Pion mass dependence of the $K_{l3}$ semileptonic scalar form factor within finite volume

TL;DR

This paper uses chiral perturbation theory to analyze how the pion mass affects the finite volume corrections to the scalar form factor in kaon semileptonic decay at a specific momentum transfer, aiding lattice QCD extrapolations.

Contribution

It provides a detailed calculation of finite volume effects on the scalar form factor at a key momentum transfer, highlighting the importance of pion mass choices in lattice simulations.

Findings

Finite volume corrections are larger than previously estimated.

Finite size effects depend significantly on pion mass used in simulations.

Results can help improve lattice QCD extrapolations for kaon decays.

Abstract

We calculate the scalar semileptonic kaon decay in finite volume at the momentum transfer $t_{m} = (m_{K} - m_{\pi})^2$, using chiral perturbation theory. At first we obtain the hadronic matrix element to be calculated in finite volume. We then evaluate the finite size effects for two volumes with $L = 1.83 fm$ and $L= 2.73 fm$ and find that the difference between the finite volume corrections of the two volumes are larger than the difference as quoted in \cite{Boyle2007a}. It appears then that the pion masses used for the scalar form factor in ChPT are large which result in large finite volume corrections. If appropriate values for pion mass are used, we believe that the finite size effects estimated in this paper can be useful for Lattice data to extrapolate at large lattice size.