Time transients in the quantum corrected Newtonian potential induced by a massless nonminimally coupled scalar field

We calculate the one-loop graviton vacuum polarization induced by a massless, nonminimally coupled scalar field on Minkowski background. We make use of the Schwinger-Keldysh formalism, which allows us to study time dependent phenomena. As an application we compute the leading quantum correction to the Newtonian potential of a point particle. The novel aspect of the calculation is the use of the Schwinger-Keldysh formalism, within which we calculate the time transients induced by switching on the graviton-scalar coupling.

Figure 1

The graviton one-loop tadpole. Only the scalar loop contribution is shown. The scalar propagator $\Delta$ is the blue dashed line, while the graviton insertion $h_{\mu \nu }$ is the red solid wavy line, and $\kappa =\sqrt{16\pi G_N}$ is the gravitational coupling constant.

Figure 2

The scalar diagrams that contribute to the graviton one-loop vacuum polarization. $\delta Z$ stands for the one-loop counterterms.

Figure 3

The scalar and graviton one-loop bubble diagrams. $\rho \sigma {\Delta }_{\mu \nu }$ denotes the graviton propagator.

Figure 4

A one-loop vertex diagram with one scalar loop and six graviton insertions.

Figure 5

The two-loop diagrams contributing to the 2PI effective action.