The Kaon B-parameter in Mixed Action Chiral Perturbation Theory

Abstract

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 one-loop order, taste-symmetry violations in the staggered sea sector only contribute to flavor-disconnected diagrams by generating an 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 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 G-W 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.

Figures (4)

• Figure 1: Tree-level and 1-loop contributions to ${\cal M}_K$. The circle represents a vertex from the LO staggered chiral Lagrangian. Each square represents an insertion of one of the two left-handed currents in ${\cal O}_K^\chi$ and "changes" the quark flavor from $d \leftrightarrow s$.
• Figure 2: Quark flow diagram contributions to $B_K$ at 1-loop. One external meson is a $\overline{K}^0$ and the other is a $K^0$. The two boxes represent an insertion of the $B_K$ operator. Each box "changes" the valence quark flavor from $d \leftrightarrow s$. Diagrams (a)--(c) contribute to Fig. \ref{['fig:BK1Loop']}(d), diagram (d) contributes to Fig. \ref{['fig:BK1Loop']}(e), and diagram (e) contributes to Fig. \ref{['fig:BK1Loop']}(f).
• Figure 3: Percent difference between the 1-loop contributions to $B_K\;$ with and without taste-breaking discretization errors, Eq. (\ref{['eq:percdiff']}), as a function of valence light quark mass. The masses and taste splittings are those of the MILC coarse ensemble ($a=0.125$ fm) with $am_{l}=0.007$ and $am_{s}=0.05$. The vertical line shows the physical light quark mass.
• Figure 4: Percent difference in the 1-loop contributions to $B_K\;$ in finite and infinite volume, Eq. (\ref{['eq:FVratio']}), as a function of valence light quark mass. The dashed curve corresponds to $\eta_{FV}$ with $a^2\Delta_I=0$ (the continuum case), and the solid curve shows $\eta_{FV}$ with $a^2\Delta_I$ set to its value on the coarse ($a=0.125$ fm) MILC lattices. The sea quark masses are $am_{l}=0.007$ and $am_{s}=0.05$, and the spatial extent of the lattice is $L=20$.