Implications of multi-Regge limits for the Bern-Dixon-Smirnov conjecture

TL;DR

This work tests the BDS conjecture for planar ${\cal N}=4$ SYM gluon amplitudes against Regge and multi-Regge limits, motivated by stringy behavior in the AdS/CFT framework. By analyzing exact BDS expressions for $n=4,5,6$ and extending to general $n$, the authors show that in the Euclidean Regge limits these amplitudes exhibit Reggeization with a universal trajectory $\alpha(t)$ and factorized residues, while cross-ratios $u(i,j;a,b)$ remain finite and govern non-trivial logarithmic and dilogarithmic contributions without altering the leading behavior. They connect these Regge properties to dual conformal Ward identities for lightlike Wilson loops, providing evidence for the Wilson-loop–amplitude duality at strong coupling and clarifying the role of cross-ratio functions. The results establish a coherent framework for Regge constraints on ${\rm N}=4$ SYM amplitudes and highlight open questions related to analytic continuation to the physical region and subleading terms.

Abstract

Planar ${\cal N} =4$ super Yang-Mills SU(N) theory is expected to exhibit stringy behavior, anticipated by the 't Hooft genus expansion and the $AdS/CFT$ correspondence. We examine the Bern-Dixon-Smirnov (BDS) conjecture for $n$-gluon amplitudes in the context of single-Regge and multi-Regge limits and show that these amplitudes have the expected Regge form in the Euclidean region.

Figures (8)

• Figure 1: Regge amplitude for elastic 4-point amplitude defines and fixes the trajectory function $\alpha(t)$ and the Reggeon vertex, $\gamma(t)$.
• Figure 2: n-point amplitude expressed as a tree diagram of effective particle ("Reggeon") exchange in order to emphasize the parameterization of the linear multi-Regge limit.
• Figure 3: Cross ratios in the single-Regge limit
• Figure 4: Regge limits for 5-point amplitude. On the left, the single Regge limit factorizes defining a new single Regge 3-particle vertex, $G^{[3]} (t_1,\kappa_{12},s_2,t_2)$ and on the right, the double Regge limit defines a new two-Reggeon vertex, $G_2(t_1,\kappa_1,t_2)$.
• Figure 5: Linear Regge limits for 6-point gluon amplitude