Anomaly for Model Building

A simple algorithm to calculate the group theory factor entering in anomalies at four and six dimensions for SU(N) and SO(N) groups in terms of the Casimir invariants of their subgroups is presented. Explicit examples of some of the lower dimensional representations of SU(n), n ≤ 5 and SO(10) groups are presented, which could be used for model building in four and six dimensions. The consistency of any gauge theory requires that the sum of anomalies [1] due to all the fermions present in the theory should cancel. The anomaly cancellation is necessary because any classical symmetry is broken by quantum effects in the presence of anomaly. In other words, any gauge theory with non-vanishing anomaly is non-renormalizable [2]. In the standard model of electroweak interactions, the fermion content is just right to make the theory anomaly free. In any extensions of the standard model, particularly the ones which introduces additional gauge symmetry, the most severe constraints come from the anomaly cancellation conditions. Anomaly plays crucial roles in theories of dimension higher than four. Since all representations in odd dimensions contains both left and right chiral fields, there is no anomaly. But in dimensions six, eight or ten, any theory has to cancel higher dimensional anomalies. In higher dimensional theories if the space is compactified in an orbifold, then the orbifold compactification also lead to anomaly at the fixed points. Consistency of such theories then require that the brane anomalies at the fixed points should vanish. Fermions in the loop contributes to anomaly. So, in non-supersymmetric theories the fermion representations are constrained by anomaly. But in supersymmetric theories, any chiral superfield would contribute to anomalies. So, the superfields containing both the scalar and fermion representations are constrained by the anomaly cancellation requirement. The cancellation of anomaly is thus an integral part of constructing any consistent model in four or higher dimensions. It is thus important to know the group theory factor of any representation contributing to anomaly in four or higher dimensions. In four dimensions one needs to calculate the triangle anomaly, while at six dimensions it is a box anomaly and at eight dimensions it is pentagon anomaly. These group theory factors for higher groups become difficult to calculate. There are rigorous methods for searching anomanly free theories that are used usually [3]. In this article we present a simple algorithm to calculate the group theory factor appearing in the expression for anomaly