Short-distance charmonium correlator on the lattice with Möbius domain-wall fermion and a determination of charm quark mass

We calculate charmonium correlators on the lattice with $2+1$ flavors of sea quarks and charm valence quarks, both described by the Möbius domain-wall fermion. Temporal moments of the correlators are calculated and matched to perturbative QCD formulas to extract the charm quark mass $m_c(\mu )$ and strong coupling constant ${\alpha }_s(\mu )$. Lattice data at three lattice spacings, 0.044, 0.055, and 0.080 fm, are extrapolated to the continuum limit. The correlators in the vector channel are confirmed to be consistent with the experimental data for $e^{+}{e}^{-}\to c\overline{c}$, while the pseudoscalar channel is used to extract $m_c(\mu )$ and ${\alpha }_s(\mu )$. We obtain $m_{\mathrm{c}}(3\mathrm{GeV})=1.003(10)\mathrm{GeV}$ and ${\alpha }_s^{\overline{\mathrm{MS}}(4)}(3\mathrm{GeV})=0.253(13)$. The dominant source of the error is the truncation of perturbative expansion at ${\alpha }_s^3$.

Figure 1

$(t/a{)}^{n}G(t)$ on the lattice of size $64^3\times 128$ at $a=0.044\mathrm{fm}$. The function is normalized by its peak. The data for $n=4$ (filled circle), 8 (open circle), and 12 (square) are shown. The long-dashed line around $t/a\sim 48$ represents the point of $\pi /(300\mathrm{MeV})$, which is the distance that the nonperturbative effect dominates. Three vertical dashed lines show the position of peak $n/M$ ($n=4$, 8 and 12) for the single exponential function ${\mathrm{e}}^{-Mt}$. Here, $Ma=0.6656$.

Figure 2

Continuum extrapolation of the reduced moments for the vector current $R_n^V$ [$n=6$ (pluses) and 8 (squares)]. Data are plotted after correcting for the finite light quark mass effects by multiplying $1/(1+f_1(m_u+m_d+m_s)/m_c)$. Lattice data are corrected for the missing charm quark loop effect, estimated by perturbation theory, $r_n^V(n_f=4)/r_n^V(n_f=3)$. The error of the individual lattice data includes that from the renormalization factor $Z_V$, which is the dominant source of error. The points at $a^2=0$ are our estimate of the continuum limit based on two methods of continuum extrapolation. Its error includes that due to the input for $a^{-1}$ as well. Phenomenological estimates of corresponding quantities are plotted on the left: Dehnadi et al. [25] (filled circle), Kuhn et al. [7] (open circle), Kuhn et al. [26] (filled square), and Hoang et al. [27] (open square).

Figure 3

Spin-averaged mass $(m_{{\eta }_c}+3m_{J/\psi })/4$ as a function of $m_{\pi }^2$. The experimental value, 3.072 GeV, is shown by a filled circle. Data at $\beta =4.17$ (square), $\beta =4.35$ (circle), and $\beta =4.47$ (triangle) are plotted. At each $\beta$, the extrapolation to the physical pion mass slightly misses the experimental value since the input $m_c$ is not exactly tuned. This tiny difference is corrected when we analyze the temporal moments.

Figure 4

$R_6$ as a function of the spin-averaged mass $a(m_{{\eta }_c}+3m_{J/\psi })/4$ from different $m_c$. Data at $\beta =4.47$ are shown. The dashed line represents the physical spin-averaged mass. Three data points shown by open square are from the supplemental data set obtained without using the $Z_2$ noise source. The filled square is our main data point calculated with the $Z_2$ noise source.

Figure 5

Continuum extrapolation of $R_n(a)$. Data points correspond to $R_6$, $R_8$, $R_{10}$, $R_{12}$, and $R_{14}$ from top to bottom. The continuum extrapolation assuming the form (5.1) is shown by lines. The points at $a=0$ represent our estimate obtained from a mean of the extrapolated values with and without the coarsest lattice data.

Figure 6

Residual scale dependence of the ratio $r_n(\mu )/m_c(\mu )$. The case for $n=8$ is plotted as a typical example.

Figure 7

Hyperfine splitting ${\mathrm{\Delta }}_{J/\psi -{\eta }_c}$ calculated on the lattice and its continuum extrapolation. The error of lattice scale $a$ from $t^{1/2}$ is added for each data point.

Figure 8

Constraints on $m_c(\mu )$ and ${\alpha }_s(\mu )$ from the moments $R_6$ (dotted curve), $R_8$ (dashed curve), $R_{10}$ (long dashed curve), and $R_6/R_8$ (solid curve). For each curve, the band represents the error due to the truncation of perturbative expansion.

Figure 9

Constraint from the vector-current moments $R_n^V$ on the $(m_c(\mu ),{\alpha }_s(\mu ))$ plane. Dotted, dashed, long-dashed, and solid curves correspond to that of $R_6$, $R_8$, $R_{10}$, and $R_6/R_8$, respectively.

Figure 10

The charm quark mass obtained in this work is compared with previous lattice determination. The previous results are separately shown for a different number of sea quarks. HPQCD 14 [3], ETM $a$ 14 [37], and ETM $b$ 14 [38] for $n_f=2+1+1$; Maezawa et al. 16 [39], HPQCD 10 [2], HPQCD 08 [1], and $\chi \mathrm{QCD}$ 14 [40] for $n_f=2+1$; and ALPHA 13 [41], ETM $a$ 11 [42], and ETM $b$ 11 [43] for $n_f=2$.