All orders results for self-crossing Wilson loops mimicking double parton scattering

Abstract

Loop-level scattering amplitudes for massless particles have singularities in regions where tree amplitudes are perfectly smooth. For example, a $2\to 4$ gluon scattering process has a singularity in which each incoming gluon splits into a pair of gluons, followed by a pair of $2\to 2$ collisions between the gluon pairs. This singularity mimics double parton scattering because it occurs when the transverse momentum of a pair of outgoing gluons vanishes. The singularity is logarithmic at fixed order in perturbation theory. We exploit the duality between scattering amplitudes and polygonal Wilson loops to study six-point amplitudes in this limit to high loop order in planar $\mathcal{N}=4$ super-Yang-Mills theory. The singular configuration corresponds to the limit in which a hexagonal Wilson loop develops a self-crossing. The singular terms are governed by an evolution equation, in which the hexagon mixes into a pair of boxes; the mixing back is suppressed in the planar (large $N_c$) limit. Because the kinematic dependence of the box Wilson loops is dictated by (dual) conformal invariance, the complete kinematic dependence of the singular terms for the self-crossing hexagon on the one nonsingular variable is determined to all loop orders. The complete logarithmic dependence on the singular variable can be obtained through nine loops, up to a couple of constants, using a correspondence with the multi-Regge limit. As a byproduct, we obtain a simple formula for the leading logs to all loop orders. We also show that, although the MHV six-gluon amplitude is singular, remarkably, the transcendental functions entering the non-MHV amplitude are finite in the same limit, at least through four loops.

64 pages, 12 figures, 1 ancillary file; v2: references added, minor typos corrected, corrected argument of NMHV finiteness in Section 2.1 and added supporting formulae in Appendix D; v3: minor typos corrected, corrected formulae (D.13)-(D.17) and the conclusions based on those equations; updated ancillary file

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Cited by in corpus (5)

• Heptagons from the Steinmann Cluster Bootstrap
• Folding Amplitudes into Form Factors: An Antipodal Duality
• Scattering Equations: Real Solutions and Particles on a Line
• The SAGEX Review on Scattering Amplitudes, Chapter 15: The Multi-Regge Limit
• Heptagon Functions and Seven-Gluon Amplitudes in Multi-Regge Kinematics