Covariant background field calculation of ultraviolet counterterms

TL;DR

This paper uses the covariant background field method to compute ultraviolet counterterms in nonlinear sigma models, revealing their renormalization properties up to two loops and showing non-renormalizability at that order.

Contribution

It provides a detailed two-loop calculation of counterterms for nonlinear sigma models with arbitrary target manifolds, analyzing their renormalizability and finiteness.

Findings

One-loop models are renormalizable and finite on Ricci-flat manifolds.

Two-loop calculations show non-renormalizable counterterms in all cases.

Renormalizability is lost at two loops for these models.

Abstract

Based on the covariant background field method, we calculate the ultraviolet counter\-terms up to two-loop order and discuss the renormalizability of the three-dimensional non-linear sigma models with arbitrary Riemannian manifolds as target spaces. We investigate the bosonic model and its supersymmetric extension. We show that at the one-loop level these models are renormalizable and even finite when the manifolds are Ricci-flat. However, at the two-loop order, we find non-renormalizable counterterms in all cases considered, so the renormalizability and finiteness of such models are completely lost in this order.