Infrared renormalon in $SU(N)$ QCD(adj.) on $\mathbb{R}^3\times S^1$

TL;DR

This paper investigates the infrared renormalon in $SU(N)$ QCD with adjoint fermions on $R^3 imes S^1$, demonstrating that the renormalon ambiguity persists in the compactified space and analyzing its dependence on calculation order.

Contribution

It provides the first detailed analysis of the infrared renormalon in $SU(N)$ QCD(adj.) on $R^3 imes S^1$ using the large-$eta_0$ approximation, highlighting the ambiguity's persistence under compactification.

Findings

Renormalon ambiguity at $u=2$ in the large-$N$ limit.

Renormalon ambiguity in $R^3 imes S^1$ matches that in $R^4$.

The Borel singularity location depends on the order of operations.

Abstract

We study the infrared renormalon in the gluon condensate in the $SU(N)$ gauge theory with $n_W$-flavor adjoint Weyl fermions (QCD(adj.)) on~$\mathbb{R}^3\times S^1$ with the $\mathbb{Z}_N$ twisted boundary conditions. We rely on the so-called large-$\beta_0$ approximation as a conventional tool to analyze the renormalon, in which only Feynman diagrams that dominate in the large-$n_W$ limit are considered while the coefficient of the vacuum polarization is set by hand to the one-loop beta function~$\beta_0=11/3-2n_W/3$. In the large~$N$ limit within the large-$\beta_0$ approximation, the W-boson, which acquires the twisted Kaluza--Klein momentum, produces the renormalon ambiguity corresponding to the Borel singularity at~$u=2$. This provides an example that the system in the compactified space~$\mathbb{R}^3\times S^1$ possesses the renormalon ambiguity identical to that in the uncompactified space~$\mathbb{R}^4$. We also discuss the subtle issue that the location of the Borel singularity can change depending on the order of two necessary operations.