The running mass $m_s$ at low scale from the heavy-light meson decay constants

It is shown that a 25(20)% difference between the decay constants $f_{D_s}(f_{B_s})$ and $f_D(f_B)$ occurs due to large differences in the pole masses of the $s$ and $d(u)$ quarks. The values $\eta_D=f_{D_s}/f_D\approx1.23(15)$, recently observed in the CLEO experiment, and $\eta_B=f_{B_s}/f_B\approx1.20$, obtained in unquenched lattice QCD, can be reached only if in the relativistic Hamiltonian the running mass $m_s$ at low scale is $m_s(\sim0.5$ GeV)$=170-200$ MeV. Our results follow from the analytical expression for the pseudoscalar decay constant $f_{\rm P}$ based on the path-integral representation of the meson Green's function.