Peculiarities of beta functions in sigma models

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 $\beta$ 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 manifest 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. In addition to the generic K\"ahler target spaces we discuss in detail the so-called Lie-algebraic sigma models. In particular, this is the case when the perturbed field $G^{i\bar j}$ is a product of the holomorphic and antiholomorphic currents satisfying two-dimensional current algebra.