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The inclusion of the $\rho-\omega$ mixing effect is essential for a precisedescription of the pion electromagnetic form factor in the $e^+e^- \rightarrow\pi^+\pi^-$ process, which quantifies the two-pion contribution to theanomalous magnetic moment of the muon $a_\mu$. In this paper, we analyse themomentum dependence of the $\rho-\omega$ mixing by considering loopcontributions at the next-to-leading order in the $1/N_C$ expansion within theframework 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 thecontribution arising from the kaon mass splitting in the kaon loops and thelatest experimental data. We perform two kinds of fits (withmomentum-independent or momentum-dependent $\rho-\omega$ mixing amplitude) todescribe 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 decaywidth of $\omega \rightarrow \pi^+\pi^-$, and compare their results. Ourfindings 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 anomalousmagnetic moment of the muon, $a_\mu^{\pi\pi}|_{[0.6,0.9]\text{GeV}}$, theresults obtained from the fits considering the momentum-dependent $\rho-\omega$mixing amplitude agree well with those obtained without incorporating themomentum dependence of the $\rho-\omega$ mixing, within the margin of errors.Furthermore, based on the fitted values of the relevant parameters, we observethat the decay width of $\omega \rightarrow \pi^+\pi^-$ is predominantlyinfluenced by the $\rho-\omega$ mixing effect.

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