Abstract
High-energy massless gravitational scattering in mathcal{N} = 8 supergravity was recently analyzed at leading level in the deflection angle, uncovering an interesting connection between exponentiation of infrared divergences in momentum space and the eikonal exponentiation in impact parameter space. Here we extend that analysis to the first non trivial sub-leading level in the deflection angle which, for massless external particles, implies going to two loops, i.e. to third post-Minkowskian (3PM) order. As in the case of the leading eikonal, we see that the factorisation of the momentum space amplitude into the exponential of the one-loop result times a finite remainder hides some basic simplicity of the impact parameter formulation. For the conservative part of the process, the explicit outcome is infrared (IR) finite, shows no logarithmic enhancement, and agrees with an old claim in pure Einstein gravity, while the dissipative part is IR divergent and should be regularized, as usual, by including soft gravitational bremsstrahlung. Finally, using recent three-loop results, we test the expectation that eikonal formulation accounts for the exponentiation of the lower-loop results in the momentum space amplitude. This passes a number of highly non-trivial tests, but appears to fail for the dissipative part of the process at all loop orders and sufficiently subleading order in ϵ, hinting at some lack of commutativity of the relevant infrared limits for each exponentiation.
Highlights
A priori the problem of transplanckian-energy collisions of light particles or strings appears to be unrelated to the one of two coalescing black holes, it has been stressed by Damour [44] that understanding such idealized processes can bring valuable information about the parameters that enter the Effective-One-Body (EOB) potential [6,7,8,9] needed for the computation of the waveforms produced in actual black-hole mergers
As in the case of the leading eikonal, we see that the factorisation of the momentum space amplitude into the exponential of the one-loop result times a finite remainder hides some basic simplicity of the impact parameter formulation
In a recent paper [51] we have shown that the exponentiation in impact-parameter space of the leading high-energy (s → ∞) terms into a leading eikonal phase has non trivial implications for the correction terms to another exponentiation, this time in momentum space, of infrared divergences
Summary
We see that the sum of leading energy contributions of the tree and one-loop amplitudes starts to exponentiate in impact parameter space iA(L0) + iA(L1) + . It is natural to assume that the leading high energy contribution at any loop order is captured by taking the Fourier transform of the leading eikonal back to momentum space, see (2.10). [51], we showed that this equation reproduces the leading terms at two- and three-loop order by using the full results for these amplitudes obtained in refs. S2) , which agrees with the second line of eq (6.5) of ref. [52]
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have