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 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:; 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 - 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 To obtain this expression we have subtracted the energy at A - 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 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 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 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. The figure is extracted from Réf. [13]. 3. PARTIAL AND COMPLETE CHIRAL SYMMETRY RESTORATION IN DENSE HATTER Let us consider dense natter as a Fermi sea of quarks [11,12] as shown on Fig.3. At normal nuclear natter density the fermi momentum (which is the sane for quarks as for nucléons) is kF- 0.25 GeV. Treating the Fermi sea in the same approximation as the Dirac sea, the energy per unit volume of nuclear matter is: (8) k < k. where Ec/fl is the Dirac sea contribution given by Eq.(5). The second term is the Fermi sea contribution. For where pg is the baryonic density. The Fermi sea contributes a term which is roughly proportional to the baryonic density and linear in FERMI SEA ,~KF DIRAC SEA Figure 3: Finite density baryonic matter is formed by adding quarks to a Fermi sea with momentum k < kf . One can consider either quarks or nucléons in the Fermi sea. On Fig.4 we show how the constituent quark nass is reduced in dense matter relative to its vacuum value. NcpBip Fermi sea quark* V a LU — ED/Û Dirac sea : vacuum CD T \ constituent quark vacuum constituent mass in nod. matter quark mass Figure 4: Contributions to the energy per unit volume of the Dirac sea and of the Fermi sea. In nuclear matter, the quark constituent mass HH is the value of Nucléon models (chiral solitons [21,17,27], chiral bags [22] and Skyrmions [23] for example), in which the contituent quark mass is generated by a chiral field, scale in such a way that their nass is proportional (and their size inversely proportional) to the surrounding or field (here proportional to The ratio 1.0 A = 600 MeV = 460 MeV 0.8 0.6 0.4 A = 600 MeV 0.2 0)~260MeV '0 0 1 0 0.5 1.0 1.5 kF/kFo Figure 5: The ratio V^H/VQ °^ c^e units of kf0- 0.25 GeV. The two curves correspond to two values of the vacuum constituent masses 0 . 4. NUCLEONS IN THE FERMI SEA The above description of nuclear matter as a Fermi sea of quarks can easily be criticized. The criticism should be tempered by the fact chat we are only using it to determine the ratio ED_ «^ n * ftk instead of the expression (8) which applies to quarks in the Fermi sea. Expressions (8) and (9) differ by the presence of a coupling constant g introduced in (9) and by the nucléon degeneracy VM— A which replaces the quark degeneracy v - N£i>H - 12. The coupling constant g is introduced in order to take into account the structure of the nucléon. It is given by the expression g E ED — 2; — + NcpB E ED — 2? — — + gpB The two expressions become identical identical when g - NC or, equivalently , when H — ^c^O • Th*s *-s a reasonable assumption which states that the constituent quark mass is 1/3 of the nucléon mass when Nc- 3. However the approximation The picture, given above, of nucléons in the Fermi sea, is closely related to Ualecka's theory of nuclear natter [25]. Indeed, what we have done is to couple nucléons to the signa field produced by Figure 6: The energy per unit volume of nuclear matter is plotted as a function of the constituent quark mass relative to its vacuum value. Curves are given for various nuclear matter densities. For each density two curves are given which are labelled Q (full curves) ««nd N (dashed curves). They correspond respectively to the cases where quarks and nucléons fill the Fermi sea. When a curve has a minimum close to zero chiral symmetry is restored. The curves are calculated with a cut-off A - 600 MeV and are taken from Réf.J12J. the quark condensate (i.e. by the quarks in the Dirac sea) instead of coupling them to an explicit sigma field as in Walecka's theory. Why is it then that Walecka did not obtain chiral symmetry restoration at normal nuclear natter density? This is explained in Fig.7 where the energy per unit volume of the vacuum is compared in Walecka's theory and in the Nambu Jona-Lasinio model. The sigma meson mass of Walecka's theory is adjusted so as to give the same curvature about the point 1 Fermi sea x Walecka contribution v a LU 0 Figure 7: In Walecka's theory the energy per unit volume of the vacuun is given by the a-meson quadratic mass term which is proportional to Attempts have been made [17,27] to form a bound state of NC quarks starting from the Narabu Jona-Lasinio hamiltonian (1). Such a state is called a chiral soliton and the procedure to obtain it is very similar to that used with the cr-model [21] . A chiral soliton describes the nucléon in much the same way as a skyrmion although it does not rely, as a skyrmion does, on a gradient expansion of the chiral field. A chiral soliton is a stationary solution of the equations of motion in which the chiral field differs from its vacuum value in a localized region of space. When the chiral field is not translationally invariant, the energy of the system is given by the expression [20] : where ex and ek are the eigenvalues of the Dirac equations: t.V \ Ax.V N r- + fXpUjlA) - exU> ^— + p The first Eq.(ll) determines the orbits of the quarks in the localised chiral field and the second Eq.(ll) determines the orbits of free quarks with constituent quark mass i9.TY 5c -+ ~ -» U - e ea(r) - rae(r) It is also found [17] that the field o . The spectrum of quark orbits in a hedgehog field (12) with an exponential profile function is shown on Fig. 8 on which one can see that considerable binding can be gained by putting NC quarks into the bound 0* orbit. POSITIVE ENERGY CONTINUUM NEGATIVE ENERGY CONTINUUM Figure 8: The spectrum of the quark orbits in the hedgehog field (12) in units of However, it costs energy to subject the Dirac sea to a hedgehog-shaped chiral field. In the calculations published so far [17,27] this energy loss compensates the energy gained by putting Nc quarks into the 0* orbit. Fig.9 shows the way in which the energy of the chiral soliton depends on its size. At zero size, the energy is equal to the unperturbed energy NC^P|J of free quarks. Therefore the minimum of Che energy curve on Fig.9 does not represent a bound state of quarks. The present lecturer is not yet convinced that such a metastable state can describe the nucléon and he recommends the reader to look into Refs.[17,27] in order to form his own opinion. 1.5 1.0 o in LU 0.5 I 0.5 Size /fm Figure 9: Qualitative behavior of the soliton energy versus its size obtained in Refs[17,27]. ACKNOWLEDGMENTS Alnost all of the work I have nade on the Nambu Jona-Lasinio nodel is part of a continued collaboration with Martine Jaminon and Pierre Stassart from the Université de Liège. I hope that this lecture does not betray the results and ideas we have developped together. 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