Gauge theories with non-unitary parallel transporters : a soluble Higgs model

It has been proposed to abandon the requirement that parallel transporters in gauge theories are unitary (or pseudo-orthogonal). This leads to a geometric interpretation of Vierbein fields as parts of gauge fields, and non-unitary parallel transport in extra directions yields Higgs fields. In such theories, the holonomy group H is larger than the gauge group G. Here we study a one-dimensional model with fermions which retains only the extra dimension, and which is soluble in the sense that its renormalization group flow may be exactly computed, with G = SU(2) and non-compact \(H\subseteq GL(2,\mathbf{C})\), or G = U(2), H = GL(2,C). In all cases the asymptotic behavior of the Higgs potential is computed, and with one possible exception for G = SU(2), H = GL(2,C), there is a flow of the action from a UV fixed point which describes a SU(2)-gauge theory with unitary parallel transporters, to a IR fixed point. We explain how exponential mass ratios of fermions of different flavor can arise through spontaneous symmetry breaking, within the general framework.