Determination of quark masses from $n_f=4$ lattice QCD and the RI-SMOM intermediate scheme

We determine the charm and strange quark masses in the $\overline{\mathrm{MS}}$ scheme, using $n_f=2+1+1$ lattice QCD calculations with highly improved staggered quarks and the regularization-independent symmetric momentum-subtraction intermediate scheme to connect the bare lattice quark masses to continuum renormalization schemes. Our study covers the analysis of systematic uncertainties from this method, including nonperturbative artifacts and the impact of the nonzero physical sea quark masses. We find $m_c^{\overline{\mathrm{MS}}}(3\mathrm{GeV})=0.9896(61)\mathrm{GeV}$ and $m_s^{\overline{\mathrm{MS}}}(3\mathrm{GeV})=0.08536(85)\mathrm{GeV}$, where the uncertainties are dominated by the tuning of the bare lattice quark masses. These results are consistent with, and of similar accuracy to, those using the current-current correlator approach coupled to high-order continuum QCD perturbation theory, implemented in the same quark formalism and on the same gauge field configurations. This provides a strong test of the consistency of methods for determining the quark masses to high precision from lattice QCD. We also give updated lattice QCD world averages for $c$ and $s$ quark masses.

Figure 1

Scatter plots from bootstrap samples of $Z_m^{\mathrm{SMOM}}$ at $\mu =2\mathrm{GeV}$ and Landau gauge trace link values on coarse set 4, for a quark mass value in lattice units of 0.0153. On the left results are for a gauge-fixing tolerance on ${10}^{-7}$ (the tolerance we use); on the right the tolerance is successively tightened to ${10}^{-10}$. Mean values for $Z_m^{\mathrm{SMOM}}$ are indicated by dashed lines of matching color.

Figure 2

The impact of the gauge fixing tolerance on $Z_m^{\mathrm{SMOM}}(\mu )$ as a function of $\mu$. Results are for coarse set 2, for a quark mass in lattice units of 0.0204, and show a steep fall with a value of $\mu$ consistent with sensitivity to gauge-noninvariant condensates.

Figure 3

The difference of vertex functions ${\mathrm{\Lambda }}_P$ and ${\mathrm{\Lambda }}_S$ (proportional to $Z_P-Z_S$) as a function of valence quark mass and as a function of $\mu$. Upper plot: $\mathrm{Tr}({\mathrm{\Lambda }}_P-{\mathrm{\Lambda }}_S)/48$ at $\mu =3\mathrm{GeV}$ plotted against the square of the valence quark mass in units of the tuned $s$ quark mass. Results are shown for coarse and fine lattices (sets 3 and 10). Lower plot: $\mathrm{Tr}({\mathrm{\Lambda }}_P-{\mathrm{\Lambda }}_S)/48$ extrapolated to zero valence quark mass, following the upper plot, now plotted against $\mu$ on a log-log scale. For comparison, the line shows a constant divided by ${\mu }^6$. This plot shows that, in the RI-SMOM scheme, the nonperturbative contributions to ${\mathrm{\Lambda }}_P-{\mathrm{\Lambda }}_s$ are much more strongly suppressed than in the RI-MOM scheme, falling as ${\mu }^{-6}$.

Figure 4

Results for $Z_m^{\mathrm{SMOM}}$ for coarse ensembles with different values of the sea $s$ quark mass, with $u/d$ and $c$ sea quark masses fixed in lattice units at 0.00507 and 0.628 respectively (sets 4, 6, 7 and 8; see Table 2). The upper plot shows results for $Z_m^{\mathrm{SMOM}}$ as a function of $\mu$ in GeV for the four sets. Dashed lines join the points. In all cases the valence quark mass is set to 0.0051 in lattice units. The lower plot gives more detail for results at $\mu =2.24\mathrm{GeV}$, showing no visible variation (using the horizontal dotted lines as guides) in $Z_m^{\mathrm{SMOM}}$ even at a level below 0.1%.

Figure 5

Results for $Z_m^{\mathrm{SMOM}}$ for coarse ensembles with different values of the sea $c$ quark mass from sets 1, 2 and 3 (see Table 2). These sets have slightly different $u/d$ and $s$ sea quark masses, but by an amount which is much smaller than the change in the $c$ sea mass from set 1 to sets 2 and 3. The upper plot shows results for $Z_m^{\mathrm{SMOM}}$ as a function of $\mu$ in GeV for the two sets. The results shown here are obtained for a valence quark mass in lattice units of 0.0051. The lower plot shows more detail for results at $\mu =2.4\mathrm{GeV}$, showing $\mathcal{O}(0.1\%)$ variation for a very substantial change in $a{m}_{c,\mathrm{sea}}$. Dashed lines simply join the points.

Figure 6

Results for $Z_m^{\mathrm{SMOM}}$ for fine ensembles with different values of the sea $u/d$ quark mass from sets 10 and 11 (see Table 2). These sets also have slightly different $s$ and $c$ sea quark masses and spatial volume. The plot shows results for $Z_m^{\mathrm{SMOM}}$ as a function of $\mu$ in GeV for the two sets, for a valence mass in lattice units of 0.0074.

Figure 7

The upper plot shows the dependence of $Z_m^{\mathrm{SMOM}}$ on valence quark mass in lattice units for coarse, fine and superfine lattices (sets 2, 10 and 12) at $\mu =3\mathrm{GeV}$. The dashed line gives the simple fit described in the text. The lower plot is a zoomed-in version of the superfine (set 12) results for which fit parameters are shown in Fig. 8. The lighter rightmost point corresponds to a quark mass equal to that of the strange quark. This point was not included in the valence quark mass fit, but lies on top of the fitted line.

Figure 8

The linear and quadratic slopes [$d_1$ and $d_2$ of Eq. (19)] of $Z_m^{\mathrm{SMOM}}$ with valence quark mass on superfine lattices (set 12), plotted against $\mu$ in GeV. The curve with the error band is a fit to the form $C/{\mu }^4$ through the $d_2$ points.

Figure 9

Dependence of $Z_m^{\mathrm{SMOM}}$ on the spatial volume of the lattice (sets 3, 4 and 5) as a function of $\mu$. The valence quark mass in lattice units is fixed to 0.0051.

Figure 10

(Upper) ${\overline{m}}_c(3\mathrm{GeV})$ and (lower) ${\overline{m}}_s(3\mathrm{GeV})$ obtained from nonperturbative lattice QCD calculations of $Z_m^{\mathrm{SMOM}}$ at three different values of $\mu$, followed by perturbative matching to the $\overline{\mathrm{MS}}$ scheme and running to 3 GeV. The results are plotted for $\mu =2$, 2.5, 3, 4 and 5 GeV against the square of the lattice spacing. Extrapolations to the continuum limit for each value of $\mu$, as discussed in the text, are shown as dashed lines (these give the fit result at ${\delta }_{\ell }^{\mathrm{sea}}={\delta }_c^{\mathrm{sea}}=0$). The error bars on the data points show only the uncorrelated uncertainties. The point plotted as a dark grey filled circle, offset to the left, is the final physical result, $\overline{m}(3\mathrm{GeV})$, from the fit described in the text. The inner error bar for this point is the uncorrelated uncertainty and the outer error bar is the full uncertainty. The point plotted as a white circle, further offset to the left, is from the current-current correlator method [12] (along with a nonperturbative determination of $m_c/m_s$ in the $m_s$ case).

Figure 11

A graphical representation of the different tests that we have done to check the robustness of our fits to obtain the physical results for ${\overline{m}}_s$ and ${\overline{m}}_c$. The different rows give variations on the fit described in the text [Eq. (26)].

Figure 12

Comparison of lattice QCD results for ${\overline{m}}_c({\overline{m}}_c,n_f=4)$. Note that results are determined at a higher energy scale and then run down to ${\overline{m}}_c$ using perturbation theory. Calculations are listed chronologically and divided into those obtained on gluon configurations that include three (blue symbol) or four (red symbol) flavors of quarks in the sea. Results obtained on three-flavor configurations have in all cases been adjusted to $n_f=4$ using perturbation theory (see the references for details of how this is done in each case). Different symbols denote different methods: open triangles use the current-current correlator (JJ) method (“a,” “b” and “c” variants; see text), open diamonds use the RI-MOM intermediate scheme, open circles use the RI-SMOM intermediate scheme and open squares use the MRS scheme. Labels on the right indicate collaboration name, quark formalism and method. The result denoted “HPQCD HISQ RI-SMOM” is this work, “FNAL/MILC/TUM HISQ MRS” is from Ref. [22], “HPQCD HISQ JJ” (red) is from Ref. [12], “ETMC RI-MOM” is from Ref. [20], “JLQCD DW JJ” is from Ref. [15], “MP(hotQCD) HISQ JJ” is from Ref. [16], “$\chi \mathrm{QCD}$ overlap RI-MOM” is from Ref. [68], ‘HPQCD HISQ JJ” (blue) is from Ref. [13] and “$\mathrm{HPQCD}+$ HISQ JJ” is from Ref. [4]. The grey shaded band indicates the world average (taking correlations into account) of the $n_f=4$ results and the lighter shaded band shows the evaluation of 1.28(3) GeV from the Particle Data Group [66].

Figure 13

Comparison of lattice QCD results for ${\overline{m}}_s(2.0\mathrm{GeV},n_f)$. Results are listed chronologically and divided into those obtained on gluon configurations that include $n_f=3$ (blue symbol) or $n_f=4$ (red symbol) flavors of quarks in the sea. The $n_f=4$ results should be 0.2 MeV smaller than $n_f=3$ from QCD perturbation theory matching for adding/removing a $c$ sea quark. Different symbols denote different methods: open triangles use the current-current correlator (JJ) method combined with a nonperturbative determination of $m_c/m_s$, open diamonds use the RI-MOM intermediate scheme, open circles use the RI-SMOM intermediate scheme and open squares use the MRS scheme combined with a nonperturbative determination of a mass ratio between heavy and $s$ quarks. Labels on the right indicate collaboration name, quark formalism and method. The result denoted “HPQCD HISQ RI-SMOM” is this work, “FNAL/MILC/TUM HISQ MRS $+$ $m_h/m_s$” is from Ref. [22], “HPQCD HISQ JJ $+$ $m_c/m_s$” (red) is from Ref. [12], ‘ETMC RI-MOM” is from Ref. [20], “MP(hotQCD) HISQ JJ $+$ $m_c/m_s$” is from Ref. [16], “RBC/UKQCD DW RI-SMOM” is from Ref. [69] (note that this uses a different method to fix $Z_q$ than is used here), “BMW clover RI-MOM” is from Refs. [70, 71], and “HPQCD HISQ JJ $+$ $m_c/m_s$” (blue) is from Refs. [13, 17]. The light grey shaded band shows the evaluation of $96(+8,-4)\mathrm{GeV}$ from the Particle Data Group [66]. The dark grey shaded band gives the weighted average (allowing for correlations) of the $n_f=2+1+1$ results as described in the text.

Figure 14

The coefficient $C_{n_f}$ which enters the $Z_m^{\overline{\mathrm{MS}}/\mathrm{SMOM}}$ factor at $\mathcal{O}({\alpha }_s^2)$ as a function of ${\mu }^2/m^2$ for a massive sea quark of mass $m$. The dashed, blue line shows the $m=0$ result. The solid, red line shows the exact result of a numerical evaluation of the contribution which is valid across the full range of $m$ values. The vertical lines denote the location of the different ${\mu }^2/m^2$ values, which are used in this section.