Light quark mass dependence of the $D$ and $D_s$ decay constants

We study the light quark mass dependence of the $D$ and $D_s$ meson decay constants, $f_D$ and $f_{D_s}$, using a covariant formulation of chiral perturbation theory ($\chi$PT) at next-to-next-to-leading order (NNLO). Using the HPQCD lattice results for the $D$($D_s$) decay constants as a benchmark we show that covariant $\chi$PT can describe the HPQCD results better than heavy meson $\chi$PT (HM$\chi$PT) at both NLO and NNLO. Within the same framework, taking into account sub-leading ($1/m_Q$, with $m_Q$ the heavy quark mass) corrections to the values of the low-energy constants and employing the lattice QCD results for $g_{BB^*\pi}$, we estimate the ratio of $f_{B_s}/f_B$ to be $1.22^{+0.05}_{-0.04}$, which agrees well with the HPQCD result $1.226(26)$.