Using recent precise hadronic $\tau$-decay data on the $V-A$ spectral function, and general properties of QCD such as analyticity, the operator product expansion, and chiral perturbation theory, we get accurate values for the QCD chiral order parameters $L_{10}^r(M_{\rho })$ and $C_{87}^r(M_{\rho })$. These two low-energy constants appear at order $p^4$ and $p^6$, respectively, in the chiral perturbation theory expansion of the $V-A$ correlator. At order $p^4$ we obtain $L_{10}^r(M_{\rho })=-(5.22\pm 0.06)\times {10}^{-3}$. Including in the analysis the two-loop (order $p^6$) contributions, we get $L_{10}^r(M_{\rho })=-(4.06\pm 0.39)\times {10}^{-3}$ and $C_{87}^r(M_{\rho })=(4.89\pm 0.19)\times {10}^{-3}{\mathrm{GeV}}^{-2}$. In the SU(2) chiral effective theory, the corresponding low-energy coupling takes the value ${\overline{l}}_5=13.30\pm 0.11$ at order $p^4$, and ${\overline{l}}_5=12.24\pm 0.21$ at order $p^6$.

• Received 14 October 2008

DOI:https://doi.org/10.1103/PhysRevD.78.116012