Constraints on the Dirac spectrum from chiral symmetry restoration

I derive constraints on the Dirac spectrum in the chirally symmetric phase of a gauge theory with two massless fermion flavors. Using only general properties of correlation functions of scalar and pseudoscalar bilinears, I prove that in the chiral limit of vanishing fermion mass $m$ the corresponding susceptibilities and all their derivatives with respect to $m^2$ must be finite. I then use the resulting spectral constraints to show that effective breaking of the anomalous $\mathrm{U}(1)_A$ symmetry is allowed in the $\mathrm{SU}(2)_A$ symmetric phase in the chiral limit, and leads to distinctive spectral features: (i) the spectral density must develop a singular $O(m^4)/\lambda$ peak as $m\to 0$, (ii) the two-point eigenvalue correlator of near-zero modes must be singular, and (iii) near-zero modes cannot be localized. Moreover, in the symmetric phase the topological charge distribution must be indistinguishable from that of an ideal gas of instantons and anti-instantons of vanishing density, to leading order in $m$.