Relating amplitude and PDF factorisation through Wilson-line geometries

Abstract

We study long-distance singularities governing different physical quantities involving massless partons in perturbative QCD by using factorisation in terms of Wilson-line correlators. By isolating the process-independent hard-collinear singularities from quark and gluon form factors, and identifying these with the ones governing the elastic limit of the perturbative Parton Distribution Functions (PDFs)δ(1 − x) in the large-x limit of DGLAP splitting functions -we extract the anomalous dimension controlling soft singularities of the PDFs, verifying that it admits Casimir scaling. We then perform an independent diagrammatic computation of the latter using its definition in terms of Wilson lines, confirming explicitly the above result through two loops. By comparing our eikonal PDF calculation to that of the eikonal form factor by Erdogan and Sterman and the classical computation of the closed parallelogram by Korchemsky and Korchemskaya, a consistent picture emerges whereby all singularities emerge in diagrammatic configurations localised at the cusps or along lightlike lines, but where distinct contributions to the anomalous dimensions are associated with finite (closed) lightlike segments as compared to infinite (open) ones. Both are relevant for resumming large logarithms in physical quantities, notably the anomalous dimension controlling Drell-Yan or Higgs production near threshold on the one hand, and the gluon Regge trajectory controlling the high-energy limit of partonic scattering on the other. Isolating hard-collinear singularitiesThe contribution of Γ J in eq. (2.18) is associated to the soft singularities of J i , which cancel in the ratio of J i and J i eq. (2.9). It is therefore convenient to focus on the poles of pure hard-collinear origin, defined aswhere J i | pole means only the poles of the jet function. We extract the function J i/J for i = q and i = g from the form factor of the quark and of the gluon, respectively, thus providing the process-independent components containing the purely collinear singularities associated with massless external partons. In order to determine J i/J , we isolate the pole part of the jet function J i , by replacing in eq. (2.18) the function G(1, α s , ǫ) with γ G , according to the

Soft corrections to inclusive deep-inelastic scattering at four loops and beyond

“…Given its relation to the anomalous dimension of the eikonal form factor, we assume here that f q 4 exhibits the same generalized maximal non-Abelian property in the presence of quartic colour factors as the cusp anomalous dimension A 4 , see [34]. This assumption, supported also by recent studies of amplitude factorization in [61], implies the absence of the colour coefficients…”

Subleading power resummation of rapidity logarithms: the energy-energy correlator in $$ \mathcal{N} $$ = 4 SYM

“…For the particular case of a conformal theory, this expression simplifies considerably, both due to the fact that the coupling doesn't run, and also since in a conformal field theory there is an equivalence between the rapidity anomalous dimension and the soft anomalous dimension [87,92]. Combining this equivalence with the relations for the soft anomalous dimension derived in [93] (see also [94] for recent work on relations between different soft functions), we have…”

Wilson-line geometries and the relation between IR singularities of form factors and the large-x limit of DGLAP splitting functions

“…(1.9) by factorising form factors and PDFs separately? In this contribution we provide an affirmative answer to the above question and establish that γ G − 2B δ = Γ /4 which are the main results of [6]. In Section 2 and 3 we factorise a form factor and a PDF respectively.…”

“…The calculations work much the same way as the one-loop case, deriving the same integral representation over one infinite line and a finite one. The procedure is outlined in [20] and details of the calculation of logW are given in [6]. (2)…”