The Coming Decade of $h\to \tau \mu$ and $\tau \to \mu \gamma$ Interplay in $\tau$ Flavor Violation Search

We explore the interplay between $h(125)\to \tau \mu$ search at the LHC and $\tau \to \mu \gamma$ at the up and coming Belle~II experiment, in context of the general two Higgs doublet model with extra Yukawa couplings such as ${\rho }_{\tau \mu }$. The search for $h\to \tau \mu$ constrains $\rho_{\tau\mu} \cos\gamma$,where$\cos\gamma$isthe$h$--$H$mixingangleof$h$ with the exotic $CP$-even scalar $H$. For $\tau \to \mu \gamma$, we define the “BSM-benchmark” by setting the extra top Yukawa coupling $\rho_{tt} = 1\ (\cong \lambda_t)$intwo-loopdiagrams,and$\cos\gamma = 0$todecouple$h$. We show that this leading effect due to $H$ and $CP$-odd scalar $A$ can be readily probed by Belle~II, even for the conservative value of $\rho_{\tau\mu} = 0.7\lambda_\tau$\mathrm{.}Wedefinethesubleading\mathrm{"}$h$-benchmark\mathrm{"}bysetting$\rho_{tt} = 0$ in two-loop diagrams, and take the conservative maximal value of $\cos \gamma =0.2$. We show that it falls beyond Belle~II reach, but can interfere with the BSM-benchmark effect, which in principle probes the phase of ${\rho }_{tt}$. We further show that the one-loop $H$, $A$ effect, proportional to ${\rho }_{\tau \mu }{\rho }_{\tau \tau }$ in amplitude, is beyond the sensitivity of Belle~II to probe, even for ${\rho }_{\tau \tau }$ as large as $3{\lambda }_{\tau }$. With the working assumption that ${\rho }_{32}^f,{\rho }_{33}^f=\mathcal{O}({\lambda }_3^f)$ for all charged fermions $f$, we find good discovery potential for both $\tau$ lepton flavor violation searches in the coming decade.