Charm quark system at the physical point of $2\mathbf{+}1$ flavor lattice QCD

We investigate the charm quark system using the relativistic heavy quark action on $2+1$ flavor PACS-CS configurations previously generated on ${32}^3\times 64$ lattice. The dynamical up, down, and strange quark masses are set to the physical values by using the technique of reweighting to shift the quark-hopping parameters from the values employed in the configuration generation. At the physical point, the lattice spacing equals $a^{-1}=2.194(10)\mathrm{GeV}$ and the spatial extent $L=2.88(1)\mathrm{fm}$. The charm quark mass is determined by the spin-averaged mass of the $1S$ charmonium state, from which we obtain $m_{\mathrm{charm}}^{\overline{\mathrm{MS}}}(\mu =m_{\mathrm{charm}}^{\overline{\mathrm{MS}}})=1.260(1)(6)(35)\mathrm{GeV}$, where the errors are due to our statistics, scale determination and renormalization factor. An additional systematic error from the heavy quark is of order ${\alpha }_s^{2}f(m_{Q}a)(a{\Lambda }_{\mathrm{QCD}})$, $f(m_{Q}a)(a{\Lambda }_{\mathrm{QCD}}{)}^2$, which are estimated to be a percent level if the factor $f(m_{Q}a)$ analytic in $m_{Q}a$ is of order unity. Our results for the charmed and charmed-strange meson decay constants are $f_D=226(6)(1)(5)\mathrm{MeV}$, $f_{D_s}=257(2)(1)(5)\mathrm{MeV}$, again up to the heavy quark errors of order ${\alpha }_s^{2}f(m_{Q}a)(a{\Lambda }_{\mathrm{QCD}})$, $f(m_{Q}a)(a{\Lambda }_{\mathrm{QCD}}{)}^2$. Combined with the CLEO values for the leptonic decay widths, these values yield $|V_{cd}|=0.205(6)(1)(5)(9)$, $|V_{cs}|=1.00(1)(1)(3)(3)$, where the last error is because of the experimental uncertainty of the decay widths.

Figure 1

Dispersion relations for $1S$ states of charmonium in pseudoscalar (left panel) and vector channel (right panel).

Figure 2

Effective masses for charmonium.

Figure 3

Effective masses for charmed mesons.

Figure 4

Effective masses for charmed-strange mesons.

Figure 5

Our results for the charmonium mass spectrum.

Figure 6

Hyperfine splitting of the charmonium with different number of flavors.

Figure 7

Comparison of the charm quark mass. The charm quark mass is obtained at $\mu =a^{-1}$, and evolved to $\mu =m_{\mathrm{charm}}^{\overline{\mathrm{MS}}}$ using four-loop beta function [16]. We employ $N_f=4$ running in this plot.

Figure 8

Our results for charmed meson masses (left panel) and charmed-strange meson masses (right panel) normalized by the experimental values.

Figure 9

Our results for the hyperfine splittings of charmed meson (left panel) and charmed-strange meson (right panel).

Figure 10

Mass difference between a charmed meson and charmonium, as well as that between a charmed-strange meson and charmonium. $m(\overline{D})$ is the spin-averaged mass of $D$ and $D^{*}$ mesons, and $m(\overline{D_s})$ is that of $D_s$ and $D_s^{*}$ mesons.

Figure 11

Comparison of pseudoscalar decay constants for the charmed meson (left panel) and charmed-strange meson (right panel).

Figure 12

Ratios of pseudoscalar decay constants for the charmed meson and charmed-strange meson.

Figure 13

Comparison of the CKM matrix elements, $|V_{cd}|$ (left panel) and $|V_{cs}|$ (right panel).

Figure 14

Ratio of the CKM matrix elements, $|V_{cs}|$ and $|V_{cd}|$.