Trace anomaly for chiral fermions via Hadamard subtraction

Abstract

We calculate the trace (conformal) anomaly for chiral fermions in a general curved background using Hadamard subtraction. While in intermediate steps of the calculation imaginary terms proportional to the Pontryagin density appear, imposing a vanishing divergence of the stress tensor these terms completely cancel, and we recover the wellknown result equal to half the trace anomaly of a Dirac fermion. We elaborate in detail on the advantages of the Hadamard method for the general definition of composite operators in general curved spacetimes, and speculate on possible causes for the appearance of the Pontryagin density in other calculations.

Axial gravity and anomalies of fermions

“…These results confirm the absence of a Pontryagin term in the trace anomaly of a Weyl fermion, as calculated in [4] and confirmed in [20]. The Pontryagin term sits only in the chiral anomaly.…”

“…Our explicit calculation within the MAT background shows indeed that such terms are absent, thereby confirming the findings of refs. [4] and [20]. The analogous case of a Weyl fermion in a gauge background has also been studied more recently in [5,12], where it was found that parity-odd terms do not contribute to the trace anomaly in that context as well.…”

Weyl fermions in a non-abelian gauge background and trace anomalies

“…The analogous case of a Weyl fermion on a curved spacetime background has been debated more extensively in the literature, where a topological term proportional to the Pontryagin density had been reported in [10], and confirmed in [11,12], where the concept of a MAT background, that extends the Bardeen construction to curved space, has been developed. However, the topological term was found to be absent in [13], as confirmed also in [14]. We believe that the latter are the correct results.…”

Trace anomaly for Weyl fermions using the Breitenlohner-Maison scheme for γ*

“…Employing the Breitenlohner-Maison scheme [25] for the treatment of the chiral matrix γ * , we did not find any parity-odd contributions to the expectation value. Therefore, also the trace anomaly does not contain any parity-odd term proportional to the Pontryagin density, and the result for the trace anomaly is half the one of a Dirac fermion, as also confirmed using other methods [7][8][9]. This result reaffirms the validity of the equivalence principle for the coupling of matter to gravity in the present case, since there is no difference between left-or right-handed fermions.…”

“…These results were derived in dimensional regularization, both using a standard perturbative computation around Minkowski spacetime using Feynman diagrams and introducing an "metric-axial-tensor gravity" approach similar to JHEP03(2021)271 the axial vector potential in gauge theory. However, with strictly four-dimensional regularization methods such as Pauli-Villars regularization [7,8] employed by Bastianelli et al, or Hadamard subtraction used by Zahn and the last author [9], CP-odd terms in the trace anomaly do not arise at all. In between, this has given rise to more general discussions involving gauge anomalies [10,11] and general CP-violating theories [12,13].…”