$L_{10}^r$ From a Combined NNLO Lattice, Continuum Analysis of the Light Quark $V-A$ Correlator

A combination of lattice and continuum data for the light-quark V-A correlator, supplemented by results from a chiral sum-rule analysis of the flavor-breaking flavor $ud$-$us$ V-A correlator difference, is shown to make possible a high-precision NNLO determination of the renormalized NLO chiral low-energy constant $L_{10}^r$. Key to this determination is the ability to simultaneously fix the two combinations of NNLO low-energy constants also entering the analysis. With current versions of the strange hadronic $\tau$ branching fractions required as input to the flavor-breaking V-A sum rule, we find $L_{10}^r(m_\rho ) = -0.00346(29)$. This represents both the best current precision for $L_{10}^r$, and the first NNLO determination having all errors under full control.