Analyticity for Multi-Regge Limits of the Bern-Dixon-Smirnov Amplitudes

TL;DR

This work probes analyticity and Regge behavior in planar ${ m N}=4$ SYM via Bern–Dixon–Smirnov amplitudes, contrasting them with flat-space open-string Regge theory. It develops a two-step analytic continuation framework for planar amplitudes, examines 4- and 5-point Regge structures, and analyzes multi-Regge factorization in higher-point functions, emphasizing signature-based factorization. A central finding is that BDS amplitudes reproduce some Regge-like features but violate Steinmann constraints and signature factorization in several multi-Regge limits, notably lacking missing-mass ($M^2$) discontinuities in triple-Regge and related regimes. The results suggest that the IR-regulated, dual-conformal structure of ${ m N}=4$ SYM departs from flat-space string expectations, and point to the necessity of additional cross-ratio–dependent or higher-order terms to restore full analyticity and unitarity in the MR regime.

Abstract

As a consequence of the AdS/CFT correspondence, planar ${\cal N} =4$ super Yang-Mills SU(N) theory is expected to exhibit stringy behavior and multi-Regge asymptotic. In this paper we extend our recent investigation to consider issues of analyticity, a central feature of Regge asymptotics. We contrast flat-space open string theory in the planar limit with the ${\cal N}=4$ super Yang-Mills theory, as represented by the Bern, Dixon and Smirnov \cite{Bern:2005iz} (BDS) conjecture for n-gluon scattering, believed to be exact for $n=4,5$ and modified only by a function of cross-ratios for $n\geq 6$. It is emphasized that multi-Regge factorization should be applied to trajectories with definite signature. A variety of analyticity and factorization constraints realized in flat space string theory are not satisfied by the BDS conjecture, at least when the exponential factors are truncate in the infra-red regulator below $O(ε)$.

Figures (17)

• Figure 1: The 5-point gluonic amplitude, ${\cal A}_5(k_1,k_2,k_3,k_4,k_5)$ evaluated on shell ($k^2_i =0$, $\sum_i k_i =0$) with all BDS invariants ($t^{[r]}_i <0$) space-like.
• Figure 2: The 6-point gluonic amplitude, ${ A}_6(k_1,k_2,k_3,k_4,k_5,k_6)$, evaluated on shell ($k^2_i =0$, $\sum_i k_i =0$) with all BDS invariants ($t^{[r]}_i <0$) space-like. Note here it was possible to set $i k^{(1)}_i$ to a constant.
• Figure 3: The 5-point gluonic amplitude, ${\cal A}_5(k_1,k_2,k_3,k_4,k_5)$ in the double-Regge region, evaluated on shell ($k^2_i =0$, $\sum_i k_i =0$) with all BDS invariants ($t^{[r]}_i <0$) space-like.
• Figure 4: Multiperipheral limit for the 2 to n-2 gluon scattering amplitude in the tree approximation.
• Figure 5: The Regge limit for the elastic planar 4-point amplitude ${A}_4(s,t)$ with thresholds for $s \ge 0, t \ge 0$ with a "twisted' Regge limit" is $s \simeq -u \rightarrow - \infty$ and $t < 0$.