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

Abstract

We investigate some aspects of 2-flavor QCD with $m_u\not= m_d$ at low-energy, using the leading order chiral perturbation theory including anomaly effects. While nothing special happens at $m_u=0$ for the fixed $m_d\not=0$, the neutral pion mass becomes zero at two critical values of $m_u$, between which the neutral pion field condenses, leading to a spontaneously CP broken phase, the so-called Dashen phase. We also show that the "topological susceptibility" in the ChPT diverges at these two critical points. We briefly discuss a possibility that $m_u=0$ can be defined by the vanishing the "topological susceptibility. We finally analyze the case of $m_u=m_d=m$ with $\theta=\pi$, which is equivalent to $m_u= - m_d=-m$ with $\theta = 0$ by the chiral rotation. In this case, the $\eta$ condensation occurs at small $m$, violating the CP symmetry spontaneously. Deep in the $\eta$ condensation phase, three pions become Nambu-Goldstone bosons, but they show unorthodox behavior at small $m$ that $m_\pi^2 = O(m^2)$, which, however, is shown to be consistent with the chiral Ward-Takahashi identities.