Arbitrariness in the gravitational Chern-Simons-like term induced   radiatively
  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. In a similar fashion as for the extended-QED model of Carroll-Field-Jackiw, we explicitly show that neither gauge invariance nor the more stringent momentum routing invariance condition on underlying Feynman diagrams fix the arbitrariness inherent to such induced term at one loop order. We present the calculation in a nonperturbative expansion in $b_\mu$ and within a framework which besides operating in the physical dimension (and thus avoiding $\gamma_5$ matrix Clifford algebra ambiguities), judiciously parametrizes regularization dependent arbitrary parameters usually fixed by symmetries. 
J. C. C. Felipe, A. R. Vieira, A. L. Cherchiglia, A. P. Ba\^eta   Scarpelli and Marcos Sampaio
hep-th