Collinear and soft divergences in perturbative quantum gravity

Collinear and soft divergences in perturbative quantum gravity are investigated to arbitrary orders in amplitudes for wide-angle scattering, using methods developed for gauge theories. We show that collinear singularities cancel when all such divergent diagrams are summed over, by using the gravitational Ward identity that decouples unphysical polarizations from the S-matrix. This analysis generalizes a result previously demonstrated in the eikonal approximation. We also confirm that the only virtual graviton corrections that give soft logarithmic divergences are of the ladder and crossed ladder types.

Figure 1

A general reduced diagram at the pinched singular point. $J$ and $S$ denote the jet and soft subdiagrams and $H$ is a hard vertex. Here each line may represent any number of soft propagators connecting $S$ to the remainder of the diagram, or the jets $J_i$ to the hard part $H$.

Figure 2

A diagram with a graviton jet attached to an external leg. We show the case where there are two gravitons being emitted from the external line and two gravitons attaching to the rest, but these can be any numbers. The analysis in Sec. 2 shows that the numbers of gravitons emitted and attached do not need to be the same.

Figure 3

A typical ladder diagram.

Figure 4

A diagram with no internal jet loops and only three-point vertices.

Figure 5

An example of an external leg correction. These do not result in collinear singularities.

Figure 6

A graviton jet turning into two gluons.

Figure 7

A graviton jet turning into four gluons.

Figure 8

Arbitrary diagram with virtual soft-graviton corrections to a hard vertex. The finite-momentum lines are drawn with a solid line.

Figure 9

An example of the type of diagram that we discuss in Sec. 3. In this diagram $L_S=4$, $N_S=8$, and $N_E=4$.

Figure 10

An example of a diagram with only three-point couplings to hard lines that will lead to a soft divergence.

Figure 11

The Ward identity for the case $n=1$.

Figure 12

The Ward identity for the case $n=2$. We do not explicitly show the diagrams where the longitudinal gravitons are interchanged.