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We study regularization scheme dependence of K\"ahler ($N=2$) supersymmetricsigma models. At the one-loop order the metric $\beta$ function is the same asin non-supersymmetric case and coincides with the Ricci tensor. Firstcorrection in MS scheme is known to appear in the fourth loop. We show that forcertain integrable K\"ahler backgrounds, such as complete $T-$dual of$\eta$-deformed $\mathbb{CP}(n)$ sigma models, there is a scheme in which thefourth loop contribution vanishes.

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