KaonB-parameter in mixed action chiral perturbation theory

We calculate the kaon $B$-parameter, $B_K$, in chiral perturbation theory for a partially quenched, mixed-action theory with Ginsparg-Wilson valence quarks and staggered sea quarks. We find that the resulting expression is similar to that in the continuum, and in fact has only two additional unknown parameters. At 1-loop order, taste-symmetry violations in the staggered sea sector only contribute to flavor-disconnected diagrams by generating an $\mathcal{O}(a^2)$ shift to the masses of taste-singlet sea-sea mesons. Lattice discretization errors also give rise to an analytic term which shifts the tree-level value of $B_K$ by an amount of $\mathcal{O}(a^2)$. This term, however, is not strictly due to taste breaking, and is therefore also present in the expression for $B_K$ for pure Ginsparg-Wilson lattice fermions. We also present a numerical study of the mixed $B_K$ expression in order to demonstrate that both discretization errors and finite volume effects are small and under control on the MILC improved staggered lattices.