Roles of axial anomaly on neutral quark matter with color superconducting phase

# Roles of axial anomaly on neutral quark matter with color superconducting phase

Zhao Zhang School of Mathematics and Physics, North China Electric Power University, Beijing 102206, China    Teiji Kunihiro Department of Physics, Kyoto University, Kyoto 606-8502, Japan
###### Abstract

We investigate effects of the axial anomaly term with a chiral-diquark coupling on the phase diagram within a two-plus-one-flavor Nambu-Jona-Lasinio (NJL) model under the charge-neutrality and -equilibrium constraints. We find that when such constraints are imposed, the new anomaly term plays a quite similar role as the vector interaction does on the phase diagram, which the present authors clarified in a previous work. Thus, there appear several types of phase structures with multiple critical points at low temperature , although the phase diagrams with intermediate- critical point(s) are never realized without these constraints even within the same model Lagrangian. This drastic change is attributed to an enhanced interplay between the chiral and diquark condensates due to the anomaly term at finite temperature; the u-d diquark coupling is strengthened by the relatively large chiral condensate of the strange quark through the anomaly term, which in turn definitely leads to the abnormal behavior of the diquark condensate at finite , inherent to the asymmetric quark matter. We note that the critical point from which the crossover region extends to zero temperature appears only when the strength of the vector interaction is larger than a critical value. We also show that the chromomagnetic instability of the neutral asymmetric homogenous two-flavor color superconducting(2CSC) phase is suppressed and can be even completely cured by the enhanced diquark coupling due to the anomaly term and/or by the vector interaction.

###### pacs:
12.38.Aw, 11.10.Wx, 11.30.Rd, 12.38.Gc

## I Introduction

It is generally believed that the strongly interacting matter exhibits a rich phase structure in extreme environment such as at high temperature and high baryon chemical potential. Experimentally, RHIC (Relativistic Heavy Ion Collider) and LHC (Large Hadron Collider) may provide more information on this topics. Theoretically, some results have been already obtained on a sound basis: First, the lattice simulations of quantum chromodynamics (QCD) indicate that, for physical quark masses, the transition from the hadronic phase to the quark gluon plasma (QGP) is a smooth crossover at finite temperature and vanishing baryon chemical potential  Cheng:2009be (); Borsanyi:2010bp (), whereas in the low temperature and very high density region, the techniques of perturbation QCD can be used and the color flavor locking (CFL) Alford:1998mk () phase is proved to be the ground state of QCD  Son:1998uk (); Schafer:1999jg (); Shovkovy:1999mr (); Schafer:1999fe ().

However, the above methods based on the first principle fail at the low temperature and moderate density region, due to the sign problem or the non-perturbative effect. Phenomenologically, such a region in the - plane is more relevant to reality and hence interesting since it is directly related to the physics of compact stars. On that account, chiral models of QCD such as the NJL model Nambu:1961tp (); Vogl:1991qt (); Klevansky:1992 (); Hatsuda:1994pi () that embody the basic low-energy characteristics of QCD such as symmetry properties have been extensively used to explore the - phase diagram of strongly interacting matter. In particular, such model calculations suggest that CSC phase may occur at low temperature and large chemical potential (for reviews, see Rajagopal:2000wf (); Rischke:2003mt (); Buballa:2003qv (); Alford:2007xm ()). In addition, a popular result from the model studies is that the chiral phase transition always keeps first order at the low-temperature region  Asakawa:1989bq (); Barducci:1989 (); Kunihiro:1991hp (); Berges:1998rc (); Ruester:2005jc (); Abuki:2005ms (). Combined with the crossover transition confirmed by lattice QCD at zero baryon chemical potential, usually, a schematic - phase diagram with one chiral critical point(CP) is widely adopted in the literature Stephanov:2007fk (). Such a CP may be located at relatively high temperature and low baryon chemical potential, which has attracted considerable attention as it is potentially detectable in heavy-ion experiments Stephanov:1998dy (); Minami:2009hn ().

Generally, there is no reason to rule out the possibility that the QCD phase diagram may contain more than one chiral CP, especially when the chiral and diquark condensates are considered simultaneously; some rich structures with multiple CP’s may be expected for the phase diagram owing to somehow enhanced interplay between the two types of condensates111Note that multiple critical points had also been found in two-flavor models of QCD without considering diquark paringBowman:2008kc (); Ferroni:2010ct ().. It has been shown on the basis of the NJL model that it is indeed the case  Kitazawa:2002bc (); Addenda (); Zhang:2008wx (); Zhang:2009mk (); the QCD phase diagram can admit multiple CP’s when the repulsive vector interaction  Kitazawa:2002bc (); Addenda () or the charge-neutrality and -equilibrium Zhang:2008wx () or both of these two ingredients Zhang:2009mk () are included 222 Note that the renormalization group theory for deducing low-energy effective vertex favors the presence of the vector-type interaction ref:EHS (); ref:SW-reno (). . This is because the two ingredients act so as to enhance the competition between the chiral and diquark condensates and thus the would-be first-order boundary line in the low-temperature region of the - plane can be turned into a smooth crossover or multiply-cut crossover lines with new CP(’s). Indeed it has been found Zhang:2009mk () that the number of the chiral CP’s may vary from zero to four with the joint effect of these two ingredients. Moreover, the present authors have shown Zhang:2009mk () that the vector interaction can effectively suppress the chromomagnetic instability Huang:2004bg () in the asymmetric homogeneous CSC phase.

It is noteworthy that a direct coupling term between the chiral and diquark condensates can be supplied by the axial anomaly Alford:1998mk (); Rapp:1999qa (); Steiner:2005jm (), which thus might lead to a new CP in the low-temperature region, as first conjectured in Hatsuda:2006ps () on the basis of an analysis using the Ginzburg-Landau (GL) theory in the chiral limit; see also subsequent detailed analyses Yamamoto:2007ah (); Baym:2008me () though still in the chiral limit. It is, however, to be noted that the GL theory assumes that both the diquark and chiral condensates are sufficiently small around the phase boundaries provided that the phase transitions are of second-order. In addition, the GL theory itself can not determine the coefficients in the action, and some microscopic model or theory is necessary for such a determination.

Recently, a microscopic calculation has been done Abuki:2010jq () with use of a three-flavor NJL model for massive quarks incorporating the axial anomaly term with a form of a six-quark interaction Alford:1998mk (); Rapp:1999qa (); Steiner:2005jm (); Yamamoto:2007ah () , the coupling constant of which is denoted by : It was claimed in  Abuki:2010jq () that the low-temperature CP can exist owing to the axial anomaly for an appropriate range of the model parameters even off the chiral limit but still with a flavor symmetry as in the GL approach in the chiral limit. It should be noticed, however, that the SU(3)-flavor symmetry may lead to a special type of CSC phase, i.e., the CFL phase, as is taken for granted in Hatsuda:2006ps (); Yamamoto:2007ah (); Baym:2008me (); Abuki:2010jq (), which automatically satisfies the charge-neutrality and -equilibrium constraints.

Then one may suspect that the possible emergence of the new CP might be an artifact of such an ideal situation with the three-flavor symmetry. Nevertheless, it is a very interesting possibility that the axial anomaly-induced interplay between the chiral and diquark condensates would lead to a new CP in the low-temperature region. Thus it is worth exploring to see whether a new low-temperature CP is induced by the axial anomaly in a dynamical model of QCD by considering the realistic situation with the broken flavor symmetry by the hierarchical current quark masses.We note that once the quark mass difference of different flavors is taken into account, it becomes a complicated dynamical problem to make the charge-neutrality and -equilibrium constraints satisfied.

More recently, such a realistic calculation in the framework of a two-plus-one-flavor NJL model has been done by Basler et al. Basler:2010xy (); they have shown that such a new low-temperature CP is not found in such a model even with the axial anomaly term, because an unusual interplay between the chiral and diquark condensates induced by the anomaly term actually leads favorably to the 2CSC phase Alford:1997zt (); Rapp:1997zu () rather than the CFL phase near the chiral phase boundary, even in the case with the equal quark mass limit  Basler:2010xy ().

It is worth emphasizing here that the constraints by the charge-neutrality and -equilibrium are not taken into consideration in Abuki:2010jq () nor Basler:2010xy () in contrast to Zhang:2008wx (); Zhang:2009mk () where various types of multiple-CP structures are found in the phase diagram. Thus the following two questions arise naturally: Will the results found in Basler:2010xy () be altered or not when the charge-neutrality and -equilibrium constraints and/or the vector interaction are taken into account? Or will the phase structure with multiple CP’s found in  Zhang:2009mk () rather persist when taking into account the coupling between the chiral and diquark condensates induced by such a six-quark interaction? The main purpose of this paper is to answer these questions by incorporating the anomaly term that breaks the symmetry as well as the vector interaction under the constraints of the charge-neutrality and -equilibrium in the two-plus-one-flavor NJL model. The present work may be regarded as either an extension of the paper  Basler:2010xy () by incorporating the charge-neutrality, -equilibrium and the vector interaction, or an extension of the paper Zhang:2009mk () by including the -interaction.

The main conclusion we reach is that the key results on the phase structure obtained in Ref. Zhang:2008wx (); Zhang:2009mk () persist even when the attractive -interaction is incorporated. That is, there appear new CP(’s) at the intermediate temperature owing to charge-neutrality constraint and then the transition in the low temperature region extending to zero becomes a crossover when the strength of the vector interaction becomes larger than a critical value: Thus the number of the CP’s can be even more than two, depending on the values of some related coupling constants. Strikingly enough, we find that the interplay between the chiral and the diquark condensates induced by the anomaly term even acts toward realizing the multi-CP structure of the phase diagram under the neutrality and -equilibrium constraints even without the help of the vector interaction. Accordingly, the results in Ref. Basler:2010xy () are modified by considering these constraints. Even though the chiral boundary in the low- region extending to zero also remains first order in our case in the absence of the vector interaction, which shrinks and vanishes eventually as becomes large and exceeds a critical value.

We shall also examine the chromomagnetic (in)stability under the influence of the axial anomaly, as was done in Zhang:2009mk (). It is well known that the asymmetric homogenous 2CSC phase suffers from the chromomagnetic instability. At zero temperature, the calculation based on the hard-dense-loop (HDL) method Huang:2004bg () suggests that the Meissner mass squared of the 8th gluon becomes negative for while the 4th-7th gluons acquire negative Meissner masses squared for ; here and denote the difference of the chemical potentials of u and d quarks and the gap, respectively. Note that is just equal to a half of the electron chemical potential when the vector interaction is absent, and this quantity is to be replaced by an effective chemical potential (see below) when the vector interaction is present, as shown in Zhang:2009mk (). The instability of the asymmetric homogenous CSC phase should imply the existence of a yet unknown but stable phase in this region of the - plane. Candidates of such a stable phase include the Larkin-Ovchinnikov-Fulde-Ferrel (LOFF) phase Giannakis:2004pf () and gluonic phase Gorbar:2005rx (). Besides developing the possible new phases, the instability problem may also be totally or partially gotten rid of by some other mechanisms. For instance, the instability problem becomes less severe simply at finite temperature because the smeared Fermi surface relaxes the mismatch of the Fermi spheres of the asymmetric quark matter Kiriyama:2006jp (); He:2007cn (); Fukushima:2005cm (). Furthermore, it is known that the larger the quark mass and the stronger the diquark coupling, more suppressed the instability even at zero temperatureKitazawa:2006zp (). Recently, the present authorsZhang:2009mk () have shown that the repulsive vector interaction can also resolve the instability problem totally or partially. The stability by the vector interaction is realized due to the following two ingredients: (1) the density difference between the u and d quarks reduces the mismatch in the effective chemical potentials; (2) the nonzero vector interaction suppresses the formation of high density and hence larger quark masses than those obtained without the interaction are realized. We shall show that the new anomaly term play a quite similar role as the vector interaction and the interplay between the chiral and diquark condensates induced by the anomaly term acts toward suppressing the unstable region of the homogeneous 2CSC phase in the - plane; the neutral 2CSC phase can become even free from the chromomagnetic instability if is larger than a critical value , which can be reduced significantly when the vector interaction is incorporated.

This paper is organized as follows. In Sec.II, the two-plus-one-flavor NJL model with the extended flavor-mixing six-quark interaction is introduced under the constraints of the charge-neutrality and -equilibrium. The phase diagram of the neutral strongly interacting matter with the influence of the axial anomaly is presented in Sec.III. Sec.IV focuses on the role of the axial anomaly on the chromomagnetic (in)stability. The conclusion and outlook are given in Sec.V.

## Ii NJL Model With Axial Anomaly and Vector Interaction

### ii.1 The model Lagrangian

We start from the following two-plus-one-flavor NJL model with the vector interaction Klimt:1989pm (); Zhang:2009mk () and two types of six-quark anomaly terms,

 L=¯ψ(i∂to0.0pt$╱$−^m)ψ+L(4)χ+L(4)d+L(6)χ+L(6)χd, (1)

where denotes the current-quark mass matrix and

 L(4)χ=GS8∑i=0[(¯ψλfiψ)2+(¯ψiγ5λfiψ)2]−GV8∑i=0[(¯ψγμλfiψ)2+(¯ψγμγ5λfiψ)2], (2) L(4)d=GD3∑i,j=1[(¯ψiγ5tfitcjψC)(¯ψCiγ5tfitcjψ)+(¯ψtfitcjψC)(¯ψCtfitcjψ)], (3) (4) L(6)χd=K′83∑i,j,k=1∑±[(ψtfitck(1±γ5)ψC)(¯ψtfjtck(1±γ5)¯ψC)(¯ψi(1±γ5)ψj)]. (5)

Here the four-fermion interactions are all invariant under the -transformation in the flavor space. In our notations, the Gell-Mann matrices in flavor (color) space are with , and , and the antisymmetric one is denoted by with :

 tf(c)1=λf(c)7,tf(c)2=λf(c)5,tf(c)3=λf(c)2. (6)

The scalar interaction in is responsible for the dynamical chiral symmetry breaking in the vacuum with the formation of the chiral condensate, while the vector interaction can be used to investigate the effect of density-density interaction on the chiral phase transition  Zhang:2009mk ()333We remark that the effects of the vector interaction on the baryon-number susceptibility and the chiral transition in the two-flavor case are examined in Kunihiro:1991qu () and Asakawa:1989bq (); Kitazawa:2002bc (), respectively.. In Eqs.(3) and (5), stands for and is the Dirac charge conjugation matrix. We remark that the suffix in denotes the channel for the ud pairing, for example. For lower temperature and large enough baryon chemical potential , leads to the formation of diquark condensate in the color-anti-triplet channel  Alford:1997zt (); Rapp:1997zu (); Alford:1998mk (). Besides the four-fermion interactions, Lagrangian (1) also contains two types of six-quark interactions, and : the former is the traditional Kobayashi-Maskawa-’tHooft (KMT) interaction  Kobayashi:1970ji (); 't Hooft:1976fv () and its effect on the phase diagram in - plane is fully examinedKunihiro:1989my (); Kunihiro:1991hp (); Hatsuda:1994pi (); Buballa:2003qv (); Fu:2007xc (); Kunihiro:2009ds (), whereas the latter could be obtained by a Fierz transformation of the former and induces the coupling between the chiral and diquark condensates  Alford:1998mk (); Rapp:1999qa (); Steiner:2005jm (); Yamamoto:2007ah (); Abuki:2010jq (). We remark that both interactions respect the flavor symmetry of while violating the symmetry as mentioned above. The former is responsible for accounting for the abnormally large mass of beyond the Weinberg inequalityWeinberg:1975ui () (in contrast to other pseudo Nambu-Goldstone bosons ) in the effective chiral model and can be identified as an induced quark interaction from instantons't Hooft:1976fv (); ref:SS (). The introduction of the latter to the Lagrangian expands the study of CSC to the six-fermion level Steiner:2005jm ().

### ii.2 The model parameters

The numerical values of some model parameters are given in Table 1. In contrast to Abuki:2010jq () , we only consider the case with realistic quark masses. The choice of the model parameters is the same as that in  Ruester:2005jc (); Zhang:2009mk (); Basler:2010xy () (all following Ref. Rehberg:1995kh () ), where , the coupling constant for the scalar meson channel, and , the coupling constant of the KMT term, are fixed by the vacuum physical observables (meson masses and decay constants). We shall work in the isospin symmetric limit in two-flavor space with , and a sharp three-momentum cut-off is adopted.

In contrast to and , no definite observables in the vacuum are available for determining the coupling constants and in such a quark model, although we could read off their values from a Fierz transformation of known vertices: the coupling constant , for instance, can be related to through the Fierz transformation of the instanton vertex, and is found to be identical to  Abuki:2010jq (). Since we are mainly interested in the roles of and on the chiral phase transition and the chromomagnetic instability, both these coupling constants are treated as free parameters in the present work. Following Abuki:2010jq (); Basler:2010xy (), we only consider the attractive interaction between the chiral condensate and the diquark condensate. Namely, the coupling is kept positive. As for the ratio of , we adopt the standard value from Fierz transformation in this paper. Due to the contribution from the KMT interaction, the ratio from Fierz transformation should be 0.95 rather than 0.75 obtained by only considering the four-quark interaction  Buballa:2003qv (). Such a choice of the coupling has also been used in Refs. Zhang:2009mk () and  Basler:2010xy (). In the literature, the diquark-diquark interaction near the standard value from Fierz transformation is usually called the intermediate coupling.

### ii.3 Thermodynamic potential with the constraints of charge-neutrality and β-equilibrium

The grand partition function of the NJL model is given by

 Z≡e−ΩV/T=∫D¯ψDψei∫dx4(L+ψ†^μψ), (7)

where is the thermodynamic potential density and is the quark chemical potential matrix. In general, the quark chemical potential matrix takes the form Alford:2002kj ()

 ^μ=μ−μeQ+μ3T3+μ8T8, (8)

where is the quark chemical potential (i.e. one third of the baryon chemical potential), the chemical potential associated with the (negative) electric charge, and and represent the color chemical potentials corresponding to the Cartan subalgebra in the SU(3)-color space. The explicit form of the electric charge matrix is in flavor space, and the color charge matrices are and in the color space. The chemical potentials for the quarks with respective flavor and color charges are listed below:

 μru=μ−23μe+12μ3+13μ8,μrd=μ+13μe+12μ3+13μ8,μrs=μ+13μe+12μ3+13μ8,μgu=μ−23μe−12μ3+13μ8,μgd=μ+13μe−12μ3+13μ8,μgs=μ+13μe−12μ3+13μ8,μbu=μ−23μe−23μ8,μbd=μ+13μe−23μ8,μbs=μ+13μe−23μ8. (9)

Corresponding to the chiral and diquark interactions in Eq. (1), we assume that the following condensates are formed in the system, namely, the scalar quark-antiquark condensate

 σi=⟨¯ψiψi⟩, (10)

and the scalar diquark condensate

 si=⟨¯ψCiγ5tfitciψ⟩. (11)

In addition, we remark that the quark-number (or baryon-number) density

 ρi=⟨¯ψiγ0ψi⟩, (12)

has a finite value for finite . Note that the indices 1,2 and 3 in Eqs. (10) and (12) represent u, d and s quarks, respectively, whereas in Eq. (11), the indices 1, 2 and 3 stand for the diquark condensate in d-s, s-u and u-d pairing channels, respectively. Here we have assumed the condensates and the density are all homogeneous; the study of the phase structure with inhomogeneous condensates and/or baryon-number density Nakano:2004cd (); Nickel:2009ke (); Nickel:2009wj (); Carignano:2010ac (); Giannakis:2004pf () is surely intriguing but beyond the scope of the present work.

The constituent quark masses and the dynamical Majarona masses are expressed in terms of these condensates as follows:

 Mi=mi−4GSσi+K|εijk|σjσk+K′4|si|2, (13)

and

 Δi=2(GD−K′4σi)si. (14)

Similarly, it is convenient to define the dynamical quark chemical potential for flavor by

 ~μi=μi−4GVρi. (15)

A few remarks are in order here:

1. Both types of the anomaly terms and contribute to the constituent quark masses in Eq. (13), and thus if and the diquark condensate are finite, chiral symmetry is dynamically broken even when the usual chiral condensates are absent.

2. The new anomaly term also modifies the formula for the Majarona mass for the CSC phase so that the chiral condensates affects the Majorana mass, and hence induce an interplay between the two condensates: Indeed the ‘bare’ diquark-diquark coupling is replaced by an effective one, , as shown in Eq. (14), which is dependent on the chiral condensates. Thus the flavor-dependent effective coupling is now dependent on and through .

3. Equations. (13) and (14) clearly show that the flavor-mixing occurs not only in the usual chiral condensates due to but also in the diquark condensates owing to , which would lead to interesting physical consequences.

4. It is also to be noted that the dynamical quark chemical potential for u and d quarks are different from each other because of the constraint of electric charge-neutrality () in 2CSC; notice also, however, that they are dependent only on the respective density and hence the dynamical chemical potentials tend to come closer because with the common coupling constant Zhang:2009mk ().

In the mean field level, the thermodynamic potential for the two-plus-one-flavor NJL with the charge-neutrality constraints reads

 Ω = Ωl+2GS3∑i=1σ2i−2GV3∑i=1ρ2i+3∑i=1(GD−K′2σi)|si|2 (16) − 4Kσ1σ2σ3−T2V∑PlndetS−1MFT.

Notice the presence of the new cubic-mixing terms among the chiral and diquark condensates. In Eq. (16), denotes the contribution from free leptons. Note that should include the contributions from both electrons and muons for completeness. Since and , ignoring the contribution of muons has little effect on the phase structure. Therefore, only electrons are considered in our calculation and the corresponding reads

 Ωl=−112π2(μ4e+2π2T2μ2e+7π415T4). (17)

Due to the large mass difference between s and u [d] quarks, the most favored phase at low temperature and moderate density tends to be the 2CSC rather than CFL phase, as demonstrated in the two-plus-one-flavor NJL model Ruester:2005jc (); Abuki:2005ms (). Surprisingly enough, if the anomaly term is incorporated, the 2CSC phase turns to be still favored in the intermediate density region even when the three flavors have the equal mass Basler:2010xy (). Needless to say, the dominance of the 2CSC phase over the CFL one is more robust when the realistic mass hierarchy for the three flavors is adopted. Moreover, it is worth mentioning here that the mass disparity favors the 2CSC phase with the u-d pairing also through the inequality of the effective diquark coupling when the anomaly coupling is present; see Eq. (14). Since the main purpose of the present work is to explore how the axial anomaly term affect the phase boundary involving the chiral transition at moderate densities, under the constraints of the charge-neutrality and -equilibrium, we only consider the 2CSC phase in the following. Note that in (8) vanishes in the 2CSC phase because the color SU(2) symmetry for the red and green quarks are left unbroken.

The inverse quark-propagator matrix in the Nambu-Gor’kov formalism takes the following form in the mean-field approximation,

 S−1MF(iωn,→p)=([G+0]−1Δγ5tf3tc3−Δ∗γ5tf3tc3[G−0]−1), (18)

with

 [G±0]−1=γ0(iωn±^~μ)−→γ⋅→p−^M, (19)

where , and is the Matsubara frequency. Taking the Matsubara sum, the last part of the thermodynamic potential (16) is expressed as

 −T2V∑PlndetS−1MFT=−18∑i=1∫d3p(2π)3{(Ei−E0i)+2Tln(1+e−Ei/T)}, (20)

with the dispersion relations for nine quasi-particles (that is, three flavors three colors; the spin degeneracy is already taken into account in Eq. (20)) and nine quasi-antiparticles. In Eq. (20), represents . The s quark and unpaired blue u and d quarks have twelve energy dispersion relations with a similar form. For example, the dispersion relations for the blue u quark and anti blue u quark are

 Ebu=E−~μbuand¯Ebu=E+~μbu, (21)

respectively, with . In the - quark sector with pairing we can find the four dispersion relations,

 (22)

and the - sector has another four as

 (23)

where and ; from now on stands for . The average chemical potential is defined by

 ¯~μ=~μrd+~μgu2=~μru+~μgd2=μ−μe6−2GV(ρ1+ρ2)+μ83, (24)

and the effective mismatch between the chemical potentials of u and d quarks takes the form

 δ~μ=12(μe−4GV(ρ2−ρ1)). (25)

Ignoring the the mass difference between u and d quarks, the determinantal term in Eq. (16) has an analytical form which greatly simplifies the numerical calculation. Adopting the variational method, we get the eight non-linear coupling equations

 ∂Ω∂σ1=∂Ω∂σ3=∂Ω∂s3=∂Ω∂ρ1=∂Ω∂ρ2=∂Ω∂ρ3=∂Ω∂μe=∂Ω∂μ8=0. (26)

Since is tiny around the chiral transition region Ruester:2005jc (); Abuki:2005ms (), we shall set it zero with little effect in the numerical results Zhang:2009mk (). Thus, Eq. (26) is then simplified to a system of seven coupled equations.

## Iii Phase Structure With The Axial Anomaly

In this section, we show numerical results of the effects of the new six-quark interaction (5) on the chiral phase transition under the charge-neutrality and -equilibrium constraints with or without the vector interaction. Since we are mainly interested in the phase diagram involving chiral transition at low temperatures, all the phase diagrams will be plotted for the range where the chiral transition is expected to be relevant. As for the type of the CSC phase, Ref. Basler:2010xy () indicates that the CFL phase is only realized for even when with the same model parameters as ours, and an increase of pushes the CFL phase to even higher region. Therefore, we exclusively consider the 2CSC phase near the chiral boundary without a loss of generality.

In the following, we use the same notations as in Ref. Hatsuda:2006ps (); Zhang:2008wx (); Zhang:2009mk () to distinguish the different regions in the - phase diagram. Namely, NG, CSC, COE, and NOR refer to the hadronic (Nambu-Goldstone) phase with and , color-superconducting phase with and , coexisting phase with and , and normal phase with , respectively, though they have exact meanings only in the chiral limit.

### iii.1 The case without vector interaction

We first show the numerical results in the case without the vector interaction. The phase diagrams with varying coupling constant are displayed in Fig. 1. In contrast to Fig. 8 in Ref. Basler:2010xy (), the multi-CP structure can still appear with a choice of in the phase diagram when the charge-neutrality, -equilibrium and the new axial anomaly term are simultaneously taken into account. For , Fig. 1a shows that there exists only one usual chiral CP even though the COE emerges: We remark that the COE region does not exist when , which is not displayed in Fig. 1. Figure 1b shows that when is increased to 2.25, the chiral transition turns to a crossover at relatively lower temperatures, and hence there appear two new chiral CP’s. With a further increase of , the boundary line for first-order transition at higher temperature shrinks and thus the two crossover boundary lines in Fig. 1b join with each other, and eventually only one CP is left in the phase diagram, as shown in Fig. 1c. When is large enough, Fig. 1d indicates that the first order boundary vanishes completely and there is no chiral CP in the phase diagram.

We note that the emergence of the three CP’s in Fig. 1b comes from a joint effect of the interplay between the chiral and diquark condensates and the electric charge-neutrality constraint. First of all, we recall that the abnormal thermal behavior of the diquark condensate that it has a maximum at a finite temperature in the COE is responsible for the emergence of the multiple chiral CP structure Kitazawa:2002bc (); Addenda (); Zhang:2008wx (); Zhang:2009mk (). Such a behavior is also observed in the present case, as displayed in Fig. 1b. As first indicated in Ref. Zhang:2008wx (), when is positive, the boundary of the chiral transition is shifted towards higher region, and leads to the formation of the COE at low-temperature region, in which the chiral phase transition is significantly weakened by the smearing of the Fermi surface inherent in the CSC phase. In this regard, plays a role of an effective vector interaction Kitazawa:2002bc (); Addenda (); Zhang:2009mk (). On the other hand, the chiral anomaly term with positive intensifies the competition between the chiral and diquark condensates due to the enhanced effective diquark-diquark interaction. Thus when is increased, the CSC region expands towards lower region in the - plane. Consequently, the COE region tends to be more easily formed when both and take effects. Therefore the chiral transition is significantly weakened and the smooth crossover gets to appear with new CP’s in the intermediate temperature owing to the abnormal thermal behavior of the diquark condensate.

The -dependence of and for fixed and several values of is shown in Fig. 2. One can see that, with increasing , the constituent quark masses decrease persistently while the Majarona mass for CSC first increases, has the maximum value and then decreases in the COE region and nearby. Let us see the details for each value of . For a small MeV, the diquark pairing is weak and the gap does not appear at lower temperature region. Thus the chiral phase transition keeps the nature of the first order at MeV. At a lager MeV, the diquark pairing becomes significant and the shows the abnormal thermal behavior with a maximum value around   MeV, and hence the chiral phase transition turns to a smooth crossover owing to the competition with the diquark condensate in the COE region. For even larger MeV, the diquark pairing becomes more significant and the still shows the abnormal thermal behavior. However, the competition between the two condensates is not strong enough to qualitatively change the nature of the chiral restoration and a first order transition happens at MeV. The reason why the crossover does not occur at MeV but happens at MeV can be understood as follows: Starting from the same point (, MeV) in the COE region, an increase of affects the nature of the chiral transition more significantly than that of does, since .

We have seen that the abnormal thermal behavior of the gap plays an essential role in realizing the multi-CP structure of the phase diagram. Such an unusual -dependence of the can be attributed to the following two mechanisms: (i) the mismatch between the chemical potentials of u and d quarks owing to the charge-neutrality and -equilibrium constraints and (ii) the small Fermi spheres of the quarks in the COE region due to the relatively large quark masses: First, the difference in the chemical potentials or the mismatch of the Fermi momenta disfavors the u-d pairing at zero or small temperature. However, as the temperature is raised in the low- region, more and more u and d quarks tends to participate in the pairing due to the smearing of the Fermi surfaces, especially that of the u quark. Of course, when is raised too much, the pairing will be gradually destroyed. Thus the will have the maximum value at a finite and then disappears eventually when is further raised. These dual effects of the temperature on the diquark pairing lead to the abnormal behavior of the . This behavior becomes more prominent in the 2CSC phase for a weak diquark coupling Shovkovy:2004me () or in the COE region for a moderate or strong diquark coupling Zhang:2008wx (); Zhang:2009mk (). Second, when is raised, the dynamical quark masses decrease and hence the Fermi spheres or momenta of u and d quarks grow significantly for a fixed , which means that the density of states at the Fermi surface increases with , and thus the diquark pairing is enhanced in the COE region. The increased diquark condensates in turn tend to further suppress the dynamical quark masses.

Notice that such an increase of the diquark condensate along with increasing is expected to be most prominent around the phase boundary of the chiral transition, including the COE region, where the chiral condensates change most significantly. For the neutral 2CSC, once the COE is formed, both of these mechanisms take effects simultaneously and are mutually enhanced, and thus the formation of the multiple-CP structure is readily made.

This may explain why no intermediate-temperature CP is realized in Ref. Basler:2010xy () where the chiral-diquark interplay is embodied by the anomaly term but without the charge-neutrality and -equilibrium constraints; the anomaly term solely is insufficient for realizing the abnormal thermal behavior of the . It should be stressed that the mechanism for the emergence of the intermediate-temperature CP’s in Fig. 1b is apparently similar to that in the two-flavor case found in  Zhang:2008wx (). However, the strange quark plays an important role in the present case since the chiral condensate of the strange quark contributes positively to the effective diquark-diquark coupling for u and d quarks through the axial anomaly. We should stress that apart from the appearance of intermediate-temperature CP’s, there is a common feature with and without charge-neutrality constraint: the chiral transition in the low- region extending zero temperature keeps first order provided that does not exceed a critical value at which the first-order line completely disappears.

Last but not least, we remark that the - region where the two new low-temperature CP’s are located in Fig. 1b is free from the chromomagnetic instability, which is obvious from Fig. 5a; a detailed discussion on this point will be given in Sec.IV.

### iii.2 The case for nonzero vector interaction

In this subsection, we will investigate the phase diagram when both the vector and the new six-quark interactions are present under the charge-neutrality and -equilibrium constraints.

There are some choices for the value of the vector coupling: The chiral instanton-anti-instanton molecule model  ref:SS () gives the ratio , while the Fierz transformation of the vertex given in the truncated Dyson-Schwinger model  ref:RWP () gives the ratio 0.5. Thus we rather treat the as a free parameter in the range, -.

We first explore the phase diagram in the - plane by varying the ratio but with being fixed as 0.25, the value given in the instanton-anti-instanton molecule model. When is small and less than , only the usual phase structure with single CP is obtained. When exceeds , other four different types of the CP structures appear, as displayed in Fig. 3. At , a phase diagram similar to that in Fig. 1b is obtained, as shown in Fig. 3a, where two new intermediate-temperature CP’s emerge. When is slightly increased to 0.57, the chiral transition becomes crossover in the lower-temperature region which extends to zero temperature; thus the total number of the CP’s becomes four, which indicates stronger competition between the chiral and diquark condensates at relatively larger . Further increasing , the low-temperature chiral boundary totally turns into a crossover one and only one first-order transition line with two CP’s attached remains in the phase diagram, as displayed in Fig. 3c. In this case, the number of the CP’s is reduced to two accordingly. When is large enough, Fig. 3d shows that only chiral crossover transition exists in the phase diagram with no CP.

In comparison with Fig. 1 where the vector interaction is not included, Fig. 3 indicates that the phase structures with multiple CP’s can be realized with relatively small owing to the vector interaction. We remark that all the types of the chiral CP structures displayed in Fig. 3 by varying are obtained by varying without the anomaly term Zhang:2009mk (). In the present case, the number of the critical points changes as 1  3  4  2  0 when is increased.

As is mentioned before, the Fierz transformation of the instanton vertex leads to the identity , so it is of special interest to investigate the phase diagram in the case of . A series of phase diagram with fixed but varied are shown in Fig. 4. One finds that all the chiral CP structures in Fig. 3 still appear in the phase diagrams, and moreover, even as large as five CP’s can exist in the phase diagram, as shown in Fig. 4c. This suggests that the interplay between the chiral and the diquark condensates in the COE region becomes complicated once the charge-neutrality, the vector interaction and the axial anomaly are all taken into account. A comparison with the case of vanishing , which is given in Fig. 7 in  Zhang:2009mk (), shows that the parameter range of for realizing the low-temperature CP’s moves towards lower , which is actually natural because gives the same effect as on the chiral transition. It is noteworthy that such lower values of are also close to the standard value of derived from the instanton model. The number of the CP’s changes as 1   3  5  4  2    0 with increasing .

The anomaly terms in Eq.(1) are supposed to originate from the instantons, which are to be screened at finite chemical potential and temperatureref:SS (). Accordingly, both the coupling constants and are expected to diminish around the phase boundary. However, we emphasize that the main effect of is to enhance the chiral condensate of the strange quark and the u-d diquark condensate by each other through the cubic coupling among them for the realistic quark masses, and Figs.3 and 4 tell us that even smaller values of expected at low temperature and moderate density can still lead to a quite different phase structure with multiple CP’s when the vector interaction is present under the charge-neutrality constraint.

## Iv The Influence On The Chromomagnetic Instability

In this section, we investigate effect of the new axial-anomaly term on the chromomagnetic instability of the asymmetric homogeneous 2CSC phase by varying . We shall show that the anomaly-induced interplay between the chiral and diquark condensates acts toward suppressing the unstable region of the homogeneous 2CSC phase in the - plane. Thus the 2CSC phase can become even free from the chromomagnetic instability provided that is larger than a critical value , which can be reduced significantly when the vector interaction is incorporated.

The magnetic instability region in the - plane is determined by calculating the Meissner masses squared which can be negative when the charge-neutrality constraint is imposed. Here we adopt the same method as that in Kiriyama:2006jp () to evaluate the Meissner mass squared

 m2M=∂2∂B2[Ω(Δ)−Ω(Δ=0)]B=0, (27)

where has the same meaning as that in Kiriyama:2006jp (). Since the strange quark does not take part in the diquark pairing in the present case, we can directly use the formula for two-flavor NJL model to calculate the Meissner mass squared.

The effect of the coupling constant on the chromomagnetic instability is shown in Fig.5. We have adopted the model parameters in Table 1 to calculate the Meissner mass squared. Figure 5a displays the change of the unstable region of the chromomagnetic instability with varying when the vector interaction is not included. One can see that the instability region tends to shrink with increasing and eventually vanishes for . This suggests that the neutral homogenous 2CSC phase will be totally free from the chromomagnetic instability if that is derived by the Fierz transformation from the usual instanton vertex. When taking the vector interaction with , the unstable region shrinks more significantly with increasing and eventually disappears in the - plane for , as shown in Fig.5b. This could be an expected result because of the effect of the vector interaction on the instability problem found in Zhang:2009mk ().

The reason for the suppression of the chromomagnetic instability by is understood as follows. First of all, Eq. (14) tells us that the u-d diquark coupling is enhanced by the presence of the s quark chiral condensate due to the coupling between the chiral and diquark condensates induced by the term. On the other hand, as was first shown in Kitazawa:2006zp () through changing the diquark coupling by hand, the chromomagnetic instability tends to be suppressed in the strong coupling region and can be completely gotten rid of when the diquark coupling is strong enough. Thus we see that the coupling between the u-d diquark and the chiral s-quark condensates by the term leads to the suppression of the chromomagnetic instability. This is a new mechanism of the stabilization of the gapless 2CSC phase, found in the present work. We here emphasize the important role of the strange quark and the anomaly term in suppressing the instability: In contrast to the pure two-flavor case, the rather large chiral condensate of the strange quark enhances the diquark coupling between the u and d quarks owing to the axial anomaly in the two-plus-one flavor case, and this enhancement of the diquark coupling causes the stabilization.

Due to their common effects on the chromomagnetic (in)stability, Fig.5 suggests that the instability may be totally cured in the asymmetric homogeneous 2CSC phase when the coupling constants of the vector interaction and the extended six-quark interaction are in an appropriate range. Admittedly, the present work has only dealt with the case of the so called intermediate diquark coupling. Nevertheless, for a weaker diquark coupling, it is expected that the system can be still free from the chromomagnetic instability only with larger couplings for both the vector interaction and the anomaly -term.

## V Conclusions and Outlook

We have explored the phase structure and the chromomagnetic instability of the strongly interacting matter under the charge-neutrality constraint within a two-plus-one-flavor NJL model by incorporating a new anomaly term as well as the conventional KMT interaction. The anomaly terms have the forms of six-quark interactions and violate the symmetry as a reflection of the axial anomaly of QCD. Similarly to the KMT term, the new anomaly interaction with the coupling constant also induces a flavor-mixing which leads to a direct coupling between the chiral and diquark condensates.

We first investigated the role of the axial anomaly on the emergence of the low or intermediate-temperature CP(’s) without the vector interaction. Owing to the large strange quark mass, the favored CSC phase near the chiral boundary is 2CSC rather than CFL, where the electric chemical potential required by the charge-neutrality plays an important role on the chiral phase transition Zhang:2008wx (). The once-declared low-temperature CP in the symmetric three-flavor limit Abuki:2010jq () was ruled out in Ref. Basler:2010xy () due to the actual dominance of the 2CSC over the CFL. We have shown that this is true under charge-neutrality constraint without the vector interaction; the chiral transition in the low- region extending zero temperature keeps first order provided that does not exceed a critical value at which the first-order line completely disappears. However, the new chiral anomaly term enhances the competition between the chiral and diquark condensates under charge-neutrality constraint, and gives rise to the intermediate-temperature CP’s for an appropriate range of . No such intermediate-temperature CP’s had been found in the same model when only either the charge-neutrality constraint or the axial anomaly is exclusively included, as shown in Ruester:2005jc () and Basler:2010xy (); both of which did not take into account the vector interaction, either.

We then investigated the - phase diagram by incorporating the repulsive vector interaction as well: We remark that this task may be viewed as an extension of the work Zhang:2009mk (), in which the effect of the vector interaction on the phase diagram is fully explored under the charge-neutrality constraint, to incorporate the anomaly term. We have found that the cubic coupling between the chiral and diquark condensates induced by the axial anomaly does not affect the qualitative results obtained in Zhang:2009mk (). Rather, the vector interaction and the anomaly term jointly act so that the multiple CP’s are realized. Indeed, by varying with fixed vector coupling or vise verse, we have shown that all the types of multiple-CP structures obtained in Zhang:2009mk () can be reproduced. In particular, the phase transition in the low- region extending zero temperature becomes a crossover only when the vector interaction is present with a strength larger than a critical value. Furthermore, the maximum number of the CP’s can reach as large as five when both the interactions are put on. In this case, the low- and intermediate-temperature CP’s can appear even with small values of owing to the help by the vector interaction. This is very welcome because in the realistic situation at moderate and high density should be weaker than that in the vacuum, since the anomaly term is supposed to originate from the instanton configuration which is expected to be suppressed at finite density.

Besides the influence on the chiral phase transition, we have shown that the axial anomaly also plays an important role on the suppression of the chromomagnetic instability for the asymmetric homogenous 2CSC phase, which is first disclosed in the present work: With an increase of the extended six-quark interaction, the - region with the chromomagnetic instability shrinks and eventually vanishes when the coupling is sufficiently large. In particular, when taking into account the vector interaction simultaneously, the chromomagnetic instability is suppressed more significantly and can be completely gotten rid of by the axial anomaly.

A general and remarkable message obtained from the present investigation is that the strange quark can significantly affect the properties of the neutral strongly interacting matter in which the 2CSC phase with u-d pairing is realized: Even though the strange quark does not directly participate in the Cooper pairing in the 2CSC, the interplay between the u-d diquark condensate and the strange chiral condensate induced by the anomaly term can lead to drastically different phase structure in the - plane under charge-neutrality constraint.

It should be remarked here that the contribution of other possible cubic flavor-mixing terms composed of different condensates, such as

 σρ2=ϵijkσiρjρk, (28)

which arise from another type of six-quark interaction

 L(6)χρ∼ϵijkϵlmn(¯ψiγμ(1±γ5)ψl)(¯ψjγμ(1±γ5)ψm)(¯ψk(1±γ5)ψn), (29)

are all neglected in Eq. (16) for simplicity. The interaction (29) can be derived from the KMT interaction, which may or may not affect the phase structure. Beside their direct contribution to the thermodynamic potential, these flavor-mixing terms also modify the dispersion relations of the quasi-quarks: For example, the dynamical quark mass becomes dependent on the quark-number density through the term . It is certainly an interesting problem to explore the possible effects of these cubic coupling terms on the phase diagram, we leave such a task to a future work.

Even apart from the neglect of the above vertex (29), there are some caveats with the present study based on a chiral model that does not embody the confinement effect, and is relied on the mean-field approximation. The results obtained in the current study are largely parameter dependent and bears the shortcomings inherent in the mean-field approximation. For instance, the result that there can be multiple CP’s associated with the chiral transition and the CSC actually may merely mean that the QCD matter is very soft for a simultaneous formation of the diquark and chiral condensates coupled with the baryonic density along the phase boundary. Of course, a study which incorporates these fluctuations should be performed, say, by means of the nonperturbative/functional renormalization group methodAoki:2000wm (); Berges:2000ew () with the present model used as a bare model. More profoundly, the effect of the confinement should be incorporated even in an effective model approach, which is a more challenging problem since the mechanism of confinement is still unclear. Anyway, further studies based on different models and/or methods are needed to determine whether the low-temperature CP(’s) exists. One of the tasks of future is exploring whether the low-temperature CP(’s) persists or not when the inhomogeneous phases are taken into consideration such as the chiral crystalline phase Nakano:2004cd (); Nickel:2009ke (); Nickel:2009wj (); Carignano:2010ac () or the LOFF phase Giannakis:2004pf ().

###### Acknowledgements.
Z.Z. was supported by the Fundamental Research Funds for the Central Universities of China. T.K. was partially supported by a Grant-in-Aid for Scientific Research by the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan (No.20540265), by Yukawa International Program for Quark-Hadron Sciences, and by the Grant-in-Aid for the global COE program “ The Next Generation of Physics, Spun from Universality and Emergence ” from MEXT.

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