Semileptonic kaon decay in staggered chiral perturbation theory

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 staggered chiral perturbation theory expressions that may be employed to analyze staggered simulations of ${f}_{+}({q}^{2})$ with three light flavors. We consider both the case of a mixed action, where the valence and sea sectors have different staggered actions, and the standard case, where these actions are the same. The momentum transfer ${q}^{2}$ of the form factor is allowed to have an arbitrary value. We give results for the generic situation where the $u$, $d$, and $s$ quark masses are all different, ${N}_{f}=1+1+1$, and for the isospin limit, ${N}_{f}=2+1$. The expression we obtain for ${f}_{+}({q}^{2})$ is independent of the mass of the (valence) spectator quark. In the limit of vanishing lattice spacing, our results reduce to the one-loop continuum partially quenched expression for ${f}_{+}({q}^{2})$, which has not previously been reported in the literature for the ${N}_{f}=1+1+1$ case. Our expressions have already been used in staggered lattice analyses of ${f}_{+}(0)$ and should prove useful in future calculations as well.