Coulomb energy and gluon distribution in the presence of static sources

We compute the energy of the ground state and a low-lying excitation of the gluonic field in the presence of static quark-anti-quark ($q\overline{q}$) sources. We show that for separation between the sources less then a few fm the gluonic ground state of the static $q\overline{q}$ system can be well described in terms of a mean field wave functional with the excited states corresponding to a single quasiparticle excitation of the gluon field. We also discuss the role of many particle excitations relevant for large separation between sources.

Figure 1

The $O(g^4)$, one loop diagrams contributing to the leading log term in the expansion of the $\beta$ function in YM theory with heavy sources. (a) Antiscreening dressing of the Coulomb potential by transverse gluons, (b) Debye screening of the Coulomb potential by transverse gluons. The Coulomb potentials are represented by the dashed lines, and sources by thick lines.

Figure 2

Comparison between $V(R)=C_F{V}_C(r)$ from Eq. (24) (solid line) and $V(R)=E_0(R)$ lattice data from [1] ($r_0=1/450{\mathrm{MeV}}^{-1}$).

Figure 3

Matrix elements, $⟨R,{\mathbf{k}}^{\prime },{\lambda }^{\prime }|H|R,\mathbf{k},\lambda ⟩$. (a) and (b) represent gluon and quark self-energies, respectively. (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, (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. The short dashed vertical lines in (c)–(e) represent the dressed (blob) Coulomb potentials.

Figure 4

The matrix element $⟨R|H|R,\mathbf{k},\lambda ⟩$. It originates from expansion of the Coulomb kernel $K[A]$ to first order in $A$.

Figure 5

Comparison between $V(R)=C_F{V}_C(r)$ from Eq. (24) (lower solid line) and the $V(R)=E_0(R)$ lattice data from 1 ($r_0=1/450{\mathrm{MeV}}^{-1}$). The upper solid line represents the first excited state.

Figure 6

Comparison between $V(R)=C_F{V}_C(r)$ from Eq. (24) (lower solid line) and the $V(R)=E_0(R)$ lattice data from 1 ($r_0=1/450{\mathrm{MeV}}^{-1}$). The (dashed) line labeled $qq$ without $qqg$ mixing is only slightly above the (lower solid) line labeled $qq$ with $qqg$ mixing. The upper solid line represents the first excited state. For the excited state the effect of mixing with the $qqg$ component is even smaller than for the ground state and invisible on the scale of this graph.

Figure 7

Normalized probability of finding the bare $|q\overline{q}⟩$ state in the full ground state of the $|qq,N=0⟩$ (which is also equal to the probability of finding the $|q\overline{q}g⟩$ state in the first excited $|qq,N=1⟩$ state).

Figure 8

Matrix elements $⟨qq|H|q\overline{q},2g⟩$ leading to the $|q\overline{q},2g⟩$ component in the ground state ${\Sigma }_g^{+}$ potential, (a) interaction mediated via the Coulomb line coupled to quark sources, (b) interaction between a single quark and the gluon charge density.

Figure 9

Typical diagrams contribution to mixing between $n$ and $m$ gluon states. Vertical dots represent any number of gluons not affected by the interaction. (a) mixing mediated by the Coulomb potential, (b) same as in (a) with the rearrangement of gluons, (c) long-range Coulomb interaction between gluon charge densities, (d) same as in (c) but with the charge density of the quark sources.

Figure 10

Shift in the ground state ${\Sigma }_g^{+}$ energy due to coupling with mutigluon states of the model Hamiltonian of Eqs. (36, 37). The maximum number of states was taken to be $n_{g,\max }=40$. The other parameters are $b=0.21{\mathrm{GeV}}^{-2}$ and $m_g=0.65\mathrm{GeV}$.

Figure 11

Average number of quasigluons in the full eigenstate of the model Hamiltonian of Eqs. (36, 37).