Is the ground state of Yang-Mills theory Coulombic?

We study trial states modelling the heavy quark-antiquark ground state in SU(2) Yang-Mills theory. A state describing the flux tube between quarks as a thin string of glue is found to be a poor description of the continuum ground state; the infinitesimal thickness of the string leads to UV artifacts which suppress the overlap with the ground state. Contrastingly, a state which surrounds the quarks with non-Abelian Coulomb fields is found to have a good overlap with the ground state for all charge separations. In fact, the overlap increases as the lattice regulator is removed. This opens up the possibility that the Coulomb state is the true ground state in the continuum limit.

Figure 1

Overlap between the axial state and the ground state, at various lattice spacings, plotted against the number of lattice points $n=r/a$ between charges (for simulation details, see the text). The overlap is independent of $\beta$.

Figure 2

Overlap between the axial state and the ground state, plotted against the physical separation of the charges. Some lines have been added to guide the eye. It is seen that, for all $r$, the overlap is smaller for finer lattice spacings (larger $\beta$).

Figure 3

Smeared state overlap with the ground state at various lattice spacings. For a fixed smearing level $S$ the results lie on the same curve, they are $\beta$ independent. We have added data for the axial state ($S=0$) for comparison, and some lines to guide the eye. Lattice: $500$ configurations, $T=4a$, relative error of Wilson line correlator $<0.5$, ${\chi }^2/\nu <3$ and $Q>{10}^{-2}$.

Figure 4

Smeared state overlap with the ground state, plotted against physical separation. Axial state data ($S=0$) is displayed for comparison and some lines have been added to guide the eye. Legend as in Fig. 3.

Figure 5

Improved action simulation of the Coulomb overlap at fixed $\beta =1.55$, ${20}^4$ lattice. As $T$ increases contributions from excited states decouple, and we see a decrease in the overlap, which nevertheless remains above 0.7 for the $T$ values plotted—however, the difference between results at different $T$ is decreasing.

Figure 6

Improved action simulation of the overlap between the Coulomb and ground states. At a fixed time, here $T=3a$, we observe an increase in the overlap in the continuum limit ($\beta \to \infty$ or $a\to 0$).

Figure 7

A summary of the improved action results for the Coulomb overlap, plotted at a range of $\beta$ and $T$ values.

Figure 8

Wilson action overlap of the Coulomb state and ground state, plotted against the number of lattice sites $n=r/a$ between the charges. For lattice data, see the text. Measurements were taken from $T\ge 3a$. Notably, the overlap increases as we go to the continuum, $\beta \to \infty$.

Figure 9

Wilson action overlap of the Coulomb state and ground state, plotted against the physical separation of the charges. The overlap increases as we increase $\beta$.