Adelaide University ADPT February A quarkmeson coupling mo del for nuclear and neutron matter K Saito Physics Division Tohoku College of Pharmacy Sendai Japan and y A W Thomas Department of Physics and Mathematical Physics University of Adelaide South Australia Australia March Abstract nucl-th/9403015 18 Mar 1994 An explicit quark mo del based on a mean eld description of nonoverlapping nucleon bags b ound by the selfconsistent exchange of ! and mesons is used to investigate the prop erties of b oth nuclear and neutron matter We establish a clear understanding of the relationship b etween this mo del which incorp orates the internal structure of the nucleon and QHD Finally we use the mo del to study the density dep endence of the quark condensate inmedium Corresp ondence to Dr K Saito email ksaitonuclphystohokuacjp y email athomasphysicsadelaideeduau Recently there has b een considerable interest in relativistic calculations of innite nuclear matter as well as dense neutron matter A relativistic treatment is of course essential if one aims to deal with the prop erties of dense matter including the equation of state EOS The simplest relativistic mo del for hadronic matter is the Walecka mo del often called Quantum Hadro dynamics ie QHDI which consists of structureless nucleons interacting through the exchange of the meson and the time comp onent of the meson in the meaneld approximation MFA Later Serot and Walecka extended the mo del to incorp orate the isovector mesons and QHDI I and used it to discuss systems like neutron stars with N Z QHD has proven to b e a very p owerful to ol for the treatment of a wide range of nuclear phenomena However it should b e noted that a strong excitation of nucleonantinucleon pairs due to the meson elds is somewhat unlikely from the p oint of view of the quark mo del b ecause the meson elds create only one quarkantiquark pair at a time An extension of such a mo del to include the structure of the nucleon may provide some new insight into the prop erties of hadronic matter A quarkmeson coupling QMC mechanism for the saturation of nuclear matter was initially prop osed by Guichon In his simple mo del nuclear matter consists of non overlapping nucleon bags b ound by the selfconsistent exchange of scalar and vector mesons in the MFA This mo del has b een generalised to include nucleon Fermi motion as well as the centerofmass cm correction to the bag energy Using this mo del it has proven p ossible to investigate b oth the change of nucleon prop erties in nuclear matter and nuclear structure functions An alternative version of this mo del based on the color dielectric soliton mo del has b een developed by Naar and Birse who also applied it to deepinelastic scattering To treat asymmetric nuclear matter ie N Z the mo del requires some extension In this work we show that it is p ossible to include the contribution of the isovector vector meson in the study of such systems We shall see that it is very interesting to compare this new mo del with QHDI I and to investigate the prop erties of hadronic matter in terms of quark degrees of freedom Let the meaneld values for the the time comp onent and the time comp onent in the third direction of isospin elds in uniformly distributed nuclear matter with N Z b e and b resp ectively The nucleon is describ ed by the static spherical MIT bag in which quarks interact with those mean elds The Dirac equation for a quark 1 eld in a bag is then given by q 0 z i m V V V q q q q q q q q where V g V g and V g b with the quarkmeson coupling constants g g q z and g The bare quark mass is denoted by m and is the third comp onent of the Pauli q q matrix Here we deal with u and d quarks qu or d only The normalized ground state for a quark in the nucleon is then given by j xr R q 0 i tR q p r t N e q i r j xr R q 1 where u quark V for RV q q d 2 3 2 2 N R j x Rm x q q 0 q v u u Rm q q t q Rm q q q 2 2 with x Rm and the quark spinor The eective quark mass m is q q q q dened by q m V m g m q q q The b oundary condition at the surface provides the equation for the eigenvalue x j x j x 0 q 1 E p Using the SU spinavor nucleon wave function the nucleon energy is given by n X p p z E V E V bag N q n n q proton V for E V bag neutron 1 The sign of the eld value in this pap er is opp osite to that in Ref in order to simplify the comparison with QHD Here the bag energy is P z q 0 q 3 E B R bag R with B the bag constant and z the usual parameter which accounts for zerop oint motion 0 In order to correct for spurious cm motion in the bag the mass of the nucleon at rest is taken to b e q 2 2 M E hp i N bag cm P 2 2 2 hp i and hp i is the exp ectation value of the quark momentum squared where hp i q q q cm 2 xR Inside nuclear matter the eective nucleon mass M is also given by minimizing N eq with resp ect to R To test the sensitivity of our results to the radius of the free nucleon R we vary B and z to x R f m and the free nucleon mass 0 0 0 M MeV The values of B and z for several choices of radius for the free nucleon N 0 14 Table B and z for some bag radii m MeV 0 q R f m 0 14 B MeV z 0 are summarized in table The Pauli principle induces Fermi motion of the nucleon Since the energy of a moving bag with momentum k is q proton 2 2 k M k V V for N neutron the total energy p er nucleon at the nuclear density is given by B q Z Z k k F F n p V 3 2 2 2 2 2 2 2 2 k V M dk m m m b E tot N 3 B B B Here is the dierence b etween the proton and neutron densities and k and 3 p n F p k are the Fermi momenta for protons and neutrons resp ectively F n 3 3 and k k n B p n p F F p n 3 3 The eld created by uniformly distributed nucleons is determined by baryon number conservation to b e q V g g B B q 2 2 g m m q where we dene a new coupling constant g g while the and meanelds are given by the thermo dynamic conditions E E tot tot and b R R B B Since the eld value is expressed by g b 3 2 m q where we take g g the total energy p er nucleon can b e rewritten as Z Z q 2 2 2 k k F F g p n m g 2 2 2 2 dk k M E B 3 tot N 3 2 2 m m B B B Therefore using eq the value of the eld is given by Z Z k k F F p n M M N N q dk 2 3 m 2 2 R M k N We should note that the expression for the total energy eq is identical to that of QHDI I The eect of the internal quark structure enters entirely through the eective nucleon mass eqs and and the selfconsistency condition SCC for the eld eq Let us consider the SCC further The eective quark mass m is given by eq If q the nucleon were simply made of three massive constituent quarks the nucleon mass in vacuum and that in matter would satisfy M m and M m N q N q and hence using eq the eective nucleon mass could b e M M g N N q where we dened g g Since one nds M N g from eq the SCC gives q g g Z Z 2 k k F F p n g M N q dk 2 3 m 2 2 M k N Together with eq we then nd the SCC in the heavy quark mass limit M g M N N Z Z 2 k k F F p n M g N q dk M N 2 3 m 2 2 k M N This is exactly the SCC for QHDI I However in the more realistic case of light quarks the internal structure of the nucleon signicantly alters the SCC In particular the u and d quarks in the bag mo del like current quarks have very small masses and must therefore b e treated in a fully relativistic manner Using eqs and one nds E V M q bag N g S M E R E bag bag N R g C where S is the scalar density of the nucleon bag in matter Z Rm q q q S dr q q Rm q q q Hence we nd the SCC for the eld in the case where the nucleon has the quark structure Z Z 2 k k F F n p M g N q dk C g 2 3 m 2 2 M k N Clearly the eect of the internal quark structure is completely absorb ed into the scalar density factor C Thus we see that it is p ossible to unify the present QMC mo del and QHD in the MFA the energy p er nucleon and the SCC for the eld are resp ectively given by eq and eq with M QHD N for g QMC C R The scalar density factor is shown in g as a function of the scalar eld strength for 0.45 0.4 0.35 0.3 C 0.25 0.2 0.15 0.1 0 0.1 0.2 0.3 0.4 0.5 RVσ Figure Scalar density factors for various bag radii as a function of RV m q MeV The solid dotted and dashed curves show the results for R and f m 0 resp ectively three bag radii We see that it is much smaller than unity and that the dep endence on the bag radius is not strong Now we are in a p osition to present numerical results First we take m MeV and q 2 2 determine the coupling constants g and g so as to t the binding energy MeV 3 and the saturation density f m for equilibrium nuclear matter Furthermore 0 2 we choose the meson coupling constant g so as to repro duce the bulk symmetry energy MeV The coupling constants and some calculated prop erties of nuclear Table Coupling constants and calculated prop erties of equilibrium nuclear matter The eective nucleon mass M the nuclear compressibility K and the symmetry energy a 4 N are quoted in MeV The b ottom row is for QHDI I We take m MeV m MeV and m MeV x R 2 2 2 K a M g g R g 4 0 N R x 0 0 QHDI I matter at saturation