Isospin breaking in $\rho\pi\pi$ couplings: $\rho^{+,0}$ width difference and muon $g-2$

Abstract

The dominant contribution to the hadronic vaccum polarization (HVP) of the muon $g-2$ can be calculated using the hadronic spectrum in $\tau^- \to \pi^-\pi^0\nu_{\tau}$ decay, provided isospin breaking (IB) corrections are properly accounted for. Among these corrections, IB in the $\rho\pi\pi$ coupling has been pointed out as a missing ingredient. In this work, we calculate this IB effect within the impulse approximation using a symmetry-preserving framework based on Dyson-Schwinger and Bethe-Salpeter equations, incorporating fully dressed quark propagators and meson Bethe-Salpeter amplitudes. We then estimate its effect in the HVP contribution to the muon $g-2$. As a natural extension, we also compute the IB in the ratio of $\phi K^+K^-/\phi K^0\bar{K^0}$ couplings and compare with existing data. We obtain a coupling ratio $g_{\rho^0\pi^+\pi^-}/g_{\rho^+\pi^+\pi^0} = 1.0024(17)(1)$, which yields an isospin-breaking correction of $\Delta a_\mu^{\mathrm{HVP,LO}}[\rho\pi\pi] = -2.32(1.64)\times 10^{-10}$ to the two-pion contribution to the muon anomalous magnetic moment extracted from $\tau$ decay data.