A four-dimensional non-linear σ-model

Abstract We study a four-dimensional version of the non-linear O( n ) σ -model in Euclidean space, introducing a suitable kinetic term in order to have a theory with many of the properties of the two-dimensional model. The outcoming theory is shown to be ultraviolet free for n > 2. Its behaviour is then discussed in the large- n expansion. It is found that in general two phases exist, in one symmetry is broken, in the other preserved, and both are possible in four dimensions. For a range of values of the coupling constant both asymptotic freedom and the perturbative expansion are valid, contrary to the situation of the two-dimensional σ-model. The infrared behaviour is found to be regular and a singularity of the running coupling constant to be of no relevance for the Green functions.