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Lorentz violation through a radiatively induced Chern-Simons-like term in afermionic theory embedded in linearized quantum gravity with a Lorentz- andCPT-violating axial-vector term in the fermionic sector proportional to aconstant field $b_\mu$ has been recently studied. In a similar fashion as forthe extended-QED model of Carroll-Field-Jackiw, we explicitly show that neithergauge invariance nor the more stringent momentum routing invariance conditionon underlying Feynman diagrams fix the arbitrariness inherent to such inducedterm at one loop order. We present the calculation in a nonperturbativeexpansion in $b_\mu$ and within a framework which besides operating in thephysical dimension (and thus avoiding $\gamma_5$ matrix Clifford algebraambiguities), judiciously parametrizes regularization dependent arbitraryparameters usually fixed by symmetries.

Arbitrariness in the gravitational Chern-Simons-like term induced radiatively