Abstract
A representation of the two-loop contribution to the pion decay constant in SU(3) chiral perturbation theory is presented. The result is analytic up to 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 we perform the matching of the chiral SU(2) and SU(3) low-energy constants. A numerical analysis demonstrates the high accuracy of our representation, and the strong dependence of the pion decay constant upon the values of the low-energy constants, especially in the chiral limit. Finally, we present a simplified representation that is particularly suitable for fitting with available lattice data.
Highlights
There is, a need for an analytic study of the observables in chiral perturbation theory (ChPT) since one would like to have an intuitive sense for the results appearing therein
[5,6,7] are steps in that direction, but, as the results presented there are in the chiral limit, mu = md = 0, the need for more general expressions is left unfulfilled
Kaiser [8] studied the problem of the pion mass in the analytic framework, and was able to employ well known properties of sunset integrals to reduce a large number of expressions to analytic ones
Summary
There is, a need for an analytic study of the observables in ChPT since one would like to have an intuitive sense for the results appearing therein. The analytic studies of SU (3) amplitudes in the strange-quark mass expansion of [5,6,7] are steps in that direction, but, as the results presented there are in the chiral limit, mu = md = 0, the need for more general expressions is left unfulfilled. C (2017) 77:497 the case when the square of the external momentum vanishes [10] This is the case for most of the sunset integrals appearing in the expressions for the mass and decay constants. 7, we discuss several possible ways of expressing the results of this paper, and present a simplified representation that is suitable for performing fittings with available lattice data.
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