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

TL;DR

This paper uses combined continuum and lattice methods to improve the determination of the CKM matrix element V_us from tau decay data, addressing previous systematic issues and achieving more stable, accurate results.

Contribution

It introduces a new implementation of the flavor-breaking sum rule approach that fits both V_us and higher-dimensional condensates to data, reducing unphysical dependencies.

Findings

New implementation yields V_us with no unphysical s_0 or w dependence.

Results are approximately 0.0020 higher than conventional methods.

V_us results agree with K_{l3} measurements and unitarity expectations.

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 $\tau$ 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 $\vert V_{us}\vert$, 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 $\vert V_{us}\vert$ 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 $K\pi$ branching fraction, the resulting $\vert V_{us}\vert$ is in excellent agreement with that obtained from $K_{\ell 3}$, and compatible within errors with expectations from three-family unitarity.