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Predicting the Aoki Phase using the Chiral Lagrangian

This work is concerned with the phase diagram of Wilson fermions in the mass and coupling constant plane for two-flavor (unquenched) QCD. We show that as the continuum limit is approached, one can study the lattice theory using the continuum chiral Lagrangian, supplemented by additional terms proportional to powers of the lattice spacing. We find two possible phase structures at non-zero lattice spacing: (1) There is an Aoki phase of spontaneously broken flavor and parity, with two massless Goldstone-pions, and a width $Δm_0 \sim a^3$; (2) There is no spontaneous symmetry breaking, and all three pions have equal mass of order $a$. Present numerical simulations suggest that the former option is realized.