Symmetry breaking patterns of the 3-3-1 model at finite temperature

We consider the minimal version of an extension of the standard electroweak model based on the \(SU(3)_c \times SU(3)_L \times U(1)_X\) gauge symmetry (the 3-3-1 model). We analyze the most general potential constructed from three scalars in the triplet representation of \(SU(3)_L\), whose neutral components develop nonzero vacuum expectation values, giving mass for all the model's massive particles. {}For different choices of parameters, we obtain the particle spectrum for the two symmetry breaking scales: one where the \(SU(3)_L \times U(1)_X\) group is broken down to \(SU(2)_L\times U(1)_Y\) and a lower scale similar to the standard model one. Within the considerations used, we show that the model encodes two first-order phase transitions, respecting the pattern of symmetry restoration. The last transition, corresponding to the standard electroweak one, is found to be very weak first-order, most likely turning second-order or a crossover in practice. However, the first transition in this model can be strongly first-order, which might happen at a temperature not too high above the second one. We determine the respective critical temperatures for symmetry restoration for the model.