Abstract
We calculate the quantum corrections to the gauge-invariant gravitational potentials of spinning particles in flat space, induced by loops of both massive and massless matter fields of various types. While the corrections to the Newtonian potential induced by massless conformal matter for spinless particles are well-known, and the same corrections due to massless minimally coupled scalars [Class. Quant. Grav. 27 (2010) 245008], massless non-conformal scalars [Phys. Rev. D 87 (2013) 104027] and massive scalars, fermions and vector bosons [Phys. Rev. D 91 (2015) 064047] have been recently derived, spinning particles receive additional corrections which are the subject of the present work. We give both fully analytic results valid for all distances from the particle, and present numerical results as well as asymptotic expansions. At large distances from the particle, the corrections due to massive fields are exponentially suppressed in comparison to the corrections from massless fields, as one would expect. However, a surprising result of our analysis is that close to the particle itself, on distances comparable to the Compton wavelength of the massive fields running in the loops, these corrections can be enhanced with respect to the massless case.
Highlights
Studied by many authors [1, 7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25]
While the corrections to the Newtonian potential induced by massless conformal matter for spinless particles are well known, and the same corrections due to massless minimally coupled scalars [23], massless non-conformal scalars [25] and massive scalars, fermions and vector bosons [99] have been recently derived, spinning particles receive additional corrections which are the subject of the present work
The results are too tiny to be measured experimentally in the foreseeable future, but they are important in principle, especially for providing unambiguous results for low-energy quantum gravitational predictions which must be reproduced in any full theory of quantum gravity
Summary
The quantum corrections to the gravitational potentials are obtained by solving the field equations coming from an effective action which includes the vacuum polarisation due to quantum matter. The second point can be shown nicely using the background field formalism [49,50,51,52]: the basic argument is that, since the gauge invariance of the metric perturbations (following from diffeomorphism invariance of the full theory) is unbroken at the quantum level, the counterterms in any regularisation which respects the gauge symmetry, such as dimensional regularisation, must be invariant as well, i.e., scalars constructed out of curvature tensors. (see [55] for a scalar field with general mass and curvature coupling) These works were done in momentum space, where the extraction of the differential operators (2.16) just corresponds to a reordering of the pμ, and the spin-2 and spin-0 parts are the coefficients of the two tensor structures one can form out of the pμ and the flat metric ημν which are transverse and have the correct symmetries. + R−(x)R−(y) KR−2−(x − y) − βδ4(x − y) d4x d4y , understood to second order in the perturbation hμν
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