  We consider the renormalization group flow equation for the two-dimensional
sigma models with the K\"ahler target space. The first-order formulation allows
us to treat perturbations in these models as current-current deformations. We
demonstrate, however, that the conventional first-order formalism misses
certain anomalies in the measure, and should be amended. We reconcile beta
functions obtained within the conformal perturbation theory for the
current-current deformations with traditional ``geometric" results obtained in
the background field methods, in this way resolving the peculiarities pointed
out in [JHEP10(2023)097]. The result is achieved by the supersymmetric
completion of the first-order sigma model.
