On the consistency of the Aoki-phase

Lattice QCD with two flavors of Wilson fermions can exhibit spontaneous breaking of flavor and parity, with the resulting "Aoki phase" characterized by the non-zero expectation value $<\bar\psi \gamma_5 \tau_3 \psi>\ne0$. This phenomenon can be understood using the chiral effective theory appropriate to the Symanzik effective action. Within this standard analysis, the flavor-singlet pseudoscalar expectation value vanishes: $<i \bar\psi \gamma_5 \psi>=0$. A recent reanalysis has questioned this understanding, arguing that either the Aoki-phase is unphysical, or that there are additional phases in which $<i \bar\psi \gamma_5 \psi>\ne0$. The reanalysis uses the properties of probability distribution functions for observables built of fermion fields and expansions in terms of the eigenvalues of the hermitian Wilson-Dirac operator. Here I show that the standard understanding of the Aoki-phase is, in fact, consistent with the approach used in the reanalysis. Furthermore, if one assumes that the standard understanding is correct, one can use the methods of the reanalysis to derive lattice generalizations of the continuum sum rules of Leutwyler and Smilga.