\begin{document}$\rho-\omega$\end{document} mixing effect is essential for a precise description of the pion electromagnetic form factor in the \begin{document}$e^+e^- \rightarrow\pi^+\pi^-$\end{document} process, which quantifies the two-pion contribution to the anomalous magnetic moment of muon \begin{document}$a_\mu$\end{document}. In this study, we analyze the momentum dependence of \begin{document}$\rho-\omega$\end{document} mixing by considering loop contributions at the next-to-leading order in \begin{document}$1/N_C$\end{document} expansion within the framework of resonance chiral theory. We revisit a previous study [Y. H. Chen, D. L. Yao, and H. Q. Zheng, Commun. Theor. Phys. 69, 1 (2018)] and consider the contribution arising from the kaon mass splitting in the kaon loops and latest experimental data. We perform two types of fits (with momentum-independent or momentum-dependent \begin{document}$\rho-\omega$\end{document} mixing amplitude) to describe \begin{document}$e^+e^-\rightarrow \pi^+\pi^-$\end{document} and \begin{document}$\tau\rightarrow \nu_{\tau}2\pi$\end{document} data within the energy region of 600\begin{document}$-$\end{document}900 MeV and decay width of \begin{document}$\omega \rightarrow \pi^+\pi^-$\end{document}. Furthermore, we compare their results. Our findings indicate that the momentum-independent and momentum-dependent \begin{document}$\rho-\omega$\end{document} 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 \begin{document}$\Pi_{\rho\omega}$\end{document} obtained. Regarding the contribution to the anomalous magnetic moment of the muon, \begin{document}$a_\mu^{\pi\pi}|_{[0.6,0.9]\text{GeV}}$\end{document}, the results obtained from the fits considering the momentum-dependent \begin{document}$\rho-\omega$\end{document} mixing amplitude are in good agreement with those obtained without incorporating the momentum dependence of \begin{document}$\rho-\omega$\end{document} mixing, within the margin of errors. Furthermore, based on the fitted values of the relevant parameters, we observe that the decay width of \begin{document}$\omega \rightarrow \pi^+\pi^-$\end{document}is predominantly influenced by the \begin{document}$\rho-\omega$\end{document} mixing effect."> Momentum dependence of <inline-formula><tex-math id="M1">\begin{document}$ {\boldsymbol{\rho-\omega}} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="//www.macurncorp.com/hepnp/article/app/id/e9dd28bc-7e2b-47fa-9633-ed49d25db895/CPC-2023-0096_M1.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="//www.macurncorp.com/hepnp/article/app/id/e9dd28bc-7e2b-47fa-9633-ed49d25db895/CPC-2023-0096_M1.png"/></alternatives></inline-formula> mixing in the pion vector form factor and its effect on <inline-formula><tex-math id="M2">\begin{document}${\boldsymbol{(g-2)_\mu}} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="//www.macurncorp.com/hepnp/article/app/id/e9dd28bc-7e2b-47fa-9633-ed49d25db895/CPC-2023-0096_M2.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="//www.macurncorp.com/hepnp/article/app/id/e9dd28bc-7e2b-47fa-9633-ed49d25db895/CPC-2023-0096_M2.png"/></alternatives></inline-formula> -
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