Renormalization of anSU(3)⊗SU(3)Chiral-InvariantσModel

Abstract

We examine the single-loop charge renormalization of an $\mathrm{SU}(3)\ensuremath{\bigotimes}\mathrm{SU}(3)$ chiral-invariant $\ensuremath{\sigma}$ model. This requires that a systematic way of treating charge renormalization for models with the added complexity of $\mathrm{SU}(3)$ be developed. The model we consider has two nonderivative four-point couplings and one nonderivative three-point coupling. We find that two of these three independent chiral-invariant couplings are not individually renormalizable, as their higher-order $\mathrm{SU}(3)$ structure is not expressible in terms of the $\mathrm{SU}(3)$ structure of their respective tree diagrams. Only when all three invariant couplings are considered simultaneously is the resultant $\mathrm{SU}(3)$ structure given by the tree diagrams. Hence, each of the three charge renormalization constants is a function of all three coupling constants.