Charm and strange quark masses and $f_{D_s}$ from overlap fermions

We use overlap fermions as valence quarks to calculate meson masses in a wide quark mass range on the $2+1$-flavor domain-wall fermion gauge configurations generated by the RBC and UKQCD Collaborations. The well-defined quark masses in the overlap fermion formalism and the clear valence quark mass dependence of meson masses observed from the calculation facilitate a direct derivation of physical current quark masses through a global fit to the lattice data, which incorporates $O(a^2)$ and $O(m_c^4{a}^4)$ corrections, chiral extrapolation, and quark mass interpolation. Using the physical masses of $D_s$, $D_s^{*}$ and $J/\psi$ as inputs, Sommer’s scale parameter $r_0$ and the masses of charm quark and strange quark in the $\overline{\mathrm{MS}}$ scheme are determined to be $r_0=0.465(4)(9)\mathrm{fm}$, $m_c^{\overline{\mathrm{MS}}}(2\mathrm{GeV})=1.118(6)(24)\mathrm{GeV}$ (or $m_c^{\overline{\mathrm{MS}}}(m_c)=1.304(5)(20)\mathrm{GeV}$), and $m_s^{\overline{\mathrm{MS}}}(2\mathrm{GeV})=0.101(3)(6)\mathrm{GeV}$, respectively. Furthermore, we observe that the mass difference of the vector meson and the pseudoscalar meson with the same valence quark content is proportional to the reciprocal of the square root of the valence quark masses. The hyperfine splitting of charmonium, $M_{J/\psi }-M_{{\eta }_c}$, is determined to be 119(2)(7) MeV, which is in good agreement with the experimental value. We also predict the decay constant of $D_s$ to be $f_{D_s}=254(2)(4)\mathrm{MeV}$. The masses of charmonium $P$-wave states ${\chi }_{c0}$, ${\chi }_{c1}$ and $h_c$ are also in good agreement with experiments.

Figure 1

The plateau of heavy quark potential in coarse/fine lattice ensembles with lightest sea quark masses. The upper panel is for the ${24}^3\times 64$ lattice with $m_l^{(s)}a=0.005$, the lower panel for the ${32}^3\times 64$ lattice with $m_l^{(s)}a=0.004$.

Figure 2

Renormalized sea quark mass $m_q^{R}a$ dependence of $r_0/a$. Square points for $\beta =2.13$ with lattice spacing $a_1$, and circular points for $\beta =2.25$ with lattice spacing $a_2$.

Figure 3

Effective mass plot of the pseudoscalar meson mass ($Ma$) at fixed quark masses ($ma$). The plateau in the $ma=0.7$ case with point source is not reliable. The one with coulomb gauge fixed wall source is better, while the plateau still drops down for the $t>25$ region. This plot is based on the result from the gauge ensemble at $\beta =2.13$ with the light sea quark mass $m_l^{(\mathrm{sea})}a=0.05$.

Figure 4

The masses of the pseudoscalar and vector $\overline{c}s$ mesons are plotted with respect to the renormalized $m_c^R+m_s^R$ (tentatively taking $r_0=0.46$) for the six RBC/UKQCD gauge ensembles, where the linear behaviors in $m_c^R+m_s^R$ are clearly seen. The horizontal lines in the plot are the physical value of $D_s$ in the upper panel and $D_s^{*}$ in the lower panel.

Figure 5

The quark mass dependence of the masses of the pseudoscalar and vector $c\overline{c}$ mesons is illustrated in the plots for the six RBC/UKQCD gauge ensembles, where the linear behaviors in $m_c^R$ (tentatively taking $r_0=0.46$) are clearly seen. The horizon lines in the plot are the physical value ${\eta }_c$ in the upper panel and $J/\mathrm{\Psi }$ in the lower panel.

Figure 6

The quark mass dependence of the hyperfine splittings ${\mathrm{\Delta }}_{\mathrm{HFS}}=m_V-m_{\mathrm{PS}}$ times $\sqrt{m_{q_1}^R+m_{q_2}^R}$ for $\overline{c}s$ systems from the gauge ensemble with the lightest sea quark mass at $\beta =2.13/2.25$. One can see that such a combination is consistent with a constant within one sigma for each ensemble, indicated by a solid line in the plot, obtained from an independent constant fit for each of the two ensembles.

Figure 7

The $m_{q_1}+m_{q_2}$ dependence of the ${\mathrm{\Delta }}_{\mathrm{HFS}}^{-2}$, vs the physical quarks mass coming from PDG values [28] renormalized in $\overline{\mathrm{MS}}$ scheme at 2 GeV. From left to right, the HFS for $M_{\rho }-M_{\pi }$, $M_{K^{*}}-M_K$, $M_{D_{(s)}^{*}}-M_{D_{(s)}}$, $M_{J/\psi }-M_{{\eta }_c}$, $M_{B_{(s)}^{*}}-M_{B_{(s)}}$ and $M_{{\mathrm{\Psi }}_b}-M_{{\eta }_b}$ are plotted for comparison. The solid line in the plot is based on the correlated fit of the simulation data points with $\delta m=0.068\mathrm{GeV}$.

Figure 8

The interpolated values of the $M_{D_s}$ and $M_{J/\psi }$ from the data points of the two charm quark mass values which bracket the physical one, for each ensemble with different $\beta$, versus the renormalized sea quark mass. The lattice spacing dependence of all the three quantities is manifest given the precision of the data. At the same time, the sea quark dependence of them is not negligible and trends similarly for each $\beta$.

Figure 9

The prediction of $m_s^{\overline{\mathrm{MS}}}(2\mathrm{GeV})$ (upper panel) and $m_c^{\overline{\mathrm{MS}}}(m_c)$ (lower panel) from this work, compared to those of other works.

Figure 10

The running charm quark mass $m_c(\mu )$ versus the scale $\mu$. Since the mass $m_c(m_c)$ is fully correlated to that scale, the uncertainty of $m_c(m_c)$ by the running from a given scale will be suppressed by approximately $\sqrt{2}$.

Figure 11

The prediction of the hyperfine splitting of charmonium in this work, compared to these of other works and experiment. Note the lattice results have not included the $\overline{c}c$ annihilation diagram which is expected to lower HFS by 1–5 MeV [29, 47, 48].

Figure 12

The interpolated values of ${\mathrm{\Delta }}_{\mathrm{HFS},\overline{\mathrm{c}}\mathrm{c}}$ on the data points of the two charm quark masses which bracket the physical one, for each ensemble with different $\beta$, versus the renormalized sea quark mass. Note that the $O(m^4{a}^4)$ effect is large so that then the continuum limit based on our functional form is between the data at the two finite lattice spacings.

Figure 13

The charm quark mass dependence (upper panel) and the strange quark mass dependence (lower panel) of $f_{D_s}$ with the other quark mass close to the physical point. This plot is based on the $\beta =2.13$ ensemble with lightest sea quark mass as an illustration.

Figure 14

The prediction of $f_{D_s}$ in this work, compared to those of other works.

Figure 15

We list the ratio of our simulation results to PDG averages. Note that the numbers of PDG averages are in italic type, and all the numbers are in unit of GeV, except $r_0$. For $r_0$, we list its value from HPQCD (0.4661(38) fm) and RBC-UKQCD (0.48(1) fm) for reference. Note that the values of the renormalized charm and strange quark masses are those at 2 GeV in $\overline{\mathrm{MS}}$ scheme.