(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.