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

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 $\phi K^{+}{K}^{-}\mathrm{/}\phi K^0\overline{K^0}$ couplings and compare with existing data. We obtain a coupling ratio $g_{{\rho }^0{\pi }^{+}{\pi }^{-}}\mathrm{/}{g}_{{\rho }^{+}{\pi }^{+}{\pi }^0}=1.0024(17)(1)$, which yields an isospin-breaking correction of $\mathrm{\Delta }{a}_{\mu }^{\mathrm{H}\mathrm{V}\mathrm{P},\mathrm{L}\mathrm{O}}[\rho \pi \pi ]=-2.32(1.64)\times {10}^{-10}$ to the two-pion contribution to the muon anomalous magnetic moment extracted from $\tau$ decay data.