Correspondence between Soft and Rapidity Anomalous Dimensions

Abstract

We establish a correspondence between ultraviolet singularities of soft factors for multiparticle production and rapidity singularities of soft factors for multiparton scattering. This correspondence is a consequence of the conformal mapping between scattering geometries. The correspondence is valid to all orders of perturbation theory and in this way provides one with a proof of rapidity renormalization procedure for multiparton scattering (including the transverse momentum dependent (TMD) factorization as a special case). As a by-product, we obtain an exact relation between the rapidity anomalous dimension and the well-known soft anomalous dimension. The three-loop expressions for TMD and a general multiparton scattering rapidity anomalous dimension are derived.Introduction. Factorization theorems are an effective tool for the description of hadron reactions within the perturbative Quantum Chromodynamics (pQCD) [1][2][3][4][5]. Factorization formulas have a common structure which includes a hard part, parton distributions, jet functions, and the soft factor(s). However, the operator structure of these ingredients can differ drastically for different processes, that leads to significant fragmentation of theoretical results. In this Letter, we discuss a correspondence between soft factors (SFs) typical for different kinematics and consequences of this correspondence.Generally, SFs represent the soft part of the betweenparton interaction. A typical SF is given by a vacuum matrix element of a configuration of Wilson lines that reflects the classical picture of scattering. Being in many aspects artificial, SFs contain a set of infrared divergences and are defined only within an appropriate regularization. Studying the structure of divergences, one gets access to the scaling equations and corresponding anomalous dimensions. In turn, it allows one to resum large logarithms of factorization scales and obtain scattering cross-sections in a wide kinematic range.Considering the processes where several partons participate in single hard interaction one often deals with SFs of the form

Systematic analysis of double-scale evolution

“…Despite the fact that the TMD evolution is known up to the third order in the strong coupling expansion [23,24], the error coming from the evolution is the largest among all other theoretical inputs, as it has been shown in ref. [1].…”

“…Nowadays, these anomalous dimensions are known up to three-loop order (i.e. including term a 3 s or up to NNLO), see [23,24,[34][35][36]. In figures 3 we show the function R for different orders of perturbation theory and for different explicit path solutions given in eq.…”

“…On the other side, in conformal field theory, where the coupling constant is independent on µ, the rapidity anomalous dimension is linear in logarithms of µ b and coincides with the soft anomalous dimension [22,24].…”

The principle of maximal transcendentality and the four-loop collinear anomalous dimension

“…This result was explained in ref. [38] by mapping the appropriate configurations of Wilson lines for the two computations into each other using a conformal transformation. Hence we will obtain the four-loop values of both the threshold soft and rapidity anomalous dimensions in planar N = 4 SYM from…”

“…The collinear anomalous dimension also enters the Regge trajectory for forward scattering [10,30,31]. An eikonal (Wilson line) version of it enters both the threshold soft anomalous dimension for threshold resummation [32][33][34][35] and the rapidity anomalous dimension for transverse momentum resummation [35][36][37][38].…”