Influence of chiral chemical potential ${\mu }_5$ on phase structure of the two-color quark matter

In this paper the influence of chiral chemical potential ${\mu }_5$ on the phenomenon of diquark condensation and phase structure of dense quark matter altogether is contemplated in the framework of the effective 2-color and 2-flavor Nambu–Jona-Lasinio model. The nonzero values of baryon ${\mu }_B$, isospin ${\mu }_I$ and chiral isospin ${\mu }_{I5}$ chemical potentials are also taken into account. We show that the duality relations between diquark condensation, charged pion condensation, and chiral symmetry breaking phenomena, found in the case of zero ${\mu }_5$, are also valid for any value of ${\mu }_5\ne 0$. In terms of dualities and the influence on the phase diagram, chiral imbalance ${\mu }_5$ stands alone from other chemical potentials. Indeed, in comparison with other chemical potentials, ${\mu }_5$ has two interesting features. (i) In the region of moderate values of ${\mu }_B$, ${\mu }_I$, and ${\mu }_{I5}$ it manifests itself as a universal catalyst, since it enhances just the phase that is realized in the system at ${\mu }_5=0$. (ii) In the second regime, when several other chemical potentials reach rather large values, one could observe a rather complicated and rich phase structure, and chiral chemical potential ${\mu }_5$ can be a factor that not so much catalyzes as triggers rather peculiar phases.

Figure 1

(a) $({\nu }_5,{\mu }_5)$-phase diagram at $\nu =\mu =0\mathrm{GeV}$. (b) $(\nu ,{\mu }_5)$-phase diagram at $\mu ={\nu }_5=0\mathrm{GeV}$. (c) $(\mu ,{\mu }_5)$-phase diagram at ${\nu }_5=\nu =0\mathrm{GeV}$. In these diagrams BSF means the baryon superfluid phase, PC—the charged pion condensation phase, CSB—the chiral symmetry breaking, and SYM denotes the symmetrical phase of the model.

Figure 2

The behaviors of condensates vs chemical potentials: (a) $M$ as a function of ${\mu }_5$ at ${\nu }_5=0.1\mathrm{GeV}$, $\mu =\nu =0\mathrm{GeV}$. (b) $M$ and ${\pi }_1$ as a functions of $\mu$ at $\nu =0$, ${\nu }_5=0.25\mathrm{GeV}$ and ${\mu }_5=0.5\mathrm{GeV}$ (c) $\mathrm{\Delta }$, ${\pi }_1$ and $M$ as a functions of $\mu$ at $\nu =0$, ${\nu }_5=0.2\mathrm{GeV}$ and ${\mu }_5=0.5\mathrm{GeV}$. The notations are the same as in Fig. 1.

Figure 3

$(\mu ,\nu )$-phase diagrams at: (a) ${\mu }_5={\nu }_5=0\mathrm{GeV}$. (b) at ${\mu }_5=0\mathrm{GeV}$, ${\nu }_5=0.2\mathrm{GeV}$. (c) at ${\mu }_5=0\mathrm{GeV}$, ${\nu }_5=0.4\mathrm{GeV}$. The notations are the same as in Fig. 1.

Figure 4

Dually conjugated phase portraits: (a) $(\nu ,\mu )$-phase diagram at ${\mu }_5=0.4\mathrm{GeV}$, ${\nu }_5=0\mathrm{GeV}$. (b) $(\nu ,{\nu }_5)$-phase diagram at ${\mu }_5=0.4\mathrm{GeV}$, $\mu =0\mathrm{GeV}$. It is a ${\mathcal{D}}_2$ mapping of Fig. 4. (c) $({\nu }_5,\mu )$-phase diagram at ${\mu }_5=0.4\mathrm{GeV}$, $\nu =0\mathrm{GeV}$. It is a ${\mathcal{D}}_3$ mapping of the diagram Fig. 4. The notations are from Fig. 1.

Figure 5

Dually conjugated phase portraits: (a) $(\nu ,{\mu }_5)$-phase diagram at $\mu =0.1\mathrm{GeV}$, ${\nu }_5=0\mathrm{GeV}$. (b) $(\mu ,{\mu }_5)$-phase diagram at ${\nu }_5=0$, $\nu =0.1\mathrm{GeV}$. It is a ${\mathcal{D}}_1$ mapping of Fig. 5. (c) $({\nu }_5,{\mu }_5)$-phase diagram at $\mu =0.1\mathrm{GeV}$, $\nu =0\mathrm{GeV}$. It is a ${\mathcal{D}}_3$ mapping of Fig. 5. The notations are from Fig. 1.

Figure 6

$(\mu ,\nu )$-phase diagrams: (a) at ${\nu }_5=0.25\mathrm{GeV}$, ${\mu }_5=0.5\mathrm{GeV}$, (b) at ${\nu }_5=0.4\mathrm{GeV}$, ${\mu }_5=-0.3\mathrm{GeV}$, (c) at ${\nu }_5=0.2\mathrm{GeV}$, ${\mu }_5=0.5\mathrm{GeV}$. The notations are from Fig. 1.