  We consider the two-point function of the gauge field in Lorentz-breaking
theories with higher-derivative extension of the Dirac Lagrangian. We show that
the Carroll-Field-Jackiw term naturally arises in this theory as a quantum
correction being perfectly finite and thus displaying no ambiguities. Also, the
finiteness of this term at low energy limit and the absence of large Lorentz
violating corrections allows to avoid the fine-tuning problem.
