Momentum dependence of mixing in the pion vector form factor and its effect on *

Abstract

The inclusion of the $\rho-\omega$ mixing effect is essential for a precise
description of the pion electromagnetic form factor in the $e^+e^- \rightarrow
\pi^+\pi^-$ process, which quantifies the two-pion contribution to the anomalous magnetic moment of the
muon $a_\mu$. In this paper, we aim to analyse the impact of the momentum dependence of the $\rho-\omega$ mixing within the framework of resonance chiral theory. The momentum dependence of the $\rho-\omega$ mixing
is incorporated by considering loop contributions at the next-to-leading order in the $1/N_C$ expansion. We revisit the work {[}Y. H. Chen, D. L. Yao, and H. Q. Zheng,
Commun. Theor. Phys. 69 (2018) 1{]}, considering the contribution arising from the kaon mass splitting in the kaon loops and the latest experimental data.
We perform two kinds of fits (with momentum-independent or momentum-dependent $\rho-\omega$ mixing amplitude) to describe the $e^+e^-\rightarrow \pi^+\pi^-$
and $\tau\rightarrow \nu_{\tau}2\pi$ data within the energy region of 600$\sim$900 MeV and the decay width of $\omega \rightarrow \pi^+\pi^-$, and compare their results.
Our findings indicate that both the momentum-independent and momentum-dependent $\rho-\omega$ mixing schemes provide appropriate descriptions of the data. However, the momentum-dependent scheme exhibits greater self-consistency, considering the reasonable imaginary part of the mixing matrix element $\Pi_{\rho\omega}$ obtained.
Regarding the contribution to the anomalous magnetic moment of the muon, $a_\mu^{\pi\pi}|_{[0.6,0.9]\text{GeV}}$, the results obtained from the fits considering the momentum-dependent $\rho-\omega$ mixing amplitude agree well with those obtained without incorporating the momentum dependence of the $\rho-\omega$ mixing, within the margin of errors.
Furthermore, based on the fitted values of the relevant parameters, we observe that the decay width of $\omega \rightarrow \pi^+\pi^-$ is predominantly influenced by the $\rho-\omega$ mixing effect. Additionally, we find that including the direct $\omega_I\rightarrow \pi^+\pi^-$ coupling is crucial in the analysis for extracting the mass of the $\omega$ meson from the $e^+e^- \rightarrow
\pi^+\pi^-$ process. Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Article funded by SCOAP3 and published under licence by Chinese Physical Society and the Institute of High Energy Physics of the Chinese Academy of Science and the Institute of Modern Physics of the Chinese Academy of Sciences and IOP Publishing Ltd