# There is No Ambiguity in the Radiatively Induced Gravitational Chern-Simons Term

## TL;DR

This paper demonstrates that in a gravitational context, radiative corrections to the Chern-Simons term are unambiguous and must vanish when Lorentz symmetry is spontaneously broken, resolving previous ambiguities.

## Contribution

It shows that gauge invariance constraints in gravity eliminate ambiguities in the radiative Chern-Simons term when Lorentz symmetry is spontaneously broken.

## Key findings

- Radiative corrections in gravitational theories are unambiguous.
- The Chern-Simons term must vanish under spontaneous Lorentz symmetry breaking.
- Gauge invariance enforces the absence of ambiguity in the gravitational case.

## Abstract

Quantum corrections to Lorentz- and CPT-violating QED in flat spacetime produce unusual radiative corrections, which can be finite but of undetermined magnitude. The corresponding radiative corrections in a gravitational theory are even stranger, since the term in the fermion action involving a preferred axial vector $b^{\mu}$ would give rise to a gravitational Chern-Simons term that is proportional $b^{\mu}$, yet which actually does not break Lorentz invariance. Initially, the coefficient of this gravitational Chern-Simons term appears to have the same ambiguity as the coefficient for the analogous term in QED. However, this puzzle is resolved by the fact that the gravitational theory has more stringent gauge invariance requirements. Lorentz symmetry in a metric theory of gravity can only be broken spontaneously, and when the vector $b^{\mu}$ arises from spontaneous symmetry breaking, these specific radiative corrections are no longer ambiguous but instead must vanish identically.