Pion Masses in Two-Flavor QCD with $\eta$ Condensation

We investigate some aspects of two-flavor QCD with $m_u\ne m_d$ at low energy, using the leading order chiral perturbation theory including anomaly effects. While nothing special happens at $m_u=0$ for the fixed $m_d\ne 0$, 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 chiral perturbation theory 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.

Figure 1

Phase structure in $m_u\text{−}{m}_d$ plain, where the $CP$ breaking Dashen phase are shaded in gray (blue), while the $CP$ preserving phase with $U_0={\tau }^3$ (lower right) or $U_0=-{\tau }^3$ (upper left) are shaded in light gray (red).

Figure 2

$m_{\pi }^2$ [gray (blue)] and $m_{\eta }^2$ [light gray (red)] in unit of $\frac{\mathrm{\Delta }}{f^2}$ as a function of $m$.