Toward a proof of long range order in 4-d SU(N) lattice gauge theory

Abstract

An extended version of 4-d SU(2) lattice gauge theory is considered in which different inverse coupling parameters are used, $\beta_H=4/g_{H}^2$ for plaquettes which are purely spacelike, and $\beta_V$ for those which involve the Euclidean timelike direction. It is shown that when $\beta_H = \infty$ the partition function becomes, in the Coulomb Gauge, exactly that of a set of non-interacting 3-d O(4) classical Heisenberg models. Long range order at low temperatures (weak coupling) has been rigorously proven for this model. It is shown that the correlation function demonstrating spontaneous magnetization in the ferromagnetic phase is a continuous function of $g_H$ at $g_H =0$ and therefore that the spontaneously broken phase enters the ($\beta_H$, $\beta_V$) phase plane (no step discontinuity at the edge). Once the phase transition line has entered, it can only exit at another identified edge, which requires the SU(2) gauge theory within also to have a phase transition at finite $\beta$. A phase exhibiting spontaneous breaking of the remnant symmetry left after Coulomb gauge fixing, the relevant symmetry here, is non-confining. Easy extension to the SU(N) case implies that the continuum limit of zero-temperature 4-d SU(N) lattice gauge theories is not confining, in other words gluons by themselves do not produce a confinement.