High Precision determination of the pi, K, D and D_s decay constants from lattice QCD

We present a determination of the decay constants of the $D$ and $D_s$ mesons from lattice QCD, each with a total error of less than 2%, a factor of 4--5 better than previous calculations and existing experiment. Our results are: $f_{D_s}$ = 241(3) MeV, $f_D$ = 208(4) MeV and an even more accurate result for the ratio: $f_{D_s}/f_D$ = 1.162(9). As part of this analysis we also determine $f_K$ and $f_{\pi}$. We obtain $f_K/f_{\pi}$ = 1.189(7) and the double ratio $(f_{D_s}/f_D)/(f_K/f_{\pi})$ = 0.977(10). Our value for $f_K/f_{\pi}$ can be combined with experimental leptonic branching fractions to determine $V_{us}$=0.2262(14). We use existing experimental measurements of the leptonic decay rate of the $D_s$ and $D$ to $\mu \nu$ to obtain a direct determination of $V_{cs}/V_{cd}$ of 4.42(4)(41), where the first error is theoretical and the second experimental. We have been able to achieve these results through the use of a highly improved discretisation of QCD for quarks (HISQ) coupled to numerical simulations of gluon fields by the MILC collaboration that include the full effect of sea $u$, $d$ and $s$ quarks. We have results for a range of $u/d$ masses down to $m_s/8$ and three different values of the lattice spacing and perform accurate continuum and chiral extrapolations. We fix the $u/d, s$ and $c$ masses from the $\pi$, $K$, and $\eta_c$ meson masses respectively. A stringent test of our calculation is then our result for the mass of the $D$ and $D_s$ mesons: $m_{D_s}$ = 1.963(5) GeV (experiment = 1.968 GeV), $m_{D^+}$ = 1.869(6) GeV (experiment = 1.869 GeV) and $\Delta m_{D_s}/\Delta m_D$ = 1.249(14) where $\Delta m_{D_x} = 2m_{D_x}-m_{\eta_c}$ (experiment = 1.260(2)).