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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

After the first observation of the

Bulk of the events in the Drell-Yan reaction are from the region where the transverse momentum of the dilepton

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 [

Another important observable in the Drell-Yan process is the

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

The remained content of the article is organised as follows. In Section

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

In general, it is more convenient to solve the evolution equations for the TMD distributions in the coordinate space (

The energy evolution for the

As the

In the small

The Sudakov-like form factor

The general form of

There are several extractions for

The original BLNY fit parameterized

Inspired by [

Since the original BLNY fit fails to simultaneously describe Drell-Yan process and SIDIS process, in [

In [

To study the pion-nucleon Drell-Yan data, it is also necessary to know the nonperturbative Sudakov form factor for the pion meson. In [

Figure

The fitted cross section (solid line) of pion-nucleon Drell-Yan as functions of

From the fitted result, we find that the value of the parameter

After solving the evolution equations and incorporating the Sudakov form factor, the scale-dependent TMD distribution function

Similarly, the evolution formalism of the proton transversity distribution in

The Sivers function and Boer-Mulders function, which are T-odd, can be expressed as follows in

Similar to what has been done to the unpolarized distribution function and transversity distribution function, in the low

The TMD evolution formalism in (

In this section we will set up the necessary framework for physical observables in

In Drell-Yan process

The differential cross section formulated in TMD factorization is usually expressed in the

The general differential cross section for the unpolarized Drell-Yan process can be written as [

In general, TMD factorization [

The hard

One can absorb the scheme-dependent hard factors

With the new

In the Drell-Yan process with a

The structure function

The spin-dependent differential cross section in (

The angular differential cross section for unpolarized Drell-Yan process has the following general form

The coefficients

The

Based on the TMD evolution formalism for the distributions set up in Section

In [

Subtracted unpolarized TMD distribution of the pion meson for valence quarks in

The scale dependence of the T-odd distributions, such as the Sivers function and the Boer-Mulders function, is more involved than that of the T-even distributions. This is because their collinear counterparts are the twist-3 multiparton correlation functions [

In [

Subtracted Sivers function for the up quarks in Drell-Yan in

The evolution of the Boer-Mulders function for the valence quark inside

We plot the

The Boer-Mulders function for

In conclusion, we find that the tendency of the distributions is similar: the distribution is dominated by perturbative region in

Based on the general TMD factorization framework provided in Section

QCD predicts that the T-odd PDFs present generalized universality, i.e., the sign of the Sivers function measured in Drell-Yan process should be opposite to its sign measured in SIDIS [

The very first step to understand the Sivers asymmetry in the

The transverse spectrum of lepton pair production in the unpolarized pion-nucleon Drell-Yan process, with an

In [

The TMD evolution effect of the Sivers asymmetry in SIDIS and

With the numerical results of the TMD distributions in (

The Sivers asymmetry within TMD factorization for a

Using (

The

It has been a broad consensus that the study on the TMD observables will provide information on the partons’ intrinsic transverse motions inside a hadron. In the previous sections we have tried to substantiate this statement mainly focusing on the unpolarized and single-polarized

The extraction of nonperturbative Sudakov form factor from the

The precise measurement on the single-spin asymmetry in the kinematical region of COMPASS can provide great opportunity to access the Sivers function. Besides the TMD evolution effect, the choice of the scale dependence of the Qiu-Sterman function can affect the shape of the asymmetry and should be considered in the future extraction of the Sivers function.

Sizable

Although a lot of progress on the theoretical framework of the TMD factorization and TMD evolution has been made, the improvement is still necessary both from the perturbative and nonperturbative aspects. In the future, the study of

The authors declare that they have no conflicts of interest.

This work is partially supported by the NSFC (China) Grant 11575043 and by the Fundamental Research Funds for the Central Universities of China. X. Wang is supported by the NSFC (China) Grant 11847217 and the China Postdoctoral Science Foundation under Grant no. 2018M640680.

^{↑}+p→W

^{±}/Z

^{0}at RHIC

_{T}, transverse parton distributions and the collinear anomaly

_{T}and transverse-momentum distributions on-the-light-cone

_{F}(x,x)

^{3}He Target at Q

^{2}=1.4–2.7 GeV

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