Constraints on the relativistic mean field of $Delta$-isobar in nuclear matter

a r X i v :n u c l -t h /9704012v 2 12 M a y 1998Constraints on the relativistic mean ?eld of

?-isobar in nuclear matter

D.S.Kosov 1,C.Fuchs 1,B.V.Martemyanov 1,2,Amand Faessler 11Institut f¨u r Theoretische Physik,Universit¨a t T¨u bingen,Auf der Morgenstelle 14,D-72076T¨u bingen,Germany 2Institute for Theoretical and Experimental Physics,B.Cheremushkinskaay 25,117259Moscow,Russia

The role of the?-isobars in nuclear matter has been discussed in various investigations in the framework of e?ective hadron?eld theories,i.e.Quantum Hadron Dynamics(QHD)[1–3].In the framework of QHD[4]nucleons as well as?-isobars interact with the surrounding nuclear medium by the exchange of e?ective scalarσand vector Vμ-mesons.However,the coupling strengths of the respective baryon-meson vertices are supposed to be di?erent for nucleons and deltas.A lot of interesting physical observable depend crucially on the choice of the respective coupling strengths[1–3].In the framework of e?ective QHD the coupling constants for the nucleon-meson vertices are e?ective quantities which are normally adjusted to the nuclear matter saturation properties[4]. They can,however,be also taken from di?erent sources,e.g.from chiral models [5]or from Dirac-Brueckner-Hartree-Fock calculations[6].There is up to now no de?nite information about the scalar and vector delta coupling strengths. Recent studies based on the QCD?nite density sum-rule[7]have shown that the?vector self-energy is considerably weaker than that of the nucleon,while its scalar self-energy is somewhat stronger.However,the QCD sum-rule pre-dictions for the scalar self-energy are sensitive to the unknown density depen-dence of four-quark condensates and due to this there is no reliable information about the coupling constant of the?-isobar with scalar mesons.Thus phe-nomenological constraints on scalar and vector coupling constants of?-isobar are of general interest,in particular because they give new restrictions on the dependence of four-quark condensates on the nucleon density.Further such considerations show up the range for possible values of the respective ver-tices.This is also of importance for phenomenological applications such as, e.g.,heavy ion collisions where the decay and rescattering of?-isobars are the prominent sources of meson production[8,9].In relativistic heavy ion collisions the excess of deltas can reach values comparable to the saturation density of nuclear matter(so called resonance matter)[8].In the theoretical description within transport models the?-isobars are commonly treated with the same coupling strengths as the nucleons,i.e.the scalar and vector components of the relativistic mean?eld are assumed to be identical for nucleons and?-isobars[9,10].This has of course in?uence on the delta dynamics as well as on the description of inelastic collisions,e.g.,the thresholds for the production of mesons would be a?ected[10].In order to use,e.g.,pions or kaons as a source of information on the hot and compressed phase of nuclear matter the understanding of the properties of?-isobars in the nuclear medium appears to be indispensable.

In the present work we investigate the mixed phase of nuclear and?-matter within the framework of e?ective QHD.The model system we are considering consists of nucleonsΨN,?-isobars which are treated as a Rarita-Schwinger particleΨ?α[11],e?ective scalarσand vector Vμmeson?elds which are both isoscalars.The inclusion of these mesons is su?cient for the description of symmetric nuclear matter with a vanishing isovector density and also in the

mixed phase where the total(zero)isospin should be conserved.We adopt the extension of the Walecka model[4]by including?-isobars as an additional degree of freedom in the Lagrangian[1]

L=L f?g vˉΨNγμVμΨN?g′vˉΨα?γμVμΨ?α+

gσˉΨNσΨN+gσˉΨα?σΨ?α+U(σ).(1) Here L f collects a free Lagrangian of the baryon?eldsΨN,Ψ?αand the Vμ,σmeson?elds.The interaction terms of the nucleons and?-isobars with the meson?elds are explicitly given in Eq.(1).The respective coupling constants of nucleons gσ(g v)and deltas g′σ(g′v)with the scalar(vector)mesons can be di?erent in magnitude.The phenomenological self-interaction of the scalar ?eld U(σ)speci?ed as[12]

U(σ)=?1

4

Cσ4(2)

introduces an additional non-linear density dependence into the model which improves the description of nuclear matter bulk properties.The respective parameters taken from Ref.[13]yield a saturation densityρ0=0.167fm?3, a binding energy of-15.8MeV and a compressibility of290MeV.To keep the model as general as possible no restrictions are imposed on the?-meson vertices from the beginning,i.e.,g′σand g′v are treated as free parameters.The range of their possible values is?nally obtained by the variation of g′σ,v and the consideration of the resulting equation of state.

In the mean?eld approximation the meson?elds in Eq.(1)are given by their classical expectation values over the ground state of the nuclear system. Thus one obtains the Lagrangian density in the mean-?eld approximation. The Dirac?eld equations follow from mean?eld Lagrangian density as [iγμ?μ?g vγμVμ?M?N]ΨN(x)=0(3) iγμ?μ?g′vγμVμ?M?? ΨαN(x)=0.(4)

The e?ective masses are given as

M?N=M N?gσσ,M??=M??g′σσ(5) where M N,M?stand for the bare nucleon and?-masses.In in?nite nuclear matter the mean?eld approximation yields the scalarσand vector Vμ=δμ0V0 meson?elds which are time and space independent and directly proportional to the respective source terms,i.e.

V0=g v

m2v

ρB(?)(6)

m2σσ+Bσ2+Cσ3=gσρs(N)+g′σρs(?)(7) withρB(N),ρs(N),ρB(?),ρs(?)being the nucleon and?-isobar baryon and scalar densities.The equation for the e?ective nucleon mass M?N is given by

M?N=M N?g2σ

m2σ

ρs(?)+

B

g2σm2σ

(M N?M?N)3(8)

which has to be solved selfconsistently.The e?ective?-mass can be found as a function of M?N and the scalar coupling constant g′σ,gσfrom Eq.(5). Chemical stability of the nucleons and?-isobars requires that the chemical potentials(Fermi energies)are equal:

E F(N)=E F(?),(9) with

E F(N)=g v V0+

k F(?)2+M??2.(11)

The Fermi momentum k F(?)of the?-isobar matter is determined from Eq.(9) and k F(?)is related to the density by usual relation,thereby the?-isobars density is governed by the requirement of a chemical equilibrium Eq.(9).

In Fig.1we show the equation of state,i.e.,the energy per baryon as a function of baryon density for di?erent ratios of scalar and vector coupling constants of the?over those of the nucleon r s=g′σ/gσand r v=g′v/g v.It is found that the presence of?-isobars becomes energetically favorable at higher densities (ρ>2ρ0)where the mixed?-nucleon phase appears.The second minimum corresponds to new metastable state of nuclear matter as already found in Ref.[1].

We obtain the?rst constraint on the delta coupling constants,i.e.,on ratios r s and r v by the requirement that the second minimum should lie above the saturation energy of normal nuclear matter(a).This means that in the mixed ?-nucleon phase only a metastable state can occur.An additional restriction on the coupling constant can be obtained if one requires that there are no

?-isobars present at saturation density.In other words,the inclusion of a?-isobar?eld in the Lagrangian(1)should not e?ect the value for the binding energy of nuclear matter at saturation density,i.e.k F(?)=0at saturation density(b).This requirement which is based on the fact that there are no ?-isobars in the ground state of?nite nuclei imposes a”soft”restriction on the coupling constants[2]

r s≤0.82r v+0.71.(12) Recent QCD?nite density sum-rule calculations[7]have shown that there exists a larger net attraction for a?-isobar than for a nucleon in the nuclear medium.Thus we can further restrict our consideration to the cases where r s≥1and r v≤1(c).

In Fig.2we show the”window”for the possible values of ratios which are allowed under the restrictions(a-c).The constraints(a,b)(under the condition (c))can thereby be summarized a s

r s≤1.01r v+0.38(13) which imposes a more restrictive constraint on the ratios then Eq.(12).

The QCD sum-rule calculations predict[7]that the?-isobar vector coupling is two times smaller(r v～0.4?0.5)than the corresponding nucleon coupling and the magnitude of the?-isobar scalar coupling is larger than that for nucleon(r s～1.3).We checked these ratios between the?-isobar and nucleon coupling constants by our phenomenological model and we found in this case that the depth of the second minimum is of more than one order of magnitude larger(280MeV)than the minimum at saturation density.This means that for the QCD predictions of Ref.[7]the real ground state of nuclear matter would occur to be a?-isobar matter at a density of about～3ρ0and a binding energy of～280MeV.

To summarize we have investigated the properties of?-isobars in nuclear matter within the framework of Quantum Hadron Dynamics.Based on the constraints that there are no?-isobars present at saturation density,the ap-pearance of?-isobars at higher density leads to a metastable state which is not the ground state of nuclear matter and that the scalar interaction is more attractive and the vector interaction is less repulsive for deltas than for nu-cleons we have been able to derive a window for possible values of mean?eld scalar and vector coupling constants for the delta.It turned out that present QCD sum-rule predictions do not fall into that window and seem to yield too strong deviations of the?-isobar self-energy from the nucleon self-energy in the nuclear medium.

Acknowledgements

The authors acknowledge fruitful discussions with Drs.M.I.Krivoruchenko and L.Sehn.B.V.M.is grateful to the Institute for Theoretical Physics of University of Tuebingen for hospitality and?nancial support.This work was partially supported by the Deutsche Forschungsgemeinschaft(contract No FA67/20-1).

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0246

ρ/ρ0?100?50

50100

E /N ?938 (M e V

)Fig.1.Equation of state,i.e.binding energy per baryon versus density for nuclear matter in the mixed nucleon and ?-isobar phase.The calculation are performed for various values of r s and r v (r s =1.35,r v =1for the dashed curve,and r s =1.35,r v =0.9for the dash-dotted curve).The solid curve shows the nuclear matter equation of state without ?-isobars.

0.8 1.0 1.2 1.4 1.6

g’σ / g σ0.40.6

0.8

1.0

1.2

1.4

g ’v / g

v Fig.2.Values of ratios of scalar and vector coupling constant for a ?-isobar to that for a nucleon which are allowed under the assumption:(a)The mixed ?-nucleon state is metastable,i.e.the binding energy at the second minimum is smaller than in ground state nuclear matter.(b)There are no deltas present at the saturation density.(c)The scalar mean ?eld is more attractive and the vector potential is less repulsive for deltas than for nucleons.