A resolution of the inclusive flavor-breaking sum rule $\ddot{I}„$ $V_{us}$ puzzle

Abstract

A combination of continuum and lattice methods is used to investigate systematic issues in the finite-energy-sum-rule determination of $V_{us}$ based on flavor-breaking combinations of hadronic $\ddot{I}„$ decay data. Results for $V_{us}$ obtained using assumptions for $D>4$ OPE contributions employed in previous conventional implementations of this approach are shown to display significant unphysical dependences on the choice of sum rule weight, $w$, and upper limit, $s_0$, of the relevant experimental spectral integrals. Continuum and lattice results suggest the necessity of a new implementation of the flavor-breaking sum rule approach, in which not only $|V_{us}|$, but also $D>4$ effective condensates are fit to data. Lattice results also provide a means of quantifying the truncation error for the slowly converging $D=2$ OPE series. The new implementation is shown to produce $|V_{us}|$ results free of unphysical $s_0$- and $w$-dependences and typically $\sim 0.0020$ higher than the (unstable) results found using the conventional implementation. With preliminary new experimental results for the $K\ddot{I}€$ branching fraction, the resulting $|V_{us}|$ is in excellent agreement with that obtained from $K_{\ell 3}$, and compatible within errors with expectations from three-family unitarity.

7 pages, 4 figures. Presented at the 33rd International Symposium on Lattice Field Theory, Kobe, Japan