  Conceivable Lorentz-violating effects in the neutrino sector remain a
research area of great general interest, as they touch upon the very
foundations on which the Standard Model and our general understanding of
fundamental interactions is laid. Here, we investigate the relation of Lorentz
violation in the neutrino sector in light of the fact that neutrinos and
corresponding left-handed charged leptons form SU(2)_L doublets under the
electroweak gauge group. Lorentz-violating effects thus cannot be fully
separated from questions related to gauge invariance. The model dependence of
the effective interaction Lagrangians used in various recent investigations is
investigated, with a special emphasis on neutrino splitting, otherwise known as
neutrino-pair Cerenkov radiation, NPCR, and vacuum pair emission
(electron-positron-pair Cerenkov radiation, LPCR). We investigate two scenarios
in which Lorentz violating effects do not necessarily also break electroweak
gauge invarianceThe first of these involves a restricted set of gauge
transformation, a subgroup of SU(2)_L x U(1)_Y, while in the second,
differential Lorentz violation is exclusively introduced by the mixing of the
neutrino flavor and mass eigenstates. Our study culminates in a model which
fully preserves SU(2)_L x U(1)_Y gauge invariance, involves flavor-dependent
Lorentz-breaking parameters, and still allows for NPCR and LPCR decays to
proceed.
