The Bethe Roots of Regge Cuts in Strongly Coupled N=4 SYM Theory

TL;DR

This work develops a general algorithm to compute the remainder function $R_n$ for $n$-gluon scattering in multi-Regge kinematics at strong coupling in planar $\mathcal{N}=4$ SYM, by mapping the problem to the infrared limit of a thermodynamic Bethe Ansatz/Y-system describing strings in $AdS_5\times S^5$. The MRK limit reduces the Y-system to a set of algebraic Bethe Ansatz equations whose solutions (Bethe roots) encode Regge-cut contributions; the cross ratios are reconstructed from these solutions to yield $R_n$. Explicit results are obtained for $n=6$ and $n=7$, including detailed analyses of the regions $(--+)$, $(---)$, and $(-+-)$, with the first two showing Regge-cut structures that match weak-coupling expectations, while the last region exhibits a potential path-dependence issue. Overall, the study demonstrates that strong-coupling Regge physics in $\mathcal{N}=4$ SYM can be captured by integrable-system techniques, and provides concrete, all-orders (in principle) predictions for seven-point amplitudes that align with perturbative insights in several regions. These results pave the way for higher-point explorations and tighter connections to proposed all-loop formulations.

Abstract

We describe a general algorithm for the computation of the remainder function for n-gluon scattering in multi-Regge kinematics for strongly coupled planar N=4 super Yang-Mills theory. This regime is accessible through the infrared physics of an auxiliary quantum integrable system describing strings in AdS5xS5. Explicit formulas are presented for n=6 and n=7 external gluons. Our results are consistent with expectations from perturbative gauge theory. This paper comprises the technical details for the results announced in arXiv:1405.3658 .

Figures (9)

• Figure 1: Graphical representation of the choice of Mandelstam variables as defined in Eq. (\ref{['eq:def_mandelstam']}).
• Figure 2: Graphical representation of the cross ratios for the $7$-gluon amplitude. The cross ratios $u_{as}$ are chosen to be independent (cf. Eqs. (\ref{['eq:cr1']})-(\ref{['eq:cr3']})), while $\tilde{u}$ depends on those cross ratios through a conformal Gram relation.
• Figure 3: Paths of the driving terms during the analytic continuation for the path Eq. (\ref{['eq:7pt_mmp']}). The starting values for the parameters chosen here are $|m_1|=10$, $|m_2|=9$, $C_1=\mathrm{arccosh}\left(\frac{3}{5}\right)$, $C_2=\mathrm{arccosh}\left(\frac{4}{7}\right)$. Note that some axes have been shifted and rescaled. The direction of growing $\varphi$ is indicated by the arrows.
• Figure 4: Left: Crossing solutions of $\tilde{{\rm Y}}_{3,2}(\theta)=-1$ during the continuation Eq. (\ref{['eq:7pt_mmp']}). We find that two solutions cross the real axis and approach the endpoints $\pm i\frac{\pi}{4}$. Right: Towards the end of the continuation, a pair of solutions of $\tilde{{\rm Y}}_{2,2}(\theta)=-1$ approaches the real axis, but does not contribute to the remainder function as is argued in the main text. The direction of growing $\varphi$ is indicated by the arrows.
• Figure 5: Solution of the driving terms during the continuation Eq. (\ref{['eq:7pt_mmm']}). The direction of growing $\varphi$ is indicated by the arrows. For simplicity, the sets of parameters are identified as $|m|=|m_1|=|m_2|=10$, $C=C_1=-C_2=\mathrm{arccosh}\left(\frac{3}{5}\right)$ at the starting point.