Efficient resummation of high post-Newtonian contributions to the binding energy

Stefano Foffa,William J Torres Bobadilla,Riccardo Sturani

doi:10.1007/jhep02(2021)165

Abstract

A factorisation property of Feynman diagrams in the context the Effective Field Theory approach to the compact binary problem has been recently employed to efficiently determine the static sector of the potential at fifth post-Newtonian (5PN) order. We extend this procedure to the case of non-static diagrams and we use it to fix, by means of elementary algebraic manipulations, the value of more than one thousand diagrams at 5PN order, that is a substantial fraction of the diagrams needed to fully determine the dynamics at 5PN. This procedure addresses the redundancy problem that plagues the computation of the binding energy with respect to more “efficient” observables like the scattering angle, thus making the EFT approach in harmonic gauge at least as scalable as the others methods.

Highlights

A factorisation property of Feynman diagrams in the context the Effective Field Theory approach to the compact binary problem has been recently employed to efficiently determine the static sector of the potential at fifth post-Newtonian (5PN) order. We extend this procedure to the case of non-static diagrams and we use it to fix, by means of elementary algebraic manipulations, the value of more than one thousand diagrams at 5PN order, that is a substantial fraction of the diagrams needed to fully determine the dynamics at 5PN
A collaboration including the same authors as [32] has extended their results to cover some sectors of the 6PN order [33], while the contribution of hereditary effects to the conservative dynamics have been determined at 5PN in [34] along the same path established for the 4PN case within the EFT approach [35, 36]
We have observed the emergence of novel ideas to provide precision calculations of the two-body dynamics at increasingly higher PN orders, even though to bootstrap the available techniques one needs to overcome three kinds of obstacles

Summary

EFT and factorisation of non-static diagrams

The details of the procedure for computing the near-zone (i.e. at distances from the source smaller than the radiation wavelength) contribution to the effective potential have been outlined and discussed in several works [9,10,11,12, 31, 45, 46]. With, i) V and V1,2 are, respectively, the values of the factorisable diagram and of the two sub-diagrams, ii) K accounts for the new matter interaction vertex of V (emerging from the sewing) out of matter interaction vertices of the two sub-diagrams and can be determined using equation (2.6), and iii) C = Cfactorisable/(C1 × C2) where the C’s are the combinatoric factors associated with each graph This procedure, introduced in [31], where it was used to obtain all the contributions to the static part of the 5PN effective action, can be extended to the non-static case with only minor adjustments, the main difference being that non-static diagrams can contain time derivatives which can “propagate” across factors a feature that can be naturally implemented in the codes usually employed to write the amplitude corresponding to a given graph. After inserting the appropriate factors K and C (see the appendix A for more details) one obtains the result,

Proof of concept

Conclusion

A Detailed computation of a sample diagram