We argue that the gauge symmetry which includes $SU(3{)}_L$ as a higher weak-isospin symmetry is manifestly given by $SU(3{)}_C\otimes SU(3{)}_L\otimes U(1{)}_X\otimes U(1{)}_N$, where the last two factors determine the electric charge and $B-L$, respectively. This theory not only provides a consistent unification of the electroweak and $B-L$ interactions, but also gives insights in dark matter, neutrino masses, and inflation. The dark matter belongs to a class of new particles that have wrong $B-L$ numbers, and is stabilized due to a newly realized $W$-parity as residual gauge symmetry. The $B-L$ breaking field is important to define the $W$-parity, seesaw scales, and the inflaton. Furthermore, the number of fermion generations and the electric charge quantization are explained naturally. We also show that the previous 3-3-1 models are only an effective theory, as the $B-L$ charge and the unitarity argument are violated. This work substantially generalizes our recently proposed 3-3-1-1 model.

• Received 25 May 2015

DOI:https://doi.org/10.1103/PhysRevD.92.055026