2PM waveform from loop corrected soft theorems

Abstract

We introduce a classical version of the loop corrected soft graviton theorem and we use it to compute the universal part of the one-loop (2PM) waveform up to sub- subleading order in the energy ω of the emitted graviton for spinless black holes scat- tering. In particular, we compute the action of the soft operators on the classically resummed four-point amplitude, that can be written in terms of the exponential of the eikonal phase (and is therefore non-perturbative in the Newton’s constant GN) and then we perform the usual PM expansion in powers of GN. We find perfect agree- ment with the existing 2PM literature at the orders ω−1, log ω and ω log2 ω, which are universal. Furthermore, we use this method to compute the universal part of the ω log ω contribution to the 2PM waveform. Even if in the present analysis we limit ourselves to compute the soft 2PM waveform, our general formulae can be used to ex- tract all universal PM orders of the terms connected with the infrared divergences up to non-linear memory contributions, once the impulse at the corresponding precision is known. Our approach, based on the resummed eikonal amplitude, gives a unified picture of the various computations of the classical soft graviton behaviour that are present in the literature since the seminal paper by Weinberg in 1965 [1].