Beyond scalar QED radiative corrections: the $ρ^{\pm}-ρ^0$ width difference, FSR corrections and their impact on $Δa_μ^{\rm HVP, LO}[τ]$

TL;DR

This work advances the treatment of isospin-breaking effects in the rho meson system relevant to the muon g-2 by moving beyond scalar QED to a generalized vector-meson-dominance (GVMD) description of photon-hadron interactions. It computes the photon-inclusive width difference $\Delta\Gamma_{\rho}$ by including the full $\rho\pi\pi$ and $\rho\rho\gamma$ vertices, and evaluates structure-dependent final-state radiation for $e^+e^-\to\pi^+\pi^-$, finding modest but non-negligible corrections that reduce as energy grows above the resonance. The results yield $\Delta\Gamma_{\rho}=(-0.62\pm0.22)$ MeV and indicate a 30% reduction in FSR corrections relative to the pure $s$QED case above $\sqrt{s}\gtrsim1.2$ GeV, with a net impact of $+3.1\times10^{-10}$ on the tau-based determination of $\Delta a_\mu^{\rm HVP, LO}$, giving $\Delta a_\mu^{\rm HVP, LO}[\tau,2\pi]=(-11.9\pm1.7)\times10^{-10}$. These refinements help reduce IB uncertainties in the two-pion channel and enhance the precision of dispersive HVP evaluations for muon g-2.

Abstract

In a previous paper arXiv:0708.3256 [hep-ph] we have calculated the radiative corrections to $ρ\to ππ$ decays, aiming to estimate the width difference between charged and neutral rho mesons. There, we have used the scalar QED approximation and taken the convection terms to keep the loop contributions finite in the case of charged rho meson decays. Here we compute the radiative corrections by considering the electromagnetic structure of charged mesons and we also include the full Lorentz structure of the electromagnetic vertices. We re-evaluate the width difference of $ρ^{\pm}-ρ^0$ vector mesons and calculate the structure-dependent contributions to Final State Radiation terms in the $e^+e^-\to π^+π^-$ cross section. Both effects are important inputs for evaluating the isospin breaking corrections in the dominant hadronic vacuum polarization contributions to the muon $g-2$ when using $τ$ lepton data.

Figures (3)

• Figure 1: Virtual photonic corrections to $\rho^0 \to \pi^+\pi^-$ (upper half) and $\rho^+\to \pi^+\pi^0$ (lower half) decays considering the structure of hadrons in the VMD model. We expect that other model-dependent contributions where virtual mesons can be replaced by $a_1, \omega, \pi$, are very suppressed owing to their small electromagnetic couplings Flores-Baez:2008owd.
• Figure 2: Radiative correction to $\rho^0\to \pi^+\pi^-(\gamma)$ decay as a function of energy $\sqrt{s}$: scalar QED approximation of Ref.Flores-Baez:2007vnd (dotted-line) and the full correction including structure-dependent effects via vector dominance model (VMD) computed in this work (blue solid-line). The FSR$(s)-1$($=\delta_0(s)$) corrections to the $e^+e^-\to \pi^+\pi^-$ cross section computed in Refs. Schwinger:2019zjkDrees:1990te is shown for comparison (red dashed-line).
• Figure 3: Radiative corrections to $\rho^+(s)\to \pi^+\pi^0(\gamma)$ decay computed in this work using VMD (blue solid-line) as a function of energy $\sqrt{s}$. The dashed-line corresponds to the correction in the convection approximation used in Ref. Flores-Baez:2007vnd.