We consider QCD-like theories with one massless fermion in various representations of the gauge group $\mathrm{SU}(N)$. The theories are formulated on $R_3\times S_1$. 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 ($\chi \mathrm{SB}$). 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 $\chi \mathrm{SB}$. 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 $R_4$.

• Received 31 March 2008

DOI:https://doi.org/10.1103/PhysRevD.78.065004