$_{s}$/m$_{H}$ [GeV −1 ]  c $_{µ}$e s 1 1.01 1.02 1.03 1.04 1.05 0.99 0.98 0.97 0.96 0.95 0.2 0.4 0.6 0.8 1 −2 −1 0 1 2 FIG. 2: Contour for Γ( K ± e $_{2}$) / Γ( K ± µ $_{2}$) as a function of η$_{s}$/m$_{H}$ ± and Δ c µe s , where the values in the figure are R Exp K /R SM K . Similarly, the same discussions could be applied to R$_{π}$ = Γ$_{π}$$_{e}$$_{2}$ / Γ$_{π}$$_{µ}$$_{2}$, where the current world average is R$_{π}$ = (1 . 230 ± 0 . 004) × 10 − 4 [14] while the SM prediction is R SM π = (1 . 2352 ± 0 . 0001) × 10 − 4 [19]. By the results, it seems that R$_{π}$ could give a more strict constraint on free parameters. For clarifying this point, we use the result of Eq. (18) and choose some typical values of parameters as the illustration. With m$_{u}$ ∼ m$_{d}$ , m$_{π}$ = 0 . 14 GeV, m$_{H}$ ± = 200 GeV and η$_{d}$ = 50, Eq. (18) could be estimated to be As a result, the charged Higgs effect on R$_{π}$ is of order of 10 − $^{3}$, which is smaller than that on R$_{K}$ by a factor m 2 $_{K}$/m 2 π ≈ 12. Hence, R$_{K}$ is more sensitive to the charged Higgs effects. Although P → ℓ $^{+}$νγ will contribute to the measurement of P → ℓ $^{+}$ν and contaminate the extraction of P -meson decay constant, however, it has been studied that the radiative corrections to D + s → µ $^{+}$( τ $^{+}$) ν are around (below) 1% [2, 26]. Consequently, if we regard the corresponding CKM matrix element as a certain input, the decay constant of charmed meson could be taken as the physical quantity to test the SM. For D + s,d → ℓ $^{+}$ν decays, since the H $^{±}$-mediated effects are proportional to the masses of down type quarks, from Eq. (16) one can understand that Lattice calculations and data have consistent results in f$_{D}$$_{d}$ ; however due to r ℓ D$_{s}$ being not negligible, a sizable difference in f$_{D}$$_{s}$ between Lattice [9, 10] and data can occur. Hence, for displaying the H ± effects, the relationship in decay constant of D$_{s}$ r ℓ π ≈ 1 − 1 . 2 × 10 − $^{3}$e iφ ℓ d . (20) 8