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 quark mass m all the corresponding susceptibilities must be power series in m² with finite coefficients, from which spectral constraints follow. I then use them to show that effective breaking of the anomalous U(1)A symmetry is allowed in the SU(2)A symmetric phase in the chiral limit, and leads to distinctive spectral features: (i) the spectral density must develop a singular m⁴/λ peak as m→ 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 the fermion mass.