# Twist-3 Contributions in Semi-Inclusive DIS in the Target Fragmentation
Region

K.B. Chen<EMAIL_ADDRESS>School of Science, Shandong Jianzhu
University, Jinan, Shandong 250101, China  J.P. Ma<EMAIL_ADDRESS>CAS Key
Laboratory of Theoretical Physics, Institute of Theoretical Physics, P.O. Box
2735, Chinese Academy of Sciences, Beijing 100190, China  School of Physical
Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
School of Physics and Center for High-Energy Physics, Peking University,
Beijing 100871, China  X.B. Tong<EMAIL_ADDRESS>School of Science and
Engineering, The Chinese University of Hong Kong, Shenzhen, Shenzhen,
Guangdong, 518172, China  University of Science and Technology of China,
Hefei, Anhui, 230026, China

###### Abstract

We present the complete results up to twist-3 for hadron production in the
target fragmentation region of semi-inclusive deep inelastic scattering with a
polarized lepton beam and polarized nucleon target. The non-perturbative
effects are factorized into fracture functions. The calculation up to twist-3
is non-trivial since one has to keep gauge invariance. By applying collinear
expansion, we show that the hadronic tensor can be expressed by gauge-
invariant fracture functions. We also present the results for the structure
functions and azimuthal asymmetries.

##  I Introduction

Semi-Inclusive Deep Inelastic Scattering (SIDIS) is an important process for
hadronic physics. It provides a cleaner environment for detecting the inner
structure of the initial hadron than inclusive processes in hadron-hadron
collisions. The kinematic region of SIDIS can roughly be divided into two
parts (see e.g., Boglione:2022gpv ; Boglione:2019nwk ; Gonzalez-
Hernandez:2018ipj ; Boglione:2016bph ; Mulders:2000jt  for more discussions).
One is called the Current Fragmentation Region (CFR) where the observed hadron
in the final state moves into the forward region of the virtual photon.
Another one is the Target Fragmentation Region (TFR), where the measured
hadron predominantly travels in the forward direction of the incoming target.
Events in both regions can be used to comprehend the internal structure of
hadrons and the properties of strong interactions.

So far, the bulk of research on SIDIS has focused on the CFR, where hadron
production can be understood as the fragmentation of a parton emitted from the
target and struck by the virtual photon. This allows us to investigate various
parton distributions functions (PDFs) Collins:1981uw ; Ethier:2020way ;
Jimenez-Delgado:2013sma ; Forte:2013wc  and fragmentation functions (FFs)
Collins:1981uw ; Metz:2016swz ; Chen:2023kqw  within the transverse-momentum-
dependent (TMD) Collins:1981uk ; Ji:2004wu ; Ji:2004xq ; Rodini:2023plb ;
Boussarie:2023izj  or collinear factorization formalisms Collins:1989gx ;
Meng:1991da ; Daleo:2004pn ; Kniehl:2004hf ; Wang:2019bvb  at small or large
hadron transverse momentum. While there have been significant developments in
recent years for physics in the CFR Barone:2010zz ; Boussarie:2023izj , the
physics in the TFR has received less attention.

The early analysis of the experimental data at HERA ZEUS:1994mgw ; H1:1995cha
indicates a surprisingly high number of events in the TFR and has stimulated
the introduction of fracture functions Trentadue:1993ka ; Berera:1995fj ;
Grazzini:1997ih . Physically, fracture functions describe the distributions of
the struck parton inside the target when the remnant spectators fragment
inclusively into the detected hadron. They encompass intricate initial-final
state correlations and provide a unique perspective into the partonic dynamics
and hadronization, complementing PDFs and FFs. Most of our current knowledge
about fracture functions comes from the analysis of proton diffraction (see
e.g., Frankfurt:2022jns  for a recent review), where the final hadron
coincides with the target proton, and the fracture function is conventionally
called as diffractive PDF Berera:1995fj . Phenomenological fittings of
diffractive PDFs from HERA data H1:1994ahk ; H1:1995cha ; H1:1997bdi ;
H1:2006zyl ; H1:2006uea ; H1:2011jpo ; H1:2012pbl ; H1:2012xlc  have been
conducted in deFlorian:1998rj ; Goharipour:2018yov ; Khanpour:2019pzq ;
Maktoubian:2019ppi ; Salajegheh:2022vyv ; Salajegheh:2023jgi . Fracture
functions for other leading baryon production, such as neutron and
$\Lambda$-hyperon, are also constrained with parameterization assumptions in
deFlorian:1997wi ; Ceccopieri:2014rpa ; Shoeibi:2017lrl ; Shoeibi:2017zha ;
Mohsenabadi:2021bhs  and Ceccopieri:2012rm ; Ceccopieri:2015kya ,
respectively. Fracture functions are also utilized in hadron collisions in
Ceccopieri:2008fq ; Ceccopieri:2010zu ; Chai:2019ykk .

From a theoretical point of view, most of the aforementioned investigations
about TFR hadron production are based on the factorization at twist-2 in terms
of collinear fracture functions. This factorization has been proven to hold to
all orders of $\alpha_{s}$ Collins:1997sr  and has been confirmed through the
explicit calculations up to ${\cal O}(\alpha_{s}^{2})$ Graudenz:1994dq ;
deFlorian:1995fd ; Daleo:2003jf ; Daleo:2003xg . Initially, the fracture
functions in the factorization included an integration over the final-hadron
transverse momentum $P_{h\perp}$ Trentadue:1993ka , however, without this
integration the momentum transfer Berera:1995fj ; Grazzini:1997ih  and
azimuthal-angle distribution Anselmino:2011ss ; Chen:2021vby  can be studied.
To further probe the spin Anselmino:2011ss ; deFlorian:1995fd  and parton-
transverse-momentum Anselmino:2011ss ; Ceccopieri:2005zz ; Ceccopieri:2007ek
dependence of fracture functions, several observables and factorization
assumptions are proposed in Anselmino:2011bb ; Anselmino:2011vkz ;
Yang:2020sos . CLAS collaboration at JLab has recently reported the first
measurement of these dependences CLAS:2022sqt . More detailed discussions on
the factorization with TMD fracture functions and relevant evolution is
presented in Chai:2019ykk . Furthermore, recent investigations have also
addressed the factorization properties of fracture functions in different
kinematic regions Chai:2019ykk ; Chen:2021vby ; Hatta:2022lzj . The small-$x$
behavior of fracture functions is studied in Hatta:2022lzj . In Chen:2021vby
the large-$P_{h\perp}$ behavior is explored in detail, aiming to understand
the transition of production mechanisms between the TFR and CFR in SIDIS.

Despite these progresses, the contributions of SIDIS in the TFR beyond the
leading twist are still unknown, and the theoretical framework for embracing
higher-twist effects in the TFR has not yet been established. The importance
of higher-twist effects in improving the description of the experimental data
has already been emphasized by recent phenomenological studies of fracture
functions Golec-Biernat:2007mao ; Maktoubian:2019ppi ; Salajegheh:2022vyv .

Moreover, it is known that the absence of higher-twist effects results in the
loss of predictions for fourteen SIDIS structure functions in the TFR at the
tree level. At the leading twist, only four structure functions are nonzero
Anselmino:2011ss  for the case of unpolarized hadron production and a spin-1/2
target Diehl:2005pc ; Bacchetta:2006tn . These higher-twist contributions are
responsible for various intriguing azimuthal and spin asymmetries. Especially,
some of these asymmetries are already within the reach of the ongoing
experimental program by CLAS12 Burkert:2020akg  at JLab due to the
availability of a longitudinally polarized target Arrington:2021alx ;
Accardi:2023chb . For instance, a preliminary investigation utilizing CLAS12
data has revealed the significance of the beam-spin asymmetry in the TFR,
suggesting that its sign and magnitude could serve as a novel indicator for
tracking the transition between the TFR and the CFR (Sec. 5.3 in
Accardi:2023chb ). Furthermore, the potential JLab@22GeV program
Accardi:2023chb  and the planned Electron-ion colliders in the U.S.
Boer:2011fh ; Accardi:2012qut ; AbdulKhalek:2021gbh ; AbdulKhalek:2022hcn ;
Abir:2023fpo  and China Anderle:2021wcy  are poised to provide additional
exciting opportunities for exploring new frontiers in TFR physics. Given these
experimental progresses, it is important to undertake an evaluation of the
higher-twist contributions of SIDIS in the TFR.

The objective of this paper is to present a first analysis of twist-$3$
contributions to SIDIS in the TFR within the framework of collinear
factorization at the tree level of quantum chromodynamics (QCD) perturbation
theory. Our focus is on the scenario where the target is spin-1/2 and the
polarization of the final hadron is unobserved. The framework can be easily
extended to the case of a spin-1 target. By employing the collinear expansion
technique Ellis:1982wd ; Ellis:1982cd ; Qiu:1988dn ; Qiu:1990xxa ; Qiu:1990xy
; Liang:2006wp ; Song:2010pf ; Wei:2014pma ; Yang:2017sxz , we demonstrate
that the hadronic tensor of SIDIS in the TFR can be expressed in terms of
three distinct types of twist-3 collinear fracture functions. We discuss the
classification of these fracture functions and show that they are not
independent due to the constraints imposed by the QCD Equation of Motion
(EOM). With the EOM, the twist-3 contributions can be expressed with two-
parton fracture functions at the considered order. Our findings also have
significant phenomenological implications.

The rest of this paper is organized as follows. In section II, we discuss the
kinematics for the polarized SIDIS in TFR and present the general form for the
cross-section in terms of the structure functions. In section III, we present
detailed calculations of the hadronic tensor up to twist-3. In section IV, we
give the final results for the structure functions and azimuthal or spin
asymmetries expressed by fracture functions. A short summary is given in
section V.

##  II Kinematics and Structure Functions of SIDIS in the TFR

Through out this paper, we use the light-cone coordinate system, in which a
vector $a^{\mu}$ is expressed as
$a^{\mu}=(a^{+},a^{-},\vec{a}_{\perp})=\bigl{(}(a^{0}+a^{3})/{\sqrt{2}},(a^{0}-a^{3})/{\sqrt{2}},a^{1},a^{2}\bigr{)}$.
With the light-cone vectors $n^{\mu}=(0,1,0,0)$ and $\bar{n}^{\mu}=(1,0,0,0)$,
the transverse metric is defined as
$g_{\perp}^{\mu\nu}=g^{\mu\nu}-\bar{n}^{\mu}n^{\nu}-\bar{n}^{\nu}n^{\mu}$, and
the transverse antisymmetric tensor is given as
$\varepsilon_{\perp}^{\mu\nu}=\varepsilon^{\mu\nu\alpha\beta}\bar{n}_{\alpha}n_{\beta}$
with $\varepsilon_{\perp}^{12}=1$. We also use the notation
$\tilde{a}_{\perp}^{\mu}\equiv\varepsilon_{\perp}^{\mu\nu}a_{\perp\nu}$.

We consider the SIDIS process with a polarized electron beam and nucleon
target as follows:

$e(l,\lambda_{e})+h_{A}(P,S)\to e(l^{\prime})+h(P_{h})+X,$ (1)

where $l$, $l^{\prime}$, $P$ and $P_{h}$ are the 4-momenta of the incident,
the outgoing electron, the nucleon target and the detected final state hadron,
respectively. At the leading order of quantum electrodynamics, there is an
exchange of one virtual photon between the electron and the nucleon. The
momentum of the virtual photon is given by $q=l-l^{\prime}$. The helicity of
the electron is denoted by $\lambda_{e}$, and $S$ is the polarization vector
of the nucleon. We consider the production of a spin-0 or unpolarized final
state hadron $h$. The Lorentz invariant variables of SIDIS are conventionally
defined by

$\displaystyle Q^{2}=-q^{2},~{}x_{B}=\frac{Q^{2}}{2P\cdot q},~{}y=\frac{P\cdot
q}{P\cdot k_{e}},~{}z_{h}=\frac{P\cdot P_{h}}{P\cdot q}.$ (2)

We are interested in the TFR, where $P_{h}$ is almost collinear with $P$ and
$z_{h}\ll 1$. As discussed in Graudenz:1994dq ; Anselmino:2011ss , $z_{h}$ is
not convenient for us to describe the hadron production in TFR, because one
can not differentiate the scenario of TFR considered here from the soft-hadron
production. Instead, we will use Anselmino:2011ss ; Boglione:2019nwk

$\displaystyle\xi_{h}=\frac{P_{h}\cdot q}{P\cdot q}.$ (3)

Figure 1: The kinematics for SIDIS in the TFR

We work in the reference frame shown in Fig. 1, where the nucleon $h_{A}$
moves along the $+z$-direction and the virtual photon moves in the
$-z$-direction. In this frame, the momenta of the particles are given by

$\displaystyle P^{\mu}\approx(P^{+},0,0,0),$ (4) $\displaystyle
P_{h}^{\mu}=(P_{h}^{+},P_{h}^{-},\vec{P}_{h\perp}),$ (5) $\displaystyle
l^{\mu}=\Big{(}\frac{1-y}{y}x_{B}P^{+},~{}\frac{Q^{2}}{2x_{B}yP^{+}},~{}\frac{Q\sqrt{1-y}}{y},~{}0\Big{)},$
(6) $\displaystyle
q^{\mu}=\Big{(}-x_{B}P^{+},~{}\frac{Q^{2}}{2x_{B}P^{+}},0,0\Big{)}.$ (7)

For the case that the produced hadron $h$ has small transverse momentum and in
the TFR, we have $P_{h}^{+}\gg|\vec{P}_{h\perp}|\gg P_{h}^{-}$ and
$\xi_{h}\approx P_{h}^{+}/P^{+}$, which specifies the longitudinal momentum
fraction of the nucleon taken by the final state hadron $h$. The polarization
vector of the nucleon with mass $M$ can be decomposed by

$\displaystyle
S^{\mu}=S_{L}\frac{P^{+}}{M}\bar{n}^{\mu}+S_{\perp}^{\mu}-S_{L}\frac{M}{2P^{+}}n^{\mu},$
(8)

where $S_{L}$ is the longitudinal polarization of the nucleon and
$S_{\perp}^{\mu}=(0,0,\vec{S}_{\perp})$ the transverse polarization vector.

The incoming- and outgoing electron span the lepton plane. We define the
azimuthal angle $\phi_{h}$ for $\vec{P}_{h\perp}$ with respect to the lepton
plane, and $\phi_{S}$ is that for $\vec{S}_{\perp}$. The azimuthal angle of
the outgoing lepton around the lepton beam with respect to the spin vector is
denoted by $\psi$. In the kinematic region of SIDIS with large $Q^{2}$, one
has $\psi\approx\phi_{S}$ Diehl:2005pc . With these specifications, the
differential cross-section is given by

$\displaystyle\frac{d\sigma}{dx_{B}dyd\xi_{h}d\psi
d^{2}P_{h\perp}}=\frac{\alpha^{2}y}{4\xi_{h}Q^{4}}L_{\mu\nu}(l,\lambda_{e},l^{\prime})W^{\mu\nu}(q,P,S,P_{h}),$
(9)

where $\alpha$ is the fine structure constant. The leptonic tensor is

$\displaystyle
L^{\mu\nu}(l,\lambda_{e},l^{\prime})=2(l^{\mu}l^{\prime\nu}+l^{\nu}l^{\prime\mu}-l\cdot
l^{\prime}g^{\mu\nu})+2i\lambda_{e}\epsilon^{\mu\nu\rho\sigma}l_{\rho}l^{\prime}_{\sigma}.$
(10)

The hadronic tensor is defined by

$\displaystyle
W^{\mu\nu}(q,P,S,P_{h})=\sum_{X}\int\frac{d^{4}x}{(2\pi)^{4}}e^{iq\cdot
x}\langle S;h_{A}|J^{\mu}(x)|hX\rangle\langle Xh|J^{\nu}(0)|h_{A};S\rangle,$
(11)

where $J^{\mu}(x)=e_{q}\bar{\psi}(x)\gamma^{\mu}\psi(x)$ is the
electromagnetic current. A summation over quark favors is implicit in Eq.
(11).

In general, the hadronic tensor can be decomposed into a sum of basic Lorentz
tensors constructed by the kinematic variables of the process. After
contracting with the leptonic tensor, one can get the differential cross
section in terms of the structure functions. It has been shown that the
differential cross section of SIDIS at small transverse momentum with the
polarized lepton beam and nucleon target is described by eighteen structure
functions Bacchetta:2006tn . We have the same number of structure functions
for SIDIS in the TFR, and the general form of the differential cross section
can be expressed as

$\displaystyle\frac{d\sigma}{dx_{B}dyd\xi_{h}d\psi
d^{2}P_{h\perp}}=\frac{\alpha^{2}}{x_{B}yQ^{2}}\Bigl{\\{}A(y)F_{UU,T}+E(y)F_{UU,L}+B(y)F_{UU}^{\cos\phi_{h}}\cos\phi_{h}+E(y)F_{UU}^{\cos
2\phi_{h}}\cos 2\phi_{h}$
$\displaystyle+\lambda_{e}D(y)F_{LU}^{\sin\phi_{h}}\sin\phi_{h}+S_{L}\Bigl{[}B(y)F_{UL}^{\sin\phi_{h}}\sin\phi_{h}+E(y)F_{UL}^{\sin
2\phi_{h}}\sin
2\phi_{h}\Bigr{]}+\lambda_{e}S_{L}\Bigl{[}C(y)F_{LL}+D(y)F_{LL}^{\cos\phi_{h}}\cos\phi_{h}\Bigr{]}$
$\displaystyle+|\vec{S}_{\perp}|\Bigl{[}\bigl{(}A(y)F_{UT,T}^{\sin(\phi_{h}-\phi_{S})}+E(y)F_{UT,L}^{\sin(\phi_{h}-\phi_{S})}\bigr{)}\sin(\phi_{h}-\phi_{S})+E(y)F_{UT}^{\sin(\phi_{h}+\phi_{S})}\sin(\phi_{h}+\phi_{S})$
$\displaystyle\qquad+B(y)F_{UT}^{\sin\phi_{S}}\sin\phi_{S}+B(y)F_{UT}^{\sin(2\phi_{h}-\phi_{S})}\sin(2\phi_{h}-\phi_{S})+E(y)F_{UT}^{\sin(3\phi_{h}-\phi_{S})}\sin(3\phi_{h}-\phi_{S})\Bigr{]}$
$\displaystyle+\lambda_{e}|\vec{S}_{\perp}|\Bigl{[}D(y)F_{LT}^{\cos\phi_{S}}\cos\phi_{S}+C(y)F_{LT}^{\cos(\phi_{h}-\phi_{S})}\cos(\phi_{h}-\phi_{S})+D(y)F_{LT}^{\cos(2\phi_{h}-\phi_{S})}\cos(2\phi_{h}-\phi_{S})\Bigr{]}\Bigr{\\}}.$
(12)

Here we have defined several functions of $y$ for convenience, i.e.,

$\displaystyle A(y)=y^{2}-2y+2,$ $\displaystyle B(y)=2(2-y)\sqrt{1-y},$
$\displaystyle C(y)=y(2-y),$ $\displaystyle D(y)=2y\sqrt{1-y},$ $\displaystyle
E(y)=2(1-y).$ (13)

All the structure functions in Eq. (12) are scalar functions depending on
$x_{B}$, $\xi_{h}$, $Q^{2}$ and $\vec{P}_{h\perp}^{2}$. The first and second
subscripts of the structure functions denote the polarization of the electron
and the nucleon, respectively. The third subscript, if any, specifies the
polarization of the virtual photon. Note that the normalization of the
structure functions adopted here is different from that in Bacchetta:2006tn
by a Jacobian since we have used $\xi_{h}$ instead of $z_{h}$.

##  III The hadronic tensor results up to twist-3

###  III.1 Collinear expansion for the hadronic tensor

Figure 2: Diagrams for the hadronic tensor in TFR at tree level.

Now we perform the collinear expansion for the hadronic tensor in Eq. (11) up
to twist-3. At the tree level of QCD perturbation theory, the hadronic tensor
in the TFR can be represented by the diagrams in Fig. 2. The gray boxes
represent the parton correlation matrices with a hadron $h$ identified in the
final state, which we call fracture matrices in the following. The
contributions for each diagram in Fig. 2 are

$\displaystyle W^{\mu\nu}\Big{|}_{2a}=$
$\displaystyle\int\frac{d^{3}k}{(2\pi)^{3}}\left[\left(\gamma^{\mu}(\not{k}+\not{q})\gamma^{\nu}\right)_{ij}2\pi\delta\bigl{(}(k+q)^{2}\bigr{)}\right]\sum_{X}\int\frac{d^{3}\eta}{(2\pi)^{4}}e^{-ik\cdot\eta}\langle
h_{A}|\bar{\psi}_{i}(\eta)|hX\rangle\langle Xh|\psi_{j}(0)|h_{A}\rangle,$ (14)
$\displaystyle W^{\mu\nu}\Big{|}_{2b}=$
$\displaystyle\int\frac{d^{3}k_{1}d^{3}k_{2}}{(2\pi)^{6}}\left[\left(\gamma^{\mu}(\not{k}_{1}+\not{q})\gamma_{\alpha}\frac{i(\not{k}_{2}+\not{q})}{(k_{2}+q)^{2}+i\epsilon}\gamma^{\nu}\right)_{ij}2\pi\delta\bigl{(}(k_{1}+q)^{2}\bigr{)}\right]$
$\displaystyle\times(-ig_{s})\sum_{X}\int\frac{d^{3}\eta
d^{3}\eta_{1}}{(2\pi)^{4}}e^{-ik_{1}\cdot\eta}e^{i(k_{1}-k_{2})\cdot\eta_{1}}\langle
h_{A}|\bar{\psi}_{i}(\eta)|hX\rangle\langle
Xh|G^{\alpha}(\eta_{1})\psi_{j}(0)|h_{A}\rangle,$ (15) $\displaystyle
W^{\mu\nu}\Big{|}_{2c}=$
$\displaystyle\int\frac{d^{3}k_{1}d^{3}k_{2}}{(2\pi)^{6}}\left[\left(\gamma^{\mu}\frac{i(\not{k}_{1}+\not{q})}{(k_{1}+q)^{2}-i\epsilon}\gamma_{\alpha}(\not{k}_{2}+\not{q})\gamma^{\nu}\right)_{ij}2\pi\delta\bigl{(}(k_{2}+q)^{2}\bigr{)}\right]$
$\displaystyle\times(-ig_{s})\sum_{X}\int\frac{d^{3}\eta
d^{3}\eta_{1}}{(2\pi)^{4}}e^{-ik_{1}\cdot\eta}e^{i(k_{1}-k_{2})\cdot\eta_{1}}\langle
h_{A}|\bar{\psi}_{i}(\eta)G^{\alpha}(\eta_{1})|hX\rangle\langle
Xh|\psi_{j}(0)|h_{A}\rangle,$ (16)

where $ij$ are the Dirac and color indices. The summation over quark flavors
$\sum_{q}e_{q}^{2}$ is implied in the expressions. The integration variables
take the following forms:

$\displaystyle k^{\mu}=(k^{+},0,\vec{k}_{\perp}),\quad
k_{1}^{\mu}=(k_{1}^{+},0,\vec{k}_{1\perp}),\quad
k_{2}^{\mu}=(k_{2}^{+},0,\vec{k}_{2\perp}),$ (17)
$\displaystyle\eta^{\mu}=(0,\eta^{-},\vec{\eta}_{\perp}),\quad\eta_{1}^{\mu}=(0,\eta_{1}^{-},\vec{\eta}_{1\perp}),\quad\eta_{2}^{\mu}=(0,\eta_{2}^{-},\vec{\eta}_{2\perp}).$
(18)

$k$ is the momentum carried by the quark line leaving the box of Fig. 2(a).
$k_{1}$, $k_{2}$ are the momenta carried by the quark lines flowing into and
out of the boxes of Figs. 2(b) or 2(c). These momenta follow the collinear
scaling, e.g., $k^{\mu}\sim Q(1,\lambda^{2},\lambda)$ with
$\lambda=\Lambda_{\text{QCD}}/Q$. To obtain the contributions up to twist-3,
one has to expand the contributions in Figs. 2(a)-2(c) in powers of $\lambda$
up to ${\cal O}(\lambda)$. Here we have already neglected the minus components
of $k$, $k_{1}$ and $k_{2}$ in $[\cdots]$ of Eqs. (14)-(16), since these
components only yield the corrections beyond twist-3.

For $W^{\mu\nu}|_{2a}$, if we further neglect the quark transverse momentum
and take $k^{\mu}\approx(k^{+},0,\vec{0}_{\perp})$ in $[\cdots]$ of Eq. (14),
one can obtain the contribution with the collinear fracture matrix involving a
non-local operator of quark and anti-quark fields. To obtain a gauge-invariant
form, we should sum over the contributions from the $G^{+}$-gluon exchange in
Fig. 2(b) and Fig. 2(c) as well as those with the exchange of any number of
$G^{+}$-gluons. Here the gluon field $G^{\mu}$ scales like
$(1,\lambda^{2},\lambda)$, and hence the $G^{+}$-gluon does not induce any
power suppression. After this summation, we can obtain the following gauge-
invariant contribution

$\displaystyle W^{\mu\nu}\Big{|}_{\text{q}}=$
$\displaystyle(\gamma^{\mu}\gamma^{+}\gamma^{\nu})_{ij}\sum_{X}\int\frac{d\eta^{-}}{2(2\pi)^{4}}e^{-ix_{B}P^{+}\eta^{-}}\langle
h_{A}|\bar{\psi}_{i}(\eta^{-}){\cal
L}_{n}^{\dagger}(\eta^{-})|hX\rangle\langle Xh|{\cal
L}_{n}(0)\psi_{j}(0)|h_{A}\rangle,$ (19)

where the gauge link is defined as

$\displaystyle{\cal L}_{n}(x)={\cal
P}\exp\biggr{\\{}-ig_{s}\int_{0}^{\infty}d\lambda~{}G^{+}(\lambda
n+x)\biggr{\\}}.$ (20)

The above contribution yields the gauge-invariant collinear quark fracture
matrix. As we will present later in Sec. III.2, by parametrization of this
matrix up to ${\cal O}(\lambda)$, one can obtain the hadronic tensor in terms
of twist-2 and twist-3 quark collinear fracture functions.

To derive the other twist-3 contributions from $W^{\mu\nu}|_{2a}$, we need to
take into account the $k_{\perp}$-dependence in $[\cdots]$ of Eq. (14) by the
collinear expansion to ${\cal O}(\lambda)$. We notice that there is no
contribution from the partial derivatives acting on the delta function in the
expansion since $\partial\delta\bigl{(}(\hat{k}+q)^{2}\bigr{)}/\partial
k_{\perp}^{\alpha}\propto q_{\perp\alpha}=0$. After this expansion, we get the
contribution from the fracture matrix with the transverse partial derivative
acting on the (anti-)quark fields. Again, after the combination with the
relevant gauge-link contributions from $W^{\mu\nu}|_{2b+2c}$, we obtain, up to
${\cal O}(g_{s})$,

$\displaystyle W^{\mu\nu}\Big{|}_{\partial}=$
$\displaystyle\frac{-i}{2q^{-}}(\gamma^{\mu}\gamma^{+}\gamma_{\perp\alpha}\gamma^{-}\gamma^{\nu})_{ij}\sum_{X}\int\frac{d\eta^{-}}{2(2\pi)^{4}}e^{-ix_{B}P^{+}\eta^{-}}\langle
h_{A}|\bar{\psi}_{i}(\eta^{-}){\cal L}_{n}^{\dagger}(\eta^{-})|hX\rangle$
$\displaystyle\qquad\times\langle Xh|\partial_{\perp}^{\alpha}({\cal
L}_{n}\psi_{j})(0)]|h_{A}\rangle+(\mu\leftrightarrow\nu)^{*}~{},$ (21)

where $(\mu\leftrightarrow\nu)^{*}$ stands for exchanging $\mu\nu$ indices and
taking complex conjugate of the first term. Due to the presence of the
transverse derivative, the leading contribution of the fracture matrix in Eq.
(21) is at twist-3.

In addition, after subtracting the gauge-link contributions to Eqs. (19) and
(21) from the collinear expansion of $W^{\mu\nu}|_{2b+2c}$, we find the
remaining part can be expressed by the fracture matrix with the gluon field
strength tensor
$g_{s}F^{+\alpha}=g_{s}[\partial^{+}G_{\perp}^{\alpha}-\partial_{\perp}^{\alpha}G^{+}]+{\cal
O}(g_{s}^{2})$. This gives another contribution that starts from twist-3,
which up to ${\cal O}(g_{s})$ can be summarized as

$\displaystyle W^{\mu\nu}\Big{|}_{F}=$
$\displaystyle\frac{-i}{2q^{-}}(\gamma^{\mu}\gamma^{+}\gamma_{\perp\alpha}\gamma^{-}\gamma^{\nu})_{ij}\int
dx_{2}\Bigl{[}{\rm P}\frac{1}{x_{2}-x_{B}}-i\pi\delta(x_{2}-x_{B})\Bigr{]}$
$\displaystyle\times\sum_{X}\int\frac{d\eta^{-}d\eta_{1}^{-}}{4\pi(2\pi)^{4}}e^{-ix_{B}P^{+}\eta^{-}-i(x_{2}-x_{B})P^{+}\eta_{1}^{-}}\langle
h_{A}|\bar{\psi}_{i}(\eta^{-})|hX\rangle\langle
Xh|g_{s}F^{+\alpha}(\eta_{1}^{-})\psi_{j}(0)|h_{A}\rangle+(\mu\leftrightarrow\nu)^{*}~{}.~{}$
(22)

Here $\rm P$ in $[\cdots]$ of Eq. (22) stands for the principle-value
prescription. The $\delta$-function term in Eq. (22) comes from the absorptive
part of the quark propagator that connects the electromagnetic current to the
quark-gluon vertex in Figs. 2(b) and 2(c). In this term, the gluon has zero
momentum and generates the so-called soft-gluon-pole contributions, see e.g.,
Chen:2015uqa  and references therein.

The total contribution of the hadronic tensor is given by the sum of the
results in Eqs. (19), (21) and (22). The following gauge-invariant collinear
fracture matrices are relevant:

$\displaystyle{\cal
M}_{ij}(x)=\int\frac{d\eta^{-}}{2\xi_{h}(2\pi)^{4}}e^{-ixP^{+}\eta^{-}}\sum_{X}\langle
h_{A}|\bar{\psi}_{j}(\eta^{-}){\cal
L}_{n}^{\dagger}(\eta^{-})|hX\rangle\langle Xh|{\cal
L}_{n}(0)\psi_{i}(0)|h_{A}\rangle,$ (23) $\displaystyle{\cal
M}_{\partial,ij}^{\alpha}(x)=\int\frac{d\eta^{-}}{2\xi_{h}(2\pi)^{4}}e^{-ixP^{+}\eta^{-}}\sum_{X}\langle
h_{A}|\bar{\psi}_{j}(\eta^{-}){\cal
L}_{n}^{\dagger}(\eta^{-})|hX\rangle\langle Xh|\partial_{\perp}^{\alpha}({\cal
L}_{n}\psi_{i})(0)]|h_{A}\rangle,$ (24) $\displaystyle{\cal
M}_{F,ij}^{\alpha}(x_{1},x_{2})=\int\frac{d\eta^{-}d\eta_{1}^{-}}{4\pi\xi_{h}(2\pi)^{4}}e^{-ix_{1}P^{+}\eta^{-}-i(x_{2}-x_{1})P^{+}\eta_{1}^{-}}\sum_{X}\langle
h_{A}|\bar{\psi}_{j}(\eta^{-})|hX\rangle\langle
Xh|g_{s}F^{+\alpha}(\eta_{1}^{-})\psi_{i}(0)|h_{A}\rangle.$ (25)

Here we have suppressed the gauge links for brevity in Eq. (25). Besides the
partonic momentum fractions, the fracture matrices also depend on the momentum
of the observed hadron ($\xi_{h}$, $P_{h\perp}$) and the spin vector of the
target, which are not shown explicitly in the arguments. With the above
notations, the hadronic tensor can be written as

$\displaystyle W^{\mu\nu}$
$\displaystyle=\xi_{h}(\gamma^{\mu}\gamma^{+}\gamma^{\nu})_{ij}{\cal
M}_{ji}(x_{B})+\left[\frac{-i\xi_{h}}{2q^{-}}(\gamma^{\mu}\gamma^{+}\gamma_{\perp\alpha}\gamma^{-}\gamma^{\nu})_{ij}{\cal
M}_{\partial,ji}^{\alpha}(x_{B})+(\mu\leftrightarrow\nu)^{*}\right]$
$\displaystyle+\left\\{\frac{-i\xi_{h}}{2q^{-}}(\gamma^{\mu}\gamma^{+}\gamma_{\perp\alpha}\gamma^{-}\gamma^{\nu})_{ij}\int
dx_{2}\Bigl{[}{\rm
P}\frac{1}{x_{2}-x_{B}}-i\pi\delta(x_{2}-x_{B})\Bigr{]}{\cal
M}_{F,ji}^{\alpha}(x_{B},x_{2})+(\mu\leftrightarrow\nu)^{*}\right\\}.$ (26)

It is obvious that only the chirality-even parts of the fracture matrices will
contribute to the hadronic tensor. We discuss the parameterization of these
fracture matrices in the next subsection.

###  III.2 Parameterization of the fracture matrices and the hadronic tensor
in terms of fracture functions

The collinear fracture matrices in Eqs. (23)-(25) can be decomposed using
Dirac $\Gamma$-matrices. From the constraints of parity invariance, the
fracture matrices can be generally parameterized as follows:

$\displaystyle{\cal M}_{ij}(x)$
$\displaystyle=\frac{(\gamma_{\rho})_{ij}}{2N_{c}}\Bigl{[}\bar{n}^{\rho}\Bigl{(}u_{1}-\frac{P_{h\perp}\cdot\tilde{S}_{\perp}}{M}u_{1T}^{h}\Bigr{)}+\frac{1}{P^{+}}\Bigl{(}P_{h\perp}^{\rho}u^{h}-M\tilde{S}_{\perp}^{\rho}u_{T}-S_{L}\tilde{P}_{h\perp}^{\rho}u_{L}^{h}-\frac{P_{h\perp}^{\langle\rho}P_{h\perp}^{\beta\rangle}}{M}\tilde{S}_{\perp\beta}u_{T}^{h}\Bigr{)}\Bigr{]}$
$\displaystyle-\frac{(\gamma_{\rho}\gamma_{5})_{ij}}{2N_{c}}\Bigl{[}\bar{n}^{\rho}\Bigl{(}S_{L}l_{1L}-\frac{P_{h\perp}\cdot
S_{\perp}}{M}l_{1T}^{h}\Bigr{)}+\frac{1}{P^{+}}\Bigl{(}\tilde{P}_{h\perp}^{\rho}l^{h}+MS_{\perp}^{\rho}l_{T}+S_{L}P_{h\perp}^{\rho}l_{L}^{h}-\frac{P_{h\perp}^{\langle\rho}P_{h\perp}^{\beta\rangle}}{M}S_{\perp\beta}l_{T}^{h}\Bigr{)}\Bigr{]}+\cdots,$
(27) $\displaystyle{\cal M}_{\partial,ij}^{\alpha}(x)$
$\displaystyle=\frac{(\gamma^{-})_{ij}}{2N_{c}}i\Bigl{(}-P_{h\perp}^{\alpha}u_{\partial}^{h}+M\tilde{S}_{\perp}^{\alpha}u_{\partial
T}+S_{L}\tilde{P}_{h\perp}^{\alpha}u_{\partial
L}^{h}+\frac{P_{h\perp}^{\langle\alpha}P_{h\perp}^{\beta\rangle}}{M}\tilde{S}_{\perp\beta}u_{\partial
T}^{h}\Bigr{)}$
$\displaystyle+\frac{(\gamma^{-}\gamma_{5})_{ij}}{2N_{c}}i\Bigl{(}\tilde{P}_{h\perp}^{\alpha}l_{\partial}^{h}+MS_{\perp}^{\alpha}l_{\partial
T}+S_{L}P_{h\perp}^{\alpha}l_{\partial
L}^{h}-\frac{P_{h\perp}^{\langle\alpha}P_{h\perp}^{\beta\rangle}}{M}S_{\perp\beta}l_{\partial
T}^{h}\Bigl{)}+\cdots,$ (28) $\displaystyle{\cal
M}_{F,ij}^{\alpha}(x_{1},x_{2})$
$\displaystyle=\frac{(\gamma^{-})_{ij}}{2N_{c}}\Bigl{(}P_{h\perp}^{\alpha}w^{h}-M\tilde{S}_{\perp}^{\alpha}w_{T}-S_{L}\tilde{P}_{h\perp}^{\alpha}w_{L}^{h}-\frac{P_{h\perp}^{\langle\alpha}P_{h\perp}^{\beta\rangle}}{M}\tilde{S}_{\perp\beta}w_{T}^{h}\Bigr{)}$
$\displaystyle-\frac{(\gamma^{-}\gamma_{5})_{ij}}{2N_{c}}i\Bigl{(}\tilde{P}_{h\perp}^{\alpha}v^{h}+MS_{\perp}^{\alpha}v_{T}+S_{L}P_{h\perp}^{\alpha}v_{L}^{h}-\frac{P_{h\perp}^{\langle\alpha}P_{h\perp}^{\beta\rangle}}{M}S_{\perp\beta}v_{T}^{h}\Bigr{)}+\cdots,$
(29)

where $\cdots$ denote the contributions beyond twist-3 or the chirality-odd
parts. In the above, we have used the shorthand notations
$P_{h\perp}^{\langle\alpha}P_{h\perp}^{\beta\rangle}\equiv
P_{h\perp}^{\alpha}P_{h\perp}^{\beta}+g_{\perp}^{\alpha\beta}\vec{P}_{h\perp}^{2}/2$
for simplicity. As pointed out in Anselmino:2011ss , the fracture matrix is
not constrained by time reversal invariance, as it identifies a hadron in the
out state. Additionally, we note that the collinear fracture matrices formally
share similar parametrization forms with those of the conventional TMD PDFs
(see e.g., Wei:2016far ).

The functions $u$’s and $l$’s in Eqs. (27) and (28) are quark collinear
fracture functions, they are functions of $x$, $\xi_{h}$ and
$\vec{P}_{h\perp}^{2}$. $w$’s and $v$’s in Eq. (29) are quark-gluon collinear
fracture functions, they depend on $x_{1}$ and $x_{2}$ besides of $\xi_{h}$
and $\vec{P}_{h\perp}^{2}$. We have suppressed all these arguments for
simplicity. From hermiticity, the fracture functions defined in Eqs. (27) and
(28) are real, while those in Eq. (29) are complex in general.

The naming rules for these fracture functions we have used are as follows:
Four fracture functions in Eq. (27) with “1” in the subscript are of twist-2.
The remaining is of twist-3. The “$\partial$” in the subscript denote that the
fracture functions are defined via the fracture matrix with the partial
derivative operator. The “$L$” or “$T$” in the subscript denotes the
dependence on the longitudinal or transverse polarization of the nucleon. The
superscript “$h$” denotes the explicit dependence on the transverse momentum
of the final state hadron $h$ in the decomposition of the matrix elements. We
note that the TMD quark fracture functions at twist-2 have been classified for
a polarized nucleon target in Anselmino:2011ss ; Anselmino:2011bb . After
integrating over the transverse momentum of the parton, they are equivalent to
the twist-2 collinear quark fracture functions defined in Eq. (27).

We further note that the twist-3 fracture functions defined above are not
independent of each other. From the QCD equation of motion $i\gamma\cdot
D\psi=0$, one can show that their relations can be written in a unified form
as follows:

$\displaystyle x[u_{S}^{K}(x)+il_{S}^{K}(x)]=u_{\partial
S}^{K}(x)+il_{\partial S}^{K}(x)+i\int dy\Bigl{[}{\rm
P}\frac{1}{y-x}-i\pi\delta(y-x)\Bigr{]}[w_{S}^{K}(x,y)-v_{S}^{K}(x,y)],$ (30)

where $(S,~{}K)=(\text{null},~{}h)$, $(L,~{}h)$, $(T,~{}\text{null})$ or
$(T,~{}h)$. I.e., we have four sets of relations in the unified form of Eq.
(30). With these relations, we find that the hadronic tensor in Eq. (26) can
be expressed only with the fracture functions defined via ${\cal M}_{ij}$ in
Eq. (27). We obtain

$\displaystyle W^{\mu\nu}$
$\displaystyle=-2g_{\perp}^{\mu\nu}\Bigl{(}u_{1}-\frac{P_{h\perp}\cdot\tilde{S}_{\perp}}{M}u_{1T}^{h}\Bigr{)}+2i\varepsilon_{\perp}^{\mu\nu}\Bigl{(}S_{L}l_{1L}-\frac{P_{h\perp}\cdot
S_{\perp}}{M}l_{1T}^{h}\Bigr{)}$ $\displaystyle+\frac{2}{P\cdot
q}P_{h\perp}^{\\{\mu}\bar{q}^{\nu\\}}\Bigl{(}u^{h}-\frac{P_{h\perp}\cdot\tilde{S}_{\perp}}{M}u_{T}^{h}\Bigr{)}+\frac{2i}{P\cdot
q}P_{h\perp}^{[\mu}\bar{q}^{\nu]}\Bigl{(}l^{h}-\frac{P_{h\perp}\cdot\tilde{S}_{\perp}}{M}l_{T}^{h}\Bigr{)}-\frac{2M}{P\cdot
q}\tilde{S}_{\perp}^{\\{\mu}\bar{q}^{\nu\\}}\Bigl{(}u_{T}-\frac{\vec{P}_{h\perp}^{2}}{2M^{2}}u_{T}^{h}\Bigr{)}$
$\displaystyle-\frac{2iM}{P\cdot
q}\tilde{S}_{\perp}^{[\mu}\bar{q}^{\nu]}\Bigl{(}l_{T}-\frac{\vec{P}_{h\perp}^{2}}{2M^{2}}l_{T}^{h}\Bigr{)}-\frac{2S_{L}}{P\cdot
q}\tilde{P}_{h\perp}^{\\{\mu}\bar{q}^{\nu\\}}u_{L}^{h}-\frac{2iS_{L}}{P\cdot
q}\tilde{P}_{h\perp}^{[\mu}\bar{q}^{\nu]}l_{L}^{h},$ (31)

where $A^{\\{\mu}B^{\nu\\}}\equiv A^{\mu}B^{\nu}+A^{\nu}B^{\mu}$ and
$A^{[\mu}B^{\nu]}\equiv A^{\mu}B^{\nu}-A^{\nu}B^{\mu}$. We have also used the
shorthand notation $\bar{q}^{\mu}\equiv q^{\mu}+2x_{B}P^{+}\bar{n}^{\mu}$. The
first line in Eq. (31) is of twist-2 contributions, and the remains are of
twist-3 contributions. Because $q^{\mu}$ has only longitudinal components and
also $q\cdot\bar{q}=0$, we see explicitly that the hadronic tensor of Eq. (31)
satisfies the $U(1)$-gauge invariance or the current conservation, i.e.,
$q_{\mu}W^{\mu\nu}=q_{\nu}W^{\mu\nu}=0$.

##  IV The results of structure functions and azimuthal or spin asymmetries

###  IV.1 The results of structure functions

Substituting the hadronic tensor result of Eq. (31) into Eq. (9), we obtain
the differential cross section. Comparing with the cross section expressed by
structure functions in Eq. (12), we obtain the results of structure functions
in terms of the collinear fracture functions. Four structure functions are at
twist-2, which are expressed in terms of the four twist-2 fracture functions,
i.e.,

$\displaystyle F_{UU,T}=x_{B}u_{1},\qquad
F_{UT,T}^{\sin(\phi_{h}-\phi_{S})}=\frac{|\vec{P}_{h\perp}|}{M}x_{B}u_{1T}^{h},$
(32) $\displaystyle F_{LL}=x_{B}l_{1L},\qquad
F_{LT}^{\cos(\phi_{h}-\phi_{S})}=\frac{|\vec{P}_{h\perp}|}{M}x_{B}l_{1T}^{h}.$
(33)

The summation over quark flavors, i.e., $\sum_{q}e_{q}^{2}\cdots$, is implicit
on the right-hand side of the equations. This twist-2 result has been obtained
in Anselmino:2011bb . There are eight structure functions that have
contributions starting from twist-3. They are expressed with eight different
twist-3 fracture functions, i.e.,

$\displaystyle
F_{UU}^{\cos\phi_{h}}=-\frac{2|\vec{P}_{h\perp}|}{Q}x_{B}^{2}u^{h},\qquad
F_{LU}^{\sin\phi_{h}}=\frac{2|\vec{P}_{h\perp}|}{Q}x_{B}^{2}l^{h},$ (34)
$\displaystyle
F_{UL}^{\sin\phi_{h}}=-\frac{2|\vec{P}_{h\perp}|}{Q}x_{B}^{2}u_{L}^{h},\qquad
F_{LL}^{\cos\phi_{h}}=-\frac{2|\vec{P}_{h\perp}|}{Q}x_{B}^{2}l_{L}^{h},$ (35)
$\displaystyle F_{UT}^{\sin\phi_{S}}=-\frac{2M}{Q}x_{B}^{2}u_{T},\qquad\quad
F_{LT}^{\cos\phi_{S}}=-\frac{2M}{Q}x_{B}^{2}l_{T},$ (36) $\displaystyle
F_{UT}^{\sin(2\phi_{h}-\phi_{S})}=-\frac{\vec{P}_{h\perp}^{2}}{QM}x_{B}^{2}u_{T}^{h},\quad
F_{LT}^{\cos(2\phi_{h}-\phi_{S})}=-\frac{\vec{P}_{h\perp}^{2}}{QM}x_{B}^{2}l_{T}^{h}.$
(37)

The remaining six structure functions are all zero up to twist-3. We see that
half of the eight twist-3 structure functions are related to the transverse
polarization-dependent fracture functions.

###  IV.2 Azimuthal or spin asymmetries

In addition to structure functions, one can also construct various azimuthal
or spin asymmetries by

$\displaystyle\langle{\cal F}\rangle_{{\cal P}_{e}{\cal
P}_{N}}\equiv\int\frac{d\sigma}{dxdyd\xi_{h}d\psi d^{2}P_{h\perp}}{\cal
F}d\phi_{h}d\psi\bigg{/}\int\frac{d\sigma}{dxdyd\xi_{h}d\psi
d^{2}P_{h\perp}}d\phi_{h}d\psi,$ (38)

where the subscripts ${\cal P}_{e}=U$ or $L$ and ${\cal P}_{N}=U$, $L$ or $T$
denote the polarization states of the electron and the nucleon target. From
our results of the structure functions, we see clearly that there are two
spin-dependent azimuthal asymmetries at twist-2. They both depend on the
nucleon transverse polarization and are given by

$\displaystyle\langle\sin(\phi_{h}-\phi_{S})\rangle_{UT}=\frac{|\vec{P}_{h\perp}|}{2M}\frac{u_{1T}^{h}(x_{B},\xi_{h},P_{h\perp})}{u_{1}(x_{B},\xi_{h},P_{h\perp})},$
(39)
$\displaystyle\langle\cos(\phi_{h}-\phi_{S})\rangle_{LT}=\frac{|\vec{P}_{h\perp}|C(y)}{2MA(y)}\frac{l_{1T}^{h}(x_{B},\xi_{h},P_{h\perp})}{u_{1}(x_{B},\xi_{h},P_{h\perp})}.$
(40)

Here and in the below, a summation over quark flavors, i.e.,
$\sum_{q}e_{q}^{2}\cdots$, is implicit both in the numerators and the
denominators. We note that the asymmetry
$\langle\sin(\phi_{h}-\phi_{S})\rangle_{UT}$ is of Sivers-type Sivers:1989cc
and it does not depend on $y$ because of the cancellation of the common $A(y)$
factors associated with $F_{UT,T}^{\sin(\phi_{h}-\phi_{S})}$ and $F_{UU,T}$ in
the cross-section.

We have in particular eight azimuthal or spin asymmetries at twist-3
associated with the eight twist-3 structure functions in Eqs. (34)-(37), i.e.,

$\displaystyle\langle\cos\phi_{h}\rangle_{UU}=-\frac{|\vec{P}_{h\perp}|}{Q}\frac{B(y)}{A(y)}\frac{x_{B}u^{h}(x_{B},\xi_{h},P_{h\perp})}{u_{1}(x_{B},\xi_{h},P_{h\perp})},$
(41)
$\displaystyle\langle\sin\phi_{h}\rangle_{LU}=\frac{|\vec{P}_{h\perp}|}{Q}\frac{D(y)}{A(y)}\frac{x_{B}l^{h}(x_{B},\xi_{h},P_{h\perp})}{u_{1}(x_{B},\xi_{h},P_{h\perp})},$
(42)
$\displaystyle\langle\sin\phi_{h}\rangle_{UL}=-\frac{|\vec{P}_{h\perp}|}{Q}\frac{B(y)}{A(y)}\frac{x_{B}u_{L}^{h}(x_{B},\xi_{h},P_{h\perp})}{u_{1}(x_{B},\xi_{h},P_{h\perp})},$
(43)
$\displaystyle\langle\cos\phi_{h}\rangle_{LL}=-\frac{|\vec{P}_{h\perp}|}{Q}\frac{D(y)}{A(y)}\frac{x_{B}l_{L}^{h}(x_{B},\xi_{h},P_{h\perp})}{u_{1}(x_{B},\xi_{h},P_{h\perp})},$
(44)
$\displaystyle\langle\sin\phi_{S}\rangle_{UT}=-\frac{M}{Q}\frac{B(y)}{A(y)}\frac{x_{B}u_{T}(x_{B},\xi_{h},P_{h\perp})}{u_{1}(x_{B},\xi_{h},P_{h\perp})},$
(45)
$\displaystyle\langle\cos\phi_{S}\rangle_{LT}=-\frac{M}{Q}\frac{D(y)}{A(y)}\frac{x_{B}l_{T}(x_{B},\xi_{h},P_{h\perp})}{u_{1}(x_{B},\xi_{h},P_{h\perp})},$
(46)
$\displaystyle\langle\sin(2\phi_{h}-\phi_{S})\rangle_{UT}=-\frac{\vec{P}_{h\perp}^{2}}{2MQ}\frac{B(y)}{A(y)}\frac{x_{B}u_{T}^{h}(x_{B},\xi_{h},P_{h\perp})}{u_{1}(x_{B},\xi_{h},P_{h\perp})},$
(47)
$\displaystyle\langle\cos(2\phi_{h}-\phi_{S})\rangle_{LT}=-\frac{\vec{P}_{h\perp}^{2}}{2MQ}\frac{D(y)}{A(y)}\frac{x_{B}l_{T}^{h}(x_{B},\xi_{h},P_{h\perp})}{u_{1}(x_{B},\xi_{h},P_{h\perp})}.$
(48)

One can see that at the order we are considering, each azimuthal or spin
asymmetry in the TFR is only generated by a specific fracture function. This
suggests that interpretations of these functions from experimental data may be
simpler and more straightforward compared to the CFR at small $P_{h\perp}$,
where multiple TMD PDFs and FFs are typically involved and intertwined in the
asymmetry Bacchetta:2006tn . Some of the twist-3 asymmetries, such as
$\langle\sin\phi_{h}\rangle_{UL}$ and $\langle\sin\phi_{h}\rangle_{LU}$, have
already been measured in the TFR by CLAS12 at JLab Harut . Of particular
interest is the beam-spin asymmetry $\langle\sin\phi_{h}\rangle_{LU}$ in Eq.
(42), which is related to a twist-3 longitudinal quark fracture function
$l^{h}$. A preliminary analysis shows that it undergoes a clear sign flip from
the TFR to the CFR and could serve as an efficient tool to understand the
transition between the production mechanisms (Sec. 5.3 in Accardi:2023chb ).
Further experimental measurements will provide us with more information about
the relevant fracture functions, especially the twist-3 ones.

##  V Summary

In summary, we have derived the hadronic tensor up to twist-3 level for SIDIS
with hadron production in the target fragmentation region. The hadronic tensor
at the considered order is shown to be expressed by gauge-invariant fracture
functions defined with two-parton correlations. Based on the obtained hadronic
tensor, the results for structure functions are derived for both polarized
lepton beam and polarized nucleon target. At the tree level, there are four
structure functions at twist-2 and eight structure functions at twist-3.
Azimuthal or spin asymmetries are given based on the results of the structure
functions. These observables are all expressed using twist-2 or twist-3
collinear fracture functions. Possible connections to experimental
measurements are discussed. Future SIDIS experiments measuring these azimuthal
or spin asymmetries will provide opportunities to extract the corresponding
fracture functions.

## Acknowledgements

We would like to thank Harut Avakian for bringing our attention to this
project and insightful discussions. We also thank Timothy Hayward for useful
discussions. The work is supported by National Natural Science Foundation of
P.R. China (Nos.12075299,11821505, 11847612 and 11935017) and by the Strategic
Priority Research Program of Chinese Academy of Sciences, Grant No.
XDB34000000. K.B. Chen is supported by National Natural Science Foundation of
China (Nos. 12005122, 11947055) and Shandong Province Natural Science
Foundation (No. ZR2020QA082). X.B. Tong is supported by the CUHK-Shenzhen
university development fund (No. UDF01001859) and the China Postdoctoral
Science Foundation (No. 2022M723065).

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