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 scattering. 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mrow><mml:msub><mml:mi>G</mml:mi><mml:mtext>N</mml:mtext></mml:msub></mml:mrow></mml:math> ) and then we perform the usual Post-Minkowskian expansion in powers of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mrow><mml:msub><mml:mi>G</mml:mi><mml:mtext>N</mml:mtext></mml:msub></mml:mrow></mml:math> . We find perfect agreement with the existing 2PM literature at the orders ω −1 , <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mrow><mml:mi>log</mml:mi><mml:mo>⁡</mml:mo><mml:mi>ω</mml:mi></mml:mrow></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mrow><mml:mi>ω</mml:mi><mml:msup><mml:mi>log</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>⁡</mml:mo><mml:mi>ω</mml:mi></mml:mrow></mml:math> , which are universal. Furthermore, we use this method to compute the universal part of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mrow><mml:mi>ω</mml:mi><mml:mi>log</mml:mi><mml:mo>⁡</mml:mo><mml:mi>ω</mml:mi></mml:mrow></mml:math> 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 extract 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 (1965 Phys. Rev. 140 B516–24).