Center-stabilized Yang-Mills theory: Confinement and large $N$ volume independence

We examine a double trace deformation of $SU(N)$ Yang-Mills theory which, for large $N$ and large volume, is equivalent to unmodified Yang-Mills theory up to $O(1/N^2)$ corrections. In contrast to the unmodified theory, large $N$ volume independence is valid in the deformed theory down to arbitrarily small volumes. The double trace deformation prevents the spontaneous breaking of center symmetry which would otherwise disrupt large $N$ volume independence in small volumes. For small values of $N$, if the theory is formulated on ${\mathbb{R}}^3\times S^1$ with a sufficiently small compactification size $L$, then an analytic treatment of the nonperturbative dynamics of the deformed theory is possible. In this regime, we show that the deformed Yang-Mills theory has a mass gap and exhibits linear confinement. Increasing the circumference $L$ or number of colors $N$ decreases the separation of scales on which the analytic treatment relies. However, there are no order parameters which distinguish the small and large radius regimes. Consequently, for small $N$ the deformed theory provides a novel example of a locally four-dimensional pure-gauge theory in which one has analytic control over confinement, while for large $N$ it provides a simple fully reduced model for Yang-Mills theory. The construction is easily generalized to QCD and other QCD-like theories.

Figure 1

Large $N$ equivalences relating ordinary and deformed $SU(N)$ Yang-Mills theories, as a function of the size $L$ of the periodic volume in which the theories are defined. In deformed YM volume independence holds for all $L$, while in ordinary YM volume independence fails below a critical size, $L<L_{\mathrm{c}}$, (shaded region) due to spontaneous breaking of center symmetry. This prevents reduction all the way down to a single-site matrix model for ordinary YM. Large $N$ equivalence holds between ordinary and deformed YM theories as long as center symmetry is unbroken in ordinary YM. Large $N$ volume independence is a type of orbifold equivalence, as discussed in Ref. 14. The combination of volume changing orbifold projections in the deformed theory, along with the deformation equivalence in sufficiently large volume, provides a useful equivalence between deformed YM in small volume and ordinary YM in large volumes. In particular, a single-site matrix model of the deformed theory will reproduce properties of ordinary Yang-Mills theory in infinite volume. The construction can be generalized to QCD, with the deformed theory providing a fully reduced matrix model for QCD.