$R(D^{(*)})$ in a general two Higgs doublet model

Motivated by an anomaly in $R(D^{(*)})={\rm BR}(\bar{B}\rightarrow D^{(*)} \tau^-\bar{\nu})/{\rm BR}(\bar{B}\rightarrow D^{(*)} l^-\bar{\nu})$ reported by BaBar, Belle and LHCb, we study $R(D^{(*)})$ in a general two Higgs doublet model (2HDM). Although it has been suggested that it is difficult for the 2HDM to explain the current world average for $R(D^{(*)})$, it would be important to clarify how large deviations from the standard model predictions for $R(D^{(*)})$ are possible in the 2HDM. We investigate possible corrections to $R(D^{(*)})$ in the 2HDM by taking into account various flavor physics constraints (such as $B_c^-\rightarrow \tau^- \bar{\nu}$, $b\rightarrow s\gamma$, $b\rightarrow s l^+l^-$, $\Delta m_{B_{d,s}}$, $B_s\rightarrow \mu^+\mu^-$ and $\tau^+\tau^-$, and $B^-\rightarrow \tau^- \bar{\nu}$), and find that it would be possible (impossible) to accommodate the 1$\sigma$ region suggested by the Belle's result when we adopt a constraint ${\rm BR}(B_c^-\rightarrow \tau^- \bar{\nu})\le30~\%$ (${\rm BR}(B_c^-\rightarrow \tau^- \bar{\nu})\le10~\%$). We also study productions and decays of heavy neutral and charged Higgs bosons at the Large Hadron Collider (LHC) experiment and discuss the constraints and implications at the LHC. We show that in addition to well-studied production modes $bg\rightarrow tH^-$ and $gg\rightarrow H/A$, exotic productions of heavy Higgs bosons such as $cg\rightarrow bH^+,t+H/A$ and $c\bar{b}\rightarrow H^+$ would be significantly large, and the search for their exotic decay modes such as $H/A\rightarrow t\bar{c}+c\bar{t},~\mu^\pm\tau^\mp$ and $H^+\rightarrow c\bar{b}$ as well as $H/A\rightarrow \tau^+\tau^-$ and $H^+\rightarrow \tau^+\nu$ would be important to probe the interesting parameter regions for $R(D^{(*)})$.