A perturbed lepton-specific two-Higgs-doublet model facing experimental hints for physics beyond the Standard Model

The BaBar, Belle, and LHCb collaborations have reported evidence for new physics in $B\to D\tau\nu$ and $B\to D^*\tau\nu$ of approximately $3.8\sigma$. There is also the long lasting discrepancy of about $3\sigma$ in the anomalous magnetic moment of the muon, and the branching ratio for $\tau\to\mu\nu\nu$ is $1.8\sigma$ ($2.4\sigma$) above the Standard Model expectation using the HFAG (PDG) values. Furthermore, CMS found hints for a non-zero decay rate of $h\to\mu\tau$. Interestingly, all these observations can be explained by introducing new scalars. In this article we consider these processes within a lepton-specific two-Higgs doublet model (i.e. of type X) with additional non-standard Yukawa couplings. It is found that one can accommodate $\tau\to\mu\nu\nu$ with modified Higgs--$\tau$ couplings. The anomalous magnetic moment of the muon can be explained if the additional neutral CP-even Higgs $H$ is light (below 100 GeV). Also $R(D)$ and $R(D^*)$ can be easily explained by additional $t$--$c$--Higgs couplings. Combining these $t$--$c$ couplings with a light $H$ the decay rate for $t\to H c$ can be in a testable range for the LHC. Effects in $h\to\mu\tau$ are also possible, but in this case a simultaneous explanation of the anomalous magnetic moment of the muon is difficult due to the unavoidable $\tau\to\mu\gamma$ decay.