Gravitational cubic-in-spin interaction at the next-to-leading post-Newtonian order

Abstract

In this work we derive for the first time the complete gravitational cubic-in-spin effective action at the next-to-leading order in the post-Newtonian (PN) expansion for the interaction of generic compact binaries via the effective field theory for gravitating spinning objects, which we extend in this work. This sector, which enters at the fourth and a half PN (4.5PN) order for rapidly-rotating compact objects, completes finite-size effects up to this PN order, and is the first sector completed beyond the current state of the art for generic compact binary dynamics at the 4PN order. At this order in spins with gravitational nonlinearities we have to take into account additional terms, which arise from a new type of worldline couplings, due to the fact that at this order the Tulczyjew gauge for the rotational degrees of freedom, which involves the linear momentum, can no longer be approximated only in terms of the four-velocity. One of the main motivations for us to tackle this sector is also to see what happens when we go to a sector, which corresponds to the gravitational Compton scattering with quantum spins larger than one, and maybe possibly also get an insight on the inability to uniquely fix its amplitude from factorization when spins larger than two are involved. A general observation that we can clearly make already is that even-parity sectors in the order of the spin are easier to handle than odd ones. In the quantum context this corresponds to the greater ease of dealing with bosons compared to fermions.

Figures (5)

• Figure 1: The gravitational Compton scattering relevant as of the one-loop level. The gravitational Compton amplitude involves two massive spinning particles and two gravitons, where factorization constraints do not uniquely determine the amplitude for $s>2$Arkani-Hamed:2017jhn.
• Figure 2: The Feynman graphs of one-graviton exchange, which contribute to the NLO cubic-in-spin interaction at the 4.5PN order for maximally rotating compact objects. The graphs should be included together with their mirror images, i.e. with the worldline labels $1\leftrightarrow2$ exchanged. At the linear level of one-graviton exchange we only have two kinds of interactions contributing, similar to the LO in Levi:2014gsa, namely either a quadrupole-dipole or an octupole-monopole interaction. As noted in Levi:2014gsa there are nice analogies among these interactions according to the parity of the multipole moments involved. Following these analogies the relevant graphs here are easily constructed. Notice that we have here the four graphs that appeared at the LO with the quadratic time insertions on the propagators at graphs (a7)-(a10), and a new octupole coupling involving the KK tensor field at graph (a3).
• Figure 3: The Feynman graphs of two-graviton exchange, which contribute to the NLO cubic-in-spin interaction at the 4.5PN order for maximally-rotating compact objects. The graphs should be included together with their mirror images, i.e. with the worldline labels $1\leftrightarrow2$ exchanged. These graphs include all relevant interactions among the spin-induced quadrupole, octupole, and the mass and spin, in particular here at the nonlinear level there are also interactions involving the various multipoles on two different points of the worldline, which add up to interactions that are cubic in the spin, such as a spin dipole and a spin-induced quadrupole or two spin dipoles, on the same worldline, which can already be seen as of the NLO spin-squared sector Levi:2015msaLevi:2015ixa. Consequently notice that there are nonlinearities originating from gravitons sourced strictly from minimal coupling to the worldline as shown in graphs (b13)-(b15). We also have here two new two-graviton octupole couplings in graphs (b1), (b2).
• Figure 4: The Feynman graphs at one-loop level, i.e. with cubic self-gravitational interaction, which contribute to the NLO cubic-in-spin interaction at the 4.5PN order for maximally-rotating compact objects. The graphs should be included together with their mirror images, i.e. with the worldline labels $1\leftrightarrow2$ exchanged. Similar to the nonlinear graphs of two-graviton exchange, these graphs include all relevant interactions among the spin-induced quadrupole, octupole, and the mass and spin, and we have here nonlinearities originating from gravitons sourced strictly from minimal coupling to the worldline, as shown in graphs (c4)-(c8). We also have here cubic vertices containing time derivatives, similar to what we have in the NLO odd-parity spin-orbit sector Levi:2010zuLevi:2015msaLevi:2015uxa.
• Figure 5: The extra Feynman graphs of one- and two-graviton exchange, which appear at the NLO cubic-in-spin interaction at the 4.5PN order for maximally-rotating compact objects. The graphs should be included together with their mirror images, i.e. with the worldline labels $1\leftrightarrow2$ exchanged. These graphs contain a new type of worldline-graviton couplings, which we refer to as "composite" octupole ones, and obviously yield similar graphs to the corresponding ones with the "elementary" spin-induced octupole couplings in figure \ref{['cubspin1g0loop']}(a1),(a2) and in figure \ref{['cubspin2g0loop']}(b1),(b2).