Leading quantum gravitational corrections to QED

We treat the coupled Dirac-Einstein system in the framework of effective field theories in order to quantize the gravitational field at long distances in a consistent manner. In the Dirac-Einstein system we consider the leading post-Newtonian and quantum corrections to the nonrelativistic scattering amplitude of charged spin-$\frac{1}{2}$ fermions. We extract the relevant vertex rules from the action appropriate to the one-loop level calculations and find the nonrelativistic scattering matrix potential for two massive charged spin-$\frac{1}{2}$ fermions. Our focus is kept only on the nonanalytic parts contributing to the potential which are known to generate the long-range interactions.

Figure 1

The set of tree diagrams.

Figure 2

The set of box diagrams contributing to the potential.

Figure 3

The set of triangular diagrams contributing nonanalytically to the potential.

Figure 4

The circular diagram with nonanalytic contributions.

Figure 5

The first set of 1PR diagrams contributing nonanalytically to the potential.

Figure 6

The second set of 1PR diagrams contributing to the potential.

Figure 7

The third set of 1PR diagrams contributing to the potential.

Figure 8

The vacuum polarization diagram contribution to the nonrelativistic potential.