Abstract
In this work we propose to use leading singularities to obtain the classical pieces of amplitudes of two massive particles whose only interaction is gravitational. Leading singularities are generalizations of unitarity cuts. At one-loop we find that leading singularities obtained by multiple discontinuities in the t-channel contain all the classical information. As the main example, we show how to obtain a compact formula for the fully relativistic classical one-loop contribution to the scattering of two particles with different masses. The non-relativistic limit of the leading singularity agrees with known results in the post-Newtonian expansion. We also compute a variety of higher loop leading singularities including some all-loop families and study some of their properties.
Highlights
Successful Effective One Body approach [23,24,25,26]
In this work we propose to use leading singularities to obtain the classical pieces of amplitudes of two massive particles whose only interaction is gravitational
At one-loop we find that leading singularities obtained by multiple discontinuities in the t-channel contain all the classical information
Summary
Scattering amplitudes possess a very intricate analytic structure in perturbation theory as can be seen from imposing unitarity [7]. The discontinuities in a given channel possess an intricate analytic structure and the process can be repeated leading to what is known as generalized unitarity constraints [7]. Most quantum field theory textbooks present discontinuities in a given channel from two-particle exchanges and refer to them as unitarity cuts. These can be thought of as residues of the amplitude by taking two propagators 1/(L21 − m21 + i ) and 1/(L22 − m22 + i ) to define variables 1/u1 and 1/u2 and integrate over contours |ua| = ε that encircle ua = 0 in the corresponding complex planes. Hereafter we denote by ki the momenta associated to massless particles, while Pi will denote the external momenta for massive ones
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