# Exploring orbital angular momentum and spin-orbit correlation for gluons
at the Electron-Ion Collider

Shohini Bhattacharya<EMAIL_ADDRESS>Theoretical Division, Los Alamos
National Laboratory, Los Alamos, New Mexico 87545, USA  RIKEN BNL Research
Center, Brookhaven National Laboratory, Upton, NY 11973, USA  Renaud Boussarie
<EMAIL_ADDRESS>CPHT, CNRS, Ecole Polytechnique, Institut
Polytechnique de Paris, 91128 Palaiseau, France  Yoshitaka Hatta
<EMAIL_ADDRESS>Physics Department, Brookhaven National Laboratory, Upton, NY
11973, USA  RIKEN BNL Research Center, Brookhaven National Laboratory, Upton,
NY 11973, USA

###### Abstract

In our previous work Bhattacharya et al. (2022a), we introduced a pioneering
observable aimed at experimentally detecting the orbital angular momentum
(OAM) of gluons. Our focus was on the longitudinal double spin asymmetry
observed in exclusive dijet production during electron-proton scattering. We
demonstrated the sensitivity of the $\cos\phi$ angular correlation between the
scattered electron and proton as a probe for gluon OAM at small-$x$ and its
intricate interplay with gluon helicity. This current work provides a
comprehensive exposition, diving further into the aforementioned calculation
with added elaboration and in-depth analysis. We reveal that, in addition to
the gluon OAM, one also gains access to the spin-orbit correlation of gluons.
We supplement our work with a detailed numerical analysis of our observables
for the kinematics of the Electron-Ion Collider. In addition to dijet
production, we also consider the recently proposed semi-inclusive diffractive
deep inelastic scattering process which potentially offers experimental
advantages over dijet measurements. Finally, we investigate quark-channel
contributions to these processes and find an unexpected breakdown of collinear
factorization.

††preprint: LA-UR-24-22343

##  I Introduction

The investigation of nucleon spin structure, prompted by the discovery of the
“spin crisis”, has evolved into a captivating research field over the past
three decades. A primary objective of this field is to comprehend the spin
decomposition of the nucleon. As per the Jaffe-Manohar decomposition Jaffe and
Manohar (1990), the proton spin originates from four distinct sources

$\displaystyle\frac{1}{2}=\frac{1}{2}\Delta\Sigma+\Delta G+L_{q}+L_{g}\,.$ (1)

The contribution of quark spin, $\Delta\Sigma\sim 0.3$, obtained from the
integration of the quark helicity distribution, is relatively well
constrained. However, the gluon spin contribution, $\Delta G$, is anticipated
to be significant Adamczyk et al. (2015); de Florian et al. (2014); Nocera et
al. (2014); Abdallah et al. (2022), although the uncertainties surrounding the
gluon helicity distribution at small $x$ are substantial. The forthcoming
Electron-Ion Collider (EIC) is expected to provide precise experimental data
that will allow for a more accurate determination of the gluon spin
contribution Abdul Khalek et al. (2022).

In addition to the ongoing exploration of the gluon helicity distribution at
small $x$, there is currently significant attention devoted to investigating
the orbital angular momentum (OAM) contributions, $L_{q/g}$, from quarks and
gluons. This pursuit holds great promise for advancing our understanding of
the partonic structure of the nucleon and shedding light on the underlying
dynamics of QCD. To date, our knowledge regarding the OAM of partons remains
limited despite much progress in formal theory Lorce and Pasquini (2011);
Hatta (2012); Lorce et al. (2012); Hatta and Yoshida (2012); Ji et al. (2013);
Rajan et al. (2016); Engelhardt (2017); Boussarie et al. (2019); Guo et al.
(2021); Kovchegov (2019); Engelhardt et al. (2020); Kovchegov and Manley
(2024); Manley (2024). The extraction of Jaffe-Manohar-type (or canonical)
parton OAM from high-energy scattering experiments remains highly challenging,
both from theoretical and experimental perspectives. Previous approaches for
probing parton OAM have relied on its connection with the Wigner distribution,
or equivalently, the generalized transverse momentum distribution function
(GTMD) Lorce and Pasquini (2011); Hatta (2012); Lorce et al. (2012),

$\displaystyle xL_{g}(x,\xi)=-\int
d^{2}k_{\perp}\frac{k_{\perp}^{2}}{M^{2}}F_{1,4}^{g}(x,k_{\perp},\xi,\Delta_{\perp}=0)\,,$
(2)

where the specific kinematical variables will be provided below. (For quarks,
the above equation remains the same, except that there is no factor of $x$ on
the left-hand side.) The parton OAM can be obtained by integrating the
$x$-dependent OAM distribution: $L_{g,q}=\int_{0}^{1}dxL_{g,q}(x,\xi=0)$.
Significant theoretical advancements have been made in investigating the
experimental signatures of the GTMD $F_{1,4}$ in recent years. In Refs. Ji et
al. (2017); Hatta et al. (2017); Bhattacharya et al. (2022a), it was
demonstrated that the single and double longitudinal spin asymmetry observed
in exclusive dijet production during $ep$ collisions provides access to the
“Compton form factor” (or $x$-moments) involving $F_{1,4}$. Furthermore, the
direct measurement of $F_{1,4}$, along with several other GTMDs, can be
accomplished through the exclusive double Drell-Yan or quarkonium production
Bhattacharya et al. (2017); Bhattacharya et al. (2022b); Boussarie et al.
(2018). An earlier notable endeavor aimed at identifying an observable for
probing OAM can be found in Ref. Courtoy et al. (2014); Rajan et al. (2016). A
recent study proposed a method to extract the imaginary component of the gluon
GTMD $F_{1,4}$ by measuring the azimuthal angular correlation of $\sin 2\phi$
in exclusive $\pi^{0}$ production during polarized $ep$ collisions
Bhattacharya et al. (2023a). This approach exploits the interference effect
between the QCD interaction and the Primakoff process, resulting in a single-
target longitudinal spin asymmetry. Additionally, in Ref. Bhattacharya et al.
(2023b), the same $\sin 2\phi$ asymmetry, which mirrors the aforementioned but
does not arise from the interference effect described there, provides access
to the quark GTMD $F_{1,4}$. Hence, there has been a consistent advancement in
identifying observables that exhibit sensitivity to OAM.

In our earlier work Bhattacharya et al. (2022a), we introduced a novel
observable to experimentally detect the OAM of gluons, which, as discussed
above, plays a pivotal role in the proton spin sum rule. We focused on the
longitudinal double spin asymmetry observed in exclusive dijet production
during $ep$ scattering, demonstrating that the $\cos\phi$ angular correlation
between the scattered electron and proton serves as a probe of gluon OAM at
small-$x$, as well as its intricate interplay with gluon helicity. In this
study, we provide a comprehensive exposition, offering further elaboration and
in-depth analysis of the aforementioned calculation. We reveal that, in
addition to gluon OAM and its helicity, one also gains access to the spin-
orbit correlation of gluons. We then provide a detailed numerical analysis of
our observable in the dijet process. Due to practical complications associated
with dijet reconstruction, we also investigate the possibility to reinterpret
our observable in terms of semi-inclusive diffractive deep inelastic
scattering (‘SIDDIS’) Hatta et al. (2022) and perform a numerical analysis.
Furthermore, we undertake the calculation of quark-channel contributions to
these processes. Contrary to our initial expectations, we discover an
unexpected breakdown of factorization compared to the case of the gluon
channel.

The paper is structured as follows: In Sec. II, we establish the definitions
of GTMDs for both gluons and quarks, elucidating their connections with OAM.
Additionally, we briefly explain the subtlety associated with the frame-
dependence of GTMDs. The primary findings regarding exclusive dijet production
are detailed in Sec. III, where we discuss the calculation of the observable
double spin asymmetry. In Sec. IV, we explore the quark channel contribution
to the dijet process, addressing factors contributing to the breakdown of
factorization. Sec. V explores the semi-inclusive diffractive deep inelastic
scattering (SIDDIS) process, similar to dijets but with a focus on tagging
inclusive hadron species within a specific rapidity interval. The final
numerical results for both dijets and SIDDIS under realistic EIC kinematics
are shown in Sec. VI. Conclusions are presented in Sec. VII.

##  II GTMDs and orbital angular momentum

###  II.1 Definitions

In this section we recapitulate the basic definition of GTMDs and their
connection to the parton orbital angular momentum (OAM). Following Meissner et
al. (2009), we parameterize the leading-twist quark and gluon GTMDs as111Our
conventions are $\gamma_{5}=-i\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3}$ and
$\epsilon^{0ij3}=\epsilon^{-+ij}=\epsilon^{ij}=\epsilon_{ij}$ ($i,j=1,2$) such
that ${\rm
Tr}[\gamma_{5}\gamma^{\mu}\gamma^{\nu}\gamma^{\rho}\gamma^{\lambda}]=4i\epsilon^{\mu\nu\rho\lambda}$
and the spin four-vector is $2s^{\mu}=\bar{u}(ps)\gamma_{5}\gamma^{\mu}u(ps)$.

$\displaystyle f_{q}(x,\xi,\widetilde{k}_{\perp},\widetilde{\Delta}_{\perp})$
$\displaystyle=$
$\displaystyle\int\frac{d^{3}z}{2(2\pi)^{3}}e^{ixP^{+}z^{-}-i\widetilde{k}_{\perp}\cdot\widetilde{z}_{\perp}}\langle
p^{\prime}|\bar{\psi}(-z/2)\gamma^{+}\psi(z/2)|p\rangle$ (3) $\displaystyle=$
$\displaystyle\frac{1}{2M}\bar{u}(p^{\prime})\left[F_{1,1}^{q}+i\frac{\sigma^{j+}}{P^{+}}(\widetilde{k}_{\perp}^{j}F_{1,2}^{q}+\widetilde{\Delta}_{\perp}^{j}F^{q}_{1,3})+i\frac{\sigma^{ij}\widetilde{k}_{\perp}^{i}\widetilde{\Delta}_{\perp}^{j}}{M^{2}}F^{q}_{1,4}\right]u(p),$
$\displaystyle\tilde{f}_{q}(x,\xi,\widetilde{k}_{\perp},\widetilde{\Delta}_{\perp})$
$\displaystyle=$
$\displaystyle\int\frac{d^{3}z}{2(2\pi)^{3}}e^{ixP^{+}z^{-}-i\widetilde{k}_{\perp}\cdot\widetilde{z}_{\perp}}\langle
p^{\prime}|\bar{\psi}(-z/2)\gamma_{5}\gamma^{+}(z/2)|p\rangle$ (4)
$\displaystyle=$
$\displaystyle\frac{-i}{2M}\bar{u}(p^{\prime})\left[\frac{\epsilon_{ij}\widetilde{k}_{\perp}^{i}\widetilde{\Delta}_{\perp}^{j}}{M^{2}}G^{q}_{1,1}-\frac{\sigma^{i+}\gamma_{5}}{P^{+}}(\widetilde{k}_{\perp}^{i}G^{q}_{1,2}+\widetilde{\Delta}_{\perp}^{i}G^{q}_{1,3})-\sigma^{+-}\gamma_{5}G^{q}_{1,4}\right]u(p),$
$\displaystyle xf_{g}(x,\xi,\widetilde{k}_{\perp},\widetilde{\Delta}_{\perp})$
$\displaystyle=$
$\displaystyle\delta_{ij}\int\frac{d^{3}z}{(2\pi)^{3}P^{+}}e^{ixP^{+}z^{-}-i\widetilde{k}_{\perp}\cdot\widetilde{z}_{\perp}}\langle
p^{\prime}|F_{a}^{+i}(-z/2)F_{a}^{+j}(z/2)|p\rangle$ (5) $\displaystyle=$
$\displaystyle\frac{1}{2M}\bar{u}(p^{\prime})\left[F^{g}_{1,1}+i\frac{\sigma^{j+}}{P^{+}}(\widetilde{k}_{\perp}^{j}F^{g}_{1,2}+\widetilde{\Delta}_{\perp}^{j}F^{g}_{1,3})+i\frac{\sigma^{ij}\widetilde{k}_{\perp}^{i}\widetilde{\Delta}_{\perp}^{j}}{M^{2}}F^{g}_{1,4}\right]u(p),$
$\displaystyle
x\tilde{f}_{g}(x,\xi,\widetilde{k}_{\perp},\widetilde{\Delta}_{\perp})$
$\displaystyle=$
$\displaystyle-i\epsilon_{ij}\int\frac{d^{3}z}{(2\pi)^{3}P^{+}}e^{ixP^{+}z^{-}-i\widetilde{k}_{\perp}\cdot\widetilde{z}_{\perp}}\langle
p^{\prime}|F_{a}^{+i}(-z/2)F_{a}^{+j}(z/2)|p\rangle$ (6) $\displaystyle=$
$\displaystyle\frac{-i}{2M}\bar{u}(p^{\prime})\left[\frac{\epsilon_{ij}\widetilde{k}_{\perp}^{i}\widetilde{\Delta}_{\perp}^{j}}{M^{2}}G^{g}_{1,1}-\frac{\sigma^{i+}\gamma_{5}}{P^{+}}(\widetilde{k}_{\perp}^{i}G^{g}_{1,2}+\widetilde{\Delta}_{\perp}^{i}G^{g}_{1,3})-\sigma^{+-}\gamma_{5}G^{g}_{1,4}\right]u(p),$

where $P^{\mu}=\frac{p+p^{\prime}}{2}$, $\Delta^{\mu}=p^{\prime\mu}-p^{\mu}$
and $\xi=(p^{+}-p^{\prime+})/2P^{+}$. $i,j=1,2$ are two-dimensional vector
indices. For the quark GTMDs, a staple-shaped gauge link $W^{\pm}$ which
connects the two points $-z/2$ and $z/2$ via the light-cone infinity
$z^{-}=\pm\infty$ is understood. For the gluon GTMDs, there are two
inequivalent ways to insert Wilson lines

$\displaystyle F_{a}(-z/2)F_{a}(z/2)\to\begin{cases}2{\rm
tr}[F(-z/2)W^{\pm}F(z/2)(W^{\pm})^{\dagger}],\\\ 2{\rm
tr}[F(-z/2)W^{\pm}F(z/2)(W^{\mp})^{\dagger}],\end{cases}$ (7)

where the matrices are in the fundamental representation. The first choice is
called the Weiszäcker-Williams gluon distribution and the second is the dipole
gluon distribution. Which gluon distribution one should adopt depends on the
process under consideration Bomhof et al. (2006); Dominguez et al. (2011). For
exclusive dijet production, we select the dipole one Altinoluk et al. (2016);
Hatta et al. (2016); Boussarie et al. (2016).

All the GTMDs are a function of
$x,\xi,\widetilde{k}_{\perp}^{2},\widetilde{\Delta}_{\perp}^{2}$ and
$\widetilde{k}_{\perp}\cdot\widetilde{\Delta}_{\perp}$. (Throughout this paper
we denote $A_{\perp}^{2}=A_{\perp}^{i}A_{\perp}^{i}$, $A_{\perp}\cdot
B_{\perp}=A_{\perp}^{i}B_{\perp}^{i}$, and $A_{\perp}\times
B_{\perp}=\epsilon^{ij}A_{\perp}^{i}B_{\perp}^{j}$ for products of two-
dimensional vectors.) The twist-two quark and gluon GPDs are obtained by
integrating over $\widetilde{k}_{\perp}$.

$\displaystyle\int d^{2}\widetilde{k}_{\perp}f_{q}$ $\displaystyle=$
$\displaystyle\frac{1}{2P^{+}}\bar{u}(p^{\prime})\left(H_{q}\gamma^{+}+E_{q}\frac{i\sigma^{+\nu}\Delta_{\nu}}{2M}\right)u(p),$
(8) $\displaystyle\int d^{2}\widetilde{k}_{\perp}\tilde{f}_{q}$
$\displaystyle=$
$\displaystyle\frac{1}{2P^{+}}\bar{u}(p^{\prime})\left(\tilde{H}_{q}\gamma_{5}\gamma^{+}-\tilde{E}_{q}\frac{\gamma_{5}\Delta^{+}}{2M}\right)u(p),$
(9) $\displaystyle\int d^{2}\widetilde{k}_{\perp}xf_{g}$ $\displaystyle=$
$\displaystyle\frac{1}{2P^{+}}\bar{u}(p^{\prime})\left(H_{g}\gamma^{+}+E_{g}\frac{i\sigma^{+\nu}\Delta_{\nu}}{2M}\right)u(p),$
(10) $\displaystyle\int d^{2}\widetilde{k}_{\perp}x\tilde{f}_{g}$
$\displaystyle=$
$\displaystyle\frac{1}{2P^{+}}\bar{u}(p^{\prime})\left(\tilde{H}_{g}\gamma_{5}\gamma^{+}-\tilde{E}_{g}\frac{\gamma_{5}\Delta^{+}}{2M}\right)u(p).$
(11)

In the forward limit, they reduce to the unpolarized and polarized quark and
gluon PDFs as $H_{q}(x)=q(x)$, $\tilde{H}_{q}(x)=\Delta q(x)$,
$H_{g}(x)=xG(x)$ and $\tilde{H}_{g}(x)=x\Delta G(x)$. Let us define

$\displaystyle L_{q}(x,\xi)\equiv-\int
d^{2}\widetilde{k}_{\perp}\frac{\widetilde{k}_{\perp}^{2}}{M^{2}}F^{q}_{1,4}(x,\xi,\widetilde{\Delta}_{\perp}=0),\qquad
xL_{g}(x,\xi)\equiv-\int
d^{2}\widetilde{k}_{\perp}\frac{\widetilde{k}_{\perp}^{2}}{M^{2}}F^{g}_{1,4}(x,\xi,\widetilde{\Delta}_{\perp}=0).$
(12)

In the limit $\xi\to 0$, these become the quark and gluon OAM parton
distribution functions Hatta and Yoshida (2012)

$\displaystyle L_{q}(x)=-\int
d^{2}\widetilde{k}_{\perp}\frac{\widetilde{k}_{\perp}^{2}}{M^{2}}F^{q}_{1,4}(x,\xi=0,\widetilde{\Delta}_{\perp}=0),\qquad
xL_{g}(x)=-\int
d^{2}\widetilde{k}_{\perp}\frac{\widetilde{k}_{\perp}^{2}}{M^{2}}F^{g}_{1,4}(x,\xi=0,\widetilde{\Delta}_{\perp}=0),$
(13)

normalized as222Our normalization of $L_{g}(x)$ differs from the one in Hatta
and Yoshida (2012) by a factor of 2.

$\displaystyle\int_{-1}^{1}dxL_{q}(x)=\int_{0}^{1}dx(L_{q}(x)+L_{\bar{q}}(x))=L_{q},\qquad\int_{0}^{1}dxL_{g}(x)=L_{g}.$
(14)

Due to $PT$-symmetry, $L_{g,q}(x)$ does not depend on the direction (future-
pointing or past-pointing) of the gauge link $W^{\pm}$ Hatta (2012). Moreover,
in the gluon case the two choices (7) result in the same function $L_{g}(x)$
Hatta et al. (2017). $L_{q,g}(x)$ can be decomposed into the Wandzura-Wilczek
part related to the twist-two PDFs and the genuine twist-three part (matrix
elements of $qqg$ and $ggg$ correlators)

$\displaystyle L_{q}(x)$ $\displaystyle=$ $\displaystyle
x\int_{x}^{\epsilon(x)}\frac{dx^{\prime}}{x^{\prime}}(H_{q}(x^{\prime})+E_{q}(x^{\prime}))-x\int_{x}^{\epsilon(x)}\frac{dx^{\prime}}{x^{\prime
2}}\tilde{H}_{q}(x^{\prime})+\cdots,\qquad(-1<x<1)$ (15) $\displaystyle
L_{g}(x)$ $\displaystyle=$ $\displaystyle
x\int_{x}^{1}\frac{dx^{\prime}}{x^{\prime
2}}(H_{g}(x^{\prime})+E_{g}(x^{\prime}))-2x\int_{x}^{1}\frac{dx^{\prime}}{x^{\prime
2}}\Delta G(x^{\prime})+\cdots,\qquad(0<x<1)$ (16)

where $\epsilon(x)$ is the sign function. The genuine twist-three parts are
omitted for simplicity. Their complete expressions can be found in Hatta and
Yoshida (2012); Hatta and Yao (2019). At small-$x$, the parton distributions
have the power-law behavior

$\displaystyle H_{g}(x)\sim E_{g}(x)\sim\frac{1}{x^{c}},\qquad\Delta
G(x)\sim\frac{1}{x^{d}}.$ (17)

[The result $H_{g}(x)\propto E_{g}(x)$ at small-$x$ has been recently
established in Hatta and Zhou (2022).] In perturbation theory,
$c\propto\alpha_{s}$ Kuraev et al. (1977); Balitsky and Lipatov (1978) and
$d\propto\sqrt{\alpha_{s}}$ Bartels et al. (1996a); Borden and Kovchegov
(2023). We thus expect that $c<d$ and the right hand side of (16) is dominated
by the $\Delta G$ term Boussarie et al. (2019); Kovchegov and Manley (2024);
Manley (2024)

$\displaystyle L_{g}(x)\approx-\frac{2}{1+d}\Delta G(x).$ (18)

This relation will play an important role in our analysis below. It means that
a significant cancellation occurs between the OAM and helicity distributions
at small-$x$ Hatta et al. (2017); More et al. (2018); Hatta and Yang (2018);
Boussarie et al. (2019). Recently, it has been shown that this feature is
robust even after including the genuine twist-three corrections Kovchegov and
Manley (2024); Manley (2024) (see also Hatta and Yao (2019); Kovchegov
(2019)).

###  II.2 Transverse boost

It is often taken for granted that GTMDs are defined in the ‘symmetric’ frame
where $P_{\perp}=0$ so that
$\widetilde{p}_{\perp}^{\prime}=\widetilde{\Delta}_{\perp}/2=-\widetilde{p}_{\perp}$.
The advantage of this frame is that one can exploit hermiticity and $PT$
(parity & time-reversal) symmetry to constrain the dependence of GTMDs on
various variables Meissner et al. (2009)

$\displaystyle\mathfrak{Re}\,F_{1,n}(x,\xi,\widetilde{k}_{\perp}\cdot\widetilde{\Delta}_{\perp},\widetilde{k}_{\perp}^{2},\widetilde{\Delta}_{\perp}^{2})=\mathfrak{Re}\,F_{1,n}(x,-\xi,-\widetilde{k}_{\perp}\cdot\widetilde{\Delta}_{\perp},\widetilde{k}_{\perp}^{2},\widetilde{\Delta}_{\perp}^{2}),$
(19)
$\displaystyle\mathfrak{Im}\,F_{1,n}(x,\xi,\widetilde{k}_{\perp}\cdot\widetilde{\Delta}_{\perp},\widetilde{k}_{\perp}^{2},\widetilde{\Delta}_{\perp}^{2})=-\mathfrak{Im}\,F_{1,n}(x,-\xi,-\widetilde{k}_{\perp}\cdot\widetilde{\Delta}_{\perp},\widetilde{k}_{\perp}^{2},\widetilde{\Delta}_{\perp}^{2}),$
(20)

for $n=1,3,4$ and

$\displaystyle\mathfrak{Re}\,F_{1,2}(x,\xi,\widetilde{k}_{\perp}\cdot\widetilde{\Delta}_{\perp},\widetilde{k}_{\perp}^{2},\widetilde{\Delta}_{\perp}^{2})=-\mathfrak{Re}\,F_{1,2}(x,-\xi,-\widetilde{k}_{\perp}\cdot\widetilde{\Delta}_{\perp},\widetilde{k}_{\perp}^{2},\widetilde{\Delta}_{\perp}^{2}),$
(21)
$\displaystyle\mathfrak{Im}\,F_{1,2}(x,\xi,\widetilde{k}_{\perp}\cdot\widetilde{\Delta}_{\perp},\widetilde{k}_{\perp}^{2},\widetilde{\Delta}_{\perp}^{2})=\mathfrak{Im}\,F_{1,2}(x,-\xi,-\widetilde{k}_{\perp}\cdot\widetilde{\Delta}_{\perp},\widetilde{k}_{\perp}^{2},\widetilde{\Delta}_{\perp}^{2}).$
(22)

Throughout this paper, we are interested in the regime where $\xi$ and
$\Delta_{\perp}$ are small. Then the above relations already imply that
$F_{1,n=1,3,4}$ are dominated by the real part and $F_{1,2}$ is dominated by
the imaginary part. In the forward limit, ${\mathfrak{I}m}F^{q/g}_{1,2}$ are
called the quark/gluon Sivers function. However, this frame is neither
convenient nor practically used when describing actual experimental processes.
We shall instead work in the so-called hadron frame, where the incoming
virtual photon and the proton move along the $-x^{3}$ and $+x^{3}$ directions,
respectively, with $p_{\perp}=0$. The two frames are related by the so-called
transverse boost, a Lorentz transformation that leaves invariant the plus
component of a four-vector $V^{\mu}=(V^{+},V^{-},V_{\perp})$

$\displaystyle V^{+}=\widetilde{V}^{+},\qquad
V_{\perp}=\widetilde{V}_{\perp}+C_{\perp}\widetilde{V}^{+},\qquad
V^{-}=\widetilde{V}^{-}+C_{\perp}\cdot\widetilde{V}_{\perp}+\frac{C_{\perp}^{2}}{2}\widetilde{V}^{+}$
(23)

where $C_{\perp}$ is an arbitrary two-dimensional vector. We start with the
definition of GTMDs in the symmetric frame (3)-(6) and apply the following
transverse boost with $C_{\perp}=\frac{\widetilde{\Delta}_{\perp}}{2p^{+}}$

$\displaystyle p^{+},p^{\prime+},$ $\displaystyle\to$ $\displaystyle
p^{+},p^{\prime+},$ $\displaystyle z^{+}=0$ $\displaystyle\to$ $\displaystyle
0,$ $\displaystyle\widetilde{p}_{\perp}=-\frac{\widetilde{\Delta}_{\perp}}{2}$
$\displaystyle\to$ $\displaystyle p_{\perp}=0,$
$\displaystyle\widetilde{p}^{\prime}_{\perp}=\frac{\widetilde{\Delta}_{\perp}}{2}$
$\displaystyle\to$ $\displaystyle
p^{\prime}_{\perp}=\frac{\widetilde{\Delta}_{\perp}}{2}+\frac{\widetilde{\Delta}_{\perp}}{2p^{+}}p^{\prime+}=\frac{\widetilde{\Delta}_{\perp}}{1+\xi}\equiv\Delta_{\perp},$
$\displaystyle z_{\perp}$ $\displaystyle\to$ $\displaystyle z_{\perp},$
$\displaystyle z^{-}$ $\displaystyle\to$ $\displaystyle
z^{-}+\frac{z_{\perp}\cdot\widetilde{\Delta}_{\perp}}{2p^{+}}+{\cal
O}(\Delta^{2}_{\perp}).$ (24)

Then in (3)-(6) one can write

$\displaystyle\int
dz^{-}d^{2}z_{\perp}e^{ixP^{+}\left(z^{-}-\frac{z_{\perp}\cdot\widetilde{\Delta}_{\perp}}{2p^{+}}\right)-i\widetilde{k}_{\perp}\cdot
z_{\perp}}\bigl{\langle}(1-\xi)P^{+},\Delta_{\perp}\bigr{|}\cdots\left|(1+\xi)P^{+},0_{\perp}\right\rangle,$
$\displaystyle=\int
dz^{-}d^{2}z_{\perp}e^{ixP^{+}z^{-}-i\left(\widetilde{k}_{\perp}+\frac{x\widetilde{\Delta}_{\perp}}{2(1+\xi)}\right)\cdot
z_{\perp}}\bigl{\langle}(1-\xi)P^{+},\Delta_{\perp}\bigr{|}\cdots\left|(1+\xi)P^{+},0_{\perp}\right\rangle.$
(25)

This means that if one considers scattering processes in the hadron frame
where $p_{\perp}=0$, transverse momentum transfer
$p^{\prime}_{\perp}=\Delta_{\perp}$ and $t$-channel gluons with transverse
momentum $k_{\perp}$, the GTMDs
$F_{1,n}(x,\xi,\widetilde{k}_{\perp},\widetilde{\Delta}_{\perp})$ should be
evaluated at

$\displaystyle\widetilde{k}_{\perp}=k_{\perp}-\frac{x\widetilde{\Delta}_{\perp}}{2(1+\xi)}=k_{\perp}-\frac{x}{2}\Delta_{\perp},\qquad\widetilde{\Delta}_{\perp}=(1+\xi)\Delta_{\perp}.$
(26)

Many of the previous phenomenological applications of GTMDs, including this
work, have focused on the small-$x$ kinematics $x\ll 1$, or more precisely,
$\xi\ll 1$. Moreover, in the subsequent calculations we shall only keep the
linear terms in $\Delta_{\perp}$, and the ${\cal O}(\Delta_{\perp})$ shift in
$\widetilde{k}_{\perp}$ turns out to be a higher order effect. Therefore, for
the purpose of the present work, the differences (26) can be neglected. We
however envisage more general reactions involving GTMDs where the difference
must be taken into account. We finally note that the longitudinally polairized
spin vector of the incoming proton

$\displaystyle
s^{\mu}=(s^{+},s^{-},0_{\perp})=h_{p}\left((1+\xi)P^{+},\frac{-M^{2}}{2(1+\xi)P^{+}},0_{\perp}\right),$
(27)

in the hadron frame corresponds in the symmetric frame to

$\displaystyle\widetilde{s}^{\mu}=\left(s^{+},s^{-}+\frac{\Delta_{\perp}^{2}s^{+}}{8(p^{+})^{2}},-\frac{s^{+}}{2p^{+}}\Delta_{\perp}\right).$
(28)

##  III Double spin asymmetry in exclusive dijet production

###  III.1 General considerations

Figure 1: Exclusive dijet production in electron-proton scattering.

In this section, we calculate the cross section of longitudinal double spin
asymmetry in exclusive dijet production. A brief summary of the result has
been reported in our previous publication Bhattacharya et al. (2022a). The
process is depicted in Fig. 1. At high center-of-mass energies, the dominant
contribution comes from the two-gluon exchange diagrams. The quark exchange
diagrams will be discussed in a later section.

We assume that both the incoming proton and lepton are longitudinally
polarized with helicity $h_{p}=\pm 1$ and $h_{l}=\pm 1$, respectively. We
write the cross section $d\sigma^{h_{p}h_{l}}$ in the form

$\displaystyle d\sigma^{h_{p}h_{l}}=d\sigma_{0}+h_{p}d\sigma_{\rm
SSA}+h_{l}d\sigma_{\rm BSA}+h_{p}h_{l}d\sigma_{\rm DSA},$ (29)

where the last three terms represent single spin asymmetry (SSA), beam spin
asymmetry (BSA) and double spin asymmetry (DSA). The previous works Ji et al.
(2017); Hatta et al. (2017) looked for signals of OAM in SSA, while our main
focus here is DSA which can be isolated by forming the linear combination

$\displaystyle d\sigma_{\rm
DSA}=\frac{1}{4}(d\sigma^{++}-d\sigma^{+-}-d\sigma^{-+}+d\sigma^{--}).$ (30)

The incoming proton has momentum $p^{\mu}\approx((1+\xi)P^{+},0,0_{\perp})$,
where $\xi$ is the skewness variable, and scatters elastically with momentum
transfer $\Delta^{\mu}=(-2\xi P^{+},0,\Delta_{\perp})$ with
$\Delta_{\perp}=|\Delta_{\perp}|(\cos\phi_{\Delta_{\perp}},\sin\phi_{\Delta_{\perp}})$.
The virtual photon’s momentum is taken in the form

$\displaystyle
q^{\mu}=\left(-\frac{Q}{\sqrt{2}},\frac{Q}{\sqrt{2}},0_{\perp}\right),$ (31)

with $q^{2}=-Q^{2}$. In the present frame, the incoming lepton momentum is
given by

$\displaystyle
l^{\mu}=\left(\frac{Q(1-y)}{\sqrt{2}y},\frac{Q}{\sqrt{2}y},\frac{Q\sqrt{1-y}}{y}n_{\perp}\right),$
(32)

where $y=p\cdot q/p\cdot l$ is the usual variable in DIS and
$n_{\perp}=(\cos\phi_{l_{\perp}},\sin\phi_{l_{\perp}})$ is a unit vector in
the transverse plane. The lepton mass is neglected so that the lepton spin
vector is $s^{\mu}\approx h_{l}l^{\mu}$. The center-of-mass energies of the
$ep$ and $\gamma^{*}p$ collisions are denoted by $s_{ep}=(p+l)^{2}$ and
$W^{2}=(p+q)^{2}=M^{2}-Q^{2}+ys_{ep}$, respectively.

The signal of OAM proposed in Bhattacharya et al. (2022a) is an azimuthal
correlation of the form

$\displaystyle d\sigma_{\rm DSA}\propto\Delta_{\perp}\cdot
l_{\perp}=|\Delta_{\perp}||l_{\perp}|\cos(\phi_{\Delta_{\perp}}-\phi_{l_{\perp}}),$
(33)

measured in coincidence with the production of two jets (dijet) whose
azimuthal angle is integrated over. We assume $\Delta_{\perp}$ is small and
keep only linear terms in $\Delta_{\perp}$. Under this assumption, we
parameterize the momenta of the final state quark and antiquark as

$\displaystyle\begin{split}&q_{1}^{\mu}=\frac{q_{\perp}^{2}+\mu^{2}}{4\xi\bar{z}}n^{\mu}+\frac{2\xi\bar{z}q_{\perp}^{2}}{q_{\perp}^{2}+\mu^{2}}\left(1-2z\frac{q_{\perp}\cdot\Delta_{\perp}}{q_{\perp}^{2}}\right)P^{\mu}+\delta^{\mu}_{i}(q_{\perp}^{i}-z\Delta^{i}_{\perp}),\\\
&q_{2}^{\mu}=\frac{q_{\perp}^{2}+\mu^{2}}{4\xi z}n^{\mu}+\frac{2\xi
zq_{\perp}^{2}}{q_{\perp}^{2}+\mu^{2}}\left(1+2\bar{z}\frac{q_{\perp}\cdot\Delta_{\perp}}{q_{\perp}^{2}}\right)P^{\mu}+\delta^{\mu}_{i}(-q_{\perp}^{i}-\bar{z}\Delta^{i}_{\perp}),\end{split}$
(34)

where $\mu^{2}\equiv z\bar{z}Q^{2}$. $n^{\mu}=\delta^{\mu}_{-}/P^{+}$ and
$P^{\mu}=\delta^{\mu}_{+}P^{+}$ with $n\cdot P=1$ are orthogonal light-like
vectors. The condition $q_{1}^{2}=q_{2}^{2}=0$ is satisfied to linear order in
$\Delta_{\perp}$. We have introduced the common variables $z=P\cdot
q_{1}/P\cdot q=q_{1}^{-}/q^{-}$ and $\bar{z}\equiv 1-z=q_{2}^{-}/q^{-}$
denoting the longitudinal momentum fractions of the virtual photon carried by
the quark and the antiquark, respectively. The skewness variable is related to
$q_{\perp}$ as

$\displaystyle\xi(q^{2}_{\perp})$
$\displaystyle=\dfrac{q^{2}_{\perp}+\mu^{2}}{-q^{2}_{\perp}+\mu^{2}+2z\overline{z}W^{2}}.$
(35)

At high center-of-mass energy ($W\sim 100$ GeV at the EIC), typically $\xi\ll
1$.

The spin-dependent part of the lepton tensor is

$\displaystyle
L^{\mu\nu}=\sum_{s^{\prime}}\bar{u}(ls)\gamma^{\mu}u(l^{\prime}s^{\prime})\bar{u}(l^{\prime}s^{\prime})\gamma^{\nu}u(ls)\sim-2i\epsilon^{\rho\tau\mu\nu}s_{\rho}q_{\tau}=-2ih_{l}\epsilon^{\rho\tau\mu\nu}l_{\rho}q_{\tau}.$
(36)

Since the observable (33) depends on $\phi_{l_{\perp}}$, the index $\rho$ has
to be transverse. On the other hand, the index $\tau$ is longitudinal (see
(31)). This means that either $\mu$ or $\nu$ must be transverse and the other
longitudinal. In other words, we are after the interference effect between
amplitudes with longitudinally polarized and transversely polarized virtual
photons. At the same time, the cross section also encodes interference between
twist-two and twist-three amplitudes because the dependence on
$\Delta_{\perp}$ shows up only at the twist-three level. Let us thus denote
the longitudinal ($L$) and transverse ($T$) amplitudes as

$\displaystyle A_{L}\equiv\epsilon_{\mu}^{L}A^{\mu}=A_{L}^{2}+A_{L}^{3},\qquad
A_{T\lambda}\equiv\epsilon^{i}_{T\lambda}A^{i}=\epsilon_{T\lambda}^{i}(A_{T}^{2i}+A_{T}^{3i}),$
(37)

where $\lambda=1,2$ is a label for the two transverse polarization vectors and

$\displaystyle\epsilon_{\mu}^{L}=\frac{1}{Q}\left(q_{\mu}+\frac{Q^{2}}{p\cdot
q}p_{\mu}\right),$ (38)

is the longitudinal photon polarization vector. The term proportional to
$q^{\mu}$ can be omitted due to gauge invariance.333Strictly speaking, the QED
Ward identity does not hold after a twist expansion. However, the corrections
either do not affect the $\cos(\phi_{\Delta_{\perp}}-\phi_{l_{\perp}})$
dependence or are of higher order in $\Delta_{\perp},k_{\perp}$.  The
superscripts 2 and 3 denote the twist at which the amplitude is evaluated. The
relevant part of the cross section is then given by

$\displaystyle L^{\mu\nu}A^{*}_{\mu}A_{\nu}$ $\displaystyle\sim$
$\displaystyle-\sum_{\lambda=1,2}L^{\mu\nu}(\epsilon^{L}_{\mu}\epsilon_{\nu}^{T\lambda*}A^{*}_{L}A^{\lambda}_{T}+\epsilon^{T\lambda}_{\mu}\epsilon_{\nu}^{L}A_{T}^{\lambda*}A_{L})$
(39) $\displaystyle=$ $\displaystyle
2ih_{l}|l_{\perp}|\epsilon_{ij}(A_{L}^{2*}A^{3i}_{T}+A_{L}^{3*}A^{2i}_{T}-A^{2i*}_{T}A^{3}_{L}-A^{3i*}_{T}A^{2}_{L})n_{\perp}^{j}$
$\displaystyle=$ $\displaystyle 4h_{l}|l_{\perp}|\epsilon_{ij}{\rm
Re}(iA_{L}^{2*}A^{3i}_{T}-iA^{2i*}_{T}A^{3}_{L})n_{\perp}^{j}.$

Dirac traces involving the incoming and outgoing nucleon spinors, as well as
the produced $q\bar{q}$ spinors are understood. The cosine angular correlation
(33) originates from the following structure

$\displaystyle{\rm Re}(iA_{L}^{2*}A^{3i}_{T}-iA^{2i*}_{T}A^{3}_{L})\sim
h_{p}\epsilon^{ik}\Delta^{k}_{\perp}.$ (40)

As observed in Bhattacharya et al. (2022a), there are two different but
equally important (at least parametrically) contributions to (40). A factor of
$\Delta_{\perp}$ can arise either from the soft (GTMD) part or the hard
scattering part. The former is directly related to the gluon OAM which we are
mainly interested in, and the latter is the so-called kinematical twist-three
effect which involves only twist-two GPDs (hence unrelated to GTMDs). We shall
discuss these contributions in the next subsection.

###  III.2  Outline of the calculation

Figure 2: Presented are the leading-order Feynman diagrams depicting exclusive
dijet production initiated by gluons. The complementary set of three diagrams,
with the quark lines exchanged, is implicit.

In the two-gluon exchange approximation, the hadronic part of the scattering
amplitude can be written as

$\displaystyle
iA_{L/T}=\frac{-ig^{2}e_{em}e_{q}C_{F}}{\left(N_{c}^{2}-1\right)P^{+}}\int\frac{dz^{-}d^{2}z_{\perp}}{\left(2\pi\right)^{3}}\int\frac{dx}{x^{2}-\xi^{2}}\int
d^{2}k_{\perp}e^{ixP^{+}z^{-}-ik_{\perp}\cdot
z_{\perp}}D^{L/T}_{\mu\nu}\left\langle
p^{\prime}\left|F^{+\mu}\left(-\frac{z}{2}\right)F^{+\nu}\left(\frac{z}{2}\right)\right|p\right\rangle.$
(41)

Here and below, $\frac{1}{x^{2}-\xi^{2}}$ is a shorthand notation for

$\displaystyle\frac{1}{(x-\xi+i\epsilon)(x+\xi-i\epsilon)}.$ (42)

The hard part $D_{\mu\nu}$ consists of six diagrams as shown in Fig. 2.
Explicitly,

$\displaystyle D^{L/T}_{\mu\nu}$ $\displaystyle=$
$\displaystyle\epsilon^{\rho}_{L/T}\bar{u}(q_{1})\Biggl{[}\gamma_{\nu}\frac{1}{{\ooalign{\hfil/\hfil\crcr$q$}}_{1}+{\ooalign{\hfil/\hfil\crcr$k$}}+{\ooalign{\hfil/\hfil\crcr$\Delta$}}/2}\gamma_{\mu}\frac{1}{{\ooalign{\hfil/\hfil\crcr$q$}}_{1}+{\ooalign{\hfil/\hfil\crcr$\Delta$}}}\gamma_{\rho}+\gamma_{\rho}\frac{1}{{\ooalign{\hfil/\hfil\crcr$q$}}_{2}+{\ooalign{\hfil/\hfil\crcr$\Delta$}}}\gamma_{\mu}\frac{1}{{\ooalign{\hfil/\hfil\crcr$q$}}_{2}+{\ooalign{\hfil/\hfil\crcr$k$}}+{\ooalign{\hfil/\hfil\crcr$\Delta$}}/2}\gamma_{\nu}$
(109)
$\displaystyle\qquad\quad+\gamma_{\mu}\frac{1}{{\ooalign{\hfil/\hfil\crcr$q$}}_{1}-{\ooalign{\hfil/\hfil\crcr$k$}}+{\ooalign{\hfil/\hfil\crcr$\Delta$}}/2}\gamma_{\nu}\frac{1}{{\ooalign{\hfil/\hfil\crcr$q$}}_{1}+{\ooalign{\hfil/\hfil\crcr$\Delta$}}}\gamma_{\rho}+\gamma_{\rho}\frac{1}{{\ooalign{\hfil/\hfil\crcr$q$}}_{2}+{\ooalign{\hfil/\hfil\crcr$\Delta$}}}\gamma_{\nu}\frac{1}{{\ooalign{\hfil/\hfil\crcr$q$}}_{2}-{\ooalign{\hfil/\hfil\crcr$k$}}+{\ooalign{\hfil/\hfil\crcr$\Delta$}}/2}\gamma_{\mu}$
$\displaystyle\qquad\quad-\gamma_{\mu}\frac{1}{{\ooalign{\hfil/\hfil\crcr$q$}}_{1}-{\ooalign{\hfil/\hfil\crcr$k$}}+{\ooalign{\hfil/\hfil\crcr$\Delta$}}/2}\gamma_{\rho}\frac{1}{{\ooalign{\hfil/\hfil\crcr$q$}}_{2}+{\ooalign{\hfil/\hfil\crcr$k$}}+{\ooalign{\hfil/\hfil\crcr$\Delta$}}/2}\gamma_{\nu}-\gamma_{\nu}\frac{1}{{\ooalign{\hfil/\hfil\crcr$q$}}_{1}+{\ooalign{\hfil/\hfil\crcr$k$}}+{\ooalign{\hfil/\hfil\crcr$\Delta$}}/2}\gamma_{\rho}\frac{1}{{\ooalign{\hfil/\hfil\crcr$q$}}_{2}-{\ooalign{\hfil/\hfil\crcr$k$}}+{\ooalign{\hfil/\hfil\crcr$\Delta$}}/2}\gamma_{\mu}\Biggr{]}v(q_{2}),$

where $k^{\mu}=(xP^{+},0,k_{\perp})$. One then projects onto the symmetric
(unpolarized) and antisymmetric (polarized) parts in the gluon transverse
indices

$\displaystyle D_{\mu\nu}\langle F^{+\mu}F^{+\nu}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{2}g_{\mu\nu}^{\perp}D^{\mu\nu}\langle
g_{\alpha\beta}^{\perp}F^{+\alpha}F^{+\beta}\rangle+\frac{1}{2}\epsilon^{\perp}_{\mu\nu}D^{\mu\nu}\langle\epsilon^{\perp}_{\alpha\beta}F^{+\alpha}F^{+\beta}\rangle$
(110) $\displaystyle\sim$
$\displaystyle\frac{1}{2}\delta_{ij}D_{ij}xf_{g}+\frac{1}{2}\epsilon_{ij}D_{ij}ix\tilde{f}_{g},$

where $g_{\mu\nu}^{\perp}=g_{\mu\nu}-P_{\mu}n_{\nu}-P_{\nu}n_{\mu}$ and
$\epsilon_{\mu\nu}^{\perp}=\epsilon_{\mu\nu\alpha\beta}P^{\alpha}n^{\beta}$.

Our basic strategy is to expand $D_{ij}$ in powers of $k_{\perp}$ and
$\Delta_{\perp}$ to linear order

$\displaystyle D_{ij}(k_{\perp},\Delta_{\perp})\approx
D_{ij}(0_{\perp},0_{\perp})+\left.\frac{\partial
D_{ij}(k_{\perp},0_{\perp})}{\partial
k_{\perp}^{l}}\right|_{k=0}k_{\perp}^{l}+\left.\frac{\partial
D_{ij}(0_{\perp},\Delta_{\perp})}{\partial\Delta_{\perp}^{l}}\right|_{\Delta=0}\Delta_{\perp}^{l}.$
(111)

To do so, we need to expand each building block of (109). For example, the
first denominator in (109) becomes

$\displaystyle\frac{1}{(q_{1}+k+\Delta/2)^{2}}$ $\displaystyle\approx$
$\displaystyle\frac{1}{\frac{q_{\perp}^{2}+\mu^{2}}{2\xi\bar{z}}(x-\xi)-2q_{\perp}\cdot
k_{\perp}-q_{\perp}\cdot\Delta_{\perp}}$ (112) $\displaystyle\approx$
$\displaystyle\frac{2\xi\bar{z}}{(q_{\perp}^{2}+\mu^{2})(x-\xi)}+\frac{(2\xi\bar{z})^{2}}{(q_{\perp}^{2}+\mu^{2})^{2}(x-\xi)^{2}}\left(2q_{\perp}\cdot
k_{\perp}+q_{\perp}\cdot\Delta_{\perp}\right).$

When (111) is substituted into (110), the integral $\int
d^{2}k_{\perp}k_{\perp}^{l}xf_{g}\sim\epsilon^{lm}\Delta_{\perp}^{m}L_{g}$
establishes a connection to the gluon OAM (13). On the other hand, the first
and last terms in (111) are independent of $k_{\perp}$, so these terms lead to
the twist-two gluon GPDs $\int d^{2}k_{\perp}xf_{g}\sim H_{g},E_{g}$ and $\int
d^{2}k_{\perp}x\tilde{f}_{g}\sim\tilde{H}_{g},\tilde{E}_{g}$. Further squaring
the amplitude as in (39), we encounter a number of interference terms. We have
considered all the possibilities and found that the angular dependence (33)
can come from the following sources

$\displaystyle H_{g}\otimes\int d^{2}k_{\perp}k^{l}_{\perp}xf_{g},\qquad
H_{g}\otimes\tilde{H}_{g},\qquad\tilde{H}_{g}\otimes\int
d^{2}k_{\perp}k^{l}_{\perp}x\tilde{f}_{g}.$ (113)

The first term involves the gluon OAM and the second term represents the
kinematical twist-three effect mentioned above. In Bhattacharya et al.
(2022a), we neglected the third term assuming that, at high energy $\xi\ll 1$,
the product of the polarized GPD and the polarized GTMD would be subleading.
In the following, we shall also calculate the third term to check the validity
of this assumption.

First, consider the twist-two ($k_{\perp}=\Delta_{\perp}=0$) term in (111).
Applying the symmetric projection $\delta_{ij}D_{ij}$ (denoted by a
superscript $\delta$), one finds Braun and Ivanov (2005)444 The following
formulas which can be derived from the Dirac equations
$\bar{u}(q_{1}){\ooalign{\hfil/\hfil\crcr$q$}}_{1}={\ooalign{\hfil/\hfil\crcr$q$}}_{2}v(q_{2})=0$
are useful when relating seemingly different expressions in the literature
$\displaystyle\bar{u}(q_{1})\left[(z-\bar{z})q_{\perp}\cdot\epsilon_{\perp}\gamma^{-}-i\epsilon_{\perp}\times
q_{\perp}\gamma_{5}\gamma^{-}+\epsilon_{\perp}\cdot\gamma_{\perp}\frac{q_{\perp}^{2}+\mu^{2}}{2\xi
P^{+}}\right]v(q_{2})\approx 0,$ (114)
$\displaystyle\bar{u}(q_{1})\left[iq_{\perp}\cdot\epsilon_{\perp}\gamma_{5}\gamma^{-}+(z-\bar{z})\epsilon_{\perp}\times
q_{\perp}\gamma^{-}+\epsilon_{\perp}\times\gamma_{\perp}\frac{q_{\perp}^{2}+\mu^{2}}{2\xi
P^{+}}\right]v(q_{2})\approx 0,$ (115) where $\approx$ means neglecting terms
of order $\Delta_{\perp}$.

$\begin{split}&iA^{2\delta}_{T}=-\dfrac{ig^{2}e_{em}e_{q}}{N_{c}}\frac{\bar{u}(q_{1})\epsilon_{\perp}\cdot\gamma_{\perp}v(q_{2})}{q_{\perp}^{2}+\mu^{2}}\int
dx\frac{1}{x^{2}-\xi^{2}}\left(1+\frac{2\xi^{2}(1-2\beta)}{x^{2}-\xi^{2}}\right)\int
d^{2}k_{\perp}xf_{g}(x,\xi,k_{\perp},\Delta_{\perp}),\\\
&iA^{2\delta}_{L}=\dfrac{ig^{2}e_{em}e_{q}}{N_{c}}\frac{Q}{(q_{\perp}^{2}+\mu^{2})q^{-}}\bar{u}(q_{1})\gamma^{-}v(q_{2})\int
dx\frac{1}{x^{2}-\xi^{2}}\left(1+\frac{4\xi^{2}(1-\beta)}{x^{2}-\xi^{2}}\right)\int
d^{2}k_{\perp}xf_{g}(x,\xi,k_{\perp},\Delta_{\perp}),\end{split}$ (116)

where

$\displaystyle\beta\equiv\frac{\mu^{2}}{q_{\perp}^{2}+\mu^{2}}.$ (117)

As we shall discuss later, $\beta$ is related to the invariant mass of the
dijet. It is actually a commonly used variable in the context of diffractive
DIS. The following relation is useful throughout the calculation

$\displaystyle P^{+}q^{-}=\frac{q_{\perp}^{2}+\mu^{2}}{4\xi
z\bar{z}}=\frac{q_{\perp}^{2}}{4\xi z\bar{z}(1-\beta)}.$ (118)

The antisymmetric contraction (denoted by a superscript $\epsilon$) gives

$\begin{split}&iA_{T}^{2\epsilon}=\dfrac{ig^{2}e_{em}e_{q}}{N_{c}}\frac{\bar{u}(q_{1})\epsilon_{\perp}\times\gamma_{\perp}v(q_{2})}{q_{\perp}^{2}+\mu^{2}}\int
dx\frac{2\xi x}{(x^{2}-\xi^{2})^{2}}\int
d^{2}k_{\perp}x\tilde{f}_{g}(x,\xi,k_{\perp},\Delta_{\perp}),\\\
&iA_{L}^{2\epsilon}=0.\end{split}$ (119)

The $k_{\perp}$-integrals in (116) and (119) convert the gluon GTMDs into the
gluon GPDs $H_{g},E_{g},\tilde{H}_{g},\tilde{E}_{g}$. Further integration over
$x$ yields the following moments of the GPDs

$\begin{split}&{\cal
H}^{(1)}_{g}(\xi)=\int^{1}_{-1}dx\frac{H_{g}(x,\xi)}{x^{2}-\xi^{2}},\qquad{\cal
E}^{(1)}_{g}(\xi)=\int^{1}_{-1}dx\frac{E_{g}(x,\xi)}{x^{2}-\xi^{2}},\\\ &{\cal
H}^{(2)}_{g}(\xi)=\int^{1}_{-1}dx\frac{\xi^{2}H_{g}(x,\xi)}{(x^{2}-\xi^{2})^{2}},\qquad{\cal
E}^{(2)}_{g}(\xi)=\int^{1}_{-1}dx\frac{\xi^{2}E_{g}(x,\xi)}{(x^{2}-\xi^{2})^{2}},\\\
&\tilde{{\cal
H}}^{(2)}_{g}(\xi)=\int^{1}_{-1}dx\frac{x\tilde{H}_{g}(x,\xi)}{(x^{2}-\xi^{2})^{2}},\qquad\tilde{{\cal
E}}^{(2)}_{g}(\xi)=\int^{1}_{-1}dx\frac{x\tilde{E}_{g}(x,\xi)}{(x^{2}-\xi^{2})^{2}}.\end{split}$
(120)

These integrals are well defined as long as the GPDs and their first
derivatives are continuous at $x=\pm\xi$, which is usually considered to be
the case. Squaring the amplitudes (116), one obtains the cross section of
unpolarized dijet production Braun and Ivanov (2005).

We now turn to the twist-three contributions that give rise to the asymmetry
(33). The first entry in (113) related to the OAM originates from the second
term in (111). The calculation of the amplitude has been done in Ji et al.
(2017) in the context of SSA, and the same results are relevant to DSA:

$\begin{split}iA^{3\delta}_{T}&=-\dfrac{ig^{2}e_{em}e_{q}}{N_{c}}\dfrac{4\xi(\overline{z}-z)}{(q^{2}_{\perp}+\mu^{2})^{2}}\bar{u}(q_{1})\epsilon_{\perp}\cdot\gamma_{\perp}v(q_{2})\\\
&\qquad\qquad\times\int
dx\dfrac{x}{(x^{2}-\xi^{2})^{2}}\bigg{(}1+\dfrac{4\xi^{2}(1-2\beta)}{x^{2}-\xi^{2}}\bigg{)}\int
d^{2}k_{\perp}q_{\perp}\cdot
k_{\perp}\,xf_{g}(x,\xi,k_{\perp},\Delta_{\perp})\\\\[5.69046pt]
&\quad-\dfrac{ig^{2}e_{em}e_{q}}{N_{c}}\dfrac{2\xi}{(q^{2}_{\perp}+\mu^{2})q^{-}}\bar{u}(q_{1})\gamma^{-}v(q_{2})\int
dx\dfrac{x}{(x^{2}-\xi^{2})^{2}}\int d^{2}k_{\perp}\epsilon_{\perp}\cdot
k_{\perp}\,xf_{g}(x,\xi,k_{\perp},\Delta_{\perp}),\\\\[8.5359pt]
iA^{3\delta}_{L}&=\dfrac{ig^{2}e_{em}e_{q}}{N_{c}}\dfrac{4\xi(\overline{z}-z)Q}{(q^{2}_{\perp}+\mu^{2})^{2}q^{-}}\bar{u}(q_{1})\gamma^{-}v(q_{2})\\\
&\qquad\qquad\times\int
dx\dfrac{x}{(x^{2}-\xi^{2})^{2}}\bigg{(}1+\dfrac{8\xi^{2}(1-\beta)}{x^{2}-\xi^{2}}\bigg{)}\int
d^{2}k_{\perp}\,q_{\perp}\cdot
k_{\perp}\,xf_{g}(x,\xi,k_{\perp},\Delta_{\perp}).\end{split}$ (121)

For this contribution, the factor $\Delta_{\perp}$ necessary for the angular
dependence (33) emerges from the Dirac trace over the nucleon spinors. Indeed,
taking the interference effect between (116) and (121) and summing over the
outgoing nucleon spins while fixing the spin of the incoming nucleon, we find

$\displaystyle\frac{1}{4P^{+}M}{\rm
tr}\left[\left(H_{g}\gamma^{+}-\frac{i\sigma^{+\nu}\widetilde{\Delta}_{\nu}}{2M}E_{g}\right)\left({\ooalign{\hfil/\hfil\crcr$p$}}^{\prime}+M\right)\left(F^{g}_{1,1}+i\frac{\sigma^{j+}}{P^{+}}(\widetilde{k}_{\perp}^{j}F^{g}_{1,2}+\widetilde{\Delta}_{\perp}^{j}F^{g}_{1,3})+i\frac{\sigma^{ij}\widetilde{k}_{\perp}^{i}\widetilde{\Delta}_{\perp}^{j}}{M^{2}}F^{g}_{1,4}\right)\left({\ooalign{\hfil/\hfil\crcr$p$}}+M\right)\frac{{\ooalign{\hfil/\hfil\crcr$s$}}\gamma_{5}}{2M}\right]$
(128)
$\displaystyle=ih_{p}\frac{1+\xi}{2M^{2}}\epsilon^{ij}k_{\perp}^{i}\Delta_{\perp}^{j}\left[2\left(H_{g}-\frac{\xi^{2}}{1-\xi^{2}}E_{g}\right)F_{1,4}^{g}-E_{g}F_{1,2}^{g}\right].$
(129)

As indicated by the notations
$\widetilde{k}_{\perp},\widetilde{\Delta}_{\perp}$, this can be conveniently
evaluated in the symmetric frame where the factor of $1+\xi$ comes from
$\tilde{\Delta}_{\perp}=(1+\xi)\Delta_{\perp}$. The product of
$k_{\perp}^{i}F_{1,4}^{g}$ from (129) and $k_{\perp}$ from (121) leads to the
moment (13) after the $k_{\perp}$ integration. From the
$k_{\perp}^{i}F_{1,2}^{g}$ term in (129), we get another moment

$\displaystyle xO(x,\xi)\equiv\int
d^{2}k_{\perp}\frac{k_{\perp}^{2}}{M^{2}}F^{g}_{1,2}(x,\xi).$ (130)

The notation $O$ is a reminder that the imaginary part of $F_{1,2}^{g}$ is
related to the so-called spin-dependent Odderon Zhou (2014); Szymanowski and
Zhou (2016). Its second moment at $\xi=0$ is proportional to the $C$-odd
three-gluon correlator $\langle
P|d_{abc}F_{a}^{+i}F_{b}^{+j}F_{c}^{+i}|P\rangle$ Ji (1992) relevant to
transverse single spin asymmetry Zhou (2014).555We note that if one chooses
the Weiszäcker-Williams gluon distribution in (7), the moment (130) is instead
related to the $C$-even three-gluon correlator $\langle
P|if_{abc}F_{a}^{+i}F_{b}^{+j}F_{c}^{+i}|P\rangle$. The real part of
$F_{1,2}^{g}$ is proportional to $\xi$ (at $\Delta_{\perp}=0$, see (21)) so it
is suppressed when $\xi\ll 1$.666Recently it has been shown in Agrawal et al.
(2023) that the real part of $F_{1,2}^{g}$ has a singular small-$x$ behavior
$\sim 1/x^{c}$ with $c>0$ unlike the imaginary part. It may become relevant
once higher order effects in $\Delta_{\perp}$ are considered.

The subsequent integration over $x$ in (121) yields the following moments

$\displaystyle{\cal L}^{(2)}_{g}(\xi)$ $\displaystyle=$
$\displaystyle\int^{1}_{-1}dx\frac{x^{2}L_{g}(x,\xi)}{(x^{2}-\xi^{2})^{2}},\qquad{\cal
O}^{(2)}(\xi)=\int^{1}_{-1}dx\frac{x^{2}O(x,\xi)}{(x^{2}-\xi^{2})^{2}},$ (131)
$\displaystyle{\cal L}^{(3)}_{g}(\xi)$ $\displaystyle=$
$\displaystyle\int^{1}_{-1}dx\frac{\xi^{2}x^{2}L_{g}(x,\xi)}{(x^{2}-\xi^{2})^{3}},\qquad{\cal
O}^{(3)}(\xi)=\int^{1}_{-1}dx\frac{\xi^{2}x^{2}O(x,\xi)}{(x^{2}-\xi^{2})^{3}}.$
(132)

We immediately notice triple poles at $x=\pm\xi$ which were absent in the
twist-two amplitudes (120). In order for the $x$-integrals in (132) to be
well-defined, the second derivative of $L_{g}(x,\xi)$ and $O(x,\xi)$ must be
continuous at $x=\pm\xi$. It is known that such a condition can be violated
for gluonic GPDs, and this in fact signals the violation of collinear
factorization. We postpone this issue until the next subsection where the
problem becomes sharper. In the meantime, let us just point out that all the
terms in (121) which contain triple poles can be eliminated by setting
$z=\bar{z}=\frac{1}{2}$, that is, when the produced jets are symmetric.

Next, the second entry in (113) comes from the last term in (111) obtained by
setting $k_{\perp}=0$ and keeping one factor of $\Delta_{\perp}$ in the hard
part. Accordingly, in the soft part one can set $\Delta_{\perp}=0$ and
integrate over $k_{\perp}$, after which GTMDs reduce to GPDs in both the
amplitude and the complex-conjugate amplitude. The structure (40) then arises
from interference between the unpolarized and polarized amplitudes. The
asymmetry is therefore proportional to

$\displaystyle\frac{1}{4(P^{+})^{2}}{\rm
tr}\left[\left(H_{g}\gamma^{+}-\frac{i\sigma^{+\nu}\Delta_{\nu}}{2M}E_{g}\right)\left({\ooalign{\hfil/\hfil\crcr$p$}}^{\prime}+M\right)\left(i\gamma_{5}\gamma^{+}\widetilde{H}_{g}-i\frac{\gamma_{5}\Delta^{+}}{2M}\widetilde{E}_{g}\right)\left({\ooalign{\hfil/\hfil\crcr$p$}}+M\right)\frac{{\ooalign{\hfil/\hfil\crcr$s$}}\gamma_{5}}{2M}\right]$
(139)
$\displaystyle=ih_{p}\left(1-\xi^{2}\right)\left(H_{g}-\frac{\xi^{2}}{1-\xi^{2}}E_{g}\right)\left(\widetilde{H}_{g}-\frac{\xi^{2}}{1-\xi^{2}}\widetilde{E}_{g}\right).$
(140)

The calculation of the hard part is straightforward but quite cumbersome,
since one has to consider many cases: longitudinal/transverse photons; twist-
two/twist-three amplitudes; symmetric/antisymmetric projections (110).
Moreover, in addition to the explicit $\Delta_{\perp}$-dependence coming from
the expansion of the propagators, cf., (112), the factor of $\Delta_{\perp}$
can also come from the Dirac trace over the final state quark spinors
$\bar{u}(q_{1}),v(q_{2})$ since $q_{1,2}$ depend on $\Delta_{\perp}$, see
(34). Therefore, in practice it is more advantageous to expand the squared
amplitude in $\Delta_{\perp}$ instead of the amplitude itself. Schematically,
(39) should be generalized as

$\displaystyle{\rm Re}\left[iA_{L}^{2\delta*}A_{T}^{3\epsilon
i}+iA_{L}^{3\delta*}A_{T}^{2\epsilon i}-iA_{T}^{2\delta
i*}A_{L}^{3\epsilon}-iA_{T}^{3\delta
i*}A_{L}^{2\epsilon}+i(A_{L}^{\delta*}A_{T}^{\epsilon i})^{3}-i(A_{T}^{\delta
i*}A_{L}^{\epsilon})^{3}\right].$ (141)

In the last two terms, the factor of $\Delta_{\perp}$ comes from the squared
hard amplitude including the final state $q\bar{q}$ spinors. In Bhattacharya
et al. (2022a) we have performed this calculation analytically for the special
case $z=\frac{1}{2}$. For generic values of $z$, we have developed a
Mathematica program to facilitate the calculation.

Finally, we turn to the third entry in (113). This can be calculated in the
same way as the OAM contribution except that $\delta_{ij}\to\epsilon_{ij}$
everywhere. The twist-three hard amplitudes read

$\begin{split}iA^{3\epsilon}_{T}&=-\dfrac{ig^{2}e_{em}e_{q}}{N_{c}}\int\frac{dx}{(x^{2}-\xi^{2})^{2}}\Biggl{[}\left\\{\bar{u}(q_{1})\epsilon_{\perp}\times\gamma_{\perp}v(q_{2})(z-\bar{z})\left(3+\frac{4\xi^{2}}{x^{2}-\xi^{2}}\right)+\bar{u}(q_{1})\gamma^{-}v(q_{2})\frac{2q_{\perp}\times\epsilon_{\perp}}{q^{-}}\right\\}\\\
&\qquad\qquad\qquad\qquad\qquad\times\frac{4\xi^{2}}{(q_{\perp}^{2}+\mu^{2})^{2}}\int
d^{2}k_{\perp}q_{\perp}\cdot
k_{\perp}\,ix\tilde{f}_{g}(x,\xi,k_{\perp},\Delta_{\perp})\\\\[5.69046pt]
&\qquad\qquad\qquad\quad+\bar{u}(q_{1})\gamma^{-}v(q_{2})\frac{2\xi^{2}}{(q_{\perp}^{2}+\mu^{2})q^{-}}\int
d^{2}k_{\perp}\epsilon_{\perp}\times
k_{\perp}\,ix\tilde{f}_{g}(x,\xi,k_{\perp},\Delta_{\perp})\Biggr{]},\\\\[8.5359pt]
iA^{3\epsilon}_{L}&=-\dfrac{ig^{2}e_{em}e_{q}}{N_{c}}\frac{4\xi^{2}Q}{q^{-}(q_{\perp}^{2}+\mu^{2})^{2}}\bar{u}(q_{1})\gamma_{5}\gamma^{-}v(q_{2})\int\dfrac{dx}{(x^{2}-\xi^{2})^{2}}\int
d^{2}k_{\perp}\,q_{\perp}\cdot
k_{\perp}\,x\tilde{f}_{g}(x,\xi,k_{\perp},\Delta_{\perp}).\end{split}$ (142)

In the soft part, the nucleon spinor sum gives

$\displaystyle\frac{-i}{4P^{+}M}{\rm
tr}\left[\left(\tilde{H}_{g}\gamma_{5}\gamma^{+}+\tilde{E}_{g}\frac{\gamma_{5}\Delta^{+}}{2M}\right)\left({\ooalign{\hfil/\hfil\crcr$p$}}^{\prime}+M\right)\left(\frac{\epsilon_{ij}\widetilde{k}_{\perp}^{i}\widetilde{\Delta}_{\perp}^{j}}{M^{2}}G^{g}_{1,1}-\frac{\sigma^{i+}\gamma_{5}}{P^{+}}(\widetilde{k}_{\perp}^{i}G^{g}_{1,2}+\widetilde{\Delta}_{\perp}^{i}G^{g}_{1,3})-\sigma^{+-}\gamma_{5}G^{g}_{1,4}\right)\left({\ooalign{\hfil/\hfil\crcr$p$}}+M\right)\frac{{\ooalign{\hfil/\hfil\crcr$s$}}\gamma_{5}}{2M}\right]$
(149)
$\displaystyle=ih_{p}\frac{1+\xi}{2M^{2}}\left(2\left(\tilde{H}_{g}-\frac{\xi^{2}}{1-\xi^{2}}\tilde{E}_{g}\right)G^{g}_{1,1}-\xi\tilde{E}_{g}G^{g}_{1,2}\right)\epsilon_{ij}k_{\perp}^{i}\Delta^{j}_{\perp}.$
(150)

The $k_{\perp}$ integrals then lead to the moments

$\displaystyle xC_{g}(x,\xi)\equiv\int
d^{2}k_{\perp}\frac{k_{\perp}^{2}}{M^{2}}G_{1,1}^{g}(x,\xi),\qquad
xC^{\prime}_{g}(x,\xi)\equiv-\int
d^{2}k_{\perp}\frac{k_{\perp}^{2}}{M^{2}}G_{1,2}^{g}(x,\xi).$ (151)

With these definitions, $C_{g}(x,\xi)$ and $C^{\prime}_{g}(x,\xi)$ are odd
functions in $x$. [Note that $H_{g}(x,\xi),x\tilde{H}_{g}(x,\xi),L_{g}(x,\xi)$
and $O(x,\xi)$ are all even functions in $x$.] Similarly to the quark case
Lorce and Pasquini (2011), we interpret $C_{g}(x,\xi=0)$ as the gluons’ spin-
orbit correlation. On the other hand, $C^{\prime}_{g}(x,\xi=0)$ is related to
another $C$-odd three-gluon correlator $\langle
d_{abc}F_{a}^{+i}F_{b}^{+j}\tilde{F}_{c}^{+i}\rangle$. Very little is known
about these distributions except that they appear in specific exclusive
reactions Bhattacharya et al. (2022b); Boussarie et al. (2018).

Let us provide quick insights into the characteristics of $C_{g}(x)$ which
allows for a quantitative estimate sufficient for the present purpose, leaving
a more detailed analysis for future work. First, despite being associated with
the polarized gluon operator $\tilde{F}^{+\mu}F^{+}_{\ \mu}$, $C_{g}(x)$ has
little to do with the parent hadron spin and exists even for spinless hadrons.
Second, just like $L_{g}(x)$ (see (16)), using the equations of motion one can
express $C_{g}(x)$ as the sum of the Wandzura-Wilczek (WW) part and the
genuine twist-three part. Adapting the method in Hatta and Yoshida (2012), we
have computed the WW part

$\displaystyle C_{g}(x)$ $\displaystyle=$ $\displaystyle
x\int_{x}^{\epsilon(x)}\frac{dx^{\prime}}{x^{\prime
2}}\tilde{H}_{g}(x^{\prime})-2x\int_{x}^{\epsilon(x)}\frac{dx^{\prime}}{x^{\prime
2}}G(x^{\prime})+\cdots.$ (152)

(152) allows us to infer the small-$x$ behavior of $C_{g}(x)$. Clearly, the
second term dominates at small-$x$ because, roughly, $\tilde{H}_{g}(x)=x\Delta
G(x)\sim x^{2}G(x)$. Assuming the power-law (17), we find

$\displaystyle C_{g}(x)\approx-\frac{2}{2+c}G(x)\approx-G(x),$ (153)

where the last expression follows if the exponent is perturbatively small
$c\propto\alpha_{s}\ll 1$. Since $G(x)$ is positive at small-$x$, $C_{g}(x)$
is negative, meaning that the helicity and OAM of individual gluons are anti-
aligned (have opposite signs) Lorce and Pasquini (2011) at small-$x$, in
accordance with the theoretical prediction (18). (153) also shows that
$C_{g}(x)$ grows strongly in the small-$x$ region. This is consistent with an
independent analysis based on an effective theory of small-$x$ QCD Boer et al.
(2018); S. Bhattacharya, R. Boussarie, and Y. Hatta (2024).

###  III.3 The result

Let us introduce the convenient notation

$\displaystyle F_{g}\equiv
H_{g}-\frac{\xi^{2}}{1-\xi^{2}}E_{g},\qquad\tilde{F}_{g}\equiv\tilde{H}_{g}-\frac{\xi^{2}}{1-\xi^{2}}\tilde{E}_{g},$
(154)

for the linear combinations of GPDs which appear in the Dirac traces (129),
(140) (150). Similarly to (120), we define ($n=1,2,3$)

$\displaystyle{\cal
F}^{(n)}_{g}(\xi)=\int_{-1}^{1}dx\frac{\xi^{2(n-1)}F_{g}(x,\xi)}{(x^{2}-\xi^{2})^{n}},\qquad\tilde{\cal
F}^{(n)}_{g}(\xi)=\int_{-1}^{1}dx\frac{\xi^{2(n-2)}x\tilde{F}_{g}(x,\xi)}{(x^{2}-\xi^{2})^{n}},$
(155)

and

$\displaystyle{\cal
C}_{g}^{(2)}(\xi)=\int_{-1}^{1}dx\frac{\xi^{2}xC_{g}(x,\xi)}{(x^{2}-\xi^{2})^{2}},\qquad{\cal
C}^{\prime(2)}_{g}(\xi)=\int_{-1}^{1}dx\frac{\xi^{2}xC^{\prime}_{g}(x,\xi)}{(x^{2}-\xi^{2})^{2}}.$
(156)

Using these moments, our final result for the cross section can be expressed
as

$\displaystyle\dfrac{d\sigma_{\rm
DSA}}{dydQ^{2}d\phi_{l_{\perp}}dzdq_{\perp}^{2}d^{2}\Delta_{\perp}}=\dfrac{e_{q}^{2}\alpha_{em}^{2}\alpha_{s}^{2}y}{2\pi^{3}Q^{2}N_{c}}\dfrac{\xi(1+\xi)|l_{\perp}||\Delta_{\perp}|\cos(\phi_{l_{\perp}}-\phi_{\Delta_{\perp}})}{z\overline{z}(W^{2}+Q^{2})(W^{2}-M_{J}^{2})(q_{\perp}^{2}+\mu^{2})^{2}}{\rm
Re}(\Sigma_{L}+\Sigma_{O}+\Sigma_{h}+\Sigma_{C}),$ (157)

where the contribution from the gluon OAM is

$\displaystyle\Sigma_{L}$ $\displaystyle=$ $\displaystyle-\left({\cal
F}^{(1)*}_{g}+4(1-\beta){\cal F}^{(2)*}_{g}\right){\cal
L}_{g}^{(2)}+(z-\bar{z})^{2}{\cal F}_{g}^{(1)*}\left({\cal
L}_{g}^{(2)}+8(1-\beta){\cal L}_{g}^{(3)}\right).$ (158)

There are also contributions from the ‘odderon’

$\displaystyle\Sigma_{O}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\left({\cal E}^{(1)*}_{g}+4(1-\beta){\cal
E}^{(2)*}_{g}\right){\cal O}_{g}^{(2)}+\frac{(z-\bar{z})^{2}}{2}{\cal
E}_{g}^{(1)*}\left({\cal O}_{g}^{(2)}+8(1-\beta){\cal O}_{g}^{(3)}\right),$
(159)

and from the gluon helicity

$\displaystyle\Sigma_{h}$ $\displaystyle=$ $\displaystyle(1-\xi){\cal
F}_{g}^{(1)*}\tilde{\cal
F}_{g}^{(2)}-(1-\xi)(z-\bar{z})^{2}\Biggl{[}8(1-\beta)\left({\cal
F}_{g}^{(1)*}+4(1-\beta){\cal F}_{g}^{(2)*}\right)\tilde{\cal F}^{(3)}_{g}$
(160) $\displaystyle\qquad\qquad+\left((4\beta-3){\cal
F}_{g}^{(1)*}+16(1-\beta){\cal F}^{(2)*}_{g}+32(1-\beta)^{2}{\cal
F}_{g}^{(3)*}\right)\tilde{\cal F}^{(2)}_{g}\Biggr{]},$

and from the spin-orbit correlation

$\displaystyle\Sigma_{C}=2(1-\beta)\left(2{\cal C}_{g}^{(2)*}\tilde{\cal
F}_{g}^{(2)}+\xi{\cal C}^{\prime(2)*}_{g}\tilde{\cal E}_{g}^{(2)}\right).$
(161)

Importantly $\Sigma_{h}$ contains integrals with triple poles

$\begin{split}&{\cal
H}_{g}^{(3)}(\xi)=\int_{-1}^{1}dx\frac{\xi^{4}H_{g}(x,\xi)}{(x^{2}-\xi^{2})^{3}},\qquad{\cal
E}_{g}^{(3)}(\xi)=\int_{-1}^{1}dx\frac{\xi^{4}E_{g}(x,\xi)}{(x^{2}-\xi^{2})^{3}},\\\
&\tilde{\cal
H}_{g}^{(3)}(\xi)=\int_{-1}^{1}dx\frac{\xi^{2}x\tilde{H}_{g}(x,\xi)}{(x^{2}-\xi^{2})^{3}},\qquad\tilde{\cal
E}_{g}^{(3)}(\xi)=\int_{-1}^{1}dx\frac{\xi^{2}x\tilde{E}_{g}(x,\xi)}{(x^{2}-\xi^{2})^{3}},\end{split}$
(162)

These are similar to the integrals which we encountered before (132) and are
contained in $\Sigma_{L},\Sigma_{O}$. They are well-defined if the second
derivative of $H_{g}$, etc, are continuous at $x=\pm\xi$. Unfortunately,
however, this condition is known to be violated for $H_{g}$ and $E_{g}$ at
least for asymptotically large values of the renormalization scale $\mu_{r}$

$\displaystyle
H_{g}(x,\xi,\mu_{r}),E_{g}(x,\xi,\mu_{r})\sim\theta(\xi-|x|)(x^{2}-\xi^{2})^{2}\qquad(\mu_{r}\to\infty)\,.$
(163)

Essentially the same problem was previously encountered in the exclusive
production of $P$-wave charmonium states Cui et al. (2019) where it was
concluded that collinear factorization is violated due to this ‘end-point’
singularity.

In the present problem, clearly the origin of this divergence is the expansion
(112) in $k_{\perp}$ and $\Delta_{\perp}$. Had we not done this, the cross
section would have been perfectly finite and given in terms of GTMDs (not
GPDs). But then we would not be able to exploit the formula (13) to establish
a (direct) connection to gluon OAM. Without the detailed knowledge of the
behavior of gluon GPDs at $x=\pm\xi$, it is not clear to us whether the
integrals (162) are always divergent for realistic values of $\mu_{r}$.
However, as already noticed in Bhattacharya et al. (2022a), the integrals with
triple poles (162) and (132) always accompany the prefactor $(z-\bar{z})^{2}$,
hence they all vanish at $z=\frac{1}{2}$. Since by symmetry $z=\frac{1}{2}$ is
a stationary point of the dijet cross section, we can expect that the ${\cal
O}(z-\bar{z})^{2}$ ‘curvature’ corrections around $z=\frac{1}{2}$ is moderate
in practice and may be neglected to first approximation once the divergence at
$z\neq\frac{1}{2}$ is regularized by the the $k_{\perp}$-dependent (GTMD)
approach. Further neglecting the odderon contribution and the terms which are
explicitly suppressed by powers of $\xi\ll 1$, we arrive at a practically
useful formula valid near $z\sim\frac{1}{2}$

$\displaystyle\Sigma_{L}+\Sigma_{O}+\Sigma_{h}+\Sigma_{C}\approx-\left({\cal
H}^{(1)*}_{g}+4(1-\beta){\cal H}^{(2)*}_{g}\right){\cal
L}_{g}^{(2)}+\left({\cal H}_{g}^{(1)*}+4(1-\beta){\cal
C}_{g}^{(2)*}\right)\tilde{\cal H}_{g}^{(2)}.$ (164)

In Bhattacharya et al. (2022a), we did not calculate the last term related to
the spin-orbit correlation assuming that the contribution from the polarized
GTMDs and GPDs are suppressed compared to the unpolarized ones when $\xi\ll
1$. However, the present complete calculation, together with the new formula
(152), indicates that this is not the case. From (18) and (152), we expect
that ${\cal H}_{g}^{(1)}\sim{\cal H}_{g}^{(2)}\sim{\cal C}_{g}^{(2)}$ and
$\tilde{\cal H}_{g}^{(2)}\sim{\cal L}_{g}^{(2)}$ (see Appendix A). We then
immediately see that all the terms in (164) are parametrically of the same
order.

Let us briefly discuss the impact of the spin-orbit correlation, deferring a
numerical estimate to a later section. Previously in Bhattacharya et al.
(2022a), we have observed an interesting interplay between the gluon helicity
and the gluon OAM in (164). As we show in Appendix A, ${\cal H}_{g}^{(1,2)}$
are dominated by the imaginary part at small-$\xi$ and there is an approximate
relation ${\rm Im}{\cal H}_{g}^{(1)}(\xi)\approx-2{\rm Im}{\cal
H}^{(2)}_{g}(\xi)$ (cf. Fig. 10). Plugging this into (164) (without the ${\cal
C}_{g}^{(2)}$ term), we find that the cross section is roughly proportional to

$\displaystyle d\sigma_{\rm DSA}\sim{\cal H}^{(1)}_{g}\left(\tilde{\cal
H}^{(2)}_{g}+\frac{q_{\perp}^{2}-z\bar{z}Q^{2}}{q_{\perp}^{2}+z\bar{z}Q^{2}}{\cal
L}_{g}^{(2)}\right).$ (165)

Since the helicity and OAM have opposite signs at small-$x$ (18), when
$q^{2}_{\perp}\gg z\bar{z}Q^{2}\approx\frac{Q^{2}}{4}$, the two terms in (165)
tend to cancel each other, and when $q^{2}_{\perp}\ll z\bar{z}Q^{2}$, they add
up. Such an expectation was borne out by the numerical analysis performed in
Bhattacharya et al. (2022a). However, this scenario is modified by the spin-
orbit term. Due to (153), we expect that ${\cal C}_{g}^{(2)}\approx-{\cal
H}_{g}^{(2)}$, and this effectively enhances the coefficient of $\tilde{\cal
H}_{g}^{(2)}$ in (164) by a factor $1+2(1-\beta)$. As a result, the impact of
the cancellation when $q_{\perp}^{2}\gg Q^{2}/4$ is diminished unless $\beta$
is close to unity.

##  IV Quark exchange channel

Figure 3: Displayed are the leading-order Feynman diagrams illustrating
exclusive dijet production initiated by quarks. The corresponding set of two
diagrams, with the quark lines exchanged, is implied.

Figure 4: Presented here are the leading-order Feynman diagrams showing
exclusive gluon-jet production initiated by quarks. The corresponding pair of
diagrams, wherein the quark lines are exchanged, is implied.

We now discuss the quark exchange diagrams for exclusive dijet production. Our
original goal was to obtain a formula similar to (157) but this time featuring
the quark OAM (9). However, we have encountered an unexpected problem whose
resolution is beyond the scope of this work. Therefore, instead of presenting
the complete calculation, we will focus on explaining the origin and nature of
the problem which may be an interesting observation by itself.

The relevant diagrams are shown in Fig. 3. Their sum is proportional to

$\displaystyle A^{q}_{L/T}$ $\displaystyle\propto$
$\displaystyle\frac{\epsilon_{L/T}^{\rho}}{(k+\Delta/2+q_{2})^{2}}\bar{u}(q_{1})\Biggl{[}\gamma^{\alpha}\frac{{\ooalign{\hfil/\hfil\crcr$k$}}-\frac{{\ooalign{\hfil/\hfil\crcr$\Delta$}}}{2}+{\ooalign{\hfil/\hfil\crcr$q$}}}{(k-\Delta/2+q)^{2}}\gamma_{\rho}\langle
p^{\prime}|\psi\bar{\psi}|p\rangle\gamma_{\alpha}+\gamma_{\rho}\frac{{\ooalign{\hfil/\hfil\crcr$q$}}_{1}-{\ooalign{\hfil/\hfil\crcr$q$}}}{(q_{1}-q)^{2}}\gamma^{\alpha}\langle
p^{\prime}|\psi\bar{\psi}|p\rangle\gamma_{\alpha}\Biggr{]}v(q_{2})$ (187)
$\displaystyle+\frac{\epsilon_{L/T}^{\rho}}{(k-\Delta/2-q_{1})^{2}}\bar{u}(q_{1})\Biggl{[}\gamma^{\alpha}\langle
p^{\prime}|\psi\bar{\psi}|p\rangle\gamma_{\rho}\frac{{\ooalign{\hfil/\hfil\crcr$k$}}+\frac{{\ooalign{\hfil/\hfil\crcr$\Delta$}}}{2}-{\ooalign{\hfil/\hfil\crcr$q$}}}{(k+\Delta/2-q)^{2}}\gamma_{\alpha}+\gamma^{\alpha}\langle
p^{\prime}|\psi\bar{\psi}|p\rangle\gamma_{\alpha}\frac{-{\ooalign{\hfil/\hfil\crcr$q$}}_{2}+{\ooalign{\hfil/\hfil\crcr$q$}}}{(-q_{2}+q)^{2}}\gamma_{\rho}\Biggr{]}v(q_{2}).$

We use the Fierz identity

$\displaystyle\langle\psi\bar{\psi}\rangle=-\frac{1}{4}\langle\bar{\psi}\gamma^{+}\psi\rangle\gamma^{-}+\frac{1}{4}\langle\bar{\psi}\gamma_{5}\gamma^{+}\psi\rangle\gamma_{5}\gamma^{-}+\cdots,$
(188)

to project onto the leading twist unpolarized (3) and polarized (4) quark
GTMDs. We then extract the twist-two and twist-three parts by expanding to
linear order in $k_{\perp}$ and $\Delta_{\perp}$. The twist-two part can be
found in Braun and Ivanov (2005). As for the twist-three part, consider for
definiteness the longitudinally polarized case
${\ooalign{\hfil/\hfil\crcr$\epsilon$}}_{L}\to\epsilon_{L}^{+}\gamma^{-}$.777
QED gauge invariance is known to be more subtle in quark exchange diagrams,
see e.g., Anikin et al. (2000). However, we do not expect this issue to
resolve the problem that we report here.  The unpolarized and polarized
amplitudes are given by

$\displaystyle A^{q,{\rm unpol}}_{L}$ $\displaystyle\propto$
$\displaystyle\frac{\langle\bar{\psi}\gamma^{+}\psi\rangle}{8z\bar{z}\xi(P^{+})^{2}q^{-}}(\bar{u}{\ooalign{\hfil/\hfil\crcr$\epsilon$}}_{L}v)\Biggl{[}\frac{z}{x-\xi}+\frac{\bar{z}}{x+\xi}+\frac{(-z^{2}+\bar{z}^{2})(x^{2}+\xi^{2})-2x\xi(z^{2}+\bar{z}^{2})}{z\bar{z}(x^{2}-\xi^{2})^{2}}\frac{k_{\perp}\cdot
q_{\perp}}{P^{+}q^{-}}$ (192)
$\displaystyle\qquad\qquad+\frac{(z^{2}-\bar{z}^{2})(x^{3}-3\xi^{2}x)-2\xi^{3}(z^{2}+\bar{z}^{2})}{2\xi
z\bar{z}(x^{2}-\xi^{2})^{2}}\frac{q_{\perp}\cdot\Delta_{\perp}}{P^{+}q^{-}}\Biggr{]},$
$\displaystyle A_{L}^{q,{\rm pol}}$ $\displaystyle\propto$
$\displaystyle-\frac{\langle\bar{\psi}\gamma_{5}\gamma^{+}\psi\rangle}{8z\bar{z}\xi(P^{+})^{2}q^{-}}(\bar{u}{\ooalign{\hfil/\hfil\crcr$\epsilon$}}_{L}\gamma_{5}v)\Biggl{[}\frac{z}{x-\xi}+\frac{\bar{z}}{x+\xi}+\frac{(-z^{2}+\bar{z}^{2})(x^{2}+\xi^{2})-2x\xi(z^{2}+\bar{z}^{2})}{z\bar{z}(x^{2}-\xi^{2})^{2}}\frac{k_{\perp}\cdot
q_{\perp}}{P^{+}q^{-}}$ (196)
$\displaystyle\qquad\qquad+\frac{(z^{2}-\bar{z}^{2})(x^{3}-3\xi^{2}x)-2\xi^{3}(z^{2}+\bar{z}^{2})}{2\xi
z\bar{z}(x^{2}-\xi^{2})^{2}}\frac{q_{\perp}\cdot\Delta_{\perp}}{P^{+}q^{-}}\Biggr{]}.$

Integrating the ${\cal O}(k_{\perp})$ term of (192) and (196) over
$k_{\perp}$, we obtain the GPD $L_{g}(x,\xi)$ associated with the quark OAM
(12) and the quarks’ spin-orbit correlation Lorce and Pasquini (2011), cf.,
(151)

$\displaystyle C_{q}(x,\xi)\equiv\int
d^{2}k_{\perp}\frac{k_{\perp}^{2}}{M^{2}}G_{1,1}^{q}(x,\xi).$ (197)

On the other hand, from the ${\cal O}(\Delta_{\perp})$ term of (192) and
(196), we obtain the unpolarized and polarized quark GPDs $H_{q}$,
$\tilde{H}_{q}$. Up to this point, the discussion is completely parallel with
the gluon exchange channel in the previous section. The difference starts to
appear after the subsequent integration over $x$ which gives two types of
moments with double poles at $x=\pm\xi$

$\displaystyle(z^{2}-\bar{z}^{2})\int_{-1}^{1}dx\frac{L_{q}(x,\xi)}{(x^{2}-\xi^{2})^{2}}=\frac{z^{2}-\bar{z}^{2}}{2}\int_{-1}^{1}dx\frac{L^{(+)}_{q}(x,\xi)}{(x^{2}-\xi^{2})^{2}},$
(198)
$\displaystyle(z^{2}+\bar{z}^{2})\int_{-1}^{1}dx\frac{xL_{q}(x,\xi)}{(x^{2}-\xi^{2})^{2}}=\frac{z^{2}+\bar{z}^{2}}{2}\int_{-1}^{1}dx\frac{xL^{(-)}_{q}(x,\xi)}{(x^{2}-\xi^{2})^{2}},$
(199)
$\displaystyle(z^{2}-\bar{z}^{2})\int_{-1}^{1}dx\frac{xH_{q}(x,\xi)}{(x^{2}-\xi^{2})^{2}}=\frac{z^{2}-\bar{z}^{2}}{2}\int_{-1}^{1}dx\frac{xH^{(+)}_{q}(x,\xi)}{(x^{2}-\xi^{2})^{2}},$
(200)
$\displaystyle(z^{2}+\bar{z}^{2})\int_{-1}^{1}dx\frac{H_{q}(x,\xi)}{(x^{2}-\xi^{2})^{2}}=\frac{z^{2}+\bar{z}^{2}}{2}\int_{-1}^{1}dx\frac{H^{(-)}_{q}(x,\xi)}{(x^{2}-\xi^{2})^{2}},$
(201)

where we defined the $C$-parity even $(+)$ and odd $(-)$ quark GPD
combinations as

$\displaystyle L_{q}^{(\pm)}(x,\xi)=L_{q}(x,\xi)\pm L_{q}(-x,\xi),$
$\displaystyle H^{(\pm)}_{q}(x,\xi)=H_{q}(x,\xi)\mp H_{q}(-x,\xi).$ (202)

The $C$-odd contributions arise because the nucleon is not an eigenstate of
$C$-parity and the outgoing $q\bar{q}$ pair can be in both $C$-even and
$C$-odd states. $L_{q}^{(+)}$ is directly related to the nucleon spin sum
rule, while $L_{q}^{(-)}$ can be understood as the OAM of valence quarks.

The problem is that, in the quark channel, it is the double pole that is
dangerous for factorization. Indeed, the above integrals are finite provided
the first derivative of $H_{q}^{(\pm)},L^{(\pm)}_{q}$ is continuous at
$x=\pm\xi$. This condition is known to be violated for $H_{q}^{(\pm)}$ (see
Diehl (2003) and references therein), and presumably also for $L_{q}^{(\pm)}$.
Note that (200) is exactly the same integral that violates the collinear
factorization of the exclusive production of a transversely polarized vector
meson Mankiewicz and Piller (2000); Anikin and Teryaev (2003). This integral,
together with (198), can be eliminated by setting $z=\frac{1}{2}$ as in the
gluon case. Surprisingly, however, the $C$-odd integrals (199), (201) survive
even after this approximation and contribute to the cosine modulation (33)
after interfering with the twist-two transverse amplitude $A_{T}^{q}$ Braun
and Ivanov (2005). We have seen in the previous section that in the gluon case
the $C$-odd odderon exchange does not break factorization at least around
$z\approx\frac{1}{2}$. Thus the problem is unique to the quark exchange (in
contrast to Nabeebaccus et al. (2023)) and exemplifies a novel pattern of
factorization breaking.

Another class of diagrams with quark exchange in the $t$ channel could also
contribute: diagrams where both jets are initiated by hard gluons (Fig. 4). In
that case, by color conservation the gluon pair is on a singlet state and it
is thus $C$-even. As a result, the $C$-odd part of the quark distribution is
probed and we face the same non-cancellation of double poles even for
$z\approx\frac{1}{2}$.

It is expected that, at high energy where $\xi\ll 1$, the quark contribution
is subleading compared to the gluon one calculated in the previous section. We
therefore neglect it in the numerical analysis below. However, the problem
just described appears to pose a challenge if one would like to probe the
quark OAM in the large-$x$ region.

##  V Semi-inclusive diffractive cross section

In a previous publication Bhattacharya et al. (2022a), we have presented a
numerical evaluation of the formula (164) (without the last term). While the
result cleanly demonstrated the salient features of the cross section such as
the interplay between the gluon OAM and helicity contributions, a serious
practical challenge was the overall smallness of the cross section.
Considering the luminosity of the Electron-Ion Collider, one has to probe the
region of jet transverse momentum around $q_{\perp}=1\sim 2$ GeV. The problem
is that it is quite difficult to reconstruct jets at such low transverse
momentum. One possibility to circumvent this difficulty is to consider
dihadron or heavy-quark pair production instead of light-quark dijet
production (see, e.g., Reinke Pelicer et al. (2019); Bergabo and Jalilian-
Marian (2022); Linek et al. (2023); Fucilla et al. (2023)). In this section,
we adopt a different strategy inspired by a recent work Hatta et al. (2022).

One of the advantages of our proposal (as opposed to the earlier proposals Ji
et al. (2017); Hatta et al. (2017)) is that the angular modulation
$\cos(\phi_{l_{\perp}}-\phi_{\Delta_{\perp}})$ we look for does not depend on
the jet azimuthal angle $\phi_{q_{\perp}}$. In fact, $\phi_{q_{\perp}}$ has
been integrated over to arrive at the formula (157). It is then tempting to
integrate also over $q_{\perp}$. Let us explore this possibility.
Experimentally, the parameter $\beta$ introduced in (117), or the invariant
mass of the diffractively produced system

$\displaystyle
M_{X}^{2}=\frac{q_{\perp}^{2}}{z\bar{z}}=\frac{1-\beta}{\beta}Q^{2},$ (203)

is easy to measure since it is not strongly affected by the hadronization
effects and does not require jet reconstruction. Moreover, at fixed energy and
$Q^{2}$, $\beta$ is in one-to-one correspondence with the skewness parameter

$\displaystyle\xi=\frac{Q^{2}}{(2\beta-1)Q^{2}+2\beta W^{2}},$ (204)

upon which the Compton amplitudes, ${\cal H}_{g}^{(1)}(\xi)$ etc., depend. The
simplest observable would then be the inclusive cross section $d\sigma/d\beta$
in which $\beta$ is fixed by inserting the constraint

$\displaystyle
1=\int_{0}^{1}d\beta\delta\left(\beta-\frac{z(1-z)Q^{2}}{q_{\perp}^{2}+z(1-z)Q^{2}}\right),$
(205)

and $q_{\perp}$ and $z$ are integrated over

$\displaystyle\frac{d\sigma}{dydQ^{2}d\phi_{l_{\perp}}d^{2}\Delta_{\perp}d\beta}$
$\displaystyle=$
$\displaystyle\int_{0}^{1}dz\int_{0}^{\infty}dq_{\perp}^{2}\delta\left(\beta-\frac{z(1-z)Q^{2}}{q_{\perp}^{2}+z(1-z)Q^{2}}\right)f(z,q_{\perp})$
(206) $\displaystyle=$
$\displaystyle\left.\int_{0}^{1}dz\frac{z(1-z)Q^{2}}{\beta}f(z,q_{\perp})\right|_{q_{\perp}^{2}=z(1-z)\frac{1-\beta}{\beta}Q^{2}},$

where $f$ is the right-hand-side of (157). This is similar to the inclusive
diffractive DIS cross section. However, the integral probes the so-called
‘aligned jet’ region $z,\bar{z}\to 0$ where our approximation
$z\approx\frac{1}{2}$ breaks down. Besides, at fixed $\beta$, $z,\bar{z}\to 0$
means $q^{2}_{\perp}\to 0$ and $\mu^{2}=z\bar{z}Q^{2}\to 0$, so the twist
expansion such as (112) becomes invalid. Furthermore, already the unpolarized
cross section integrated over $z,q_{\perp}$ receives a significant
contribution from this nonperturbative region, hence the calculation is not
reliable Bartels et al. (1996b); Forshaw and Ross (1998).

To remedy this, we need to impose a lower cutoff in $q_{\perp}$. Since we try
to avoid having to measure jets, in practice, we shall impose this cutoff for
the transverse momentum $P_{h\perp}$ of certain hadron species $h$ by
introducing the corresponding fragmentation function. We are thus led to
consider the cross section

$\displaystyle\frac{d\sigma}{dydQ^{2}d\phi_{l_{\perp}}dz_{f}dP_{h\perp}^{2}d^{2}\Delta_{\perp}}=\int_{z_{f}}^{1}\frac{dz}{z}\frac{z^{2}}{z_{f}^{2}}D_{q}^{h}\left(\frac{z_{f}}{z}\right)f\left(z,\frac{z}{z_{f}}P_{h}\right)+\int_{z_{f}}^{1}\frac{d\bar{z}}{\bar{z}}\frac{\bar{z}^{2}}{z_{f}^{2}}D_{\bar{q}}^{h}\left(\frac{z_{f}}{\bar{z}}\right)f\left(z,\frac{\bar{z}}{z_{f}}P_{h}\right),$
(207)

where

$\displaystyle z_{f}=\frac{P\cdot P_{h}}{P\cdot q},$ (208)

is the standard variable in semi-inclusive DIS. We then insert (205) and
integrate over $P_{h}^{2}$ with a lower cutoff $\Lambda\sim 1$ GeV.

$\displaystyle\frac{d\sigma}{dydQ^{2}d\phi_{l_{\perp}}dz_{f}d\beta
d^{2}\Delta_{\perp}}$ $\displaystyle=$
$\displaystyle\int_{\Lambda^{2}}^{\infty}dP_{h}^{2}\int_{z_{f}}^{1}\frac{dz}{z}\frac{z^{2}}{z_{f}^{2}}D_{q}^{h}\left(\frac{z_{f}}{z}\right)f\left(z,\frac{z}{z_{f}}P_{h}\right)\delta\left(\beta-\frac{z(1-z)Q^{2}}{q_{\perp}^{2}+z(1-z)Q^{2}}\right)+(q\to\bar{q})$
(209) $\displaystyle=$
$\displaystyle\int^{1}_{z_{f}}\frac{dz}{z}\int^{\infty}_{\Lambda^{2}\frac{z^{2}}{z_{f}^{2}}}dq_{\perp}^{2}D_{q}^{h}\left(\frac{z_{f}}{z}\right)f\left(z,q_{\perp}\right)\delta\left(\beta-\frac{z(1-z)Q^{2}}{q_{\perp}^{2}+z(1-z)Q^{2}}\right)+(q\to\bar{q})$
$\displaystyle=$
$\displaystyle\left.\int^{z_{max}}_{z_{f}}\frac{dz}{z}\frac{z(1-z)Q^{2}}{\beta^{2}}\left(D_{q}^{h}\left(\frac{z_{f}}{z}\right)+D_{\bar{q}}^{h}\left(\frac{z_{f}}{z}\right)\right)f(z,q_{\perp})\right|_{q_{\perp}^{2}=z(1-z)\frac{1-\beta}{\beta}Q^{2}},$

where the upper limit of the $z$-integral is fixed by the condition

$\displaystyle
q_{\perp}^{2}=z(1-z)\frac{1-\beta}{\beta}Q^{2}>\Lambda^{2}\frac{z^{2}}{z_{f}^{2}}\qquad\to\qquad
z_{f}<z<\frac{\frac{1-\beta}{\beta}Q^{2}}{\frac{1-\beta}{\beta}Q^{2}+\frac{\Lambda^{2}}{z_{f}^{2}}}\equiv
z_{max}\,.$ (210)

Requiring $z_{max}>z_{f}$, we find the following constraints on the parameters
involved

$\displaystyle\frac{\beta\Lambda^{2}}{(1-\beta)Q^{2}}<z_{f}(1-z_{f})<\frac{1}{4}.$
(211)

Another constraint specific to the present work is that we have to make sure
that the $z$-integral is limited to the region $z\sim\frac{1}{2}$. This can be
easily done. For example, taking $Q^{2}=3$ GeV2, $\beta=0.2$, $\Lambda^{2}=1$
GeV2, $z_{f}=0.4$, we find

$\displaystyle 0.4<z<0.66.$ (212)

The formula (209) is an example of semi-inclusive diffractive DIS, or ‘SIDDIS’
introduced recently in Hatta et al. (2022) (see also Fucilla et al. (2024);
Guo and Yuan (2023)). It is a combination of semi-inclusive DIS (SIDIS) and
diffractive DIS. One measures an elastically scattered nucleon in the final
state followed by a rapidity gap $\Delta Y=\ln\frac{1}{2\xi}$ devoid of
particles, and then by a diffractively produced system with invariant mass
$M_{X}$ which spans a rapdity interval $\Delta Y=\ln\frac{1}{\beta}$. Out of
this diffractive system, one inclusively measures the transverse momentum of
one hadron species (above some cutoff $P_{h\perp}>\Lambda$ in our case). For
the present purpose, one may consider tagging several hadron species, or even
all charged hadrons, in order to increase the cross section. Experimentally,
this is a crucial simplification because measuring hadrons is easier than
measuring jets at low transverse momentum.

##  VI Numerical results

Finally, in this section we make predictions for the EIC. The numerical result
for the dijet cross section has been reported in Bhattacharya et al. (2022a).
Here we explain the details of the computation and study the impact of the
spin-orbit contribution which was not included in Bhattacharya et al. (2022a).
We also make new predictions for the ‘SIDDIS’ observable discussed in the
previous section.

###  VI.1 Non-perturbative input: Double distribution approach

In order to evaluate the various Compton form factors in the formula (164),
first we have to model the GPDs
$f(x,\xi)=\\{H_{g}(x,\xi),\tilde{H}_{g}(x,\xi),xL_{g}(x,\xi),xC_{g}(x,\xi)\\}$.
We use the so-called double distribution approach Radyushkin (1999, 2000)
wherein a GPD $f(x,\xi)$ is reconstructed from its forward limit $f(x)$ (PDF)
via the formula

$\displaystyle f(x,\xi)$
$\displaystyle=\dfrac{1}{\xi}\theta(x-\xi)\int^{x_{1}}_{x_{2}}d\beta
K(\dfrac{x-\beta}{\xi},\beta)+\dfrac{1}{\xi}\theta(\xi-x)\left[\int^{x_{1}}_{0}d\beta
K(\dfrac{x-\beta}{\xi},\beta)-(x\to-x)\right],$ (213)

where

$K(\alpha,\beta)=\frac{15}{16}\frac{((1-|\beta|)^{2}-\alpha^{2})^{2}}{(1-|\beta|)^{5}}\times
f(\beta),$ (214)

and the limits of the integrals $(x_{1},x_{2},x_{3})$ are:

$\displaystyle x_{1}=\dfrac{x+\xi}{1+\xi},\quad
x_{2}=\dfrac{x-\xi}{1-\xi},\quad x_{3}=-\dfrac{\xi-x}{1+\xi}.$ (215)

Once the GPDs are obtained this way, the associated Compton form factors are
evaluated using the formulas collected in Appendix A. We use $H_{g}(x)=xG(x)$
from JAM 19 (NLO) Sato et al. (2020) and $\tilde{H}_{g}(x)=x\Delta G(x)$ from
JAM 17 (NLO) Ethier et al. (2017). As for $xL_{g}(x)$ and $xC_{g}(x)$, we use
the Wandzura-Wilczek approximation (see Hatta and Yoshida (2012) and (152))

$\begin{split}&L_{g}(x)\,\,\approx\,\,x\int^{1}_{x}\dfrac{dx^{\prime}}{x^{\prime
2}}(H_{g}(x^{\prime})+E_{g}(x^{\prime}))\,-\,2x\int^{1}_{x}\dfrac{dx^{\prime}}{x^{\prime
2}}\Delta G(x^{\prime}),\\\\[5.69046pt]
&C_{g}(x)\,\,\approx\,\,x\int_{x}^{1}\frac{dx^{\prime}}{x^{\prime
2}}\tilde{H}_{g}(x^{\prime})-2x\int_{x}^{1}\frac{dx^{\prime}}{x^{\prime
2}}G(x^{\prime}),\end{split}$ (216)

though of course, we are eventually interested in constraining the total OAM
distribution including the genuine twist-three terms. In Bhattacharya et al.
(2022a), we neglected $E_{g}(x)$ in the above formula. However, a recent study
has revealed a rapid growth of $E_{g}(x)$ at small-$x$, with the ratio of
$E_{g}(x)/H_{g}(x)$ approaching a constant as $x\to 0$ Hatta and Zhou (2022).
We thus include it assuming the form $E_{g}(x)\approx-0.15H_{g}(x)$ where the
coefficient is from the light-cone spectator model Tan and Lu (2023). We
however find that the impact of $E_{g}(x)$ is minor, since $L_{g}(x)$ in (216)
is dominated by the last term as mentioned already.

We integrate the cross section over $|\Delta_{\perp}|$ assuming a Gaussian
form $d\sigma\propto e^{-b\Delta_{\perp}^{2}}$ with $b=5$ GeV-2 Braun and
Ivanov (2005) and plot

$\displaystyle\frac{d\sigma_{\rm DSA}}{dydQ^{2}d\Delta\phi dz(d\xi\ {\rm or}\
q_{\perp}^{2})},\qquad\frac{d\sigma_{\rm DSA}}{dydQ^{2}d\Delta\phi
dz_{f}d\beta},$ (217)

for the dijet and SIDDIS cross sections, respectively. The conversion
$\xi\leftrightarrow q_{\perp}^{2}$ is done according to (35). We fix $y=0.7$,
$\sqrt{s}=120$ $\rm{GeV}$,
$\Delta\phi=\phi_{l_{\perp}}-\phi_{\Delta_{\perp}}=0$, and employ three values
of photon virtuality $Q^{2}=2.7$ GeV2, 4.8 GeV2 and 10 GeV2. Typically, with
these values we probe $\xi\big{|}_{\rm min.}\sim 10^{-3}$ corresponding to
$q_{\perp}\big{|}_{\rm min.}=1$ $\rm{GeV}$.

###  VI.2 Results for dijet process

Figure 5: The DSA part of the differential cross-section, both with and
without $E_{g}$, is presented as a function of $\xi$ at
$Q^{2}=2.7\,\textrm{GeV}^{2}$ (left), $Q^{2}=4.8\,\textrm{GeV}^{2}$ (middle),
and $Q^{2}=10\,\textrm{GeV}^{2}$ (right). The results with $E_{g}=0$
correspond to those reported in our previous work Bhattacharya et al. (2022a).
These plots suggest that the incorporation of contributions from $E_{g}$ into
the orbital angular momentum does not result in a significant alteration of
our previous results.

Figure 6: The DSA part of the differential cross-section with $E_{g}\neq 0$ is
presented as a function of $\xi$ at $Q^{2}=2.7\,\textrm{GeV}^{2}$ (left),
$Q^{2}=4.8\,\textrm{GeV}^{2}$ (middle), and $Q^{2}=10\,\textrm{GeV}^{2}$
(right). These plots include contributions from spin-orbit correlation,
explicitly demonstrating how the results in the above figure are
quantitatively modified in the presence of this contribution.

Figure 7: The DSA part of the differential cross-section with $E_{g}\neq 0$ is
presented as a function of $q_{\perp}$ at $Q^{2}=2.7\,\textrm{GeV}^{2}$
(left), $Q^{2}=4.8\,\textrm{GeV}^{2}$ (middle), and
$Q^{2}=10\,\textrm{GeV}^{2}$ (right).

In Fig. 5 and Fig. 6, we present Eq. (217) as functions of $\xi$. In the
former, we display the results of our previous work with $E_{g}$ activated,
contrasting them with our published findings. In the latter, we introduce the
spin-orbit correlation term. Our analysis reveals that the OAM and helicity
contributions tend to either cancel or reinforce each other, depending on
$Q^{2}$, as discussed earlier. Notably, the spin-orbit correlation and
helicity terms exhibit comparable magnitudes and are substantial. Since the
spin-orbit correlation is solely dictated by the unpolarized gluon
distribution, it is relatively better constrained than OAM, which requires
precise knowledge of both unpolarized and polarized gluon distributions. The
latter has significant uncertainties, particularly at small $x$. Subtracting
the contributions from the helicity and spin-orbit correlation terms could
provide direct sensitivity to OAM.

In Fig. 7, we present the same plot as in Fig. 6, but now as a function of the
dijet transverse momentum $q_{\perp}$. Note that the units of the vertical
axis have changed pb $\to$ fb (apart from the trivial dimensional factor
GeV-2). This is because of the typical value $\xi\sim 10^{-3}$ probed in this
kinematics.

It is important to emphasize that our numerical estimations incorporated
several approximations, including systematic errors inherent in the double
distribution method and the Wandzura-Wilczek approximation. However, the
primary source of uncertainty lies in the unpolarized and especially polarized
gluon distributions which are currently not well constrained at small-$x$.
Moreover, ideally the next-to-leading order corrections should be included
Boussarie et al. (2016). Due to these reasons, we did not attempt to include
error bars in the above plots. Therefore, our numerical results provide only a
rough estimate of the overall magnitudes and signs of various contributions.
In the future, as more accurate determinations of $\Delta G(x)$ become
available, our predictions can be correspondingly improved.

###  VI.3 Results for SIDDIS process

Figure 8: Plot depicting the SIDDIS cross section as a function of the $\beta$
parameter for $\Lambda=1$ GeV and at two different $Q^{2}$ values:
$2.7\,\textrm{GeV}^{2}$ (left) and $10\,\textrm{GeV}^{2}$ (right). The maximum
$\beta$ value is restricted by the condition $q_{\perp}<1$ GeV.

Figure 9: Plot depicting the SIDDIS cross section as a function of the $\beta$
parameter for $\Lambda=0.5$ GeV and at two different $Q^{2}$ values:
$2.7\,\textrm{GeV}^{2}$ (left) and $10\,\textrm{GeV}^{2}$ (right). The change
of OAM sign, resulting from the interplay between OAM and helicity at small
$x$, is once again clearly observable in the high $\beta$ region.

In Figs. 8 and 9, we present the SIDDIS cross section as a function of the
$\beta$ parameter for two distinct cutoff values: $\Lambda=1.0$ and $0.5$ GeV.
The calculation considers tagged hadron species, such as charged pions and
kaons, while the fragmentation quarks are assumed to be
$u,\bar{u},d,\bar{d},s$ and $\bar{s}$. The change in the sign mechanism of the
OAM, as observed in the dijet process, is also evident here for $\Lambda=0.5$
GeV and in the higher regions of $\beta$. While the obtained number values
suggest that measuring this process might still be challenging, our main point
is that there is no need to reconstruct the dijet transverse momentum, yet
sensitivity to OAM is retained. Therefore, this approach may offer a promising
route for measurement at the EIC to constrain OAM.

##  VII Conclusion

In conclusion, our overarching goal has been to identify and establish
‘practical’ physical observables capable of directly measuring parton orbital
angular momentum (OAM). In this endeavor, we have expanded upon our previous
work Bhattacharya et al. (2022a) by offering a comprehensive analysis of the
longitudinal double spin asymmetry in exclusive dijet production during
electron-proton scattering. Our investigation reaffirms the sensitivity of the
$\cos\phi$ angular correlation as a crucial indicator of gluon OAM at
small-$x$, and its intricate interplay with gluon helicity. Furthermore, our
study uncovers an additional aspect unexplored in our prior research: the
spin-orbit correlation of gluons $C_{g}(x)$. For the first time, we shed light
on the small-$x$ behavior of this distribution (see also S. Bhattacharya, R.
Boussarie, and Y. Hatta (2024)) and find that it is approximately proportional
to the unpolarized gluon distribution but with an opposite sign. Consequently,
its contribution to the present observable is unsuppressed, contrary to naive
expectations. The determination of OAM therefore necessitates accurate
knowledge of the unpolarized gluon distribution at small-$x$, in addition to
the polarized gluon distribution $\Delta G(x)$. Alternatively, since the spin-
orbit correlation exists already in unpolarized hadrons, one may be able to
constrain $C_{g}(x)$ (including the genuine twist-three contributions) from
other spin-independent exclusive processes, see, e.g., Boussarie et al.
(2018). Based on the analytical results, we have performed a detailed
numerical analysis in both dijet and semi-inclusive diffractive deep inelastic
scattering processes. The latter may offer experimental advantages as it does
not require the reconstruction of dijet transverse momenta; instead, it
involves the inclusive tagging of one or more hadron species.

We have also undertaken the first exploration of quark-channel contributions
to double spin asymmetry, revealing an unexpected breakdown of factorization.
Indeed, even in the gluon case, we encountered an ‘end-point’ singularity at
$x=\pm\xi$ which we circumvented by restricting to $z\sim 1/2$. In the quark
case, we observed a similar singularity across all $z$ values due to the
$C$-odd exchange in the $t$-channel. We recognize the importance of revisiting
the present calculation using a dedicated $k_{T}$ factorization approach. This
approach holds promise to remedy the problems with factorization and make the
amplitude manifestly finite. But the connection to parton OAMs becomes
necessarily less direct.

Overall, our work contributes significantly to the ongoing efforts aimed at
unraveling the orbital angular momentum of partons and its role in shaping the
spin structure of the nucleon. We anticipate that our findings will serve as a
valuable foundation for inspiring and guiding further studies in this field.

###### Acknowledgements.

We thank Saad Nabeebaccus and Feng Yuan for discussion. The work of S. B. has
been supported by the Laboratory Directed Research and Development program of
Los Alamos National Laboratory under project number 20240738PRD1. S. B. has
also received support from the U. S. Department of Energy through the Los
Alamos National Laboratory. Los Alamos National Laboratory is operated by
Triad National Security, LLC, for the National Nuclear Security Administration
of U. S. Department of Energy (Contract No. 89233218CNA000001). S. B. and Y.
H. were 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. Y. H. is also supported by the
framework of the Saturated Glue (SURGE) Topical Theory Collaboration.

##  Appendix A Compton form factors

In this appendix we collect formulas for the Compton form factors introduced
in (120), (131) and (155) that are useful for the numerical evaluation of
(164).

$\displaystyle{\cal H}^{(1)}_{g}(\xi)$
$\displaystyle=\frac{1}{\xi}\left(H_{g}(\xi,\xi)\ln\frac{1-\xi}{1+\xi}+\int_{-1}^{1}dx\frac{H_{g}(x,\xi)-H_{g}(\xi,\xi)}{x-\xi}\right)-i\dfrac{\pi}{\xi}H_{g}(\xi,\xi),$
(218) $\displaystyle{\cal H}^{(2)}_{g}(\xi)$
$\displaystyle=\dfrac{1}{2}\bigg{(}H^{\prime}_{g}(\xi,\xi)\ln\dfrac{1-\xi}{1+\xi}+\int_{-1}^{1}dx\frac{H^{\prime}_{g}(x,\xi)-H^{\prime}_{g}(\xi,\xi)}{x-\xi}\bigg{)}$
$\displaystyle-\dfrac{1}{2\xi}\left(H_{g}(\xi,\xi)\ln\frac{1-\xi}{1+\xi}+\int_{-1}^{1}dx\frac{H_{g}(x,\xi)-H_{g}(\xi,\xi)}{x-\xi}\right)-i\frac{\pi}{2}\bigg{(}H^{\prime}_{g}(\xi,\xi)-\dfrac{1}{\xi}H_{g}(\xi,\xi)\bigg{)},$
(219) $\displaystyle\tilde{\cal H}^{(2)}_{g}(\xi)$
$\displaystyle=\dfrac{1}{2}\bigg{(}\tilde{H}^{\prime}_{g}(\xi,\xi)\ln\dfrac{1-\xi}{1+\xi}+\int_{-1}^{1}dx\frac{\tilde{H}^{\prime}_{g}(x,\xi)-\tilde{H}^{\prime}_{g}(\xi,\xi)}{x-\xi}\bigg{)}-i\frac{\pi}{2\xi}\tilde{H}^{\prime}_{g}(\xi,\xi),$
(220) $\displaystyle{\cal L}^{(2)}_{g}(\xi)$
$\displaystyle=\frac{1}{2\xi}\bigg{(}Y_{g}(\xi,\xi)\ln\frac{1-\xi}{1+\xi}+\int_{-1}^{1}dx\frac{Y_{g}(x,\xi)-Y_{g}(\xi,\xi)}{x-\xi}\bigg{)}-i\dfrac{\pi}{2\xi}Y_{g}(\xi,\xi),$
(221)

where we defined $H^{\prime}_{g}(x,\xi)\equiv\dfrac{d}{dx}{H_{g}(x,\xi)}$,
$\tilde{H}^{\prime}_{g}(x,\xi)\equiv\dfrac{d}{dx}{\tilde{H}_{g}(x,\xi)}$, and
$Y_{g}(x,\xi)\equiv\dfrac{d}{dx}xL_{g}(x,\xi)$. $C_{g}^{(2)}(\xi)$ is obtained
via the same formula as (219) with $H_{g}(x,\xi)\to xC_{g}(x,\xi)$.

Note that, when $\xi\ll 1$, ${\cal H}_{g}^{(1,2)}$ are dominated by the
imaginary parts

$\displaystyle{\mathfrak{I}m}{\cal
H}_{g}^{(1)}(\xi)=-\frac{\pi}{\xi}H_{g}(\xi,\xi),\qquad{\mathfrak{I}m}{\cal
H}^{(2)}_{g}(\xi)=-\frac{\pi}{2}\left(\left.\frac{dH_{g}(x,\xi)}{dx}\right|_{x=\xi}-\frac{1}{\xi}H_{g}(\xi,\xi)\right).$
(222)

Moreover, assuming a Regge-like behavior $H_{g}(x,\xi)\sim\frac{1}{x^{c}}$
with $c\propto\alpha_{s}$, we see that the second term in ${\cal H}_{g}^{(2)}$
is parametrically larger than the first term. This leads to the relation

$\displaystyle{\rm Im}{\cal H}_{g}^{(1)}(\xi)\approx-2{\rm Im}{\cal
H}_{g}^{(2)}(\xi).$ (223)

In Figs. 10, 11, and 12 we present plots depicting both the real and imaginary
components of the Compton form factors as functions of $\xi$. As anticipated,
the imaginary components of the Compton factors ${\cal{H}}^{(1,2)}_{g}$
significantly outweigh their real counterparts at small $\xi$ values. We also
see that the relation (223) is approximately satisfied. However, these
properties do not hold for the helicity and OAM Compton factors
${\cal{\tilde{H}}}^{(2)}_{g}$ and ${\cal L}_{g}^{(2)}$. From the plots, we
observe that the real parts are nearly as prominent as the imaginary ones for
small $\xi$ values and they roughly satisfy

$\displaystyle\tilde{\cal H}_{g}^{(2)}\approx-{\cal L}_{g}^{(2)},$ (224)

as expected (see a comment after (165)). Somewhat unexpectedly, however, we
find that $|\tilde{\cal H}_{g}^{(2)}|\sim|{\cal L}_{g}^{(2)}|$ are about an
order of magnitude larger than $|{\cal H}_{g}^{(1,2)}|$, cf., a comment below
(17).

Figure 10: CFFs of the unpolarized gluon GPDs.

Figure 11: CFFs of the helicity (left) and OAM (right) distributions.  Figure
12: CFFs of the spin-orbit correlation.

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