Scattering amplitudes in the Regge limit and the soft anomalous dimension through four loops

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.

The Infrared Structure of Perturbative Gauge Theories

“…More recently, further constraints on the infrared matrix, arising from the high-energy limit, were uncovered in Refs. [423][424][425][426][427][428]: the high-energy limit of the infrared matrix is now known to all-orders at next-to-leading logarithmic (NLL) accuracy, while NNLL contributions constrain the quartic Casimir component at four loops.…”

“…[422] in a formulation allowing for a much more direct comparison with infrared factorisation: this made possible the first direct calculation of a contribution to the soft anomalous dimension matrix going beyond the dipole formula, at the four-loop level. A systematic development of the framework proposed in [422] has led to the determination of the soft anomalous dimension matrix for 2 → 2 scattering amplitudes at NLL accuracy to all orders in perturbation theory [423][424][425], and at NNLL accuracy up to four loops [426][427][428]. This result, in turn, establishes for the first time the presence of quartic Casimir contributions to the soft anomalous dimension matrix, beyond those induced by the cusp anomalous dimension.…”

The Soft Anomalous Dimension at four loops in the Regge Limit

“…This will form part of what is called the Regge-limit basis, which involves writing the colour operators in terms of T 2 s−u and T 2 t in nested commutators where possible and can be seen in eqs. (21,24,25). The quartic Casimir which contains a fully symmetrised trace, first appears in the soft anomalous dimension at four loops.…”

“…with all other functions contributing at NNLL accuracy in the Regge limit having a similar expansion. The colour structures are expressed in a Regge-limit basis with the steps elaborated in [24].…”

“…There has been much study of scattering amplitudes in the highenergy (Regge) limit in QCD [10][11][12][13][14][15] including recent work on 2 → 2 scattering [16][17][18][19][20][21]. The new constraints are found by using results from newly calculated amplitudes in the Regge limit to Next-to-Leading Logarithmic (NLL) accuracy for the even signature amplitude [10,16,18,22] and NNLL accuracy for the odd signature amplitude [21,23,24]. These results are available in full colour for arbitrary representations of the scattered partons.…”