Wilson loop OPE, analytic continuation and multi-Regge limit

TL;DR

We connect the collinear expansion of the hexagon remainder function in ${\mathcal N}=4$ SYM, computed via the Wilson loop OPE, to the multi-Regge limit obtained from BFKL by analytic continuation to the Mandelstam region. The collinear-Regge expansion reproduces the BFKL results order by order, including NNLLA and higher predictions up to five loops, and yields an all-loop imaginary-part structure. The framework leverages the coefficient relation ${[F_{m,n}^{(\ell)}]}^{\mathcal{C}} = F_{m,n}^{(\ell)} + 2\pi i \Delta F_{m,n}^{(\ell)}}$ to obtain continued data without the full analytically continued remainder. This work strengthens the duality between MHV amplitudes and null polygonal Wilson loops and provides a practical route to interpolate between weak and strong coupling in the multi-Regge limit. Overall, the paper demonstrates a consistent, cross-validated bridge between OPE-based collinear physics and high-energy MR behavior in ${\mathcal N}=4$ SYM, with concrete high-loop predictions.

Abstract

We explore a direct connection between the collinear limit and the multi-Regge limit for scattering amplitudes in the N=4 super Yang-Mills theory. Starting with the collinear expansion for the six-gluon amplitude in the Euclidean kinematic region, we perform an analytic continuation term by term to the so-called Mandelstam region. We find that the result coincides with the collinear expansion of the analytically continued amplitude. We then take the multi-Regge limit, and conjecture that the final result precisely reproduces the one from the BFKL approach. Combining this procedure with the OPE for null polygonal Wilson loops, we explicitly compute the leading contribution in the "collinear-Regge" limit up to five loops. Our results agree with all the known results up to four loops. At five-loop, our results up to the next-to-next-to-leading logarithmic approximation (NNLLA) also reproduce the known results, and for the N^3LLA and the N^4LLA give non-trivial predictions. We further present an all-loop prediction for the imaginary part of the next-to-double-leading logarithmic approximation. Our procedure has a possibility of an interpolation from weak to strong coupling in the multi-Regge limit with the help of the OPE.

Figures (5)

• Figure 1: The six-gluon $2 \to 4$ scattering in the multi-Regge limit. We show the kinematics (a) in the Euclidean region and (b) in the Mandelstam region. The variables $s_{i \dots j}$ are defined by $s_{i \dots j}=(p_i+\cdots+p_j)^2$. These kinematic regions are connected by the analytic continuation \ref{['eq:AC-0']} or \ref{['eq:AC-1']}.
• Figure 2: Schematic relation in this work. There are several routes to reach the collinear and multi-Regge behavior in the Mandelstam region. The thick red arrows show our strategy.
• Figure 3: The paths of the analytic continuation $\mathcal{C}$ for $S$ (Left) and $c=\cos \phi$ (Right). To avoid confusion, we denote the initial point by $S_0$ and $c_0=\cos \phi_0$, respectively. The path of $c$ intersects with the imaginary axis at the point $i(c_0+S_0^{-1}(T_0+T_0^{-1}))$.
• Figure 4: (a) Pentagon decomposition of the null hexagonal Wilson loop. (b) The intermediate state $\psi$ in \ref{['eq:OPE1']} is an excitation over the flux tube vacuum, which propagates from the bottom line to the top line of the square. In general, multi-particle states are also allowed.
• Figure 5: The paths for (a) $x_{1,2}^+$, (b) $x_{1,2}^-$, (c) $x_3^+$ and (d) $x_3^-$ along the analytic continuation $\mathcal{C}$.