Heptagon Amplitude in the Multi-Regge Regime

TL;DR

The paper develops a framework to compute the strong-coupling multi-Regge regime of scattering amplitudes in ${\cal N}=4$ SYM by mapping to a 1D quantum integrable system that solves the AdS$_5$ minimal area problem. It provides explicit $n=7$ gluon results across multiple MRK regions, using a NLIE/Bethe Ansatz approach and analytic continuation to produce Regge-cut contributions built from a universal function ${\cal R}^{\infty}$, with results matching weak-coupling expectations. This work demonstrates a strong-weak coupling consistency for high-energy amplitudes and lays out a general algorithm that can be extended to more external legs, with detailed derivations to appear in forthcoming work. The findings have implications for understanding the Regge limit in holographic gauge theories and for connecting Wilson loop OPE insights to MRK expansions at strong coupling.

Abstract

As we have shown in previous work, the high energy limit of scattering amplitudes in N=4 supersymmetric Yang-Mills theory corresponds to the infrared limit of the 1-dimensional quantum integrable system that solves minimal area problems in AdS5. This insight can be developed into a systematic algorithm to compute the strong coupling limit of amplitudes in the multi-Regge regime through the solution of auxiliary Bethe Ansatz equations. We apply this procedure to compute the scattering amplitude for n=7 external gluons in different multi-Regge regions at infinite 't Hooft coupling. Our formulas are remarkably consistent with the expected form of 7-gluon Regge cut contributions in perturbative gauge theory. A full description of the general algorithm and a derivation of results will be given in a forthcoming paper.

Figures (6)

• Figure 1: Kinematics of the scattering process $2 \to n-2$. On the right-hand side we show a graphical representation of the dual variables $x_i$.
• Figure 2: Graphical representation of two Regge regions of the $7$-point amplitude.
• Figure 3: In order to describe the path $P_{--+}$ we vary the system parameters $m_1,m_2$ and $C_1,C_2$ along the curves shown above. As start values we choose $m_1(0)= 10, m_2(0)= 9$ and $C_1(0) \sim .93i, C_2(0)=.96i$. These values determine the values of the kinematic variables $w_i$. The qualitative features of the curves do not depend on the precise values of the initial conditions. Note that the curves for $m_2$ and $C_2$ are very similar to those of $m$ and $C$ found in the case of 6-gluon scattering, see Bartels:2010ej.
• Figure 4: During the continuation along $P_{--+}$ some solutions of $Y_{as}(\theta_\ast) = -1$ approach or cross the real line. These are shown in the plots. We change the color of the plot once the pair of solutions crosses the real axis. Solutions of $Y_{a1}(\theta_\ast)=-1$ move very little and stay away from the real axis. This is related to the small changes we see in the parameter $C_1$, see figure \ref{['fig:mmpC1']}. The pattern of pole crossings is very similar to that found for 6-gluon scattering, see Bartels:2010ej.
• Figure 5: In order to describe the path $P_{---}$ we vary the system parameters $m_s$ and $C_s$ along the curves shown in this figure. For these figures we have chosen symmetric initial conditions $m(0) = m_1(0) = m_2(0) = 10$ and $C(0) = C_1(0) = - C_2(0) \sim .93i$. For other (non-symmetric) initial conditions the qualitative features of these curves remain the same. Initial conditions must be varied in order to explore the whole range of dynamical variables $w_1$ and $w_2$.