Chiral Sum Rules to Second Order in Quark Mass

A new calculation of the isospin and hypercharge axial-vector current propagators [ ${\Delta }_{\mathrm{A}33}^{\mu \nu }\left(q^2\right)$ and ${\Delta }_{\mathrm{A}88}^{\mu \nu }\left(q^2\right)$] to two loops in $\mathrm{SU}\left(3\right)\times \mathrm{SU}\left(3\right)$ chiral perturbation theory is used to derive chiral spectral function sum rules valid to second order in the light quark masses. Explicit forms are given for the three-pion isospin axial-vector spectral functions at low energy, and application of the sum rules to the determination of counterterms of the chiral Lagrangian is discussed.