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Probing quark orbital angular momentum at EIC and EicC

Shohini Bhattacharya RIKEN BNL Research Center, Brookhaven National Laboratory, Upton, NY 11973, USA    Duxin Zheng Shandong Institute of Advanced Technology, Jinan, Shandong, 250100, China    Jian Zhou School of Physics and Key Laboratory of Particle Physics and Particle Irradiation (MOE), Shandong University, QingDao, Shandong, 266237, China
Abstract

We propose to detect signals from quark orbital angular momentum (OAM) through exclusive π0\pi^{0} production in electron-(longitudinally-polarized) proton collisions. Our analysis demonstrates that the sin2ϕ\sin 2\phi azimuthal angular correlation between the transverse momentum of the scattered electron and the recoil proton serves as a sensitive probe of quark OAM. Additionally, we present a numerical estimate of the asymmetry associated with this correlation for the kinematics accessible at EIC and EicC. This study aims to pave the way for the first experimental study of quark OAM in relation to the Jaffe-Manohar spin sum rule.

1. Introduction—The exploration of nucleon spin structure, sparked by the revelation of the “spin crisis”, has developed into a captivating research field over the past three decades. A central goal of this field is to comprehend the nucleon’s spin in terms of contributions from its underlying partons. Significant progress has been made in deciphering this partonic content of nucleon spin, particularly in constraining contributions from quark and gluon spins in the moderate and large xx regions through measurements of parton helicity distributions at accelerator facilities worldwide Adamczyk et al. (2015); de Florian et al. (2014); Nocera et al. (2014); Abdallah et al. (2022). The upcoming Electron-Ion Collider in the US and China (EIC and EicC) Abdul Khalek et al. (2022); Anderle et al. (2021) is expected to play a crucial role in precisely determining the gluon helicity distribution at small xx. While parton helicities represent a significant fraction of nucleon spin, there remains ample opportunity to investigate the contribution of parton Orbital Angular Momentum (OAM) to nucleon spin, constituting another key objective of the EIC and EicC.

Refer to caption
Figure 1: An illustration of exclusive π0\pi^{0} production.

In an interacting theory like Quantum Chromodynamics (QCD), two types of OAMs exist: the kinetic type (Ji’s type) and the canonical type (Jaffe-Manohar’s type). The difference between the two definitions of OAM in QCD can be attributed to the gauge potential term. In practice, the kinetic OAM of quarks and gluons is determined by subtracting their helicity contributions from the total angular momentum contributions, which can be accessed through hard exclusive processes Ji (1997a, b). However, extracting Jaffe-Manohar type parton OAM Jaffe and Manohar (1990), or equivalently canonical OAM, in high-energy scattering processes poses a significant experimental challenge. Progress in this direction was limited until a connection between parton OAM and Wigner distribution functions Belitsky et al. (2004), or equivalently, Generalized Transverse Momentum-dependent Distributions (GTMDs) Meissner et al. (2009), was revealed about a decade ago. For the quark case, this connection is given by Lorce and Pasquini (2011); Hatta (2012); Lorce et al. (2012),

Lq(x,ξ)=d2kk2M2F1,4q(x,k,ξ,Δ=0).\displaystyle L^{q}(x,\xi)=-\int d^{2}k_{\perp}\frac{k_{\perp}^{2}}{M^{2}}F_{1,4}^{q}(x,k_{\perp},\xi,\Delta_{\perp}=0)\,. (1)

All quantities appear in the above equation will be specified below. The quark OAM can be reconstructed by integrating over the xx-dependent OAM distribution: Lq=01𝑑xLq(x,ξ=0)L_{q}=\int_{0}^{1}dxL_{q}(x,\xi=0). This relation, coupled with Eq. (1), thus opens a new avenue to directly access the parton canonical OAM contribution to the nucleon spin through GTMDs. Note that this relation is expected to hold beyond the tree level up to some power corrections Ebert et al. (2022); Bertone (2022); Echevarria et al. (2023). In recent years, theoretical efforts have primarily centered on investigating the experimental signals of the gluon GTMD F1,4F_{1,4} Ji et al. (2017); Hatta et al. (2017); Bhattacharya et al. (2022a, b); Boussarie et al. (2018). Conversely, the exclusive double Drell-Yan process, the sole known process providing access to quark GTMDs, mainly offers sensitivity to quark GTMD F1,4F_{1,4} in the Efremov-Radyushkin-Brodsky-Lepage (ERBL) region Bhattacharya et al. (2017). This poses a challenge when extrapolating the distribution to the forward limit.

In this paper, we introduce a novel observable to experimentally detect the quark GTMD F1,4F_{1,4} in the Dokshitzer–Gribov–Lipatov–Altarelli–Parisi (DGLAP) region, establishing a direct link to quark OAM through Eq. (1). Our proposal involves the exclusive π0\pi^{0} production process in electron-proton collisions(see Fig. 1): epepπ0ep\rightarrow e^{\prime}p^{\prime}\pi^{0}, with a longitudinally polarized proton target. Our analysis demonstrates that the longitudinal single target-spin asymmetry results in a sin2(ϕlϕΔ)\sin 2(\phi_{l_{\perp}}-\phi_{\Delta_{\perp}}) azimuthal angular correlation, where ϕl\phi_{l_{\perp}} and ϕΔ\phi_{\Delta_{\perp}} denote the azimuthal angles of the transverse momentum of the scattered electron and the recoil proton. This correlation exhibits a direct sensitivity to quark OAM.

The proposed observable stands out as an ideal probe for quark OAM from both theoretical and practical perspectives. Firstly, the background for this process remains clean, free from contamination by final-state soft gluon radiation effects Hatta et al. (2021a, b); Zhang et al. (2020); Tong et al. (2023a); Gao et al. (2023); Tong et al. (2023b). Additionally, our observable, akin to the unpolarized cross section, constitutes a twist-3 contribution (or equivalently, a sub-leading power correction). This characteristic enables the maximal enhancement of the asymmetry without being washed out by the unpolarized cross section.

2. Probing the quark GTMD F1,4F_{1,4} in exclusive π0\pi^{0} production—First, let us define the kinematics of the process under consideration,

e(l)+p(p,λ)π0(lπ)+e(l)+p(p,λ).\displaystyle e(l)+p(p,\lambda)\longrightarrow\pi^{0}(l_{\pi})+e(l^{\prime})+p(p^{\prime},\lambda^{\prime})\,. (2)

The standard kinematic variables are defined as follows: Q2=q2=(ll)2Q^{2}=-q^{2}=-(l-l^{\prime})^{2}, representing the photon’s virtuality; and the incoming electron’s momentum is parameterized as lμ=(l+,l,l)=(Q(1y)2y,Q2y,Q1yy)l^{\mu}=(l^{+},l^{-},l_{\perp})=(\tfrac{Q(1-y)}{\sqrt{2}y},\tfrac{Q}{\sqrt{2}y},\tfrac{Q\sqrt{1-y}}{y}). Here, ‘+/+/-’ denotes the light-cone plus/minus components. The y=pq/ply=p\cdot q/p\cdot l represents the usual momentum fraction. The γp\gamma^{*}p center-of-mass energy is given by W2=(p+q)2W^{2}=(p+q)^{2}. The pion mass in our calculation is neglected (lπ20l_{\pi}^{2}\approx 0), simplifying the analysis. We work in the symmetric frame where the initial state and the final state proton carry the transverse momenta p=Δ/2p_{\perp}=-\Delta_{\perp}/2 and p=Δ/2p_{\perp}^{\prime}=\Delta_{\perp}/2, respectively. The skewness variable is given by ξ=(p+p+)/(p++p+)=Δ+/(2P+)=xB/(2xB)\xi=(p^{+}-p^{\prime+})/(p^{+}+p^{\prime+})=-\Delta^{+}/(2P^{+})=x_{B}/(2-x_{B}), xB=Q2/2pqx_{B}=Q^{2}/2p\cdot q, and the momentum transfer squared can be expressed as t=(pp)2=4ξ2M2+Δ21ξ2t=(p-p^{\prime})^{2}=-\tfrac{4\xi^{2}M^{2}+\Delta_{\perp}^{2}}{1-\xi^{2}}, with MM being the proton mass.

In the near forward region, the leading power contribution to the exclusive transversely polarized virtual photon production of π0\pi^{0} emerges at the twist-3 level. This suppression of the leading power contribution is a result of the conservation of angular momentum along the direction of the virtual-nucleon beam. Since the unpolarized cross section starts at twist 3, the investigated longitudinal-spin asymmetry is not power-suppressed. In the region where the momentum transfer tt is exceedingly small, the exclusive π0\pi^{0} production process becomes susceptible to being dominated by the Primakoff process Primakoff (1951); Gasparian (2016); Liping et al. (2014); Kaskulov and Mosel (2011); Lepage and Brodsky (1980); Khodjamirian (1999); Jia et al. (2022), and the interference between the Primakoff process and the contribution from the gluon GTMD F1,4F_{1,4} Bhattacharya et al. (2023). In this work, we specifically concentrate on the valence quark region, where ξ0.1\xi\sim 0.1, thereby permitting the neglect of contributions from both the Primakoff process and the gluon-initiated process Bhattacharya et al. (2023).

We will perform the calculation within the framework of collinear higher-twist expansion. This technique, first developed in Refs.  Ellis et al. (1982, 1983), was applied to the study of the canonical OAM (see Ref. Ji et al. (2017)’s Eq. (5)), which we closely follow in this work. In this approach, the hard factor H(k,Δ)H(k_{\perp},\Delta_{\perp}) is expanded in terms of k/Qk_{\perp}/Q and Δ/Q\Delta_{\perp}/Q, where kk_{\perp} denotes the relative transverse momentum carried by the exchanged quarks,

H(k,Δ)=H(k=0,Δ=0)+\displaystyle\!\!\!\!\!\!\!\!H(k_{\perp},\Delta_{\perp})=H(k_{\perp}=0,\Delta_{\perp}=0)+ (3)
H(k,Δ=0)kμ|k=0kμ+H(k=0,Δ)Δμ|Δ=0Δμ+\displaystyle\!\!\!\!\!\frac{\partial H(k_{\perp},\Delta_{\perp}\!\!=0)}{\partial k_{\perp}^{\mu}}\Big{|}_{k_{\perp}\!\!=0}\!\!k_{\perp}^{\mu}+\frac{\partial H(k_{\perp}\!\!=0,\Delta_{\perp})}{\partial\Delta_{\perp}^{\mu}}\Big{|}_{\Delta_{\perp}\!\!=0}\!\!\Delta_{\perp}^{\mu}+...

The zeroth-order expansion of kk_{\perp} and Δ\Delta_{\perp} yields a null result for both the spin-averaged cross section and the longitudinal polarization-dependent cross section. Following this expansion, the subsequent step involves integrating over kk_{\perp}. Consequently, the scattering amplitudes are expressed as the convolution of the next-to-leading power of Eq. (3) with the GPDs or the first kk_{\perp}-moment of certain GTMDs, including the kk_{\perp}-moment of the quark GTMD F1,4F_{1,4}—in other words, the quark OAM distribution.

The leading twist quark GTMDs for nucleons are parameterized as the off-forward quark-quark correlator Meissner et al. (2009); Lorcé and Pasquini (2013),

Wλ,λ[Γ]=d3z2(2π)3eikzp,λ|q¯(z2)Γq(z2)|p,λ|z+=0,W_{\lambda,\lambda^{\prime}}^{[\Gamma]}=\!\!\int\frac{d^{3}z}{2(2\pi)^{3}}\,e^{ik\cdot z}\,\langle p^{\prime},\lambda^{\prime}|\,\bar{q}(-\tfrac{z}{2})\Gamma q(\tfrac{z}{2})\,|p,\lambda\rangle\Big{|}_{z^{+}=0}\,, (4)

where Γ\Gamma indicates a generic gamma matrix. The Wilson line in Eq. (4) is suppressed for brevity. In the notation of Meissner et al. (2009); Lorce and Pasquini (2011), they are expressed as follows:

Wλ,λ[γ+]\displaystyle W_{\lambda,\lambda^{\prime}}^{[\gamma^{+}]}\! =\displaystyle= 12Mu¯(p,λ)[F1,1+iσi+P+(kiF1,2+ΔiF1,3)\displaystyle\!\frac{1}{2M}\bar{u}(p^{\prime},\lambda^{\prime})\bigg{[}F_{1,1}+\frac{i\sigma^{i+}}{P^{+}}(k_{\perp}^{i}F_{1,2}+\Delta_{\perp}^{i}F_{1,3}) (5)
+iσijkiΔjM2F1,4]u(p,λ),\displaystyle\!+\frac{i\sigma^{ij}k_{\perp}^{i}\Delta_{\perp}^{j}}{M^{2}}\,F_{1,4}\bigg{]}u(p,\lambda)\,,
Wλ,λ[γ+γ5]\displaystyle W_{\lambda,\lambda^{\prime}}^{[\gamma^{+}\gamma_{5}]}\! =\displaystyle= 12Mu¯(p,λ)[iεijkiΔjM2G1,1+iσi+γ5kiP+G1,2\displaystyle\!\frac{1}{2M}\bar{u}(p^{\prime},\lambda^{\prime})\bigg{[}\frac{-i\varepsilon_{\perp}^{ij}k_{\perp}^{i}\Delta_{\perp}^{j}}{M^{2}}G_{1,1}+\frac{i\sigma^{i+}\gamma_{5}k_{\perp}^{i}}{P^{+}}G_{1,2} (6)
+iσi+γ5ΔiP+G1,3+iσ+γ5G1,4]u(p,λ),\displaystyle\!+\frac{i\sigma^{i+}\gamma_{5}\Delta_{\perp}^{i}}{P^{+}}G_{1,3}+i\sigma^{+-}\gamma_{5}G_{1,4}\bigg{]}u(p,\lambda)\,,

where εij=ε+ij\varepsilon_{\perp}^{ij}=\varepsilon^{-+ij} with ε0123=1\varepsilon^{0123}=1. The arguments of the GTMDs depend on (x,ξ,k,Δ,kΔ)(x,\xi,\vec{k}_{\perp},\vec{\Delta}_{\perp},\vec{k}_{\perp}\cdot\vec{\Delta}_{\perp}) but have been omitted in the above formulas for the sake of notation convenience. In addition to F1,4F_{1,4}, the quark GTMD G1,1G_{1,1} is particularly intriguing. The real part of G1,1G_{1,1} encodes information about the quark’s spin-orbital angular momentum correlation inside an unpolarized nucleon Meissner et al. (2009); Lorce and Pasquini (2011). These GTMDs have been explored in various models Meissner et al. (2008, 2009); Lorce and Pasquini (2011); Kanazawa et al. (2014); Mukherjee et al. (2014); Hagiwara et al. (2016); Zhou (2016); Courtoy and Miramontes (2017); Boer and Setyadi (2023, 2021); Hatta and Zhou (2022); Tan and Lu (2023); Xu et al. (2022); Ojha et al. (2022, 2023), and studied in the small xx limit Hatta and Zhou (2022); Agrawal et al. (2023).

There are a total of four diagrams contributing to the exclusive π0\pi^{0} production amplitude. One of these diagrams is shown in Fig. 2. Our explicit calculation has confirmed that the contributions from all four diagrams vanish at the leading power. To isolate the twist-3 contribution, we perform an expansion in kk_{\perp} and Δ\Delta_{\perp}. In doing so, it is essential to handle the kk_{\perp} and Δ\Delta_{\perp} dependencies from the exchanged quark legs with utmost care. To address this, we employ a technique known as the special propagator technique, first introduced in Ref. Qiu (1990). The inclusion of the special propagator contribution is crucial to ensure electromagnetic gauge invariance. It is noteworthy that an alternative approach, which also maintains electromagnetic gauge invariance at twist-3 accuracy, has been developed in Refs. Anikin et al. (2000); Radyushkin and Weiss (2001).

Depending on the various vector structures, the scattering amplitude can be organized into three terms,

1\displaystyle{\cal M}_{1}\!\! =\displaystyle= gs2efπ22(Nc21)2ξNc21ξ2δλλϵ×ΔQ2{1,1+𝒢1,1},\displaystyle\!\!\frac{g_{s}^{2}ef_{\pi}}{2\sqrt{2}}\frac{(N_{c}^{2}-1)2\xi}{N_{c}^{2}\sqrt{1-\xi^{2}}}\delta_{\lambda\lambda^{\prime}}\frac{\epsilon_{\perp}\times\Delta_{\perp}}{Q^{2}}\left\{{\cal F}_{1,1}+{\cal G}_{1,1}\right\},
2\displaystyle{\cal M}_{2}\!\! =\displaystyle= gs2efπ22(Nc21)2ξNc21ξ2δλ,λMϵSQ2{1,2+𝒢1,2},\displaystyle\!\!\frac{g_{s}^{2}ef_{\pi}}{2\sqrt{2}}\frac{(N_{c}^{2}-1)2\xi}{N_{c}^{2}\sqrt{1-\xi^{2}}}\delta_{\lambda,-\lambda^{\prime}}\frac{M\epsilon_{\perp}\cdot S_{\perp}}{Q^{2}}\left\{{\cal F}_{1,2}+{\cal G}_{1,2}\right\},
4\displaystyle{\cal M}_{4}\!\! =\displaystyle= igs2efπ22(Nc21)2ξNc21ξ2λδλλϵΔQ2{1,4+𝒢1,4},\displaystyle\!\!\frac{ig_{s}^{2}ef_{\pi}}{2\sqrt{2}}\frac{(N_{c}^{2}-1)2\xi}{N_{c}^{2}\sqrt{1-\xi^{2}}}\lambda\delta_{\lambda\lambda^{\prime}}\frac{\epsilon_{\perp}\cdot\Delta_{\perp}}{Q^{2}}\left\{{\cal F}_{1,4}+{\cal G}_{1,4}\right\}, (7)

where fπ=131f_{\pi}=131 MeV represents the π0\pi^{0} decay constant, ϵ\epsilon_{\perp} denotes the virtual photon’s transverse polarization vector, and SS_{\perp} is defined as Sμ=(0+,0,i,λ)S_{\perp}^{\mu}=(0^{+},0^{-},-i,\lambda). i,j{\cal F}_{i,j} and 𝒢i,j{\cal G}_{i,j} serve as shorthand notations for complex convolutions involving the GTMDs Fi,jF_{i,j}, Gi,jG_{i,j}, and the π0\pi^{0} distribution amplitude (DA) ϕπ(z)\phi_{\pi}(z). They are expressed as follows,

1,1\displaystyle{\cal F}_{1,1}\!\! =\displaystyle= 𝑑x𝑑zF~1,1(0)(x,ξ,Δ)x2ϕπ(z)(1+z2z)z2(1z)2,\displaystyle\!\!\int dxdz\tilde{F}_{1,1}^{(0)}(x,\xi,\Delta_{\perp})x^{2}\frac{\phi_{\pi}(z)(1+z^{2}-z)}{z^{2}(1-z)^{2}}\,, (8)
𝒢1,1\displaystyle{\cal G}_{1,1}\!\! =\displaystyle= 𝑑x𝑑zG~1,1(1)(x,ξ,Δ)ϕπ(z)(x2+2x2z+ξ2)z2,\displaystyle\!\!\int dxdz\tilde{G}_{1,1}^{(1)}(x,\xi,\Delta_{\perp})\frac{\phi_{\pi}(z)(x^{2}+2x^{2}z+\xi^{2})}{z^{2}}\,, (9)
1,2\displaystyle{\cal F}_{1,2}\!\! =\displaystyle= 𝑑x𝑑zF~1,2(1)(x,ξ,Δ)\displaystyle\!\!\int dxdz\tilde{F}_{1,2}^{(1)}(x,\xi,\Delta_{\perp}) (10)
×xξ(1ξ2)ϕπ(z)(1+z2z)z2(1z)2,\displaystyle\times x\xi(1-\xi^{2})\frac{\phi_{\pi}(z)(1+z^{2}-z)}{z^{2}(1-z)^{2}}\,,
𝒢1,2\displaystyle{\cal G}_{1,2} =\displaystyle= 𝑑x𝑑zG~1,2(1)(x,ξ,Δ)\displaystyle\int dxdz\tilde{G}_{1,2}^{(1)}(x,\xi,\Delta_{\perp}) (11)
×ϕπ(z)(x2+2x2z+ξ2)(1ξ2)z2,\displaystyle\times\frac{\phi_{\pi}(z)(x^{2}+2x^{2}z+\xi^{2})(1-\xi^{2})}{z^{2}}\,,
1,4\displaystyle{\cal F}_{1,4}\!\! =\displaystyle= 𝑑x𝑑zF~1,4(1)(x,ξ,Δ)xξϕπ(z)(1+z2z)z2(1z)2,\displaystyle\!\!\int dxdz\tilde{F}_{1,4}^{(1)}(x,\xi,\Delta_{\perp})x\xi\frac{\phi_{\pi}(z)(1+z^{2}-z)}{z^{2}(1-z)^{2}}\,, (12)
𝒢1,4\displaystyle{\cal G}_{1,4}\!\! =\displaystyle= 𝑑x𝑑zG~1,4(0)(x,ξ,Δ)\displaystyle\!\!\int dxdz\tilde{G}_{1,4}^{(0)}(x,\xi,\Delta_{\perp}) (13)
×x(4ξ2z+ξ22x2z+x2)z2ξϕπ(z),\displaystyle\times\frac{x(4\xi^{2}z+\xi^{2}-2x^{2}z+x^{2})}{z^{2}\xi}\phi_{\pi}(z)\,,

where

f~(n)(x,ξ,Δ)=d2k(k2M2)n12(23fu+13fd)(x+ξiϵ)2(xξ+iϵ)2\displaystyle\tilde{f}^{(n)}(x,\xi,\Delta_{\perp})\!=\!\int\!\!d^{2}k_{\perp}\!\left(\frac{k_{\perp}^{2}}{M^{2}}\right)^{\!\!n}\!\frac{{\frac{1}{\sqrt{2}}\left(\frac{2}{3}f^{u}+\frac{1}{3}f^{d}\right)}}{(x+\xi-i\epsilon)^{2}(x-\xi+i\epsilon)^{2}}\,\, (14)

with n=0,1n=0,1, and 𝑑x𝑑z11𝑑x01𝑑z\int dxdz\equiv\int_{-1}^{1}dx\int_{0}^{1}dz. The superscript on the GTMDs `f`f’, whose arguments have been suppressed for brevity, indicates the summation of up and down quark contributions. Here, zz represents the longitudinal momentum fraction of π0\pi^{0} carried by the quark. The derivation of the above expressions involves the repeated use of the symmetry property: 𝑑zzϕπ(z)z2(1z)2=𝑑z(1z)ϕπ(z)z2(1z)2\int dz\frac{z\phi_{\pi}(z)}{z^{2}(1-z)^{2}}=\int dz\frac{(1-z)\phi_{\pi}(z)}{z^{2}(1-z)^{2}}.

Refer to caption
Figure 2: A diagram contributing to exclusive π0\pi^{0} production.

A few remarks are now in order. First, we obtain the terms 1,2{\cal F}_{1,2}, 1,4{\cal F}_{1,4}, 𝒢1,1{\cal G}_{1,1}, and 𝒢1,2{\cal G}_{1,2} by performing kk_{\perp} expansion, while the Δ\Delta_{\perp} expansion gives rise to the contributions 1,1{\cal F}_{1,1} and 𝒢1,4{\cal G}_{1,4}. Second, the amplitudes 1{\cal M}_{1}, 2{\cal M}_{2}, and 4{\cal M}_{4} exhibit distinct Δ\Delta_{\perp}-dependent behaviors. Notably, 2{\cal M}_{2} persists as Δ\Delta_{\perp} approaches zero, even when averaging over SS_{\perp} in the unpolarized cross section. This persistence is attributed to the helicity flip mechanism provided by the quark GTMDs F1,2F_{1,2} and G1,2G_{1,2}, akin to what the gluon GTMD F1,2F_{1,2} does Boussarie et al. (2020). The last point to emphasize is that exclusive π0\pi^{0} production selects a C-odd exchange. This implies that the hard factors associated with F1,1F_{1,1}, G1,1G_{1,1}, and G1,2G_{1,2} must be even functions of xx, while those proportional to F1,2F_{1,2}, F1,4F_{1,4}, and G1,4G_{1,4} must be odd functions of xx. This property is explicitly satisfied by our results. Note that our treatment of the twist-3 contribution to spin independent amplitudes differs from the approach advocated in Ref. Goloskokov and Kroll (2008); Duplančić et al. (2023) which involves a twist-3 pion DA.

Assembling all the pieces, we derive the following spin-averaged and single target longitudinal polarization-dependent cross section:

dσTdtdQ2dxBdϕ=(Nc21)2αem2αs2fπ2ξ3Δ22Nc4(1ξ2)Q10(1+ξ)[1+(1y)2]\displaystyle\!\!\!\!\!\!\!\!\!\!\frac{d\sigma_{T}}{dtdQ^{2}dx_{B}d\phi}\!=\!\frac{(N_{c}^{2}-1)^{2}\alpha_{em}^{2}\alpha_{s}^{2}f_{\pi}^{2}\xi^{3}\Delta_{\perp}^{2}}{2N_{c}^{4}(1-\xi^{2})Q^{10}(1+\xi)}\left[1+\!(1\!-\!y)^{2}\right] (15)
×\displaystyle\times {[|1,1+𝒢1,1|2+|1,4+𝒢1,4|2+2M2Δ2|1,2+𝒢1,2|2]\displaystyle\!\!\!\left\{\left[|{\cal F}_{1,1}+{\cal G}_{1,1}|^{2}+|{\cal F}_{1,4}+{\cal G}_{1,4}|^{2}+\!2\frac{M^{2}}{\Delta_{\perp}^{2}}|{\cal F}_{1,2}+{\cal G}_{1,2}|^{2}\right]\right.\
+cos(2ϕ)a[|1,1+𝒢1,1|2+|1,4+𝒢1,4|2]\displaystyle+\cos(2\phi)a\left[-|{\cal F}_{1,1}+{\cal G}_{1,1}|^{2}+|{\cal F}_{1,4}+{\cal G}_{1,4}|^{2}\right]
+λsin(2ϕ) 2aRe[(i1,4+i𝒢1,4)(1,1+𝒢1,1)]},\displaystyle+\lambda\sin(2\phi)\,2a\,{\rm Re}\left[\left(i{\cal F}_{1,4}+i{\cal G}_{1,4}\right)\left({\cal F}^{*}_{1,1}+{\cal G}^{*}_{1,1}\right)\right]\Big{\}},

where ϕ=ϕlϕΔ\phi=\phi_{l_{\perp}}-\phi_{\Delta_{\perp}} and a=2(1y)1+(1y)2a=\frac{2(1-y)}{1+(1-y)^{2}}. Eq. (15) stands as the central result of our paper. The real part of the quark GTMD F1,4F_{1,4}, and consequently, the quark OAM, leaves a distinct signature through an azimuthal angular correlation of sin2ϕ\sin 2\phi in the longitudinal single target-spin asymmetry.

Refer to caption
Figure 3: The unpolarized cross section, as given by Eq. (15), is displayed in the top plot for EIC kinematics with Q2=10GeV2Q^{2}=10\,\textrm{GeV}^{2} and sep=100GeV\sqrt{s_{ep}}=100\,\textrm{GeV}, as well as for EicC kinematics with Q2=3GeV2Q^{2}=3\,\textrm{GeV}^{2} and sep=16GeV\sqrt{s_{ep}}=16\,\textrm{GeV}. The unpolarized cross section for the EIC case is re-scaled by a factor of 100. The bottom plot shows the average value of sin(2ϕ)\langle\sin(2\phi)\rangle given by Eq. (16). The variable tt is integrated over the range [0.5GeV2-0.5\,\textrm{GeV}^{2}, 4ξ2M21ξ2-\frac{4\xi^{2}M^{2}}{1-\xi^{2}}].The error bands are obtained by varying the value of p2\sqrt{\langle p_{\perp}^{2}\rangle} from 150150 MeV to 250250 MeV and the value of α\alpha^{\prime}, which determines the tt-dependence of the various distributions in the double distribution approach (see supplementary material), from 1.2 to 1.4.

3. Numerical results—We now present the numerical results for both the unpolarized cross section and the sin2ϕ\sin 2\phi asymmetry. It is noteworthy that two of the kk_{\perp}-integrated GTMDs, F1,1F_{1,1} and G1,4G_{1,4}, can be linked to the standard unpolarized GPD and the helicity GPD Meissner et al. (2009). In the forward limit, the GTMD F1,2F_{1,2} is related to the Sivers function f1Tf^{\perp}_{1T} Meissner et al. (2009); Boussarie et al. (2020); Boer et al. (2016); Zhou (2014), and the GTMD G1,2G_{1,2} reduces to the worm-gear function g1Tg_{1T} Meissner et al. (2009). In our first attempt at a numerical study, we choose to neglect contributions from the GTMD G1,1G_{1,1}, which lacks a GPD or TMD counterpart 111After submitting this work, a preprint was released providing information about the small-xx behavior of the GTMD G11G_{11} Bhattacharya et al. (2024a, b). We intend to update our numerical analysis in the future to incorporate the contribution of this GTMD.. Regarding the 1,2{\cal F}_{1,2}, 1,4{\cal F}_{1,4}, and 𝒢1,4{\cal G}_{1,4} terms, we only consider their pole contributions from their imaginary parts. However, for the term 1,1{\cal F}_{1,1}, 𝒢1,2{\cal G}_{1,2}, we include both its imaginary and real parts in the numerical estimation, as they dominate the cross section at high and low tt respectively.

Note that the hard part becomes divergent as zz approaches 0 or 1. This behavior, known as the endpoint singularity, typically signals factorization breaking. From a phenomenological standpoint, regularization is achievable by considering the transverse momentum dependence of the pion DA Goloskokov and Kroll (2005, 2007); Sun et al. (2021). An effective way to introduce transverse momentum dependence is to modify the upper and lower integration limits of zz to p2/Q21p2/Q2𝑑z\int_{\langle p_{\perp}^{2}\rangle/Q^{2}}^{1-\langle p_{\perp}^{2}\rangle/Q^{2}}dz Goloskokov and Kroll (2008), where p2\langle p_{\perp}^{2}\rangle is the mean squared transverse momentum of the quark inside the pion. Its central value is chosen to be p2=0.04GeV2\langle p_{\perp}^{2}\rangle=0.04\ \textrm{GeV}^{2} in our numerical calculation, based on a fit to the CLAS data (see supplementary material for brief discussion, which includes Refs. Bedlinskiy et al. (2014); Hand (1963); Defurne et al. (2016); Mankiewicz et al. (1998); Guichon and Vanderhaeghen (1998); Berthou et al. (2018)). For simplicity, we consider the asymptotic form for the pion’s DA, ϕπ(z)=6z(1z)\phi_{\pi}(z)=6z(1-z). On the other hand, the discontinuity of the derivative of quark GPDs at the endpoints x=±ξx=\pm\xi (as seen in, for example, Refs. Bhattacharya et al. (2019, 2020)), coupled with the double poles at x=±ξx=\pm\xi, may also potentially lead to a divergent component in the cross section. To address this potential issue, we employ a shift of the double pole from 1(xξ+iϵ)2\tfrac{1}{(x-\xi+i\epsilon)^{2}} to 1(xξp2/Q2+iϵ)2\tfrac{1}{(x-\xi-\langle p_{\perp}^{2}\rangle/Q^{2}+i\epsilon)^{2}} (and similarly for the negative xx region). A similar shift was introduced in Ref. Anikin and Teryaev (2003) to handle the aforementioned divergence. More phenomenological input Radyushkin (1999, 2000); Goloskokov and Kroll (2008); Hatta and Yoshida (2012); Goloskokov and Kroll (2009, 2010, 2011); Bhattacharya et al. (2022c); Yang et al. (2024); Echevarria et al. (2014); Bacchetta et al. (2022); Echevarria et al. (2021); Bury et al. (2021) is detailed in the supplemental material.

We now present numerical predictions for the EIC and EicC kinematics. The tt-integrated unpolarized cross section is shown as a function of ξ\xi in the top panel of Fig. 3. The asymmetry, quantified by the average value of sin(2ϕ)\sin(2\phi) and depicted as a function of ξ\xi in the bottom plot of Fig. 3, is defined as:

sin(2ϕ)\displaystyle\langle\sin(2\phi)\rangle =\displaystyle= dΔσd𝒫.𝒮.sin(2ϕ)d𝒫.𝒮.dσd𝒫.𝒮.d𝒫.𝒮.,\displaystyle\frac{\int\frac{d\Delta\sigma}{d{\cal P.S.}}\sin(2\phi)\ d{\cal P.S.}}{\int\frac{d\sigma}{d{\cal P.S.}}d{\cal P.S.}}\,, (16)

where dΔσ=σ(λ=1)σ(λ=1)d\Delta\sigma=\sigma(\lambda=1)-\sigma(\lambda=-1). The unpolarized cross section exhibits a notable magnitude at EicC energy, whereas it is relatively small at EIC energy. Note that at EIC, the cross section for low Q2Q^{2} would be similar to that at EicC. However, EIC’s smaller ξ\xi for the same Q2Q^{2} might offer a greater leverage in constraining quark OAM in the small xx region. Additionally, the asymmetries are substantial for both EIC and EicC kinematics. Consequently, our numerical results signify that the azimuthal asymmetry sin2ϕ\sin 2\phi in exclusive π0\pi^{0} production stands out as a promising avenue for probing the quark OAM distribution.

4. Summary—We propose extracting the quark OAM associated with the Jaffe-Manohar spin sum rule by measuring the azimuthal angular correlation sin2ϕ\sin 2\phi in exclusive π0\pi^{0} production at EIC and EicC. This observable serves as a clean and sensitive probe of quark OAM for several reasons. Firstly, the azimuthal asymmetry is not a power correction, as both the unpolarized and longitudinal polarization-dependent cross sections contribute at twist-3. Secondly, the produced π0\pi^{0} transverse momentum Δ-\Delta_{\perp} remains unaffected by final state QCD radiations. Detecting π0\pi^{0} makes it less challenging to experimentally measure Δ\Delta_{\perp}, in contrast to the diffractive di-jet production case where reconstructing Δ\Delta_{\perp} from the total transverse momentum of the di-jet system is impossible due to the contamination of final-state soft gluon radiations. Most importantly, this process enables the direct access to the quark GTMD F1,4F_{1,4} in the DGLAP region for the first time. In addition to unveiling access to quark OAM, our work highlights another significant finding that the quark component of F1,2F_{1,2} and G1,2G_{1,2}, or equivalently the Sivers function and the worm-gear function respectively, contribute to the unpolarized cross-section of this process. This result is particularly noteworthy since conventionally, the Sivers function and the worm-gear function are understood to be probed only through transversely polarized targets.

We computed the differential cross section within the collinear higher-twist expansion framework. Despite the substantial uncertainties associated with the model inputs, our numerical results reveal a sizable azimuthal asymmetry, which critically relies on the quark OAM distribution. In the kinematic range accessible to the EIC and EicC, our observable can be thoroughly investigated, paving the way for the first experimental extraction of the canonical quark OAM distribution in the future.

Acknowledgements: The authors would like to express their gratitude to C. Cocuzza for providing the LHAPDF tables of JAM22 PDFs as referenced in Cocuzza et al. (2022), and to Ya-ping Xie for sharing code with us from Refs. Goloskokov et al. (2022, 2023); Xie et al. (2024). We also thank Y. Hatta and F. Yuan for their insightful discussions. This work has been supported by the National Natural Science Foundation of China under Grant No. 12175118 and under Contract No. PHY-1516088(J. Z.). S. B. has been supported by the U.S. Department of Energy under Contract No. DE-SC0012704, and also by Laboratory Directed Research and Development (LDRD) funds from Brookhaven Science Associates.

SUPPLEMENTARY MATERIAL

In the supplementary material of our paper, we present more details of numerical estimations and compare our theoretical calculations with the CLAS measurement of the unpolarized exclusive π0\pi^{0} production cross section Bedlinskiy et al. (2014).

.1 Phenomenological inputs for numerical estimations

To provide a model input for the xx-dependent quark OAM distribution, specifically the kk_{\perp}-moment of F1,4F_{1,4}, we employ the Wandzura-Wilczek (WW) approximation Hatta and Yoshida (2012),

Lq(x)xx1dxx(Hq(x)+Eq(x))xx1dxx2H~q(x),\displaystyle L_{q}(x)\approx x\!\int_{x}^{1}\!\frac{dx^{\prime}}{x^{\prime}}\big{(}H_{q}(x^{\prime})+E_{q}(x^{\prime})\big{)}-x\!\int_{x}^{1}\!\frac{dx^{\prime}}{x^{\prime 2}}{\tilde{H}_{q}(x^{\prime})}\,, (1)

where Hq(x)=q(x)H_{q}(x^{\prime})=q(x^{\prime}) and H~q(x)=Δq(x)\tilde{H}_{q}(x^{\prime})=\Delta q(x^{\prime}) denote the usual unpolarized quark PDF and quark helicity PDF, respectively. This approximation, neglecting genuine twist-3 terms, is often employed for convenience in the absence of reliable experimental estimations. We use the JAM (valence) quark PDFs q(x)q(x) and Δq(x)\Delta q(x) as inputs in Eq. (1) from Ref. Cocuzza et al. (2022). As for GPD Eq(x)E_{q}(x^{\prime}), we use the parameterization from Goloskokov and Kroll (2009) and Goloskokov and Kroll (2010). It is a common practice to reconstruct the ξ\xi-dependence for xLq(x,ξ)xL_{q}(x,\xi) from its PDF counterpart xLq(x)xL_{q}(x) using the double distribution method Radyushkin (1999, 2000); Goloskokov and Kroll (2008). Specifically, we use:

Fq(x,ξ)=11𝑑β1+|β|1|β|𝑑αδ(β+ξαx)fq(α,β)\displaystyle F_{q}(x,\xi)=\int_{-1}^{1}d\beta\int_{-1+|\beta|}^{1-|\beta|}d\alpha\delta(\beta+\xi\alpha-x)f_{q}(\alpha,\beta) (2)

where

fq(α,β)\displaystyle f_{q}(\alpha,\beta) =\displaystyle= 34[(1|β|)2α2](1|β|)3βLq(β).\displaystyle\frac{3}{4}\frac{\left[(1-|\beta|)^{2}-\alpha^{2}\right]}{(1-|\beta|)^{3}}\beta L_{q}(\beta)\,. (3)

As for its tt-dependence, we adopt a Gaussian form factor, represented by et/Λe^{t/\Lambda} with Λ=0.5GeV2\Lambda=0.5\,\text{GeV}^{2}.

The Compton form factors in Eqs. (10)-(11) involve the kk_{\perp} moments of GTMDs F1,2F_{1,2} and G1,2G_{1,2}. These moments can be determined in the forward limit by connecting them to the corresponding TMDs. Specifically, in this limit, GTMD F1,2F_{1,2} is related to the Sivers function f1Tf^{\perp}_{1T}, while GTMD G1,2G_{1,2} reduces to the worm-gear function g1Tg_{1T}. For numerical estimations, we rely on the parametrizations for the quark Sivers function from Ref. Echevarria et al. (2014) and the parametrizations for the worm-gear function from Ref. Bhattacharya et al. (2022c). For G1,2G_{1,2} contribution, we utilize the relation,

d2kk22M2Re[G1,2(x,ξ=0,Δ=0,k)]\displaystyle\int d^{2}k_{\perp}\frac{k_{\perp}^{2}}{2M^{2}}{\text{R}e}[G_{1,2}(x,\xi=0,\Delta_{\perp}=0,k_{\perp})]
=d2kk22M2g1T(x,k)=g1T(1)(x)\displaystyle=\int d^{2}k_{\perp}\frac{k_{\perp}^{2}}{2M^{2}}g_{1T}(x,k_{\perp})=g_{1T}^{(1)}(x) (4)

where g1T(1)(x)g_{1T}^{(1)}(x) is parameterized as Bhattacharya et al. (2022c),

g1T(1)(x)=NqN~qxαq(1x)βqq(x)\displaystyle g_{1T}^{(1)}(x)=\frac{N_{q}}{\tilde{N}_{q}}x^{\alpha_{q}}(1-x)^{\beta_{q}}q(x) (5)

with αu=αd=1.9\alpha_{u}=\alpha_{d}=1.9, βu=βd=1\beta_{u}=\beta_{d}=1 and Nu=0.033,Nd=0.002N_{u}=0.033,N_{d}=-0.002. N~\tilde{N} is determined through N~=01𝑑xxα+1(1x)βq(x)\tilde{N}=\int_{0}^{1}dxx^{\alpha+1}(1-x)^{\beta}q(x). We notice that other fitting for g1T(1)(x)g_{1T}^{(1)}(x) exist too Yang et al. (2024). Meanwhile, the kk_{\perp} moment of F1,2F_{1,2} can be related to the Qiu-Sterman function,

d2kk2MIm[F1,2(x,ξ=0,Δ=0,k)]\displaystyle\int d^{2}k_{\perp}\frac{k_{\perp}^{2}}{M}{\text{I}m}[F_{1,2}(x,\xi=0,\Delta_{\perp}=0,k_{\perp})]
=d2kk2Mf1T(x,k)=TF(x,x)\displaystyle=-\int d^{2}k_{\perp}\frac{k_{\perp}^{2}}{M}f_{1T}^{\perp}(x,k_{\perp})=T_{F}(x,x) (6)

where the Qiu-Sterman function is parametrized as Echevarria et al. (2014),

TF(x,x)=Nq(αq+βq)(αq+βq)αqαqβqβqxαq(1x)βqq(x)\displaystyle T_{F}(x,x)=N_{q}\frac{(\alpha_{q}+\beta_{q})^{(\alpha_{q}+\beta_{q})}}{\alpha_{q}^{\alpha_{q}}\beta_{q}^{\beta_{q}}}x^{\alpha_{q}}(1-x)^{\beta_{q}}q(x) (7)

with αu=1.051,αd=1.552\alpha_{u}=1.051,\alpha_{d}=1.552, βu=βd=4.857\beta_{u}=\beta_{d}=4.857, and Nu=1.06,Nd=0.163N_{u}=1.06,N_{d}=-0.163. See also Refs. Bacchetta et al. (2022), Echevarria et al. (2021), and Bury et al. (2021) for the state-of-the-art extractions of the Sivers functions. Once the xx-dependence of the kk_{\perp} moments of F1,2F_{1,2} and G1,2G_{1,2} is reconstructed as explained above, we reconstruct their (ξ,t)(\xi,t)-dependence in accordance with the double distribution method. Specifically, we use the parameterization:

Fq(x,ξ,t)=11𝑑β1+|β|1|β|𝑑αδ(β+ξαx)fq(β,α,t)\displaystyle F_{q}(x,\xi,t)=\int_{-1}^{1}d\beta\int_{-1+|\beta|}^{1-|\beta|}d\alpha\delta(\beta+\xi\alpha-x)f_{q}(\beta,\alpha,t) (8)

where

fq(α,β,t)\displaystyle f_{q}(\alpha,\beta,t) =\displaystyle= 34|β|αt[(1|β|)2α2](1|β|)3{g1T(1)(|β|)TF(|β|,|β|)\displaystyle\frac{3}{4}|\beta|^{-\alpha^{\prime}t}\frac{\left[(1-|\beta|)^{2}-\alpha^{2}\right]}{(1-|\beta|)^{3}}\begin{cases}g^{(1)}_{1T}(|\beta|)\\ T_{F}(|\beta|,|\beta|)\end{cases}

with α=1.3\alpha^{\prime}=1.3.

Finally, we reconstruct the ξ\xi and tt-dependence of the GPDs H(x,ξ,t)H(x,\xi,t) and H~(x,ξ,t)\tilde{H}(x,\xi,t) entering Eqs. (8) and (13) from their PDF counterparts using the double distribution method.

.2 Comparison with the CLAS data

We now compare our theoretical calculations with the CLAS measurement of the unpolarized exclusive π0\pi^{0} production cross section Bedlinskiy et al. (2014). The CLAS measurements primarily cover the large skewness region, making it challenging to extrapolate the functional form of GPDs to the forward limit. Nevertheless, the comparison with CLAS data serves as a valuable test for our theoretical calculations and allows us to fine-tune relevant parameters.

The CLAS data is provided for the reduced or “virtual photon” cross sections. By adopting the hand convention Hand (1963) for the virtual photon flux definition, the reduced cross section dσT/dtd\sigma_{T}/dt can be extracted directly from Eq. (15) in the main text,

dσTdt=2π2αs2αemfπ2(Nc21)2ξ3Δ2xBNc4(1ξ2)(1+ξ)(1xB)Q8\displaystyle\!\!\!\!\!\!\!\!\!\!\!\frac{d\sigma_{T}}{dt}=\frac{2\pi^{2}\alpha_{s}^{2}\alpha_{em}f_{\pi}^{2}(N_{c}^{2}-1)^{2}\xi^{3}\Delta_{\perp}^{2}x_{B}}{N_{c}^{4}(1-\xi^{2})(1+\xi)(1-x_{B})Q^{8}} (9)
×[|1,1+𝒢1,1|2+|1,4+𝒢1,4|2+2M2Δ2|1,2+𝒢1,2|2].\displaystyle\!\!\!\!\!\!\!\!\!\times\!\left[|{\cal F}_{1,1}+{\cal G}_{1,1}|^{2}+|{\cal F}_{1,4}+{\cal G}_{1,4}|^{2}+\!2\frac{M^{2}}{\Delta_{\perp}^{2}}|{\cal F}_{1,2}+{\cal G}_{1,2}|^{2}\right]\,.

After accounting for the mass correction which is necessary at the CLAS energy, the above formula is modified as,

dσTdt=2π2αs2αemfπ2(Nc21)2ξ3Δ2Nc4(1ξ2)(1ξ)Q6Λ(W2,Q2,m2)\displaystyle\!\!\!\!\!\!\!\!\!\!\!\frac{d\sigma_{T}}{dt}=\frac{2\pi^{2}\alpha_{s}^{2}\alpha_{em}f_{\pi}^{2}(N_{c}^{2}-1)^{2}\xi^{3}\Delta_{\perp}^{2}}{N_{c}^{4}(1-\xi^{2})(1-\xi)Q^{6}\sqrt{\Lambda(W^{2},-Q^{2},m^{2})}} (10)
×[|1,1+𝒢1,1|2+|1,4+𝒢1,4|2+2M2Δ2|1,2+𝒢1,2|2]\displaystyle\!\!\!\!\!\!\!\!\!\times\!\left[|{\cal F}_{1,1}+{\cal G}_{1,1}|^{2}+|{\cal F}_{1,4}+{\cal G}_{1,4}|^{2}+\!2\frac{M^{2}}{\Delta_{\perp}^{2}}|{\cal F}_{1,2}+{\cal G}_{1,2}|^{2}\right]

where Λ(x,y,z)\Lambda(x,y,z) is defined as Λ(x,y,z)=(x2+y2+z2)2xy2xz2yz\Lambda(x,y,z)=(x^{2}+y^{2}+z^{2})-2xy-2xz-2yz and W2W^{2} is the center of mass energy of γp\gamma^{*}p system. The reduced cross sections measured by the CLAS Collaboration is given by the combination dσTdt+adσLdt\frac{d\sigma_{T}}{dt}+a\frac{d\sigma_{L}}{dt} where σL\sigma_{L} denotes the cross section of the π0\pi^{0} production initiated by the longitudinally polarized virtual photons. The parameter aa, defined in the main text, represents the ratio of fluxes between longitudinally and transversely polarized virtual photons. It is important to emphasize that the contributions from σL\sigma_{L} and σT\sigma_{T} can be disentangled by examining the combined cross section at different values of yy Defurne et al. (2016).

Refer to caption
Figure 1: The unpolarized cross section is plotted as the function of |t||t^{\prime}| with Q2=2.21GeV2Q^{2}=2.21\text{GeV}^{2} and xB=0.28x_{B}=0.28. The CLAS data are displayed for comparison.

In contrast to exclusive meson production initiated by transversely polarized virtual photons, the factorization for longitudinally polarized photon production has been rigorously proven  Collins et al. (1997). While σT\sigma_{T} receives the leading contribution from the twist-3 level, the calculation of σL\sigma_{L} is well formulated within the leading twist collinear factorization framework Mankiewicz et al. (1998); Guichon and Vanderhaeghen (1998). However, due to the cancellation between contributions from uu-quark and dd-quark GPDs Goloskokov and Kroll (2010, 2011), the yield of the longitudinal photon production of π0\pi^{0} was shown to be negligibly small Goloskokov et al. (2022, 2023); Xie et al. (2024).

To evaluate σL\sigma_{L}, one can utilize existing code such as the PARtonic Tomography Of Nucleon Software (PARTONS) Berthou et al. (2018), or the package developed in the Refs. Goloskokov et al. (2022, 2023); Xie et al. (2024), which we employ in this study. We plot dσTdt\frac{d\sigma_{T}}{dt} and dσLdt\frac{d\sigma_{L}}{dt} as functions of |t|=|t+4ξ2M21ξ2||t^{\prime}|=|t+\frac{4\xi^{2}M^{2}}{1-\xi^{2}}| in Fig. 1 separately. Additionally, we compare the combined cross section dσTdt+adσLdt\frac{d\sigma_{T}}{dt}+a\frac{d\sigma_{L}}{dt} with the CLAS measurement, as shown in Fig. 1. Our calculation is in reasonable agreement with the experimental data. We observe that G1,2G_{1,2} term dominates the cross section as Δ\Delta_{\perp} approaches zero. In an extended version of this paper, we will conduct a more thorough analysis of the JLab measurements Bedlinskiy et al. (2014); Dlamini et al. (2021) by taking into account another type twist-3 contributions as well Goloskokov and Kroll (2008); Duplančić et al. (2023).

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