Single-loop counterterm for 4-dimensional Sigma model with higher derivatives

The most general action without dimensional parameters for the nonlinear sigma model with higher derivatives (of fourth order) is formulated in 4-dimensional space-time. A generalized Schwinger–DeWitt technique is used to calculate the single-loop counterterm up to terms proportional to the equations of motion. Conditions of single-loop finiteness are established, and renormalization-group equations for the multiplicatively renormalizable $n$-sphere case are obtained. Solutions of the renormalization-group equations with asymptotic freedom in the ultraviolet region are found.