MIT-CTP/5632aainstitutetext: Department of Physics and Astronomy, University
of California, Los Angeles, CA 90095, USAbbinstitutetext: Mani L. Bhaumik
Institute for Theoretical Physics, University of California, Los Angeles, CA
90095, USAccinstitutetext: Center for Frontiers in Nuclear Science, Stony
Brook University, Stony Brook, NY 11794, USAddinstitutetext: Nuclear Science
Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720,
USAeeinstitutetext: Center for Theoretical Physics, Massachusetts Institute of
Technology, Cambridge, MA 02139ffinstitutetext: Department of Physics, Center
for Field Theory and Particle Physics and Key Laboratory of Nuclear Physics
and Ion-beam Application (MOE), Fudan University, Shanghai 200433,
Chinagginstitutetext: Shanghai Research Center for Theoretical Nuclear
Physics, NSFC and Fudan University, Shanghai 200438, China

# Probing Transverse Momentum Dependent Structures with Azimuthal Dependence
of Energy Correlators

Zhong-Bo Kang d,e , Kyle Lee f,g , Ding Yu Shao e,a , Fanyi Zhao
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We study the azimuthal angle dependence of the energy-energy correlators
$\langle\mathcal{E}(\hat{n}_{1})\mathcal{E}(\hat{n}_{2})\rangle$ in the back-
to-back region for $e^{+}e^{-}$ annihilation and deep inelastic scattering
(DIS) processes with general polarization of the proton beam. We demonstrate
that the polarization information of the beam and the underlying partons from
the hard scattering is propagated into the azimuthal angle dependence of the
energy-energy correlators. In the process, we define the Collins-type EEC jet
functions and introduce a new EEC observable using the lab-frame angles in the
DIS process. Furthermore, we extend our formalism to explore the two-point
energy correlation between hadrons with different quantum numbers
$\mathbb{S}_{i}$ in the back-to-back limit
$\langle\mathcal{E}_{\mathbb{S}_{1}}(\hat{n}_{1})\mathcal{E}_{\mathbb{S}_{2}}(\hat{n}_{2})\rangle$.
We find that in the Operator Product Expansion (OPE) region the
nonperturbative information is entirely encapsulated by a single number. Using
our formalism, we present several phenomenological studies that showcase how
energy correlators can be used to probe transverse momentum dependent
structures.

##  1 Introduction

The energy-energy correlator (EEC) Basham:1978bw ; Basham:1978zq  is one of
the earliest infrared and collinear (IRC) safe observables Kinoshita:1962ur ;
Lee:1964is . It measures the energy correlations as a function of the angle
$\theta$ between two detectors. This allows EEC to be represented in terms of
the correlation function of energy flow operators Sveshnikov:1995vi ;
Tkachov:1995kk ; Korchemsky:1999kt ; Bauer:2008dt ; Hofman:2008ar ;
Belitsky:2013xxa ; Belitsky:2013bja ; Kravchuk:2018htv  defined as

$\displaystyle\mathcal{E}(\vec{n})=\int_{0}^{\infty}dt\lim_{r\rightarrow\infty}r^{2}n^{i}T_{0i}(t,r\vec{n})\,,$
(1)

making them a subject of extensive study in conformally invariant
$\mathcal{N}=4$ super-Yang-Mills (SYM) theory Hofman:2008ar ; Belitsky:2013xxa
; Belitsky:2013bja ; Belitsky:2013ofa ; Chicherin:2023gxt . The EEC as event
shape observables has been extensively measured in $e^{+}e^{-}$ collisions
SLD:1994idb ; L3:1992btq ; OPAL:1991uui ; TOPAZ:1989yod ; TASSO:1987mcs ;
JADE:1984taa ; Fernandez:1984db ; Wood:1987uf ; CELLO:1982rca ; PLUTO:1985yzc
; OPAL:1990reb ; ALEPH:1990vew ; L3:1991qlf ; SLD:1994yoe  and most recently
has been a target of study in electron-hadron scatterings for the future
Electron-Ion Collider (EIC) Li:2021txc ; Neill:2022lqx . In hadron-hadron
collisions, a generalization of the EEC called transverse EEC (TEEC) has been
measured and studied at the Large Hadron Collider (LHC) kinematics Gao:2019ojf
; ATLAS:2015yaa ; ATLAS:2017qir ; ATLAS:2020mee ; Alvarez:2023fhi . The study
of EEC has played a crucial role in advancing our understanding of fundamental
particles and their interactions. For example, these measurements of EEC
provide one of the sharpest determination of the strong coupling constant for
both hadron and lepton collider environments ParticleDataGroup:2020ssz ;
deFlorian:2004mp ; Burrows:1995vt ; Kardos:2018kqj ; Alvarez:2023fhi .

Over the last several decades, with particularly reinvigorated efforts in
recent years, there have been significant advances in the theoretical studies
of the EEC. There have been rigorous computations at fixed orders for
arbitrary $\theta$ angles Schneider:1983iu ; Falck:1988gb ; Glover:1994vz ;
Kramer:1996qr ; Ali:1982ub ; Ali:1984gzn ; Richards:1982te ; Richards:1983sr ;
Catani:1996jh ; Tulipant:2017ybb ; Belitsky:2013xxa ; Belitsky:2013bja , where
the state-of-the-art computations provide an analytical expressions up to NLO
Dixon:2018qgp ; Luo:2019nig  and numerical results at NNLO accuracy
DelDuca:2016ily  in QCD, and analytical evaluations at both NLO
Belitsky:2013ofa  and NNLO Henn:2019gkr  for $\mathcal{N}=4$ SYM. There has
also been many important works to better understand the nonperturbative
structure of the EEC Schindler:2023cww ; Nason:1995np ; Korchemsky:1999kt ;
Belitsky:2001ij . Furthermore, the factorization structure of the singular
regions of the EEC in the back-to-back ($\theta\to\pi$) Kodaira:1981nh ;
Kodaira:1982az ; deFlorian:2004mp ; Tulipant:2017ybb ; Moult:2018jzp ;
Ebert:2020sfi ; Duhr:2022yyp ; Moult:2019vou ; Collins:1981uk  and collinear
limits ($\theta\to 0$) Dixon:2019uzg ; Korchemsky:2019nzm ; Konishi:1978yx ;
Konishi:1978ax  have been better understood in recent years using the soft
collinear effective field theory (SCET) Bauer:2001yt ; Bauer:2000yr ;
Bauer:2001ct , allowing them to be studied to unprecedented accuracy in both
regions.111The collinear limit has been computed at NNLL accuracy. For rich
interplay and wide-ranging applications of EEC in the collinear limit with the
jet substructure program, see Lee:2023npz ; Devereaux:2023vjz ; Craft:2022kdo
; Lee:2022ige ; Yang:2023dwc ; Andres:2022ovj ; Procura:2022fid ; Chen:2020vvp
. In particular, connection to the Transverse-Momentum Dependent (TMD)
observables Boussarie:2023izj  in the back-to-back limit has been observed and
studied in the context of unpolarized processes in both $e^{+}e^{-}$ and
appropriately modified definition of the EEC for the $ep$ collisions. With the
state-of-the-art computation of the $4$-loop rapidity anomalous dimension
Moult:2022xzt ; Duhr:2022yyp , the EEC in the back-to-back limit has been
calculated at very impressive N4LL accuracy Duhr:2022yyp . Within the
$\mathcal{N}=4$ SYM, even the subleading power resummation in the back-to-back
limit has been carried out Chen:2023wah  using techniques from conformal
bootstrap. Many developments of EEC like this in the context of
$\mathcal{N}=4$ SYM give hope of importing valuable techniques from the CFTs
to improve and better our understanding for QCD as well.

Figure 1: Illustration of EEC for $e^{+}e^{-}$ annihilation. Here $\theta$ is
the angular separation between the two detectors pointing along $\hat{n}_{1}$
and $\hat{n}_{2}$ and $\phi$ is the azimuthal angle between the plane formed
by these detectors (referred to as “detector plane”, shown as yellow) and the
plane generated by the direction $\hat{n}_{2}$ and the beam (referred to as
“lepton plane”, shown as gray).

While significant progress has been made for the EEC, almost all the existing
work concentrate on studying the EEC as a function of the angle $\theta$
only.222Refer to Chen:2020adz ; Barata:2023zqg ; Li:2023gkh  for several
important recent examples that are exceptions, notably considering the
azimuthal angle dependence in the collinear limit of the EEC, as well as for
the nucleon EEC. However, as shown already in the original paper where the EEC
was first introduced Basham:1978bw ; Basham:1978zq , when considering the
incoming beam direction in $e^{+}e^{-}$ collisions, there would be two
additional independent angles one has to introduce in order to fully describe
the EEC. As shown in Fig. 1, in addition to the usual angle $\theta$ between
the two detectors, one also has an azimuthal angle difference $\phi$ between
the plane formed by the detectors (referred to as “detector plane”, shown as
yellow) and the plane generated by $\hat{n}_{2}$ and the beam (referred to as
“lepton plane”, shown as gray). The very first proposal of energy correlators
in Basham:1978bw  included an LO calculation of the EEC as a function of the
angle $\theta$ and the azimuthal angle $\phi$. However, at the end of the
paper, the authors decided to integrate over the azimuthal angle $\phi$,
indicating that the importance of this azimuthal dependence in the EEC was not
realized back then. Since then, higher-order computations of energy
correlators have consistently integrated such azimuthal dependence.

In recent years, spin asymmetries have been widely studied in the hadron
physics community. Because these asymmetries provide nontrivial quantum
correlations and imaging for the hadron structure, their studies have been one
of the most important scientific thrusts for the future EIC Accardi:2012qut ;
AbdulKhalek:2021gbh ; AbdulKhalek:2022hcn . Motivated by the fact that many
spin asymmetries arise from the azimuthal correlations, we study the azimuthal
dependent EEC. We will develop the theoretical formalism to demonstrate the
full potential of such azimuthal dependent EEC observables, in particular in
terms of exploring the nucleon structure.

In this paper we concentrate on the EEC in the back-to-back region where the
angle between the detectors approaches $\theta\to\pi$ and study its azimuthal
angle dependence. For convenience, we also often use the variable $\tau$
defined as333Note that $z$ variable defined as $z=1-\tau$ is also often used
in the literature Moult:2018jzp ; Ebert:2020sfi .

$\displaystyle\tau=\frac{1+\cos\theta}{2}\,,$ (2)

and thus the back-to-back region corresponds to $\tau\to 0$. Then the
azimuthal dependent EEC in the $e^{+}e^{-}$ collisions is defined as

$\displaystyle\mathrm{{{EEC}}}_{e^{+}e^{-}}(\tau,\phi)\equiv\frac{d\Sigma_{e^{+}e^{-}}}{d\tau
d\phi}=\frac{1}{2}\sum_{1,2}\int d\sigma
z_{1}z_{2}\,\delta\left(\tau-\frac{1+\cos\theta_{12}}{2}\right)\delta(\phi-\phi_{12})\,,$
(3)

where the weighted sum over pairs of hadrons produced in the final state give
correlation between the energy flown into the detectors in the form of hadrons
detected. We also use $z_{1,2}=2E_{1,2}/Q$ to denote the energy fractions of
the final-state hadrons $1,2$ flown into the detectors $1,2$, respectively.
This definition of the EEC can also be recasted using the operator definition
of the energy flow given in Eq. (1) as

$\displaystyle\frac{1}{\sigma_{\rm tot}}\frac{d\Sigma_{\rm e^{+}e^{-}}}{d\tau
d\phi}=\frac{\left\langle\mathcal{O}\mathcal{E}\left(\vec{n}_{1}\right)\mathcal{E}\left(\vec{n}_{2}\right)\mathcal{O}^{\dagger}\right\rangle}{\left\langle\mathcal{OO}^{\dagger}\right\rangle}\,,$
(4)

where $\mathcal{O}$ is a source operator that creates the excitation that is
detected in the form of energy carried by the hadrons in the asymptotic
detector.

Below, we will demonstrate the similarity between the azimuthal dependent EEC
and the usual TMD factorization formalism. In particular, we show in the back-
to-back region that a new term $\propto\cos(2\phi)$ will arise for
$\mathrm{{{EEC}}}_{e^{+}e^{-}}(\tau,\phi)$, which is related to the Collins
fragmentation function Collins:1992kk , one of the most extensively discussed
polarized transverse momentum dependent functions (TMDs), describing the
fragmentation of a transversely polarized quark into an unpolarized hadron.
This fragmentation process correlates the transverse momentum of the outgoing
hadron with the transverse polarization of the quark, giving rise to non-
trivial azimuthal angular asymmetries. For the $e^{+}e^{-}$ annihilation, two
transversely polarized quarks can be produced from the unpolarized
$e^{+}e^{-}$ without violating spin conservation. The Collins asymmetries then
manifest itself with the azimuthal asymmetry $\cos(2\phi)$ in the azimuthal
dependent EEC observable.

In addition, we introduce a similar version of azimuthal dependent EEC in deep
inelastic $ep$ scattering (DIS), extending the unpolarized case considered in
Li:2021txc  using the Breit frame. We show that the azimuthal dependence in
the EEC would allow us to probe the transverse momentum dependent parton
distribution functions (TMD PDFs). In particular, in the context of $ep$
collisions employing polarized electron and/or proton beams, the EEC in DIS in
the back-to-back region exhibits a remarkable correlation between the EEC jet
functions defined below and the polarized and unpolarized TMD PDFs,
highlighting the immense potential of azimuthal dependent EEC as a novel tool
for probing TMD PDFs and advancing our knowledge of the internal structure of
nucleons. Furthermore, building on the insights presented in Gao:2022bzi , we
introduce a new EEC observable in DIS defined using the lab-frame angle
$q_{*}$. By solely relying on angles defined within the Lab frame, this
observable offers an order-of-magnitude improvement in the anticipated
experimental resolution at the EIC.

The rest of the paper is structured as follows. Section 2 presents a
comprehensive analysis of the factorization framework governing azimuthal
dependent energy-energy correlators in $e^{+}e^{-}$ and $ep$ collisions. We
introduce the Collins-type EEC jet function, which has a close relation to the
Collins fragmentation function. In Section 3, we study the properties of the
unpolarized and Collins-type EEC jet functions. Section 4 presents our
phenomenological study to demonstrate the potential of azimuthal dependent EEC
observables for probing nucleon structures. Finally, we conclude our work in
Section 5.

##  2 Factorization Formalism for Azimuthal Dependent EEC

In this section, we study the azimuthal dependent EEC observables for both
$e^{+}e^{-}$ annihilation and DIS processes. We show that in the back-to-back
region, they can be related to the TMD factorization framework. We demonstrate
that in the factorization formalism, beside the usual unpolarized EEC jet
function which is related to the unpolarized TMD fragmentation functions, the
Collins-type EEC jet function arises that is closely connected with the
Collins fragmentation functions. We write down all the azimuthal angle
dependent correlations for the EEC$(\tau,\phi)$ observables for both
$e^{+}e^{-}$ and DIS processes. In our study of the EEC in DIS, we extend the
version adapted from the Breit frame as cited in Li:2021txc . Drawing
inspiration from Gao:2022bzi , we also introduce a new EEC utilizing lab-frame
angles. Furthermore, we demonstrate the applicability of EEC in DIS in probing
both polarized and unpolarized nucleon structures.

###  2.1 Azimuthal dependent EEC for $e^{+}e^{-}$ annihilation

We will first discuss the azimuthal dependent EEC for the $e^{+}e^{-}$
annihilation. Let us start with specifying the details of the coordinate frame
in which the observables will be measured, shown already in Fig. 1. As often
done in the TMD factorization for $e^{+}e^{-}$, we adopt the so-called
Gottfried-Jackson (GJ) frame Gottfried:1964nx ; Pitonyak:2013dsu  by aligning
the detector 2 along the $\hat{n}_{2}=\hat{n}_{z}$. Therefore, the angle
$\theta$ between the two detectors is the polar angle of the detector 1 in the
GJ frame. With this setup, we can easily measure the azimuthal angle of the
detector plane with respect to the lepton plane, as mentioned already in the
Introduction and shown clearly in Fig. 1. This coordinate frame is slightly
different from what was introduced in the original EEC paper Basham:1978bw
where the $+z$ axis is defined to be along the incoming lepton $e^{-}$ beam,
while the azimuthal angle $\phi$ is kept the same. In general, one can also
parameterize the difference between the two frame choices by keeping track of
the angle between the $e^{-}$ beam and the detector $2$. However, as we
integrate over this angle and only measure $\theta$ and $\phi$, the two
coordinate frame choice give the same results. This demonstrates that our
choice of using GJ frame was not unique, though it is useful to visualize the
measurements.

In the back-to-back limit of the two detectors where the angle approaches
$\theta\to\pi$ (or $\tau\to 0$), we find up to power corrections,

$\displaystyle\tau=\frac{\bm{P}_{h_{1}\perp}^{2}}{z_{1}^{2}Q^{2}}\,,$ (5)

for a given pair of hadrons in the two detectors. Here, $\bm{P}_{h_{1}\perp}$
is the transverse momentum of the hadron $h_{1}$ in the GJ frame, namely with
respect to $\hat{n}_{2}$ in the pair production. The azimuthal angle
associated with $\bm{P}_{h_{1}\perp}$ is simply $\phi$ in the GJ frame, as
shown in Fig. 1. The energy fractions $z_{i=1,2}$ are given by

$\displaystyle z_{i}=\frac{2P_{hi}\cdot q}{Q^{2}}=\frac{2E_{i}}{Q}\,,$ (6)

where $q=\ell_{e^{+}}+\ell_{e^{-}}$ with $Q^{2}=q^{2}$. For convenience, we
also introduce $\bm{q}_{T}=-\bm{P}_{h_{1}\perp}/z_{1}$.

With the relation between $\tau$ and ${\bm{q}}_{T}$ at hand in the back-to-
back region, the azimuthal dependent EEC for $e^{+}e^{-}$ defined in Eq. (3)
can then be related to the $\bm{q}_{T}$-differential cross section as

$\displaystyle\mathrm{{{EEC}}}_{e^{+}e^{-}}(\tau,\phi)$
$\displaystyle=\frac{d\Sigma_{e^{+}e^{-}}}{d\tau d\phi}$
$\displaystyle=\frac{1}{2}\sum_{1,2}\int
d^{2}\bm{q}_{T}\,dz_{1}\,dz_{2}\,z_{1}\,z_{2}\,\frac{d\sigma}{dz_{1}dz_{2}d^{2}\bm{q}_{T}}\,\delta\left(\tau-\frac{\bm{q}_{T}^{2}}{Q^{2}}\right)\delta(\phi-\phi_{12})\,,$
(7)

where the standard TMD factorization for the back-to-back di-hadron process in
$e^{+}e^{-}$ is given by Boer:2008fr ; Pitonyak:2013dsu

$\displaystyle\frac{d\sigma}{dz_{i}dz_{j}d^{2}{\bm{q}}_{T}}=$
$\displaystyle\,\sigma_{0}\,H(Q,\mu)\sum_{q}e_{q}^{2}\int
d^{2}{\bm{p}}_{1\perp}d^{2}{\bm{p}}_{2\perp}d^{2}{\bm{\lambda}}_{\perp}\delta^{2}\left(\frac{{\bm{p}}_{1\perp}}{z_{1}}+\frac{{\bm{p}}_{2\perp}}{z_{2}}-{\bm{\lambda}}_{\perp}+{\bm{q}}_{T}\right){S}({\bm{\lambda}}_{\perp}^{2},\mu,\nu)$
$\displaystyle\times\bigg{[}D_{1,h_{1}/q}^{(u)}(z_{1},{\bm{p}}_{1\perp}^{2},\mu,\zeta/\nu^{2})D^{(u)}_{1,h_{2}/\bar{q}}(z_{2},{\bm{p}}_{2\perp}^{2},\mu,\zeta/\nu^{2})+\cos(2\phi_{12})\left({\bm{\hat{}}{\bm{q}}}_{T,\alpha}{\hat{\bm{q}}}_{T,\beta}-\frac{1}{2}g_{\perp,{\alpha\beta}}\right)$
$\displaystyle\times\frac{\bm{p}_{1\perp}^{\alpha}}{z_{1}M_{1}}H_{1,h_{1}/q}^{\perp(u)}(z_{1},{\bm{p}}_{1\perp}^{2},\mu,\zeta/\nu^{2})\frac{\bm{p}_{2\perp}^{\beta}}{z_{2}M_{2}}H_{1,h_{2}/\bar{q}}^{\perp(u)}(z_{2},{\bm{p}}_{2\perp}^{2},\mu,\zeta/\nu^{2})\bigg{]}\,.$
(8)

Here we have integrated over the angle between the hadron $h_{2}$ and the
incoming beam direction from the conventional TMD factorization expression. It
is natural to concentrate on the dependence on the angle $\theta$ and $\phi$
only from the EEC perspective as it is an inclusive measurement over all the
hadron pairs as a function of angular separation of detectors in which they
are found. The born cross-section is given by $\sigma_{0}=4\pi
N_{c}\alpha_{\rm em}^{2}/3Q^{2}$. On the other hand,
$D_{1,h/q}^{(u)}(z,{\bm{p}}_{\perp}^{2},\mu,\zeta/\nu^{2})$ and
$H_{1,h/q}^{\perp(u)}(z,{\bm{p}}_{\perp}^{2},\mu,\zeta/\nu^{2})$ are the
unsubtracted ($u$) unpolarized and Collins transverse momentum dependent
fragmentation functions (TMD FFs) Boussarie:2023izj , with $\bm{p}_{\perp}$
the hadron transverse momentum with respect to the fragmenting parton. As we
have mentioned already, Collins fragmentation function describes the process
where a transversely polarized quark fragments into an unpolarized hadron with
the quark transverse spin correlated with the hadron transverse momentum. Here
$\mu$ and $\nu$ are the usual renormalization and rapidity renormalization
scales, and $\zeta$ is the Collins-Soper scale, for details see Ref.
Boussarie:2023izj . We also have $H(Q,\mu)$, the hard function, which at the
next-to-leading order is (see e.g. Becher:2008cf )

$\displaystyle
H(Q,\mu)=1+\frac{\alpha_{s}}{2\pi}C_{F}\left[3\ln\frac{Q^{2}}{\mu^{2}}-\ln^{2}\frac{Q^{2}}{\mu^{2}}-8+\frac{7\pi^{2}}{6}\right]\,.$
(9)

Finally, ${S}({\bm{\lambda}}_{\perp}^{2},\mu,\nu)$ is the soft function with
$\bm{\lambda}_{\perp}$ describing the soft recoil of the fragmenting partons
from being exactly back-to-back. The TMD factorization formalism can be
transformed into the coordinate $b$-space as

$\displaystyle\frac{d\sigma}{dz_{1}dz_{2}d^{2}{\bm{q}}_{T}}=$
$\displaystyle\,\sigma_{0}\,H(Q,\mu)\sum_{q}e_{q}^{2}\int\frac{d^{2}{\bm{b}}}{(2\pi)^{2}}e^{-i{\bm{b}}\cdot{\bm{q}}_{T}}S({\bm{b}}^{2},\mu,\nu)$
$\displaystyle\times\bigg{[}\tilde{D}^{(u)}_{1,h_{1}/q}(z_{1},b,\mu,\zeta/\nu^{2})\tilde{D}^{(u)}_{1,h_{2}/q}(z_{2},b,\mu,\zeta/\nu^{2})+\cos(2\phi_{12})\left({\bm{\hat{}}{\bm{q}}}_{T,\alpha}{\hat{\bm{q}}}_{T,\beta}-\frac{1}{2}g_{\perp,{\alpha\beta}}\right)$
$\displaystyle\hskip
14.22636pt\times\tilde{H}_{1,h_{1}/{q}}^{\perp(u)\,\alpha}(z_{1},{\bm{b}},\mu,\zeta/\nu^{2})\tilde{H}_{1,h_{2}/{q}}^{\perp(u)\,\beta}(z_{2},{\bm{b}},\mu,\zeta/\nu^{2})\bigg{]}\,,$
(10)

where the TMD FFs in the $b$-space are defined as

$\displaystyle\tilde{D}^{(u)}_{1,h/{q}}(z,b,\mu,\zeta/\nu^{2})=$
$\displaystyle\int
d^{2}{\bm{p}}_{\perp}e^{-i{\bm{b}}\cdot{{\bm{p}}_{\perp}}/{z}}D^{(u)}_{1,h/q}(z,{\bm{p}}_{\perp}^{2},\mu,\zeta/\nu^{2})\,,$
(11)
$\displaystyle\tilde{H}_{1,h/{q}}^{\perp(u)\,\alpha}(z,{\bm{b}},\mu,\zeta/\nu^{2})=$
$\displaystyle\int
d^{2}{\bm{p}}_{\perp}e^{-i{\bm{b}}\cdot{{\bm{p}}_{\perp}}/{z}}\frac{\bm{p}_{\perp}^{\alpha}}{zM}H_{1,h/q}^{\perp(u)}(z,{\bm{p}}_{\perp}^{2},\mu,\zeta/\nu^{2})\,,$
(12) $\displaystyle\equiv$
$\displaystyle\left(-\frac{i{\bm{b}}^{\alpha}}{2}\right)\tilde{H}_{1,h/{q}}^{\perp(u)}(z,b,\mu,\zeta)$
(13)

and likewise for the soft function ${S({\bm{b}}^{2},\mu,\nu)}$ in the
$b$-space, which has been computed to the three-loop order in Li:2016ctv . To
simplify the notation, here we introduce the “subtracted” TMD FFs as,

$\displaystyle\tilde{F}(z,{\bm{b}},\mu,\zeta)=\tilde{F}^{(u)}(z,{\bm{b}},\mu,\zeta/\nu^{2})\sqrt{S({\bm{b}}^{2},\mu,\nu)}\,,$
(14)

where $\tilde{F}$ can be unpolarized or the Collins TMD FFs. As a result, Eq.
(10) is further written as

$\displaystyle\frac{d\sigma}{dz_{1}dz_{2}d^{2}{\bm{q}}_{T}}=$
$\displaystyle\,\sigma_{0}\,H(Q,\mu)\sum_{q}e_{q}^{2}\int\frac{d^{2}{\bm{b}}}{(2\pi)^{2}}e^{-i{\bm{b}}\cdot{\bm{q}}_{T}}\bigg{[}\tilde{D}_{1,h_{1}/q}(z_{1},b,\mu,\zeta)\tilde{D}_{1,h_{2}/q}(z_{2},b,\mu,\zeta)$
$\displaystyle+\cos(2\phi_{12})\left({\bm{\hat{}}{\bm{q}}}_{T,\alpha}{\hat{\bm{q}}}_{T,\beta}-\frac{1}{2}g_{\perp,{\alpha\beta}}\right)\tilde{H}_{1,h_{1}/{q}}^{\perp\,\alpha}(z_{1},{\bm{b}},\mu,\zeta)\tilde{H}_{1,h_{2}/{q}}^{\perp\,\beta}(z_{2},{\bm{b}},\mu,\zeta)\bigg{]}\,.$
(15)

To proceed, we will also have a “subtracted” TMD Collins FF
$\tilde{H}_{1,h/{q}}^{\perp}(z,b,\mu,\zeta)$ following Eq. (13),

$\displaystyle\tilde{H}_{1,h_{1}/{q}}^{\perp\,\alpha}(z,{\bm{b}},\mu,\zeta)\equiv\left(-\frac{i{\bm{b}}^{\alpha}}{2}\right)\tilde{H}_{1,h/{q}}^{\perp}(z,b,\mu,\zeta)\,.$
(16)

Next, we carry out the Lorentz contractions on $\alpha$ and $\beta$ and then
perform the energy-fraction-weighted integrals in Eq. (2.1) to obtain the
azimuthal-dependent EEC in the back-to-back limit and arrive at

$\displaystyle\mathrm{{{EEC}}}_{e^{+}e^{-}}(\tau,\phi)$
$\displaystyle=\frac{d\Sigma_{e^{+}e^{-}}}{d\tau d\phi}$
$\displaystyle=\frac{1}{2}\sigma_{0}\,H(Q,\mu)\sum_{q}e_{q}^{2}\int\frac{b\,db}{2\pi}\bigg{[}J_{0}(b\sqrt{\tau}Q){J}_{q}(b,\mu,\zeta){J}_{\bar{q}}(b,\mu,\zeta)$
$\displaystyle\hskip
71.13188pt+\cos(2{\phi})\,\frac{b^{2}}{8}J_{2}(b\sqrt{\tau}Q)\left.{J}_{q}^{\perp}(b,\mu,\zeta){J}_{\bar{q}}^{\perp}(b,\mu,\zeta)\right]\,.$
(17)

This is one of the key results of our paper, which shows that a new term
$\propto\cos(2\phi)$ arises in the azimuthal dependent EEC observable for
$e^{+}e^{-}$ annihiliation. Here it is important to emphasize that beside
${J}_{q}(b,\mu,\zeta)$, the so-called unpolarized EEC jet function introduced
previously in Moult:2018jzp , we have defined a new EEC jet function
${J}_{q}^{\perp}(b,\mu,\zeta)$. The unpolarized EEC jet function has a close
relation to the unpolarized TMD FFs, while ${J}_{q}^{\perp}(b,\mu,\zeta)$ is
closely connected with the Collins fragmentation functions:

$\displaystyle{J}_{q}(b,\mu,\zeta)\equiv$
$\displaystyle\sum_{h}\int_{0}^{1}dz\,z\,\tilde{D}_{1,h/q}(z,b,\mu,\zeta)\,,$
(18) $\displaystyle{J}_{q}^{\perp}(b,\mu,\zeta)\equiv$
$\displaystyle\sum_{h}\int_{0}^{1}dz\,z\,\tilde{H}_{1,h/q}^{\perp}(z,b,\mu,\zeta)\,.$
(19)

Because of this close connection, we name the new EEC jet function
${J}_{q}^{\perp}(b,\mu,\zeta)$ as the “Collins-type” EEC jet function. Note
that the left-hand side in Eqs. (18) and (19) no longer carries hadron label
as we sum over the final state hadrons as appropriate from the inclusive
nature of the EEC. We will defer the discussion on the properties of these jet
functions, especially the Collins-type EEC jet function in Sec. 3. We will now
study the azimuthal dependent EEC in $ep$ collisions.

###  2.2 Azimuthal dependent EEC in DIS

A modified version of EEC in the deep inelastic $ep$ scattering was first
introduced in Li:2021txc  in the Breit frame. It measures the energy
correlation as a function of the angle $\theta$ (we also often use $\tau$
defined similarly as Eq. (2) for convenience) between a detector and the
incoming proton $p$ beam, as shown in the left panel of Fig. 2.444Technically,
since we detect energy flowing into only a single detector, calling it an
‘energy-energy’ correlator is somewhat of a misnomer. As in Li:2021txc , we
begin by generalizing the DIS version of EEC in the Breit frame, which we
define using the Trento conventions. In this frame, the exchanged virtual
photon has no temporal component and points along the $+z$ direction and the
proton beam moves along the $-z$ axis. We can then form two planes: the
detector plane that is generated by the detector direction and the initial
proton $p$ and the lepton plane that is formed by the incoming (outgoing)
leptons and the initial proton. We denote the azimuthal angle difference
between these two planes as $\phi$. We will be particularly interested in
studying the consequence of such an azimuthal dependent EEC for DIS process,
especially when both the incoming lepton and proton beams can be polarized.

Figure 2: Left: Illustration of EEC for DIS in the Breit frame using the
Trento conventions. Here the exchanged virtual photon is along the $+z$
direction and the incoming proton moves along the $-z$ axis, together with the
incoming lepton, they form the lepton plane (shown as gray in the figure). The
detector direction $\hat{n}$ and the initial proton forms the detector plane
(shown as yellow), which has an azimuthal angle $\phi$ with respect to the
lepton plane. The angle between the detector and the proton $p$ is given by
$\theta$. Right: EEC for DIS in the EIC lab frame. Here the incoming proton is
along the $+z^{\rm lab}$ direction and the incoming electron is along the
$-z^{\rm lab}$ direction.

Let us start with the kinematics. Since the EEC will be closely related to the
TMD factorization for producing a final-state hadron, $e(\ell)+p(P)\to
e(\ell^{\prime})+h(P_{h})+X$, we define the usual semi-inclusive DIS kinematic
variabbles,

$\displaystyle x_{B}=\frac{Q^{2}}{2P\cdot q},\qquad
y=\frac{Q^{2}}{x_{B}S},\qquad z_{h}=\frac{P\cdot P_{h}}{P\cdot q},$ (20)

where $q=\ell^{\prime}-\ell$ is the momentum of the exchanged virtual photon
with $Q^{2}=-q^{2}$ and $S=(\ell+P)^{2}$ is the electron-proton center-of-mass
energy. Denoting the corresponding opening and azimuthal angles of a given
final-state hadron measured in the detector as $\theta_{hp}$ and $\phi_{h}$,
we generalize the EEC definition in Li:2021txc  to include the azimuthal angle
dependence as

$\displaystyle\mathrm{{{EEC}}_{DIS}}(\tau,\phi)$
$\displaystyle\equiv\frac{d\Sigma_{\rm DIS}}{dx_{B}dyd\tau d\phi}$ (21)
$\displaystyle=\sum_{h}\int
d\theta_{hp}\,d\phi_{h}\,dz_{h}\,z_{h}\,\frac{d\sigma}{dx_{B}dydz_{h}d\theta_{hp}d\phi_{h}}\,\delta\left(\tau-\frac{1+\cos\theta_{hp}}{2}\right)\delta(\phi-\phi_{h})\,,$

where we have suppressed $x_{B}$ and $y$ dependence in the azimuthal dependent
EEC definition, $\mathrm{{{EEC}}_{DIS}}(\tau,\phi)$.

Similar to the discussion above for $e^{+}e^{-}$ collisions, in the back-to-
back limit of the incoming proton and the detector, i.e. $\theta\to\pi$ (or
$\tau\to 0$), we find

$\displaystyle\tau=\frac{\bm{P}_{hT}^{2}}{z_{h}^{2}Q^{2}}\,,$ (22)

for a given outgoing hadron inside the detector up to power corrections. Here
$\bm{P}_{hT}$ is the transverse momentum of the final hadron $h$, measured
using the Trento convention Bacchetta:2004jz  with azimuthal angle $\phi_{h}$.
For convenience, we introduce the shorthand notation
$\bm{q}_{T}\equiv-\bm{P}_{hT}/z_{h}$, which corresponds to the virtual
photon’s transverse momentum in the center-of-mass frame where the initial
proton and final hadron align along the $z$-axis. From the Breit frame we use
here, $\bm{q}_{T}$ should just be thought of as a convenient auxiliary
variable.

With the relation between $\tau$ and hadron’s transverse momentum, we can then
relate the azimuthal dependent EEC in Eq. (21) in the back-to-back limit to
the ${\bm{q}}_{T}$-differential cross section as

$\displaystyle\mathrm{{{EEC}}}_{\rm DIS}(\tau,\phi)=$
$\displaystyle\frac{d\Sigma_{\rm DIS}}{dx_{B}dyd\tau d\phi}$ $\displaystyle=$
$\displaystyle\sum_{h}\int
d^{2}\bm{q}_{T}\,dz_{h}\,z_{h}\,\frac{d\sigma}{dx_{B}dydz_{h}d^{2}\bm{q}_{T}}\,\delta\left(\tau-\frac{\bm{q}_{T}^{2}}{Q^{2}}\right)\delta(\phi-\phi_{h})\,,$
(23)

where $\phi_{h}$ is the azimuthal angle of $\bm{P}_{hT}$. The standard TMD
factorization in the Breit frame tells us that the transverse momentum can be
sourced from different contributions as Collins:2011zzd ; Bacchetta:2022awv

$\displaystyle\bm{q}_{T}=-\frac{\bm{p}_{\perp}}{z_{h}}-\bm{k}_{\perp}-\bm{\lambda}_{\perp}\,,$
(24)

where $\bm{p}_{\perp}$ is the hadron transverse momentum with respect to the
fragmenting parton (quark or anti-quark, in this case), $\bm{k}_{\perp}$ is
the transverse momentum of the parton inside the incoming proton, and
$\bm{\lambda}_{\perp}$ is the soft-radiation describing the recoil and is
measured with respect to the photon-beam axis.

Then in the back-to-back limit, using this known TMD factorization in the
Breit frame (e.g. see Kotzinian:1994dv ; Diehl:2005pc ; Bacchetta:2006tn ;
Anselmino:2008sga ; Kang:2015msa ), the azimuthal angle dependent EEC for DIS
defined in the Eq. (21) can be presented in the back-to-back limit as

$\displaystyle\mathrm{{{EEC}}}_{\rm DIS}(\tau,\phi)$
$\displaystyle=\frac{d\Sigma_{\rm DIS}}{dx_{B}dyd\tau d\phi}=\sigma_{0}\int
d^{2}{\bm{q}}_{T}\delta(\tau-\frac{{\bm{q}}_{T}^{2}}{Q^{2}})\delta(\phi-\phi_{h})\int\frac{db\
b}{2\pi}\bigg{\\{}\mathcal{F}_{UU}$
$\displaystyle+\cos(2\phi_{h})\frac{2(1-y)}{1+(1-y)^{2}}\mathcal{F}_{UU}^{\cos(2\phi_{h})}+S_{\parallel}\sin(2\phi_{h})\frac{2(1-y)}{1+(1-y)^{2}}\mathcal{F}_{UL}^{\sin(2\phi_{h})}$
$\displaystyle+|{\bm{S}}_{\perp}|\bigg{[}\sin(\phi_{h}-\phi_{s})\mathcal{F}_{UT}^{\sin(\phi_{h}-\phi_{s})}+\sin(\phi_{h}+\phi_{s})\frac{2(1-y)}{1+(1-y)^{2}}\mathcal{F}_{UT}^{\sin(\phi_{h}+\phi_{s})}$
$\displaystyle\hskip
39.83368pt+\sin(3\phi_{h}-\phi_{s})\frac{2(1-y)}{1+(1-y)^{2}}\mathcal{F}_{UT}^{\sin(3\phi_{h}-\phi_{s})}\bigg{]}$
$\displaystyle+\lambda_{e}\bigg{[}S_{\parallel}\frac{y(2-y)}{1+(1-y)^{2}}\mathcal{F}_{LL}+|{\bm{S}}_{\perp}|\cos(\phi_{h}-\phi_{s})\mathcal{F}_{LT}^{\cos(\phi_{h}-\phi_{s})}\bigg{]}\bigg{\\}}\,,$
(25)

where the indices $A$ and $B$ of the structure functions $\mathcal{F}_{AB}$
represent the polarization of the incoming electron and incoming proton,
respectively. The born cross-section is given by

$\displaystyle\sigma_{0}=\frac{2\pi\alpha_{\rm
em}^{2}}{Q^{2}}\frac{1+(1-y)^{2}}{y}\,.$ (26)

Also, $S_{\parallel}$ and $|{\bm{S}}_{\perp}|$ respectively denote the
helicity and transverse spin of the incoming proton, whereas $\lambda_{e}$
denotes the helicity of the incoming electron. The angle $\phi_{s}$ is the
azimuthal angle of the transverse spin of the beam. The exact expression of
each structure functions in terms of different nuclear TMDs and the EEC jet
functions defined above in Eqs. (18) and (19) are given in App. A. Unlike the
EEC in $e^{+}e^{-}$ annihilation, the DIS version of the EEC has only one EEC
jet function per term as it measures correlation between one outgoing hadron
and the beam.

###  2.3 Azimuthal dependent EEC in DIS using the lab-frame angles

As demonstrated in Gao:2022bzi , a new angular observable $q_{*}$ defined in
the Lab frame was proposed for precisely extracting the transverse momentum
distribution of hadrons in DIS experiments, by taking advantage of the near-
perfect resolution on the _angles_ of charged particle tracks as opposed to
momentum. Although energy correlators are inherently angular observables,
boosting to the Breit frame in Sec. 2.2 requires precise determination of the
virtual photon momentum, which compromises precision. To this end, we study
how energy correlator in DIS can be formulated using the angle defined in the
lab frame using the proposed $q_{*}$ observable.

More specifically, as discussed in Gao:2022bzi , a high-precision
reconstruction of polar and azimuthal angles in the EIC lab frame was
developed to exploit the acoplanarity angle $\phi_{\rm acop}^{\mathrm{EIC}}$
as a precision probe of hadron transverse momentum $\bm{P}_{hT}$ measured in
Trento convention discussed above. In the EIC lab frame as given in right
panel of Fig. 2, the acoplanarity angle $\phi_{\rm acop}^{\mathrm{EIC}}$ are
defined as $P_{hy}^{\rm EIC}/P_{hx}^{\rm EIC}$, where $P_{hx}^{\rm EIC}$ and
$P_{hy}^{\rm EIC}$ is the $x$ and $y$ component of the produced hadron
momentum in the EIC frame. At the leading power kinematics, with
$\sqrt{\bm{P}_{hT}^{2}}=P_{hT}$, one can relate this angle with quantities in
the Breit frame as

$\displaystyle\tan\phi_{\rm
acop}^{\mathrm{EIC}}=-\frac{P_{hT}\sin\phi_{h}}{z_{h}Q\sqrt{1-y}}+\mathcal{O}\left({P_{hT}^{2}}/{(z_{h}Q)}^{2}\right)=-\sqrt{\frac{\tau}{1-y}}\sin\phi_{h}+\mathcal{O}(\tau)\,,$
(27)

where $Q^{2}=-q^{2}=-(l-l^{\prime})^{2}$, and $y$ and $z_{h}$ have been
defined in Eq. (20). To probe $P_{hT}$ more straightforwardly with quantities
that can be directly measured in the EIC frame, an optimized observable
$q_{*}$ is constructed and given by

$\displaystyle q_{*}\equiv
2P_{\mathrm{EIC}}^{0}\frac{e^{\eta_{h}}}{1+e^{\Delta\eta}}\tan\phi_{\mathrm{acop}}^{\mathrm{EIC}}\,.$
(28)

Here $P_{\rm EIC}^{0}$, $\eta_{h}$ and $\eta_{e}$ are energy and
pseudorapidities measured in the EIC frame. At the leading-power limit
$P_{hT}\ll z_{h}Q$, one can apply the relation between $\phi_{\rm
acop}^{\mathrm{EIC}}$ and $P_{hT}$ in Eq. (27) and obtain the approximation
$q_{*}\approx-P_{hT}\sin\phi_{h}$ to simplify the analytical calculation. An
illustration of $\phi_{h}$ in the $x$-$y$ plane of Fig. 2 is shown in Fig. 3.
Although $\phi_{h}$ here, the azimuthal angle of ${\bm{P}}_{hT}$, is measured
in the Trento frame, the observable $q_{*}$ itself is purely a measurable
observable in the lab frame and designed to be maximally resilient against
resolution effects while delivering the same sensitivity to TMD dynamics as
$P_{hT}\ll Q$. Moreover, this resolves the issue of accurately reconstructing
small transverse momentum $P_{hT}$ in 3d measurements of confinement and
hadronization.

Figure 3: Illustration of $\phi_{h}$ in the $x-y$ plane of the left panel in
Fig. 2.

Now we will construct a new EEC observable called $\mathrm{{{EEC}}}_{\rm
DIS}^{*}$ as a function of $q_{*}$,

$\displaystyle\mathrm{{{EEC}}}_{\rm DIS}^{*}(q_{*})=\frac{d\Sigma_{\rm
DIS}}{dx_{B}dydq_{*}}=$ $\displaystyle\sum_{h}\int
dz_{h}z_{h}\frac{d\sigma}{dx_{B}dydz_{h}dq^{h}_{*}}\delta(q_{*}-q_{*}^{h})\,.$
(29)

Using the factorization theorem presented for
$\frac{d\sigma}{dxdydz_{h}dq^{h}_{*}}$ given in Gao:2022bzi , the
$\mathrm{{{EEC}}}_{\rm DIS}^{*}(q_{*})$ is given as

$\displaystyle\frac{d\Sigma_{\rm DIS}}{dx_{B}dydq_{*}}=$
$\displaystyle\sigma_{0}\int
db\bigg{\\{}\cos\left(bq_{*}\right)\bigg{[}\mathcal{C}[\tilde{f}_{1}{J}]+\frac{2(1-y)}{1+(1-y)^{2}}\mathcal{C}^{\perp}[\tilde{h}_{1}^{\perp(1)}{J}^{\perp}]+\lambda_{e}S_{\parallel}\frac{y(2-y)}{1+(1-y)^{2}}\mathcal{C}[\tilde{g}_{1L}{J}]\bigg{]}$
$\displaystyle+|{\bm{S}}_{\perp}|\sin\left(bq_{*}\right)\cos\phi_{s}\bigg{[}-\mathcal{C}[\tilde{f}^{\perp(1)}_{1T}{J}]+\frac{2(1-y)}{1+(1-y)^{2}}\mathcal{C}^{\perp}[\tilde{h}_{1}{J}^{\perp}]$
$\displaystyle+\frac{2(1-y)}{1+(1-y)^{2}}\frac{1}{4}\mathcal{C}^{\perp}[\tilde{h}^{\perp(2)}_{1T}{J}^{\perp}]\bigg{]}+\lambda_{e}|{\bm{S}}_{\perp}|\sin\left(bq_{*}\right)\sin\phi_{s}\mathcal{C}[\tilde{g}^{(1)}_{1T}{J}]\bigg{\\}}\,,$
(30)

where the $\tilde{f}^{(m)}$ is $m$-moment TMD PDFs in the Fourier $b$-space
“subtracted” with squared root of the soft function
$\sqrt{S({\bm{b}}^{2},\mu,\nu)}$, similar to the “subtracted” TMD FFs defined
in Eq. (14). ${J}$ and ${J}^{\perp}$ are the “subtracted” unpolarized and
Collins-type EEC jet function, respectively, given in Eq. (18) and (19). For
convenience, we define the notations
$\mathcal{C}^{(\perp)}[\tilde{f}^{(m)}{J}^{(\perp)}]$ as

$\displaystyle\mathcal{C}[\tilde{f}^{(m)}{J}]=$
$\displaystyle\sum_{q}e_{q}^{2}H(Q,\mu){J}_{q}(b,\mu,\zeta)b^{m}{M^{m}}\tilde{f}^{(m)}_{q}(x,b,\mu,\zeta)\,,$
(31) $\displaystyle\mathcal{C}^{\perp}[\tilde{f}^{(m)}{J}^{\perp}]=$
$\displaystyle\sum_{q}e_{q}^{2}H(Q,\mu){J}_{q}^{\perp}(b,\mu,\zeta)\frac{b^{m+1}}{2}{M^{m}}{(-M_{h})}\tilde{f}^{(m)}_{q}(x,b,\mu,\zeta)\,.$
(32)

Using the new EEC observable defined in Eq. (30), one can get access to
probing 7 leading-twist TMD PDFs with $q_{*}$ measured, which is constructed
as a high-precision probe in the EIC lab frame. Note that the reason
$\tilde{h}_{1L}^{\perp(1)}$ is not present in Eq. (30) is that its coefficient
$\sin(2\phi_{h})$ has an odd symmetry thus the contribution vanishes after
integrating over $\phi_{h}$.

##  3 Properties of Collins-type EEC jet functions

In this section, we derive the intricate behavior of the EEC jet functions
using the operator product expansion (OPE) as done for the unpolarized case in
Moult:2018jzp , and extend their analysis to the Collins-type EEC jet function
defined in Eq. (19). We find that the Collins-type EEC jet function becomes
null in the OPE region upon neglecting the off-diagonal matching terms, which
is $\alpha_{s}$ suppressed relative to the diagonal terms. Finally, we
conclude by discussing the application of considering less inclusive EEC jet
function by restricting to a subset of hadrons.

###  3.1 Operator product expansion region

In order to elucidate the OPE of the EEC jet functions, we start with the
concept of “subtracted” TMD FFs as defined in Eq. (14), which absorbs the
square-root of the soft-function. This serves as a vital starting point for
our analysis, enabling us to uncover the intricate interplay between the OPE
and the azimuthal angle dependence of the EEC jet functions.

The OPE of the subtracted unpolarized and Collins TMD FFs then gives
Koike:2006fn ; Yuan:2009dw ; Bacchetta:2013pqa ; Echevarria:2014rua ;
Kang:2015msa ; Boussarie:2023izj

$\displaystyle\tilde{D}_{1,h/q}(z,b,\mu,\zeta)=$
$\displaystyle\left[C_{j\leftarrow q}\otimes
D_{1,h/{j}}\right]\left(z,b,\mu,\zeta\right)+\mathcal{O}(b^{2}\Lambda_{\rm
QCD}^{2}),$ (33) $\displaystyle\tilde{H}_{1,h/{q}}^{\perp}(z,b,\mu,\zeta)=$
$\displaystyle\left[\delta C_{j\leftarrow q}^{\rm
Collins}\otimes\hat{H}_{1,h/j}^{\perp(3)}+A_{j\leftarrow
q}\tilde{\otimes}\hat{H}_{F,h/j}\right]\left(z,b,\mu,\zeta\right)+\mathcal{O}(b^{2}\Lambda_{\rm
QCD}^{2})\,,$ (34)

where the usual convolution $\otimes$ is given by

$\displaystyle\left[C_{j\leftarrow{q}}\otimes
F_{h/j}\right]\left(z,b,\mu,\zeta\right)=\int_{z}^{1}\frac{dx}{x}C_{j\leftarrow{q}}\left(\frac{z}{x},b,\mu,\zeta\right)F_{h/j}\left(x,\mu\right)\,,$
(35)

and the convolution $\tilde{\otimes}$ is a double convolution

$\displaystyle\left[A_{j\leftarrow
q}\tilde{\otimes}\hat{H}_{F,h/j}\right]\left(z,b,\mu,\zeta\right)$
$\displaystyle\quad=\int_{z}^{1}\frac{dx}{x}\,\int\frac{dz_{1}}{z_{1}^{2}}\,{\rm
PV}\left(\frac{1}{\frac{1}{x}-\frac{1}{z_{1}}}\right)A_{j\leftarrow{q}}\left(\frac{z}{x},z_{1},b,\mu,\zeta\right)\hat{H}_{F,h/j}\left(x,z_{1},\mu\right)\,,$
(36)

where $A_{q^{\prime}\leftarrow q}\left(z,z_{1},\mu,\zeta\right)$ starts at the
order $\mathcal{O}(\alpha_{s})$ and will be ignored in our analysis relative
to the $\delta C_{j\leftarrow q}^{\rm Collins}$ term. Also, ${\rm PV}$
represents principle value and Eq. (36) comes from the two-variable twist-3
fragmentation function $\hat{H}_{F,h/q}(z_{1},z_{2},\mu)$ that appears in the
collinear factorization region in $b\ll 1/\Lambda_{\rm QCD}$ Yuan:2009dw .
Such two-variable dependence arise from different longitudinal momentum flow
on the amplitude and conjugate amplitude side.

Here, $D_{1,h/j}(z,\mu)$ in Eq. (33) describes the unpolarized collinear
fragmentation functions. On the other hand,
$\hat{H}_{1,h/j}^{\perp(3)}(z,\mu)$ is a twist-3 fragmentation function, which
can be related to the first ${\bm{k}}_{\perp}$-moment of the Collins TMD FF
through an equation of motion relation Yuan:2009dw ; Kang:2015msa . Matching
of the unpolarized TMD FF and the Collins TMD FF given in Eqs. (33) and (34)
can be derived through the usual OPE in the small-$b$ region,
$1/b\gg\Lambda_{\rm QCD}$. To one-loop, these matching coefficients of Eqs.
(33) and (34) are given by (see e.g. Bacchetta:2013pqa ; Echevarria:2014rua ;
Kang:2015msa ; Echevarria:2016scs )

$\displaystyle C_{q^{\prime}\leftarrow q}\left(z,b,\mu,\zeta\right)=$
$\displaystyle\delta_{qq^{\prime}}\bigg{\\{}\delta\left(1-z\right)+\frac{\alpha_{s}}{\pi}\bigg{[}C_{F}\delta\left(1-z\right)\left(-\frac{L_{b}^{2}}{4}+\frac{L_{b}}{2}\left(\frac{3}{2}+\ln\frac{\mu^{2}}{\zeta^{2}}\right)-\frac{\pi^{2}}{24}\right)$
$\displaystyle\qquad+\frac{C_{F}}{2}\left(1-z\right)+\left(\ln
z-\frac{L_{b}}{2}\right)P_{q\leftarrow
q}\left(z\right)\bigg{]}\bigg{\\}}+\mathcal{O}(\alpha_{s}^{2})\,,$ (37)
$\displaystyle\delta C_{q^{\prime}\leftarrow q}^{\rm
Collins}\left(z,b,\mu,\zeta\right)=$
$\displaystyle\delta_{qq^{\prime}}\bigg{\\{}\delta\left(1-z\right)+\frac{\alpha_{s}}{\pi}\bigg{[}C_{F}\delta\left(1-z\right)\left(-\frac{L_{b}^{2}}{4}+\frac{L_{b}}{2}\left(\frac{3}{2}+\ln\frac{\mu^{2}}{\zeta^{2}}\right)-\frac{\pi^{2}}{24}\right)$
$\displaystyle\qquad+\left(\ln
z-\frac{L_{b}}{2}\right)\hat{P}^{c}_{q\leftarrow
q}\left(z\right)\bigg{]}\bigg{\\}}+\mathcal{O}(\alpha_{s}^{2})\,,$ (38)

where $L_{b}=\ln(\mu^{2}/\mu_{b}^{2})$ with $\mu_{b}=2e^{-\gamma_{E}}/b$. The
splitting functions $P_{q\leftarrow q}(z)$ and $\hat{P}_{q\leftarrow
q}^{c}(z)$ is given as

$\displaystyle P_{q\leftarrow q}(z)$
$\displaystyle=C_{F}\left[\frac{1+z^{2}}{(1-z)_{+}}+\frac{3}{2}\delta(1-z)\right]\,,$
(39) $\displaystyle\hat{P}_{q\leftarrow q}^{c}(z)$
$\displaystyle=C_{F}\left[\frac{2z}{(1-z)_{+}}+\frac{3}{2}\delta(1-z)\right]\,.$
(40)

The matching coefficients $\delta C_{q^{\prime}\leftarrow q}^{\rm Collins}$
from (anti-)quark to gluon or different flavored (anti-)quark are zero at
$1$-loop for the Collins TMD FFs. And other matching coefficients for the
unpolarized TMD FFs have been provided in Kang:2015msa .

The collinear functions in the matching obey the sum rules Schafer:1999kn ;
Meissner:2010cc ; Metz:2016swz

$\displaystyle\sum_{h}\int_{0}^{1}dz\,z\,D_{1,h/j}\left(z,\mu\right)=1\,,$
(41)
$\displaystyle\sum_{h}\int_{0}^{1}dz\,\hat{H}_{1,h/q}^{\perp(3)}\left(z,\mu\right)=0\,.$
(42)

The first sum rule is just the longitudinal momentum conservation, i.e. sum
over longitudinal momentum fraction carried by the hadron $h$ is $1$. The
second sum rule for the twist-$3$ fragmentation function is called Schäfer-
Teryaev (ST) sum rule, which is related to the transverse momentum
conservation of the Collins TMD FF Schafer:1999kn . A general proof of the ST
sum rule in QCD was given in Meissner:2010cc . It was intuitively understood
as the fact that the transverse momentum carried by the final hadron must sum
to $0$ as the fragmenting parton has a $0$ transverse momentum.

With the “subtracted” EEC jet function defined in Eqs. (18) and (19), we
provide the parameterization of our TMD FFs using the $b^{*}$ prescription
Collins:2011zzd ; Aybat:2011zv ; Prokudin:2015ysa ; Kang:2015msa , which
ensures $\mu_{b_{*}}\gg\Lambda_{\rm QCD}$ and gives

$\displaystyle\tilde{D}_{1,h/q}(z,b,\mu,\zeta_{f})=$
$\displaystyle\,\tilde{D}_{1,h/q}(z,b,\mu_{b_{*}},\zeta_{i})e^{-S_{\rm
pert}(\mu,\mu_{b_{*}})-S_{\rm
NP}^{D_{1}}(b,Q_{0},\zeta_{f})}\left(\sqrt{\frac{\zeta_{f}}{\zeta_{i}}}\right)^{\kappa\left(b,\mu_{b_{*}}\right)}\,,$
(43) $\displaystyle\tilde{H}_{1,h/{q}}^{\perp}(z,b,\mu,\zeta_{f})=$
$\displaystyle\,\tilde{H}_{1,h/{q}}^{\perp}(z,b,\mu_{b_{*}},\zeta_{i})e^{-S_{\rm
pert}(\mu,\mu_{b_{*}})-S_{\rm
NP}^{H^{\perp}_{1}}(b,Q_{0},\zeta_{f})}\left(\sqrt{\frac{\zeta_{f}}{\zeta_{i}}}\right)^{\kappa\left(b,\mu_{b_{*}}\right)}\,,$
(44)

where $\kappa\left(b,\mu_{b_{*}}\right)$ is the Collins-Soper (CS) kernel
Collins:1981va , denoting the universal rapidity evolution Chiu:2011qc  for
TMDs, and can be perturbatively calculated at small $b$ region
Echevarria:2016scs . When $b$ is large, the CS kernel is non-perturbative and
can be computed in lattice QCD Avkhadiev:2023poz ; LatticePartonLPC:2023pdv ;
Shu:2023cot ; Schlemmer:2021aij  or extracted from experimental data
Moos:2023yfa ; Bacchetta:2022awv ; Bertone:2019nxa ; Bacchetta:2019sam ;
Scimemi:2019cmh . The non-perturbative Sudakov factors $S_{\rm NP}^{D_{1}}$
and $S_{\rm NP}^{H^{\perp}_{1}}$ in general depend on the hadron species and
we will explore it in the section below.

In Moult:2018jzp , interesting observation was made about simplification of
unpolarized EEC jet function when the non-perturbative Sudakov effects are
ignored, i.e. $\frac{1}{b}\gg\Lambda_{\rm QCD}$. That is, using the sum rule
in Eq. (41), we arrive at

$\displaystyle{J}_{q}(b,\mu,\zeta)=$
$\displaystyle\sum_{h}\int_{0}^{1}z\,dz\,\tilde{D}_{1,h/q}(z,b,\mu,\zeta)$
$\displaystyle=$
$\displaystyle\sum_{h}\int_{0}^{1}z\,{dz}\int_{z}^{1}\frac{dx}{x}\,C_{j\leftarrow
q}\left(\frac{z}{x},b,\mu_{b_{*}},\zeta\right)D_{1,h/j}\left(x,\mu_{b_{*}}\right)e^{-S_{\rm
pert}(\mu,\mu_{b_{*}})}$ $\displaystyle=$
$\displaystyle\int_{0}^{1}\tau\,{d\tau}\,{C}_{j\leftarrow
q}(\tau,b,\mu_{b_{*}},\zeta)\left[\sum_{h}\int_{0}^{1}{dx}\,x\,D_{1,h/j}\left(x,\mu_{b_{*}}\right)\right]e^{-S_{\rm
pert}(\mu,\mu_{b_{*}})}$ $\displaystyle=$
$\displaystyle\int_{0}^{1}\tau\,{d\tau}\,{C}_{j\leftarrow
q}(\tau,b,\mu_{b_{*}},\zeta)e^{-S_{\rm pert}(\mu,\mu_{b_{*}})}\,,$ (45)

and thus reproduce the result from Moult:2018jzp  that unpolarized EEC jet
function is purely perturbative object in the OPE region, given by the
matching coefficients $C_{j\leftarrow q}$ alone.

On the other hand, the Collins-type EEC jet function can be simplified
analogously using the ST sum rule described in Eq. (42), with the contribution
from twist-3 function $\hat{H}_{F,h/j}$ in Eq. (34) neglected. By making this
simplification, we can achieve the same level of accuracy as in the previous
scenario. In the OPE region, the Collins-type EEC jet functions can be further
simplified as follows:

$\displaystyle{J}_{q}^{\perp}(b,\mu,\zeta)=$
$\displaystyle\,\sum_{h}\int_{0}^{1}dz\,z\,\tilde{H}_{1,h/q}^{\perp}(z,b,\mu,\zeta)$
$\displaystyle=$
$\displaystyle\,\sum_{h}\int_{0}^{1}dz\int_{z}^{1}\frac{dx}{x}\,\delta
C_{q\leftarrow q}^{\rm
Collins}(\frac{z}{x},b,\mu_{b_{*}},\zeta)\hat{H}_{1,h/q}^{\perp(3)}\left(x,\mu_{b_{*}}\right)e^{-S_{\rm
pert}(\mu,\mu_{b_{*}})}$ $\displaystyle=$
$\displaystyle\,\int_{0}^{1}d\tau\delta C_{q\leftarrow q}^{\rm
Collins}(\tau,b,\mu_{b_{*}},\zeta)\left[\sum_{h}\int_{0}^{1}dx\,\hat{H}_{1,h/q}^{\perp(3)}\left(x,\mu_{b_{*}}\right)\right]e^{-S_{\rm
pert}(\mu,\mu_{b_{*}})}$ $\displaystyle=$ $\displaystyle\,\,0\,,$ (46)

which demonstrates that the Collins-like EEC jet function ${J}_{q}^{\perp}$
becomes $0$ in the OPE region. Both of these results only hold true when non-
perturative effects can be ignored.

###  3.2 Collins-type EEC with subsets of hadrons

In this section, we explore a less inclusive version of EEC in the back-to-
back limit that is only sensitive to the energy flow of subset of hadrons
$\langle\mathcal{E}_{\mathbb{S}_{1}}(\hat{n}_{1})\mathcal{E}_{\mathbb{S}_{2}}(\hat{n}_{2})\rangle$.555See
Lee:2023tkr ; Lee:2023npz ; Li:2021zcf ; Chen:2020vvp ; Jaarsma:2023ell  for
similar consideration in the context of EEC in the collinear limit, where the
track function formalism was used to study energy correlation between hadrons
with specific quantum number. Here, we denote some subset of hadron sharing
some quantum number as $\mathbb{S}$. By considering the $\tau,\
\phi$-differential and $z$-weighed cross section, while summing over this
subset of hadrons $\mathbb{S}$, we arrive at the modified jet functions from
Eqs. (18) and (19),

$\displaystyle{J}_{q/\mathbb{S}}(b,\mu,\zeta)\equiv$
$\displaystyle\sum_{h\in\mathbb{S}}\int_{0}^{1}dz\,z\,\tilde{D}_{1,h/q}(z,b,\mu,\zeta)\,,$
(47) $\displaystyle{J}_{q/\mathbb{S}}^{\perp}(b,\mu,\zeta)\equiv$
$\displaystyle\sum_{h\in\mathbb{S}}\int_{0}^{1}dz\,z\,\tilde{H}_{1,h/q}^{\perp}(z,b,\mu,\zeta)\,.$
(48)

In the OPE region, these subset jet functions can then be matched to the
collinear functions as

$\displaystyle{J}_{q/\mathbb{S}}(b,\mu,\zeta)=$
$\displaystyle\sum_{j}F_{j\to\mathbb{S}}\int_{0}^{1}{d\tau}\,\tau\,{C}_{j\leftarrow
q}(\tau,b,\mu_{b_{*}},\zeta)e^{-S_{\rm pert}(\mu,\mu_{b_{*}})}\,,$ (49)
$\displaystyle{J}_{q/\mathbb{S}}^{\perp}(b,\mu,\zeta)=$
$\displaystyle\sum_{j}F^{\perp}_{j\to\mathbb{S}}\int_{0}^{1}d\tau\,\delta
C_{j\leftarrow q}^{\rm Collins}(\tau,b,\mu_{b_{*}},\zeta)e^{-S_{\rm
pert}(\mu,\mu_{b_{*}})}\,,$ (50)

Physically, $F_{j\to\mathbb{S}}$ is the average fraction of longitudinal
momentum of parton $j$ carried by the subset $\mathbb{S}$ of hadrons
Li:2021txc , which is thus between $0$ and $1$. On the other hand,
$F^{\perp}_{j\to\mathbb{S}}$ is related to the average transverse momentum
carried by the subset $\mathbb{S}$ of the hadrons, which is thus able to take
any real value. These two functions are defined in terms of the moments of NP
collinear objects as

$\displaystyle F_{j\to\mathbb{S}}$
$\displaystyle=\sum_{h\in\mathbb{S}}\int_{0}^{1}{dz}\,z\,D_{1,h/j}\left(z,\mu_{b_{*}}\right)\,,$
(51) $\displaystyle F^{\perp}_{j\to\mathbb{S}}$
$\displaystyle=\sum_{h\in\mathbb{S}}\int_{0}^{1}dz\,\hat{H}_{1,h/j}^{\perp(3)}\left(z,\mu_{b_{*}}\right)\,.$
(52)

It is worth noting that since $\delta C_{j\leftarrow q}^{\rm Collins}$ is only
non-zero when $j=q$, the only relevant term is $F^{\perp}_{q\to\mathbb{S}}$.
Additionally, as demonstrated in Eqs. (41) and (42), when $\mathbb{S}=$ all
hadrons, one will obtain $F_{j\to\mathbb{S}}=1$ and
$F^{\perp}_{j\to\mathbb{S}}=0$.

Although the EEC jet functions are now sensitive to the non-perturbative
information carried by the subset of hadrons, it is worth noting that all of
the non-perturbative information are captured by a single number in the OPE
region. As pointed out in Li:2021txc , considering EEC with subsets of hadrons
can be interesting from the point of view of considering a set of all charged
particles $\mathbb{S}=\mathbb{C}$ or a set of a single hadron type
$\mathbb{S}=h$. From the perspective of Collins-type EEC jet function, we are
motivated to consider the so-called favored and unfavored subset as well
Vogelsang:2005cs ; Kang:2015msa ; Metz:2016swz . The vanishing value of
Collins-type EEC jet function in the OPE region can be understood as
$F^{\perp}_{j\to\text{fav}}\approx-F^{\perp}_{j\to\text{unfav}}$, and thus it
is also interesting to consider
$\mathbb{S}=\\{\pi^{+}\\},\\{\pi^{-}\\},\\{\pi^{0}\\},\\{\pi^{+},\pi^{-}\\},\\{\pi^{+},\pi^{-},\pi^{0}\\}$
for Collins-type EEC in phenomenology.

##  4 Phenomenology

The present study focuses on the Collins azimuthal asymmetry in $e^{+}e^{-}$
annihilation with subsets of hadrons, namely $\mathbb{S}=\\{\pi^{+},\
\pi^{-}\\}$, $\\{\pi^{+}\\}$, and $\\{\pi^{-}\\}$, as the hadron component of
the EEC jet using EIC kinematics. Furthermore, we provide predictions for both
the Collins and Sivers asymmetry in the Breit frame for the DIS process. It is
worth noting that the EEC jet function is also a powerful tool for probing the
internal structure of nucleons in the lab frame, as evidenced by our example
of an azimuthal asymmetry related to the worm-gear TMD PDFs and EEC jet
functions.

###  4.1 The EEC Collins asymmetry in $e^{+}e^{-}$ annihilation

With the EEC derived for $e^{+}e^{-}$ annihilation in the back-to-back region
in terms of the EEC jet functions in Eq. (17), we now carry out
phenomenological study of the asymmetry associated with EEC. To achieve this,
we first rewrite the Eq. (17) as

$\displaystyle\mathrm{{{EEC}}}_{e^{+}e^{-}}(\tau,\phi)=\frac{d\Sigma_{e^{+}e^{-}}}{d\tau
d\phi}=$ $\displaystyle\frac{1}{2}\sigma_{0}\sum_{q}e_{q}^{2}\int
d{\bm{q}}_{T}^{2}\,\delta(\tau-\frac{{\bm{q}}_{T}^{2}}{Q^{2}})\,Z_{uu}\,\left[1+\cos(2\phi)\,\frac{Z_{\rm
Collins}}{Z_{uu}}\right]$ $\displaystyle\equiv$
$\displaystyle\frac{1}{2}\sigma_{0}\sum_{q}e_{q}^{2}\,Z_{uu}\,\left[1+\cos(2\phi)\,A_{e^{+}e^{-}}(\tau
Q^{2})\right]\,,$ (53)

where

$\displaystyle Z_{uu}=$
$\displaystyle\int\frac{bdb}{2\pi}J_{0}(bq_{T}){J}_{q}(b,\mu,\zeta){J}_{\bar{q}}(b,\mu,\zeta)\,,$
(54) $\displaystyle Z_{\rm Collins}=$
$\displaystyle\int\frac{bdb}{2\pi}\frac{b^{2}}{8}J_{2}(bq_{T}){J}_{q}^{\perp}(b,\mu,\zeta){J}_{\bar{q}}^{\perp}(b,\mu,\zeta)\,.$
(55)

Recall that subtracted EEC jet functions were defined in Eqs. (18) and (19).
As pions are the primary measurements available for the Collins-type
asymmetry, we now study the ratio
$A_{e^{+}e^{-}}^{\mathbb{S}_{1}\times\mathbb{S}_{2}}(\bm{q}_{T}^{2})=A_{e^{+}e^{-}}^{\mathbb{S}_{1}\times\mathbb{S}_{2}}(\tau
Q^{2})$ with different subsets $\mathbb{S}_{1}\times\mathbb{S}_{2}$ of
produced pion pairs. This amounts to simply using EEC jet functions with a
subset in Eqs. (47) and (48) defined in terms of TMD FFs of the pions. As
explained in the section 4.1, TMD FFs can be matched to the collinear
functions in the OPE region. We now use the parametrization of Kang:2015msa
to generate $A_{e^{+}e^{-}}^{\mathbb{S}_{1}\times\mathbb{S}_{2}}(\tau Q^{2})$.
For the TMD Collins FFs in Eq. (44), at next-to-leading logarithm (NLL), we
have

$\displaystyle\tilde{H}_{1,h/{q}}^{\perp}(z,b,\mu,\zeta)=$
$\displaystyle\left[\delta C_{j\leftarrow q}^{\rm
Collins}\otimes\hat{H}_{1,h/j}^{\perp(3)}\right]\left(x,b,\mu_{b_{*}},\mu_{b_{*}}^{2}\right)e^{-\frac{1}{2}S_{\rm
pert}\left(\mu,\mu_{b_{*}}\right)-S_{\rm
NP}^{H_{1}^{\perp}}\left(b,Q_{0},\zeta\right)}\,.$ (56)

The corresponding twist-3 collinear fragmentation functions
$\hat{H}^{(3)}(z,Q_{0})$ were parametrized as

$\displaystyle\hat{H}_{fav}^{(3)}(z,Q_{0})$
$\displaystyle=N_{u}^{c}z^{\alpha_{u}}(\tau)^{\beta_{u}}D_{1,\pi^{+}/u}(z,Q_{0})\,,$
(57) $\displaystyle\hat{H}_{unfav}^{(3)}(z,Q_{0})$
$\displaystyle=N_{d}^{c}z^{\alpha_{d}}(\tau)^{\beta_{d}}D_{1,\pi^{+}/d}(z,Q_{0})\,,$
(58) $\displaystyle\hat{H}_{s/\bar{s}}^{(3)}(z,Q_{0})$
$\displaystyle=N_{d}^{c}z^{\alpha_{d}}(\tau)^{\beta_{d}}D_{1,\pi^{+}/s,\bar{s}}(z,Q_{0})\,,$
(59)

and fitting parameters $N_{u}^{c},\ N_{d}^{c},\ \alpha_{u},\ \alpha_{d},\
\beta_{u},\ \beta_{d},$ are provided in Kang:2015msa . The collinear FFs
$D_{1,h/i}$ are the NLO $\mathtt{DSS}$ fragmentation functions
deFlorian:2014xna . Note here the contribution from twist-3 function
$\hat{H}_{F,h/j}$ in Eq. (34) has been neglected following what was done in
Kang:2015msa  and the coefficient $\delta\hat{C}^{\rm
Collins}_{q^{\prime}\leftarrow q}$ is given in Eq. (38). Here we also provide
the parametrization of the non-perturbative Sudakov factors for unpolarized
and Collins TMD FFs,

$\displaystyle S_{\rm NP}^{D_{1}}(b,Q_{0},\zeta)$
$\displaystyle=\frac{g_{2}}{2}\ln\left(\frac{b}{b_{*}}\right)\ln\left(\frac{\sqrt{\zeta}}{Q_{0}}\right)+\frac{g_{h}}{z^{2}}b^{2}\,,$
(60) $\displaystyle S_{\rm NP}^{H_{1}^{\perp}}(b,Q_{0},\zeta)$
$\displaystyle=\frac{g_{2}}{2}\ln\left(\frac{b}{b_{*}}\right)\ln\left(\frac{\sqrt{\zeta}}{Q_{0}}\right)+\frac{g_{h}-g_{c}}{z^{2}}b^{2}\,,$
(61)

where $g_{2}=0.84$, $Q_{0}^{2}=2.4$ GeV2, $g_{h}=0.042$ GeV2 and $g_{c}$ is
extracted in Kang:2015msa .

Figure 4: $A_{e^{+}e^{-}}^{\mathbb{S}_{1}\times\mathbb{S}_{2}}$ for
$\mathbb{S}_{1}\times\mathbb{S}_{2}=\\{\pi^{+},\ \pi^{-}\\}\times\\{\pi^{+},\
\pi^{-}\\}$, $\\{\pi^{+}\\}\times\\{\pi^{-}\\}$ and
$\\{\pi^{-}\\}\times\\{\pi^{-}\\}$ as a function of $\tau$ at $\sqrt{s}=10.6$
GeV with Collins functions fitted in Kang:2015msa . As discussed in Sec. 3.2,
we find
$|A_{e^{+}e^{-}}^{\\{\pi^{+},\pi^{-}\\}\times\\{\pi^{+},\pi^{-}\\}}|\ll|A_{e^{+}e^{-}}^{\\{\pi^{+}\\}\times\\{\pi^{-}\\}}|\approx|A_{e^{+}e^{-}}^{\\{\pi^{-}\\}\times\\{\pi^{-}\\}}|$
due to $J^{\perp}_{{\it fav}}\approx-J^{\perp}_{{\it unfav}}$.

In Fig. 4, we present the results for
$A_{e^{+}e^{-}}^{\mathbb{S}_{1}\times\mathbb{S}_{2}}(\tau Q^{2})$ with
different subsets of pions $\mathbb{S}_{1}\times\mathbb{S}_{2}=\\{\pi^{+},\
\pi^{-}\\}\times\\{\pi^{+},\ \pi^{-}\\}$ (red curve),
$\\{\pi^{+}\\}\times\\{\pi^{-}\\}$ (green curve) and
$\\{\pi^{-}\\}\times\\{\pi^{-}\\}$ (blue curve) at $\sqrt{s}=10.6$ GeV with
the error bands given by the uncertainties in the global fitting of the
Collins function in Kang:2015msa . Although the subset with
$\mathbb{S}_{1}\times\mathbb{S}_{2}=\\{\pi^{+},\ \pi^{-}\\}\times\\{\pi^{+},\
\pi^{-}\\}$ is suppressed due to the sum rule, one can have sizable
asymmetries when restricting to a subset of either positively or negatively
charged pions in EEC.

###  4.2 The EEC Collins asymmetry in DIS

To define the asymmetry associated with EEC for DIS, we begin by rewriting
Eq.(25) as

$\displaystyle\mathrm{{{EEC}}}_{\rm DIS}(\tau,\phi)=\frac{d\Sigma_{\rm
DIS}}{dx_{B}dyd\tau d\phi}$ $\displaystyle=\sigma_{0}\int
d^{2}{\bm{q}}_{T}\delta(\tau-\frac{{\bm{q}}_{T}^{2}}{Q^{2}})\int\frac{db\
b}{2\pi}\bigg{\\{}\mathcal{F}_{UU}+\sin(\phi_{h}+\phi_{s})\frac{2(1-y)}{1+(1-y)^{2}}\mathcal{F}_{UT}^{\sin(\phi_{h}+\phi_{s})}+\cdots\bigg{\\}}\,,$
(62)

where the ellipsis represents other spin-dependent structures given in Eq.
(25) and

$\displaystyle\mathcal{F}_{UU}$ $\displaystyle=\int
d^{2}{\bm{q}}_{T}\delta(\tau-\frac{{\bm{q}}_{T}^{2}}{Q^{2}})\int\frac{bdb}{2\pi}J_{0}(bq_{T})\,\mathcal{C}\left[\tilde{f}_{1}{J}\right]\,,$
(63) $\displaystyle\mathcal{F}_{UT}^{\sin(\phi_{h}+\phi_{s})}$
$\displaystyle=\int
d^{2}{\bm{q}}_{T}\delta(\tau-\frac{{\bm{q}}_{T}^{2}}{Q^{2}})\int\frac{bdb}{2\pi}J_{1}(bq_{T})\,\mathcal{C}^{\perp}\left[\tilde{h}_{1}{J}^{\perp}\right]\,,$
(64)

where the notations $\mathcal{C}^{(\perp)}[\cdots]$ has been defined in Eqs.
(31) and (32). Note that we absorbed $\sqrt{S}$ to define the subtracted
versions of TMD PDFs $\tilde{f}_{1}^{q}$ and $\tilde{h}_{1}^{q}$ as we did for
the EEC jet functions. Then we define the asymmetry as the ratio of the above
structure functions

$\displaystyle\mathcal{A}_{\rm DIS}(\tau Q^{2})$
$\displaystyle=\frac{2(1-y)}{1+(1-y)^{2}}\frac{\mathcal{F}_{UT}^{\sin(\phi_{h}+\phi_{s})}}{\mathcal{F}_{UU}}\,.$
(65)

As discussed for the $e^{+}e^{-}$ case, we can analogously define
$\mathcal{A}_{\rm DIS}^{\mathbb{S}}$ for various subsets of hadrons. We use
the same parameterization of the twist-3 fragmentation functions discussed
around Eq. (57)-(59). As for the TMD PDFs, we follow the $b^{*}$ prescription
and write them as

$\displaystyle\tilde{f}_{1,q/p}(x,b,\mu,\zeta)$
$\displaystyle=\left[\hat{C}_{q\leftarrow i}\otimes
f_{1,i/{p}}\right]\left(x,b,\mu_{b_{*}},\mu_{b_{*}}^{2}\right)e^{-\frac{1}{2}S_{\rm
pert}\left(\mu,\mu_{b_{*}}\right)-S_{\rm
NP}^{f_{1}}\left(b,Q_{0},\zeta\right)}\,,$ (66)
$\displaystyle\tilde{h}_{1,q/p}(x,{b},\mu,\zeta)$ $\displaystyle=\left[\delta
C_{q\leftarrow i}\otimes
h_{1,i/{p}}\right]\left(x,b,\mu_{b_{*}},\mu_{b_{*}}^{2}\right)e^{-\frac{1}{2}S_{\rm
pert}\left(\mu,\mu_{b_{*}}\right)-S_{\rm
NP}^{h_{1}}\left(b,Q_{0},\zeta\right)}\,,$ (67)

where the coefficients can be found in e.g. Kang:2015msa ; Boussarie:2023izj
as

$\displaystyle\hat{C}_{q\leftarrow i}(x,b,\mu_{b_{*}},\mu_{b_{*}}^{2})$
$\displaystyle=\delta_{qi}\left[\delta(1-x)+\frac{\alpha_{s}}{\pi}\left(-C_{F}\frac{\pi^{2}}{24}\delta\left(1-x\right)+\frac{C_{F}}{2}(1-x)\right)\right]+\mathcal{O}(\alpha_{s}^{2})\,,$
(68) $\displaystyle\delta C_{q\leftarrow i}(x,b,\mu_{b_{*}},\mu_{b_{*}}^{2})$
$\displaystyle=\delta_{qi}\big{[}\delta(1-x)\big{]}+\mathcal{O}(\alpha_{s}^{2})\,,$
(69)

and the convolution $\otimes$ has been defined in Eq. (35). To parametrize the
non-perturbative form factors for TMD PDFs, one has Kang:2015msa

$\displaystyle S_{\rm NP}^{h_{1}}(b,Q_{0},\zeta)$
$\displaystyle=S_{\mathrm{NP}}^{f_{1}}(b,Q_{0},\zeta)=\frac{g_{2}}{2}\ln\left(\frac{b}{b_{*}}\right)\ln\left(\frac{\sqrt{\zeta}}{Q_{0}}\right)+g_{q}b^{2}\,.$
(70)

where $g_{2}=0.84$, $Q_{0}^{2}=2.4$ GeV2 and $g_{q}=0.106$ GeV2. For the quark
transversity distribution $h_{1}^{q}\left(x,\mu\right)$, we use the
parametrization from Kang:2015msa . On the other hand, we use
$\mathtt{CT10nlo}$ Lai:2010vv  for the unpolarized PDFs $f_{1,q/p}(x,\mu)$.

In Fig. 5, we present the results for $A_{\rm DIS}^{\mathbb{S}}(\tau Q^{2})$
using the future EIC kinematics with different subsets of pions
$\mathbb{S}=\\{\pi^{+},\ \pi^{-}\\}$ (red curve), $\\{\pi^{+}\\}$ (green
curve) and $\\{\pi^{-}\\}$ (blue curve). Here the error bands correspond to
the uncertainties in the global extractions Kang:2015msa  of transversity TMD
PDFs $h_{1,q/p}$ and Collins fragmentation functions $H_{1}^{\perp q}$. Again,
as expected, we find $|A_{\rm DIS}^{\\{\pi^{+},\pi^{-}\\}}|\ll|A_{\rm
DIS}^{\\{\pi^{+}\\}}|$ and $|A_{\rm DIS}^{\\{\pi^{-}\\}}|$. The asymmetry is
overall much smaller than the $e^{+}e^{-}$ case due to the cancellation
between $h_{1}^{u}$ and $h_{1}^{d}$ as discussed in the Sec. 3.2. And the
asymmetry is measurable when choosing a subset of $\pi^{+}$ or $\pi^{-}$.

Figure 5: $A_{\rm DIS}^{\mathbb{S}}(\tau Q^{2})$ for
$\mathbb{S}=\\{\pi^{+}\\},\ \\{\pi^{-}\\}$, and $\\{\pi^{+},\ \pi^{-}\\}$
using the future EIC kinematics with fittings from Kang:2015msa . Here one
also has $|A_{\rm DIS}^{\\{\pi^{+},\pi^{-}\\}}|\ll|A_{\rm
DIS}^{\\{\pi^{+}\\}}|$ and $|A_{\rm DIS}^{\\{\pi^{-}\\}}|$.

###  4.3 The EEC Sivers asymmetry in DIS

To define the asymmetry associated with the Sivers asymmetry, we begin by
noting that the Eq.(25) has Sivers function dependent part

$\displaystyle\mathrm{{{EEC}}}_{\rm DIS}(\tau,\phi)=\frac{d\Sigma_{\rm
DIS}}{dx_{B}dyd\tau d\phi}$ $\displaystyle=\sigma_{0}\frac{1}{2}\int
d^{2}{\bm{q}}_{T}\delta(\tau-\frac{{\bm{q}}_{T}^{2}}{Q^{2}})\int\frac{db\
b}{2\pi}\bigg{\\{}\mathcal{F}_{UU}+\sin(\phi_{h}-\phi_{s})\frac{2(1-y)}{1+(1-y)^{2}}\mathcal{F}_{UT}^{\sin(\phi_{h}-\phi_{s})}+\cdots\bigg{\\}}\,,$
(71)

where the ellipsis represents other spin-dependent structures given in Eq.
(25) and

$\displaystyle\mathcal{F}_{UU}=$ $\displaystyle\int
d^{2}{\bm{q}}_{T}\delta(\tau-\frac{{\bm{q}}_{T}^{2}}{Q^{2}})\int\frac{bdb}{2\pi}J_{0}(bq_{T})\,\mathcal{C}\left[\tilde{f}_{1}{J}\right]\,,$
(72) $\displaystyle\mathcal{F}_{UT}^{\sin(\phi_{h}-\phi_{s})}=$
$\displaystyle\int
d^{2}{\bm{q}}_{T}\delta(\tau-\frac{{\bm{q}}_{T}^{2}}{Q^{2}})\int\frac{bdb}{2\pi}J_{1}(bq_{T})\,\mathcal{C}\left[\tilde{f}_{1T}^{\perp(1)}{J}\right]\,.$
(73)

Here the notation $\mathcal{C}[\cdots]$ is provided in Eq. (31). Note that we
absorbed $\sqrt{S}$ to define the subtracted versions of TMD PDFs
$\tilde{f}_{1}$ and $\tilde{f}_{1T}^{\perp(1)}(x,b,\mu,\zeta)$ as we did for
the EEC jet functions. Then we define the asymmetry as the ratio of the above
structure functions

$\displaystyle\mathcal{A}^{\rm Sivers}_{\rm DIS}(\tau Q^{2})$
$\displaystyle=\frac{2(1-y)}{1+(1-y)^{2}}\frac{\mathcal{F}_{UT}^{\sin(\phi_{h}-\phi_{s})}}{\mathcal{F}_{UU}}\,.$
(74)

Now we can study $\mathcal{A}^{\rm Sivers}_{\rm DIS}$ for the final hadron
subset $\\{\pi^{+},\pi^{-}\\}$ as an example. As for the TMD PDFs, we follow
the $b^{*}$ prescription and write them as Echevarria:2020hpy

$\displaystyle\tilde{f}_{1T,q/p}^{\perp(1)}(x,b,\mu,\zeta)=$
$\displaystyle-\frac{1}{2M}\int_{x}^{1}\frac{d\hat{x}_{1}}{\hat{x}_{1}}\frac{d\hat{x}_{2}}{\hat{x}_{2}}\bar{C}_{q\leftarrow
i}\left(x/\hat{x}_{1},x/\hat{x}_{2},b,\mu,\zeta\right)$ $\displaystyle\hskip
51.21504pt\times
T_{Fi/p}\left(\hat{x}_{1},\hat{x}_{2};\mu\right)e^{-\frac{1}{2}S_{\rm
pert}\left(\mu,\mu_{b_{*}}\right)-S_{\rm
NP}^{f^{\perp}_{1T}}\left(b,Q_{0},\zeta\right)}\,,$ (75)

where

$\displaystyle\bar{C}_{q\leftarrow
i}\left(x_{1},x_{2},b,\mu_{b_{*}},\mu_{b_{*}}^{2}\right)=$
$\displaystyle\delta_{qi}\bigg{[}\delta\left(1-x_{1}\right)\delta\left(1-x_{2}\right)-\frac{\alpha_{s}}{2\pi}\frac{1}{2N_{C}}\delta\left(1-x_{2}/x_{1}\right)\left(1-x_{1}\right)$
$\displaystyle\hskip
14.22636pt-\frac{\alpha_{s}}{2\pi}C_{F}\frac{\pi^{2}}{12}\delta\left(1-x_{1}\right)\delta\left(1-x_{2}\right)\bigg{]}\,.$
(76)

And the non-perturbative form factors of TMD PDFs are given by

$\displaystyle S_{\rm NP}^{f_{1T}^{\perp}}(b,Q_{0},Q^{2})$
$\displaystyle=\frac{g_{2}}{2}\ln\left(\frac{b}{b_{*}}\right)\ln\left(\frac{Q}{Q_{0}}\right)+g_{q}^{f_{1T}^{\perp}}b^{2}\,,$
(77)

where $g_{2}=0.84$, $Q_{0}^{2}=2.4$ GeV2 and the value of
$g_{q}^{f_{1T}^{\perp}}$ given by the fitting in Echevarria:2020hpy , where
the Qiu-Sterman function $T_{F\,q/p}(x,x,\mu)$ has the following
parametrization form

$\displaystyle
T_{F\,q/p}(x,x,\mu)=N_{q}\frac{\left(\alpha_{q}+\beta_{q}\right)^{\left(\alpha_{q}+\beta_{q}\right)}}{\alpha_{q}^{\alpha_{q}}\beta_{q}^{\beta_{q}}}x^{\alpha_{q}}(1-x)^{\beta_{q}}f_{1,q/p}(x,\mu)\,.$
(78)

where all fitting parameters are presented in Kang:2015msa , $f_{1,q/p}$ are
the NLO unpolarized PDF set from $\mathtt{HERAPDF20\\_NLO\\_ALPHAS\\_118}$
H1:2015ubc , which is applied in the Sivers fitting in Echevarria:2020hpy .

In Fig. 6, we present the results for the Sivers asymmetry $A^{\rm
Sivers}_{\rm DIS}(\tau Q^{2})$ in terms of EEC jet functions convoluted with
TMD PDFs using the parametrization of the twist-3 Qiu-Sterman function
fittings from Echevarria:2020hpy , using the future EIC kinematics with all
pions measured in the final state. The magnitude is about a few percents and
indicates that this asymmetry is a reasonable measurement for constraining the
Sivers function.

Figure 6: $A_{\rm DIS}^{\rm Sivers}(\tau Q^{2})$ for $\mathbb{S}=\\{\pi^{+},\
\pi^{-}\\}$ using the future EIC kinematics with fittings from
Echevarria:2020hpy . The error band corresponds to the uncertainties in the
global extractions of Sivers function in Echevarria:2020hpy .

###  4.4 The azimuthal asymmetry in DIS using the lab-frame angles

In this subsection, we provide an azimuthal asymmetry prediction related to
the worm-gear function $g_{1T}$, which has been recently extracted from
experimental data (see e.g. Bhattacharya:2021twu ; Horstmann:2022xkk ). In the
factorization formalism, one has the EEC defined using the lab frame angular
observable $q_{*}$ Gao:2022bzi  as we have introduced in Sec. 2.3. As we have
shown in Eq. (30), the two structure functions for unpolarized and double-spin
polarized (worm-gear) incoming proton are given by

$\displaystyle\int db\cos\left(bq_{*}\right)\mathcal{C}[\tilde{f}_{1}{J}]=$
$\displaystyle\int
db\cos\left(bq_{*}\right)\sum_{q}e_{q}^{2}H(Q,\mu){J}_{q}(b,\mu,\zeta)\,\tilde{f}_{1,q}(x,b,\mu,\zeta)\,,$
(79) $\displaystyle\int
db\sin\left(bq_{*}\right)\mathcal{C}[\tilde{g}^{(1)}_{1T}{J}]=$
$\displaystyle\int
db\sin\left(bq_{*}\right)\sum_{q}e_{q}^{2}H(Q,\mu){J}_{q}(b,\mu,\zeta)\,b\,{M}\,\tilde{g}^{(1)}_{1T,q}(x,b,\mu,\zeta)\,.$
(80)

For numerical results, using the $b_{*}$ prescription Collins:1984kg  as we
have applied in previous sections, we combine TMD evolution with the recent
Gaussian fit Bhattacharya:2021twu  for the worm-gear function,

$\displaystyle g_{1T,q}^{(1)}\left(x,b,\mu,\zeta\right)=$
$\displaystyle\,g_{1T,q}^{(1)}\left(x,\mu_{b_{*}}\right)e^{-\frac{1}{2}S_{\mathrm{pert}}\left(\mu,\mu_{b_{*}}\right)-S_{\mathrm{NP}}^{g_{1T}}(b,Q_{0},\zeta)}\,,$
(81)

where $S_{\rm pert}(\mu,\mu_{b_{*}})$ is the same as in Eqs. (66) and (67).
The parametrization of the non-perturbative Sudakov factor is given by

$\displaystyle S_{\rm NP}^{g_{1T}}(b,Q_{0},\zeta)$
$\displaystyle=\frac{g_{2}}{2}\ln\left(\frac{b}{b_{*}}\right)\ln\left(\frac{\sqrt{\zeta}}{Q_{0}}\right)+\frac{\langle
k_{\perp}^{2}\rangle|_{g_{1T}^{q}}}{4}b^{2}\,,$ (82)

and the fitted function $g_{1T,q}^{(1)}$ has the following functional form

$\displaystyle
g_{1T,q}^{(1)}\left(x,Q^{2}\right)=\frac{N_{q}}{\widetilde{N}_{q}}x^{\alpha_{q}}(1-x)^{\beta_{q}}f_{1}\left(x,Q^{2}\right)\,,$
(83)

with the same DGLAP evolution as $f_{1}(x,Q^{2})$ as adopted in the fitting
Bhattacharya:2021twu . Here $\langle k_{\perp}^{2}\rangle|_{g_{1T}^{q}}$,
$N_{q}$, $\alpha_{q}$ and $\beta_{q}$ are provided in Bhattacharya:2021twu ,
$\tilde{N}_{q}$ is defined in terms of the unpolarized collinear PDFs
$f_{1}(x,Q_{0}^{2})$ with $Q_{0}=2$ GeV,

$\displaystyle\tilde{N}_{q}\equiv\int_{0}^{1}dxx^{\alpha_{q}+1}(1-x)^{\beta_{q}}f_{1}(x,Q_{0}^{2})\,.$
(84)

Figure 7: The EEC related to unpolarized TMD PDFs $f_{1}$ (blue) and worm-gear
TMD PDFs $g_{1T}$ (green) for all pions in the final state at the EIC
kinematics are shown as a function of $q_{*}$ in the upper panel. The ratio
$\mathcal{A}_{\rm DIS}^{\rm WG}$ is shown at the bottom. The error bands in
both panels correspond to the uncertainties in the global extractions of worm-
gear function in Bhattacharya:2021twu .

In the upper panel of Fig. 7, we present the unpolarized structure function in
Eq. (79) (divided by 10) with a blue curve as a function of $q_{*}$. We choose
the central fit for the unpolarized TMDPDFs, with the collinear PDFs
$f_{1}(x,Q^{2})$ given by $\mathtt{CT10nlo}$ Bhattacharya:2021twu . In the
lower panel, we present the structure function depending on the worm-gear
functions as given in Eq. (80) (multiplied by 100), plotted as a green curve
along with the error band. This error band is again given by the global
extraction of the worm-gear functions Bhattacharya:2021twu . In the lower
panel of Fig. 7, we use Eq. (80) divided by Eq. (79) and plot the ratio
$\mathcal{A}^{\rm WG}_{\rm DIS}$ as a function of $q_{*}$ as well. As given by
the extraction in Bhattacharya:2021twu , the worm-gear functions of $u$ quark
and $d$ quark have opposite signs and similar magnitudes, and convoluted with
unpolarized EEC jet functions, which is always positive. Thus for the
production of all pions as final hadrons, the EEC related to worm-gear
function are suppressed. This observable can be an inspiration for us to
explore new directions for studying 3-dimensional nucleon structures encoded
in TMD PDFs.

##  5 Conclusion

In summary, we present a comprehensive study of the azimuthal angle dependence
of the energy-energy correlators (EEC) in the back-to-back region to both
$e^{+}e^{-}$ and deep inelastic scattering (DIS) processes with unpolarized
hadron production. Notably, we introduce the Collin-type EEC jet function for
the first time in this context. Using the unpolarized and the Collins-type EEC
jet function, we demonstrate that all of the leading-twist Transverse-
Momentum-Dependent Parton Distribution Functions (TMD PDFs) can be extracted
from the DIS process. This finding points towards a new approach for gaining a
deeper understanding of the complex structure of nucleons. Furthermore, we
introduce a new EEC observable defined only using the lab-frame angle first
introduced in Gao:2022bzi , which offers much better experimental resolution.

To illustrate phenomenological applications, we provide predictions for the
Collins asymmetry in $e^{+}e^{-}$ annihilation and DIS, within the contexts of
Belle and Electron-Ion Collider (EIC) kinematics, respectively. Furthermore,
we present predictions for the Sivers asymmetry using the Breit frame version
of the EEC, as well as the worm-gear asymmetry using the new EEC observable
based on lab-frame angles, both with future EIC kinematics. These examples
underscore the potential of azimuthal angle dependent EEC as a unique tool for
a deeper understanding of nucleon structures.

###### Acknowledgements.

We thank Anjie Gao, Xiaohui Liu, Johannes Michel, Ian Moult, Iain Stewart,
Zhiquan Sun, and Feng Yuan for helpful discussions. This work is supported by
the National Science Foundation under Grant No. PHY-1945471 (Z.K. and F.Z.),
the U.S. Department of Energy, Office of Science, Office of Nuclear Physics
under grant Contract Number DE-SC0011090 (F.Z.), Co-design Center for Quantum
Advantage (C2QA) under contract number DE-SC0012704 (F.Z.), the U.S.
Department of Energy under Contract No. DE-SC0011090 (K.L.), the National
Science Foundations of China under Grant No. 12275052 and No. 12147101 and the
Shanghai Natural Science Foundation under Grant No. 21ZR1406100 (D.Y.S.).

##  Appendix A DIS structure functions with hadrons and EEC jets

In this appendix, we provide the relevant azimuthal-angle-dependent
$\bm{q}_{T}$ factorization in the Breit frame and $q_{*}$ factorization in the
lab frame for the DIS process as well as the structure functions factorized
with EEC jet functions.

###  A.1 Breit frame adaptations

As shown in Gourdin:1972kro ; Kotzinian:1994dv ; Diehl:2005pc ;
Bacchetta:2006tn ; Anselmino:2008sga ; Collins:2011zzd ; Kang:2015msa , the
differential cross section for the DIS process up to twist-2 functions is
factorized by

$\displaystyle\frac{d\sigma}{dxdydz_{h}d^{2}{\bm{q}}_{T}}=$
$\displaystyle\sigma_{0}\int\frac{bdb}{2\pi}\bigg{\\{}F_{UU}+\cos(2\phi_{h})\frac{2(1-y)}{1+(1-y)^{2}}F_{UU}^{\cos(2\phi_{h})}$
$\displaystyle+S_{\parallel}\sin(2\phi_{h})\frac{2(1-y)}{1+(1-y)^{2}}F_{UL}^{\sin(2\phi_{h})}+|{\bm{S}}_{\perp}|\bigg{[}\sin(\phi_{h}-\phi_{s})F_{UT}^{\sin(\phi_{h}-\phi_{s})}$
$\displaystyle+\sin(\phi_{h}+\phi_{s})\frac{2(1-y)}{1+(1-y)^{2}}F_{UT}^{\sin(\phi_{h}+\phi_{s})}$
$\displaystyle+\sin(3\phi_{h}-\phi_{s})\frac{2(1-y)}{1+(1-y)^{2}}F_{UT}^{\sin(3\phi_{h}-\phi_{s})}\bigg{]}$
$\displaystyle+\lambda_{e}\bigg{[}S_{\parallel}\frac{y(2-y)}{1+(1-y)^{2}}F_{LL}+|{\bm{S}}_{\perp}|\cos(\phi_{h}-\phi_{s})F_{LT}^{\cos(\phi_{h}-\phi_{s})}\bigg{]}\bigg{\\}}\,,$
(85)

where for structure functions $F_{AB}$, the indices $A$ and $B$ represent the
polarization of the incoming electron and incoming proton. $S_{\parallel}$ and
$|{\bm{S}}_{\perp}|$ are the helicity and transverse spin of the incoming
proton, $\lambda_{e}$ is the helicity of the incoming electron. The variables
$x$ and $y$ represent the longitudinal momentum fraction of the incoming
parton inside the beam and the inelasticity variable, respectively, which we
will additionally be differential in. The azimuthal angles $\phi_{s}$ and
$\phi_{h}$ are the azimuthal angles of the transverse spin of the beam and the
transverse momentum of the produced hadron, respectively. By defining the
notation ${C}[\tilde{f}^{(m)}\tilde{D}^{(n)}]$ as

$\displaystyle{C}[\tilde{f}^{(m)}\tilde{D}^{(n)}]=$
$\displaystyle\sum_{q}e_{q}^{2}b^{m+n}J_{m+n}(b\,q_{T}){(M)^{m}}{(-zM_{h})^{n}}$
$\displaystyle\times\tilde{f}^{(m)}(x,b,\mu,\zeta)\tilde{D}^{(n)}(z,b,\mu,\hat{\zeta})H(Q,\mu)\,,$
(86)

where $\tilde{f}^{(m)}(x,b,\mu,\zeta)$ and
$\tilde{D}^{(n)}(z,b,\mu,\hat{\zeta})$ are the Fourier transform of TMD PDFs
and TMD FFs in $b$-space666To simplify the notations, we apply $n=0$ by
default, namely $\tilde{f}_{1}=\tilde{f}_{1}^{(0)}$ and
$\tilde{D}_{1}=\tilde{D}_{1}^{(0)}$. and defined by Boer:2011xd ; Kang:2021ffh

$\displaystyle\tilde{f}^{(m)}(x,b,\mu,\zeta)=$ $\displaystyle\frac{2\pi
m!}{(M^{2})^{m}}\int
dp_{T}p_{T}\left(\frac{p_{T}}{b}\right)^{m}J_{m}(bp_{T})f(x,{\bm{p}}_{T}^{2},\mu,\zeta)$
(87) $\displaystyle\tilde{D}^{(n)}(z,b,\mu,\hat{\zeta})=$
$\displaystyle\frac{2\pi n!}{(zM_{h}^{2})^{n}}\int
dk_{T}k_{T}\left(\frac{k_{T}}{b}\right)^{n}J_{n}(bk_{T})D(z,{\bm{k}}_{T}^{2},\mu,\hat{\zeta})\,.$
(88)

Thus one obtains the factorization of the spin-averaged structure function
$F_{UU}$ and the all possible spin-dependent structure functions shown in Eq.
(85) as

$\displaystyle F_{UU}=$ $\displaystyle\,C[\tilde{f}_{1}\tilde{D}_{1}]\,,$ (89)
$\displaystyle F_{UU}^{\cos(2\phi_{h})}=$
$\displaystyle\,-C[\tilde{h}^{\perp(1)}_{1}\tilde{H}^{\perp(1)}_{1}]\,,$ (90)
$\displaystyle F_{UL}^{\sin(2\phi_{h})}=$
$\displaystyle\,-C[\tilde{h}^{\perp(1)}_{1L}\tilde{H}^{\perp(1)}_{1}]\,,$ (91)
$\displaystyle F_{UT}^{\sin(\phi_{h}-\phi_{s})}=$
$\displaystyle\,-C[\tilde{f}^{\perp(1)}_{1T}\tilde{D}_{1}]\,,$ (92)
$\displaystyle F_{UT}^{\sin(\phi_{h}+\phi_{s})}=$
$\displaystyle\,-C[\tilde{h}_{1}\tilde{H}^{\perp(1)}_{1}]\,,$ (93)
$\displaystyle F_{UT}^{\sin(3\phi_{h}-\phi_{s})}=$
$\displaystyle\,-\frac{1}{4}C[\tilde{h}^{\perp(2)}_{1T}\tilde{H}^{\perp(1)}_{1}]\,,$
(94) $\displaystyle F_{LL}=$
$\displaystyle\,C[\tilde{g}_{1L}\tilde{D}_{1}]\,,$ (95) $\displaystyle
F_{LT}^{\cos(\phi_{h}-\phi_{s})}=$
$\displaystyle\,C[\tilde{g}^{\perp(1)}_{1T}\tilde{D}_{1}]\,,$ (96)

where we encounter unpolarized and Collins TMD FF from Eq. (8) again. Here,
summation over $q$ includes the anti-quarks as well. Then using the EEC jet
functions defined above in Eqs. (18) and (19), namely

$\displaystyle{J}_{q}(b,\mu,\zeta)\equiv$
$\displaystyle\sum_{h}\int_{0}^{1}dz\,z\,\tilde{D}_{1,h/q}(z,b,\mu,\zeta)$
(18) $\displaystyle{J}_{q}^{\perp}(b,\mu,\zeta)\equiv$
$\displaystyle\sum_{h}\int_{0}^{1}dz\,z\,\tilde{H}_{1,h/q}^{\perp}(z,b,\mu,\zeta)\,.$
(19)

Then the azimuthal angle dependent EEC at DIS in Eq. (2.2) can be written as
777Here the “subtracted” TMD PDFs $\tilde{f}_{q}^{(n)}(x,b,\mu,\zeta)$, EEC
jet functions ${J}_{q}(b,\mu,\zeta)$ and ${J}_{q}^{\perp}(b,\mu,\zeta)$ are
written as $\tilde{f}^{(n)}_{q},\ {J}_{q}$ and ${J}_{q}^{\perp}$ respectively,
for simplification of the notations.

$\displaystyle\frac{d\Sigma_{\rm DIS}}{dxdyd\tau d\phi}=$
$\displaystyle\sum_{h}\int
d^{2}\bm{q}_{T}dz_{h}z_{h}\frac{d\sigma}{dxdydz_{h}d^{2}\bm{q}_{T}}\delta\left(\tau-\frac{\bm{q}_{T}^{2}}{Q^{2}}\right)\delta(\phi-\phi_{h})$
$\displaystyle=$ $\displaystyle\sigma_{0}\sum_{q}e_{q}^{2}\int
d^{2}{\bm{q}}_{T}\delta(\tau-\frac{{\bm{q}}_{T}^{2}}{Q^{2}})\delta(\phi-\phi_{h})\int\frac{db\
b}{2\pi}H(Q,\mu)$
$\displaystyle\times\bigg{\\{}J_{0}(bq_{T})\tilde{f}_{1,q}{J}_{q}+\cos(2\phi_{h})\frac{2(1-y)}{1+(1-y)^{2}}J_{2}(bq_{T})\frac{b^{2}}{2}\tilde{h}_{1,q}^{\perp(1)}{J}^{\perp}_{q}$
$\displaystyle\quad+S_{\parallel}\sin(2\phi_{h})\frac{2(1-y)}{1+(1-y)^{2}}J_{2}(bq_{T})\frac{b^{2}}{2}\tilde{h}_{1L,q}^{\perp(1)}{J}^{\perp}_{q}$
$\displaystyle\quad+|{\bm{S}}_{\perp}|\bigg{[}-\sin(\phi_{h}-\phi_{s})J_{1}(bq_{T}){b}\tilde{f}_{1T,q}^{\perp(1)}{J}_{q}$
$\displaystyle\quad+\sin(\phi_{h}+\phi_{s})\frac{2(1-y)}{1+(1-y)^{2}}\frac{b}{2}J_{1}(bq_{T})\tilde{h}_{1,q}{J}^{\perp}_{q}$
$\displaystyle\quad+\sin(3\phi_{h}-\phi_{s})\frac{2(1-y)}{1+(1-y)^{2}}\frac{M^{2}b^{3}}{8}J_{3}(bq_{T})\tilde{h}_{1T,q}^{\perp(2)}{J}^{\perp}_{q}\bigg{]}$
$\displaystyle+\lambda_{e}\bigg{[}S_{\parallel}\frac{y(2-y)}{1+(1-y)^{2}}J_{0}(bq_{T})\tilde{g}_{1L,q}{J}_{q}+|{\bm{S}}_{\perp}|\cos(\phi_{h}-\phi_{s})J_{1}(bq_{T}){b}\tilde{g}_{1T,q}^{\perp(1)}{J}_{q}\bigg{]}\bigg{\\}}\,,$
(97)

where the first term appeared already in Li:2021txc .

Unlike the EEC in $e^{+}e^{-}$ annihilation, DIS version of the EEC has only
one EEC jet function per term as it measures correlation between one outgoing
hadron and the beam. By defining the notation
$\hat{\mathcal{C}}^{(\perp)}[\tilde{f}^{(m)}{J}^{(\perp)}]$ similar to
$\mathcal{C}^{(\perp)}[\tilde{f}^{(m)}{J}^{(\perp)}]$ in Eqs. (31) and (31)
but with an extra Bessel function, namely

$\displaystyle\mathcal{C}[\tilde{f}^{(m)}{J}]=$
$\displaystyle\sum_{q}e_{q}^{2}H(Q,\mu){J}_{q}(b,\mu,\zeta)b^{m}{M^{m}}\tilde{f}^{(m)}_{q}(x,b,\mu,\zeta)\,,$
(31) $\displaystyle\mathcal{C}^{\perp}[\tilde{f}^{(m)}{J}^{\perp}]=$
$\displaystyle\sum_{q}e_{q}^{2}H(Q,\mu){J}_{q}^{\perp}(b2,\mu,\zeta)\frac{b^{m+1}}{2}{M^{m}}{(-M_{h})}\tilde{f}^{(m)}_{q}(x,b,\mu,\zeta)\,,$
(32) $\displaystyle\hat{\mathcal{C}}[\tilde{f}^{(m)}{J}]=$
$\displaystyle\mathcal{C}[\tilde{f}^{(m)}{J}]J_{m}(bq_{T})\,,$ (98)
$\displaystyle\hat{\mathcal{C}}^{\perp}[\tilde{f}^{(m)}{J}^{\perp}]=$
$\displaystyle\mathcal{C}^{\perp}[\tilde{f}^{(m)}{J}^{\perp}]J_{m+1}(bq_{T})\,.$
(99)

Finally, we can define the structure functions $\mathcal{F}_{AB}$ as

$\displaystyle\mathcal{F}_{UU}=$
$\displaystyle\,\hat{\mathcal{C}}[\tilde{f}_{1}{J}]\,,$ (100)
$\displaystyle\mathcal{F}_{UU}^{\cos(2\phi_{h})}=$
$\displaystyle\,-\hat{\mathcal{C}}^{\perp}[\tilde{h}^{\perp(1)}_{1}{{J}^{\perp}}]\,,$
(101) $\displaystyle\mathcal{F}_{UL}^{\sin(2\phi_{h})}=$
$\displaystyle\,-\hat{\mathcal{C}}^{\perp}[\tilde{h}^{\perp(1)}_{1L}{{J}^{\perp}}]\,,$
(102) $\displaystyle\mathcal{F}_{UT}^{\sin(\phi_{h}-\phi_{s})}=$
$\displaystyle\,-\hat{\mathcal{C}}[\tilde{f}^{\perp(1)}_{1T}{J}]\,,$ (103)
$\displaystyle\mathcal{F}_{UT}^{\sin(\phi_{h}+\phi_{s})}=$
$\displaystyle\,-\hat{\mathcal{C}}^{\perp}[\tilde{h}_{1}{{J}^{\perp}}]\,,$
(104) $\displaystyle\mathcal{F}_{UT}^{\sin(3\phi_{h}-\phi_{s})}=$
$\displaystyle\,-\frac{1}{4}\hat{\mathcal{C}}^{\perp}[\tilde{h}^{\perp(2)}_{1T}{{J}^{\perp}}]\,,$
(105) $\displaystyle\mathcal{F}_{LL}=$
$\displaystyle\,\hat{\mathcal{C}}[\tilde{g}_{1L}{J}]\,,$ (106)
$\displaystyle\mathcal{F}_{LT}^{\cos(\phi_{h}-\phi_{s})}=$
$\displaystyle\,\hat{\mathcal{C}}[\tilde{g}^{\perp(1)}_{1T}{J}]\,,$ (107)

and obtains the azimuthal angle-dependent EEC at DIS as given in Eq. (25).

###  A.2 EEC using the lab-frame angles

As introduced in Gao:2022bzi , an optimized observable $q_{*}$ is defined for
probing the transverse momentum of the produced hadron with respect to the
photon momentum direction and at leading power $q_{*}=q_{T}\sin\phi_{h}$.
Namely one has

$\displaystyle\frac{d\sigma}{dxdydz_{h}d{q_{*}}}=$ $\displaystyle\int
d^{2}{\bm{q}}_{T}\delta\left(q_{*}-q_{T}\sin\phi_{h}\right)\frac{d\sigma}{dxdydz_{h}d^{2}{\bm{q}}_{T}}\,.$
(108)

Using the identities

$\displaystyle J_{0}\left({bq_{T}}\right):\ $ $\displaystyle\int
d^{2}{\bm{q}}_{T}\delta\left(q_{*}-q_{T}\sin\phi_{h}\right)J_{0}(bq_{T})=\int
q_{T}dq_{T}\frac{d\phi_{h}}{\sin\phi_{h}}\delta\left(\frac{q_{*}}{\sin\phi_{h}}-{q_{T}}\right)J_{0}\left({bq_{T}}\right)$
$\displaystyle=$
$\displaystyle\int\frac{\phi_{h}}{\sin^{2}\phi_{h}}q_{*}\Theta\left(\frac{q_{*}}{\sin\phi_{h}}\right)J_{0}\left(\frac{bq_{*}}{\sin\phi_{h}}\right)=\,\frac{2\cos\left(bq_{*}\right)}{b}\,,$
(109) $\displaystyle J_{1}\left({bq_{T}}\right):\ $ $\displaystyle\int
d^{2}{\bm{q}}_{T}\delta\left(q_{*}-q_{T}\sin\phi_{h}\right)\sin\left(\phi_{h}\right)J_{1}({bq_{T}})$
$\displaystyle=$
$\displaystyle\int\frac{\phi_{h}}{\sin^{2}\phi_{h}}q_{*}\Theta\left(\frac{q_{*}}{\sin\phi_{h}}\right)\sin\left(\phi_{h}\right)J_{1}\left(\frac{bq_{*}}{\sin\phi_{h}}\right)=\,\frac{2\sin\left(bq_{*}\right)}{b}\,,$
(110) $\displaystyle\int
d^{2}{\bm{q}}_{T}\delta\left(q_{*}-q_{T}\sin\phi_{h}\right)\cos\left(\phi_{h}\right)J_{1}(bq_{T})=0\,,$
(111) $\displaystyle J_{2}\left({bq_{T}}\right):\ $ $\displaystyle\int
d^{2}{\bm{q}}_{T}\delta\left(q_{*}-q_{T}\sin\phi_{h}\right)\sin\left(2\phi_{h}\right)J_{2}({bq_{T}})=0\,,$
(112) $\displaystyle\int
d^{2}{\bm{q}}_{T}\delta\left(q_{*}-q_{T}\sin\phi_{h}\right)\cos\left(2\phi_{h}\right)J_{2}({bq_{T}})$
$\displaystyle=$
$\displaystyle\int\frac{\phi_{h}}{\sin^{2}\phi_{h}}q_{*}\Theta\left(\frac{q_{*}}{\sin\phi_{h}}\right)\cos\left(2\phi_{h}\right)J_{2}\left(\frac{bq_{*}}{\sin\phi_{h}}\right)=\,\frac{2\cos\left(bq_{*}\right)}{b}\,,$
(113) $\displaystyle J_{3}\left({bq_{T}}\right):\ $ $\displaystyle\int
d^{2}{\bm{q}}_{T}\delta\left(q_{*}-q_{T}\sin\phi_{h}\right)\sin\left(3\phi_{h}\right)J_{3}(bq_{T})$
$\displaystyle=$
$\displaystyle\int\frac{\phi_{h}}{\sin^{2}\phi_{h}}q_{*}\Theta\left(\frac{q_{*}}{\sin\phi_{h}}\right)\sin\left(3\phi_{h}\right)J_{3}\left(\frac{bq_{*}}{\sin\phi_{h}}\right)=\,\frac{2\sin\left(bq_{*}\right)}{b}\,,$
(114) $\displaystyle\int
d^{2}{\bm{q}}_{T}\delta\left(q_{*}-q_{T}\sin\phi_{h}\right)\cos\left(3\phi_{h}\right)J_{3}(bq_{T})=0\,,$
(115)

following the DIS differential cross section introduced in App. A and Eq.
(108), one obtains

$\displaystyle\frac{d\sigma}{dxdydz_{h}d{q_{*}}}=$
$\displaystyle\sigma_{0}\int
db\sum_{q}e_{q}^{2}H(Q,\mu)S({\bm{b}}^{2},\mu,\nu)\bigg{\\{}\cos\left(bq_{*}\right)\bigg{[}\tilde{f}_{1}\tilde{D}_{1}$
$\displaystyle+\frac{2(1-y)}{1+(1-y)^{2}}zMM_{h}b^{2}\tilde{h}_{1}^{\perp(1)}\tilde{H}_{1}^{\perp(1)}+\lambda_{e}S_{\parallel}\frac{y(2-y)}{1+(1-y)^{2}}\tilde{g}_{1L}\tilde{D}_{1}\bigg{]}$
$\displaystyle+|{\bm{S}}_{\perp}|\sin\left(bq_{*}\right)\cos\phi_{s}\bigg{[}-Mb\tilde{f}_{1T}^{\perp(1)}\tilde{D}_{1}+\frac{2(1-y)}{1+(1-y)^{2}}zM_{h}b\tilde{h}_{1}\tilde{H}_{1}^{\perp(1)}$
$\displaystyle+\frac{2(1-y)}{1+(1-y)^{2}}\frac{zM^{2}M_{h}b^{3}}{4}\tilde{h}_{1T}^{\perp(2)}\tilde{H}_{1}^{\perp(1)}\bigg{]}+\lambda_{e}|{\bm{S}}_{\perp}|\sin\left(bq_{*}\right)\sin\phi_{s}M\tilde{g}_{1T}^{(1)}\tilde{D}_{1}\bigg{\\}}\,.$
(116)

Note that as indicated in Eq. (112), the contribution from $h_{1L}^{\perp}$
will vanish when measuring $q_{*}$. Now we define the EEC with $q_{*}$
measured as

$\displaystyle\frac{d\Sigma_{\rm DIS}}{dxdydq_{*}}=$
$\displaystyle\sum_{h}\int
d^{2}\bm{q}_{T}dz_{h}z_{h}\frac{d\sigma}{dxdydz_{h}d^{2}\bm{q}_{T}}\delta(q_{*}-q_{T}\sin\phi_{h})$
$\displaystyle=$ $\displaystyle\sigma_{0}\int
db\bigg{\\{}\cos\left(bq_{*}\right)\bigg{[}\mathcal{C}[\tilde{f}_{1}{J}_{q}]$
$\displaystyle+\frac{2(1-y)}{1+(1-y)^{2}}\mathcal{C}^{\perp}[\tilde{h}^{\perp(1)}{J}_{q}^{\perp}]+\lambda_{e}S_{\parallel}\frac{y(2-y)}{1+(1-y)^{2}}\mathcal{C}[\tilde{g}_{1L}{J}_{q}]\bigg{]}$
$\displaystyle+|{\bm{S}}_{\perp}|\sin\left(bq_{*}\right)\cos\phi_{s}\bigg{[}-\mathcal{C}[\tilde{f}^{\perp(1)}_{1T}{J}_{q}]+\frac{2(1-y)}{1+(1-y)^{2}}\mathcal{C}^{\perp}[\tilde{h}_{1}{J}_{q}^{\perp}]$
$\displaystyle+\frac{2(1-y)}{1+(1-y)^{2}}\frac{1}{4}\mathcal{C}^{\perp}[\tilde{h}^{\perp(2)}_{1T}{J}_{q}^{\perp}]\bigg{]}+\lambda_{e}|{\bm{S}}_{\perp}|\sin\left(bq_{*}\right)\sin\phi_{s}\mathcal{C}[\tilde{g}^{(1)}_{1T}{J}_{q}]\bigg{\\}}\,,$
(117)

where

$\displaystyle\mathcal{C}[\tilde{f}^{(m)}{J}]=$
$\displaystyle\sum_{q}e_{q}^{2}H(Q,\mu){J}_{q}(b,\mu,\zeta)b^{m}{M^{m}}\tilde{f}^{(m)}_{q}(x,b,\mu,\zeta)\,,$
(31) $\displaystyle\mathcal{C}^{\perp}[\tilde{f}^{(m)}{J}^{\perp}]=$
$\displaystyle\sum_{q}e_{q}^{2}H(Q,\mu){J}_{q}^{\perp}(b,\mu,\zeta)\frac{b^{m+1}}{2}{M^{m}}{(-M_{h})}\tilde{f}^{(m)}_{q}(x,b,\mu,\zeta)\,.$
(32)

Compared to $\hat{\mathcal{C}}[\cdots]$ defined in Eqs. (98) and (99), the
notation here $\mathcal{C}[\cdots]$ is not equipped with a Bessel function,
since $J_{m(+1)}(bq_{T})$ has been integrated out in Eqs. (109)-(115) for
measuring $q_{*}$.

Upon closer inspection of Eq. (117) and Eq.(10) in Gao:2022bzi , the disparity
in sign arises from different definitions of the $m$-th order TMD PDFs in
$b$-space. Nonetheless, one can successfully harmonize the two works by
appropriately factoring in $(-1)^{m}$ for each structure function in Eq. (117)
that corresponds to $\tilde{f}^{(m)}$, and thereby cancel the divergence in
convention.

## References

  * (1) C. L. Basham, L. S. Brown, S. D. Ellis, and S. T. Love, Energy Correlations in electron - Positron Annihilation: Testing QCD, Phys. Rev. Lett. 41 (1978) 1585.
  * (2) C. L. Basham, L. S. Brown, S. D. Ellis, and S. T. Love, Energy Correlations in electron-Positron Annihilation in Quantum Chromodynamics: Asymptotically Free Perturbation Theory, Phys. Rev. D 19 (1979) 2018.
  * (3) T. Kinoshita, Mass singularities of Feynman amplitudes, J. Math. Phys. 3 (1962) 650–677.
  * (4) T. D. Lee and M. Nauenberg, Degenerate Systems and Mass Singularities, Phys. Rev. 133 (1964) B1549–B1562.
  * (5) N. Sveshnikov and F. Tkachov, Jets and quantum field theory, Phys. Lett. B 382 (1996) 403–408, [hep-ph/9512370].
  * (6) F. V. Tkachov, Measuring multi - jet structure of hadronic energy flow or What is a jet?, Int. J. Mod. Phys. A 12 (1997) 5411–5529, [hep-ph/9601308].
  * (7) G. P. Korchemsky and G. F. Sterman, Power corrections to event shapes and factorization, Nucl. Phys. B 555 (1999) 335–351, [hep-ph/9902341].
  * (8) C. W. Bauer, S. P. Fleming, C. Lee, and G. F. Sterman, Factorization of e+e- Event Shape Distributions with Hadronic Final States in Soft Collinear Effective Theory, Phys. Rev. D 78 (2008) 034027, [arXiv:0801.4569].
  * (9) D. M. Hofman and J. Maldacena, Conformal collider physics: Energy and charge correlations, JHEP 05 (2008) 012, [arXiv:0803.1467].
  * (10) A. V. Belitsky, S. Hohenegger, G. P. Korchemsky, E. Sokatchev, and A. Zhiboedov, From correlation functions to event shapes, Nucl. Phys. B 884 (2014) 305–343, [arXiv:1309.0769].
  * (11) A. V. Belitsky, S. Hohenegger, G. P. Korchemsky, E. Sokatchev, and A. Zhiboedov, Event shapes in $\mathcal{N}=4$ super-Yang-Mills theory, Nucl. Phys. B 884 (2014) 206–256, [arXiv:1309.1424].
  * (12) P. Kravchuk and D. Simmons-Duffin, Light-ray operators in conformal field theory, JHEP 11 (2018) 102, [arXiv:1805.00098].
  * (13) A. V. Belitsky, S. Hohenegger, G. P. Korchemsky, E. Sokatchev, and A. Zhiboedov, Energy-Energy Correlations in N=4 Supersymmetric Yang-Mills Theory, Phys. Rev. Lett. 112 (2014), no. 7 071601, [arXiv:1311.6800].
  * (14) D. Chicherin, G. P. Korchemsky, E. Sokatchev, and A. Zhiboedov, Energy correlations in heavy states, arXiv:2306.14330.
  * (15) SLD Collaboration, K. Abe et al., Measurement of $\alpha_{s}(m_{Z}^{2})$ from hadronic event observables at the $Z^{0}$ resonance, Phys. Rev. D 51 (1995) 962–984, [hep-ex/9501003].
  * (16) L3 Collaboration, O. Adrian et al., Determination of alpha-s from hadronic event shapes measured on the $Z^{0}$ resonance, Phys. Lett. B 284 (1992) 471–481.
  * (17) OPAL Collaboration, P. D. Acton et al., An Improved measurement of $\alpha_{s}(m_{Z^{0}})$ using energy correlations with the OPAL detector at LEP, Phys. Lett. B 276 (1992) 547–564.
  * (18) TOPAZ Collaboration, I. Adachi et al., Measurements of $\alpha_{s}$ in $e^{+}e^{-}$ Annihilation at $\sqrt{s}=53.3$ GeV and $59.5$-GeV, Phys. Lett. B 227 (1989) 495–500.
  * (19) TASSO Collaboration, W. Braunschweig et al., A Study of Energy-energy Correlations Between 12-GeV and 46.8-GeV CM Energies, Z. Phys. C 36 (1987) 349–361.
  * (20) JADE Collaboration, W. Bartel et al., Measurements of Energy Correlations in $e^{+}e^{-}\to$ Hadrons, Z. Phys. C 25 (1984) 231\.
  * (21) E. Fernandez et al., A Measurement of Energy-energy Correlations in $e^{+}e^{-}\to$ Hadrons at $\sqrt{s}=29$ GeV, Phys. Rev. D 31 (1985) 2724\.
  * (22) D. R. Wood et al., Determination of $\alpha_{s}$ From Energy-energy Correlations in $e^{+}e^{-}$ Annihilation at $29$ GeV, Phys. Rev. D 37 (1988) 3091.
  * (23) CELLO Collaboration, H. J. Behrend et al., Analysis of the Energy Weighted Angular Correlations in Hadronic $e^{+}e^{-}$ Annihilations at 22 GeV and 34 GeV, Z. Phys. C 14 (1982) 95.
  * (24) PLUTO Collaboration, C. Berger et al., A Study of Energy-energy Correlations in $e^{+}e^{-}$ Annihilations at $\sqrt{s}=34$.6-GeV, Z. Phys. C 28 (1985) 365.
  * (25) OPAL Collaboration, M. Z. Akrawy et al., A Measurement of energy correlations and a determination of $\alpha_{s}(M_{Z^{0}}^{2})$ in $e^{+}e^{-}$ annihilations at $\sqrt{s}=91$ GeV, Phys. Lett. B 252 (1990) 159–169.
  * (26) ALEPH Collaboration, D. Decamp et al., Measurement of alpha-s from the structure of particle clusters produced in hadronic Z decays, Phys. Lett. B 257 (1991) 479–491.
  * (27) L3 Collaboration, B. Adeva et al., Determination of alpha-s from energy-energy correlations measured on the $Z^{0}$ resonance., Phys. Lett. B 257 (1991) 469–478.
  * (28) SLD Collaboration, K. Abe et al., Measurement of alpha-s from energy-energy correlations at the $Z^{0}$ resonance, Phys. Rev. D 50 (1994) 5580–5590, [hep-ex/9405006].
  * (29) H. T. Li, Y. Makris, and I. Vitev, Energy-energy correlators in Deep Inelastic Scattering, Phys. Rev. D 103 (2021), no. 9 094005, [arXiv:2102.05669].
  * (30) D. Neill, G. Vita, I. Vitev, and H. X. Zhu, Energy-Energy Correlators for Precision QCD, in Snowmass 2021, 3, 2022.  arXiv:2203.07113.
  * (31) A. Gao, H. T. Li, I. Moult, and H. X. Zhu, Precision QCD Event Shapes at Hadron Colliders: The Transverse Energy-Energy Correlator in the Back-to-Back Limit, Phys. Rev. Lett. 123 (2019), no. 6 062001, [arXiv:1901.04497].
  * (32) ATLAS Collaboration, G. Aad et al., Measurement of transverse energy-energy correlations in multi-jet events in $pp$ collisions at $\sqrt{s}=7$ TeV using the ATLAS detector and determination of the strong coupling constant $\alpha_{\mathrm{s}}(m_{Z})$, Phys. Lett. B 750 (2015) 427–447, [arXiv:1508.01579].
  * (33) ATLAS Collaboration, M. Aaboud et al., Determination of the strong coupling constant $\alpha_{\mathrm{s}}$ from transverse energy–energy correlations in multijet events at $\sqrt{s}=8~{}\hbox{TeV}$ using the ATLAS detector, Eur. Phys. J. C 77 (2017), no. 12 872, [arXiv:1707.02562].
  * (34) ATLAS Collaboration, Determination of the strong coupling constant and test of asymptotic freedom from Transverse Energy-Energy Correlations in multijet events at $\sqrt{s}=13$ TeV with the ATLAS detector, .
  * (35) M. Alvarez, J. Cantero, M. Czakon, J. Llorente, A. Mitov, and R. Poncelet, NNLO QCD corrections to event shapes at the LHC, JHEP 03 (2023) 129, [arXiv:2301.01086].
  * (36) Particle Data Group Collaboration, P. A. Zyla et al., Review of Particle Physics, PTEP 2020 (2020), no. 8 083C01.
  * (37) D. de Florian and M. Grazzini, The Back-to-back region in e+ e- energy-energy correlation, Nucl. Phys. B 704 (2005) 387–403, [hep-ph/0407241].
  * (38) P. N. Burrows, H. Masuda, D. Muller, and Y. Ohnishi, Application of ’optimized’ perturbation theory to determination of $\alpha_{s}(M_{Z}^{2})$ from hadronic event shape observables in $e^{+}e^{-}$ annihilation, Phys. Lett. B 382 (1996) 157–164, [hep-ph/9602210].
  * (39) A. Kardos, S. Kluth, G. Somogyi, Z. Tulipánt, and A. Verbytskyi, Precise determination of $\alpha_{S}(M_{Z})$ from a global fit of energy–energy correlation to NNLO+NNLL predictions, Eur. Phys. J. C 78 (2018), no. 6 498, [arXiv:1804.09146].
  * (40) H. N. Schneider, G. Kramer, and G. Schierholz, Higher Order QCD Corrections to the Energy-energy Correlation Function, Z. Phys. C 22 (1984) 201.
  * (41) N. K. Falck and G. Kramer, Theoretical Studies of Energy-energy Correlation in $e^{+}e^{-}$ Annihilation, Z. Phys. C 42 (1989) 459\.
  * (42) E. W. N. Glover and M. R. Sutton, The Energy-energy correlation function revisited, Phys. Lett. B 342 (1995) 375–380, [hep-ph/9410234].
  * (43) G. Kramer and H. Spiesberger, A New calculation of the NLO energy-energy correlation function, Z. Phys. C 73 (1997) 495–504, [hep-ph/9603385].
  * (44) A. Ali and F. Barreiro, An O ($\alpha^{-}s^{2}$) Calculation of Energy-energy Correlation in $e^{+}e^{-}$ Annihilation and Comparison With Experimental Data, Phys. Lett. B 118 (1982) 155–161.
  * (45) A. Ali and F. Barreiro, Energy-energy Correlations in $e^{+}e^{-}$ Annihilation, Nucl. Phys. B 236 (1984) 269.
  * (46) D. G. Richards, W. J. Stirling, and S. D. Ellis, Second Order Corrections to the Energy-energy Correlation Function in Quantum Chromodynamics, Phys. Lett. B 119 (1982) 193–197.
  * (47) D. G. Richards, W. J. Stirling, and S. D. Ellis, Energy-energy Correlations to Second Order in Quantum Chromodynamics, Nucl. Phys. B 229 (1983) 317–346.
  * (48) S. Catani and M. H. Seymour, The Dipole formalism for the calculation of QCD jet cross-sections at next-to-leading order, Phys. Lett. B 378 (1996) 287–301, [hep-ph/9602277].
  * (49) Z. Tulipánt, A. Kardos, and G. Somogyi, Energy–energy correlation in electron–positron annihilation at NNLL + NNLO accuracy, Eur. Phys. J. C 77 (2017), no. 11 749, [arXiv:1708.04093].
  * (50) L. J. Dixon, M.-X. Luo, V. Shtabovenko, T.-Z. Yang, and H. X. Zhu, Analytical Computation of Energy-Energy Correlation at Next-to-Leading Order in QCD, Phys. Rev. Lett. 120 (2018), no. 10 102001, [arXiv:1801.03219].
  * (51) M.-X. Luo, V. Shtabovenko, T.-Z. Yang, and H. X. Zhu, Analytic Next-To-Leading Order Calculation of Energy-Energy Correlation in Gluon-Initiated Higgs Decays, JHEP 06 (2019) 037, [arXiv:1903.07277].
  * (52) V. Del Duca, C. Duhr, A. Kardos, G. Somogyi, Z. Szőr, Z. Trócsányi, and Z. Tulipánt, Jet production in the CoLoRFulNNLO method: event shapes in electron-positron collisions, Phys. Rev. D 94 (2016), no. 7 074019, [arXiv:1606.03453].
  * (53) J. M. Henn, E. Sokatchev, K. Yan, and A. Zhiboedov, Energy-energy correlation in $N$=4 super Yang-Mills theory at next-to-next-to-leading order, Phys. Rev. D 100 (2019), no. 3 036010, [arXiv:1903.05314].
  * (54) S. T. Schindler, I. W. Stewart, and Z. Sun, Renormalons in the energy-energy correlator, arXiv:2305.19311.
  * (55) P. Nason and M. H. Seymour, Infrared renormalons and power suppressed effects in e+ e- jet events, Nucl. Phys. B 454 (1995) 291–312, [hep-ph/9506317].
  * (56) A. V. Belitsky, G. P. Korchemsky, and G. F. Sterman, Energy flow in QCD and event shape functions, Phys. Lett. B 515 (2001) 297–307, [hep-ph/0106308].
  * (57) J. Kodaira and L. Trentadue, Summing Soft Emission in QCD, Phys. Lett. B 112 (1982) 66.
  * (58) J. Kodaira and L. Trentadue, Single Logarithm Effects in electron-Positron Annihilation, Phys. Lett. B 123 (1983) 335–338.
  * (59) I. Moult and H. X. Zhu, Simplicity from Recoil: The Three-Loop Soft Function and Factorization for the Energy-Energy Correlation, JHEP 08 (2018) 160, [arXiv:1801.02627].
  * (60) M. A. Ebert, B. Mistlberger, and G. Vita, The Energy-Energy Correlation in the back-to-back limit at N3LO and N3LL’, JHEP 08 (2021) 022, [arXiv:2012.07859].
  * (61) C. Duhr, B. Mistlberger, and G. Vita, Four-Loop Rapidity Anomalous Dimension and Event Shapes to Fourth Logarithmic Order, Phys. Rev. Lett. 129 (2022), no. 16 162001, [arXiv:2205.02242].
  * (62) I. Moult, G. Vita, and K. Yan, Subleading power resummation of rapidity logarithms: the energy-energy correlator in $\mathcal{N}$ = 4 SYM, JHEP 07 (2020) 005, [arXiv:1912.02188].
  * (63) J. C. Collins and D. E. Soper, Back-To-Back Jets in QCD, Nucl. Phys. B 193 (1981) 381. [Erratum: Nucl.Phys.B 213, 545 (1983)].
  * (64) L. J. Dixon, I. Moult, and H. X. Zhu, Collinear limit of the energy-energy correlator, Phys. Rev. D 100 (2019), no. 1 014009, [arXiv:1905.01310].
  * (65) G. P. Korchemsky, Energy correlations in the end-point region, JHEP 01 (2020) 008, [arXiv:1905.01444].
  * (66) K. Konishi, A. Ukawa, and G. Veneziano, A Simple Algorithm for QCD Jets, Phys. Lett. B 78 (1978) 243–248.
  * (67) K. Konishi, A. Ukawa, and G. Veneziano, On the Transverse Spread of QCD Jets, Phys. Lett. B 80 (1979) 259–264.
  * (68) C. W. Bauer, D. Pirjol, and I. W. Stewart, Soft collinear factorization in effective field theory, Phys. Rev. D 65 (2002) 054022, [hep-ph/0109045].
  * (69) C. W. Bauer, S. Fleming, D. Pirjol, and I. W. Stewart, An Effective field theory for collinear and soft gluons: Heavy to light decays, Phys. Rev. D 63 (2001) 114020, [hep-ph/0011336].
  * (70) C. W. Bauer and I. W. Stewart, Invariant operators in collinear effective theory, Phys. Lett. B 516 (2001) 134–142, [hep-ph/0107001].
  * (71) K. Lee and I. Moult, Energy Correlators Taking Charge, arXiv:2308.00746.
  * (72) K. Devereaux, W. Fan, W. Ke, K. Lee, and I. Moult, Imaging Cold Nuclear Matter with Energy Correlators, arXiv:2303.08143.
  * (73) E. Craft, K. Lee, B. Meçaj, and I. Moult, Beautiful and Charming Energy Correlators, arXiv:2210.09311.
  * (74) K. Lee, B. Meçaj, and I. Moult, Conformal Colliders Meet the LHC, arXiv:2205.03414.
  * (75) Z. Yang, Y. He, I. Moult, and X.-N. Wang, Probing the Short-Distance Structure of the Quark-Gluon Plasma with Energy Correlators, arXiv:2310.01500.
  * (76) C. Andres, F. Dominguez, R. Kunnawalkam Elayavalli, J. Holguin, C. Marquet, and I. Moult, Resolving the Scales of the Quark-Gluon Plasma with Energy Correlators, Phys. Rev. Lett. 130 (2023), no. 26 262301, [arXiv:2209.11236].
  * (77) M. Procura, J. Holguin, I. Moult, and A. Pathak, Weighing the Top with Energy Correlators, in Strong interactions from QCD to new strong dynamics at LHC and Future Colliders, pp. 60–66, 2022.
  * (78) H. Chen, I. Moult, X. Zhang, and H. X. Zhu, Rethinking jets with energy correlators: Tracks, resummation, and analytic continuation, Phys. Rev. D 102 (2020), no. 5 054012, [arXiv:2004.11381].
  * (79) R. Boussarie et al., TMD Handbook, arXiv:2304.03302.
  * (80) I. Moult, H. X. Zhu, and Y. J. Zhu, The four loop QCD rapidity anomalous dimension, JHEP 08 (2022) 280, [arXiv:2205.02249].
  * (81) H. Chen, X. Zhou, and H. X. Zhu, Power Corrections to Energy Flow Correlations from Large Spin Perturbation, arXiv:2301.03616.
  * (82) H. Chen, I. Moult, and H. X. Zhu, Quantum Interference in Jet Substructure from Spinning Gluons, Phys. Rev. Lett. 126 (2021), no. 11 112003, [arXiv:2011.02492].
  * (83) J. a. Barata, J. G. Milhano, and A. V. Sadofyev, Picturing QCD jets in anisotropic matter: from jet shapes to Energy Energy Correlators, arXiv:2308.01294.
  * (84) X. L. Li, X. Liu, F. Yuan, and H. X. Zhu, Illuminating Nucleon Gluon Interference via Calorimetric Asymmetry, arXiv:2308.10942.
  * (85) A. Accardi et al., Electron Ion Collider: The Next QCD Frontier: Understanding the glue that binds us all, Eur. Phys. J. A 52 (2016), no. 9 268, [arXiv:1212.1701].
  * (86) R. Abdul Khalek et al., Science Requirements and Detector Concepts for the Electron-Ion Collider: EIC Yellow Report, arXiv:2103.05419.
  * (87) R. Abdul Khalek et al., Snowmass 2021 White Paper: Electron Ion Collider for High Energy Physics, arXiv:2203.13199.
  * (88) J. C. Collins, Fragmentation of transversely polarized quarks probed in transverse momentum distributions, Nucl. Phys. B 396 (1993) 161–182, [hep-ph/9208213].
  * (89) A. Gao, J. K. L. Michel, I. W. Stewart, and Z. Sun, Better angle on hadron transverse momentum distributions at the Electron-Ion Collider, Phys. Rev. D 107 (2023), no. 9 L091504, [arXiv:2209.11211].
  * (90) K. Gottfried and J. D. Jackson, On the Connection between production mechanism and decay of resonances at high-energies, Nuovo Cim. 33 (1964) 309–330.
  * (91) D. Pitonyak, M. Schlegel, and A. Metz, Polarized hadron pair production from electron-positron annihilation, Phys. Rev. D 89 (2014), no. 5 054032, [arXiv:1310.6240].
  * (92) D. Boer, Angular dependences in inclusive two-hadron production at BELLE, Nucl. Phys. B 806 (2009) 23–67, [arXiv:0804.2408].
  * (93) T. Becher and M. D. Schwartz, A precise determination of $\alpha_{s}$ from LEP thrust data using effective field theory, JHEP 07 (2008) 034, [arXiv:0803.0342].
  * (94) Y. Li and H. X. Zhu, Bootstrapping Rapidity Anomalous Dimensions for Transverse-Momentum Resummation, Phys. Rev. Lett. 118 (2017), no. 2 022004, [arXiv:1604.01404].
  * (95) A. Bacchetta, U. D’Alesio, M. Diehl, and C. A. Miller, Single-spin asymmetries: The Trento conventions, Phys. Rev. D 70 (2004) 117504, [hep-ph/0410050].
  * (96) J. Collins, Foundations of perturbative QCD, vol. 32.  Cambridge University Press, 11, 2013.
  * (97) MAP (Multi-dimensional Analyses of Partonic distributions) Collaboration, A. Bacchetta, V. Bertone, C. Bissolotti, G. Bozzi, M. Cerutti, F. Piacenza, M. Radici, and A. Signori, Unpolarized transverse momentum distributions from a global fit of Drell-Yan and semi-inclusive deep-inelastic scattering data, JHEP 10 (2022) 127, [arXiv:2206.07598].
  * (98) A. Kotzinian, New quark distributions and semiinclusive electroproduction on the polarized nucleons, Nucl. Phys. B 441 (1995) 234–248, [hep-ph/9412283].
  * (99) M. Diehl and S. Sapeta, On the analysis of lepton scattering on longitudinally or transversely polarized protons, Eur. Phys. J. C 41 (2005) 515–533, [hep-ph/0503023].
  * (100) A. Bacchetta, M. Diehl, K. Goeke, A. Metz, P. J. Mulders, and M. Schlegel, Semi-inclusive deep inelastic scattering at small transverse momentum, JHEP 02 (2007) 093, [hep-ph/0611265].
  * (101) M. Anselmino, M. Boglione, U. D’Alesio, A. Kotzinian, S. Melis, F. Murgia, A. Prokudin, and C. Turk, Sivers Effect for Pion and Kaon Production in Semi-Inclusive Deep Inelastic Scattering, Eur. Phys. J. A 39 (2009) 89–100, [arXiv:0805.2677].
  * (102) Z.-B. Kang, A. Prokudin, P. Sun, and F. Yuan, Extraction of Quark Transversity Distribution and Collins Fragmentation Functions with QCD Evolution, Phys. Rev. D 93 (2016), no. 1 014009, [arXiv:1505.05589].
  * (103) Y. Koike, J. Nagashima, and W. Vogelsang, Resummation for polarized semi-inclusive deep-inelastic scattering at small transverse momentum, Nucl. Phys. B 744 (2006) 59–79, [hep-ph/0602188].
  * (104) F. Yuan and J. Zhou, Collins Fragmentation and the Single Transverse Spin Asymmetry, Phys. Rev. Lett. 103 (2009) 052001, [arXiv:0903.4680].
  * (105) A. Bacchetta and A. Prokudin, Evolution of the helicity and transversity Transverse-Momentum-Dependent parton distributions, Nucl. Phys. B 875 (2013) 536–551, [arXiv:1303.2129].
  * (106) M. G. Echevarria, A. Idilbi, and I. Scimemi, Unified treatment of the QCD evolution of all (un-)polarized transverse momentum dependent functions: Collins function as a study case, Phys. Rev. D 90 (2014), no. 1 014003, [arXiv:1402.0869].
  * (107) M. G. Echevarria, I. Scimemi, and A. Vladimirov, Unpolarized Transverse Momentum Dependent Parton Distribution and Fragmentation Functions at next-to-next-to-leading order, JHEP 09 (2016) 004, [arXiv:1604.07869].
  * (108) A. Schafer and O. V. Teryaev, Sum rules for the T - odd fragmentation functions, Phys. Rev. D 61 (2000) 077903, [hep-ph/9908412].
  * (109) S. Meissner, A. Metz, and D. Pitonyak, Momentum sum rules for fragmentation functions, Phys. Lett. B 690 (2010) 296–303, [arXiv:1002.4393].
  * (110) A. Metz and A. Vossen, Parton Fragmentation Functions, Prog. Part. Nucl. Phys. 91 (2016) 136–202, [arXiv:1607.02521].
  * (111) S. M. Aybat and T. C. Rogers, TMD Parton Distribution and Fragmentation Functions with QCD Evolution, Phys. Rev. D 83 (2011) 114042, [arXiv:1101.5057].
  * (112) A. Prokudin, P. Sun, and F. Yuan, Scheme dependence and transverse momentum distribution interpretation of Collins–Soper–Sterman resummation, Phys. Lett. B 750 (2015) 533–538, [arXiv:1505.05588].
  * (113) J. C. Collins and D. E. Soper, Back-To-Back Jets: Fourier Transform from B to K-Transverse, Nucl. Phys. B 197 (1982) 446–476.
  * (114) J.-y. Chiu, A. Jain, D. Neill, and I. Z. Rothstein, The Rapidity Renormalization Group, Phys. Rev. Lett. 108 (2012) 151601, [arXiv:1104.0881].
  * (115) A. Avkhadiev, P. Shanahan, M. Wagman, and Y. Zhao, Collins-Soper kernel from lattice QCD at the physical pion mass, arXiv:2307.12359.
  * (116) Lattice Parton (LPC) Collaboration, M.-H. Chu et al., Lattice calculation of the intrinsic soft function and the Collins-Soper kernel, JHEP 08 (2023) 172, [arXiv:2306.06488].
  * (117) H.-T. Shu, M. Schlemmer, T. Sizmann, A. Vladimirov, L. Walter, M. Engelhardt, A. Schäfer, and Y.-B. Yang, Universality of the Collins-Soper kernel in lattice calculations, arXiv:2302.06502.
  * (118) M. Schlemmer, A. Vladimirov, C. Zimmermann, M. Engelhardt, and A. Schäfer, Determination of the Collins-Soper Kernel from Lattice QCD, JHEP 08 (2021) 004, [arXiv:2103.16991].
  * (119) V. Moos, I. Scimemi, A. Vladimirov, and P. Zurita, Extraction of unpolarized transverse momentum distributions from fit of Drell-Yan data at N4LL, arXiv:2305.07473.
  * (120) V. Bertone, I. Scimemi, and A. Vladimirov, Extraction of unpolarized quark transverse momentum dependent parton distributions from Drell-Yan/Z-boson production, JHEP 06 (2019) 028, [arXiv:1902.08474].
  * (121) A. Bacchetta, V. Bertone, C. Bissolotti, G. Bozzi, F. Delcarro, F. Piacenza, and M. Radici, Transverse-momentum-dependent parton distributions up to N3LL from Drell-Yan data, JHEP 07 (2020) 117, [arXiv:1912.07550].
  * (122) I. Scimemi and A. Vladimirov, Non-perturbative structure of semi-inclusive deep-inelastic and Drell-Yan scattering at small transverse momentum, JHEP 06 (2020) 137, [arXiv:1912.06532].
  * (123) K. Lee and I. Moult, Joint Track Functions: Expanding the Space of Calculable Correlations at Colliders, arXiv:2308.01332.
  * (124) Y. Li, I. Moult, S. S. van Velzen, W. J. Waalewijn, and H. X. Zhu, Extending Precision Perturbative QCD with Track Functions, Phys. Rev. Lett. 128 (2022), no. 18 182001, [arXiv:2108.01674].
  * (125) M. Jaarsma, Y. Li, I. Moult, W. J. Waalewijn, and H. X. Zhu, Energy Correlators on Tracks: Resummation and Non-Perturbative Effects, arXiv:2307.15739.
  * (126) W. Vogelsang and F. Yuan, Single-transverse spin asymmetries: From DIS to hadronic collisions, Phys. Rev. D 72 (2005) 054028, [hep-ph/0507266].
  * (127) D. de Florian, R. Sassot, M. Epele, R. J. Hernández-Pinto, and M. Stratmann, Parton-to-Pion Fragmentation Reloaded, Phys. Rev. D 91 (2015), no. 1 014035, [arXiv:1410.6027].
  * (128) H.-L. Lai, M. Guzzi, J. Huston, Z. Li, P. M. Nadolsky, J. Pumplin, and C.-P. Yuan, New parton distributions for collider physics, Phys. Rev. D 82 (2010) 074024, [arXiv:1007.2241].
  * (129) M. G. Echevarria, Z.-B. Kang, and J. Terry, Global analysis of the Sivers functions at NLO+NNLL in QCD, JHEP 01 (2021) 126, [arXiv:2009.10710].
  * (130) H1, ZEUS Collaboration, H. Abramowicz et al., Combination of measurements of inclusive deep inelastic ${e^{\pm}p}$ scattering cross sections and QCD analysis of HERA data, Eur. Phys. J. C 75 (2015), no. 12 580, [arXiv:1506.06042].
  * (131) S. Bhattacharya, Z.-B. Kang, A. Metz, G. Penn, and D. Pitonyak, First global QCD analysis of the TMD g1T from semi-inclusive DIS data, Phys. Rev. D 105 (2022), no. 3 034007, [arXiv:2110.10253].
  * (132) M. Horstmann, A. Schafer, and A. Vladimirov, Study of the worm-gear-T function g1T with semi-inclusive DIS data, Phys. Rev. D 107 (2023), no. 3 034016, [arXiv:2210.07268].
  * (133) J. C. Collins, D. E. Soper, and G. F. Sterman, Transverse Momentum Distribution in Drell-Yan Pair and W and Z Boson Production, Nucl. Phys. B 250 (1985) 199–224.
  * (134) M. Gourdin, Semiinclusive reactions induced by leptons, Nucl. Phys. B 49 (1972) 501–512.
  * (135) D. Boer, L. Gamberg, B. Musch, and A. Prokudin, Bessel-Weighted Asymmetries in Semi Inclusive Deep Inelastic Scattering, JHEP 10 (2011) 021, [arXiv:1107.5294].
  * (136) Z.-B. Kang, K. Lee, D. Y. Shao, and F. Zhao, Spin asymmetries in electron-jet production at the future electron ion collider, JHEP 11 (2021) 005, [arXiv:2106.15624].