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 $C_{ik}$ 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 $\mathcal{O}({\epsilon }^2)$ at two loops, and through $\mathcal{O}({\epsilon }^0)$ 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.

34 pages; v2: clarification added, minor typos corrected, published version

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