Precise determination of the lattice spacing in full lattice QCD

Abstract

We compare three different methods to determine the lattice spacing in lattice QCD and give results from calculations on the MILC ensembles of configurations that include the effect of $u$, $d$ and $s$ sea quarks. It is useful, for ensemble to ensemble comparison, to express the results as giving a physical value for $r_1$, a parameter from the heavy quark potential. Combining the three methods gives a value for $r_1$ in the continuum limit of 0.3133(23)(3) fm. Using the MILC values for $r_0/r_1$, this corresponds to a value for the $r_0$ parameter of 0.4661(38) fm. We also discuss how to use the $η_s$ for determining the lattice spacing and tuning the $s$-quark mass accurately, by giving values for $m_{η_s}$ (0.6858(40) GeV) and $f_{η_s}$ (0.1815(10) GeV).

Figures (5)

• Figure 1: Results for highly excited states from our fit to the $3\times 3$ matrix of $\Upsilon$ correlators from the coarse 005/05 (set 4) ensemble as a function of the number of exponentials included in the fit. The $\chi^2/dof$ is also shown - the fit had 208 degrees of freedom. The results are stable from 9 to 12 exponentials.
• Figure 2: Results for $m_{\eta_c}$ vs $m_{\eta_s}$ for different charm and strange quark masses on the coarse 01/05 ensemble (set 4). Points corresponding to different charm quark masses are given in different colors. Several different strange quark masses are given for each charm quark mass. The lattice spacing is determined from $m_{D_s}-m_{\eta_c}/2$. Note that the experimental point is shifted to allow for electromagnetic effects missing from our calculation, as described in the text.
• Figure 3: Simulation results for the effective $r_1$ obtained from $m_{D_s}-m_{\eta_c}/2$ (top), $f_{\eta_s}$ (middle), and $m_{\Upsilon^\prime}-m_\Upsilon$ (bottom) are plotted versus $(a/r_1)^2$ for various values of the sea-quark mass. The lines show the tuned fit functions from our simultaneous fit to all three sets of simulation results. We used the fit functions to correct the simulation data points for the sea-quark masses; data points and lines are for $\delta m_q^\mathrm{sea}=0$. The gray band is the continuum value obtained from the fit: $r_1=0.3133(23)$ fm.
• Figure 4: The pseudoscalar decay constants plotted versus quark mass; $m_\pi^2/(2m_K^2-m_\pi^2)$ is approximately the ratio of the $u/d$ to $s$ quark masses: $m_l/m_s$. The fit data is from lattice simulations with three different lattice spacings; results decrease with decreasing lattice spacing. The data have been adjusted to correspond to points where the sea-quark masses correspond to the valence masses. The lines are from the tuned fit function for each of the three lattice spacings. The bottom line in each group is the extrapolation to $a=0$. The gray bands indicate final values from the fit for the physical decay constants for all three mesons; the leftmost data points for $f_\pi$ and $f_K$ are the current experimental values.
• Figure 5: Fits to two different sets of fake data for pion and kaon decay constants with very different $a^2$ behavior from each other and from the real simulation data (Figure \ref{['fig:fmpi2']}). The "experimental" points indicated in each case correspond to the exact results, extracted from the formulas used to generate the fake data.