Splittings of low-lying charmonium masses at the physical point

We present high-precision results from lattice QCD for the mass splittings of the low-lying charmonium states. For the valence charm quark, the calculation uses Wilson-clover quarks in the Fermilab interpretation. The gauge-field ensembles are generated in the presence of up, down, and strange sea quarks, based on the improved staggered (asqtad) action, and gluon fields, based on the one-loop, tadpole-improved gauge action. We use five lattice spacings and two values of the light sea-quark mass to extrapolate the results to the physical point. An enlarged set of interpolating operators is used for a variational analysis to improve the determination of the energies of the ground states in each channel. We present and implement a continuum extrapolation within the Fermilab interpretation, based on power-counting arguments, and thoroughly discuss all sources of systematic uncertainty. We compare our results for various mass splittings with their experimental values, namely, the 1S hyperfine splitting, the 1P-1S splitting and the P-wave spin-orbit and tensor splittings. Given the uncertainty related to the width of the resonances, we find excellent agreement between our results and the mass splittings derived from the hadron masses measured in the laboratory.

Figure 1

Shapes and size of the expected heavy-quark discretization uncertainties for charmonium splittings (NRQCD power counting [60]) in the Fermilab approach (using $v^2=0.3$ and $m{v}^2\approx 420\mathrm{MeV}\approx 1\mathrm{P}\text{−}1\mathrm{S}$ splitting). These are as in Figs. 3 and 4 of Ref. [59], and the notation for the terms follows that reference. Values of ${\alpha }_s$ consistent with those in Table 5 have been used. The terms arising from mass mismatches are denoted in the plot by the masses in the short-distance coefficients. In addition a rotational symmetry breaking term (with coefficient $w_4$) is important for the 1P-1S splitting. Expressions for the short-distance coefficients can be found in Ref. [59].

Figure 2

Chiral and continuum fit for the 1S hyperfine splitting. The black circles denote the lattice data. Curves for physical (black), $0.1m_s$ (red), and $0.2m_s$ (blue) light-quark masses are plotted. Due to the mistuning of the strange quark in the sea, which differs from ensemble to ensemble, the data points appear away from the curves. To illustrate that the data are well described by the fit, the black crosses show the fit results evaluated at the lattice parameters of the gauge ensemble. The magenta symbol indicates the result in the combined chiral and continuum limit.

Figure 3

Chiral and continuum fit for the 1P-1S splitting. The black circles denote the lattice data. Curves for physical (black), $0.1m_s$ (red), and $0.2m_s$ (blue) light-quark masses are plotted. The black crosses show the fit results evaluated at the lattice parameters of the gauge ensemble. The magenta symbol indicates the result in the combined chiral and continuum limit.

Figure 4

Chiral and continuum fit for the 1P spin-orbit splitting. The black circles denote the lattice data. Curves for physical (black), $0.1m_s$, (red) and $0.2m_s$ (blue) light-quark masses are plotted. The black crosses show the fit results evaluated at the lattice parameters of the gauge ensemble. The magenta symbol indicates the result in the combined chiral and continuum limit.

Figure 5

Chiral and continuum fit for the 1P tensor splitting. The black circles denote the lattice data. Curves for physical (black), $0.1m_s$, (red) and $0.2m_s$ (blue) light-quark masses are plotted. The black crosses show the fit results evaluated at the lattice parameters of the gauge ensemble. The magenta symbol indicates the result in the combined chiral and continuum limit.

Figure 6

Chiral and continuum fit for the P-wave hyperfine splitting. The black circles denote the lattice data. Curves for physical (black), $0.1m_s$ (red), and $0.2m_s$ (blue) light-quark masses are plotted. The black crosses show the fit results evaluated at the lattice parameters of the gauge ensemble. The magenta symbol indicates the result in the combined chiral and continuum limit.

Figure 7

Systematic variation of the charmonium mass splittings when varying the details of the chiral-continuum extrapolation. The label “default” indicates our final result described in detail for each splitting in Sec. 4. The variations (letters A to G) are described in Table 6.

Figure 8

The connected part of the 1S hyperfine splitting from lattice-QCD calculations. Results that include a continuum limit (and an extrapolation to physical sea-quark masses where appropriate) are shown alongside two recent preliminary results. For comparison we also show the 1S hyperfine splitting from the PDG [27]. The sum in quadrature of the statistical and systematic uncertainties quoted by the authors of the respective publication are shown. Effects from charm-anticharm annihilation (of valence quarks) are not included in any of these calculations. Note that calculations neglecting disconnected contributions and treating the ${\eta }_c$ as stable need not result in the same value as the PDG.

Figure 9

Sample plot of the $t_0$ dependence for the $J/\psi$ bound state on the finest lattice. The actual dependence for our choice of fit range is shown in red with square symbols (slightly displaced on the $x$ axis; the value used for the final results is highlighted magenta). The black circles result from artificially emphasizing residual excited-state contaminations due to the choice of $t_0$ by truncating the (upper) fit range. While a clear $t_0$ dependence is visible in this artificial case, this small effect is insignificant for all P-wave states (where the fit plateaus are short) and tends to get further canceled in energy differences.