Semileptonic kaon decay in staggered chiral perturbation theory

The determination of vertical bar V-us vertical bar 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.