We determine the next-to-leading order chiral low-energy constant $L_{10}^r$, and combinations $C_{12}^r\pm C_{61}^r+C_{80}^r$, $C_{13}^r-C_{62}^r+C_{81}^r$, $C_{61}^r$, and $C_{87}^r$, of the next-to-next-to-leading order (NNLO) chiral low-energy constants incorporating recently revised ALEPH results for the nonstrange vector ($V$) and axial-vector ($A$) hadronic $\tau$ decay distributions and recently updated RBC/UKQCD lattice data for the nonstrange $V-A$ two-point function. In the $\overline{\mathrm{MS}}$ scheme, at renormalization scale $\mu =770\mathrm{MeV}$, we find $L_{10}^r=-0.00350(17)$, $C_{12}^r+C_{61}^r+C_{80}^r=0.00237(16){\mathrm{GeV}}^{-2}$, $C_{12}^r-C_{61}^r+C_{80}^r=-0.00056(15){\mathrm{GeV}}^{-2}$, $C_{13}^r-C_{62}^r+C_{81}^r=0.00046(9){\mathrm{GeV}}^{-2}$, $C_{61}^r=0.00146(15){\mathrm{GeV}}^{-2}$, and $C_{87}^r=0.00510(22){\mathrm{GeV}}^{-2}$. With errors here at or below the level expected for contributions of yet higher order in the chiral expansion, the analysis exhausts the possibilities of what can be meaningfully achieved in an NNLO analysis. We also consider the dimension-six and dimension-eight coefficients in the operator product expansion in the $V-A$ channel.

• Received 21 March 2015

DOI:https://doi.org/10.1103/PhysRevD.92.114501