A $L_{\mu }-L_{\tau }$ theory of Higgs flavor violation and $(g-2{)}_{\mu }$

Several experiments reported hints for the violation of lepton flavor or lepton flavor universality in processes involving muons. Most prominently, there is the hint for a non-zero rate of the flavor violating Higgs decay $h \to \tau\mu$ at the LHC, as well as the hint for lepton flavor universality violation in rare $B$ meson decays at LHCb. In addition, also the long standing discrepancy in the anomalous magnetic moment of the muon motivates new physics connected to muons. A symmetry which violates lepton flavor universality, is $L_{\mu }-L_{\tau }$: the difference of muon-number and tau-number. We show that adding vector-like fermions to a $L_{\mu }-L_{\tau }$ theory generates naturally an effect in the anomalous magnetic moment of the muon and $h\to \tau \mu$, while effects in other $\tau \to \mu$ transitions are systematically suppressed by symmetry arguments. We find that if $L_{\mu }-L_{\tau }$ is gauged it is possible to also accommodate the discrepant $b\to s\mu \mu$ data while predicting a $\tau \to 3\mu$ and a modified $h\to \mu \mu$ rate within reach of upcoming experiments.