High-precision $c$ and $b$ masses, and QCD coupling from current-current correlators in lattice and continuum QCD

We extend our earlier lattice-QCD analysis of heavy-quark correlators to smaller lattice spacings and larger masses to obtain new values for the $c$ mass and QCD coupling, and, for the first time, values for the $b$ mass: $m_c(3\mathrm{GeV},n_f=4)=0.986(6)\mathrm{GeV}$, ${\alpha }_{\overline{\mathrm{MS}}}(M_Z,n_f=5)=0.1183(7)$, and $m_b(10\mathrm{GeV},n_f=5)=3.617(25)\mathrm{GeV}$. These are among the most accurate determinations by any method. We check our results using a nonperturbative determination of the mass ratio $m_b(\mu ,n_f)/m_c(\mu ,n_f)$; the two methods agree to within our 1% errors and taken together imply $m_b/m_c=4.51(4)$. We also update our previous analysis of ${\alpha }_{\overline{\mathrm{MS}}}$ from Wilson loops to account for revised values for $r_1$ and $r_1/a$, finding a new value ${\alpha }_{\overline{\mathrm{MS}}}(M_Z,n_f=5)=0.1184(6)$; and we update our recent values for light-quark masses from the ratio $m_c/m_s$. Finally, in the Appendix, we derive a procedure for simplifying and accelerating complicated least-squares fits.

Figure 1

Function $z(\mu /m_h=3,m_{{\eta }_h})\equiv m_{{\eta }_h}/(2m_h)$ as a function of $m_{{\eta }_h}$. The solid line, plus gray error envelope, shows the $a=0$ extrapolation obtained from our fit. This is compared with simulation results for $R_n/r_n$ for $n=6$, 8, 10 from our 5 different lattice spacings, together with the best fits (dashed lines) corresponding to those lattice spacings. Dashed lines for smaller lattice spacings extend farther to the right. The points marked by an × are for the largest mass we tried (last line in Table 2); these are not included in the fit because $a{m}_{{\eta }_h}$ is too large. Finite-$a$ errors become very small for the larger-$n$ moments, causing points from different lattice spacings to overlap.

Figure 2

QCD coupling ${\alpha }_{\overline{\mathrm{MS}}}(\mu ,n_f=3)$ as a function of $m_{{\eta }_h}$ where $\mu =3m_h$. The solid line, plus gray error envelope, shows the best-fit coupling from our fit when perturbative evolution is assumed. The data points are values of ${\alpha }_{\overline{\mathrm{MS}}}$ extracted from individual simulation results for $R_n$ after extrapolating to $a=0$ and dividing out $z(3,m_{{\eta }_h})$ ($n>4$). Results are given for moments $n=4–10$ and all 5 lattice spacings. Several points from different lattice spacings overlap in these plots.

Figure 3

Lattice-spacing dependence of $R_n$ for masses $m_{{\eta }_h}$ within 5% of $m_{{\eta }_c}$ and moments $n=\mathbf{4}$, 6, 8, and 10. The dashed lines show our fit for the average of these masses, and the points at $a=0$ are the continuum extrapolations of our data.

Figure 4

Ratio $m_{0h}/m_{{\eta }_h}$ divided by $m_{0c}/m_{{\eta }_c}$ [which we approximate by $w(m_{{\eta }_c},a)/2$ from our fit] as a function of $m_{{\eta }_h}$. The solid line shows the $a=0$ extrapolation obtained from our fit. This is compared with simulation results for our 4 smallest lattice spacings, together with the best fits (dashed lines) corresponding to those lattice spacings. The gray band shows the $a=0$ extrapolation for this quantity obtained from our best fit to the moments in the perturbative analysis of the previous section. The point marked by an × is for the largest mass we tried (last line in Table 2); this was not included in the fit because $a{m}_{{\eta }_h}$ is too large.

Figure 5

Updated values for the 5-flavor ${\alpha }_{\overline{\mathrm{MS}}}$ at the $Z$-meson mass from each of 22 different short-distance quantities built from Wilson loops. The gray band indicates a composite average, 0.1184(6). ${\chi }^2$ per data point is 0.3.

Figure 6

$z(\mu /m_h,m_{{\eta }_h})$ versus $m_{{\eta }_h}$ (in GeV) for three different values of $\mu /m_h$. The curve for $\mu =3m_h$ comes from the best fit to the moments. The other curves are obtained by evolving perturbatively from $\mu =3m_h$.

Figure 7

Simulation results for reduced moments $R_n$ with $n=6$, 8, and 10 as functions of $m_{{\eta }_h}$ for 5 different lattice spacings. The dashed lines show the corresponding behavior of our fit function, with the best-fit parameters. The curves for smaller lattice spacings extend farther to the right. The solid lines show the $a=0$ limit of our best fit.

Figure 8

The 5-flavor QCD coupling ${\alpha }_{\overline{\mathrm{MS}}}$ at the $Z$ mass as determined by a variety of different methods. The nonlattice numbers used here are from the review in [31].

Figure 9

$\overline{\mathrm{MS}}$ masses, for the 5 lightest quarks, from this paper compared with the Particle Data Group’s current estimates [41]. Each mass is quoted at its conventional scale: 2 GeV for $u$, $d$, $s$ ($n_f=3$); $m_c$ for $c$ ($n_f=4$); and $m_b$ for $b$ ($n_f=5$).

Figure 10

Same as Fig. 7 but with $N_{am}=80$ and ${\overline{N}}_{am}=3$, instead of $N_{am}={\overline{N}}_{am}=30$. The error bars are almost entirely due to systematic errors caused by $a{m}_{{\eta }_h}/2$ corrections omitted from the fit function.