Renormalization of the $N=1$ supersymmetric 4-dimensional nonlinear sigma model with higher derivatives

The general action for the $N=1$ supersymmetric nonlinear sigma model with derivatives of fourth order in 4-dimensional space-time is formulated in terms of chiral $N=1$ superfields. The generalized Schwinger–DeWitt technique in flat $N=1$ superspace is used to calculate the geometrical single-loop counterterm. Conditions on the geometry of the field manifold of the sigma model sufficient for single-loop finiteness on the mass shell are found. Multiplicatively renormalizable sigma models of fourth order on Kählerian spaces of constant holomorphic sectional curvature are considered. The phenomenon of asymptotic freedom for theories on the complex projective spaces $CP(k)$ is found.