We present a lattice QCD calculation of the pseudoscalar decay constants $f_K$, $f_D$ and $f_{D_s}$ performed using the gauge configurations produced by the European Twisted Mass Collaboration with $N_f=2+1+1$ dynamical quarks, which include in the sea, besides two light mass degenerate quarks, also the strange and charm quarks with masses close to their values in the real world. The simulations are based on a unitary setup for the two light mass-degenerate quarks and on a mixed action approach for the strange and charm quarks. We use data simulated at three different values of the lattice spacing in the range $0.06–0.09\mathrm{fm}$ and at pion masses in the range $210–450\mathrm{MeV}$. Our main results are $f_{K^{+}}/f_{{\pi }^{+}}=1.184(16)$, $f_{K^{+}}=154.4(2.0)\mathrm{MeV}$, which incorporate the leading strong isospin breaking correction due to the up and down quark mass difference, and $f_K=155.0(1.9)\mathrm{MeV}$, $f_D=207.4(3.8)\mathrm{MeV}$, $f_{D_s}=247.2(4.1)\mathrm{MeV}$, $f_{D_s}/f_D=1.192(22)$ and $(f_{D_s}/f_D)/(f_K/f_{\pi })=1.003(14)$ obtained in the isospin symmetric limit of QCD. Combined with the experimental measurements of the leptonic decay rates of kaon, pion, $D$ and $D_s$ mesons our results lead to the following determination of the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements: $|V_{us}|=0.2269(29)$, $|V_{cd}|=0.2221(67)$ and $|V_{cs}|=1.014(24)$. Using the latest value of $|V_{ud}|$ from superallowed nuclear $\beta$ decays the unitarity of the first row of the CKM matrix is fulfilled at the per mill level.

• Received 6 December 2014

DOI:https://doi.org/10.1103/PhysRevD.91.054507