Radiative corrections to $\tau\to\pi\pi\nu_\tau$

Hadronic $\tau$ decays present an opportunity to determine the isovector part of the hadronic-vacuum-polarization contribution to the anomalous magnetic moment of the muon in a way complementary to $e^+e^-\to\text{hadrons}$ cross sections. However, the required isospin rotation is only exact in the isospin limit, and corrections need to be under control to draw robust conclusions, most notably for $\tau\to\pi\pi\nu_\tau$ decays to determine the two-pion contribution, $a_\mu^\text{HVP, LO}[\pi\pi,\tau]$. In this work, we present a novel analysis of the required radiative corrections using dispersion relations, thereby extending in a model-independent way the previous analysis in chiral perturbation theory (ChPT) beyond the threshold region. In particular, we include the dominant structure-dependent virtual corrections from pion-pole diagrams, leading to sizable changes in the vicinity of the $\rho(770)$ resonance. Moreover, we work out the matching to ChPT and devise a strategy for a stable numerical evaluation of real-emission contributions near the two-pion threshold, which proves important to capture isospin-breaking corrections enhanced by the threshold singularity. For the numerical analysis, we use a dispersive representation of the pion form factor including the $\rho'$, $\rho''$ resonances, perform fits to the available data sets for the $\tau\to\pi\pi\nu_\tau$ spectral function, and calculate the corresponding radiative correction factor $G_\text{EM}(s)$ in a self-consistent manner. Based on these results, we evaluate the $\tau$-specific isospin-breaking corrections to $a_\mu^\text{HVP, LO}[\pi\pi,\tau]$.