is bothersome, but it is consistent with the scenario of “no confinement without Coulomb confinement” of Zwanziger. The Coulomb potential leads to a variational (stronger) upper bound to the true confining interaction.

Similar behavior at large distances is also true for the three quark sources, except that here we find the emergence of the “Y”-shape junction. This is consistent with lattice simulations, but is remarkable in our case as it arises from two-body forces. It will be interesting to examine field distributions which include transverse field excitations. In that case the only lattice results available are for the potential, not for the field distributions. Finally we note that, since the mean field calculation provides a variational upper bound, the long range behavior of the field distribution falls-off more slowly than expected for the Van der Waals force. Certainly as the complete string develops this is expected to disappear and it would be interesting to build a string-like model for the ansatz ground state to verify this assertion.

The authors wish to thank J. Greensite, H. Reinhardt, Y. Simonov, F. Steffen, H. Suganuma and D. Zwanziger for helpful feedback. This work was supported in part by the US Department of Energy 