Non-abelian infrared divergences on the celestial sphere

Lorenzo Magnea

doi:10.1007/jhep05(2021)282

Abstract

We consider the infrared factorisation of non-abelian multi-particle scattering amplitudes, and we study the form of the universal colour operator responsible for infrared divergences, when expressed in terms of coordinates on the ‘celestial sphere’ intersecting the future light-cone at asymptotic distances. We find that colour-dipole contributions to the infrared operator, to all orders in perturbation theory, have a remarkably simple expression in these coordinates, with scale and coupling dependence factorised from kinematics and colour. Generalising earlier suggestions in the abelian theory, we then show that the infrared operator can be computed as a correlator of vertex operators in a conformal field theory of Lie-algebra-valued free bosons on the celestial sphere. We verify by means of the OPE that the theory correctly predicts the all-order structure of collinear limits, and the tree-level factorisation of soft real radiation.

Highlights

All the developments briefly summarised above are based on standard, if advanced, techniques of quantum field theory in four dimensions, including diagrammatic tools, renormalisation group equations, Ward identities and effective field theories
We consider the infrared factorisation of non-abelian multi-particle scattering amplitudes, and we study the form of the universal colour operator responsible for infrared divergences, when expressed in terms of coordinates on the ‘celestial sphere’ intersecting the future light-cone at asymptotic distances
We find that colour-dipole contributions to the infrared operator, to all orders in perturbation theory, have a remarkably simple expression in these coordinates, with scale and coupling dependence factorised from kinematics and colour

Summary

Non-abelian soft anomalous dimensions: a primer

The structure of infrared divergences for on-shell scattering amplitudes in massless nonabelian gauge theories is understood in remarkable generality. This approximation is known to fail at the four-loop level [34, 35, 86,87,88,89], where contributions proportional to quartic Casimir eigenvalues have been shown to appear, with expected consequences on the structure of Γn at four loops and beyond [90, 91] With this single simplifying approximation, the soft anomalous dimension matrix admits the all-order representation. Where μ is a fixed scale, while λ is the integration variable in eq (2.3) In this form, one can show that (aside from poles originating from the running of the coupling) the first term, which contains dipole colour correlations, generates only single poles of soft origin, while the remaining terms, which are colour singlets, generate single poles of hard-collinear origin, as well as double poles of soft-collinear origin. The correlator in eq (2.14) is invariant under the rescalings βi → κiβi, this invariance is broken by collinear divergences in the massless case: the dipole contribution to the anomalous dimension matrix arises as a solution to the corresponding ‘anomaly equation’, where the ‘anomaly’ is expressed by the cusp anomalous dimension [18, 19]

Colour correlations on the celestial sphere

A theory of Lie-algebra-valued free bosons

Some open questions

Perspectives