It has been shown in

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3 \Lambda_{\rm QCD} \sim 650$--$700$\,MeV}, where $\mu=\mu_B/3$ is the quark number chemical potential (maybe in reality, it should be much higher), because below this line the QCD system is certainly a strongly-coupled and there perturbative calculations cannot be trusted. As a rule, perturbative QCD (without large $N_c$ limit) starts to work at very high values of chemical potential, $\mu \gg 1$\,GeV~\cite{Schafer-ml-1999jg}. But at that large values of chemical potentials the NJL model is not expected to give trustworthy results, as it is an effective low-energy model for QCD, which is adequate only up to the cut-off, which is around 650\,MeV\@. And the region of the phase diagram with values of the larger $\mu$ is out of scope of the consideration. Let us also notice that at large values of $\mu$, even not necessarily as large as 650--700\,MeV, the system is probably in the color superconducting phase. DGR noticed that color superconductivity is suppressed at large $N_c$ due to the fact that the Cooper pair is not a color singlet (the diagram responsible for color superconductivity is non-planar) and only in the large $N_{c}$ limit the DGR phase is predicted. So summarizing, DGR is an asymptotic phase (at high $\mu$) and cannot be obtained at low values of $\mu$, whereas NJL model describe only the region of comparatively low values~of~$\mu$. To find the TDP of the system, we use a semibosonized version of the Lagrangian~(\ref{1}), which contains composite bosonic fields $\sigma (x)$ and $\pi_a (x)$ $(a=1,2,3)$ (in what follows, we use the notations $\mu\equiv\mu_B/3$, $\nu\equiv\mu_I/2$ and $\nu_{5}\equiv\mu_{I5}/2$): {\rdmathspace \begin{align} \widetilde L & = \bar q\Big [\gamma^\rho\mathrm{i}\partial_\rho +\mu\gamma^0 + \nu\tau_3\gamma^0+\mu_{5}\gamma^0\gamma^5+\nu_{5}\tau_3\gamma^0\gamma^5-\sigma -\mathrm{i}\gamma^5\pi_a\tau_a\Big ]q -\frac{N_c}{4G}\Big [\sigma\sigma+\pi_a\pi_a\Big ]. \label{2} \end{align}}\relax In~(\ref{2}) and below the summation over repeated indices is implied. From the auxiliary Lagrangian~(\ref{2}) one gets the equations for the bosonic fields \begin{align} \sigma(x)=-2\frac G{N_c}(\bar qq);\qquad \pi_a (x)=-2\frac G{N_c}(\bar q \mathrm{i}\gamma^5\tau_a q). \label{200} \end{align} Note that the composite bosonic field $\pi_3 (x)$ can be identified with the physical $\pi^0(x)$-meson field, whereas the physical $\pi^\pm (x)$-meson fields are the following combinations of the composite fields, $\pi^\pm (x)=(\pi_1 (x)\mp i\pi_2 (x))/\sqrt{2}$. Obviously, the semibosonized Lagrangian $\widetilde L$ is equivalent to the initial Lagrangian~(\ref{1}) when using the equations~(\ref{200}). Furthermore, it is clear from~(\ref{2001}) and footnote~\ref{f1,1} that the composite bosonic fields~(\ref{200}) are transformed under the isospin $\U_{I_3}(1)$ and axial isospin $\U_{AI_3}(1)$ groups in the following manner: {\rdmathspace \begin{align} \U_{I_3}(1):&&\sigma& \to\sigma; & \pi_3& \to\pi_3; & \pi_1& \to\cos (\alpha)\pi_1+\sin (\alpha)\pi_2; & \pi_2& \to\cos (\alpha)\pi_2-\sin (\alpha)\pi_1,\nonumber\\ \U_{AI_3}(1):&&\pi_1& \to\pi_1; & \pi_2& \to\pi_2; & \sigma& \to\cos (\alpha)\sigma+\sin (\alpha)\pi_3; & \pi_3& \to\cos (\alpha)\pi_3-\sin (\alpha)\sigma. \label{201} \end{align}}\relax Starting from the theory~(\ref{2}), one obtains in the leading order of the large $N_c$-expansion (i.e.\ in the one-fermion loop approximation) the following path integral expression for the effective action ${\cal S}_{\rm {eff}}(\sigma,\pi_a)$ of the bosonic $\sigma (x)$ and $\pi_a (x)$ fields: $$ \exp(\mathrm{i}{\cal S}_{\rm {eff}}(\sigma,\pi_a))= N'\int[d\bar q][dq]\exp\Bigl(\mathrm{i}\int\widetilde L\,d^4x\Bigr), $$ where \begin{equation} {\cal S}_{\rm {eff}} (\sigma,\pi_a) =-N_c\int d^4x\left [\frac{\sigma^2+\pi^2_a}{4G} \right ]+\tilde {\cal S}_{\rm {eff}} \label{3} \end{equation} and $N'$ is a normalization constant. The quark contribution to the effective action, i.e.\ the term $\tilde {\cal S}_{\rm {eff}}$ in eq.~(\ref{3}), is given by: {\rdmathspace \begin{align} \scalemathtyp{\exp(\mathrm{i}\tilde {\cal S}_{\rm {eff}})=N'\int [d\bar q][dq]\exp\left(\mathrm{i}\int\Big\{\bar q\big [\gamma^\rho\mathrm{i}\partial_\rho +\mu\gamma^0+\nu\tau_3\gamma^0+ \mu_5\gamma^0\gamma^5+\nu_5\tau_3\gamma^0\gamma^5-\sigma -\mathrm{i}\gamma^5\pi_a\tau_a\big ]q\Big\}d^4 x\right)}. \label{4} \end{align}}\relax The ground state expectation values $\vev{\sigma(x)}$ and $\vev{\pi_a(x)}$ of the composite bosonic fields are determined by the saddle point equations, \begin{align} \frac{\delta {\cal S}_{\rm {eff}}}{\delta\sigma (x)}=0,\qquad \frac{\delta {\cal S}_{\rm {eff}}}{\delta\pi_a (x)}=0, \label{05} \end{align} where $a=1,2,3$. It is clear from eq.~(\ref{201}) that if $\vev{\sigma(x)}\ne 0$ and/or $\vev{\pi_3(x)}\ne 0$, then the axial isospin $\U_{AI_3}(1)$ symmetry of the model is spontaneously broken down, whereas at $\vev{\pi_1(x)}\ne 0$ and/or $\vev{\pi_2(x)}\ne 0$ we have a spontaneous breaking of the isospin $\U_{I_3}(1)$ symmetry. Since in the latter case the ground state expectation values, or condensates, both of the field $\pi^+(x)$ and of the field $\pi^-(x)$ are nonzero, this phase is usually called the CPC phase. It is easy to see from eq.~(\ref{200}) that the nonzero condensates $\vev{\pi_{1,2}(x)}$ (or $\vev{\pi^\pm(x)}$) are not invariant with respect to the electromagnetic $\U_Q(1)$ transformations~(\ref{2002}) of the flavor quark doublet. Hence in the CPC phase the electromagnetic $\U_Q(1)$ invariance of the model~\eqref{1} is also broken spontaneously, so superconductivity is an unavoidable property of the CPC phase. \looseness=-1 In vacuum, i.e.\ in the state corresponding to an empty space with zero particle density and zero values of the chemical potentials $\mu$, $\nu$, $\mu_5$ and $\nu_5$, the quantities $\vev{\sigma(x)}$ and $\vev{\pi_a(x)}$ do not depend on space coordinate $x$. However, in dense medium, when some of the chemical potentials are nonzero quantities, the ground state expectation values of bosonic fields might have a nontrivial dependence on spatial coordinates. In particular, in this paper we use the following spatially inhomogeneous CDW ansatz for chiral condensate and the single-plane-wave LOFF ansatz for charged pion condensates (for simplicity we suppose that wavevectors of the inhomogeneous condensates are directed along the $x^1$ coordinate axis): \begin{alignat}{2} \vev{\sigma(x)}& =M\cos (2kx^{1}),& \qquad \vev{\pi_3(x)}& =M\sin (2kx^{1}),\nonumber\\ \vev{\pi_1(x)}& =\Delta\cos(2k'x^{1}),& \qquad \vev{\pi_2(x)}& =\Delta\sin(2k'x^{1}), \label{06} \end{alignat} where gaps $M,\Delta$ and wavevectors $k,k'$ are constant dynamical quantities. In fact, they are coordinates of the global minimum point of the TDP $\Omega (M,k,k',\Delta)$. In the leading order of the large-$N_c$ expansion it is defined by the following expression: \begin{align} \int d^4x \Omega (M,k,k',\Delta) =-\frac{1}{N_c}{\cal S}_{\rm {eff}}\{\sigma(x),\pi_a(x)\}\big|_{\sigma (x)=\vev{\sigma(x)},\pi_a(x)=\vev{\pi_a(x)}} ,\label{080} \end{align} which gives \begin{align} \int d^4x\Omega (M,k,k',\Delta)=\int d^4x\frac{M^2+\Delta^2}{4G}+\frac{\mathrm{i}}{N_c}\ln\left ( \int [d\bar q][dq]\exp\Bigl(\mathrm{i}\int d^4 x\bar q \widetilde{D} q \Bigr)\right ), \label{08} \end{align} where \begin{align} \bar q \widetilde{D} q & = \bar q\big [\gamma^\rho\mathrm{i}\partial_\rho +\mu\gamma^0+\nu\tau_3\gamma^0+\mu_5\gamma^0 \gamma^5+\nu_5\tau_3\gamma^0 \gamma^5-M\exp(2\mathrm{i}\gamma^5\tau_3kx^1)\big ]q\nonumber \\ & \quad - \Delta\big (\bar q_u\mathrm{i}\gamma^5 q_d\big )\e^{-2\mathrm{i}k'x^1}-\Delta\big (\bar q_d\mathrm{i}\gamma^5 q_u\big )\e^{2\mathrm{i}k'x^1}.\label{09} \end{align} (Remember that in this formula $q$ is indeed a flavor doublet, i.e.\ $q=(q_u,q_d)^T$.) To proceed, let us introduce in eqs.~(\ref{08})--(\ref{09}) the new quark doublets, $\psi$ and $\bar\psi$, by the so-called Weinberg (or chiral) transformation of these fields~\cite{kkzz,weinberg}, $\psi=\exp(\mathrm{i}\tau_3k'x^{1}+\mathrm{i}\tau_3\gamma^5kx^{1})q$ and $\bar\psi = \bar q\exp(\mathrm{i}\tau_3\gamma^5kx^{1}-\mathrm{i}\tau_3k'x^{1})$. Since this transformation of quark fields does not change the path integral measure in eq.~(\ref{08}),\footnote{Strictly speaking, performing Weinberg transformation of quark fields in eq.~(\ref{08}), one can obtain in the path integral measure a factor, which however does not depend on the dynamical variables $M$, $\Delta$, $k$, and $k'$. Hence, we ignore this unessential factor in the following calculations. Note that only in the case when there is an interaction between spinor and gauge fields there might appear a nontrivial, i.e.\ dependent on dynamical variables, path integral measure, generated by Weinberg transformation of spinors. This unobvious fact follows from the investigations by Fujikawa~\cite{fujikawa}.} the expression~(\ref{08}) for the TDP is easily transformed to the following one: {\rdmathspace \begin{align} \int d^4x\Omega (M,k,k',\Delta)=\int d^4x\frac{M^2+\Delta^2}{4G}+\frac{\mathrm{i}}{N_c}\ln\left ( \int [d\bar\psi][d\psi]\exp\Bigl(\mathrm{i}\int d^4 x\bar\psi D\psi \Bigr)\right ), \label{010}\end{align}}\relax where instead of the $x$-dependent Dirac operator $\widetilde{D}$ a new $x$-independent operator \linebreak $D=i\gamma^\mu \partial_{\mu} +\mu\gamma^0 + \nu\tau_3\gamma^0+ \mu_{5}\gamma^0\gamma^5+\nu_{5}\tau_3\gamma^0\gamma^5 +\tau_3\gamma^1 \gamma^5 k+\tau_3\gamma^1 k^{\prime}-M -\mathrm{i}\gamma^5\Delta\tau_1$ appears. In this case path integral can be evaluated and one get for the TDP~(\ref{08}) an expression that reads \begin{align} \Omega (M,\Delta,k,k')~ =\frac{M^2+\Delta^2}{4G}+\mathrm{i}\frac{{\rm Tr}_{sfx}\ln D}{\int d^4x}=\frac{M^2+\Delta^2}{4G}+\mathrm{i}\int\frac{d^4p}{(2\pi)^4}\ln\Det\overline{D}(p), \label{07} \end{align} where \begin{equation} \overline{D}(p)=\not\!p +\mu\gamma^0 + \nu\tau_3\gamma^0+\mu_{5}\gamma^0\gamma^5 + \nu_{5}\tau_3\gamma^0\gamma^5 +\tau_3\gamma^1 \gamma^5 k+\tau_3\gamma^1 k^{\prime}-M -\mathrm{i}\gamma^5\Delta\tau_1\equiv\left (\begin{array}{cc} A\,, & U\\ V\,, & B \end{array}\right ) \label{500} \end{equation} is the momentum space representation of the Dirac operator $D$. The quantities $A,B,U,V$ in~(\ref{500}) are really the following 4$\times$4 matrices in the spinor space, \begin{align} A& =\not\!p +\mu\gamma^0 + \nu\gamma^0+ \mu_{5}\gamma^0\gamma^5+\nu_{5}\gamma^0\gamma^5-M+\gamma^1 \gamma^5 k+\gamma^1 k^{\prime};\nonumber\\ B& =\not\!p +\mu\gamma^0 - \nu\gamma^0+\mu_{5}\gamma^0\gamma^5- \nu_{5}\gamma^0\gamma^5-M-\gamma^1 \gamma^5 k-\gamma^1 k^{\prime};\nonumber\\ U=V& =-\mathrm{i}\gamma^5\Delta, \label{80} \end{align} so the quantity $\overline{D}(p)$ from~(\ref{500}) is indeed a 8$\times$8 matrix whose determinant in eq.~(\ref{07}) can be calculated on the basis of the following general relations {\rdmathspace\setlength{\arraycolsep}{1pt} \begin{align} \Det\overline{D}(p)\equiv\det\left (\!\!\begin{array}{cc} A, & U\\ V, & B \end{array}\right )\!=\det [-VU+VAV^{-1}B]=\det [\Delta^{2}+ \gamma^5 A \gamma^5 B]=\det [BA-BUB^{-1}V]. \label{9} \end{align}}\relax ]]>

m_{\pi}/2$, there is
homogeneous
CPC
phase at rather small values of $\mu$. But if $\mu_I$ is less than pion mass, $\nu

\mu_{I5}$. Knowing condensates and other dynamical and thermodynamical quantities of the system, e.g., in the CSB phase, one can then obtain the corresponding quantities in the dually conjugated CPC phase of the model, by simply performing there the duality transformation, $\mu_{I}\leftrightarrow\mu_{I5}$. Two other symmetries of the TDP, ${\cal D}_{HM}$ and ${\cal D}_{H\Delta}$, can also impose some restrictions on the shape of the CSB and CPC phases, respectively (see in~\cite{kkz2}). Note that similar dualities have been also considered in the framework of universality principle (large-$N_{c}$ orbifold equivalence) of phase diagrams in QCD and QCD-like theories in the limit of large $N_{c}$~\cite{hanada,hanada2}. In particular, it was noted there that in the chiral limit QCD at $\mu_{I5}\ne 0$ might be equivalent to QCD at $\mu_I\ne 0$, etc (see remarks in section 4 of~\cite{hanada}). Since the Lagrangian~\eqref{1} itself does not have a symmetry that would automatically lead to the dual symmetries of its phase portrait, an interesting question arise whether the duality of the phase portrait obtained in the large-$N_c$ limit is a deep property of the theory described by Lagrangian~\eqref{1} or it is just an accidental feature.\footnote{As a counterexample, we can bring the duality between CSB and superconductivity in some (1 + 1)- and (2 + 1)-dimensional theories~\cite{ekkz2,ekkz2.m001,ebert.m001,ekkz21.m001}. But there the original Lagrangians are invariant with respect to the so-called dual symmetry, which includes both the transformation of chemical potentials and coupling constants as well as the Pauli-Gursey transformation of spinor fields (that transforms the chiral interaction channel into a superconducting one, and vice versa). As a result, a duality between these phenomena appears on the phase portrait.} In order to get some hints in this direction, the discussed duality between CSB and CPC phenomena has been established both within the homogeneous and inhomogeneous approaches to condensates, but only in the framework of the NJL$_{2}$ model~\cite{kkz,kkzz} (but in this case the dualities in inhomogeneous and homogeneous approaches are quite similar in terms of proving them, and duality in inhomogeneous case is much easier to show). To confirm these results and to ensure that the duality and related phenomena are intrinsic also to (3+1)-dimensional variant of the model~\eqref{1}, in the present paper we study the possibility of dual symmetries of its thermodynamic potential using, in contrast to ref.~\cite{kkz2}, a more extended approach based on the spatially inhomogeneous condensates. \relax In this paper we have obtained in the leading $1/N_c$ order an exact expression~(\ref{07}) of the thermodynamic potential of the model~\eqref{1}, when for chiral and charged pion condensates the CDW and single-plane-wave LOFF ansatzes are used, respectively (see eq.~(\ref{06})). A detailed analysis of the phase structure of the model was not carried out in this case. However, we were able to establish that the TDP~(\ref{07}) of the system possesses the dual symmetry ${\cal D}_{I}$~(\ref{160}), which necessarily leads to a duality between CSB and CPC phenomena. Hence, in the chiral limit both in the homogeneous and more extended spatially inhomogeneous approaches to the ground state of the NJL$_4$ system~\eqref{1}, we observe the duality between CSB and CPC phenomena. So, in our opinion, this type of duality is not an artifact of the method of investigation, but the true property of a chirally and isotopically asymmetric dense medium described by the NJL$_4$ Lagrangian~\eqref{1}. \looseness=-1 Moreover, it is known that when nonzero current quark masses is included in the consideration (at the physical point) the duality is not exact, though it is a good approximation~\cite{Khunjua-ml-2018jmn}. When one considers the duality in the inhomogeneous case (between inhomogeneous phases) then the duality is exact even at the physical point (see in section IIIC). It is necessary to bear in mind that in the model~\eqref{1} an arbitrary dual invariance of its TDP, calculated in the approach with homogeneous condensates, is not automatically transferred to the case of inhomogeneous condensates. Indeed, the duality between CSB and CPC are realized in the model~\eqref{1} in both approaches, however other dual symmetries, ${\cal D}_{HM}$ and ${\cal D}_{H\Delta}$, of the TDP~(\ref{07}) at $k=0$ and $k'=0$ are not observed in the case with inhomogeneous condensates, i.e.\ at $k\ne 0$ and $k'\ne 0$. In this paper we have not studied the phase portrait in the framework of (3+1)-dimensional massless NJL model itself, but we have shown that even if the phase diagram contains phases with nonzero inhomogeneous condensates, it possesses the property of duality (dual symmetry). We demonstrate this fact in terms of TDP in the leading order of the large-$N_c$ approximation. In the literature, the $(\nu, \mu)$-QCD phase diagram has been studied very intensively and it is understood well in homogeneous case. The various aspects of the $(\nu, \mu)$-phase diagram with possible inhomogeneous condensates were investigated in~\cite{Nowakowski,Nowakowski-ml-2015ksa,he}, etc. It has been shown that it is possible to use these shreds and combine them into one unified picture and draw full $(\nu, \mu)$-phase diagram in inhomogeneous case (see in figure~\ref{fig1}). When this interesting thing has been completed, from this assembled phase diagram the phase diagram in a completely different situation has been obtained, namely $(\nu_5, \mu)$-diagram of chirally asymmetric QCD matter (see in figure~\ref{fig2}). It has been shown that at nonzero values of $\nu_{5}$ there is a very rich phase diagram featuring both the ICSB phase at rather high values of $\nu_{5}$ and the ICPC phase at small and moderate values of $\nu_{5}$. The phase diagram of figure~\ref{fig2} was obtained only by using the duality between CSB and CPC phenomena. Moreover, as it was established in ref.~\cite{Khunjua-ml-2018jmn}, the existence of the duality between CSB and CPC phenomena is supported by lattice QCD results. These instances show that the duality is an inherent property of dense quark matter. So, it is not just entertaining mathematical gaud and interesting but useless mathematical property. In our opinion, it is a potent instrument with very high predictivity power. It has also been hinted that in inhomogeneous case CPC phase is generated in dense quark matter even by infinitesimally small values of chiral isospin chemical potential $\nu_{5}$. Qualitatively, the same behaviour has been predicted in the framework of (1+1)-dimensional NJL model, this concurrence once more consolidates the confidence that NJL$_{2}$ model can be used as a legit laboratory for the qualitative simulation of specific properties of QCD. ]]>

0$, the density, e.g., $n_{uR}\equiv\vev{\bar q_{uR}\gamma^0 q_{uR}}$ of the right-handed $u$ quarks is greater than the density $n_{uL}\equiv\vev{\bar q_{uL}\gamma^0 q_{uL}}$ of the left-handed $u$ quarks. Hence, in this case we have at $z>0$ the positive values of the chiral density $n_{u5}\equiv n_{uR}-n_{uL}$ for $u$ quarks. (It is evident that at $z<0$ the chiral density of $u$ quarks is negative.) On the contrary, since the axial current $\vec j_{5d}$ of $d$ quarks differs in its direction from the axial current $\vec j_{5u}$ of $u$ quarks, one can see that in this case at $z>0$ (at $z<0$) the chiral density $n_{d5}$ of $d$ quarks is negative (positive). Consequently, we have at $z>0$ the positive values of the quantity $n_{I5}\equiv n_{u5}-n_{d5}$, which is the ground state expectation value of the chiral isospin density. Whereas at $z<0$ the chiral isospin density is negative. In summary, we can say that in dense quark medium under the influence of a strong magnetic field (as an example we can mention neutral stars), regions with a nonzero chiral isospin density $n_{I5}$ might appear. Therefore physical processes inside these regions can be described, e.g., in the framework of the Lagrangians of the form~\eqref{1}, containing chiral isospin chemical potential $\mu_{I5}$. ]]>