Pion masses in 2-flavor QCD with $\eta$ condensation

Abstract

We investigate the 2-flavor QCD with non-degenerate quark masses at low-energy, using the chiral perturbation theory including the $\eta$ meson and anomaly effects. For the fixed $m_d\not=0$, the neutral pion becomes massless at two values of $m_u$, between which a spontaneously CP broken phase appears with the neutral pion condensation. We then show that the topological susceptibility diverges at these two critical points. We also consider the case of $m_u=m_d$ but $\theta=\pi$, equivalently $m_u= - m_d$ with $\theta = 0$ by the chiral rotation and show that the CP symmetry is spontaneously broken by the $\eta$ condensation at small $m$. Around $m=0$, three pions become Nambu-Goldstone modes, showing non-standard behavior that $m_\pi^2 = O(m^2)$, which, however, is consistent with the chiral Ward-Takahashi identities.}