The dual properties of chiral and isospin asymmetric dense quark matter formed of two-color quarks

Abstract: In this paper the phase structure of dense baryon matter composed of $u$ and $d$ quarks with two colors has been investigated in the presence of baryon $\mu_B$, isospin $\mu_I$ and chiral isospin $\mu_{I5}$ chemical potentials in the framework of Nambu--Jona-Lasinio model. In the chiral limit, it has been shown that the duality between phases with spontaneous chiral symmetry breaking and condensation of charged pions, found in the three color case, remains valid in the two color case. In addition, it has been shown that there are two more dualities in the phase diagram in two color case, namely (as in the case with $\mu_{I5}=0$), at $\mu_{I5}\ne 0$ the general $(\mu,\mu_I,\mu_{I5})$-phase portrait of the model has dual symmetry between the phase with condensation of charged pions and the phase with diquark condensation. This duality stays exact even in the physical point, $m_0\ne 0$. And at $m_0=0$ the $(\mu,\mu_I,\mu_{I5})$-phase portrait becomes even more symmetrical, since dual symmetry between phases with spontaneous chiral symmetry breaking and diquark condensation appears. It is shown that due to the dualities the phase diagram is extremely symmetric and has interlacing structure. One can show that the phase portrait of two-color NJL model can be obtained just by duality properties from the results of investigations of three-color NJL model (it was noticed only after the numerical calculations have been performed). Three-color case shares only one duality of the two color one, and one can only see a facet of this enormously symmetric picture in the case of three colors. Using dualities only, it is possible to show that there are no mixed phases (phases with two non-zero condensates). This prediction of dualities is of great use, because for sure it can be shown by the direct calculations but it would be enormously more complicated and time-consuming numerically.