All-loop-orders relation between Regge limits of mathcal{N} = 4 SYM and mathcal{N} = 8 supergravity four-point amplitudes

Abstract

We examine in detail the structure of the Regge limit of the (nonplanar) mathcal{N} = 4 SYM four-point amplitude. We begin by developing a basis of color factors Cik suitable for the Regge limit of the amplitude at any loop order, and then calculate explicitly the coefficients of the amplitude in that basis through three-loop order using the Regge limit of the full amplitude previously calculated by Henn and Mistlberger. We compute these coefficients exactly at one loop, through mathcal{O}left({upepsilon}^2right) at two loops, and through mathcal{O}left({upepsilon}^0right) at three loops, verifying that the IR-divergent pieces are consistent with (the Regge limit of) the expected infrared divergence structure, including a contribution from the three-loop correction to the dipole formula. We also verify consistency with the IR-finite NLL and NNLL predictions of Caron-Huot et al. Finally we use these results to motivate the conjecture of an all-orders relation between one of the coefficients and the Regge limit of the mathcal{N} = 8 supergravity four-point amplitude.