIONS
It is a formidable task to calculate systematically the exchanges of more than one pion. Such calculations have been performed for the meson theoretic N-N potential with reasonable success. They were, however, limited to two-pion exchanges. The two-pion exchange graphs are usually supplemented by exchanges of resonances such as p, w. Apart from the complexity, clearly there would be a need for care because of the double-counting. There have been calculations in fourth-order perturbation theory for the GT [ref. ’ 6)] and MM [ref. 2‘)I exchange operators, but the reliability of such calculations is not clear. In our approach, we shall neglect the uncorrelated multipion exchanges, but assume as in nuclear-force calculations that the correlated part of multipion
GT
%)Yp m,)/x,
fmr= 3gI, h,, = AI
= -$f&kdl+
hI = j,=
W/M;)
TABLE 5
h
mI = -4
(GT) and magnetic moment in sect. 3
= 3mII =
to the Gamow-Teller
Pair excitation current
Heavy meson exchange contributions
(
2
j,
1
=
defined by eqs. (5), (6) and (8)
=
-
2 K,(X,)
( 1 5
* r2 + Ml)
+
~&)lIY,
&hJ
2C+(1+ZKS)KO(Xp)-lgXpKI(Xp)l
-h,
Nucleon recoil current
(MM) exchange operators
7E
3 0
MESON EXCHANGE
+
z II
uosam-m
4-4
I II
CURRENTS
37
38
M. CHEMTOB
AND M. RHO
exchanges is important. Thus we shall consider only the w- and p-exchanges t and use the vector meson dominance model for their evaluations. The results are given in table 5. A strict adherence to the procedures used in the calculation of nuclear potentials would require that VNN coupling constants and rnv be determined phenomenologitally by fitting the scattering data etc. We have already touched upon this question in connection with the pionic form factor of nucleons. In view of rather large discrepancies between parameters used by different people, we shall not do the (in principle) correct procedure. We shall just use the coupling constants already determined in the previous subsection. Even restricted to the vector meson exchanges, a complete calculation is difficult because of the vector nature of the particles. For the nucleon-recoil graphs (equivalent to figs. 9a and 9b with the pion replaced by the vector mesons), the time-time components of the vector meson vertices give the leading terms, while those coming from the space components are of order M -’ relative to the former. However, our calculation of other contributions in the MM case is not complete for the following reasons. We have disregarded the mesonic-current processes in which the currents interact with the intermediate mesons (i.e. graphs equivalent to fig. 9c with rt replaced by the vector mesons), partly because of the complicated structure of the radiative vertices and partly because the relevant form factors (associated with monopole, dipole, and quadrupole interactions) are not at present available. It is worth while to note here that a part of the contributions from these processes must be related to the NN potentials due to the vector meson exchanges in the same way as the Sachs moment [eq. (4.41)] is related to the OPEP. For the pair excitation graphs in the MM case, both the time-time and space-space vertices of the vector mesons give contributions of the same order. The results given in table 5 refer only to the latter, where only the spin currents were taken (neglecting the convection current part).
5. Applications to 3H and 3He
The isospin doublet 3H and 3He is the simplest nuclear system with all the manybody characteristics, such as off-shell effects, etc. Next to the deuteron, its wave function is the best known, and it has a complete set of experiments we are interested in - i.e., magnetic moments and B-decay. Deuteron possesses only an isoscalar moment, which is too tiny and cannot make a bound-to-bound P-decay transition, making a clear-cut calculation unfeasible. Thus the trinucleon system is better suited + We have also looked into the q-meson exchange graphs but found them to have unimportant effects. They have essentially the same radial dependence as the OPE diagrams but do not compete significantly with them because of the smaller nNN coupling constant [exact SU(3) with a ratio f/d = 0.6 8ives gsNN/ffzNN% 0.31 and the shorter range. They also turn out to be small compared to the vector meson exchange diagrams. The role that the q plays in meson-exchange currents appears therefore to be very similar to the one it plays in the NN potentials.
MESON EXCHANGE
39
CURRENTS
for our purpose. Historically, the anomaly in its magnetic moment was one of the first pieces of evidence of meson-exchange effects 28). Before going into details of calculation, let us review both theoretical and experimental situations. 5.1. MAGNETIC 5.1.1.
MOMENT
AND GAMOW-TELLER
MATRIX
EJ,EMENTS
One-body operators
MM: GT:
CLI= tkp+ Y”)ai+3(YP-Yn)birAi) +orbitaL I-I!” = -gAqr(+)(i),
(5.1)
where 3(r, *XI) =
;*;;;
(.
1
.
The classification of allowed states: (Jp = if, T = 4) of three nucleons gives 10 distinct states ” ) corresponding to the spectroscopic terms: three 2S,, ‘P, and 4D, and one 4P,. If we assume that of all these only the S (full space-symmetry), S’ (mixed space-symmetry), D and T = 4 ( mixed space-symmetry) states are significant which seems to be borne out by most recent analyses 30), then the GT matrix elements and the MM are given in terms of the state probabilities Ps . . . and the amplitudes A s. . . by ,I)
MA
_
-
p$” =
l(3J+41~!“l13WI = J~[P,-~P,,+~P,+~P~+~A,A~],
J2
~(Yp+Yn)(Ps+Ps-P,+tP*)+tPn,
(5.2)
CL? = JL(r, - Y”)(PS - 3Pk + 3PD + 4Pb) - 6PD 3
where all contributions are independent of the radial forms chosen for the states +. The last terms in &$ are the small contributions from the orbital magnetic moment. The isoscalar and isovector moments (ps and pv) are related to 3H and 3He magnetic moments by I+ = t(A3H) + A3He)), (5.3) PV =
+(d3H)
-d3J--Wh
so that P =
Ps-wPv;
2t = +1(-l)
for
3He(3H).
5.1.2. Two-body operators. In evaluating the exchange current correction, we shall not use the full wave function, but retain only the symmetric S-state. This brings with it a considerable simplification in the calculation. t Similar radial forms were assumed for T = 4 and S’ states. Note that breakdown of mirror symmetry which may be due to the T = 4 state or to different state probabilities in 3H and 3He has only a slight effect. We find, for example, that a 10 ‘A difference in Ps between aH and 3He results in a change in pv and ps by only 1 % and 20 %, respectively.
M. CHEMTOB AND M. RHO
40
The spin-isospin part of the S-state wave function is given by
4” = x0(1, 2)~(3),
(5.4)
where m, t are the z-components of the total spin and isospin, respectively. A more explicit form for 4”=* and $m=3 is convenient for calculation;
$Jm=+ = 1-[(+-+)-(-++)I,
42
p=+ = &g<-
(5.5)
+ +)+(+ - +))-(+ + -)],
and similarly for q’=* and $=*, where ( + ) and (- ) represent spin up and down for #I and proton and neutron for q. The simplifications that eq. (5.4) brings are that because the S-state is fully space-symmetric any operator odd under interchange of space coordinates has vanishing expectation values; thus the c.m. dependent operators [eq. (3.8)] do not contribute. Moreover, tensor operators have vanishing matrix elements. Note that the vertex corrections with an intermediate state N:+s cannot contribute in fully space-symmetric states by virtue of spin-isospin selection rules. Unlike the one-body operators of eq. (5.1), the matrix elements of the exchange current operators depend on radial wave functions. There are various forms involving the coordinates rl, rz, rg which satisfy the symmetry. We shall work exclusively with the analytically simplest Gaussian form exp (-const (rF2 + r:s + &}) where rii = Ir,-- rj]. The fact that some of the Born terms and also all heavy meson exchanges are very sensitive to the part of wave functions where rij is small requires that the shortrange correlation be handled with care. Again for simplicity, we shall use the Jastrow type correlation function nix j (1 -e -yZrtjz)* to simulate the short-distance behaviour. For completeness, in appendix B we examine what happens when a correlation function of different nature used frequently in nuclear matter calculation is used. Although this cannot be a quantitative assessment of what we do below, it gives an idea as to how different correlation functions behave at short distances. The complete form of our radial function is then (5.6) where the range parameter c1 takes the value 0.384 fm-’ when fitted to Coulomb energy and is taken in our calculation to be 0.337, 0.384 (CE) and 0.440 fm-‘. The parameter y is interpreted as indicative of repulsive core radius r,, the rough relation being r, z (7,/2)-l. Th e values used in the calculation are r, = 0.0
y=cO
0.2
3.536
0.4
0.6
0.8
1.768
1.179
0.884
fm fm-’
(5.7)
MESON EXCHANGE
CURRENTS
41
We can now write down the non-vanishing matrix elements with the wave function eq. (5.4), and these are given in table 6. They comprise all matrix elements needed, since operators like [r(l) x r(Z)], (crl x e2) reduce to [r(l)-r(2)]j (ul -g2) in the S-state, and all others vanish. Using table 6, and eqs. (3.5) and (3.6), we have
TABLE 6
Spin-isospin
matrix elemknts of exchange current operators for a trinucleon system in a symmetric S-state Operator
MM
U12
Matrix element
[t3(1)-t3(2)l(ul--(T2)3 [r(l) * C9l(o,+o2)3
-16mt
-12m +4m
(al+u2)3 ts(l)t3(2)(uI+ud GT
MM:
GT:
(J/‘“‘] c O,,]~“‘);
-4m
t~(l)--s(2)1+(~1--oz) m = *3,
i
r=
443
+t,
i
where by definition the one-body matrix element in 82) is evaluated with the S-state only. We use (St> = N”Jlf(r
12, r23 prd2gdr1ddr,
dr2 dr3,
and superscripts M and p for magnetic moment and P-decay. The functions g, h, . . . , are given in terms of amplitudes Y and A in tables 2 and 3 for the case of NB term, and in sect. 4 for other contributions. In order to compute j.+, s and fiA, it suffices to collect all contributions to g, h, etc. For example, Sr = (SI)OPE+(gI) HME
= (gI)~~+(gl)~~~+(gI)HME,
(5.9)
and likewise for the others. 5.1.3. Wave function normalization correction. As we have discussed in sect. 4, there is a correction quite distinctive from the two-body operators we have been considering and any other one-body correction such as relativistic corrections etc. -
42
M. CHEMTOB
AND M. RHO
it is the normalization in the wave function due to an explicit dependence on meson degrees of freedom. For the trinucleon system, it adds the following corrections: APG” = I&AZ>, 6n0rm=
(5.10)
where Z =
IfnZNNi;k(Si
’
(
K+] + Ki?,(xz)1>
[
z/c)C”i . uk) KO(x*)-
Sik
II
and (2) means an expectation value which is independent of the spin and isospin projections. Since
we find
There are two points we must emphasize here. Firstly, we are only considering the normalization correction coming from the pion exchanges whereas vector meson exchanges are also included in our calculation. In principle, we could do a similar calculation for the exchanges of other mesons, but it is numerically small and perhaps the meaning of such a calculation is not clear because of the difficulty of extending renormalization procedure to two-particle systems. We find it more appropriate in presenting results to lump the normalization correction into the OPE Born term. The second point is that practical calculation of (2) gives a positive result instead of negative as the interpretation of a normalization would suggest. In other words, it enhances rather than reduces the matrix elements. We also note here that this normalization correction cancels only partially the nucleon recoil term. The exact cancellation occurs only in the extreme case of the neutral scalar meson theory. 5.2. EXPERIMENTAL
SITUATIONS
The magnetic moments are known very accurately and are given by PL(~H) = 2.97893,
p(3He) = -2.12815,
(5.12)
or in terms of the isoscalar and isovector moments, /I;‘~ = 0.426,
pvp = +2.553.
(5.13)
The discrepancies between the experimental values [eq. (5.13)] and the single-particle
MESON EXCHANGE
43
CURRENTS
values [eq. (5.2)] as a function of various P’s are given in table 7, where 6~ denotes (5.14)
&$v = &Py-$?.
The experimental situation for MA is quite controversial, because of the various experimental quantities involved in its extraction. This is seen in the formula
IK”I
~H+“l-lc
_
3
-
(fal+, 2(fr)140,14N-(ff)‘H-r3He C 2(ft)‘~O-rw-(fon+p tI%zi
(5.15)
1
TABLE 7 Discrepancies between experiments and single-particle operator values inps, v and MA for ‘H and 3He Single-particle values
96 94 92.8 92 “) 91”)8 89 d,
4 6 6 6 9
0 0 1.2 2 1 2
0.423 0.417 0.417 0.417 0.410 0.406
2.284 2.249 2.211 2.185 2.183 2.134
1.686 1.663 1.635 1.616 1.617 1.582
Expected corrections
0.003(0.70 %) 0.269(10.5%) -0.001 0.009(2.11) 0.304(11.9) 0.022 0.009(2.11) 0.342(13.4) 0.050 0.009(2.11) 0.368(14.4) 0.069 0.013(3.05) 0.370(14.5) 0.068 0.020(4.69) 0.419(16.4) 0.103
-0.059% 1.306 2.967 4.094 4.036 6.112
Probablhtres . . . Ps, PD . . . are given in %. *) Values in parentheses are % deviations. b, We have used Salgo and Staub’s 35) value IMrpj = 1.685f0.018. ‘) Corresponds to Gibson wave function 30) (setting P+ = 0). d, Correspond to wave functions determined variationally by Blatt and his co-workers 41*6).
which shows that, besides information on (ft)3H+SHe, the super-allowed decay 140 + 14N needs to be used to determine Gv (Fermi constant) and neutron p-decay to determine gA E GA/G,. For all these &cays, uncertainties are also introduced by radiative and Coulomb corrections. The former correction, though small, is not unambiguously known. The major uncertainties lie however in fr values for n and 3H decays. Prev’rous determinations based on the neutron lifetime measurement by Sosnovsky et al. 31), (t,), = 11.7 kO.3 mm, gave IMA12 ranging between 3.30 and 3.17. The more recent value reported by Christensen er al. 32), (t+). = 10.80+_0.16 min, decreases lMA12 down to 2.88. All these estimates are done with the same namely the value quoted by Goldhaber 33), 1132 +90, which is close to w)%x-~Hc~ Porter’s 34) 1137 f 20. In recent re-examinations of the triton /?-spectrum by Salgo and Staub 3‘) and by Bergkvist 3 ‘), the end-point of the spectrum was found to extend farther to the right than in the older data. This has the effect of increasing theft values and therefore of reducing still further lMAi2. Using almost similar inputs they find
(ft)aH= 1159+11 set, (ff)BH= 1143f3 set,
lMA12= 2.84kO.06 Salgo-Staub, IMAI’ = 2.94 f 0.05 Bergkvist.
(5.16)
M. CHEMTOB AND M. RHO
44
A summary of the early and recent status of this important question is given in table 8. The discrepancy between the value of Salgo and Staub and the single-particle operator value is tabulated as a function of P’s in table 7. We also give in table 9 the variations in d(Gibson) G (I~.&,,IM~‘Ioibson)/M~bibson and similarly in G(Blatt-Delves) corresponding to four values of lMAlexpgiven in table 8 and probability amplitudes listed in table 7. TABLE 8 fr values and lMajsB-,3Be determined therefrom W),.+,
(frb0-‘4N
1190237 set’) 1213135 b, 1099&34 “) 1108f16.45 “)
3075f15”) 3127,t31 b, 3075*15 3117f18 ‘)
(J”~)JH-P~~
9A
1132+40 “) 1132&40 “) 1132+40 1159,tll ‘)
1.18 ho.026 1.18 ho.028 1.24 ho.02 1.243*0.011
Pfal
Pw
3.17*0.19 3.25f0.19 2.88*0.11 2.84hO.06
1.78 kO.054 1.80 10.052 1.70 *0.034 1.685f0.0178
“) and b, Taken from ref. 36) corresponding to values respectively without and with radiative corrections. Neutron data correspond to Sosnovsky ef al. 31). ‘) Obtained with the samef-value as in “) but with the new neutron data of Christensen ef al. d, Value quoted by Goldhaber 33) in 1958, close to Porter’s a4) value of Cft)aB = 1137&20 sec. “) New analysis by Salgo and Staub 35) based on Christensen neutron data 32) and their own 3H data.
TABLE 9 The discrepancies
between the various experimental MA’s of table 8 and the values obtained with Gibson and Blatt-Delves wave functions G(Gibson) “)
IK?I 1.78 50.054 1.80 ho.052 1.70 10.034 1.685f0.0178
0.107~0.033 0.120*0.032 0.054+0.021 0.047*0.011
G(Blatt-Delves) ‘) 0.101 kO.033 0.114&0.032 0.048 AO.021 0.042hO.011
“) 1Mil)lGlbao,,= 1.609.
b, IM~‘%,at,-ucrvcs = 1.617.
5.3. NUMERICAL
RESULTS
We now write down the necessary formulae explicitly and evaluate them:
where subscripts n, p, w represent the mesons exchanged, the superscripts the kinds of terms: pionic current, pair excitation current, normalization correction, non-Born
MESON EXCHANGE
CURRENTS
(NB) term, nucleon recoil term etc., p$,yic+pr$-_4M y-$w
- l)YO(x,))
= 0.7257<(2x, - l)YO(x,)),
R
= - [9.897(&(x,))
Pyz= pg
=:
-0.1964(x,
2
K&&j,
= o.01551l
‘f,“NN(l +~V)= o*9755(yl(x,)>, mP
= 0.3785
= “;
-0.08300(~~ K,(x,)>,
~j~NN(l+~s)(Y&,)) 0
= -0.4523(Y,(x,)),
45
M. CHEMTOB
46 P;‘,“”
6::
=
-4
=
-0.8954(&(x,))
(
yy
AND M. RHO
2
-6%l
1
= ; z&(
Y&J)
+0.1964(x,
Jx%Jl>
K,(x,)),
= 0.03136( Y,(x,)),
recoil 6ASS = ~/.‘NN(Ko(x.)-KI(x.)Ix.)
= O.l019
KI(X,)/JE,>,
6n*rm A,% =
sNB
-_-- 1 &NN
mnm,2[2A~-)-A~-)--)AS+)](Y,(x,))
gA M
A’s-6n
=
-0.0144(Y,(x,,)+,
where the numerical values are obtained from coupling constants listed in table 10. Other quantities needed are the nucleon mass M =,.. 938.9 MeV, uv = 2.70, rcs = -0.12, the matrix elements (,~:l’)~ = 2.353, (,uY’)~ = 0.440. We have taken positive sign for gpny gpNN and gony &NN following Adler and Drell 38), and Abarbanel et al, ““). W e h ave also used $fh,,-4/r,,] = 0.8197x lO-‘/m~ obtained by using the Roper resonance parametriza:ion 21). TABLE 10
Coupling constants used to evaluate eq. (5.18)
?G
P UJ
9flNN
m&feV)
@my
13.64 2.56 6.82
138 750 783
0.448 0.758
0.080 0.0829 0.643
The results for ,u:“,kand SF1 are given in table 11 in terms of the range parameter CL[eq. (5.6)], the hard-core radius r, [eq. (5.7)], and individually for the OPE and HME contributions. For reasonable values of u and r,, a = 0.384 and r, z 0.4, t This formula was quoted erroneously in eq. (5) of our previous paper ‘) and elsewhere 37) with a misprint of a factor of two and a sign mistake inside the brackets. Due to the sign mistake the conclusions drawn there about the NB contributions are not correct.
contains Born graphs+normalization
“Born”
0.61 LE-2 0.544E-2 0.395E-2 0.285E-2 0.212E-2
0.404
0.418E-2 0.398E-2 0.357E-2 0.313E-2 0.276E-2
0.674E-2 0.491E-2 0.308E-2 0.202E-2 O.l34E-2
0.0 0.2 0.4 0.6 0.8
OS32E-2 0.475B-2 0.346E-2 0.250E-2 0.1868-2
0.3128-2 0.334E-2 0.245E-2 O.l77E-2 O.l31E-2
0.384
0.379E-2 O,362E-2 0.326E-2 0.287E-2 0.253E-2
0.2928-Z 0.281E-2 0.256E-2 0.227E-2 0.201E-2
NB
HME
pk2’
0.578E-2 0.42OE-2 0.262E-2 O.l70E-2 O.llOE-2
0.386E-2 0.278E-2 O.l70E-2 O.l06E-2 0.632E-3
Born
OPE
TABLE 11
correction.
O.l70E-1 O.l43E-1 O.l06E-1 0.8OOE-2 0.622B2
O.l49E-1 0.126G1 0.934E-2 0.7073%2 0.549E-2
O.l05E-1 0.892E-2 0.671E-2 0.510E-2 0.395E-2
Total
-0.281B3 -0.784E-3 --0.706E-3 -0.622E-3 -0.547E-3
-0.633E-3 -0608E-3 --0.554E-3 -0.492E-3 -0.435E-3
NB
0.288EO -0.905E-3 0.261EO -0.862E-3 0.234EO -0.774E-3 0.215EO -0.679E-3 0.2OOEO -0.597P3
0.266EO 0.243EO 0.219EO 0.203EO 0.189EO
0.217EO 0.2OlEO 0.185EO 0.173EO 0.161EO
BORN
OPE HME
-OS34E-1 -0.415E-1 -0.294E-1 -O.lQQE-3 -O.l42E-1
-0.4648-l -0.261E-1 -0.257E-1 -O.l73E-1 -O.l23E-1
-0.321E-I -0.251E-1 -0.179E-1 -0.121E-1 -0.856E-2
(2) Pv
0.234EO 0.218EO 0.203EO 0.195EO 0.185EO
-
0.219EO 0.206EO 0.193EO 0.185EO 0.176EO
0.185EO 0.175EO 0.166EO 0.160EO 0.152EO
Total
0.295E-1 0.234E-1 O.l68E-1 0.126JS1 0.975E-2
0.257E-1 0.204E-I O.l48E-1 O.lllE-1 0.855E-2
O.l8OE-1 O.l44E-1 O.l05E-1 0.790E-2 0.603E-2
BORN
NB
-0.388B2 -0.370E-2 -0.332E-2 -0.291E-2 -0.256E-2
-0.352B2 -0.336E-2 -0.303E-2 -0.267E-2 -0.235E-2
-0.27lE-2 -0.261E-2 -0.237E-2 -0.211E-2 -O.l86E-2
OPE
-
HME
-0.233E-1 -O.lSlE-1 -0.129E-1 -0869E-2 -0.619E2
-0.202E-1 -O.l57E-1 -O.l12E-1 -0.757E-2 -0.538E-2
-O.l4OE-1 -O.lOYE-1 -0.784E-2 -0.529E-2 -0.374E-2
Sk"'
0.230E-2 O.l%E-2 0.631B3 0.104E2 O.lOOB-2
O.l96E-2 0.1358-2 0.5348-3 0.879B-3 0.826E-3
O.l24E-2 0.827E-3 0.276%3 0.49633-3 0.431E-3
Total
to magnetic moments and CT matrix element in trinucleon system calculated using eqs. (5.17) and (5.18) as a function of range parameter a and hard-core radius r,
0.0 0.2 0.4 0.6 0.8
--
0.337
0.0 0.2 0.4 0.6 0.8
(fmy-l $I)
Exchange contributions
CA
2 P g 3
8
5
2
5 g z x”
48
M. CHEMTOB
AND
M. RHO
our results are &?’ = 0.193(+::;::), ‘@ = 0.0093( :;:;;33>, sa”) = 0.053(+;:&)
(5.19)
%,
where we have indicated in parenthesis the range of values they can take when c1and r, are varied. We may summarize the results in the following way. pv(2)** The low-energy theorem implied that the pion-exchange contribution is dominated by the Born term. This is confirmed explicitly. Even taking some sizeable errors in the method of calculation, the NB term is definitely negligible, even less than 4 % of the Born term. The heavy meson exchange is small (about 10 % of the Born term), but needs to be treated carefully since it subtracts. As one can see from eq. (5.18) there is a large cancellation between p- and o-terms; thus the result could be quite sensitive to the relative size of the coupling constants gpNNand gwNN. In our calculation g~nN/g& M 7. However, the more recent value quoted by Ting 39) has the ratio 8.6-J 1.0, while Partovi and Lomon 40) prefer in their nuclear force calculation a ratio of about 12. Furthermore, the normalization correction which we made only for the pion may be necessary for other mesons if more accuracy is needed. Thus there is room for improvement in the HME contributions. It should be emphasized that the normalization correction has roughly the same magnitude as the OPE contribution, but is more strongly suppressed when rc is increased. (2)* Ins * Clearly the low-energy theorem is useless here as all the model-dependent corrections are as large as the Born term. The non-Born term coming from the vertex correction due to the p-meson is of about the same size as the Born term. Here not only the magnitude but also the sign of the coupling constants gpNNgpny gnNN are uncertain. Although the positive sign is favoured by other evidences, there is no fundamental reason why it cannot be negative. If it is negative there will be almost complete cancellation within the pion-exchange term, leaving everything to the HME contribution. The same discussion as for the #’ leaves ,up’ in considerable doubt. Clearly more refined and if possible model-independent calculations are called for. 62’: The Born terms are small as expected, but in principle the NB term could be substantial. In triton p-decay, the latter calculated in S-state is very small because of the fact that Nz, 3 cannot contribute. Table 4 shows that the Adler values are mostly accounted for by Nt,3, the remaining part presumably coming from NT, 1, N:, 1 and p-terms. The small NB contribution is most probably a fortuitous situation due to the cancellation of N& terms, typical of the S-state of triton, but perhaps the situation could be different in other systems. Because of this cancellation, the OPE term is as small as the HME term, with the resulting value consistent with zero exchange contribution.
MESON EXCHANGE 5.4. COMPARISON
WITH
CURRENTS
49
EXPERIMENTS
In table 7, we have given the expected corrections for different choic.es of state probabilities (S, S’ and D). Let us compare these values with our results for #‘, ,@ and My’ given in table 11. For the isovector moment and the GT matrix element, our results would be compatible with the wave functions with anomalously small PD.The Blatt-Delves 41) wave functions (P, = 9 %, Ps8= 2 o/,), for example, would require much larger corrections than our calculated meson exchange corrections would provide. With the Gibson wave function 30) (P, = 6 %, Ps = 2 y*), the corrections needed are smaller. With the GT matrix element, and isovector and isoscalar magnetic moments, we have three linear relations for the state probabilities:
& =
3typfYn)(Ps+PS-PD+tPt)+~P,+P,~~2’,
CL,
~(Yp-yn)(Ps-3Ps+3Pn+gP~)-ifPD+Ps~L:2!
=
(5.20)
MA= J3(p,-3P,,+3P,+3P,+3A,.A)+6~f). One may ask what conclusions about the state probabilities can be drawn from our A ; pv and MA appear in a similar combination of P’s: therefore values of ,ut\ and 8”’ we may consider ps and pv only. Neglecting P+ z 0.25 % which has negligible effect, wefindwith PD+Ps+Ps = 1,
PD = 3(1-4, Ps*= $[l-b-+P,],
(5.21)
where (5.22) Note that whereas Ps depends on the isovector and isoscalar moments, PD depends only on the isoscalar moment. Thus, a precise calculation of pi2’ is needed since it would unambiguously fix Pn. (However, the isoscalar moment is a difficult object to calculate.) Taking our calculated values for ,u$, the solutions of eq. (5.21), for the range of a and r, considered, are Ps.zz0,PD 5 6 %. 5.5. REMARKS
To summarize, we have found that our calculated values for $\ are consistent with the wave functions different from what the presently accepted nuclear force dictates; yp’ directly gives PD , but its calculation is model-dependent and uncertain because of ambiguities in the coupling constants and normalization correction; &’ is fairly model-independent, although the exact magnitude can also be affected by the model-dependent correction terms. As for the Sy), the corrections considered are small, consistent with zero. As one can see in table 12 there are, however, some differences between the PCAC and the phenomenological Lagrangian methods. If in the latter one treats the ~-propagator as suggested by the PCAC [i.e. replacement
50
M. CHEMTOB
AND M. RHO
TABLE 12 Comparison
between the PCAC and phenomenological Lagrangian methods for 82) (in %) calculated with the S-state with c( = 0.384 fm
rc
0.0 fm
0.4 fm
Adler
-0.35
-0.30
0
-i
w.
‘aI c
WN.1
0 0.29
s .$! ;$ E&I 23 2 a
%, 1 P”)
-0.44 2.4(-3.7)
0.25 -0.38 2.05( -0.64)
Total
2.4(-4.0)
1.9 (-0.77)
3
“) The numbers given outside of the parentheses are calculated by setting q2 = 0 in the p-propagator whereas those in the parentheses by taking the q2 dependence into account.
of l/(q2fm~) by l/nzz], the difference in the two methods is substantial. However, if one keeps (4’ +mi)- 1 in Fourier transforming, the p-contribution becomes negligible for r, w 0.4 and the net result is small, close to the PCAC value. Since a 3-5 % reduction is expected from the relativistic corrections to the GT one-body operator 42), a still larger exchange correction than given in table 11 would be needed to eliminate the discrepancy. There are a number of effects that can resolve the discrepancies. Firstly, although the D-states may enter with small probabilities, they can compete substantially through the S-D matrix elements. The reason is that the S-D matrix elements can receive contributions from the vertex correction with an intermediate N:, 3 and in general from tensor-like operators [see eq. (3.6a)]. Recent calculations by Blomqvist 43), and Brown and Riska “) which include the D-states by approximate methods indeed suggest the importance of the S-D terms. Secondly, a possibility of having a substantial contribution from three-body currents cannot be excluded as suggested by the calculation of the isovector MM by Padgett et al. 44). Finally, there is the dependence of the two-body operators on the shapes of the radial wave function and on the types of core correlations (soft-core or hardcore). The Gaussian function is perhaps not very adequate in view of its too rapid fall-off. The dependence is insignificant for two-body operators of long-range character, but may not be for operators of the forms [Y,(x,)-(m,/m,)3 Yo(~,)] and eir matrix elements depend also on whether softor hard-core &(+&(Qlx,ITh correlations are used. These matters are further discussed in appendix B. We are very grateful to Professor G. E. Brown for numerous discussions on the subject and for having made possible the completion of this paper. One of us (M.R.) has benefited largely from conversations with Professors M. Ericson, T. E. 0. Ericson, A. M. Green and B. W. Lee. He also wishes to thank Professor J. Prentki for the hospitality at the Theory Division, CERN. The other (M.C.) is grateful to the Centre
MESON EXCHANGE
CURRENTS
51
National de la Recherche Scientifique (France) for financial support and to the Niels Bohr Institute for its warm hospitality. He also thanks Dr. D. 0. Riska for useful discussions. Appendix A MESON
THEORY
ln order to have a concise definition of the two-body electromagnetic or weak exchange operators in meson theory, one may follow the same line of reasoning as in the definition of the two-body nuclear potential. Some approximation scheme is set up for the evaluation of the field-theoretic S-matrix in the scattering of two nucleons involving interaction with an external field (here, absorption of photon or /?-decay). The hypothesis which is then made is that the mutual interaction energy of the nucleons and their interaction energy with the external field have representations in terms of non-relativistic, energy-independent potentials. The identification of transition amplitudes obtained in the meson theory description and the non-relativistic equivalent operator description give then prescriptions of how to define the latter. The non-unique character of the prescriptions arise, in general, from the different choices that can be made for the representation space of the two-nucleon system. The method used in this work, called the S-matrix method, rests on the representation of the two-nucleon states in the unperturbed Fock basis. It has some specific features which are not shared by others, the unitary transformation method being one example. We shall, therefore, sketch them briefly and refer the reader interested in more details to ref. “‘). Consider the decomposition of the field theory Hamiltonian of the two-nucleon system H = Ho + V, + V, into parts associated to the free fields (Ho), to the pionnucleon interaction (V,) and interactions of the pion and nucleon fields with the external field (V,), all taken to be additive. In the quantum mechanical description the interactions of the nucleons are represented by a potential. We denote its stationary states by 14,). The index a is a label for the state of the two nucleons. If 1Y,) denote the corresponding stationary states of the field-theory Hamiltonian H, it can be shown that 14,) are just their projections on the meson vacuum. More precisely, one has a
where a = E-Ho, A is a projection operator off the meson vacuum and Ri is the reaction matrix associated to V, but with all intermediate states containing at least one pion or NN pair. The transition amplitude between states labelled by m and n due to one interaction with V, can be written as
52
M. CHEMTOB
AND M. RHO
The first equation is the definition of H,, the interaction energy of the system with the external field. In the second equation, the first term stands for the normal onebody operator and the others for the operators describing virtual meson effects. The projection operator ll shows why processes described by the diagrams of fig. 8 had to be excluded. In the perturbation theory expansion of eq. (A.2) there are a number of terms representing self-action effects which we assumed are already taken into account in using the renormalized forms of the one-body operators and in evaluating the twobody diagrams in renormalized perturbation theory. In principle renormalized quantities are here “system” quantities in the sense that they depend on the state of the system as a whole even though they may refer to the properties of a single nucleon, such as mass, form factors, etc. It was, however, shown by Frantz 45) within the framework of the static model that this system dependence disappears when the adiabatic limit is taken. There is still another important feature of this approach which deserves some attention - the wave function normalization correction, The state vectors lYE) as given by eq. (A.l) are not normalized. Assuming 14,) to be normalized then the normalization is taken care of by the substitution l!P#) -P 1Y,)/Z~* where 2L3 = (c#J..J!P,).From eq. (A.l) we get (A-3) which, if solved to lowest order in V,, gives
(A-4) The one-body parts of the operator V,(LI/U)~V, are assumed to be incorporated into the renormalization effects mentioned above. The remainder is a two-body part which can be given an operator representation by using essentially the same methods as for the calculation of nuclear potentials. By taking the static limit and transforming to coordinate space, we fmd
= 1+ -$&(l)
* z(2)] [(CT1* u2) (K&J-
K+)
+f31z K2(x.)]
3
(A.9
where the derivative is with respect to x,. The correction to the matrix elements <~,lH,l~,> amounts to multiplying them by the factor 12~Z~13, where ZL(ZA) denote the expectation value of eq. (A.5) in the state m(n). In a second-order perturbation treatment of the normalization correction, it seems adequate to introduce the correction only for the one-body operator, in which case it just adds on the r.h.s. of
MESON EXCHANGE CURRENTS
53
eq. (A.2) the term
m9
11.
<&II KlUC(Z:,ZP--
Meson-exchange effects reflect the extent to which meson degrees of freedom are eliminated from the nuclear wave functions or, stated equivalently, the extent to which more diagrams appear in a field-theory description than in a non-relativistic description. The normalization correction eq. (A.6) accounts for these effects as much as the two-body operators. To conclude, we should mention that the question of normalization has raised much controversy in the past in relation to the definition of nuclear potentials. The difference between the fourth-order parts of the Taketani-Machida-Onuma (TMO) and Brueckner-Watson (BW) potentials lies precisely in its omission in the BW potential 46). The question is not whether it should be included or excluded but rather how it should be handled. The procedure that we used stays essentially on perturbation theory arguments. It is therefore not very satisfactory, but can perhaps be trusted if Z: stays close to one. This turns out to be the case in our applications to the three-nucleon systems. Appendix B TRINUCLEON WAVE FUNCTIONS FOR THE S-STATE
We briefly discuss here other forms of wave function which are available and how these different wave functions modify our results. In our calculations, we have used the Gaussian form for the radial wave function of the S-state. The others frequently used are the exponential, Irving, and Irving-Gunn wave functions 6), all of which fall off somewhat less rapidly than the Gaussian form. We have calculated the matrix elements of Y,-,(x,), [K,(x,)-K,(x,)/x,] and YO(xP) with the Irving wave function (multiplied with a Jastrow factor) and found that they were not significantly different from those of Gaussian wave function over the range of r, we have considered. Let us now examine the correlation function. We choose a hard-core function frequently used in nuclear matter calculation. Assume that we can write
f(r 12,
r13)
rz3,
=
~e-~2f(~122+r13’+r232)
iIJ(1-4*j).
(B.1)
which seems to be valid when on- and off-shell “defect” wave functions are taken to be the same, and
= 1 =
0
for
c 5 rij 5
for for
r,j < C,
d, (B-2)
rlj > d.
Here, d is the separation distance which is about 1 fm for acceptable potentials, and c is the hard-core radius of order of 0.4-0.5 fm. For the CIgiven by Coulomb energy
54
M. CHEMTOB AND M. RHO
a = 0.384,eq. (B.l) and eq. (5.6) (the soft-core), when properly normalized, are found to give the same matrix element of Ye(x,) for each c = (742)-l if we choose d = 1 fm. For our discussion we may take this as some sort of normalization. Then we have evaluated the matrix elements of [K&V,) -K,(x,)/x,] and Ye(x,) with both wave functions, and plotted them as a function of c = r, = (~42)~~ in fig. 10.
0
0.1
0.2
0.4
0.3 c:r,
0.5
0.6
0.7
in fm
Fig. 10. Plot of the variation of the expectation values of exchange operators having short-range character as a function of core radius. Comparison is given between soft-core and hard-core correlations built with Jastrow factors into the same radial function: Gaussian exp [-+$aZ(r~2+r:3 -&)I with a = 0.384 fm-‘.
Note that the correlation function eq. (B.2) suppresses their matrix elements more than the soft-core factor does. As a consequence, it is possible that our calculations with the soft-core factor obtain somewhat overestimated values for the nucleon recoil, the normalization correction, and HME terms, the bulk contributions of which come from the inner part of the wave function. References 1) 2) 3) 4) 5) 6) 7)
M. Chemtob and M. Rho, Phys. Lett. 29B (1969) 540 R. G. Sachs, Nuclear theory (Addison Wesley, Cambridge, 1953) P. Signell, Adv. in Nucl. Phys. 2 (1969) 223 S. L. Adler and R. Dashen, Current algebras (Benjamin, New York, 1968) G. E. Brown and A. M. Green, Nucl. Phys. Al37 (1969) 1 L. M. Delves and A. C. Phillips, Rev. Mod. Phys. 41 (1969) 497 H. Ohtsubo, J. I. Fujita and G. Takeda, Phys. Lett. 32B (1970) 82; R. J. Blin-Stoyle and M. Tint, Phys. Rev. 160 (1967) 803 ; W. K. Cheng and E. Fischbach, Phys. Rev. 188 (1969) 1530 8) S. L. Adler, Ann. of Phys. 50 (1968) 189
MESON EXCHANGE 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23) 24) 25) . 26) 27) 28) 29) 30) 31) 32) 33) 34) 35)
36) 37) 38) 39) 40) 41) 42) 43) 44) 45) 46)
CURRENTS
55
S. L. Adler and Y. Dothan, Phys. Rev. 151 (1966) 1267 G. W. Gaffney, Phys. Rev. 161 (1967) 1599 G. Furlan, N. Paver and C. Verzegnassi, Nuovo Cim. 62A (1969) 519 R. K. Osborne and L. L. Foldy, Phys. Rev. 79 (1950) 795 G. J. Kynch, Phys. Rev. 81 (1951) 1060 R. G. Sachs and N. Austem, Phys. Rev. 81 (1951) 705; 81 (1951) 710 B. W. Lee, Proceedings of 13th Int. Conf. on high-energy physics (University of California Press, 1967) W. K. Cheng and C. W. Kim, Phys. Rev. 154 (1967) 1525 J. Bernstein, Elementary particles and their currents (W. H. Freeman and Co., San Francisco and London, 1968) S. L. Adler and F. J. Gilman, Phys. Rev. 152 (1966) 1460 M. Ericson, Contribution to Third Int. Conf. on high-energy physics and nuclear structure (Plenum Publishing Corporation, 1969) G. F. Chew, M. L. Goldberger, F. E. Low and Y. Nambu, Phys. Rev. 106 (1957) 1345 L. D. Roper, Phys. Rev. Lett. 12 (1964) 340 H. D. I. Abarbanel, C. G. Callan and D. H. Sharp, Phys. Rev. 143 (1966) 1225 J. D. Walecka and P. A. Zucker, Phys. Rev. 167 (1968) 1479 2. C. T. Guiragossian and A. Levy, SLAC Pub-535 (1969) M. Chemtob, Les courants ~intera~ion nucleaires a deux corps (thesis, Uuiversito do Paris, 1969) unpublished R. J, Blin-Stoyle, V. Gupta and H. Primakoff, Nucl. Phys. 11 (1959) 444 S. Hatano and T. Kaneno, Progr. Theor. Phys. 15 (1956) 63 F. Villars, Phys. Rev. 72 (1947) 257 B. F. Gibson and L. I. Schiff, Phys. Rev. 138 (1965) B26 B. F. Gibson, Phys. Rev. 139 (1965) B11.53; Nuci. Phys. B2 (1967) 501 A, N. Sosnovsky et al., Nucl. Phys. 10 (1959) 395 C. 3. Christensen et al., Phys. Lett. 263 (1967) 11; 28B (1969) 411 M. Goldhaber, Report at the Annual int. Conf. on high-energy physics, CERN, 1958 F. T. Porter, Phys. Rev. 115 (1959) 450 R. C. Salgo and H. H. Staub, Nucl. Phys. Al38 (1969) 417; K. E. Bergkvist, private communication C. P. Bhalla, Phys. Lett. 19 (1966) 691 M. Rho, Review paper in third Int. Conf. on high-energy physics and nuclear structure (Plenum Publishing Corporation, 1969) R. Adler and S. D. Drell, Phys. Rev. Lett. 13 (1964) 349 S. C. Ting, Seminar given at CERN, 1970 H. Partovi and E. L. Lomon, Phys. Rev. D2 (1970) 1999 J. M. Blatt and L. M. Delves, Phys. Rev. Lett. 12 (1964) 544 H. J. Strubbe and D. K. Collebaut, Nucl. Phys. Al43 (1970) 537 R. J. Blin-Stoyie and S. Papageorgiou, Nucl. Phys. 64 (1965) 1 3. Blomqvist, Phys. Lett. 32B (1970) 1; G. E. Brown and D. 0. Riska, Phys. Lett. 32B (1970) 662 D. W. Padgett, W. M. Frank and J. G. Brennan, Nucl. Phys. 73 (1965) 424,445 L. M. Frantz, Phys. Rev. 111 (1957) 346 K. Nisbijima, Suppl. Progr. Theor. Phys. (Tokyo) 2 (1955) 138
be reprocked
~o~f~-~5i~~
P~~~ki~~ CO., Amsterdam
by photoprint or microfilm without written permission from the publisher
MESON EXCHANGE CURRENTS IN NUCLEAR WEAK AND ELECTROMAGNETIC INTERACTIONS M. CHBMTOB t
Received 6 October
1970
Atrstract: Meson theory is appIied to the study of the exchange magnetic moment and GamowTeller operators. We consider contributions due to the exchange of one pion and of vector mesons and give their detailed expressions in terms of a general classification of the operators. The one-pion-exchange process is separated into Born and non-Born parts corresponding respectively to the Born and non-Born parts in the process of weak or photoproduction of a pion by the nucleon. The results of low-energy theorems for this process enable us to reach a modeiindependent description of the non-Born parts. Corrections to the soft-pion limit are studied by means of simple dynamical models: a phenomeno~ogi~a~ Lagrangian for the weak current and the Chew-Low model for the e-m. current. The exchange operators due to the exchaage, of P- and w-mesons are evaluated in the vector dominance model. The applications are concerned with the exchange effect in the isovector and isoscalar magnetic moments of 3He and “H and in the Gamow-Teller matrix element for 3H /?-decay. The results are obtained with the dominant Sstate with a Gaussian radial function. We also discuss their dependence on short-range correlations described in terms of&&row factors, The exchange corrections are found to be too smslf for the current wave functions of the trinucleons. We discuss briefiY other agencies that may account for the remaining discrepancies.
1. Introduction In a previous paper ‘1, we made use of the low-energy theorem based otl the PCAC hypothesis to obtain the complete onepion-exchange (OPE) two-body contribution to the &mow-Teller matrix element in p-decay and applied it to the process 3H -+ ‘He f e- + &. We have suggested the approach as a successful means of tackling the difficult problem of meson exchange effects in nuclei, In this article, we would like to expand that theme to both weak and electromagnetic interactions by systematically discussing the low-energy theorems, defining as clearty as possible what we mean by the meson exchange corrections, pointing out which pieces of information are model-independent, and emphasizing the basic differences in the meson exchange currents of weak and electromagnetic processes. The problem of exchange currents in the weak and electromagnetic processes is an old one ‘>. However, it still remains unclarified because of the difliculties in t On leave from SPT, CEN Sactay, France.
2
M. CHEMTOB
AND M. RHO
defining interactions in a composite system starting from the field-theoretic description. We do not pretend to resolve these difficulties. The solution to the problem would involve beyond
a fully relativistic
equation
for many-body
systems, which is at present
our capability.
Our approach in obtaining the equivalent two-body current (we neglect many-body currents) relies on the conventional S-matrix method which has the advantage of being both simple and consistent. This has been used before. A renewal of interest in it stems from two sources: one, a recent development in deriving the two-nucleon potential, notably the one-boson-exchange (OBE) models 3), and the other, the success of the low-energy theorems associated with the PCAC and current algebra in describing the X-N interaction “). Within the conventional S-matrix framework, the former tells us how to split the totality of the relevant Feynman graphs, and the latter allows us to treat the OPE term correctly to all orders in the strong interaction coupling constant. In our studies, we encounter both the soft-current and soft-pion theorems. The former is applicable to the Gamow-Teller matrix element where the four-momentum transfer k, is small. The soft-pion theorem supplements the former when we deal with the OPE graph. For the magnetic moment,. however, it is more convenient to take the soft-pion limit, as will become clear in sects. 2 and 3. Let us point out here that the soft-pion theorem has proven to be an extremely powerful technique in understanding both the nuclear force and n-nucleus interaction, thus being applicable to the virtual as well as the real pions. The pion exchanged in nuclei is not really soft; the pion four-momentum qn does not vanish. But the extrapolation to the relevant value of q2 has been studied, and shows that the effect is not significant, at least for the OPE term “). From these considerations emerges an important and hitherto unappreciated fact that the exchange current in the weak process is quite different from that in the electromagnetic process, that is, the sources of the corrections are different. In treating the exchanges of meeons other than a single pion, the situation is not simple. The exchange of two soft pions may be treated, but it cannot represent the true situation. Because of the tensor nature, one or two of the pions can carry considerable momentum. There are also the practical difficulties in the choice of graphs. So we shall not consider the multi-pion exchanges here. We include however the vector meson (p and 0) exchanges in an approximate way with the hope that they simulate, at least partially, the effect of multi-pion exchanges. We apply our theory to the bound three-nucleon systems. Aside from the deuteron, the wave functions of these nuclei are the best known in nuclear physics. There have been innumerable studies, both theoretical and experimental, on the ground state properties of these nuclei: the Coulomb entrgy difference, the electromagnetic and weak properties such as electric and magnetic form factors, n-d radiative capture, photo- and electro-disintegration, P-decay and muon capture. A lucid discussion on the status of the three-body problem can be found in the review article of Delves and
MESON EXCHANGE
CURRENTS
3
Phillips “). Despite such a long history and considerable efforts, the understanding of the three-nucleon system is far short of the extent to which one knows of the deuteron. The reason is, of course, the extra degrees of freedom an additional nucleon brings in. A reliable calculation of meson exchange corrections in triton b-decay, and in the magnetic moments of 3H and 3He would be valuable since, once we know the corrections, a set of linear equations describing the P-decay and the magnetic moments can be used to draw conclusions about the structure of the bound three-nucleon systems. Our calculation is limited in that only the dominant S-state component of the trinucleon wave functions is used. This in only a first step towards more accurate calculations using a more realistic wave function. Furthermore, since our exchange current operators (except for the isoscalar magnetic moment) should be reliable, an extended application to other nuclei would be just an exercise in nuclear structure calculations. In view of the length of the paper, we summarize the essential points here. The twobody meson exchange operator is divided into a one-pion exchange (OPE) term and a heavy-meson exchange (HME) term. We apply the low-energy theorems to the OPE term. Sect. 2 deals with this question, We give the theorems relevant to the Gamow-Teller transition, and the isovector and isoscalar magnetic moments. The principal conclusions are that (i) for the Gamow-Teller operator, the theorem specifies the exchange current in terms of two off-shell nN amplitudes, available from Adler’s works, (ii) for the isovector magnetic moment, it leads to a reliable expression given in terms of generalized Born graphs, and (iii) for the isoscalar magnetic moment, the theorem has no predictive power and hence one needs to go beyond it. Sect. 3 is devoted to the classifi~tion of two-body meson exchange operators based on general invariance considerations, Explicit expressions are given for the OPE operators when certain assumptions are made. We also indicate how the operators are extracted from Feynman graphs. In sect. 4, the one-boson-exchange contributions are explicitIy cal&lated. For both the p-decay and the magnetic moments, models are used to get the HME contribution. For the magnetic moments, the vertex correction coming from N* is calculated by means of the Chew-Low model. The shortcomings in our treatment of the HME graphs are emphasized, and should be kept in mind in assessing the results of sect. 5. Sect. 5 deals with the details of ex~riment~ situations and theoretical calculations of the Gamow-Teller matrix element and the isovector and isoscalar magnetic moments in the three-nucleon system. We use for simplicity the dominant S-component of the wave functions. For this reason, the exchange Gamow-Teller matrix element is small as the Nz3 cannot contribute. The importance of the D-state is pointed out. For the isovector magnetic moment where the Nj, is expected to play a minor role, our calculated value should be reliable, but there still remains some
4
M. CHEMTOB AND M. RHO
sizeable discrepancy. Although the exchange isoscalar moment fills up the existing discrepancy, this may be fortuitous because of its model dependence. Appendix A deals with the problem of defining the meson exchange current from a field theory, and appendix B deals with the model dependence in the nuclear wave function. 2. Low-energy theorems The study of the two-body exchange currents amounts to the description of the process given in fig. 1. We follow the usual procedure used in the nuclear force problem, and split the contributions into a one-pion exchange (OPE) graph and a
Fig. 1. Diagram representing the transition amplitude of a two-nucleon system interacting with the axial or vector currents (denoted by the wavy line).
b)
0) Fig. 2. One-pion exchange (a) and one-boson-exchange
(b) contributions
to the exchange currents.
heavy-meson exchange (HME) graph as given by figs. 2a and 2b. As was pointed out by Blin-Stoyle and Tint ‘) for the triton B-decay and by Ohtsubo et al. ‘) for the axial current in general, a model-independent way would be to look at the two-body exchange contribution by means of the PCAC hypothesis applied directly to the two-body processes N, +N, -+ N,+N,+y, N,+N,
-+ N,+N,+e-+7,,
(2.1)
which may be given in the low-energy limit (that is, when the current momentum k approaches zero) by a non-radiative amplitude T(p,, p4; pz, pl) describing the twoparticle reaction in the absence of a current. We shall not consider this approach here.
MESON EXCHANGE
5
CURRENTS
In this section, we discuss the low-energy theorems relevant to the OPE graph of fig. 2a, although by now these are folk-lores in particle physics. Our philosophy is that tbis is the dominant contribution in most of the exchange current contributions and therefore should be treated as accurately as possible. The HME terms are treated as corrections, and hence are more model dependent. Let us consider the process given by fig. 2a. The matrix element for such a process is 1
---6(Pl+Pz+~--P’,-P;)
(243
x W(dN(p;)lJilN(~d)
dz(q2) q2+mz
R
(N(P;)IJW(P,))+(~
*
9,
(2.2)
where superscripts n and j are the isospin indices for pions and current, dz(q2) is the pion propagator form factor, and Ji is the pion source current. The nNN vertex d,(q2)(NlJ.lN> will be taken to be known from dispersion-theoretic analysis (see subsect. 3.2 for more details). Therefore we need only to know the left vertex which is essentially a matrix element for the production of a pion by a current Ji, which is either J:“* or Jy. In allowed /&transition we are only interested in the soft-current limit (k 3 0). In all cases, however, the pion is assumed to be soft, since it carries a mass (- q2) which is small in the process where a nucleon makes transitions with little momentum transfer, as in the case of B-decay or magnetic moment. On the average, q2 x m,2. Thus the well-known soft-pion theorem is ideally suited to this kind of process occurring in nuclei, in contradistinction to elementary particle physics where the soft-pion limit is only a mathematical limit. We shall argue later that q2 = 0 is a reliable limit, the extrapolation to q2 w ma modifying the results unappreciably.
Fig. 3. Pion-production
amplitude from a single nucleon by the axial or vector currents.
2.1. DEFINITIONS
We shall follow the notations + of Adler “) with minor modifications. The process given by fig. 3 which represents the reaction
J&9 + NP,) + n”(q)+ N(d),
(2.3)
7 We use the natural units 6 = c = 1, e2/4n = &, and the Pauli metric, where (I . b = a$, = a. b =a& = a * b--a&. The Dirac matrices are all Hermitian and satisfy {yp,yy} = 26,,. We also use I+ = YIYZYJY.+ and upY = (1/2i)(y,,y, -IV&
6
M. CHEMTOB AND M. RHO
is described by the set of kinematic invariants k2, q2, v = -(p,
+p;)
* k/2M,
I+, = q * k/2M,
in which M is the nucleon mass. We denote the axial current current by Ji’“’ = Jy + Ji, where Jp and JI are the isovector nents, respectively. When we write simply Jj, it means both otherwise specified. The transition amplitude for the process terms of Mij E i(q2+mz)
s
where T is the usual time-ordering MA by the decompositions
d4xe
by J,“’ and the e.m. and isoscalar compoJy and Ji.“. unless (2.3) is described in
-iq’“
operator. Let us specify the isospin structure of
MA”’ = a’t’A:+‘+J$)A:-), A M :‘“’ E j,&j+MS,”
(2.4)
= a~if)V:+)+a~T)r;,(-)+alO)v,(o),
(2.6)
where (2.7)
and the space-spin structure by
A:” =
vj*“) =
&4iyv, VB)
k2,q2)~Mj4hW~,j,
$)/*‘“‘(V, v,, k2,q2)~(P;)o,(%)fJ(P,j.
(2.8)
Ineq. (2.8), A!*)(. . .j and V/*“)(. . .) are amplitudes which depend on the invariants eq. (2.4), and O,(A,) and O,(VJ are covariants given in table 1. The aim of any theory is to determine the amplitudes Ai*) and V/*,‘). We shall often use the matrix elements of the currents between nucleon states:
iu(p;)[gA(Q2)y,Y,-ihA(Q2)Q,YS13zju(pl),(2.9a)
and the form factors at Q2 = 0 have the values gA(O) = 1.23)0.01, F;(O) = F;(O) = 1, F;(O) = 2% = _ 0.12, 2M
(2.10)
MESON EXCHANGE 2.2. AXIAL
CURRENTS
7
CURRENT
In this section, we discuss in a concise manner what we mean by the low-energy theorem relevant to the /?-decay. In the limit as k + 0 which is valid in b-decay where both energy and three-momentum transfers are small, the Adler-Dothan theorem *#‘) provides an expression for i@’ valid up to order k. The derivation is given in the papers of Adler and Dothan 9), and Adler s); so we shall simply give the results. Let us separate IV; into a “Born term’, (Mf)B and a non-Born term Mf, M; = (Mf)B+m;.
(2.11)
Then the Born term is defined by (M?)B = f(P;)[igpr,Ys
S&r + k)ig.4(k2)3rjYAYs + ig,(k2)~zjYnysSF(~1-q)igrZnYS]U(P1),
Fig. 4. Nucleon pole diagrams in the amplitude for pion production current.
(2.12)
from the nucleon by the axial
where S,(P) = (iy++M)-‘. This corresponds to fig. 4. Rearranging the non-Born term a little from the original expression of refs. **‘) we may write a;
= ~~(p;){~~i”r(~2>~~+~(,~‘[-2P(~2)P~-2~~(q2)M~~])~(~~),
where a($)
= [a~-‘(q”>-~~~-‘cq”)]Y~ve~~2~~,
PCq2) = -[A:-)(q2)+~(;)(qZ)]y’vB’k2’0,
(2.13)
y(q2) = &+)(q2)ly,ve=k2=0. Thanks to the PCAC on which the low-energy theorem is based: (2.14) c(, j?, and y can be given in terms of the invariant nN scattering amplitudes AnN, BnN : U(p;)WN’*h(pl)
= - il(p;)[AnN’*’
- iy * kBxN(*)]u(p,),
8
M. CHEMTOB AND M. RHO
(2.15)
The a and fi are related at q2 = 0 by the Adler-Weisberger relation “) - ~
2M2
[/?(O)+a(O)]
= 1 -gi’.
(2.16)
%(%A
It is thus advantageous to work at q2 = 0, in which case only two off-shell nN amplitudes completely specify Mf since the Born term eq. (2.12) is given unambiguously. One may, however, ask at this point how CI,j3 and y extrapolate from their values at q2 = 0 to q2 z ml. Such an extrapolation necessarily involves models, since it deals with momenta which cannot be reached directly by experiments. It has been shown in particular by Brown and Green that numerically the different models yield similar results up to q2 z 10 rnz [ref. 5)]. Thus, from a practical point of view, the model dependence is not an obstacle. We have applied a method similar to the one used by Brown and Green for threebody force, and found that the extrapolation in q2 changes insignificantly from the values at q2 = 0. More specifically, using the same arguments and procedures [i.e. saturation of the dispersion integral by (3,3) resonance, zero-width approximation, etc.], we found that a(q2) = o(q2)a(0), (2.17)
B(s2) = %12)8(0)9 y(q2) = +12)Y(0)P with a(q2)z
(l+-$$)g9
b(q2)=
(l+&)gy
c(q2) z (l+ &)
$
where the magnitude of each of &, ;1,, Iz, is greater than unity. It is thus consistent with
a(q2) x b(f.p) E c(q2)x
Ag,
MESON EXCHANGE
CURRENTS
9
for q2 < M2. If the pionic form factor of nucleon K(q’) is assumed to extrapolate to q2 = rnz as smoothly as it does from q2 = -m: to q2 = 0, then we would expect that a(m2)
x b(mi)w c(mz) z 0.95,
which gives a 5 y0 variation from q2 = 0 value. We shall neglect this variation from now on. There is a variant of the low-energy theorems that can be obtained by considering directly the soft-pion limit (q + 0). In this limit, the only surviving amplitudes are 2’;’ and Ai-’ for which the PCAC places the constraints
&--)
= - ;?A
[F:(k’)-g,g,(k2)+2MF:(k2)].
One can also take advantage of the relations (2.15) valid at k + 0, arbitrary q, and the above relations valid at q + 0, arbitrary k, describing the same quantity i@, to obtain the combined constraints which would specify Is;if up to 0 (qk, q2, k’). The results are given in Adler-Dothan ‘). The advantage of this method is that it allows ;I\-’ and A:+) to be specified separately in addition A:-‘, AL-‘. 2.3. ELECTROMAGNETIC
CURRENT
Unlike the axial case where the k -+ 0 limit is appropriate, we shall instead take the soft-pion limit for the electromagnetic current. The reason is that the k + 0 limit is not quite appropriate for the magnetic moment, and furthermore we want to rely on the experiments of the threshold production of pions for which the soft-pion theorem is known to be reliable for charged pions. A shortest derivation of the theorem is through the modified PCAC relation in the presence of electromagnetic field Al [ref. ‘“)I a,Jt(*)T
ieA,Q*)
= LX#J~*),
A
=
J;’
JAW) A
=
J;‘3,
JA'*'
c =
(2.18)
T iJt2,
@f&h
gm
= f,%t,
where (+, 0) stand for the charge states of the pion. For charged pions, the photo-
10
production
M. CHEMTOB
AND M. RHO
amplitude is
where aAis the photon polarization four-vector, the r.h.s. means that a pion-pole term
and the bar in the second term on
has been separated out. Similar equations can be written for the neutral pion. Now imposing the gauge condition k,Mi’m’ = 0, one finds that A4Fm*is given to order (m,/M) by the B orn terms of fig. 5.
Fig. 5. Born diagrams representing the amplitude for pion production by the vector current. The separation into the seagull diagram (a), the nucleon-pole diagrams (b) and (c) and the pion-pole diagram (d) provides, when appropriate expressions for the vertices are used, a book-keeping of the terms describing the full amplitude in the soft-pion limit. These modified vertices also follow from the chiral Lagrangians.
The first term in eq. (2.19) yields the seagull term fig. (5a) and a piece of fig. 5d. (Note that the seagull term is equivalent to the Born term with NN intermediate state. It contributes only to the charged pion production.) One can see this by examining the matrix element of Jf in eq. (2.9a) and using the Goldberger-Treiman relation to express hA ill terms of gA and letting Q2 = 0 in the form factors. The figs. 5b, 5c and a part of fig. 5d come from the second term of eq. (2.19). Note that we are to take the full electromagnetic vertex as defined in eq. (2.9b) and F,(q’) -+ F,(O) = 1 to ensure the gauge condition. It is important to note that to the order (m,/M), no non-Born terms contribute, in contradistinction to the axial current where non-trivial contributions come from the non-Born terms. 2.3.1. Thresholdproduction of pions. In order to have an idea how good eq. (2.19) (and a similar equation for the neutral pion) described by lig. 5 can be for the isovector V(‘) and isoscalar V(O) amplitudes for our purpose, it is instructive to look at the experjments on the photoproduction of pions at threshold. We shall assume that the
MESON EXCHANGE
CURRENTS
11
features observed at q2 = -mz (physical pions) can give us some information about the q2 z rni point. It is now well established that fig. 5 describes very reliably the charged pion (n’) production, whereas it appears to be insufficient for no production where the main terms figs. 5a and 5d vanish. Also theoretically, the corrections to the soft-pion theorems are found to be very small for X* , while important for no [ref. I’)]. Now to translate these observations in terms of the isovector and isoscalar amplitudes, we note from eqs. (2.6) and (2.7) that M>“jyp + z’n) =
$ pp+ Vn’O’],
Mk”‘(yn + z-p) =
-
$
[Vp-V,‘“‘],
(2.20)
$[v,‘+‘+ VjO’], Mpm’(yn + non) = p,‘+‘- v,1’O’-J.
Mtm.(yp --f “Op) = Or inversely v,‘-) = $
[M’,‘“‘(yp -+ n+n)-M;m.(yn
-+ n-p)],
vi+’ = [M”;“QJp + ~“p)+M~“~(yn + n’p)], T/(O) E. = $
[Mtm.(yp -+ n+n)+Mtm$yn
(2.21)
3 n-p)],
where M2.m*(yp + z+n) stands for the amplitude for y +p + z+ + n, etc. Substituting into eq. (2.21) the experimental values for z* production and the theoretical estimate of no production by Furlan et al. ‘l), we have found that
(2.22) w 7x 10-2.
Thus, in the isovector part we may safely ignore Vi+‘. The amplitude V,$-’ being the sum of numerically reliable quantities is thus correctly given by the low-energy theorem. The same would hold for the amplitudes we are interested in if the extrapolation from q2 = -mi to q2 z rnz is as smooth as is assumed. On the other hand, Vi” so obtained would not be good, since it is a difference between two large numbers, and any further corrections to it can become significant. 3. Two-body exchange operators 1.1. GENERAL
CLASSIFICATION
Tn this section, we discuss how the non-relativistic two-body operators for the netic moment (MM) and the Gamow-Teller (GT) P-decay transitions are
12
M. CHEMTOB
AND M. RHO
extracted, given the appropriate meson theories (next section) to calculate the Feynman graphs. Let us first classify the functional forms for these operators based on general invarjance requirements. We assume that GT and MM exchange operators have the same transforma~on properties as the corresponding one-body operators. They are both axial vectors The GT operator is an isovector, whereas the MM has an isovector as well as an isoscalar component. For their construction we have at our disposal position coordinates of the two nucleons x1, x2 their conjugate momenta pl, pz, the Pauli spin matrices bi , rr2 and isospin matrices r(l), r(2). For the isospin parts of the operators, al1 possible two-body terms formed with z( I ) and r(2) must reduce to three isoscalars:
and three isovectors: Cz(i)x
2(2)1j~
0.2)
llzj(1)4Tj(2)j~
where the MM operators involve the isovector componentj = 3 and the GT operatorsj = (1)&i(2). For the space-spin parts it is convenient to work with the relative and cm. coordinates: r = x1-x2,
R = +(x1 +x2).
Operators depending on R (i.e. translationally non-invariant terms) or on the momentum operators pr, pz (non-local terms) are, in principle, allowed but we shall not consider them in our classification. The space-spin parts must then be axial vectors formed with ul, b2 and r. They must be of the following seven forms (9 z r/lrl): (01 x @A (@I _+%I, (a, x (r2) * 69, (#I 5%)
(3.3)
+M,
That no new terms can occur upon introducing the isospin and spin exchange operators Pf and PC2 can be seen by using the foIlowing identities among the isospin matrices: Z
[z(l) X @>]PIz Cl -
= i[+)-+)],
Mhw1~;2 = 41) *42)--m42),
(3.4)
and Iikewise for the spin matrices. Because of the antisymmetry of nuclear wave functions Pfl, Pyz Pi2 = - 1, the Majorana operator (Prz) would therefore bring
MESON EXCHANGE
13
CURRENTS
nothing new. Thus the operators (3.3) give the full space-spin dependence within multiplicative scalar functions of the relative coordinate. The complete operators are formed by combining these with the isospin parts (3.1) and (3.2) in all possible ways under the restriction that they are fully symmetric under the interchange of all particle coordinates. We thus arrive at the general classifications of the isovector and isoscalar components:
where we have used the definitions: f;,z = +4-(~1 * f)(u2 Tp
=
[(u
1 0
x P) +(a,
u 2 ) * w-+(u
x P)(@2 1 0
u 2 )] 9
* q-j, 0
=
i-,
x.
(3.6a)
The operators T12 are the D-wave parts of (ui 0 u2)3 * 3, obtained by subtracting out their average over orientations of P. The functions gr, g,r, h,, hi, etc., are arbitrary scalar functions of 1~1= r, which must all be real by virtue of the transformation properties of the operators under time reversal [see appendix 3 of ref. ‘)I. The constant G is defined by e/2M (MM) G= (3.7) ( -gA (GT). We note that of the 20 terms entering our classification, only 12 of them would be allowed in the phenomenological theory of Osborne and Foldy I’). The additional 8 terms are those involving the functions hi,,, , hi,, , and also those involving functions with subscript III, namely j,ir, haI, etc. They had to be rejected in the phenomenological theory of Osborne-Foldy as a consequence of the assumptions used in its construction and the restrictions imposed on the two-body operators by time reversal 13). To be more specific, their assumption that the exchange operators are proportional to the NN potential imposes reality conditions on the radial functions, which for the above-mentioned terms are incompatible with conditions imposed by time reversal. For example, a term such as
transforms properly under time reversal only if we allow G(r) to become pure imaginary. On general grounds there is, however, no reason to discard these terms, which in fact will appear in our explicit meson-theory descriptions. The classification of operators non-invariant under translation is a rather fastidious task which we do not wish to undertake here. We simply write down some typical
14
M. CHEMTOB
AND M. RHO
terms we shall encounter in the calculations: H’ = GI[(a,+a,).iip-(bl~bZ)(P.~j]~~+[(~1-a,).~3-((a,--a,)(p.~)]F,, + C(@l=P)(q x i()+(q
x i&Q2 * P)]&
+ i(P x ii)&),
(3.8)
where FI;*-, F,, are real scalar functions of r which may depend on spin and isospin operators, and have appropriate symmetries so that H’ is fully symmetric under interchange of two coordinates. As for non-local operators, there exists a phenomenological classification of them by Sachs and Austern r4 ). In our calculation where the static approximation is made for nucleons, non-local operators are met only infrequently, so we shall not go into them; they do not contribute in our application to the three-nucleon systems. 3.2. DEFINITION
OF EXCHANGE
OPERATORS
FROM
FEYNMAN
DIAGRAMS
The most convenient way of looking at the meson exchanges is to divide them into an OPE term and a multipion exchange contribution which is to be simulated partly by heavy meson exchanges (HME). We have given the corresponding Feynman graphs in fig. 2. They represent two-nucleon scattering from initial momenta pi, pz to final momenta pi, pi while undergoing an interaction with the external field (electromagnetic or weak), The relevant currents are the e.m. or weak currents. The deduction of exchange operators from the Feynman amplitudes associated with those graphs will now be discussed. Let us first consider the OPE graph, fig. 2a. The matrix element associated with it is denoted by (p;p;lJjlplp2). If the vertex corresponding to the pion production by a current JA is written as (~“(q)N(p~)l~~iN(~~)) where the superscript n stands for the isospin index for pion, then using the conventional form of the zNN vertex
= igrKmfq%(P;)~z ~414,
(3.9)
we may write
J; =
(2~)3s(P,+P2$.k-P;-P;)
= (2n)-36(p,+P2+k-P;-P;)
dff(q2){N(~~)l~~l~(~~)> +(I * 2) <~“(~)N(P:)IJ~IN(P~)> q2 lr
(3.10)
in which (1 Z$ 2) stands for the term obtained by interchange of nucleons 1 and 2 in initial and final states. The 1: can be considered as a matrix element in momentum space of the two-body current operator for two-nucleon systems. Eq. (3.10) represents an amplitude expressed in renormalized perturbation theory, thus containing a pionic form factor of nucleon K,,, and the pionic propagator form factor d,. These quantities are normalized at the pion pole q2 = -mz. In principle, the off-shell behaviour in these form factors can be very important. However, if we want to pursue the calculation in the line adopted in nuclear force theory, we need to be careful. The reason is that when obtaining two-nucleon potentials by using phenon-
MESON EXCHANGE
CURRENTS
15
enological coupling constants and masses for pion and heavier mesons, the form factor effect is assumed to be automatically taken care of. This fact was duly recognized by Brown and Green “) in their treatment of three-body force. Since we are going to use correlated nuclear wave functions, and put in also the heavy meson exchanges, it is deemed necessary and consistent to put K(q2) s KnNN(q2)d,(q2) on the mass shell K( - mz) = 1. This argument does not apply to the weak or photoproduction vertex (nNI.@N), whose off-shell-dependence cannot be ignored. In our case, we work with the amplitude at the soft-pion limit q2 = 0 which is sufficiently reliable for small q2. As it stands, the representation (3.10) is not suitable for nuclear physics applications. The reason is that conventional description of nuclear structure is usually given in terms of non-relativistic, single-time wave functions depending on the space and intrinsic coordinates of nucleons. The transformations required for such a descrption are standard ones. Matrix elements with respect to Dirac spinors must, firstly, be expressed in terms of Pauli spinors and, secondly, transformed to position space. In the non-relativistic reduction of Dirac to Pauli spinors, the static limit is taken. This is achieved by expressing matrix elements in a particular representation for the Dirac matrices and retaining leading terms in the limit where the nucleon mass M --f co. The procedure reproduces the lowest-order terms in a Foldy-Wouthuysen transformation. The representation in configuration space is then obtained by doing a Fourier transform on each nucleon momentum
where Ji(xl, x2) is the diagonal part of the matrix element. It represents the Fourier transform of the current density of the two-nucleon system as the momentum k of the external field is held fixed. Any momentum-dependent term would come from a derivative contained in Jj(x, , x2) operating on the delta functions. In table 1, non-relativistic expressions in the static limit are given for each covariant O,(A) and O,(V). The nNN vertex is, on the other hand,
and in the static limit q2 w q2. As an illustation, let us consider the axial current. To make an unambiguous separation of local and non-local terms, one should be careful in performing the integrals in eq. (3.11). Of the four integrals over nucleon momenta, one, say pi, is used to eliminate the delta function expressing momentum conservation. The remaining integrals are then expressed in terms of the independent momenta pl, *(pi + PZ) and
M. CHEMTOB
16
AND
M. RHO
TABLE 1 Explicit forms of covariants
O(A,)
{a, bjA z aA b
0,
4 A2 A3
where
NR limits
-(@X4) 2iP
ih
ici
-(Y - W’,
expressions,
rZ= 1,2,3,4
2iPA
A5
A8
* kbA, P z +(p,+p;),
LTllrqr
-MY,%
4
. k-u
Covariants
A4
A6
and O( V,) and their non-relativistic
-4~ - kk, ik, -(Y * k)h
v,
v3 v,
-$w(k-g)[k(q*kj-qk2]
v5
-i[(uxk)xk]
v,
(pi -p2)
E q. The argument of the exponent is written as
The integrals over the first two momenta introduce the delta functions displayed in eq. (3.11) so that any dependence on them is equivalent to the terms Pl -+
3(P;+Pd
$1
+
B2+4t
where hr and F2 are momentum operators acting on the ket wave functions. Putting
MESON EXCHANGE
17
CURRENTS
eq. (3.10) into (3.11), we can write
-iq.r e x -9,~(P;)~Y,~“~(P,)+(l q’+rn:
~5 2)) ,
(3.12)
where s = 1,. . ., 8, and a$) = a[ z,, Zj]* as defined before. After expressing all the dependences on pion momentum by a derivative, i.e. qr + ii?,, I = 1,2, 3, the integration over q brings in the familiar Yukawa function
s
-iq.r dq -%---
= 23czmnY,(q),
(3.13)
q’+m;f
Fig. 6. Representation of the pion-production amplitude by the axial or vector currents in terms of pole diagrams. Graphs (a) and (b) are for the intermediate nucleon isobars (s-channel singularities) and graph (c) is for the vector meson intermediate states (t-channel singularites).
Fig. 7. One-pion-exchange diagrams representing the vertex corrections due to vector meson intermediate states. Due to its odd G-parity, the o-meson (J PG = 1 - -) contributes only to second-class currents.
where Y,,(X) = eWX/x, with X, = m,r. If the invariant amplitudes AL” have no dependence on q, the integrand in eq. (3.12) depends through O,(A,) and ys on the pion momentum, at most quadratically. As we have seen in the previous section, the amplitudes A,(*I have that feature in low-energy limits. But suppose we consider explicitly the vertex correction due to p (fig. 6c), giving rise to the two-body operator of fig. 7. In the limit k + 0, the amplitude relevant to this contains the factor 1 q2’
(3.14)
if the Feynman graph is evaluated in the usual way. In such a case, the integration is
18
M. CHEMTOB
AND M. RHO
MESON EXCHANGE
19
CURRENTS
not eq. (3.13) but
s 4
e (q2 +
-iq.r
m,2)(q2 + m;) =
m4i?^) %2 2_1 m,
[Y&J-
(t)
y0 (?%)I
. (3.15)
1
If, furthermore, pions can propagate with either positive or negative frequencies alone as is the case in some of the Born terms (see the next subsection), then eq. (3.12) does not hold, and radial functions different from the Yukawa types [eq. (3.13) or eq. (3.15)] appear. Now given si, what are the relevant operators for the magnetic moment and the P-decay? To answer this, we recall that only the space part of the current four-vector is needed. The Gamow-Teller operator is simply the monopole part of the axial current, and the magnetic operator is the dipole part of the current, i.e. JAj]k,lJ M = -+iV,x
Je.m.lk=O
for
B-decay
for
MM.
(3.16)
We conclude this section by giving the expressions of the radial dependences of the operators in the case where the amplitudes A$*), VJ*“) are independent of the invariants v, vn and q’. In table 2, we give the functions g, h, j for the GT operators, and in table 3 the functions g, h, j, k, Z, m for the MM operators. We note that this condition is met for the NB parts of the amplitudes found in the application of the low-energy theorems. It does not, however, apply to the Born parts which have singular terms in the soft-pion or soft-current limits. For this and other reasons to which we will return in the next section, the Born parts must be treated separately. Note that for the GT operator where k + 0, A,, . . ., A8 do not contribute, and for the MM operator, V, and V, vanish since k2 = 0 for real photon. Furthermore, in the non-relativistic (NR) limit, VI and V, are negligible compared to V3 and V,, since the matrix elements of O,( V,) and O,(V,) are proportional to k, or M-l. These are the reasons why table 2 contains only A,, s = 1, . . .4 and table 3 only v,, s = 3, 4. We also note that at the limit v = vn = k2 = q2 = 0 which comes in the low-energy theorem, some amplitudes vanish due to crossing symmetry, and these are indicated therein. 4. One-boson exchanges In this chapter we calculate the two-body exchange operators due to OPE and heavy-meson exchange (HME) contributions. For the OPE term, we use the lowenergy theorem information, and supplement the calculation by using other models: phenomenological Lagrangian model for the GT case and the Chew-Low model for the MM case. For the HME, we consider the p- and o-exchange graphs.
M. CHEMTOB
20 4.1.
ONE-PION
EXCHANGE
AND M. RHO
CONTRIBUTION
For the benefit of those readers who have not quite appreciated the power and relevance of low-energy theorems, we summarize here some important points to bear in mind. For both the e.m. and axial currents, we have divided the pion-production amplitudes (and therefore the exchange current) into a well-defined Born term, and the rest into the non-Born term. Then the low-energy theorems tell us essentially the following: (i) for the isovector e.m. current, the NB term is negligible; (ii) for the isoscalar e.m. current, the Born term is small, while the NB term is undetermined; and (iii) for the axial current, the NB term is given in terms of off-shell nN scattering amplitudes, not necessarily small. These results rest on the applicability of the soft-pion theorems, and the question is then: to what extent can the exchanged pions in the OPE process be considered as soft? In practice, there are limitations in eq. (3.12) on the range of integration over the pion momentum. The long-range parts of the interactions are only sensitive to small values of 4. The longer the range, the smaller the q involved. From energymomentum conservation at the rcNN vertex, we also have: q = pi -p2. Thus for uncorrelated nucleons near the Fermi surface, q0 = p;,, -pzO GZ0 and q2 roughly in the range between ma and 4rni with the peak at ml. This is not far from the softpion limit, and thus renders the PCAC applicable. Once this is established, the power of the theorems is that they give us non-trivial relations exact to all orders in strong interaction coupling constants connecting the required amplitudes to either calculable quantities or measurable quantities. 4.1 .l. Non-Born contribution. We reverse the order and treat the non-Born part first. The reason is that the NB part permits the full pion propagator and has no danger of double-counting which the Born term does. Thus the discussion of sect. 3 is directly applicable. In following this part of the section, tables 2 and 3 are of practical importance to us, since once the amplitudes are calculated explicitly, the associated GT and MM operators can be read off from the tables. The assumption that the amplitudes are independent of the invariants applies to most of the situations discussed below, as the relevant point corresponds to v = vn = k2 = q2 = 0. (i) Axial current. Let us consider the NB parts (denoted by a bar) of the pion production matrix element given in eq. (2.16) in the limit k -+ 0. They are regular, so we have m; - (n”(q)N(p;jlJ:“IN(pl))NB
= ~(p;){[A!-‘+A:-)]a~7)ol(A2)
+A:+‘~~~)O,(A,)+[A~-)-~A(,-)JU!~)O~(A~)}U(P~)+ = fi(d){
-B(42)@qA2)
+Y(42)Q01(A3)
O(k)
-2cr(4’)al7’o,(A,)}u(p,)+
O(k)
*
(4.1)
This equation defines CL,j? and y for arbitrary q 2. Note that a and B are related by
MESON EXCHANGE
21
CURRENTS
the Adler-Weisberger sum rule at q2 = 0, as can be seen from eq. (3.19). The softpion limit is a good approximation for the case we are considering; hence we need only a(O), j(O) and y(O). The corrections for the case q2 w rnz can be safely neglected. From eq. (3.5) we have Hy’ = S’(Cr(l) Xf(2)IjCCcl X@2h+ G2X)GI +Cz(1)-2(2>lj[(bl-~2)(h,+h~PI2)+T~(;’(h,,+h~~P~2)1 +[~:z(1)+z(2)ljC(u1+a2)j,+T,(:)j,,l+HrL),
(4.2)
where Sn = - &40Y2(4
91 = 3MO)Y&), 4 = h = -aw-o~~,)~ h; = - HdO>
h, = h = -3wv2(4,
+ B&01xd4A
where 5 =
k E
z&
Y,(x)= There is no translationally
(4.3)
hi, = - tC40) + B(O)1Y2C4,
Y,(x)
=
e:,
(l+;+;)Yo(.)r
non-invariant
CNj)NL= -[~(1)x2(2)lj2[a(o)+Bio)l
term, but there is a non-local term: ;[(@I 7.
‘P)P2+(a2’
p)Pll (l+
:) yO(xJ* x (4.4)
The PCAC implies that a(0)
=
g_!k (o) CB”N(-% 3 I
p(o) _
gA
““1:,I’ 1 gso)
10’
(4.5)
where the subscript 0 means that v = vn = k2 = q2 = 0. These values have been evaluated by Adler “) by extrapolating from on-shell nN scattering amplitudes. They are given in table 4, row 1, denoted for short as “Adler”. The errors involved in these numbers are hard to assess, but we would guess that they are of the same magnitudes as the errors usually quoted in the test of the Adler consistency condition and the Alder-Weisberger sum rule. We shall now turn to an alternative calculation of the amplitudes which may be considered as some sort of book-keeping of the global predictions of the low-energy theorems. The low-energy theorems can be viewed as sum rules derived from current algebra by insertion of a complete set of intermediate states in the equal-time com-
Total
2.62 0.38 1.51
“) Amplitudes are given in units of
PI
P
OprJ e9
2.13 5.2 1.23
rnse2.
0.509
0.578 -0.0693
N4.3
%.I xSl
0.28K(O)
I
4
0.512
0.442
0.0699
0,47K(O)
2.519
2.31 0.139 0.0699
3MC(O)
- 1.297
--1.123
-0.174
-0.83KiOI
1.158
0.562
0.578 -- 0.087 0.0693
0,7OK(O)
-1.021
-0.442
-0.578 -0.0699 0.0693
-0.75X(O)
between Lagrangian and PCAC methods for calculating the amplitudes Ai*) and
Adler
jes
PCAC
Comparisons
TABLE a(O),
p(O),
y(0)
0.0910
0.0970
0 -0.0180 0.0120
-0.0144
“)
MESON EXCHANGE
23
CURRENTS
mutators “). Cheng and Kim 16) exploited this idea in relation to the AdlerWeisberger sum rule. We shall apply here a similar idea and approximate the pionproduction process by pole diagrams with intermediate states corresponding to nucleon isobars (N*) and vector mesons (V). These are illustrated by the diagrams given in figs. 6a to 6c. Nucleon isobars: We describe the intermediate resonances in particle representation. Effective couplings are used for the vertices nNN* and ANN*, where A is a shorthand notation for the axial current. Coupling constants for the former vertices are determined from available experimental widths and for the latter by using one-pion-pole dominance in the axial form factors. The isobars with angular momentum and parity Jp = (n++)*, n > 0, are described by Rarita-Schwinger spinors z+, . . . ,JP). For n = 0, the usual Dirac spinors are used. We give the rcNN* vertices in terms of the matrix elements of the Lagrangian density ’ “):
<~~)N(P’)I~IN*(P)) [(JP)pp
= gnNN8i%,, =
(n++)*,
’ ( 05
. . . c&+fi(~‘)
n
=
ulrl
. ...,(P),
)
1,2, . . .],
(44
(j;5) U(P), WI)N(P’GW*(P)>= gnNWfi(P’) (4.7)
NJPh*= &'I.
Introduction of isospin is straightforward: let T be the isospin of the isobar, s’ its projection along isospin z-axis and let cc (spherical component) and s specify the isospin projections of rc and N, respectively. Inserting the Clebsch-Gordan coefficient (~$~s~Ts’) in the r.h.s. of eqs. (4.6) and (4.7) takes care of the isospin dependence by making them isoscalars, as they should be. The ANN* couplings involve, in general, a number of invariant operators. In the limit k --t 0, relevant to our problem only one invariant operator survives. We write the vertices in terms of the matrix elements of the axial current:
WP’)IJ?IN*(PD = +d in- ‘aA,,, qlrz. . . qPn fi(~‘) i:5 Q~,,~. . JP) + O(k), (
[J’ = (n+$)‘,
>
n = 1,2,. . .],
(N(P’P?IN*(P))
= tid~(p’)y, [J’
= 4’1.
(4.8)
(‘;) u(p)+ O(k), (4.9)
The requirement of time-reversal invariance implies that all the coupling constants introduced so far must be real. The inclusion of isospin dependence is done in a similar way to that for the nNN* vertices.
M. CHEMTOB AND M. RHO
24
The PCAC hypothesis places constraints on the coupling constants gz that we shall use in the following. They are connected to the coupling constants gnNN, through the relation ’ “)
m,2<+~)Nb’)l~lN*(p>>.
(4.10)
g,(O) 4’ + mi
These Goldberger-Treiman-type
relations may be written in general form
2MgA
92
(4.11)
(gnNN,[JP= (n++)*, n = ‘9 2Y *I ’
f
nNN’
=
gnNN*
[J’
M*fM
*
(4.12)
= +*I.
In particular, they fix relative signs of the coupling constants which would otherwise be difficult to know. We have now the necessary ingredients for calculating the diagrams given in figs. 6a and 6b. We restrict to the lowest-excited nucleon resonances Nz+,(1236, 120), N:+,(1400, 120), and NT- 1(1525,40), where the numbers in parentheses specify masses and partial widths (in MeV) for N* --+N + n decay modes, and the subscripts stand for (2Jp, 2T). Corresponding to the Feynman propagator for spin-4 particles we use the following expression for the spin-3 propagator: (iy * Q+M)-‘,
*Q+Mh
(&,+Y,Y,)/(~
where terms of order Q/M which vanish in the static limit and unknown off-massshell terms have been dropped. The results will be expressed in terms of the contribution to the amplitudes A$*‘. Terms of order q2/M2 appear in the calculations but will not be written down: NT+,:
a(0) = -/3(O) = *y(o) = ;
N:+I: $-’
= A$+’
a(0) =
M
=
M*+M
;$y$, $3
=
2&NN*g: M*2-M2
(4.13)
’
gnNN* d
M(M*-M)
’
P(O) = -y(O) = =p$.
(4.14)
MESON EXCHANGE NT_,.
;I;-’
=
_-)&+’
=
_
CURRENTS
25
2 gnNN*g: ~M*+M'
a(0) = -g(O)
= -*y(o)
= - ;
g=&--.
(4.15)
The numerical values are given in table 4 where the low-energy theorem values are also given for comparison. In the last column of the table, we consider a particular combination of the amplitudes given by [2A~-‘-Z~-‘-3A5+‘] within a factor. This is precisely the combination which determines the GT matrix element of triton betadecay if one restricts to the S-state. Note that the contribution from the N:+, is exactly cancelled. Vector mesons: A strict polology treatment of the vertex correction given by fig. 6c presents delicate points because of the vector-meson propagators (q’+m:)-’ and their non-trivial dependence on the pion mass (-4’). Adler “) has given a general treatment of these vector meson exchange amplitudes by a method implementing the vector meson dominance (VMD) hypothesis with the current algebra. It includes the effects of all vector mesons with quantum numbers JPG = l-+. We shall not enter into the details of his method, but shall simply enumerate some definitions and quote his results. They are strictly valid in the soft-pion limit q + 0, but we shall also retain the q2 dependence originating from the vector meson propagator. The VNN vertex is taken to be
iii(p;)[yJ, -o,,(k-q),f233zeU(Pl),
(4.16)
where fi and f2 are form factors, functions of (k-q)2. The p-propagator has the form [&, + (k - q)l(k - qX/rn$]/ [(k - q)2 + m:]. The matrix element of the axial current between vector meson and pion states is found to be (up to order k) (4.17j
where unknown off-shell terms are neglected’. The factor F, is the quantity used in the field-current identity “) to relate the current Ji.“‘. to the vector-meson fields VA, namely: e.m.
JA
=
F, V,
E
f-6V, .
(4.18)
2Yv
t The result (17) is related through the KSFR relation to the result obtained by the application of the PCAC hypothesis to the V,A vertex. The PCAC constraint specifies the coupling constant gvnn in terms off, 3 -\/2[MgJg,(O)] (pion decay constant) and g vRll (the coupling constant for V -+ nn decay) by the relation gvnn = &!gv,,f,. The KSFR relation gives on the other hand g& = (mdf,)’ which leads to gvnA = [g,(O)/g&f](n?:/2yv).
26
The amplitudes,
M. CHEMTOB
AND
M. RHO
as k + 0 are Ap(q2)
-
g,(O)+
mf)
2M2gA q2 + rnp”
(4.19)
where we have used the hypothesis of p-dominance of the isovector e.m. form factor Fz 2(k2) =f1,2FJ(k2+$). If one neglects q2 as compared to rn;, one obtains Ai-,
_
gr(Oh
Ai-’
-2M29,’
= -
Mc&) (If&/), A
or
a(0) =
do) -(l+Kv),
2M2g,
(4.20)
which are almost identical to the results of the low-energy theorems except for an additional term in 2i-j +. The numerical values corresponding to this assumption are given in table 4, fifth row. If the model used to get eq. (4.19) were correct, then the q2 dependence would just reflect the off-shell extrapolation of the amplitudes Ai-’ and A$--‘. However, even though the q2 -+ 0 limit is in agreement with the PCAC prediction, it is not clear whether the q2 dependence in eq. (4.19) is correct. For example, the hard-pion theories give quite different behaviour, with a disagreement + with the PCAC results in the q2 = 0 limit. As already noted, when transformation made, keeping the factor m~(q2+m~)’ in fig. 7 amounts function e-mn’/m,r by
into coordinate space is to replacing the Yukawa
(4.21) Note that this is dominated by the p-meson term at small r( < 1 fm), and for r >> 1 fm by the pion term: it is negative for r 2 1 fm, and changes sign to positive for r 2 1 fm. This is quite different from what one would expect from taking eq. (4.20). For a sensible nuclear wave function, eq. (4.21) gives expectation values with signs opposite to the PCAC results, and is very sensitive to its inner parts. Thus if the extrapolation from q2 = 0 to q2 # 0 is indeed smooth, then we must conclude that a calculation based on eq. (4.21) 1s . a suspect. It might be appropriate at this point to note that this sort of ambiguities also appears in treating K(q2) extrapolation in coordinate space. 7 Note that the PCAC
value
for Ai-)
= -(gr(0)/MZg,)(l
+~v)+g~g,JM’.
MESON EXCHANGE
CURRENTS
27
(ii) Vector current. Unlike the axial case, the limit k + 0 cannot be taken directly in the case of magnetic moment, since the latter may depend on terms linear in k in the matrix elements of the vector current [see eq. (3.16)]. Regardless of what k takes, we can choose to work with the soft-pion limit (q + 0), in which case, as we have seen in sect. 2, only the generalized Born terms survive. This does not mean that the non-Born terms cannot be given by the soft-pion technique. According to Adler and Gilman la), the terms determined in this way are jp+)
1
~(0)
1
_
g@h
--Tgjr’ _
-x
gr(O)Ks
(4.22)
while the other amplitudes are not determined, since they are associated with the invariants on(Vs) linear in q. But these contributions are already of order MV2 and, in accordance with the general results of table 1, the exchange MM (expressed in units of nuclear magneton) corresponding to these terms is of order Me3. Thus we conclude that the NB terms are small. The question remains, however, as to what happens when q2 moves away from zero. In sect. 2 we have given an argument that for the axial current, an extrapolation to some finite value q2 << M2 introduces only a small modification; but what about the e.m. current? There is a systematic way of resolving this question as discussed by Furlan et al. ‘l), and extensively used by Ericson 19) in photoproduction of pion on nuclei. We shall not do such analysis for the reason that even extrapolated to 4 2%m 2, the NB terms remain small for the isovector e.m. current, and hence an elaborate theory is not warranted. Therefore we shall use models to evaluate them. This is perhaps dangerous for the isoscalar moment for the reason that the NB terms are not small corrections any more. Our analysis for this, therefore, may not be as meaningful as for the isovector case. In the same way as for the axial vector current, we shall assume that the NB terms can be described by the N* graphs, and the vector meson (p, w) graphs. Nucleon isobars: Unlike the p-decay case, coupling constants are not readily available for the phenomenological Lagrangian approach. We shall instead use the celebrated dispersion theoretical method of Chew, Goldberger, Low andNambu2’) (CGLN) for treating the pion photoproduction amplitudes. We take the static limit for the nucleons and the Ml approximation. The latter amounts to neglecting all partial waves other than the magnetic dipole coming from the nucleon magnetic moment and is expected to take into account the contributions from low-lying even-parity nN resonances with (2Jp, 2T) 5 (3+, 3). The procedure goes as follows. We use the expressions connecting the invariant amplitudes V, to the c.m. amplitudes $J. These are given in appendix 1 of Adler’s
28
M. CHEMTOB
AND M. RHO
paper “). We then take the static limit for the nucleon M --, co, and find
(4.23)
v, = - 1 2% Iqllkl
where Ikl and 141are the c-m. momenta of the current and the pion, respectively, and o = W-M, where W is the total energy. From partial wave decomposition of the amplitudes fy, keeping only the magnetic dipole M,j, j = _t, 3, we get J%
3(k*q)M,+
/“j 5?1:-3M,,,
f;
= 2M,,-tM,,,
(4.24)
$1 x 0.
The amplitudes Mzj of course depend upon o. It is a well-known result of the CGLN theory that the solution for M,j is
(4.25)
where h,j is related to the partial wave amplitude for elastic n-N scattering fIj by (4.26) Here, Sj is the p-wave phase shift for j = $ or 3. To avoid double-counting, the Born contributions to Mrj must be subtracted out. However, the same eq. (4.25) which connects the full M,j and hi, also connects their Born parts. Thus the NB part of M,j is connected to the NB part of hj by the same equation. To state the results for the functions 9 and h of eq. (3,5), it is convenient to introduce the following combinations of p-wave scattering amplitudes
MO) = (~11-~1,-~,1+~,,)l,=,, la,(O) = #,I
+2h,3+2h,,
(4.27)
+4h33)lw=0,
where the subscripts on h stand for (2J, 27’). We evaluate these quantities at w = 0
MESON EXCHANGE
CURRENTS
29
since as v, vg -+ 0, o = [2M(v -vg) + A4213- M -+ 0. Explicitly we get h,(O) =
s
2_‘mdp
7T 0 wp2pz
[sin2 6,,(p) + sin2 6, ,(p)] z O.O74,/m~, (4.28)
dp [sin2 6,,(p) ++ sin’ 6, ,(p)] h,(O) = 2 m __ 71s 0 o;p2
w O.O658/m~,
where the numerical values are obtained by putting in the known values for 6,, and the Roper 21) parametrization for 6,, . In terms of A,,, 2T, the desired amplitudes are + V’f’ = 0, $1 =
Jq+’ = vi-’ = 35[h,,+h,3-2h,,]l,,o)
35h, - h~~-hs~+hdo=o,
(4.29)
j7(+) = 35~h11+2h,3+2h3,+4h331lo=o, 4 (E
-2h,,)l,,, = 0, Vi:+’ = Vi-’ = 0; so we need the combinations Since (h,,+h,, h, and h2 only. Now from table 3, the following contributions to the MM operator arz obtained:
91 = %3(r,-y,)h,(0)Yo(x,)m,3,
grr = - f Hy,-
h, = _it = -$ 3(r,-y,)4h2(0)Yo(x,)m;l h
y&(O) Y2(4m~
2
= .h = - f Hr, - ~nPUO)Y2k)m~.
Note that we have not considered N* contributions to the isoscalar amplitudes F’(O), since the dispersion integral cannot contribute much because of the factor &P+y.) and of the fact that only (3, 3) resonance (Roper) is allowed. Vector mesons
(p and CD):The CGLN theory based on fixed momentum-transfer
dispersion relations overlooks a piece of the pion photoproduction amplitude associated with the t-channel singularities. It is known that the neglected part is well approximated by a resonating Jp = 1- wave in the t-channel, provided the Ml approximation is made and the dispersion integrals are restricted to the (3, 3) resonance region. We shall study these effects in an approach similar to the one developed in the axial case. We assume them to be described by the vector meson exchange processes given by fig, 6(c). Both p and o are allowed intermediate states and contribute to the matrix elements of the isoscalar and isovector currents, respectively. The V7ryvertices, 7 Note that P, is proportional to l/w but that, since the associated invariant 0~(Vi) is itself proportional to w (see table l), the contribution On( VI) V1 to the matrix element of the current is regular in the limit 6.1+ 0 as it should be. In view of this fact, there may be a contribution to the MM operator associated with P,. However, the Ml approximation gives exactly P1 = 0.
30
M. CHEMTOB
AND
M. RHO
where V is p or o, are taken to be
<~“(q)lJ;lp;(k’)) = - i F
elw,,, k: qc,a”“,
(4.31) (d’(q)JJj;jlcop(k’))
= -i
gz m,
Earn,,k: q,,@,
where .Q,,~~is the fully antisymmetric tensor (&I234 = + 1). These are the only possible gauge-invariant couplings if gradient terms are ignored. In a recent dispersion theory analysis of the nucleon magnetic moments, made by Abarbanel, Callan and Sharp 22), the p and o pole contributions were included and vertices of the form (4.31) used. This study , as well as the analysis of the inelastic form factors of the nucleon by Walecka and Zucker 23), give us some clue to the signs of the coupling constants. The VNN vertices are written as in eq. (4.16) for the axial vector current W(P;)IJ$NP,))
= @,NNz(Pi)zj
(N(p;)lJ;IN($%))
=
hNN
[
Yp-
2%cjtv(Pi-Pl)v1
u(Pl)9
1
qp;)
(4.32)
44.
Note that gpNN
2fl(o)9 Kv=2Mf&,
=
flP)
wheref, and fiare the form factors used before [see eq. (4.16)]. From radiative widths of the vector mesons r(p --f ny) = 0.5 MeV and r(~ + ~7) = 1.3 MeV, =
gPnY
The one-pole approximation
0.448,
gmny= 0.758.
(4.33)
to the e.m. form factors of nucleons gives gVNN
=
YV,
where yv is the coupling constant in the direct V-y coupling [see eq. (4.18)J. The coupling constants yv can be ontained from leptonic decays of the vector mesons or storage ring experiments or photoreactions on hadrons. There is some disagreement among various experiments. We shall use the values deduced from leptonic decays 24), g = 0.52kO.03, 4n
d = 3.7040.7, 4n
which give gpNN
= 2.56,
gmNN
=
6.82.
(4.34)
MESON EXCHANGE CURRENTS
31
With the vertices eqs. (4.31) and (4.32), the V-amplitudes Y(*“) are easily computed: v;~)
_ --
2
v(+) 1
~(cbJ2pNNl.m,)
t+m,2 qo,
=
_ -
2 G7wnyf2aNN/mm) t+mt,
’ vi+)
%bJ2PNNlmfJ
t+m,2
=
2h7any.f2~NN/mJ p-p)
’
vy’ = 2(9,,, s,dm,) t+m,2
’
(4.35)
t+m2, V;+) _ - %dh.JmJ t+mi
’
’
where we use the abbreviations t = (q-k)‘, f2pNN= gpNN(+/2&f), f2mNN= gwNN (tcJ2M). At the symmetry point and to the lowest order in M-‘, we have j7i.o’ = 2g,NNg,n, 3 mP
vi+’
=
@oNN&ony
(4.36)
’ mS
sign
The gVNN gv.y > 0 is favoured in various analyses 22*23). Referring to table 3, the exchange MM operators coming from the vector mesons are
(4.37)
3 hlI
=
j,,
=
35,
J!!F ( m,
y2(x,),
)
where CV
=
-
h’NN
gVny
d12n.
Let us emphasize that the results (4.37) are simple since q2 is taken to be zero. However, if we were to keep the q2 dependence, we would have a situation analogous to that discussed near eq. (4.21). If one keeps the q2 dependence, then the following modification should be made to eq. (4.37): yo,
2kJ
+
m&
[Yo,2hJ-
(5)3yo,24
*
(4.38j
4.1.2. Born contributions. In principle, the Born contributions to the amplitudes are uniquely defined, as it is through the definition of the Born terms as in figs. 4 and 5 that the non-Born terms are given their precise meaning. However, in the spirit of deriving an equivalent operator which is to be sandwiched between nuclear wave functions, there is a problem of double-counting, which is a unique feature of a many-body system. In as far as the nuclear wave function already contains some parts of meson exchanges through the N-N potential, not all the graphs of figs. 4 and 5 can be ascribed to the additional meson exchange effects we are talking about. This can be best illustrated in terms of the time-dependent picture. The pieces already included in the wave function are given in fig. 8, in which emission and reabsorption
M. CHEMTOB AND M. RHO
32
of pions is completed before or after the interaction of a nucleon with the external field. Subtracting these graphs from the Born terms, one obtains the legitimate nucleon Born terms of fig. 9. One notes that in figs. 9a and 9b a neutron /?-decays (for example) to a proton before the pion gets reabsorbed by the spectator nucleon. Figs. 9c and 9d correspond to NH pair intermediate states. Here of course, the pion can propagate with both positive and negative frequencies and has the Feynman propagator. Fig. 9e contributes only to the isovector e.m. current.
Fig. 8. Time-ordered Feynman diagrams representing mechanisms where the interaction with the current occurs after emission of the pion and its absorption. Together with the diagrams where the current interacts before the pion is exchanged, they represent effects already included in the wave functions.
____ 4I ____ 41 d 1 Tc
a)
cl
b)
/i ____ ____
lT
d)
IT
TL
e)
Fig. 9. The parts of the Born contributions to the OPE process that remain after excluding the unwanted parts given in fig. 8. The time-ordered diagrams (a) and (b) give the nucleon-recoil exchange current, and (c) and (d) the pair-excitation exchange current. The Feynman diagram (e) gives the pionic exchange current.
Because the pion is present in the intermediate states in figs. 9a and 9b, an introduction of such terms as corrections would necessitate a normalization correction also. We do not know at this moment how to treat this satisfactorily, the trouble being that our formulation is not a perturbation theory. (It is important to realize that despite the terminology “Born term”, our Born terms are not perturbative
MESON EXCHANGE
33
CURRENTS
objects.) In perturbation theory, one may be able to formulate a systematic theory to take into account terms such as those given by figs. 9a and 9b and corresponding normalization corrections. These questions are discussed in appendix A from a general point of view. The practical aspects pertinent to our application will be left to the next section. We divide the Born terms into three groups. Referring to fig. 9, we shall call the current associated with fig. 9e a “pionic current”, figs. 9c and 9d a “pair excitation current”, and figs. 9a and 9b a “nucleon recoil current”. Both the axial and e.m. currents will be discussed simultaneously. Most of the arguments having been presented already 25), the discussion will be kept as brief as possible. (i) Pionic current. All the time-ordered diagrams are allowed here so that the Feynman rules are to be used in getting the matrix element of the exchange current in momentum space. There cannot be any contributions to the Gamow-Teller operator because of the Lorentz invariance, nor to the isoscalar magnetic moment operator because of the G-parity invariance. The isovector matrix element in momentum space is J;' =
-” 6(p,+p2+k--pi-p;)
1
-
(P1 -Pi)‘+
(2n)3
1
[~(P;)r5Z,U(Pl>liEj*“[~~P~)~5
x(P,-PP;+P;-P2), b2
-A)’
m,2
+
zn”(P2)lm (403g)
d
After some standard calculations, the two-body MM operator associated with pionic current &?T2 is found to be [in units of nuclear magneton (e/2M)] “&f2 = -+NH) -M 2 z
x ,wd~I
- Vl)(fl2
* V2)k
x
mJ(41~
(4.40)
where
Carrying out the differentiation indicated by VI and V, in eq. (4.40), we find among other terms the well-known Sachs’ space-exchange moment: JZ Sachs
=
&f”l”“[‘(‘)
x +)1&
x R)[(%
. a2)YOk)+S12
y2(xz)l,
(4.41) s12 = 3(c: . P)(rQ * 3)-(61
* u2),
which can be identified with the fourth term in eq. (3.8); i.e.,
In addition to &?sa&, we have other operators which can be written in terms of
34
M. CHEMTOB AND M. RHO
the classifications (3.8) and (3.5), Sr = - 5 ~L%2--x,)Y&,)9 x 4m4~yI(&J~
Fn, = -fAN[:(l) Y,(x) z (l+x)Y,(x)
911= - ~~A~(1 x
+x.)Yo(xx), (4.43)
= (l+x)eG.
(ii) Pair excitation current. The relevant diagrams include all possible timeorderings for the lrNN vertex, but the current is allowed to create or annihilate a nucleon-antinucleon pair only. Thus in writing down the contributions to the exchange current, we can still make use of the Feynman rules except that in the nucleon propagator (iy - Q + M)-’ only the part which propagates with negative frequency must be kept. This part is the second term in
(jy . Q+M)-’
=
_
2+
Q+M _
(74EQ-i7Q
3’4~;;.
i&
Qo-EQ+
4’“).
0
Q-=
This term contributes to both the axial and vector current. The contribution isovector e.m. vertex leads to the large seagull term which is just fig. 5a +, (ff”(q)N(p;)IJ~IN(p,)),*,
+ - $
snjkrk 01,
1 = 1,2,3,
to the
(4.44)
whereas there is no contribution to the isoscalar vertex to the order l/M. On the other hand, due to the fact that the axial vector current of the nucleons has an odd Dirac operator, we have to go to one more power of M-‘, with the result
We shall keep this, since terms of that order also occur in the NB parts. It should be clear to everyone at this point that the exchange currents must be basically different between the B-decay and the MM. The notion which is still invoked, that one can determine one matrix element from another independently of exchange corrections, is clearIy erroneous.
The pair contributions
expressed in terms of our classification are:
axial current 91 = 2hf = 5 ~f&Y,(x.), (4.46) gn = 2hp, = -3i,r1
= - ZfzW
Yz(xJ.
t The equivalence between the seagull term (sometimes called Kroll-Ruderman term) and the pair term in the NR limit can be checked explicitly by noting that the negative frequency propagator becomes - [(/I-l)+@ * Q/iU]/4M + -(/?-1)/4M to the lowest order in M-l.
35
MESON EXCHANGE CURRENTS
vector current
Comparing with eq. (4.43), we see that the sum of R-dependent pionic and pair currents vanishes, i.e. F;;I + Firs” = 0.
terms from the
(iii) Nucleon recoil current. The graphs to be considered are the time-ordered ones given by fig. 8. These contribute in the adiabatic limit, since the nucleon recoil is due to the emission or absorption of the exchanged virtual pion. The matrix element of the exchange current can be calculated in the non-covariant perturbation theory, which we do not reproduce here. We simply quote the results: axial current h, = j, = -+g,
= - 3$,&N
I(( ) Kk)1, & o x, -
-
(4.48)
vector current m,
=2jr=2hr=
-$g,=
m,, = 21,~= 2hn = -2gu
- $L, 2
[K&d-
yq
x
3 (4.49)
2
= - --fn~~ K2(4 7r
where K,(x,) are the Bessel functions of the second kind.
4.2. HEAVY MESON EXCHANGE
CONTRIBUTIONS
It is a formidable task to calculate systematically the exchanges of more than one pion. Such calculations have been performed for the meson theoretic N-N potential with reasonable success. They were, however, limited to two-pion exchanges. The two-pion exchange graphs are usually supplemented by exchanges of resonances such as p, w. Apart from the complexity, clearly there would be a need for care because of the double-counting. There have been calculations in fourth-order perturbation theory for the GT [ref. ’ 6)] and MM [ref. 2‘)I exchange operators, but the reliability of such calculations is not clear. In our approach, we shall neglect the uncorrelated multipion exchanges, but assume as in nuclear-force calculations that the correlated part of multipion
GT
%)Yp m,)/x,
fmr= 3gI, h,, = AI
= -$f&kdl+
hI = j,=
W/M;)
TABLE 5
h
mI = -4
(GT) and magnetic moment in sect. 3
= 3mII =
to the Gamow-Teller
Pair excitation current
Heavy meson exchange contributions
(
2
j,
1
=
defined by eqs. (5), (6) and (8)
=
-
2 K,(X,)
( 1 5
* r2 + Ml)
+
~&)lIY,
&hJ
2C+(1+ZKS)KO(Xp)-lgXpKI(Xp)l
-h,
Nucleon recoil current
(MM) exchange operators
7E
3 0
MESON EXCHANGE
+
z II
uosam-m
4-4
I II
CURRENTS
37
38
M. CHEMTOB
AND M. RHO
exchanges is important. Thus we shall consider only the w- and p-exchanges t and use the vector meson dominance model for their evaluations. The results are given in table 5. A strict adherence to the procedures used in the calculation of nuclear potentials would require that VNN coupling constants and rnv be determined phenomenologitally by fitting the scattering data etc. We have already touched upon this question in connection with the pionic form factor of nucleons. In view of rather large discrepancies between parameters used by different people, we shall not do the (in principle) correct procedure. We shall just use the coupling constants already determined in the previous subsection. Even restricted to the vector meson exchanges, a complete calculation is difficult because of the vector nature of the particles. For the nucleon-recoil graphs (equivalent to figs. 9a and 9b with the pion replaced by the vector mesons), the time-time components of the vector meson vertices give the leading terms, while those coming from the space components are of order M -’ relative to the former. However, our calculation of other contributions in the MM case is not complete for the following reasons. We have disregarded the mesonic-current processes in which the currents interact with the intermediate mesons (i.e. graphs equivalent to fig. 9c with rt replaced by the vector mesons), partly because of the complicated structure of the radiative vertices and partly because the relevant form factors (associated with monopole, dipole, and quadrupole interactions) are not at present available. It is worth while to note here that a part of the contributions from these processes must be related to the NN potentials due to the vector meson exchanges in the same way as the Sachs moment [eq. (4.41)] is related to the OPEP. For the pair excitation graphs in the MM case, both the time-time and space-space vertices of the vector mesons give contributions of the same order. The results given in table 5 refer only to the latter, where only the spin currents were taken (neglecting the convection current part).
5. Applications to 3H and 3He
The isospin doublet 3H and 3He is the simplest nuclear system with all the manybody characteristics, such as off-shell effects, etc. Next to the deuteron, its wave function is the best known, and it has a complete set of experiments we are interested in - i.e., magnetic moments and B-decay. Deuteron possesses only an isoscalar moment, which is too tiny and cannot make a bound-to-bound P-decay transition, making a clear-cut calculation unfeasible. Thus the trinucleon system is better suited + We have also looked into the q-meson exchange graphs but found them to have unimportant effects. They have essentially the same radial dependence as the OPE diagrams but do not compete significantly with them because of the smaller nNN coupling constant [exact SU(3) with a ratio f/d = 0.6 8ives gsNN/ffzNN% 0.31 and the shorter range. They also turn out to be small compared to the vector meson exchange diagrams. The role that the q plays in meson-exchange currents appears therefore to be very similar to the one it plays in the NN potentials.
MESON EXCHANGE
39
CURRENTS
for our purpose. Historically, the anomaly in its magnetic moment was one of the first pieces of evidence of meson-exchange effects 28). Before going into details of calculation, let us review both theoretical and experimental situations. 5.1. MAGNETIC 5.1.1.
MOMENT
AND GAMOW-TELLER
MATRIX
EJ,EMENTS
One-body operators
MM: GT:
CLI= tkp+ Y”)ai+3(YP-Yn)birAi) +orbitaL I-I!” = -gAqr(+)(i),
(5.1)
where 3(r, *XI) =
;*;;;
(.
1
.
The classification of allowed states: (Jp = if, T = 4) of three nucleons gives 10 distinct states ” ) corresponding to the spectroscopic terms: three 2S,, ‘P, and 4D, and one 4P,. If we assume that of all these only the S (full space-symmetry), S’ (mixed space-symmetry), D and T = 4 ( mixed space-symmetry) states are significant which seems to be borne out by most recent analyses 30), then the GT matrix elements and the MM are given in terms of the state probabilities Ps . . . and the amplitudes A s. . . by ,I)
MA
_
-
p$” =
l(3J+41~!“l13WI = J~[P,-~P,,+~P,+~P~+~A,A~],
J2
~(Yp+Yn)(Ps+Ps-P,+tP*)+tPn,
(5.2)
CL? = JL(r, - Y”)(PS - 3Pk + 3PD + 4Pb) - 6PD 3
where all contributions are independent of the radial forms chosen for the states +. The last terms in &$ are the small contributions from the orbital magnetic moment. The isoscalar and isovector moments (ps and pv) are related to 3H and 3He magnetic moments by I+ = t(A3H) + A3He)), (5.3) PV =
+(d3H)
-d3J--Wh
so that P =
Ps-wPv;
2t = +1(-l)
for
3He(3H).
5.1.2. Two-body operators. In evaluating the exchange current correction, we shall not use the full wave function, but retain only the symmetric S-state. This brings with it a considerable simplification in the calculation. t Similar radial forms were assumed for T = 4 and S’ states. Note that breakdown of mirror symmetry which may be due to the T = 4 state or to different state probabilities in 3H and 3He has only a slight effect. We find, for example, that a 10 ‘A difference in Ps between aH and 3He results in a change in pv and ps by only 1 % and 20 %, respectively.
M. CHEMTOB AND M. RHO
40
The spin-isospin part of the S-state wave function is given by
4” = x0(1, 2)~(3),
(5.4)
where m, t are the z-components of the total spin and isospin, respectively. A more explicit form for 4”=* and $m=3 is convenient for calculation;
$Jm=+ = 1-[(+-+)-(-++)I,
42
p=+ = &g<-
(5.5)
+ +)+(+ - +))-(+ + -)],
and similarly for q’=* and $=*, where ( + ) and (- ) represent spin up and down for #I and proton and neutron for q. The simplifications that eq. (5.4) brings are that because the S-state is fully space-symmetric any operator odd under interchange of space coordinates has vanishing expectation values; thus the c.m. dependent operators [eq. (3.8)] do not contribute. Moreover, tensor operators have vanishing matrix elements. Note that the vertex corrections with an intermediate state N:+s cannot contribute in fully space-symmetric states by virtue of spin-isospin selection rules. Unlike the one-body operators of eq. (5.1), the matrix elements of the exchange current operators depend on radial wave functions. There are various forms involving the coordinates rl, rz, rg which satisfy the symmetry. We shall work exclusively with the analytically simplest Gaussian form exp (-const (rF2 + r:s + &}) where rii = Ir,-- rj]. The fact that some of the Born terms and also all heavy meson exchanges are very sensitive to the part of wave functions where rij is small requires that the shortrange correlation be handled with care. Again for simplicity, we shall use the Jastrow type correlation function nix j (1 -e -yZrtjz)* to simulate the short-distance behaviour. For completeness, in appendix B we examine what happens when a correlation function of different nature used frequently in nuclear matter calculation is used. Although this cannot be a quantitative assessment of what we do below, it gives an idea as to how different correlation functions behave at short distances. The complete form of our radial function is then (5.6) where the range parameter c1 takes the value 0.384 fm-’ when fitted to Coulomb energy and is taken in our calculation to be 0.337, 0.384 (CE) and 0.440 fm-‘. The parameter y is interpreted as indicative of repulsive core radius r,, the rough relation being r, z (7,/2)-l. Th e values used in the calculation are r, = 0.0
y=cO
0.2
3.536
0.4
0.6
0.8
1.768
1.179
0.884
fm fm-’
(5.7)
MESON EXCHANGE
CURRENTS
41
We can now write down the non-vanishing matrix elements with the wave function eq. (5.4), and these are given in table 6. They comprise all matrix elements needed, since operators like [r(l) x r(Z)], (crl x e2) reduce to [r(l)-r(2)]j (ul -g2) in the S-state, and all others vanish. Using table 6, and eqs. (3.5) and (3.6), we have
TABLE 6
Spin-isospin
matrix elemknts of exchange current operators for a trinucleon system in a symmetric S-state Operator
MM
U12
Matrix element
[t3(1)-t3(2)l(ul--(T2)3 [r(l) * C9l(o,+o2)3
-16mt
-12m +4m
(al+u2)3 ts(l)t3(2)(uI+ud GT
MM:
GT:
(J/‘“‘] c O,,]~“‘);
-4m
t~(l)--s(2)1+(~1--oz) m = *3,
i
r=
443
+t,
i
where by definition the one-body matrix element in 82) is evaluated with the S-state only. We use (St> = N”Jlf(r
12, r23 prd2gdr1ddr,
dr2 dr3,
and superscripts M and p for magnetic moment and P-decay. The functions g, h, . . . , are given in terms of amplitudes Y and A in tables 2 and 3 for the case of NB term, and in sect. 4 for other contributions. In order to compute j.+, s and fiA, it suffices to collect all contributions to g, h, etc. For example, Sr = (SI)OPE+(gI) HME
= (gI)~~+(gl)~~~+(gI)HME,
(5.9)
and likewise for the others. 5.1.3. Wave function normalization correction. As we have discussed in sect. 4, there is a correction quite distinctive from the two-body operators we have been considering and any other one-body correction such as relativistic corrections etc. -
42
M. CHEMTOB
AND M. RHO
it is the normalization in the wave function due to an explicit dependence on meson degrees of freedom. For the trinucleon system, it adds the following corrections: APG” = I&AZ>, 6n0rm=
(5.10)
where Z =
IfnZNNi;k(Si
’
(
K+] + Ki?,(xz)1>
[
z/c)C”i . uk) KO(x*)-
Sik
II
and (2) means an expectation value which is independent of the spin and isospin projections. Since
we find
There are two points we must emphasize here. Firstly, we are only considering the normalization correction coming from the pion exchanges whereas vector meson exchanges are also included in our calculation. In principle, we could do a similar calculation for the exchanges of other mesons, but it is numerically small and perhaps the meaning of such a calculation is not clear because of the difficulty of extending renormalization procedure to two-particle systems. We find it more appropriate in presenting results to lump the normalization correction into the OPE Born term. The second point is that practical calculation of (2) gives a positive result instead of negative as the interpretation of a normalization would suggest. In other words, it enhances rather than reduces the matrix elements. We also note here that this normalization correction cancels only partially the nucleon recoil term. The exact cancellation occurs only in the extreme case of the neutral scalar meson theory. 5.2. EXPERIMENTAL
SITUATIONS
The magnetic moments are known very accurately and are given by PL(~H) = 2.97893,
p(3He) = -2.12815,
(5.12)
or in terms of the isoscalar and isovector moments, /I;‘~ = 0.426,
pvp = +2.553.
(5.13)
The discrepancies between the experimental values [eq. (5.13)] and the single-particle
MESON EXCHANGE
43
CURRENTS
values [eq. (5.2)] as a function of various P’s are given in table 7, where 6~ denotes (5.14)
&$v = &Py-$?.
The experimental situation for MA is quite controversial, because of the various experimental quantities involved in its extraction. This is seen in the formula
IK”I
~H+“l-lc
_
3
-
(fal+, 2(fr)140,14N-(ff)‘H-r3He C 2(ft)‘~O-rw-(fon+p tI%zi
(5.15)
1
TABLE 7 Discrepancies between experiments and single-particle operator values inps, v and MA for ‘H and 3He Single-particle values
96 94 92.8 92 “) 91”)8 89 d,
4 6 6 6 9
0 0 1.2 2 1 2
0.423 0.417 0.417 0.417 0.410 0.406
2.284 2.249 2.211 2.185 2.183 2.134
1.686 1.663 1.635 1.616 1.617 1.582
Expected corrections
0.003(0.70 %) 0.269(10.5%) -0.001 0.009(2.11) 0.304(11.9) 0.022 0.009(2.11) 0.342(13.4) 0.050 0.009(2.11) 0.368(14.4) 0.069 0.013(3.05) 0.370(14.5) 0.068 0.020(4.69) 0.419(16.4) 0.103
-0.059% 1.306 2.967 4.094 4.036 6.112
Probablhtres . . . Ps, PD . . . are given in %. *) Values in parentheses are % deviations. b, We have used Salgo and Staub’s 35) value IMrpj = 1.685f0.018. ‘) Corresponds to Gibson wave function 30) (setting P+ = 0). d, Correspond to wave functions determined variationally by Blatt and his co-workers 41*6).
which shows that, besides information on (ft)3H+SHe, the super-allowed decay 140 + 14N needs to be used to determine Gv (Fermi constant) and neutron p-decay to determine gA E GA/G,. For all these &cays, uncertainties are also introduced by radiative and Coulomb corrections. The former correction, though small, is not unambiguously known. The major uncertainties lie however in fr values for n and 3H decays. Prev’rous determinations based on the neutron lifetime measurement by Sosnovsky et al. 31), (t,), = 11.7 kO.3 mm, gave IMA12 ranging between 3.30 and 3.17. The more recent value reported by Christensen er al. 32), (t+). = 10.80+_0.16 min, decreases lMA12 down to 2.88. All these estimates are done with the same namely the value quoted by Goldhaber 33), 1132 +90, which is close to w)%x-~Hc~ Porter’s 34) 1137 f 20. In recent re-examinations of the triton /?-spectrum by Salgo and Staub 3‘) and by Bergkvist 3 ‘), the end-point of the spectrum was found to extend farther to the right than in the older data. This has the effect of increasing theft values and therefore of reducing still further lMAi2. Using almost similar inputs they find
(ft)aH= 1159+11 set, (ff)BH= 1143f3 set,
lMA12= 2.84kO.06 Salgo-Staub, IMAI’ = 2.94 f 0.05 Bergkvist.
(5.16)
M. CHEMTOB AND M. RHO
44
A summary of the early and recent status of this important question is given in table 8. The discrepancy between the value of Salgo and Staub and the single-particle operator value is tabulated as a function of P’s in table 7. We also give in table 9 the variations in d(Gibson) G (I~.&,,IM~‘Ioibson)/M~bibson and similarly in G(Blatt-Delves) corresponding to four values of lMAlexpgiven in table 8 and probability amplitudes listed in table 7. TABLE 8 fr values and lMajsB-,3Be determined therefrom W),.+,
(frb0-‘4N
1190237 set’) 1213135 b, 1099&34 “) 1108f16.45 “)
3075f15”) 3127,t31 b, 3075*15 3117f18 ‘)
(J”~)JH-P~~
9A
1132+40 “) 1132&40 “) 1132+40 1159,tll ‘)
1.18 ho.026 1.18 ho.028 1.24 ho.02 1.243*0.011
Pfal
Pw
3.17*0.19 3.25f0.19 2.88*0.11 2.84hO.06
1.78 kO.054 1.80 10.052 1.70 *0.034 1.685f0.0178
“) and b, Taken from ref. 36) corresponding to values respectively without and with radiative corrections. Neutron data correspond to Sosnovsky ef al. 31). ‘) Obtained with the samef-value as in “) but with the new neutron data of Christensen ef al. d, Value quoted by Goldhaber 33) in 1958, close to Porter’s a4) value of Cft)aB = 1137&20 sec. “) New analysis by Salgo and Staub 35) based on Christensen neutron data 32) and their own 3H data.
TABLE 9 The discrepancies
between the various experimental MA’s of table 8 and the values obtained with Gibson and Blatt-Delves wave functions G(Gibson) “)
IK?I 1.78 50.054 1.80 ho.052 1.70 10.034 1.685f0.0178
0.107~0.033 0.120*0.032 0.054+0.021 0.047*0.011
G(Blatt-Delves) ‘) 0.101 kO.033 0.114&0.032 0.048 AO.021 0.042hO.011
“) 1Mil)lGlbao,,= 1.609.
b, IM~‘%,at,-ucrvcs = 1.617.
5.3. NUMERICAL
RESULTS
We now write down the necessary formulae explicitly and evaluate them:
where subscripts n, p, w represent the mesons exchanged, the superscripts the kinds of terms: pionic current, pair excitation current, normalization correction, non-Born
MESON EXCHANGE
CURRENTS
(NB) term, nucleon recoil term etc., p$,yic+pr$-_4M y-$w
- l)YO(x,))
= 0.7257<(2x, - l)YO(x,)),
R
= - [9.897(&(x,))
Pyz= pg
=:
-0.1964(x,
2
K&&j,
= o.01551
‘f,“NN(l +~V)
= 0.3785
= “;
-0.08300(~~ K,(x,)>,
~j~NN(l+~s)(Y&,)) 0
= -0.4523(Y,(x,)),
45
M. CHEMTOB
46 P;‘,“”
6::
=
-4
=
-0.8954(&(x,))
(
yy
AND M. RHO
2
-6%l
1
= ; z&(
Y&J)
+0.1964(x,
Jx%Jl>
K,(x,)),
= 0.03136( Y,(x,)),
recoil 6ASS = ~/.‘NN(Ko(x.)-KI(x.)Ix.)
= O.l019
KI(X,)/JE,>,
6n*rm A,% =
sNB
-_-- 1 &NN
mnm,2[2A~-)-A~-)--)AS+)](Y,(x,))
gA M
A’s-6n
=
-0.0144(Y,(x,,)+,
where the numerical values are obtained from coupling constants listed in table 10. Other quantities needed are the nucleon mass M =,.. 938.9 MeV, uv = 2.70, rcs = -0.12, the matrix elements (,~:l’)~ = 2.353, (,uY’)~ = 0.440. We have taken positive sign for gpny gpNN and gony &NN following Adler and Drell 38), and Abarbanel et al, ““). W e h ave also used $fh,,-4/r,,] = 0.8197x lO-‘/m~ obtained by using the Roper resonance parametriza:ion 21). TABLE 10
Coupling constants used to evaluate eq. (5.18)
?G
P UJ
9flNN
m&feV)
@my
13.64 2.56 6.82
138 750 783
0.448 0.758
0.080 0.0829 0.643
The results for ,u:“,kand SF1 are given in table 11 in terms of the range parameter CL[eq. (5.6)], the hard-core radius r, [eq. (5.7)], and individually for the OPE and HME contributions. For reasonable values of u and r,, a = 0.384 and r, z 0.4, t This formula was quoted erroneously in eq. (5) of our previous paper ‘) and elsewhere 37) with a misprint of a factor of two and a sign mistake inside the brackets. Due to the sign mistake the conclusions drawn there about the NB contributions are not correct.
contains Born graphs+normalization
“Born”
0.61 LE-2 0.544E-2 0.395E-2 0.285E-2 0.212E-2
0.404
0.418E-2 0.398E-2 0.357E-2 0.313E-2 0.276E-2
0.674E-2 0.491E-2 0.308E-2 0.202E-2 O.l34E-2
0.0 0.2 0.4 0.6 0.8
OS32E-2 0.475B-2 0.346E-2 0.250E-2 0.1868-2
0.3128-2 0.334E-2 0.245E-2 O.l77E-2 O.l31E-2
0.384
0.379E-2 O,362E-2 0.326E-2 0.287E-2 0.253E-2
0.2928-Z 0.281E-2 0.256E-2 0.227E-2 0.201E-2
NB
HME
pk2’
0.578E-2 0.42OE-2 0.262E-2 O.l70E-2 O.llOE-2
0.386E-2 0.278E-2 O.l70E-2 O.l06E-2 0.632E-3
Born
OPE
TABLE 11
correction.
O.l70E-1 O.l43E-1 O.l06E-1 0.8OOE-2 0.622B2
O.l49E-1 0.126G1 0.934E-2 0.7073%2 0.549E-2
O.l05E-1 0.892E-2 0.671E-2 0.510E-2 0.395E-2
Total
-0.281B3 -0.784E-3 --0.706E-3 -0.622E-3 -0.547E-3
-0.633E-3 -0608E-3 --0.554E-3 -0.492E-3 -0.435E-3
NB
0.288EO -0.905E-3 0.261EO -0.862E-3 0.234EO -0.774E-3 0.215EO -0.679E-3 0.2OOEO -0.597P3
0.266EO 0.243EO 0.219EO 0.203EO 0.189EO
0.217EO 0.2OlEO 0.185EO 0.173EO 0.161EO
BORN
OPE HME
-OS34E-1 -0.415E-1 -0.294E-1 -O.lQQE-3 -O.l42E-1
-0.4648-l -0.261E-1 -0.257E-1 -O.l73E-1 -O.l23E-1
-0.321E-I -0.251E-1 -0.179E-1 -0.121E-1 -0.856E-2
(2) Pv
0.234EO 0.218EO 0.203EO 0.195EO 0.185EO
-
0.219EO 0.206EO 0.193EO 0.185EO 0.176EO
0.185EO 0.175EO 0.166EO 0.160EO 0.152EO
Total
0.295E-1 0.234E-1 O.l68E-1 0.126JS1 0.975E-2
0.257E-1 0.204E-I O.l48E-1 O.lllE-1 0.855E-2
O.l8OE-1 O.l44E-1 O.l05E-1 0.790E-2 0.603E-2
BORN
NB
-0.388B2 -0.370E-2 -0.332E-2 -0.291E-2 -0.256E-2
-0.352B2 -0.336E-2 -0.303E-2 -0.267E-2 -0.235E-2
-0.27lE-2 -0.261E-2 -0.237E-2 -0.211E-2 -O.l86E-2
OPE
-
HME
-0.233E-1 -O.lSlE-1 -0.129E-1 -0869E-2 -0.619E2
-0.202E-1 -O.l57E-1 -O.l12E-1 -0.757E-2 -0.538E-2
-O.l4OE-1 -O.lOYE-1 -0.784E-2 -0.529E-2 -0.374E-2
Sk"'
0.230E-2 O.l%E-2 0.631B3 0.104E2 O.lOOB-2
O.l96E-2 0.1358-2 0.5348-3 0.879B-3 0.826E-3
O.l24E-2 0.827E-3 0.276%3 0.49633-3 0.431E-3
Total
to magnetic moments and CT matrix element in trinucleon system calculated using eqs. (5.17) and (5.18) as a function of range parameter a and hard-core radius r,
0.0 0.2 0.4 0.6 0.8
--
0.337
0.0 0.2 0.4 0.6 0.8
(fmy-l $I)
Exchange contributions
CA
2 P g 3
8
5
2
5 g z x”
48
M. CHEMTOB
AND
M. RHO
our results are &?’ = 0.193(+::;::), ‘@ = 0.0093( :;:;;33>, sa”) = 0.053(+;:&)
(5.19)
%,
where we have indicated in parenthesis the range of values they can take when c1and r, are varied. We may summarize the results in the following way. pv(2)** The low-energy theorem implied that the pion-exchange contribution is dominated by the Born term. This is confirmed explicitly. Even taking some sizeable errors in the method of calculation, the NB term is definitely negligible, even less than 4 % of the Born term. The heavy meson exchange is small (about 10 % of the Born term), but needs to be treated carefully since it subtracts. As one can see from eq. (5.18) there is a large cancellation between p- and o-terms; thus the result could be quite sensitive to the relative size of the coupling constants gpNNand gwNN. In our calculation g~nN/g& M 7. However, the more recent value quoted by Ting 39) has the ratio 8.6-J 1.0, while Partovi and Lomon 40) prefer in their nuclear force calculation a ratio of about 12. Furthermore, the normalization correction which we made only for the pion may be necessary for other mesons if more accuracy is needed. Thus there is room for improvement in the HME contributions. It should be emphasized that the normalization correction has roughly the same magnitude as the OPE contribution, but is more strongly suppressed when rc is increased. (2)* Ins * Clearly the low-energy theorem is useless here as all the model-dependent corrections are as large as the Born term. The non-Born term coming from the vertex correction due to the p-meson is of about the same size as the Born term. Here not only the magnitude but also the sign of the coupling constants gpNNgpny gnNN are uncertain. Although the positive sign is favoured by other evidences, there is no fundamental reason why it cannot be negative. If it is negative there will be almost complete cancellation within the pion-exchange term, leaving everything to the HME contribution. The same discussion as for the #’ leaves ,up’ in considerable doubt. Clearly more refined and if possible model-independent calculations are called for. 62’: The Born terms are small as expected, but in principle the NB term could be substantial. In triton p-decay, the latter calculated in S-state is very small because of the fact that Nz, 3 cannot contribute. Table 4 shows that the Adler values are mostly accounted for by Nt,3, the remaining part presumably coming from NT, 1, N:, 1 and p-terms. The small NB contribution is most probably a fortuitous situation due to the cancellation of N& terms, typical of the S-state of triton, but perhaps the situation could be different in other systems. Because of this cancellation, the OPE term is as small as the HME term, with the resulting value consistent with zero exchange contribution.
MESON EXCHANGE 5.4. COMPARISON
WITH
CURRENTS
49
EXPERIMENTS
In table 7, we have given the expected corrections for different choic.es of state probabilities (S, S’ and D). Let us compare these values with our results for #‘, ,@ and My’ given in table 11. For the isovector moment and the GT matrix element, our results would be compatible with the wave functions with anomalously small PD.The Blatt-Delves 41) wave functions (P, = 9 %, Ps8= 2 o/,), for example, would require much larger corrections than our calculated meson exchange corrections would provide. With the Gibson wave function 30) (P, = 6 %, Ps = 2 y*), the corrections needed are smaller. With the GT matrix element, and isovector and isoscalar magnetic moments, we have three linear relations for the state probabilities:
& =
3typfYn)(Ps+PS-PD+tPt)+~P,+P,~~2’,
CL,
~(Yp-yn)(Ps-3Ps+3Pn+gP~)-ifPD+Ps~L:2!
=
(5.20)
MA= J3(p,-3P,,+3P,+3P,+3A,.A)+6~f). One may ask what conclusions about the state probabilities can be drawn from our A ; pv and MA appear in a similar combination of P’s: therefore values of ,ut\ and 8”’ we may consider ps and pv only. Neglecting P+ z 0.25 % which has negligible effect, wefindwith PD+Ps+Ps = 1,
PD = 3(1-4, Ps*= $[l-b-+P,],
(5.21)
where (5.22) Note that whereas Ps depends on the isovector and isoscalar moments, PD depends only on the isoscalar moment. Thus, a precise calculation of pi2’ is needed since it would unambiguously fix Pn. (However, the isoscalar moment is a difficult object to calculate.) Taking our calculated values for ,u$, the solutions of eq. (5.21), for the range of a and r, considered, are Ps.zz0,PD 5 6 %. 5.5. REMARKS
To summarize, we have found that our calculated values for $\ are consistent with the wave functions different from what the presently accepted nuclear force dictates; yp’ directly gives PD , but its calculation is model-dependent and uncertain because of ambiguities in the coupling constants and normalization correction; &’ is fairly model-independent, although the exact magnitude can also be affected by the model-dependent correction terms. As for the Sy), the corrections considered are small, consistent with zero. As one can see in table 12 there are, however, some differences between the PCAC and the phenomenological Lagrangian methods. If in the latter one treats the ~-propagator as suggested by the PCAC [i.e. replacement
50
M. CHEMTOB
AND M. RHO
TABLE 12 Comparison
between the PCAC and phenomenological Lagrangian methods for 82) (in %) calculated with the S-state with c( = 0.384 fm
rc
0.0 fm
0.4 fm
Adler
-0.35
-0.30
0
-i
w.
‘aI c
WN.1
0 0.29
s .$! ;$ E&I 23 2 a
%, 1 P”)
-0.44 2.4(-3.7)
0.25 -0.38 2.05( -0.64)
Total
2.4(-4.0)
1.9 (-0.77)
3
“) The numbers given outside of the parentheses are calculated by setting q2 = 0 in the p-propagator whereas those in the parentheses by taking the q2 dependence into account.
of l/(q2fm~) by l/nzz], the difference in the two methods is substantial. However, if one keeps (4’ +mi)- 1 in Fourier transforming, the p-contribution becomes negligible for r, w 0.4 and the net result is small, close to the PCAC value. Since a 3-5 % reduction is expected from the relativistic corrections to the GT one-body operator 42), a still larger exchange correction than given in table 11 would be needed to eliminate the discrepancy. There are a number of effects that can resolve the discrepancies. Firstly, although the D-states may enter with small probabilities, they can compete substantially through the S-D matrix elements. The reason is that the S-D matrix elements can receive contributions from the vertex correction with an intermediate N:, 3 and in general from tensor-like operators [see eq. (3.6a)]. Recent calculations by Blomqvist 43), and Brown and Riska “) which include the D-states by approximate methods indeed suggest the importance of the S-D terms. Secondly, a possibility of having a substantial contribution from three-body currents cannot be excluded as suggested by the calculation of the isovector MM by Padgett et al. 44). Finally, there is the dependence of the two-body operators on the shapes of the radial wave function and on the types of core correlations (soft-core or hardcore). The Gaussian function is perhaps not very adequate in view of its too rapid fall-off. The dependence is insignificant for two-body operators of long-range character, but may not be for operators of the forms [Y,(x,)-(m,/m,)3 Yo(~,)] and eir matrix elements depend also on whether softor hard-core &(+&(Qlx,ITh correlations are used. These matters are further discussed in appendix B. We are very grateful to Professor G. E. Brown for numerous discussions on the subject and for having made possible the completion of this paper. One of us (M.R.) has benefited largely from conversations with Professors M. Ericson, T. E. 0. Ericson, A. M. Green and B. W. Lee. He also wishes to thank Professor J. Prentki for the hospitality at the Theory Division, CERN. The other (M.C.) is grateful to the Centre
MESON EXCHANGE
CURRENTS
51
National de la Recherche Scientifique (France) for financial support and to the Niels Bohr Institute for its warm hospitality. He also thanks Dr. D. 0. Riska for useful discussions. Appendix A MESON
THEORY
ln order to have a concise definition of the two-body electromagnetic or weak exchange operators in meson theory, one may follow the same line of reasoning as in the definition of the two-body nuclear potential. Some approximation scheme is set up for the evaluation of the field-theoretic S-matrix in the scattering of two nucleons involving interaction with an external field (here, absorption of photon or /?-decay). The hypothesis which is then made is that the mutual interaction energy of the nucleons and their interaction energy with the external field have representations in terms of non-relativistic, energy-independent potentials. The identification of transition amplitudes obtained in the meson theory description and the non-relativistic equivalent operator description give then prescriptions of how to define the latter. The non-unique character of the prescriptions arise, in general, from the different choices that can be made for the representation space of the two-nucleon system. The method used in this work, called the S-matrix method, rests on the representation of the two-nucleon states in the unperturbed Fock basis. It has some specific features which are not shared by others, the unitary transformation method being one example. We shall, therefore, sketch them briefly and refer the reader interested in more details to ref. “‘). Consider the decomposition of the field theory Hamiltonian of the two-nucleon system H = Ho + V, + V, into parts associated to the free fields (Ho), to the pionnucleon interaction (V,) and interactions of the pion and nucleon fields with the external field (V,), all taken to be additive. In the quantum mechanical description the interactions of the nucleons are represented by a potential. We denote its stationary states by 14,). The index a is a label for the state of the two nucleons. If 1Y,) denote the corresponding stationary states of the field-theory Hamiltonian H, it can be shown that 14,) are just their projections on the meson vacuum. More precisely, one has a
where a = E-Ho, A is a projection operator off the meson vacuum and Ri is the reaction matrix associated to V, but with all intermediate states containing at least one pion or NN pair. The transition amplitude between states labelled by m and n due to one interaction with V, can be written as
52
M. CHEMTOB
AND M. RHO
The first equation is the definition of H,, the interaction energy of the system with the external field. In the second equation, the first term stands for the normal onebody operator and the others for the operators describing virtual meson effects. The projection operator ll shows why processes described by the diagrams of fig. 8 had to be excluded. In the perturbation theory expansion of eq. (A.2) there are a number of terms representing self-action effects which we assumed are already taken into account in using the renormalized forms of the one-body operators and in evaluating the twobody diagrams in renormalized perturbation theory. In principle renormalized quantities are here “system” quantities in the sense that they depend on the state of the system as a whole even though they may refer to the properties of a single nucleon, such as mass, form factors, etc. It was, however, shown by Frantz 45) within the framework of the static model that this system dependence disappears when the adiabatic limit is taken. There is still another important feature of this approach which deserves some attention - the wave function normalization correction, The state vectors lYE) as given by eq. (A.l) are not normalized. Assuming 14,) to be normalized then the normalization is taken care of by the substitution l!P#) -P 1Y,)/Z~* where 2L3 = (c#J..J!P,).From eq. (A.l) we get (A-3) which, if solved to lowest order in V,, gives
(A-4) The one-body parts of the operator V,(LI/U)~V, are assumed to be incorporated into the renormalization effects mentioned above. The remainder is a two-body part which can be given an operator representation by using essentially the same methods as for the calculation of nuclear potentials. By taking the static limit and transforming to coordinate space, we fmd
= 1+ -$&(l)
* z(2)] [(CT1* u2) (K&J-
K+)
+f31z K2(x.)]
3
(A.9
where the derivative is with respect to x,. The correction to the matrix elements <~,lH,l~,> amounts to multiplying them by the factor 12~Z~13, where ZL(ZA) denote the expectation value of eq. (A.5) in the state m(n). In a second-order perturbation treatment of the normalization correction, it seems adequate to introduce the correction only for the one-body operator, in which case it just adds on the r.h.s. of
MESON EXCHANGE CURRENTS
53
eq. (A.2) the term
m9
11.
<&II KlUC(Z:,ZP--
Meson-exchange effects reflect the extent to which meson degrees of freedom are eliminated from the nuclear wave functions or, stated equivalently, the extent to which more diagrams appear in a field-theory description than in a non-relativistic description. The normalization correction eq. (A.6) accounts for these effects as much as the two-body operators. To conclude, we should mention that the question of normalization has raised much controversy in the past in relation to the definition of nuclear potentials. The difference between the fourth-order parts of the Taketani-Machida-Onuma (TMO) and Brueckner-Watson (BW) potentials lies precisely in its omission in the BW potential 46). The question is not whether it should be included or excluded but rather how it should be handled. The procedure that we used stays essentially on perturbation theory arguments. It is therefore not very satisfactory, but can perhaps be trusted if Z: stays close to one. This turns out to be the case in our applications to the three-nucleon systems. Appendix B TRINUCLEON WAVE FUNCTIONS FOR THE S-STATE
We briefly discuss here other forms of wave function which are available and how these different wave functions modify our results. In our calculations, we have used the Gaussian form for the radial wave function of the S-state. The others frequently used are the exponential, Irving, and Irving-Gunn wave functions 6), all of which fall off somewhat less rapidly than the Gaussian form. We have calculated the matrix elements of Y,-,(x,), [K,(x,)-K,(x,)/x,] and YO(xP) with the Irving wave function (multiplied with a Jastrow factor) and found that they were not significantly different from those of Gaussian wave function over the range of r, we have considered. Let us now examine the correlation function. We choose a hard-core function frequently used in nuclear matter calculation. Assume that we can write
f(r 12,
r13)
rz3,
=
~e-~2f(~122+r13’+r232)
iIJ(1-4*j).
(B.1)
which seems to be valid when on- and off-shell “defect” wave functions are taken to be the same, and
= 1 =
0
for
c 5 rij 5
for for
r,j < C,
d, (B-2)
rlj > d.
Here, d is the separation distance which is about 1 fm for acceptable potentials, and c is the hard-core radius of order of 0.4-0.5 fm. For the CIgiven by Coulomb energy
54
M. CHEMTOB AND M. RHO
a = 0.384,eq. (B.l) and eq. (5.6) (the soft-core), when properly normalized, are found to give the same matrix element of Ye(x,) for each c = (742)-l if we choose d = 1 fm. For our discussion we may take this as some sort of normalization. Then we have evaluated the matrix elements of [K&V,) -K,(x,)/x,] and Ye(x,) with both wave functions, and plotted them as a function of c = r, = (~42)~~ in fig. 10.
0
0.1
0.2
0.4
0.3 c:r,
0.5
0.6
0.7
in fm
Fig. 10. Plot of the variation of the expectation values of exchange operators having short-range character as a function of core radius. Comparison is given between soft-core and hard-core correlations built with Jastrow factors into the same radial function: Gaussian exp [-+$aZ(r~2+r:3 -&)I with a = 0.384 fm-‘.
Note that the correlation function eq. (B.2) suppresses their matrix elements more than the soft-core factor does. As a consequence, it is possible that our calculations with the soft-core factor obtain somewhat overestimated values for the nucleon recoil, the normalization correction, and HME terms, the bulk contributions of which come from the inner part of the wave function. References 1) 2) 3) 4) 5) 6) 7)
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MESON EXCHANGE 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23) 24) 25) . 26) 27) 28) 29) 30) 31) 32) 33) 34) 35)
36) 37) 38) 39) 40) 41) 42) 43) 44) 45) 46)
CURRENTS
55
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