Renormalon structure in compactified spacetime

We point out that the location of renormalon singularities in theory on a circle-compactified spacetime $\mathbb{R}^{d-1} \times S^1$ (with a small radius $R \Lambda \ll 1$) can differ from that on the non-compactified spacetime $\mathbb{R}^d$. We argue this under the following assumptions, which are often realized in large-$N$ theories with twisted boundary conditions: (i) a loop integrand of a renormalon diagram is volume independent, i.e. it is not modified by the compactification, and (ii) the loop momentum variable along the $S^1$ direction is not associated with the twisted boundary conditions and takes the values $n/R$ with integer $n$. We find that the Borel singularity is generally shifted by $-1/2$ in the Borel $u$-plane, where the renormalon ambiguity of $\mathcal{O}(\Lambda^k)$ is changed to $\mathcal{O}(\Lambda^{k-1}/R)$ due to the circle compactification $\mathbb{R}^d \to \mathbb{R}^{d-1} \times S^1$. The result is general for any dimension $d$ and is independent of details of the quantities under consideration. As an example, we study the $\mathbb{C} P^{N-1}$ model on $\mathbb{R} \times S^1$ with $\mathbb{Z}_N$ twisted boundary conditions in the large-$N$ limit.