$L_{\mu }-L_{\tau }$ 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 nonzero 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 the 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 vectorlike 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.

Figure 1

Example diagrams giving rise to $h\to \tau \mu$ (left), $\tau \to \mu \gamma$ (center) and $(g-2{)}_{\mu }$ (right) at leading order in $v$ and $v_{\varphi }$ after symetry breaking, i.e. after the external $H$ and $\varphi$ fields are replaced by their vacuum expectation values.

Figure 2

Left: Correlations between $h\to \tau \mu$ and $\tau \to \mu \gamma$ for $\sin \alpha =0.2$. The couplings $Y_{EL},Y_{LE},{\lambda }_{\mu L},{\lambda }_{\mu E},{\lambda }_{\tau L},{\lambda }_{\tau E}$ are scanned in the range 0.5–2 and $M_E=M_L$ in the range 1 TeV–3 TeV as well as $0.1v<v_{\varphi }<2v$. The gray region is excluded by the current bound on $\mathrm{BR}(\tau \to \mu \gamma )$. The horizontal dashed line indicates the experimental central value of $\mathrm{BR}(h\to \tau \mu )$. Center: Regions in the ${\lambda }_{\mu E}$ vs ${\lambda }_{\tau E}={\lambda }_{\tau L}$ plane where both the $h\to \tau \mu$ signal and the $(g-2{)}_{\mu }$ discrepancy can be explained simultaneously. In the plot, $M_E=M_L=1\mathrm{TeV}$, $Y_{LE}=Y_{EL}=2$, $\tan \alpha =0.2$, and we consider three scenarios specified in the text. Right: Allowed regions in the $v_{\varphi }/v$ vs $m_{Z^{\prime }}/g^{\prime }$ plane from $h\to \tau \mu$ ($1\sigma$) and $\tau \to 3\mu$ (95% C.L.) for $\alpha =0.1$ (blue) and $\alpha =0.2$ (red). In the gray region, the $b\to s\mu \mu$ data cannot be explained by NP without violating $B_s$ mixing bounds.