Closed-form resummation of the interacting waveform

Derive a closed-form expression for the Post-Minkowskian-resummed interacting component of the 1SF gravitational waveform, replacing the all-orders series expressed through the integrals \(S_{\alpha_1,\alpha_2}(k,b)\).

Background

The interacting waveform is computed to every Post-Minkowskian order and represented as a series involving the family of transverse integrals $S_{\alpha_1,\alpha_2}(k,b)$. The paper supplies parametric and soft-series representations for these integrals, which are sufficient for several checks and asymptotic analyses.

However, the authors explicitly state that they could not resum the resulting Post-Minkowskian series for the interacting waveform. A closed-form resummation would extend the nonperturbative control already achieved for the shockwave propagator and would facilitate a more complete analysis of the waveform outside the soft and strong-field asymptotic regimes.