Two-loop gravity amplitudes from four dimensional unitarity

We compute the polylogarithmic parts of the two-loop four- and five-graviton amplitudes where the external helicities are positive and express these in a simple compact analytic form. We use these to extract the $\ln ({\mu }^2)$ terms from both the four- and five-point amplitudes and show that these match the same $R^3$ counterterm.

Figure 1

Four dimensional cuts of the two-loop all-plus amplitude involving an all-plus one-loop vertex (indicated by the solid circles). Contributions can arise from (a) boxes, (b) two-mass triangles, (c) one-mass triangles, and (d) bubbles.

Figure 2

The labeling and internal helicities of the quadruple cut.

Figure 3

Contributions to the two-loop amplitudes involving an all-plus loop (indicated by the solid circle). Contributions arise from (a) five-point and (b) four-point all-plus one-loop insertions.