Kaon semi-leptonic form factor at zero momentum transfer in finite volume

Abstract

Using Chiral Perturbation Theory, we obtain the kaon semi-leptonic vector form factor in finite volume at a generic momentum transfer, $q^2$, up to one loop order. At first we confirm the lattice observation that the contribution of the heavy Pseudo-Goldstone boson in the finite volume corrections at zero momentum transfer is unimportant. We then evaluate the form factor at $q^2=0$ numerically and compare our results with the present lattice data. It turns out that our ChPT results are comparable with the lattice data to some extend. The formula for the finite volume corrections obtained for the form factor at momentum transfer $q^2$, provides a tool for lattice data in order to extrapolate at large lattice size.