QCD-like theories on R-3 x S-1: A smooth journey from small to large r(S-1) with double-trace deformations

We consider QCD- like theories with one massless fermion in various representations of the gauge group SU(N). The theories are formulated on R-3 x S-1. In the decompactification limit of large r(S-1) all these theories are characterized by confinement, mass gap, and spontaneous breaking of a (discrete) chiral symmetry (chi SB). At small r(S-1), in order to stabilize the vacua of these theories at a center- symmetric point, we suggest to perform a double- trace deformation. With this deformation, the theories at hand are at weak coupling at small r(S-1) and yet exhibit basic features of the large r(S-1) limit: confinement and chi SB. We calculate the string tension, mass gap, bifermion condensates, and theta dependence. The double- trace deformation becomes dynamically irrelevant at large r(S-1). Despite the fact that at small r(S-1) confinement is Abelian, while it is expected to be non- Abelian at large r(S-1), we argue that small and large r(S-1) physics are continuously connected. If so, one can use small r(S-1) laboratory to extract lessons about QCD and QCD- like theories on R-4.