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

Abstract

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 $\cos \gamma$ is the $h$--$H$ mixing angle of $h$ with the exotic $CP$-even scalar $H$. For $\tau \to \mu \gamma$, we define the "BSM-benchmark" by setting the extra top Yukawa coupling ${\rho }_{tt}=1(\cong {\lambda }_t)$ in two-loop diagrams, and $\cos \gamma =0$ to decouple $h$. 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 ${\rho }_{\tau \mu }=0.7{\lambda }_{\tau }$. We define the subleading "$h$-benchmark" by setting ${\rho }_{tt}=0$ 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.

Comments: 7 pages, 3 figures