Radiative corrections to $\ddot{I}„\to \ddot{I}€\ddot{I}€\hat{I}{½}_{\ddot{I}}„$

Abstract

Hadronic $\ddot{I}„$ 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 $\ddot{I}„\to \ddot{I}€\ddot{I}€\hat{I}{½}_{\ddot{I}}„$ decays to determine the two-pion contribution, $a_{\hat{I}}{¼}^{\text{HVP, LO}}[\ddot{I}€\ddot{I}€,\ddot{I}„]$. 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 $\ddot{I}(770)$ 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 $\ddot{I}{}^{\prime }$, $\ddot{I}{}^{\prime \prime }$ resonances, perform fits to the available data sets for the $\ddot{I}„\to \ddot{I}€\ddot{I}€\hat{I}{½}_{\ddot{I}}„$ 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 $\ddot{I}„$-specific isospin-breaking corrections to $a_{\hat{I}}{¼}^{\text{HVP, LO}}[\ddot{I}€\ddot{I}€,\ddot{I}„]$.

54 pages, 19 figures, results for the long-range radiative correction factor $G_{\text{EM}}(s)$ included as ancillary material; notation improved, comment on imaginary part corrected, numerical results unchanged; journal version