Abstract
In the context of effective field theory, we consider quantum gravity with minimally coupled massless particles. Fixing the background geometry to be of the Kerr-Schild type, we fully determine the one-loop effective action of the theory whose finite non-local part is induced by the long-distance portion of quantum loops. This is accomplished using the non-local expansion of the heat kernel in addition to a non-linear completion technique through which the effective action is expanded in gravitational curvatures. Via Euclidean methods, we identify a logarithmic correction to the Bekenstein-Hawking entropy of Schwarzschild black hole. Using dimensional transmutation the result is shown to exhibit an interesting interplay between the UV and IR properties of quantum gravity.
Highlights
General relativity is a well-behaved quantum theory at low energies [1, 2]
Fixing the background geometry to be of the KerrSchild type, we fully determine the one-loop effective action of the theory whose finite nonlocal part is induced by the long-distance portion of quantum loops
This is accomplished using the non-local expansion of the heat kernel in addition to a non-linear completion technique through which the effective action is expanded in gravitational curvatures
Summary
General relativity is a well-behaved quantum theory at low energies [1, 2]. Treated as an effective field theory (EFT), quantum predictions can systematically be quantified. The clear separation of scales provided by the EFT framework enables the extraction of the leading quantum effects The latter are precisely due to the low-energy portion of the theory which is dictated by the symmetries of general relativity. The effective Lagrangian of quantum gravity is arranged according to the energy or derivative expansion and only local polynomials of curvature invariants appear. This is the typical story when one integrates out a heavy field from the path integral of the theory. Quantum loops of massless fields yield a non-local effective theory. Focusing on the Schwarzschild solution, we use the effective action to construct the partition function From the latter, the entropy is determined and our main result reads. In appendix C we collect useful formulas used throughout the paper
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