Trans-series from condensates in the non-linear sigma model

TL;DR

This paper develops a perturbative framework for the 2D non-linear sigma model using a limit of the linear sigma model, enabling the calculation of operator condensates and revealing the nature of renormalons.

Contribution

It introduces a massless perturbative approach based on the linear sigma model that reproduces non-perturbative effects and clarifies the role of UV renormalons in the NLSM.

Findings

Reproduces perturbative and condensate contributions at next-to-leading order in 1/N.

Shows decoupling of UV physics from IR in the weak-coupling limit.

Identifies the first renormalon as an UV renormalon canceling condensate ambiguities.

Abstract

In this work we provide a massless perturbative framework for the two dimensional non-linear sigma model (NLSM), that allows the computation of the perturbative series attached to the operator condensates in the operator product expansion (OPE). It is based on a limit of the quartic linear sigma model (LSM) and is manifestly $O(N)$ symmetric. We show, at next-to-leading order in the $1/N$ expansion, how this framework reproduces the perturbative contribution to the two-point function, as well as its first exponentially small correction due to the condensate of the Lagrangian operator, in full agreement with the exact non-perturbative large $N$ solution. We also show that, in the full LSM, the physics at the natural UV cutoff indeed decouples from the NLSM in the IR, in the weak-coupling limit. In particular, we show that the perturbative framework for the LSM at the cutoff scale is connected to the one in the NLSM. The structure of power divergences in the LSM regularization also reveals that the first renormalon on the positive Borel axis of the NLSM perturbative self-energy is an UV renormalon, which cancels against the ambiguity in the condensate.