A new renormalon in two dimensions

TL;DR

This paper reveals a new IR renormalon singularity in two-dimensional super-renormalizable scalar theories with broken symmetry, challenging the standard belief that such theories only have instanton singularities.

Contribution

It demonstrates the existence of an IR renormalon in 2D super-renormalizable theories with broken symmetry and connects it to non-perturbative effects like Goldstone bosons.

Findings

Identification of an IR renormalon in 2D scalar theories with broken symmetry

Perturbative calculation of Lieb--Liniger energy up to two loops

Agreement of perturbative results with Bethe ansatz predictions

Abstract

According to standard lore, perturbative series of super-renormalizable theories have only instanton singularities. In this paper we show that two-dimensional scalar theories with a spontaneously broken $O(N)$ symmetry at the classical level, which are super-renormalizable, have an IR renormalon singularity at large $N$. Since perturbative expansions in these theories are made around the "false vacuum" in which the global symmetry is broken, this singularity can be regarded as a manifestation of the non-perturbative absence of Goldstone bosons. We conjecture that the Borel singularity in the ground state energy of the Lieb--Liniger model is a non-relativistic manifestation of this phenomenon. We also provide {\it en passant} a detailed perturbative calculation of the Lieb--Liniger energy up to two-loops, and we check that it agrees with the prediction of the Bethe ansatz.