Extraction of $|V_{cd}|$ and $|V_{cs}|$ from experimental decay rates using lattice QCD $D \to \pi(K) \ell \nu$ form factors

Abstract

We present a determination of the Cabibbo-Kobayashi-Maskawa matrix elements $|V_{cd}|$ and $|V_{cs}|$ obtained by combining the momentum dependence of the semileptonic vector form factors $f_+^{D \to \pi}(q^2)$ and $f_+^{D \to K}(q^2)$, recently determined from lattice QCD simulations, with the differential rates measured for the semileptonic $D \to \pi \ell \nu$ and $D \to K \ell \nu$ decays. Our analysis is based on the results for the semileptonic form factors produced by the European Twisted Mass Collaboration with $N_f = 2 + 1 + 1$ flavors of dynamical quarks in the whole range of values of the squared 4-momentum transfer accessible in the experiments. The statistical and systematic correlations between the lattice data as well as those present in the experimental data are properly taken into account. With respect to the standard procedure based on the use of only the vector form factor at zero 4-momentum transfer, we obtain more precise and consistent results: $|V_{cd} |= 0.2341 ~ (74)$ and $|V_{cs} |= 0.970 ~ (33)$. The second-row CKM unitarity is fulfilled within the current uncertainties: $|V_{cd}|^2 + |V_{cs}|^2 + |V_{cb}|^2 = 0.996 ~ (64)$. Moreover, using for the first time hadronic inputs determined from first principles, we have calculated the ratio of the semileptonic $D \to \pi(K)$ decay rates into muons and electrons, which represent a test of lepton universality within the SM, obtaining in the isospin-symmetric limit of QCD: ${\cal{R}}_{LU}^{D\pi} = 0.985~(2)$ and ${\cal{R}}_{LU}^{DK} = 0.975~(1)$.