Dual properties of dense quark matter with color superconductivity phenomenon

In this paper the massless NJL model extended by the diquark interaction channel is considered. We study its phase structure at zero temperature and in the presence of baryon $\mu_B$, isospin $\mu_I$, chiral $\mu_{5}$ and chiral isospin $\mu_{I5}$ chemical potentials in the mean-field approximation. It is shown that the model thermodynamic potential, which depends on three order parameters, $M$, $\pi_1$ and $\Delta$ (where $M$, $\pi_1$, and $\Delta$ -- are, respectively, chiral, charged pion and diquark condensates of the model), is symmetric with respect to the (dual) transformation when $M\leftrightarrow\pi_1$ and simultaneously $\mu_I\leftrightarrow\mu_{I5}$. As a result, on the mean-field phase portrait of the model the chiral symmetry breaking (CSB) and charged pion condensation (PC) phases turn out to be dually conjugate with each other, which greatly simplifies the study of the phase portrait of the model. In particular, the duality between CSB and charged PC phases means that in the $(\mu_I,\mu_{I5})$-phase portrait these phases are mirror-symmetrical with respect to the line $\mu_I=\mu_{I5}$, which at the same time is the symmetry axis of the color superconducting (CSC) phase. Moreover, it follows from our analysis that chiral $\mu_5$ chemical potential promotes the formation of CSC phase in dense quark matter. And together with $\mu_{I5}$ it can generates the charged PC phase even at $\mu_I=0$.


I. INTRODUCTION
It is well known that quantum chromodynamics (QCD) is the theoretical basis for the study of strongly interacting matter.However, to consider the properties of dense quark (baryonic) matter, which can exist in the cores of compact stars or arise in collisions of heavy ions, perturbative QCD methods are inapplicable, since the coupling constant of strong interactions is quite large.In this case, effective QCD-like models, including the simplest Nambu-Jona-Lasinio (NJL) model which describes four-fermion interaction of u and d quarks [1][2][3][4], are often used to study dense baryonic medium composed of u and d quarks.
Extending this model by the terms with baryon µ B and isospin µ I chemical potentials, one can investigate the properties of quark matter with nonzero baryon n B and nonzero isospin n I densities (which is typical for neutron stars) [5][6][7][8][9][10].More recently, it has become clear that chiral asymmetry (or chiral imbalance), i.e. unequal densities n L and n R of all left-and all right-handed quarks, is also one of the properties of dense quark matter.Usually the chiral asymmetry is characterized by the quantity n 5 , called the chiral density, n 5 ≡ n R −n L .It can be generated dynamically at high temperatures, such as in a fireball after a heavy ion collision, due to the Adler-Bell-Jackiw anomaly and the interaction of quarks with gauge (gluon) field configurations with non-trivial topology, called sphalerons.However, in the most general case the chiral densities of u and d quarks, i.e. the values n u5 ≡ n uR − n uL and n d5 ≡ n dR − n dL , are not equal and the quantity n I5 ≡ n u5 − n d5 is also nonzero (it is the so-called chiral isospin density of quark matter).As it was discussed in Ref. [11], the nonzero n I5 can be created in dense quark matter in the presence of rather strong magnetic fields and even not so large values of temperature, i.e. chiral isospin density can be observed in magnetars.As a result, we see that baryon medium composed of u and d quarks can be characterized, in addition to temperature, by four more physical parameters, i.e. by densities n B , n I , n 5 and n I5 .Its thermodynamics can be studied, in particular, on the basis of the simplest NJL model extended with four chemical potentials µ B , µ I , µ 5 and µ I5 , which are thermodynamically conjugated to the corresponding densities.
In the mean-field approximation, the (µ B , µ I , µ 5 , µ I5 )-phase portrait of this massless NJL model has been investigated in the papers [12,13], where it was shown that only two non-trivial phases are allowed for dense quark matter: (i) the chiral symmetry breaking (CSB) phase and (ii) the charged pion condensation (PC) one.Moreover, it was established that CSB and charged PC phases are dually conjugated to each other.It means that at fixed µ B and µ 5 these phases are arranged mirror-symmetrically with respect to the line µ I = µ I5 on the mean-field (µ I , µ I5 )-phase portrait of the model, and so on.A more detailed influence of this dual symmetry on the phase structure of quark matter considered in the framework of the simplest massless NJL model has been investigated in Refs.[4,[12][13][14][15].As a result, it became clear that duality greatly simplifies the study of the phase structure of quark matter, especially when it is under the influence of several external factors (chemical potentials).However, it must be borne in mind that the duality between CSB and charged PC is exact (in the large-N c or mean-field approximations) only in the chiral limit, i.e. when the bare quark mass m is zero.At m = 0 the duality between CSB and charged PC is only approximative [15].Nevertheless, the NJL analysis at the physical point and observed approximative duality between these phases are in fairly good agreement with lattice QCD.In addition, it is important to be aware that duality has been proven only within the framework of the simplest NJL model, which is equivalent effectively to QCD only in the region of rather small baryon densities, i.e. at µ B < 1 GeV.At higher energies or values of chemical potentials a more complicated NJL models, whose Lagrangians contain other quark interaction channels, should be taken into account.As a result, the set of possible ground states of dense quark matter can be expanded, which (in the absence of dual symmetry between some phases of matter) leads to significant difficulties in studying the phase structure of real quark matter, especially in the presence of four chemical potentials.The solution of the task could be greatly simplified if we had confidence in the existence of duality between some phases of matter even at rather large values of µ B , i.e. in the framework of a more complex NJL models, which have a rather diverse structure of four-quark interactions.The proposed work is devoted to the consideration of this problem.
An illustration of the above can be the situation with quark matter inside neutron stars, whose baryon density is several times greater than the density of ordinary nuclear matter, i.e. in this case µ B 1 GeV.Moreover, there exist an evident isospin and chiral asymmetries.Under such extrim conditions, not only quark-antiquark pairing is possible, but also the formation of diquark pairs, as well as their condensation.It means that for large values of µ B the phenomenon of color superconductivity (CSC) can be observed in the system.Therefore, in order to make it more adequate an investigation of quark matter properties, an extension of the standard NJL model with the help of four-fermion terms responsible for the diquark interaction channel is usually used (see, e.g., the reviews [3,16] or Eq. ( 1) below).The main goal of the present paper is to show that in this case, despite the complication of the model, its thermodynamic potential as a function of order parameters and four above mentioned chemical potentials has in the mean-field approximation a dual symmetry between the CSB and the charged PC phases.Thereby, the study of the phase structure of the model is simplified, and it is possible to shed new light on the role of both isospin and chiral (isospin) asymmetries in the formation of diquark condensation.Moreover, our results can serve as an argument in favor of the fact that duality is a characteristic feature of the QCD itself, and not only of its different effective models.
The paper is organized as follows.In Sec.II a (3+1)-dimensional NJL model with two massless quark flavors (u and d quarks) that includes four kinds of chemical potentials, µ B , µ I , µ I5 , µ 5 , is introduced.In addition to usual quark-antiquark channels, the model contains the diquark interaction one and is intended to describe the phenomenon of color superconductivity in a dense quark medium.Furthermore, the symmetries of the model are discussed and its thermodynamic potential (TDP) is presented in the mean-field approximation.In Sec.III the definition of the dual symmetry of the TDP is given.It means that TDP is invariant under some interchange of chemical potentials as well as, in some cases, simultaneous interchange of condensates.In particular, it was established, and this is one of the main results of the paper, that in the mean-field approximation the TDP of the model is invariant under the following two simultaneous transformations, µ I ↔ µ I5 and chiral condensate↔charged PC one.As a result, in the different (µ I , µ I5 )-phase portraits of the model CSB and charged PC phases are dually conjugated to each other, i.e. they locate mirror symmetrically with respect to the line µ I = µ I5 .In Sec.IV summary and conclusions are given.Some technical details are relegated to Appendices A and B.

II. THE MODEL AND ITS THERMODYNAMIC POTENTIAL
Our investigation is based on the NJL type model with two quark flavors.Its Lagrangian describes the interaction in the quark-antiquark as well as scalar diquark channels, where the quark field q ≡ q iα is a flavor doublet (i = 1, 2; alternatively we use the notations q 1 = u and q 2 = d) and color triplet (α = 1, 2, 3 or α = r, g, b) as well as a four-component Dirac spinor; q c = C qT and q c = q T C are charge-conjugated spinors, and C = iγ 2 γ 0 is the charge conjugation matrix (the symbol T denotes the transposition operation).It is supposed that u and d quarks have an equal current (bare) mass m.Furthermore, τ a stands for Pauli matrices, and λ A for Gell-Mann matrices in flavor and color space, respectively.Clearly, the Lagrangian L is invariant under transformations from color SU(3) c as well as baryon U(1) B groups.In addition, at m = 0 this Lagrangian is invariant under the chiral SU(2) L ×SU(2) R group.At m = 0 the chiral symmetry is broken to the diagonal isospin subgroup SU(2) I with the generators 2,3).Moreover, in our system the electric and baryonic charges are conserved quantities, too, since , where I 3 is the third generator of the isospin group SU(2) I , Q is the electric charge generator, and B is the baryon charge generator (evidently, these quantities are unit matrices in color space, but in flavor space they are: 1) is obtained from the QCD one-gluon exchange approximation, then the quantity H/G should not differ too much from the value of 0.75 [3,16].Note that the Lagrangian (1) describes physical processes in vacuum.In order to use the model (1) to study properties of dense quark medium, it is necessary to modify Lagrangian (1) by adding terms with chemical potentials, where the chemical potential matrix M contains in the most general case four different chemical potential terms responsible for the description of quark matter with nonzero baryon (µ B ), isospin (µ I ), chiral (µ 5 ) and chiral isospin (µ I5 ) densities, respectively.If all chemical potentials in ( 2) are nonzero quantities, then SU (2) I at m = 0 is not the symmetry group of this Lagrangian.Instead, due to the µ I term, it is symmetric under the flavor U (1) I3 group, q → exp(iατ 3 /2)q.Note however, that in the chiral limit (m = 0) an additional symmetry of the Lagrangian (2) appears, U (1) AI3 : q → exp(iαγ 5 τ 3 )q.
To study the phase diagram of the system (2), we need to get its thermodynamic potential (in the mean-field approximation).For this purpose, let us consider the linearized version L of Lagrangian ( 2) that contains auxiliary bosonic fields, In Eq. ( 3) and later a summation over repeated indices a = 1, 2, 3 and A, A ′ = 2, 5, 7 is implied.Clearly, the Lagrangians ( 2) and ( 3) are equivalent, as can be seen by using the equations of motion for bosonic fields, which take the form It follows from ( 4) that the mesonic fields σ(x), π a (x) are real quantities, i. e. (σ(x)) † = σ(x), (π a (x)) † = π a (x) (the superscript symbol † denotes the hermitian conjugation), but all diquark fields ∆ A (x) are complex scalars, so (∆ A (x)) † = ∆ * A (x). Clearly, the real σ(x) and π a (x) fields are color singlets, whereas scalar diquarks ∆ A (x) form a color antitriplet 3c of the SU(3) c group.Note that the auxiliary bosonic field π 3 (x) corresponds to real π 0 (x) meson, whereas the physical π ± (x)-meson fields are the following combinations of the composite fields ( 4), If some of the scalar diquark fields have a nonzero ground state expectation value, i. e. ∆ A (x) = 0, the color symmetry of the model ( 2) is spontaneously broken down.
In the one fermion-loop approximation (or in the mean-field approximation) and in the presence of dense quark medium, the effective action S eff (σ, π a , ∆ A , ∆ * A ′ ) of the model ( 1) is expressed by means of the path integral over quark fields, where and N ′ is a normalization constant.The quark contribution to the effective action, i. e. the term Seff in (6), is the following one, where we have used the notation where 1 I 3c is the unit operator in the tree-dimensional color space.Starting from Eqs. ( 6) and ( 7), one can define the thermodynamic potential (TDP) Ω(σ, π a , ∆ A , ∆ * A ′ ) of the model (1).Indeed, this quantity is defined by the following relation, The ground state expectation values (mean values) σ(x) , π a (x) , ∆ A (x) , ∆ * A ′ (x) of the auxiliary bosonic fields (4) are solutions of the gap equations for the TDP Ω(σ, π a , ∆ A , ∆ * A ′ ) (in our approach all ground state expectation values do not depend on coordinates x), and usually they are the coordinates of the global minimum point (GMP) of Ω vs. σ, π a , ∆ A , ∆ * A ′ .Note that at nonzero bare quark mass, m = 0, the quantity σ(x) , called chiral condensate, is always nonzero.However, in the chiral limit, m = 0, there can exists a region of chemical potentials in which σ(x) = 0.In this case it is the chiral symmetric phase of the model.In contrast, at m = 0 the region with σ(x) = 0 is called the chiral symmetry breaking (CSB) phase.Moreover, both at m = 0 and m = 0 the GMP of the TDP with π ± (x) = 0 corresponds to the so-called charged pion condensation (PC) phase.But in the case when one of the scalar diquark fields has a nonzero ground state expectation value, ∆ A (x) = 0, we have a phase with spontaneously broken color SU (3) c symmetry.It is called the color superconducting (CSC) one.
Recall that both the Lagrangian ( 2) and the effective action ( 6) are invariant under the color SU (3) c group.As a consequence of this fact, all ∆ A and ∆ * A ′ -dependence of the TDP Ω(σ, π a , ∆ A , ∆ * A ′ ) ( 9) is shown in its form depending on the combination , where ∆ is a real quantity.Let us note also that in the chiral limit (due to a U I3 (1) × U AI3 (1) invariance of the model ( 2)) the TDP Ω ( 9) depends effectively only on the combinations σ2 + π 2 3 and π 2 1 + π 2 2 (in addition to ∆).So at m = 0 one can put π 3 = 0 and π 2 = 0 without loss of generality.Whereas at the physical point (i.e. at m = 0) it depends effectively on the combination π 2 1 + π 2 2 as well as on σ and π 3 .Since in this case the relations σ(x) = 0 and π 3 (x) = 0 are always satisfied in the NJL model ( 2) at H = 0 2 , at m = 0 one can made a rather plausible assumption that in Eq. ( 9) π 2 = π 3 = 0 is also valid and study this TDP as a function of only three variables, σ, π 1 and ∆, i.e.Ω ≡ Ω(σ, π 1 , ∆).It is clear that in order to calculate Ω(σ, π 1 , ∆) it is enough to suppose that in Eqs. ( 6) and ( 7) ∆ 2 = ∆ * 2 = ∆, ∆ 5 = ∆ 7 = 0 and π 2 = π 3 = 0 (note that in the following we also suppose that all auxiliary bosonic fields do not depend on space coordinate x).Since where σ 2 is the corresponding Pauli matrix acting in the two-dimensional fundamental representation of the SU (2) c subgroup of the SU (3) c , it is clear that uder this assumption the contribution of the blue q b quarks in the expression ( 7) is factorized.Then where q b is the flavor doublet of the blue quarks and Q is the flavor (u and d) and color (red and green) quark doublet.
Moreover, here we use the notation D + for the operator that is in the round brackets of Eq. ( 8) at π 2 = π 3 = 0 (see below in Eq. ( 14)) and 1 I 2c is the unit operator in the two-dimensional color space.Performing in Eq. ( 20) the functional integrations over q b (which is a trivial one) and over Q (see Appendix A), we find where and Note that matrix elements of the matrix Z (13) are the operators in two-dimensional color and flavor spaces as well as in four-dimensional spinor and coordinate spaces.Then, it follows from Eqs. ( 6) and ( 12) that where we have introduced the gap M ≡ σ + m.The last term of Eq. ( 15), which does not depend on ∆, was calculated in our recent paper [12] (see there Eqs. ( 16)-( 21)), where (we also use in Eq. ( 17) and below the notations µ = µ B /3, ν = µ I /2 and ν 5 = µ I5 /2).The next undefined term of Eq. ( 15) is the following where L(p) is given by Eq. (A20), i.e. it is the 4×4 matrix in spinor space.Taking into account the relations ( 15)-( 18) as well as the definition (9), it is easy to get the TDP of the model in the mean-field approximation, In what follows, when performing numerical calculations or finding symmetry properties of the TDP Ω(M, π 1 , ∆), it is very convenient to represent det L(p) in Eq. ( 19) as a production of four eigenvalues of the matrix L(p), Here and where Exact expressions for the eigenvalues λi (p) were obtained earlier in the study of the phase structure of the two-color QCD (see, e.g., the section III of Ref. [18]).We are now ready to discuss in the chiral limit, i.e. at m = 0, the dual properties of the NJL model ( 1) in the mean-field approximation.
Throughout the paper we use in numerical investigations of the TDP (19) the soft cut-off regularization scheme when d 4 p ≡ dp 0 d 3 p → dp 0 d 3 pf Λ ( p).Here the cut-off function is and the parameter fit used is G = 4.79 GeV −2 , Λ = 638.8MeV and N = 5.

III. DUAL SYMMETRIES OF THE TDP (19) AND PHASE PORTRAITS OF THE MODEL
First of all, we note once again that in order to find the phase structure of the NJL model ( 2), it is necessary to study its TDP (19) as a function of three variables M, π 1 , ∆ for an absolute minimum, and then see how the properties of its global minimum point (GMP) change depending on the values of chemical potentials µ, ν, ν I5 , µ 5 .
In this regard, we remark that in the chiral limit, m = 0, for sufficiently low values of the chemical potentials (say at µ, ν, ν 5 , µ 5 < 1 GeV) at the GMP (M, π 1 , ∆) of the TDP (19) there can be no more than one nonzero coordinates.(This conclusion can be done based on the arguments presented earlier in our papers [12,18].)Therefore, with such a restriction on chemical potentials, in the chiral limit only four different phases can be realized in the system.(I) If GMP has the form (M = 0, π 1 = 0, ∆ = 0), then the chiral symmetry breaking (CSB) phase appears in the model.(II) If it has the form (M = 0, π 1 = 0, ∆ = 0), the charged pion condensation (PC) phase is realized.(III) When the GMP looks like (M = 0, π 1 = 0, ∆ = 0), it corresponds to the color superconducting (CSC) or diquark condensation phase.And finally, (IV) the GMP of the form (M = 0, π 1 = 0, ∆ = 0) corresponds to a symmetrical phase with all zero condensates.Then a phase portrait of the model is no more than a one-to-one correspondence between any point (µ, ν, ν 5 , µ 5 ) of the four-dimensional space of chemical potentials and possible model phases (CSB, charged PC, CSC and symmetric phase).However, it is clear that this four-dimensional phase portrait (diagram) is quite bulky and it is rather hard to imagine it as a whole.So in order to obtain a more deep understanding of the phase diagram as well as to get a greater visibility of it, it is very convenient to consider different low-dimensional cross-sections of this general (µ, µ 5 , ν, ν 5 )-phase portrait, defined by the constraints of the form µ = const and µ 5 = const, etc.
In addition, note that the study of the phase structure of any model is greatly simplified if the so-called dual symmetries of its TDP is taken into account.Recall that by dual symmetry of the TDP we understand its symmetry (invariance) with respect to any discrete transformations as order parameters (in our case, these are M , π 1 and ∆ condensates) and free external parameters of the system (these may be chemical potentials, coupling constants, etc).The presence of the dual symmetry of the model TDP means that in its phase portrait there is some symmetry between phases with respect to the transformation of external parameters.And this circumstance can simplify the construction of the full phase diagram of the system.
For example, taking into account the Eqs.( 17)-( 20), it is possible to establish the invariance of the TDP ( 19) with respect to each of the following six transformations, in each of them two chemical potentials change their sign simultaneously: The invariance of the TDP (19) under the transformations (i)-(vi) is one of the simplest examples of its dual symmetries that can help us to simplify the analysis of the phase portrait of the model.In particular, due to the symmetry of the TDP (19) under transformations (i)-(vi), it is sufficient to study the phase structure of the NJL model ( 2) only, e.g., in the case when arbitrary three of the four chemical potentials have positive signs, whereas the sign of the rest chemical potential is not fixed.Then, applying to a phase diagram with this particular distribution of the chemical potential signs one or several transformations (i)-(vi), it is possible to find a phase portrait of the model at an arbitrary distribution of chemical potential signs.Hence, by agreement, in the following we will study the phase diagram of the model ( 2) only at µ ≥ 0, ν ≥ 0, ν 5 ≥ 0 and for arbitrary sign of µ 5 .
Another and more non-trivial example of the dual symmetry of the TDP Ω(M, π 1 , ∆) ( 19) is its invariance under the two discrete transformations D, in the chiral limit.The invariance of the last term of Eq. ( 19) under the transformation D follows directly from Eqs. ( 16)-( 17).However, checking the D-invariance of its det L(p) term ( 20) is not such a simple task.However, this is true, since this fact is easy to establish taking into account the relations ( 21)-( 25) and using any program of analytical calculations.
To understand the physical meaning of the dual symmetry ( 27), let us suppose that m = 0 and that at the point (µ = a, ν = b, ν 5 = c, µ 5 = d) of the mean-field phase portrait the GMP of the TDP (19) lies, e.g., at the point of the condensate space of the form (M = A, π 1 = B, ∆ = C).Then, due to a dual symmetry (27) of the TDP, at the dually D-conjugated point of the phase portrait, i.e. at the point (µ = a, ν = c, ν 5 = b, µ 5 = d), the dually D-conjugated phase should be located.Its condensate structure has the form (M = B, π 1 = A, ∆ = C).Hence, we see that if, e.g., M = A = 0, π 1 = B = 0, ∆ = C = 0, then CSB phase corresponding to a GMP of the form M = B = 0, π 1 = A = 0, ∆ = C = 0 should be dually conjugated to the initial charged PC phase, and vice versa.But if at the original point there is symmetric or CSC phase, then it is also realized at the dually conjugated point.Thus, knowing the phase of the model, which is realized at some point of its (µ, ν, ν 5 , µ 5 )-phase portrait, we can predict which phases are arranged at the dually conjugated points of this phase diagram.Moreover, the order parameter of the initial CSB phase of the point (µ = a, ν = b, ν 5 = c, µ 5 = d), i.e. the quantity M = A, is equal to the order parameter π 1 = A of the D-dually conjugated charged PC phase of the point (µ = a, ν = c, ν 5 = b, µ 5 = d) of the model phase portrait, etc.
As a consequence, we see that at some fixed values of µ and µ 5 , in the (ν, ν 5 )-phase portrait of the model the CSB and charged PC phases should be mirror symmetrical to each other with respect to the ν = ν 5 line.And each of the remaining phases, i.e. symmetric or CSC one, must occupy an area symmetrical about this line.This fact is well illustrated by four (ν, ν 5 )-phase portraits depicted in Figs. 1 and 2 at µ = 0.3 GeV and different values of µ 5 .Moreover, it is clear from Fig. 1 (left panel) that at µ 5 = 0 and at rather small values of ν and ν 5 the CSC phase is absent.But when µ 5 is nonzero, this phase appears in a small neighborhood of the point ν = 0, ν 5 = 0 (see other phase diagrams of Figs.1,2).Thus, chiral µ 5 chemical potential promotes the formation of the CSC phase in a dense baryonic medium.In addition, the phase diagram of Fig. 2 (right panel) confirms one more interesting property of µ 5 (as well as of ν 5 chemical potential) noted earlier in the framework of the ordinary NJL model [12] (see there Figs. 10,11): If µ 5 is nonzero, then chiral isospin ν 5 chemical potential generates charged pion condensation in dense quark matter (it is the PC phase in the right panel of Fig. 2) even if isospin ν chemical potential equals to zero (see also Figs. 10,11 of Ref. [12]).We emphasize once again that in order for this generation to take place one needs to have nonzero chiral chemical potential µ 5 .In contrast, as it was discussed in [13], this generation requires nonzero values of ν in the case of µ 5 = 0. Hence, in the ν = 0 case chiral µ 5 chemical potential can take the role of ν and allow this generation to happen.Finally, we would like to draw attention to the fact that at m = 0 the duality transformation D (27) of the TDP (19) can also be applied to a mean-field phase portrait of the model as a whole.Namely, in this case, i.e. when acting on the phase diagram, it is necessary to rename both the diagram axes and phases in such a way, that ν ↔ ν 5 and CSB↔charged PC.At the same time the µ-and µ 5 -axes and CSC and symmetrical phases should not change their names and positions.It is evident that after such D transformation the full (µ, ν, ν 5 , µ 5 )-phase diagram is mapped to itself, i.e. the most general (µ, ν, ν 5 , µ 5 )-phase portrait of the model is self-D-dual.In a similar way it is clear that various (ν, ν 5 )-phase diagrams at fixed µ and µ 5 values are transformed into themselves after applying to them the dual operation (27), i.e. they are also self- e.g.,in Figs. 1,2).But other cross-sections of the full meanfield (µ, ν, ν 5 , µ 5 )-phase diagram, e.g., the (µ, ν)-phase portrait at some fixed values of ν 5 and µ 5 , are not invariant, in general, under the action of dual transformation D. As a result, a completely different phase portrait can be obtained.Hence, based on this mechanism, it is possible, having a well-known cross-section of the full phase diagram of the model, to obtain its phase portrait in a less studied range of values of chemical potentials.For example, In Fig. 3 (left panel) one can see the (µ, ν)-phase portrait at fixed ν 5 = 0.05 GeV and µ 5 = −0.05GeV.Applying to it the dual operation D according to the rule described above, one can obtain (without any numerical calculations) its dual conjugation, i.e. the (µ, ν 5 )-phase portrait at fixed ν = 0.05 GeV and µ 5 = −0.05GeV (see the right panel of Fig. 3).

IV. SUMMARY AND CONCLUSIONS
In this paper the phase structure of the generalized massless NJL model (1), which describes interactions both in quark-antiquark and diquark channels, is discussed at zero temperature and in the presence of four chemical potentials, baryon µ B = 3µ, isospin µ I = 2ν, chiral isospin µ I5 = 2ν 5 , and chiral µ 5 chemical potentials, in the mean-field approximation.The model is intended to be a theoretical basis for considering the properties of dense quark matter, the ground state of which can be realized as one of the following phases: CSB, charged PC, CSC, and symmetric phase.To find out which of the phases is implemented in the model, we have considered its thermodynamic potential in the mean field approximation (19).But, it is obvious that even in this approximation, the study of this TDP to an absolute minimum is a rather difficult problem, especially in the presence of four above mentioned chemical potentials.(By the way, we note that these chemical potentials are thermodynamically conjugated with the real physical characteristics of dense quark matter that can exist in the cores of neutron stars (see the Introduction).) Previously, a similar problem arose when studying the phase structure of the simplest NJL model (without taking into account the diquark interaction channel) with the same four chemical potentials [12,13].However, it turns out that in the latter case, the TDP of the model in the mean-field approximation has (i) a more simple form since it is a function of only two order parameters, and (ii) it is invariant under the so-called dual transformation D (its form is given by Eq. ( 27)).Due to these circumstances, and in particular the property (ii), the solution of the problem is greatly simplified.
In the present work, we show that in the more complex four-fermion NJL model ( 1) the thermodynamic potential in the chiral limit and in the mean-field approximation also has dual discrete symmetry D (27), which greatly simplifies the study of the phase structure of the model.And it is the main result of the paper.This dual symmetry between the CSB and the charged PC phases is especially clearly manifested in the so-called (ν, ν 5 )-phase diagrams by the mirror-symmetric arrangement of these phases relative to the line ν = ν 5 .At the same time, the symmetric and CSC phases in these diagrams are symmetrical with respect to the same line (see Figs. 1 and 2).In addition, acting by the dual transformation D on more well-known phase portraits, one can get an idea of the phase structure of the model in the region of less studied values of chemical potentials (compare the two diagrams in Fig. 3).
Hence, we see that in two qualitatively different QCD-like NJL models there is a duality between CSB and charged PC phases in the mean-field approximation.The conclusion suggests itself that the dual symmetry D is inherent in these models not only in the mean-field approximation, but is also a characteristic property of both their microscopic Lagrangians and their full thermodynamic potentials.Moreover, we believe that the full massless two-flavor QCD Lagrangian also has a dual symmetry between CSB and charged PC phenomena.In the future, these issues will be discussed in more detail.
We hope that our results might shed some new light on phase structure of dense quark matter with isospin and chiral (isospin) imbalances and hence could be important for describing physics, for example, in an interior of the compact stars.