BKin staggered chiral perturbation theory

Abstract

We calculate the kaon $B$ parameter, ${B}_{K}$, to next-to-leading order in staggered chiral perturbation theory. We find expressions for partially quenched QCD with three sea quarks, quenched QCD, and full QCD with ${m}_{u}={m}_{d}\ensuremath{\ne}{m}_{s}$. We extend the usual power counting to include the effects of using perturbative (rather than nonperturbative) matching factors. Taste breaking enters through the $O({a}^{2})$ terms in the effective action, through $\mathcal{O}({a}^{2})$ terms from the discretization of operators, and through the truncation of matching factors. These effects cause mixing with several additional operators, complicating the chiral and continuum extrapolations. In addition to the staggered expressions, we present ${B}_{K}$ at next-to-leading order in continuum $\mathrm{PQ}\ensuremath{\chi}\mathrm{PT}$ for ${N}_{f}=3$ sea quarks with ${m}_{u}={m}_{d}\ensuremath{\ne}{m}_{s}$.