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Lorentz violation through a radiatively induced Chern-Simons-like term in a fermionic theory embedded in linearized quantum gravity with a Lorentz- and CPT-violating axial-vector term in the fermionic sector proportional to a constant field $b_\mu$ has been recently studied.
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G. Bonneau, Lorentz and CPT violations in QED: A Short comment on recent controversies
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