Abstract
We provide a 0.8%-accurate determination of ${V}_{cs}$ from combining experimental results for the differential rate of $D\ensuremath{\rightarrow}K$ 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 $|{V}_{cs}|=0.9663(53{)}_{\mathrm{latt}}(39{)}_{\mathrm{exp}}(19{)}_{{\ensuremath{\eta}}_{\mathrm{EW}}}(40{)}_{\mathrm{EM}}$, where the uncertainties come from lattice QCD, experiment, short-distance electroweak, and electromagnetic corrections, respectively. This last uncertainty, neglected for $D\ensuremath{\rightarrow}K\ensuremath{\ell}\ensuremath{\nu}$ 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 flavor universality violation. We find the ratio of branching fractions for ${D}^{0}\ensuremath{\rightarrow}{K}^{\ensuremath{-}}$ with $\ensuremath{\mu}$ and $e$ in the final state to be ${R}_{\ensuremath{\mu}/e}=0.9779(2{)}_{\mathrm{latt}}(50{)}_{\mathrm{EM}}$ in the Standard Model, with the uncertainty dominated by that from electromagnetic corrections.
Highlights
IntroductionThe flavor changing weak interactions between quarks via emission of W bosons can be parametrized in terms of the unitary Cabbibo-Kobayashi-Maskawa (CKM) matrix in the Standard Model, given by [1,2]
We provide a 0.8%-accurate determination of Vcs from combining experimental results for the differential rate of D → K semileptonic decays with precise form factors that we determine from lattice QCD
We describe how the form factor results are extrapolated to the physical continuum limit
Summary
The flavor changing weak interactions between quarks via emission of W bosons can be parametrized in terms of the unitary Cabbibo-Kobayashi-Maskawa (CKM) matrix in the Standard Model, given by [1,2]. VCKM 1⁄4 64 Vcd Vcs Vcb 75: ð1Þ Vtd Vts Vtb. Precise and independent determination of each of the CKM matrix elements from multiple processes is crucial to test the Standard Model stringently. Current accuracy varies from 0.014% for Vud to 6% for Vub with several reviews in [3] discussing different aspects of their determination. For a recent review of the impact of lattice QCD on this endeavor see [4]. We will focus on the determination of Vcs and provide a significant improvement in its accuracy that expands the range of tests we can perform of the CKM matrix
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