Abstract
An explicit formula to compute the multiplicative anomaly or defect of ζ-regularized products of linear factors is derived, by using a Feynman parametrization, generalizing Shintani-Mizuno formulas. Firstly, this is applied on n-spheres, reproducing known results in the literature. Then, this framework is applied to a closed Einstein universe at finite temperature, namely \( {S}_{\beta}^1\times {S}^{n-1} \). In doing so, it is shown that the standard Casimir energy (as computed via ζ regularization) for GJMS operators coincides with the accumulated multiplicative anomaly for the shifted Laplacians that build them up. This equivalence between Casimir energy and multiplicative anomaly within ζ regularization, unnoticed so far to our knowledge, brings about a new turn regarding the physical significance of the multiplicative anomaly, putting both now on equal footing. An emergent improved Casimir energy, that incorporates the multiplicative anomaly among the building Laplacians, is also discussed.
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Acknowledgments
We thank F. Bastianelli, L. Casarin, J.S. Dowker, A. Monin, and especially E. Friedman for valuable conversations and comments. We are also grateful to the anonymous referee for helpful suggestions and clarifications. This work was partially funded through FONDECYT-Chile 1220335. D.E.D. wishes to salute Harald Dorn and Hans-Jörg Otto on the occasion of the 30th anniversary of the DOZZ formula.
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Aros, R., Bugini, F., Díaz, D.E. et al. Multiplicative anomaly matches Casimir energy for GJMS operators on spheres. J. High Energ. Phys. 2023, 142 (2023). https://doi.org/10.1007/JHEP12(2023)142
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DOI: https://doi.org/10.1007/JHEP12(2023)142