More loops and legs in Higgs-regulated N=4 SYM amplitudes

TL;DR

This work extends Higgs-regulated, dual-conformal approaches to planar N=4 SYM by computing the four-loop four-point amplitude and the two-loop five-point amplitude, testing a BDS-like exponentiation beyond dimensional regularization. It demonstrates that the IR-finite part of log amplitudes maintains the expected kinematic structure, fixes cusp-related constants at high loop order, and analyzes Regge limits where vertical ladder diagrams dominate at LL and include H-shaped insertions at NLL. The results provide precise predictions for the four-loop cusp anomalous dimension, support the universality of IR exponentiation across n-point amplitudes, and propose a generalized exponentiation framework for Higgs-regulated planar MHV amplitudes. Together, these findings reinforce the utility of the Higgs regulator and extended dual conformal symmetry in high-loop amplitude computations and Regge analyses.

Abstract

We extend the analysis of Higgs-regulated planar amplitudes of N=4 supersymmetric Yang-Mills theory to four loops for the four-gluon amplitude and to two loops for the five-gluon amplitude. Our calculations are consistent with a proposed all-loop ansatz for planar MHV n-gluon amplitudes that is the analog of the BDS ansatz in dimensional regularization. In all cases considered, we have verified that the IR-finite parts of the logarithm of the amplitudes have the same dependence on kinematic variables as the corresponding functions in dimensionally-regulated amplitudes (up to overall additive constants, which we determine). We also study various Regge limits of N=4 SYM planar n-gluon amplitudes. Euclidean Regge limits of Higgs-regulated n \geq 4 amplitudes yield results similar in form to those found using dimensional regularization, but with different expressions for the gluon trajectory and Regge vertices resulting from the different regulator scheme. We also show that the Regge limit of the four-gluon amplitude is dominated at next-to-leading-log order by vertical ladder diagrams together with the class of vertical ladder diagrams with a single H-shaped insertion.

Figures (3)

• Figure 1: The eight diagrams contributing to the four-loop four-point amplitude. We use the standard dual variable notation, labeling the external faces by $x_1$ through $x_4$ and the internal faces by $x_a$ through $x_d$. The former are related to the external momenta via $p_i = x_i - x_{i+1}$ (where $i$ is understood mod 4) while the latter are each integrated with the measure $d^4 x/(i \pi^2)$. Under each diagram is shown the numerator factor for the corresponding integral. To avoid clutter, we omit an overall factor of $x_{13}^2 x_{24}^2 = s t$ from each diagram (where $x_{ab} \equiv x_a - x_b$), and we do not label internal faces not appearing in numerator factors. As an illustrative example we demonstrate how to assemble all ingredients of the integral $I_{4b}$ in eq. (\ref{['twopointnine']}).
• Figure 2: Factorization of the leading-log and next-to-leading-log contributions to the Regge limit $s \gg t$ of the $L$-loop vertical ladder integral $I_{L\,a}(v,u)$ into simpler integrals. Factorization of the NLL contribution of the vertical ladder integral with H-shaped insertion $I_{L\,H}$. The dotted line indicates a loop-momentum-dependent numerator.
• Figure :