High-precision $f_{B_s}$ and heavy quark effective theory from relativistic lattice QCD

We present a new determination of the $B_s$ leptonic decay constant from lattice QCD simulations that use gluon configurations from MILC and a highly improved discretization of the relativistic quark action for both valence quarks. Our result, $f_{B_s}=0.225(4)\mathrm{GeV}$, is almost 3 times more accurate than previous determinations. We analyze the dependence of the decay constant on the heavy quark’s mass and obtain the first empirical evidence for the leading $1/\sqrt{m_h}$ dependence predicted by heavy quark effective theory. As a check, we use our analysis technique to calculate the $m_{B_s}-m_{{\eta }_b}/2$ mass difference. Our result agrees with experiment to within errors of 11 MeV (better than 2%). We discuss how to extend our analysis to other quantities in $B_s$ and $B$ physics, making 2% precision possible for the first time.

Figure 1

The leptonic decay constant $f_{H_s}$ for pseudoscalar $h\overline{s}$ mesons $H_s$, plotted on the left versus the $H_s$ mass as the $h$-quark’s mass is varied. The solid line and gray band show our best-fit estimates for the decay constants extrapolated to zero lattice spacing. Best-fit results (dashed lines) and simulation data are also shown for five different lattice spacings, with results for smaller lattice spacings extending to higher masses (since we restrict $a{m}_h<1$). The simulation data points have been corrected for small mistunings of the $s$ quark’s mass. On the right the same simulation data and fits are plotted for $\sqrt{m_{H_s}}{f}_{H_s}$ versus $1/m_{H_s}$.

Figure 2

The $H_s-{\eta }_h/2$ mass difference plotted versus $m_{H_s}$ as the $h$ quark’s mass is varied. The solid line and gray band show our best-fit estimates for the mass differences extrapolated to zero lattice spacing. Best-fit results (dashed lines) and simulation data are also shown for five different lattice spacings, with results for smaller lattice spacings extending to larger masses (since we require $a{m}_h<1$). The simulation data points have been corrected for small mistunings in the $s$ quark’s mass. Data points (in black) at $m_{D_s}$ and $m_{B_s}$ are the experimental values after correcting for small effects from electromagnetism, ${\eta }_b$ annihilation, and $c$ quarks in the sea, none of which are included in the simulation.