Axial gravity: a non-perturbative approach to split anomalies

Abstract

In a theory of a Dirac fermion field coupled to a metric-axial-tensor (MAT) background, using a SchwingerDeWitt heat kernel technique, we compute non-perturbatively the two (odd parity) trace anomalies. A suitable collapsing limit of this model corresponds to a theory of chiral fermions coupled to (ordinary) gravity. Taking this limit on the two computed trace anomalies we verify that they tend to the same expression, which coincides with the already found odd parity trace anomaly, with the identical coefficient. This confirms our previous results on this issue.

Axial gravity and anomalies of fermions

“…Nevertheless, they are independent symmetries of the massless and massive actions. They acquire a clear geometrical meaning once the spacetime point x µ is extended to have an axial partner [2], but we do not need to do that for the scope of the present investigation. These symmetries imply that the stress tensor and its axial partner satisfy suitable covariant conservation laws.…”

“…A metric-axial-tensor (MAT) background for Dirac fermions has been recently constructed in [1,2], with the main purpose of addressing anomalies, especially in a suitable chiral limit. It generalizes to curved space the approach used by Bardeen to study vector and axial couplings of Dirac fermions to gauge fields and analyze their anomalies [3].…”

“…All quantities with a hat contain an axial extension with γ 5 and appear always sandwiched between the Dirac spinors ψ and ψ. For details we refer to [1,2], especially appendix B of the latter.…”

“…This happens since the γ 5 matrices acting on chiral fermions are substituted by the corresponding eigenvalues. One could be more general, allowing also for the spacetime points to have an axial extension, as outlined in [2], but the present formulation is sufficient for our purposes.…”

“…Setting g µν = f µν → 1 2 g µν produces instead a left handed chiral fermion λ coupled to the final metric g µν , while the other chirality is projected out (it remains coupled to the singular metric g − µν = 0). A less singular limit, which keeps a free propagating right-handed fermion, is to consider the "collapsing limit" [2], which consists in setting g + µν = g µν and g − µν = η µν , with η µν the flat Minkowski metric.…”