A tale of two exponentiations in N=8 supergravity

Abstract

High-energy massless gravitational scattering in 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 threeloop 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 ǫ, hinting at some lack of commutativity of the relevant infrared limits for each exponentiation.

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

“…In ref. [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. [52,55].…”

“…In section 2 we discuss the two types of exponentiations and the distinction between classical and quantum contributions at arbitrary loop order. In section 3 we summarize, for completeness, the check presented in [51] that the scattering data up to three loops are consistent with the leading eikonal exponentiation. In section 4 we extend the procedure to subleading terms at high energy and then concentrate on the first subleading correction.…”

“…Hopefully, in this highly supersymmetric context, one will be able to enter even the gravitational collapse regime: after all the famous microscopic understanding of black-hole entropy in string theory [50] does make crucial use of supersymmetry! 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 (the so-called remainders) to another exponentiation, this time in momentum space, of infrared divergences. And indeed the two-and threeloop remainders of [52] are found to be fully consistent with those implications.…”

“…The remainder has to be a complicated function of b 2 s so that the full amplitude does not depend on it! Or, turning things around, we can say that a simple physical requirement determines a very non trivial structure for the remainder (in analogy with the consequences of exponentiation discussed in [51]).…”

Scattering of tidally interacting bodies in post-Minkowskian gravity

“…There is a current debate about the recently computed 3PM O(G 3 ) contribution [21,22] which yields a divergent G 3 contribution, O(α 3 ln(γ) → ∞ as γ → ∞, while the computation of Ref. [66] (see also [30]) got a finite O(α 3 ) correction in the massless limit. [The massless limit m 1 , m 2 → 0 corresponds to γ = −(P 1 · P 2 )/(m 1 m 2 ) → ∞.]…”