Momentum-dependence of $\rho -\omega$ mixing in the pion vector form factor and its effect on $(g-2{)}_{\mu }$

Abstract

The inclusion of the $\rho -\omega$ mixing effect is essential for a precise description of the pion electromagnetic form factor in the $e^{+}{e}^{-}\to {\pi }^{+}{\pi }^{-}$ process, which quantifies the two-pion contribution to the anomalous magnetic moment of the muon $a_{\mu }$. In this paper, we analyse the momentum dependence of the $\rho -\omega$ mixing by considering loop contributions at the next-to-leading order in the $1/N_C$ expansion within the framework of resonance chiral theory. 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 $e^{+}{e}^{-}\to {\pi }^{+}{\pi }^{-}$ and $\tau \to {\nu }_{\tau }2\pi$ data within the energy region of 600$\sim$900 MeV and the decay width of $\omega \to {\pi }^{+}{\pi }^{-}$, 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 $\omega \to {\pi }^{+}{\pi }^{-}$ is predominantly influenced by the $\rho -\omega$ mixing effect.

Comments: 24 pages, 4 figures