Running charm quark mass versus $M_D$ and the $D_{(s)}\to \mu\nu$ decay

I study (for the first time) the dependence of M_D and of the leptonic decay constant f_D on the variation of the running charm quark mass m_c(\nu). I conclude that the present data on f_{D_s} from D_s into \mu\nu decay give a weaker constraint than M_D, where the latter leads to the result, m_c(M_c)=(1.08\pm 0.11) GeV, to two-loop accuracy in the {MS}-bar scheme, in good agreement with the value m_c(M_c)=(1.23^{+0.04}_{-0.05}) GeV extracted directly, within the same approximation, from M_{J/\psi}. The agreement of the corresponding perturbative pole mass with the one extracted directly from the data can indicate that the non-perturbative effects to the pole mass are negligible. Inversely, injecting the average value of m_c(M_c) from M_D and M_{J/\psi} into the m_c behaviour of f_D, I obtain f_D\simeq (1.52\pm 0.16)f_\pi, which combined with the sum rule prediction for f_{D_s}/f_D, gives f_{D_s}\simeq (1.75\pm 0.18)f_\pi in good agreement with the data. The extension of the analysis to the case of f_B and f_{D^*} is discussed.