Analytic representation of FK/Fπ in two loop chiral perturbation theory

Abstract

We present an analytic representation of F K =F π as calculated in three-flavor two-loop chiral perturbation theory, which involves expressing three mass scale sunsets in terms of Kampé de Fériet series. We demonstrate how approximations may be made to obtain relatively compact analytic representations. An illustrative set of fits using lattice data is also presented, which shows good agreement with existing fits.

Analytic representations of mK , FK , mηAnanthanarayan1, Bijnens2, Friot3 et al. 2018Phys. Rev. DSelf Cite160201View full textAdd to dashboardCite

“…For the quantities of interest, there are then two mass ratios present and one may wish to provide a double series representation in these mass ratios. In this work we have carried out precisely this exercise, by introducing MB representation for the sunset diagrams at hand (see [23] for details). Whereas for problems with a single MB parameter, a simple approach exists which allows one to carry out the evaluation and summation of residues of poles, closing the contour in the complex plane to the left or to the right, and then using simple ratio tests to figure out the regions of convergence in the single parameter, a more sophisticated analysis is required when two or (especially) more MB parameters appear.…”

“…In this work, we therefore present the full analytic expressions valid for the kaon and eta masses and decay constants in terms of double infinite series involving two mass ratios, thus completing the program first initiated in [20] of evaluating the three mass scale sunset diagrams in SUð3Þ ChPT. 2 Here, we present only the results, and the complete derivation will be given in an upcoming paper [23]. An overview of the method used is given in [2], and detailed descriptions can be found in [24,25].…”

“…For special cases, with up to two distinct mass scales, closed form results are available [13][14][15]. 1 With two mass scales not falling into the threshold or pseudothreshold configurations [2], or with three distinct mass scales, the sunsets cannot be written in terms of elementary functions. In fact, the sunset diagram with three different masses and arbitrary p 2 cannot even be expressed entirely in terms of multiple polylogarithms.…”

“…The lattice expressions given in Sec. VI are suitable for fits with the data given in [27], and an illustrative fit with these expressions is presented in [1].…”

“…In a recent publication, the important ratio F K =F π was evaluated in a scheme that allows for the derivation of compact analytic approximations in two loop chiral perturbation theory (ChPT) [1], based on the Mellin-Barnes (MB) approach detailed in [2]. In a prior work, a different scheme was employed to obtain analytic approximations of m π and F π in SUð3Þ ChPT at two loops [3].…”