On β-function of N = 2 supersymmetric integrable sigma-models

We study the regularization scheme dependence of Kähler (N = 2) supersymmetric sigma models. At the one-loop order the metric β function is the same as in the non-supersymmetric case and it coincides with the Ricci tensor. The first correction in the MS scheme is known to appear in the fourth loop [1, 2]. We show that for certain integrable Kähler backgrounds, such as the complete T-dual of η-deformed $\mathbb{CP}\left(n\right)$ \( \mathbbm{CP}(n) \) sigma models, there is a scheme in which the fourth loop contribution vanishes.