Next-to-leading order gravitational spin1-spin2 coupling with Kaluza-Klein reduction

We use the recently proposed Kaluza-Klein (KK) reduction over the time dimension, within an effective field theory (EFT) approach, to calculate the next-to-leading order gravitational spin1-spin2 interaction between two spinning compact objects. It is shown here that to next-to-leading order in the spin1-spin2 interaction, the reduced KK action within the stationary approximation is sufficient to describe the gravitational interaction, and that it simplifies calculation substantially. We also find here that the gravitomagnetic vector field defined within the KK decomposition of the metric mostly dominates the mediation of the interaction. Our results coincide with those calculated in the Arnowitt-Deser-Misner Hamiltonian formalism, and we provide another explanation for the discrepancy with the result previously derived within the EFT approach, thus demonstrating clearly the equivalence of the Arnowitt-Deser-Misner Hamiltonian formalism and the EFT action approach.

Figure 1

Feynman diagram of the leading order spin1-spin2 interaction. The thick solid lines represent the time evolution of the point particles worldlines. The blobs represent spin insertions on the worldline. The dashed line represents the $A_i$ gravitomagnetic vector field propagator.

Figure 2

Next-to-leading order spin1-spin2 interaction Feynman diagrams of one-graviton exchange. The solid line represents the scalar field propagator. The double line represents the 2-tensor field propagator. The cross vertex corresponds to an insertion of the graviton kinetic term. (b) should be included together with its mirror image.

Figure 3

Feynman diagrams of nonlinear spin1-spin2 interaction in next-to-leading order. The black blobs represent mass vertices on the worldline. These diagrams should be included together with their mirror images.