Abstract
We compute the trace, diffeomorphism and Lorentz anomalies of a free Weyl fermion in a gravitational background field by path integral methods. This is achieved by regularising the variation of the determinant of the Weyl operator building on earlier work by Leutwyler. The trace anomaly is found to be one half of the one of a Dirac fermion. Most importantly we establish that the potential parity-odd curvature term \( R\overset{\sim }{R} \), corresponding to the Pontryagin density, vanishes. This is to the contrary of some recent findings in the literature which gave rise to a controversy. We verify, that the regularisation does not lead to (spurious) anomalies in the Lorentz and diffeomorphism symmetries. We argue that in d = 2 (mod 4) P- and CP-odd terms cannot appear and that for d = 4 (mod 4) they are absent at least at leading order.
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Acknowledgments
The authors are grateful to Luis Álvarez-Gaumé, Bill Bardeen, Fiorenzo Bastianelli, Loriano Bonora, Mike Duff, Sebastián Franchino-Viñas for helpful discussions or comments on the manuscript. The authors are particularly thankful to Sebastián Franchino-Viñas for rising their curiosity on the subject. The work of RL and JQ is supported by the IN2P3 Master projects A2I and BSMGA, by the project AFFIRM of the Programme National GRAM of CNRS/INSU with INP and IN2P3 co-funded by CNES and by the project EFFORT supported by the programme IRGA from UGA. JQ thanks LAPTh Annecy for hospitality while this work was completed. JQ and RZ acknowledge the support of CERN associateships. The work of RZ is supported by the STFC Consolidated Grant, ST/P0000630/1. Many manipulations were carried out with the help of Mathematica and the package xAct [57].
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Larue, R., Quevillon, J. & Zwicky, R. Trace anomaly of weyl fermions via the path integral. J. High Energ. Phys. 2023, 64 (2023). https://doi.org/10.1007/JHEP12(2023)064
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DOI: https://doi.org/10.1007/JHEP12(2023)064