PHYSICAL REVIEW C VOLUME 47, NUMBER 4 APRIL 1993

Radiative corrections to the nucleon axial vector coupling constantin the chiral soliton quark model

Ian DuckBonner Nuclear Laboratory, Rice University, Houston, Texas 77251

(Received 4 May 1992)

Second-order radiative corrections to the nucleon axial vector coupling constant from gluon, pion,and sigma meson exchange are calculated in the chiral soliton quark model. Many apparent processesare found not to contribute. The soliton is elastically decoupled from meson radiative corrections whichare dominated by a gluon exchange contribution equivalent to a gluonic hybrid component of the nu-

cleon. A 30% radiative reduction of the axial coupling strength is indicated.

PACS number(s): 21.60.—n, 12.40.Aa

I. INTRODUCTION

Two recent major developments make it important toreexamine critically the quark model nucleon structurecalculations, especially their predictions for the axial vec-tor coupling constant. The first of these developments isthe EMC experiment [1] which radically contradicts thenaive quark model expectation that the angular momen-tum of the nucleon is in large part due to the quark spins.As is well known, the EMC analysis leads to a nucleonquark spin close to zero [2]. In a second development,Weinberg [3] has raised the question of the reason for thesuccess of the quark model in predicting the nucleon axi-al vector coupling constant G~ (= 1.254). In quark mod-els of the nucleon, there are two (at least) competingpoints of view, characterized as the constituent quarkmodel [4] and the current quark model. In the constitu-ent quark model, quarks are dressed in a chiral phasetransition and acquire a mass taken to be about one-thirdthe nucleon mass. Nucleon structure calculations theninvolve weak binding of these constituent quarks [5]. Inthe current quark model, exemplified by the MIT bagmodel [6], relativistic quarks are confined by an ad hocprescription and subject to meson interactions to restorechiral invariance violated by the confinement mechanism.Even here there is a multiplicity of opinion on, for exam-ple, the degree of dominance of the meson interactionsand even on the necessity of including quarks in the firstplace [7].

In spite of the ambiguities, we choose the currentquark point of view because it is a Lagrangian-basedmodel, including basic QCD degrees of freedom supple-mented with various meson interactions, for which wecan do field theoretic perturbation calculations and,hopefully, calculate corrections to the model predictions.Furthermore, the relativistic chiral soliton quark model iswell developed, especially in the work of Birse and Baner-jee [8], and it is remarkably successful in its predictions ofnucleon parameters including mass, magnetic moment,axial, and pion couplings. It is the quantitative success ofthis model which we examine by calculating perturbativecorrections to the axial vector coupling constant. Weconcentrate specifically on the axial vector coupling, but

radiative corrections to the magnetic moment are also in-teresting. As we will see in the conclusions, perturbationtheory is quite encouraging. It gives sensible resultswhich appear to converge where testable, but show needof improvement where otherwise suspect.

We will make frequent reference to the MIT bag mod-el, whose relevant quark wave functions are listed in Ap-pendix A. Together with the prescription of the SU(6)spin-isospin symmetric wave function for the (XA )

baryon multiplet [9], it produces an axial vector couplingconstant

G&(MIT)=(P1~o r ~&T)g&(q)

=—,'gz (q) = 1.09,

in terms of the 1Squark mode value of the axial coupling

g„(q)= f (U —V /3)d r =0.653 (1.2)

L (x)=q(x) [iy„R+g [o(x)+ir ~(x)y, ]]q (x)

+,'a„~(x)a~~(x)+-,'a„~(x) a~~(x)

—A, /4[cr(x)2+m. (x) —f ] —f m o (x) .

(1.3)

We refer to the extensive discussion of the meson-quark soliton contained in Birse s paper for technical de-tails, and just briefly describe the results. The equationsof motion for the sigma-pion-quark system separate forthe hedgehog configuration where the spin and isospinare in the radial direction. The resulting soliton is a su-

for zero quark mass. G„(MIT) increases to 1.25 for aquark mass equal to 1/R (=250 MeV for R =0.8 fm),and to —', if the quark mass is taken large and the relativis-tic current quark model is replaced by the nonrelativisticconstituent quark model. Clearly, there is no di%culty inchoosing a quark mass to produce the required result forGz. Vfe ignore this tactic, however, and use a masslesscurrent quark as a QCD degree of freedom interacting ina chirally symmetric way with the sigma-pion mesonswhich occupy all space in accord with the Lagrangianused by Birse [10]

47 1751 1993 The American Physical Society

1752 IAN DUCK 47

perposition of degenerate states with total angularmomentum J equal to total isospin I, with J, +I, =O.The N(J =

—,', I =—,'

) nucleon and the b,(J =—', , I =—,'

) 7rN

resonance dominate the hedgehog soliton. Matrix ele-ments of observables, including the axial vector couplingconstant, must be evaluated by projecting the hedgehogstate onto the appropriate (J,I) component. Birse devel-ops the technique formally, and the intuitive results ofCohen and Broniowski [11]are very helpful also.

The hedgehog soliton is characterized by radial profilefunctions corresponding to (U, V) of the quark states inthe MIT bag, and a sigma meson profile g(r) and pionprofile h (r). These are reproduced from Birse's results inFig. 1. The key features of these profile functions are thefollowing.

(1) Asymptotically g(r)= —f (the pion decay con-stant equal to 93 MeV), and h (r) =0, corresponding to anasymptotic, external quark mass M~=g ~ f and thecoupling parameter g =5 chosen for a fit by the nu-cleon parameters. In Birse's calculation, this asymptoticquark mass is responsible for a pseudoconfinement of thequarks so that bag confinement is rendered irrelevant.(Note, however, that even though confinement is ir-relevant for the soliton ground state, it cannot be ignoredwhen discussing excited quark states or gluon states. Inthese situations the soliton binding must be supplementedwith a confinement mechanism. Also in so-called weakcoupling, or one-phase, soliton models, the quarks are notbound by the mesons and must be confined by an MITbag or some other mechanism. )

(2) A sigma field that turns over from f at large r to-+f„at r =0, with g (r =0.5 fm) =0 corresponding to aninternal quark mass that is zero at r =0.5 fm, giving riseto a relativistic quark bound state with Dirac radial func-tions ( U, V) similar to those of the MIT bag model.

(3) A strong potential which constrains o2+~2 to be

practically equal to f„. We will make extensive use ofthe soliton calculations as a zero-order result upon whichwe base our perturbative corrections.

The axial vector current of this Lagrangian is

A"=qy"y —q +o.B"m.—m.d"cr .52 (1.4)

The axial vector coupling constant defined by the matrixelement

(1.5)

gets quark and meson contributions which must be calcu-lated from the hedgehog configuration by projection ontothe proton spin-isospin eigenstate. The result followingfrom Birse's Eqs. (A10) and (A18) is a quark contributionthat depends on the mean number of pions in the soliton,X, approximately as

G„(q)=—',g„(q)tl —,', N + —,"—,, N (1.6)

evaluated with the soliton profile functions for sigma andpion. Birse s soliton configuration is conveniently sum-marized by the profile functions

There is also an implicit dependence upon the mesonconfiguration, hidden in the quark wave functions U andV determining g~ (q). Birse obtains a value N = 1 and a

7%%uo reduction of the quark contribution to G~ from thenominal bag model result.

The sigma-pion fields in the soliton also contributedirectly to G~. The projection of the hedgehog matrixelement of this piece follows from Birse's Eq. (A37) as afactor of (

——,'

) to give

Gs (meson)= ——'(2 f A (z, 3;meson)dsrl

o (r) =f„cosP=g (r)

and a radial-isospin pion soliton

m(r)=rf sin(}I)=h (r)r

where the profile angle

P(r) =~ tanh(ar)

(1.8)

(1.9)

(1.10)

varies from zero to ~ for 0 & r & infinity, with a =0.916for a radius 0.8 fm. The sigma-pion contribution to G~ is

G~(meson)= —, o m"z —n"z (T d r=2dz dz

f — h (r)r'dr9 dr

FIG. 1. (A) Birse profile functions for the hedgehog pionh =f„sin(t) and for the sigma meson g =f cos(t) with$=2r tanh(0. 916r}.(B) Birse quark radial wave functions U andV (solid lines) and MIT bag model, massless quark radial wavefunctions (dashed lines) for a bag radius 0.8 fm.

which vanishes for a constant sigma field. In terms of theangle P

Gz(meson) = f f sin~(}}2 r dr .16m 2 . p d9 dr

(1.12)

Birse obtains Gz(q)=0. 98 and Gz(m) =0.81 for a veryunsatisfactory result. It was conjectured at the time thatradiative corrections might improve the model predic-

47 RADIATIVE CORRECTIONS TO THE NUCLEON AXIAL. . . 1753

tion, whose possibility we examine here. In a subsequentextension of the chiral soliton model to include vectormesons, Broniowski and Banerjee [12] obtained goodagreement and a much reduced meson component. Sternand Clement [13] have also achieved an improved fit us-

ing different parameters for the sigma and a semiclassicalprojection of the hedgehog.

Before we describe the radiative corrections to themodel Gz, let us discuss further the role of perturbativecalculations in this problem. There are essential featuresof the problem —confinement and the spontaneously bro-ken chiral symmetry of the vacuum state —which must beincorporated in the basis states on which the perturbationtheory is erected. A rigorous theory of meson field Auc-tuations on the soliton background [14] seems prohibi-tively difficult even in the simplest model problems. Theevaluation of radiative corrections for meson field modesother than simple plane waves is an intimidating task.Nonetheless, it is interesting to make even the most naivestart at evaluating corrections to the underlying solitonmodel if for no other reason than to see perturbationtheory fail explicitly, and then perhaps to fix it. Further-more, many perturbative processes are not contained atall in the soliton results, so that even a crude estimate oftheir importance is useful. For example, radiative pro-cesses with quark excitation, or with antiquarks, gluons,L, = 1 sigmas and I, =0 pions, various non-wave-functiontime orderings, and so on are not included in the solitoncalculation. The perturbative processes which are includ-ed in the soliton results can be identified and require spe-cial attention. We consider all but these "double-counted" processes to be legitimate radiative correctionsto the soliton results. The double-counted perturbativeprocesses are among the largest, but an improved pertur-bation scheme on the soliton background eliminatesmany of these contributions. The reason is that the soli-ton configuration is "immunized" against elastic radia-tive processes. The source term for radiative modeswhich leave the soliton unchanged are just the solitonequations of motion, which vanish. This conclusion ismade clear in Eqs. (36)—(39) of Ren and Banerjee [14].The radiative processes that remain necessarily have exci-tations of the internal degrees of freedom of the soliton.However, no calculations exist on excited states of themeson-quark soliton although the required equations arealso presented in Ren and Banerjee. Even if we were tosolve these equations for excited states we would have toinvoke a confinement mechanism for sensible results forquark and gluon modes. We avoid this quagmire in ourfirst explorations of the problem and, without furtherado, evaluate such contributions using MIT bag modes.For reference we first state results of meson radiative pro-cesses calculated with the MIT bag as a source, and theneliminate those terms which are decoupled from the soli-ton. In the event that the perturbative predictions weredisastrous, we would be provided with strong motivationto face the real difficulties without compromise. In fact,the perturbative results seem well under control. One isreminded of the similar experience in nuclear matter cal-culations, where a modified perturbative approach hasled to an understanding of the properties of the strongly

TABLE I. Contributions of amplitudes of Figs. 2(A) —10(B)to dG„ /G~ and dZ sq of Eq. (2.3). The QCD couplinga=g /4~=0. 8. Small quantities in parentheses are contribu-tions which survive elastic decoupling of mesons from the soli-ton, described in Sec. III E.

Amplitude

(2A)BCD, E,F, GHIJKTotal 2 (TE)3ABCTotal 3 (TM)4D

FTotal 4 (Coul)5BC,D

Total 5 (Nh~)6ABCDTotal 6 (vr, L =0)7A, BC, D

FGHTotal 7 (m.,L =1)Total 6,78A, BC,D,ETotal 8 (u, m)

9C,D, G

FH, IJKTotal 9 (0L =0)10ABTotal 10 (o.,L =1)Total 9,10

dGA/GA

(0.653)—0.043+0.017—0.004—0.0008—0.005+0.036—0.067—0.066a+0.0012+0.00006—0.0014—0.00015a—0.0225+0.0165—0.0199—0.0259a+0.019(0)+0.132(0)+0.057(0)+0.208(0)—0.003+0.015—0.006+0.008+0.014+0.002 (0.0007)+0.013 ( —0.008)+0.031 (0)+0.018 (0)+0.057 (0)—0.004+0.127 ( —0.011)+0.141 (0.003)+0.232 (0)+0.049+0.281 (0.049)—0.0009 (0)+0.0012—0.0017—0.0863 (0)—0.0050+0.0028—0.0899 ( —0.003)—0.0023—0.0003—0.0026—0.0925 ( —0.005)

dz s9'

(1.0)0.1290.0860.004

0.0167

+0.236a0.00180.0002

+0.002a

+0.173(0)

+0.102(0)+0.275(0)+0.024

+0.020

+0.044+0.0095

+0.027+0.037+0.081

+0.0019

+0.0069+0.0088+0.0118

+0.0118+0.0206

interacting many-body problem.In the following sections we discuss in order (1) gluon

exchange-, (2) nucleon-delta multiplet pion exchange-, (3)other pion exchange-, (4) sigma-pion transition-, andfinally (5) sigma exchange-radiative corrections to the axi-al vector coupling constant. All calculations are based on

1754 IAN DUCK 47

ordinary time-dependent perturbation theory using ener-

gy denominators for propagators, and all time orderingsfor the two strong interaction vertices and the one weakinteraction vertex. In a straightforward way, many suchprocesses are recognized to make contributions that canbe ignored because they are effectively absorbed intomodel wave functions. We omit as many calculationaldetails as possible and simply enumerate the perturbationterms via topologically distinct Feynman diagrams, andsummarize the numerical results in Table I. Section IIcontains the gluon radiative corrections; Sec. III containsthe meson radiative corrections calculated with the MITbag quarks as meson sources, and finally eliminates dou-ble counted meson corrections which are decoupled fromthe soliton; Sec. IV discusses other corrections and otherwork; Sec. V is a summary and conclusion.

II. GLUON RADIATIVE CORRECTIONS

A. TE gluon exchange

Following the work of Close and Horgan and others[15], we only brietly describe the elements required forcalculating the gluon radiative corrections. The pertur-bation Hamiltonian is

c

E F

FIG. 2. Feynman diagrams for quark matrix elements ofoperator X. (A) the nominal matrix element, and (B)—(K) aretransverse electric (lowest gluon bag mode) radiative correc-tions. (B) and (C) with no quark excitations from the 1S modeare gluonic nucleon contributions. In (D)—(H) heavy lines are2S quark excitations. In (I)—(K) heavy lines are 1P,&2 and 1P3/2antiquarks. All possible time orderings of vertices and all spec-tator interactions are implied. (L) and (M) make no contribu-tions.

H= —ggcD f qG y" .q d r

2(2.1)

Gz +DG& =(Gz +dG~ )/(1+dZ sq),

leading to a complete correction

DG~ /G~ =dG~ /G~ —dZ sq,

(2.2)

(2.3)

keeping terms to second order in the strong interaction.Note that a conserved vector current gets, term by term,corrections dGV/Gv equal to dZsq and no radiativecorrection, so DGz/G~=O as required. For the axialvector current which is only partially conserved, this can-cellation does not occur exactly and the problem in per-

with gQCD/4~=a, the strong coupling parameter anda= 1, . . . , 8, are the Gell-Mann SU(3) (color) ma-

trices. The quark fields q will be limited to low-lying bagmodes with quantum numbers 1S, 2S, 1P»2, and 1P3/pfor quarks and antiquarks. Our discussion will focus onthe lowest L =0 TE (i.e., magnetic dipole) gluon whosebag mode wave function is also given in Appendix A.The calculation requires (1Sor 2S to 1S or 2S) q-q-g ma-trix elements and (1S or 2S plus 1P, &2 or 1P3/2) q-q-gmatrix elements which are collected in Appendix B. Theeffect of the lowest-lying TM gluon mode has also beenexplored and found to be completely negligible comparedto TE. Later we discuss the contribution of the Coulombgluon exchange which is also very small.

We summarize the required amplitudes via the Feyn-man diagrams of Fig. 2 and their contributions to dG&,the correction to the axial vector coupling, and to dZ sq,the correction to the wave-function normalization. Thewave-function renormalization is required because theperturbation mixes a 3q +g component into the nominal3q nucleon state whose normalization must be reducedaccordingly. The corrected axial vector coupling is

turbation theory is to choose a basis in which these par-tial cancellations are recognizable. In this regard, ourwork in the following is only a preliminary step in whichthe passage to a conserved axial current is not explicitlyelucidated, but we are dependent on the underlyingchiral-invariant model for incorporating the partial con-servation.

The uncorrected matrix element of Fig. 2(A) is evalu-ated with the SU(6) spin-isospin 3q wave function withmodel independent 1S quark radial functions. The pro-cesses of Figs. 2(B) and 2(C) involving 1S quarks and aTE gluon exchange, dominate the gluon radiative correc-tions. These processses constitute a perturbative descrip-tion of the mixing of the 3q nucleon with its 3q+ggluonic-nucleon partner [16]. Similar processes involvinga 2S quark excitation are indicated in Figs. 2(D)—2(H).Figure 2(H) has all possible time orderings of the gluoninteraction with the spectator quarks. Only Fig. 2(F)contributes to the wave-function renormalization. Fig-ures 2(I)—2(K) are also a priori interesting processes in-volving 1P&/2 and 1P3/2 antiquark excitations, some ofwhich make marginally competitive contributions. Radi-ative processes involving only spectator quarks as in Fig.2(L) contribute equally to dG„ /G~ and dZ sq and can-cel, leaving the processes of Fig. 2(A)—2(K) to be calculat-ed. Figure 2(M) involves only post or prior exchangeswhich are supposed to be absorbed into the model wavefunctions [17].

G-luon and other radiative contributions can sometimesbe deduced qualitatively and, for example, major cancel-lations can be anticipated on the basis of the overall colorneutrality of the nucleon. The wave-function renormal-ization dZ sq is positive and can dominate the correctionto G~. Although it is reassuring to see these intuitive

47 RADIATIVE CORRECTIONS TO THE NUCLEON AXIAL. . . 1755

features appear, by and large we are dependent on the de-tailed calculations.

The contributions of the various TE gluon exchangeperturbation diagrams to dG& and to dZ sq are shown inTable I and sum to

G~+dG& =1.088 —0.072', and 1+dZ sq D. — E~c &

= 1+0.236', (2.4)

giving a net correction

DG„ IG„(TE)=( —0.066—0.236)a, = —0.302a, .

(2.5)

B. TM gluon exchange

FIG. 4. Coulomb gluon radiative corrections. (A) —(C) withan L =0 gluon and no quark excitation make no contribution.(D) with L =0 gluon and 2S quark, (E) with L =0 gluon, 1P&/2antiquark and 2S quark, and (F) with L =1 gluon, 1P, /& and1P3/2 quarks, and 1S,2S antiquarks make nonzero contribu-tions.

dG„ IG~ = —0.0002a, and dZ sq =+0.0020a, (2.6)

giving a net correction from this source of

DG~/G~(TM)= —0.002a, . (2.7)

C. Coulomb gluon exchange

We calculate radiative corrections for L =0 and L = 1

Coulomb gluon interactions of the quarks using theBarnes, Close, and Monaghan [18] screened Coulomb in-teraction for confined gluons. The interaction Hamiltoni-an is

H(Coul)= jd r p (r)G, (r —r')p (r')d r' (2.8)

with the quark color charge density

p (r)=q(r)y q(r) (2.9)

and the screened Coulomb Green's function

1/(r ) ) for L =0,G, (r r')= '—

[(r &)I(r ) ) +2rr']cos(r r') for L =1,(2.10)

The TM gluon exchange has many fewer diagramsshown in Fig. 3(A)—3(C), none of which compete with thedominant TE exchange contributions of Fig. 2. The con-tributions are shown in the table and sum to

where r ) (r & ) is the greater (lesser) of r and r', and lessthan the bag radius which is 1 in our units. The relevantdiagrams are shown in Fig. 4. Those with post or priorL =0 interactions with no quark excitations are absorbedinto the SU(6) wave function so Fig. 4(A) makes no con-tribution. The L =0 exchange Z diagrams of Fig. 4(B)and 4(C) contribute nothing because of the color neutrali-ty of the nucleon state. The diagrams that remain are theL =0 Coulomb gluon exchange with 2S quark excitationof Figs. 4(D) and 4(E) and also the L =1 exchange with1P»z and 1P3&2 quark excitation of Fig. 4(F). Their cal-culation is straightforward and we just state the totalcontribution

DG& /G„(Coul) = —0.026a, , (2.1 1)

for a total one gluon exchange correction to the axial-vector coupling constant of

DG~ /G„(one gluon exchange) = —0.330a, . (2.12)

Approximately three-quarters of this comes from the TEwave function renormalization, and the other one quarterfrom the explicit TE reduction of G~. As already noted,this dominance leads to an interpretation in terms of agluonic component of the nucleon which has some angu-lar momentum carried by the gluon with a resultantdepolarization of the quarks and a necessary reduction ofG~. For orientation, we use a value of e, =0.8 to give atotal gluon correction of DG~ /G„= —0.265. This valueof a, is much smaller than the frequently mentionedvalue of 2.2 originally introduced in bag calculations toremove the nucleon and delta degeneracy, but is close tothe value of a, =0.75 advocated by Barnes, Close, andMonaghan [18],and to that of Golowich, Haqq, and Karl[16], and to the bag model parametrization developed byDonoghue and Johnson [19]. Stern and Clement [13]favor an even smaller value.

III. MESON RADIATIVE CORRECTIONS

FIG. 3. Transverse magnetic gluon radiative corrections.Heavy quark lines are 1P&/2 and 1P3/2 quarks and 1S anti-quarks.

We consider the relativistic chiral quark model withconfined quarks in chiral invariant interaction withscalar-isoscalar sigma mesons and with pions, for whichwe calculate perturbative results for comparison with

1756 IAN DUCK 47

hedgehog soliton predictions. We discuss in order, pure-ly pionic, mixed pion-sigma, and purely sigma contribu-tions. The purely pionic contribution separates naturallyinto Nb, ~ (L = 1) processes which we discuss in the nextSec. III A, and other L =0 and L =1 pion reactions, in-volving quark excitations out of the 1S orbitals or excita-tion of antiquark states, which we postpone to Sec. III B.

A. Nhm radiative processes

The (Nhm) contribution is summarized by the dia-grams of Fig. 5, where the coupling strengths G„(NN)G~(NA), and G„(bh) are readily calculated from matrixelements of A (z, 3;q) with the nominal SU(6) wave func-tions. The (NN~), (Ne'er), and (b.b~) vertices are easilyexpressed [9] in terms of couplings f (NNvr), f (Nbn),and f

(Aber)

and a form factor F(k ) =3j&(kR)/kR foran L = 1 pion of momentum magnitude k.

A straightforward calculation of the axial coupling ofthe 6 produced from a spin-up proton by emission of apion with (L = 1; L, = 1,0, —1) and (I, = + 1,0, —1) pro-duces a somewhat surprising result [20]. The b, (J,=—', ,I, =

—,') has an axial coupling of 3g~(q) compared to theproton (I, =1/2, I, = 1/2) with —,'g~(q). However, the5 is produced with a probability of only —, for this stateand all other 5 states have a smaller axial coupling. Theweighted mean of the 6 axial coupling over all spin-isospin states is —, X —, indicating a quark spin-isospindepolarization from the initial nucleon due to the pionemission. The correction to the nucleon axial couplingwill be the probability I' of pion emission and reabsorp-tion, multiplied by this averaged quark depolarization of9 minus one from the wave-function renorma 1ization

DGq /G„=P ( —', —1), (3.1)

(involving g &z/4vr=13, Mz is the nucleon mass, o+ isthe m.+-proton cross section at m. laboratory energy v)would be qualitatively reAected in the perturbative result.

which is a negative contribution from the (~A) inter-mediate state. This conclusion remains valid when (NA)weak transitions are included. One might have thoughtthat the Adler-Weisberger sum rule [21] which is dom-inated by the b,(++) resonance in AN scattering, andwhich requires G~ ) 1 through the relation

1 —1/Gw = [2M~/(2~g ~~)]J (u+ u) —(3.2)

The (Nb, vr) radiative corrections of Fig. 5 are readilycalculated using the factors of Appendix B. Integralsover the virtual pion momentum get their major contri-butions from -2/R -500 MeV/c, and have convergedby -4/R. The resulting corrections, summarized inTable I, sum to

DG~ /G& (NET) =dG~ /G& dZ—sq

= +0.208 —0.275 = —0.067 (3.3)

dominated by 17% (Nm) and 10% (b~) wave-functionrenormalizations of the nominal three-quark nucleonstate. We have used a bag radius of 0.8 fm. The correc-tions get rapidly more severe as the bag radius is reducedand reach 0.52 —0.66= —0. 14 for R =0.6 fm. The indi-vidual corrections dG~ /G~ and dZ sq are no longer inany sense small and perhaps it is better to write

DGz /Gz =(1+dG& /Gz )/(1+dZ sq) —1

= 1.52/l. 66—1 = —0.08 (3.4)

which reduces the correction to Gz, and leaves a perhapsbelievable perturbation correction. The legitimacy ofthese pionic radiative corrections has already been ques-tioned because of the elastic decoupling of the pion radia-tive modes from the soliton. We will return to this ques-tion in the subsection following our calculations usingMIT bagged quarks as meson sources. We turn now topion radiative processes which are not included in the(Nb.~) system.

B. Other pion radiative corrections

The single pion emission amplitude of Fig. 5(F) inwhich the virtual pion is annihilated by the axial currentis known from Nambu's analysis [22] to contribute onlyto the induced pseudoscalar coupling, but not to the axialvector coupling because it vanishes at zero momentumtransfer where Gz is defined. The surviving purely pion-ic radiative corrections involve quark excitations, includ-ing 2S quarks or a q-q pair, plus an L =0 or L = 1 pion,and are listed in Figs. 6 and 7. All these processes con-nect the bare nucleon to intermediate quark states out-side the (Nh) multiplet and are not included in the previ-ous calculations, or in the soliton calculation. There arehazards of double-counting perturbative q-q amplitudeswhich should be included as part of meson states. Such asituation can be recognized when axial vector mesons are

IN

p ii ii E ii lp

r+

pFIG. 5. One pion exchange radiative corrections within the

N4m. (L =1) system. (8)—(E) have NN, NA, 6N, hh intermedi-ate states. (F) is the pion pole contribution to the axial vectormatrix element vanishing at k =0. Single dashed lines arepions.

FIG. 6. One pion exchange radiative corrections with L =0pions and quark excitations. Heavy lines are 1P&/2 quarks and1S,2S antiquarks.

47 RADIATIVE CORRECTIONS TO THE NUCLEON AXIAL. . . 1757

A B

0 E

FIG. 7. One pion exchange radiative corrections with L =1pions and quark excitations leading out of the NA system andnot included in Fig. 5. Heavy lines are 2S quarks and1P)/2, 1P3/2 antiquarks.

FIG. 8. Sigma-pion transition radiative corrections. (A) and(B) have L(sigma) =0, L(pion) =1, no quark excitation. (C) hasa 2S quark. (D) and (E) have L(sigma) =1, L(pion) =0, and1P

& /2 quark and 1S antiquark excitations. Double dashed linesare sigmas.

considered, as in the Broniowski and Banerjee extensionof the sigma-pion-quark soliton to include the vector andaxial vector mesons. The contributions of individual pro-cesses to dG& /Gz and to dZ sq are listed in Table I. Wedo not go into the calculational details which arestraightforward, and just report the result for all the ex-tra pion radiative corrections

DG„ /G„(extra pion) =dG„ /G~ —dZ sq

tions, we discuss the soliton radiative corrections whichsurvive elastic decoupling.

Figure 8 contains the sigma-pion diagrams that con-tribute to dG& /G~. None of these diagrams involving asigma-pion transition via the axial vector current haveaccompanying wave-function renormalizations. The pro-cesses involving all possible time-orderings and all quarkcontributions like Figs. 8(A) and 8(B), with an L =0 sig-ma transformed to an L = 1 pion and no quark excita-tions, dominate and make a large correction

=0.14—0.08 =0.06, (3.5) DG&/Go[Figs. 8(A) and 8(B)j=+0.232 . (3.6)

which almost completely cancels that contained in the(Nb, m) processes, leaving a net pionic radiative correc-tion of —1%. This is of course much smaller than ourability to calculate. As in the previous subsection, welater select those processes of Fig. 7 which survive elasticdecoupling from the soliton, as legitimate radiativecorrections. We turn next to radiative processes involv-ing both the pion and the sigma meson.

C. Sigma-pion radiative corrections

The Birse-Banerjee chiral soliton has a vacuum valuefor the sigma field equal to f . They set—tle on a cou-pling strength g =5 which gives an external quarkmass M =465 MeV. There is an essentially nonpertur-bative defect in the soliton interior where the sigma fieldswitches from +f to f giving an interior—quark massMq(r) which can be much smaller than M~, and evennegative, so the quarks are relativistic and approximatethe massless MIT bag modes. Our initial sigma-pion ra-diative corrections will take account of Auctuations of thesigma field on the vacuum background f . Such calcu-—lations are admittedly suspect, but provide necessaryorientation for later calculations where the efFect of thesoliton field will be taken into account.

Following Birse's Eqs. (1.4) and (6.2), we have the in-teraction Hamiltonian for the MIT bag quark source andthe sigma-pion transition axial-vector current. Anotheringredient of the calculation is a mass for the asymptoticsigma field Auctuations which Birse takes as 1 GeV, fixingthe meson potential strength. We now present the resultsof unmodified perturbation theory for sigma-pion andthen sigma-sigma processes. Following these two subsec-

This diagram is directly analogous to the meson contribu-tion in the soliton calculation, where the hedgehog staticsigma and pion configurations contribute a large positiveGz ( m ) = +0.81. The modification of naive perturbationtheory necessary for this particularly important processwill be discussed in a later subsection.

Other processes which have quark excitation, q-q exci-tation, or L =1 sigma and L =0 pion with 1P&&z quarkexcitation, contribute +0.049 for a total sigma-pioncorrection

DG„ /G„(o-~) = +0.281 . (3.7)

D. Sigma radiative corrections

Radiative corrections involving only a scalar sigmameson with L =0 or L =1 and all possible L =0 andL =1 quark excitations, are listed in Figs. 9 and 10. Ofthese, all diagrams with an L =0 sigma and 1S quarks,Figs. 9(A) and 9(B), can be ignored because they contrib-ute equally to dG~ /G~ and to dZ sq and therefore con-tribute nothing to the net correction DG~ /G~. This isobvious from the spin and isospin independence of thesigma coupling, or from the fact that these sigma-exchanges can be absorbed into the 1S quark wave func-tion without change of the spin or isospin or, consequent-ly, of the axial coupling. The remaining diagrams all in-volve either an L =0 sigma and 2S quark or 1P antiquarkexcitations, or an L =1 sigma plus 1P quark. Diagramswith an L = 1 sigma and 1S quark plus 1S antiquark donot contribute.

The remaining sigma diagrams are easily evaluated andwe simply state the result

1758 IAN DUCK 47

B ==:- C

0 E

Prr

FIG. 9. Sigma exchange radiative corrections withL(sigma)=0. (A) and (B) do not contribute. (C)—(K) have 2Squark and 1P&/2 antiquark excitations.

DG~/G„(cr)= —0. 113 . (3.8)

The dominant contribution of (—0.086) comes from the

I.=0 sigma exchange with excitation of a 1P& &2 anti-

quark in the Z process of Figs. 9(H) and 9(I). All the oth-er processes contribute fractions of 1%, but the directcorrections dG&/G~ are predominantly negative, andreinforce the wave-function renormalization contribu-tions which dominate. All these fragmentary contribu-tions combine to ( —0.027).

The sum of all these radiative corrections —Eqs. (2.12),(3.3), (3.5), (3.7), and (3.8)—constitutes the radiativecorrection to Gz for the MIT bag model coupled pertur-batively to QCD and to the linear (o-m) chiral restoringinteraction. The net result

DG& /Gz = —0.265(gluons) —0.067(NET)+0. 06(~)

+0.281(o7r) —0. 113(o )

= —0. 105 (3.9)

should be applied to Eq. (1.1) to reduce Gz from 1.09 to0.975. There are severe canceHations in the contributionsof Eq. (3.9) whose magnitudes sum to 0.72. The cancella-

FIG. 10. Sigma exchange radiative corrections withL(sigma)-1 and 1P,/„1P3/p 2S quark and 1S antiquark excita-tions. (C) and (D) do not contribute.

tions can be believed qualitatively but the resulting mag-nitude cannot be taken seriously for at least two reasons.One is that no direct connection has been built into thegluon and meson corrections, so their cancellations mustbe somewhat fortuitous and can be easily destroyed by adifferent choice of the QCD a, . But also, the very ex-istence of a soliton means that at least the sigma-quarkinteraction cannot be treated perturbatively.

Before we leave the MIT bag model for the Birse-Banerjee soliton model, it is interesting to see what thesuccesses and failures of perturbation theory are, basedon Birse's nonperturbative results. It is amusing that atleast the XA~ contribution is very close to the 7%%uo reduc-tion of Eq. (1.6) and, even though they have quitedifferent origins, the perturbative result is not misleading.Birse's result contains no semblance of the extra ~ contri-bution of +0.06 or of the extra o. contribution of—0. 113. The o.-m transition contribution of 0.232 fromFigs. 8(A) and 8(B) is a failure of perturbation theory toproduce Birse's meson contribution of 0.81/1.09=0.74.

In the next subsection we bring our perturbative treat-ment of meson field fIuctuations into accord with the soli-ton as source and, in the process, make a substantialreduction in the radiative corrections involving sigmasand pions.

E. Radiative corrections to the soliton

Soliton matrix elements of functional derivatives of theLagrangian with respect to the meson fields

(Soi al. /ay~Sol)

are equations of motion for the soliton and vanish for

(Soi~y~Soi) =y, ,

the soliton configuration. If we go further and investigatefluctuations 5$ on the soliton background, the same func-tional derivative serves as the source of the 5$ and van-ishes in the soliton matrix element, decoupling the Auc-tuations from the soliton. This conclusion is also con-tained in the explicit equations of motion of Ren andBanerjee [14]. Correspondingly, a large number of ampli-tudes in the radiative corrections will vanish. Every soli-ton pion or sigma vertex which leaves the soliton unexcit-ed will go to zero. Going through the figures, we see thatall the amplitudes of Fig. 5 will be zero, which is only tosay that XA~ radiative corrections are already countedin the Birse calculation. The amplitudes of Fig. 6 remainand will be taken at their MIT bag value. Amplitudes7(A), 7(D), 7(E), 7(F), and 7(G) with 1S-1SMIT bag ver-tices are set to zero, as are the previously dominant o.-~transition amplitudes of Figs. 8(A) and 8(B) now includedin Birse's (0.81), and the amplitudes 9(C), 9(D), 9(G),9(H), and 9(I).

What remains after decoupling elastic radiative transi-tions from the soliton are much reduced meson radiativecorrections and a correction to the axial vector couplingconstant replacing the MIT bag result Eq. (3.9):

47 RADIATIVE CORRECTIONS TO THE NUCLEON AXIAL. . . 1759

DG„/G& = —0.265(gluons)+0(Noir) —0.078(m)

+0.049(o-m) —0.026(cr )

= —0.32 . (3.10)

IV. DISCUSSION

A. Nonleading QCD contributions

Model predictions are also subject to corrections fromthe current quark mass and from recoil effects [23],which can easily overwhelm the radiative correctionswhich are our primary interest. We quote radiativecorrections only for zero quark mass because in our limit-ed explorations we have found them to be insensitive tothis parameter. We have included recoil only in the(NAn ) contributions, and there only in the energydenominators. Recoil effects in the energy denominatorsreduce the contribution of these diagrams to DG„/G„by thirty percent.

Recoil effects will be the only contribution to the elasti-cally decoupled radiative modes which have been com-pletely omitted in going from radiative corrections forthe MIT bag to those for the soliton. It probably is notworth any great effort to refine the small radiative correc-tions involving inelastic transitions in the source, but itwould be interesting to refine our treatment of elasticdecoupling to include leading recoil effects.

C. Constituent quark model

In the work of Weinberg [24], Peris [25], and Rosenfeldand Rosner [26], based on the chiral constituent quark

The TE gluon radiative corrections have included onlythe lowest energy magnetic dipole contribution withco(TE1, Ng =1)=2.744. The dominant contribution toDG„ /G„comes from the wave function renormalizationnecessitated by a gluonic nucleon component with a(TE1, Ng =1) gluon plus three S(Ns =1) quarks. It iseasy to check only one convergence property of the per-turbation sequence without being led into prohibitivecomputational complexity, and that is to calculate contri-butions for other radial excitations with Ng, Ns ) 1. Notsurprisingly, we find that the diagonal elements in thesearrays dominate the nondiagonal by an order of magni-tude. The convergence is even more rapid for theNg =¹=1 leading term which dominates Figs. 2(B) and2(C) by two orders. The Ng =¹=2 term dominates theNg =1, Ns =2 by one order and is the one quoted inTable I for the contribution of Fig. 2(F) to dZ sq. Similarconclusions apply to the Z graphs of Figs. 2(J) and 2(K)where the term with Ng =Ns =Np =1 gives 95% of therapidly convergent diagonal sum. To explore the contri-bution of higher multipoles would require the full angularmomentum formalism and is not feasible with our tech-nique of projection operators. Our general experience isthat the leading diagonal term dominates other contribu-tions at the 90% level, a precision which exceeds ourability to calculate.

B. Other corrections

theory of Manohar and Georgi [27], a very different viewof nucleon structure leads to a calculation of the leading1/N, correction to the axial vector coupling constant.Here we point out differences between the constituentquark model and the relativistic chiral quark bag modelof the nucleon. In their work a constituent quark Q withmass M& =360 MeV is formed in the chiral phase transi-tion in which the sigma field assumes the value ( f )—everywhere. Rosenfeld and Rosner find that the constitu-ent quark has the Dirac coupling g~(Q) =1 and a Diracmagnetic moment fit to the nucleon magnetic moment bychoice of M&. Peris calculates one-loop (o, vr) radiativecorrections to g„(Q) in leading log approximation usingthe linear o. model and finds a renormalization

Mln

Mg(4.1)

and with M =4~f„gets g„(Q)=1—0.2 and a nucleonaxial coupling

G =5/3g„(Q)=1. 3 . (4.2)

D. One phase model

The chiral soliton quark model can be brought intocloser accord with the constituent quark model ofManohar and Georgi and the work of Weinberg andPeris at small cost and substantial benefit, as indicated byRen and Banerjee [28] and in earlier work [29]. At asmaller value of g =4 rather than 5 as used by Birse,the interior o field remains near its vacuum value of f-and does not flip over to +f at r =0. As a result thequarks have almost their asymptotic mass

Weinberg uses the Adler-Weisberger sum rule for pionscattering from the constituent quark, together with thedominance of m+d over m+u scattering to argue for asimilar reduction. Peris also shows that the calculationsare equivalent provided mQ scattering is dominated by o.

production through a one pion exchange Primakoff-typeprocess. All these calculations indicate a radiative reduc-tion of g z ( Q) at the 10—20 % level. There remainconfinement effects and center-of-mass effects contribut-ing further 10% reductions. In addition, Peris states thatgluon exchange specifically does not renormalize g„(Q)in the leading-log approximation.

The two points of view could hardly be more different.The bag model starts with a reduction of g„(q) due toconfinement, then a positive contribution from themesons due to the reduction of o. inside the bag, and fol-lows with reduction due to gluon exchange involving in-dividual quarks and quarks in interaction with spectatorsdominated by Figs. 2(B) and 2(C). The chiral solitonquark bag model does not manifest the Adler-Weisbergersum rule at the quark level, but only at the nucleon level.Furthermore, by working with a static source, the bagmodel has very soft form factors which provide lowmomentum cutoffs before the logarithmic divergences ofPeris are ever encountered.

1760 IAN DUCK 47

Mq =g f =373 MeV everywhere and must beconfined by the bag. The o. and ~ fields differ little fromtheir asymptotic values and can sensibly be treated inperturbation theory. The hedgehog soliton calculation isthen replaced with a perturbatively corrected MIT bag.The radiative corrections of Eq. (3.9) are relevant with

just two modifications. The meson contributions must bereduced by a factor of (

~) to account for the changed

coupling strength, and the gluon correction must becorrected by a factor of 0.62 to go from Mq =0 toMq =373 MeV in the calculation of the dominant TE1S-1Smatrix element. The resulting correction to 6~ is

DG~ /G„(Mq =373 MeV) = —0.265(0.62)+ (—0.067+0.06+0.281 —0. 113)(0.64)

= —0.06 . (4.3)

The axial matrix element gz is increased from 0.653 to0.80 and the nucleon axial coupling becomes

G„=—',0.8(1—0.06) =1.25 . (4.4)

If true, the benefits of being able to use constituentquarks in a bag, and perturbation theory on a simple vac-uum background are enormous. The whole intractableproblem of radiative corrections to the soliton would beavoided. There are still substantial differences with thework of Weinberg and of Peris, who ignore spectator in-teractions, do not depend on confinement for cutoffs andhave no gluon radiative corrections fallowing Manoharand Georgi who recommended a gluon coupling muchsmaller than that of Donoghue and Johnson, which is theone used here. The Adler-Weisberger sum rule, whichdepends upon PCAC and the Goldberger-Treiman rela-tion, will be satisfied at the nucleon level but not at thequark level.

cates that meson corrections must be substantially re-duced, because the soliton is elastically decoupled frommeson radiative modes. The meson radiative correctionsthat remain involve inelastic transitions on meson emis-sion and absorption. Calculated in the MIT bag approxi-mation these corrections are predicted to beinsignificantly small. An improved treatment of thesecorrections to the chiral soliton quark model leads intorecoil corrections to elastic decoupling and to inelastictransition amplitudes requiring excited states of the soli-ton. We have showed that the relevant amplitudes arevery small as calculated with MIT bag sources, leavingthe dominant TE gluon correction to reduce the axialcoupling as much as 30%. Agreement with experiment isonly qualitative. Applied directly to Birse s result, G~ isreduced from

G„(q)+G„(m) =0.98+0.81

to perhaps

(0.98+0.81 )( 1 —0.3 ) = 1.25,V. CONCLUSIONS

Gluon and meson radiative corrections to the nucleonaxial vector coupling constant have been calculated in therelativistic chiral quark model. Second-order perturba-tive processes of Figs. 2—10 have been included. Thedominant contributions to the radiative corrections aremade by the processes of Figs. 2(B) and 2(C). These aretransverse electric gluon exchanges which mix a (three-quark plus gluon) component with the usual three-quarknucleon state with a probability of order 0.230.„depolar-izing the quarks and reducing the axial vector couplingby 0.30m, . Another significant process is the sigma-piontransition of Figs. 8(A) and 8(B) which is a positive con-tribution tending to cancel the gluon contribution. Thedegree of cancellation is dificult to calculate precisely notonly because of the uncertainty in the gluon couplingstrength but primarily because of a sensitivity of the sig-ma and pion couplings to the soliton backgrounds. Re-cent work by Mattingly and Stevenson [30] determinesthe gluon coupling in perturbative QCD even at very lowmomentum transfers and supports a value near 0.8 for a,in bag calculations, which puts the gluon part of the cal-culation under control. A heuristic calculation of theeffect of the soliton background on meson couplings indi-

but it is still necessary to rationalize this correction withthe Banerjee and Broniowski vector meson soliton resultwhich already agrees with experiment.

Finally, we are led to the Manohar-Georgi constituentquarks bag-confined in a model-dependent way as themost tractable basis for calculating radiative correctionsto the axial vector coupling constant. The correctionsare small and the agreement is encouraging.

ACKNOWLEDGMENTS

I am grateful to Alan Mattingly and to the late EricUmland for their early collaborations on this work. Thiswork was supported in part by the U.S. Dept. of Energy.

APPENDIX A: BAG MODEL DIRAC SPINORS

The bag model Dirac spinors for massless quark andanti-quark states, with bag radius equal unity, are the fol-lowing.

S states: q [S]= ( U, i a r V)g (for quarks), q [S]=( rVa, iU)g (for antiquarks), with y a Pauli spinor,y=ia~ an antispinor, and U(r) =Xj0(cor), V(r)=Nj &(d'or) where co[1S]=2.043, X [1S]=0.6403; andco[2S]=5.396, X [2S]= l.538.

47 RADIATIVE CORRECTIONS TO THE NUCLEON AXIAL . . ~ 1761

1Pt/z state: q [P)/z]=(U, i—cr rV}cr ry, q [PI/2]=(ct rV iU)cr rg with U[P&/z]=1. 089j, (3.81lr)and V[P~/z] =1.089jo(3.811r).

1P3/z state: q [P3/z] =( U, io.r V. )e Py and q [P3/z ]=(o rV, iU)E Pg with U[P3/z] =1.065j, (3.203r) andV[P3/z]=1. 065jz(3.203r). The Projection oPerator forthe spin- —', vector-spinor E P =(E r —ct r cr e'/3)3/ &2 isa convenient but redundant representation involving athree-component polarization vector c and a two-component Pauli spinor g. The six states must beweighted with a further factor of —', in the P3/2 propaga-tor.

In addition, we note gluon TE1 and TM1 modes:

G(TE1)=(+ctk /2)(N/t 2')&3j, (cor)(r XE ),G(TM1) =(&aA, /2)(N/&2')&3

X I E [—,' jt(cur) —

,'j z(co—r)]+r(Es.r j)z(cur) j,where a=gQCD/4~=0. 8 is the nominal value of thegluon coupling, A, are the Gell-Mann SU(3) matrices, E

is the gluon polarization vector, and

ATE& —9.097, coTE&=2.744; NTM& =21.190, coTM&

=4.493 .

APPENDIX B: MATRIX ELEMENTS

The table of matrix elementsI. TE matrix elements with a common factor V'aA, /2 removed, for massless quarks in a bag of unit radius. g are

Pauli spinors, g antispinors:

(q 1S~H(TE)~q 1S & =0.4929'+o"E y

(q2S~H(TE)~q 1S & =0.656yf+(2S)o" r. y, (1S),

(O~H(TE}~qlS, qPI/z &= —0.3255j+(P&/z)o" E~g;(1S),

(O~H(TE)~qlS, qP3/z &= —0.2093/+(1P3/z}(2E E~ icr E —XE )y;(1S) .

II. Axial vector matrix elements with H =o.,~, :

(q 1S~H ~q 1S & =O 6530'.f+cr, ~,q;,(q2S~H ~q 1S &

= —0. 1390yf+(2S)cr,r,y;(1S),(q2S H ~q2S& =0 4091'.f+cr, w, y, ,

( OiH q 1S&qP t/z &:0.4695/ + (Pi/z )ct 7 g ( 1S)

(O~H ~q2S, qP, /z & =0.4150@+(P,/z)o, r,y, (2S), .

(O~H ~q 1S,qP3/z & =0. 1985/ (P3/z)(2E z —io"e Xz)r,y(IS),(O~H q2S, qP3/z & =0. 1265/ (P3/z)(2E~ z —io E~ Xz)~,y(2S),

& qPi/2 lH lqP1/2 &= —0.2641/,+(P~/z )o,r,f f(P, /z), '

(qP~/z IH lqP3/z & =+0.2897@; (P3/z )(2e z —io .e Xz )~gf (P, /z),

& qP3/z lH. IqP„, & =I,'(P„,)( —o.7o66 ~ —o.2923m)&, gf (P„,},where

& =(E, P+o, Ef P&, .

& =&.", P+~.r~, ~.r Ef.p &,

both averaged on r.

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1762 IAN DUCK 47

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