Twist-3 Gluon Fragmentation Contribution to Hyperon Polarization
in Semi-Inclusive Deep Inelastic Scattering

Riku Ikarashi1, Yuji Koike2, Kenta Yabe1 and Shinsuke Yoshida3,4

1Graduate School of Science and Technology, Niigata University, Ikarashi 2-no-
cho, Niigata 950-2181, Japan

2Department of Physics, Niigata University, Ikarashi 2-no-cho, Niigata
950-2181, Japan

3Guangdong Provincial Key Laboratory of Nuclear Science, Institute of Quantum
Matter,
South China Normal University, Guangzhou 510006, China

4Guangdong-Hong Kong Joint Laboratory of Quantum Matter, Southern Nuclear
Science Computing Center, South China Normal University, Guangzhou 510006,
China

Abstract

We derive the twist-3 gluon fragmentation function (FF) contribution to the
transversely polarized hyperon production in semi-inclusive deep inelastic
scattering, $ep\to e\Lambda^{\uparrow}X$, in the leading order (LO) with
respect to the QCD coupling in the framework of the collinear twist-3
factorization. Together with the known result for the contribution from the
twist-3 distribution in the proton and the twist-3 quark FFs for the hyperon,
this completes the LO cross section for this process. The constraint relations
among the twist-3 FFs are taken into account. The formula is relevant to
large-$P_{T}$ hyperon production in the future Electron-Ion-Collider
experiment.

##  1 Introduction

In a recent paper[1], three of the present authors studied the transverse
polarization of hyperons produced in semi-inclusive deep inelastic scattering,
$ep\to e\Lambda^{\uparrow}X$. For a large-$P_{T}$ hyperon production, this
process can be analyzed in the framework of the collinear factorization, in
which the polarization appears as a twist-3 observable in the absence of a
leading twist-2 effect. For $ep\to e\Lambda^{\uparrow}X$, the responsible
twist-3 effects are (i) the twist-3 distribution functions (DFs) in the
initial proton combined with the twist-2 transversity fragmentation function
(FF) for $\Lambda$ and (ii) the twist-3 FFs for the polarized hyperon combined
with the twist-2 unpolarized parton DFs in the proton. The twist-3 FFs in (ii)
are chiral-even, and both (a) quark and (b) gluon types of twist-3 FFs
contribute. In [1], the twist-3 polarized cross section for $ep\to
e\Lambda^{\uparrow}X$ from the above (i) and (ii)(a) was derived in the
leading order (LO) with respect to the QCD coupling constant. As a sequel to
[1], we will derive in this paper the LO cross section from (ii)(b), which
completes the LO twist-3 cross section for this process. Since the gluons are
ample in the collision environment and the twist-3 quark and gluon FFs mix
under renormalization, the effect of (ii)(b) could be as important as (ii)(a).
We also remind that the twist-3 fragmentation effect is important to
understand the single transverse-spin asymmetry in $p^{\uparrow}p\to\pi X$[2,
3], which shows a similar rising asymmetry at large $x_{F}$ as the
polarization in $pp\to\Lambda^{\uparrow}X$. Our present study has a direct
relevance to the hyperon polarization phenomenon in the future Electron-Ion-
Collider (EIC) experiment.

Here we make some remarks on the phenomenological use of the twist-3 cross
section. As we will see, it contains several unknown nonperturbative
functions, determination of which requires global analysis of data for various
processes such as $ep\to e\Lambda^{\uparrow}X$,
$e^{+}e^{-}\to\Lambda^{\uparrow}X$ and $pp\to\Lambda^{\uparrow}X$, etc,
combined with an appropriate modelling of those functions. We also recall that
in the small-$P_{T}$ region the transverse-momentum-dependent (TMD)
factorization holds for $ep\to e\Lambda^{\uparrow}X$ and
$e^{+}e^{-}\to\Lambda^{\uparrow}X$, and we anticipate that the two frameworks
match in the intermediate region of $P_{T}$111Study on this matching will be
reported elsewhere. as for the case of $p^{\uparrow}p\to\ell^{+}\ell^{-}X$[4]
and $ep^{\uparrow}\to e\pi X$ [5, 6, 7]. Information on the TMD functions
obtained from the analysis of those small-$P_{T}$ data will also help to
constrain the twist-3 functions owing to the relations between the TMD
functions and the twist-3 functions[8, 9]. In this connection we mention the
recent data on $e^{+}e^{-}\to\Lambda^{\uparrow}X$ at Belle[10] and the
phenomenological analyses of the data in terms of the TMD factorization[11,
12, 13, 14]. These studies will be useful to analyze the EIC data at
large-$P_{T}$ in terms of the twist-3 cross section derived in this work.

The formalism of calculating the twist-3 gluon FFs contribution is very
complicated and was completed only recently for a similar process in the pp
collision, $pp\to\Lambda^{\uparrow}X$[15]. Here we apply the method to $ep\to
e\Lambda^{\uparrow}X$. Since the kinematics for this process was described in
[1] and the method is in parallel to the case for
$pp\to\Lambda^{\uparrow}X$[15], our presentation in this paper will be brief,
referring to those papers for the details.

The remainder of this paper is organized as follows: In section 2, we
introduce the twist-3 gluon FFs relevant in our study. In section 3, we
briefly describe the formalism for calculating the twist-3 gluon FF
contribution to $ep\to e\Lambda^{\uparrow}X$ and present the LO cross section.
Section 4 is devoted to a brief summary.

##  2 Twist-3 gluon fragmentation functions

###  2.1 Three types of twist-3 gluon FFs and $q\bar{q}g$ FFs

Here we list the twist-3 gluon FFs for spin-1/2 hyperon which are necessary to
derive the twist-3 cross section for $ep\to e\Lambda^{\uparrow}X$ [9, 15].
They are classified into the intrinsic, kinematical and dynamical FFs. First,
the intrinsic gluon FFs are defined as the lightcone correlators of the
gluon’s field strength $F^{\mu\nu}_{a}$ with color index $a$ [9, 16]:

$\displaystyle\widehat{\Gamma}^{\alpha\beta}(z)=\frac{1}{N^{2}-1}\sum_{X}\int\frac{d\lambda}{2\pi}\mathrm{e}^{-i\lambda/z}\langle
0|(\left[\infty w,0\right]F^{w\beta}(0))_{a}|hX\rangle\langle
hX|(F^{w\alpha}(\lambda w)\left[\lambda w,\infty w\right])_{a}|0\rangle{}$
$\displaystyle=-g^{\alpha\beta}_{\perp}\widehat{G}(z)-i\epsilon^{P_{h}w\alpha\beta}(S\cdot
w)\Delta\widehat{G}(z)+M_{h}\epsilon^{P_{h}wS_{\perp}\\{\alpha}w^{\beta\\}}\Delta\widehat{G}_{3\bar{T}}(z)+iM_{h}\epsilon^{\alpha\beta
wS_{\perp}}\Delta\widehat{G}_{3T}(z)+\cdots,$ (1)

where $P_{h}$ is the four momentum of the hyperon with its mass $M_{h}$.
$P_{h}^{\mu}$ can be regarded as lightlike in the twist-3 accuracy and
$w^{\mu}$ is another lightlike vector satisfying $P_{h}\cdot w=1$. $S^{\mu}$
is the spin vector of the hyperon normalized as $S^{2}=-M_{h}^{2}$ and can be
decomposed as $S^{\mu}=(S\cdot w)P_{h}^{\mu}+(S\cdot
P_{h})w^{\mu}+M_{h}S_{\perp}^{\mu}$ with the transverse spin vector
$S_{\perp}^{\mu}$ ($S_{\perp}^{2}=-1$). $g^{\alpha\beta}_{\perp}\equiv
g^{\alpha\beta}-P_{h}^{\alpha}w^{\beta}-P_{h}^{\beta}w^{\alpha}$, $N=3$ is the
number of colors for $SU(N)$ and the ellipsis denotes twist-4 or higher.
$|h\rangle$ denotes the hyperon state. $[\lambda w,\mu
w]\equiv\mathcal{P}\exp{\left[ig\int_{\mu}^{\lambda}d\tau w\cdot A(\tau
w)\right]}$ is the gauge-link operator which guarantees gauge invariance of
the correlation function. We use the convention for the Levi-Civita symbol as
$\epsilon^{0123}=1$. The shorthand notation
$\epsilon^{P_{h}w\alpha\beta}\equiv\epsilon^{\mu\nu\alpha\beta}{P_{h}}_{\mu}w_{\nu}$,
etc. is used, and $\\{\alpha\,\beta\\}$ denotes the symmetrization of Lorentz
indices. $\widehat{G}(z)$ and $\Delta\widehat{G}(z)$ are twist-2 unpolarized
and helicity FFs, respectively, and $\Delta\widehat{G}_{3\bar{T}}(z)$ and
$\Delta\widehat{G}_{3T}(z)$ are intrinsic twist-3 FFs. All FFs in (2.1) are
defined to be real and have a support on $0<z<1$.
$\Delta\widehat{G}_{3\bar{T}}(z)$ is naïvely T-odd, and contributes to the
hyperon polarization.

Second, the kinematical gluon FFs are defined from the derivative of the
correlation functions for the intrinsic one:

$\displaystyle\widehat{\Gamma}^{\alpha\beta\gamma}_{\partial}(z)$
$\displaystyle=\frac{1}{N^{2}-1}\sum_{X}\int\frac{d\lambda}{2\pi}\mathrm{e}^{-i\lambda/z}\langle
0|(\left[\infty w,0\right]F^{w\beta}(0))_{a}|hX\rangle\langle
hX|(F^{w\alpha}(\lambda w)\left[\lambda w,\infty
w\right])_{a}|0\rangle\overset{\leftarrow}{\partial}{}^{\gamma}{}$
$\displaystyle=-i\frac{M_{h}}{2}g_{\perp}^{\alpha\beta}\epsilon^{P_{h}wS_{\perp}\gamma}\widehat{G}_{T}^{(1)}(z)+\frac{M_{h}}{2}\epsilon^{P_{h}w\alpha\beta}S_{\perp}^{\gamma}\Delta\widehat{G}^{(1)}_{T}(z){}$
$\displaystyle-i\frac{M_{h}}{8}\left(\epsilon^{P_{h}wS_{\perp}\\{\alpha}g_{\perp}^{\beta\\}\gamma}+\epsilon^{P_{h}w\gamma\\{\alpha}S^{\beta\\}}_{\perp}\right)\Delta\widehat{H}^{(1)}_{T}(z)+\cdots,$
(2)

where

$\displaystyle F^{w\alpha}(\lambda w)[\lambda w,\infty
w]|0\rangle\overset{\leftarrow}{\partial}{}^{\gamma}\equiv\lim_{\xi\to
0}\frac{d}{d\xi_{\gamma}}F^{w\alpha}(\lambda w+\xi)[\lambda w+\xi,\infty
w+\xi]|0\rangle.$ (3)

There are three twist-3 gluonic kinematical FFs, $\widehat{G}^{(1)}_{T}(z)$,
$\Delta\widehat{G}^{(1)}_{T}(z)$ and $\Delta\widehat{H}^{(1)}_{T}(z)$, which
are real functions and have a support on $0<z<1$. Among them,
$\widehat{G}^{(1)}_{T}(z)$ and $\Delta\widehat{H}^{(1)}_{T}(z)$ are naïvely
T-odd contributing to the hyperon polarization, while
$\Delta\widehat{G}^{(1)}_{T}(z)$ is naïvely T-even. They can also be written
as the $k^{2}_{T}/M_{h}^{2}$-moment of the TMD FFs [16].

Third, the dynamical gluon FFs are defined from the 3-gluon correlators.
Contraction of color indices with two structure constants for color $SU(N)$,
i.e. $-if_{abc}$ and $d_{abc}$, yields two types of FFs[9, 17, 18, 19]:

$\displaystyle\widehat{\Gamma}^{\alpha\beta\gamma}_{FA}\left(\frac{1}{z_{1}},\frac{1}{z_{2}}\right){}$
$\displaystyle=\frac{-if_{abc}}{N^{2}-1}\sum_{X}\iint\frac{d\lambda}{2\pi}\frac{d\mu}{2\pi}\mathrm{e}^{-i\lambda/z_{1}}\mathrm{e}^{-i\mu(1/z_{2}-1/z_{1})}\langle
0|F^{w\beta}_{b}(0)|hX\rangle\langle hX|F^{w\alpha}_{a}(\lambda
w)gF^{w\gamma}_{c}(\mu w)|0\rangle{}$
$\displaystyle=-M_{h}\left(\widehat{N}_{1}\left(\frac{1}{z_{1}},\frac{1}{z_{2}}\right)g^{\alpha\gamma}_{\perp}\epsilon^{P_{h}wS_{\perp}\beta}+\widehat{N}_{2}\left(\frac{1}{z_{1}},\frac{1}{z_{2}}\right)g^{\beta\gamma}_{\perp}\epsilon^{P_{h}wS_{\perp}\alpha}-\widehat{N}_{2}\left(\frac{1}{z_{2}}-\frac{1}{z_{1}},\frac{1}{z_{2}}\right)g^{\alpha\beta}_{\perp}\epsilon^{P_{h}wS_{\perp}\gamma}\right),$
(4)
$\displaystyle\widehat{\Gamma}^{\alpha\beta\gamma}_{FS}\left(\frac{1}{z_{1}},\frac{1}{z_{2}}\right){}$
$\displaystyle=\frac{d_{abc}}{N^{2}-1}\sum_{X}\iint\frac{d\lambda}{2\pi}\frac{d\mu}{2\pi}\mathrm{e}^{-i\lambda/z_{1}}\mathrm{e}^{-i\mu(1/z_{2}-1/z_{1})}\langle
0|F^{w\beta}_{b}(0)|hX\rangle\langle hX|F^{w\alpha}_{a}(\lambda
w)gF^{w\gamma}_{c}(\mu w)|0\rangle{}$
$\displaystyle=-M_{h}\left(\widehat{O}_{1}\left(\frac{1}{z_{1}},\frac{1}{z_{2}}\right)g^{\alpha\gamma}_{\perp}\epsilon^{P_{h}wS_{\perp}\beta}+\widehat{O}_{2}\left(\frac{1}{z_{1}},\frac{1}{z_{2}}\right)g^{\beta\gamma}_{\perp}\epsilon^{P_{h}wS_{\perp}\alpha}+\widehat{O}_{2}\left(\frac{1}{z_{2}}-\frac{1}{z_{1}},\frac{1}{z_{2}}\right)g^{\alpha\beta}_{\perp}\epsilon^{P_{h}wS_{\perp}\gamma}\right),$
(5)

where the gauge-link operators are suppressed for simplicity. There are four
purely gluonic dynamical FFs,
$\widehat{N}_{1,2}\left(\frac{1}{z_{1}},\frac{1}{z_{2}}\right)$ and
$\widehat{O}_{1,2}\left(\frac{1}{z_{1}}.\frac{1}{z_{2}}\right)$, which are
complex functions and have a support on $1/z_{2}>1$ and $1/z_{2}>1/z_{1}>0$.
Their real parts are naïvely $T$-even, while their imaginary parts are naïvely
$T$-odd. $\widehat{N}_{1}\left(\frac{1}{z_{1}},\frac{1}{z_{2}}\right)$ and
$\widehat{O}_{1}\left(\frac{1}{z_{1}},\frac{1}{z_{2}}\right)$ satisfy the
symmetry relations

$\displaystyle\widehat{N}_{1}\left(\frac{1}{z_{1}},\frac{1}{z_{2}}\right)=-\widehat{N}_{1}\left(\frac{1}{z_{2}}-\frac{1}{z_{1}},\frac{1}{z_{2}}\right),\qquad\widehat{O}_{1}\left(\frac{1}{z_{1}},\frac{1}{z_{2}}\right)=\widehat{O}_{1}\left(\frac{1}{z_{2}}-\frac{1}{z_{1}},\frac{1}{z_{2}}\right).$
(6)

Finally, we introduce other dynamical FFs defined from the quark-antiquark-
gluon correlators[9], which are necessary for the derivation of the twist-3
cross section for $ep\to e\Lambda^{\uparrow}X$,

$\displaystyle\widetilde{\Delta}^{\alpha}_{ij}\left(\frac{1}{z_{1}},\frac{1}{z_{2}}\right)$
$\displaystyle=\frac{1}{N}\sum_{X}\iint\frac{d\lambda}{2\pi}\frac{d\mu}{2\pi}\mathrm{e}^{-i\lambda/z_{1}}\mathrm{e}^{-i\mu(1/z_{2}-1/z_{1})}\langle
0|gF^{w\alpha}_{a}(\mu w)|hX\rangle\langle hX|\bar{\psi}_{j}(\lambda
w)t^{a}\psi_{i}(0)|0\rangle{}$ $\displaystyle=M_{h}\left(\epsilon^{\alpha
P_{h}wS_{\perp}}({\ooalign{\hfil/\hfil\crcr$P$}}_{h})_{ij}\widetilde{D}_{FT}\left(\frac{1}{z_{1}},\frac{1}{z_{2}}\right)+iS^{\alpha}_{\perp}(\gamma_{5}{\ooalign{\hfil/\hfil\crcr$P$}}_{h})_{ij}\widetilde{G}_{FT}\left(\frac{1}{z_{1}},\frac{1}{z_{2}}\right)\right),$
(11)

where $t^{a}$ is the generators of SU(N) and the spinor indices $i,j$ are
shown explicitly. These two functions
$\widetilde{D}_{FT}\left(\frac{1}{z_{1}},\frac{1}{z_{2}}\right)$ and
$\widetilde{G}_{FT}\left(\frac{1}{z_{1}},\frac{1}{z_{2}}\right)$ are complex
functions and have a support on $1/z_{1}>0,1/z_{2}<0$ and $1/z_{1}-1/z_{2}>1$.
Their real parts are naïvely $T$-even, while the imaginary parts are naïvely
$T$-odd.

###  2.2 Constraint relations among twist-3 gluon FFs

The gluon FFs introduced above are not independent but are subject to the QCD
equation-of-motion (EOM) relations and the Lorentz invariance relations
(LIRs). The complete set of those relations were derived in [9]. Here we quote
those relations which are useful to simplify the twist-3 cross section for
$ep\to e\Lambda^{\uparrow}X$. The relevant EOM relation allows us to express
the intrinsic FF in terms of the kinematical and dynamical FFs as

$\displaystyle\frac{1}{z}\Delta\widehat{G}_{3\bar{T}}(z)=-\Im\widetilde{D}_{FT}(z)+\frac{1}{2}\left(\widehat{G}_{T}^{(1)}(z)+\Delta\widehat{H}_{T}^{(1)}(z)\right){}$
$\displaystyle\qquad+\int
d\left(\frac{1}{z^{\prime}}\right)\frac{1}{1/z-1/z^{\prime}}\Im\left(2\widehat{N}_{1}\left(\frac{1}{z^{\prime}},\frac{1}{z}\right)+\widehat{N}_{2}\left(\frac{1}{z^{\prime}},\frac{1}{z}\right)-\widehat{N}_{2}\left(\frac{1}{z}-\frac{1}{z^{\prime}},\frac{1}{z}\right)\right),$
(12)

where $\widetilde{D}_{FT}(z)$ is defined as

$\displaystyle\widetilde{D}_{FT}(z)\equiv\frac{2}{C_{F}}\int_{0}^{1/z}d\left(\frac{1}{z^{\prime}}\right)\widetilde{D}_{FT}\left(\frac{1}{z^{\prime}},\frac{1}{z^{\prime}}-\frac{1}{z}\right),\qquad{\rm
with}\ C_{F}=\frac{N^{2}-1}{2N}.$ (13)

Other relations derived from the LIRs and the EOM relations represent the
derivative of the kinematical FFs in terms of other FFs as

$\displaystyle\frac{1}{z}\frac{\partial\widehat{G}_{T}^{(1)}(z)}{\partial(1/z)}=-2\left(\Im\widetilde{D}_{FT}(z)-\widehat{G}_{T}^{(1)}(z)\right){}$
$\displaystyle\qquad+4\int
d\left(\frac{1}{z^{\prime}}\right)\frac{1}{1/z-1/z^{\prime}}\Im\left(\widehat{N}_{1}\left(\frac{1}{z^{\prime}},\frac{1}{z}\right)-\widehat{N}_{2}\left(\frac{1}{z}-\frac{1}{z^{\prime}},\frac{1}{z}\right)\right){}$
$\displaystyle\qquad+2\int
d\left(\frac{1}{z^{\prime}}\right)\frac{1/z}{(1/z-1/z^{\prime})^{2}}\Im\left(\widehat{N}_{1}\left(\frac{1}{z^{\prime}},\frac{1}{z}\right)+\widehat{N}_{2}\left(\frac{1}{z^{\prime}},\frac{1}{z}\right)-2\widehat{N}_{2}\left(\frac{1}{z}-\frac{1}{z^{\prime}},\frac{1}{z}\right)\right),$
(14)

and

$\displaystyle\frac{1}{z}\frac{\partial\Delta\widehat{H}_{T}^{(1)}(z)}{\partial(1/z)}=-4\left(\Im\widetilde{D}_{FT}(z)-\Delta\widehat{H}_{T}^{(1)}(z)\right){}$
$\displaystyle\qquad+8\int
d\left(\frac{1}{z^{\prime}}\right)\frac{1}{1/z-1/z^{\prime}}\Im\left(\widehat{N}_{1}\left(\frac{1}{z^{\prime}},\frac{1}{z}\right)+\widehat{N}_{2}\left(\frac{1}{z^{\prime}},\frac{1}{z}\right)\right){}$
$\displaystyle\qquad+4\int
d\left(\frac{1}{z^{\prime}}\right)\frac{1/z}{(1/z-1/z^{\prime})^{2}}\Im\left(\widehat{N}_{1}\left(\frac{1}{z^{\prime}},\frac{1}{z}\right)+\widehat{N}_{2}\left(\frac{1}{z^{\prime}},\frac{1}{z}\right)\right).$
(15)

The relations (2.2), (2.2) and (2.2) show that the purely gluonic twist-3 FFs
are related to the quark-antiquark-gluon FFs, which implies the contribution
to $ep\to e\Lambda^{\uparrow}X$ from the latter needs to be considered
together. It’s been shown that the above three relations (2.2), (2.2) and
(2.2) are crucial to guarantee the frame independence of the cross section for
$pp\to\Lambda^{\uparrow}X$. Using these relations, we will express the cross
section in terms of $\widehat{G}^{(1)}_{T}$, $\Delta\widehat{H}^{(1)}_{T}$,
$\Im\widehat{N}_{1,2}$, $\Im\widehat{O}_{1,2}$, $\Im\widetilde{D}_{FT}$ and
$\Im\widetilde{G}_{FT}$ (see eq. (3.3) below), which gives the most concise
expression for the cross section. We also note that, in principle, the twist-3
kinematical FFs, $\widehat{G}^{(1)}_{T}$ and $\Delta\widehat{H}^{(1)}_{T}$,
can be also eliminated in terms of the twist-3 dynamical FFs (see eqs. (74)
and (75) of [9]).

##  3 Twist-3 gluon FF contribution to $ep\to e\Lambda^{\uparrow}X$

###  3.1 Kinematics

Here we briefly summarize the kinematics for the process[1],

$\displaystyle e(\ell)+p(p)\to
e(\ell^{\prime})+\Lambda^{\uparrow}(P_{h},S_{\perp})+X,$ (16)

where $\ell$, $\ell^{\prime}$, $p$ and $P_{h}$ are the momenta of each
particle and $S_{\perp}$ is the transverse spin vector for $\Lambda$. With the
virtual photon’s momentum $q=\ell-\ell^{\prime}$, we introduce the five
Lorentz invariants as

$\displaystyle S_{ep}\equiv(p+\ell)^{2}\simeq 2p\cdot\ell,\qquad
Q^{2}\equiv-q^{2},$ $\displaystyle x_{bj}\equiv\frac{Q^{2}}{2p\cdot q},\qquad
z_{f}\equiv\frac{p\cdot P_{h}}{p\cdot q},\qquad q_{T}\equiv\sqrt{-q_{t}^{2}},$
(17)

where

$\displaystyle q_{t}^{\mu}\equiv q^{\mu}-\frac{P_{h}\cdot q}{p\cdot
P_{h}}p^{\mu}-\frac{p\cdot q}{p\cdot P_{h}}P_{h}^{\mu}$ (18)

is a space-like momentum satisfying $q_{t}\cdot p=q_{t}\cdot P_{h}=0$.

Figure 1: Hadron frame and the transverse spin vector $\vec{S}_{\perp}$. To
make clear the convention for $\Phi_{S}$, rotate the $Z$ and $X$ axes around
the $Y$ axis by $\theta$ (polar angle of $\vec{P}_{h}$) so that the new $Z$
axis becomes parallel to $\vec{P}_{h}$. $\Phi_{S}$ is defined to be the
azimuthal angle of $\vec{S}_{\perp}$ around $\vec{P}_{h}$ measured from the
new $X$ axis, just like $\phi$ and $\chi$ are measured from the $x$ axis
around the $z$ axis.

As in [1], we work in the hadron frame[20] (See Fig. 1), where $p^{\mu}$ and
$q^{\mu}$ are collinear and take the following form:

$\displaystyle p^{\mu}=\frac{Q}{2x_{bj}}(1,0,0,1),$ (19) $\displaystyle
q^{\mu}=(0,0,0,-Q).$ (20)

Defining the azimuthal angles for the hadron plane and the lepton plane as
$\chi$ and $\phi$, respectively, as shown in Fig. 1, $P_{h}^{\mu}$ and
$\ell^{\mu}$ can be written as

$\displaystyle
P_{h}^{\mu}=\frac{z_{f}Q}{2}\left(1+\frac{q_{T}^{2}}{Q^{2}},\frac{2q_{T}}{Q}\cos{\chi},\frac{2q_{T}}{Q}\sin{\chi},-1+\frac{q_{T}^{2}}{Q^{2}}\right),$
(21)
$\displaystyle\ell^{\mu}=\frac{Q}{2}(\cosh{\psi},\sinh{\psi}\cos{\phi},\sinh{\psi}\sin{\phi},-1),$
(22)

where $\psi$ is defined by

$\displaystyle\cosh{\psi}\equiv\frac{2x_{bj}S_{ep}}{Q^{2}}-1.$ (23)

With this parameterization, the transverse momentum of the hyperon $P_{hT}$ is
given by $P_{hT}=z_{f}q_{T}$.

For the calculation of the cross section, we introduce four axes by

$\displaystyle T^{\mu}\equiv\frac{1}{Q}(q^{\mu}+2x_{bj}p^{\mu})=(1,0,0,0),{}$
$\displaystyle Z^{\mu}\equiv-\frac{q^{\mu}}{Q}=(0,0,0,1),$ $\displaystyle
X^{\mu}\equiv\frac{1}{q_{T}}\left[\frac{P_{h}^{\mu}}{z_{f}}-q^{\mu}-\left(1+\frac{q_{T}^{2}}{Q^{2}}\right)x_{bj}p^{\mu}\right]=(0,\cos{\chi},\sin{\chi},0),{}$
$\displaystyle
Y^{\mu}\equiv\epsilon^{\mu\nu\rho\sigma}Z_{\nu}T_{\rho}X_{\sigma}=(0,-\sin{\chi},\cos{\chi},0),$
(24)

where the actual form in the hadron frame is given after the last equality in
each equation. The final hyperon resides in the $XZ$-plane and the transverse
spin vector of the hyperon can be written as

$\displaystyle
S_{\perp}^{\mu}=\cos{\theta}\cos{\Phi_{S}}X^{\mu}+\sin{\Phi_{S}}Y^{\mu}-\sin{\theta}\cos{\Phi_{S}}Z^{\mu},$
(25)

where $\theta$ is the polar angle of $\vec{P_{h}}$ as measured from the
$Z$-axis and $\Phi_{S}$ is the azimuthal angle of $\vec{S}_{\perp}$ around
$\vec{P}_{h}$ as measured from the $XZ$-plane. From (21), the polar angle
$\theta$ is written as

$\displaystyle\cos{\theta}$
$\displaystyle=\frac{P_{hz}}{|\vec{P}_{h}|}=\frac{q_{T}^{2}-Q^{2}}{q_{T}^{2}+Q^{2}},$
(26) $\displaystyle\sin{\theta}$
$\displaystyle=\frac{P_{hT}}{|\vec{P}_{h}|}=\frac{2q_{T}Q}{q_{T}^{2}+Q^{2}}.$
(27)

With the kinematical variables defined above, the polarized differential cross
section for (16), $\sigma\equiv\sigma(p,\ell,\ell^{\prime},P_{h},S_{\perp})$,
takes the following form:

$\displaystyle\frac{d^{6}\sigma}{dx_{bj}dQ^{2}dz_{f}dq_{T}^{2}d\phi
d\chi}=\frac{\alpha_{em}^{2}}{128\pi^{4}x_{bj}^{2}S_{ep}^{2}Q^{2}}z_{f}L^{\rho\sigma}(\ell,\ell^{\prime})W_{\rho\sigma}(p,q,P_{h}),$
(28)

where $\alpha_{em}=e^{2}/(4\pi)$ is the QED coupling constant,
$L^{\rho\sigma}=2(\ell^{\rho}\ell^{\prime\sigma}+\ell^{\sigma}\ell^{\prime\rho})-Q^{2}g^{\rho\sigma}$
is the leptonic tensor and $W_{\rho\sigma}$ is the hadronic tensor. Although
there are two azimuthal angles, $\phi$ and $\chi$, the cross section depends
on the relative angle $\varphi\equiv\phi-\chi$ only. Therefore it can be
expressed in terms of $S_{ep}$, $Q^{2}$, $x_{bj}$, $z_{f}$, $q_{T}^{2}$,
$\varphi$ and $\Phi_{S}$.

###  3.2 Hadronic tensor

We now calculate the twist-3 gluon FF contribution to (28) following the
formalism developed for $pp\to\Lambda^{\uparrow}X$[15]. It occurs as a nonpole
contribution from the hard part as in the case of other twist-3 fragmentation
contributions in $ep^{\uparrow}\to e\pi X$ [21] and $pp\to\Lambda^{\uparrow}X$
[22, 15]. We first factorize the twist-2 unpolarized quark DFs $f_{1}(x)$ from
the hadronic tensor $W_{\rho\sigma}(p,q,P_{h})$:

$\displaystyle
W_{\rho\sigma}(p,q,P_{h})=\int\frac{dx}{x}f_{1}(x)w_{\rho\sigma}(xp,q,P_{h}),$
(29)

where $x$ is the momentum fraction of the quark in the proton, and we have
omitted the factor associated with the quark’s fractional electric charge as
well as summation over quark flavors. Up to twist-3, $w_{\rho\sigma}$ receives
contribution from the 2-gluon, 3-gluon and quark-antiquark-gluon correlation
functions corresponding to (a)-(e) of Fig. 2:

$\displaystyle w_{\rho\sigma}\equiv
w_{\rho\sigma}^{\rm(a)}+w_{\rho\sigma}^{\rm(b)}+w_{\rho\sigma}^{\rm(c)}+w_{\rho\sigma}^{\rm(d)}+w_{\rho\sigma}^{\rm(e)},$
(30)

where each term can be written as

$\displaystyle w_{\rho\sigma}^{\rm(a)}$
$\displaystyle=\int\frac{d^{4}k}{(2\pi)^{4}}\Gamma^{(0)\mu\nu}_{ab}(k)S^{ab}_{\mu\nu,\rho\sigma}(k),$
(31) $\displaystyle w_{\rho\sigma}^{\rm(b)}$
$\displaystyle=\frac{1}{2}\iint\frac{d^{4}k}{(2\pi)^{4}}\frac{d^{4}k^{\prime}}{(2\pi)^{4}}\Gamma^{(1)\mu\nu\lambda}_{\mathrm{L}abc}(k,k^{\prime})S^{\mathrm{L}abc}_{\mu\nu\lambda,\rho\sigma}(k,k^{\prime}),$
(32) $\displaystyle w_{\rho\sigma}^{\rm(c)}$
$\displaystyle=\frac{1}{2}\iint\frac{d^{4}k}{(2\pi)^{4}}\frac{d^{4}k^{\prime}}{(2\pi)^{4}}\Gamma^{(1)\mu\nu\lambda}_{\mathrm{R}abc}(k,k^{\prime})S^{\mathrm{R}abc}_{\mu\nu\lambda,\rho\sigma}(k,k^{\prime}),$
(33) $\displaystyle w_{\rho\sigma}^{\rm(d)}$
$\displaystyle=\mathrm{Tr}{}\iint\frac{d^{4}k}{(2\pi)^{4}}\frac{d^{4}k^{\prime}}{(2\pi)^{4}}\widetilde{\Delta}^{(1)\alpha}_{\mathrm{L}a}(k,k^{\prime})\widetilde{S}^{\mathrm{L}a}_{\alpha,\rho\sigma}(k,k^{\prime}),$
(34) $\displaystyle w_{\rho\sigma}^{\rm(e)}$
$\displaystyle=\mathrm{Tr}{}\iint\frac{d^{4}k}{(2\pi)^{4}}\frac{d^{4}k^{\prime}}{(2\pi)^{4}}\widetilde{\Delta}^{(1)\alpha}_{\mathrm{R}a}(k,k^{\prime})\widetilde{S}^{\mathrm{R}a}_{\alpha,\rho\sigma}(k,k^{\prime}).$
(35)

Here $S^{ab}_{\mu\nu,\rho\sigma}(k)$,
$S^{\mathrm{L}(\mathrm{R})abc}_{\mu\nu\lambda,\rho\sigma}(k,k^{\prime})$, and
$\widetilde{S}^{\mathrm{L}(\mathrm{R})a}_{\alpha,\rho\sigma}(k,k^{\prime})$
represent the partonic hard parts with $k$ and $k^{\prime}$ the momenta of
partons fragmenting into the final hyperon, and the dependence on $q$ is
suppressed for simplicity. $\Gamma^{(0)\mu\nu}_{ab}$,
$\Gamma^{(1)\mu\nu\lambda}_{\mathrm{L}(\mathrm{R})abc}$ and
$\widetilde{\Delta}^{(1)\alpha}_{\mathrm{L}(\mathrm{R})a}$ denote the
fragmentation matrix elements defined as

$\displaystyle\Gamma^{(0)\mu\nu}_{ab}(k)$ $\displaystyle=\sum_{X}\int
d^{4}\xi\mathrm{e}^{-ik\cdot\xi}\langle 0|A^{\nu}_{b}(0)|hX\rangle\langle
hX|A^{\mu}_{a}(\xi)|0\rangle,$ (36)
$\displaystyle\Gamma^{(1)\mu\nu\lambda}_{\mathrm{L}abc}(k,k^{\prime})$
$\displaystyle=\sum_{X}\iint d^{4}\xi
d^{4}\eta\mathrm{e}^{-ik\cdot\xi}\mathrm{e}^{-i(k^{\prime}-k)\cdot\eta}\langle
0|A^{\nu}_{b}(0)|hX\rangle\langle
hX|A^{\mu}_{a}(\xi)gA^{\lambda}_{c}(\eta)|0\rangle,$ (37)
$\displaystyle\Gamma^{(1)\mu\nu\lambda}_{\mathrm{R}abc}(k,k^{\prime})$
$\displaystyle=\sum_{X}\iint d^{4}\xi
d^{4}\eta\mathrm{e}^{-ik\cdot\xi}\mathrm{e}^{-i(k^{\prime}-k)\cdot\eta}\langle
0|A^{\nu}_{b}(0)gA^{\lambda}_{c}(\eta)|hX\rangle\langle
hX|A^{\mu}_{a}(\xi)|0\rangle,$ (38)
$\displaystyle\widetilde{\Delta}^{(1)\alpha}_{\mathrm{L}a,ij}(k,k^{\prime})$
$\displaystyle=\sum_{X}\iint d^{4}\xi
d^{4}\eta\mathrm{e}^{-ik\cdot\xi}\mathrm{e}^{-i(k^{\prime}-k)\cdot\eta}\langle
0|gA^{\alpha}_{a}(\eta)|hX\rangle\langle
hX|\psi_{i}(0)\bar{\psi}_{j}(\xi)|0\rangle,$ (39)
$\displaystyle\widetilde{\Delta}^{(1)\alpha}_{\mathrm{R}a,ij}(k,k^{\prime})$
$\displaystyle=\sum_{X}\iint d^{4}\xi
d^{4}\eta\mathrm{e}^{-ik\cdot\xi}\mathrm{e}^{-i(k^{\prime}-k)\cdot\eta}\langle
0|\psi_{i}(0)\bar{\psi}_{j}(\xi)|hX\rangle\langle
hX|gA^{\alpha}_{a}(\eta)|0\rangle.$ (40)

The contribution with two parton lines in the left (right) of the cut in Fig.
2 (b)-(e) are characterized by the symbol $\mathrm{L}$ ($\mathrm{R}$) in the
hard parts and the fragmentation matrix elements. The superscripts $(0)$ and
$(1)$ indicate the order of the gauge coupling $g$ corresponding,
respectively, to the 2-parton and 3-parton correlation functions. The factor
$1/2$ in (32) and (33) takes into account the exchange symmetry in the
corresponding matrix element. In (34) and (35), the hard parts and the
fragmentation matrix elements are matrices both in color and spinor spaces for
the quark and ${\rm Tr}$ indicates trace over both indices. The hard parts and
the fragmentation matrix elements satisfy
$\Gamma^{(1)\mu\nu\lambda}_{\mathrm{R}abc}(k,k^{\prime})=\Gamma^{(1)\nu\mu\lambda}_{\mathrm{L}bac}(k^{\prime},k)^{*}$
,
$\widetilde{\Delta}^{(1)\alpha}_{\mathrm{R}a}(k,k^{\prime})=\gamma^{0}\widetilde{\Delta}^{(1)\alpha}_{\mathrm{L}a}(k^{\prime},k)^{\dagger}\gamma^{0}$
,
$S^{\mathrm{R}abc}_{\mu\nu\lambda,\rho\sigma}(k,k^{\prime})=S^{\mathrm{L}bac}_{\nu\mu\lambda,\sigma\rho}(k^{\prime},k)^{*}$
and
$\widetilde{S}^{\mathrm{R}a}_{\alpha,\rho\sigma}(k,k^{\prime})=\gamma^{0}\widetilde{S}^{\mathrm{L}a}_{\alpha,\sigma\rho}(k^{\prime},k)^{\dagger}\gamma^{0}$.
We thus have

$\displaystyle
w_{\rho\sigma}=w^{\rm(a)}_{\rho\sigma}+2\Re\,w^{\rm(b)}_{\rho\sigma}+2\Re\,w^{\rm(d)}_{\rho\sigma}.$
(41)

Figure 2: Cut diagrams for the twist-3 gluon fragmentation contribution to
$ep\to e\Lambda^{\uparrow}X$. In each diagram, the lower blob represents the
unpolarized quark distribution, the middle one represents the partonic hard
cross section and the upper one represents the fragmentation matrix elements
for the final hyperon.

To extract the twist-3 contribution to $ep\to e\Lambda^{\uparrow}X$ we apply
the collinear expansion to the hard part, $S^{ab}_{\mu\nu,\rho\sigma}$,
$S^{\mathrm{L}abc}_{\mu\nu\lambda,\rho\sigma}$ and
$\widetilde{S}^{\mathrm{L}a}_{\alpha,\rho\sigma}$, with respect to the momenta
$k$ and $k^{\prime}$ around $P_{h}/z$ and $P_{h}/z^{\prime}$, respectively,
taking into account of the following Ward-Takahashi (WT) identities[15]:

$\displaystyle
k^{\mu}S^{ab}_{\mu\nu,\rho\sigma}(k)=k^{\nu}S^{ab}_{\mu\nu,\rho\sigma}(k)=0,$
(42) $\displaystyle
k^{\mu}S^{\mathrm{L}abc}_{\mu\nu\lambda,\rho\sigma}(k,k^{\prime})=\frac{if^{abc}}{N^{2}-1}S_{\lambda\nu,\rho\sigma}(k^{\prime}),$
(43) $\displaystyle
k^{\prime\nu}S^{\mathrm{L}abc}_{\mu\nu\lambda,\rho\sigma}(k,k^{\prime})=0,$
(44)
$\displaystyle(k^{\prime}-k)^{\lambda}S^{\mathrm{L}abc}_{\mu\nu\lambda,\rho\sigma}(k,k^{\prime})=\frac{-if^{abc}}{N^{2}-1}S_{\mu\nu,\rho\sigma}(k^{\prime}),$
(45)
$\displaystyle(k-k^{\prime})^{\alpha}\widetilde{S}^{\mathrm{L}a}_{\alpha,\rho\sigma}(k,k^{\prime})=0,$
(46)

where
$S_{\mu\nu,\rho\sigma}(k)\equiv\delta^{ab}S^{ab}_{\mu\nu,\rho\sigma}(k)$. We
note that unlike the case for $pp\to\Lambda^{\uparrow}X$ no ghost-like terms
appear in the WT identities (42)-(46) for the present case. This way, one
obtains the hadronic tensor $w_{\rho\sigma}$ in terms of the gauge invariant
FFs as (See eq. (51) of [15] and eq. (56) of [21])

$\displaystyle
w_{\rho\sigma}=\Omega^{\alpha}_{\;\;\mu}\Omega^{\beta}_{\;\;\nu}\int
d\left(\frac{1}{z}\right)z^{2}{\widehat{\Gamma}^{\mu\nu}(z)}{S_{\alpha\beta,\rho\sigma}\left({1\over
z}\right)}{}$
$\displaystyle\qquad-i\Omega^{\alpha}_{\;\;\mu}\Omega^{\beta}_{\;\;\nu}\Omega^{\gamma}_{\;\;\lambda}\int
d\left(\frac{1}{z}\right)z^{2}{\widehat{\Gamma}^{\mu\nu\lambda}_{\partial}(z)}\left.\frac{\partial{S_{\alpha\beta,\rho\sigma}(k)}}{\partial
k^{\gamma}}\right|_{k=P_{h}/z}{}$
$\displaystyle\qquad+\Re\left[i\Omega^{\alpha}_{\;\;\mu}\Omega^{\beta}_{\;\;\nu}\Omega^{\gamma}_{\;\;\lambda}\iint
d\left(\frac{1}{z}\right)d\left(\frac{1}{z^{\prime}}\right)zz^{\prime}\frac{1}{1/z-1/z^{\prime}}\right.{}$
$\displaystyle\qquad\qquad\times\left.\left\\{-\frac{if^{abc}}{N}{\widehat{\Gamma}^{\mu\nu\lambda}_{FA}\left(\frac{1}{z^{\prime}},\frac{1}{z}\right)}+\frac{Nd^{abc}}{N^{2}-4}{\widehat{\Gamma}^{\mu\nu\lambda}_{FS}\left(\frac{1}{z^{\prime}},\frac{1}{z}\right)}\right\\}{S^{\mathrm{L}abc}_{\alpha\beta\gamma,\rho\sigma}\left(\frac{1}{z^{\prime}},\frac{1}{z}\right)}\right]{}$
$\displaystyle\qquad+\Re\left[i\Omega^{\alpha}_{\;\;\mu}\iint
d\left(\frac{1}{z}\right)d\left(\frac{1}{z^{\prime}}\right)z\,{\rm Tr}_{\rm
s}\left\\{{\widetilde{\Delta}^{\mu}\left(\frac{1}{z^{\prime}},\frac{1}{z^{\prime}}-\frac{1}{z}\right)}{\widetilde{S}^{\mathrm{L}}_{\alpha,\rho\sigma}\left(\frac{1}{z^{\prime}},\frac{1}{z^{\prime}}-\frac{1}{z}\right)}\right\\}\right],$
(47)

where $\widehat{\Gamma}^{\mu\nu}(z)$,
$\widehat{\Gamma}^{\mu\nu\lambda}_{\partial}(z)$,
$\widehat{\Gamma}^{\mu\nu\lambda}_{FA}\left(\frac{1}{z^{\prime}},\frac{1}{z}\right)$,
$\widehat{\Gamma}^{\mu\nu\lambda}_{FS}\left(\frac{1}{z^{\prime}},\frac{1}{z}\right)$
and
$\widetilde{\Delta}^{\mu}\left(\frac{1}{z^{\prime}},\frac{1}{z^{\prime}}-\frac{1}{z}\right)$
are given by (2.1), (2.1), (2.1), (2.1) and (2.1). For the hard part we have
used the notation $S_{\alpha\beta,\rho\sigma}\left({1\over z}\right)$ for
$S_{\alpha\beta,\rho\sigma}\left({P_{h}\over z}\right)$ and
$S^{\mathrm{L}abc}_{\alpha\beta\gamma,\rho\sigma}\left(\frac{1}{z^{\prime}},\frac{1}{z}\right)$
for
$S^{\mathrm{L}abc}_{\alpha\beta\gamma,\rho\sigma}\left(\frac{P_{h}}{z^{\prime}},\frac{P_{h}}{z}\right)$,
etc, suppressing $P_{h}$ for short. In the last term of (47),
$\widetilde{S}^{\mathrm{L}}_{\alpha,\rho\sigma}$ is defined from
$\widetilde{S}^{\mathrm{L}a}_{\alpha,\rho\sigma}$ in (34) by
$\displaystyle\left(\widetilde{S}^{\mathrm{L}a}_{\alpha,\rho\sigma}\right)_{rs}={1\over
2N}t^{a}_{rs}\widetilde{S}^{\mathrm{L}}_{\alpha,\rho\sigma}$, where $r,s$
indicates the color indices for the quark, and ${\rm Tr}_{\rm s}$ denotes the
trace in the spinor space. The LO diagrams for the hard parts of Figs. 2(a),
(b) and (d) are, respectively, shown in Figs. 3, 4 and 5. It is easy to show
that the hadronic tensor $w_{\rho\sigma}$ satisfies the electromagnetic gauge
invariance, $q^{\rho}w_{\rho\sigma}=q^{\sigma}w_{\rho\sigma}=0$, owing to the
WT identity in QED.

Figure 3: The lowest order Feynman diagrams for
$S_{\alpha\beta,\rho\sigma}\left({1\over z}\right)$ in (47). We set
$\hat{p}\equiv xp$ and $\hat{P}_{h}\equiv P_{h}/z$. The symbol $\otimes$
indicates the fragmentation to the final hadron and $p_{d}$ is the momentum of
an unobserved parton in the final state.  Figure 4:  The lowest order Feynman
diagrams for
$S^{\mathrm{L}abc}_{\alpha\beta\gamma,\rho\sigma}\left(\frac{1}{z^{\prime}},\frac{1}{z}\right)$
in (47). We set $\hat{P}^{\prime}_{h}\equiv P_{h}/z^{\prime}$. Three crosses
($\times$) on the quark line in the upper diagrams indicates that the virtual
photon line with a cross at one end needs to be attached to one of these
crosses, and all three diagrams have to be included. Thus the number of
diagrams for $S^{\mathrm{L}abc}_{\alpha\beta\gamma,\rho\sigma}$ is
$(3+3)\times 2=12$. The meaning of the other symbols are the same as that in
Fig. 3.  Figure 5: The lowest order Feynman diagrams for
$\widetilde{S}^{\mathrm{L}}_{\alpha,\rho\sigma}\left(\frac{1}{z^{\prime}},\frac{1}{z^{\prime}}-\frac{1}{z}\right)$.
The meaning of the symbols is the same as that in Fig. 4. The total number of
diagrams for $\widetilde{S}^{\mathrm{L}}_{\alpha,\rho\sigma}$ is $(4+2)\times
2=12$.

###  3.3 Spin dependent cross section

The calculation of $L^{\rho\sigma}W_{\rho\sigma}$ in (28) can be done in the
same way as [1]: $W^{\rho\sigma}$ can be expanded in terms of the six
tensors[20] $\mathscr{V}_{k}^{\rho\sigma}$ ($k=1,\cdots,4,8,9)$ defined by

$\displaystyle\mathscr{V}^{\mu\nu}_{1}=X^{\mu}X^{\nu}+Y^{\mu}Y^{\nu},$
$\displaystyle\mathscr{V}^{\mu\nu}_{2}=g^{\mu\nu}+Z^{\mu}Z^{\nu},$
$\displaystyle\mathscr{V}^{\mu\nu}_{3}=T^{\mu}X^{\nu}+X^{\mu}T^{\nu},{}$
$\displaystyle\mathscr{V}^{\mu\nu}_{4}=X^{\mu}X^{\nu}-Y^{\mu}Y^{\nu},$
$\displaystyle\mathscr{V}^{\mu\nu}_{8}=T^{\mu}Y^{\nu}+Y^{\mu}T^{\nu},$
$\displaystyle\mathscr{V}^{\mu\nu}_{9}=X^{\mu}Y^{\nu}+Y^{\mu}X^{\nu}.$ (48)

By introducing the inverses of $\mathscr{V}_{k}^{\rho\sigma}$,
$\tilde{\mathscr{V}}_{k}^{\rho\sigma}$ satisfying
$\mathscr{V}_{k}^{\rho\sigma}\tilde{\mathscr{V}}_{k^{\prime}\,\rho\sigma}=\delta_{kk^{\prime}}$,
as

$\displaystyle\tilde{\mathscr{V}}^{\mu\nu}_{1}=\frac{1}{2}(2T^{\mu}T^{\nu}+X^{\mu}X^{\nu}+Y^{\mu}Y^{\nu}),\quad\tilde{\mathscr{V}}^{\mu\nu}_{2}=T^{\mu}T^{\nu},$
$\displaystyle\tilde{\mathscr{V}}^{\mu\nu}_{3}=-\frac{1}{2}(T^{\mu}X^{\nu}+X^{\mu}T^{\nu}),{}$
$\displaystyle\tilde{\mathscr{V}}^{\mu\nu}_{4}=\frac{1}{2}(X^{\mu}X^{\nu}-Y^{\mu}Y^{\nu}),\quad\tilde{\mathscr{V}}^{\mu\nu}_{8}=-\frac{1}{2}(T^{\mu}Y^{\nu}+Y^{\mu}T^{\nu}),$
$\displaystyle\tilde{\mathscr{V}}^{\mu\nu}_{9}=\frac{1}{2}(X^{\mu}Y^{\nu}+Y^{\mu}X^{\nu}),$
(49)

$W^{\mu\nu}$ can be expanded as

$\displaystyle
W^{\mu\nu}=\sum_{k=1,\cdots,4,8,9}\mathscr{V}^{\mu\nu}_{k}[W_{\rho\sigma}\tilde{\mathscr{V}}^{\rho\sigma}_{k}].$
(50)

Then one obtains

$\displaystyle L^{\mu\nu}W_{\mu\nu}=\sum_{k=1,\cdots
4,8,9}[L_{\mu\nu}\mathscr{V}^{\mu\nu}_{k}][W_{\rho\sigma}\tilde{\mathscr{V}}^{\rho\sigma}_{k}]=Q^{2}\sum_{k=1,\cdots
4,8,9}\mathscr{A}_{k}(\phi-\chi)[W_{\rho\sigma}\tilde{\mathscr{V}}^{\rho\sigma}_{k}],$
(51)

where $\mathscr{A}_{k}(\varphi)\equiv
L_{\mu\nu}\mathscr{V}^{\mu\nu}_{k}/Q^{2}$ are given by

$\displaystyle\mathscr{A}_{1}(\varphi)=1+\cosh^{2}{\psi},$
$\displaystyle\mathscr{A}_{2}(\varphi)=-2,$
$\displaystyle\mathscr{A}_{3}(\varphi)=-\cos{\varphi}\sinh{2\psi},{}$
$\displaystyle\mathscr{A}_{4}(\varphi)=\cos{2\varphi}\sinh^{2}{\psi},$
$\displaystyle\mathscr{A}_{8}(\varphi)=-\sin{\varphi}\sinh{2\psi},$
$\displaystyle\mathscr{A}_{9}(\varphi)=\sin{2\varphi}\sinh^{2}{\psi},$ (52)

with $\psi$ defined in (23). From (51) and (52), one sees the cross section
can be decomposed into the five structure functions with different azimuthal
dependences which are carried by the $\mathscr{A}_{k}(\varphi)$s. Substituting
(2.1), (2.1), (2.1), (2.1) and (2.1) into (47), we find that the cross section
(28) takes the following structure:

$\displaystyle\frac{d^{6}\sigma}{dx_{bj}dQ^{2}dz_{f}dq_{\mathrm{T}}^{2}d\phi
d\chi}{}$
$\displaystyle=\frac{\alpha_{em}^{2}\alpha_{s}M_{h}}{16\pi^{2}x_{bj}^{2}S_{ep}^{2}Q^{2}}\sum_{k}\mathscr{A}_{k}(\varphi)\mathcal{S}_{k}\iint
dxd\left(\frac{1}{z}\right)\frac{z^{3}}{x}f_{1}(x)\delta\left(\frac{q_{T}^{2}}{Q^{2}}-\left(1-\frac{1}{\hat{x}}\right)\left(1-\frac{1}{\hat{z}}\right)\right){}$
$\displaystyle\times\left\\{\frac{1}{z}\Delta\widehat{G}_{3\bar{T}}(z)\hat{\sigma}^{k}_{int}+\widehat{G}^{(1)}_{T}(z)\hat{\sigma}^{k}_{NDG}+\frac{1}{z}\frac{\partial\widehat{G}_{T}^{(1)}(z)}{\partial(1/z)}\hat{\sigma}^{k}_{DG}+\Delta\widehat{H}^{(1)}_{T}(z)\hat{\sigma}^{k}_{NDH}+\frac{1}{z}\frac{\partial\Delta\widehat{H}_{T}^{(1)}(z)}{\partial(1/z)}\hat{\sigma}^{k}_{DH}\right.{}$
$\displaystyle+\frac{1}{2}\int
d\left(\frac{1}{z^{\prime}}\right)\left[\sum_{i=1}^{3}\Im\widehat{N}_{i}\left(\frac{1}{z^{\prime}},\frac{1}{z}\right)\left(\frac{1}{1/z-1/z^{\prime}}\hat{\sigma}^{-(1)}_{i,k}+\frac{1}{z}\left(\frac{1}{1/z-1/z^{\prime}}\right)^{2}\hat{\sigma}^{-(2)}_{i,k}+z^{\prime}\hat{\sigma}^{-(3)}_{i,k}+\frac{z^{\prime
2}}{z}\hat{\sigma}^{-(4)}_{i,k}\right)\right.{}$
$\displaystyle\left.+\sum_{i=1}^{3}\widehat{O}_{i}\left(\frac{1}{z^{\prime}},\frac{1}{z}\right)\left(\frac{1}{1/z-1/z^{\prime}}\hat{\sigma}^{+(1)}_{i,k}+\frac{1}{z}\left(\frac{1}{1/z-1/z^{\prime}}\right)^{2}\hat{\sigma}^{+(2)}_{i,k}+z^{\prime}\hat{\sigma}^{+(3)}_{i,k}+\frac{z^{\prime
2}}{z}\hat{\sigma}^{+(4)}_{i,k}\right)\right]{}$ $\displaystyle+\int
d\left(\frac{1}{z^{\prime}}\right)\frac{2}{C_{F}}\left[\Im{\widetilde{D}_{FT}\left(\frac{1}{z^{\prime}},\frac{1}{z^{\prime}}-\frac{1}{z}\right)}\right.\left({\hat{\sigma}^{k}_{DF}}+\frac{1}{z}\frac{1}{1/z-1/z^{\prime}}{\hat{\sigma}^{k}_{DF2}}+\frac{z^{\prime}}{z}{\hat{\sigma}^{k}_{DF3}}\right.{}$
$\displaystyle\left.\qquad\qquad+\frac{1}{1-(1-q_{T}^{2}/Q^{2})z_{f}/z^{\prime}}{\hat{\sigma}^{k}_{DF4}}+\frac{1}{1-(1-q_{T}^{2}/Q^{2})z_{f}(1/z-1/z^{\prime})}{\hat{\sigma}^{k}_{DF5}}\right){}$
$\displaystyle\qquad\qquad+\Im{\widetilde{G}_{FT}\left(\frac{1}{z^{\prime}},\frac{1}{z^{\prime}}-\frac{1}{z}\right)}\left({\hat{\sigma}^{k}_{GF}}+\frac{1}{z}\frac{1}{1/z-1/z^{\prime}}{\hat{\sigma}^{k}_{GF2}}+\frac{z^{\prime}}{z}{\hat{\sigma}^{k}_{GF3}}\right.{}$
$\displaystyle\left.\qquad\qquad\left.\left.+\frac{1}{1-(1-q_{T}^{2}/Q^{2})z_{f}/z^{\prime}}{\hat{\sigma}^{k}_{GF4}}+\frac{1}{1-(1-q_{T}^{2}/Q^{2})z_{f}(1/z-1/z^{\prime})}{\hat{\sigma}^{k}_{GF5}}\right)\right]\right\\},$
(53)

where

$\displaystyle\mathcal{S}_{1,2,3,4}\equiv\sin{\Phi_{S}},\quad\mathcal{S}_{8,9}\equiv\cos{\Phi_{S}},\quad\hat{x}=x_{bj}/x,\quad\hat{z}=z_{f}/z,$
(54)

and we have set

$\displaystyle\widehat{N}_{3}\left(\frac{1}{z^{\prime}},\frac{1}{z}\right)\equiv-\widehat{N}_{2}\left(\frac{1}{z}-\frac{1}{z^{\prime}},\frac{1}{z}\right),\qquad\widehat{O}_{3}\left(\frac{1}{z^{\prime}},\frac{1}{z}\right)\equiv\widehat{O}_{2}\left(\frac{1}{z}-\frac{1}{z^{\prime}},\frac{1}{z}\right)$
(55)

for convenience. Partonic hard parts for each FF can be computed from the
corresponding diagrams, Figs. 3, 4 and 5. We have reached the form (53) based
on the observation that the $z^{\prime}$-dependence of the hard parts for the
dynamical FFs appears in the cross section only through the factors explicitly
shown in (53) (See Appendix C of [15]), and hence we can define all the
partonic hard cross sections $\hat{\sigma}$’s in (53) as the functions of
$\hat{x}$, $\hat{z}$, $Q$ and $q_{T}$. In addition we found by explicit
calculation of the LO diagrams that

$\displaystyle\hat{\sigma}^{\pm(3)}_{i,k}=\hat{\sigma}^{\pm(1)}_{i,k},$ (56)
$\displaystyle\hat{\sigma}^{k}_{DF}=\hat{\sigma}^{k}_{GF}=0.$ (57)

In order to transform the cross section (53) into a more concise form, we note
the following points: (I) Owing to the symmetry property under
$1/z^{\prime}\leftrightarrow 1/z-1/z^{\prime}$ of $\widehat{N}_{1}$ and
$\widehat{O}_{1}$ (6) and the relations (55), the terms of
$\hat{\sigma}^{\pm(3)}_{i,k}$ and $\hat{\sigma}^{\pm(4)}_{i,k}$ can be
combined, respectively, with those of $\hat{\sigma}^{\pm(1)}_{i,k}$ and
$\hat{\sigma}^{\pm(2)}_{i,k}$, taking into account the relation (56). (II)
Using (2.2), (2.2) and (2.2), one can eliminate the intrinsic FF and the
derivative of the kinematical FFs in favor of the kinematical and the
dynamical FFs. This way we finally obtain the twist-3 gluon FF contribution to
$ep\to e\Lambda^{\uparrow}X$ as222As was noted at the end of Sec. 2, the
kinematical FFs $\widehat{G}^{(1)}_{T}(z)$ and
$\Delta\widehat{H}^{(1)}_{T}(z)$ can, in principle, be eliminated in terms of
the dynamical FFs.

$\displaystyle\frac{d^{6}\sigma}{dx_{bj}dQ^{2}dz_{f}dq_{\mathrm{T}}^{2}d\phi
d\chi}{}$
$\displaystyle=\frac{\alpha_{em}^{2}\alpha_{s}M_{h}}{16\pi^{2}x_{bj}^{2}S_{ep}^{2}Q^{2}}\sum_{k}\mathscr{A}_{k}(\phi-\chi)\mathcal{S}_{k}\int^{1}_{x_{min}}{dx\over
x}\int^{1}_{z_{min}}{dz\over
z}z^{2}f_{1}(x)\delta\left(\frac{q_{T}^{2}}{Q^{2}}-\left(1-\frac{1}{\hat{x}}\right)\left(1-\frac{1}{\hat{z}}\right)\right){}$
$\displaystyle\times\left\\{{\widehat{G}^{(1)}_{T}(z)}{\hat{\sigma}^{k}_{G}}+{\Delta\widehat{H}^{(1)}_{T}(z)}{\hat{\sigma}^{k}_{H}}\right.{}$
$\displaystyle+\int
d\left(\frac{1}{z^{\prime}}\right)\left[\frac{1}{1/z-1/z^{\prime}}\Im\left({\widehat{N}_{1}\left(\frac{1}{z^{\prime}},\frac{1}{z}\right)}{\hat{\sigma}^{k}_{N1}}+{\widehat{N}_{2}\left(\frac{1}{z^{\prime}},\frac{1}{z}\right)}{\hat{\sigma}^{k}_{N2}}+{\widehat{N}_{2}\left(\frac{1}{z}-\frac{1}{z^{\prime}},\frac{1}{z}\right)}{\hat{\sigma}^{k}_{N3}}\right)\right.{}$
$\displaystyle+\frac{1}{z}\left(\frac{1}{1/z-1/z^{\prime}}\right)^{2}\Im\left({\widehat{N}_{1}\left(\frac{1}{z^{\prime}},\frac{1}{z}\right)}{\hat{\sigma}^{k}_{DN1}}+{\widehat{N}_{2}\left(\frac{1}{z^{\prime}},\frac{1}{z}\right)}{\hat{\sigma}^{k}_{DN2}}+{\widehat{N}_{2}\left(\frac{1}{z}-\frac{1}{z^{\prime}},\frac{1}{z}\right)}{\hat{\sigma}^{k}_{DN3}}\right){}$
$\displaystyle+\frac{1}{1/z-1/z^{\prime}}\Im\left({\widehat{O}_{1}\left(\frac{1}{z^{\prime}},\frac{1}{z}\right)}{\hat{\sigma}^{k}_{O1}}+{\widehat{O}_{2}\left(\frac{1}{z^{\prime}},\frac{1}{z}\right)}{\hat{\sigma}^{k}_{O2}}+{\widehat{O}_{2}\left(\frac{1}{z}-\frac{1}{z^{\prime}},\frac{1}{z}\right)}{\hat{\sigma}^{k}_{O3}}\right){}$
$\displaystyle\left.+\frac{1}{z}\left(\frac{1}{1/z-1/z^{\prime}}\right)^{2}\Im\left({\widehat{O}_{1}\left(\frac{1}{z^{\prime}},\frac{1}{z}\right)}{\hat{\sigma}^{k}_{DO1}}+{\widehat{O}_{2}\left(\frac{1}{z^{\prime}},\frac{1}{z}\right)}{\hat{\sigma}^{k}_{DO2}}+{\widehat{O}_{2}\left(\frac{1}{z}-\frac{1}{z^{\prime}},\frac{1}{z}\right)}{\hat{\sigma}^{k}_{DO3}}\right)\right]{}$
$\displaystyle+\int
d\left(\frac{1}{z^{\prime}}\right)\frac{2}{C_{F}}\left[\Im{\widetilde{D}_{FT}\left(\frac{1}{z^{\prime}},\frac{1}{z^{\prime}}-\frac{1}{z}\right)}\right.\left({\hat{\sigma}^{k}_{DF1}}+\frac{1}{z}\frac{1}{1/z-1/z^{\prime}}{\hat{\sigma}^{k}_{DF2}}+\frac{z^{\prime}}{z}{\hat{\sigma}^{k}_{DF3}}\right.{}$
$\displaystyle\left.+\frac{1}{1-(1-q_{T}^{2}/Q^{2})z_{f}/z^{\prime}}{\hat{\sigma}^{k}_{DF4}}+\frac{1}{1-(1-q_{T}^{2}/Q^{2})z_{f}(1/z-1/z^{\prime})}{\hat{\sigma}^{k}_{DF5}}\right){}$
$\displaystyle+\Im{\widetilde{G}_{FT}\left(\frac{1}{z^{\prime}},\frac{1}{z^{\prime}}-\frac{1}{z}\right)}\left(\frac{1}{z}\frac{1}{1/z-1/z^{\prime}}{\hat{\sigma}^{k}_{GF2}}+\frac{z^{\prime}}{z}{\hat{\sigma}^{k}_{GF3}}\right.{}$
$\displaystyle\left.\left.\left.+\frac{1}{1-(1-q_{T}^{2}/Q^{2})z_{f}/z^{\prime}}{\hat{\sigma}^{k}_{GF4}}+\frac{1}{1-(1-q_{T}^{2}/Q^{2})z_{f}(1/z-1/z^{\prime})}{\hat{\sigma}^{k}_{GF5}}\right)\right]\right\\},$
(58)

where the lower limits of $x$ and $z$ are, respectively, given by
$x_{min}=x_{bj}\left(1+\frac{z_{f}}{1-z_{f}}\frac{q_{T}^{2}}{Q^{2}}\right)$
and
$z_{min}=z_{f}\left(1+\frac{x_{bj}}{1-x_{bj}}\frac{q_{T}^{2}}{Q^{2}}\right)$.
The partonic hard cross sections which appear newly in (3.3) are defined from
those in (53) as

$\displaystyle\hat{\sigma}^{k}_{G}=\frac{1}{2}\hat{\sigma}^{k}_{int}+\hat{\sigma}^{k}_{NDG}+2\hat{\sigma}^{k}_{DG},$
(59)
$\displaystyle\hat{\sigma}^{k}_{H}=\frac{1}{2}\hat{\sigma}^{k}_{int}+\hat{\sigma}^{k}_{NDH}+4\hat{\sigma}^{k}_{DH},$
(60)
$\displaystyle\hat{\sigma}^{k}_{N1}=2\hat{\sigma}^{k}_{int}+4\hat{\sigma}^{k}_{DG}+8\hat{\sigma}^{k}_{DH},$
(61)
$\displaystyle\hat{\sigma}^{k}_{N2}=\hat{\sigma}^{k}_{int}+8\hat{\sigma}^{k}_{DH}+\frac{1}{2}(\hat{\sigma}^{-(1)}_{2,k}-\hat{\sigma}^{-(1)}_{3,k}),$
(62)
$\displaystyle\hat{\sigma}^{k}_{N3}=-\hat{\sigma}^{k}_{int}-4\hat{\sigma}^{k}_{DG}+\frac{1}{2}(\hat{\sigma}^{-(1)}_{2,k}-\hat{\sigma}^{-(1)}_{3,k}),$
(63)
$\displaystyle\hat{\sigma}^{k}_{DN1}=2\hat{\sigma}^{k}_{DG}+4\hat{\sigma}^{k}_{DH}+\frac{1}{2}(\hat{\sigma}^{-(2)}_{1,k}-\hat{\sigma}^{-(4)}_{1,k}),$
(64)
$\displaystyle\hat{\sigma}^{k}_{DN2}=2\hat{\sigma}^{k}_{DG}+4\hat{\sigma}^{k}_{DH}+\frac{1}{2}(\hat{\sigma}^{-(2)}_{2,k}-\hat{\sigma}^{-(4)}_{3,k}),$
(65)
$\displaystyle\hat{\sigma}^{k}_{DN3}=-4\hat{\sigma}^{k}_{DG}+\frac{1}{2}(\hat{\sigma}^{-(4)}_{2,k}-\hat{\sigma}^{-(2)}_{3,k}),$
(66) $\displaystyle\hat{\sigma}^{k}_{O1}=\hat{\sigma}^{+(1)}_{1,k},$ (67)
$\displaystyle\hat{\sigma}^{k}_{O2}=\frac{1}{2}(\hat{\sigma}^{+(1)}_{2,k}+\hat{\sigma}^{+(1)}_{3,k}),$
(68)
$\displaystyle\hat{\sigma}^{k}_{O3}=\frac{1}{2}(\hat{\sigma}^{+(1)}_{2,k}+\hat{\sigma}^{+(1)}_{3,k}),$
(69)
$\displaystyle\hat{\sigma}^{k}_{DO1}=\frac{1}{2}(\hat{\sigma}^{+(2)}_{1,k}+\hat{\sigma}^{+(4)}_{1,k}),$
(70)
$\displaystyle\hat{\sigma}^{k}_{DO2}=\frac{1}{2}(\hat{\sigma}^{+(2)}_{2,k}+\hat{\sigma}^{+(4)}_{3,k}),$
(71)
$\displaystyle\hat{\sigma}^{k}_{DO3}=\frac{1}{2}(\hat{\sigma}^{+(4)}_{2,k}+\hat{\sigma}^{+(2)}_{3,k}),$
(72)
$\displaystyle\hat{\sigma}^{k}_{DF1}=-\hat{\sigma}^{k}_{int}-2\hat{\sigma}^{k}_{DG}-4\hat{\sigma}^{k}_{DH},$
(73)

and others are the same as those appearing in (53). Although
$\hat{\sigma}^{k}_{DF}=0$ as shown in (57), $\hat{\sigma}^{k}_{DF1}$ term
appears in (3.3) due to the relations (2.2), (2.2) and (2.2).

To write down the partonic hard cross sections in (3.3), we further take into
account the following relations:

$\displaystyle\hat{\sigma}^{k}_{O2}=\hat{\sigma}^{k}_{O3},$ (74)
$\displaystyle\hat{\sigma}^{k}_{DF1}=-{1\over 2}\hat{\sigma}^{k}_{N1},$ (75)
$\displaystyle\hat{\sigma}^{k}_{DN1}=\hat{\sigma}^{k}_{DN2}=\hat{\sigma}^{k}_{DO1}=\hat{\sigma}^{k}_{DO2},$
(76) $\displaystyle\hat{\sigma}^{k}_{DN3}=-\hat{\sigma}^{k}_{DO3}.$ (77)

The relations (74) and (75) are obvious from (68), (69), (61) and (73), and
(76) and (77) are obtained by explicit calculation of the LO diagrams.

Then the independent hard cross sections are given as follows.

$\displaystyle\begin{dcases}\hat{\sigma}^{1}_{G}=C_{F}\frac{2Q^{2}}{q_{T}^{3}}\frac{(-1+\hat{z})^{2}(-1-\hat{x}^{2}+\hat{z}^{2}(1-6\hat{x}+6\hat{x}^{2}))}{\hat{x}\hat{z}^{3}},\\\\[7.0pt]
\hat{\sigma}^{2}_{G}=C_{F}\frac{8}{q_{T}}{\hat{x}(-1+\hat{z})},\\\\[7.0pt]
\hat{\sigma}^{3}_{G}=C_{F}\frac{2Q}{q_{T}^{2}}\frac{(-1+\hat{z})(\hat{z}-\hat{x}-2\hat{z}\hat{x}+\hat{z}^{2}(-2+4\hat{x}))}{\hat{z}^{2}},\\\\[7.0pt]
\hat{\sigma}^{4}_{G}=\frac{1}{2}\hat{\sigma}^{2}_{G},\\\\[7.0pt]
\hat{\sigma}^{8}_{G}=C_{F}\frac{2Q}{q_{T}^{2}}\frac{(-1+\hat{z})(-\hat{x}+\hat{z}(-1+2\hat{x}))}{\hat{z}^{2}},\\\\[7.0pt]
\hat{\sigma}^{9}_{G}=C_{F}\frac{4}{q_{T}}\frac{\hat{x}(-1+\hat{z})}{\hat{z}},\end{dcases}$
(78)
$\displaystyle\begin{dcases}\hat{\sigma}^{1}_{H}=-C_{F}\frac{4Q^{2}}{q_{T}^{3}}\frac{(-1+\hat{z})^{2}}{\hat{z}^{2}},\\\\[7.0pt]
\hat{\sigma}^{2}_{H}=0,\\\\[7.0pt]
\hat{\sigma}^{3}_{H}=-C_{F}\frac{2Q}{q_{T}^{2}}\frac{(-1+\hat{z})(-1+2\hat{z})}{\hat{z}^{2}},\\\\[7.0pt]
\hat{\sigma}^{4}_{H}=-C_{F}\frac{4}{q_{T}}\frac{(-1+\hat{z})}{\hat{z}},\\\\[7.0pt]
\hat{\sigma}^{8}_{H}=\hat{\sigma}^{3}_{H},\\\\[7.0pt]
\hat{\sigma}^{9}_{H}=\hat{\sigma}^{4}_{H},\end{dcases}$ (79)
$\displaystyle\begin{dcases}\hat{\sigma}^{1}_{N1}=C_{F}\frac{8Q^{2}}{q_{T}^{3}}\frac{(-1+\hat{z})^{2}(3\hat{z}(1-2\hat{x})\hat{x}+\hat{x}(1+\hat{x})+\hat{z}^{2}(1-6\hat{x}+6\hat{x}^{2}))}{\hat{x}\hat{z}^{3}},\\\\[7.0pt]
\hat{\sigma}^{2}_{N1}=C_{F}\frac{32}{q_{T}}\frac{\hat{x}(-1+\hat{z})^{2}}{\hat{z}},\\\\[7.0pt]
\hat{\sigma}^{3}_{N1}=C_{F}\frac{8Q}{q_{T}^{2}}\frac{(-1+\hat{z})^{2}(-1-2\hat{x}+\hat{z}(-2+4\hat{x}))}{\hat{z}^{2}},\\\\[7.0pt]
\hat{\sigma}^{4}_{N1}=-C_{F}\frac{8}{q_{T}}\frac{(-1+\hat{z})(1-2(-1+\hat{z})\hat{x})}{\hat{z}},\\\\[7.0pt]
\hat{\sigma}^{8}_{N1}=-C_{F}\frac{8Q}{q_{T}^{2}}\frac{(-1+\hat{z})^{2}}{\hat{z}^{2}},\\\\[7.0pt]
\hat{\sigma}^{9}_{N1}=-C_{F}\frac{8}{q_{T}}\frac{(-1+\hat{z})}{\hat{z}},\end{dcases}$
(80)
$\displaystyle\begin{dcases}\hat{\sigma}^{1}_{N2}=-C_{F}\frac{4Q^{2}}{q_{T}^{3}}\frac{(-1+\hat{z})^{2}(-1+(-1+3\hat{z})\hat{x})}{\hat{z}^{3}},\\\\[7.0pt]
\hat{\sigma}^{2}_{N2}=-C_{F}\frac{8}{q_{T}}\frac{\hat{x}(-1+\hat{z})}{\hat{z}},\\\\[7.0pt]
\hat{\sigma}^{3}_{N2}=-C_{F}\frac{2Q}{q_{T}^{2}}\frac{(-1+\hat{z})(-3(1+\hat{x})+\hat{z}(3+4\hat{x}))}{\hat{z}^{2}},\\\\[7.0pt]
\hat{\sigma}^{4}_{N2}=-C_{F}\frac{4}{q_{T}}\frac{(-1+\hat{z})(2+\hat{x})}{\hat{z}},\\\\[7.0pt]
\hat{\sigma}^{8}_{N2}=-C_{F}\frac{2Q}{q_{T}^{2}}\frac{(-1+\hat{z})(-3-\hat{x}+\hat{z}(3+2\hat{x}))}{\hat{z}^{2}},\\\\[7.0pt]
\hat{\sigma}^{9}_{N2}=\hat{\sigma}^{4}_{N2},\end{dcases}$ (81)
$\displaystyle\begin{dcases}\hat{\sigma}^{1}_{N3}=-C_{F}\frac{4Q^{2}}{q_{T}^{3}}\frac{(-1+\hat{z})^{2}(3\hat{z}(2-3\hat{x})\hat{x}+\hat{x}(1+\hat{x})+2\hat{z}^{2}(1-6\hat{x}+6\hat{x}^{2}))}{\hat{x}\hat{z}^{3}},\\\\[7.0pt]
\hat{\sigma}^{2}_{N3}=-C_{F}\frac{8}{q_{T}}\frac{\hat{x}(-1+\hat{z})(-3+4\hat{z})}{\hat{z}},\\\\[7.0pt]
\hat{\sigma}^{3}_{N3}=-C_{F}\frac{2Q}{q_{T}^{2}}\frac{(-1+\hat{z})(1+\hat{z}(7-20\hat{x})+5\hat{x}+8\hat{z}^{2}(-1+2\hat{x}))}{\hat{z}^{2}},\\\\[7.0pt]
\hat{\sigma}^{4}_{N3}=\frac{1}{2}\hat{\sigma}^{2}_{N3},\\\\[7.0pt]
\hat{\sigma}^{8}_{N3}=-C_{F}\frac{2Q}{q_{T}^{2}}\frac{(-1+\hat{z})(1-\hat{x}+\hat{z}(-1+2\hat{x}))}{\hat{z}^{2}},\\\\[7.0pt]
\hat{\sigma}^{9}_{N3}=-C_{F}\frac{4}{q_{T}}\frac{\hat{x}(-1+\hat{z})}{\hat{z}},\end{dcases}$
(82)
$\displaystyle\begin{dcases}\hat{\sigma}^{1}_{DN1}=C_{F}\frac{2Q^{2}}{q_{T}^{3}}\frac{(-1+\hat{z})^{2}(2\hat{z}(1-3\hat{x})\hat{x}+(1+\hat{x})^{2}+\hat{z}^{2}(1-6\hat{x}+6\hat{x}^{2}))}{\hat{x}\hat{z}^{3}},\\\\[7.0pt]
\hat{\sigma}^{2}_{DN1}=\frac{1}{4}\hat{\sigma}^{2}_{N1},\\\\[7.0pt]
\hat{\sigma}^{3}_{DN1}=C_{F}\frac{4Q}{q_{T}^{2}}\frac{(-1+\hat{z})^{2}(-1-\hat{x}+\hat{z}(-1+2\hat{x}))}{\hat{z}^{2}},\\\\[7.0pt]
\hat{\sigma}^{4}_{DN1}=-C_{F}\frac{4}{q_{T}}\frac{(-1+\hat{z})(1+\hat{x}-\hat{x}\hat{z})}{\hat{z}},\\\\[7.0pt]
\hat{\sigma}^{8}_{DN1}=\frac{1}{2}\hat{\sigma}^{8}_{N1},\\\\[7.0pt]
\hat{\sigma}^{9}_{DN1}=\frac{1}{2}\hat{\sigma}^{9}_{N1},\end{dcases}$ (83)
$\displaystyle\begin{dcases}\hat{\sigma}^{1}_{DN3}=-C_{F}\frac{4Q^{2}}{q_{T}^{3}}\frac{(-1+\hat{z})^{2}(1+2\hat{z}(2-3\hat{x})\hat{x}+\hat{x}^{2}+\hat{z}^{2}(1-6\hat{x}+6\hat{x}^{2}))}{\hat{x}\hat{z}^{3}},\\\\[7.0pt]
\hat{\sigma}^{2}_{DN3}=-2\hat{\sigma}^{2}_{DN1},\\\\[7.0pt]
\hat{\sigma}^{3}_{DN3}=-C_{F}\frac{8Q}{q_{T}^{2}}\frac{(-1+\hat{z})^{2}(-\hat{x}+\hat{z}(-1+2\hat{x}))}{\hat{z}^{2}},\\\\[7.0pt]
\hat{\sigma}^{4}_{DN3}=-\hat{\sigma}^{2}_{DN1},\\\\[7.0pt]
\hat{\sigma}^{8}_{DN3}=0,\\\\[7.0pt] \hat{\sigma}^{9}_{DN3}=0,\end{dcases}$
(84)
$\displaystyle\begin{dcases}\hat{\sigma}^{1}_{O1}=-C_{F}\frac{8Q^{2}}{q_{T}^{3}}\frac{(-1+\hat{z})^{2}(-1-\hat{x}+\hat{z}(-2+3\hat{x}))}{\hat{z}^{3}},\\\\[7.0pt]
\hat{\sigma}^{2}_{O1}=-C_{F}\frac{16}{q_{T}}\frac{\hat{x}(-1+\hat{z})}{\hat{z}},\\\\[7.0pt]
\hat{\sigma}^{3}_{O1}=-C_{F}\frac{4Q}{q_{T}^{2}}\frac{(-1+\hat{z})(-1-3\hat{x}+\hat{z}(-1+4\hat{x}))}{\hat{z}^{2}},\\\\[7.0pt]
\hat{\sigma}^{4}_{O1}=\frac{1}{2}\hat{\sigma}^{2}_{O1},\\\\[7.0pt]
\hat{\sigma}^{8}_{O1}=-C_{F}\frac{4Q}{q_{T}^{2}}\frac{(-1+\hat{z})(-1-\hat{x}+\hat{z}(-1+2\hat{x}))}{\hat{z}^{2}},\\\\[7.0pt]
\hat{\sigma}^{9}_{O1}=\frac{1}{2}\hat{\sigma}^{2}_{O1},\end{dcases}$ (85)
$\displaystyle\begin{dcases}\hat{\sigma}^{1}_{O2}=C_{F}\frac{4Q^{2}}{q_{T}^{3}}\frac{(-1+\hat{z})^{2}(1+\hat{x}+3\hat{z}(2-3\hat{x})\hat{x}+2\hat{x}^{2}+\hat{z}^{2}(1-6\hat{x}+6\hat{x}^{2}))}{\hat{x}\hat{z}^{3}},\\\\[7.0pt]
\hat{\sigma}^{2}_{O2}=-C_{F}\frac{8}{q_{T}}\frac{\hat{x}(-1+\hat{z})(3-2\hat{z})}{\hat{z}},\\\\[7.0pt]
\hat{\sigma}^{3}_{O2}=C_{F}\frac{2Q}{q_{T}^{2}}\frac{(-1+\hat{z})(1+\hat{z}(5-16\hat{x})+7\hat{x}+\hat{z}^{2}(-4+8\hat{x}))}{\hat{z}^{2}},\\\\[7.0pt]
\hat{\sigma}^{4}_{O2}=\frac{1}{2}\hat{\sigma}^{2}_{O2},\\\\[7.0pt]
\hat{\sigma}^{8}_{O2}=\frac{1}{2}\hat{\sigma}^{8}_{O1},\\\\[7.0pt]
\hat{\sigma}^{9}_{O2}=\frac{1}{2}\hat{\sigma}^{9}_{O1},\end{dcases}$ (86)
$\displaystyle\begin{dcases}\hat{\sigma}^{1}_{DF2}=\frac{C_{F}}{N}\frac{Q^{2}}{q_{T}^{3}}\frac{(-1+\hat{z})(1+\hat{x}-\hat{z}\hat{x})}{\hat{x}\hat{z}^{2}},\\\\[7.0pt]
\hat{\sigma}^{2}_{DF2}=0,\\\\[7.0pt]
\hat{\sigma}^{3}_{DF2}=-\frac{C_{F}}{N}\frac{Q}{q_{T}^{2}}\frac{(-1+\hat{z})}{\hat{z}},\\\\[7.0pt]
\hat{\sigma}^{4}_{DF2}=-\frac{C_{F}}{N}\frac{1}{q_{T}},\\\\[7.0pt]
\hat{\sigma}^{8}_{DF2}=\hat{\sigma}^{3}_{DF2},\\\\[7.0pt]
\hat{\sigma}^{9}_{DF2}=\hat{\sigma}^{4}_{DF2},\end{dcases}$ (87)
$\displaystyle\begin{dcases}\hat{\sigma}^{1}_{DF3}=-\frac{C_{F}}{N}\frac{1}{q_{T}}\frac{3\hat{z}(1-2\hat{x})\hat{x}+\hat{x}(1+\hat{x})+\hat{z}^{2}(1-6\hat{x}+6\hat{x}^{2})}{\hat{z}^{2}},\\\\[7.0pt]
\hat{\sigma}^{2}_{DF3}=-\frac{C_{F}}{N}\frac{4q_{T}}{Q^{2}}\hat{x}^{2},\\\\[7.0pt]
\hat{\sigma}^{3}_{DF3}=-\frac{C_{F}}{N}\frac{1}{Q}\frac{\hat{x}(-1-2\hat{x}+\hat{z}(-2+4\hat{x}))}{\hat{z}},\\\\[7.0pt]
\hat{\sigma}^{4}_{DF3}=-\frac{C_{F}}{N}\frac{1}{q_{T}}\frac{(-1+\hat{x})(-1+2(-1+\hat{z})\hat{x})}{\hat{z}},\\\\[7.0pt]
\hat{\sigma}^{8}_{DF3}=\frac{C_{F}}{N}\frac{1}{Q}\frac{\hat{x}}{\hat{z}},\\\\[7.0pt]
\hat{\sigma}^{9}_{DF3}=\frac{C_{F}}{N}\frac{1}{q_{T}}\frac{-1+\hat{x}}{\hat{z}},\end{dcases}$
(88)
$\displaystyle\begin{dcases}\hat{\sigma}^{1}_{DF4}=\frac{C_{F}}{N}\frac{Q^{2}}{q_{T}^{3}}\frac{(-1+\hat{z})(-1+\hat{z}+5\hat{x}-6\hat{z}\hat{x}-6\hat{x}^{2}+6\hat{z}\hat{x}^{2})}{\hat{x}^{2}}\\\
\ \
+\frac{C_{F}}{q_{T}}\frac{(\hat{x}-1)^{2}(1+\hat{x})-\hat{z}(\hat{x}-1)^{2}(1+6\hat{x})-\hat{z}^{3}(1-6\hat{x}+6\hat{x}^{2})+\hat{z}^{2}(1+\hat{x}-6\hat{x}^{2}+6\hat{x}^{3})}{\hat{z}\hat{x}(-1+\hat{z}+\hat{x})},\\\\[7.0pt]
\hat{\sigma}^{2}_{DF4}=\frac{C_{F}}{N}\frac{4}{q_{T}}{\hat{z}(-1+\hat{z})}-C_{F}\frac{4q_{T}}{Q^{2}}\frac{\hat{z}(1+\hat{z}-\hat{x})\hat{x}}{(-1+\hat{z}+\hat{x})},\\\\[7.0pt]
\hat{\sigma}^{3}_{DF4}=\frac{C_{F}}{N}\frac{2Q}{q_{T}^{2}}\frac{(-1+\hat{z})(-\hat{x}+\hat{z}(-1+2\hat{x}))}{\hat{x}}+\frac{2C_{F}}{Q}\frac{(\hat{z}+\hat{z}^{2}+\hat{x}-2\hat{z}\hat{x}-2\hat{z}^{2}\hat{x}-\hat{x}^{2}+2\hat{z}\hat{x}^{2})}{(-1+\hat{z}+\hat{x})},\
\\\\[7.0pt]
\hat{\sigma}^{4}_{DF4}=\frac{C_{F}}{N}\frac{1}{q_{T}}\frac{(1+\hat{x}+2\hat{z}^{2}\hat{x}-\hat{z}(1+2\hat{x}))}{\hat{x}}-C_{F}\frac{2}{q_{T}}\frac{(-1+\hat{x})(1+\hat{z}^{2}\hat{x}+\hat{x}^{2}-\hat{z}(1+\hat{x}^{2}))}{\hat{x}(-1+\hat{z}+\hat{x})},\\\\[7.0pt]
\hat{\sigma}^{8}_{DF4}=\frac{C_{F}}{N}\frac{2}{Q}\hat{z}-C_{F}\frac{4}{Q}\frac{\hat{z}(-1+\hat{x})}{(-1+\hat{z}+\hat{x})},\\\\[7.0pt]
\hat{\sigma}^{9}_{DF4}=\frac{C_{F}}{N}\frac{1}{q_{T}}\frac{(1-\hat{x}+\hat{z}(-1+2\hat{x}))}{\hat{x}}-C_{F}\frac{2}{q_{T}}\frac{(-1+\hat{x})(1-\hat{x}+\hat{z}(-1+2\hat{x}))}{\hat{x}(-1+\hat{z}+\hat{x})},\end{dcases}$
(89)
$\displaystyle\begin{dcases}\hat{\sigma}^{1}_{DF5}=\frac{C_{F}}{N}\frac{1}{q_{T}}\frac{(-1+\hat{x})^{2}(1+\hat{x}+6\hat{z}^{2}\hat{x}-\hat{z}(1+6\hat{x}))}{\hat{z}^{2}\hat{x}}\\\
\quad-\frac{C_{F}}{q_{T}}\frac{(\hat{x}-1)^{2}(1+\hat{x})-\hat{z}(\hat{x}-1)^{2}(1+6\hat{x})-\hat{z}^{3}(1-6\hat{x}+6\hat{x}^{2})+\hat{z}^{2}(1+\hat{x}-6\hat{x}^{2}+6\hat{x}^{3})}{\hat{z}\hat{x}(-1+\hat{z}+\hat{x})},\\\\[7.0pt]
\hat{\sigma}^{2}_{DF5}=\frac{C_{F}}{N}\frac{4q_{T}}{Q^{2}}{(-1+\hat{x})\hat{x}}+C_{F}\frac{4q_{T}}{Q^{2}}\frac{\hat{z}(1+\hat{z}-\hat{x})\hat{x}}{(-1+\hat{z}+\hat{x})},\\\\[7.0pt]
\hat{\sigma}^{3}_{DF5}=\frac{C_{F}}{N}\frac{2}{Q}\frac{(-1+\hat{x})(-\hat{x}+\hat{z}(-1+2\hat{x}))}{\hat{z}}-\frac{2C_{F}}{Q}\frac{(\hat{z}+\hat{z}^{2}+\hat{x}-2\hat{z}\hat{x}-2\hat{z}^{2}\hat{x}-\hat{x}^{2}+2\hat{z}\hat{x}^{2})}{(-1+\hat{z}+\hat{x})},\\\\[7.0pt]
\hat{\sigma}^{4}_{DF5}=\frac{C_{F}}{N}\frac{1}{q_{T}}\frac{(-1+\hat{x})(-1+\hat{z}+\hat{x}-2\hat{z}\hat{x}-2\hat{x}^{2}+2\hat{z}\hat{x}^{2})}{\hat{x}\hat{z}}\\\
\qquad\quad+C_{F}\frac{2}{q_{T}}\frac{(-1+\hat{x})(1+\hat{z}^{2}\hat{x}+\hat{x}^{2}-\hat{z}(1+\hat{x}^{2}))}{\hat{x}(-1+\hat{z}+\hat{x})},\\\\[7.0pt]
\hat{\sigma}^{8}_{DF5}=-\frac{C_{F}}{N}\frac{2}{Q}{(-1+\hat{x})}+C_{F}\frac{4}{Q}\frac{\hat{z}(-1+\hat{x})}{(-1+\hat{z}+\hat{x})},\\\\[7.0pt]
\hat{\sigma}^{9}_{DF5}=-\frac{C_{F}}{N}\frac{1}{q_{T}}\frac{(-1+\hat{x})(1-\hat{x}+\hat{z}(-1+2\hat{x}))}{\hat{x}\hat{z}}+C_{F}\frac{2}{q_{T}}\frac{(-1+\hat{x})(1-\hat{x}+\hat{z}(-1+2\hat{x}))}{\hat{x}(-1+\hat{z}+\hat{x})},\end{dcases}$
(90)
$\displaystyle\begin{dcases}\hat{\sigma}^{1}_{GF2}=-\frac{C_{F}}{N},\frac{Q^{2}}{q_{T}^{3}}\frac{(-1+\hat{z})(1-\hat{x}+\hat{z}\hat{x})}{\hat{x}\hat{z}^{2}},\\\\[7.0pt]
\hat{\sigma}^{2}_{GF2}=0,\\\\[7.0pt]
\hat{\sigma}^{3}_{GF2}=\hat{\sigma}^{3}_{DF2},\\\\[7.0pt]
\hat{\sigma}^{4}_{GF2}=\hat{\sigma}^{4}_{DF2},\\\\[7.0pt]
\hat{\sigma}^{8}_{GF2}=\hat{\sigma}^{8}_{DF2},\\\\[7.0pt]
\hat{\sigma}^{9}_{GF2}=\hat{\sigma}^{9}_{DF2},\end{dcases}$ (91)
$\displaystyle\begin{dcases}\hat{\sigma}^{1}_{GF3}=-\frac{C_{F}}{N}\frac{1}{q_{T}}\frac{\hat{z}(5-6\hat{x})\hat{x}+\hat{x}(-1+\hat{x})+\hat{z}^{2}(1-6\hat{x}+6\hat{x}^{2})}{\hat{z}^{2}},\\\\[7.0pt]
\hat{\sigma}^{2}_{GF3}=\hat{\sigma}^{2}_{DF3},\\\\[7.0pt]
\hat{\sigma}^{3}_{GF3}=-\frac{C_{F}}{N}\frac{1}{Q}\frac{\hat{x}(-1+2\hat{z})(-1+2\hat{x})}{\hat{z}},\\\\[7.0pt]
\hat{\sigma}^{4}_{GF3}=-\frac{C_{F}}{N}\frac{1}{q_{T}}\frac{(-1+\hat{x})(1+2(-1+\hat{z})\hat{x})}{\hat{z}},\\\\[7.0pt]
\hat{\sigma}^{8}_{GF3}=-\hat{\sigma}^{8}_{DF3},\\\\[7.0pt]
\hat{\sigma}^{9}_{GF3}=-\hat{\sigma}^{9}_{DF3},\end{dcases}$ (92)
$\displaystyle\begin{dcases}\hat{\sigma}^{1}_{GF4}=\frac{C_{F}}{N}\frac{Q^{2}}{q_{T}^{3}}\frac{(-1+\hat{z})(-1+\hat{z}+7\hat{x}-6\hat{z}\hat{x}-6\hat{x}^{2}+6\hat{z}\hat{x}^{2})}{\hat{x}^{2}}\\\
\qquad\qquad-
C_{F}\frac{1}{q_{T}}\frac{(-1+\hat{x})^{2}-6\hat{z}(-1+\hat{x})\hat{x}+\hat{z}^{2}(1-6\hat{x}+6\hat{x}^{2})}{\hat{z}\hat{x}},\\\\[7.0pt]
\hat{\sigma}^{2}_{GF4}=\frac{C_{F}}{N}\frac{4}{q_{T}}{\hat{z}(-1+\hat{z})}-C_{F}\frac{4q_{T}}{Q^{2}}{\hat{z}\hat{x}},\\\\[7.0pt]
\hat{\sigma}^{3}_{GF4}=\frac{C_{F}}{N}\frac{2Q}{q_{T}^{2}}\frac{(-1+\hat{z})(1-\hat{x}+\hat{z}(-1+2\hat{x}))}{\hat{x}}-C_{F}\frac{2}{Q}{(1-\hat{x}+\hat{z}(-1+2\hat{x}))},\\\\[7.0pt]
\hat{\sigma}^{4}_{GF4}=\frac{C_{F}}{N}\frac{1}{q_{T}}\frac{(-1+\hat{z}+\hat{x}-2\hat{z}\hat{x}+2\hat{z}^{2}\hat{x})}{\hat{x}}-C_{F}\frac{2}{q_{T}}\frac{(-1+\hat{x})(1+(-1+\hat{z})\hat{x})}{\hat{x}},\\\\[7.0pt]
\hat{\sigma}^{8}_{GF4}=\frac{C_{F}}{N}\frac{2Q}{q_{T}^{2}}{(-1+\hat{z})}-C_{F}\frac{2}{Q},\\\\[7.0pt]
\hat{\sigma}^{9}_{GF4}=\frac{C_{F}}{N}\frac{1}{q_{T}}\frac{(-1+\hat{z}-\hat{x}+2\hat{z}\hat{x})}{\hat{x}}-C_{F}\frac{2}{q_{T}}\frac{(-1+\hat{x})}{\hat{x}},\end{dcases}$
(93)
$\displaystyle\begin{dcases}\hat{\sigma}^{1}_{GF5}=\frac{C_{F}}{N}\frac{1}{q_{T}}\frac{(-1+\hat{x})^{2}(-1+\hat{z}+\hat{x}-6\hat{z}\hat{x}+6\hat{z}^{2}\hat{x})}{\hat{z}^{2}\hat{x}}\\\
\qquad\qquad-
C_{F}\frac{1}{q_{T}}\frac{(-1+\hat{x})^{2}-6\hat{z}(-1+\hat{x})\hat{x}+\hat{z}^{2}(1-6\hat{x}+6\hat{x}^{2})}{\hat{z}\hat{x}},\\\\[7.0pt]
\hat{\sigma}^{2}_{GF5}=\frac{C_{F}}{N}\frac{4q_{T}}{Q^{2}}{(-1+\hat{x})\hat{x}}-C_{F}\frac{4q_{T}}{Q^{2}}{\hat{z}\hat{x}},\\\\[7.0pt]
\hat{\sigma}^{3}_{GF5}=\frac{C_{F}}{N}\frac{2}{Q}\frac{(-1+\hat{x})(1-\hat{x}+\hat{z}(-1+2\hat{x}))}{\hat{z}}-C_{F}\frac{2}{Q}{(1-\hat{x}+\hat{z}(-1+2\hat{x}))},\\\\[7.0pt]
\hat{\sigma}^{4}_{GF5}=\frac{C_{F}}{N}\frac{1}{q_{T}}\frac{(-1+\hat{x})(-1+\hat{z}+3\hat{x}-2\hat{z}\hat{x}-2\hat{x}^{2}+2\hat{z}\hat{x}^{2})}{\hat{x}\hat{z}}\\\
\qquad\qquad-
C_{F}\frac{2}{q_{T}}\frac{(-1+\hat{x})(1+(-1+\hat{z})\hat{x})}{\hat{x}},\\\\[7.0pt]
\hat{\sigma}^{8}_{GF5}=-\frac{C_{F}}{N}\frac{2}{Q}\frac{(-1+\hat{z})(-1+\hat{x})}{\hat{z}}-C_{F}\frac{2}{Q},\\\\[7.0pt]
\hat{\sigma}^{9}_{GF5}=-\frac{C_{F}}{N}\frac{1}{q_{T}}\frac{(-1+\hat{x})(1-3\hat{x}+\hat{z}(-1+2\hat{x}))}{\hat{x}\hat{z}}-C_{F}\frac{2}{q_{T}}\frac{(-1+\hat{x})}{\hat{x}}.\end{dcases}$
(94)

Eqs. (78)-(94) and the relations (74), (75), (76), (77) specify all the
partonic cross sections in the final formula (3.3).

##  4 Summary

In this paper we have studied the transversely polarized spin-1/2 hyperon
production in SIDIS, $ep\to e\Lambda^{\uparrow}X$. Specifically, we have
derived the LO twist-3 gluon FF contribution to the polarized cross section.
Since the twist-3 gluon FFs are related to the $q\bar{q}g$-FFs through the EOM
relations and the LIRs, we consistently took into account the latter
contribution together. This has completed the twist-3 LO cross section for
this process together with the results for the contribution from the twist-3
DF and the twist-3 quark FFs derived in [1]. The final result for the cross
section is given in (3.3). It consists of five components with different
azimuthal structures as

$\displaystyle\frac{d^{6}\sigma}{dx_{bj}dQ^{2}dz_{f}dq_{T}^{2}d\phi d\chi}=$
$\displaystyle\mathcal{F}_{1}\sin\Phi_{S}+\mathcal{F}_{2}\sin\Phi_{S}\cos\varphi+\mathcal{F}_{3}\sin\Phi_{S}\cos
2\varphi{}$
$\displaystyle+\mathcal{F}_{4}\cos\Phi_{S}\sin\varphi+\mathcal{F}_{5}\cos\Phi_{S}\sin
2\varphi,$ (95)

where $\varphi=\phi-\chi$ is the relative azimuthal angle between the lepton
($\phi$) and the hadron ($\chi$) planes and $\Phi_{S}$ is the azimuthal angle
of the transverse spin vector of $\Lambda^{\uparrow}$ measured from the hadron
plane with the structure functions $\mathcal{F}_{1,2,3,4,5}$ written as
convolution of the twist-3 FFs and the quark DF in the proton and the partonic
hard cross sections. The LO cross section given in [1] and the present study
contains several unknown nonperturbative functions, and their determination
requires global anayses of many twist-3 processes in which the same twist-3
functions appear. Information from analyses of small-$P_{T}$ data in terms of
the TMD factorization is also of great help to constrain some of the twist-3
functions. In any case our twist-3 cross section formula is the starting point
of analyzing the large-$P_{T}$ hyperon polarization in SIDIS which we hope to
be measured in the future EIC experiment.

## Acknowledgments

This work has been supported by the establishment of Niigata university
fellowships towards the creation of science technology innovation (R.I.), the
Grant-in-Aid for Scientific Research from the Japanese Society of Promotion of
Science under Contract Nos. 19K03843 (Y.K.) and 18J11148 (K.Y.), National
Natural Science Foundation in China under grant No. 11950410495, Guangdong
Natural Science Foundation under No. 2020A1515010794 and research startup
funding at South China Normal University (S.Y.).

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