Pion masses in 2-flavor QCD with η condensation

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We investigate the 2-flavor QCD with non-degenerate quark masses at low-energy, using the chiral perturbation theory including the η meson and anomaly effects. For the fixed md=0, the neutral pion becomes massless at two values of mᵤ, 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 mᵤ=md but θ=π, equivalently mᵤ= - md with θ= 0 by the chiral rotation and show that the CP symmetry is spontaneously broken by the η condensation at small m. Around m=0, three pions become Nambu-Goldstone modes, showing non-standard behavior that mπ² = O(m²), which, however, is consistent with the chiral Ward-Takahashi identities.