# Infrared renormalon in the supersymmetric $\mathbb{C}P^{N-1}$ model on $\mathbb{R}\times S^1$

## TL;DR

This paper investigates the infrared renormalon structure in the large-N limit of the supersymmetric ^N-1 model on Rs1, revealing a novel singularity at u=3/2 that challenges existing semi-classical interpretations.

## Contribution

It uncovers a new renormalon singularity at u=3/2 in the compactified model, differing from the uncompactified case, and questions the semi-classical bion picture of infrared renormalons.

## Key findings

- The u=2 renormalon singularity disappears on Rs1.
- A new singularity at u=3/2 emerges in the compactified model.
- The semi-classical interpretation of the u=3/2 singularity remains unclear.

## Abstract

In the leading order of the large-$N$ approximation, we study the renormalon ambiguity in the gluon (or, more appropriately, photon) condensate in the 2D supersymmetric $\mathbb{C}P^{N-1}$ model on~$\mathbb{R}\times S^1$ with the $\mathbb{Z}_N$ twisted boundary conditions. In our large~$N$ limit, the combination $\Lambda R$, where $\Lambda$ is the dynamical scale and $R$~is the $S^1$ radius, is kept fixed (we set $\Lambda R\ll1$ so that the perturbative expansion with respect to the coupling constant at the mass scale~$1/R$ is meaningful). We extract the perturbative part from the large-$N$ expression of the gluon condensate and obtain the corresponding Borel transform~$B(u)$. For~$\mathbb{R}\times S^1$, we find that the Borel singularity at~$u=2$, which exists in the system on the uncompactified~$\mathbb{R}^2$ and corresponds to twice the minimal bion action, disappears. Instead, an unfamiliar renormalon singularity \emph{emerges\/} at~$u=3/2$ for the compactified space~$\mathbb{R}\times S^1$. The semi-classical interpretation of this peculiar singularity is not clear because $u=3/2$ is not dividable by the minimal bion action. It appears that our observation for the system on~$\mathbb{R}\times S^1$ prompts reconsideration on the semi-classical bion picture of the infrared renormalon.