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We present recent progress in the lattice calculation of leptonic decay constants for $B_{(s)}$ and $D_{(s)}$ mesons using the Oktay-Kronfeld (OK) action for charm and bottom valence quarks, whose masses are tuned non-perturbatively. The calculations are done on 6 HISQ ensembles generated by the MILC collaboration with $N_f=2+1+1$ flavors. We also use the HISQ action for the light spectator quarks. Results are presented for the ratios $f_{B_s}/f_B$ and $f_{D_s}/f_D$, which reflect $SU(3)$ flavor symmetry breaking, and are independent of the renormalization constants of the axial currents.
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We present a calculation of the form factors, $f_0$ and $f_+$, for the $B_{(s)} to D_{(s)}$ semileptonic decays. Our work uses the MILC $n_f=2+1$ AsqTad configurations with NRQCD and HISQ valence quarks at four values of the momentum transfer $q^2$. We provide results for the chiral-continuum extrapolations of the scalar and vector form factors.
We present results for the $SU(3)$ breaking ratios of decay constants $f_{D_s}/f_D$ and $f_{B_s}/f_B$ and - for the first time with physical pion masses - the ratio of bag parameters $B_{B_s}/B_{B_d}$, as well as the ratio $xi$, forming the ratio of the nonpeturbative contributions to neutral $B_{(s)}$ meson mixing. Our results are based on Lattice QCD simulations with chirally symmetric 2+1 dynamical flavors of domain wall fermions. Eight ensembles at three different lattice spacing in the range $a = 0.11 - 0.07,mathrm{fm}$ enter the analysis two of which feature physical light quark masses. Multiple heavy quark masses are simulated ranging from below the charm quark mass to half the bottom quark mass. The $SU(3)$ breaking ratios display a very benign heavy mass behaviour allowing for extrapolation to the physical bottom quark mass. The results in the continuum limit including all sources of systematic errors are $f_{D_s}/f_D = 1.1740(51)_mathrm{stat}(^{+68}_{-68})_mathrm{sys}$, $f_{B_s}/f_B = 1.1949(60)_mathrm{stat}(^{+hphantom{0}95}_{-175})_mathrm{sys}$, $B_{B_s}/B_{B_d} = 0.9984(45)_mathrm{stat}(^{+80}_{-63})_mathrm{sys}$ and $xi = 1.1939(67)_mathrm{stat}(^{+hphantom{0}95}_{-177})_mathrm{sys}$. Combining these with experimentally measured values we extract the ratios of CKM matrix elements $|V_{cd}/V_{cs}| = 0.2164(57)_mathrm{exp}(^{+12}_{-12})_mathrm{lat}$ and $|V_{td}/V_{ts}| = 0.20329(41)_mathrm{exp}(^{+162}_{-301})_mathrm{lat}$.
The branching fraction of the decay $B_{s}^{0} rightarrow D_{s}^{(*)+}D_{s}^{(*)-}$ is measured using $pp$ collision data corresponding to an integrated luminosity of $1.0fb^{-1}$, collected using the LHCb detector at a centre-of-mass energy of $7$TeV. It is found to be begin{align*} {mathcal{B}}(B_{s}^{0}rightarrow~D_{s}^{(*)+}D_{s}^{(*)-}) = (3.05 pm 0.10 pm 0.20 pm 0.34)%, end{align*} where the uncertainties are statistical, systematic, and due to the normalisation channel, respectively. The branching fractions of the individual decays corresponding to the presence of one or two $D^{*pm}_{s}$ are also measured. The individual branching fractions are found to be begin{align*} {mathcal{B}}(B_{s}^{0}rightarrow~D_{s}^{*pm}D_{s}^{mp}) = (1.35 pm 0.06 pm 0.09 pm 0.15)%, ewline{mathcal{B}}(B_{s}^{0}rightarrow~D_{s}^{*+}D_{s}^{*-}) = (1.27 pm 0.08 pm 0.10 pm 0.14)%. end{align*} All three results are the most precise determinations to date.
Measurements are presented of the branching fractions of the decays $B_{s}^{0} to D_{s}^{mp} K^{pm}$ and $B^{0} to D_{s}^{-} K^{+}$ relative to the decays $B_{s}^{0} to D_{s}^{-} pi^{+}$ and $B^{0} to D^{-} pi^{+}$, respectively. The data used correspond to an integrated luminosity of 3.0 fb$^{-1}$ of proton-proton collisions. The ratios of branching fractions are $dfrac{mathcal{B}(B_{s}^{0} to D_{s}^{mp} K^{pm})}{mathcal{B}(B_{s}^{0} to D_{s}^{-} pi^{+})} = 0.0752 pm 0.0015 pm 0.0019$ and $dfrac{mathcal{B}(B^{0} to D_{s}^{-} K^{+})}{mathcal{B}(B^{0} to D^{-} pi^{+})} = 0.0129 pm 0.0005 pm 0.0008,$ where the uncertainties are statistical and systematic, respectively.
We review recent progress in the calculation of the decay constants $f_{D}$ and $f_{D_s}$ from lattice QCD. We focus particularly on simulations with $N_f=2+1$ and $N_f=2+1+1$ and simulations with close to physical light quark masses.
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