COMMISSARIAT A L'ENERGIE ATOMIQUE CENTRE D'ETUDES NUCLEAIRES DE 5ACLAY CEA-CONF —10051 Service de Documentation F9I19I GIF SUR YVETTE CEDEX L3 NAMBU JONA-LASINIO MODELS APPLIED TO DENSE HADRONIC MATTER RIPKA G. CEA Centre d'Etudes Nucléaires de Soclay, 91 - Gif-sur-Yvette (FR)< Service de Physique Théorique Communication présentée à : 12. Workshop on Nuclear Physics Iguazu Falls 28 Aug - 1 Sep 1989 HAMBU JONA-LASINIO MODELS APPLIED TO DENSE HADRONIC HATTER. Georges Ripka Service de Physique Théorique de Saclay Laboratoire de l'Institut de Recherche Fondamentale du Commissariat à l'Energie Atomique F-91191 Gif-sur-Yvette Cedex ABSTRACT The Nambu Jona-Lasinio model is briefly introduced and applied to dense baryonic matter. The effects of the regularization on the model parameters are discussed. Dense nuclear matter is discussed in the cases where either quarks or nucléons fill the Fermi sea. In the latter case the theory is compared to Walecka's relativistic mean field theory of nuclear matter. 1. INTRODUCTION The Nambu Jona-Lasinio model has been introduced to explain the spontaneous chiral symmetry breaking of the physical vacuum. It is a theory of Dirac particles with a local A-fermion interaction and, as such, it belongs to the same class of effective theories as Landau's theory of Fermi liquids or the BCS theory of superconducting metals. Like the latter, it is not renormalisable and it requires the introduction of a momentum cut-off which is an important parameter of the theory. Since its original introduction by Nambu and Jona-Lasinio in 1961 [1] , it has been applied with variable success to the calculation of a wide variety of hadronic matter properties. The fermions of the theory are quarks with NC colors and, by construction, it reproduces correctly the Nc dependence of various observables. Besides explaining the chiral symmetry breaking of the physical vacuum together with the Cell-Mann-Oakes-Renner [8] formula f*m^- -mdw/), the XII Workshop on Nuclear Physics SPhT 'So liiuniii Falls. Argentine, August 28 -Sept. 1er. !«>*»« Nambu Jona-Lasinio model has been extensively used Co calculate properties of low-lying mesons in terms of qq excitations of the vacuum in the random phase approximation. In Réf.[2,3] for example, the Nambu Jona-Lasinio model is used to give a unified description of soft pion theorems, PCAC, the Golberger Treiman relation f1tg-MgA [A], the KSFR relation [5], the Weinberg relation [6], quark-loop effects such as anomalies, anomalous decays and Wess-Zumino terms [7], hidden symmetries [9] , and p meson dominance of electro-magnetic interactions [10] . We will not expand on these successful applications. Instead we will discuss more recent work concerning mesons and chiral symmetry restoration in dense hadronic matter. One can use the model to study meson progators in dense hadronic matter [11,12,13] as well as chiral symmetry restoration at high baryonic density. Such studies are useful because the equation of state of cold dense matter is an input to the codes which attempt to explode supernovae [14] and lattice QCD calculations are unable, at present, to cope with dense matter. We shall also discuss attempts to calculate the nucléon as a soliton consisting of a bound state of quarks [17,27]. Last but not least, the Nambu Jona-Lasinio model has very recently been applied to the standard model [15] . In this application the Higgs meson is a tt top quark mass excitation and, as a result, the Higgs mass is between 1 and 2 times the top quark mass. There have been several attempts to derive the Nambu Jona-Lasinio model from QCD. In this context two approaches should be mentioned. That of the Diakonov and Petrov who relate the model to the instanton structure of the QCD vacuum (16] and that of R.Ball [3] who rewrites the QCD lagrangian in terns of new non-local fields which, at low energy, reduce to colorless mesons and glueballs. We expect the Nambu Jona-Lasinio theory to be part of an effective theory for the low energy (perhaps 1-2 GeV) properties of hadronic matter. An ideal effective theory should account for the chiral symmetry breaking of the vacuum, its restoration at finite density and temperature, the structure, decays and electro-weak properties of low energy mesons, the structure and low energy properties of baryons (N,A,Z,A,...) together with their couplings to the mesons, the modification of hadrons propagating in nornal and dense nuclear matter and it should form the basis for a calculation of the equation of state of dense natter which is encountered in star evolution. In the present state of the art, the Nanbu Jona-Lasinio nodel of hadronic matter stands in opposition to constituent quark models in which quarks are confined by string-like or flux-tube forces [18] or color-dielectric [19] fields. In these models it is not possible to explain the roughly 300 MeV constituent quark mass. The Nambu Jona-Lasinio model does yield a constituent quark mass of this order and provides a Dirac sea from which to form qq excitations. However it lacks the confinement properties of the constituent quark models. The merging of these two approaches would constitute a major step towards the formulation of a useful effective theory of hadronic matter. 2. DETERMINATION OF THE FARAMETERS The spontaneous chiral symmetry breaking of the physical vacuum is most easily expressed in the Hamiltonian formulation. The hamiltonian of the system is: . In t7 \ *2<£\ (1) J«y A ,_ where <pU = <pe 5 is a chiral field coupled to the quark field t|s. In the following we neglect the small current quark mass (about 7 MeV) which explicitely breaks chiral symmetry and gives mass to the pion. The- chiral field has no associated conjugate momentum so that it appears as a constraint. It can be trivially eliminated in which case the Hamiltonian acquires the original form [1]: P ( ^.V 1-2 1 r- -» 1 X H - d3r Kj/ -.— * - — r(«W) + - (i|/V5T«|/) (2) J ^ x 2a2 2az so that a is the inverse coupling constant. The mean field approximation is however obtained by using the form (1) of the hamiltonian with a classical chiral field and a quantized quark field. For the trans lationally invariant vacuum, the U field can be eliminated by the rotation ty -» U"1/2 i|/ in which case the hamiltonian (1) describes free quarks of mas:; <p which we shall call the constituent quark mass. The vacuum is then a Oirac sea (Fig.l) with quarks filling negative energy orbits. The vacuum energy per unit volume is : n (3) where v is the degeneracy of the quark plane wave orbits {v—12 for two flavor quarks). k^A DIRAI SEA Figure 1: In the Narabu Jona-Lasinio model, the physical vacuum is described by quarks filling a Dirac sea with momenta k ^ A. In the physical vacuum the constituent quark mass takes the value <p0 which makes the energy stationary: - 0 <?=•?„ This is the gap equation. Ue use it to eliminate the unphysical coupling constant a in favor of the more physical constituent quark mass <p0 . The vacuum energy per unit volume becomes: <P2- v- -n ks (5) To obtain this expression we have subtracted the energy at <p - <p^ so as to set the vacuum energy to zero and we have added a Pauli-Villars type regularizing term so as to cut-off the momenta larger than A. The energy is a function of 2 parameters: The constituent 2 quark mass cp0 (which replaces the inverse coupling constant a ) and the cut-off A. The two parameters can be reduced to one by fitting the pion decay constant fn- 93 MeV. This leads to the following relation between the 2 parameters [20]: A - <p0e (6) The relation is dispayed on Fig.2. Most applications have used small values of the cut-off in the region where the constituent quark mass is close to 300 MeV. This is because they insist in fitting the estimated [21] vacuum expectation value <iW/> = -(250 MeV)3of the quark condensate. Calculated values of the condensate are shown in parentheses on Fig.2. They are however very sensitive to the cut-off profile [12.13] because the quark condensate depends quadratically on the cut-off while observables such as frt depend only logarithmically on the cut-off. It is dangerous to nix such quantities and it makes more sense to calculate quadratically divergent quantities with two subtractions using two cut-off parameters A, and A^ : (7) (210*. A2)] This allows one to fit both the pion decay constant and the quark condensate along any point of the curve on Fig.2. Two parameter cut-off profiles have also been used in Refs.[16,17]. (The numbers on Fig.2 are obtained from (7) by setting A,- /^.) Equation (7) establishes the scale for the cut-off. In the standard model the quantity which plays the role of fn is v — 257 GeV and <pjj plays the role of the top quark mass (with i>-6). For a top quark mass of ] 75 GeV the cut-off would be of the order of the GUT scale !015GeV, used in Réf.[15]. 1.5 9-° 0.5 (U3) I I I I I 0.0 0.6 0.8 1.0 1.2 U A /GeV Figure 2: The relation between the cut-off A and the vacuum constituent quark mass <p>0 obtained by fitting the pion decay constant fv— 93 MeV. The values <«W/> of the quark condensa te, obtained from Eq.(7) with A,- Aj are given in parentheses.