Confinement for all values of the coupling in four-dimensional SU(2) gauge theory

E. T. Tomboulis

Published 2007-07-15Version 1

A derivation is given from first principles of the fact that the SU(2) gauge theory is in a confining phase for all values of the coupling $0 < g < \infty$ defined at lattice spacing (UV regulator) $a$, and space-time dimension $d \leq 4$. The strategy is to employ approximate RG decimation transformations of the potential moving type which give both upper and lower bounds on the partition function at each successive decimation step. By interpolation between these bounds an exact representation of the partition function is obtained on progressively coarser lattices. In the same manner, one obtains a representation of the partition function in the presence of external center flux. Under successive decimations the flow of the effective action in these representations is constrained by that in the upper and lower bounds which are easily explicitly computable. Confining behavior for the vortex free energy order parameter (ratio of partition functions with and without external flux), hence `area law' for the Wilson loop, is the result for any initial coupling. Keeping the string tension fixed determines the dependence $g(a)$, which is such that $g(a) \to 0$ for $a \to 0$.