Abstract
We construct chiral theories with the smallest number $n_\chi$ of Weyl fermions that form an anomaly-free set under various Abelian gauge groups. For the $U(1)$ group, where $n_\chi = 5$, we show that the general solution to the anomaly equations is a set of charges given by cubic polynomials in three integer parameters. For the $U(1) \times U(1)$ gauge group we find $n_\chi = 6$, and derive the general solution to the anomaly equations, in terms of 6 parameters. For $U(1) \times U(1) \times U(1)$ we show that $n_\chi = 8$, and present some families of solutions. These chiral gauge theories have potential applications to dark matter models, right-handed neutrino interactions, and other extensions of the Standard Model. As an example, we present a simple dark sector with a natural mass hierarchy between three dark matter components.
Highlights
Gauge theories with chiral fermions have remarkable properties
For the Uð1Þ × Uð1Þ gauge theories we find that the smallest number of chiral fermions is nχ 1⁄4 6, and we construct the general solution to the anomaly equations for this chiral theory, in terms of six integer parameters
In this article we have analyzed the solutions to the anomaly equations for various Abelian gauge theories with the smallest number of Weyl fermions necessary to form anomaly-free chiral sets
Summary
Gauge theories with chiral fermions have remarkable properties. The fermion masses can be generated only if the chiral gauge group is spontaneously broken. That result shows that the most general Uð1Þ charges of n Weyl fermions are given by certain quartic polynomials in n − 2 integer parameters No such general solution is known for larger Abelian gauge symmetries, which are products of Uð1Þ groups. For the Uð1Þ × Uð1Þ gauge theories we find that the smallest number of chiral fermions is nχ 1⁄4 6, and we construct the general solution to the anomaly equations for this chiral theory, in terms of six integer parameters.
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