Mesons and diquarks in neutral color superconducting quark matter with $\beta$ equilibrium

The spectrum of meson and diquark excitations in cold color superconducting (2SC) quark matter is investigated under local color and electric neutrality constraints with $\beta$ equilibrium. A two-flavored Nambu-Jona-Lasinio–type model including baryon ${\mu }_B$, color ${\mu }_8$, and electric ${\mu }_Q$ chemical potentials is used. The contribution from free electrons to the free energy is added to take into account the $\beta$ equilibrium. The sensitivity of the model to the tuning of the interaction constants in the diquark ($H$) and quark-antiquark ($G$) channels is examined for two different parametrization schemes by choosing the ratio $H/G$ to be $3/4$ and 1, respectively. The gapless neutral color superconductivity is realized at $H=3G/4$, and the gapped neutral color superconductivity at $H=G$. It is shown that color and electrical neutrality together with $\beta$ equilibrium lead to a strong mass splitting within the pion isotriplet in the 2SC phase (both gapped and gapless), in contrast with non-neutral matter. The $\pi$- and $\sigma$-meson masses are evaluated to be $\sim 300\mathrm{MeV}$. It is also shown that the properties of the physical $SU(2{)}_c$-singlet diquark excitation in the 2SC ground state vary for different parametrization schemes. Thus, for $H=3G/4$ one finds a heavy resonance with mass $\sim 1100\mathrm{MeV}$ in the non-neutral (gapped) case, whereas, if neutrality is imposed, a stable diquark with mass $\sim |{\mu }_Q|\sim 200\mathrm{MeV}$ appears in the gapless 2SC environment. For a stronger attraction in the diquark channel ($H=G$), there is again a resonance (with the mass $\sim 300\mathrm{MeV}$) in the neutral gapped 2SC phase. Hence, the existence of the stable massive $SU(2{)}_c$-singlet diquark excitation is a new peculiarity of the gapless 2SC. In addition, the behavior of the diquark mass in vacuum, i.e., at ${\mu }_B=0$, as a function of $H$ has been investigated.

Figure 1

The behavior of $M$ vs ${\mu }_B$ in neutral matter.

Figure 2

The behavior of $\Delta$ vs ${\mu }_B$ in neutral matter.

Figure 3

The behavior of ${\mu }_8$ vs ${\mu }_B$ in neutral matter.

Figure 4

The behavior of ${\mu }_Q$ vs ${\mu }_B$ in neutral matter.

Figure 5

The behavior of meson masses vs ${\mu }_B$ in the gapless ($H=3G/4$) color superconducting neutral matter (${\mu }_B>{\mu }_B^c=1.08\mathrm{GeV}$).

Figure 6

The behavior of meson masses vs ${\mu }_B$ in the gapped ($H=G$) color superconducting neutral matter (${\mu }_B>{\mu }_B^c=1.04\mathrm{GeV}$).

Figure 7

The sketch of the $\mathrm{SU}(2{)}_c$-singlet diquark resonance in the gapped ($H=G$) 2SC neutral matter (${\mu }_B>{\mu }_B^c=1.04\mathrm{GeV}$). The solid line is for its mass $M_D$ vs ${\mu }_B$, whereas the width of the strip between dashed lines $M_D\pm \Gamma /2$ is its width $\Gamma$ vs ${\mu }_B$.

Figure 8

The behavior of the stable $\mathrm{SU}(2{)}_c$-singlet diquark mass $M_D$ vs ${\mu }_B$ in the gapless ($H=3G/4$) color superconducting neutral matter (${\mu }_B>{\mu }_B^c=1.08\mathrm{GeV}$).

Figure 9

The behavior of the stable diquark mass $M_D^o$ vs $\eta \equiv H/G$ in the vacuum (at ${\mu }_B=0$). Here ${\eta }^{*}=H^{*}/G\approx 0.698$, ${\eta }^{**}=H^{**}/G\approx 1.525$, $M\approx 0.350\mathrm{GeV}$. At $\eta <{\eta }^{*}$ there are no stable diquarks in the normal phase, and these particles are resonances. At $\eta >{\eta }^{**}$ an $\mathrm{SU}(3{)}_c$ symmetric ground state of the normal phase (including the vacuum) is unstable in favor of color superconducting phases.