Improved <math><msub><mi>V</mi><mrow><mi>c</mi><mi>s</mi></mrow></msub></math> determination using precise lattice QCD form factors for <math><mi>D</mi><mo>→</mo><mi>K</mi><mo>ℓ</mo><mi>ν</mi></math>

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. This is the first time that Vcs 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 q2 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 Vcs from a correlated weighted average over q2 bins. We obtain |Vcs|=0.9663(53)latt(39)exp(19)ηEW(40)EM, where the uncertainties come from lattice QCD, experiment, short-distance electroweak, and electromagnetic corrections, respectively. This last uncertainty, neglected for D→Kℓν hitherto, now needs attention if the uncertainty on Vcs is to be reduced further. We also determine Vcs values in good agreement using the measured total branching fraction and the rates extrapolated to q2=0. Our form factors enable tests of lepton flavor universality violation. We find the ratio of branching fractions for D0→K− with μ and e in the final state to be Rμ/e=0.9779(2)latt(50)EM in the Standard Model, with the uncertainty dominated by that from electromagnetic corrections.