QCD-like Theories on R_3\times S_1: a Smooth Journey from Small to Large r(S_1) with Double-Trace Deformations

TL;DR

The paper analyzes QCD-like theories on $\mathbb{R}_3\times S_1$ with one fermion flavor across representations by introducing a double-trace deformation to stabilize center symmetry at small circle size. This enables analytic control at small $L$, where confinement, a mass gap, and discrete chiral symmetry breaking arise via magnetic bion mechanisms, while reproducing key large-$L$ features of confinement. The authors show a continuous connection between Abelian confinement at small $L$ and non-Abelian confinement at large $L$, arguing for a smooth interpolation without phase transitions and suggesting that small-$L$ physics can yield insights into QCD on $\mathbb{R}_4$. They also discuss theta-dependence and planar equivalence to ${\cal N}=1$ SYM, with implications for lattice tests and extensions to more flavors to illuminate QCD-like dynamics on $R_4$.

Abstract

We consider QCD-like theories with one massless fermion in various representations of the gauge group SU$(N)$. The theories are formulated on $R_3\times S_1$. 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 ($χ$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 these 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 $χ$SB. We calculate the string tension, mass gap, bifermion condensates and $θ$ 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 $R_4$.

Figures (2)

• Figure 1: Quantum chromodynamics as a function of compactified direction circumference before and after surgery (QCD and QCD$^*$, respectively). $L_c$ is the point of a phase transition.
• Figure 2: $Z_N$ symmetric vacuum fields $v_k$.