O. Gamayun

A. Losev

M. Shifman

The beta function for a class of sigma models is not found to be geometric, but rather has an elegant form in the context of algebraic data.

In quantum field theory, beta functions describe the flow of coupling constants, which for sigma models exist as a metric of the target space. Generally, the flow of a metric is expressed by geometric structures. Surprisingly, we find an exception to this in the second order of perturbation theory. The expression we derive has an elegant form in the context of algebraic data, which may be explained by infrared anomalies.

In this paper we consider perturbation theory in generic two-dimensional sigma models in the so-called first order formalism, using the coordinate regularization approach. Our goal is to analyze the first-order formalism in application to β functions and compare its results with the standard geometric calculations. Already in the second loop, we observe deviations from the geometric results that cannot be explained by the regularization/renormalization scheme choices. Moreover, in certain cases the first-order calculations produce results that are not symmetric under the classical diffeomorphisms of the target space. Although we could not present the full solution to this remarkable phenomenon, we found some indirect arguments indicating that an anomaly similar to that established in supersymmetric Yang-Mills theory might manifests itself starting from the second loop. We discuss why the difference between two answers might be an infrared effect, similar to that in beta functions in supersymmetric Yang-Mills theories.

Peculiarities of beta functions in sigma models