On $\beta$-function of $N=2$ supersymmetric integrable sigma-models

We study regularization scheme dependence of Kähler ($N=2$) supersymmetric sigma models. At the one-loop order the metric $\beta$ function is the same as in non-supersymmetric case and coincides with the Ricci tensor. First correction in MS scheme is known to appear in the fourth loop. We show that for certain integrable Kähler backgrounds, such as complete $T-$dual of $\eta$-deformed $\mathbb{C}\mathbb{P}(n)$ sigma models, there is a scheme in which the fourth loop contribution vanishes.