Momentum-dependence of $ρ-ω$ mixing in the pion vector form factor and its effect on $(g-2)_μ$
The inclusion of the $ρ-ω$ mixing effect is essential for a precise description of the pion electromagnetic form factor in the $e^+e^- \rightarrow π^+π^-$ process, which quantifies the two-pion contribution to the anomalous magnetic moment of the muon $a_μ$. In this paper, we analyse the momentum dependence of the $ρ-ω$ 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 $ρ-ω$ mixing amplitude) to describe the $e^+e^-\rightarrow π^+π^-$ and $τ\rightarrow ν_τ2π$ data within the energy region of 600$\sim$900 MeV and the decay width of $ω\rightarrow π^+π^-$, and compare their results. Our findings indicate that both the momentum-independent and momentum-dependent $ρ-ω$ 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 $Π_{ρω}$ obtained. Regarding the contribution to the anomalous magnetic moment of the muon, $a_μ^{ππ}|_{[0.6,0.9]\text{GeV}}$, the results obtained from the fits considering the momentum-dependent $ρ-ω$ mixing amplitude agree well with those obtained without incorporating the momentum dependence of the $ρ-ω$ mixing, within the margin of errors. Furthermore, based on the fitted values of the relevant parameters, we observe that the decay width of $ω\rightarrow π^+π^-$ is predominantly influenced by the $ρ-ω$ mixing effect.