Higgs-regularized three-loop four-gluon amplitude in N=4 SYM: exponentiation and Regge limits

TL;DR

This work computes the three-loop planar four-gluon amplitude in ${\cal N}=4$ SYM using the Higgs IR regulator, testing a BDS-like exponential ansatz and demonstrating consistency with dual conformal symmetry. By restricting to extended dual-conformal integrals and employing Mellin-Barnes techniques, the authors confirm the Higgs-regulated exponentiation and extract the three-loop cusp-related constants, while comparing Regge-limit behavior across regulators. A key finding is that, in Higgs (and cutoff) regularization, the leading Regge logarithms are dominated by the vertical ladder diagram, yielding Regge-exact results up to $O(m^2)$ and linking the trajectory to cusp anomalous dimensions. The study highlights the practical advantages of the Higgs regulator for IR-sensitive amplitudes and provides a coherent bridge between Regge physics and Wilson-line cusp data, with implications for higher-loop amplitudes and non-MHV structures.

Abstract

We compute the three-loop contribution to the N=4 supersymmetric Yang-Mills planar four-gluon amplitude using the recently-proposed Higgs IR regulator of Alday, Henn, Plefka, and Schuster. In particular, we test the proposed exponential ansatz for the four-gluon amplitude that is the analog of the BDS ansatz in dimensional regularization. By evaluating our results at a number of kinematic points, and also in several kinematic limits, we establish the validity of this ansatz at the three-loop level. We also examine the Regge limit of the planar four-gluon amplitude using several different IR regulators: dimensional regularization, Higgs regularization, and a cutoff regularization. In the latter two schemes, it is shown that the leading logarithmic (LL) behavior of the amplitudes, and therefore the lowest-order approximation to the gluon Regge trajectory, can be correctly obtained from the ladder approximation of the sum of diagrams. In dimensional regularization, on the other hand, there is no single dominant set of diagrams in the LL approximation. We also compute the NLL and NNLL behavior of the L-loop ladder diagram using Higgs regularization.

Figures (9)

• Figure 1: One-loop scalar box diagram representing the on-shell integral (\ref{['int-1loop']}) together with its dual diagram (\ref{['int-1loopdual']}). The numerator factor $(p_{1}+p_{2})^2 (p_{2}+p_{3})^2 = x_{13}^2 x_{24}^2$ is not displayed.
• Figure 2: Mass assignment of a generic planar diagram in the Higgs setup. The external lines correspond to on-shell particles with $p_{i}^2 = -(m_{i}-m_{i+1})^2$. The internal dashed lines correspond to massless particles. We call particles of mass $| m_{i} - m_{i+1} |$ 'light' and particles of mass $m_{i}$ 'heavy' since the former become massless when we consider the equal mass case $m_{i}=m$.
• Figure 3: Two- and three-loop four-point dual conformal diagrams. $I_{2}$ and $I_{3a}$ are ladder diagrams; $I_{3b}$ is the tennis-court diagram. Numerator factors (including a loop-momentum-dependent factor for the tennis court) are omitted. See eqs. (\ref{['defladders']}) and (\ref{['deftenniscourt']}) for explicit formulae for the integrals.
• Figure 4: Three-loop four-point dual conformal diagrams, with numerator factors omitted. Both integrals require a factor of $m^2$ in the numerator in order to be dual-conformally-invariant.
• Figure 5: (a) One-loop scattering of massive particles. (b) Sample higher-loop diagram. Fat lines and thin lines denote particles of mass $M$ and $m$, respectively. Dashed thin lines denote massless particles.