\begin{document}$ a_\gamma $\end{document}) and weak magnetic moment (\begin{document}$ a_W $\end{document}) of charged leptons and quarks are sensitive to quantum effects of new physics heavy resonances. In effective field theory, \begin{document}$ a_\gamma $\end{document} and \begin{document}$ a_W $\end{document} are induced by two independent operators. Therefore, one has to measure both \begin{document}$ a_\gamma $\end{document} and \begin{document}$ a_W $\end{document} to shed light on new physics. The \begin{document}$ a_W $\end{document}'s of the SM fermions are measured at the LEP. In this work, we analyze the contributions from magnetic and weak magnetic moment operators in the processes of \begin{document}$ pp\to H \gamma $\end{document} and \begin{document}$ gg\to H \to \tau^+ \tau^- \gamma $\end{document} at the High-Luminosity Large Hadron Collider. We demonstrate that the two processes can cover most of the parameter space that cannot be probed at the LEP."> Probing magnetic moment operators in <inline-formula><tex-math id="M1-4">\begin{document}${\boldsymbol H \gamma }$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="//www.macurncorp.com/hepnp/article/app/id/bc537c1c-ff02-461c-94d9-b461892124ba/CPC-2021-0272_M1-4.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="//www.macurncorp.com/hepnp/article/app/id/bc537c1c-ff02-461c-94d9-b461892124ba/CPC-2021-0272_M1-4.png"/></alternatives></inline-formula> production and <inline-formula><tex-math id="M2">\begin{document}${\boldsymbol H \to \tau^+ \tau^- \gamma }$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="//www.macurncorp.com/hepnp/article/app/id/bc537c1c-ff02-461c-94d9-b461892124ba/CPC-2021-0272_M2.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="//www.macurncorp.com/hepnp/article/app/id/bc537c1c-ff02-461c-94d9-b461892124ba/CPC-2021-0272_M2.png"/></alternatives></inline-formula> rare decay -
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