Abstract
Investigations of deconfined quark matter within NJLtype models are reviewed, focusing on the regime of low temperatures and “moderate” densities, which is not accessible by perturbative QCD. Central issue is the interplay between chiral symmetry restoration and the formation of color superconducting phases. In order to lay a solid ground for this analysis, we begin with a rather detailed discussion of two and threeflavor NJL models and their phase structure, neglecting the possibility of diquark pairing in a first step. An important aspect of this part is a comparison with the MIT bag model. The NJL model is also applied to investigate the possibility of absolutely stable strange quark matter. In the next step the formalism is extended to include diquark condensates. We discuss the role and mutual influence of several conventional and less conventional quarkantiquark and diquark condensates. As a particularly interesting example, we analyze a spin1 diquark condensate as a possible pairing channel for those quarks which are left over from the standard spin0 condensate. For threeflavor systems, we find that a selfconsistent calculation of the strange quark mass, together with the diquark condensates, is crucial for a realistic description of the 2SCCFL phase transition. We also study the effect of neutrality constraints which are of relevance for compact stars. Both, homogeneous and mixed, neutral phases are constructed. Although neutrality constraints generally tend to disfavor the 2SC phase we find that this phase is again stabilized by the large values of the dynamical strange quark mass which follow from the selfconsistent treatment. Finally, we combine our solutions with existing hadronic equations of state to investigate the existence of quark matter cores in neutron stars.
PACS: 12.39.Ki, 12.38.AW, 11.30.Qc, 25.75.Nq, 26.60.+c
Key words: dense quark matter, QCD phase diagram,
color superconductivity, compact stars
NJLmodel analysis
[2mm] of dense quark matter
Michael Buballa
[10mm] Institut für Kernphysik, Technische Universität Darmstadt,
Schlossgartenstr. 9, D64289 Darmstadt, Germany
email:
phone: +49 6151 16 3272 fax: +49 6151 16 6076
Contents
 1 Introduction
 2 NJLmodel description of color nonsuperconducting twoflavor quark matter
 3 Threeflavor systems
 4 Twoflavor color superconductors
 5 Threeflavor color superconductors
 6 Neutral quark matter
 7 Application: Neutron stars with color superconducting quark cores
 8 Summary and discussion
 A Fierz transformations
 B Twoflavor NambuGorkov propagator
Chapter 1 Introduction
Exploring the phase structure of quantum chromodynamics (QCD) is certainly one of the most exciting topics in the field of strong interaction physics. Already in the 70s, rather soon after it had become clear that hadrons consist of confined quarks and gluons, it was argued that the latter should become deconfined at high temperature or density when the hadrons strongly overlap and loose their individuality [1, 2]. In this picture, there are thus two distinct phases, the “hadronic phase” where quarks and gluons are confined, and the socalled quarkgluon plasma (QGP) where they are deconfined. This scenario is illustrated in the upper left panel of Fig. 1.1 by a schematic phase diagram in the plane of (quark number) chemical potential and temperature. A diagram of this type has essentially been drawn already in Ref. [2] and can be found, e.g., in Refs. [3, 4].
In nature, the QGP surely existed in the early universe, a few microseconds after the Big Bang when the temperature was very high. It is less clear whether deconfined quark matter also exists in the relatively cold but dense centers of neutron stars. Experimentally, the creation and identification of the QGP is the ultimate goal of ultrarelativistic heavyion collisions. First indications of success have been reported in press releases at CERN (SPS) [5] and BNL (RHIC) [6], although the interpretation of the data is still under debate. There is little doubt that the QGP will be created at the Large Hadron Collider (LHC), which is currently being built at CERN.
At least on a schematic level, the phase diagram shown in the upper left panel of Fig. 1.1 remained the standard picture for about two decades. In particular the possibility of having more than one deconfined phase was not taken into account. Although Cooper pairing in cold, dense quark matter (“color superconductivity”) had been mentioned already in 1975 [1] and had further been worked out in Refs. [7, 8, 9], the relevance of this idea for the QCD phase diagram was widely ignored until the end of the 90s. At that time, new approaches to color superconductivity revealed that the related gaps in the fermion spectrum could be of the order of 100 MeV [10, 11], much larger than expected earlier. Since larger gaps are related to larger critical temperatures, this would imply a sizeable extention of the color superconducting region into the temperature direction. Hence, in addition to the two standard phases, there should be a nonnegligible region in the QCD phase diagram where strongly interacting matter is a color superconductor. (For reviews on color superconductivity, see Refs. [12, 13, 14, 15].)
Once color superconductivity was on the agenda, the door was open for many new possibilities. This is illustrated by the remaining three phase diagrams of Fig. 1.1, which are taken from the literature. It is expected that at large chemical potentials up, down, and strange quarks are paired in a socalled colorflavor locked (CFL) condensate [16]. However, this might become unfavorable at lower densities, where the strange quarks are suppressed by their mass. It is thus possible that in some intermediate regime there is a second color superconducting phase (2SC) where only up and down quarks are paired. This scenario is depicted in the upper right diagram of Fig. 1.1, taken from Ref. [17]. More recently, further phases, like threeflavor color superconductors with condensed kaons (CFLK) [18, 19, 20] or crystalline color superconductors (“LOFF phase”) [21, 22] have been suggested, which might partially (lower right diagram [14]) or even completely (lower left diagram [23]) replace the 2SC phase.
Fig. 1.1, which is only an incomplete compilation of recent suggestions, illustrates the potential richness of the phase structure, which has not been appreciated for a long time. At the same time, it makes obvious that the issue is not at all settled. Note that all phase diagrams shown in the figure are only “schematic”, i.e., educated guesses, based on certain theoretical results or arguments. In this situation, and since exact results from QCD are rather limited, model calculations may provide a useful tool to test the robustness of these ideas and to develop new ones.
In the present report we discuss the phase diagram and related issues which result from studies with Nambu–JonaLasinio (NJL) type models. These are schematic models with pointlike quark (anti)quark vertices, but no gluons. As a consequence, NJLtype models have several wellknown shortcomings, most important, they do not have the confinement property of QCD. This is certainly a major drawback in the hadronic phase, where constituent quarks are not the proper quasiparticle degrees of freedom. At high temperatures, confinement becomes less relevant but obviously a realistic description of the quarkgluon plasma requires explicit gluon degrees of freedom. On the other hand, the use of NJLtype models seems to be justified – at least on a schematic level – to study cold deconfined quark matter where both, confinement and gluon degrees of freedom, are of minor importance.
In any case, every model calculation should be confronted with the “facts”, as far as available. To that end, we briefly list the main features of QCD in Sec. 1.1 and summarize what is currently known about the QCD phase diagram in Sec. 1.2. The present work will then be outlined in more details in Sec. 1.3.
1.1 Basics of QCD
Quantum chromodynamics is defined by the Lagrangian [24, 25]
(1.1) 
where denotes a quark field with six flavor () and three color degrees of freedom, and is the corresponding mass matrix in flavor space. The covariant derivative
(1.2) 
is related to the gluon field , and
(1.3) 
is the gluon field strength tensor. and denote the generators of (GellMann matrices) and the corresponding antisymmetric structure constants, respectively. is the QCD coupling constant.
The QCD Lagrangian is by construction symmetric under gauge transformations in color space. Because of the nonAbelian character of the gauge group, underlined by the presence of the term in Eq. (1.3), the theory has several nontrivial features which are not present in Abelian gauge theories, like quantum electrodynamics:

contains gluonic selfcouplings (three and fourgluon vertices), i.e., gluons carry color.

QCD is an asymptotically free theory [26, 27], i.e., the coupling becomes weak at short distances or, equivalently, large Euclidean momenta . To oneloop order,
(1.4) where is the number of relevant flavors and is the QCD scale parameter which can be determined, e.g., by fitting Eq. (1.4) (or improved versions thereof) to experimental data at large . In this way, one finds MeV for five flavors in the scheme.

In turn, Eq. (1.4) implies that the coupling becomes strong at low momenta. In particular, perturbative QCD is not applicable to describe hadrons with masses below GeV. This may or may not be related to the phenomenon of “confinement”, i.e., to the empirical fact that colored objects, like quarks and gluons, do not exist as physical degrees of freedom in vacuum. There are interesting attempts to relate confinement to particular topological objects in the QCD vacuum, like monopoles or vortices, but it is fair to say that confinement is not yet fully understood (see, e.g., Refs. [29, 30] and references therein).
Another important feature of is its (approximate) chiral symmetry, i.e., its symmetry under global transformations. This is equivalent to be invariant under global vector and axialvector transformations,
(1.5) 
where are the generators of flavor . These symmetries would be exact in the limit of massless flavors. (For it is sufficient to have degenerate flavors.) In reality all quarks have nonvanishing masses. Still, chiral symmetry is a useful concept in the up/down sector () and even, although with larger deviations, when strange quarks are included as well (). The sector of heavy quarks (charm, bottom, top) is governed by the opposite limit, corresponding to an expansion in inverse quark masses, but this will not be of further interest for us.
The is also an (approximate) symmetry of the QCD vacuum, reflected by the existence of nearly degenerate multiplets in the hadron spectrum. If this was also true for the axial symmetry each hadron should have an approximately degenerate “chiral partner” of opposite parity. Since this is not the case, one concludes that chiral symmetry is spontaneously broken in vacuum. This is closely related to the existence of a nonvanishing quark condensate , which is not invariant under and therefore often deals as an order parameter for spontaneous chiral symmetry breaking^{1.1}^{1.1}1.1 Note, however, that does not necessarily mean that chiral symmetry is restored since it could still be broken by other condensates. This is for instance the case for three massless flavors in the CFL phase, see Sec. 5.1.2..
Another hint for the spontaneously broken chiral symmetry is the low mass of the pion which comes about quite naturally if the pions are interpreted as the corresponding Goldstone bosons in the twoflavor case. If chiral symmetry was exact on the Lagrangian level (“chiral limit”) they would be massless, while the small but finite pion mass reflects the explicit symmetry breaking through the up and down quark masses. In the analogous way, the pseudoscalar meson octet corresponds to the Goldstone bosons in the threeflavor case. The Goldstone bosons obey several lowenergy theorems which provide the basis for chiral perturbation theory (PT) [31, 32, 33]. Unlike ordinary perturbation theory, corresponds to an expansion in quark masses and momenta and can be applied in regions where is large.
Since the perturbative vacuum is chirally symmetric for massless quarks, one expects that chiral symmetry gets restored at high temperature. This would also be the case at high density if the matter was in a trivial rather than in a color superconducting state (see footnote 1.1). The “partial restoration” of chiral symmetry, i.e., the inmedium reduction of at small temperatures or densities can be studied within PT. For two flavors one finds to leading order in temperature and density [34, 35]
(1.6) 
where MeV is the pion decay constant and 45 MeV is the pionnucleon sigma term. The two correction terms on the r.h.s. describe the effect of thermally excited noninteracting pions and of nucleons, respectively. (Interaction effects are of higher order.) To be precise, the behavior is only correct in the chiral limit, i.e., for massless pions, which are otherwise exponentially suppressed. Since in the hadronic phase the physical pion mass can never be assumed to be small against the temperature, Eq. (1.6) is more of theoretical rather than practical interest. Systematic mass corrections are of course possible [34]. It remains at least qualitatively correct that pions, being the lightest hadrons, dominate the lowtemperature behavior.
On the classical level, is also invariant under global
(1.7) 
in the limit of massless quarks. This symmetry is, however, broken on the quantum level and therefore not a real symmetry of QCD [36]. Most prominently, this is reflected by the relatively heavy meson which should be much lighter (lighter than the meson) if it was a Goldstone boson of a spontaneously broken symmetry.
In this context instantons play a crucial role. These are semiclassical objects which originate in the fact that the classical YangMills action gives rise to an infinite number of topologically distinct degenerate vacuum solutions. Instantons correspond to tunneling events between these vacua. When quarks are included, the instantons mediate an interaction which is symmetric, but explicitly breaks the symmetry (“’t Hooft interaction”) [36]. In particular it is repulsive in the channel.
In the instanton liquid model [37], the gauge field contribution to the QCD partition function is replaced by an ensemble of instantons, characterized be a certain size and density distribution in Euclidean space. It turned out that hadronic correlators obtained in this way agree remarkably well with the “exact” solutions on the lattice (see Ref. [38] for review).
1.2 The QCD phase diagram
We have pointed out that the phase diagrams presented in Fig. 1.1 are schematic conjectures, which are constrained only by a relatively small number of safe theoretical or empirical facts. Before discussing them in more detail, we should note that in general we are not restricted to a single chemical potential, but there is a chemical potential for each conserved quantity. Hence, the QCD phase diagram has in general more dimensions than shown in Fig. 1.1. If we talk about one chemical potential only, we thus have to specify under which conditions this is fixed. For instance, it is often simply assumed that the chemical potential is the same for all quark species. On the other hand, in a neutron star we should consider neutral matter in beta equilibrium whereas in heavyion collisions we should conserve isospin and strangeness. This means, the different sources of information we are going to discuss below describe different phase diagrams (or different slices of the complete multidimensional phase diagram) as far as they correspond to different physical situations. Moreover, the results are often obtained in or extrapolated to certain unphysical limits, like vanishing or unrealistically large quark masses or the neglect of electromagnetism.
Direct empirical information about the phase structure of strongly interacting matter is basically restricted to two points at zero temperature, both belonging to the “hadronic phase” where quarks and gluons are confined and chiral symmetry is spontaneously broken. The first one corresponds to the vacuum, i.e., , the second to nuclear matter at saturation density (baryon density ), which can be inferred from systematics of atomic nuclei^{1.2}^{1.2}1.2The numbers quoted in Ref. [39] are a binding energy of MeV per nucleon and a Fermi momentum of fm. Since the binding energy of nuclear matter is about 16 MeV per nucleon, it follows a baryon number chemical potential of 923 MeV, corresponding to a quark number chemical potential 308 MeV. Unless there exists a sofar unknown exotic state which is bound more strongly (like absolutely stable strange quark matter [40, 41]), this point marks the onset of dense matter, i.e., the entire regime at and 308 MeV belongs to the vacuum^{1.3}^{1.3}1.3The authors of Ref. [42] distinguish between “QCD”, which is a theoretical object with strong interactions only, and “QCD+”, which corresponds to the real world with electromagnetic effects included. In this terminology, our discussion refers to QCD. In QCD+ the ground state of matter is solid iron, i.e., a crystal of iron nuclei and electrons. Here the onset takes place at 930 MeV and the density is about 13 orders of magnitude smaller than in symmetric nuclear matter.. The onset point is also part of a firstorder phase boundary which separates a hadron gas at lower chemical potentials from a hadronic liquid at higher chemical potentials if one moves to finite temperature. This phase boundary is expected to end in a critical endpoint, which one tries to identify within multifragmentation experiments. Preliminary results seem to indicate a corresponding critical temperature of about 15 MeV [43]. Of course both, the gas and the liquid are part of the hadronic phase. They have been mentioned for completeness but are not subject of this report. There might be further hadronic phases, e.g., due to the onset of hyperons or to superfluidity^{1.4}^{1.4}1.4 We are using the word “phase” in a rather lose sense. If there is a firstorder phase boundary which ends in a critical endpoint one can obviously go around this point without meeting a singularity. Hence, in a strict sense, both sides of the “phase boundary” belong to the same phase. However, from a practical point of view it often makes sense to be less strict, since the properties of matter sometimes change rather drastically across the boundary (see, e.g., liquid water and vapor)..
Obtaining empirical information about the QGP is the general aim of ultrarelativistic heavyion collisions. As mentioned earlier, there are some indications that this phase has indeed been reached at SPS [5] and at RHIC [6]. There are also claims that the chemical freezeout points, which are determined in a thermalmodel fit to the measured particle ratios, must be very close to the phase boundary [44]. However, since the system cannot be investigated under static conditions but only integrating along a trajectory in the phase diagram, an interpretation of the results without theoretical guidance is obviously very difficult.
There is also only little hope that information about color superconducting phases can be obtained from ultrarelativistic heavyion collisions, which are more suited to study high temperatures rather than high densities. For instance, at chemical freezeout one finds MeV and MeV at AGS and MeV and MeV for PbPb collisions at SPS [45]. Even though the CBM project at the future GSI machine is designed to reach higher densities [46], it is very unlikely that the corresponding temperatures are low enough to allow for diquark condensation. Of course, no final statement can be made as long as reliable predictions for the related critical temperatures are missing.
On the theoretical side, most of our present knowledge about the QCD phase structure comes from abinitio Monte Carlo calculations on the lattice (see Refs. [47, 48] for recent reviews). For a long time, these were restricted to zero chemical potential, i.e., to the temperature axis of the phase diagram, and only recently some progress has been made in handling nonzero chemical potentials. Before summarizing the main results, let us mention that twenty years ago, the chiral phase transition at finite temperature has been analyzed on a more general basis, applying universality arguments related to the symmetries of the problem [49]. It was found that the order of the phase transition depends on the number of light flavors: If the strange quark is heavy, the phase transition is second order for massless up and down quarks and becomes a smooth crossover if up and down have nonvanishing masses. On the other hand, if is small enough the phase transition becomes firstorder. The question which scenario corresponds to the physical quark masses cannot be decided on symmetry arguments but must be worked out quantitatively. In principle, this can be done on the lattice. In practice, there is the problem that it is not yet possible to perform lattice calculations with realistic up and down quark masses. The calculations are therefore performed with relatively large masses and have to be extrapolated down to the physical values. Although this imposes some uncertainty, the result is that the transition at is most likely a crossover [47, 48].
While there is thus no real phase transition, the crossover is sufficiently rapid that the definition of a transition temperature makes sense. This can be defined as the maximum of the chiral susceptibility, which is proportional to the slope of the quark condensate. One finds a transition temperature of about 170 MeV. It is remarkable that the susceptibility related to the Polyakov loop – which deals as order parameter of the deconfinement transition – peaks at the same temperature, i.e., chiral and deconfinement transition coincide. It is usually expected that this is a general feature which also holds at finite chemical potential, but this is not clear.
When extrapolated to the chiral limit, the critical temperature is found to be MeV for two flavors and MeV for three flavors [50]. This is considerably lower than in the pure gauge theory without quarks where MeV [47, 48]. According to the symmetry arguments mentioned above, the phase transition in the twoflavor case is expected to belong to the universality class, thus having critical exponents. Present lattice results seem to be consistent with this, but they are not yet precise enough to rule out other possibilities.
The extension of lattice analyses to (real) nonzero chemical potentials is complicated by the fact that in this case the fermion determinant in the QCD partition function becomes complex. As a consequence the standard statistical weight for the importance sampling is no longer positive definite which spoils the convergence of the procedure. Quite recently, several methods have been developed which allow to circumvent this problem, at least for not too large chemical potentials (). One possibility is to perform a Taylor expansion in terms of and to evaluate the corresponding coefficients at [51, 52]. The second method is a reweighting technique where the ratio of the fermion determinants at and at is taken as a part of the operator which is then averaged over an ensemble produced at [53, 54, 55]. The third way is to perform a calculation at imaginary chemical potentials [56, 57, 58]. In this case the fermion determinant remains real and the ensemble averaging can be done in the standard way. The results are then parametrized in terms of simple functions and analytically continued to real chemical potentials.
These methods have been applied to study the behavior of the phase boundary for nonzero . From the Taylor expansion one finds for the curvature of the phase boundary at [51], , i.e., . Within error bars this result is consistent with the two other methods. However, all these calculations have been performed with relatively large quark masses, and the curvature is expected to become larger for smaller masses.
Another important result is the lattice determination of a critical endpoint [52, 54, 55]. We have seen that at the phase transition should be second order for two massless flavors and most likely is a rapid crossover for realistic quark masses. On the other hand, it has been argued some time ago that at low temperatures and large chemical potentials the phase transition is probably first order. Hence, for two massless flavors there should be a “tricritical point”, where the secondorder phase boundary turns into a firstorder one [42, 59, 60]. Similarly, for realistic quark masses one expects a firstorder phase boundary at large , which ends in a (secondorder) critical endpoint, as indicated in the three last phase diagrams in Fig. 1.1.
It has been suggested that this point could possibly be detected in heavyion experiments through eventbyevent fluctuations in the multiplicities and mean transverse momenta of charged pions. These fluctuations should arise as a result of critical fluctuations in the vicinity of the endpoint [61, 62]. To that end and should be measured as a function of a control parameter which determines the trajectory of the evolving system in the phase diagram. This parameter could be, e.g., the beam energy or the centrality of the collision. If for some value of the trajectory comes very close to the critical point this should show up as a maximum in the above eventbyevent fluctuations. In this case the system is expected to freeze out close to the critical point because of critical slowing down. For a more detailed discussion of possible signatures under realistic conditions (including finite size and finite time effects), see Ref. [62].
On the lattice, the position of the endpoint has been determined first by Fodor and Katz, employing the reweighting method [54]. The result was MeV and MeV. However, these calculation suffered from the fact that they have been performed on a rather small lattice with relatively large up and down quark masses. More recently, the authors have performed an improved calculation on a larger lattice and with physical quark masses, shifting the endpoint to MeV and MeV. This result is consistent with estimates using the Taylor expansion method, indicating a critical endpoint at , i.e., [52].
The techniques described above do not allow to study the expected transition to color superconducting phases at large chemical potentials but low temperatures. So far, the only controlled way to investigate these phases is a weakcoupling expansion, which becomes possible at very large densities because of asymptotic freedom. These analyses show that strongly interacting matter at asymptotic densities is indeed a color superconductor, which for three flavors is in the CFL phase [63, 64]. Unfortunately, the weakcoupling expansion breaks down at densities several orders of magnitude higher than what is of “practical” relevance [65], e.g., for the interior of compact stars, and it can, of course, not be applied to study the hadronquark phase transition itself.
1.3 Scope and outline of this report
The above discussion has shown that the detailed phase structure of strongly interacting matter at (not asymptotically) high density and low temperature is largely unknown. In particular, there is practically no exact information about the density region just above the hadronquark phase transition which might be relevant for the interiors of compact stars.
In this situation models may play an important role in developing and testing new ideas on a semiquantitative basis and checking the robustness of older ones. Since models are simpler than the fundamental theory (QCD) – otherwise they are useless – they often allow for studying more complex situations than accessible by the latter. The price for this is, of course, a reduced predictive power due to dependencies on model parameters or certain approximation schemes. The results should therefore always be confronted with model independent statements or empirical facts, as far as available. In turn, models can help to interprete the latter where no other theory is available. Also, “model independent results” are often derived by an expansion in parameters which are assumed to be small. Thus, although mathematically rigorous, they do not necessarily describe the real physical situation. Here models can give hints about the validity of these assumptions or even uncover further assumptions which are hidden.
An example which will be one of the central points in this report is the effect of the strange quark mass on the phase structure. Starting from the idealized case of three massless flavors, this is often studied within an expansion in terms of , assuming that the strange quark mass is much smaller than the chemical potential. Obviously, this can always be done for large enough values of , but the expansion eventually breaks down, when becomes of the order of . However, the crucial point is that itself should be considered to be a dependent effective (“constituent”) quark mass. This does not only imply that the expansion breaks down earlier than one might naively expect (because can be considerably larger than the perturbative quark mass which is listed in the particle data book), but also that effects due to the dependence of are missed completely. In particular, there can be strong discontinuities in across firstorder phase boundaries which, of course, cannot be described by a Taylor expansion.
In the present work we investigate this kind of questions within models of the Nambu–JonaLasinio type, i.e., schematic quark models with simple fourfermion (sometimes sixfermion) interactions. Historically, the NJL model [66, 67] has been introduced to describe spontaneous chiral symmetry breaking in vacuum in analogy to the BCS mechanism for superconductivity [68] and has later been extended to study its restoration at nonvanishing temperatures or densities. On the other hand, NJLtype models are straightforwardly used in the original BCS sense to calculate color superconducting pairing gaps. In fact, the renewed interest in color superconductivity was caused by analyses in such models [10, 11]. For the obvious next step, namely considering chiral quark condensates and diquark condensates simultaneously and studying their competition and mutual influence, NJLtype models are therefore the natural choice [69]. Going further, the relatively simple interaction allows to attack quite involved problems. For instance, in order to describe the transition from two to threeflavor color superconductors including dynamical mass effects, we have to allow for six different condensates which can be all different if we consider beta equilibrated matter.
The details of the results of these investigations are of course model dependent. In particular, it is not clear whether the model parameters, which are usually fitted to vacuum properties, can still be applied at large densities. In fact, it seems to be quite natural that the fourpoint couplings are and dependent quantities, just like the effective quark masses we compute. Nevertheless, at least qualitatively, our analysis can give important hints which of the sofar neglected effects could be important and which are indeed negligible. In the first place it should therefore be viewed in conjunction with and relative to other approaches.
There are few exceptions, where we make definite predictions for – in principle – observable quantities, like absolutely stable strange quark matter or quark matter cores in neutron stars. Here, although we try to estimate the robustness of the results with respect to the parameters, we are forced to assume that these are at least not completely different from the vacuum fit. In these cases the arguments could be turned around: If at some stage the predictions turn out to be wrong, this would mean that the model parameters must change drastically.
The focus of the present report is an NJLmodel study of the
phase diagram at large densities,
with special emphasis on color superconducting phases and their properties.
We begin, however, with a rather detailed discussion of the model
and its phase structure without taking into account diquark pairing.
Besides defining the basic concepts of the model, this is done because we
think that in order to understand the influence of color superconductivity
one should know how the model behaves without.
In particular, some detailed knowledge about the mechanism of the chiral
phase transition in the model will be helpful when this
is combined with the pairing transition.
This prediscussion will also allow us to point out the main limitations
of the model.
Many of them, like artifacts of missing confinement or of the
meanfield treatment, are not restricted to the NJL model,
and this discussion could also be useful for other approaches.
The remainder of this report is organized as follows.
In Chap. 2 we review the thermal properties of the NJL model, concentrating on color nonsuperconducting quark matter with two flavors. Besides setting up the formalism, a central aspect will be a comparison with the MIT bag model, which is often used to describe dense matter. In Chap. 3 the model is extended to three quark flavors. As a first application, we investigate the possibility of absolutely stable strange quark matter.
Color superconducting phases are included in the Chapters 47. In Chap. 4 we again restrict ourselves to twoflavor systems. We begin with a general overview about the basic concepts of color superconductivity and briefly discuss other approaches. We then extend our formalism to include diquark pairing and discuss the role and mutual influence of several conventional and less conventional quarkantiquark and diquark condensates. A point of particular interest will be the discussion of a spin1 diquark condensate as a possible pairing channel for those quarks which are left over from the standard spin0 condensate. Next, in Chap. 5 we consider color superconductivity in a threeflavor system. Central issue will be the description of the 2SCCFL phase transition, taking into account and dependent constituent quark masses together with the diquark condensates. For simplicity, we consider a single chemical potential for all quarks. This restriction is relaxed in Chap. 6, where we construct neutral quark matter in beta equilibrium which is of possible relevance for compact stars. To that end we have to introduce up to four independent chemical potentials. Both, homogeneous and mixed neutral phases are discussed. In Chap. 7 we use the resulting homogeneous quark matter equation of state to investigate the possibility of a quark matter core in neutron stars. In order to model the hadronic phase we take existing hadronic equations of state from the literature. Finally, in Chap. 8 we summarize what we have done and, more important, what remains to be done.
Chapter 2 NJLmodel description of color nonsuperconducting twoflavor quark matter
In this chapter we give a general introduction to the use of Nambu–JonaLasinio (NJL) type models for analyzing quark matter at nonzero density or temperature. To that end, we concentrate on the – technically simpler – twoflavor version of the model and neglect the possibility of color superconducting phases. The NJL model will be introduced in Sec. 2.2 where we briefly summarize its vacuum properties, before the analysis is extended to hot and dense matter in Sec. 2.3. Of course, this is not meant to be exhaustive, and the interested reader is referred to Refs. [70, 71, 72, 73] for reviews. Here, our main intention is to lay a solid ground for our later investigations, including a critical discussion of the limits of the model.
A central aspect of the present chapter will be a comparison with the MIT bag model, which is the most frequently used model to describe quarkgluon matter at large temperature or density. Originally, both models have been developed to analyze hadron properties. Focusing on two different aspects of QCD, they are almost complementary: Whereas the MIT bag model is based on a phenomenological realization of confinement, the main characteristics of the NJL model is chiral symmetry and its spontaneous breakdown in vacuum. On the other hand the NJL model does not confine and the MIT bag model violates chiral symmetry. Nevertheless, the models behave quite similarly when they are employed to calculate the equation of state of deconfined quark matter at high density. In this regime the essential feature of both models is the existence of a nonvanishing vacuum pressure (‘‘bag constant’’) whereas confinement and chiral symmetry are of course less important in the deconfined, chirally restored regime^{2.1}^{2.1}2.1In fact, the MIT bag model is chirally symmetric in the thermodynamic limit, because chiral symmetry is broken only at the bag surface.. In the MIT bag model the bag constant is an external parameter, while in the NJL model it is dynamically generated. In order to work out this correspondence in some detail, we begin with a brief summary of the basic features of the MIT bag model.
2.1 MIT bag model
The MIT bag model has been suggested in the midseventies as a microscopic model for hadrons [74, 75, 76]. At that time, QCD had already been formulated and the MIT bag model was one of the first quark models where the notions of confinement and asymptotic freedom have been implemented in a constitutive way^{2.2}^{2.2}2.2 Strictly speaking, rather than on asymptotic freedom, the model was based on the experimental fact of Bjorken scaling. Since QCD was not yet generally accepted, at least at the beginning [74], the model was presented in a more general way, referring to QCD only as one of several possibilities. .
In the MIT bag model, hadrons consist of free (or only weakly interacting) quarks which are confined to a finite region of space: the ‘‘bag’’. The confinement is not a dynamical result of the underlying theory, but put in by hand, imposing the appropriate boundary conditions^{2.3}^{2.3}2.3 However, if the quarks are coupled to a nonAbelian gauge field, one finds the nontrivial result that the boundary conditions can only be fulfilled if the system is a color singlet [74]. . The bag is stabilized by a term of the form which is added to the energymomentum tensor inside the bag. Recalling the energymomentum tensor of a perfect fluid in its rest frame,
(2.1) 
the bag constant is immediately interpreted as positive contribution to the energy density and a negative contribution to the pressure inside the bag. Equivalently, we may attribute a term to the region outside the bag. This leads to the picture of a nontrivial vacuum with a negative energy density and a positive pressure . The stability of the hadron then results from balancing this positive vacuum pressure with the pressure caused by the quarks inside the bag.
The MIT bag model says nothing about the origin of the nontrivial vacuum, but treats as a free parameter. Evaluating the energymomentum tensor in QCD, one finds
(2.2) 
which is dominated by the contribution of the gluon condensate (first term on the r.h.s.). In the second term denotes a quark with flavor , and is the corresponding current quark mass. Employing the QCD sumrule result of Ref. [77], Shuryak obtained 455 MeV [78], while a modern value of the gluon condensate would yield a somewhat larger result. In any case, as we will see below, this is much larger than the values of one obtains in a typical bag model fit.
2.1.1 Hadron properties
Assuming a static spherical bag of radius , the mass of a hadron in the MIT bag model is given by the sum [76]
(2.3) 
The first term on the r.h.s. corresponds to the volume energy, required to replace the nontrivial vacuum by the trivial one inside the bag. The second term was introduced in Ref. [76] to parametrize the finite part of the zeropoint energy of the bag. The constant was treated as a free parameter, whose theoretical determination was left for future work. In a later analysis, however, it turned out that not all singularities arising from the zeropoint energy could be absorbed in a renormalization of the model parameters, like the bag constant [79] (also see [80] for a recent discussion). Therefore the definition of the finite part is ambiguous and remained an undetermined fitting parameter in the literature. The third term in Eq. (2.3) is the (rest + kinetic) energy of the quarks. For massless quarks in the lowest state one finds as solutions of the eigenvalue problem. Finally, corresponds to perturbative corrections due to lowestorder gluon exchange. This term gives rise, e.g., to the mass splitting.
Fit  [MeV]  [MeV]  [MeV/fm]  [MeV]  [fm]  

[76]  0  279  57.5  1.84  2.2  280  1.0 
[76]  108  353  31.8  1.95  3.0  175  1.1 
[87]  0  288  351.7  0.00  (running)  (tachyonic)  0.6 
[88]  5  354  44.7  1.17  0  not given  1.0 
[88]  5  356  161.5  2.04  0  not given  0.6 
Eq. (2.3) contains the following parameters: The bag constant , the parameter , the quark masses (entering ), and the strong coupling constant (entering ). The bag radius is not a parameter but is separately fixed for each hadron to minimize its mass. The parameters of the original fit [76] are listed in Table 2.1 (first two lines). They have been adjusted to fit the masses of the nucleon, the , the , and the meson. The light quark masses () have not been fitted, but have been set to zero (fit A) and to 108 MeV (fit B) to test the sensitivity of the fit to its variation. With these parameters the authors of Ref. [76] obtained a good overall fit of the light hadron spectra (baryon octet and decuplet, and vector meson nonet), magnetic moments, and charge radii.
There are, however, a couple of wellknown problems: Since chiral symmetry is explicitly broken on the bag surface and the anomaly is not included, the pion mass comes out too large, whereas the is too light. Also, the values of needed to reproduce the mass splitting are extremely large and obviously inconsistent with the idea that the corrections are perturbative. Finally, the nucleon radii of 1.0  1.1 fm, as shown in Table 2.1, would mean that the bags overlap with each other in a nucleus, which is clearly inconsistent with the success of mesonexchange models.
Some of these problems are related to each other. For instance, in the socalled chiral bag models, where chiral symmetry is restored by coupling external pion fields (introduced as elementary fields) to the bag surface [81], the bag radii can be considerably smaller than in the MIT bag (“little bag” [82] fm, “cloudy bag” [83] fm). This also leads to smaller values of needed to fit the mass splitting [82, 83]. In the chiral bag models the nucleon mass is the sum of the bag contribution and a selfenergy contribution from the pion cloud. Since the latter is negative, the former can be (much) larger than the physical nucleon mass. This is the reason why the radii can be smaller than in the MIT fit. In fact, many observables are quite insensitive to the bag radius, which can even be taken to be zero (“Cheshire cat principle”, for review see, e.g., Ref. [84]).
Eq. (2.3) contains effects of spurious c.m. motion, which can approximately be projected out if one replaces by the mass [85, 86, 87]. This correction has not been performed in the original MIT fit [76] but is standard in modern bag model calculations. It has the appreciable effect that the bag radius is reduced by about 30% and might also remove some of the other problems discussed above. In fact, instead of a too large pion mass, the authors of Ref. [87] obtain slightly negative, and reasonable values for and , once c.m. corrections are included (third line of Table 2.1). Note, however, that this model also differs from the original MIT bag model in the treatment of the perturbative corrections, bringing in additional parameters, while the parameter was not employed in the fit. On the other hand, the authors of Ref. [88] did not include perturbative corrections and restricted their fit to the baryon octet. They also considered and mesons, but to that end they introduced independent zeropoint energies as additional parameters. In this way the (baryonic) zeropoint energy was left undetermined and could be employed to vary the bag radius at will (cf. last two lines of Table 2.1).
For the thermodynamic description of hot or dense quarkgluon matter, many problems discussed above are not of direct relevance (see next section). We have mentioned them, however, because they have a strong effect on the parameter fit, and thereby indirectly on the thermodynamics. In this context, the most important parameter is the bag constant. As obvious from Table 2.1, the fitted values of vary by more than a factor of six, although they are all smaller than the value derived from the gluon condensate, Eq. (2.2). From a modern point of view it is also remarkable that the strange quark mass in all fits listed in the table is considerably larger than today accepted values of its “current quark mass” (see Table 3.1). Later we will see that similar masses appear in NJLmodel quark matter at comparable densities.
2.1.2 Thermodynamics
When we consider a large number of quarks and gluons in a large MIT bag we can replace the exact solutions of the boundary problem by plane waves, while zeropoint energy and c.m. correction terms drop out. Hence the energy density at temperature and a set of chemical potentials reduces to
(2.4) 
where is the energy density of a free relativistic gas of quarks, antiquarks, and gluons, while corresponds to perturbative corrections. Equivalently, we can write for the pressure
(2.5) 
Here the earlier mentioned role of as a negative pressure inside the bag relative to the nontrivial vacuum is obvious. The free part is given by
(2.6) 
where the second integral corresponds to the gluons (2 spin and 8 color degrees of freedom) and the first integral corresponds to the quarks and antiquarks (2 spin and 3 color degrees of freedom and a sum over flavors). is the onshell energy of a quark of flavor with threemomentum .
In the following we consider the case of two massless quark flavors with a common chemical potential . The integrals in Eq. (2.6) are then readily evaluated and the bag model pressure becomes
(2.7) 
where we have neglected the perturbative contribution. The famous factor 37 = 16+21 in front of the term is the sum of the 16 gluonic degrees of freedom and the product of the 24 quark and antiquark degrees of freedom with a factor due to Fermi statistics.
In order to be stable, the pressure in the quarkgluon phase must not be negative. For this leads to a minimum temperature
(2.8) 
while for we obtain a minimum chemical potential
(2.9) 
It is interesting that has already been derived by the MIT group in their first paper about the bag model [74]. There was identified with the “limiting temperature” of a highly excited hadron, i.e., the temperature a hadronic bag must not exceed in order to remain stable. Thus, although formally equivalent to the above derivation of , the perspective is rather opposite.
For arbitrary chemical potentials smaller than one can easily solve for the temperature for which the pressure vanishes. The result is displayed in Fig. 2.1 (dashed line). For later purposes we also show the corresponding curve one obtains with quark degrees of freedom only (dotted). In this case the factor 37 in Eq. (2.7) is replaced by 21. Accordingly the minimum temperature is enhanced by a factor , while the minimum chemical potential remains unchanged.
Eq. (2.7) is the most simple example for a bagmodel description of the quarkgluon plasma. In order to construct a phase transition, we also need an equation of state for the hadronic phase. Usually, the latter is not described within the bag model as well but taken from models with hadronic degrees of freedom. As a prototype, we consider a gas of noninteracting massless pions, which should dominate the lowtemperature regime. The resulting phase boundary, i.e., the line where becomes equal to the pressure of the pion gas, is indicated by the solid line in Fig. 2.1. Since the pions have an isospin degeneracy of three, the pressure difference is given by Eq. (2.7), but with 37 replaced by 373=34. Hence, the critical temperature at is given by .
So far we have not set the absolute scale in our phase diagram. Taking the bag constant from the original MIT fit, MeV/fm ( MeV) [76], we obtain MeV, whereas in order to get the lattice value, MeV, we need a seven times larger bag constant MeV/fm ( MeV). For , which in the present model is the critical chemical potential for the phase transition at , this variation of leads to values roughly ranging from 300 to 500 MeV.
For several reasons, however, it is clear that a model of this type is too simplistic:

Since mesons are not sensitive to the (quark number) chemical potential they do not influence the phase transition in the large low regime. Thus, for a more realistic description, we need baryonic degrees of freedom. In the above example, it would be most natural to add the contribution of a free nucleon gas to the hadronic phase. It is a wellknown fact, however, that this would lead to the unphysical situation that the hadronic phase “wins” at large chemical potentials. This is obvious from the contribution of a fermion of type to the pressure at ,
(2.10) where the ellipsis indicates corrections due to the fermion mass which can be neglected at very large chemical potentials. Here is a degeneracy factor which is three times larger for quarks than for nucleons because of color. On the other hand, since nucleons consist of three valence quarks, their chemical potential is three times larger than the quark number chemical potential. Altogether, this means , which leads to the above mentioned unphysical result.

As we have seen, the phase transition at is a result of the larger number of quarkgluon degrees of freedom (37) compared with the hadronic ones (3 for the pion gas), i.e., the larger coefficient of the term. This necessarily means, that the phase transition is firstorder with a latent heat per volume
(2.11) This is in strong contrast to the universality arguments mentioned in the Introduction, according to which we would expect a secondorder phase transition in QCD with two massless flavors [49]. It is possible to reduce by introducing additional hadrons, but obviously it is very difficult to get without inhibiting the phase transition.
It is quite plausible that these problems can be traced back to the hybrid nature of the above model, i.e., the fact that the hadronic and the quarkgluon degrees of freedom are not derived from the same Lagrangian. In a more consistent picture one should start from a gas of hadronic bags to describe the hadronic phase. Since the bags have finite sizes which can only be reduced at the expense of energy, it is obvious that the system will not stay in the hadronic phase up to arbitrarily large densities. In the most naive picture the bags would simply unite to form a uniform phase when the average quark number density exceeds the density inside a single bag. (In fact, it is more difficult to prevent the bags from forming one large bag already at low densities. For this one would need to introduce a negative surface tension or some repulsive force between separate bags.) In this way also the connection between the chiral and the deconfinement phase transition appears quite natural if one attributes the nontrivial vacuum outside the bags to the spontaneous breakdown of chiral symmetry, which is restored inside the bags. We will come back to this point of view in Sec. 2.3.3.
While qualitatively the picture drawn in the previous paragraph looks quite attractive, its quantitative realization is of course very difficult. An interesting step in this direction is the quarkmeson coupling model where nuclear matter and even finite nuclei are described by MIT bags which interact by exchange of (elementary) scalar and vector mesons [88, 89, 90, 91]. A somewhat “cheaper” alternative is to employ hybrid models with finite volume corrections on the hadronic side (see, e.g., Ref. [92]).
2.1.3 Equation of state at zero temperature
Since we are mostly interested in quark matter at low temperature we would like to discuss a few more details of the bag model equation of state in the zero temperature limit. This will also provide a basis for our later comparison with the NJL model.
For , Eq. (2.6) simplifies, and the total pressure is given by
(2.12) 
where denotes the Fermi momentum of flavor . For simplicity, we have dropped the temperature argument.
In the following, we concentrate again on the case of a uniform chemical potential for two massless flavors. Neglecting the perturbative term and applying standard thermodynamic relations, we obtain for the pressure, energy density, and quark number density
(2.13) 
This is simply a free gas behavior, modified by the bag constant. From these expressions we immediately get for the function , which, e.g., determines the massradius relation of neutron stars,
(2.14) 
Another quantity of interest, which follows from Eq. (2.13), is the energy per particle as a function of density,
(2.15) 
For later convenience, we rephrase this as the energy per baryon number in terms of the baryon number density ,
(2.16) 
This function is plotted in Fig. 2.2 (solid line).
Its general structure is easily understood if we recall that . Thus the first term on the r.h.s. is just the volume energy of the MIT bag, while the second term reflects the behavior of the quark energy in Eq. (2.3). Of course, the coefficient of the latter depends on the number of occupied states and therefore it is different in the thermodynamic limit from the case of a single hadron.
Because of the interplay of the two terms, diverges for both, and , and has a minimum at
(2.17) 
Note that this is just the point where the pressure vanishes, reflecting the general thermodynamic relation
(2.18) 
The physical meaning of this relation becomes clear if we consider a finite lump of quark matter (large enough that the thermodynamic treatment is valid). For (i.e., ), the pressure is negative and the lump shrinks, thereby increasing the density. On the other hand, for (), the pressure is positive and the lump tends to expand, unless this is prevented by external forces. Hence, the only stable point is , where the pressure vanishes. In a canonical (instead of grand canonical) treatment, which would be more appropriate for this example with fixed particle number, this stability becomes manifest as a minimum in the energy. (Recall that and hence the Helmholtz free energy .)
The absolute scale of Fig. 2.2 is again set by the value of the bag constant. As discussed above, varying between the two “extreme” values, MeV/fm from the original MIT fit and MeV/fm from fitting MeV within the simple hybrid model, leads to values of between about 300 and 500 MeV. This corresponds to ranging from about 900 MeV at to 1500 MeV at . Of course, values for lower than the energy per nucleon in atomic nuclei must be excluded, since otherwise nuclei should be able to decay into a large bag of deconfined quarks. We will come back to this in Sec. 3.3.
Finally, we discuss briefly the influence of perturbative corrections, which we have neglected so far. To order the freegas terms in Eq. (2.13) are scaled by a factor [93],
(2.19) 
Obviously, this correction makes only sense for . This demonstrates again that the value of the original MIT fit, , is completely out of range for a perturbative treatment. Note that it would even change the sign of the freegas terms.
As a consequence of the correction, the chemical potential at which the pressure vanishes and thus are enhanced by a factor , whereas the corresponding baryon number density is reduced by a factor . Accordingly, the minimum of is shifted to a larger value at a lower density. This is illustrated by the dashed line in Fig. 2.2, which corresponds to . On the other hand, the relation between and , Eq. (2.14), remains unaffected by the correction.
2.2 Nambu–JonaLasinio model in vacuum
As pointed out earlier, the Nambu–JonaLasinio (NJL) model is to some extent complementary to the MIT bag model. Historically, it goes back to two papers by Nambu and JonaLasinio in 1961 [66, 67], i.e., to a time when QCD and even quarks were still unknown. In its original version, the NJL model was therefore a model of interacting nucleons, and obviously, confinement – the main physics input of the MIT bag model – was not an issue. On the other hand, even in the preQCD era there were already indications for the existence of a (partially) conserved axial vector current (PCAC), i.e., chiral symmetry. Since (approximate) chiral symmetry implies (almost) massless fermions on the Lagrangian level, the problem was to find a mechanism which explains the large nucleon mass without destroying the symmetry. It was the pioneering idea of Nambu and JonaLasinio that the mass gap in the Dirac spectrum of the nucleon can be generated quite analogously to the energy gap of a superconductor in BCS theory, which has been developed a few years earlier [68]. To that end they introduced a Lagrangian for a nucleon field with a pointlike, chirally symmetric fourfermion interaction [67],
(2.20) 
Here is a small bare mass of the nucleon, is a Pauli matrix acting in isospin space, and a dimensionful coupling constant. As we will discuss in more detail in Sec. 2.2.1, the selfenergy induced by the interaction generates an effective mass which can be considerably larger than and stays large, even when is taken to zero (“chiral limit”). At the same time there are light collective nucleonantinucleon excitations which become massless in the chiral limit: The pion emerges as the Goldstone boson of the spontaneously broken chiral symmetry. In fact, this discovery was an important milestone on the way to the general derivation of the Goldstone theorem in the same year [94].
After the development of QCD, the NJL model was reinterpreted as a schematic quark model [95, 96, 97]. At that point, of course, the lack of confinement became a problem, severely limiting the applicability of the model. On the other hand, there are many situations where chiral symmetry is the relevant feature of QCD, confinement being less important. The most prominent example is again the Goldstone nature of the pion. In this aspect the NJL model is superior to the MIT bag, which, as we have seen, fails to explain the low pion mass.
End of the nineties, a third era of the NJL model began when the model was employed to study color superconducting phases in deconfined quark matter. There, by definition, lack of confinement is again of minor relevance. As outlined in the Introduction, color superconductivity will be discussed in great detail in Chap. 4 and thereafter.
After the reinterpretation of the NJL model as a quark model, many authors kept the original form of the Lagrangian, Eq. (2.20), with now being a quark field with two flavor and three color degrees of freedom. However, this choice is not unique and we can write down many other chirally symmetric interaction terms. For instance, from a modern point of view, local 4point interactions between quarks in a twoflavor system can be thought of to be abstracted from instanton induced interactions [36]. In this case the interaction Lagrangian should have the form
(2.21) 
Here and in the following we denote quark fields by . In general we will call all these models “NJLtype models” (or just “NJL models”) as long as they describe quarks interacting via 4point vertices (or sometimes higher point vertices).
2.2.1 Constituent quarks and mesons
In this section we briefly review the vacuum properties of quarks and mesons described within the NJL model. For simplicity, we restrict this discussion to the standard NJLLagrangian, Eq. (2.20), for quarks with two flavor and three color degrees of freedom. Most of this can easily be generalized to other NJLtype Lagrangians with two degenerate flavors. The discussion of the threeflavor case, which has some additional features, is deferred to the next chapter.
In most publications (including the original papers by Nambu and JonaLasinio [66, 67]) the quark selfenergy which arises from the interaction term has been calculated within Hartree or HartreeFock approximation. The corresponding Dyson equation is depicted in Fig. 2.3.
Since in this approximation the selfenergy is local, it only gives rise to a constant shift in the quark mass,
(2.22) 
Here is the dressed quark propagator, underlining the nonperturbative character of the approximation. The trace is to be taken in color, flavor, and Dirac space. One finds
(2.23) 
where and are the number of flavors and colors, respectively. For a sufficiently strong coupling , this allows for a nontrivial solution , even in the chiral limit , producing a gap of in the quark spectrum. In analogy to BCS theory, Eq. (2.23) is therefore often referred to as “gap equation”. is often called “constituent quark mass”. A closely related quantity is the quark condensate, which is generally given by
(2.24) 
and thus in the present case
(2.25) 
Iterating the 4point vertex as shown in Fig. 2.4 yields the quarkantiquark Tmatrix in random phase approximation (RPA),
(2.26) 
where
(2.27) 
is the quarkantiquark polarization in the channel with the quantum numbers . For the Lagrangian Eq. (2.20) we have the sigma channel () and three pion channels (, ). Evaluating the traces and employing the gap equation for one finds
(2.28) 
where
(2.29) 
In order to determine meson properties one interpretes as an effective meson exchange between the external quark legs in Fig. 2.4 and parametrizes the pole structure as
(2.30) 
Thus
(2.31) 
Using Eq. (2.28), one immediately finds that if , in accordance with the Goldstone theorem. For , also becomes nonzero (see below).
The pion decay constant can be obtained from the onepiontovacuum matrix element visualized in Fig. 2.5,
(2.32) 
It is straight forward to show that in the chiral limit the generalized GoldbergerTreiman relation [98],
(2.33) 
holds. Moreover, in first nonvanishing order in , the pion mass satisfies the GellMann Oakes Renner relation [99],
(2.34) 
In the brief discussion presented above, we have ignored several problems:

In general, the gap equation has more than one solution. For instance, in the chiral limit, , there is always a trivial solution , but there can be nontrivial solutions as well. In this case, one has to find out which solution minimizes the vacuum energy.
One way is to go back to the underlying mechanism of the gap equation, which is a BogoliubovValatin rotation of the fields. Starting point is a variational ansatz for the nontrivial vacuum of the model [10, 66, 71],
(2.35) where is the perturbative vacuum, and and are the corresponding annihilation operators for a quark or antiquark, respectively, with momentum , helicity , flavor , and color . According to this ansatz, is a coherent state composed of quarkantiquark pairs with zero total momentum. This underlines once more the analogy to the BCS ground state. Minimizing the ground state energy,
(2.36) ( Hamiltonian of the model) with respect to variations of the functions and leads to a selfconsistency equation, which is equivalent to the HartreeFock gap equation discussed above. (For the difference between Hartree and HartreeFock, see below.) It turns out that the vacuum energy is minimized by the solution with the largest . In the chiral limit this means that the nontrivial solution is stable whenever it exists.
An alternative way to calculate the ground state energy will be presented in Sec. 2.3 in the context of the thermodynamics of the model. There we will discuss further details.

The NJL model is not renormalizable. Since the above expressions contain divergent integrals, e.g., Eqs. (2.23), (2.29), and (2.32), we have to specify how to regularize these divergencies. This prescription is then part of the model. There are several regularization schemes which have been used in the literature, and each of them have certain advantages and disadvantages [71, 73]. When the model is applied to thermodynamics, most authors prefer to regularize the integrals by a (sharp or smooth) 3momentum cutoff. Besides being relatively simple, this has the advantage that it preserves the analytical structure, necessary, e.g., for the analytical continuation of functions given on imaginary Matsubara frequencies. Of course, 3momentum cutoffs violate the Lorentz covariance of the model. It is often argued that this problem is less severe at finite temperature or density where manifest covariance is anyway broken by the medium. Although this argument is questionable, since it makes a difference whether the symmetry is broken by physical effects or by hand, it is perhaps true that a 3momentum cutoff has the least impact on the medium parts of the regularized integrals, in particular at . This will become more clear in Sec. 2.3. In this report we will therefore regularize the model using a sharp 3momentum cutoff , unless stated otherwise.

We already mentioned that the NJL model does not confine. Formally, this is reflected by the fact that the integral , and hence the polarization functions , get an imaginary part above the threshold, i.e., for . As a consequence, mesons with a mass larger than have a finite width, which indicates that they are unstable against decay into a quarkantiquark pair. The pion is of course not affected by this problem. However, as can be seen from Eqs. (2.28) and (2.31), if and it moves above the threshold if . If vector mesons are included, it depends on the parameters whether they have masses above or below the threshold, while axial vector mesons always decay into pairs in the model.

The formulae given above correspond to the Hartree approximation and to RPA without Pauli exchange terms, respectively. However, because of the local 4point interaction, exchange diagrams can always be cast in the form of direct diagrams via a Fierz transformation (see App. A). This means, the HartreeFock approximation is equivalent to the Hartree approximation with appropriately redefined coupling constants. In this sense, Hartree is as good as HartreeFock, as long as the interaction terms in the Lagrangian are not fixed by some underlying theory.
Extensions of the approximation scheme beyond HartreeFock + RPA are much more difficult. This topic will briefly be discussed in Sec. 2.3.5.
2.2.2 Parameter fit
As a basis for the subsequent discussions we perform a first parameter fit for the simple twoflavor model, Eq. (2.20), within the Hartree + RPA scheme. As mentioned above, we will regularize the integrals by a sharp 3momentum cutoff. We thus have three parameters, the bare quark mass , the coupling constant , and the cutoff . These parameters are usually fixed by fitting the pion mass, the pion decay constant, and the quark condensate. Whereas the pion mass, MeV [100]^{2.4}^{2.4}2.4This corresponds to the mass of the , which is not affected by electromagnetic corrections [101]. For the NJL model, this was explicitly proven in Ref. [102]. Of course, having the other uncertainties in mind, fitting to the charged pion mass would not cause any practical difference. and the pion decay constant, MeV [103], are known quite accurately, the uncertainties for the quark condensate are rather large. Limits extracted from sum rules are 190 MeV 260 MeV at a renormalization scale of 1 GeV [104], while lattice calculations yield MeV [105].
In this situation we first fix and for arbitrary values of by fitting and to their empirical values. The corresponding solutions of the gap equation are displayed in Fig. 2.6. In the left panel the constituent mass is shown as a function of the cutoff. Obviously, must be larger than some critical value (about 568 MeV 6.1 ) to find a solution. Above this value, there is a “lowmass” branch ( MeV) and a “highmass” branch of solutions. This kind of behavior is typical for the model and is also found within other regularization schemes [73].
On the r.h.s. of Fig. 2.6 we show the corresponding values of the quark condensate. We see that the model (together with the used regularization scheme) cannot accommodate values of smaller than 240 MeV. Taking the upper limit of Ref. [104], 260 MeV, we find 720 MeV for the lowmass branch and 585 MeV for the highmass branch. This restricts the constituent mass to lie between about 270 and 800 MeV. Taking the upper limit of to be 250 MeV, as suggested by the results of Ref. [105], would constrain to an interval between about 300 and 640 MeV.
Four parameter sets, more or less representing this interval, are listed in Table 2.2. In the last column we also list the corresponding “bag constants”, i.e., the energy gain per volume, due to the formation of the nontrivial vacuum state. Obviously, the uncertainty in (caused by the uncertainty in the quark condensate and the peculiar behavior of having two solutions for the same value of ) leads to a big uncertainty in the bag constant and thereby in the thermodynamic behavior of the model. This will be discussed in more detail in the next section.
set  [MeV]  [MeV]  [MeV]  [MeV]  [MeV/fm]  

1  664.3  2.06  5.0  300  250.8  76.3 
2  587.9  2.44  5.6  400  240.8  141.4 
3  569.3  2.81  5.5  500  242.4  234.1 
4  568.6  3.17  5.1  600  247.5  356.1 
2.3 Nonzero densities and temperatures
Soon after the reinterpretation of the NJL model as an effective quark model it has been employed to study quark and meson properties in hot or dense matter [106, 107].
Applying standard techniques of thermal field theory [108] it is straight forward to evaluate the quark loop which enters the gap equation or the mesonic polarization diagrams at nonvanishing temperature or chemical potential. The results have basically the same structure as the vacuum expressions, but are modified by thermal occupation numbers. For instance, the gap equation Eq. (2.23) becomes
(2.37) 
where is the onshell energy of the quark, selfconsistently evaluated for the constituent mass which solves the equation^{2.5}^{2.5}2.5We will not use a special notation, like, e.g., , to indicate inmedium quantities, since most quantities in this report correspond to nonzero temperature or density. Instead, we will sometimes indicate vacuum quantities by the suffix “”, if necessary.. and are Fermi occupation numbers of quarks and antiquarks, respectively,
(2.38) 
They are related to the total quark number density in the standard way,
(2.39) 
For , we have and Eq. (2.37) becomes identical to Eq. (2.23) if there the integration over is turned out.
In medium, the occupation numbers are nonzero and reduce the value of the constituent mass. For large temperatures or densities the factor goes to zero, and approaches the value of the current mass . This is illustrated in Fig. 2.7 for zero density and nonzero temperatures (left panel) and for zero temperature and nonzero densities (right panel). The solid lines indicate the (maximal) solutions of the gap equation for parameter set 2 of Table 2.2. Taking the chiral limit we arrive at the dashed lines. In this case Eq. (2.37) does no longer support nontrivial solutions for temperatures larger than MeV or densities larger than . (For a detailed discussion of the critical line in the densitytemperature plane, see, e.g., [109].) Note that the critical temperature at zero density is quite large compared with the lattice value of about 170 MeV, while the critical density at zero temperature seems to be rather low. As we will discuss in Sec. 2.3.4, this is a typical feature of an NJL meanfield calculation, which is mainly due to the fact that the phase transition is driven by the wrong degrees of freedom (unconfined quarks).
In the beginning, the smooth behavior of the constituent quark mass as a function of temperature or density lead several authors to believe that the phase transition is second order [106]. However, in order to decide on the order of the phase transition it is not sufficient to solve the gap equation. As we have seen in vacuum, the gap equation does not have a unique solution and one should look for the solution with the lowest energy. Similarly, at nonvanishing temperature or chemical potential, one should minimize the (grand canonical) thermodynamic potential. In this way it was revealed by Asakawa and Yazaki that the phase transition can indeed be first order, at least at low temperatures [60]. In the following we will basically adopt their method to calculate the thermodynamic potential.
2.3.1 Thermodynamic potential
We consider a twoflavor NJLtype Lagrangian of the form
(2.40) 
To keep the discussion rather general we have not restricted ourselves to the standard NJL interaction term in the scalar and pseudoscalarisovector channels, but we have added explicitly a term in the vectorisoscalar channel. It is known, e.g., from the Walecka model [110], that this channel is quite important at nonzero densities. In principle we allow for further channels (indicated by the ellipsis), which, however, do not contribute at meanfield level as long as we have only one common quark chemical potential.
The thermodynamic potential per volume at temperature and quark chemical potential is defined as
(2.41) 
where is the Hamiltonian density and a functional trace over all states of the system, i.e., spin, flavor, color and momentum. is the grand canonical partition function.
To calculate in meanfield (Hartree) approximation we could in principle proceed in the way sketched in Sec. 2.2.1, i.e., we define a nontrivial ground state via a Bogoliubov rotation and then evaluate the free energy in this state. This method would have the advantage that it contains the full information about the structure of the ground state. On the other hand, the derivations become quite involved and extensions to include several condensates at the same time are very difficult. Therefore we follow Ref. [60] where an equivalent but much simpler method has been applied. To that end we consider two nonvanishing “condensates”,
(2.42) 
i.e., the quark condensate and the total quark number density. As long as we assume that – apart from chiral symmetry and Lorentz invariance (which is explicitly broken by the chemical potential) – all symmetries of the Lagrangian remain intact, these are the only allowed expectation values which are bilinear in the quark fields. (Later we will encounter many other condensates, due to both, explicit and spontaneous symmetry breaking.)
Next, we linearize the interaction terms of in the presence of and ^{2.6}^{2.6}2.6 A more formal, but essentially equivalent, method is to bosonize the model. In that context, and emerge as auxiliary Bose fields which are introduced in the framework of a HubbardStratonovich transformation.,
(2.43) 
where terms quadratic in the fluctuations, like , have been neglected. In particular terms in channels without condensate, like or the space components in the vector vertex, drop out. In this approximation,
(2.44) 
where we have introduced the constituent mass and the renormalized chemical potential ,
(2.45) 
This means, apart from constant (i.e., field independent) terms, which give trivial contributions to the r.h.s. of Eq. (2.41), the problem is equivalent to a system of noninteracting particles with mass at chemical potential . Hence, the meanfield thermodynamic potential takes the form
(2.46) 
with the free Fermigas contribution