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

Understand

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.