The Four Dimensional Green-Schwarz Mechanism and Anomaly Cancellation Conditions

TL;DR

This paper revisits anomaly calculations in theories with gauge group G×U(1)_A, presenting a new solution to the descent equations, analyzing their physical equivalence, and applying the Green-Schwarz mechanism to ensure anomaly cancellation.

Contribution

It introduces an alternative solution to gauge anomaly descent equations, clarifies their physical equivalence, and extends the Green-Schwarz mechanism to four dimensions with detailed constraints.

Findings

Both anomaly solutions are physically equivalent after adding counterterms.

The Green-Schwarz mechanism can be consistently implemented in 4D theories.

Additional Lorentz invariance constraints are identified for anomaly cancellation.

Abstract

We consider a theory with gauge group $G \times U(1)_A$ containing: i) an abelian factor for which the chiral matter content of the theory is anomalous $\sum_{f} q^f_A \neq 0 \neq \sum_{f} (q^f_A)^3$ ; ii) a nonanomalous factor $G$. In these models, the calculation of consistent gauge anomalies usually found in the literature as a solution to the Zumino-Stora descent equations is reconsidered. Another solution of the descent equations that differs on the terms involving mixed gauge anomalies is presented on this paper. The origin of their difference is analysed, and using Fujikawa's formalism the second result is argued to be the divergence of the usual chiral current. Invoking topological arguments the physical equivalence of both solutions is explained, but only the second one can be technically called the consistent anomaly of a classically invariant theory. The first one corresponds to the addition of noninvariant local counterterms to the action. A consistency check of their physical equivalence is performed by implementing the four dimensional string inspired Green-Schwarz mechanism for both expressions. This is achieved adding slightly different anomaly cancelling terms to the original action, whose difference is precisely the local counterterms mentioned before. The complete anomaly free action is therefore uniquely defined, and the resulting constraints on the spectrum of fermion charges are the same. The Lorentz invariance of the fermion measure in four dimensions forces the Lorentz variation of the Green-Schwarz terms to cancel by itself, producing an additional constraint usually overlooked in the literature. This often happens when a dual description of the theory is used without including all local counterterms.