Improved $V_{cs}$ determination using precise lattice QCD form factors for $D\to K\ell \nu$

Abstract

We provide a 0.8\%-accurate determination of $V_{cs}$ from combining experimental results for the differential rate of $D\to K$ semileptonic decays with precise form factors that we determine from lattice QCD. This is the first time that $V_{cs}$ has been determined with an accuracy that allows its difference from 1 to be seen. Our lattice QCD calculation uses the Highly Improved Staggered Quark (HISQ) action for all valence quarks on gluon field configurations generated by the MILC collaboration that include the effect of $u$, $d$, $s$ and $c$ HISQ quarks in the sea. We use eight gluon field ensembles with five values of the lattice spacing ranging from 0.15 fm to 0.045 fm and include results with physical $u/d$ quarks for the first time. Our calculated form factors cover the full $q^2$ range of the physical decay process and enable a Standard Model test of the shape of the differential decay rate as well as the determination of $V_{cs}$ from a correlated weighted average over $q^2$ bins. We obtain $|V_{cs}|=0.9663(53{)}_{\text{latt}}(39{)}_{\text{exp}}(19{)}_{{\eta }_{EW}}(40{)}_{\text{EM}}$, where the uncertainties come from lattice QCD, experiment, short-distance electroweak and electromagnetic corrections, respectively. This last uncertainty, neglected for $D\to K\ell \nu$ hitherto, now needs attention if the uncertainty on $V_{cs}$ is to be reduced further. We also determine $V_{cs}$ values in good agreement using the measured total branching fraction and the rates extrapolated to $q^2=0$. Our form factors enable tests of lepton flavour universality violation. We find the ratio of branching fractions for $D^0\to K^{-}$ with $\mu$ and $e$ in the final state to be $R_{\mu /e}=0.9779(2{)}_{\text{latt}}(50{)}_{\mathrm{EM}}$ in the Standard Model, with the uncertainty dominated by that from electromagnetic corrections.

arXiv comment27 pages, 28 figures. Small changes to text to improve readability. Updated to include the BES average for the D to K branching fraction from 2104.08081

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