# Twisted finite-volume corrections to $K_{l3}$ decays with partially-quenched and rooted-staggered quarks

## TL;DR

This paper derives one-loop partially quenched chiral perturbation theory formulas for $K_{l3}$ form factors at finite volume, including staggered quarks and twisted boundary conditions, to improve lattice calculations of $|V_{us}|$.

## Contribution

It provides new finite-volume correction formulas for $K_{l3}$ form factors with staggered and twisted boundary conditions, aiding precise lattice determinations of $f_+(0)$.

## Key findings

- Finite-volume corrections are small at current lattice sizes.
- More form factors than just $f_+$ and $f_-$ are relevant at finite volume.
- Ward identity remains satisfied despite finite-volume effects.

## Abstract

The determination of $|V_{us}|$ from kaon semileptonic decays requires the value of the form factor $f_+(q^2=0)$ which can be calculated precisely on the lattice. We provide the one-loop partially quenched chiral perturbation theory expressions both with and without including the effects of staggered quarks for all form factors at finite volume and with partially twisted boundary conditions for both the vector current and scalar density matrix elements at all $q^2$. We point out that at finite volume there are more form factors than just $f_+$ and $f_-$ for the vector current matrix element but that the Ward identity is fully satisfied. The size of the finite-volume corrections at present lattice sizes is small. This will help improve the lattice determination of $f_+(q^2=0)$ since the finite-volume error is the dominant error source for some calculations. The size of the finite-volume corrections may be estimated on a single lattice ensemble by comparing results for various twist choices.