Leptonic-Decay-Constant RatiofK+/fπ+from Lattice QCD with Physical Light Quarks

A calculation of the ratio of leptonic decay constants ${f}_{{K}^{+}}/{f}_{{$\pi${}}^{+}}$ makes possible a precise determination of the ratio of Cabibbo-Kobayashi-Maskawa (CKM) matrix elements $|{V}_{\mathrm{us}}|/|{V}_{\mathrm{ud}}|$ in the standard model, and places a stringent constraint on the scale of new physics that would lead to deviations from unitarity in the first row of the CKM matrix. We compute ${f}_{{K}^{+}}/{f}_{{$\pi${}}^{+}}$ numerically in unquenched lattice QCD using gauge-field ensembles recently generated that include four flavors of dynamical quarks: up, down, strange, and charm. We analyze data at four lattice spacings $a$\approx${}0.06$, 0.09, 0.12, and 0.15 fm with simulated pion masses down to the physical value 135 MeV. We obtain ${f}_{{K}^{+}}/{f}_{{$\pi${}}^{+}}=1.1947(26)(37)$, where the errors are statistical and total systematic, respectively. This is our first physics result from our ${N}_{f}=2+1+1$ ensembles, and the first calculation of ${f}_{{K}^{+}}/{f}_{{$\pi${}}^{+}}$ from lattice-QCD simulations at the physical point. Our result is the most precise lattice-QCD determination of ${f}_{{K}^{+}}/{f}_{{$\pi${}}^{+}}$, with an error comparable to the current world average. When combined with experimental measurements of the leptonic branching fractions, it leads to a precise determination of $|{V}_{\mathrm{us}}|/|{V}_{\mathrm{ud}}|=0.2309(9)(4)$ where the errors are theoretical and experimental, respectively.