Group theory for Feynman diagrams in non-Abelian gauge theories

The increased interest in non-Abelian gauge theories has in recent years led to the computation of many higher-order Feynman diagrams. ' " Asymptotic form-factor and scattering amplitude calculations are of special interest, because they suggest that it might be possible to sum up diagrams with arbitrary numbers of soft gluons just as one can sum up soft-photon processes in QED. In such a program the analysis of the momentum integrals proceeds by the traditional techniques developed for QED calculations. The new aspect, characteristic of non-Abelian gauge theories, is the emergence of a group-theoretic weight (or weight, "for short) associated with each Feynman diagram. The dramatic cancellations among various diagrams occur through interplay of their group-theoretic weights and their momentum-space integrals. So the study of weights becomes of interest, as it might suggest cancellation patterns needed for summations of diagrams. In this paper we give a general method for computing group-theoretic weights, and give explicit rules for SU(n), SO(n), Sp(n), G„E„F„andE, symmetry groups. We restrict ourselves to the models with quarks in the defining (lowest dimensional) representation, but the method can be extended to higher representations. .4,s only global symmetry is assumed, we can compute weights not only in symmetric gauge theories, but also in those spontaneously broken gauge theories where all particles within a multiplet have the same mass. The evaluation procedure is very simple. We think of the weight itself as a Feynman integral (over a discrete lattice), and introduce Feynman diagrammatic notation to replace the unwieldy algebraic expressions. Then we give two relations; the first eliminates all three-gluon vertices, and the second eliminates all internal gluon lines. The result is a sum over a unique set of irreducible group-theoretic tensors which form a natural basis for all Lie algebras. All this is accomplished without recourse to any explicit representation of the group generators and structure constants. As a by-product, we learn how to count quickly the number of invariant couplings for arbitrary numbers of quarks and gluons, thus avoiding involved reductions of outer products of representations by Young tableaux. In most calculations, one looks for properties which arise solely from gauge invariance, and there the explicit numerical values of weights should really not be necessary. While in some such calculations' ""it is appealing to express simple diagrams in terms of quadratic Casimir operators (so that the form of the expression is independent of the particular gauge group and the particular representation), for higher-order diagrams there is no simple way of relating weights to generalized Casimir operators, '""and such an approach becomes very cumbersome. Then the explicit expressions for weights might be both suggestive and useful as checks for the cancellations among various diagrams. Another application of explicit weight expressions is 1/n expansions" for which the above evaluation method gives simple and direct estimates. " Possibly, a novel aspect of this paper is its treatment of exceptional groups. It is known" that exceptional groups arise from invariance of norms defined on octonion spaces, but the demonstration is rather difficult (it involves Jordan algebras over octonionic matrices). We skirt the complexities of this underlying structure by giving a formulation of exceptional groups purely in terms of the geometrical properties of their defining representations. Intuition so developed might be of use to quark-model builders. We give the following example: Because SU(3) has a cubic invariant &'"q,q,q„ it is possible to build a three-quark color singlet with desirable phenomenological properties. " Are there any other groups that could accommodate three-quark color singlets'P It turns out that the defining representations of G„F„and E, are among groups with such invariants. A systematic discussion of such invariants shall be given elsewhere. " In the past, most weight calculations have involved SU(n) and, even more specifically, SU(3). This has led to the development of methods specific to SU(n). 25 " For the sake of completeness and comparison, we pursue this traditional line for awhileandfind ourselves at an impasse.