6 Diagrams for Multicomponent Fields 6.1 Interactions with O(n ) and Cubic Symmetry

So far, we have considered only a single real field φ with a φ 4-interaction. The theory can, however, easily be extended to a set of N identical real fields φ α with α = 1,. .. , N. This extension does not produce any new Feynman diagrams. Additional work arises from the fact that the coupling constant g becomes now a tensor g αβγδ , and each momentum integral in a Feynman diagram is accompanied by a corresponding sum over indices. If the coupling tensor is decomposed into basis tensors which satisfy certain symmetry and completeness properties, the result of each index contraction can again be decomposed into these basis tensors, with invariant factors called symmetry factors S G. In this chapter we show how these are calculated. In Section 1.1 we have explained that many physical systems are described by an O(N)-symmetric φ 4-theory with N identical fields. The simplest example is superfluid 4 He, whose local fluctuations are described by a complex field φ = (φ 1 + iφ 2)/ √ 2, with an energy functional