There is no ambiguity in the radiatively induced gravitational Chern-Simons term

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.

Figure 1

One-loop diagrams that can contribute to the radiatively generated Lorentz-violating Chern-Simons terms. The dots represent the $b^{\mu }$ insertions appearing in the fermion propagator $S_b$. (a) The two triangle diagrams that exist in the radiative calculation of the Abelian Chern-Simons term. (b) The additional contributing diagram that appears in the gravitational theory.

Figure 2

Feynman rules for the fermion-graviton vertices in the presence of $b^{\mu }$.

Figure 3

Four diagrams that do not contribute to the radiatively induced gravitational Chern-Simons term. (a) Diagrams with a $b^{\mu }$-modified fermion-photon vertex, which leads to a term in which $b^{\mu }$ is contracted directly with one of the indices of the external graviton $h^{\mu \nu }$. (b) Pure tadpole diagrams.