# Corrections to finite-size scaling in two-dimensional O(N) sigma-models

## TL;DR

This paper analyzes finite-size scaling corrections in two-dimensional O(N) sigma-models at N=∞, revealing how these corrections depend on the type of action and discussing complex behaviors in models with four-spin interactions.

## Contribution

It provides a detailed calculation of leading finite-size correction terms for a broad class of O(N) sigma-models, highlighting differences based on action improvements.

## Key findings

- Corrections behave as (f(z) log L + g(z))/L^2, with f(z) vanishing for Symanzik improved actions.
- On-shell improved actions do not exhibit the same correction behavior as Symanzik improved actions.
- Models with four-spin interactions display more complex finite-size correction behaviors.

## Abstract

We have considered the corrections to the finite-size-scaling functions for a general class of $O(N)$ $\sigma$-models with two-spin interactions in two dimensions for $N=\infty$. We have computed the leading corrections finding that they generically behave as $(f(z) \log L + g(z))/L^2$ where $z = m(L) L$ and $m(L)$ is a mass scale; $f(z)$ vanishes for Symanzik improved actions for which the inverse propagator behaves as $q^2 + O(q^6)$ for small $q$, but not for on-shell improved ones. We also discuss a model with four-spin interactions which shows a much more complicated behaviour.