Advances in High Energy Physics

Volume 2019, Article ID 6734293, 20 pages

https://doi.org/10.1155/2019/6734293

## Drell-Yan Process in TMD Factorization

^{1}School of Physics and Engineering, Zhengzhou University, Zhengzhou, Henan 450001, China^{2}School of Physics, Southeast University, Nanjing 211189, China

Correspondence should be addressed to Zhun Lu; nc.ude.ues@ulnuhz

Received 16 November 2018; Accepted 9 January 2019; Published 22 January 2019

Guest Editor: Zhongbo Kang

Copyright © 2019 Xiaoyu Wang and Zhun Lu. 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

This article presents the review of the current understanding on the pion-nucleon Drell-Yan process from the point of view of the TMD factorization. Using the evolution formalism for the unpolarized and polarized TMD distributions developed recently, we provide the theoretical expression of the relevant physical observables, namely, the unpolarized cross section, the Sivers asymmetry, and the asymmetry contributed by the double Boer-Mulders effects. The corresponding phenomenology, particularly at the kinematical configuration of the COMPASS Drell-Yan facility, is displayed numerically.

#### 1. Introduction

After the first observation of the lepton pairs produced in collisions [1], the process was interpreted that a quark and an antiquark from each initial hadron annihilate into a virtual photon, which in turn decays into a lepton pair [2]. This explanation makes the process an ideal tool to explore the internal structure of both the beam and target hadrons. Since then, a wide range of studies on this (Drell-Yan) process have been carried out. In particular, the Drell-Yan process has the unique capability to pin down the partonic structure of the pion, which is an unstable particle and therefore cannot serve as a target in deep inelastic scattering processes. Several pion-induced experiments have been carried out, such as the NA10 experiment at CERN [3–6], the E615 [7], E444 [8], and E537 [9] experiments at Fermilab three decades ago. These experimental measurements have provided plenty of data, which have been used to considerably constrain the distribution function of the pion meson. Recently, a new pion-induced Drell-Yan program with polarized target was also proposed [10] at the COMPASS of CERN, and the first data using a high-intensity beam of 190 GeV colliding on a target has already come out [11].

Bulk of the events in the Drell-Yan reaction are from the region where the transverse momentum of the dilepton is much smaller than the mass of the virtual vector boson; thus the intrinsic transverse momenta of initial partons become relevant. It is also the most interesting regime where a lot of intriguing physics arises. Moreover, in the small region (), the fixed-order calculations of the cross sections in the collinear picture fail, leading to large double logarithms of the type . It is necessary to resum such logarithmic contributions to all orders in the strong coupling to obtain a reliable result. The standard approach for such resummation is the Collins-Soper-Sterman (CSS) formalism [12], originated from previous work on the Drell-Yan process and the annihilation three decades ago. In recent years the CSS formalism has been successfully applied to develop a factorization theorem [13–15] in which the gauge-invariant [16–19] transverse momentum dependent (TMD) parton distribution functions or fragmentation functions (collectively called TMDs) [20, 21] play a central role. From the point of view of TMD factorization [12, 13, 15, 22], physical observables can be written as convolutions of a factor related to hard scattering and well-defined TMDs. After solving the evolution equations, the TMDs at fixed energy scale can be expressed as a convolution of their collinear counterparts and perturbatively calculable coefficients in the perturbative region, and the evolution from one energy scale to another energy scale is included in the exponential factor of the so-called Sudakov-like form factors [12, 15, 23, 24]. The TMD factorization has been widely applied to various high energy processes, such as the semi-inclusive deep inelastic scattering (SIDIS) [14, 15, 22, 23, 25, 26], annihilation [15, 27, 28], Drell-Yan [15, 29], and W/Z production in hadron collision [12, 15, 30]. The TMD factorization can be also extended to the moderate region where an equivalence [31, 32] between the TMD factorization and the twist-3 collinear factorization is found.

One of the most important observables in the polarized Drell-Yan process is the Sivers asymmetry. It is contributed by the so-called Sivers function [33], a time-reversal-odd (T-odd) distribution describing the asymmetric distribution of unpolarized quarks inside a transversely polarized nucleon through the correlation between the quark transverse momentum and the nucleon transverse spin. Remarkably, QCD predicts that the sign of the Sivers function changes in SIDIS with respect to the Drell-Yan process [16, 34, 35]. The verification of this sign change [36–41] is one of the most fundamental tests of our understanding of the QCD dynamics and the factorization schemes, and it is also the main pursue of the existing and future Drell-Yan facilities [10, 11, 42–45]. The advantage of the Drell-Yan measurement at COMPASS is that almost the same setup [11, 46] is used in SIDIS and Drell-Yan processes, which may reduce the uncertainty in the extraction of the Sivers function. In particular, the COMPASS Collaboration measured for the first time the transverse-spin-dependent azimuthal asymmetries [11] in the Drell-Yan process.

Another important observable in the Drell-Yan process is the angular asymmetry, where corresponds to the azimuthal angle of the dilepton. The fixed-target measurements from the NA10 and E615 collaborations showed that the unpolarized cross section possesses large asymmetry, which violates the Lam-Tung relation [47]. Similar violation has also been observed in the colliders at Tevatron [48] and LHC [49]. It has been explained from the viewpoints of higher-twist effect [50–53], the noncoplanarity effect [30, 54], and the QCD radiative effects at higher order [55, 56]. Another promising origin [57] for the violation of the Lam-Tung relation at low transverse momentum is the convolution of the two Boer-Mulders functions [58] from each hadron. The Boer-Mulders function is also a TMD distribution. As the chiral-odd partner of the Sivers function, it describes the transverse-polarization asymmetry of quarks inside an unpolarized hadron [57, 58], thereby allowing the probe of the transverse spin physics from unpolarized reaction.

This article aims at a review on the current status of our understanding on the Drell-Yan dilepton production at low transverse momentum, especially from the collision, based on the recent development of the TMD factorization. We will mainly focus on the phenomenology of the Sivers asymmetry as well as the asymmetry from the double Boer-Mulders effect. In order to quantitatively understand various spin/azimuthal asymmetries in the Drell-Yan process, a particularly important step is to know in high accuracy the spin-averaged differential cross section of the same process with azimuthal angles integrated out, since it always appears in the denominator of the asymmetries’ definition. Thus, the spin-averaged cross section will be also discussed in great details.

The remained content of the article is organised as follows. In Section 2, we will review the TMD evolution formalism of the TMDs, mostly following the approach established in [15]. Particularly, we will discuss in detail the extraction of the nonperturbative Sudakov form factor for the unpolarized TMD distribution of the proton/pion as well as that for the Sivers function. In Section 3, putting the evolved result of the TMD distributions into the TMD factorization formulae, we will present the theoretical expression of the physical observables, such as the unpolarized differential cross section, the Sivers asymmetry, and the asymmetry contributed by the double Boer-Mulders effect. In Section 4, we present the numerical evolution results of the unpolarized TMD distributions and the Boer-Mulders function of the pion meson, as well as that of the Sivers function of the proton. In Section 5, we display the phenomenology of the physical observables (unpolarized differential cross section, the Sivers asymmetry, and the asymmetry) in the Drell-Yan with TMD factorization at the kinematical configuration of the COMPASS experiments. We summarize the paper in Section 6.

#### 2. The TMD Evolution of the Distribution Functions

In this section, we present a review on the TMD evolution of the distribution functions. Particularly, we provide the evolution formalism for the unpolarized distribution function , transversity , Sivers function , and the Boer-Mulders function of the proton, as well as and of the pion meson, within the Collins-11 TMD factorization scheme [15].

In general, it is more convenient to solve the evolution equations for the TMD distributions in the coordinate space ( space) other than that in the transverse momentum space, with conjugate to via Fourier transformation [12, 15]. The TMD distributions in space have two kinds of energy dependence, namely, is the renormalization scale related to the corresponding collinear PDFs, and is the energy scale serving as a cutoff to regularize the light-cone singularity in the operator definition of the TMD distributions. Here, is a shorthand for any TMD distribution function and the tilde denotes that the distribution is the one in space. If we perform the inverse Fourier transformation on , we recover the distribution function in the transverse momentum space , which contains the information about the probability of finding a quark with specific polarization, collinear momentum fraction , and transverse momentum in a specifically polarized hadron .

##### 2.1. TMD Evolution Equations

The energy evolution for the dependence of the TMD distributions is encoded in the Collins-Soper (CS) [12, 15, 63] equation:while the dependence is driven by the renormalization group equation aswith being the strong coupling at the energy scale , being the CS evolution kernel, and , being the anomalous dimensions. The solutions of these evolution equations were studied in detail in [15, 63, 64]. Here, we will only discuss the final result. The overall structure of the solution for is similar to that for the Sudakov form factor. More specifically, the energy evolution of TMD distributions from an initial energy to another energy is encoded in the Sudakov-like form factor by the exponential form where is the factor related to the hard scattering. Hereafter, we will set and express as .

As the -dependence of the TMDs can provide very useful information regarding the transverse momentum dependence of the hadronic 3D structure through Fourier transformation, it is of fundamental importance to study the TMDs in space. In the small region, the dependence is perturbatively calculable, while in the large region, the dependence turns to be nonperturbative and may be obtained from the experimental data. To combine the perturbative information at small with the nonperturbative part at large , a matching procedure must be introduced with a parameter serving as the boundary between the two regions. The prescription also allows for a smooth transition from perturbative to nonperturbative regions and avoids the Landau pole singularity in . A -dependent function is defined to have the property at low values of and at large values. In this paper, we adopt the original CSS prescription [12]:The typical value of is chosen around to guarantee that is always in the perturbative region. Besides the CSS prescription, there were several different prescriptions in literature. In [65, 66] a function decreasing with increasing was also introduced to match the TMD factorization with the fixed-order collinear calculations in the very small region.

In the small region , the TMD distributions at fixed energy can be expressed as the convolution of the perturbatively calculable coefficients and the corresponding collinear PDFs or the multiparton correlation functions [22, 67]Here, stands for the convolution in the momentum fraction and is the corresponding collinear counterpart of flavor in hadron at the energy scale . The latter one could be a dynamic scale related to by , with and the Euler Constant [22]. The perturbative hard coefficients , independent of the initial hadron type, have been calculated for the parton-target case [23, 68] as the series of and the results have been presented in [67] (see also Appendix A of [23]).

##### 2.2. Sudakov Form Factors for the Proton and the Pion

The Sudakov-like form factor in (4) can be separated into the perturbatively calculable part and the nonperturbative part According to the studies in [26, 39, 69–71], the perturbative part of the Sudakov form factor has the same result among different kinds of distribution functions, i.e., is spin-independent. It has the general formThe coefficients and in(9) can be expanded as the series of :Here, we list to and to up to the accuracy of next-to-leading-logarithmic (NLL) order [12, 23, 26, 69, 72, 73]:For the nonperturbative form factor , it can not be analytically calculated by the perturbative method, which means it has to be parameterized to obtain the evolution information in the nonperturbative region.

The general form of was suggested as [12]The nonperturbative functions and are functions of the impact parameter and depend on the choice of . To be more specific, contains the information on the large behavior of the evolution kernel . Also, according to the power counting analysis in [74], shall follow the power behavior as at small- region, which can be an essential constraint for the parameterization of . The well-known Brock-Landry-Nadolsky-Yuan (BLNY) fit parameterizes as with a free parameter [72]. We note that is universal for different types of TMDs and does not depend on the particular process, which is an important prediction of QCD factorization theorems involving TMDs [15, 23, 39, 75]. The nonperturbative function contains information on the intrinsic nonperturbative transverse motion of bound partons, namely, it should depend on the type of hadron and the quark flavor as well as for TMD distributions. As for the TMD fragmentation functions, it may depend on , the type of the produced hadron, and the quark flavor. In other words, depends on the specific TMDs.

There are several extractions for in literature, we review some often-used forms below.

The original BLNY fit parameterized as [72]where and are the longitudinal momentum fractions of the incoming hadrons carried by the initial state quark and antiquark. The BLNY parameterization proved to be very reliable to describe Drell-Yan data and boson production [72]. However, when the parameterization is extrapolated to the typical SIDIS kinematics in HERMES and COMPASS, the transverse momentum distribution of hadron can not be described by the BLNY-type fit [76, 77].

Inspired by [72, 78], a widely used parameterization of for TMD distributions or fragmentation functions was proposed [39, 67, 72, 78–80]where the factor in front of comes from the fact that only one hadron is involved for the parameterization of , while the parameter in [78] is for collisions. The parameter in (17) depends on the type of TMDs, which can be regarded as the width of intrinsic transverse momentum for the relevant TMDs at the initial energy scale [23, 73, 81]. Assuming a Gaussian form, one can obtainwhere and represent the relevant averaged intrinsic transverse momenta squared for TMD distributions and TMD fragmentation functions at the initial scale , respectively.

Since the original BLNY fit fails to simultaneously describe Drell-Yan process and SIDIS process, in [77] the authors proposed a new form for which releases the tension between the BLNY fit to the Drell-Yan (such as , and low energy Drell-Yan pair productions) data and the fit to the SIDIS data from HERMES/COMPASS in the CSS resummation formalism. In addition, the -dependence in (16) was separated with a power law behavior assumption: , where and are the fixed parameters as and . The two different behaviors (logarithmic in (16) and power law) will differ in the intermediate regime. Reference [76] showed that a direct integration of the evolution kernel from low to high led to the form of term as and could describe the SIDIS and Drell-Yan data with values ranging from a few GeV to 10 GeV. Thus, the term was modified to the form of and the functional form of extracted in [77] turned to the formAt small region ( is much smaller than ), the parameterization of the term can be approximated as , which satisfied the constraint of the behavior for . However, at large region, the logarithmic behavior will lead to different predictions on the dependence, since the Gaussian-type parameterization suggests that it is strongly suppressed [82]. This form has been suggested in an early research by Collins and Soper [83], but has not yet been adopted in any phenomenological study until the study in [77]. The comparison between the original BLNY parameterization and this form with the experimental data of Drell-Yan type process has shown that the new form of can fit with the data as equally well as the original BLNY parameterization.

In [66], the function was parameterized as , following the BLNY convention. Furthermore, in the function , the Gaussian width also depends on . The authors simultaneously fit the experimental data of SIDIS process from HERMES and COMPASS Collaborations, the Drell-Yan events at low energy, and the boson production with totally 8059 data points. The extraction can describe the data well in the regions where TMD factorization is supposed to hold.

To study the pion-nucleon Drell-Yan data, it is also necessary to know the nonperturbative Sudakov form factor for the pion meson. In [59], we extended the functional form for the proton TMDs [77] to the case of the pion TMDs:with and the free parameters. Adopting the functional form of in (20), for the first time, we performed the extraction [59] of the nonperturbative Sudakov form factor for the unpolarized TMD PDF of pion meson using the experimental data in the Drell-Yan process collected by the E615 Collaboration at Fermilab [7, 84]. The data fitting was performed by the package MINUIT [85, 86], through a least-squares fit:The total number of data in our fit is . Since the TMD formalism is valid in the region , we did a simple data selection by removing the data in the region . We performed the fit by minimizing the chi-square in (21), and we obtained the following values for the two parameters: with .

Figure 1 plots the -dependent differential cross section (solid line) calculated from the fitted values for and in (22) at the kinematics of E615 at different bins. The full squares with error bars denote the E615 data for comparison. As Figure 1 demonstrates, a good fit is obtained in the region .