Renormalon structure in compactified spacetime

Abstract

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.

15 pages, 1 figure, version to appear in PTEP

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Cited by in corpus (12)

• Non-Perturbative Completion of Hopf-Algebraic Dyson-Schwinger Equations
• Renormalons in integrable field theories
• Resurgence and renormalons in the one-dimensional Hubbard model
• Semiclassical Trans-Series from the Perturbative Hopf-Algebraic Dyson-Schwinger Equations: ${\varphi }^3$ QFT in 6 Dimensions
• Lattice $\mathbb{C}{P}^{N-1}$ model with ${\mathbb{Z}}_N$ twisted boundary condition: bions, adiabatic continuity and pseudo-entropy
• A new renormalon in two dimensions
• Quantum phase transition and Resurgence: Lessons from 3d $\mathcal{N}=4$ SQED
• Identification of perturbative ambiguity canceled against bion
• Renormalons beyond the Borel plane
• Confinement as Analytic Continuation Beyond Infinity
• Remarks on the large-$N$ $\mathbb{C}{P}^{N-1}$ model
• More on the infrared renormalon in $SU(N)$ QCD(adj.) on ${\mathbb{R}}^3\times S^1$