Chiral imbalanced hot and dense quark matter: NJL analysis at the physical point and comparison with lattice QCD
Abstract
Hot and dense quark matter with isospin and chiral imbalances is investigated in the framework of the (3+1)dimensional Nambu–JonaLasinio model (NJL) in the large limit ( is the number of quark colors). Its phase structure is considered in terms of baryon – , isospin – and chiral isospin – chemical potentials. It is shown in the paper that (i) in the chiral limit there is a duality between chiral symmetry breaking (CSB) and charged pion condensation (PC) phenomena. (ii) At the physical point, i.e. at nonzero bare quark mass , and temperature this duality relation is only approximate, although rather accurate. (iii) We have shown that the chiral isospin chemical potential in dense quark matter generates charged pion condensation both at zero and nonzero , and at this phase might be observed up to temperatures as high as 100 MeV. (iv) Pseudocritical temperature of the chiral crossover transition rises in the NJL model with increasing . (v) It has been found an agreement between particular sections of the phase diagram in the framework of NJL model and corresponding ones in lattice QCD simulations. Two different plots from different lattice simulations that are completely independent and are not connected at the first sight are in reality dual to each other, it means that lattice QCD simulations support the hypothesis that in real quark matter there exists the (approximate) duality between CSB and charged PC. Moreover, we can reverse the logic and we can predict the increase of pseudocritical temperature with chiral chemical potential, the much debated effect recently, just by the duality notion, hence bolster confidence in this result (lattice QCD showed this feature for unphysically large pion mass) and put it on the considerably more solid ground.
I Introduction
At normal (Earth) conditions, protons and neutrons form atomic nuclei, and the latter, together with their orbital electrons, form the ordinary matter of our environment. If matter is subjected to extreme compression, eventually all chemical and nuclear bonds are broken, and the matter is squeezed from the molecular scale to the subparticle scale with density higher than 0.15 baryon per . Experimental creation of such dense matter is a very hard problem but such conditions can take place inside compact stars due to compression by gravity into a stable and extremely dense state. As a rule neutron stars have comparatively low temperatures and one can assume that it is zero. What happens at high temperature is hard to probe studying the physics of neutron stars. Nevertheless, due to technology advances, modern accelerators of elementary particles are now able to collide not only single high energy protons, but also heavy ions consisting of many coupled protons and neutrons. It is believed that in the fireball just after heavyion collisions there emerges a droplet of quark gluon plasma with very high temperature. Physics of heavy ion collision experiments can shed some light on the conditions that existed a few microseconds after the Big Bang and provide answers to several other questions.
The fundamental theory of matter in such extreme conditions is quantum chromodynamics (QCD) which is a gauge field theory associated with group, where gauge bosons (gluons) play the role of interaction carriers of quarks. The main method of QCD analysis is the perturbative technique on the basis of coupling constant. However, it is not always possible to use this technique, as QCD calculations can be too complex or in the low energy region when the coupling constant is too large. In particular, QCD perturbative technique is not applicable in a consideration of physically reachable dense matter, etc. In these cases, nonperturbative methods, such as effective theories or lattice calculations, are usually used.
Thus, the entire QCD phase diagram could not be described currently in the framework of a unified theory. Lattice calculations are very useful for description of the region of zero density and high temperature. However, the socalled sign problem still presents insurmountable difficulties for lattice calculations in the nonzero density region. On the other hand effective theories do not have fundamental background and as a result do not share the main prominent features with QCD such as a gauge invariance, renormalizability, etc. Nevertheless, at this moment, effective models are the best tool for investigating dense quark matter. At this time one of the most widely used effective model is the Nambu–JonaLasinio (NJL) model Nambu:1961fr (); Klevansky:1992qe (); Hatsuda:1994pi (); Buballa:2003qv ().
It is well known that usually dense baryonic matter in compact stars obeys an isospin asymmetry, i.e. where the densities of up and down quarks are different (it is characterised by isospin chemical potential ). In experiments on heavyion collisions, we also have to deal with quark matter which has an evident isospin asymmetry because of different neutron and proton contents of colliding ions. In early 70th Sawyer Sawyer:1972cq () and independently Migdal Migdal:1973 () have shown that there might be phase transition from pure neutron matter to mixed hadron matter with protons, neutrons and pions at superdense matter in the compact stars. Later, using the chiral perturbation theory, it was shown that there is a threshold of a phase transition to the charged pion condensation (PC) phase Son:2000 (). This result was ultimately proved in the framework of random matrix model Klein:2003 (), LadderQCD model Barducci (), resonance gas model Toublan (), quarkmeson model Fraga (), NJL model ek1 () (including (1+1)dimensional version of the NJL model ek2 ()) and lattice simulations Gupta (). Nevertheless, the whole picture is still a matter of debate.
Now the main question is whether the charged pion condensation exists in the real world and how this phenomenon behaves under influence of various external factors. And different factors can have a completely different effect on this phase. For example, in the framework of NJL model the finitesize effects, spatial inhomogeneity of the pion condensate Khunjua () or chromomagnetic background field Fedotov () could promote the charged PC phase. On the other hand, if the electric charge neutrality and equilibrium constraints are imposed, the charged PC phenomenon in quark matter depends strongly on the bare (current) quark mass values. In particular, it turns out that the charged PC phase with nonzero baryonic density is not realized within NJL models, if the bare quark mass reaches the physically acceptable values of MeV abuki (), i.e. at the physical point. In addition, temperature and different model parameters such as coupling constants, etc, as well strongly influence on this phase Ebert (). It is also worth to note that the phase structure of the isospin imbalanced quark matter below the threshold () is an important question because even small nonzero could double the critical endpoint of a phase diagram and affects the results of heavyion collision experiments Klein:2003 (); Kogut ().
Recently, it has been shown in the framework of the massless (3+1)dimensional NJL model (and in the leading large order, where is the number of colors of quarks) that chiral imbalance promotes charged PC phase in dense matter at zero temperature Khunjua:2017mkc (); Khunjua:2018sro () and responsible for the existence of the duality between chiral symmetry breaking (CSB) and charged PC phases. The imbalance between densities of lefthanded and righthanded quarks (chiral imbalance) is a highly anticipated phenomenon that could occur both in compact stars and heavy ion collisions. This effect could stem from nontrivial interplay of axial anomaly and the topology of gluon configurations.^{1}^{1}1It is predicted that there is an electrical current in the chiral imbalanced quark matter under strong magnetic field Fukushima:2008xe (). This phenomenon is called chiral magnetic effect and it could be an evidence of the chiral imbalance in QCD. Also, there is another mechanism of its origin – chiral separation effect which can be realized in dense matter in the presence of a strong magnetic field. In this case lefthanded and righthanded quarks tend to move in opposite directions along the magnetic field, thereby creating regions with chiral imbalance. Moreover, in the case of twoflavored quark matter the chiral separation effect could promote (see below in Appendix A) both nonzero chiral density and nonzero isotopic chiral density , and quark matter can be described using the corresponding chemical potentials and .
It was already mentioned above that nonzero bare quark mass and nonzero temperature could destroy charged PC phase in the physically adequate circumstances. So one of the aims of our present work is to check the robustness of the charged PC phase generated by chiral imbalance under the influence of these destructive factors. Another purpose is to study in the framework of the NJL model the fate of the duality observed in the chiral limit Khunjua:2017mkc () (where it is an exact symmetry) between CSB and charged PC phenomena in the leading large order: we investigate the influence of the bare quark mass and temperature on this effect, etc. In particular, it is shown in our paper that duality correspondence between CSB and charged PC still is a very good approximate symmetry of a phase portrait of the NJL model even at and .
It is interesting to investigate not only the charged PC phase but also hot quark matter itself with chiral asymmetry only. In this case at zero baryon chemical potential, , there is no signproblem and we have solid results from lattice simulations Braguta:2015zta (). Nevertheless, some key properties of chirally imbalanced quark matter are still under debate. So, in addition to charged PC phase, in the present paper we also investigate in the framework of the NJL model at the dependence of the (pseudo)critical temperature, which characterizes the chiral crossover region of the phase diagram, on the chiral isospin chemical potential and compare our results with other effective model investigations and lattice simulations on this topic. Note also that at and the  and phase diagrams have been obtained both using lattice QCD simulations and in the framework of the NJL model, and the results are in good agreement. Moreover, in the present paper we show that just these phase diagrams are dually conjugated (with a good precision) to each other, so there is a good reason to argue that duality between CSB and charged PC phenomena is confirmed by lattice QCD calculations.
The paper is organized as follows. In Sec. II a (3+1)dimensional NJL model with two massive quark flavors ( and quarks) that includes three kinds of chemical potentials, , is introduced. Furthermore, the symmetries of the model are discussed and its thermodynamic potential is presented in the leading order of the large expansion both at zero and nonzero temperature . In particular, it is shown in this section that in the chiral limit () the phase structure of the model (in the leading order over ) has a dual symmetry between CSB and charged PC phenomena. In the next section we formulate the main consequences of the exact dual symmetry (Sec. III A), using which it is possible to decide that dual symmetry is performed approximately in the NJL model at and , but with good accuracy (Sec. III B). It Sec. III C we show that at nonzero values of the chiral isospin chemical potential the charged PC phase with nonzero quark density can be realized in the model up to rather high values of temperature, MeV. Moreover, here we show that duality is also fulfilled approximately at . In Sec. III D the plot of the pseudocritical temperature of the chiral crossover transition as a function of at is obtained. Here it is compared with results of other effective models and lattice QCD approaches. Sec. IV presents summary and discussion leading to the conclusion that duality between CSB and charged PC observed in the NJL model is supported by some phase diagrams obtained by lattice QCD simulations at . Some technical details and issues not directly related to this work are relegated to Appendices A and B.
Ii The model and its thermodynamic potential
ii.1 Lagrangian and symmetries
It is well known that in the framework of effective fourfermion field theories dense and isotopically asymmetric quark matter, composed of and quarks, can be described by the following (3+1)dimensional NJL Lagrangian
(1) 
Here is a flavor doublet, , where and are fourcomponent Dirac spinors as well as color plets of the and quark fields, respectively (the summation in Eq. (1) over flavor, color, and spinor indices is implied); () are Pauli matrices; is the bare quark mass (for simplicity, we assume that and quarks have the same mass); and are chemical potentials which are introduced in order to study quark matter with nonzero baryon and isospin densities, respectively.
The symmetries of the Lagrangian (1) depends essentially on wether the bare quark mass and chemical potentials take zero or nonzero values. For example, in the most particular case, when the Lagrangian (1) is invariant under transformations from chiral group, which is also inherent in 2flavor QCD in the chiral limit. This symmetry is reduced to group if all chemical potentials are nonzero, and . In this case the abelian baryon , isospin and chiral isospin subgroups act on flavor doublet in the following way
(2) 
As a result, we see that in the chiral limit () the quantities , and are the density operators of the conserved baryon, isospin and chiral isospin charges of the system (1), respectively. Introducing the particle density operators for and quarks, and , we have
(3) 
One can also introduce the particle density operators and for right and lefthanded quarks of each flavor (see in Appendix A). In this case the density operator of the chiral isospin charge looks like
(4) 
where the quantity is usually called the density operator of the chiral charge for the quark flavor . Below, in Appendix A, we discuss the possibility of the appearance of a nonzero chiral isotopic density in quark matter inside neutron stars. It can be explained on the basis of the chiral separation effect in the presence of a strong magnetic field in a dense baryonic medium.
However, at the physical point () the symmetry of the Lagrangian (1) under transformations from axial isotopic group is explicitly broken. So in the most general case with , and the initial model (1) is invariant only under the group. (We would like also to remark that Lagrangian (1) is invariant with respect to the electromagnetic group, , at arbitrary values of , where .)
The ground state (the state of thermodynamic equilibrium) of quark matter with and , where , ,^{2}^{2}2The notation means the ground state expectation value of the operator . both at zero and nonzero values of has been investigated in the framework of the NJL model (1), e.g., in Refs. ek1 (); Ebert (). However, the fact that quark matter may have a nonzero chiral isotopic charge was ignored in those papers. Recently, this gap in researches was filled in the paper Khunjua:2017mkc (), where we have studied the properties of equilibrium quark matter at , as well as at nonzero chiral isospin charge density in the framework of the massless (3+1)dimensional twoflavor NJL model (temperature was taken to be zero in Ref. Khunjua:2017mkc ()). In contrast to this, in the present paper we consider the properties of a more realistic quark matter, i.e. at and , for which all densities , and are also nonzero. The solution of this problem can be most conveniently carried out in terms of chemical potentials , and , which are the quantities, thermodynamically conjugated to corresponding charge densities , and presented in Eqs. (3) and (4). Therefore, when solving this problem, one can rely on the Lagrangian of the form
(5)  
(Generally speaking, in this case the chiral isospin charge is no more a conserved quantity of our system. Therefore, chiral isospin chemical potential is not conjugated to a strictly conserved charge. However, denoting by the typical time scale in which all chirality changing processes take place, one can treat as the chemical potential that describes a system in thermodynamic equilibrium with a fixed value of on a time scale much larger than .)
Our goal is the investigation of the ground state properties (or phase structure) of the system, described by the Lagrangian (5), and its dependence on the chemical potentials , and (both at zero and nonzero temperature). It is well known that all information on the phase structure of the model is contained in its thermodynamic potential (TDP). Namely, in the behavior of its global minimum point vs. chemical potentials. Moreover, the values of charge densities , and in equilibrium quark matter can be found by differentiating the TDP in the global minimum point with respect to the corresponding chemical potentials , and , etc. In order to find the TDP of the model, we start from a semibosonized version of the Lagrangian (5), which contains composite bosonic fields and :
(6) 
Here, and also we introduced the notations , and . From the auxiliary Lagrangian (6) one gets the equations for the bosonic fields:
(7) 
Note that the composite bosonic field can be identified with the physical meson field, whereas the physical meson fields are the following combinations of the composite fields, . Obviously, the semibosonized Lagrangian is equivalent to the initial Lagrangian (5) when using the equations (7). Furthermore, the composite bosonic fields (7) change under the influence of transformations from the isospin and axial isospin groups in the following manner:
(8) 
ii.2 Thermodynamical potential. Zero temperature case.
Starting from the auxiliary Lagrangian (6), one obtains in the leading order of the large expansion (i.e. in the onefermion loop approximation) the following path integral expression for the effective action of the bosonic and fields:
where
(9) 
The quark contribution to the effective action, i.e. the term in (9), is given by:
(10)  
where is a normalization constant. Moreover, in (10) we have introduced the notation ,
(11) 
for the Dirac operator, which acts in the flavor, spinor as well as coordinate spaces only. Using the general formula , one obtains for the effective action (9) the following expression
(12) 
where the Troperation stands for the trace in spinor (), flavor () as well as fourdimensional coordinate () spaces, respectively.
The ground state expectation values and of the composite bosonic fields are determined by the saddle point equations,
(13) 
where . Just the knowledge of and and, especially, of their behaviour vs. chemical potentials supplies us with a phase structure of the model. In the present paper we suppose that in the ground state of the system the quantities and do not depend on spacetime coordinates ,
(14) 
where and () are already spatially independent constant quantities. In fact, they are coordinates of the global minimum point of the thermodynamic potential (TDP) . In the leading order of the large expansion it is defined by the following expression:
(15) 
In what follows we are going to investigate the dependence of the global minimum point of the function vs . Let us note that in the chiral limit (due to a invariance of the model) the TDP (15) depends effectively only on the combinations and . Whereas at the physical point (i.e. at ) it depends effectively on the combination as well as on and . Since in this case the relations and are always satisfied (see, e.g., in Ref. Fedotov ()), at one can put without loss of generality in Eq. (15), and study the TDP as a function of only two variables. For simplicity, we introduce the following and notations, and throughout the paper use the ansatz
(16) 
If in the global minimum point of the TDP we have , then isospin symmetry of the model is spontaneously broken down. Moreover, since at chiral symmetry is explicitly broken down in the model, the coordinate of the global minimum is always a nonzero quantity. Note also that is a dynamical or constituent quark mass. In terms of and the TDP (15) reads
(17)  
where
(18) 
is the momentum space representation of the Dirac operator (11) under the constraint (16). The quantities in Eq. (18) are really the following 44 matrices,
(19) 
so the quantity from Eq. (18) is indeed a 88 matrix whose determinant appears in the expression (17). Based on the following general relations
(20) 
and using any program of analytical calculations, one can find from Eqs. (19) and (20)
(21) 
where , and
(22) 
It is evident from Eq. (22) that the TDP (17) is an even function over the variable , and parameters and . In addition, it is invariant under the transformation . ^{3}^{3}3Indeed, if simultaneously with we perform in the integral (17) the change of variables, then one can easily see that the expression (17) remains intact. Hence, without loss of generality we can consider in the following only , , , and values of these quantities. Moreover in the chiral limit, the TDP (17) is invariant with respect to the socalled duality transformation:
(23) 
(It is interesting to note that the dual symmetry (23) is also an inherent property of the TDP of the model (5) in the chiral limit and at , but in the (1+1)dimensional spacetime 2dim ().) One can find roots of the polynomials (21) analytically, the procedure is relegated to Appendix B. Four roots of have the following form
(24) 
The roots of can be obtained by changing (changing is equivalent to ),
(25) 
where , and has quite complicated form, but could be always chosen as a real one (all the details can be found in Appendix B). As a result, we have from Eq. (21) that
(26) 
where each root is invariant with respect to the duality transformation (23). So, it is evident from Eqs. (17) and (21) that for the TDP one can obtain the following expression
(27) 
Then, taking in account a general formula
(28) 
and using the fact that each root of Eqs. (24) and (25) has a counterpart with opposite sign as well as the relation , one gets
(29)  
To obtain the second line of Eq. (29), where and is a threemomentum cutoff parameter, we have integrated in the first line of it over angle variables. If we are interested in knowing the phase structure of the model at zero temperature, we should study just the TDP (29) vs and on the global minimum point (GMP). It is clear that at the GMP of the TDP has the form , where is always a nonzero quantity. If in this case , then we are in the charged PC phase with spontaneous breaking of the isospin symmetry.
ii.3 Thermodynamical potential. Nonzero temperature case.
Though, the effect of nonzero temperatures is quite predictable (one can expect that the temperatures just restore all the broken symmetries of the model), here we include nonzero temperatures into consideration because it is important in a number of applications. In heavy ion collisions and early Universe the temperatures are huge and its account looks inevitable, but it even makes sense in other not so apparent situations. We know that compact stars are cold and one can consider their temperatures as zero. But probably there could be scenarios in which the temperatures could be important even in the context of compact stars. For example, their temperatures right after they are born in a supernova explosion can be as high as MeV. So it is instructive to know how robust the charged PC phase under temperature.
To introduce finite temperature into consideration, it is very convenient to use the zero temperature expression (27) for the TDP. Then, to find the temperature dependent TDP one should replace in Eq. (27) the integration over in favor of the summation over Matsubara frequencies by the rule
(30) 
In the expression obtained, it is possible to sum over Matsubara frequencies using the general formula (the corresponding technique is presented, e.g., in jacobs ())
(31) 
where . As a result, one can obtain the following expression for the TDP
(32) 
where is the TDP (29) of the system at zero temperature. Since each root in Eq. (32) is a dually invariant quantity (see in Eq. (23)), it is clear that in the chiral limit the temperature dependent TDP (32) is also symmetric with respect to the duality transformation .
ii.4 Technical details
Technically, to define the ground state of the system one should find the coordinates of the global minimum point (GMP) of the TDP (29). Since the NJL model is a nonrenormalizable theory we have to use fitting parameters for the quantitative investigation of the system. We use the following, widely used parameters:
(33) 
In this case at one gets for constituent quark mass the value . Moreover, we suppose that quark chemical potentials are varied in the region MeV, MeV and MeV. At higher values of , and the NJL model (5) no longer describes a phase structure of real quark matter. The reason is that in this case it is necessary to take into account the condensation of mesons, color superconductivity phenomenon, etc.
As our main goal of the present paper is to prove the possibility of the charged PC phenomenon in hot dense quark matter with chiral imbalance, i.e. in the framework of the NJL model (5), the consideration of the physical quantity , called quark number density, is now in order. This quantity is a very important characteristic of the ground state, especially in dynamical phenomena such as superfluidity. It is related to the baryon number density as because . In the general case this quantity is defined by the relation
(34) 
where and are coordinates of the GMP of a thermodynamic potential. In addition, one can find also the density of isospin, , as well as the chiral isospin density , .
We distinguish the following phases that could be realized in the chirally asymmetric system under different external circumstances (the quantities and below are the coordinates of the GMP of the TDP (29) in the corresponding phase):

– symmetrical phase. It could be realized only in the chiral limit, . Usually, in this phase at .

– chiral symmetry breaking phase (we use for it the notation CSB). Since quark number (baryon) density is zero in this phase, sometimes it is called the ordinary baryonic vacuum.

– chiral symmetry breaking phase with nonzero quark density (below it is CSB phase).
In the CSB phase the order parameter is usually greater than quark number chemical potential . Moreover, is of order of the gap in the energy spectrum of quarks. Due to this reasons quarks cannot be created in this phase and . However, with increasing of chemical potentials, it is advantageous for the system to abruptly decrease the parameter (see, e.g., the right panel of Fig. 5) and move into a new CSB phase. In this case, the gap in the energy spectrum of quarks significantly decreases, which makes it possible to create quarks in the ground state. As a result, the quark number density is nonzero in the CSB phase.

– charged pion condensation phase with zero quark density (below in all phase diagrams we use for it the notation PC) ( in the chiral limit). In the charged PC phase symmetry is spontaneously broken down. Since in this phase , sometimes it is called the charged pion gas phase.

– charged pion condensation phase with nonzero quark density (PC). In the PC phase symmetry is also spontaneously broken down. ^{4}^{4}4The transition between PC and PC phases is also a firstorder phase transition, as in this case the order parameter decreases by a jump (see, e.g., the left panel of Fig. 5), and the possibility for the creation of quarks appears. Therefore, in the PC phase the quark number density is nonzero. Moreover, in both phases the isospin density is nonzero.

We use the notation ApprSYM for the approximate symmetrical phase. In the literature this phase is usually called WignerWeyl phase Klevansky:1992qe (); Zong (). It also corresponds to a GMP of the TDP (29), in which and . But in contrast to the CSB and CSB phases, dynamical quark mass in the ApprSYM phase drops rapidly and continuously to the current quark mass with increasing temperature or chemical potentials. As it follows from Eqs. (7) and (16), under such conditions the chiral condensate is almost zero, and the chiral symmetry is approximately restored in the model. Moreover, at this phase turns into an exactly symmetrical phase with . These are the reasons why we use the name ApprSYM in all phase portraits below.
Note that at zero temperature changes its value by a jump when there is a phase transition from different CSB or charged PC phases to the ApprSYM phase (see, e.g., in Figs 4, 5). However, at nonzero temperature there is usually a chiral crossover transition between CSB and ApprSYM phases (see in Fig. 8).
Below we present different phase portraits of the model as well as its properties in terms of this notations.
Iii Phase structure of the model
iii.1 Exact duality in the chiral limit at zero temperature
Let us first consider some equilibrium properties of the model starting from the TDP (17) or (29), i.e. at zero temperature, and in the chiral limit (). Although this case has been investigated in details in the article Khunjua:2017mkc (), it is useful to recall the main features of the model phase structure obtained in the leading order of the large expansion.
It was already noted above that in the chiral limit the TDP (17) is invariant under the socalled duality transformation , where , which could be strictly seen from Eq. (22). It means that if at some fixed values of the chemical potentials the TDP has a GMP of the form , then at the transposed values of the isospin chemical potentials, i.e. at , but at the unaltered value of , the GMP of the TDP (17) lies already at the point . As a result, we see that if at , e.g., the CSB phase is realized with , then at the permuted (we say dually conjugated) values of chemical potentials the charged PC phase should be realized with , and vice versa. Hence, in the phase portrait all charged PC phases should be arranged mirror symmetrically to all CSB phases with respect to the line . However, the symmetrical phase turns into itself under the duality transformation, and on the plane the line is the axis of symmetry of this phase. Just these facts are well illustrated by the phase diagrams of Fig. 1. There one can see three phase portraits of the model: the left panel corresponds to , at the central panel and at the right one . Moreover, It is clear from the phase diagrams of Fig. 1 that in dense quark matter, i.e. at , chemical potential does promote the charged PC phase with nonzero quark density (there it is PC phase).
So, in the presence of duality the knowledge of a phase of the model (5) at some fixed values of external free model parameters (and at ) is sufficient to understand what a phase (we call it a dually conjugated) is realized at rearranged values of isospin chemical potentials, , at fixed . Furthermore, different physical parameters such as condensates, densities, etc, which characterize both the initial phase and the dually conjugated one, are connected by the main duality transformation . For example, the chiral condensate of the initial CSB phase at some fixed is equal to the chargedpion condensate of the dually conjugated charged PC phase. The quark number density (34) of the initial CSB phase is equal to the quark number density in the dually conjugated charged PC phase, etc.
Perhaps, the duality between CSB and charged PC phases is valid in the framework of the NJL model under consideration only in the leading large order (and at ). However, we think that some signs of this duality remain at the physical point of the full theory and can be observed, e.g., using lattice calculations. What gives us duality? If exact or approximate dual symmetry between different phenomena exists in the model, then, knowing the phase structure or other thermodynamic characteristics of the model in a certain region of chemical potentials, one can predict its properties in the dualconjugated domain. For example, due to the duality between CSB and charge PC phenomena, there was no need to investigate numerically the TDP (29) at each point of the plane in order to find the phase diagrams of Fig. 1 (or the similar diagrams at other values of ). Instead, it would be sufficient to obtain a phase portrait in a more narrow region, e.g., at . In this case it is composed of PC, PC and symmetrical phases (see in Fig. 1). Then one should transform each phase of it, using the mapping , into a dually conjugated phase, which is already located in the region . At the same time we should change the name of the phase according to the rule: PCCSB, PCCSB and the name of the symmetric phase under the dual transformation does not change. Thus, the duality property of the model can help to save not only the time of numerical calculations but also immediately imagine the properties of the model in previously unexplored regions of the values of chemical potentials.
There is an even more interesting use of duality. So, if we know, for example, the phase portrait of the model at fixed , there is no need to perform detailed calculations in order to obtain its phase portrait at fixed . To do this, it is enough to rename the axis of the initial phase diagram to the axis and change the name of the phases according to the rule: PCCSB, PCCSB (symmetrical phase remains intact). We call this technical procedure as the dual conjugation of a phase diagram. Hence, the and phase portraits are mutually conjugate to each other. However, any phase portrait (such as in Fig. 1) is selfdual, i.e. it is transformed into itself by the dual conjugation.
Finally note that there is another kind of duality, the duality between chiral symmetry breaking and superconductivity phenomena, which is realized in some (1+1) and (2+1)dimensional fourfermion theories thies (); ekkz2 (). But in these models the duality is a consequence of Pauli–Gürsey symmetry of initial Lagrangians.
iii.2 Approximate duality in the case of and
In the present section we study the influence of a nonzero value (33) of the bare quark mass on the charged PC phase. Moreover, since at the TDP (17) is no more invariant with respect to the dual symmetry (23), which is exact only in the chiral limit, we will examine the question whether there are some formal signs indicating that the dual symmetry is at least an approximate symmetry of the NIL model at . Among these signs are the following features of the NJL model at the physical point, when MeV,

At some reliable values of the chemical potentials each phase portrait of the model (at some fixed ) is approximately selfdual, i.e. approximately all charged PC phases of it are arranged mirror symmetrically to all CSB phases with respect to the line .

Each phase diagram has a phase (it is the ApprSYM phase), which is approximately symmetric under the transformation , i.e. it is arranged symmetrically with respect to the line .

Under the dual transformation, when , the order parameter of CSB or CSB phase is approximately equal to the order parameter of the dually conjugated charged PC or PC phase.

The quark number density in any phase, corresponding to the chemical potential point , is approximately equal to quark number density of its dually conjugated phase that lies at the point .

Each phase portrait (at some fixed ) of the model is approximately the dual mapping of a corresponding phase portrait (at some fixed ) and vice versa.
If these properties are inherent in the model or theory, then we say that in the model (theory) there is an approximate duality between its chiral properties and charged pion condensation phenomena.
Bearing this in mind, let us look at the phase portraits of Fig. 2, which are depicted for the same values of the quark number chemical potential as in Fig. 1. First of all note that at in all diagrams of Fig. 2 there is a threshold MeV of a second order phase transition to the PC phase, which is also predicted by all known investigations Son:2000 () (including lattice calculations Gupta ()). Moreover, it is easily seen from these diagrams that promotes the charged PC phase in dense quark matter (it is the phase PC in Fig. 2) even in the case of . (see also in Fig. 3).
Concerning the abovelisted duality signs (i)(v), we see that in the region , where is of the order of the pion mass , there is no sence to say about duality (even approximate), because the point (i) of this list is not fulfilled. However, as it follows from Figs. 2 and 3, outside the region and for all values of , and restricted by the conditions GeV, GeV and GeV (the duality is even better symmetry in the region of larger values of chemical potentials but the results of NJL model in this region are not trustworthy) we see that the items (i) and (ii) are satisfied.
To have a more precise picture, let us take a look at the Figs. 4 and 5, where the Gaps and baryon density vs. and are depicted. It follows from these pictures that if we go from the phase, corresponding, e.g., to a chemical potential set MeV, to the dually conjugated phase with MeV (or vice versa), then pion condensate in the charged PC phase is approximately the same as dynamical quark mass in the dually conjugated CSB phase (compare the left and right panels of Fig. 5) and baryon density is not changed (approximately). In the dually conjugated points of the ApprSYM phase both and dynamic quark mass are not changed, approximately. The same conclusions one can obtain from Fig. 4 for MeV when two phases, CSB and PC, are present. Hence, the items (iii) and (iv) of the list of duality signs are also satisfied.
Finally, comparing, e.g., the phase diagram at fixed MeV and the phase diagram at fixed MeV (see in Fig. 6), we see that qualitatively they are dually conjugated to each other at a rather low values of MeV, i.e in this region of each diagram of Fig. 6 one can perform the following axis and phase renaming, , CSBPC and CSBPC (the ApprSYM phase does not change its name by the duality transformation), in order to obtain (approximately) the corresponding region of another diagram of Fig. 6. This conclusion agrees with phase portraits of Fig. 2 for moving along the lines MeV (or MeV) of these diagrams we intersect just the phases shown in Fig. 6 at low . In addition, it is easy to see that there is a duality between diagrams of Fig. 6 in the regions, where MeV (left panel) and MeV (right panel). So the item (v) of the list of duality signs is also satisfied.
In conclusion of this section, we can say that the duality between the phenomena of CSB and a charged PC, inherent for this model in the chiral limit at , is approximately fulfilled even at , but only for the points of the chemical potential space from the region, in which