Abstract
Using rapidity evolution equations we study two-to-two gauge-theory scattering amplitudes in the Regge limit. We carry out explicit computations at next-to-next-to-leading logarithmic accuracy through four loops and present new results for both infrared-singular and finite contributions to the amplitude. New techniques are devised in order to derive the colour structure stemming from three-Reggeon exchange diagrams in terms of commutators of channel operators, obtaining results that are valid for any gauge group, and apply to scattered particles in any colour representation. We also elucidate the separation between contributions to the Regge cut and Regge pole in the real part of the amplitude to all loop orders. We show that planar contributions due to multiple-Reggeon exchange diagrams can be factorised as a Regge pole along with the single-Reggeon exchange, and when this is done, the singular part of the gluon Regge trajectory is directly determined by the cusp anomalous dimension. We explicitly compute the Regge cut component of the amplitude through four loops and show that it is non-planar. From a different perspective, the new results provide important information on soft singularities in general kinematics beyond the planar limit: by comparing the computed corrections to the general form of the four-loop soft anomalous dimension we derive powerful constraints on its kinematic dependence, opening the way for a bootstrap-based determination.
Highlights
Introduction to the soft anomalous dimension7.2 The soft anomalous dimension in the high-energy limit7.2.1 Four-generator four-line term (4T − 4L)7.2.2 Quartic Casimir four-generator four-line term (Q4T − 4L)7.2.3 Five-generator four-line term (5T − 4L)7.2.4 The Regge limit of the soft anomalous dimension7.3 Constraints on the kinematic functions in the soft anomalous dimension7.4 The soft anomalous dimension at four loopsA Coefficients of the Regge pole amplitudeB Anomalous dimensionsC Computing colour factors for arbitrary representationsD The reduced amplitude in an explicit colour basis
We show that planar contributions due to multiple-Reggeon exchange diagrams can be factorised as a Regge pole along with the single-Reggeon exchange, and when this is done, the singular part of the gluon Regge trajectory is directly determined by the cusp anomalous dimension
The behaviour of amplitudes at high energy was described in terms of fundamental objects in Regge theory associated with specific exchanges in the scattering process, which are characterised by their singularities in the complex angular momentum plane [3, 5, 6], namely Regge cuts and Regge poles
Summary
The high-energy limit of scattering amplitudes has been a fascinating avenue in exploring the strong interactions already before the discovery of QCD, see e.g. [1–6]. This tower of corrections has been recently computed by solving the BFKL equation iteratively through 13 loops [23–26] This example makes it clear that the special features of the Regge limit allow to uncover structures of the perturbative expansion of gauge theories, far beyond the reach of fixed-order calculations in general kinematics. In this work we focus on the tower of Next-to-Next-to-Leading Logarithms (NNLL) in the real part of the amplitude Contributions of both a single-Reggeon exchange and a triple-Reggeon exchange become important and their interplay generates both a Regge pole and a Regge cut.
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