Effective field theory of gravity: Leading quantum gravitational corrections to Newton’s and Coulomb’s laws

In this paper we consider general relativity and its combination with scalar quantum electrodynamics (QED) as an effective quantum field theory at energies well below the Planck scale. This enables us to compute the one-loop quantum corrections to the Newton and Coulomb potentials induced by the combination of graviton and photon fluctuations. We derive the relevant Feynman rules and compute the nonanalytical contributions to the one-loop scattering matrix for charged scalars in the nonrelativistic limit. In particular, we derive the post-Newtonian corrections of order $Gm/{\mathrm{c}}^{2}r$ from general relativity and the genuine quantum corrections of order $G\hslash /{\mathrm{c}}^3{r}^2$.

Figure 1

The two tree diagrams.

Figure 2

Vertex corrections from purely gravitational effects. (a) and (b) contain the ${\varphi }^{2}h$ coupling, (c) and (d) the ${\varphi }^2{h}^2$ coupling.

Figure 3

Vertex corrections involving gravitational and electromagnetic interactions (graviton 1-particle reducible).

Figure 4

Vertex corrections involving gravitational and electromagnetic interactions (photon 1-particle reducible).

Figure 5

Triangle diagrams with only gravitational interactions.

Figure 6

Triangle diagrams with gravitational and electromagnetic interactions.

Figure 7

Box and crossed box diagrams from purely gravitational interactions.

Figure 8

Box and crossed box diagrams with gravitational and electromagnetic interactions.

Figure 9

Circular diagrams: (a) purely gravitational; (b) gravitational and electromagnetic interactions.

Figure 10

The set of vacuum polarization diagrams which contribute to the potential.

Figure 11

The one-loop contribution to the photon self-energy in SQED.

Figure 12

The set of diagrams which only give analytical contributions to the scattering amplitude.