Determination of $$f_+^K(0)$$ and extraction of $$|V_{cs}|$$|Vcs| from semileptonic $$D$$D decays

By globally analyzing all existing measured branching fractions and partial rates in different four momentum transfer-squared \(q^2\) bins of \(D\rightarrow Ke^+\nu _e\) decays, we obtain the product of the form factor and magnitude of Cabibbo–Kobayashi–Maskawa matrix element \(V_{cs}\) to be \(f_+^K(0)|V_{cs}|=0.717\pm 0.004\). With this product, we determine the \(D\rightarrow K\) semileptonic form factor \(f_+^K(0)=0.737\pm 0.004\pm 0.000\) in conjunction with the value of \(|V_{cs}|\) determined from the standard model global fit. Alternately, with the product together with the input of the form factor \(f_+^K(0)\) calculated in lattice quantum chromodynamics (LQCD) recently, we extract \(|V_{cs}|^{D\rightarrow Ke^+\nu _e}=0.962\pm 0.005\pm 0.014\), where the error is still dominated by the uncertainty of the form factor calculated in LQCD. Combining the \(|V_{cs}|^{D_s^+\rightarrow \ell ^+\nu _\ell }=1.012\pm 0.015\pm 0.009\) extracted from all existing measurements of \(D^+_s\rightarrow \ell ^+\nu _\ell \) decays and \(|V_{cs}|^{D\rightarrow Ke^+\nu _e}=0.962\pm 0.005\pm 0.014\) together, we find the most precisely determined \(|V_{cs}|\) to be \(|V_{cs}|=0.983\pm 0.011\), which improves the accuracy of the PDG’2014 value \(|V_{cs}|^\mathrm{PDG'2014}=0.986\pm 0.016\) by \(45\,\%\).