Dark leptophilic scalar with the updated muon $g-2$ anomaly

The muon $g-2$ anomaly was strengthened by recent experimental results at Fermilab, which may be signatures of new physics. A scenario of leptophilic scalar $\phi_L$ accounting for the muon $g-2$ anomaly is investigated in this paper. Though a light $\phi_L$ mainly decaying into standard model (SM) particles has been excluded by experiments, a dark leptophilic scalar $\phi_L$ predominantly decaying into invisible fermionic dark matter (DM) $\chi \bar{\chi}$ is still allowed. Considering the decay mode $\phi_L \to \chi \bar{\chi}$ opened (here $m_{\phi_L} = 3 m_\chi$), the coupling preferred by the muon $g-2$ and the $\phi_L-\chi$ coupling are derived. The light/heavy $\phi_L$ (roughly 1 GeV as a benchmark value) can be tested by future experiments via DM/SM decay modes, and $\phi_L$'s contribution to the anomalous magnetic moment of tau lepton could be investigated at lepton colliders. The search of $\chi$ via $\chi-$electron scattering in DM direct detection is not sensitive due to a tiny $\phi_L-$electron coupling, especially for $m_\chi \gtrsim m_\mu$.