Lattice results for $D/D_s$ leptonic and semileptonic decays

This review article summarizes recent lattice QCD results for $D$ and $D_s$ meson leptonic and semileptonic decays. Knowing the meson decay constants and semileptonic form factors from theory, one can extract CKM elements $V_{cd}$ and $V_{cs}$ from experimental results. At present, the most accurate results for decay constants are from the Fermilab Lattice and MILC Collaborations: $f_D = 212.5 \pm 0.5_{\mathrm{stat}} {}^{+0.6}_{-1.5} |_{\mathrm{syst}}$ MeV and $f_{D_s} = 248.9 \pm 0.2_{\mathrm{stat}} {}^{+0.5}_{-1.6}|_{\mathrm{syst}}$ MeV, giving $V_{cd} = 0.2184 \pm 0.009_{\mathrm{expt}} {}^{+0.0008}_{-0.0016} |_{\mathrm{lattice}}$ and $V_{cs} = 1.017 \pm 0.02_{\mathrm{expt}} {}^{+0.002}_{-0.007} |_{\mathrm{lattice}}$. The shapes of the semileptonic form factors from lattice QCD agree very well with experiment, and the accuracy is currently at the 2-5% level for $D \to \pi\ell\nu$ and 1-2% for $D \to K\ell\nu$. Extracting the CKM elements from the semileptonic decays yields $V_{cd} = 0.225(6)_{\mathrm{expt}}(10)_{\mathrm{lattice}}$ (HPQCD Collaboration) and $V_{cs} = 0.963(5)_{\mathrm{expt}}(14)_{\mathrm{lattice}}$ (HPQCD Collaboration). These lattice calculations also revealed that the semileptonic form factors are insensitive to whether the spectator quark is a light or strange quark.