New high-precision $b$, $c$, and $s$ masses from pseudoscalar-pseudoscalar correlators in $n_f=4$ lattice QCD

Abstract

We extend an earlier lattice QCD analysis of heavy-quark current-current correlators to obtain new values for the $\overline{\mathrm{MS}}$ masses of the $b$, $c$, and $s$~quarks. The analysis uses gluon configurations from the MILC collaboration with vacuum polarization contributions from $u$, $d$, $s$, and~$c$ quarks ($n_f=4$), and lattice spacings down to~0.032~fm. We find that $\overline{m}_b(\overline{m}_b, n_f=5)=4.1923(63)$~GeV, $\overline{m}_c(3~\mathrm{GeV}, n_f=4)=0.9813(34)$~GeV, and $\overline{m}_s(3~\mathrm{GeV}, n_f=4)=83.39(26)$~MeV. These results are corrected for QED by including (quenched) QED in the simulations. They are among the most accurate values by any method to date. We give a detailed analysis of finite lattice-spacing errors that shows why the HISQ discretization of the quark action is particularly useful for $b$-quark simulations even for lattices where~$am_b\approx1$. We also calculate QED and isospin corrections to the (fictitious) $η_s$-meson mass, which is used to tune $s$-quark masses in lattice simulations.

Figures (10)

• Figure 1: Average three momentum $p$, divided by the bare mass $m_h$, of the heavy valence quarks for moment $G_n$ in lowest-order perturbation theory (solid line). The average is calculated by inserting a factor of $\log({\bf p}^2)/2$ into the Feynman integral over Euclidean four-momentum $p_\mu$Lepage:1992xa. $p/m_h\approx v/c$, where $v$ is the typical heavy-quark velocity, is well approximated by function $3.15/n^{0.75}$ (dashed line).
• Figure 2: Plots of $c^2({\bf p}\to 0)$ for the $\eta_b$ (red, triangles) and $\eta_c$ (blue, circles) mesons versus the valence-quark mass $am$ in lattice units. Fits of these data to a formula of the form $c_2 (am)^2 + c_4 (am)^4$ are also shown (short dashes). Results for the $\eta_b$ are from simulations with gluon ensembles EF-5, UF-5, and SF-5. Results for the $\eta_c$ are from coarser lattices, with lattice spacings of 0.09 fm, 0.12 fm and 0.15 fm.
• Figure 3: Fits to lattice values for the reduced moments $R_n$. Results for $R_n/m_b$ are plotted for a range of (large) valence-quark masses $m_h$. The data points are lattice results for moments with $n=6$, 14, and 22. These are calculated using gluon ensembles SF-5 (blue), SF-P (orange), UF-5 (green), and EF-5 (red), with lattice spacings of 0.059, 0.057, 0.044, and 0.032 fm, respectively. The lattice results are compared with fit results (dashed lines) based on Eq. (\ref{['eq:fit1']}). The dotted grey lines and 1-sigma grey bands are fit results with $am_h$ set to zero.
• Figure 4: Fit to results for the $b$-quark masses $\overline{m}_b^{(s)}(\overline{m}_b^{(s)})$ from each gluon ensemble $s$. Fit results (dashed line and grey bar), using Eq. (\ref{['eq:fit2']}), are shown for tuned sea quarks ($\delta m_{uds}^\mathrm{sea}\to0$) and for zero lattice spacing. The first, second, and fourth masses are from simulations where the light sea-quark mass $m_\ell$ was too large (ensembles EF-5, UF=5, and SF-5 where $m_s/m_\ell=5$); the third result has the correct mass (SF-P).
• Figure 5: Results for the $b$-quark mass $\overline{m}_b(\overline{m}_b,n_f=5)$ obtained from each moment $n$ separately. The dashed line and grey band correspond to the result obtained from the joint analysis of all moments from $n=6$ through $n=22$ together (red data points, Eq. (\ref{['eq:mbmb5']})).