muon $g-2$ anomaly and $\mu$-$\tau$-philic Higgs doublet with a light CP-even component

We examine the possibilities of accommodating the muon $g-2$ anomaly reported by Fermilab in the 2HDM with a discrete $Z_4$ symmetry in which an inert Higgs doublet field ($H,~A,~H^\pm$) has the lepton flavor violation $\mu$-$\tau$ interactions. We study the case of light $H$ (5 GeV $<m_H<$ 115 GeV) and assume the Yukawa matrices to be real and symmetrical. Considering relevant theoretical and experimental constraints, especially for the multi-lepton searches at the LHC, we find the muon $g-2$ anomaly can be explained within $2\sigma$ range in the region of 5 GeV $<m_H<20$ GeV, 130 GeV $< m_A~(m_{H^\pm})<$ 610 GeV, and 0.005 $<\rho<$ 0.014. Meanwhile, the $\chi^2_\tau$ fitting the data of lepton flavour universality in the $\tau$ decays approaches to the SM prediction.