Advances in High Energy Physics

Volume 2019, Article ID 3142510, 17 pages

https://doi.org/10.1155/2019/3142510

## A Short Review on Recent Developments in TMD Factorization and Implementation

Departamento de Física Teórica and IPARCOS, Universidad Complutense de Madrid, Ciudad Universitaria, 28040 Madrid, Spain

Correspondence should be addressed to Ignazio Scimemi; se.mcu@soizangi

Received 20 December 2018; Accepted 7 April 2019; Published 13 May 2019

Guest Editor: Zhongbo Kang

Copyright © 2019 Ignazio Scimemi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### Abstract

In the latest years the theoretical and phenomenological advances in the factorization of several collider processes using the transverse momentum dependent distributions (TMD) have greatly increased. I attempt here a short resume of the newest developments discussing also the most recent perturbative QCD calculations. The work is not strictly directed to experts in the field and it wants to offer an overview of the tools and concepts which are behind the TMD factorization and evolution. I consider both theoretical and phenomenological aspects, some of which have still to be fully explored. It is expected that actual colliders and the Electron Ion Collider (EIC) will provide important information in this respect.

#### 1. Introduction

The knowledge of the structure of hadrons is a leitmotiv for the study of quantum chromodynamics (QCD) for decades. Apart from the notions of quarks and gluons (we call them generically “partons” in the following), the natural question is how the momenta of these particles are distributed inside the hadrons and how the spin of hadrons is generated. Phenomenologically it is possible to access at this problem only in some particular kinematical conditions, as provided, for instance, in experiments like (semi-inclusive) deep inelastic scattering, vector and scalar boson production, hadrons, or jets. I review the basic principle which supports this investigation. Let us consider, to start with, the cross section for di-lepton production in a typical Drell-Yan process where includes all particles which are not directly measured. The cross section for this process can be written formally aswhere is the virtual di-lepton invariant mass, are the parton momenta fraction along a light-cone direction or Bjorken variables, and are the parton distribution functions (PDF). The r.h.s. of (1) assumes several notions which, nowadays, can be found in textbooks. In fact a central hypothesis is a clear energy separation between the di-lepton invariant mass and the scale at which QCD cannot be treated perturbatively any more (we call it the hadronization scale GeV), that is, . Given this, one can factorize the cross section in a perturbatively calculable part and the rest. Formula (1) represents just a first term of an “operator product expansion” of the cross section. The price to pay for this separation is the introduction of a factorization scale which can be used to resume logarithms in combination with renormalization group equations [1–3]. Another aspect, which is remarkable, is that the nonperturbative part of the cross section can be also expressed as the product of two parton distribution functions. This fact has two main consequences: on the one hand, all the nonperturbative information of the process is included in the PDFs; on the other hand, the partons belonging to different hadrons are completely disentangled. In these conditions so the longitudinal momenta of quarks and gluons can be reconstructed nonperturbatively and this fact has given rise to a large investigation whose review goes beyond the purpose of this writing.

The ideal description of the process in (1) however becomes more involved in the case of more differential cross sections [32–34]. So, for instance, one can wonder whether a formula likehas any physical consistency. (I use the notation for 2-dimensional impact parameter, , is the center of mass energy of the process, ) The answer to this question is necessarily more complex than in the case of (1) for the simple fact that a new kinematic scale, , the transverse momentum of the di-lepton pair, has now appeared. In this article I will concentrate on the description of the casewhich is interesting for a number of observables. The restriction to this kinematical regime represents also a limitation of the present approach which should be overcome with further studies.

The study of factorization [25, 27, 29, 30, 35, 36] has lead finally to the conclusion that actually (2) in not completely correct because the cross section for these kind of processes should instead be of the formwith and being the rapidity scales. Formula (4) shows explicitly that the TMD functions contain nonperturbative QCD information different from the usual PDF, while they still allow completing disentangle QCD effects coming from different hadrons. These new nonperturbative QCD inputs can be written in terms of well-defined matrix elements of field operators which can be extracted from experiments or evaluated with appropriate theoretical tools. These objectives require some discussion, which I partially provide in this text.

The scale is the authentic key stone of the TMD factorization. Its origin is different from the usual factorization scale and because of this it is allowed to perform a special resummation for this scale. This leads to the fact that a consistent and efficient implementation of the evolution is crucial for the prediction and extraction of TMDs from data. A possible implementation of the TMD evolution is historically provided by Collins-Soper-Sterman (CSS) [32–34]. However a complete discussion of more efficient alternatives has started more recently [21–23, 26, 37]. The point is that the rapidity scale evolution has both a perturbative and nonperturbative input, as it is actually provided by (derivatives of) an operator matrix element (the so called soft function). An efficient implementation and scale choice so should separate as much as possible the nonperturbative inputs with different origin inside the cross sections. This target is not completely realized with the CSS implementation, while it can be achieved with the -prescription discussed in the text. This discussion is also relevant for multiple reasons. In fact various orders in perturbation theory are available already for unpolarized and polarized distribution and, in the future, one expects more results in this respect for many polarized distributions. When dealing with several perturbative orders, the convergence of the perturbative series can be seriously undermined by an inappropriate choice of scales, and this is a well-known problem that can affect the theoretical error of any result. A more subtle issue comes from the fact that the evolution corrections can also be of nonperturbative nature. It would be certainly clarifying a scheme in which the nonperturbative effects of the evolution are clearly separated from the intrinsic nonperturbative TMD effects. Such a request results to be important when several extraction of TMD from data are compared and also when a complete nonperturbative evaluation of TMD can be provided.

In the rest of this review I will try to give an idea on how all these problems can be consistently treated, which can be useful also to explore new and more efficient solutions. Several parts of this review use material that can be originally found in [4, 23, 38].

#### 2. Factorization

The factorization of the cross sections into TMD matrix elements has been provided by several authors and it has been object of many discussions [25, 27, 29, 30, 32–36]. We briefly review the main ideas here for the case of Drell-Yan. The process is characterized by two initial hadrons which come from opposite collinear directions and produce two leptons in the final state plus unmeasured radiation. We identify collinear (anticollinear) light-cone directions () and , for the momentum of colliding particles. The momentum of collinear particles is with and and . The momenta of collinear particles are characterized by the scaling where is the di-lepton invariant mass and is a small parameter being the hadronization scale. A reversed scaling of momentum is valid for anticollinear particles, say . The soft radiation which entangles collinear and anticollinear particles is homogeneous in momentum distribution (its momentum scales as ) and can be distinguished from the collinear radiation only for a different scaling of the components of the momenta. Given this, it is natural to divide the hadronic phase space in regions as in Figure 1. In this picture, the collinear and soft regions are necessarily separated by rapidity and they all share the same energy .