Understanding Transversely Polarized Quarks Inside Longitudinally Polarized Nucleons

Abstract

We review the current understanding of transversely polarized quarks inside longitudinally polarized nucleons from the perspective of the longi-transversity distribution function and transverse-momentum-dependent (TMD) factorization. Employing the recently developed TMD evolution formalism for the unpolarized and longi-transversity distribution functions, we present the theoretical formalism for the relevant physical observables, namely the $A_{LL}^{\cos 2\varphi }$ asymmetry in the Drell–Yan process and the $A_{UL}^{\sin 2{\varphi }_h}$ asymmetry in the semi-inclusive deep inelastic scattering process. The phenomenological results are presented numerically.

1. Introduction

In Quantum Chromodynamics (QCD) [1,2,3], nucleons arise as relativistic, strongly interacting bound states composed of quarks and gluons—collectively referred to as partons. Characterizing the quark–gluon composition of hadrons and investigating how hadrons emerge from their underlying quark and gluon constituents pose intellectual challenges due to color confinement. The established collinear QCD factorization framework captures one-dimensional snapshots of the longitudinal momentum distributions of quarks and gluons within a colliding nucleon and represents them through universal parton distribution functions (PDFs). Over decades of deep inelastic scattering (DIS) [4,5] experiments, these collinear PDFs have provided essential insights into the longitudinal momentum distributions of partons within nucleons [6,7,8].

The confined motion and spatial distribution of quarks and gluons within bound nucleons are key features of the nucleon’s three-dimensional internal structure, arising directly from the QCD dynamics. To explore the three-dimensional internal structure of the nucleons, one needs a new kind of two-scale probe that is distinct from the DIS process. These are physical observables defined by a large momentum transfer, $Q_1\gg {\Lambda }_{\mathrm{QCD}}\sim 1/R$, which effectively localizes the probe and allows it to resolve the particle-like nature of quarks and gluons. Moreover, a distinct soft momentum scale, $Q_2\ll Q_1$, is needed, which is particularly sensitive to the details of a hadron’s internal structure. For instance, the cross section in the Drell–Yan process [9,10] serves as a typical two-scale observable when the differential cross section is expressed in terms of the transverse momentum of the lepton pair, $q_T$, in addition to its invariant mass, Q. This is because the production rate is mainly controlled by the kinematic domain where $q_T\ll Q$. The semi-inclusive deep inelastic scattering (SIDIS) process [11,12,13,14] provides another well-defined two-scale process, characterized by a hard scale $Q\gg {\Lambda }_{\mathrm{QCD}}$ and a soft scale given by the transverse momentum $P_{hT}$ of the detected final-state hadron in the photon-hadron frame, where the exchanged virtual photon and the initial hadron define the z-axis. In this virtual photon–hadron frame, the distribution of $P_{hT}$ in the lepton–hadron SIDIS process, in the region $P_{hT}/z\ll Q$, thus constitutes an additional two-scale observable.

In recent years, theoretical progress has led to the establishment of the transverse momentum dependent (TMD) factorization framework [15]. Over the last two decades, TMD factorization has become a key method for investigating the three-dimensional structure of the nucleon and has been extensively applied to various high-energy reactions. The TMD factorization theorem states that, in the region of small transverse momentum, $P_{hT}/z\ll Q$ for SIDIS or $q_T\ll Q$ for Drell–Yan, the differential cross section can be written as a convolution of short-distance hard scattering coefficients with long-distance dynamics, the latter being encoded in well-defined TMD parton distribution functions (PDFs) or fragmentation functions (FFs). This framework provides access to the three-dimensional motion of quarks and gluons inside a nucleon participating in a collision and permits the extraction of universal TMD PDFs [13,14,15,16,17,18,19,20,21,22,23,24,25]. TMD PDFs [11,26,27,28] provide essential information on the transverse-momentum structure of nucleons, allowing for a comprehensive characterization of the momentum space of nucleon constituents.

Formally, the quark–quark correlator of a nucleon characterized by momentum P and spin vector S can be written as

$${\Phi }_{ij}(x,{\mathit{k}}_T;S)=\int {\displaystyle \frac{d{\xi }^{-}{d}^2{\mathsf{\xi }}_T}{{\left(2\pi \right)}^3}}{e}^{ik·\xi }\langle P,S|{\overline{\psi }}_j\left(0\right){\mathcal{U}}_{[0,\xi ]}{\psi }_i\left(\xi \right)|P,S\rangle {|}_{{\xi }^{+}=0},$$

(1)

where ${\mathcal{U}}_{[0,\xi ]}$ represents the gauge link that guarantees color gauge invariance. By performing the Dirac-structure decomposition [26,29], one obtains eight leading-twist TMD PDFs, each associated with a different type of spin–momentum correlation. Among these, three TMD PDFs—the unpolarized distribution $f_1(x,{\mathit{k}}_T)$, the helicity distribution $g_1(x,{\mathit{k}}_T)$, and the transversity distribution $h_1(x,{\mathit{k}}_T)$—reduce to their corresponding collinear PDFs after integrating over the transverse momentum. The other five TMD PDFs, which depend on the polarization of the parent proton and/or the quarks, describe genuinely transverse-momentum-dependent phenomena.

Among the eight leading-twist TMD PDFs, the distribution function $h_{1L}^{\perp }$ is arguably the least studied, as it encodes the correlation between a longitudinally polarized nucleon and transversely polarized quarks via the matrix element

$$h_{1L}^{\perp }\propto {\displaystyle \frac{S_L{p}_T^{\alpha }}{M}}\langle \overline{\psi }i{\sigma }^{\alpha +}{\gamma }_5\psi \rangle ,$$

(2)

where $S_L$ is a scalar representing the longitudinal spin of the nucleon. Consequently, $h_{1L}^{\perp }$ is commonly known as longitudinal transversity (often shortened to longi-transversity). A further distinctive feature of $h_{1L}^{\perp }$ is its chiral-odd character, which requires it to be accompanied by other chiral-odd functions (such as the Collins fragmentation function [30,31]) to be probed experimentally through the single-spin asymmetry $A_{UL}^{\sin 2\varphi }$ in SIDIS processes $l+N^{\to }\to l+h+X$. Predictions for this asymmetry have been presented in Refs. [32,33,34,35,36,37], and it has been measured in fixed-target experiments such as HERMES and CLAS [38,39,40]. These studies indicate that the magnitude of the asymmetry is at the level of a few percent, though the associated systematic uncertainties are still sizable. The distribution $h_{1L}^{\perp }$ can also be probed via the Drell–Yan process. As shown in Refs. [10,41], the convolution of the $h_{1L}^{\perp }$ distributions from the two colliding protons induces a $\cos 2\varphi$ azimuthal modulation in the doubly longitudinally polarized Drell–Yan process, where $\varphi$ is the azimuthal angle of the dilepton pair with respect to the hadronic plane. This provides a theoretically clean observable that is especially sensitive to sea-quark contributions to $h_{1L}^{\perp }$ [10,13,14,15,16,17,18,19,20,21,22,23,24,25,41,42].

Two complementary processes, doubly polarized Drell–Yan and SIDIS with a longitudinally polarized target, serve as the primary experimental avenues for accessing the longi-transversity $h_{1L}^{\perp }$. Recent comprehensive studies have provided detailed phenomenological predictions for the corresponding azimuthal asymmetries ($A_{LL}^{\cos 2\varphi }$ and $A_{UL}^{\sin 2{\varphi }_h}$) within TMD factorization frameworks [20,22].

The theoretical treatment of these azimuthal asymmetries fundamentally depends on TMD factorization, which is valid in the kinematic regions where the transverse momentum is much smaller than the hard scale of the process ($q_T\ll Q$). A key element of this framework is the QCD evolution of TMDs, which is dictated by the Collins–Soper equation [15,16,17,43] and realized via the Sudakov form factor [13,15,17,44]. The evolution incorporates both perturbatively calculable components and non-perturbative elements that must be constrained by experimental data. The development of various parametrizations (such as the EIKV [20] and BDPRS approaches [22]) reflects ongoing efforts to understand the universal aspects of TMD evolution while accounting for process-dependent characteristics [13,15,17,18,20,22,45,46,47,48,49,50,51,52,53,54,55].

Phenomenological analyses face significant challenges due to limited experimental constraints on $h_{1L}^{\perp }$. The Wandzura–Wilczek approximation provides a practical solution by relating $h_{1L}^{\perp }$ to the better-understood transversity distribution $h_1$, although this approach introduces theoretical uncertainties that require careful evaluation. Recent comprehensive studies have advanced our understanding through detailed investigations of both $\sin 2{\varphi }_h$ asymmetries of pion production in SIDIS and $\cos 2\varphi$ asymmetries in double-polarized Drell–Yan processes. These studies have produced valuable comparisons with available experimental data and have also generated predictions at facilities like RHIC and NICA. The findings reveal a marked sensitivity to sea quark contributions, with forecasts indicating potentially observable effects in specific conditions.

This review synthesizes significant developments, examining the theoretical foundations, phenomenological results, and future prospects for understanding $h_{1L}^{\perp }$ through measurements of azimuthal asymmetry. By comparing findings from SIDIS and Drell–Yan processes, we aim to highlight the complementary nature of these experimental approaches and identify promising directions for future research into the three-dimensional structure of the nucleon.

The rest of this review is structured as follows. Section 2 derives the main physical observables within TMD factorization, focusing on the $\cos 2\varphi$ asymmetry in polarized Drell–Yan and the $\sin 2{\varphi }_h$ asymmetry in SIDIS. In Section 3, we introduce the TMD evolution framework for distribution functions, covering the Collins–Soper equation and the Sudakov form factor. Section 4 offers an extensive overview of the numerical results, confronting theoretical predictions with experimental measurements and presenting novel projections for experiments. Finally, in Section 5, we provide a summary and discuss future perspectives.

2. Physical Observables in Drell–Yan and SIDIS in the Framework of TMD Factorization

In this section, we establish the formalism for the physical observables in polarized Drell–Yan and SIDIS processes that investigate the longi-transversity, $h_{1L}^{\perp }$, within the TMD factorization approach.

2.1. The $\cos 2\varphi$ Asymmetry in the Double-Polarized Proton–Proton Drell–Yan Process

We first consider the doubly longitudinally polarized $pp$ Drell–Yan process [10]:

$$\begin{array}{c}\hfill p^{\to }\left(P_1,S_1\right)+p^{\to }\left(P_2,S_2\right)\to {\gamma }^{*}\left(q\right)+X\to l^{+}(\mathsf{\ell })+l^{-}\left({\mathsf{\ell }}^{\prime }\right)+X.\end{array}$$

(3)

Here, $P_{1\left(2\right)}$ and $S_{1\left(2\right)}$ denote the four-momenta and spin vectors of the incoming protons, respectively. The virtual photon carries momentum $q=(\mathsf{\ell }+{\mathsf{\ell }}^{\prime })$, which is time-like, and $Q^2=q^2=m_{ll}^2$ represents the invariant mass squared of the final-state dilepton pair. The arrow → indicates that the proton is longitudinally polarized.

As usual, the kinematics of the process are described by the squared center-of-mass energy $s={(P_1+P_2)}^2$, the longitudinal momentum fractions $x_{1\left(2\right)}={\displaystyle \frac{Q^2}{2P_{1\left(2\right)}·q}}$, the Feynman variable $x_F=2q_L/s=x_1-x_2$, and the dilepton rapidity $y={\displaystyle \frac{1}{2}}\ln {\displaystyle \frac{q^{+}}{q^{-}}}={\displaystyle \frac{1}{2}}\ln {\displaystyle \frac{x_1}{x_2}}$. Moreover, $x_{1/2}$ can be written in terms of $x_F$ and $\tau =Q^2/s=x_1{x}_2$, or alternatively in terms of y and $\tau$, as

$$\begin{array}{c}\hfill x_{1/2}={\displaystyle \frac{\pm x_F+\sqrt{x_F^2+4\tau }}{2}},x_{1/2}=\sqrt{\tau }{e}^{\pm y}.\end{array}$$

(4)

The Drell–Yan mechanism serves as a prototypical “two-scale” process, and its observables can be written within the framework of TMD factorization in the kinematic region where the transverse momentum of the dilepton pair fulfills ${\mathit{q}}_T\ll Q$ [13,15,20,22]. In this regime, the leading-twist differential cross section takes the form [41]

$$\begin{array}{cc}\hfill & {\displaystyle \frac{d\sigma }{d{x}_{1}d{x}_2{d}^2{\mathit{q}}_{T}d\Omega }}\hfill \\ & ={\displaystyle \frac{{\alpha }_{em}^2}{4Q^2}}\left\{\left(1+{\cos }^2\theta \right)F_{UU}+S_{1L}{S}_{2L}{\sin }^2\theta \cos 2\varphi F_{LL}^{\cos 2\varphi }\right.\hfill \\ & +\left|{\mathit{S}}_{1T}\right|S_{2L}\left(1+{\cos }^2\theta \right)\cos {\varphi }_1{F}_{TL}^{\cos {\varphi }_1}\hfill \\ & +S_{1L}\left|{\mathit{S}}_{2T}\right|{\sin }^2\theta \left[\cos \left(2\varphi +{\varphi }_2\right)F_{LT}^{\cos \left(2\varphi +{\varphi }_2\right)}\right.\hfill \\ & \left.+\cos \left(2\varphi -{\varphi }_2\right)F_{LT}^{\cos \left(2\varphi -{\varphi }_2\right)}\right]\hfill \\ & +\left|{\mathit{S}}_{1T}\right|\left|{\mathit{S}}_{2T}\right|\left(1+{\cos }^2\theta \right)\left[\left(\cos \left({\varphi }_1+{\varphi }_2\right)F_{TT}^{\cos \left({\varphi }_1+{\varphi }_2\right)}\right.\right.\hfill \\ & +\cos \left({\varphi }_1-{\varphi }_2\right)\left.\left.\left.F_{TT}^{\cos \left({\varphi }_1-{\varphi }_2\right)}\right)\right]+\cdots \right\}.\hfill \end{array}$$

(5)

Here, ${\varphi }_{1/2}$ denotes the azimuthal angle of the transverse spin vector ${\mathit{S}}_{(1/2)T}$ with respect to the lepton plane, while $\varphi$ and $\theta$ represent the azimuthal and polar angles, respectively, of the lepton momentum in the Collins–Soper frame. The solid angle $\Omega$ specifies the spatial orientation of the dilepton system.

In addition, the subscripts in the structure functions $P=UU,P=LL,P=TL,P=LT,$ $P=TT$ specify the polarization configuration of the incoming protons (U for unpolarized, T for transverse polarization, and L for longitudinal polarization). The spin-dependent observables are associated with azimuthal asymmetries and can be written as the ratio of the spin-dependent structure function to the spin-averaged structure function, namely

$$\begin{array}{c}\hfill A_P^{f\left(\varphi \right)}={\displaystyle \frac{F_P^{f\left(\varphi \right)}}{F_{UU}}},\end{array}$$

(6)

which could be accessed through measurements in doubly longitudinally polarized Drell–Yan processes.

The spin-averaged/unpolarized structure function $F_{UU}$ can be represented as a convolution of the unpolarized parton distribution functions from each proton in the initial-state [10,41]

$$\begin{array}{c}\hfill F_{UU}=\mathcal{C}\left[f_1^{q/p}{f}_1^{\overline{q}/p}\right],\end{array}$$

(7)

whereas the spin-dependent structure function $F_{LL}^{\cos 2\varphi }$ is given by the convolution of the longi-transversity distributions of each of the incoming protons [10,41]

$$\begin{array}{c}\hfill F_{LL}^{\cos 2\varphi }=\mathcal{C}\left[{\displaystyle \frac{2\left(\widehat{\mathit{h}}·{\mathit{k}}_{1T}\right)\left(\widehat{\mathit{h}}·{\mathit{k}}_{2T}\right)-{\mathit{k}}_{1T}·{\mathit{k}}_{2T}}{M^2}}{h}_{1L}^{\perp ,q/p}{h}_{1L}^{\perp ,\overline{q}/p}\right],\end{array}$$

(8)

where the unit vector $\widehat{\mathit{h}}$ is given by $\widehat{\mathit{h}}\equiv {\mathit{q}}_{\perp }/\left|{\mathit{q}}_{\perp }\right|$ and M denotes the proton mass [10,56]. The vectors ${\mathit{k}}_{1\perp }$ and ${\mathit{k}}_{2\perp }$ correspond to the transverse momenta of the quark and antiquark inside the initial protons, respectively. The convolution of the TMDPDFs in transverse-momentum space is given by [10]

$$\begin{array}{cc}\hfill & \mathcal{C}\left[w\left({\mathit{k}}_{1T},{\mathit{k}}_{2T}\right)f_1{f}_2\right]\hfill \\ & ={\displaystyle \frac{1}{N_c}}\sum _q{e}_q^2\int d^2{\mathit{k}}_{1T}{d}^2{\mathit{k}}_{2T}{\delta }^{\left(2\right)}\left({\mathit{q}}_T-{\mathit{k}}_{1T}-{\mathit{k}}_{2T}\right)\hfill \\ & \times w\left({\mathit{k}}_{1T},{\mathit{k}}_{2T}\right)f_1^q\left(x_1,{\mathit{k}}_{1T}^2\right)f_2^{\overline{q}}\left(x_2,{\mathit{k}}_{2T}^2\right).\hfill \end{array}$$

(9)

Here, $N_c=3$ denotes the number of colors, and ${\mathit{q}}_T$ stands for the transverse momentum of the lepton pair. $\omega ({\mathit{k}}_{1T},{\mathit{k}}_{2T})$ is a function that depends on ${\mathit{k}}_{1T}$ and ${\mathit{k}}_{2T}$.

In practice, a more practical way to analyze the structure function is to work in $b_T$ space, where the convolution of TMDs reduces to an ordinary product of $b_T$-dependent TMDs. The corresponding physical observables are then obtained by performing an inverse Fourier transform from $b_T$ space back to $k_T$ space.

By employing the Fourier transform of the $\delta$-function

$$\begin{array}{c}\hfill {\delta }^2({\mathit{q}}_T-{\mathit{k}}_{1T}-{\mathit{k}}_{2T})={\displaystyle \frac{1}{{\left(2\pi \right)}^2}}\int d^2{\mathit{b}}_T{e}^{i{\mathit{b}}_T·({\mathit{q}}_T-{\mathit{k}}_{1T}-{\mathit{k}}_{2T})},\end{array}$$

(10)

one can derive the spin-dependent structure function $F_{LL}^{\cos 2\varphi }$, which takes the form

$$\begin{array}{cc}\hfill & F_{LL}^{cos2\varphi }={\displaystyle \frac{1}{N_c}}\sum _q{e}_q^2\int d^2{\mathit{k}}_{1T}{d}^2{\mathit{k}}_{2T}\int {\displaystyle \frac{d^2{\mathit{b}}_{\perp }}{{\left(2\pi \right)}^2}}{e}^{i\left({\mathit{q}}_{\perp }-{\mathit{k}}_{1T}-{\mathit{k}}_{2T}\right)·{\mathit{b}}_T}\hfill \\ \hfill & \times \left[{\displaystyle \frac{2(\mathit{h}·{\mathit{k}}_{1T})(\mathit{h}·{\mathit{k}}_{2T})-({\mathit{k}}_{1T}·{\mathit{k}}_{2T})}{M^2}}\right]h_{1L}^{\perp ,q/p}\left(x_1,{\mathit{k}}_{1T}^2,Q\right)h_{1L}^{\perp ,\overline{q}/p}\left(x_2,{\mathit{k}}_{2T}^2,Q\right)\hfill \\ \hfill & ={\displaystyle \frac{1}{N_c}}{\displaystyle \sum _q}{e}_q^2\int {\displaystyle \frac{d^2{\mathit{b}}_T}{{\left(2\pi \right)}^2}}{e}^{i{\mathit{q}}_{\mathit{T}}·{\mathit{b}}_T}\left(2h_{\alpha }{h}_{\beta }-g_{\alpha \beta }\right)·{\tilde{h}}_{1L}^{\perp q/p,\alpha }(x_1,{\mathit{b}}_T,Q){\tilde{h}}_{1L}^{\perp \overline{q}/p,\beta }(x_2,{\mathit{b}}_T,Q).\hfill \end{array}$$

(11)

Here, the quantities with a tilde denote their counterparts in $b_T$-space. The longi-transversity of the incoming protons in $b_T$-space is defined as

$$\begin{array}{c}\hfill {\tilde{h}}_{1L}^{\perp q/p,\alpha }(x_1,{\mathit{b}}_T;Q)=\int d^2{\mathit{k}}_{1T}{e}^{-i{\mathit{k}}_{1T}·{\mathit{b}}_T}{\displaystyle \frac{k_{1T}^{\alpha }}{M}}{h}_{1L}^{\perp q/p}(x_1,{\mathit{k}}_{1T}^2;Q),\end{array}$$

(12)

$$\begin{array}{c}\hfill {\tilde{h}}_{1L}^{\perp \overline{q}/p,\beta }(x_2,{\mathit{b}}_T;Q)=\int d^2{\mathit{k}}_{2T}{e}^{-i{\mathit{k}}_{2T}·{\mathit{b}}_T}{\displaystyle \frac{k_{2T}^{\beta }}{M}}{h}_{1L}^{\perp \overline{q}/p}(x_2,{\mathit{k}}_{2T}^2;Q).\end{array}$$

(13)

Similarly, the spin-averaged structure function $F_{UU}$ may be written as

$$\begin{array}{cc}\hfill & F_{UU}={\displaystyle \frac{1}{N_c}}\sum _q{e}_q^2\int d^2{\mathit{k}}_{1T}{d}^2{\mathit{k}}_{2T}\int {\displaystyle \frac{d^2{\mathit{b}}_T}{{\left(2\pi \right)}^2}}{e}^{i\left({\mathit{q}}_T-{\mathit{k}}_{1T}-{\mathit{k}}_{2T}\right)·{\mathit{b}}_T}\hfill \\ \hfill & \times f_1^{q/p}\left(x_1,{\mathit{k}}_{1T}^2,Q\right)f_1^{\overline{q}/p}\left(x_2,{\mathit{k}}_{2T}^2,Q\right)\hfill \\ \hfill & ={\displaystyle \frac{1}{N_c}}{\displaystyle \sum _q}{e}_q^2\int {\displaystyle \frac{d^2{\mathit{b}}_T}{{\left(2\pi \right)}^2}}{e}^{i{\mathit{q}}_{\mathit{T}}·{\mathit{b}}_T}{\tilde{f}}_1^{q/p}(x_1,{\mathit{b}}_T,Q){\tilde{f}}_1^{\overline{q}/p}(x_2,{\mathit{b}}_T,Q),\hfill \end{array}$$

(14)

where the unpolarized distribution functions for each proton in $b_{\perp }$-space are defined as follows

$$\begin{array}{c}\hfill {\tilde{f}}_1^{q/p}(x_1,{\mathit{b}}_T,Q)=\int d^2{\mathit{k}}_{1T}{e}^{-i{\mathit{k}}_{1T}·{\mathit{b}}_T}{f}_1^{q/p}(x_1,{\mathit{k}}_{1T}^2;Q),\end{array}$$

(15)

$$\begin{array}{c}\hfill {\tilde{f}}_1^{\overline{q}/p}(x_2,{\mathit{b}}_T,Q)=\int d^2{\mathit{k}}_{2T}{e}^{-i{\mathit{k}}_{2T}·{\mathit{b}}_T}{f}_1^{\overline{q}/p}(x_2,{\mathit{k}}_{2T}^2;Q).\end{array}$$

(16)

2.2. The $\sin 2{\varphi }_h$ Asymmetry of Pion Production in SIDIS

Next, we consider the SIDIS process in which a lepton scatters from a longitudinally polarized nucleon target and produces an observed pion in the final state:

$$\begin{array}{c}\hfill l(\mathsf{\ell })+N^{\to }\left(P\right)\to l\left({\mathsf{\ell }}^{\prime }\right)+\pi \left(P_{\pi }\right)+X,\end{array}$$

(17)

where a lepton beam with momentum ℓ interacts with a longitudinally polarized nucleon N carrying momentum P. In the final state, the momentum ${\mathsf{\ell }}^{\prime }$ of the scattered lepton is detected together with an unpolarized hadron h (in this work, h is taken to be a $\pi$ meson). We introduce the spacelike momentum transfer $q=\mathsf{\ell }-{\mathsf{\ell }}^{\prime }$ and define the following kinematic invariants

$$\begin{array}{c}\hfill x={\displaystyle \frac{Q^2}{2P·q}},y={\displaystyle \frac{P·q}{P·\mathsf{\ell }}}={\displaystyle \frac{Q^2}{xs}},z={\displaystyle \frac{P·P_{\pi }}{P·q}},Q^2=-q^2,s={(P+\mathsf{\ell })}^2.\end{array}$$

(18)

Here, s denotes the squared total center-of-mass energy, x represents the Bjorken scaling variable, y is the inelasticity, and z corresponds to the momentum fraction of the parent parton carried by the hadron in the final state.

To the leading order in $1/Q$, the SIDIS cross section (for a nucleon target with longitudinal polarization) can be written as [11]

$$\begin{array}{c}\hfill {\displaystyle \frac{d^5\sigma }{dxdydzd{\varphi }_{h}d{P}_{\pi T}^2}}={\displaystyle \frac{2\pi {\alpha }^2}{x_{B}y{Q}^2}}\times \left\{(1-y+{\displaystyle \frac{1}{2}}{y}^2)F_{UU,T}+(1-y)S_L\sin 2{\varphi }_h{F}_{UL}^{\sin 2{\varphi }_h}\right\}.\end{array}$$

(19)

The relevant reference frame for the process is illustrated in Figure 1. In this frame, the momentum q of the virtual photon defines the z-axis; the hadron plane is spanned by the z-axis and the momentum direction of the detected final-state hadron, while the lepton plane is spanned by the momentum directions of ℓ and ${\mathsf{\ell }}^{\prime }$. Accordingly, ${\varphi }_h$ denotes the azimuthal angle of the produced hadron measured with respect to the lepton plane; ${\mathit{P}}_{\pi T}$ is the component of ${\mathit{P}}_{\pi }$ transverse to $\mathit{q}$, with ${\mathit{P}}_{\pi T}=-z{\mathit{q}}_T$ [57]. TMD factorization for the SIDIS process is valid only in the regime where ${\mathit{P}}_{\pi T}^2/z^2\ll Q^2$ [15]. Here, $F_{UU,T}$ and $F_{UL}^{\sin 2{\varphi }_h}$ denote the spin-averaged and spin-dependent structure functions, respectively.

The $\sin 2{\varphi }_h$ azimuthal asymmetry can be written as

$$\begin{array}{c}\hfill A_{UL}^{\sin 2{\varphi }_h}(x,y,z,P_{\pi T})={\displaystyle \frac{\frac{1}{xy{Q}^2}(1-y)F_{UL}^{\sin 2{\varphi }_h}}{\frac{1}{xy{Q}^2}(1-y+\frac{1}{2}{y}^2)F_{UU,T}}}.\end{array}$$

(20)

Within the TMD factorization framework, the structure functions $F_{UU,T}$ and $F_{UL}^{\sin 2{\varphi }_{\pi }}$ are expressed through an integral that convolves the transverse parton momentum in the distribution function with that in the fragmentation function [30,31] as

$$\begin{array}{cc}\hfill F_{UU,T}(x,z,Q;P_{\pi T}^2)=& x\sum _q{e}_q^2\int d^2{\mathit{p}}_T{d}^2{\mathit{k}}_T{\delta }^2({\mathit{p}}_T-{\mathit{k}}_T-{\mathit{P}}_{\pi T}/z)f_1^{q/N}(x,{\mathit{p}}_T^2;Q)D_1^{\pi /q}(z,{\mathit{k}}_T^2;Q),\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill F_{UL}^{\sin 2{\varphi }_h}(x,z,Q;P_{\pi T})=& x\sum _q{e}_q^2\int d^2{\mathit{p}}_T{d}^2{\mathit{k}}_T{\delta }^2({\mathit{p}}_T-{\mathit{k}}_T-{\mathit{P}}_{\pi T}/z)[-{\displaystyle \frac{2(\widehat{\mathit{h}}·{\mathit{k}}_T)(\widehat{\mathit{h}}·{\mathit{p}}_T)-{\mathit{k}}_T·{\mathit{p}}_T}{M_N{M}_{\pi }}}]\hfill \\ \hfill & \times h_{1L}^{\perp q/N}(x,{\mathit{p}}_T^2;Q)H_1^{\perp \pi /q}(z,{\mathit{k}}_T^2;Q),\hfill \end{array}$$

(22)

where the unit vector $\widehat{\mathit{h}}$ is given by $\widehat{\mathit{h}}={\mathit{P}}_{\pi T}/P_{\pi T}$ and the transverse momentum ${\mathit{k}}_T$ is connected to the transverse momentum of the produced hadron with respect to the quark via the relation ${\mathit{K}}_T=-z{\mathit{k}}_T$.

Transforming to b-space yields a more tractable form. The structure functions become

$$\begin{array}{cc}\hfill & F_{UU,T}(x,z,Q;P_{\pi T})\hfill \\ \hfill =& x{\displaystyle \frac{1}{z^2}}\sum _q{e}_q^2\int {\displaystyle \frac{d^{2}b}{{\left(2\pi \right)}^2}}{e}^{i{\mathit{P}}_{\pi T}·b/z}{\tilde{f}}_1^{q/N}(x,b;Q){\tilde{D}}_1^{\pi /q}(z,b;Q)\hfill \\ \hfill =& x{\displaystyle \frac{1}{z^2}}\sum _q{e}_q^2\int {\displaystyle \frac{dbb}{2\pi }}{J}_0\left(\right|{\mathit{P}}_{\pi T}\left|\right|b|/z){\tilde{f}}_1^{q/N}(x,b;Q){\tilde{D}}_1^{\pi /q}(z,b;Q)\hfill \end{array}$$

(23)

In a similar manner, the spin-dependent structure function $F_{UL}^{\sin \left(2{\varphi }_h\right)}$ can be expressed as

$$\begin{array}{cc}\hfill & F_{UL}^{\sin 2{\varphi }_h}(x,z,Q;P_{\pi T})\hfill \\ \hfill =& x{\displaystyle \frac{1}{z^3}}\sum _q{e}_q^2\int d^2{\mathit{p}}_T{d}^2{\mathit{K}}_T\int {\displaystyle \frac{d^{2}b}{{\left(2\pi \right)}^2}}{e}^{-i({\mathit{p}}_T+{\mathit{K}}_T/z-{\mathit{P}}_{\pi T}/z)·b}[2(\widehat{\mathit{h}}·{\mathit{K}}_T)(\widehat{\mathit{h}}·{\mathit{p}}_T)-{\mathit{K}}_T·{\mathit{p}}_T]\hfill \\ \hfill & \times {\displaystyle \frac{h_{1L}^{\perp q/N}(x,{\mathit{p}}_T^2;Q)H_1^{\perp \pi /q}(z,{\mathit{K}}_T^2;Q)}{M_N{M}_{\pi }}}\hfill \\ \hfill =& x{\displaystyle \frac{1}{z^3}}\sum _q{e}_q^2\int {\displaystyle \frac{d^{2}b}{{\left(2\pi \right)}^2}}{e}^{i{\mathit{P}}_{\pi T}·b/z}(2{\widehat{h}}_{\alpha }{\widehat{h}}_{\beta }-g_{\alpha \beta }){\tilde{h}}_{1L}^{\perp \beta ,q/N}(x,b;Q){\tilde{H}}_1^{\perp \alpha ,\pi /q}(z,b;Q)\hfill \end{array}$$

(24)

Therefore, we arrive at the evolved expressions for the structure functions $F_{UU,T}(x,z,Q;P_{\pi T})$ and $F_{UL}^{\sin 2{\varphi }_h}(x,z,Q;P_{\pi T})$.

3. TMD Evolution of Parton Distribution and Fragmentation Functions

In this section, we discuss the evolution formalism for the unpolarized PDF $f_1$ and the longitudinal transversity PDF $h_{1L}^{\perp }$ of the nucleon, as well as for the unpolarized FF $D_1$ and the Collins FF $H_1^{\perp }$ of the pion, all within the framework of TMD factorization.

TMD evolution is most commonly carried out in the conjugate coordinate b-space, because in this representation the cross section takes a simpler, factorized form than in transverse-momentum $k_T$ space (the b-space is related to $k_T$-space through a Fourier transform) [15,17]. Within TMD factorization formulated in various schemes (for example, the CS-81 [16], JMY [12,58], and Collins-11 schemes [15]), the TMDs $\tilde{F}(x,b;\mu ,{\zeta }_F)$ and $\tilde{D}(z,b;\mu ,{\zeta }_D)$ are functions of two distinct energy scales [13,15,16,17,50,59]. One scale is the renormalization scale $\mu$, and the other is the energy scale ${\zeta }_F$ (or ${\zeta }_D$), which acts as a cutoff to regularize the light-cone singularity appearing in the operator definition of the TMDs. These two types of energy dependence are governed by distinct evolution equations. The dependence of the TMD PDFs (or FFs) on ${\zeta }_F$ (or ${\zeta }_D$) is described by the Collins–Soper (CS) equation [16]:

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial \ln \tilde{F}(x,b;\mu ,{\zeta }_F)}{\partial \sqrt{{\zeta }_F}}}={\displaystyle \frac{\partial \ln \tilde{D}(z,b;\mu ,{\zeta }_D)}{\partial \sqrt{{\zeta }_D}}}=\tilde{K}(b;\mu ),\end{array}$$

(25)

Here, $\tilde{K}$ denotes the Collins–Soper (CS) evolution kernel, which for small values of b can be calculated perturbatively (up to order ${\alpha }_s$) as

$$\begin{array}{c}\hfill \tilde{K}(b;\mu )=-{\displaystyle \frac{{\alpha }_s{C}_F}{\pi }}\left[\ln \left({\mu }^2{b}^2\right)-\ln 4+2{\gamma }_E\right]+\mathcal{O}\left({\alpha }_s^2\right),\end{array}$$

(26)

where ${\gamma }_E\approx 0.577$ is Euler’s constant [16].

The $\mu$ dependence of the TMDs is obtained from the renormalization group equation

$$\begin{array}{cc}\hfill & {\displaystyle \frac{d\tilde{K}}{d\ln \mu }}=-{\gamma }_K\left({\alpha }_s\left(\mu \right)\right),\hfill \end{array}$$

(27)

$$\begin{array}{cc}\hfill & {\displaystyle \frac{d\ln \tilde{F}(x,b;\mu ,{\zeta }_F)}{d\ln \mu }}={\gamma }_F({\alpha }_s\left(\mu \right);{\displaystyle \frac{{\zeta }_F^2}{{\mu }^2}}),\hfill \end{array}$$

(28)

$$\begin{array}{cc}\hfill & {\displaystyle \frac{d\ln \tilde{D}(z,b;\mu ,{\zeta }_D)}{d\ln \mu }}={\gamma }_D({\alpha }_s\left(\mu \right);{\displaystyle \frac{{\zeta }_D^2}{{\mu }^2}}),\hfill \end{array}$$

(29)

where ${\gamma }_K$, ${\gamma }_F$, and ${\gamma }_D$ denote the anomalous dimensions associated with $\tilde{K}$, $\tilde{F}$, and $\tilde{D}$, respectively,

$$\begin{array}{cc}\hfill {\gamma }_K& =2{\displaystyle \frac{{\alpha }_s{C}_F}{\pi }}+\mathcal{O}\left({\alpha }_s^2\right),\hfill \end{array}$$

(30)

$$\begin{array}{cc}\hfill {\gamma }_D& ={\gamma }_F={\alpha }_s{\displaystyle \frac{C_F}{\pi }}\left({\displaystyle \frac{3}{2}}-\ln \left({\displaystyle \frac{{\zeta }_F}{{\mu }^2}}\right)\right)+\mathcal{O}\left({\alpha }_s^2\right).\hfill \end{array}$$

(31)

Solving Equations (27)–(29), one can derive the general expression describing how $\tilde{F}$ (or $\tilde{D}$) depends on the energy [12,15,16,17,43,52,58]:

$$\begin{array}{c}\hfill \tilde{F}(x,b;Q)=\mathcal{F}\times e^{-S(Q,b)}\times \tilde{F}(x,b;{\mu }_i),\end{array}$$

(32)

$$\begin{array}{c}\hfill \tilde{D}(z,b;Q)=\mathcal{D}\times e^{-S(Q,b)}\times \tilde{D}(z,b;{\mu }_i),\end{array}$$

(33)

where $\mathcal{F}$ and $\mathcal{D}$ denote the factors associated with the hard scattering, and $S(Q,b)$ is the Sudakov form factor. In what follows, we choose $\mu =\sqrt{{\zeta }_F}=\sqrt{{\zeta }_D}=Q$, and, for brevity, we will write $\tilde{F}(x,b;\mu =Q,{\zeta }_F=Q^2)$ (or $\tilde{D}(z,b;\mu =Q,{\zeta }_D=Q^2)$) simply as $\tilde{F}(x,b;Q)$ (or $\tilde{D}(z,b;Q)$). Equation (32) (or Equation (33)) shows that the distribution $\tilde{F}$ (or $\tilde{D}$) at an arbitrary scale Q is obtained from the corresponding distribution at an initial scale ${\mu }_i$ via the evolution in the exponential factor $\exp (-S(Q,b\left)\right)$.

More precisely, the exponential factor exp($-S(Q,b)$) can be written in the following explicit form (using the expression for $\tilde{F}$ as an example)

$$\begin{array}{cc}\hfill \exp (-S(Q,b\left)\right)& =\exp \left\{\ln {\displaystyle \frac{Q}{\mu }}\tilde{K}(b_{*};\mu )+{\int }_{{\mu }_i}^{\mu }{\displaystyle \frac{d\overline{\mu }}{\overline{\mu }}}\left[{\gamma }_F(g\left(\overline{\mu }\right);1)-\ln \left({\displaystyle \frac{Q}{\overline{\mu }}}\right){\gamma }_K\left(g\left(\overline{\mu }\right)\right)\right]\right\}\hfill \\ \hfill & \times \exp \left\{g_{j/P}(x,b)+g_K\left(b\right)\ln {\displaystyle \frac{Q}{Q_0}}\right\}.\hfill \end{array}$$

(34)

By introducing the parameter $b_{\max }$ as the separation between the perturbative and nonperturbative regions, one obtains the full expression for the Sudakov form factor that appears in Equations (32) and (33) as

$$S(Q,b)=S_{\mathrm{P}}(Q,b_{*})+S_{\mathrm{NP}}(Q,b).$$

(35)

Specifically, the exponential factor in the first line of Equation (34) is obtained from the solutions of Equations (25), (27) and (28) in the perturbative domain, i.e., the small-b region ($b\ll 1/\Lambda$). This factor includes $\tilde{K}(b_{*};\mu )$, the CS evolution kernel valid at small b, and the anomalous dimensions ${\gamma }_F$ and ${\gamma }_K$. In contrast, in the nonperturbative regime (the large-b region), the evolution kernel $\tilde{K}(b;\mu )$ cannot be calculated perturbatively. To incorporate the contributions from this large-b region, the exponential appearing in the second line of Equation (34) is introduced. In this term, the function $g_{j/P}(x,b)$ encodes the nonperturbative information of the intrinsic transverse momentum distribution usually defined as the large $b_T$ correction of the perturbative matching of the TMDs onto the PDFs at small $b_T$, whereas the universal function $g_K$ describes the nonperturbative large-b behavior of the evolution kernel $\tilde{K}(b;\mu )$.

The parameter $b_{\max }$ can ensure a smooth transition of b from the perturbative regime to the nonperturbative one and to avoid encountering the Landau pole. A typical choice for $b_{\max }$ is around $1{\mathrm{GeV}}^{-1}$, which guarantees that $b_{*}$ remains within the perturbative domain. One may further define a b-dependent function $b_{*}\left(b\right)$ such that $b_{*}\approx b$ for small b and $b_{*}\approx b_{\max }$ for large b. Several functional forms for $b_{*}\left(b\right)$ have been proposed in the literature [17,22,60]. A commonly adopted choice is the Collins–Soper–Sterman (CSS) prescription [17]:

$$\begin{array}{c}\hfill b_{*}=b/\sqrt{1+b^2/b_{\max }^2},b_{\max }<1/{\Lambda }_{\mathrm{QCD}}.\end{array}$$

(36)

The perturbative contribution, $S_{\mathrm{P}}(Q,b_{*})$, has been investigated in detail in Refs. [20,45,46,50,59] and takes the following form:

$$S_{\mathrm{P}}(Q,b_{*})={\int }_{{\mu }_b^2}^{Q^2}{\displaystyle \frac{d{\overline{\mu }}^2}{{\overline{\mu }}^2}}\left[A\left({\alpha }_s\left(\overline{\mu }\right)\right)\ln {\displaystyle \frac{Q^2}{{\overline{\mu }}^2}}+B\left({\alpha }_s\left(\overline{\mu }\right)\right)\right],$$

(37)

which remains valid for various types of PDFs and FFs; in other words, $S_P$ does not depend on spin. Furthermore, the coefficients A and B in Equation (37) may be expressed as a series of ${\alpha }_s/\pi$:

$$\begin{array}{c}\hfill A={\displaystyle \sum _{n=1}^{\infty }}{A}^{\left(n\right)}{({\displaystyle \frac{{\alpha }_s}{\pi }})}^n,\end{array}$$

(38)

$$\begin{array}{c}\hfill B={\displaystyle \sum _{n=1}^{\infty }}{B}^{\left(n\right)}{({\displaystyle \frac{{\alpha }_s}{\pi }})}^n.\end{array}$$

(39)

In this analysis, we include the coefficients $A^{\left(n\right)}$ up to $A^{\left(2\right)}$ and $B^{\left(n\right)}$ up to $B^{\left(1\right)}$ at next-to-leading logarithmic (NLL) accuracy [13,17,45,47,59,61]:

$$\begin{array}{cc}\hfill A^{\left(1\right)}& =C_F,\hfill \end{array}$$

(40)

$$\begin{array}{cc}\hfill A^{\left(2\right)}& ={\displaystyle \frac{C_F}{2}}\left[C_A\left({\displaystyle \frac{67}{18}}-{\displaystyle \frac{{\pi }^2}{6}}\right)-{\displaystyle \frac{10}{9}}{T}_R{n}_f\right],\hfill \end{array}$$

(41)

$$\begin{array}{cc}\hfill B^{\left(1\right)}& =-{\displaystyle \frac{3}{2}}{C}_F,\hfill \end{array}$$

(42)

with $C_F=4/3$, $C_A=3$ and $T_R=1/2$.

The non-perturbative contribution $S_{\mathrm{NP}}$ in Equation (35) cannot be obtained using perturbative methods; instead, it is typically modeled and its parameters are determined from experimental measurements. Although more recent extractions of non-perturbative Sudakov factors exist [62,63,64,65,66], we adopt the Echevarria–Idilbi–Kang–Vitev (EIKV) [20] and Bacchetta–Delcarro–Pisano–Radici–Signori (BDPRS) [22] parametrizations as representative examples of two distinct approaches to modeling the non-perturbative TMD evolution. These two schemes have been widely used in the literature and allow a direct comparison with our previous studies [37]. Future updates using the latest parametrizations are left for subsequent work.

The EIKV non-perturbative Sudakov factor $S_{\mathrm{NP}}$ for unpolarized TMD PDFs (or TMD FFs) is given by

$$S_{\mathrm{NP}}^{\mathrm{pdf}}(b,Q)=b^2(g_1^{\mathrm{pdf}}+{\displaystyle \frac{g_2}{2}}\ln {\displaystyle \frac{Q}{Q_0}}),$$

(43)

$$S_{\mathrm{NP}}^{\mathrm{ff}}(b,Q)=b^2(g_1^{\mathrm{ff}}+{\displaystyle \frac{g_2}{2}}\ln {\displaystyle \frac{Q}{Q_0}}).$$

(44)

Here, $S_{\mathrm{NP}}^{\mathrm{pdf}}(b,Q)$ and $S_{\mathrm{NP}}^{\mathrm{ff}}(b,Q)$ denote the non-perturbative parts of the Sudakov form factor for the TMD PDF and TMD FF, respectively. $g_2$ characterizes the behavior of the evolution kernel $\tilde{K}$ at large b values ($g_K\left(b\right)=g_2{b}^2$). This function is universal across different kinds of TMDs and is independent of the specific process, which is a key outcome of QCD factorization theorems for TMDs [13,15,20,21]. The parameter $g_1$ encodes the intrinsic nonperturbative transverse motion of the bound partons. It may vary with the type of TMD and can be viewed as the intrinsic transverse-momentum width of the corresponding TMDs at the initial scale $Q_0$ [13,51,61,67,68]. In addition, $g_1^{\mathrm{pdf}}$ and $g_1^{\mathrm{ff}}$ are specified as

$$g_1^{\mathrm{pdf}}={\displaystyle \frac{{\langle k_T^2\rangle }_{Q_0}}{4}},$$

(45)

$$g_1^{\mathrm{ff}}={\displaystyle \frac{{\langle p_T^2\rangle }_{Q_0}}{4z^2}},$$

(46)

where ${\langle k_T^2\rangle }_{Q_0}$ and ${\langle p_T^2\rangle }_{Q_0}$ denote the average intrinsic squared transverse momenta associated with the TMD PDFs and FFs at the initial scale $Q_0$, respectively. In Ref. [20], the authors adjusted the empirically extracted ranges of the three parameters ${\langle k_T^2\rangle }_{Q_0},{\langle p_T^2\rangle }_{Q_0}$, and $g_2$, using $Q_0=\sqrt{2.4}\mathrm{GeV}$ as in Refs. [69,70,71], and showed that the following fixed parameter values provide a reasonable simultaneous description of SIDIS data, Drell–Yan lepton-pair production, and $W/Z$ boson production data:

$${\langle k_T^2\rangle }_{Q_0}=0.38{\mathrm{GeV}}^2,{\langle p_T^2\rangle }_{Q_0}=0.19{\mathrm{GeV}}^2,g_2=0.16{\mathrm{GeV}}^2,b_{\max }=1.5{\mathrm{GeV}}^{-1}.$$

(47)

In addition to the Sudakov form factors given in Equations (32) and (33), a further key ingredient in these equations is the TMDs evaluated at a fixed scale $\mu$. In the small-b region, $\tilde{F}(x,b;\mu )$ and $\tilde{D}(z,b;\mu )$ at a fixed scale $\mu$ can be written as convolutions of the perturbatively computable coefficients C with the corresponding collinear functions $F_{i/H}(\xi ,\mu )$ (or $D_{H/j}(\xi ,\mu )$),

$$\begin{array}{c}\hfill \tilde{F}(x,b;\mu )=\sum _i{\int }_x^1{\displaystyle \frac{d\xi }{\xi }}{C}_{q\leftarrow i}(x/\xi ,b;\mu )F_{i/H}(\xi ,\mu ),\end{array}$$

(48)

$$\begin{array}{c}\hfill \tilde{D}(z,b;\mu )=\sum _j{\int }_z^1{\displaystyle \frac{d\xi }{\xi }}{C}_{j\leftarrow q}(z/\xi ,b;\mu )D_{H/j}(\xi ,\mu ),\end{array}$$

(49)

Here, $\mu$ denotes a dynamic scale connected to $b_{*}$ via $\mu =c/b_{*}$, where $c=2e^{-{\gamma }_E}$. The functions $C_{q\leftarrow i}(x/\xi ,b;\mu )={\sum }_{n=0}^{\infty }{C}_{q\leftarrow i}^{\left(n\right)}{({\alpha }_s/\pi )}^n$ and $C_{j\leftarrow q}(z/\xi ,b;\mu )={\sum }_{n=0}^{\infty }{C}_{j\leftarrow q}^{\left(n\right)}{({\alpha }_s/\pi )}^n$ represent perturbatively calculable coefficient functions.

After obtaining the solutions of the evolution equations and including the Sudakov form factor, the scale-dependent TMDs in b space can be expressed as

$$\begin{array}{c}\hfill {\tilde{F}}_{q/H}(x,b;Q)=e^{-{\displaystyle \frac{1}{2}}{S}_{\mathrm{P}}(Q,b_{*})-S_{\mathrm{NP}}^{F_{q/H}}(Q,b)}{F}_{q/H}(x,\mu ),\end{array}$$

(50)

$$\begin{array}{c}\hfill {\tilde{D}}_{H/q}(z,b;Q)=e^{-{\displaystyle \frac{1}{2}}{S}_{\mathrm{P}}(Q,b_{*})-S_{\mathrm{NP}}^{D_{H/q}}(Q,b)}{D}_{H/q}(z,\mu )\end{array}$$

(51)

The prefactor ${\displaystyle \frac{1}{2}}$ in front of $S_{\mathrm{P}}$ arises because $S_{\mathrm{P}}$ is shared equally between the initial-state quark and the final-state quark [72]. In this study, we use the leading-order (LO) expressions for the hard coefficients C, $\mathcal{F}$ and $\mathcal{D}$ associated with $f_1$, $h_{1L}^{\perp }$, $D_1$, and $H_1^{\perp }$, namely $C_{q\leftarrow i}^{\left(0\right)}={\delta }_{iq}\delta (1-x)$, $C_{j\leftarrow q}^{\left(0\right)}={\delta }_{qj}\delta (1-z)$, $\mathcal{F}=1$, and $\mathcal{D}=1$.

With all the above components specified, we can derive the unpolarized nucleon PDFs ${\tilde{f}}_1^{q/p}$ (and, analogously, the pion FFs ${\tilde{D}}_1^{\pi /q}$) in b space

$$\begin{array}{cc}\hfill {\tilde{f}}_1^{q/p}(x,b;Q)& =e^{-{\displaystyle \frac{1}{2}}{S}_{\mathrm{P}}(Q,b_{*})-S_{\mathrm{NP}}^{pdf}(Q,b)}{f}_1^{q/p}(x,\mu ),\hfill \end{array}$$

(52)

$$\begin{array}{cc}\hfill {\tilde{D}}_1^{\pi /q}(z,b;Q)& =e^{-{\displaystyle \frac{1}{2}}{S}_{\mathrm{P}}(Q,b_{*})-S_{\mathrm{NP}}^{ff}(Q,b)}{D}_1^{\pi /q}(z,\mu ).\hfill \end{array}$$

(53)

By carrying out the Fourier transform, we can derive the unpolarized nucleon PDFs $f_1^{q/N}$ (or, analogously, the pion FFs $D_1^{\pi /q}$) in transverse-momentum space

$$\begin{array}{cc}\hfill f_1^{q/N}(x,{\mathit{p}}_T;Q)& ={\int }_0^{\infty }{\displaystyle \frac{dbb}{2\pi }}{J}_0\left(\right|{\mathit{p}}_T\left|b\right)e^{-{\displaystyle \frac{1}{2}}{S}_P(Q,b_{*})-S_{\mathrm{NP}}^{\mathrm{pdf}}(Q,b)}{f}_1^{q/p}(x,\mu ),\hfill \end{array}$$

(54)

$$\begin{array}{cc}\hfill D_1^{\pi /q}(z,{\mathit{K}}_T;Q)& ={\int }_0^{\infty }{\displaystyle \frac{dbb}{2\pi }}{J}_0\left(\right|{\mathit{K}}_T|b/z)e^{-{\displaystyle \frac{1}{2}}{S}_P(Q,b_{*})-S_{\mathrm{NP}}^{\mathrm{ff}}(Q,b)}{D}_1^{\Lambda /q}(z,\mu ).\hfill \end{array}$$

(55)

The TMD distributions are constructed using the standard small-b matching formalism. The coefficient functions $C_{q\leftarrow i}$ ensure the matching between TMD and collinear factorization. At leading order, the perturbative matching coefficients reduce to delta functions, so that the longitudinal momentum dependence is entirely inherited from the corresponding collinear distributions, while the transverse-momentum dependence is generated through perturbative and nonperturbative Sudakov evolution. In principle, integrating the transverse momentum of the TMD, one would obtain the collinear PDF, which has been demonstrated in the recent study of the helicity distribution function $g_1$ [73]. Consequently, possible intrinsic $x-k_{\perp }$ correlations beyond this factorized approximation are not included. There are extractions of TMD distribution functions [22,73] that can parametrize the initial TMD functional form as the function of x and b, which can generate the transverse momentum dependence through Fourier Transformation.

According to Equations (48) and (49), in the small-b region, the longi-transversity PDF of the nucleon target, ${\tilde{h}}_{1L}^{\perp }$, and the pion Collins FF, ${\tilde{H}}_1^{\perp }$, at a fixed energy scale $\mu$, can likewise be represented in terms of perturbatively calculable coefficient functions and the associated collinear correlation functions:

$$\begin{array}{cc}\hfill {\tilde{h}}_{1L}^{\perp \left(\beta \right)q/N}(x,b;\mu )& =i{b}^{\beta }{M}_N{\tilde{h}}_{1L}^{\perp \left(1\right)}(x,\mu ),\hfill \end{array}$$

(56)

$$\begin{array}{cc}\hfill {\tilde{H}}_1^{\perp \left(\alpha \right)\pi /q}(z,b;\mu )& ={\displaystyle \frac{i{b}^{\alpha }}{2}}{\widehat{H}}^{\left(3\right)}(z,z,\mu ),\hfill \end{array}$$

(57)

where the hard coefficients are evaluated at leading order (LO) and the nucleon target longi-transversity together with the pion Collins fragmentation function in b-space are defined as

$$\begin{array}{cc}\hfill {\tilde{h}}_{1L}^{\perp \beta ,q/N}(x,b;\mu )=& \int d^2{\mathit{p}}_T{e}^{-i{\mathit{p}}_T·b}{\displaystyle \frac{p_T^{\beta }}{M_N}}{h}_{1L}^{\perp q/N}(x,{\mathit{p}}_T^2;\mu ),\hfill \\ \hfill {\tilde{H}}_1^{\perp \alpha ,\pi /q}(z,b;\mu )=& \int d^2{\mathit{K}}_T{e}^{-i{\mathit{K}}_T·b/z}{\displaystyle \frac{K_T^{\alpha }}{M_{\pi }}}{H}_1^{\perp \pi /q}(z,{\mathit{K}}_T^2;\mu ).\hfill \end{array}$$

(58)

The collinear function ${\widehat{H}}^{\left(3\right)}(z,z,\mu )$ denotes the twist-3 quark–gluon–quark correlation function, which is connected to the first transverse-momentum moment of the Collins fragmentation function $H_1^{\perp \left(1\right)}$ as shown in Ref. [74]:

$$\begin{array}{c}\hfill {\widehat{H}}^{\left(3\right)}(z,z,\mu )=\int d^2{\mathit{K}}_T{\displaystyle \frac{|{\mathit{K}}_T^2|}{M_{\Lambda }}}{H}_1^{\perp \pi /q}(z,{\mathit{K}}_T^2,\mu )=2M_{\pi }{\tilde{H}}_1^{\perp \left(1\right)}(z,\mu ).\end{array}$$

(59)

For the non-perturbative Sudakov factors associated with $h_{1L}^{\perp }$ and $H_1^{\perp }$, we assume they are the same as those for the unpolarized TMD PDF and FF, respectively, i.e., $S_{1\mathrm{L}}^{\mathrm{h}}{\perp }_{\mathrm{NP}}=S_{\mathrm{NP}}^{\mathrm{pdf}}$ and $S_{\mathrm{NP}}^{H_1^{\perp }}=S_{\mathrm{NP}}^{\mathrm{ff}}$. This is a common simplifying assumption in the absence of direct extractions. Under this assumption, we can express the longi-transversity and the Collins function in b-space as

$$\begin{array}{cc}\hfill {\tilde{h}}_{1L}^{\perp \left(\beta \right)q/N}(x,b;Q)& =i{b}^{\beta }{M}_N{e}^{-{\displaystyle \frac{1}{2}}{S}_P(Q,b_{*})-S_{\mathrm{NP}}^{\mathrm{pdf}}(Q,b)}{\tilde{h}}_{1L}^{\perp \left(1\right)}(x,\mu ),\hfill \end{array}$$

(60)

$$\begin{array}{cc}\hfill {\tilde{H}}_1^{\perp \left(\alpha \right)\pi /q}(z,b;Q)& ={\displaystyle \frac{i{b}^{\alpha }}{2}}{e}^{-{\displaystyle \frac{1}{2}}{S}_{\mathrm{P}}(Q,b_{*})-S_{\mathrm{NP}}^{\mathrm{ff}}(Q,b)}{\widehat{H}}^{\left(3\right)}(z,z,\mu ).\hfill \end{array}$$

(61)

By applying a Fourier transform, the TMDs can be expressed in transverse momentum space

$$\begin{array}{cc}\hfill {\displaystyle \frac{p_{\perp }^{\beta }}{M_N}}{\tilde{h}}_{1L}^{\perp q/N\left(\beta \right)}(x,p_T;Q)& =M_N{\int }_0^{\infty }{\displaystyle \frac{db{b}^2}{2\pi }}{J}_1\left(\right|{\mathit{p}}_T\left|b\right)e^{-{\displaystyle \frac{1}{2}}{S}_P(Q,b_{*})-S_{\mathrm{NP}}^{\mathrm{pdf}}(Q,b)}{\tilde{h}}_{1L}^{\perp \left(1\right)}(x,\mu ),\hfill \end{array}$$

(62)

$$\begin{array}{cc}\hfill {\displaystyle \frac{K_T^{\alpha }}{M_{\pi }}}{\tilde{H}}_1^{\perp \pi /q\left(\alpha \right)}(z,K_T;Q)& ={\int }_0^{\infty }{\displaystyle \frac{db{b}^2}{4\pi }}{J}_1\left(\right|{\mathit{K}}_T|b/z)e^{-{\displaystyle \frac{1}{2}}{S}_P(Q,b_{*})-S_{\mathrm{NP}}^{\mathrm{ff}}(Q,b)}{\widehat{H}}^{\left(3\right)}(z,z,\mu ).\hfill \end{array}$$

(63)

In addition to the EIKV parametrization for the non-perturbative factor $S_{\mathrm{NP}}$ discussed above, we also employ the BDPRS parametrization of $S_{\mathrm{NP}}$ for unpolarized TMDs, which is given by

$$\begin{array}{cc}\hfill {\tilde{f}}_1^a(x,b;Q)& =f_1^a(x;{\mu }^2)e^{-S(\mu ,Q)}{e}^{{\displaystyle \frac{1}{2}}{g}_K\left(b\right)\ln (Q^2/Q_0^2)}{\tilde{f}}_{1\mathrm{NP}}^a(x,b),\hfill \end{array}$$

(64)

$$\begin{array}{cc}\hfill {\tilde{D}}_1^{a\to h}(z,b;Q)& =D_1^{a\to h}(z;{\mu }^2)e^{-S(\mu ,Q)}{e}^{{\displaystyle \frac{1}{2}}{g}_K\left(b\right)\ln (Q^2/Q_0^2)}{\tilde{D}}_{1\mathrm{NP}}^{a\to h}(z,b),\hfill \end{array}$$

(65)

where $g_K=-g_2{b}^2/2$, consistent with the convention adopted in Refs. [47,53,55]. The functions ${\tilde{f}}_{1\mathrm{NP}}^a(x,b^2)$ and ${\tilde{D}}_{1\mathrm{NP}}^{a\to h}(z,b^2)$ denote the intrinsic nonperturbative contributions to the PDFs and FFs, respectively, and are modeled as

$${\tilde{f}}_{1\mathrm{NP}}^a(x,b^2)={\displaystyle \frac{1}{2\pi }}{e}^{-g_1{\displaystyle \frac{b^2}{4}}}\left(1-{\displaystyle \frac{\lambda g_1^2}{1+\lambda g_1}}{\displaystyle \frac{b^2}{4}}\right),$$

(66)

$${\tilde{D}}_{1\mathrm{NP}}^{a\to h}(z,b^2)={\displaystyle \frac{g_3{e}^{-g_3\frac{b^2}{4z^2}}+\left(\frac{{\lambda }_F}{z^2}\right)g_4^2(1-g_4\frac{b^2}{4z^2})e^{-g_4\frac{b^2}{4z^2}}}{2\pi z^2(g_3+\left(\frac{{\lambda }_F}{z^2}\right)g_4^2)}},$$

(67)

with

$$g_1\left(x\right)=N_1{\displaystyle \frac{{(1-x)}^{\alpha }{x}^{\sigma }}{{(1-\widehat{x})}^{\alpha }{\widehat{x}}^{\sigma }}},$$

(68)

$$g_{3,4}\left(z\right)=N_{3,4}{\displaystyle \frac{(z^{\beta }+\delta ){(1-z)}^{\gamma }}{({\widehat{z}}^{\beta }+\delta ){(1-\widehat{z})}^{\gamma }}}.$$

(69)

Here, $\widehat{x}=0.1$ and $\widehat{z}=0.5$ are kept fixed, while $\alpha ,\sigma ,\beta ,\gamma ,\delta$, $N_1\equiv g_1\left(\widehat{x}\right)$, and $N_{3,4}\equiv g_{3,4}\left(\widehat{z}\right)$ are free parameters that are determined by fitting to the available data from SIDIS, Drell–Yan, and $W/Z$ boson production processes. In addition to the $b_{*}\left(b\right)$ prescription of the original CSS formalism [17], several alternative functional forms for $b_{*}\left(b\right)$ have been introduced in the literature [22,60]. In Ref. [22], a novel $b_{*}$ prescription, distinct from Equation (36), was proposed as

$$b_{*}=b_{\max }{\left({\displaystyle \frac{1-e^{-b^4/b_{\max }^4}}{1-e^{-b^4/b_{\min }^4}}}\right)}^{1/4}.$$

(70)

Here, $b_{\max }$ again denotes the separation between the nonperturbative and perturbative domains in b space, and is fixed to $b_{\max }=2e^{-{\gamma }_E}{\mathrm{GeV}}^{-1}\approx 1.123{\mathrm{GeV}}^{-1}$.

The introduction of the parameter $b_{\min }$ (or, more generally, the prescription regulating the small-$b_T$ region) is not arbitrary but is closely related to the treatment of logarithmic contributions in the perturbative Sudakov exponent. In transverse-momentum resummation, the logarithmic expansion parameter has the form of $\ln (Q^2{b}^2/b_0^2)$, with $b_0=2e^{-{\gamma }_E}$. In the limit $b\to 0$, the expansion parameter logarithms become unphysically large. To reduce the impact of unjustified higher-order contributions in the large-$q_T$ (small b) region, Ref. [75] introduces a modified logarithmic structure $L\equiv \ln \left(1+{\displaystyle \frac{Q^2{b}_T^2}{b_0^2}}\right)$ that effectively freezes the argument of the logarithm in the ultraviolet region. In addition, the authors of Ref. [22] chose to saturate $b_{*}$ at a minimum value $b_{\min }\propto 2e^{-{\gamma }_E}/Q$ that corresponds to modifying the resummation logarithms.

4. Numerical Evaluation of Physical Observables in Drell–Yan and SIDIS Processes

4.1. Common Input: The Longitudinal Transversity Distribution and Its Evolution

Both the $\cos 2\varphi$ asymmetry in the Drell–Yan process and the $\sin 2{\varphi }_h$ asymmetry in SIDIS depend on the collinear function of the $h_{1L}^{\perp }$.

Since $h_{1L}^{\perp }$ has not yet been extracted from experimental measurements, we determine $h_{1L}^{\perp \left(1\right)}(x,\mu )$ by using the WW-type approximation [76,77]. This approximation is based on the assumption [76] that the quark–quark–gluon correlations (i.e., the genuine twist-3 contributions) and the current-quark mass terms appearing in the operator decomposition of certain distributions can be neglected. It has been shown that the WW approximation provides a good description of the data for the twist-3 structure function $g_2\left(x\right)$ (equivalently, the distribution $g_T\left(x\right)$) in DIS [78,79,80,81,82,83,84,85]. This result is further corroborated by lattice QCD computations [86,87] and by theoretical analyses based on the instanton model [88,89]. Within the WW-type approximation, one can express $h_L\left(x\right)$ in the following way [77]

$$\begin{array}{c}\hfill h_L\left(x\right)=2x{\int }_x^1{\displaystyle \frac{dy}{y^2}}{h}_1\left(y\right)+{\tilde{h}}_L\left(x\right)\stackrel{WW-type}{\approx }2x{\int }_x^1{\displaystyle \frac{dy}{y^2}}{h}_1^a\left(y\right),\end{array}$$

(71)

which contains the twist-2 component $h_1\left(y\right)$ and the genuine twist-3 term ${\tilde{h}}_L\left(x\right)$ originating from quark–quark–gluon correlations; the latter is discarded in the WW approximation. At the same time, it should be emphasized that the WW approximation can be inadequate in certain situations. As discussed in Ref. [90], the genuine twist-3 contributions can violate the WW relation for the structure function $g_2$ (or $g_T\left(x\right)$) by about 15–40% of the magnitude of $g_2$. Using both lattice data and data from global fits [91], the twist-3 distribution function $g_T\left(x\right)$ has been found to be consistent with its WW approximation for $x\lesssim 0.4$, but within uncertainties, one cannot exclude its violation at levels of up to 40%. Therefore, the WW approximation of $h_{1L}^{\perp }$ can be seen as a practical estimate, while the further inclusion of the genuine the higher-twist effects may induce uncertainties in the prediction of the $\sin 2{\varphi }_h$ asymmetry.

Furthermore, one can similarly derive the following relation

$$\begin{array}{c}\hfill x{h}_L(x,p_T^2)\stackrel{WW-type}{\approx }-{\displaystyle \frac{p_T^2}{M^2}}{h}_{1L}^{\perp }(x,p_T^2).\end{array}$$

(72)

Consequently, $h_{1L}^{\perp \left(1\right)}\left(x\right)$ is related to the transversity distribution through the relation [85]

$$\begin{array}{c}\hfill h_{1L}^{\perp \left(1\right)}\left(x\right)\stackrel{WW-type}{\approx }-x^2{\int }_x^1{\displaystyle \frac{dy}{y^2}}{h}_1\left(y\right).\end{array}$$

(73)

where $h_1$ denotes the transversity distribution. For its valence-quark component, we adopt the parametrization obtained in Ref. [21], which was derived within the TMD evolution framework. At the initial scale $Q_0=\sqrt{2.4}\mathrm{GeV}$, this parametrization reads

$$\begin{array}{c}\hfill h_1^q(x,Q_0)=N_q^h{x}^{a_q}{(1-x)}^{b_q}{\displaystyle \frac{{(a_q+b_q)}^{a_q+b_q}}{a_q^{a_q}{b}_q^{b_q}}}\times {\displaystyle \frac{1}{2}}\left(f_1^q(x,Q_0)+g_1^q(x,Q_0)\right).\end{array}$$

(74)

Here, $g_1^q$ denotes the helicity distribution function [92], while $f_1^q$ represents the unpolarized distribution function. For the sea-quark contribution to the transversity distribution, we adopt the following assumption:

$$\begin{array}{c}\hfill h_1^{sea}(x,Q_0)=N{\displaystyle \frac{1}{2}}\left(f_1^q(x,Q_0)+g_1^q(x,Q_0)\right),\end{array}$$

(75)

where N denotes the fraction of the proton’s contribution arising from sea quarks. In our analysis, we examine three possible values for this sea-quark contribution ratio in the proton: $N=0$, $N=0.5$, and $N=1$.

The scale evolution of $h_{1L}^{\perp \left(1\right)}$ has been investigated in detail in Ref. [93], where it was shown to take a rather intricate form. Specifically, the corresponding evolution kernel is composed of a homogeneous and an inhomogeneous contribution. The former is associated with the diagonal component of the relevant operator, while the latter is determined by the nondiagonal component. Since, to date, neither a theoretical calculation nor a phenomenological extraction of the nondiagonal part of the operator for $h_{1L}^{\perp \left(1\right)}$ is available, we adopt the same simplifying assumption as in Refs. [21,24], retaining only the diagonal contribution in our phenomenological treatment of TMD evolution. Clearly, this constitutes an approximation to the full dynamics and introduces additional uncertainty in the predicted asymmetry, originating from the neglected nondiagonal part. In existing studies, only the nondiagonal component of the Qiu–Sterman function $T_{q,F}$ has been analyzed [94] within a model framework, which indicates that the nondiagonal terms in the evolution equation for $T_{q,F}(x,x)$ substantially affect the evolution of these correlation functions, particularly in the small-x region. The inhomogeneous term of the evolution kernel of $h_{1L}^{\perp \left(1\right)}$ may also affect the evolution behavior in the small-x region. We anticipate that an analogous behavior may arise in the evolution of $h_{1L}^{\perp \left(1\right)}$. A quantitative estimate of the uncertainty due to the nondiagonal contribution is deferred to future work.

Accordingly, we adopt a comparable strategy and focus on the homogeneous contributions to the evolution kernel of $h_{1L}^{\perp \left(1\right)}$:

$$\begin{array}{c}\hfill P_{qq}^h=C_F\left[{\displaystyle \frac{2x}{{(1-x)}_{+}}}+2\delta (1-x)\right]-{\displaystyle \frac{C_A}{2}}{\displaystyle \frac{2x}{1-x}}.\end{array}$$

(76)

For the evolution kernel of the twist-3 fragmentation function ${\widehat{H}}^{\left(3\right)}$, we use the expression given in Ref. [21]:

$$\begin{array}{c}\hfill P_{qq}^H=C_F\left[{\displaystyle \frac{2x}{{(1-x)}_{+}}}+{\displaystyle \frac{3}{2}}\delta (1-x)\right].\end{array}$$

(77)

The DGLAP equations are solved numerically using the QCDNUM evolution package [95]. We have modified the original QCDNUM code to incorporate the evolution kernel for $h_{1L}^{\perp \left(1\right)}$. In Figure 2, we present $h_{1L}^{\perp \left(1\right)}(x,Q^2)$ (multiplied by x) as a function of x for light quark flavors, comparing the initial scale $Q^2=2.4$ GeV² with the evolved scale $Q^2=100$ GeV². The left and right panels display the results for the up and down quarks, respectively. The figure indicates that the absolute size of $h_{1L}^{\perp \left(1\right)}(x,Q^2)$ is larger for the up quark than for the down quark, with the two having opposite signs. Furthermore, the evolution from the lower to the higher scale shifts the peak of the distribution toward smaller values of x. More complete assessment of theoretical uncertainties, such as those obtained from Monte Carlo replica methods for TMDs, would be highly desirable in future dedicated phenomenological studies. Such an analysis would improve the quantitative assessment of the reliability of the predictions and their direct comparison with experimental measurements.

4.2. Drell–Yan Process: Predictions for the $\cos 2\varphi$ Asymmetry

In this section, we carry out a detailed numerical study of the $\cos 2\varphi$ azimuthal asymmetry in double-longitudinally polarized proton–proton Drell–Yan processes based on the theoretical framework developed in the preceding sections. Our computations are performed for the kinematic regions at both RHIC and NICA, with particular attention given to the impact of sea quark contributions.

The computation uses the collinear functions $h_{1L}^{\perp \left(1\right)}(x,\mu )$ and $f_1(x,\mu )$ as input. For the unpolarized distribution $f_1(x,\mu )$, we utilize the NLO CT10 parametrization (central PDF set) from Ref. [96]. For the longitudinally polarized transversity distribution $h_{1L}^{\perp \left(1\right)}(x,\mu )$, we employ the Wandzura–Wilczek approximation described in Section 4.1.

We also use the QCDNUM package [95] to perform the scale evolution of $f_1$ from the initial value $Q_0=\sqrt{2.4}\mathrm{GeV}$ to higher energies.

The Drell–Yan experiment at RHIC employed two colliding proton beams with center-of-mass energies of either $\sqrt{s}=200\mathrm{GeV}$ or $500\mathrm{GeV}$ [97]. In this work, we evaluate the $A_{LL}^{\cos 2\varphi }$ asymmetry within the RHIC kinematic range [42]:

$$\begin{array}{c}\hfill 4\mathrm{GeV}<\mathrm{Q}<9\mathrm{GeV},0<q_T<1\mathrm{GeV},-1<y<2.\end{array}$$

We further evaluate the $A_{LL}^{\cos 2\varphi }$ asymmetry within the kinematic ranges relevant for NICA [42]:

$$\begin{array}{c}\hfill \sqrt{s}=27\mathrm{GeV},4\mathrm{GeV}<Q<9\mathrm{GeV},\hfill \\ \hfill 0<q_T<1\mathrm{GeV},0.1<x<0.8.\hfill \end{array}$$

Figure 3 and Figure 4 show our numerical predictions for the $\cos 2\varphi$ azimuthal asymmetry in $pp$ Drell–Yan processes under RHIC kinematic conditions. The left, middle, and right panels illustrate, respectively, how the asymmetry varies with rapidity (y), transverse momentum ($q_T$), and invariant mass (Q). Figure 3 presents the results for $\sqrt{s}=200\mathrm{GeV}$, while Figure 4 displays the corresponding predictions for $\sqrt{s}=500\mathrm{GeV}$. The dashed, solid, and dotted curves denote the asymmetry for sea-quark contribution parameters $N=0$, $N=0.5$, and $N=1$, respectively.

As displayed in Figure 3, for all values of N other than $N=0$ (for which the asymmetry vanishes), our calculation yields a positive $\cos 2\varphi$ azimuthal asymmetry in the doubly longitudinally polarized proton–proton Drell–Yan process. Our results further indicate that the asymmetry exhibits only a mild variation with y. For $N=0.5$, the y-dependent asymmetry lies between $2\%$ and $5\%$, whereas for $N=1$, it ranges from $8\%$ to $10\%$. Regarding the $q_T$-dependent asymmetry, we observe a pronounced increase with $q_T$ for $N=1$, reaching about $17.5\%$ at $q_T\approx 1\mathrm{GeV}$. For $N=0.5$, the asymmetry shows only a moderate rise with $q_T$, attaining a maximum of roughly $5\%$. For the Q dependence, $A_{LL}^{\cos 2\varphi }$ decreases moderately from $12\%$ to $5\%$ as Q grows from 4 to $9\mathrm{GeV}$ when $N=1$. The Q dependence is weaker for $N=0.5$, with the asymmetry varying from about $4\%$ to $2\%$. Similar qualitative behaviors appear in Figure 2, but with larger absolute values (up to about $25\%$).

The NICA predictions ($\sqrt{s}=27\mathrm{GeV}$) shown in Figure 5 display the same qualitative trends as the RHIC results but differ in their quantitative characteristics. Regarding the Bjorken-x dependence, we find a non-monotonic asymmetry for $N=1$, with a peak value of about $22\%$. The asymmetry first decreases in the interval $0.1<x<0.2$, and then gradually increases up to $x\approx 0.7$. This pattern reflects the intricate balance between valence and sea quark contributions in the longitudinally polarized proton.

For the TMD dependence, the $q_T$ behavior at NICA follows trends similar to those at RHIC, albeit with smaller magnitudes. The asymmetry reaches a maximum of $17\%$ for $N=1$, indicating the sensitivity of this observable to the transverse momentum distribution of quarks inside the proton.

The Q dependence at NICA is also analogous to that observed at RHIC, with the asymmetry slowly decreasing as Q increases. The maximum asymmetry is about $15\%$ for $N=1$, in line with expectations from TMD factorization. Collectively, these findings underscore the significance of measuring the asymmetry over a broad kinematic range to tightly constrain sea quark contributions and their variation with the hard scale Q.

We additionally estimate the Drell–Yan event rate expected at NICA. Given the projected luminosity of $\mathcal{L}={10}^{32}{\mathrm{cm}}^{-2}{\mathrm{s}}^{-1}$ [98] and an assumed annual running time of $3\times {10}^7$ s, the corresponding integrated luminosity is

$$\begin{array}{c}\hfill {\mathcal{L}}_{\mathrm{int}}=\mathcal{L}\times t\approx 3\times {10}^{39}{\mathrm{cm}}^{-2}.\end{array}$$

(78)

The unpolarized cross section ${\sigma }^{\mathrm{DY}}$ obtained in our framework (Equation (5)) is first expressed in natural units (${\mathrm{GeV}}^{-2}$). By converting to experimental units (${\mathrm{cm}}^2$) using $1{\mathrm{GeV}}^{-2}\approx 3.894\times {10}^{-28}{\mathrm{cm}}^2$, we find ${\sigma }^{\mathrm{DY}}\approx 5\times {10}^{-37}{\mathrm{cm}}^2$. The corresponding expected number of events per year can then be estimated as

$$\begin{array}{c}\hfill N_{\mathrm{events}}={\sigma }^{\mathrm{DY}}\times {\mathcal{L}}_{\mathrm{int}}\approx 1600.\end{array}$$

(79)

This result indicates that the proposed asymmetry $A_{LL}^{\cos 2\varphi }$ should be within experimental reach at NICA.

In conclusion, as illustrated in Figure 3, Figure 4 and Figure 5, the $\cos 2\varphi$ azimuthal asymmetries in doubly longitudinally polarized proton–proton Drell–Yan scattering are positive in all scenarios except for $N=0$, where the asymmetry disappears. In particular, the asymmetry attains a peak value of about $25\%$ for $N=1$ and up to $10\%$ for $N=0.5$, indicating a pronounced sensitivity to the longitudinally polarized transversity distributions of sea quarks. These findings emphasize that high-precision measurements at RHIC and NICA can substantially deepen our knowledge of nucleon structure and QCD dynamics. Furthermore, the doubly polarized $p^{\to }{p}^{\to }$ Drell–Yan reaction stands out as a powerful tool for probing the contribution of sea quarks to the proton’s internal structure, offering key insights into the nonperturbative domain of the strong interaction.

4.3. SIDIS Process: Calculations and Experimental Comparison for the $\sin 2{\varphi }_h$ Asymmetry

The most direct constraints on the longi-transversity distribution come from HERMES (proton and deuteron targets) and CLAS (proton target) measurements of the $\sin 2{\varphi }_h$ asymmetry in SIDIS. For the Drell–Yan process, no double-polarized measurement has been performed to date, which motivates the predictions for facilities such as NICA presented in Section 4.2.

In this section, based on the framework established above, we provide the numerical evaluation of the $\sin 2{\varphi }_h$ azimuthal asymmetry for $\pi$ production in the process $l{N}^{\to }\to l\pi X$ involving a longitudinally polarized nucleon. We compute the asymmetry for the kinematic settings of HERMES, CLAS, and CLAS12, and compare our results with the latest experimental data [38,39,40,99,100,101,102,103,104,105]. For this analysis, we require collinear functions, as input for the evolution, that enter Equations (23) and (24). For the nucleon unpolarized PDF $f_1(x,\mu )$, we use the NLO central set of the CT10 parametrization [96]. For the pion unpolarized FF $D_1(z,\mu )$, we employ the NLO set of the de Florian, Sassot, and Stratmann (DSS) fragmentation functions [106]. The twist-3 Collins FF ${\widehat{H}}^{\left(3\right)}(z,z,\mu )$ of the pion is taken from the parametrization proposed in Ref. [21]. For the longitudinally polarized transversity distribution $h_{1L}^{\perp \left(1\right)}(x,\mu )$, we also employ the Wandzura–Wilczek approximation described in Section 4.1.

The $A_{UL}^{\sin 2{\varphi }_h}$ asymmetry in pion production was determined at HERMES using both a proton target [38,99] and a deuteron target [39], within the kinematic region

$$\begin{array}{c}\hfill 1{\mathrm{GeV}}^2<Q^2<15{\mathrm{GeV}}^2,W>2\mathrm{GeV},0.023<x<0.4,0.2<y<0.85,0.2<z<0.7,\end{array}$$

(80)

while it was likewise determined for pion production from a proton target by CLAS [40,100] within the kinematic range

$$\begin{array}{c}\hfill 1{\mathrm{GeV}}^2<Q^2<5.4{\mathrm{GeV}}^2,W>2\mathrm{GeV},0.12<x<0.48,y<0.85,0.4<z<0.7,\end{array}$$

(81)

where $W^2={(P+q)}^2\approx {\displaystyle \frac{1-x}{x}}{Q}^2$ denotes the invariant mass squared of the virtual photon–nucleon system.

In Figure 6 and Figure 7, we present numerical predictions for the $\sin 2{\varphi }_h$ azimuthal asymmetry in SIDIS off longitudinally polarized proton and deuteron targets at HERMES kinematics [38,39,82,99,107]. These results are obtained within the TMD factorization framework specified in Equations (20), (23) and (24). The solid squares denote the HERMES experimental measurements [38,39,107], with the error bars indicating the corresponding statistical uncertainties. To maintain the applicability of TMD factorization, the transverse momentum of the produced pion is integrated only over the range $P_{\pi T}/z<0.5\mathrm{GeV}$. In our computation, we employ the EIKV parametrization (dark dashed curve) and the BDPRS parametrization (red solid curve) for the nonperturbative component. The yellow, green, and cyan shaded bands around the solid curves represent, respectively, the uncertainties originating from the BDPRS parametrization parameters, from the transversity distribution $h_1\left(x\right)$ used to determine $h_{1L}^{\perp \left(1\right)}\left(x\right)$, and from the twist-3 Collins FF $H^{\left(3\right)}$. For the dashed curves, only two sources—namely the transversity distribution $h_1\left(x\right)$ entering $h_{1L}^{\perp \left(1\right)}\left(x\right)$ and the twist-3 Collins FF ${\widehat{H}}^{\left(3\right)}$—contribute to the total uncertainty, so their uncertainty bands are expected to be narrower than those associated with the solid curves.

As illustrated in Figure 6 and Figure 7, the $\sin 2{\varphi }_h$ azimuthal asymmetries in SIDIS do not exceed roughly $2\%$ in any of the considered cases. The predicted $\sin 2{\varphi }_h$ asymmetry for ${\pi }^{+}$ is negative, whereas those for ${\pi }^{-}$ and ${\pi }^0$ are positive, with the magnitude of the ${\pi }^{-}$ asymmetry exceeding that of both ${\pi }^{+}$ and ${\pi }^0$. Furthermore, our results indicate that the magnitude of the asymmetries grows with increasing x. For a proton target, the predictions based on the EIKV parametrization closely match those obtained with the BDPRS parametrization; however, for a deuteron target, the two parametrizations differ quantitatively, with the EIKV-based results being about twice as large as those from BDPRS. By comparing with the HERMES measurements [38,39,82,99,107], we observe that our calculations are compatible with the data within the experimental uncertainties for all channels, except for ${\pi }^{-}$ production on a proton target. In this latter case, the predicted asymmetry disagrees with the HERMES result, most notably in the opposite sign. A plausible explanation for this discrepancy is the presence of sizable uncertainties that have not yet been incorporated into our current analysis. We will discuss the origin of these uncertainties at the end of this section. The nonvanishing $\sin 2{\varphi }_h$ azimuthal asymmetries predicted for ${\pi }^{\pm }$ suggest that the spin–orbit correlation of transversely polarized quarks inside a longitudinally polarized nucleon could be sizable.

In Figure 8, we present our numerical results for the $\sin 2{\varphi }_h$ azimuthal asymmetry on a proton target, evaluated at CLAS kinematics [40,100]. The top panels of Figure 8 display the x-dependent asymmetries for ${\pi }^{+}$ (left) and ${\pi }^{-}$ (right) production, while the bottom panels show the x-dependence (left) and the $P_{\pi T}$-dependence (right) for ${\pi }^0$ production. For clarity, each panel is magnified to better highlight the trends. Despite the relatively large experimental uncertainties at CLAS, the data suggest that the asymmetries for both ${\pi }^{+}$ and ${\pi }^{-}$ are predominantly negative. Consequently, our calculation does not reproduce the measured x-dependent asymmetry for ${\pi }^{-}$ production. For ${\pi }^0$ production, we obtain a much smaller $\sin 2\varphi$ azimuthal asymmetry at CLAS, as well as at HERMES for the deuteron target, consistent with the experimental observations. This suppression arises because ${\pi }^0$ production involves the sum of favored and unfavored Collins functions, which largely cancel each other. As before, the yellow, green, and cyan shaded regions surrounding the solid curves represent the uncertainty bands associated with the BDPRS parametrization parameters, the transversity distribution $h_1\left(x\right)$ entering the evaluation of $h_{1L}^{\perp \left(1\right)}\left(x\right)$, and the twist-3 Collins fragmentation function $H^{\left(3\right)}$, respectively.

Although the $A_{UL}^{\sin 2{\varphi }_h}$ asymmetries on longitudinally polarized targets have been investigated in HERMES (proton, deuteron) [38,39,99] and CLAS (proton) [40,100] kinematics over the past two decades, a coherent picture of how each individual contribution enters the total structure function is still lacking, likely because earlier measurements suffered from limited statistics and restricted kinematic coverage. Accordingly, we additionally evaluate the $A_{UL}^{\sin 2{\varphi }_h}$ asymmetries on a proton target for pion production under the kinematic conditions of the CLAS12 experiment [101]

$$\begin{array}{c}\hfill 1{\mathrm{GeV}}^2<Q^2<7{\mathrm{GeV}}^2,W>2\mathrm{GeV},0.13<x<0.52,y<0.75,0.18<z<0.7.\end{array}$$

(82)

The estimated asymmetries for ${\pi }^{+}$ (upper panel), ${\pi }^{-}$ (middle panel), and ${\pi }^0$ (lower panel) production as functions of x, $P_{\pi T}$, and z are shown in Figure 9, respectively. From Figure 9, we infer that the magnitude of the $\sin 2{\varphi }_h$ azimuthal asymmetries at CLAS12 exceeds that observed at CLAS. Consequently, CLAS12 offers an enhanced opportunity to measure the $\sin 2{\varphi }_h$ asymmetry. Furthermore, we note that the $P_{\pi T}$-dependent asymmetries obtained from the EIKV parametrization and the BDPRS parametrization differ to some extent. These differences should be examined further with upcoming CLAS12 data on a proton target.

Finally, we comment on the theoretical sources of uncertainty. First, one should be careful about the applicability of TMD factorization in the HERMES and CLAS kinematic ranges, where $Q^2$ is relatively low. In our analysis, we restrict ourselves to $P_{\pi T}/z<0.5\mathrm{GeV}$, but this assumption needs to be examined with more precise data in the future. Second, since the knowledge of $h_{1L}^{\perp }$ is still quite limited, we rely on the WW approximation to determine $h_{1L}^{\perp }$ from the underlying function $h_1\left(x\right)$; therefore, contributions beyond the WW approximation, which are not included in our calculation, may also play a role. Although experimental data and lattice results indicate that the WW approximation works reasonably well for the structure function $g_2$ (or $g_T\left(x\right)$), the analysis in Ref. [90] suggests that the violation of the WW relation may reach 15–40% of the magnitude of $g_2$. This implies that, for $h_{1L}^{\perp }$, higher-twist effects could likewise be significant, and a rough estimate is that the associated uncertainty might be as large as about 40% of the central asymmetry value. Third, for ${\widehat{H}}^{\left(3\right)}$ and $h_{1L}^{\perp \left(1\right)}$, we retain only the homogeneous terms in the evolution kernel. The inhomogeneous terms are highly nontrivial and are therefore omitted to keep the analysis tractable. This is, of course, an approximation to the actual situation. Fourth, in our computation, we use the NLL expression for the perturbative Sudakov form factor. Higher-order QCD corrections, which we do not include, may further modify the results and thus influence the overall precision. If all these sources of uncertainty were properly incorporated, the error bands in Figure 3, Figure 4, Figure 5 and Figure 6 would be considerably broader. Additional investigations are required to resolve these issues.

5. Summary and Prospects

This review synthesized recent phenomenological advances in the study of the longi-transversity distribution $h_{1L}^{\perp }$ within the framework of TMD factorization. By examining the complementary roles of doubly polarized Drell–Yan scattering and SIDIS, we have demonstrated how specific azimuthal asymmetries serve as unique and sensitive probes of the correlation between a longitudinally polarized nucleon and transversely polarized quarks—a cornerstone of the three-dimensional spin structure of the nucleon.

The numerical studies presented here, which consistently incorporate TMD evolution at next-to-leading logarithmic accuracy, lead to several robust conclusions. In polarized proton–proton Drell–Yan collisions, the $\cos 2\varphi$ asymmetry is predicted to be substantial, potentially reaching 25% under maximal sea quark contributions at RHIC kinematics. This pronounced signal establishes the asymmetry as a powerful direct probe of polarized sea-quark dynamics. Event yield estimates further confirm the practical measurability of this observable at forthcoming facilities such as NICA. In contrast, in SIDIS off longitudinally polarized targets, the $\sin 2{\varphi }_h$ asymmetry is typically smaller, at the level of a few percent, and exhibits clear dependencies on final-state hadron charge and target type. Theoretical predictions, relying on the Wandzura–Wilczek approximation for $h_{1L}^{\perp }$ and employing two distinct non-perturbative Sudakov parameterizations, show broad consistency with existing HERMES and CLAS data, although certain tensions, notably for ${\pi }^{-}$ production, highlight areas for further scrutiny. The projected larger asymmetries at CLAS12 promise enhanced precision for future constraints. The application of different TMD evolution schemes underscores the overall stability of the predictions while simultaneously revealing that specific choices, particularly for the non-perturbative sector, can influence outcomes for deuteron targets and the detailed $P_T$ dependence of observables.

These investigations firmly establish $h_{1L}^{\perp }$ as a pivotal element in the quest for a complete tomographic portrait of the nucleon. They also delineate clear paths for future theoretical and experimental progress. On the theoretical front, moving beyond the Wandzura–Wilczek approximation and incorporating the full inhomogeneous terms in evolution kernels are critical steps towards reducing current uncertainties and achieving precision. A global QCD analysis that simultaneously fits $h_{1L}^{\perp }$ from Drell–Yan and SIDIS data within a unified TMD framework will be essential to minimize model dependencies and rigorously test the universality of TMD functions. Experimentally, the future is bright. Upcoming and planned measurements at facilities such as the upgraded NICA, CLAS12, and EIC will deliver high-statistics, multi-dimensional data across a vast kinematic landscape. This new generation of data will be indispensable for disentangling the contributions of valence and sea quarks, providing stringent tests of TMD evolution, and ultimately yielding a definitive understanding of longitudinal transversity and its integral role in the nucleon spin puzzle.

In summary, the exploration of azimuthal asymmetries through the lens of TMD factorization constitutes a dynamic and rapidly advancing frontier in hadron physics. The theoretical formalism and phenomenological results summarized in this work provide an overview of the current status of the field and offer a framework for understanding transversely polarized quarks inside longitudinally polarized nucleons, while also highlighting opportunities for future studies at complementary high-energy facilities.