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\(N_c\) limit (\(N_c\) is the number of quark colors). Its phase structure is considered in terms of barion – \(\mu _B\), isospin – \(\mu _I\) and chiral isospin – \(\mu _{I5}\) 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 \(m_0\), and temperature this duality relation is only approximate, although rather accurate. (iii) We have shown that the chiral isospin chemical potential \(\mu _{I5}\) in dense quark matter generates charged pion condensation both at zero and nonzero \(m_0\), and at \(\mu _{I5}\ne 0\) 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 \(\mu _{I5}\). (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.
1 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 \(\mathrm {fm^3}\). 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 collision 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 SU(3) group, where gauge bosons (gluons) play the role of interaction carriers for 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 [1, 2, 3, 4, 5].
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 \(\mu _I\)). 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 [6] and independently Migdal [7] have shown that there might be phase transition from pure neutron matter to mixed hadron matter with protons, neutrons and \(\pi ^0\)pions at superdense matter in the compact stars. Later, using the chiral perturbation theory, it was shown that there is a threshold \(\mu _I^c=m_\pi \approx 140\) MeV of a phase transition to the charged pion condensation (PC) phase [8, 9, 10, 11]. This result was ultimately proved in the framework of random matrix model [12], LadderQCD model [13], resonance gas model [14], quarkmeson model [15, 16, 17], NJL model [18, 19, 20, 21] (including (\(1+1\))dimensional version of the NJL model [22, 23]) and lattice simulations [24, 25, 26, 27, 28, 29, 30, 31]. 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 [32, 33, 34] or chromomagnetic background field [35] could promote the charged PC phase. On the other hand, if the electric charge neutrality and \(\beta \)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 \(m_0\) reaches the physically acceptable values of \(5\div 10\) MeV [36, 37, 38], i.e. at the physical point. In addition, temperature T and different model parameters such as coupling constants, etc, as well strongly influence this phase [39, 40]. It is also worth to note that the phase structure of the isospin imbalanced quark matter below the threshold (\(\mu _I<m_\pi \)) is an important question because even small nonzero \(\mu _I\) could double the critical endpoint of a phase diagram and affects the results of heavyion collision experiments [12, 41, 42].
Recently, it has been shown in the framework of the massless (\(3+1\))dimensional NJL model (and in the leading large\(N_c\) order, where \(N_c\) is the number of colors of quarks) that chiral imbalance promotes charged PC phase in dense matter at zero temperature [43, 44] 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} 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 \(n_5\) and nonzero isotopic chiral density \(n_{I5}\), and quark matter can be described using the corresponding chemical potentials \(\mu _5\) and \(\mu _{I5}\).
It was already mentioned above that nonzero bare quark mass \(m_0\) and nonzero temperature T 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\(_4\) model the fate of the duality observed in the chiral limit [43] (where it is an exact symmetry) between CSB and charged PC phenomena in the leading large\(N_c\) 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\(_4\) model even at \(m_0\ne 0\) and \(T\ne 0\).
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, \(\mu _B=0\), there is no signproblem and we have solid results from lattice simulations [46, 47]. 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\(_4\) model at \(m_0\ne 0\) the dependence of the (pseudo)critical temperature, which characterizes the chiral crossover region of the phase diagram, on the chiral isospin chemical potential \(\mu _{I5}\) and compare our results with other effective model investigations and lattice simulations on this topic. Note also that at \(\mu _B=0\) and \(m_0\ne 0\) the \((\mu _I,T)\) and \((\mu _5,T)\)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 Sect. 2 a (\(3+1\))dimensional NJL model with two massive quark flavors (u and d quarks) that includes three kinds of chemical potentials, \(\mu _B,\mu _I,\mu _{I5}\), is introduced. Furthermore, the symmetries of the model are discussed and its thermodynamic potential is presented in the leading order of the large\(N_c\) expansion both at zero and nonzero temperature T. In particular, it is shown in this section that in the chiral limit (\(m_0=0\)) the phase structure of the model (in the leading order over \(1/N_c\)) has a dual symmetry between CSB and charged PC phenomena. In the next section we formulate the main consequences of the exact dual symmetry (Sect. 3.1), using which it is possible to decide that dual symmetry is performed approximately in the NJL model at \(m_0\ne 0\) and \(T=0\), but with good accuracy (Sect. 3.2). It Sect. 3.3 we show that at nonzero values of the chiral isospin chemical potential \(\nu _{I5}\) the charged PC phase with nonzero quark density can be realized in the model up to rather high values of temperature, \(T\approx 100\) MeV. Moreover, here we show that duality is also fulfilled approximately at \(T\ne 0\). In Sect. 3.4 the plot of the pseudocritical temperature of the chiral crossover transition as a function of \(\mu _{I5}\) at \(\mu =\mu _I=\mu _5=0\) is obtained. Here it is compared with results of other effective models and lattice QCD approaches. Section 4 presents summary and discussion leading to the conclusion that duality between CSB and charged PC observed in the NJL\(_4\) model is supported by some phase diagrams obtained by lattice QCD simulations at \(\mu _B=0\). Some technical details and issues not directly related to this work are relegated to “Appendices A and B”.
2 The model and its thermodynamic potential
2.1 Lagrangian and symmetries
However, at the physical point (\(m_0\ne 0\)) the symmetry of the Lagrangian (1) under transformations from axial isotopic group \(U_{AI_3}(1)\) is explicitly broken. So in the most general case with \(m_0\ne 0\), \(\mu _B\ne 0\) and \(\mu _I\ne 0\) the initial model (1) is invariant only under the \(U_B(1)\times U_{I_3}(1)\) group. (We would like also to remark that Lagrangian (1) is invariant with respect to the electromagnetic \(U_Q(1)\) group, \(U_Q(1):~q\rightarrow \exp ({\mathrm{i}}Q\alpha ) q\), at arbitrary values of \(m_0\), where \(Q=\mathrm{diag}(2/3,1/3)\).)
2.2 Thermodynamical potential: Zero temperature case
2.3 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 \(T\approx 10\) MeV. So it is instructive to know how robust the charged PC phase under temperature.
Finally, it is necessary to note that in the framework of the NJL\(_4\) model the leading order of the large\(N_c\) limit is identical to the meanfield approximation. This suggests that at \(T=0\), where fluctuations are expected to be suppressed, the results are likely to be a good approximation to the \(N_c=3\) case. To be sure that this fact is also valid at nonzero temperature, one can remember, e.g., Refs. [53, 54, 55], where it was shown that meanfield approximation is a rather good approximation both at zero and finite temperature. So not only in the limit \(N_c\rightarrow \infty \) but also at finite \(N_c\) thermal fluctuations are not that large and probably cannot destroy the results, obtained in this approximation. Hence, in the following we may compare our NJL\(_4\) results with lattice \(N_c=3\) QCD results at \(T>0\). Despite all this arguments one should be very cautious comparing the results obtained in different approaches with different setups and approximations, and it should be mentioned that the agreement can be qualitative and not very precise.
2.4 Technical details

\(M_0=0; \Delta _0=0\) – symmetrical phase. It could be realized only in the chiral limit, \(m_0=0\). Usually, in this phase \( n_q\ne 0\) at \(\mu \ne 0\).

\(M_0\ne 0; \Delta _0=0; n_q=0\) – 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.

\(M_0\ne 0; \Delta _0=0; n_q\ne 0\) – chiral symmetry breaking phase with nonzero quark density (below it is CSB\(_{d}\) phase).

\(M_0\ne 0; \Delta _0\ne 0; n_q=0\) – charged pion condensation phase with zero quark density (below in all phase diagrams we use for it the notation PC) (\(M_0=0\) in the chiral limit). In the charged PC phase \(U_{I_3}(1)\) symmetry is spontaneously broken down. Since in this phase \(n_q=0\), sometimes it is called the charged pion gas phase.

\(M_0\ne 0; \Delta _0\ne 0; n_q\ne 0\) – charged pion condensation phase with nonzero quark density (PC\(_{d}\)). In the PC\(_{d}\) phase \(U_{I_3}(1)\) symmetry is also spontaneously broken down.^{4}

We use the notation ApprSYM for the approximate symmetrical phase. In the literature this phase is usually called WignerWeyl phase [3, 57, 58, 59]. It also corresponds to a GMP of the TDP (29), in which \(M_0\ne 0\) and \(\Delta _0=0\). But in contrast to the CSB and CSB\(_d\) phases, dynamical quark mass \(M_0\) in the ApprSYM phase drops rapidly and continuously to the current quark mass \(m_0\) with increasing temperature or chemical potentials. As it follows from Eqs. (7) and (16), under such conditions the chiral condensate \(\langle \bar{q} q\rangle \) is almost zero, and the chiral symmetry is approximately restored in the model. Moreover, at \(m_0\rightarrow 0\) this phase turns into an exactly symmetrical phase with \(M_0=0\). These are the reasons why we use the name ApprSYM in all phase portraits below.
Below we present different phase portraits of the model as well as its properties in terms of this notations.
3 Phase structure of the model
3.1 Exact duality in the chiral limit \((m_0=0)\) at zero temperature \((T=0)\)
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 (\(m_0=0\)). Although this case has been investigated in details in the article [43], it is useful to recall the main features of the model phase structure obtained in the leading order of the large\(N_c\) expansion.
So, in the presence of duality the knowledge of a phase of the model (5) at some fixed values of external free model parameters \(\mu ,\nu ,\nu _{5}\) (and at \(m_0=0\)) is sufficient to understand what a phase (we call it a dually conjugated) is realized at rearranged values of isospin chemical potentials, \(\nu \leftrightarrow \nu _{5}\), at fixed \(\mu \). 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 \(\mathcal{D}\). For example, the chiral condensate of the initial CSB phase at some fixed \(\mu ,\nu ,\nu _{5}\) is equal to the chargedpion condensate of the dually conjugated charged PC phase. The quark number density \(n_q(\nu ,\nu _{5})\) (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\(_4\) model under consideration only in the leading large\(N_c\) order (and at \(m_0=0\)). 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 does the duality give? 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 \((\nu ,\nu _5)\)plane in order to find the phase diagrams of Fig. 1 (or the similar diagrams at other values of \(\mu \)). Instead, it would be sufficient to obtain a phase portrait in a more narrow region, e.g., at \(\nu \ge \nu _5\ge 0\). In this case it is composed of PC, PC\(_d\) and symmetrical phases (see in Fig. 1). Then one should transform each phase of it, using the mapping \(\nu \longleftrightarrow \nu _5\), into a dually conjugated phase, which is already located in the region \(\nu _5\ge \nu \ge 0\). At the same time we should change the name of the phase according to the rule: PC\(\,\rightarrow \,\)CSB, PC\(_d\,\rightarrow \,\)CSB\(_d\) 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 \((\nu ,\mu )\)phase portrait of the model at fixed \(\nu _5=A\), there is no need to perform detailed calculations in order to obtain its \((\nu _5,\mu )\)phase portrait at fixed \(\nu =A\). To do this, it is enough to rename the \(\nu \) axis of the initial phase diagram to the \(\nu _5\) axis and change the name of the phases according to the rule: PC \(\rightarrow \) CSB, PC\(_d\,\rightarrow \,\)CSB\(_d\) (symmetrical phase remains intact). We call this technical procedure as the dual conjugation of a phase diagram. Hence, the \((\nu _5,\mu )\) and \((\nu ,\mu )\)phase portraits are mutually conjugate to each other. However, any \((\nu ,\nu _5)\)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 [60, 61, 62]. But in these models the duality is a consequence of Pauli–Gürsey symmetry of initial Lagrangians.
3.2 Approximate duality in the case of \(m_0\ne 0\) and \(T=0\)
 (i)
At some reliable values of the chemical potentials each \((\nu ,\nu _5)\)phase portrait of the model (at some fixed \(\mu \)) 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 \(\nu =\nu _5\).
 (ii)
Each \((\nu ,\nu _5)\)phase diagram has a phase (it is the ApprSYM phase), which is approximately symmetric under the transformation \(\nu \leftrightarrow \nu _5\), i.e. it is arranged symmetrically with respect to the line \(\nu =\nu _5\).
 (iii)
Under the dual transformation, when \(\nu \leftrightarrow \nu _5\), the order parameter \(M_0\) of CSB or CSB\(_d\) phase is approximately equal to the order parameter \(\Delta _0\) of the dually conjugated charged PC or PC\(_d\) phase.
 (iv)
The quark number density \(n_q\) in any phase, corresponding to the chemical potential point \((\mu ,\nu =A,\nu _5=B)\), is approximately equal to quark number density \(n_q\) of its dually conjugated phase that lies at the point \((\mu ,\nu =B,\nu _5=A)\).
 (v)
Each \((\nu _5,\mu )\)phase portrait (at some fixed \(\nu =A\)) of the model is approximately the dual \(\mathcal{D}\) mapping of a corresponding \((\nu ,\mu )\)phase portrait (at some fixed \(\nu _5=A\)) 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.
Concerning the abovelisted duality signs (i)–(v), we see that in the region \(\omega =\{(\mu ,\nu ,\nu _5):\mu<\Gamma (m_0),\nu<\Gamma (m_0),\nu _5<\Gamma (m_0)\}\), where \(\Gamma (m_0)\) is of the order of the pion mass \(m_\pi \), there is no sense 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 \(\omega \) and for all values of \(\mu \), \(\nu \) and \(\nu _5\) restricted by the conditions \(\mu <\Lambda \), \(\nu <\Lambda \) and \(\nu _5<\Lambda \) (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 \(M_0,\Delta _0\) and baryon density \(n_B\) vs. \(\nu _5\) and \(\nu \) for are depicted. It follows from these pictures that if we go from the phase, corresponding, e.g., to a chemical potential set \((\mu =200,\nu =350,\nu _5=A)\) MeV, to the dually \(\mathcal{D}\) conjugated phase with \((\mu =200,\nu =A,\nu _5=350)\) MeV (or vice versa), then pion condensate \(\Delta _0\) in the charged PC phase is approximately the same as dynamical quark mass \(M_0\) in the dually conjugated CSB phase (compare the left and right panels of Fig. 5) and baryon density \(n_B\) is not changed (approximately). In the dually conjugated points of the ApprSYM phase both \(n_B\) and dynamic quark mass M are not changed, approximately. The same conclusions one can obtain from Fig. 4 for \(\mu =260\) 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 \((\nu ,\mu )\)phase diagram at fixed \(\nu _5=200\) MeV and the \((\nu _5,\mu )\)phase diagram at fixed \(\nu =200\) MeV (see in Fig. 6), we see that qualitatively they are dually \(\mathcal{D}\) conjugated to each other at a rather low values of \(\mu \lesssim 200\) MeV, i.e in this region of each diagram of Fig. 6 one can perform the following axis and phase renaming, \(\nu \leftrightarrow \nu _5\), CSB\(\leftrightarrow \)PC and CSB\(_d\leftrightarrow \)PC\(_d\) (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 \(\nu =200\) MeV (or \(\nu _5=200\) MeV) of these diagrams we intersect just the phases shown in Fig. 6 at low \(\mu \). In addition, it is easy to see that there is a duality \(\mathcal{D}\) between diagrams of Fig. 6 in the regions, where \(\nu \gtrsim 200\) MeV (left panel) and \(\nu _5\gtrsim 200\) MeV (right panel). So the item (v) of the list of duality signs is also satisfied.
3.3 Phase portrait at the physical point (\(m_0=5.5\) MeV) and nonzero temperature \((T\ne 0)\)
Though, the effect of nonzero temperatures is quite predictable (indeed, one can expect that the temperature just restores all the broken symmetries of the model), we investigate nonzero temperature case because it is important in a number of applications. 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. So it is instructive to know how robust the \(\hbox {PC}_{\mathrm{d}}\)phase under the influence of temperature and chiral imbalance.
To clarify this issue, we calculated two \((\nu ,T)\)phase diagrams of the model at \(\mu =0\) (in order to compare our results with lattice investigations) and at different values of \(\nu _5\). In Fig. 7 (left panel) one can see this diagram at \(\nu _5=0\), whereas in the right panel it is at \(\nu _5=200\) MeV. Note that the phase portrait at \(\mu =0\) and \(\nu _5=0\) is in accordance with the same phase portrait obtained within first principle lattice calculations [24, 25, 26, 27, 28, 29, 30, 31]. Also, as one could expect, it is clear from Fig. 7 that temperature restores broken \(U_{I_3}(1)\)symmetry at some rather high critical values \(T_c^{PC}\), where charged PC phase is disappeared.^{5} (Of course, at fixed values of \(\mu \) and \(\nu _5\) the critical temperature \(T_c^{PC}\) depends strongly on the isospin chemical potential \(\mu _I\equiv 2\nu \).) Moreover, as it follows from Fig. 7, \(T^{PC}_c\) vs. \(\nu _5\) (at \(\mu =0\)) drops from values \(T^{PC}_c\approx 200\) MeV at \(\nu _5=0\) to values \(T^{PC}_c\approx 100\) MeV at \(\nu _5=200\) MeV (compare left and right panels of Fig. 7), i.e. when \(\nu _5\) increases the region of the PC phase is shrinked in the phase portrait of the model (this fact is in accordance with the phase diagrams of Fig. 2), but nevertheless the charged PC is a quite robust effect vs temperature at \(\mu =0\).
Finally, we would like to note that in addition to the list of signs (i)–(v) (see in the previous Sect. 3.2)) indicating on the presence in the NJL model (5) at \(m_0\ne 0\) of an approximate dual symmetry between CSB and charged pion condensation at \(T=0\), the similar approximate dual correspondence between phase diagrams exists also at nonzero temperature. For example, two diagrams of Fig. 8, the \((\nu ,T)\)phase portrait at fixed \(\mu =200\) MeV and \(\nu _5=200\) MeV (left panel) and the \((\nu _5,T)\)phase portrait at fixed \(\mu =200\) MeV and \(\nu =200\) MeV (right panel), can be considered as a dually conjugated to each other. Indeed, applying to each of these diagrams the dual mapping, i.e. the following replacements \(\nu \leftrightarrow \nu _5\), CSB\(\leftrightarrow \)PC and CSB\(_d\leftrightarrow \)PC\(_d\), it is possible to obtain approximately another diagram. So dual mapping of a wellknown phase portraits can be used in order to predict (approximately) a phase structure of the model at \(m_0\ne 0\) in the dually conjugated region, i.e. at \(\nu \leftrightarrow \nu _5\).
3.4 Pseudocritical temperature \(T_c(\nu _{5})\) in the NJL\(_4\) model: comparison with lattice QCD and other approaches
Strictly speaking, so far nobody has investigated the function \(T_c(\nu _{5})\) (35) both in the NJL model and other approaches. The matter is that in the most general case, the chiral asymmetry of dense quark matter is described by two chemical potentials, chiral \(\mu _5\) and chiral isospin \(\mu _{I5}\equiv 2\nu _5\) chemical potential.^{6} The first, \(\mu _5\), is usually used when isotopic asymmetry of quark matter is absent, i.e. in the case \(\mu _I=0\) [63]. The second, \(\mu _{I5}\), might be taken into account when, in addition to chiral, there is also isotopic asymmetry of matter, in which charged PC phenomenon can be observed, etc. [43, 44]. And up to now the behavior of a pseudocritical temperature of the crossover region as a function of only the chiral chemical potential \(\mu _5\) was investigated in different approaches at fixed \(\mu =0\), \(\mu _I=0\) and \(\nu _5=0\) [64, 65, 66, 67, 68, 69]. That is the possibility of the existence of a quark system with nonzero chiral isospin imbalance was ignored in these works. (In this particular case we use for a pseudocritical temperature of the NJL\(_4\) model the notation \(T_c\equiv \widetilde{T}_c(\mu _{5})\). Do not confuse with the expression (35), which in fact corresponds to a pseudocritical temperature, obtained in another limiting case of an external parameter set of the NJL\(_4\) model, \(\mu =0\), \(\nu =0\), \(\mu _5=0\) and for arbitrary values of \(\nu _5\).)
So in order to compare our results on the pseudocritical temperature with other approaches to this quantity, we need formally to find in the model under consideration the behavior of a pseudocritical temperature vs \(\mu _5\), i.e. the quantity \(T_c\equiv \widetilde{T}_c(\mu _{5})\).
Just the relation (37) gives us the possibility to compare our results with previous predictions for the (pseudo)critical temperature \(\widetilde{T}_c(\mu _{5})\) obtained in the framework of different effective models [63, 64, 65, 66, 67, 68, 69], using the Dyson–Schwinger equations [70, 71] and lattice calculations [46, 47, 72, 73, 74]. It should be noted that all these studies do not provide a welldefined consistent prediction for the behavior of \(\widetilde{T}_c(\mu _{5})\) vs \(\mu _5\), and rather contradict each other. For example, the works [63, 64, 65, 66, 67] predict decreasing of \(\widetilde{T}_c(\mu _{5})\) with growing \(\mu _5\), whereas in the works [69, 70] there is an opposite picture. The situation has been partially clarified in the works [68, 75, 76, 77], where it has been shown that in the framework of the effective models, regularisation scheme could play a crucial role in the behaviour of the pseudocritical temperature \(\widetilde{T}_c(\mu _{5})\). There are several approaches to regularization of effective models. For example, one can regularize the whole TDP \(\Omega _T(M,\Delta )\) (32), including both the vacuum \(\Omega (M,\Delta )\) and thermal terms of Eq. (32), or regularize only the vacuum term \(\Omega (M,\Delta )\), etc. It was shown in Refs. [75, 76, 77] that the behaviour of the \(\widetilde{T}_c(\mu _{5})\) in the NJL model depends strongly on the scheme that used. And in the present paper, as it follows from Eq. (32), we regularize the whole TDP \(\Omega _T(M,\Delta )\) using in Eq. (32) the socalled hardcutoff regularization scheme when the integration region of the thermal part of the TDP is restricted by the cutoff parameter \(\Lambda \).
Taking into account the relation (37), it is easy to see from the plot of Fig. 9, that in this regularization scheme the pseudocritical temperature \(\widetilde{T}_c(\mu _{5})\) of the NJL model increases for \(\mu _5<\mu ^*_5 \lessapprox 400\) MeV. Above this value it drops down, but at \(\mu _5>\mu ^*_5\) the NJL\(_4\) model, in our opinion, does not provide very reliable predictions, because \(\mu _5\) is near the cutoff \(\Lambda \). The similar quantity has been investigated in the NJL\(_4\) model with the same regularization scheme but in the chiral limit [75]. In this case \(\widetilde{T}_c(\mu _{5})\) is no more a pseudocritical but rather a critical temperature of a 2nd order phase transition from CSB to symmetrical phase. In contrast to Ref. [75], we study the NJL\(_4\) model at the physical point (\(m_0\ne 0\)). However, the behavior of \(\widetilde{T}_c(\mu _{5})\) vs \(\mu _5\) in both cases is qualitatively the same.
The selection of such a regularization scheme is supported and justified by several things. Namely, using the first principle lattice calculations, it was shown that the \(\widetilde{T}_c(\mu _{5})\) increases with \(\mu _5\). Then, there is a qualitative description of the mechanism leading to an increase in the pseudocritical temperature \(\widetilde{T}_c(\mu _{5})\). It is based on the Fermisphere treatment [78] and backed up with the results achieved in the framework of different nonperturbative methods [70, 71]. And finally, and maybe most importantly in the context of the present work, we are guided by predictions for \(\widetilde{T}_c(\mu _{5})\), which follow from the duality symmetries (23) and (36) of the model. Indeed, it is well established that in the \((\nu ,T)\)phase portrait the critical temperature \(T_c^{PC}\) at which isospin symmetry is restored increases with the chemical potential \(\nu \) (see, e.g., the left panel in Fig. 7). Applying to this diagram an (approximate) duality transformation \(\mathcal{D}\) (23), we obtain a \((\nu _5,T)\)phase diagram corresponding to \(\nu =0\), \(\mu =0\) and \(\mu _5=0\) with horizontal \(\nu _5\) axis as well as with the CSB phase at \(\nu _5\gtrapprox 0.1\) GeV. On the boundary between the ApprSYM and CSB phases there will most likely be a crossover region with a pseudocritical temperature \(T_c(\nu _5)\) (35) that, due to the approximate dual symmetry \(\mathcal{D}\), should increase vs \(\nu _5\), as it does \(T_c^{PC}\) vs \(\nu \) in Fig. 7. And finally, applying to this phase diagram the constrained duality transformation \(\mathcal{D}_M\) (36), we obtain a \((\mu _5,T)\)phase diagram corresponding to \(\nu =0\), \(\mu =0\) and \(\nu _5=0\) from which it is clear that \(\widetilde{T}_c(\mu _{5})\) also rises vs \(\mu _5\). It is this qualitative analysis based on the duality properties of the NJL\(_4\) model that is confirmed by Eq. (37) along with the plot of Fig. 9. The duality is only approximate but we also saw in the previous sections that it is a good approximation for values of chemical potential larger than approximately pion mass.
Note that in lattice approach to QCD the simplest \((\nu ,T)\) and \((\mu _5,T)\)phase diagrams at \(\mu =0\) are well investigated. Moreover, they are in accordance with the similar phase diagrams, obtained in the framework of the NJL model (although it should be mentioned that the agreement can be not very precise because in NJL model the results were obtained in the large\(N_c\) limit and in lattice QCD with \(N_c=3\)). But in the last approach, as it follows from above consideration, these phase portraits are (approximately) dually conjugated to each other. Consequently, the same connection can exist between these phase diagrams in real QCD. So there is a solid foundation, the lattice QCD simulations, which allows us to hope that duality between CSB and charged PC phenomena is one of the properties of real dense quark matter.
4 Summary and discussion
In this paper the influence of isotopic and chiral imbalance on phase structure of hot/cold dense quark matter has been investigated at the physical point (i.e. at nonzero current quark mass \(m_0\)) in the framework of the (\(3+1\))dimensional NJL model with two quark flavors in the large\(N_{c}\) limit (\(N_{c}\) is the number of colors). Dense matter means that our consideration has been performed at nonzero baryon \(\mu _B\) chemical potential. Isotopic and chiral imbalance in the system were accounted for by introducing isospin \(\mu _I\) and chiral isospin \(\mu _{I5}\) chemical potentials (see Lagrangian (5)). Of course, one knows that current quark masses of u and d quarks (the ones that we considered in the paper) are rather small and, in general, the chiral limit is a very good approximation. But sometimes although small but nonzero masses can change some aspects of the phase diagram. For example, charged pion condensation (PC) phase in the chiral limit and at \(T=0\) starts from infinitesimally small values of isospin chemical potential \(\mu _I\), but when one takes into account quark masses, then it shifts the charged PC to the values of \(\mu _I\) larger than pion mass \(m_\pi \approx 140\) MeV (compare diagrams of Figs. 1 and 2). The phase structure of cold dense quark matter in the chiral limit has been obtained in [43, 44], where it has been shown that chiral isospin chemical potential \(\mu _{I5}\) generates charged PC in dense quark matter and there is a duality correspondence between CSB and charged PC phenomena in the leading order of the large\(N_{c}\) approximation. The goal of the present paper is the extension of this consideration to a more physical case of the NJL\(_4\) model with nonzero current quark masses. This allows us to draw more accurate phase diagram and perform comparison with lattice QCD. Moreover, we take into account finite temperatures, which give us a chance to consider the results in the context of heavy ion collision experiments in which temperatures are always rather large. Even in the context of neutron stars it can be interesting to consider the case of finite temperature. It has been found that the duality between CSB and charged PC phenomena observed in [43, 44] in the chiral limit (where it was exact) is valid with good accuracy even in the physical point. It has been also shown that temperature does not spoil the duality as well.
We have studied the full (\(\mu _B\), \(\mu _I\), \(\mu _{I5}\), T)phase diagram of quark matter in terms of the NJL\(_{4}\) model with \(m_0\ne 0\). This general consideration is not feasible in the lattice QCD simulations, mainly due to the famous sign problem which does not allow for the consideration of finite baryon densities (nonzero baryon chemical potential \(\mu _B\)). But contrary to the case of nonzero baryon chemical potential, simulations with nonvanishing isospin \(\mu _I\) and chiral \(\mu _{5}\) chemical potentials are not hampered by a sign problem and some particular cases have been considered on the lattice. For example, the (\(\mu _I\), T)phase diagram at zero values of \(\mu _B, \mu _{I5}, \mu _5\) chemical potentials is well established as in lattice QCD as well as in different effective models and a rather good agreement can be observed between this different approaches. And there are lattice QCD simulations of the quark matter with only nonzero chiral chemical potential \(\mu _{5}\) in terms of as \(SU_c(2)\) QCD (twocolour QCD) [46] as well as real \(SU_c(3)\) QCD (threecolour QCD) [47], where the catalysis of chiral symmetry breaking by chiral chemical potential has been established. Namely, it has been shown that chiral condensate and (pseudo)critical temperature (the temperature at which the chiral condensate drops) grows with increase of chiral chemical potential \(\mu _{5}\). In this paper, as well as in Ref. [44], we have supported these conclusions by effective NJL model considerations. In paricular, the plot of the pseudocritical temperature \(\widetilde{T}_c(\mu _{5})\) vs chiral chemical potential [see Fig. 9 and take into account Eq. (37)] has been drawn and it was shown that this quantity rises with the raise of \(\mu _{5}\) and the behaviour is rather similar to the results of the lattice QCD simulations.
So let us gaze at all this from the general picture viewpoint. We have two lattice simulation results, (\(\mu _I\), T) and (\(\mu _5\), T)phase diagrams. These phase diagrams have been also obtained in the NJL model and the results are in a good agreement with lattice QCD simulations. But in terms of NJL model we can consider the general case and we know that there is the duality \(\mathcal{D}\) (23) (it is exact in the chiral limit only) between CSB and charged PC phenomena in the leading order of the large\(N_{c}\) approximation. Moreover, there is also the socalled constrained \(\mathcal{D}_M\) (36) duality of the NJL\(_4\) model phase diagram, which is valid even in the physical point [44]. So the significant regions (at \(\nu _I,\mu _5\gtrapprox m_\pi /2\)) of the particular (\(\mu _I\), T) and (\(\mu _5\), T)phase diagrams should be dually conjugated to each other with respect to a sequential action of two mappings, \(\mathcal{D}\) and \(\mathcal{D}_M\) (see the discussion at the end of Sec. 3.4), but the duality \(\mathcal{D}\) in the case of the physical point is only approximate (see in Sects. 3.2 and 3.3), although it is valid with a good precision. Since the particular (\(\mu _I\), T) and (\(\mu _5\), T)phase diagrams in these two approaches agree, one can conclude that the duality can be observed in the lattice QCD simulations. And this put the notion of the duality on another level of confidence, for it is observed in terms of the toy (\(1+1\))dimensional NJL model [48, 49, 50], effective (\(3+1\))dimensional NJL model [43, 44], lattice QCD simulations and similar dualities has been observed in the large\(N_{c}\) orbifold equivalences approach [79, 80]. Comparison to the lattice QCD is important not only due to the fact that it is ab initio method for dealing with QCD, but because it does not make use of, for example, large\(N_{c}\) approximation (as in NJL models or in large \(N_{c}\) orbifold equivalences approaches).
The question of catalysis of chiral symmetry breaking by chiral chemical potential, i.e. the growth of \(T_c\) vs \(\mu _5\), is a rather debated one and there are a number of papers that predicted that (pseudo)critical temperature decrease with increase of chiral chemical potential \(\mu _5\) [63, 64, 65, 66, 67], as well there are a number of papers that support our results [69, 70, 75, 76]. Different regularization schemes have been applied in [68, 75, 76, 77] and it has been stated that if the right one is used there is no catalysis, but lattice QCD results is probably more trustworthy and it disagrees with them. But the catalysis of chiral symmetry breaking by chiral chemical potential \(\mu _5\) can be established in terms of duality notion, let us elaborate on that. As we have talked about, the (\(\mu _I\), T)phase diagram is well established one and the duality fails only in the region of small isospin and chiral isospin chemical potentials (smaller than half of the pion mass), but works quite well for the larger values (see, e.g., in Figs. 2, 3). But at the (\(\mu _I\), T)phase diagram in the region of isospin chemical potential larger than half of the pion mass the critical temperature increases when \(\mu _{I }\) is raised and the duality here is a good approximation, so the critical temperature at the duality conjugated (\(\mu _5\), T)phase diagram should increase with rising of \(\mu _{5}\) as well (see at the end of Sect. 3.4).

It has been demonstrated that the duality correspondence between CSB and charged PC phenomena observed in Refs. [43, 44] in the chiral limit (where it was exact) is a very good approximate symmetry of the phase diagram even in the physical point in the framework of the NJL\(_{4}\) model in the leading order of the large\(N_c\) approximation (see in Sect. 3.2). And it stays a very instructive feature of the phase diagram that can be used in different situations.

It has been also shown that temperature does not spoil the duality correspondence between CSB and charged PC phenomena and it stays exact at finite temperature in the chiral limit (see the comment at the end of Sect. 2.3) and it is a good approximate symmetry at the physical point (see in Sect. 3.3).

We have shown that there is a huge PC\(\mathrm{_d}\) phase region in the phase portrait of the model (1) promoted by \(\nu _5\) even in the physical point (see, e.g., in Fig. 3). And it has been revealed that PC\(\mathrm{_d}\) phase can exist at rather large temperatures up to even about 100 MeV (see in Fig. 8).

The particular cross sections of the obtained phase portraits are in qualitative accordance with the recent lattice simulations [24, 25, 26, 27, 28, 29, 30, 31, 46, 47]. And it has been established that lattice QCD results support the existence of the duality.

The rise of pseudocritical temperature with increase of chiral chemical potential \(\mu _{5}\) has been established in terms of duality notion and the well explored results of lattice QCD and different approaches on phase structure of isotopicaly imbalanced quark matter. It gives additional argument in favour of this behaviour of the pseudocritical temperature and it is of importance because, although the lattice results confirming this behaviour are conclusive, the value of the pion mass that is used in these simulations is still quite high and well above the physical pion mass and our results are made at the physical point with the right value of the pion mass.
This work is intended to generalize and refine the previously obtained results of Refs. [43, 44] to a more physically motivated situation (physical point and finite temperatures). These generalizations require much more computing resources and technically is rather challenging, but it pays off when you can compare the results with lattice QCD and it supports them. Moreover, we hope that our results might shed new light on phase structure of dense quark matter with isotopic and chiral imbalance and hence could be of interest in the context of the heavy ion collision experiments and neutron stars interiors.
Footnotes
 1.
It is predicted that there is an electrical current in the chiral imbalanced quark matter under strong magnetic field [45]. This phenomenon is called chiral magnetic effect and it could be an evidence of the chiral imbalance in QCD.
 2.
The notation \(\langle \hat{O}\rangle \) means the ground state expectation value of the operator \(\hat{O}\).
 3.
 4.
The transition between PC and PC\(_d\) phases is also a firstorder phase transition, as in this case the order parameter \(\Delta _0\) 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\(_d\) phase the quark number density \(n_q\) is nonzero. Moreover, in both phases the isospin density \(n_I\) is nonzero.
 5.
In the points of the boundary between ApprSYM and charged PC phases in Figs. 7, 8 there is a secondorder phase transition, whereas the dashed line in each of these figures represents the socalled pseudocritical temperature, which characterizes the socalled crossover region between CSB and ApprSYM phases.
 6.
In general, chiral imbalance of dense quark matter is characterized by two densities, chiral isospin \(\hat{n}_{I5}=\frac{1}{2}\left( \hat{n}_{u5}\hat{n}_{d5}\right) \) and chiral density \(\hat{n}_5=\hat{n}_{u5}+\hat{n}_{d5}\) (see Introduction for notations). Alternatively, it can be described by corresponding chemical potentials \(\mu _{I5}\) and \(\mu _5\), which are the quantities thermodynamically conjugated to \(\hat{n}_{I5}\) and \(\hat{n}_{5}\), respectively.
Notes
Acknowledgements
The authors are thankful to Igor Shovkovy for the idea of the possibility of generation of chiral isospin imbalance in dense matter due to chiral separation effect.
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