A tale of two exponentiations in ${\cal N}=8$ supergravity at subleading level

TL;DR

This work extends the analysis of high-energy gravitational scattering in $\mathcal N=8$ supergravity to the first subleading level in the deflection-angle expansion, i.e., the $3PM$ regime for massless external states. By comparing IR exponentiation in momentum space with eikonal exponentiation in impact parameter space, the authors identify a nontrivial classical correction to the eikonal at $3PM$ and reveal an IR-divergent dissipative piece tied to soft graviton radiation. The results largely support the proposed subleading eikonal framework and its consistency with known three-loop amplitudes, but reveal a mismatch in the IR-finite imaginary part at $\mathcal O(\epsilon^0)$ and suggest subtle regime- and regularization-dependent aspects of the two exponentiations. The analysis points to a universal classical piece for the real part of the subleading eikonal, a richer structure in the remainder function, and the potential need for IR-finite observables to fully resolve the exponentiation correspondence.

Abstract

High-energy massless gravitational scattering in ${\cal N}=8$ 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.