the H $^{±}$-mediated transition matrix element combined with SM contribution for ¯ P → ℓ ¯ ν$_{ℓ}$ is obtained as with We learn that the H $^{±}$-mediated contribution is only associated with the factor r ℓ P and it depends on the species of lepton due to the appearance of η$_{ℓ}$ . Since η$_{U}$$_{i}$ , η$_{D}$$_{j}$ and η$_{ℓ}$ are all free parameters, in order to make the results be more predictive, we can adopt a simple scenario. As mentioned earlier, η$_{U}$$_{i}$ and η$_{D}$$_{i}$$_{,ℓ}$ play the role of cot β and tan β in the type-II THDM, respectively. If the new H $^{±}$-mediated effects would like to satisfy the constraints of current data such as b → sγ , it is plausible to set | η$_{U}$$_{i}$ | ≪ | η$_{D}$$_{i}$ | ≈ | η$_{ℓ}$ | [25]. As a consequence, r ℓ P could be simplified by Intriguingly, in this plain scenario we see that the dependence of lepton flavor in r ℓ P can be ascribed to the phase factor φ ℓ $_{D$_{j}$}$. Since φ ℓ D$_{j}$ are the new physical phases, in general, they cannot be rotated away. If we enforce η$_{D}$$_{i}$ = tan β , we see that the magnitude of charged Higgs effects is the same as that in type-II THDM. In other words, apart from the new phase factor φ ℓ $_{D$_{j}$}$, we do not introduce a new enhanced factor. In order to display the new physics effects numerically, we investigate the influence of charged Higgs on R$_{K}$ for K$_{ℓ}$$_{2}$ , on f$_{D}$$_{s}$r ℓ D$_{s}$ for D$_{s}$ → ℓ $^{+}$ν$_{ℓ}$ decays and on BR for B → ℓ $^{+}$ν$_{ℓ}$ , respectively. Using Eqs. (9) and (16), the ratio of Γ( K ± e $_{2}$) to Γ( K ± µ $_{2}$) can be expressed by with Δ c µe s = cos( φ µ $_{s}$) − cos( φ e $_{s}$), where because of the second term in the brackets being much smaller than unity, we have neglected the terms whose the order is higher than m 2 $_{K}$η 2 $_{s}$/m 2 H ± . The resulted numerical values as a function of η$_{s}$/m$_{H}$ ± and Δ c µe s are presented in Fig. 2. The values in the figure denote the ratio R Exp K /R SM K . We see clearly that due to the lepton flavor dependent phases, H $^{±}$-mediated contributions could modify the SM prediction and be still consistent with current data. M SM + H ± ¯ P → ℓ ¯ ν$_{ℓ}$ = − i G$_{F}$ √$_{2}$ V$_{ij}$m$_{ℓ}$f$_{P}$ r ℓ $_{P}$( ¯ ℓν$_{ℓ}$ )$_{S}$$_{−}$$_{P}$ (16) r ℓ P = 1 + η ∗ $_{ℓ}$m 2 P m 2 H ± η ∗ $_{U$_{i}$}$m$_{U}$$_{i}$ − η$_{D}$$_{j}$m$_{D}$$_{j}$ m$_{U}$$_{i}$ + m$_{D}$$_{j}$ . (17) r ℓ P ≈ 1 − m 2 P m 2 H ± m$_{D}$$_{j}$ m$_{U}$$_{i}$ + m$_{D}$$_{j}$ | η$_{D}$$_{j}$ | $^{2}$e iφ ℓ D j . (18) R$_{K}$ = R SM K ( 1 + m 2 K m 2 H ± m$_{s}$ m$_{u}$ + m$_{s}$ η 2 $_{s}$Δ c µe s ) (19) 7