l l PHYSICAL REVIEW C, VOLUME 64, 025202 Chiral corrections to baryon properties with composite pions P. J. A. Bicudo Departamento de Fı´sica and CFIF-Edifı´cio Ciência, Instituto Superior Te´cnico Avenida Rovisco Pais, 1096 Lisboa Codex, Portuga G. Krein Instituto de Fı´sica Teo´rica, Universidade Estadual Paulista, Rua Pamplona, 145, 01405-900 Sa˜o Paulo, Sa˜o Paulo, Brazil J. E. F. T. Ribeiro Departamento de Fı´sica and CFIF-Edifı´cio Ciência, Instituto Superior Te´cnico, Avenida Rovisco Pais, 1096 Lisboa Codex, Portuga ~Received 22 March 2001; published 9 July 2001! A calculational scheme is developed to evaluate chiral corrections to properties of composite baryons with composite pions. The composite baryons and pions are bound states derived from a microscopic chiral quark model. The model is amenable to standard many-body techniques such as the BCS and random phase approxi- mation formalisms. An effective chiral model involving only hadronic degrees of freedom is derived from the macroscopic quark model by projection onto hadron states. Chiral loops are calculated using the effective hadronic Hamiltonian. A simple microscopic confining interaction is used to illustrate the derivation of the pion-nucleon form factor and the calculation of pionic self-energy corrections to the nucleon andD(1232) masses. DOI: 10.1103/PhysRevC.64.025202 PACS number~s!: 13.75.Gx, 12.39.Fe, 12.39.Ki, 24.85.1p is t - k on re e h io n ar ra ra io lyi e ls s la a m n e d ak on CD ext - at a e able n, nsor rs he ter- ave eon t rk- of is the vel- to ent n ble ter- ec- ob- ng - il- onic the I. INTRODUCTION The incorporation of chiral symmetry in quark models an important issue in hadronic physics. The subject da back to the early works@1–6# aimed at restoring chiral sym metry to the MIT bag model@7#. The early attempts were based on coupling elementary pion fields directly to quar A great variety of chiral quark-pion models have been c structed since then and the subject continues to be of inte in the recent literature@8–10#. Despite the long history, ther are many important open questions in this field. In t present paper, we are concerned with one of such quest namely the coupling of the pion as a quark-antiquark bou state to the baryons. Starting from a model chiral qu Hamiltonian, we construct an effective low-energy chi pion-baryon Hamiltonian appropriate for calculating chi loop corrections to hadron properties. The composite p and baryon states are determined by the same under quark chiral dynamics. The model we use belongs to a class of quark mod inspired in the Coulomb gauge QCD Hamiltonian@11# and generalizes the Nambu–Jona-Lasinio model@12# to include confinement and asymptotic freedom. This class of mode amenable to standard many-body techniques such as BCS formalism of superconductivity. The initial studie within these models were aimed at studying the interp between confinement and dynamical chiral symmetry bre ing (DxSB), and concentrated on critical couplings@13# for DxSB and light-meson spectroscopy@14#. The model has been extended to study the pion beyond BCS level and son resonant decays in the context of a generalized reso ing group method@15#. Since the model is formulated on th basis of a Hamiltonian, it provides a natural way to stu finite temperature and chemical potential quark matter@16#. The model and the many-body techniques to solve it m direct contact with first-principle developments such as n 0556-2813/2001/64~2!/025202~11!/$20.00 64 0252 es s. - st e ns, d k l l n ng ls is the y k- e- at- y e - perturbative renormalization-group treatments of the Q Hamiltonian@17# and Hamiltonian lattice QCD@18#. One important development of the model, in the cont of the present paper, was its extension in Ref.@19# to baryon structure. In Ref.@19# a variational calculation was imple mented for the masses of baryons and it was shown th sizeableD(1232)-N mass difference is obtained from th same underlying hyperfine interaction that gives a reason value for thep-r mass difference. This hyperfine interactio along with other spin-dependent interactions such as te and spin-orbit, stem from Bogoliubov-Valatin rotated spino that depend on the ‘‘chiral angle.’’ The chiral angle gives t extent of the chiral condensation in the vacuum and de mines the chiral condensate. The very same variational w function was used later for studyingS-wave kaon-nucleon @20# scattering and the repulsive core of the nucleon-nucl force @21#. Both calculations obtainS-wave phase shifts tha compare reasonably well with experimental data. A rema able feature of all these results for the low-lying spectrum mesons and baryons andS-wave scattering phase shifts that they are obtained with a single free parameter, strength of the confining potential. In the present paper we go one step forward in the de opment of the model and set up a calculational scheme treat chiral corrections in hadron spectroscopy. In a rec publication@22#, two of us have calculated the pion-nucleo coupling constant in this model and obtained reasona agreement with its experimental value. Here, we are in ested in developing a scheme for calculating chiral corr tions to hadron properties. We study the requirements to tain in the context of the model the correct leadi nonanalytic behavior~LNA ! of chiral loops. Our scheme fol lows the standard practice@6,23# of constructing an effective baryon-pion Hamiltonian by projecting the quark Ham tonian onto a Fock-space basis of single composite hadr states. Chiral loop corrections are then calculated with ©2001 The American Physical Society02-1 ry th a c- s t th g ge s ey t t e o f re n th tio r rk ic m w el th he a a in ia es l- y h u u n V r s, in- he tep um his v- s ra- ix ion s, ap P. J. A. BICUDO, G. KREIN, AND J. E. F. T. RIBEIRO PHYSICAL REVIEW C64 025202 effective Hamiltonian in time-ordered perturbation theo The difference here is that while in the previous works pion is an elementary particle, in our approach the pions composites described by a Salpeter amplitude. A difficulty appears in the implementation of the proje tion of the microscopic quark Hamiltonian onto the compo ite hadron states, which is not present when the pion treated as an elementary particle. The difficulty is related the two-component nature of the Salpeter amplitude of pion. The two components correspond to positive and ne tive energies~forward and backward moving, in the langua of of time-ordered perturbation theory!, they are 232 matri- ces in spin space and are called energy-spin (E-spin! wave functions. For the pion, the negative-energy component i important as the positive-energy one—in the chiral limit th are equal—because of the Goldstone-boson nature of pion. Because of this, the Fock-space representation of pion state is not simple. We overcome the difficulty by r phrasing the formalism of the Salpeter equation in terms the random-phase approximation~RPA! equations of many- body theory. The single-pion state is obtained in terms o creation operator acting on the RPA vacuum. The pion c ation operator is a linear combination of creation and an hilation operators of pairs of quark-antiquark operators; positive-energy Salpeter component comes with the crea operator of the quark-antiquark pair, and the negative-ene one comes with the annihilation operator of the qua antiquark pair. In this way, the projection of the microscop quark Hamiltonian onto the single hadron states beco feasible and simple. The paper is organized as follows. In the next section review the basic equations of the model. We show the r tionship of the formalisms of the Salpeter equation and of many-body technique of the RPA. In Sec. III we derive t pion-baryon vertex function in terms of the bound-state S peter amplitudes for the pion and the baryon. We obtain expression that is valid for a general microscopic quark teraction, not restricted to a specific form of the potent Given the pion-baryon vertex function, we derive the expr sion for the baryon self-energy correction in Sec. IV. A though the derivation of the expression for the self-energ well-known in the literature, we repeat it here to make t paper easier to read. In Sec. V we obtain numerical res for the pion-baryon form factor and coupling constants. N merical results and the discussion of the LNA contributio to the baryon masses are presented in Sec. VI. Section presents our conclusions and future directions. II. THE MODEL The Hamiltonian of the model is of the general form H5H01HI , ~1! whereH0 is the Dirac Hamiltonian H05E dx c†~x!~2 i a•“1bmq!c~x!, ~2! 02520 . e re - is o e a- as he he - f a - i- e n gy - es e a- e l- n - l. - is e lts - s II with c(x) the Dirac field operator, andHI a chirally sym- metric four-fermion interaction HI5 1 2E dxE dyc†~x!TaGc~x!VG~x2y!c†~y!TaGc~y!. ~3! Here,Ta51/2la,a51, . . . ,8 are thegenerators of the colo SU~3! group,G is one, or a combination of Dirac matrice and VG contains a confining interaction and other sp dependent interactions. One example ofVG will be presented in Sec. V, when we make a numerical application of t formalism. Once the model Hamiltonian is specified, the next s consists in constructing an explicit but approximate vacu state of the Hamiltonian in the form of a pairing ansatz. T is most easily implemented in the form of a Bogoliubo Valatin transformation~BVT!. The transformation depend on a pairing function, or chiral anglew that determines the strength of the pairing in the vacuum. The quark field ope tor is expanded as c~x!5E dq ~2p!3/2 @us~q!b~q!1vs~q!d†~2q!#eiq•x, ~4! where the quark and antiquark annihilation operatorsb andd annihilate the paired vacuum, or BCS stateu0BCS&. Here the spinorsus(q) andvs(q) depend upon the chiral anglew as us~q!5 1 A2 $@11sinw~q!#1/21@12sinw~q!#1/2a•q̂%us 0 , ~5! vs~q!5 1 A2 $@11sinw~q!#1/22@12sinw~q!#1/2a•q̂%vs 0 , ~6! whereus 0 andvs 0 are the spinor eigenvectors of Dirac matr g05b with eigenvalues61, respectively. The chiral angle can be determined from the minimizat of the vacuum energy density Evac V 5E dq ~2p!3 Tr@a•qL2~q!#1 1 2E dq ~2p!3 dq8 ~2p!3 3Ṽ~q2q8!Tr@TaGL1~q!TaGL2~q8!#, ~7! where Tr is the trace over color, flavor, and Dirac indice and L1~q!5( s us~q!us †~q!, L2~q!5( s vs~q!vs †~q!. ~8! The minimization of the vacuum energy leads to the g equation A~q!cosw~q!2B~q!sinw~q!50, ~9! 2-2 et n ts n n a- s. of - - lu- ood uch n an g ss s ( do ent all ive ly he at - ud ve me yon re but CHIRAL CORRECTIONS TO BARYON PROPERTIES . . . PHYSICAL REVIEW C 64 025202 with A~q!5m1 1 2E dq8 ~2p!3 ṼG~q2q8!Tr@bTaG~L1~q8! 2L2~q!!TaG#, ~10! B~q!5q1 1 2E dq8 ~2p!3 ṼG~q2q8!Tr@a•q̂TaG~L1~q8! 2L2~q!!TaG#, ~11! whereṼG(q) is the Fourier transform ofVG(x), ṼG~q!5E dx eiq•xVG~x!. ~12! The pion bound-state equation is given by the field-theor Salpeter equation. The wave function has two compone f1 andf2, the positive- and negative-energy componen Each of thef ’s is a 232 matrix in spin spacefs1 ,s2 1 and fs1 ,s2 2 . For this reason, thef ’s are also called energy-spi (E-spin! wave functions. Thef ’s satisfy the following coupled integral equations: @M ~q!2E~q1!2E~q2!#fk 1~q!5u†~q1!Kf~k,q!v~q2!, ~13! @M ~q!1E~q1!1E~q2!#fk 2T~q!5v†~q1!Kf~k,q!u~q2!, ~14! whereq65q6k/2, and the kernelKf(k,q) is given by Kf~k,q!5E dq8 ~2p!3 ṼG~q2q8!TaG@u~q18 !fk 1~q8!v†~k28 ! 1v~k18 !fk 2T~q8!u†~q28 !#TaG. ~15! Here, the superscriptT on f2T means spin transpose off2, fs1 ,s2 2T 5fs2 ,s1 2 . The amplitudes are normalized as E dq ~2p!3 @fk 1* ~q!fk8 1 ~q!2fk 2* ~q!fk8 2 ~q!#5d~k2k8!. ~16! In the language of many-body theory, these equations ca identified with the RPA equations@24#. In the RPA formula- tion, one writes for the one-pion state~suppressing isospin quantum numbers! up~k!&5Mp † ~k!u0RPA&, ~17! where u0RPA& is the RPA vacuum, which contains correl tions beyond the BCS pairing vacuumu0BCS&, andMp † (k) is the pion creation operator 02520 ic ts . be Mp † ~k!5E dq ~2p!3/2 @fk 1~q!b†~q1!d†~q2! 2fk 2~q!b~q1!d~q2!#, ~18! where theq6 were defined above. In this formulation, Eq ~13! and ~14! above are obtained from the RPA equation motion ^p~k!u@H,Mp † #u0RPA&5@Ep~k!2Evac#^p~k!uM†u0RPA&. ~19! The normalization is such that ^p~k!up~k8!&5^0RPAu@Mp~k!,Mp † ~k8!#u0RPA&5d~k2k8!. ~20! The verification of DxSB consists in finding nontrivial solu tions to the gap equation, Eq.~9!, and the existence of solu tions for the pion wave function. References@11,15# show that for a confining force, there is always a nontrivial so tion for the gap and pion Salpeter equations. Moreover, g numerical values are obtained for the chiral parameters s as the pion decay constant and the chiral condensate whe appropriate spin-dependent potential is used@25#. The inclusion of RPA correlations beyond BCS pairin was shown in Ref.@24# to have a dramatic effect on the ma spectrum of the pseudoscalar mesons (p andh), while it has almost no effect on the mass spectrum of vector mesonr andv). One can trace this effect to the fact that the pseu scalar mesons have a sizeable ‘‘negative energy’’ compon wave function, while the vector mesons have a very sm negative energy component@15#. For baryons ~such as nucleon andD), since they do not have a sizeable negat energy component@19#, one expects that they can be reliab obtained from the BCS vacuum. We write therefore for t one-baryon state uB(0)~p!&5B(0)†~p!u0BCS&, ~21! where the baryon creation operatorB(0)†(p) is given by B(0)†~p!5E dq1dq1dq1d~p2q12q22q3! 3Cp~q1q2q3!ec1c2c3x f 1f 2f 3s1s2s3 3bc1s1f 1 † ~q1!bc2s2f 2 † ~q2!bc3s3f 3 † ~q3!. ~22! Hereec1c2c3 is the Levi-Civita tensor, which guarantees th the baryon is a color singlet andx f 1f 2f 3s1s2s3 are the spin- isospin coefficients. The wave functionCp(q1q2q3) is deter- mined variationally@19#. The index (0) on the baryon opera tors indicates a bare baryon, i.e., a baryon without pion clo corrections. The important fact to notice here is that the baryon wa function depends on the chiral anglew and as such, spin splittings and other properties are determined by the sa physics that determines vacuum properties. The pion-bar vertex, that we will discuss in the next section, will therefo depend on the chiral angle not only because of the pion, 2-3 u - rs a . o- n n su io e an a ns e- io re in e- yo am rk b in e yo ta lv n in- ure con- we lf- lly fol- e of y, P. J. A. BICUDO, G. KREIN, AND J. E. F. T. RIBEIRO PHYSICAL REVIEW C64 025202 also because of the baryon wave function. Numerical res for the masses of the the nucleon andD(1232) have been obtained previously@19# and are of the right order of mag nitude as compared with experimental values. Of cou fine-tuning with different spin-dependent interactions c improve the numerical values of the calculated quantities III. PION-BARYON VERTEX FUNCTION We obtain an effective baryon-pion Hamiltonian by pr jecting the model quark Hamiltonian onto the one-pion a one-baryon states, Eqs.~17! and ~21!. We use a shorthand notation. For the bare baryons, i.e., baryons without pio corrections, we use the indicesa,b, . . . , to indicate all the quantum numbers necessary to specify the baryon state, as spin, flavor, and center-of-mass momentum. For the p we usej ,k, . . . , to specify all the quantum numbers of th pion state. With this notation, the effective Hamiltonian c be obtained as H5( ab ua&^auHub&^bu1( jk u j &^ j uHuk&^ku 1( j ab ~ ua&u j &^a, j uHI ub&^bu1ub&^buHI u j ,a&^au^ j u!. ~23! This leads to an effective Hamiltonian that can be written the sum of the single-baryon and single-pion contributio and the pion-baryon vertex H5H01W, ~24! whereH05HB1Hp contains the single-baryon and singl pion contributions HB5( a Ea (0)Ba (0)†Ba (0) , Hp5( j EjM j †M j , ~25! andW is the pion-baryon vertex W5( j ab Wab j Bb (0)†Ba (0)M j1H.c. ~26! Here, Ba (0)† and Ba (0) (M j † and M j ) are the baryon~pion! creation and annihilation operators, discussed in the prev section. Note that we have assumed that states with diffe quantum numbers are orthogonal. Note also that in writ the stateu j ,a& we have implicitly assumed that the negativ energy component of the baryon is negligible and the bar creation operator acting on the RPA vacuum has the s effect as acting on the BCS vacuum. We note that the projection of the microscopic qua Hamiltonian to an effective hadronic Hamiltonian can be o tained in a systematic and controlled way using a mapp procedure@26#. We do not follow such a procedure here b cause we are mainly concerned with tree-level pion-bar coupling and the projection we are using is enough to ob the desired effective coupling. For processes that invo 02520 lts e, n d ic ch n, s , us nt g n e - g - n in e quark exchange, such as baryon-meson or baryon-baryo teractions, a mapping procedure would be useful. In a fut publication, we intend to address such processes in the text of the present model. The single-hadron HamiltoniansH0 andHp give H0uBa (0)&5EB (0)uB(0)&, Hpup&5Ep (0)up&, ~27! with EB (0)5MB (0) andEp (0)5mp (0) in the rest frame. The vertex W gives the coupling of the pion to the baryon and, as will show later, leads to loop corrections to the baryon se energy. The pion-nucleon vertex function can be written genera as W5( i 51 3 ~Wi 11Wi 2!, ~28! where theWi 6’s are of the general form~for simplicity we suppress the color and spin-flavor wave functions in the lowing! Wi 6~p,p8;k!5E dqdq8 dq9 ~2p!9 VG~q2q8!Cp8 * ~q18q28q38! 3W i 6@G,fk#Cp~q1q2q3!, ~29! where theW i 6’s involve the Dirac spinors and the pion wav functions. In Fig. 1 we present a pictorial representation the different contributions to the vertex function. Explicitl the W i 6’s are given by W 1 15@u†~q18!TaGu~q4!#fk 1~p4!@v†~2p4!TaGu~q1!#, ~30! FIG. 1. Graphical representation of the functionsW i 6 , i 51,2,3. 2-4 k s n d tie is th s n to on - tin ei th rk e e n s m- to lf- CHIRAL CORRECTIONS TO BARYON PROPERTIES . . . PHYSICAL REVIEW C 64 025202 W 2 15fk 1~p4!@v†~2p4!TaGu~q1!#@u†~q28!TaGu~q2!#, ~31! W 3 15W 2 1 , ~32! W 1 25@u†~q18!TaGv~2q4!#fk 2T~p4!@u†~p4!TaGu~q1!#, ~33! W 2 25@u†~q28!TaGu~q2!#@u†~q18!TaGv~2q4!#fk 2T~p4!, ~34! W 3 25W 2 2 . ~35! In these formulas, the quark momenta in the initial~final! nucleonq1 ,q2 ,q3 (q18 ,q28 ,q38) and the momenta of the quar and antiquark in the pion,p4 andq4, are expressed in term of the loop momentaq,q8,q9 by momentum conservatio ~see Fig. 1!. Once the effective baryon-pion Hamiltonian is obtaine one can calculate the pionic corrections to baryon proper as in the CBM and in the traditional Chew-Low model. Th will be done in the next section. Before leaving this section, we recall that the use of Breit frame is essential in calculations of form factors~vertex functions! in static models@27#, like the present one. This i true for composite models for which approximate solutio that maintain relativistic covariance are very difficult implement. This was the case for all old, static source, pi nucleon models such as the Chew-Low model@28#. In par- ticular, as explained in Ref.@27#, electromagnetic gauge in variance is respected in this frame. Therefore, in calcula loop corrections to baryon properties, we employ the Br frame vertex functions. In the Breit frame, we denote incoming pion and nucleon momenta byp and2p/2, respec- tively, and the outgoing nucleon momentum byp/2. In this frame, the internal momenta of quarks and antiqua q18 ,q28 , . . . are given in terms of the loop momentaq, q8, andq9 as q15p/21q1q9, q185p/21q81q9, q25q2852q81q9, q35q38522q9, p45p/22q2q9, q45p/22q2q9, ~36! for the vertexW 1 and q152p/21q81q9, q185p/21q81q9, q25q2852q81q9, q35q38522q9, p452p/21q1q9, q452p/22q2q9, ~37! for the vertexW 2. In following equations, we also denot the vertex function asW(2p/2, p/2;p)[W(p). IV. SELF-ENERGY CORRECTION TO BARYON MASSES For completeness we review the derivation of the expr sion for the self-energy correction from the effective baryo pion Hamiltonian of Eq.~24! in the ‘‘one-pion-in-the air’’ 02520 , s e s - g t- e s s- - approximation@28,6#. The baryon self-energy is defined a the difference of bare- and dressed-baryon energies S~EB!5EB2EB (0) . ~38! The physical baryon massMB is given by the~in general, nonlinear! equation MB5MB (0)1S~MB!, ~39! where MB (0) is the bare baryon mass~i.e., without pionic corrections! andS(EB) is the self-energy function. Let uB& denote the physical baryon state, anduB0& the ‘‘bare’’ undressed state. LetZ2 B be the probability of finding uB0& in uB&. Then one can write uB&5AZ2 BuB0&1LuB&, ~40! whereL is a projection operator that projects out the co ponentuB0& from uB&, L512uB0&^B0u. ~41! We have that ^BuWuB0&5^Bu~H2H0!uB0&5~EB2EB (0)!^BuB0& 5AZ2 B~EB2EB (0)!5AZ2 BS~EB!. ~42! We can now expressuB& in terms of uB0& and the pion- baryon interaction HamiltonianW as uB&5AZ2 BF12 1 EB2H02LWL WG uB0&. ~43! On the other hand, since ^BuWuB0&5AZ2 B^B0uW 1 EB2H02LWL WuB0&, ~44! we have that the self-energy is given by S~EB!5 1 AZ2 B ^BuWuB0&5^B0uW 1 EB2H02LWL WuB0&. ~45! This expression can be further approximated so as avoid solving complicated integral equations for the se energy. We can manipulate the expression forS to obtain ~for details, see Ref.@6#! S~EB!5^B0uW 1 EB2H02S0~EB! WuB0&, ~46! with S0~EB!5WL 1 EB2H0 LW. ~47! The approximation consists in absorbingS0(EB) into H0 such that 2-5 e on in nd ‘‘ d n on ls E t e p in e- e re le, ari- ry n l on ry an ctly I. es ro- hat uon vary not l gths, ical ple: to be ion ak- by - n of na, ta- ear nly els ac- ely e it l e to ro- ch. P. J. A. BICUDO, G. KREIN, AND J. E. F. T. RIBEIRO PHYSICAL REVIEW C64 025202 H01S0~EB![H̃0 , ~48! with H̃05( a EaBa (0)†Ba (0)1( j EpM j †M j , ~49! whereEa and Ep are thephysicalenergies. Therefore, th baryon self-energy can be written as S~EB!5^B0uW 1 EB2H̃0 WuB0&. ~50! Finally, insertion of a sum over intermediate baryon-pi states in Eq.~50! leads to S~EB!5( n ^B0uWun& 1 EB2En ^nuWuB0&. ~51! The structure vertex-propagator-vertexW(E2H̃0)21W in Eq. ~50! is an effective baryon-pion interaction. The ma difference here with the hybrid approaches@1–6# is that we do not have a pointlike pion coupling to pointlike quarks a antiquarks. The pion-baryon vertex arises through theZ graphs’’ in which the antiquark of the pion is annihilate with a quark of the ‘‘initial’’ baryon and the quark of the pio appears in the ‘‘final’’ baryon. Therefore, the vertex functi incorporates not only the extension of the baryons, but a the extension of the pion. We truncate the sum over the intermediate states in ~51! to the lowest mass states, namely, the nucleon and D(1232). In this case, we obtain for the on-shellN and D(1232) self-energies the coupled set of equations SN~MN!5E dk ~2p!3 F uWNN~k!u2 MN2@MN1Ep~k!# 1 WND~q!WDN~k! MN2@MD1Ep~k!#G , ~52! SD~MD!5E dk ~2p!3 F WDN~k!WND~k! MD2@MN1Ep~k!# 1 uWDD~k!u2 MD2@MD1Ep~k!#G . ~53! This is the final result for the pion loop correction for th nucleon andD(1232). One important consequence of projecting the microsco quark interaction onto hadronic states is that the lead nonanalytic~LNA ! contributions in the pion mass as pr dicted by chiral perturbation theory are correctly obtain @9#. In particular, as we discuss in the next section, Eq.~52! leads to an LNA contribution to the nucleon mass as p dicted by QCD@29#, namely, 02520 o q. he ic g d - MN LNA52 3 16p2f p 2 gA 2mp 2 . ~54! In models where the pion is treated as a pointlike partic this result follows trivially@9# from Eq. ~52!. In the context of the present model, where the pion is not treated cov antly, such a result does not follow in general for an arbitra interaction. The difficulty is related to the fact the the pio dispersion relationEp5Ak21mp 2 is not obtained in genera in a noncovariant model. In the CBM for example, the pi is point like and the normalization is correct from the ve beginning. However, the microscopic quark interaction c be chosen such that the pion dispersion relation is corre obtained@14,25#. These issues will be discussed in Sec. V V. THE PION-NUCLEON AND PION- D„1232… FORM FACTORS Our aim is to obtain an estimate for the numerical valu of the pionic self-energies. It happens that nature has p duced a sort of low energy filter~chiral symmetry! for the details of strong interactions. Indeed it is remarkable t although intermediate theoretical concepts such as gl propagators, quark effective masses and so on, might ~in fact they are not gauge invariant and hence they are physical observables!, chiral symmetry contrives for the fina physical results, e.g., hadronic masses and scattering len to be largely insensitive to the above mentioned theoret uncertainties. The pion mass furnishes the ultimate exam In the case of massless quarks, the pion mass is bound zero, regardless of the form of the effective quark interact provided it supports the mechanism of spontaneous bre down of chiral symmetry. The other example is provided the p-p scattering lengths@30# which are equally indepen dent of the form of the quark kernel@31#. Furthermore it has become more and more evident through the accumulatio theoretical calculations on low-energy hadronic phenome ranging from calculations on Euclidean space to instan neous approximations and from harmonic kernels to lin confinement, that low-energy hadronic phenomenology o seems to depend mildly on the details of the quark kern used. To this extent, we will use for the quark-quark inter tion a kernel of the form G5g0, ~55! V~k!5 3 4 ~2p!3K0 3Dkd~k!, ~56! whereK0 is a free parameter. This potential has been wid used in the context of chiral symmetry breaking becaus allows a great deal of simple analytic calculations~which is not the case for the linear potential!. The harmonic potentia basically differs from the linear potential in domains of th baryon-pion-baryon overlap kernel which contribute little the total geometrical overlap so that, at least for results p portional to these overlaps, they should not differ too mu 2-6 ud ar se i s re ti s ll io e li- the he ion nc- m on - ef. n lcu- CHIRAL CORRECTIONS TO BARYON PROPERTIES . . . PHYSICAL REVIEW C 64 025202 The momentum-dependent part of the Salpeter amplit for the baryonCp(q1q2q3) in Eq. ~22!, is taken to be of a Gaussian form Cp~q1q2q3!5 e2(r21l2)/2aB 2 NB~p! ; r5 p12p2 A2 ; l5 p11p222p3 A6 , ~57! whereaB 2 is the variational parameter andNB(p) is the nor- malization. Notice that since the integrations of the qu momenta in the functionsWi 6 in Eq. ~29! are made through a Monte Carlo integration, the Gaussian ansatz is not es tial and does not simplify our calculations, but we still use to make contact with previous calculations. As in our previous calculation@22# for the pion-nucleon coupling constant, the Salpeter amplitudesfk 6(q) up to first order ink are given by fk 1~q!.N~k!21@1sinw~q!1E1~k! f 1~q! 1 ig1~q!k~ q̂3s!#xpScolor, ~58! fk 2~q!.N~k!21@2sinw~q!1E1~k! f 1~q! 2 ig1~q!k~ q̂3s!#xpScolor, ~59! wherew is the chiral angle andE1(k) is the first-order cor- rection to the pion energy. The normalizationN(k) is pro- portional toE1(k) and is given as N 2~k!54E1~k!E dq ~2p!3 sinw~q! f 1~q![E1a2. ~60! The energyE1(k) is given in terms of the second derivative of the diagonal components of the Salpeter kernel with spect tok and its explicit form is given in Eq.~24! of Ref. @22#. Note that the truncation up to first order ink of the Salpeter amplitude constitutes a reasonable approxima due to the fact that c.m. momenta-dependent distortion the pion and nucleon wave functions are geometrica damped because of the geometric overlap kernel integrat for the functionsWi 6 in Eq. ~29!—see Ref.@32#. Explicit numerical solutions were obtained in Ref.@22# for the func- tions f 1(q) andg1(q). For completeness, we initially repeat the results of R @22# for the coupling constantsf pNN and f pND . In Ref. @22#, they were obtained as f pNN mp sN•p5 5 3A3 Of s~p! 2a sN•p̂, ~61! f pND mp S•p5F2A2 A3 Of s~p! 2a 1A2 Of s8 ~p! 2a GS•p̂, ~62! where the isospin matrix is omitted and 02520 e k n- t - on of y ns f. Of s8 ~p!.0, Of s~p!52 E @dq#~a11a21b11b2! E @dq#C in* Cout , ~63! where @dq# means integration overq, q8, and q9 @see Eq. ~29!# and the set of functionsa1,a2,b1,b2 is given by a15f1H w8~q1!s•“Cout 1Fw8~q1!1 cosw~q1! q1 Gs•q̂13~ q̂13“Cout!siJ C in , ~64! a25CoutH w8~q18!s•“C in 1Fw8~q18!1 cosw~q18! q18 Gs•q̂183~ q̂183“C in!J f2, ~65! b15f1 12sinw~q18! 2q18 H 2 cosw~p1! p1 s•q̂18 1Fw8~q1!1 cosw~q1! q1 Gs•q̂13~ q̂13q̂18!J CoutC in , ~66! b25 12sinw~q1! 2q1 H 2 cosw~q18! q18 s•q̂1 1Fw8~q1!1 cosw~q18! q18 Gs•q̂18 3~ q̂183q̂1!J CoutC inf 2. ~67! Here,C in,out stand for the baryon in and out Salpeter amp tudes andf1,2 represent the pion Salpeter amplitudes. The baryon-pion coupling constants are obtained as zero limit of the nucleon~or D) momentump→0 of the above overlap functions. For simplicity, we are defining t couplings at zero momentum, and not at the physical p mass. In order to facilitate the integration, in Ref.@22# a Gaussian parametrization for the@cosw(k)#/k and @1 2sinw(k)#/k2 was used. Here, since we need the vertex fu tion for pÞ0, we use a Monte Carlo integration to perfor the multidimensional integral that gives the overlap functi and use the full numerical solution for the gap function~not the Gaussian parametrization!. We first checked the correct ness of our Monte Carlo integration with the result of R @22# for the special case ofp50 using the same Gaussia parametrization as was used there. This was done by ca 2-7 f w eo s % tu th d ian e rgy rec- nic t by he is ne ay d ay ur- nt , to is- q. P. J. A. BICUDO, G. KREIN, AND J. E. F. T. RIBEIRO PHYSICAL REVIEW C64 025202 lating Of s(p) for p5(0,0,pz) and finding the limit ofO 5Of s /pz whenpz→0 to obtainf pNN . As in Ref. @22# we have usedK05247 MeV for the strength of the potential. The variational determination oa of the baryon amplitude, Eq.~57!, leads toaN51.2K0. For D(1232), the result is not much different and therefore useaN5aD . Introducing the quantities F15 5 3A12 , F2A2 3 , F35 1 3A12 , ~68! we can summarize the couplings of the pion to the nucl andD(1232) as follows: f pNN5F1O~0! mp a , f pND5F2O~0! mp a , f pDD5F3O~0! mp a . ~69! For the value ofK0 given above, we havemp /a53.47. The numerical values for the couplings are then f pNN51.19, f pND52.02, f pDD50.24. ~70! The effect of the Gaussian parametrization can be asse by comparing with the corresponding numbers of Ref.@22#. For example,f pNN.1.0 andf pND51.8 in Ref.@22#; the ef- fect of the parametrization is therefore of the order of 20 Next, we calculated the full overlap function forpÞ0. In Fig. 2 we plot the functionu(p)5O(p)/O(0) for the param- eters given above. It is instructive to compare the momen dependence of this form factor with the one given by CBM @6,23#: u~p!53 j 1~pR! pR , ~71! FIG. 2. The functionu(p). The solid line is the form factor obtained with the baryon amplitude of Eq.~57! and pion Salpeter amplitudes of Eqs.~58! and~59!. The dashed line is the CBM form factor of Eq.~71! for R51 fm. 02520 e n sed . m e where j 1 is the spherical Bessel function andR is the radius of the underlying MIT bag. The solid line is our result an the dashed one is the CBM result forR51 fm. The faster fall-off of our result is clearly a consequence of our Gauss ansatz. As we will discuss soon, this rapid falloff will hav the consequence of giving a smaller value of the self-ene correction to the nucleon mass, as compared to the cor tions obtained with the CBM. VI. SELF-ENERGY CORRECTIONS TO THE NUCLEON AND D„1232… MASSES In this section we present numerical results for the pio self-energy corrections to the nucleon andD(1232) masses and discuss the LNA contribution to the masses. We star rewriting the vertex function in a manner to make clear t problem with the pion dispersion relation. The pion energy given, for lowk, as@14,25# E1 2~k!5mp 2 1k2Af p (s) f p (t) , ~72! where mp 2 52 2mq^c̄c& ~ f p (t)!2 , ^c̄c&526E dq ~2p!3 sinw~q!. ~73! The point is that for an arbitrary quark-quark interaction o obtains in general two different values for the pion dec constantf p (t) and f p (s) ~the explicit calculations can be foun in Refs. @14,25#!, depending on how one defines the dec constant. When using the time component of the axial c rent, one obtainsf p (t) , and when using the space compone one obtainsf p (s) . However, as suggested in Ref.@14#, and explicitly demonstrated in Ref.@25#, this problem can be cured by adding a transverse gluon interaction. Therefore illustrate the point of obtaining the correct LNA term from Eq. ~52! with composite pions, we use the correct pion d persion relation and assumef p (t)5 f p (s)[ f p and denote E1(k)5v(k). The normalization of the pion Salpeter amplitude, E ~60!, can be rewritten as N 2~p!54v~p!E dk ~2p!3 sinw~q! f 1~q!5 2 3 v~p! f p 2 . ~74! That is,a2 from Eq. ~60! is 2/3f p 2 . We next extract from the vertex function~we concentrate on theNN form factor! this normalization in the following way: WNN i ~p!5 1 A2v~p! GA~p! 2 f p tN i sN•p. ~75! The relation of the functionGA(k2) to the overlap function O(p) can be trivially obtained by comparing with Eq.~61!. 2-8 r ie ti o bt ba e s u lt ni on r eV, ng elf- nic of id- o ive. an- be ic uch ar- ing ect iral ith are del iza- e- nic ark site ob- bes ark the on. osi- ace os- in of the tur- dy- e ill en- de- re CHIRAL CORRECTIONS TO BARYON PROPERTIES . . . PHYSICAL REVIEW C 64 025202 Inserting Eq.~75! in the expression for theN andD self- energies, Eqs.~52! and ~53!, and after performing rathe straightforward spin-isospin algebra one obtains MN5MN (0)2 f 0 2E 0 ` dp p4u2~p! v2~p! 2 32 25 f 0 2E 0 ` dp p4u2~p! v~p!@DM1v~p!# , ~76! MD5MD (0)1 8 25 f 0 2E 0 ` dp p4u2~p! v~p!@DM2v~p!# 2 f 0 2E 0 ` dp p4u2~p! v2~p! , ~77! where DM5MD2MN , v~p!5Ap21mp 2 ~78! and u~p!5 O~p! O~0! , f 0 25 108 mp 2 f pNN 2 4p . ~79! Note that in principle we have different spatial dependenc for the NN, ND, . . . , vertices, but for simplicity we have written them here as being equal. A schematic representa of Eqs.~76! and~77! is presented in Fig. 3. It is important t note that these equations are not the ones one would o by simple perturbation theory; they are actually nonpertur tive, because of the dependence onDM5MD2MN on the right-hand side. It is easy now to obtain the LNA contributions to th masses@9#. For the nucleon, the LNA contribution come from the first term in Eq.~76! by performing the integral. The integral can be done by transforming it into a conto integral and making use of Cauchy’s theorem. The resu Eq. ~54!. For theD, the LNA contribution follows in a simi- lar way from the last term in Eq.~77!. To conclude, we discuss numerical results for the pio corrections. Initially we solve variationally the bare nucle case. As discussed above, usingK05247 MeV, we obtain FIG. 3. Schematic representation of the pion self-energy cor tions to the nucleon~N! and delta (D) masses. 02520 s on ain - r is c for the variational size parameter the valueaN51.2K0. We also use hereaN5aD . This leads to the following values fo the bareN andD masses: MN (0)51174 MeV, MD (0)51373 MeV. ~80! The difference between the masses, of the order of 200 M comes from the hyperfine splitting induced by the confini interaction. Given these values, we solve the two s consistent equations given in Eqs.~76! and ~77!. They are solved by iteration. We obtain for the masses MN51125 MeV, MD51342 MeV. ~81! Comparing with the values above, we see that the pio effect is relatively small, as it should be, and of the order 50 MeV for theN and 30 MeV for theD. The pionic effect is smaller for theD, as one expects from spin-isospin cons erations@9#. The results obtained with the CBM for aR 51 fm are a bit larger@23#. The difference can be traced t the rapid falloff of the form factor in our model. We certainly do not expect these numbers to be definit Once more realistic microscopic quark interactions and satze for the baryon wave function are used, they might improved. However, independently of the microscop model, our scheme is general and able to incorporate s interactions and new baryon amplitudes. It would be of p ticular interest to have the numbers for a linear confin interaction with short range gluonic interactions that resp asymptotic freedom. VII. CONCLUSIONS AND FUTURE PERSPECTIVES We developed a calculational scheme to calculate ch loop corrections to properties of composite baryons w composite pions. The composite baryons and pions bound states derived from a microscopic chiral quark mo inspired in Coulomb gauge QCD and provides a general tion of the Nambu–Jona-Lasinio model to include confin ment and asymptotic freedom. An effective chiral hadro model is constructed by projecting the microscopic qu Hamiltonian onto a Fock-space basis of single compo hadronic states. The composite pions and baryons are tained from the same microscopic Hamiltonian that descri the chiral vacuum condensate. The projection of the qu Hamiltonian onto the pion states is nontrivial because of two-component nature of the Salpeter amplitude of the pi As explained before, the two components correspond to p tive and negative energies which complicates the Fock-sp representation of the pion state. The projection is made p sible by rephrasing the formalism of the Salpeter equation terms of the RPA equations. The development of models and calculational methods the sort described in the present paper are relevant in context of a phenomenological understanding of nonper bative phenomena of strong QCD-like confinement and namical chiral symmetry breaking. Eventually full lattic QCD simulations aimed at studying hadronic structure w be available and phenomenological models will play a c tral role in the interpretation of the data generated. The c- 2-9 t f i ra er a on s n ob pi D ar e ith ear nd the m- de al- as de- r pre- P. J. A. BICUDO, G. KREIN, AND J. E. F. T. RIBEIRO PHYSICAL REVIEW C64 025202 velopments of the present paper are of particular interes the first-principle developments based on the QCD Ham tonian, such as the nonperturbative renormalization prog for the QCD Hamiltonian@17# and Hamiltonian lattice QCD @18#. We intend to implement the technique developed h to such first-principle QCD calculations. We illustrated the applicability of the formalism with numerical calculation using a simple microscopic interacti namely a confining harmonic potential, and a simple Gau ian ansatz for the baryon amplitude. This very sameS-wave interaction has been used in a variety of earlier calculatio such as meson and baryon spectroscopy andS-wave nucleon-nucleon interaction. Numerical results were tained here for the pion-nucleon form factor and for the onic self-energy corrections to the nucleon andD(1232) masses in the nonperturbative one-loop approximation. spite the simplicity of the interaction, the results obtained very reasonable. For the future, the most pressing development would b nt F la, l. tt. . 02520 or l- m e , s- s, - - e- e to use a microscopic interaction that is consistent w asymptotic freedom and describes confinement by a lin potential. The calculation of the pion wave function beyo lowest order in momentum must be implemented and variational ansatz for the baryon amplitude must be i proved. A more ambitious development would be to inclu explicit gluonic degrees of freedom. In this case renorm ization issues will show up and the new techniques such discussed in Ref.@17# will certainly be useful. Another very interesting direction would be to employ the techniques veloped here in Hamiltonian lattice QCD. ACKNOWLEDGMENTS This work was partially supported by CNPq~Brazil! and ICCTI ~Portugal!. The authors thank Nathan Berkovits fo reading the manuscript and making suggestions on the sentation. . a, iak, . e, . C ys. s. @1# A. Chodos and C. B. Thorn, Phys. Rev. D12, 2733~1975!; T. Inoue and T. Maskawa, Prog. Theor. Phys.54, 1833~1975!. @2# M. V. Barnhill, W. K. Cheng, and A. Halprin, Phys. Rev. D20, 727 ~1979!. @3# G. E. Brown and M. Rho, Phys. Lett.82B, 177 ~1979!. @4# V. Vento, M. Rho, and G. E. Brown, Nucl. Phys.A345, 413 ~1980!. @5# R. L. Jaffe, Lectures at the 1979 Erice School, MIT prepri 1980. @6# S. Théberge, A. W. Thomas, and G. A. Miller, Phys. Rev. D22, 2838~1980!; 23, 2106~E! ~1981!; A. W. Thomas, S. The´berge, and G. A. Miller, ibid. 24, 216 ~1981!. @7# A. Chodos, R. L. Jaffe, K. Johnson, C. B. Thorn, and V. Weisskopf, Phys. Rev. D9, 3471 ~1974!; A. Chodos, R. L. Jaffe, K. Johnson, and C. B. Thorn,ibid. 10, 2599~1974!; T. A. de Grand, R. L. Jaffe, K. Johnson, and J. Kiskis,ibid. 12, 2060 ~1975!. @8# L. Ya. Glozman and D. O. Riska, Phys. Rep.268, 263 ~1996!. @9# A. W. Thomas and G. Krein, Phys. Lett. B456, 5 ~1999!; 481, 21 ~2000!. @10# N. Isgur, nucl-th/9908028. @11# M. Finger, D. Horn, and J. E. Mandula, Phys. Rev. D20, 3253 ~1979!; M. Finger, J. Weyers, D. Horn, and J. E. Mandu Phys. Lett.62B, ~1980!; A. J. C. Hey and J. E. Mandula, Nuc Phys.B198, 237 ~1980!; B198, 3253~1979!; J. R. Finger and J. E. Mandula,ibid. B199, 168 ~1982!; A. Le Yaouanc, L. Oliver, O. Péne, and J.-C. Raynal, Phys. Lett.134B, 249 ~1984!; Phys. Rev. D29, 1233 ~1984!; A. Amer, A. Le Ya- ouanc, L. Oliver, O. Pe´ne, and J.-C. Raynal, Phys. Rev. Le 50, 87 ~1984!; S. L. Adler and A. C. Davis, Nucl. Phys.B224, 469 ~1984!. @12# Y. Nambu and G. Jona-Lasinio, Phys. Rev.122, 345 ~1961!; 124, 246 ~1961!. @13# A. Le Yaouanc, L. Oliver, O. Pe´ne, and J.-C. Raynal, Phys Rev. D29, 1233~1984!. , . @14# A. Le Yaouanc, L. Oliver, O. Pe´ne, and J.-C. Raynal, Phys Rev. D31, 137 ~1985!. @15# P. J. A. Bicudo and J. E. F. T. Ribeiro, Phys. Rev. D42, 1635 ~1990!; 42, 1625~1990!; 42, 1635~1990!. @16# A. Mishra, H. Mishra, and S. P. Misra, Z. Phys. C59, 159 ~1993!; A. Mishra, H. Mishra, P. K. Panda, and S. P. Misr ibid. 63, 681 ~1994!. @17# A. P. Szczepaniak and E. S. Swanson, Phys. Rev. D55, 1578 ~1997!; D. G. Robertson, E. S. Swanson, A. P. Szczepan C.-R. Ji, and S. R. Cotanch,ibid. 59, 074019 ~1999!; E. Gubankova, C.-R. Ji, and S. R. Cotanch,ibid. 62, 074001 ~2000!; A. P. Szczepaniak and E. S. Swanson,ibid. 62, 094027 ~2000!. @18# A. Le Yaouanc, L. Oliver, O. Pe`ne, and J.-C. Raynal, Phys Rev. D 33, 3098 ~1986!; A. Le Yaouanc, L. Oliver, O. Pe`ne, J.-C. Raynal, M. Jarfi, and O. Lazrak,ibid. 37, 3691 ~1988!; 37, 3702 ~1988!; E. B. Gregory, S.-H. Guo, H. Kro¨ger, and X.-Q. Luo, ibid. 62, 054508~2000!; Y. Umino, Phys. Lett. B 492, 385 ~2000!; hep-lat/0007356; hep-ph/0101144. @19# P. J. A. Bicudo, G. Krein, J. E. F. T. Ribeiro, and J. E. Villat Phys. Rev. D45, 1673~1992!. @20# P. Bicudo, J. Ribeiro, and J. Rodrigues, Phys. Rev. C52, 2144 ~1995!. @21# P. Bicudo, L. Ferreira, C. Placido, and J. Ribeiro, Phys. Rev 56, 670 ~1997!. @22# P. Bicudo and J. Ribeiro, Phys. Rev. C55, 834 ~1997!. @23# A. W. Thomas, Adv. Nucl. Phys.13, 1 ~1984!; G. A. Miller, Int. Rev. Nucl. Phys.2, 190 ~1984!. @24# F. J. Llanes-Estrada and S. R. Cotanch, Phys. Rev. Lett.84, 1102 ~2000!. @25# P. J. A. Bicudo, Phys. Rev. C60, 035209~1999!. @26# D. Hadjimichef, G. Krein, S. Szpigel, and J. S. da Veiga, Ph Lett. B 367, 317 ~1996!; Ann. Phys.~N.Y.! 268, 105 ~1998!; M. D. Girardeau, G. Krein, and D. Hadjimichef, Mod. Phy Lett. A 11, 1121~1996!. 2-10 on us- CHIRAL CORRECTIONS TO BARYON PROPERTIES . . . PHYSICAL REVIEW C 64 025202 @27# G. A. Miller and A. W. Thomas, Phys. Rev. C56, 2329~1997!. @28# G. F. Chew and F. E. Low, Phys. Rev.101, 1570~1955!. @29# P. Langacker and H. Pagels, Phys. Rev. D8, 4595~1973!; 10, 2904 ~1974!. @30# S. Weinberg, Phys. Rev. Lett.17, 616 ~1966!. 02520 @31# J. E. Ribeiro, Proceedings of IV International Conference Quark Confinement and the Hadron Spectrum, Vienna, A tria, 2000~World Scientific, Singapore, in press!. @32# J. E. Ribeiro, Phys. Rev. D25, 2406~1982!; E. van Beveren, Z. Phys. C17, 135 ~1982!. 2-11