  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.
