Abstract
We analyze scattering amplitudes with one soft external graviton and arbitrary number of other finite energy external states carrying arbitrary mass and spin to sub-subleading order in the momentum of the soft graviton. Our result can be expressed as the sum of a universal part that depends only on the amplitude without the soft graviton and not the other details of the theory and a non-universal part that depends on the amplitude without the soft graviton, and the two and three point functions of the theory. For tree amplitudes our results are valid in all space-time dimensions while for loop amplitudes, infrared divergences force us to restrict our analysis to space time dimensions five or more. With this restriction the results are valid to all orders in perturbation theory. Our results agree with known results in quantum field theories and string theory.
Highlights
Carrying momenta p1, · · · pN and a soft graviton carrying momentum k and polarization ε, the correction term takes the form
For tree amplitudes our results are valid in all space-time dimensions while for loop amplitudes, infrared divergences force us to restrict our analysis to space time dimensions five or more
As our analysis is based on general properties of the 1PI effective action, our results are valid for any general coordinate invariant theory of gravity coupled to other fields, including string theory
Summary
The leading contribution in the soft limit k → 0 comes from figure 1 due to the pole associated with the propagator carrying momentum pi + k. We shall describe separately the evaluation of these two classes of diagrams In doing this we shall follow the strategy of [54, 55], i.e. first choose a covariant gauge fixing of the 1PI effective action of finite energy fields (including gravitons), expanded in a power series in the fields around flat space-time background, and determine the coupling of the soft graviton to the finite energy fields by replacing the background flat metric by soft graviton background metric and ordinary derivatives by covariant derivatives computed using the soft graviton background metric. As in [55], the finite energy fields will be assumed to carry flat tensor indices associated with the tangent space group so that their covariant derivatives involve the spin connection and not the Christoffel symbol
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