B - and D -meson leptonic decay constants from four-flavor lattice QCD

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We calculate the leptonic decay constants of heavy-light pseudoscalar mesons with charm and bottom quarks in lattice quantum chromodynamics on four-flavor QCD gauge-field configurations with dynamical $u$, $d$, $s$, and $c$ quarks. We analyze over twenty isospin-symmetric ensembles with six lattice spacings down to $a\ensuremath{\approx}0.03\text{ }\text{ }\mathrm{fm}$ and several values of the light-quark mass down to the physical value $\frac{1}{2}({m}_{u}+{m}_{d})$. We employ the highly-improved staggered-quark (HISQ) action for the sea and valence quarks; on the finest lattice spacings, discretization errors are sufficiently small that we can calculate the $B$-meson decay constants with the HISQ action for the first time directly at the physical $b$-quark mass. We obtain the most precise determinations to-date of the $D$- and $B$-meson decay constants and their ratios, ${f}_{{D}^{+}}=212.7(0.6)\text{ }\text{ }\mathrm{MeV}$, ${f}_{{D}_{s}}=249.9(0.4)\text{ }\text{ }\mathrm{MeV}$, ${f}_{{D}_{s}}/{f}_{{D}^{+}}=1.1749(16)$, ${f}_{{B}^{+}}=189.4(1.4)\text{ }\text{ }\mathrm{MeV}$, ${f}_{{B}_{s}}=230.7(1.3)\text{ }\text{ }\mathrm{MeV}$, ${f}_{{B}_{s}}/{f}_{{B}^{+}}=1.2180(47)$, where the errors include statistical and all systematic uncertainties. Our results for the $B$-meson decay constants are three times more precise than the previous best lattice-QCD calculations, and bring the QCD errors in the standard model predictions for the rare leptonic decays $\overline{\mathcal{B}}({B}_{s}\ensuremath{\rightarrow}{\ensuremath{\mu}}^{+}{\ensuremath{\mu}}^{\ensuremath{-}})=3.64(11)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}9}$, $\overline{\mathcal{B}}({B}^{0}\ensuremath{\rightarrow}{\ensuremath{\mu}}^{+}{\ensuremath{\mu}}^{\ensuremath{-}})=\phantom{\rule{0ex}{0ex}}1.00(3)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}10}$, and $\overline{\mathcal{B}}({B}^{0}\ensuremath{\rightarrow}{\ensuremath{\mu}}^{+}{\ensuremath{\mu}}^{\ensuremath{-}})/\overline{\mathcal{B}}({B}_{s}\ensuremath{\rightarrow}{\ensuremath{\mu}}^{+}{\ensuremath{\mu}}^{\ensuremath{-}})=0.0273(9)$ to well below other sources of uncertainty. As a byproduct of our analysis, we also update our previously published results for the light-quark-mass ratios and the scale-setting quantities ${f}_{p4s}$, ${M}_{p4s}$, and ${R}_{p4s}$. We obtain the most precise lattice-QCD determination to date of the ratio ${f}_{{K}^{+}}/{f}_{{\ensuremath{\pi}}^{+}}=1.1950({\text{ }}_{\ensuremath{-}23}^{+16})\text{ }\text{ }\mathrm{MeV}$.

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