Paper detail

The Coming Decade of $h \to τμ$ and $τ\to μγ$ Interplay in $τ$ Flavor Violation Search

We explore the interplay between $h(125) \to τμ$ search at the LHC and $τ\to μγ$ at the up and coming Belle~II experiment, in context of the general two Higgs doublet model with extra Yukawa couplings such as $ρ_{τμ}$. The search for $h \to τμ$ constrains $ρ_{τμ} \cosγ$, where $\cosγ$ is the $h$--$H$ mixing angle of $h$ with the exotic $CP$-even scalar $H$. For $τ\to μγ$, we define the "BSM-benchmark" by setting the extra top Yukawa coupling $ρ_{tt} = 1\ (\cong λ_t)$ in two-loop diagrams, and $\cosγ= 0$ 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 $ρ_{τμ} = 0.7λ_τ$. We define the subleading "$h$-benchmark" by setting $ρ_{tt} = 0$ in two-loop diagrams, and take the conservative maximal value of $\cosγ= 0.2$. We show that it falls beyond Belle~II reach, but can interfere with the BSM-benchmark effect, which in principle probes the phase of $ρ_{tt}$. We further show that the one-loop $H$, $A$ effect, proportional to $ρ_{τμ}ρ_{ττ}$ in amplitude, is beyond the sensitivity of Belle~II to probe, even for $ρ_{ττ}$ as large as $3λ_τ$. With the working assumption that $ρ_{32}^f, ρ_{33}^f = {\cal O}(λ_3^f)$ for all charged fermions $f$, we find good discovery potential for both $τ$ lepton flavor violation searches in the coming decade.