Enhancement of Chiral Symmetry Breaking from the Pion condensation at finite isospin chemical potential in a holographic QCD model

Abstract

We study the pion condensation at finite isospin chemical potential using a holographic QCD model. By solving the equations of motion for the pion fields together with those for the iso-singlet scalar and iso-triplet vector meson fields, we show that the phase transition from the normal phase to the pion condensation phase is second order with the mean field exponent, and that the critical value of the isospin chemical potential ${\mu }_I$ is equal to the pion mass, consistently with the result obtained by the chiral effective Lagrangian at $O(p^2)$. For higher chemical potential, we find a deviation, which can be understood as a higher order effect in the chiral effective Lagrangian. We investigate the ${\mu }_I$-dependence of the chiral condensate defined by $\tilde{\sigma }\equiv \sqrt{⟨\sigma {⟩}^2+⟨{\pi }^a{⟩}^2}$. We find that $\tilde{\sigma }$ is almost constant in the small ${\mu }_I$ region, while it grows with ${\mu }_I$ in the large ${\mu }_I$ region. This implies that the strength of the chiral symmetry breaking is not changed for small ${\mu }_I$: The isospin chemical potential plays a role to rotate the "vacuum angle" of the chiral circle ${\tan }^{-1}\sqrt{⟨{\pi }^a{⟩}^2/⟨\sigma {⟩}^2}$ with keeping the "radius" $\tilde{\sigma }$ unchanged for small ${\mu }_I$. For large ${\mu }_I$ region, on the other hand, the chiral symmetry breaking is enhanced by the existence of ${\mu }_I$.