Static Two-Body Potential at Fifth Post-Newtonian Order

We determine the gravitational interaction between two compact bodies up to the sixth power in Newton’s constant, $G_N$, in the static limit. This result is achieved within the effective field theory approach to general relativity, and exploits a manifest factorization property of static diagrams which allows us to derive static post Newtonian (PN) contributions of ($2n+1$) order in terms of lower order ones. We recompute in this fashion the 1PN and 3PN static potential, and present the novel 5PN contribution.

Figure 1

Examples of a prime 4PN graph (left) and of a factorizable 5PN graph (right): the latter can be obtained by sewing the former and the Newton potential diagram.

Figure 2

5PN graphs contributing to the test-particle limit. The last graph (bottom right) does not contribute to the 5PN potential, because its 4PN subdiagram vanishes.