Bootstrapping the QCD soft anomalous dimension

Abstract

The soft anomalous dimension governs the infrared singularities of scattering amplitudes to all orders in perturbative quantum field theory, and is a crucial ingredient in both formal and phenomenological applications of non-abelian gauge theories. It has recently been computed at three-loop order for massless partons by explicit evaluation of all relevant Feynman diagrams. In this paper, we show how the same result can be obtained, up to an overall numerical factor, using a bootstrap procedure. We first give a geometrical argument for the fact that the result can be expressed in terms of singlevalued harmonic polylogarithms. We then use symmetry considerations as well as known properties of scattering amplitudes in collinear and high-energy (Regge) limits to constrain an ansatz of basis functions. This is a highly non-trivial cross-check of the result, and our methods pave the way for greatly simplified higher-order calculations.

Infrared singularities of QCD scattering amplitudes in the Regge limit to all orders

“…These results also provide a valuable input for determining the structure of long-distance singularities for general kinematics using a bootstrap approach, as done at the three-loop order in ref. [32].…”

“…As an application of the interplay between these limits, it was recently demonstrated [32] that the functional form of the three-loop soft anomalous dimension in general kinematics can in fact be fully recovered via a bootstrap procedure using the high-energy limit of 2 → 2 scattering, alongside other information, as input. The bootstrap programme of the soft anomalous dimension can be extended beyond three loops, provided that information from special kinematic limits is available.…”

“…The soft anomalous dimension can be further constrained by the behaviour of scattering amplitudes in special kinematic limits, such as the Regge limit [21,22,24] and collinear limits [30,36]. Furthermore, it was recently shown [32] that the space of functions in terms of which the non-dipole correction is expressed (single-valued multiple polylogarithms) can, in fact, be deduced from general considerations. A bootstrap procedure was then set up, which remarkably completely fixes the functional form of the non-dipole correction at three loops (up to an overall rational numerical factor) based on known information from the kinematic limits mentioned above, reproducing the result of the Feynman-diagram computation of refs.…”

The High-Energy Limit of 2 to 2 Partonic Scattering Amplitudes

“…Despite the complexity of the calculation the result is remarkably simple; we refer the reader to the original publication for details. Subsequently, this result was reproduced (up to a single rational normalization factor) by a so-called bootstrap procedure [49], starting with a general ansatz in terms of the relevant type of functions (single-valued harmonic polylogarithms [50][51][52][53][54][55]) and imposing on it all of the known constraints based on the theory of soft gluon exponentiation [56], symmetries, and information from special kinematic limits. This work was reviewed in the 2017 edition of the present conference [57].…”

Wilson Lines and Webs in Higher-Order QCD

“…As shown in ref. [25], the form of the correction function is then completely fixed up to an overall constant. Furthermore, this is the first time such a procedure has been used in a fully non-planar QCD application!…”