PHYSICAL REVIEW C VOLUME 44, NUMBER 5 NOVEMBER 1991 Quark mean-field theory and consistency with nuclear matter Jishnu Dey* and Lauro Tomiof Insti tuto de I'Isica Teoriea, Uni Uersidade Estadual Paulista, Rua Pamplona 145, 01405 Sdo Paulo, Brazil Mira Dey Department ofPhysics, Maulana Azad College, Calcutta 700013, India T. Frederico~ Institute for Nuclear Theory, Department ofPhysics, FM 15, U-niversity of Washington, Seattle, Washington 98195 (Received 21 March 1991) 1/N, expansion in QCD (with N, the number of colors) suggests using a potential from meson sector (e.g., Richardson) for baryons. For light quarks a 0. field has to be introduced to ensure chiral symmetry breaking (ySB). It is found that nuclear matter properties can be used to pin down the ySB modeling. All masses, M&, m, m„, are found to scale with density. The equations are solved self-consistently. Low-energy physics is essentially controlled by the Goldstone particle, i.e., the pion. The relevant parameter is the pion decay constant, f . This is related to the quark condensate (qq ) through the Weinberg sum rule. In the QCD sum-rule approach also, the nucleon mass is determined predominantly by the odd-dimensional opera- tor (qq ) (Ioffe [1],Reinders, Rubinstein, and Yazaki [2]). At higher density the condensate decreases in magnitude (Bailin, Cleymans, and Scadron [3], Dey, Dey, and Ghose [4]) and f also decreases correspondingly (Dey and Dey [5]). This results in changes in the nucleon property, for example, in the increase in the radius expected from the European Muon Collaboration experimental data through rescaling (Close, Roberts, and Ross [6]). In the formalism of relativistic Hartree-Fock theory (Dey, Dey, and Le Tourneux [7]), justified by the large N, theory of t'Hooft [8) and Witten [9], one gets a weakening of the confinement (Dey et al. [10]) at higher density. If one uses the potential due to Richardson [11], this means a decrease in A, the only parameter present in the interac- tion. We wish to recall that for light quarks one has to use a running quark mass m(r) to get correct nucleon ra- dius and other properties [7]. Relativistic description of nuclear structure and reac- tions within quantum hadrodynamics (QHD) has been greatly developed during the last several years (Serot and Walecka [12], Celenza, Rozenthal, and Shakin [13]). Can one show the compatibility of this model with the mean- field quark model'7 In a crude way this was done by Guichon [14] and Frederico et al. [15] where they as- sumed that there is a scalar o. and a vector cu field cou- pled to the quark. As clearly stated by Guichon, this is a very strong assumption since neither the o. nor the co are fundamental at the quark level. One can think of the running quark mass m(x) as being due to a o field. There are two problems in doing this. First one has to take care of the Goldstone pion since one is breaking chiral symmetry. This is hard to do since the pion-quark coupling is a highly nonlinear one and only in the Zahed model [16] in lower dimension can one treat this exactly. But we need not worry about this here since we can intro- duce the well-established one-pion-exchange potential (OPEP) at the QHD level in nuclear matter and thus correct for the deficiency of the quark model. In some models, pions may indeed couple to the quarks them- selves, but we do not consider such models. The problem is that the pions have a dual character, being Goldstone particles as well as quark-antiquark composites; their role in QCD-inspired models is still ambiguous. For the present we prefer to ignore the pion-quark interaction and replace this by the nucleon-pion interaction, which does not introduce any additional parameters in the theory. The energy due to the OPEP in nuclear matter has been estimated in second-order perturbation theory by Cenni, Conte, and Dillon [17] using wave functions in nuclear matter derived from the Reid soft-core potential [18]. This is almost model independent since the OPEP tail is the same in all modern nucleon-nucleon interac- tions. And the second-order OPEP contribution is in- sensitive to the finer details of the nuclear matter wave function —it only depends on the wound in the wave function in a crude sort of way. One can see that the er- ror due to this cannot be very large even if pions couple to the quarks themselves as Goldstones, since the cou- pling must be cut off for a radius of around 0.35 fm or so. This is because from deep-inelastic scattering we know that at short distances the quarks are free and massless. In fact, in models like the o.-m soliton bag this restoration of chiral symmetry at short distance is hard to achieve. In some sense the Reid soft-core wound cuts off the pion from the nucleon, and therefore also from the quarks at about this distance, since in the nuclear-matter wave function, the nucleon is taken to be pointlike, apart from this wave-function effect. At the present moment, exact treatment of the nucleon-nucleon interaction with pions in the quark picture, is an insoluble problem. The o. fits in nicely with the running quark mass, but again there is problem with the vector mesons. In the Walecka model 2181 1991 The American Physical Society 2182 JISHNU DEY, LAURO TOMIO, MIRA DEY, AND T. FREDERICO there is the co meson and this also we couple to the nu- cleon rather than the quarks themselves. Now, of course, extra parameters are introduced and the purpose of the paper is to report that the nuclear matter constraints are enough to uniquely determine these parameters. We will next look at the problem from the QCD point of view. Starting from the action for a system of interact- ing quarks and gluons one can obtain, after a series of ap- proximations, a Dirac Hamiltonian with a two-body stat- ic potential. It was shown by t'Hooft [8] that such a clas- sical (as opposed to field theoretic where qq loops essen- tially introduces infinite degrees of freedom) two-body in- teraction may be derived by summing all the gluon loops that one can draw on a plane. Witten [9] further showed that this interaction, which is appropriate for the meson (essentially a two-body system), can also be used in the mean-field approximation for a baryon in the same order. Present-day techniques do not permit summing up all the planar gluon diagrams which would yield such a poten- tial unambiguously. As an alternative one can borrow a potential from the meson calculation, for example, that of Crater and van Alstine [19] mentioned before and test it for a baryon (Dey, Dey, and Le Tourneux [7]). The po- tential used in this case is due to Richardson [11]and it passes the test very well. For the sake of completeness we present this potential here: 6' A, (1)A,(2) & f(Ar) 4 33 2NI — Ar where we have the scalar product of the color SU(3) ma- trices As for the two interacting quarks, N& is the number S=S,„,„+f d x[q(i8 —m)q+j„'3'"], (3) where S &„,„denotes, collectively, the action of gluons, the gauge-fixing terms and the action of the unphysical or ghost fields, A the Geld potential, while j' is the quark current j„'=—qy„A'q . (4) The connected Green's functions of gluons are generat- ed by the functional e'~'&'= f dp[a ], exp i S,,„.„+fJ'.a "d'x where dp includes the ghost fields. The full generating functional is given by Z= f dan[A][dq dq]e (6) and using (3) one can formally integrate out the gluons and ghosts and write Z = f [dq dq ] exp i f d xq(ir( m)q+—IV(j ) (7) thus obtaining an eA'ective action for the quarks: of flavors taken to be three, and 4f ~ dq exp( qy ) ln(q —I )+~ The action of a system of interacting quarks and gluons can be written as S,s.= f d x q(i8 m)q —— —, ' fj 'I'(x)V„'"„(x,y)j (y)d y ——' fj'"(x)V„' '(x,y, z)j (y )j't'(z)d yd z— where the Vs are connected Green s functions. This equation is an infinite expansion and it is absolutely essential to have a truncation scheme if we want to extract meaningful numbers. For N, -quark systems like the baryon, the 1/N, expansion provides such a scheme. This can be seen by going back to the formalism of canonical quantization and con- sidering S,z as a function of the Geld operators q and q. For an N, -body baryon, the expectation value of the N currents (N )N, ) that are contracted with V„'.. .„"involves 1 I N (N, q'&~qq qq(N times)~N, q's) =%'~ (1, ,N, )(O~qq qq(N N, times)~0)—V~ (1, ,N, ) . C C (9) Now the factor (O~qq . qq(N N, times) ~0) c—orre- sponds to the production of virtual qq pairs and quark loops. It is suppressed by 1/N, and all terms involving more than N, currents can be dropped in a similar manner. In spite of this restriction one cannot actually compute even the two-point Green's function to all or- ders. One therefore takes the static limit and use for V00 a standard vector potential like Richardson's, for exam- ple. For light (u, d ) quarks one has to split the potential into a vector and a scalar part arbitrarily to take care of the chiral symmetry breaking [7]. An alternative has been tried recently by Dey et al. [20], where forms are m,„„„(x) = —4~'x'(4%'), (10) where (%%)=( —2S5 MeV) and the Eq. (10) is valid up to about 0.35 fm from which point the quark mass is tak- en to be Aat and constant. The interesting thing is that the form of the Hartree-Fock potential is not sensitively dependent on the form of the mass chosen, i.e., there is not too much difference between the Shuryak [21] form taken for the mass following Shuryak [21] or Brevik [22]. We will discuss the form given by Shuryak briefiy (details may be found in pp. 123—126 of his book). This gives QUARK MEAN-FIELD THEORY AND CONSISTENCY WITH. . . 2183 given above and the form given by Brevik [22]. In a sense this effective mass term is due to the qq terms of Eq. (9) taken in a heuristic manner. So it is comforting to know that while the quark mass has to be taken in a form like in Eq. (10), the results are not sensitive on its precise form chosen. It may be relevant at this point to recall that qq degrees of freedom may be more important in baryons than people thought before. We refer to only two sugges- tions in the current literature: (a) Jaffe [23] has suggested large coupling of the nucleon to the P and (b) Preparata and Soffer [24] have suggested large coupling to q', both of which are present in the original analysis of Nagels, Rijken, and de Swart [25] and have been neglected in nu- clear physics for a long time. To return to the problem we recall that one cannot use a coordinate-dependent form for the m(x ) in a Lagrang- ian formalism, but has to introduce a field o (x ) instead. This field is an effective field and its properties may there- fore change with the ambient density. This relates the quark problem to the nuclear matter problem and it is possible to find consistent solutions for the nuclear binding-energy curve from this. One has to invoke a vec- tor cu field also and, as stated by Guichon [14] and Frederico, Carlson, Rego, and Hussein [15] who first sug- gested this kind of treatment, these are not fundamental at the quark level. We find that if one couples this ap- proach with the relativistic HF at the quark level, the stringent condition of matching the o (r ) with the quark gg determines the energy-density function of the o'U(o ), and mass of the o scales as f with density. U(o ) looks like that found in soliton bag models. We will elaborate on this interesting point. One can start with a parabolic form of U for zero density but then finds that cubic and quartic terms in o have to be added. This is not unex- pected since at high density if chiral symmetry is to be re- stored one must have a second minimum in U(o ). This is not possible with a parabolic form since the dU/do is then linear in o. This type of nonlinearity has been known in the soliton bag model for a long time [26] but it is nice to find numerical analysis compelling one to simi- lar solutions when the starting point is quite different. One can fit nuclear matter in a satisfactory manner if one takes an co mass which also scales with density like f and takes a standard form for the one-pion-exchange potential contribution to nuclear matter (from, for exam- ple, Cenni, Conte, and Dillon [17]). This latter contribu- tion from the OPEP is quite justified, since QCD sum rules predict essentially the correct value for the pion- nucleon coupling constant (Reinders, Rubinstein, and Yazaki [2]) and this coupling constant has very little den- sity dependence around normal nuclear matter density (Dey, Dey, and Ghose [4]). The scaling of the mass of o with density was expected by Brown and Rho [27] from the missing strength of the longitudinal electron response. It is interesting to find that the same behavior is essentIal to get a fit to nuclear matter starting from a quark mean-field model. As stated before, one can identify m(r) with a cr field in the spirit of Friedberg and Lee model [26]. The difficulty is that one has to solve coupled equations for the cr fields and the quark field g as done by Goldfiam 0 2 do dr r dr dU(cr ) d0 (12) u(r ) = f d r'Pt(r') V(r —r')tP(r'), (13) where V(r —r') is the two-body quark-quark potential. g is the quark-o coupling constant. The mean field u(r) is totally confined inside the nucleon. U(o ) is given by 2 U(o)= (o' cr,—) + (cr —o'„) + (cr —o, ) c7 2 2 3 3 4 2 3o. 4 (14) where the scalar o. field attains its vacuum expectation value o., so that go. , =m (15) m being the constituent quark mass taken to be 300 MeV (1.5 fm '). Observe that we wrote U(o ) in such a way that m is the effective cr mass: d2U m~ = (16) dcT o =o„ g o (r ) can be fitted to an analytic form, go(r)=m(r)=m&[1 —(1+ar+a r /3) exp( ar)] —. (17) This form for the o. field was reached by a step in the pro- cess of solving consistently the equations (11)—(14). We found this as the simple analytical solution that gives the best approach for the coupled equations, and can fit well the numerical results we obtain for each parameter A, corresponding to some specific nuclear density. The pa- rameter A of the Richardson potential was fitted to vari- ous nuclear densities, as explained in Refs. [11]and [20]. In Eq. (17), we let a be a parameter to be adjusted to have the consistency fulfilled. We have used a tedious but reli- able process to obtain this part of the self-consistency. We have used a graphical package to plot the o. field in order to adjust the analytical form to our numerical re- sults, obtained from the Hartree-Pock calculation. This gives us a new input for the o. field, until the consistency is reached. The parameter o. we have obtained in this process was equal to 5 fm ', consistent with the model of Ref. [21]. It is possible that the quark mass may vary with densi- ty, but we found that if we incorporate such an effect directly in this stage of our approach, such procedure can face problems with double counting, and may be not compatible with the model. We mean "double counting" in the following sense: the effective mass has already in- and Wilets [26], but with one more complication. There is now a mean field obtained from two-body quark-quark potential self-consistently. We have [a p+Pgo(r)]/=[@ —u(r)]g, with 2184 JISHNU DEY, LAURO TOMIO, MIRA DEY, AND T. FREDERICO 2 + p +— (M —M')g 1 I 2w2 B 2 g N kF+ 3 f d k+k +M* (2~)' =8 +6 +8,+6~ . 2 (19) Here pB is the nuclear-matter density, kF is the Fermi momentum, and M* is the effective nucleon mass in the medium to be found self-consistently from corporated in it the effect of the o. mean field, as we are assuming the o.-co model for the nucleon propagation in nuclear matter. The constituent quark masses are com- ponents of the nucleon mass; and, in case we consider that also the constituent quark mass has the effect of the o. field, such effect will be doubled in the effective mass of the nucleon. Then in the present formulation of our model we favor the first option; it means only the nucleon in the sense of the o.-co model has an effective mass. We expect the possible variation with the density in the con- stituent quark mass will not change our results qualita- tively. The formulation of such a model avoiding the double counting will be pursued in a future work. With the form given in Eq. (17) we calculate the nu- cleon mass from the relativistic Hartree-Fock equation self-consistently [7]. The center-of-mass correction is also done as in Ref. [7]. The energy contribution of the o. field to the nucleon mass is obtained by integrating U(o. ) and the kinetic-energy density [—,'(Vo. ) ] of the cr field. The resulting mass MN is a bit high but we do not worry about this since pion cloud correction could bring this down. Fortunately, the nuclear-matter binding is in- dependent of this mass since this is subtracted out from the energy. As seen from this equation, for large r, o. be- comes a constant and a constant o. is what is used in the Walecka model. The coupling with the nucleon is given by g~~=3& f d "A'. The co field is introduced at this level following Serot and Walecka [13] so that the energy density of the nuclear matter is We would like to emphasize here the way the self- consistency is obtained in this calculation. We solve the equations for g and o., Eqs. (11)—(17), using the Richard- son potential [11][see Eqs. (1) and (2)] for diff'erent values of the parameter A, for the corresponding nuclear densi- ties. This parameter was fitted for different nuclear den- sities, as in Ref. [11]and [20], and we reproduce in Table I the corresponding variation. From this consistency we obtain the density dependence of the quark-0. coupling constant g and the corresponding coupling with the nu- cleon given by Eq. (18). Next we calculate the nucleon mass from the relativistic Hartree-Fock equation self- consistently. We also have included the center-of-mass contribution as has been done in Ref. [7]. The o. energy contribution to the nucleon mass E is added at this point, after integration of the density energies (kinetic and potential). After this we have the running nucleon mass MN, that varies with the density, and the effective nucleon mass in the medium is obtained self-consistently through Eqs. (20) and (21). The energy density of the nu- clear matter is obtained through Eq. (19) and the numeri- cal constant present in the co energy, Eq. (22), adjusts the saturation at the normal nuclear density. In Table I for each density we list the corresponding A, the parameters of U(o ), the quark-cr coupling, and the nucleon-0. coupling. In Table II, again for the same den- sities, we give the cr contribution for the nucleon mass E, the nucleon mass MN, the self consistent mass M*, the energy contributions to the binding and the final binding energy (BE) per nucleon. This final binding ener- gy is obtained after summing all the energy contributions and subtracting the nucleon energy at zero density. A nice feature is that the saturation is obtained with a con- sistent set of parameters. Only a large o. mass can give us the saturation in the right position and with approxi- mately the correct amount of binding energy per nucleon. The relation connecting t2 and t3, given by (t2/2m ), ensures that we have just one minimum of the potential U at o.=o and that the other two extremes are in the same position. We can also produce another minimum with a smaller value of t3, but to keep the absolute minimum at o.=o one cannot change t3 too much. The expressions for the energies E„are defined by goN 2 kFEF M* ln kF +EF M* where (20) TABLE I. Density dependence of the parameters and cou- pling constants. E~=QM* +k (21) p~ /po A(MeV) m (MeV) tz(fm 2) RaN To obtain the saturation at p~ equal to po=(0. 17 fm )g is adjusted to 243. 5 ci7 2M 2 PB N (22) The number used by Serot and Walecka [12] was 193.7, the experimental one, given in Ref. [14], being 251 is closer to 243.5. 0 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 225 217 209 202 194 187 180 173 166 2565 2488 2389 2293 2197 2118 2025 1933 1848 0 15 27 37 46 53 ~ 5 59 63.5 69.5 13.418 12.844 12.157 11.529 10.919 10.402 9.806 9.243 8.656 19.861 19.733 19.110 18.659 18.213 17.854 17.152 16.500 15.835 QUARK MEAN-FIELD THEORY AND CONSISTENCY WITH. . . 2185 TABLE II. Density dependence of the energies. All energies are given in MeV. E, BE 0 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 89.90 89.89 90.36 90.29 90.25 90.17 90.60 90.76 91.53 1241 1223 1197 1176 1155 1138 1117 1097 1078 1241 1203 1156 1112 1067 1025 980.0 935.9 894.0 1241 1210 1168 1128 1087 1049 1008 968.8 931.5 0 10.15 20.48 31.54 43.24 55.38 66.40 77.72 88.42 0 26.56 55.47 86.19 119.2 153.4 191.3 231.4 273.6 0 —7.431 —14.91 —19.96 —24.02 —27. 52 —30.58 —33.33 —35.83 0 —1.729 —12.536 —15.209 —15.717 —10.726 —5.701 3.278 16.432 (23) where the 8's are given in Eq. (19). To summarize, our main finding has been to observe that all the relevant masses scale with nuclear-matter density, in a self-consistent calculation. In particular, the scaling nz„ is expected from the missing strength of the longitudinal electron response [27]. We were led to fit nuclear matter more to constrain our model of chiral symmetry breaking. We fit U(tr), using the ansatz for the cr field given by Eq. (17), in the framework of the rela- tivistic Hartree-Fock calculation [7]. We use the Richardson potential. Calculations with other potentials like, for example, the one suggested by Hansson, Johnson, and Peterson or Ding, Huang, and Chen [28] and more detailed study of the solutions should be pur- sued and are under way. We find M& which are very reasonable and our o is like that of a glueball of mass more than 2 GeV. This is similar in spirit to Bayer and Weise [29). %'e acknowledge discussions with Professor P. Leal Ferreira and Professor J. Soffer. This work was support- ed by Conselho Nacional de Desenvolvimento Cientifico e Tecnologico, by Fundaqao de Amparo a Pesquisa do Estado de Sao Paulo (FAPESP), J.D., and also by Grant No. SP/S2/K01/87 from the Department of Science and Technology, Government of India, to M.D. 'On leave from Hoogly Mohsin College, Chinsurah 712101, India. ~On leave at School of Physical Sciences, The Flinders Uni- versity of South Australia, Bedford Park S.A. 5042, Aus- tralia. &On leave from Instituto de Estudos Avanqados, Centro Tecnico Aeroespacial, 12231 Slo Jose dos Campos, SXo Paulo, Brazil. [1]B. L. Ioffe, Nucl. Phys. B118, 317 (1981); B191, 59(E) (1981). [2] L. J. Reinders, H. R. Rubinstein, and S. Yazaki, Phys. Rep. C 127, 1 (1985). [3] D. Bailin, J. Cleymans, and M. D. Scadron, Phys. Rev. D 31, 164 (1985). [4] J. Dey, M. Dey, and P. Ghose, Phys. Lett B 165, 181 (1985). [5] J. Dey and M. Dey, Phys. Lett. B 176, 469 (1986). [6] F. E. Close, R. G. Roberts, and CJ. G Ross, Phys. Let.t. 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[20] J. Dey, M. Dey, G. Mukhopadhyay, and B. C. Samanta, Report No. IFT/P-30/89, 1989, Sao Paulo, Brazil (unpub- lished). [21]E. V. Shuryak, The QCD Vacuum, Hadrons and the Super dense Matter (%orld Scientific, Singapore, 1988). [22] I. Brevik, Phys. Rev. D 33, 290 (1986). [23] R. L. Jaffe, phys. Lett. B 229, 275 (1989}. In this paper the p(1030) is suggested as a possible second pole in the form factor of the nucleon and correspondingly there is a large g»& coupling implied. Such a coupling was suggested long back in Ref. [25] as Jaffe points out. By implication there is large ss component in the nucleon or a violation of the OZI rule [S. Okubo, Phys. Lett. 5, 1975 (1963); G. Zweig, CERN Report No. 8419/Th. 412, 1964 (unpub- lished); J. Iizuka, Progr. Theor. Phys. Suppl. 37-38, 21 2186 JISHNU DEY, LAURO TOMIO, MIRA DEY, AND T. FREDERICO (1966)] for baryons or a combination of both effects. [24] J. Soffer (private communication). The large G&z„sug- gests the same mechanism as in Ref. [23]. [25] M. M. Nagels, T. A. Rijken, and J. J. de Swart, Phys. Rev. D 12, 744 (1975);D 15, 2547 (1977);20, 1633 (1979). [26] R. Friedberg and T. D. Lee, Phys. Rev. D 15, 1694 (1977); 16, 1096 (1977); R. GoldAam and L. Wilets, ibid. 25, 1951 (1982). For a review see L. Wilets, The Nontopological Sol- iton Model (World Scientific, Singapore, 1989). [27] G. E. Brown and M. Rho, Phys. Lett. B 222, 324 (1989). [28] One can have extra parameters to simulate the medium range interpolating between the asymptotic freedom re- gion and confinement in the Richardson potential. This was suggested in T. H. Hansson, K. Johnson, and C. Peterson, Phys. Rev. D 26, 2069 (1989), in their Eq. (48) in a paper where a glueball condensate was suggested as a {( field, the equivalent of our o. field. Another possible unified potential for the meson sector was suggested by Y. Ding, T. Huang, and Z. Chen, Phys. Lett. B 196, 191 (1987). [29] L. Bayer and W. Weise, Regensburg Report No. TPR-88- 37.