Gluons in charmoniumlike states

The mass components of charmoniumlike states are investigated through the decomposition of QCD energy-momentum tensor (EMT) on lattice. The quark mass contribution $⟨H_m⟩$ and the momentum fraction $⟨x⟩$ of valence charm quark and antiquark are calculated for conventional $1S$, $1P$, $1D$ charmonia and the exotic $1^{-+}$ charmoniumlike state, based on the $N_f=2+1$ gauge configurations generated by the RBC/UKQCD collaboration. It is found that $⟨H_m⟩$ is close to each other and around 2.0 to 2.2 GeV for these states, which implies that the mass splittings among these states come almost from the gluon contribution of QCD trace anomaly. The $⟨x⟩$ of the $1^{-+}$ state is only around 0.55, while that in conventional charmonia is around 0.7 to 0.8. This difference manifests that the proportion of light quarks and gluons in the $1^{-+}$ charmoniumlike state is significantly larger than conventional states.

Figure 1

Effective mass as defined in Eq. (5) from simulation data (points) and from $C_2(t)$ of Eq. (10) with best fit parameters (color band) for various $J^{PC}$ quantum numbers on ${32}^3\times 64$ (top) and ${48}^3\times 96$ (bottom) configuration. The points with error bar are simulated data with jackknife estimated error, the light color band shows the fitted result with best fit parameters, and the dark color band shows the fitting range.

Figure 2

Effective matrix element of valence charm quark mass contribution $⟨H_m⟩(t)$ as defined in Eq. (9) from simulation data (points) and from $C_3(t,H_m)$ of Eq. (10) with best fit parameters (color band) for various $J^{PC}$ quantum numbers on ${32}^3\times 64$ (top) and ${48}^3\times 96$ (bottom) configuration. The points with errorbar are simulated data with jackknife estimated error, the light color band shows the fitted result with best fit parameters, and the dark color band shows the fitting range.

Figure 3

Effective matrix element of valence charm quark momentum fraction $⟨x{⟩}_q(t)$ as defined in Eq. (9) from simulation data (points) and from $C_3(t,x)$ of Eq. (10) with best fit parameters for various $J^{PC}$ quantum numbers on ${32}^3\times 64$ (top) and ${48}^3\times 96$ (bottom) configuration. The points with error bar are simulated data with jackknife estimated error, the light color band shows the fitted result with best fit parameters, and the dark color band shows the fitting range.

Figure 4

Ground state mass $M$ of various $J^{PC}$ quantum numbers on ${32}^3\times 64$ and ${48}^3\times 96$ configurations, the height of color boxes indicates the statistical errors and the experimental value for $1S$ and $1P$ states from PDG are shown for comparison.

Figure 5

Ground state $⟨H_m⟩$ and $⟨x{⟩}_q$ of valence charm quark of various $J^{PC}$ quantum numbers on ${32}^3\times 64$ and ${48}^3\times 96$ configuration.

Figure 6

Combined valence charm quark contribution $⟨H_q⟩$ in ground state of various $J^{PC}$ quantum numbers on ${32}^3\times 64$ and ${48}^3\times 96$ configuration, the extracted mass $M$ are also shown for comparison.