Isospin-breaking effects in the two-pion contribution to hadronic vacuum polarization

TL;DR

This paper investigates isospin-breaking effects in the two-pion contribution to hadronic vacuum polarization, focusing on the $ ho$-$ ext{omega}$ mixing and its impact on the muon g-2, considering small phase effects from radiative channels.

Contribution

It introduces a detailed analysis of the phase induced by radiative channels on the $ ho$-$ ext{omega}$ mixing parameter and performs fits to $e^+e^- o ext{pi}^+ ext{pi}^-$ data to assess its effects.

Findings

The phase $ ilde{ ext{epsilon}}$ is estimated as 3.5(1.0)° from radiative channels.

Global fits favor a non-zero phase of about 4.5(1.2)°, close to narrow-resonance estimates.

Systematic differences among data sets affect the $ ho$-$ ext{omega}$ interference analysis.

Abstract

Isospin-breaking (IB) effects in the two-pion contribution to hadronic vacuum polarization (HVP) can be resonantly enhanced, if related to the interference of the $\rho(770)$ and $\omega(782)$ resonances. This particular IB contribution to the pion vector form factor and thus the line shape in $e^+e^-\to \pi^+\pi^-$ can be described by the residue at the $\omega$ pole - the $\rho$-$\omega$ mixing parameter $\epsilon_\omega$. Here, we argue that while in general analyticity requires this parameter to be real, the radiative channels $\pi^0\gamma$, $\pi\pi\gamma$, $\eta\gamma$ can induce a small phase, whose size we estimate as $\delta_\epsilon=3.5(1.0)^\circ$ by using a narrow-width approximation for the intermediate-state vector mesons. We then perform fits to the $e^+e^-\to \pi^+\pi^-$ data base and study the consequences for the two-pion HVP contribution to the anomalous magnetic moment of the muon, its IB part due to $\rho$-$\omega$ mixing, and the mass of the $\omega$ resonance. We find that the global fit does prefer a non-vanishing value of $\delta_\epsilon=4.5(1.2)^\circ$, close to the narrow-resonance expectation, but with a large spread among the data sets, indicating systematic differences in the $\rho$-$\omega$ region.