Casimir Energy of Confining Large $N$ Gauge Theories

Four-dimensional asymptotically free large $N$ gauge theories compactified on $S_R^3\times \mathbb{R}$ have a weakly coupled confining regime when $R$ is small compared to the strong scale. We compute the vacuum energy of a variety of confining large $N$ nonsupersymmetric gauge theories in this calculable regime, where the vacuum energy can be thought of as the $S^3$ Casimir energy. The $N=\infty$ renormalized vacuum energy turns out to vanish in the class of theories we have examined. This matches an implication of a recently observed temperature-reflection symmetry of such systems.

Figure 1

Structure of singularities (red dots) coming from the first 45 terms in Eq. (4) in the large $N$ confining-phase partition functions of gauge theories with adjoint matter on $S^3\times S^1$, in the complex plane for $y=e^{-\beta /(2R)}$. The blue curve is an example of a path from $y=0$ to $y=1$ that does not pass through any singularities. Left: Yang Mills theory ($n_F=0,n_S=0$). Right: Gauge theory with $n_F=1$, $n_S=2$.

Figure 2

Visualization of Eq. (9) for pure $N=\infty$ Yang-Mills theory as a function of the UV cutoff $\epsilon =(\mu R{)}^{-1}$ such that $y=e^{-e^{-i\alpha }\epsilon }$, for $k_{\max }=500$ (solid red curve) and $k_{\max }=100$ (dashed blue curve), with fixed $\alpha =\pi /4$. The finiteness of ${\epsilon }^2{\partial }_{\epsilon }{Z}_{\mathrm{ST}}$ as $\epsilon \to 0$ implies that in the $N=\infty$ theory the leading divergence in the vacuum energy density calculation is ${\mu }^2$, rather than the ${\mu }^4$ familiar from generic 4D QFTs. The deviation from linearity at very large $\mu$ is due to the finiteness of $k_{\max }$.