The σ-model and its renormalization in a dimensionally regularised normal product formulation

The σ-model is re-examined in the light of the reformulation of the normal product method in dimensional regularisation. It is considered both as a prime example of, and testing ground for, these methods, and to display the dimensionally regularised renormalisation of the model. Explicit and spontaneous symmetry breakings are considered, and the PCAC relation and Goldstone theorem are examined and proved in the one loop approximation. The renormalisation group equation coefficients are also obtained to this order. Both arbitrary (renormalised) parametrisations, and the mass shell parametrisation with finite counterterms are considered. It is found that, in contrast with other treatments, these counter terms remain finite in the infra-red limit of a massless symmetric Lagrangian.