Lepton number violation in 331 models

Different models based on the extended $SU(3{)}_C\times SU(3{)}_L\times U(1{)}_X$ (331) gauge group have been proposed over the past four decades. Yet, despite being an active research topic, the status of lepton number in 331 models has not been fully addressed in the literature, and furthermore many of the original proposals can not explain the observed neutrino masses. In this paper we review the basic features of various 331 models, focusing on potential sources of lepton number violation. We then describe different modifications which can be made to the original models in order to accommodate neutrino (and charged lepton) masses.

Figure 1

One loop contribution to neutrino masses in the original SVS model. There are in total four diagrams, since (on top of exchanging the internal ${\psi }_L$ and $l^c$) one can exchange everywhere ${\varphi }_2$ with ${\varphi }_3$.

Figure 2

The only interactions of the ${\psi }_{\ell }$ multiplet in the PPF model are the ones with the gauge bosons (which do not change flavor) and those of the form ${\psi }_{\ell }{\psi }_{\ell }{\varphi }_3$, which are antisymmetric in flavor space. Since an odd number of these latter interactions are needed to build a mass diagram for ${\psi }_{\ell }$, such a mass must necessarily be flavor antisymmetric, and hence it cannot generate (Majorana) neutrino masses.

Figure 3

One-loop diagrams in the PPF model, extended with a vectorlike lepton.

Figure 4

Gauge loops in the extended SVS model. The full neutrino propagator with the mass eigenstates ${\mathrm{\Psi }}_n^{(\nu )}$ can be expanded in a series of mass insertions. The first nonzero term involves three mass insertions (see text).

Figure 5

Scalar loop for neutrino masses in the Pleitez-Özer model extended with a scalar triplet ${\varphi }_X$. In fact, two distinct diagrams are possible, depending on which components inside the square brackets are picked (either the ones on top, or the ones at the bottom). Analogous loops can be made for model X with an added ${\varphi }_X$ scalar triplet.