Group theory for unified model building

Richard Slansky

The results gathered here on simple Lie algebras have been selected with attention to the needs of unified model builders who study Yang-Mills theories based on simple, local-symmetry groups that contain as a subgroup the SU” x UT x SU§ symmetry of the standard theory of electromag- electromagnetic, weak, and strong interactions. The major topics include, after a brief review of the standard model and its unification into a simple group, the use of Dynkin diagrams to analyze the structure of the group generators and to keep track of the weights (quantum numbers) of the representation vectors; an analysis of the subgroup structure of simple groups, including explicit coordinatizations of the projections in weight space; lists of representations, tensor products and branching rules for a number of simple groups; and other details about groups and their representations that are often helpful for surveying unified models, including vector-coupling coefficient calculations. Tabulations of representations, tensor products, and branching rules for E6, SO,o, SU6, F4, SO9, SUs, SO8, SO7, SU4, E7, Eg, SU8, SOH, SO,8, SO22, and for completeness, SU3 are included. (These tables may have other applications.) Group-theoretical techniques for analyzing symmetry breaking are described in detail and many examples are reviewed, including explicit parameterizations of mass matrices.