An Analytic Analysis of the Pion Decay Constant in Three-Flavoured Chiral Perturbation Theory

Understand

A representation of the two-loop contribution to the pion decay constant in $SU(3)$ chiral perturbation theory is presented.

• The result is analytic upto the contribution of the three (different) mass sunset integrals, for which an expansion in their external momentum has been taken.
• We also give an analytic expression for the two-loop contribution to the pion mass based on a renormalized representation and in terms of the physical eta mass.
• We find an expansion of $F_{\pi}$ and $M_{\pi}^2$ in the strange quark mass in the isospin limit, and perform the matching of the chiral SU(2) and SU(3) low energy constants.