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We examine the lepton-specific 2HDM as a solution of muon $g-2$ anomaly undervarious theoretical and experimental constraints, especially the direct searchlimits from the LHC and the requirement of a strong first-order phasetransition in the early universe. We find that the muon g-2 anomaly can beexplained in the region of 32 $<\tan\beta<$ 80, 10 GeV $<m_A<$ 65 GeV, 260 GeV$<m_H<$ 620 GeV and 180 GeV $<m_{H^\pm}<$ 620 GeV after imposing the jointconstraints from the theory, the precision electroweak data, the 125 GeV Higgsdata, the leptonic/semi-hadronic $\tau$ decays, the leptonic $Z$ decays andBr$(B_s \to \mu^+ \mu^-)$. The direct searches from the $h\to AA$ channels canimpose stringent upper limits on Br$(h\to AA)$ and the multi-lepton eventsearches can sizably reduce the allowed region of $m_A$ and $\tan\beta$ (10 GeV$<m_A<$ 44 GeV and 32 $<\tan\beta<$ 60). Finally, we find that the model canproduce a strong first-order phase transition in the region of 14 GeV $<m_A<$25 GeV, 310 GeV $<m_H<$ 355 GeV and 250 GeV $<m_{H^\pm}<$ 295 GeV, allowed bythe explanation of the muon $g-2$ anomaly.

Revisiting lepton-specific 2HDM in light of muon g-2 anomaly