Matter quantum corrections to the graviton self-energy and the Newtonian potential

We revisit the calculation of matter quantum effects on the graviton self-energy on a flat Minkowski background, with the aim to acquire a deeper understanding of the mechanism that renders the graviton massless. To this end, we derive a low-energy theorem which directly relates the radiative corrections of the cosmological constant to those of the graviton mass to all orders in perturbation theory. As an illustrative example, we consider an Abelian Higgs model with minimal coupling to gravity and show explicitly how a suitable renormalization of the cosmological constant leads to the vanishing of the graviton mass at the one-loop level. In the same Abelian Higgs model, we also calculate the matter quantum corrections to the Newtonian potential and present analytical formulas in terms of modified Bessel and Struve functions of the particle masses in the loop. We show that the correction to the Newtonian potential exhibits an exponential fall-off dependence on the distance $r$, once the nonrelativistic limit with respect to the nonzero loop mass is carefully considered. For massless scalars, fermions, and gauge bosons in the loops, we recover the well-known results presented in the literature.

Figure 1

Diagrammatic representation of the Ward identity (2.34), where the zigzag lines denote gravitons.

Figure 2

The Higgs contribution to the graviton self-energy.

Figure 3

The fermion contribution to the graviton self-energy.

Figure 4

Gauge- and ghost-field contributions to the graviton self-energy.

Figure 5

The tree-level scattering diagram.

Figure 6

The class of diagrams corresponding to matter effects.

Figure 7

The contour used for the complex integral in (5.22) to compute the number of poles and roots for a single loop mass $m$. This contour covers the whole complex plane as $R\to \infty$, while excluding the two branch cuts indicated by the zigzag lines.

Figure 8

The contour used to compute the Fourier transform in (5.20). For a generic nonzero loop mass $m$, there is a branch cut that starts at $2mi$ and extends to $i\infty$ as illustrated by the zigzag line.

Figure 9

Estimates of the branch-cut terms $\mathrm{\Delta }{V}_H(r)$, $\mathrm{\Delta }{V}_{\psi }(r)$, and $\mathrm{\Delta }{V}_A(r)$ resulting from scalar, fermion, and gauge fields, as functions of the distance $r$, are shown in the upper left, upper right, and lower panels, respectively.