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 <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>m</a:mi></a:math> the corresponding susceptibilities and all their derivatives with respect to <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:msup><c:mi>m</c:mi><c:mn>2</c:mn></c:msup></c:math> must be finite. I then use the resulting spectral constraints to show that effective breaking of the anomalous <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi mathvariant="normal">U</e:mi><e:mo stretchy="false">(</e:mo><e:mn>1</e:mn><e:msub><e:mo stretchy="false">)</e:mo><e:mi>A</e:mi></e:msub></e:math> symmetry is allowed in the <j:math xmlns:j="http://www.w3.org/1998/Math/MathML" display="inline"><j:mi>SU</j:mi><j:mo stretchy="false">(</j:mo><j:mn>2</j:mn><j:msub><j:mo stretchy="false">)</j:mo><j:mi>A</j:mi></j:msub></j:math> symmetric phase in the chiral limit, and leads to distinctive spectral features: (i) the spectral density must develop a singular <n:math xmlns:n="http://www.w3.org/1998/Math/MathML" display="inline"><n:mi>O</n:mi><n:mo stretchy="false">(</n:mo><n:msup><n:mi>m</n:mi><n:mn>4</n:mn></n:msup><n:mo stretchy="false">)</n:mo><n:mo>/</n:mo><n:mi>λ</n:mi></n:math> peak as <r:math xmlns:r="http://www.w3.org/1998/Math/MathML" display="inline"><r:mi>m</r:mi><r:mo stretchy="false">→</r:mo><r:mn>0</r:mn></r:math>, (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 <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"><u:mi>m</u:mi></u:math>. Published by the American Physical Society 2024