  We propose a model of Majorana fermions with quartic self-couplings. These
Majorana fermions acquire masses via a type II seesaw mechanism in which the
physical eigenstates are identified as a light Majorana fermion and another
heavy Majorana fermion. On a physical basis, the quartic self-couplings involve
axial currents of these Majorana fermions, and also the interaction of the
axial current for the light particle with the heavy particle one. We introduce
two auxiliaries gauge fields in this model, and we study the stability
conditions of the correspondent effective potential of the model. The ground
state of the effective potential introduces two 4-vectors as scales of vacuum
expected values, and consequently, the dynamical Lorentz symmetry breaking
(DLSB) emerges in the model. We use the expansion of the effective action to
calculate the effective Lagrangian up to second order in the auxiliary fields
as fluctuations around the ground state. This mechanism generates dynamics for
the auxiliary gauge fields, mixed mass terms, longitudinal propagation, and
Chern-Simons term through radiative corrections. After the diagonalization, the
two gauge fields gain masses through an analogous type II seesaw mechanism in
which a gauge boson has a light mass, and the other one acquires a heavy mass.
In this scenario of Lorentz symmetry breaking, we obtain the correspondent
dispersion relations for the Majorana fermions and the gauge boson fields.
Posteriorly, we analyze the neutrino's oscillations in the presence of a DLSB
parameter, in the transition $\nu_{e} \rightarrow \nu_{\mu}$. We discuss the
parameter space of this transition and show that the DLSB can conciliate the
LSND and super-Kamiokande results.
