Energy spectrum of the low-lying gluon excitations in the Coulomb gauge

We compute the energy spectrum of low-lying gluonic excitations in the presence of static quark-antiquark sources using Coulomb gauge and the quasiparticle representation. Within the valence sector of the Fock space we reproduce both, the overall normalization and the ordering of the spin-parity multiplets. We discus how the interactions induced by the nonabelian Coulomb kernel are central in to fine structure of the spectrum.

Figure 1

Matrix elements, $⟨R,{\mathbf{k}}^{\prime },{\lambda }^{\prime }|H|R,\mathbf{k},\lambda ⟩$. Diagrams a) and b) represent gluon and quark self energies, respectively. Diagrams c) and d) represent the Coulomb interaction, $V_C$ between the gluon and one of the quarks and between the two quarks, respectively. In the bottom row, diagrams e) and f) describe matrix elements of the interaction term resulting from expansion of the Coulomb kernel $K[A]$ in up to one power in gluon field.

Figure 2

Spectrum of the $\Lambda =1$, $\Pi$ states compared to the ground state $Q\overline{Q}$ potential without the three-body quark-antiquark-gluon interaction corresponding to the diagram $e$ in Fig. 1. Lattice results are taken from 1. The energy scale is $r_0=(400\mathrm{MeV}{)}^{-1}$.

Figure 3

Same as in Fig. 2 for the $\Lambda =0$, $\Sigma$ energies.

Figure 4

Same as in Fig. 2 for the $\Lambda =2$, $\Delta$ energies.

Figure 5

Spectrum if the $\Lambda =1$, $\Pi$ states compared to the ground state $Q\overline{Q}$ potential using the complete Hamiltonian, which includes the three-body quark-antiquark-gluon interactions.

Figure 6

Same as in Fig. 5 for the $\Lambda =0$, $\Sigma$ energies.

Figure 7

Same as in Fig. 5 for the $\Lambda =2$, $\Delta$ energies.

Figure 8

Splittings between the excited energies of quasigluons and the ground state $Q\overline{Q}$ energy. The values $N$ expected from the string model are, $\mathbf{N}=\mathbf{1}$ for ${\Pi }_u$, $N=2$ for ${\Pi }_g$ and ${\Sigma }_g^{\prime }$ and $N=4$ for ${\Sigma }_g^{-}$ ($a=0.2\mathrm{fm}$).