Determination of the NNLO low-energy constant $C_{93}$

Experimental data from hadronic $\tau$ decays allow for a precision determination of the slope of the $I=1$ vacuum polarization at zero momentum. We use this information to provide a value for the next-to-next-to-leading order (NNLO) low-energy constant $C_{93}$ in chiral perturbation theory. The largest systematic error in this determination results from the neglect of terms beyond NNLO in the effective chiral Lagrangian, whose presence in the data will, in general, make the effective $C_{93}$ determined in an NNLO analysis mass dependent. We estimate the size of this effect by using strange hadronic $\tau$-decay data to perform an alternate $C_{93}$ determination based on the slope of the strange vector polarization at zero momentum, which differs from that of the $I=1$ vector channel only through $SU(3)$ flavor-breaking effects. We also comment on the impact of such higher order effects on ChPT-based estimates for the hadronic vacuum polarization contribution to the muon anomalous magnetic moment.