Gluon chain formation in presence of static charges

We consider the origins of the gluon chain model. The model serves as a realization of the dynamics of the chromoelectric flux between static quark-antiquark sources. The derivation is based on the large-$N_C$ limit of the Coulomb gauge Hamiltonian in the presence of a background field introduced to model magnetic charge condensation inducing electric confinement.

Figure 1

Interaction between quark charge (upper line) and the off-diagonal gluon charge ${\rho }_{ij}^2$ which creates two gluons. The dashed line represents the Coulomb potential given by the vev of the fully dressed Coulomb kernel.

Figure 2

Illustration of the matrix element of the one-body operator Hamiltonian, $E_i{\alpha }^i{\alpha }_i$, in the gluon chain state with $N=1$.

Figure 3

Part of the leading contribution at large $N_C$ from $H_{qq}$ which corresponds to quark or antiquark self-energy, shown here for the $N=1$ gluon chain state.

Figure 4

Quark-gluon, $H_{qg}^D$, and gluon-gluon interaction, $H_{gg}^D$, matrix elements, for $N=1$ and $N=2$ chain states, respectively.

Figure 5

Diagonal $H_{gb}^D$ and off-diagonal $H_{gb}^M$ interactions between quasigluons and the background field (shown as blobs), for $N=3$ gluon chain states.

Figure 6

A nonplanar diagram, which we do not take into account, representing direct Coulomb interaction between the quark and the antiquark in the presence of 2 gluons.

Figure 7

Ground state energy (solid line) of the chain Hamiltonian in model I. A power law fit yields $E_0^{\mathrm{I}}=0.984(R\mathrm{GeV}{)}^{0.787}\mathrm{GeV}$. The dashed line gives the energy of the bare state using for the string tension ${\overline{\sigma }}_C=0.1{\mathrm{GeV}}^2$.

Figure 8

Expectation value of the number of gluons in the chain as a function of $Q\overline{Q}$ separation, $R$. A power law fit gives $⟨N{⟩}^{\mathrm{I}}=0.985(R\mathrm{GeV}{)}^{0.623}$ and $⟨N{⟩}^{\mathrm{II}}=0.984(R\mathrm{GeV}{)}^{0.787}$ for model I (solid line) and model II (dashed line), respectively.

Figure 9

Ground state energy (solid line) of the chain Hamiltonian in model II. A power law fit yields $E_0^{\mathrm{I}}=1.867(R\mathrm{GeV}{)}^{0.379}\mathrm{GeV}$. The dashed line gives the energy of the bare state using for the string tension ${\overline{\sigma }}_C=0.1{\mathrm{GeV}}^2$.