Twist-3 Gluon Fragmentation Contribution to Polarized
Hyperon Production in Unpolarized Proton-Proton Collision

Yuji Koike1, Kenta Yabe2 and Shinsuke Yoshida3,4

1 Department of Physics, Niigata University, Ikarashi, Niigata 950-2181, Japan

2 Graduate School of Science and Technology, Niigata University, Ikarashi,
Niigata 950-2181, Japan

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

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

Abstract

Understanding the origin and mechanism of the transverse polarization of
hyperons produced in unpolarized proton-proton collision,
$pp\to\Lambda^{\uparrow}X$, has been one of the long-standing issues in high-
energy spin physics. In the framework of the collinear factorization
applicable to large-$p_{T}$ hadron productions, this phenomenon is a twist-3
observable which is caused by multi-parton correlations either in the initial
protons or in the process of fragmentation into the hyperon. We derive the
twist-3 gluon fragmentation function (FF) contribution to this process in the
leading order (LO) with respect to the QCD coupling constant. Combined with
the known results for the contribution from the twist-3 distribution function
and the twist-3 quark FF, this completes the LO twist-3 cross section. We also
found that the model independent relations among the twist-3 gluon FFs based
on the QCD equation of motion and the Lorentz invariance property of the
correlation functions guarantee the color gauge invariance and the frame-
independence of the cross section.

##  1 Introduction

It has been known that the hyperons produced in unpolarized proton-proton
collisions are polarized perpendicularly to the scattering plane,
$pp\to\Lambda^{\uparrow}X$ 111Here $\Lambda$ collectively denotes spin-1/2
hyperons such as $\Lambda$ $\Sigma$, $\Xi$, etc. [1, 2, 3, 4, 5, 6, 7, 8, 9,
10, 11]. The observed polarizations show a tendency that they become larger in
the forward rapidity region, where the asymmetry is as large as 30%. Hyperon
polarization was also observed in other reactions such as $\gamma
p\to\Lambda^{\uparrow}X$ [12, 13], quasi-real photo-production of $\Lambda$’s
in lepton scattering [14, 15] and in electron-positron collisions,
$e^{+}e^{-}\to\Lambda^{\uparrow}X$ [16, 17]. These transverse polarizations in
unpolarized collisions are the examples of the transverse single spin
asymmetries (SSAs), where only one particle participating in the scattering
process is polarized. Another well known SSA is the asymmetry with regard to
the initial (transverse) spin such as $p^{\uparrow}p\to hX$
($h=\pi,\,K,\eta,\,{\rm jet}$, etc.) [18, 19, 20, 21, 22, 23] and
$ep^{\uparrow}\to ehX$ [24, 25]. For the last several decades, lots of efforts
have been made to understand the origin and mechanism for these large SSAs,
since perturbative QCD at twist-2 level gives almost zero SSAs [26].

For a high-energy collision in which particles with large transverse momentum
are produced, the cross section can be computed in the framework of the
collinear factorization of perturbative QCD. In this framework, SSAs appear as
twist-3 observables to which nonperturbative multi-parton correlation
functions contribute instead of collinear twist-2 parton distribution
functions (PDFs) and/or fragmentation functions (FFs). Through the studies of
SSAs, the technique of calculating the twist-3 cross section has made much
progress and has been applied to many relevant processes in the leading order
(LO) with respect to the QCD coupling. For example, the complete LO cross
section for $p^{\uparrow}p\to hX$ ($h=\pi,\,D,\,\gamma,\,\gamma^{*}$) has been
derived [27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40], and the RHIC
data has been analyzed and interpreted, which suggests the main source of the
asymmetry is the twist-3 fragmentation contribution [41, 42].

In this paper we study $pp\to\Lambda^{\uparrow}X$ in the collinear twist-3
factorization. Two kinds of twist-3 cross section contribute to this process:
(i) Twist-3 unpolarized PDF in one of the initial proton convoluted with the
twist-2 “transversity” FF for the final hyperon and the twist-2 unpolarized
PDF in another proton, and (ii) Twist-3 FFs for the polarized hyperon
convoluted with the twist-2 unpolarized PDFs in the initial protons. The
complete LO cross section for (i) was derived in [43, 44, 45]. The second one
(ii) can be further classified into two, depending on whether the twist-3 FF
is of (ii-a) quark-gluon correlation type or of (ii-b) gluon correlation type.
The complete LO cross section for (ii-a) was derived in [46], while (ii-b) has
not been studied so far. In this paper we focus on the (ii-b) contribution,
and derive the corresponding cross section, which completes the LO twist-3
cross section for this process. (A short version of the present work was
presented in [47, 48].) We develop a formalism for deriving the gauge and
frame independent contribution to this twist-3 cross section from the purely
gluonic FFs, and present the result for $pp\to\Lambda^{\uparrow}X$.

The remainder of this paper is organized as follows. In section 2, we
introduce the complete set of the gluonic FFs for spin-1/2 hadron up to
twist-3 defined from correlators of two- and three- gluon field strengths,
which are necessary to derive the twist-3 cross section. We also recall from
[49] the exact relations among those FFs based on the QCD equation of motion
and the Lorentz invariance, which play a crucial role to guarantee the gauge
and Lorentz invariance of the cross section222Importance of these relations
for the frame independence of the twist-3 cross sections have been realized
for the twist-3 quark distribution functions and FFs[46, 50, 51, 52].. In
section 3, we develop a formalism to derive the twist-3 gluon FF contribution
to the twist-3 cross section, and present the corresponding LO cross section
for $pp\to\Lambda^{\uparrow}X$. We will discuss how the Lorentz invariance of
the twist-3 cross section is realized, using the relations introduced in
section 2. Gauge invariance of the cross section is discussed in Appendix B.
Section 4 is devoted to a brief summary. In other Appendices, we discuss some
technical aspects of the actual calculations.

##  2 Gluon Fragmentation Functions

###  2.1 Three types of twist-3 gluon fragmentation functions

In this section we introduce twist-3 gluon FFs for a spin-1/2 baryon relevant
to $pp\to\Lambda^{\uparrow}X$ [53, 54, 49] and summarize their basic
properties derived in [49]. They are classified into three types; intrinsic,
kinematical and dynamical FFs. The intrinsic twist-3 gluon FFs are defined as
the Fourier transform of the lightcone correlator of the gluon’s field
strength $F^{\mu\nu}_{a}$:

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

where $N=3$ is the number of colors, $|h(P_{h},S_{h})\rangle$ is the baryon
state with the four momentum $P_{h}$ ($P_{h}^{2}=M_{h}^{2}$) and the spin
vector $S_{h}$ ($S_{h}^{2}=-M_{h}^{2}$), and $[\lambda w,\infty w]$ is the
gauge link in the adjoint representation connecting $\lambda w$ and $\infty
w$. For the transversely polarized baryon, we use the spin vector $S_{\perp}$
normalized as $S_{\perp}^{2}=-1$. In the twist-3 accuracy $P_{h}$ can be
regarded as lightlike. For a baryon with large momentum,
$P_{h}\simeq(|\vec{P}_{h}|,\vec{P}_{h})$, another lightlike vector $w$ is
defined as $w=1/(2|\vec{P}_{h}|^{2})(|\vec{P}_{h}|,-\vec{P}_{h})$ which
satisfies $P_{h}\cdot w=1$. In (1), we use the notation $F^{w\beta}\equiv
F^{\mu\beta}w_{\mu}$, and $\\{\ \ \\}$ ($[\ ]$) implies the symmetrization
(anti-symmetrization) of Lorentz indices, i.e. for arbitrary four vectors
$a^{\alpha}$ and $b^{\beta}$, $a^{\\{\alpha}b^{\beta\\}}\equiv
a^{\alpha}b^{\beta}+a^{\beta}b^{\alpha}$ and $a^{[\alpha}b^{\beta]}\equiv
a^{\alpha}b^{\beta}-a^{\beta}b^{\alpha}$. $\widehat{G}(z)$ and
$\Delta\widehat{G}(x)$ are twist-2, and $\Delta\widehat{G}_{3T}(z)$ and
$\Delta\widehat{G}_{3\bar{T}}(z)$ are twist-3. We also note
$\Delta\widehat{G}_{3\bar{T}}(z)$ is naively $T$-odd, contributing to SSAs.
Each function in (1) has a support on $0<z<1$.

The kinematical FFs contain the transverse derivative of the correlation
functions of the field strengths:

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

where each function is defined to be real. The kinematical FFs are related to
the $k_{T}^{2}/M_{h}^{2}$-moment of the transverse-momentum-dependent (TMD)
FFs [53]. Each function in (2) has a support on $0<z<1$.

To define the dynamical FFs, we introduce the lightcone correlation functions
of three field strengths:

$\displaystyle\hat{\Gamma}_{F,abc}^{\alpha\beta\gamma}(\frac{1}{z_{1}},\frac{1}{z_{2}})$
$\displaystyle=\frac{1}{N^{2}-1}\sum_{X}\\!\int\\!\\!\frac{d\lambda}{2\pi}\\!\int\\!\\!\frac{d\mu}{2\pi}e^{-i{\lambda\over
z_{1}}}e^{-i\mu({1\over z_{2}}-{1\over z_{1}})}\langle
0|F^{w\beta}_{b}(0)|h(P_{h},S_{\perp})X\rangle\langle
h(P_{h},S_{\perp})X|F^{w\alpha}_{a}(\lambda w)gF^{w\gamma}_{c}(\mu
w)|0\rangle,$

where the gauge link operators are suppressed for simplicity. The color
indices of this correlator can be expanded in terms of the anti-symmetric and
symmetric structure constants of color SU(N), $-if^{abc}$ and $d^{abc}$, as

$\displaystyle\hat{\Gamma}_{F,abc}^{\alpha\beta\gamma}(\frac{1}{z_{1}},\frac{1}{z_{2}})=-\frac{if^{abc}}{N}\hat{\Gamma}_{FA}^{\alpha\beta\gamma}(\frac{1}{z_{1}},\frac{1}{z_{2}})+d^{abc}\frac{N}{N^{2}-4}\hat{\Gamma}_{FS}^{\alpha\beta\gamma}(\frac{1}{z_{1}},\frac{1}{z_{2}}).$
(4)

The dynamical FFs can be defined from
$\hat{\Gamma}_{FA}^{\alpha\beta\gamma}\left(\frac{1}{z^{\prime}},\frac{1}{z}\right)$
and
$\hat{\Gamma}_{FS}^{\alpha\beta\gamma}\left(\frac{1}{z^{\prime}},\frac{1}{z}\right)$:

$\displaystyle\hat{\Gamma}_{FA}^{\alpha\beta\gamma}(\frac{1}{z_{1}},\frac{1}{z_{2}})$
$\displaystyle=\frac{-if_{abc}}{N^{2}-1}\sum_{X}\\!\int\\!\\!\frac{d\lambda}{2\pi}\\!\int\\!\\!\frac{d\mu}{2\pi}e^{-i{\lambda\over
z_{1}}}e^{-i\mu({1\over z_{2}}-{1\over z_{1}})}\langle
0|F^{w\beta}_{b}(0)|h(P_{h},S_{\perp})X\rangle\langle
h(P_{h},S_{\perp})X|F^{w\alpha}_{a}(\lambda w)gF^{w\gamma}_{c}(\mu
w)|0\rangle$
$\displaystyle=-{M_{h}}\biggl{(}\widehat{N}_{1}\left(\frac{1}{z_{1}},\frac{1}{z_{2}}\right)g^{\alpha\gamma}_{\perp}\epsilon^{P_{h}wS_{\perp}\beta}+\widehat{N}_{2}\left(\frac{1}{z_{1}},\frac{1}{z_{2}}\right)g^{\beta\gamma}_{\perp}\epsilon^{P_{h}wS_{\perp}\alpha}-\widehat{N}_{2}\left(\frac{1}{z_{2}}-\frac{1}{z_{1}},\frac{1}{z_{2}}\right)g^{\alpha\beta}_{\perp}\epsilon^{P_{h}wS_{\perp}\gamma}\biggr{)},$
$\displaystyle\hat{\Gamma}_{FS}^{\alpha\beta\gamma}(\frac{1}{z_{1}},\frac{1}{z_{2}})$
$\displaystyle=\frac{d_{abc}}{N^{2}-1}\sum_{X}\\!\int\\!\\!\frac{d\lambda}{2\pi}\\!\int\\!\\!\frac{d\mu}{2\pi}e^{-i{\lambda\over
z_{1}}}e^{-i\mu({1\over z_{2}}-{1\over z_{1}})}\langle
0|F^{w\beta}_{b}(0)|h(P_{h},S_{\perp})X\rangle\langle
h(P_{h},S_{\perp})X|F^{w\alpha}_{a}(\lambda w)gF^{w\gamma}_{c}(\mu
w)|0\rangle$
$\displaystyle=-{M_{h}}\biggl{(}\widehat{O}_{1}\left(\frac{1}{z_{1}},\frac{1}{z_{2}}\right)g^{\alpha\gamma}_{\perp}\epsilon^{P_{h}wS_{\perp}\beta}+\widehat{O}_{2}\left(\frac{1}{z_{1}},\frac{1}{z_{2}}\right)g^{\beta\gamma}_{\perp}\epsilon^{P_{h}wS_{\perp}\alpha}+\widehat{O}_{2}\left(\frac{1}{z_{2}}-\frac{1}{z_{1}},\frac{1}{z_{2}}\right)g^{\alpha\beta}_{\perp}\epsilon^{P_{h}wS_{\perp}\gamma}\biggr{)}.$
(6)

Correlation functions (2.1) and (6), respectively, define two independent set
of the complex functions $\left\\{\widehat{N}_{1},\widehat{N}_{2}\right\\}$
and $\left\\{\widehat{O}_{1},\widehat{O}_{2}\right\\}$ due to the exchange
symmetry of the field strengths. Functions $\widehat{N}_{1}$ and
$\widehat{O}_{1}$ satisfy the relations

$\displaystyle\widehat{N}_{1}\left(\frac{1}{z_{1}},\frac{1}{z_{2}}\right)$
$\displaystyle=$
$\displaystyle-\widehat{N}_{1}\left(\frac{1}{z_{2}}-\frac{1}{z_{1}},\frac{1}{z_{2}}\right),$
$\displaystyle\widehat{O}_{1}\left(\frac{1}{z_{1}},\frac{1}{z_{2}}\right)$
$\displaystyle=$
$\displaystyle\widehat{O}_{1}\left(\frac{1}{z_{2}}-\frac{1}{z_{1}},\frac{1}{z_{2}}\right).$
(7)

The real parts of these four FFs are $T$-even and the imaginary parts are
$T$-odd, the latter being the sources of SSAs.
$\widehat{N}_{1,2}\left(\frac{1}{z_{1}},\frac{1}{z_{2}}\right)$ and
$\widehat{O}_{1,2}\left(\frac{1}{z_{1}},\frac{1}{z_{2}}\right)$ have a support
on $\frac{1}{z_{2}}>1$ and $\frac{1}{z_{2}}>\frac{1}{z_{1}}>0$.

For the derivation of the twist-3 gluon FF contribution to
$pp\to\Lambda^{\uparrow}X$, one also needs another dynamical FF,

$\displaystyle\widetilde{\Delta}_{ij}^{\alpha}\left({1\over z_{1}},{1\over
z_{2}}\right)$ $\displaystyle=$ $\displaystyle{1\over
N}\sum_{X}\int{d\lambda\over 2\pi}\int{d\mu\over 2\pi}e^{-i{\lambda\over
z_{1}}}e^{-i\mu({1\over z_{2}}-{1\over z_{1}})}\langle 0|F_{a}^{w\alpha}(\mu
w)|hX\rangle\langle hX|\bar{\psi}_{j}(\lambda w)t^{a}\psi_{i}(0)|0\rangle$ (8)
$\displaystyle=$ $\displaystyle M_{h}\left[e^{\alpha P_{h}wS_{\perp}}(\hbox
to0.0pt{/\hss}{\mkern-1.0muP}_{h})_{ij}\widetilde{D}_{FT}\left({1\over
z_{1}},{1\over z_{2}}\right)+iS_{\perp}^{\alpha}\left(\gamma_{5}\hbox
to0.0pt{/\hss}{\mkern-1.0muP}_{h}\right)_{ij}\widetilde{G}_{FT}\left({1\over
z_{1}},{1\over z_{2}}\right)\right],$

where we explicitly wrote spinor indices $i,\,j$. $\widetilde{D}_{FT}$ and
$\widetilde{G}_{FT}$ are also complex functions, with the $T$-even real part
and the $T$-odd imaginary part. They have a support on $\frac{1}{z_{1}}>0$,
$\frac{1}{z_{2}}<0$ and $\frac{1}{z_{1}}-\frac{1}{z_{2}}>1$. As was shown in
[49], the constraint relations for the twist-3 gluon FFs involve these quark-
gluon correlation functions. We collectively call the functions in (2.1), (6)
and (8) dynamical twist-3 FFs.

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

Three types of the twist-3 gluon FFs defined in the previous subsection are
not independent from each other, but are related by the QCD equation-of-motion
(EOM relations) and the Lorentz invariance properties of the correlation
functions (LIRs). In [49] the complete set of those relations have been
derived. Here we recall those relations, which, as we will see in the next
section, play a crucial role to guarantee the Lorentz- and gauge- invariance
of the twist-3 cross sections for $pp\to\Lambda^{\uparrow}X$. Here we quote
from [49] the relevant relations.

First the intrinsic FFs $\Delta\widehat{G}_{3\bar{T}}(z)$ can be written in
terms of the kinematical FFs and the dynamical ones as (See eq. (50) of [49]):

$\displaystyle{1\over
z}\Delta\widehat{G}_{3\bar{T}}(z)=-\Im\widetilde{D}_{FT}(z)$
$\displaystyle\qquad+\int_{0}^{1/z}d\left({1\over
z^{\prime}}\right)\left({1\over{1\over z}-{1\over
z^{\prime}}}\right)\Im\left\\{2\widehat{N}_{1}\left({1\over
z^{\prime}},{1\over z}\right)+\widehat{N}_{2}\left({1\over z^{\prime}},{1\over
z}\right)-\widehat{N}_{2}\left({1\over z}-{1\over z^{\prime}},{1\over
z}\right)\right\\}$ $\displaystyle\qquad+{1\over
2}\left(\widehat{G}_{T}^{(1)}(z)+\Delta\widehat{H}_{T}^{(1)}(z)\right),$ (9)

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

$\displaystyle\widetilde{D}_{FT}(z)\equiv{2\over
C_{F}}\int_{0}^{1/z}d\left({1\over
z_{1}}\right)\widetilde{D}_{FT}\left({1\over z_{1}},{1\over z_{1}}-{1\over
z}\right),$ (10)

with $C_{F}={N^{2}-1\over 2N}$ and $\Im\widetilde{D}_{FT}$ indicates the
imaginary part of $\widetilde{D}_{FT}$. The kinematical FFs can be expressed
in terms of the dynamical ones as (see eqs. (74) and (75) of [49]):

$\displaystyle\widehat{G}_{T}^{(1)}(z)=-{2\over
z^{2}}\int_{1}^{1/z}d\left({1\over
z_{2}}\right)z_{2}^{3}\,\Im\widetilde{D}_{FT}(z_{2})$
$\displaystyle\quad+{4\over z^{2}}\int_{1}^{1/z}d\left({1\over
z_{2}}\right)z_{2}^{3}\int_{0}^{1/z_{2}}d\left({1\over z_{1}}\right){1\over
1/z_{2}-1/z_{1}}\Im\left[\widehat{N}_{1}\left({1\over z_{1}},{1\over
z_{2}}\right)-\widehat{N}_{2}\left({1\over z_{2}}-{1\over z_{1}},{1\over
z_{2}}\right)\right]$ $\displaystyle\quad+{2\over
z^{2}}\int_{1}^{1/z}d\left({1\over
z_{2}}\right)z_{2}^{2}\int_{0}^{1/z_{2}}d\left({1\over
z_{1}}\right){1\over\left(1/z_{2}-1/z_{1}\right)^{2}}\Im\left[\widehat{N}_{1}\left({1\over
z_{1}},{1\over z_{2}}\right)+\widehat{N}_{2}\left({1\over z_{1}},{1\over
z_{2}}\right)\right.$ $\displaystyle\left.\hskip
256.0748pt-2\widehat{N}_{2}\left({1\over z_{2}}-{1\over z_{1}},{1\over
z_{2}}\right)\right],$ (11)

and

$\displaystyle\Delta\widehat{H}_{T}^{(1)}(z)=-{4\over
z^{4}}\int_{1}^{1/z}d\left({1\over
z_{2}}\right)z_{2}^{5}\,\Im\widetilde{D}_{FT}(z_{2})$
$\displaystyle\quad+{8\over z^{4}}\int_{1}^{1/z}d\left({1\over
z_{2}}\right)z_{2}^{5}\int_{0}^{1/z_{2}}d\left({1\over z_{1}}\right){1\over
1/z_{2}-1/z_{1}}\Im\left[\widehat{N}_{1}\left({1\over z_{1}},{1\over
z_{2}}\right)+\widehat{N}_{2}\left({1\over z_{1}},{1\over
z_{2}}\right)\right]$ $\displaystyle\quad+{4\over
z^{4}}\int_{1}^{1/z}d\left({1\over
z_{2}}\right)z_{2}^{4}\int_{0}^{1/z_{2}}d\left({1\over
z_{1}}\right){1\over\left(1/z_{2}-1/z_{1}\right)^{2}}\Im\left[\widehat{N}_{1}\left({1\over
z_{1}},{1\over z_{2}}\right)+\widehat{N}_{2}\left({1\over z_{1}},{1\over
z_{2}}\right)\right].$ (12)

From (9), (11) and (12), the intrinsic FF $\Delta\widehat{G}_{3\bar{T}}(z)$ is
also written by the dynamical ones. For the derivation of the twist-3 cross
section for $pp\to\Lambda^{\uparrow}X$, one needs derivatives of the kinematic
FFs. From (11) and (12), we can obtain those derivatives in terms of the
kinematical FFs themselves and the dynamical ones as

$\displaystyle\frac{1}{z}\frac{\partial}{\partial(1/z)}[\widehat{G}_{T}^{(1)}(z)]$
$\displaystyle=$
$\displaystyle-2\,\Im\widetilde{D}_{FT}(z)+2\widehat{G}_{T}^{(1)}(z)$
$\displaystyle+$ $\displaystyle 4\int_{0}^{1/z}d\left({1\over
z^{\prime}}\right){1\over{{1\over z}-{1\over
z^{\prime}}}}\Im\left\\{\widehat{N}_{1}\left({1\over z^{\prime}},{1\over
z}\right)-\widehat{N}_{2}\left({1\over z}-{1\over z^{\prime}},{1\over
z}\right)\right\\}$ $\displaystyle+$ $\displaystyle{2\over
z}\int_{0}^{1/z}d\left({1\over z^{\prime}}\right){1\over\left({1\over
z}-{1\over z^{\prime}}\right)^{2}}\Im\left\\{\widehat{N}_{1}\left({1\over
z^{\prime}},{1\over z}\right)+\widehat{N}_{2}\left({1\over z^{\prime}},{1\over
z}\right)-2\widehat{N}_{2}\left({1\over z}-{1\over z^{\prime}},{1\over
z}\right)\right\\},$

and

$\displaystyle\frac{1}{z}\frac{\partial}{\partial(1/z)}[\Delta\widehat{H}_{T}^{(1)}(z)]$
$\displaystyle=$
$\displaystyle-4\,\Im\widetilde{D}_{FT}(z)+4\Delta\widehat{H}_{T}^{(1)}(z)$
(14) $\displaystyle+$ $\displaystyle 8\int_{0}^{1/z}d\left({1\over
z^{\prime}}\right){1\over{{1\over z}-{1\over
z^{\prime}}}}\Im\left\\{\widehat{N}_{1}\left({1\over z^{\prime}},{1\over
z}\right)+\widehat{N}_{2}\left({1\over z^{\prime}},{1\over z}\right)\right\\}$
$\displaystyle+$ $\displaystyle{4\over z}\int_{0}^{1/z}d\left({1\over
z^{\prime}}\right){1\over\left({1\over z}-{1\over
z^{\prime}}\right)^{2}}\Im\left\\{\widehat{N}_{1}\left({1\over
z^{\prime}},{1\over z}\right)+\widehat{N}_{2}\left({1\over z^{\prime}},{1\over
z}\right)\right\\}.$

In the next section, we will use (9), (LABEL:FFrel_G) and (14) to write the
twist-3 gluon FF contribution to $pp\to\Lambda^{\uparrow}X$ in a frame-
independent form.

##  3 Twist-3 gluon fragmentation contribution to $pp\to\Lambda^{\uparrow}X$

In this section, we develop a formalism for calculating the twist-3 gluon
fragmentation contribution to

$\displaystyle p(p)+p(p^{\prime})\to\Lambda^{\uparrow}(P_{h},S_{\perp})+X,$
(15)

where $p$, $p^{\prime}$ and $P_{h}$ are the momenta of the particles and
$S_{\perp}$ is the transverse spin vector of $\Lambda^{\uparrow}$. We work in
Feynman gauge so that one can check the appearance of the gauge invariant FFs
explicitly. As in the case of the twist-3 quark FF [46, 55], naively $T$-odd
FFs give rise to the cross section as a nonpole contribution. The twist-3 FF
contribution to (15) can be written as

$\displaystyle
E_{h}\frac{d\sigma(p,p^{\prime},P_{h};S_{\perp})}{d^{3}P_{h}}=\frac{1}{16\pi^{2}S_{E}}\\!\int\\!\\!\frac{dx}{x}f_{1}(x)\\!\int\frac{dx^{\prime}}{x^{\prime}}f_{1}(x^{\prime})W_{q,g}(xp,x^{\prime}p^{\prime},P_{h}/z,S_{\perp}),$
(16)

where $S_{E}=(p+p^{\prime})^{2}$ is the center-of-mass energy squared,
$E_{h}=\sqrt{M_{h}^{2}+\vec{P}_{h}^{2}}$ is the energy of the hyperon,
$x,\,x^{\prime}$ are the momentum fractions of the partons coming out of the
initial nucleons and $f_{1}(x)$ is the unpolarized quark or gluon
distributions in the nucleon. $W_{q}$ ($W_{g}$) is the hadronic tensor
representing the partonic hard scattering followed by the fragmentation of a
quark (gluon) into the final $\Lambda^{\uparrow}$. In (16), summation over all
possible channels is implied. The LO cross section for $W_{q}$ was derived in
[46]. Here we focus on the twist-3 gluon fragmentation contribution $W_{g}$ in
(16) which is diagrammatically shown in Fig. 1.

Figure 1: Diagrams representing the twist-3 gluon fragmentation contribution
to $pp\to\Lambda^{\uparrow}X$.

$W_{g}$ consists of three terms corresponding to Figs. 1 (a), (b), (c):

$\displaystyle W_{g}(xp,x^{\prime}p^{\prime},P_{h};S_{\perp})\equiv
W^{(a)}_{g}+W^{(b)}_{g}+W^{(c)}_{g}$
$\displaystyle\qquad=\int\frac{d^{4}k}{(2\pi)^{4}}\left[\Gamma^{(0),\mu\nu}_{ab}(k)S^{(0),ab}_{\mu\nu}(k)\right]$
$\displaystyle\qquad+{1\over
2}\int\frac{d^{4}k}{(2\pi)^{4}}\int\frac{d^{4}k^{\prime}}{(2\pi)^{4}}\biggl{[}\Gamma^{(1),\mu\nu\lambda}_{L,abc}(k,k^{\prime})S^{(1),abc}_{L,\mu\nu\lambda}(k,k^{\prime})+\Gamma^{(1),\mu\nu\lambda}_{R,abc}(k,k^{\prime})S^{(1),abc}_{R,\mu\nu\lambda}(k,k^{\prime})\biggr{]},$
(17)

where the factor $1/2$ in front of $W_{g}^{(b)}$ and $W_{g}^{(c)}$ takes into
account the exchange symmetry of the gluon fields in the fragmentation matrix
elements. The four momenta $k$ and $k^{\prime}$are those of the partons
fragmenting into the final $\Lambda$. $S^{(0),ab}_{\mu\nu}(k)$,
$S^{(1),abc}_{L,\mu\nu\lambda}(k,k^{\prime})$ and
$S^{(1),abc}_{R,\mu\nu\lambda}(k,k^{\prime})$ are the partonic hard scattering
parts, and $\hat{\Gamma}^{(0),\mu\nu}_{ab}(k)$,
$\hat{\Gamma}^{(1),\mu\nu\lambda}_{L,abc}(k,k^{\prime})$ and
$\hat{\Gamma}^{(1),\mu\nu\lambda}_{R,abc}(k,k^{\prime})$ are the corresponding
hadronic matrix elements representing fragmentation of partons into $h$
($h=\Lambda^{\uparrow}$). The upper indices $(0)$ and $(1)$ represent the
number of extra gluon lines compared with the lowest order gluon fragmentation
contribution to the cross section. Hadronic matrix elements are defined as

$\displaystyle\Gamma^{(0),\mu\nu}_{ab}(k)=\sum_{X}\int{d^{4}\xi}\,{e^{-ik\xi}}{\langle
0|}A^{\nu}_{b}(0){|hX\rangle}{\langle hX|}A^{\mu}_{a}(\xi){|0\rangle},$ (18)
$\displaystyle\Gamma^{(1),\mu\nu\lambda}_{L,abc}(k,k^{\prime})=\sum_{X}\int{d^{4}\xi}\int{d^{4}\eta}\,{e^{-ik\xi}}{e^{-i(k^{\prime}-k)\eta}}{\langle
0|}A^{\nu}_{b}(0){|hX\rangle}{\langle
hX|}A^{\mu}_{a}(\xi)gA^{\lambda}_{c}(\eta){|0\rangle},$ (19)
$\displaystyle\Gamma^{(1),\mu\nu\lambda}_{R,abc}(k,k^{\prime})=\sum_{X}\int{d^{4}\xi}\int{d^{4}\eta}\,{e^{-ik\xi}}{e^{-i(k^{\prime}-k)\eta}}{\langle
0|}A^{\nu}_{b}(0)gA^{\lambda}_{c}(\eta){|hX\rangle}{\langle
hX|}A^{\mu}_{a}(\xi){|0\rangle},$ (20)

where the gauge coupling $g$ associated with the attachment of the extra gluon
line to the hard part is included in $\hat{\Gamma}^{(1)}_{L}(k,k^{\prime})$
and $\hat{\Gamma}^{(1)}_{R}(k,k^{\prime})$. Therefore the hard parts
$S^{(0)}$, $S_{L}^{(1)}$ and $S_{R}^{(1)}$ are of $O(g^{4})$ in the LO
calculation. From hermiticity, one has
$\hat{\Gamma}^{(1),\mu\nu\lambda}_{L,abc}(k,k^{\prime})^{\star}=\hat{\Gamma}^{(1)\nu\mu\lambda}_{R,bac}(k^{\prime},k)$
and
$S^{(1),abc}_{L,\mu\nu\lambda}(k,k^{\prime})^{\star}=S^{(1),bac}_{R,\nu\mu\lambda}(k^{\prime},k)$,
which guarantees $W_{g}$ in (17) is real. One can extract the twist-3 effect
from (17) by applying the collinear expansion. The collinear expansion of
$S^{(0)}$, $S_{L}^{(1)}$ and $S_{R}^{(1)}$ with respect to the parton momenta
$k^{\mu}$ and $k^{\prime\mu}$ around the parent hadron’s momentum
$P_{h}^{\mu}$ reads

$\displaystyle S^{(0),ab}_{\mu\nu}(k)$ $\displaystyle=$ $\displaystyle
S^{(0),ab}_{\mu\nu}(z)+\left.\frac{\partial S^{(0),ab}_{\mu\nu}(k)}{\partial
k^{\alpha}}\right|_{c.l.}\Omega^{\alpha}_{\ \sigma}k^{\sigma}$ (21)
$\displaystyle+$
$\displaystyle\frac{1}{2}\left.\frac{\partial^{2}S^{(0),ab}_{\mu\nu}(k)}{\partial
k^{\alpha}\partial k^{\beta}}\right|_{c.l.}\Omega^{\alpha}_{\
\sigma}k^{\sigma}\Omega^{\beta}_{\
\rho}k^{\rho}+\frac{1}{6}\left.\frac{\partial^{3}S^{(0),ab}_{\mu\nu}(k)}{\partial
k^{\alpha}\partial k^{\beta}\partial
k^{\gamma}}\right|_{c.l.}\Omega^{\alpha}_{\ \sigma}k^{\sigma}\Omega^{\beta}_{\
\rho}k^{\rho}\Omega^{\gamma}_{\ \tau}k^{\tau}+\cdots,$

and

$\displaystyle
S^{(1),abc}_{L,\mu\nu\lambda}(k,k^{\prime})=S^{(1),abc}_{L,\mu\nu\lambda}(z,z^{\prime})+\left.\frac{\partial
S^{(1),abc}_{L,\mu\nu\lambda}(k,k^{\prime})}{\partial
k^{\alpha}}\right|_{c.l.}\Omega^{\alpha}_{\
\sigma}k^{\sigma}+\left.\frac{\partial
S^{(1),abc}_{L,\mu\nu\lambda}(k,k^{\prime})}{\partial
k^{\prime\alpha}}\right|_{c.l.}\Omega^{\alpha}_{\ \sigma}k^{\prime\sigma}$
$\displaystyle+\frac{1}{2}\left.\frac{\partial^{2}S^{(1),abc}_{L,\mu\nu\lambda}(k,k^{\prime})}{\partial
k^{\alpha}\partial k^{\beta}}\right|_{c.l.}\Omega^{\alpha}_{\
\sigma}k^{\sigma}\Omega^{\beta}_{\
\rho}k^{\rho}+\frac{1}{2}\left.\frac{\partial^{2}S^{(1),abc}_{L,\mu\nu\lambda}(k,k^{\prime})}{\partial
k^{\prime\alpha}\partial k^{\prime\beta}}\right|_{c.l.}\Omega^{\alpha}_{\
\sigma}k^{\prime\sigma}\Omega^{\beta}_{\ \rho}k^{\prime\rho}$
$\displaystyle+\left.\frac{\partial^{2}S^{(1),abc}_{L,\mu\nu\lambda}(k,k^{\prime})}{\partial
k^{\alpha}\partial k^{\prime\beta}}\right|_{c.l.}\Omega^{\alpha}_{\
\sigma}k^{\sigma}\Omega^{\beta}_{\ \rho}k^{\prime\rho}$
$\displaystyle+\frac{1}{6}\left.\frac{\partial^{3}S^{(1),abc}_{L,\mu\nu\lambda}(k,k^{\prime})}{\partial
k^{\alpha}\partial k^{\beta}\partial
k^{\gamma}}\right|_{c.l.}\Omega^{\alpha}_{\ \sigma}k^{\sigma}\Omega^{\beta}_{\
\rho}k^{\rho}\Omega^{\gamma}_{\
\tau}k^{\tau}+\frac{1}{6}\left.\frac{\partial^{3}S^{(1),abc}_{L,\mu\nu\lambda}(k,k^{\prime})}{\partial
k^{\prime\alpha}\partial k^{\prime\beta}\partial
k^{\prime\gamma}}\right|_{c.l.}\Omega^{\alpha}_{\
\sigma}k^{\prime\sigma}\Omega^{\beta}_{\ \rho}k^{\prime\rho}\Omega^{\gamma}_{\
\tau}k^{\prime\tau}$
$\displaystyle+\frac{1}{2}\left.\frac{\partial^{3}S^{(1),abc}_{L,\mu\nu\lambda}(k,k^{\prime})}{\partial
k^{\alpha}\partial k^{\beta}\partial
k^{\prime\gamma}}\right|_{c.l.}\Omega^{\alpha}_{\
\sigma}k^{\sigma}\Omega^{\beta}_{\ \rho}k^{\rho}\Omega^{\gamma}_{\
\tau}k^{\prime\tau}+\frac{1}{2}\left.\frac{\partial^{3}S^{(1),abc}_{L,\mu\nu\lambda}(k,k^{\prime})}{\partial
k^{\alpha}\partial k^{\prime\beta}\partial
k^{\prime\gamma}}\right|_{c.l.}\Omega^{\alpha}_{\
\sigma}k^{\sigma}\Omega^{\beta}_{\ \rho}k^{\prime\rho}\Omega^{\gamma}_{\
\tau}k^{\prime\tau}\ +\cdots,$

where $\Omega^{\alpha}_{\ \mu}\equiv g^{\alpha}_{\
\mu}-P_{h}^{\alpha}w_{\mu}$, $S^{(0),ab}_{\mu\nu}(z)\equiv
S^{(0),ab}_{\mu\nu}(P_{h}/z)$,
$S^{(1),abc}_{L,\mu\nu\lambda}(z,z^{\prime})\equiv
S^{(1),abc}_{L,\mu\nu\lambda}(P_{h}/z,P_{h}/z^{\prime})$, “$|_{c.l.}$”
indicates taking the collinear limit, i.e., $k\to P_{h}/z$ and $k^{\prime}\to
P_{h}/z^{\prime}$ and $\cdots$ denotes the contribution of twist-4 or higher.
The collinear expansion for $S_{R}^{(1)}$ can be performed similarly. We also
decompose the gauge field $A^{\mu}$ as

$\displaystyle A^{\mu}=(A\cdot w)P_{h}^{\mu}+\Omega^{\mu}_{\ \nu}A^{\nu}.$
(23)

Inserting this decomposition into the expression for $\hat{\Gamma}^{(0)}$,
$\hat{\Gamma}^{(1),}_{L}$ and $\hat{\Gamma}^{(1)}_{R}$, one obtains

$\displaystyle\Gamma^{(0),\mu\nu}_{ab}$ $\displaystyle=$
$\displaystyle{P_{h}^{\mu}}{P_{h}^{\nu}}\Gamma^{(0),ww}_{ab}+{P_{h}^{\mu}}{\Omega^{\nu}_{\
\rho}}\Gamma^{(0),w\rho}_{ab}+{P_{h}^{\nu}}{\Omega^{\mu}_{\
\sigma}}\Gamma^{(0),\sigma w}_{ab}+{\Omega^{\mu}_{\ \sigma}}{\Omega^{\nu}_{\
\rho}}\Gamma^{(0),\sigma\rho}_{ab},$ (24)
$\displaystyle\Gamma^{(1),\mu\nu\lambda}_{L,abc}$ $\displaystyle=$
$\displaystyle{P_{h}^{\mu}}{P_{h}^{\nu}}{P_{h}^{\lambda}}\Gamma^{(1),www}_{L,abc}+{P_{h}^{\mu}}{P_{h}^{\nu}}{\Omega^{\lambda}_{\
\tau}}\Gamma^{(1),ww\tau}_{L,abc}+{P_{h}^{\mu}}{P_{h}^{\lambda}}{\Omega^{\nu}_{\
\rho}}\Gamma^{(1),w\rho
w}_{L,abc}+{P_{h}^{\nu}}{P_{h}^{\lambda}}{\Omega^{\mu}_{\
\sigma}}\Gamma^{(1),\sigma ww}_{L,abc}$ (25) $\displaystyle+$
$\displaystyle{P_{h}^{\mu}}{\Omega^{\nu}_{\ \rho}}{\Omega^{\lambda}_{\
\tau}}\Gamma^{(1),w\rho\tau}_{L,abc}+{P_{h}^{\nu}}{\Omega^{\mu}_{\
\sigma}}{\Omega^{\lambda}_{\ \tau}}\Gamma^{(1),\sigma
w\tau}_{L,abc}+{P_{h}^{\lambda}}{\Omega^{\mu}_{\ \sigma}}{\Omega^{\nu}_{\
\rho}}\Gamma^{(1),\sigma\rho w}_{L,abc}$ $\displaystyle+$
$\displaystyle{\Omega^{\mu}_{\ \sigma}}{\Omega^{\nu}_{\
\rho}}{\Omega^{\lambda}_{\ \tau}}\Gamma^{(1),\sigma\rho\tau}_{L,abc},$

and likewise for $\Gamma^{(1),\mu\nu\lambda}_{R,abc}$. Inserting the above
expansion into (17) and keeping the terms with two or three $\Omega^{\mu}_{\
\nu}$ s in the product of the hard parts $S^{(0)}$, $S_{L}^{(1)}$,
$S_{R}^{(1)}$ and the fragmentation matrix elements $\hat{\Gamma}^{(0)}$,
$\hat{\Gamma}_{L}^{(1)}$, $\hat{\Gamma}_{R}^{(1)}$, one can obtain the twist-2
and 3 contributions to the cross section. To get a gauge invariant cross
section, one needs to fully utilize the following Ward identities for the hard
parts (see Appendix A):

$\displaystyle{k}^{\mu}S^{ab}_{\mu\nu}(k)={k}^{\nu}S^{ab}_{\mu\nu}(k)=0,$ (26)
$\displaystyle{(k^{\prime}-k)}^{\lambda}S^{abc}_{L\,\mu\nu\lambda}(k,k^{\prime})$
$\displaystyle=$
$\displaystyle\frac{-if^{abc}}{N^{2}-1}S_{\mu\nu}(k^{\prime})+G^{abc}_{\mu\nu}(k,k^{\prime}),$
(27) $\displaystyle{k}^{\mu}S^{abc}_{L\,\mu\nu\lambda}(k,k^{\prime})$
$\displaystyle=$
$\displaystyle\frac{if^{abc}}{N^{2}-1}S_{\lambda\nu}(k^{\prime})+G^{cab}_{\lambda\nu}(k^{\prime}-k,k^{\prime}),$
(28) $\displaystyle{k^{\prime}}^{\nu}S^{abc}_{L\,\mu\nu\lambda}(k,k^{\prime})$
$\displaystyle=$ $\displaystyle 0,$ (29)
$\displaystyle{(k^{\prime}-k)}^{\lambda}S^{abc}_{R\,\mu\nu\lambda}(k,k^{\prime})$
$\displaystyle=$
$\displaystyle\frac{if^{abc}}{N^{2}-1}S_{\mu\nu}(k)+\left(G^{bac}_{\nu\mu}(k^{\prime},k)\right)^{\star},$
(30) $\displaystyle{k^{\prime}}^{\nu}S^{abc}_{R\,\mu\nu\lambda}(k,k^{\prime})$
$\displaystyle=$
$\displaystyle\frac{if^{abc}}{N^{2}-1}S_{\mu\lambda}(k)+\left(G^{cab}_{\lambda\mu}(k-k^{\prime},k)\right)^{\star},$
(31) $\displaystyle{k}^{\mu}S^{abc}_{R\,\mu\nu\lambda}(k,k^{\prime})$
$\displaystyle=$ $\displaystyle 0.$ (32)

Here and below we suppress the upper indices $(0)$ and $(1)$ from the hard
parts $S^{(0)}$, $S^{(1)}_{L}$ and $S^{(1)}_{R}$ for simplicity and
$S^{\mu\nu}(k)\equiv S^{\mu\nu,ab}(k)\delta_{ab}$. The $G$-terms appear due to
the off-shellness of the parton momenta entering the fragmentation matrix
elements. We present the actual forms of those “ghost terms” in Appendix A.
Here we only mention that they are proportional to $f^{abc}$ and satisfy the
relation

$\displaystyle
k^{\mu}G^{abc}_{\mu\nu}(k,k^{\prime})=k^{\prime\nu}G^{abc}_{\mu\nu}(k,k^{\prime})=0.$
(33)

We have found that the ghost terms do not contribute to the twist-3 cross
section333This was also the case for the twist-3 quark FF contribution to
$ep^{\uparrow}\to ehX$ [55] and the three-gluon distribution function
contribution to $\vec{p}p^{\uparrow}\to DX$ [56].. We thus discard them in the
following. To get the twist-3 cross section in the gauge invariant form, we
use the collinear limit of these identities as well as the first, second and
third derivatives with respect to $k$ and $k^{\prime}$ of these Ward
identities in the collinear limit.

After very lengthy calculation one eventually obtains the twist-2 and -3
contributions from Fig. 1 (a), (b) and (c) to $W_{g}$ in the following form:

$\displaystyle W^{(a)}_{g}$ $\displaystyle=$ $\displaystyle{\Omega_{\
\alpha}^{\mu}}{\Omega_{\ \beta}^{\nu}}\sum_{X}\int
d\left(\frac{1}{z}\right)z^{2}\Biggl{(}\Gamma^{(0)\alpha\beta}(z)-\Gamma^{(1)\alpha\beta}_{AA}(z)-\Gamma^{(1)\alpha\beta}_{AAR}(z)\Biggr{)}S_{\mu\nu}(z)$
(34) $\displaystyle-$ $\displaystyle i{\Omega_{\ \alpha}^{\mu}}{\Omega_{\
\beta}^{\nu}}{\Omega_{\ \gamma}^{\lambda}}\sum_{X}\int
d\left(\frac{1}{z}\right)z^{2}\Biggl{(}\Gamma^{(0)\alpha\beta\gamma}_{\partial}(z)-\Gamma^{(1)\alpha\beta\gamma}_{AA\partial}(z)-\Gamma^{(1)\alpha\beta\gamma}_{AAR\partial}(z)\Biggr{)}\left.\frac{\partial
S_{\mu\nu}(k)}{\partial k^{\lambda}}\right|_{c.l.},$ $\displaystyle
W^{(b)}_{g}$ $\displaystyle=$ $\displaystyle{\Omega_{\
\alpha}^{\mu}}{\Omega_{\ \beta}^{\nu}}\sum_{X}\int
d\left(\frac{1}{z}\right)z^{2}\Biggl{(}\Gamma^{(1)\alpha\beta}_{[A]}(z)+\Gamma^{(1)\alpha\beta}_{AA}(z)\Biggr{)}S_{\mu\nu}(z)$
(35) $\displaystyle-$ $\displaystyle i{\Omega_{\ \alpha}^{\mu}}{\Omega_{\
\beta}^{\nu}}{\Omega_{\ \gamma}^{\lambda}}\sum_{X}\int
d\left(\frac{1}{z}\right)z^{2}\Biggl{(}\Gamma^{(1)\alpha\beta\gamma}_{\partial[F]}(z)+\Gamma^{(1)\alpha\beta\gamma}_{\partial[\partial
A]}(z)+\Gamma^{(1)\alpha\beta\gamma}_{\partial[A]}(z)+\Gamma^{(1)\alpha\beta\gamma}_{AA\partial}(z)\Biggr{)}\left.\frac{\partial
S_{\mu\nu}(k)}{\partial k^{\lambda}}\right|_{c.l.}$ $\displaystyle+$
$\displaystyle{i\over 2}{\Omega_{\ \alpha}^{\mu}}{\Omega_{\
\beta}^{\nu}}{\Omega_{\ \gamma}^{\lambda}}\sum_{X}\int
d\left(\frac{1}{z}\right)\int
d\left(\frac{1}{z^{\prime}}\right)\frac{zz^{\prime}}{1/z^{\prime}-1/z}\Gamma^{(1)\alpha\beta\gamma}_{F,abc}(\frac{1}{z},\frac{1}{z^{\prime}})S^{L,abc}_{\mu\nu\lambda}(z,z^{\prime}),$
$\displaystyle W^{(c)}_{g}$ $\displaystyle=$ $\displaystyle{\Omega_{\
\alpha}^{\mu}}{\Omega_{\ \beta}^{\nu}}\int
d\left(\frac{1}{z}\right)z^{2}\Biggl{(}\Gamma^{(1)\alpha\beta}_{[AR]}(z)+\Gamma^{(1)\alpha\beta}_{AAR}(z)\Biggr{)}S_{\mu\nu}(z)$
(36) $\displaystyle-$ $\displaystyle i{\Omega_{\ \alpha}^{\mu}}{\Omega_{\
\beta}^{\nu}}{\Omega_{\ \gamma}^{\lambda}}\sum_{X}\int
d\left(\frac{1}{z}\right)z^{2}\Biggl{(}\,\Gamma^{(1)\alpha\beta\gamma}_{\partial[A\partial]}(z)+\Gamma^{(1)\alpha\beta\gamma}_{AAR\partial}(z)\Biggr{)}\left.\frac{\partial
S_{\mu\nu}(k)}{\partial k^{\lambda}}\right|_{c.l.}$ $\displaystyle+$
$\displaystyle{i\over 2}{\Omega_{\ \alpha}^{\mu}}{\Omega_{\
\beta}^{\nu}}{\Omega_{\ \gamma}^{\lambda}}\sum_{X}\int
d\left(\frac{1}{z}\right)\int
d\left(\frac{1}{z^{\prime}}\right)\frac{zz^{\prime}}{1/z^{\prime}-1/z}\Gamma^{(1)\alpha\beta\gamma}_{FR,abc}(\frac{1}{z},\frac{1}{z^{\prime}})S^{R,abc}_{\mu\nu\lambda}(z,z^{\prime}).$

Each fragmentation matrix elements $\Gamma$’s appearing in the above
expression are defined as follows:

$\displaystyle\Gamma^{(0)\alpha\beta}(z)={\delta^{ab}\over{N^{2}-1}}\sum_{X}\int\frac{d\lambda}{2\pi}e^{-i{\lambda\over
z}}\langle 0|F^{\beta w}_{b}(0)|hX\rangle\langle hX|F^{\alpha w}_{a}(\lambda
w)|0\rangle,$ (37)
$\displaystyle\Gamma^{(0)\alpha\beta\gamma}_{\partial}(z)={\delta^{ab}\over{N^{2}-1}}\sum_{X}\int\frac{d\lambda}{2\pi}e^{-i{\lambda\over
z}}\langle 0|F^{\beta w}_{b}(0)|hX\rangle\langle hX|F^{\alpha w}_{a}(\lambda
w)\overleftarrow{\partial}^{\gamma}|0\rangle,$ (38)
$\displaystyle\Gamma^{(1)\alpha\beta}_{AA}(z)={1\over{N^{2}-1}}\sum_{X}\int\frac{d\lambda}{2\pi}e^{-i{\lambda\over
z}}\langle 0|F^{\beta w}_{a}(0)|hX\rangle\langle
hX|gf_{abc}A^{\alpha}_{b}(\lambda w)A^{w}_{c}(\lambda w)|0\rangle,$ (39)
$\displaystyle\Gamma^{(1)\alpha\beta}_{AAR}(z)={1\over{N^{2}-1}}\sum_{X}\int\frac{d\lambda}{2\pi}e^{-i{\lambda\over
z}}\langle 0|gf_{abc}A^{\beta}_{b}(0)A^{w}_{c}(0)|hX\rangle\langle
hX|F^{\alpha w}_{a}(\lambda w)|0\rangle,$ (40)
$\displaystyle\Gamma^{(1)\alpha\beta\gamma}_{AA\partial}(z)={1\over{N^{2}-1}}\sum_{X}\int\frac{d\lambda}{2\pi}e^{-i{\lambda\over
z}}\langle 0|F^{\beta w}_{a}(0)|hX\rangle\langle
hX|gf_{abc}A^{\alpha}_{b}(\lambda w)A^{w}_{c}(\lambda
w)\overleftarrow{\partial}^{\gamma}|0\rangle,$ (41)
$\displaystyle\Gamma^{(1)\alpha\beta\gamma}_{AAR\partial}(z)={1\over{N^{2}-1}}\sum_{X}\int\frac{d\lambda}{2\pi}e^{-i{\lambda\over
z}}\langle 0|gf_{abc}A^{\beta}_{b}(0)A^{w}_{c}(0)|hX\rangle\langle
hX|F^{\alpha w}_{a}(\lambda w)\overleftarrow{\partial}^{\gamma}|0\rangle,$
(42)
$\displaystyle\Gamma^{(1)\alpha\beta}_{[A]}(z)={(-if^{abc})\over{N^{2}-1}}\sum_{X}\int\frac{d\lambda}{2\pi}e^{-i{\lambda\over
z}}\langle 0|F^{\beta w}_{b}(0)|hX\rangle\langle hX|F^{\alpha w}_{a}(\lambda
w)ig{\int^{\lambda}_{\infty}}d\mu A^{w}_{c}(\mu w)|0\rangle,$ (43)
$\displaystyle\Gamma^{(1)\alpha\beta\gamma}_{\partial[F]}(z)={(-if^{abc})\over{N^{2}-1}}\sum_{X}\int\frac{d\lambda}{2\pi}e^{-i{\lambda\over
z}}\langle 0|F^{\beta w}_{b}(0)|hX\rangle\langle hX|F^{\alpha w}_{a}(\lambda
w)ig{\int^{\lambda}_{\infty}}d\mu F^{\gamma w}_{c}(\mu w)|0\rangle,$ (44)
$\displaystyle\Gamma^{(1)\alpha\beta\gamma}_{\partial[\partial
A]}(z)={(-if^{abc})\over{N^{2}-1}}\sum_{X}\int\frac{d\lambda}{2\pi}e^{-i{\lambda\over
z}}\langle 0|F^{\beta w}_{b}(0)|hX\rangle\langle hX|F^{\alpha w}_{a}(\lambda
w)\overleftarrow{\partial}^{\gamma}ig{\int^{\lambda}_{\infty}}d\mu
A^{w}_{c}(\mu w)|0\rangle,$ (45)
$\displaystyle\Gamma^{(1)\alpha\beta\gamma}_{\partial[A]}(z)={(-if^{abc})\over{N^{2}-1}}\sum_{X}\int\frac{d\lambda}{2\pi}e^{-i{\lambda\over
z}}\langle 0|F^{\beta w}_{b}(0)|hX\rangle\langle hX|F^{\alpha w}_{a}(\lambda
w)igA^{\gamma}_{c}(\lambda w)|0\rangle,$ (46)
$\displaystyle\Gamma^{(1)\alpha\beta\gamma}_{F,abc}(\frac{1}{z},\frac{1}{z^{\prime}})=\int\frac{d\lambda}{2\pi}\int\frac{d\mu}{2\pi}e^{-i{\lambda\over
z}}e^{-i\mu({1\over z^{\prime}}-{1\over z})}\langle 0|F^{\beta
w}_{b}(0)|hX\rangle\langle hX|F^{\alpha w}_{a}(\lambda w)gF^{\gamma w}_{c}(\mu
w)|0\rangle,$ (47)
$\displaystyle\Gamma^{(1)\alpha\beta}_{[AR]}(z)={(-if^{abc})\over{N^{2}-1}}\sum_{X}\int\frac{d\lambda}{2\pi}e^{-i{\lambda\over
z}}\langle 0|ig{\int^{\infty}_{0}}d\mu A^{w}_{c}(\mu w)F^{\beta
w}_{b}(0)|hX\rangle\langle hX|F^{\alpha w}_{a}(\lambda w)|0\rangle,$ (48)
$\displaystyle\Gamma^{(1)\alpha\beta\gamma}_{\partial[A\partial]}(z)={(-if^{abc})\over{N^{2}-1}}\sum_{X}\int\frac{d\lambda}{2\pi}e^{-i{\lambda\over
z}}\langle 0|ig{\int^{\infty}_{0}}d\mu A^{w}_{c}(\mu w)F^{\beta
w}_{b}(0)|hX\rangle\langle hX|F^{\alpha w}_{a}(\lambda
w)\overleftarrow{\partial}^{\gamma}|0\rangle,$ (49)
$\displaystyle\Gamma^{(1)\alpha\beta\gamma}_{FR,abc}(\frac{1}{z},\frac{1}{z^{\prime}})=\sum_{X}\int\frac{d\lambda}{2\pi}\int\frac{d\mu}{2\pi}e^{-i{\lambda\over
z}}e^{-i\mu({1\over z^{\prime}}-{1\over z})}\langle 0|gF^{\gamma w}_{c}(\mu
w)F^{\beta w}_{b}(0)|hX\rangle\langle hX|F^{\alpha w}_{a}(\lambda
w)|0\rangle.$ (50)

In the LO calculation of Figs. 1 (a), (b) and (c), only $O(1)$ and $O(g)$
contributions from the hadronic matrix elements are produced. We thus note
that, in (34)-(36), one can identify the correlation functions of the field
strength
$F_{\mu\nu}^{a}=\partial_{\mu}A_{\nu}^{a}-\partial_{\nu}A_{\mu}^{a}+gf^{abc}A_{\mu}^{b}A_{\nu}^{c}$
to this accuracy. The first term in $W_{g}^{(a)}$ is the $O(1)$ contribution
from $\hat{\Gamma}(z)$ in (1), ignoring $O(g^{2})$ terms $\sim\langle
0|gf^{abc}A^{b}A^{c}|hX\rangle\langle
hX|gf^{ab^{\prime}c^{\prime}}A^{b^{\prime}}A^{c^{\prime}}|0\rangle$. The first
terms in $W_{g}^{(b)}$ and $W_{g}^{(c)}$ are the $O(g)$ terms arising from the
expansion of the gauge link and the $O(g)$ part of the field strength in
$\hat{\Gamma}(z)$. These terms contain both twist-2 and intrinsic twist-3 FFs.
Likewise the second term in $W_{g}^{(a)}$ is the $O(1)$ contribution from the
kinematical twist-3 FFs $\hat{\Gamma}_{\partial}(z)$ in (2). The second terms
in $W_{g}^{(b)}$ and $W_{g}^{(c)}$ are the $O(g)$ terms arising from the
expansion of the gauge link and the $O(g)$ part of the field strength in
$\hat{\Gamma}_{\partial}(z)$. The third term in $W_{g}^{(b)}$ is the $O(g)$
contribution from $\hat{\Gamma}_{F}$ defined in (2.1) and (6). Likewise the
third term in $W_{g}^{(c)}$ is associated with
$\hat{\Gamma}_{FR}\sim(\hat{\Gamma}_{F})^{\star}$. This way we have obtained
the sum of $W_{g}^{(a)}$, $W_{g}^{(b)}$ and $W_{g}^{(c)}$ in the color gauge
invariant form in terms of the $intrinsic$, $kinematical$ and $dynamical$ FFs.
Inserting these expressions into (16), one can eventually express the twist-3
gluon FF contribution to the cross section as

$\displaystyle
E_{h}\frac{d\sigma(p,p^{\prime},P_{h};S_{\perp})}{d^{3}P_{h}}=\frac{1}{16\pi^{2}S_{E}}\int^{1}_{0}\frac{dx}{x}f_{1}(x)\int^{1}_{0}\frac{dx^{\prime}}{x^{\prime}}f_{1}(x^{\prime})\biggl{[}{\Omega^{\mu}_{\
\alpha}}{\Omega^{\nu}_{\ \beta}}\int^{1}_{0}dz\,{\rm
Tr}\left[\hat{\Gamma}^{\alpha\beta}(z)S_{\mu\nu}(P_{h}/z)\right]$
$\displaystyle-i\,{\Omega^{\mu}_{\ \alpha}}{\Omega^{\nu}_{\
\beta}}{\Omega^{\lambda}_{\ \gamma}}\int^{1}_{0}dz\,{\rm
Tr}\left[\hat{\Gamma}_{\partial}^{\alpha\beta\gamma}(z)\left.\frac{\partial
S_{\mu\nu}(k)}{\partial
k^{\lambda}}\right|_{c.l.}\right]+{\Re}\Bigl{\\{}i\,{\Omega^{\mu}_{\
\alpha}}{\Omega^{\nu}_{\ \beta}}{\Omega^{\lambda}_{\
\gamma}}\int^{1}_{0}\frac{dz}{z}\int^{\infty}_{z}\frac{dz^{\prime}}{z^{\prime}}\,\left(\frac{1}{1/z-1/z^{\prime}}\right)$
$\displaystyle\qquad\times{\rm
Tr}\left[\left(-\frac{if^{abc}}{N}\hat{\Gamma}_{FA}^{\alpha\beta\gamma}\left(\frac{1}{z^{\prime}},\frac{1}{z}\right)+d^{abc}\frac{N}{N^{2}-4}\hat{\Gamma}_{FS}^{\alpha\beta\gamma}\left(\frac{1}{z^{\prime}},\frac{1}{z}\right)\right)S^{L}_{\mu\nu\lambda,abc}(z^{\prime},z)\right]\Bigr{\\}}\biggr{]},$
(51)

where we have used the expression (4), and $\left.\right|_{c.l.}$ implies the
collinear limit, $k\to P_{h}/z$. In writing down the contribution from the
dynamical FFs in (51), we have interchanged the role of the variables, $z$ and
$z^{\prime}$, from (35) and (36) for later convenience.

Figure 2: Diagram for the $q\bar{q}g$-type correlation function in
$pp\to\Lambda^{\uparrow}X$. Mirror diagram also contributes.

Next we substitute (1), (2), (2.1), (6) into (51). We recall that the
$q\bar{q}g$-type FFs (8) are related to the purely gluonic FFs, as was shown
in Sec. 2.2. Therefore we consider the contribution shown in Fig. 2 together.
Note also that each hard part contains the factor
$\delta\left((xp+x^{\prime}p^{\prime}-k)^{2}\right)\sim\delta\left((xp+x^{\prime}p^{\prime}-P_{h}/z)^{2}\right)$
corresponding to the on-shell condition for the final unobserved parton, and
its derivative with respect to $k$ causes the derivative of the kinematical
FFs by partial integration with respect to $1/z$. Separating various
contributions based on the $z^{\prime}$-dependence of the hard cross sections
for the dynamical FFs (See Appendix C), we can write the cross section as

$\displaystyle E_{h}\frac{d\sigma(p,p^{\prime},P_{h};S_{\perp})}{d^{3}P_{h}}$
(52) $\displaystyle=$
$\displaystyle\frac{M_{h}\alpha_{s}^{2}}{S_{E}}\int^{1}_{0}\frac{dx}{x}f_{1}(x)\int^{1}_{0}\frac{dx^{\prime}}{x^{\prime}}f_{1}(x^{\prime})\int^{1}_{0}dz\,\delta((xp+x^{\prime}p^{\prime}-P_{h}/z)^{2})$
$\displaystyle\times$
$\displaystyle\Biggl{\\{}\frac{\Delta\hat{G}_{3\bar{T}}(z)}{z}\hat{H}_{int}+\left(\hat{G}_{T}^{(1)}(z)\hat{H}_{NDG}+\frac{1}{z}\frac{\partial\,\hat{G}_{T}^{(1)}(z)}{\partial(1/z)}\hat{H}_{DG}\right)$
$\displaystyle\qquad\qquad\qquad\qquad+\left(\Delta\hat{H}_{T}^{(1)}(z)\hat{H}_{NDH}+\frac{1}{z}\frac{\partial\,\Delta\hat{H}_{T}^{(1)}(z)}{\partial(1/z)}\hat{H}_{DH}\right)$
$\displaystyle+$
$\displaystyle\int^{1/z}_{0}d\left(\frac{1}{z^{\prime}}\right)\Biggl{[}\sum_{i=1}^{3}\Im\widehat{N}_{i}\left(\frac{1}{1/z-1/z^{\prime}}\hat{H}_{1}^{N_{i}}+\frac{1}{z}\left(\frac{1}{1/z-1/z^{\prime}}\right)^{2}\hat{H}_{2}^{N_{i}}+z^{\prime}\hat{H}_{3}^{N_{i}}+\frac{z^{\prime
2}}{z}\hat{H}_{4}^{N_{i}}\right)$ $\displaystyle+$
$\displaystyle\sum_{i=1}^{3}\Im\widehat{O}_{i}\left(\frac{1}{1/z-1/z^{\prime}}\hat{H}_{1}^{O_{i}}+\frac{1}{z}\left(\frac{1}{1/z-1/z^{\prime}}\right)^{2}\hat{H}_{2}^{O_{i}}+z^{\prime}\hat{H}_{3}^{O_{i}}+\frac{z^{\prime
2}}{z}\hat{H}_{4}^{O_{i}}\right)\Biggr{]}$ $\displaystyle+$
$\displaystyle\int^{1/z}_{0}d\left(\frac{1}{z^{\prime}}\right)\,\frac{2}{C_{F}}\Bigl{[}\Im\widetilde{D}_{F}\left(\frac{1}{z^{\prime}},\frac{1}{z^{\prime}}-\frac{1}{z}\right)(\hat{H}_{DF1}+\frac{1}{z}\frac{1}{1/z-1/z^{\prime}}\hat{H}_{DF2}+\frac{z^{\prime}}{z}\hat{H}_{DF3})$
$\displaystyle+$
$\displaystyle\Im\widetilde{G}_{F}\left(\frac{1}{z^{\prime}},\frac{1}{z^{\prime}}-\frac{1}{z}\right)(\hat{H}_{GF1}+\frac{1}{z}\frac{1}{1/z-1/z^{\prime}}\hat{H}_{GF2}+\frac{z^{\prime}}{z}\hat{H}_{GF3})\Bigr{]}\Biggr{\\}},$

where $\hat{H}_{int}$, $\hat{H}_{NDG}$, $\hat{H}_{DG}$, etc. represent the
partonic hard cross sections for each FF (after separating the
$z^{\prime}$-dependence for dynamical FFs) and they are the functions of the
Mandelstam variables in the parton level,
$s=(xp+x^{\prime}p^{\prime})^{2}=2xx^{\prime}p\cdot p^{\prime}$, $t=(xp-
P_{h}/z)^{2}=-2(x/z)p\cdot P_{h}$ and
$u=(x^{\prime}p^{\prime}-P_{h}/z)^{2}=-2(x^{\prime}/z)p^{\prime}\cdot P_{h}$,
multiplied by kinematic factors with $\epsilon^{pP_{h}wS_{\perp}}$ and
$\epsilon^{p^{\prime}P_{h}wS_{\perp}}$. (See eq.(54) below as an example.) In
(52), we have used the shorthand notation,
$\widehat{N}_{3}\equiv-\widehat{N}_{2}\left(\frac{1}{z}-\frac{1}{z^{\prime}},\frac{1}{z}\right)$
and
$\widehat{O}_{3}\equiv\widehat{O}_{2}\left(\frac{1}{z}-\frac{1}{z^{\prime}},\frac{1}{z}\right)$.
Change of the variable
$\frac{1}{z^{\prime}}\leftrightarrow\frac{1}{z}-\frac{1}{z^{\prime}}$ in the
$z^{\prime}$ and $z^{\prime 2}/z$ terms in the contribution from
$\widehat{N}_{i}$ and $\widehat{O}_{i}$ leads to the following form owing to
the exchange symmetry (7) of $\widehat{N}_{1}$ and $\widehat{O}_{1}$:

$\displaystyle
E_{h}\frac{d\sigma(p,p^{\prime},P_{h};S_{\perp})}{d^{3}P_{h}}=\frac{M_{h}\alpha_{s}^{2}}{S_{E}}\int^{1}_{0}\frac{dx}{x}f_{1}(x)\int^{1}_{0}\frac{dx^{\prime}}{x^{\prime}}f_{1}(x^{\prime})\int^{1}_{0}dz\,\delta(s+t+u)$
$\displaystyle\times\Biggl{\\{}\frac{\Delta\hat{G}_{3\bar{T}}(z)}{z}\hat{H}_{int}+\hat{G}_{T}^{(1)}(z)\hat{H}_{NDG}+\frac{1}{z}\frac{\partial\,\hat{G}_{T}^{(1)}(z)}{\partial(1/z)}\hat{H}_{DG}+\Delta\hat{H}_{T}^{(1)}(z)\hat{H}_{NDH}+\frac{1}{z}\frac{\partial\,\Delta\hat{H}_{T}^{(1)}(z)}{\partial(1/z)}\hat{H}_{DH}$
$\displaystyle+\int^{1/z}_{0}d\left(\frac{1}{z^{\prime}}\right)\Biggl{[}\frac{1}{1/z-1/z^{\prime}}\Im\left\\{\widehat{N}_{1}\left(\frac{1}{z^{\prime}},\frac{1}{z}\right)\left(\hat{H}_{1}^{N_{1}}-\hat{H}_{3}^{N_{1}}\right)+\widehat{N}_{2}\left(\frac{1}{z^{\prime}},\frac{1}{z}\right)\left(\hat{H}_{1}^{N_{2}}-\hat{H}_{3}^{N_{3}}\right)\right.$
$\displaystyle\left.\qquad\qquad\qquad-\widehat{N}_{2}\left(\frac{1}{z}-\frac{1}{z^{\prime}},\frac{1}{z}\right)\left(\hat{H}_{1}^{N_{3}}-\hat{H}_{3}^{N_{2}}\right)\right\\}$
$\displaystyle\qquad+\frac{1}{z}\left(\frac{1}{1/z-1/z^{\prime}}\right)^{2}\Im\left\\{\widehat{N}_{1}\left(\frac{1}{z^{\prime}},\frac{1}{z}\right)\left(\hat{H}_{2}^{N_{1}}-\hat{H}_{4}^{N_{1}}\right)+\widehat{N}_{2}\left(\frac{1}{z^{\prime}},\frac{1}{z}\right)\left(\hat{H}_{2}^{N_{2}}-\hat{H}_{4}^{N_{3}}\right)\right.$
$\displaystyle\left.\qquad\qquad\qquad-\widehat{N}_{2}\left(\frac{1}{z}-\frac{1}{z^{\prime}},\frac{1}{z}\right)\left(\hat{H}_{2}^{N_{3}}-\hat{H}_{4}^{N_{2}}\right)\right\\}$
$\displaystyle\qquad+\frac{1}{1/z-1/z^{\prime}}\Im\left\\{\widehat{O}_{1}\left(\frac{1}{z^{\prime}},\frac{1}{z}\right)\left(\hat{H}_{1}^{O_{1}}+\hat{H}_{3}^{O_{1}}\right)+\widehat{O}_{2}\left(\frac{1}{z^{\prime}},\frac{1}{z}\right)\left(\hat{H}_{1}^{O_{2}}+\hat{H}_{3}^{O_{3}}\right)\right.$
$\displaystyle\left.\qquad\qquad\qquad+\widehat{O}_{2}\left(\frac{1}{z}-\frac{1}{z^{\prime}},\frac{1}{z}\right)\left(\hat{H}_{1}^{O_{3}}+\hat{H}_{3}^{O_{2}}\right)\right\\}$
$\displaystyle\qquad+\frac{1}{z}\left(\frac{1}{1/z-1/z^{\prime}}\right)^{2}\Im\left\\{\widehat{O}_{1}\left(\frac{1}{z^{\prime}},\frac{1}{z}\right)\left(\hat{H}_{2}^{O_{1}}+\hat{H}_{4}^{O_{1}}\right)+\widehat{O}_{2}\left(\frac{1}{z^{\prime}},\frac{1}{z}\right)\left(\hat{H}_{2}^{O_{2}}+\hat{H}_{4}^{O_{3}}\right)\right.$
$\displaystyle\left.\qquad\qquad\qquad+\widehat{O}_{2}\left(\frac{1}{z}-\frac{1}{z^{\prime}},\frac{1}{z}\right)\left(\hat{H}_{2}^{O_{3}}+\hat{H}_{4}^{O_{2}}\right)\right\\}\Biggr{]}$
$\displaystyle+\int^{1/z}_{0}d\left(\frac{1}{z^{\prime}}\right)\,\frac{2}{C_{F}}\Bigl{[}\Im\widetilde{D}_{F}\left(\frac{1}{z^{\prime}},\frac{1}{z^{\prime}}-\frac{1}{z}\right)\left(\hat{H}_{DF1}+\frac{1}{z}\frac{1}{1/z-1/z^{\prime}}\hat{H}_{DF2}+\frac{z^{\prime}}{z}\hat{H}_{DF3}\right)$
$\displaystyle\qquad\qquad+\Im\widetilde{G}_{F}\left(\frac{1}{z^{\prime}},\frac{1}{z^{\prime}}-\frac{1}{z}\right)\left(\hat{H}_{GF1}+\frac{1}{z}\frac{1}{1/z-1/z^{\prime}}\hat{H}_{GF2}+\frac{z^{\prime}}{z}\hat{H}_{GF3}\right)\Bigr{]}\Biggr{\\}}.$
(53)

Here we remind that the gauge invariance and the frame independence of the
cross section (53) are realized in a very nontrivial manner. We show that
those properties are guaranteed by the EOM relation and the LIRs introduced in
the previous section. The gauge invariance is satisfied by the EOM relation
(9), which is discussed in detail in Appendix B. Here we demonstrate how the
frame independence of the cross section is achieved.

To make clear the issue of frame dependence, we pick up the hard cross
sections $\hat{H}_{2}^{N_{1}}-\hat{H}_{4}^{N_{1}}$, $\hat{H}_{DG}$ and
$\hat{H}_{DH}$ in (53) in the $qg\to gq$-channel, as an example. They can be
computed to be

$\displaystyle\hat{H}_{2}^{N_{1}}-\hat{H}_{4}^{N_{1}}$ $\displaystyle=$
$\displaystyle{C_{F}\over
N}\Bigl{(}(2t+u)x\epsilon^{pP_{h}wS_{\perp}}+tx^{\prime}\epsilon^{p^{\prime}P_{h}wS_{\perp}}\Bigr{)}\Bigl{(}{s^{2}+t^{2}\over
s^{2}t^{2}}\Bigr{)}$ $\displaystyle-{1\over
2}\Bigl{(}u(2t+u)x\epsilon^{pP_{h}wS_{\perp}}+t(2t+3u)x^{\prime}\epsilon^{p^{\prime}P_{h}wS_{\perp}}\Bigr{)}{s^{2}+t^{2}\over
stu^{3}},$ $\displaystyle\hat{H}_{DG}$ $\displaystyle=$
$\displaystyle-{C_{F}\over
N}\Bigl{(}x\epsilon^{pP_{h}wS_{\perp}}+x^{\prime}\epsilon^{p^{\prime}P_{h}wS_{\perp}}\Bigr{)}\Bigl{(}{s^{2}+t^{2}\over
s^{2}t}\Bigr{)}$
$\displaystyle+\Bigl{(}x\epsilon^{pP_{h}wS_{\perp}}+x^{\prime}\epsilon^{p^{\prime}P_{h}wS_{\perp}}\Bigr{)}\Bigl{(}{s^{2}+t^{2}\over
su^{2}}\Bigr{)},$ $\displaystyle\hat{H}_{DH}$ $\displaystyle=$ $\displaystyle
0.$ (54)

We note that each cross section contains the lightlike vector $w^{\mu}$. On
the other hand, the physical cross section should be able to be represented in
terms of the vectors $p$, $p^{\prime}$ $P_{h}$ and $S_{\perp}$ in a Lorentz
invariant form. Since $w^{\mu}$ is defined from $P_{h}^{\mu}$, its actual form
depends on the frame. One can express the vector $w$ in terms of $p$,
$p^{\prime}$ and $P_{h}$ as[52]

$\displaystyle w^{\mu}=\alpha{p^{\mu}\over p\cdot
P_{h}}+(1-\alpha){p^{\mu}\over p^{\prime}\cdot
P_{h}}+\left\\{-\alpha(1-\alpha){p\cdot p^{\prime}\over p\cdot
P_{h}p^{\prime}\cdot P_{h}}+\beta^{2}p\cdot p^{\prime}p\cdot
P_{h}p^{\prime}\cdot P_{h}\right\\}P_{h}^{\mu}+\beta\epsilon^{\mu
pp^{\prime}P_{h}},$

which satisfies $P_{h}\cdot w=1$ and $w^{2}=0$. The values of $\alpha$ and
$\beta$ specify the frame we choose, and the above form of the hard cross
sections leads to $\alpha$\- and $\beta$-dependent cross sections. However,
use of the EOM relation, (9), and the LIRs, (LABEL:FFrel_G) and (14), leads to
the cross section independent from $\alpha$ and $\beta$ as will be seen below.
In the twist-3 cross section (53), we eliminate the intrinsic FF
$\Delta\widehat{G}_{3\bar{T}}(z)/z$ and the derivative of the two kinematical
FFs by using those relations. Then the resulting cross section is written in
terms of the (nonderivative) kinematical FFs and the dynamical FFs. If we pick
up the hard cross section for

$\displaystyle{1\over z}\int_{0}^{1/z}\,d\left({1\over
z^{\prime}}\right){1\over(1/z-1/z^{\prime})^{2}}\Im\widehat{N}_{1}\left(\frac{1}{z^{\prime}},\frac{1}{z}\right),$
(56)

we have the combination

$\displaystyle\hat{H}_{2}^{N_{1}}-\hat{H}_{4}^{N_{1}}+2\hat{H}_{DG}+4\hat{H}_{DH}={\cal
E}\hat{\sigma}_{DN1},$ (57)

where

$\displaystyle\hat{\sigma}_{DN1}\equiv-\frac{C_{F}}{N}(\frac{1}{s^{2}}+\frac{1}{t^{2}})-\frac{(2t-u)(s^{2}+t^{2})}{2stu^{3}},$
(58) $\displaystyle{\cal E}\equiv
x^{\prime}t{\,}\epsilon^{p^{\prime}P_{h}wS_{\perp}}-xu{\,}\epsilon^{pP_{h}wS_{\perp}}=-2{xx^{\prime}\over
z}\epsilon^{pp^{\prime}P_{h}S_{\perp}}.$ (59)

One should note that the kinematical factor ${\cal E}$ appearing in this
combination is free from $\alpha$ and $\beta$, which is written after the last
equal sign in (59). We have found that all the coefficient hard cross sections
for all FFs with the same $z^{\prime}$-dependence define the frame-independent
cross section with the common kinematic factor ${\cal E}$. This shows that the
frame dependence has been removed from the twist-3 cross section thanks to the
EOM relations and the LIRs. This way we can define the following set of frame
independent hard cross sections (in addition to (57)):

$\displaystyle{\cal E}\hat{\sigma}_{G}$
$\displaystyle=\hat{H}_{NDG}+\frac{1}{2}\hat{H}_{int}+2\hat{H}_{DG},$ (60)
$\displaystyle{\cal E}\hat{\sigma}_{H}$
$\displaystyle=\hat{H}_{NDH}+\frac{1}{2}\hat{H}_{int}+4\hat{H}_{DH},$ (61)
$\displaystyle{\cal E}\hat{\sigma}_{N1}$
$\displaystyle=\hat{H}_{1}^{N_{1}}-\hat{H}_{3}^{N_{1}}+2\hat{H}_{int}+4\hat{H}_{DG,}+8\hat{H}_{DH},$
(62) $\displaystyle{\cal E}\hat{\sigma}_{N2}$
$\displaystyle=\hat{H}_{1}^{N_{2}}-\hat{H}_{3}^{N_{3}}+\hat{H}_{int}+8\hat{H}_{DH},$
(63) $\displaystyle{\cal E}\hat{\sigma}_{N3}$
$\displaystyle=-\hat{H}_{1}^{N_{3}}+\hat{H}_{3}^{N_{2}}-\hat{H}_{int}-4\hat{H}_{DG},$
(64) $\displaystyle{\cal E}\hat{\sigma}_{DN2}$
$\displaystyle=\hat{H}_{2}^{N_{2}}-\hat{H}_{4}^{N_{3}}+2\hat{H}_{DG}+4\hat{H}_{DH},$
(65) $\displaystyle{\cal E}\hat{\sigma}_{DN3}$
$\displaystyle=-\hat{H}_{2}^{N_{3}}+\hat{H}_{4}^{N_{2}}-4\hat{H}_{DG},$ (66)
$\displaystyle{\cal E}\hat{\sigma}_{O1}$
$\displaystyle=\hat{H}_{1}^{O_{1}}+\hat{H}_{3}^{O_{1}},$ (67)
$\displaystyle{\cal E}\hat{\sigma}_{O2}$
$\displaystyle=\hat{H}_{1}^{O_{2}}+\hat{H}_{3}^{O_{3}},$ (68)
$\displaystyle{\cal E}\hat{\sigma}_{O3}$
$\displaystyle=\hat{H}_{1}^{O_{3}}+\hat{H}_{3}^{O_{2}},$ (69)
$\displaystyle{\cal E}\hat{\sigma}_{DO1}$
$\displaystyle=\hat{H}_{2}^{O_{1}}+\hat{H}_{4}^{O_{1}},$ (70)
$\displaystyle{\cal E}\hat{\sigma}_{DO2}$
$\displaystyle=\hat{H}_{2}^{O_{2}}+\hat{H}_{4}^{O_{3}},$ (71)
$\displaystyle{\cal E}\hat{\sigma}_{DO3}$
$\displaystyle=\hat{H}_{2}^{O_{3}}+\hat{H}_{4}^{O_{2}},$ (72)
$\displaystyle{\cal E}\hat{\sigma}_{DF1}$
$\displaystyle=\hat{H}_{DF1}-\hat{H}_{int}-2\hat{H}_{DG}-4\hat{H}_{DH},$ (73)
$\displaystyle{\cal E}\hat{\sigma}_{DF2}$ $\displaystyle=\hat{H}_{DF2},$ (74)
$\displaystyle{\cal E}\hat{\sigma}_{DF3}$ $\displaystyle=\hat{H}_{DF3},$ (75)
$\displaystyle{\cal E}\hat{\sigma}_{GF1}$ $\displaystyle=\hat{H}_{GF1},$ (76)
$\displaystyle{\cal E}\hat{\sigma}_{GF2}$ $\displaystyle=\hat{H}_{GF2},$ (77)
$\displaystyle{\cal E}\hat{\sigma}_{GF3}$ $\displaystyle=\hat{H}_{GF3}.$ (78)

With these hard cross sections, the manifestly frame-independent twist-3 cross
section is given by

$\displaystyle E_{h}\frac{d\sigma(p,p^{\prime},P_{h};S_{\perp})}{d^{3}P_{h}}$
(79) $\displaystyle=$
$\displaystyle\frac{M_{h}\alpha_{s}^{2}}{S_{E}}\int^{1}_{0}\frac{dx}{x}f_{1}(x)\int^{1}_{0}\frac{dx^{\prime}}{x^{\prime}}f_{1}(x^{\prime})\int^{1}_{0}dz\,\delta(s+t+u)\left(-2{xx^{\prime}\over
z}\epsilon^{pp^{\prime}P_{h}S_{\perp}}\right)$ $\displaystyle\times$
$\displaystyle\Biggl{\\{}\hat{G}_{T}^{(1)}(z)\hat{\sigma}_{G}+\Delta\hat{H}_{T}^{(1)}(z)\hat{\sigma}_{H}$
$\displaystyle+$
$\displaystyle\int^{1/z}_{0}d\left(\frac{1}{z^{\prime}}\right)\Biggl{[}\frac{1}{1/z-1/z^{\prime}}\Im\left(\widehat{N}_{1}\left(\frac{1}{z^{\prime}},\frac{1}{z}\right)\hat{\sigma}_{N1}+\widehat{N}_{2}\left(\frac{1}{z^{\prime}},\frac{1}{z}\right)\hat{\sigma}_{N2}+\widehat{N}_{2}\left(\frac{1}{z}-\frac{1}{z^{\prime}},\frac{1}{z}\right)\hat{\sigma}_{N3}\right)$
$\displaystyle+$
$\displaystyle\frac{1}{z}\left(\frac{1}{1/z-1/z^{\prime}}\right)^{2}\Im\left(\widehat{N}_{1}\left(\frac{1}{z^{\prime}},\frac{1}{z}\right)\hat{\sigma}_{DN1}+\widehat{N}_{2}\left(\frac{1}{z^{\prime}},\frac{1}{z}\right)\hat{\sigma}_{DN2}+\widehat{N}_{2}\left(\frac{1}{z}-\frac{1}{z^{\prime}},\frac{1}{z}\right)\hat{\sigma}_{DN3}\right)$
$\displaystyle+$
$\displaystyle\frac{1}{1/z-1/z^{\prime}}\Im\left(\widehat{O}_{1}\left(\frac{1}{z^{\prime}},\frac{1}{z}\right)\hat{\sigma}_{O1}+\widehat{O}_{2}\left(\frac{1}{z^{\prime}},\frac{1}{z}\right)\hat{\sigma}_{O2}+\widehat{O}_{2}\left(\frac{1}{z}-\frac{1}{z^{\prime}},\frac{1}{z}\right)\hat{\sigma}_{O3}\right)$
$\displaystyle+$
$\displaystyle\frac{1}{z}\left(\frac{1}{1/z-1/z^{\prime}}\right)^{2}\Im\left(\widehat{O}_{1}\left(\frac{1}{z^{\prime}},\frac{1}{z}\right)\hat{\sigma}_{DO1}+\widehat{O}_{2}\left(\frac{1}{z^{\prime}},\frac{1}{z}\right)\hat{\sigma}_{DO2}+\widehat{O}_{2}\left(\frac{1}{z}-\frac{1}{z^{\prime}},\frac{1}{z}\right)\hat{\sigma}_{DO3}\right)\Biggr{]}$
$\displaystyle+$
$\displaystyle\int^{1/z}_{0}d\left(\frac{1}{z^{\prime}}\right)\,\frac{2}{C_{F}}\Bigl{[}\Im\widetilde{D}_{F}\left(\frac{1}{z^{\prime}},\frac{1}{z^{\prime}}-\frac{1}{z}\right)\left(\hat{\sigma}_{DF1}+\frac{1}{z}\frac{1}{1/z-1/z^{\prime}}\hat{\sigma}_{DF2}+\frac{z^{\prime}}{z}\hat{\sigma}_{DF3}\right)$
$\displaystyle+$
$\displaystyle\Im\widetilde{G}_{F}\left(\frac{1}{z^{\prime}},\frac{1}{z^{\prime}}-\frac{1}{z}\right)\left(\hat{\sigma}_{GF1}+\frac{1}{z}\frac{1}{1/z-1/z^{\prime}}\hat{\sigma}_{GF2}+\frac{z^{\prime}}{z}\hat{\sigma}_{GF3}\right)\Bigr{]}\Biggr{\\}}.$

Equation (79) is the final result for the twist-3 gluon FF contribution to
$pp\to\Lambda^{\uparrow}X$.

Below we give the LO Feynman diagrams for the hard part in each channel and
present the results for hard cross sections, using the partonic Mandelstam
variables, $s$, $t$, and $u$.

(1) $qg\to gq$ channel (Figs. 4, 4)

$\displaystyle\hat{\sigma}_{G}=-\frac{C_{F}}{N}(\frac{1}{s^{2}}-\frac{1}{t^{2}})-\frac{2(s-t)}{(s+t)^{3}},$
$\displaystyle\hat{\sigma}_{H}=0,$
$\displaystyle\hat{\sigma}_{N1}=-\frac{C_{F}}{N}\frac{4}{s^{2}}-\frac{4t^{3}+10t^{2}u+4tu^{2}+u^{3}}{tu^{3}(t+u)},$
$\displaystyle\hat{\sigma}_{N2}=0,$
$\displaystyle\hat{\sigma}_{N3}=-\hat{\sigma}_{N1},$
$\displaystyle\hat{\sigma}_{DN1}=-\frac{C_{F}}{N}(\frac{1}{s^{2}}+\frac{1}{t^{2}})+\frac{(2t-u)(s^{2}+t^{2})}{2tu^{3}(t+u)},$
$\displaystyle\hat{\sigma}_{DN2}=\hat{\sigma}_{DN1},$
$\displaystyle\hat{\sigma}_{DN3}=-2\hat{\sigma}_{DN1},$
$\displaystyle\hat{\sigma}_{O1}=0,$
$\displaystyle\hat{\sigma}_{O2}=-\frac{C_{F}}{N}(\frac{2}{s^{2}}+\frac{2}{t^{2}})+\frac{3(s^{2}+t^{2})}{st(s+t)^{2}},$
$\displaystyle\hat{\sigma}_{O3}=\hat{\sigma}_{O2},$
$\displaystyle\hat{\sigma}_{DO1}=-\frac{C_{F}}{N}(\frac{1}{s^{2}}+\frac{1}{t^{2}})+\frac{3(s^{2}+t^{2})}{2st(s+t)^{2}},$
$\displaystyle\hat{\sigma}_{DO2}=\hat{\sigma}_{DO1},$
$\displaystyle\hat{\sigma}_{DO3}=2\hat{\sigma}_{DO1},$
$\displaystyle\hat{\sigma}_{DF1}=\frac{C_{F}}{N}\frac{2}{s^{2}}+\frac{4t^{3}+10t^{2}u+4tu^{2}+u^{3}}{2tu^{3}(t+u)},$
$\displaystyle\hat{\sigma}_{DF2}=\frac{C_{F}}{N}\frac{(s^{2}+t^{2})}{st(s+t)^{2}}+\frac{(s+2t)(s^{2}+t^{2})}{4st(s+t)^{2}u}+\frac{1}{N}\frac{s}{2t(s+t)^{2}}-\frac{C_{F}}{N^{2}}\frac{1}{2t^{2}},$
$\displaystyle\hat{\sigma}_{DF3}=-\frac{C_{F}}{N}\frac{(s^{2}+t^{2})}{st(s+t)^{2}}+\frac{(2s+t)(s^{2}+t^{2})}{4st(s+t)^{3}}-\frac{1}{N}\frac{t}{2s(s+t)^{2}}+\frac{C_{F}}{N^{2}}\frac{1}{2s^{2}},$
$\displaystyle\hat{\sigma}_{GF1}=0,\qquad\hat{\sigma}_{GF2}=-\hat{\sigma}_{DF2},\qquad\hat{\sigma}_{GF3}=\hat{\sigma}_{DF3}.$
(80)

Figure 3: Diagrams in the $qg\to gq$ channel for the hard part $S(k)$ and $S_{L}(z^{\prime},z)$ in eq. (51). For $S_{L}(z^{\prime},z)$, an extra gluon line connecting $\otimes$ and each of the black dots should be added for each diagram. |  Figure 4: Diagrams for the hard part in Fig. 2. For the upper middle diagram, a quark loop with the reversed arrow also needs to be included.
---|---

(2) $q\bar{q}\to gg$ channel (Figs. 6, 6)

$\displaystyle\hat{\sigma}_{G}=-\frac{C_{F}}{N^{2}}(\frac{1}{t^{2}}-\frac{1}{u^{2}})-C_{F}\frac{(t-u)(t^{2}+u^{2})(t^{2}+4tu+u^{2})}{t^{2}u^{2}(t+u)^{3}},$
$\displaystyle\hat{\sigma}_{H}=0,$
$\displaystyle\hat{\sigma}_{N1}=\frac{C_{F}}{N^{2}}\frac{4(t-u)}{tu(t+u)}-C_{F}\frac{6(t-u)(t^{2}+u^{2})}{tu(t+u)^{3}},$
$\displaystyle\hat{\sigma}_{N2}=0,$
$\displaystyle\hat{\sigma}_{N3}=-\hat{\sigma}_{N1},$
$\displaystyle\hat{\sigma}_{DN1}=-\frac{C_{F}}{N^{2}}\frac{(t-u)(t^{2}+u^{2})}{t^{2}u^{2}(t+u)}+C_{F}\frac{(t^{2}+u^{2})(t^{3}-u^{3})}{t^{2}u^{2}(t+u)^{3}},$
$\displaystyle\hat{\sigma}_{DN2}=\hat{\sigma}_{DN1},$
$\displaystyle\hat{\sigma}_{DN3}=-2\hat{\sigma}_{DN1},$
$\displaystyle\hat{\sigma}_{O1}=0,$
$\displaystyle\hat{\sigma}_{O2}=\frac{C_{F}}{N^{2}}(\frac{2}{t^{2}}+\frac{2}{u^{2}})-C_{F}\frac{2(t^{2}+u^{2})(t^{2}-tu+u^{2})}{t^{2}u^{2}(t+u)^{2}},$
$\displaystyle\hat{\sigma}_{O3}=\hat{\sigma}_{O2},$
$\displaystyle\hat{\sigma}_{DO1}=\frac{C_{F}}{N^{2}}(\frac{1}{t^{2}}+\frac{1}{u^{2}})-C_{F}\frac{(t^{2}+u^{2})(t^{2}-tu+u^{2})}{t^{2}u^{2}(t+u)^{2}},$
$\displaystyle\hat{\sigma}_{DO2}=\hat{\sigma}_{DO1},$
$\displaystyle\hat{\sigma}_{DO3}=2\hat{\sigma}_{DO1},$
$\displaystyle\hat{\sigma}_{DF1}=-\frac{C_{F}}{N^{2}}\frac{2(t-u)}{tu(t+u)}+C_{F}\frac{3(t-u)(t^{2}+u^{2})}{tu(t+u)^{3}},$
$\displaystyle\hat{\sigma}_{DF2}=-\frac{C_{F}}{N^{2}}\frac{(t^{2}+u^{2})}{tu(t+u)^{2}}+C_{F}\frac{(t^{2}+u^{2})}{2t(t+u)^{3}}-\frac{C_{F}}{N}\frac{(t^{2}+u^{2})}{2t^{2}(t+u)^{2}}+\frac{C_{F}}{N^{3}}\frac{1}{2t^{2}},$
$\displaystyle\hat{\sigma}_{DF3}=\frac{C_{F}}{N^{2}}\frac{(t^{2}+u^{2})}{tu(t+u)^{2}}-C_{F}\frac{(t^{2}+u^{2})}{2u(t+u)^{3}}+\frac{C_{F}}{N}\frac{(t^{2}+u^{2})}{2u^{2}(t+u)^{2}}-\frac{C_{F}}{N^{3}}\frac{1}{2u^{2}},$
$\displaystyle\hat{\sigma}_{GF1}=0,\qquad\hat{\sigma}_{GF2}=-\hat{\sigma}_{DF2},\qquad\hat{\sigma}_{GF3}=\hat{\sigma}_{DF3}.$
(81)

Figure 5: Diagrams in the $q\bar{q}\to gg$ channel for the hard parts $S(k)$ and $S_{L}(z^{\prime},z)$ in eq. (51). For $S_{L}(z^{\prime},z)$, an extra gluon line connecting $\otimes$ and each of the black dots should be added for each diagram. |  Figure 6: Diagrams for the hard part in Fig. 2. For the upper left diagram, a quark loop with the reversed arrow also needs to be included.
---|---

(3) $gg\to gg$ channel (Figs. 7, 8)

$\displaystyle\hat{\sigma}_{G}=-\frac{N}{C_{F}}\frac{2(s^{2}+st+t^{2})^{2}(s-t)(2s+t)(s+2t)}{s^{3}t^{3}(s+t)^{3}},$
$\displaystyle\hat{\sigma}_{H}=0,$
$\displaystyle\hat{\sigma}_{N1}=-\frac{N}{C_{F}}\frac{2(t^{2}+tu+u^{2})^{2}(t-u)(2t^{2}+7tu+2u^{2})}{t^{3}u^{3}(t+u)^{3}},$
$\displaystyle\hat{\sigma}_{N2}=0,$
$\displaystyle\hat{\sigma}_{N3}=-\hat{\sigma}_{N1},$
$\displaystyle\hat{\sigma}_{DN1}=\frac{N}{C_{F}}\frac{(t^{2}+tu+u^{2})^{2}(t-u)(2t^{2}+3tu+2u^{2})}{t^{3}u^{3}(t+u)^{3}},$
$\displaystyle\hat{\sigma}_{DN2}=\hat{\sigma}_{DN1},$
$\displaystyle\hat{\sigma}_{DN3}=-2\hat{\sigma}_{DN1},$
$\displaystyle\hat{\sigma}_{DF1}=\frac{N}{C_{F}}\frac{(t^{2}+tu+u^{2})^{2}(t-u)(2t^{2}+7tu+2u^{2})}{t^{3}u^{3}(t+u)^{3}},$
$\displaystyle\hat{\sigma}_{DF2}=-\frac{N}{C_{F}}\frac{(t^{2}+tu+u^{2})^{2}(t-u)(2t+u)(t+2u)}{4t^{3}u^{3}(t+u)^{3}},$
$\displaystyle\hat{\sigma}_{DF3}=-\frac{N}{C_{F}}\frac{(t^{2}+tu+u^{2})^{2}(t-u)(2t+u)(t+2u)}{4t^{3}u^{3}(t+u)^{3}},$
$\displaystyle\hat{\sigma}_{GF1}=0,$
$\displaystyle\hat{\sigma}_{GF2}=-\hat{\sigma}_{DF2},$
$\displaystyle\hat{\sigma}_{GF3}=\hat{\sigma}_{DF3}.$ (82)

Figure 7: Diagrams in the $gg\to gg$ channel for the hard parts $S(k)$ and
$S_{L}(z^{\prime},z)$ in eq. (51). For $S_{L}(z^{\prime},z)$, an extra gluon
line connecting $\otimes$ and each of the black dots should be added for each
diagram. A white circle represents a four-gluon vertex, making clear the
difference from the attachment of the extra gluon line. Figure 8: Diagrams
contributing to the hard part in Fig. 2 in the $gg\to gg$ channel. For all
diagrams, a quark loop with the reversed arrow also needs to be included.

##  4 Summary and conclusion

In this paper we have studied the transverse polarization of a spin-1/2
hyperon produced in the unpolarized proton-proton collision,
$pp\to\Lambda^{\uparrow}X$, within the framework of the collinear twist-3
factorization, which is relevant for the large-$p_{T}$ hyperon production. We
focused on the contribution from the twist-3 gluon FFs, which had never been
studied in previous studies. To this end we have developed a formalism to
include all effects associated with the twist-3 gluon FFs. The twist-3 cross
section receives contributions from three types of the gluon FFs, i.e.,
intrinsic, kinematical and dynamical (purely gluonic and quark-antiquark-gluon
type) ones. Applying the formalism we have calculated the LO cross section for
$pp\to\Lambda^{\uparrow}X$. This completes the LO twist-3 cross section
combined with the known results for the other contributions from the twist-3
distribution in the unpolarized proton and the twist-3 quark FFs for the
hyperon. Using the EOM relation and the LIRs for the twist-3 gluon FFs, we
have shown that the derived cross section satisfies the color gauge invariance
and the frame independence. Since the formalism developed here is a general
one, it can be applied to other processes to which the twist-3 gluon FFs
contribute.

## Acknowledgments

We thank Daniel Pitonyak and Andreas Metz for discussions and comments on the
manuscript. This work has been supported by 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.

Appendices

##  Appendix A Ward identities for the gluon fragmentation channels

###  A.1 Derivation of the Ward identities

To get a twist-3 gluon fragmentation contribution to the cross section in a
gauge invariant form, one needs to fully utilize the Ward identities for the
hard parts (26)-(32). Here we present their derivation.

####  A.1.1 $q\bar{q}\to gg$ channel

Ward identities in this channel read

$\displaystyle{(k_{2}-k_{1})}_{\lambda}S^{\mu\nu\lambda,abc}_{L}(k_{1},k_{2})$
$\displaystyle=$
$\displaystyle\frac{-if^{abc}}{N^{2}-1}S^{\mu\nu}(k_{2})+G_{q\bar{q}\to
gg}^{\mu\nu,abc}(k_{1},k_{2}),$ (83)
$\displaystyle{k_{1}}_{\mu}S^{\mu\nu\lambda,abc}_{L}(k_{1},k_{2})$
$\displaystyle=$
$\displaystyle\frac{if^{abc}}{N^{2}-1}S^{\lambda\nu}(k_{2})+G_{q\bar{q}\to
gg}^{\lambda\nu,cab}(k_{2}-k_{1},k_{2}),$ (84)
$\displaystyle{k_{2}}_{\nu}S^{\mu\nu\lambda,abc}_{L}(k_{1},k_{2})$
$\displaystyle=$ $\displaystyle 0,$ (85)
$\displaystyle{(k_{2}-k_{1})}_{\lambda}S^{\mu\nu\lambda,abc}_{R}(k_{1},k_{2})$
$\displaystyle=$
$\displaystyle\frac{if^{abc}}{N^{2}-1}S^{\mu\nu}(k_{1})+\left(G_{q\bar{q}\to
gg}^{\nu\mu,bac}(k_{2},k_{1})\right)^{\star},$ (86)
$\displaystyle{k_{2}}_{\nu}S^{\mu\nu\lambda,abc}_{R}(k_{1},k_{2})$
$\displaystyle=$
$\displaystyle\frac{if^{abc}}{N^{2}-1}S^{\mu\lambda}(k_{1})+\left(G_{q\bar{q}\to
gg}^{\lambda\mu,cab}(k_{1}-k_{2},k_{1})\right)^{\star},$ (87)
$\displaystyle{k_{1}}_{\mu}S^{\mu\nu\lambda,abc}_{R}(k_{1},k_{2})$
$\displaystyle=$ $\displaystyle 0,$ (88)

where $S^{\mu\nu\lambda,abc}_{L(R)}(k_{1},k_{2})$ represents a hard part for a
three-gluon correlation functions which has two gluon legs in the left (right)
of the final-state cut, and $S^{\mu\nu}(k)\equiv
S^{\mu\nu}_{ab}(k)\delta^{ab}$ represents a hard part for a two-gluon
correlation functions. $G$-terms in the right hand side of these equations are
ghost terms.

Figure 9:  Ward identity (83) for the $q\bar{q}\to gg$ channel. The figure
shows the left side of the cut, representing the attachment of the scalar
polarized gluon (dotted line) with the momentum $k_{2}-k_{1}$. Note that the
hard part represented in the LHS does not contain the diagram in which the
scalar polarized gluon is directly attached to the gluon line fragmenting into
the final hadron, hence it appears in the first term of the RHS. For the
meaning of the notations in the figure, see Appendix A.2.

Figure 9 shows the Ward identity (83), which states that the attachment of the
scalar polarized gluon with the momentum $k_{2}-k_{1}$ to the hard part
$S_{L\,\mu\nu\lambda}^{abc}(k_{1},k_{2})$ can be decomposed into the two-body
hard part and the ghost terms. To identify the ghost terms, we consider the
diagrams in Fig. 10. The first term in the LHS of the figure represents the
hard part $S_{L\,\mu\nu\lambda}^{abc}(k_{1},k_{2})$ and the second term
represents the attachment of the scalar polarized gluon to the gluon line
fragmenting into $\Lambda^{\uparrow}$. The LO diagrams can be classified into
three types as shown in the RHS of this figure.

Figure 10: Attachment of the scalar polarized gluon of momentum $k_{2}-k_{1}$
to $qq\to gg$ diagrams. In the RHS, it is implied that the scalar polarized
gluon is attached one of the black dots in all possible ways.

Using the tree level Ward identities (See A.2), each diagram in Fig. 10 can be
decomposed into several pieces, and some of them cancel each others owing to
the on-shell condition of the external lines (See A.3). Taking these facts
into account we rearrange each term in the RHS of Fig. 10.

First, diagrams in Fig. 10 (a) can be rewritten as in Fig. 11.

Figure 11: Decomposition of Fig. 10 (a) and the resulting terms due to the
cancellation among diagrams. Diagrams (a)-1, (a)-5 and (a)-12 vanish owing to
the on-shell condition. Combination of (a)-2, (a)-3 and (a)-7 cancels that of
(a)-4, (a)-6 and (a)-10 owing to (100).

Similarly Fig. 10 (b) can be rewritten as in Fig. 12.

Figure 12: Decomposition of Fig. 10 (b) and the resulting terms due to the
cancellation among diagrams. Diagrams (b)-1, (b)-5 and (b)-12 vanish owing to
the on-shell condition. Combination of (b)-2, (b)-4 and (b)-10 cancels that of
(b)-3, (b)-6 and (b)-7 owing to (100).

Likewise Fig. 10 (c) can be rewritten as in Fig. 13.

Figure 13:  Decomposition of Fig. 10 (c) and the resulting terms due to the
cancellation among the diagrams. Contributions from (c)-1, (c)-3, (c)-6 and
(c)-14 vanish due to the on-shell conditions. Diagrams (c)-2, (c)-4 and (c)-5
cancel each others by (100). Diagrams (c)-7, (c)-9, (c)-12 and (c)-15 cancel
each others by (101).

Using the on-shell condition, we find that diagrams (a)-8-i, (a)-11-i,
(b)-8-i, (b)-11-i and (c)-8-ii vanish, and diagrams (c)-10-ii, (c)-10-iii and
(c)-13-ii also vanish. In addition, contribution from (a)-8-ii, (b)-8-ii and
(c)-10-i cancels among them, and likewise for the combination of (a)-11-ii,
(b)-11-ii and (c)-13-i by (100). Therefore, besides (a)-9, (b)-9 and (c)-11
(see Fig. 9), remaining contributions (c)-8-i and (c)-13-iii become the ghost
term in (83). The ghost term takes the following structure in the $q\bar{q}\to
gg$ channel444In these appendices, we follow the convention of [57] for the
Feynman rule..

$\displaystyle G_{q\bar{q}\to gg}^{\mu\nu,abc}(k_{1},k_{2})$ $\displaystyle=$
$\displaystyle-\Big{\\{}(-if^{cef})(-if^{fda})d^{\mu\sigma}(k_{1})\frac{{(k_{2}-k_{1})}_{\rho}}{(xp+x^{\prime}p^{\prime}-k_{1})^{2}}$
(89) $\displaystyle+$ $\displaystyle(-if^{aef})(-if^{fdc})d^{\mu}_{\
\rho}(k_{1})\frac{{(k_{2}-k_{1})}^{\sigma}}{(xp+x^{\prime}p^{\prime}-(k_{2}-k_{1}))^{2}}\Big{\\}}H^{d}_{\sigma}(xp,x^{\prime}p^{\prime}){1\over(xp+x^{\prime}p^{\prime})^{2}}$
$\displaystyle\times$
$\displaystyle{P}^{\rho\gamma}(xp+x^{\prime}p^{\prime}-k_{2})\times\left(M^{\nu,be}_{\gamma}(k_{2})\right)^{\star},$

where $d^{\mu\sigma}(k_{1})\equiv
k_{1}^{2}g^{\mu\sigma}-k_{1}^{\mu}k_{1}^{\sigma}$,
$H^{d}_{\sigma}(xp,x^{\prime}p^{\prime})$ represents the quark-line part in
the left of the cut, which is connected to the right of the cut
$\left(M^{\nu,be}_{\gamma}(k_{2})\right)^{\star}$, and the gluon’s
polarization tensor is given by

${P}^{\rho\gamma}(xp+x^{\prime}p^{\prime}-k_{2})\equiv-g^{\rho\gamma}+{(xp+x^{\prime}p^{\prime}-k_{2})^{\rho}k_{1}^{\gamma}+(xp+x^{\prime}p^{\prime}-k_{2})^{\gamma}k_{1}^{\rho}\over(xp+x^{\prime}p^{\prime}-k_{2})\cdot
k_{1}}.$ (90)

Diagrammatically the ghost term (89) can be written as in Fig. 14.

Figure 14: Diagrams for the ghost-term (89) in the $q\bar{q}\to gg$ channel.

This completes the derivation of the Ward identity in the $q\bar{q}\to gg$
channel.

####  A.1.2  $qg\to gq$ channel

Ward identities in this channel are given by (83)-(88) except that the ghost
terms $G_{q\bar{q}\to gg}^{\mu\nu,abc}(k_{1},k_{2})$ are replaced by $G_{qg\to
gq}^{\mu\nu,abc}(k_{1},k_{2})$ which takes the following structure:

$\displaystyle G_{qg\to gq}^{\mu\nu,abc}(k_{1},k_{2})$ $\displaystyle=$
$\displaystyle-\Big{\\{}(-if^{cef})(-if^{fda})d^{\mu\alpha}(k_{1})\frac{{(k_{2}-k_{1})}^{\lambda}}{(x^{\prime}p^{\prime}-k_{1})^{2}}$
(91) $\displaystyle+$
$\displaystyle(-if^{aef})(-if^{fdc})d^{\mu\lambda}(k_{1})\frac{{(k_{2}-k_{1})}^{\alpha}}{(x^{\prime}p^{\prime}-(k_{2}-k_{1}))^{2}}\Big{\\}}H^{\prime
e}_{\lambda}(xp,xp+x^{\prime}p^{\prime}-k_{2})$ $\displaystyle\times$
$\displaystyle{1\over(x^{\prime}p^{\prime}-k_{2})^{2}}\left(M^{\nu,bd}_{\alpha}(k_{2})\right)^{\star},$

where $H^{\prime e}_{\lambda}(xp,xp+x^{\prime}p^{\prime}-k_{2})$ is the quark-
line part connected to the left of the cut
$\left(M^{\nu,bd}_{\alpha}(k_{2})\right)^{\star}$.

####  A.1.3  $gg\to gg$ channel

Ward identities in this channel are given by (83)-(88) in which
$G_{q\bar{q}\to gg}^{\mu\nu,abc}(k_{1},k_{2})$ is replaced by the following
$G_{gg\to gg}^{\mu\nu,abc}(k_{1},k_{2})$:

$\displaystyle G_{gg\to gg}^{\mu\nu,abc}(k_{1},k_{2})$ $\displaystyle\equiv$
$\displaystyle G_{1{gg\to gg}}^{\mu\nu,abc}(k_{1},k_{2})+G_{2{gg\to
gg}}^{\mu\nu,abc}(k_{1},k_{2})+G_{3{gg\to
gg}}^{\mu\nu,abc}(k_{1},k_{2})+G_{4{gg\to gg}}^{\mu\nu,abc}(k_{1},k_{2}),$
(92)

where $G_{1{gg\to gg}}$ through $G_{4{gg\to gg}}$ are the ghost terms which
occur from four types of diagrams shown in Fig. 15 for the hard scattering
amplitudes in this channel.

Figure 15: Diagrammatic representation of the ghost terms in the $gg\to gg$
channel. Depending on the position of attachment of the scalar polarized
gluon, diagrams can be classified into 4 types. Figure $G_{3}$ represents
diagrams in which the scalar polarized gluon is attached to an internal
propagator.

Each term in (92) takes the following structures:

$\displaystyle G_{1{gg\to gg}}^{\mu\nu,abc}(k_{1},k_{2})$ $\displaystyle=$
$\displaystyle(-if^{cdg})\frac{{(k_{2}-k_{1})}^{\alpha}}{(x^{\prime}p^{\prime}-(k_{2}-k_{1}))^{2}}\Big{\\{}(-if^{gfh})(-if^{hea})d^{\mu\rho}(k_{1})\frac{{(x^{\prime}p^{\prime}-(k_{2}-k_{1}))}^{\beta}}{(xp+x^{\prime}p^{\prime}-(k_{2}-k_{1}))^{2}}$
(93) $\displaystyle+$
$\displaystyle(-if^{geh})(-if^{hfa})d^{\mu\beta}(k_{1})\frac{{(x^{\prime}p^{\prime}-(k_{2}-k_{1}))}_{\rho}}{(xp-
k_{1})^{2}}$ $\displaystyle-$
$\displaystyle(-if^{gha})V^{\beta,hfe}_{\sigma\rho}d^{\mu\sigma}(k_{1})\frac{1}{(x^{\prime}p^{\prime}-k_{2})^{2}}\Big{\\}}$
$\displaystyle\times$
$\displaystyle{P}^{\rho\gamma}(xp+x^{\prime}p^{\prime}-k_{2})\times\left(M^{be,df}_{\nu\gamma,\alpha\beta}(k_{2})\right)^{\star},$
$\displaystyle G_{2{gg\to gg}}^{\mu\nu,abc}(k_{1},k_{2})$ $\displaystyle=$
$\displaystyle(-if^{cfg})\frac{{(k_{2}-k_{1})}^{\beta}}{(xp-(k_{2}-k_{1}))^{2}}\Big{\\{}(-if^{gdh})(-if^{hea})d^{\mu\rho}(k_{1})\frac{{(xp-(k_{2}-k_{1}))}^{\alpha}}{(xp+x^{\prime}p^{\prime}-(k_{2}-k_{1}))^{2}}$
(94) $\displaystyle-$
$\displaystyle(-if^{gha})V^{\alpha,hde}_{\sigma\rho}d^{\mu\sigma}(k_{1})\frac{1}{(xp-
k_{2})^{2}}$ $\displaystyle+$
$\displaystyle(-if^{geh})(-if^{hda})d^{\mu\alpha}(k_{1})\frac{{(xp-(k_{2}-k_{1}))}_{\rho}}{(x^{\prime}p^{\prime}-k_{1})^{2}}\Big{\\}}$
$\displaystyle\times$
$\displaystyle{P}^{\rho\gamma}(xp+x^{\prime}p^{\prime}-k_{2})\times\left(M^{be,df}_{\nu\gamma,\alpha\beta}(k_{2})\right)^{\star},$
$\displaystyle G_{3{gg\to gg}}^{\mu\nu,abc}(k_{1},k_{2})$ $\displaystyle=$
$\displaystyle(-if^{cgh}){(k_{2}-k_{1})}^{\sigma}\Big{\\{}-(-if^{hea})V^{\alpha\beta,dfg}_{\sigma}d^{\mu\rho}(k_{1})\frac{1}{(xp+x^{\prime}p^{\prime}-(k_{2}-k_{1}))^{2}}\frac{1}{(xp+x^{\prime}p^{\prime})^{2}}$
(95) $\displaystyle+$
$\displaystyle(-if^{gfa})V^{\alpha\rho,hde}_{\sigma}d^{\mu\beta}(k_{1})\frac{1}{(xp-
k_{1})^{2}}\frac{1}{{(xp-k_{2})}^{2}}$ $\displaystyle+$
$\displaystyle(-if^{gda})V^{\beta\rho,hfe}_{\sigma}d^{\mu\alpha}(k_{1})\frac{1}{(x^{\prime}p^{\prime}-k_{2})^{2}}\frac{1}{(x^{\prime}p^{\prime}-k_{1})^{2}}\Big{\\}}$
$\displaystyle\times$
$\displaystyle{P}_{\rho}^{\gamma}(xp+x^{\prime}p^{\prime}-k_{2})\times\left(M^{be,df}_{\nu\gamma,\alpha\beta}(k_{2})\right)^{\star},$

and

$\displaystyle G_{4{gg\to gg}}^{\mu\nu,abc}(k_{1},k_{2})$ $\displaystyle=$
$\displaystyle(-if^{ceg})\frac{{(k_{2}-k_{1})}_{\rho}}{(xp+x^{\prime}p^{\prime}-k_{1})^{2}}\Big{\\{}-(-if^{gha})V^{\alpha\beta,dfh}_{\sigma}d^{\mu\sigma}(k_{1})\frac{1}{(xp+x^{\prime}p^{\prime})^{2}}$
(96) $\displaystyle+$
$\displaystyle(-if^{gdh})(-if^{hfa})d^{\mu\beta}(k_{1})\frac{{(k_{1}-xp)}^{\alpha}}{(xp-
k_{1})^{2}}$ $\displaystyle+$
$\displaystyle(-if^{gfh})(-if^{hda})d^{\mu\alpha}(k_{1})\frac{{(k_{1}-x^{\prime}p^{\prime})}^{\beta}}{(x^{\prime}p^{\prime}-k_{1})^{2}}\Big{\\}}$
$\displaystyle\times$
$\displaystyle{P}^{\rho\gamma}(xp+x^{\prime}p^{\prime}-k_{2})\times\left(M^{be,df}_{\nu\gamma,\alpha\beta}(k_{2})\right)^{\star},$

where $V^{\beta,hfe}_{\sigma\rho}$ represents an appropriate three-gluon
vertex in each channel.

###  A.2 Decomposition of vertices

Attachment of the scalar polarized gluon to a quark or gluon line can be
decomposed as follows:

(i) quark-gluon vertex (Fig. 16):

$\displaystyle k^{\alpha}\cdot(\gamma_{\alpha})_{ij}T^{a}=(\hbox
to0.0pt{/\hss}{\mkern-1.0mup}+\hbox
to0.0pt{/\hss}{\mkern-1.0muk})_{ij}T^{a}-(\hbox
to0.0pt{/\hss}{\mkern-1.0mup})_{ij}T^{a},$ (97)

where $T^{a}$ is the generator of color SU(N) group.

Figure 16: Attachment of a scalar polarized gluon to a quark-gluon vertex
corresponding to (97).

(ii) 3-gluon vertex (Fig. 17):

$\displaystyle k^{\alpha}\cdot V^{abc}_{\alpha\beta\gamma}(k,p,-p-k)={\it
F}^{abc}d_{\beta\gamma}(p+k)+{\it F}^{acb}d_{\beta\gamma}(p),$ (98)

where $V^{abc}_{\alpha\beta\gamma}(p_{1},p_{2},p_{3})\equiv
F^{abc}\left[(p_{1}-p_{2})_{\gamma}g_{\alpha\beta}+(p_{2}-p_{3})_{\alpha}g_{\beta\gamma}+(p_{3}-p_{1})_{\beta}g_{\alpha\gamma}\right]$
and $F^{abc}\equiv-if^{abc}$.

Figure 17: Attachment of a scalar polarized gluon through a 3-gluon vertex
corresponding to (98).

Furthermore, $d^{\beta\gamma}$ in the above equations is decomposed as (Fig.
18)

$\displaystyle{\it F}^{acb}d^{\beta\gamma}(p)=p^{2}g^{\beta\gamma}{\it
F}^{acb}+(-p^{\beta}p^{\gamma}){\it F}^{acb}.$ (99)

Figure 18: Decomposition of $d^{\beta\gamma}$ in (99).

###  A.3 Cancellation among vertices

After decomposition of vertices, the following cancellation holds among
vertices.

(i) Quark-gluon vertex (Fig. 19)

$\displaystyle\left[-(T^{a}T^{b})_{lm}+(T^{b}T^{a})_{lm}+{\it
F^{alb}}(T^{l})_{lm}\right](\gamma^{\alpha})_{ij}=0.$ (100)

Figure 19: Cancelation for quark-gluon vertices (100).

(ii) Purely gluon vertex (Fig. 20)

$\displaystyle(-P_{X})^{\sigma}\cdot{\rm W}^{abcd}_{\alpha\beta\gamma\sigma}$
$\displaystyle+$ $\displaystyle{\it
F^{ald}}V^{lbc}_{\alpha\beta\gamma}(-p_{2}-p_{3},p_{2},p_{3})$ (101)
$\displaystyle+$ $\displaystyle{\it
F^{bld}}V^{alc}_{\alpha\beta\gamma}(p_{1},-p_{1}-p_{3},p_{3})+{\it
F^{cld}}V^{abl}_{\alpha\beta\gamma}(p_{1},p_{2},-p_{1}-p_{2})=0,$

where $P_{X}\equiv p_{1}+p_{2}+p_{3}$.

Figure 20: Cancellation among 3- and 4-gluon vertices (101).

##  Appendix B Color gauge invariance of the twist-3 cross section

Here we prove that the twist-3 cross section in (51) supplemented by the
quark-antiquark-gluon contribution shown in Fig. 2 is color gauge invariant
owing to the EOM relation (9). We illustrate this property for the $qg\to gq$
channel. The proof for other channels is essentially the same. The cross
section in this channel can be written as

$\displaystyle
E_{h}\frac{d\sigma(p,p^{\prime},P_{h};S_{\perp})}{d^{3}P_{h}}=\frac{1}{16\pi^{2}S_{E}}\int\frac{dx}{x}f_{1}(x)\int\frac{dx^{\prime}}{x^{\prime}}G(x^{\prime})\int{dz}\left(-{1\over
2}g_{\perp}^{\rho\sigma}(p^{\prime})\right)W_{\rho\sigma}(xp,x^{\prime}p,P_{h}/z),$

where $G(x^{\prime})$ is the unpolarized gluon distribution in the proton with
momentum $p^{\prime}$, and the Lorentz indices $\rho$ and $\sigma$ are
contracted with $g_{\perp}^{\rho\sigma}(p^{\prime})\equiv
g^{\rho\sigma}-p^{\prime\rho}n^{\prime\sigma}-p^{\prime\sigma}n^{\prime\rho}$
where $n^{\prime}$ is the usual lightlike vector satisfying $p^{\prime}\cdot
n^{\prime}=1$ to extract the contribution from $G(x^{\prime})$. From (51), we
can read off $W_{\rho\sigma}(xp,x^{\prime}p^{\prime},P_{h}/z)$ as

$\displaystyle
W_{\rho\sigma}(xp,x^{\prime}p^{\prime},P_{h}/z)={\Omega^{\mu}_{\
\alpha}}{\Omega^{\nu}_{\ \beta}}{\rm
Tr}\left[\hat{\Gamma}^{\alpha\beta}(z)S_{\mu\nu,\rho\sigma}(P_{h}/z)\right]$
$\displaystyle-i\,{\Omega^{\mu}_{\ \alpha}}{\Omega^{\nu}_{\
\beta}}{\Omega^{\lambda}_{\ \gamma}}{\rm
Tr}\left[\hat{\Gamma}_{\partial}^{\alpha\beta\gamma}(z)\left.\frac{\partial
S_{\mu\nu,\rho\sigma}(k)}{\partial
k^{\lambda}}\right|_{c.l.}\right]+{\Re}\left\\{i\,{\Omega^{\mu}_{\
\alpha}}{\Omega^{\nu}_{\ \beta}}{\Omega^{\lambda}_{\ \gamma}}{1\over
z}\int\frac{dz^{\prime}}{z^{\prime}}\,\left(\frac{1}{1/z-1/z^{\prime}}\right)\right.$
$\displaystyle\qquad\left.\times{\rm
Tr}\left[\left(-\frac{if^{abc}}{N}\hat{\Gamma}_{FA}^{\alpha\beta\gamma}\left(\frac{1}{z^{\prime}},\frac{1}{z}\right)+d^{abc}\frac{N}{N^{2}-4}\hat{\Gamma}_{FS}^{\alpha\beta\gamma}\left(\frac{1}{z^{\prime}},\frac{1}{z}\right)\right)S^{L,abc}_{\mu\nu\lambda,\rho\sigma}(z^{\prime},z)\right]\right\\}$
$\displaystyle+w^{q\bar{q}g}_{\rho\sigma}(xp,x^{\prime}p^{\prime},P_{h}/z),$
(103)

where the hard part $S^{ab}_{\mu\nu,\rho\sigma}(k)$ is related to the hard
part $S^{ab}_{\mu\nu}(k)$ in (51) by $\left(-{1\over
2}g_{\perp}^{\rho\sigma}(p^{\prime})\right)S^{ab}_{\mu\nu,\rho\sigma}(k)=S^{ab}_{\mu\nu}(k)$
and likewise for $S^{abc}_{\mu\nu\lambda,\rho\sigma}(k)$, and
$w^{q\bar{q}g}_{\rho\sigma}(xp,x^{\prime}p^{\prime},P_{h}/z)$ represents the
contribution from Fig. 2. Then the color gauge invariance of the cross section
implies

$\displaystyle
x^{\prime}p^{\prime\rho}W_{\rho\sigma}(xp,x^{\prime}p^{\prime},P_{h}/z)=x^{\prime}p^{\prime\sigma}W_{\rho\sigma}(xp,x^{\prime}p^{\prime},P_{h}/z)=0.$
(104)

To show (104), we use the EOM relation (9) and eliminate the intrinsic twist-3
FF $\Delta\widehat{G}_{3\bar{T}}(z)$ in the first term of the RHS of (103).
Then $W_{\rho\sigma}$ can be decomposed into three pieces:

$\displaystyle W_{\rho\sigma}(xp,x^{\prime}p^{\prime},P_{h}/z)=W^{({\rm
i})}_{\rho\sigma}(xp,x^{\prime}p^{\prime},P_{h}/z)+W^{({\rm
ii})}_{\rho\sigma}(xp,x^{\prime}p^{\prime},P_{h}/z)+W^{({\rm
iii})}_{\rho\sigma}(xp,x^{\prime}p^{\prime},P_{h}/z),$ (105)

where $W^{({\rm i})}_{\rho\sigma}$ represents the contribution from the
dynamical 3-body correlation function in (2.1) and (6), $W^{({\rm
ii})}_{\rho\sigma}$ represents the one from the dynamical
$q\bar{q}g$-correlation function in (8) and $W^{({\rm iii})}_{\rho\sigma}$
represents the one from the kinematical FFs in (2). To show (104), it suffices
to prove that each term of (105) separately satisfies (104).

###  B.1 Contribution from dynamical FFs: $W^{({\rm i})}_{\rho\sigma}$

We first show $W^{({\rm i})}_{\rho\sigma}$ satisfies (104). Relevant diagrams
are shown in Figs. 21 (a), (b) and (c). Here Fig. 21 (b) is meant to contain
diagrams including those of Fig. 21 (c), which is not a part of partonic cross
sections for the dynamical FFs. Therefore, after using the EOM relations, we
get the following combination of the hard cross section,

$\displaystyle W^{({\rm
i})}_{\rho\sigma}=W^{(a)}_{\rho\sigma}+\left(W^{(b)L}_{\rho\sigma}-W^{(c)L}_{\rho\sigma}\right)+\left(W^{(b)R}_{\rho\sigma}-W^{(c)R}_{\rho\sigma}\right),$
(106)

where the hard part with index $L$ indicates diagrams in Fig. 21 (b) and (c),
and that with $R$ indicates their hermitian conjugate diagrams. We define the
amplitude $M^{ad}_{\rho,\alpha}$ for $qg\to gq$ scattering and
$M^{acd}_{\rho,\alpha\gamma}$ for $qg\to ggq$ scattering as shown in Fig. 21.

Figure 21: Diagrams contributing to $W^{({\rm i})}_{\rho\sigma}$.
$M^{ad}_{\rho,\alpha}$ and $M^{acd}_{\rho,\alpha\gamma}$, respectively,
represent the $qg\to gq$ and $qg\to ggq$ scattering amplitudes, which
constitute the hard cross section as shown in the figure.

With the physical polarizations for the Lorentz indices $\alpha$ and $\gamma$,
these amplitudes satisfy the following relations,

$\displaystyle x^{\prime}p^{\prime\rho}M^{ad}_{\rho,\alpha}=0,\qquad
x^{\prime}p^{\prime\rho}M^{acd}_{\rho,\alpha\gamma}=0.$ (107)

This implies

$\displaystyle x^{\prime}p^{\prime\rho}W^{(b)L}_{\rho\sigma}=0,\qquad
x^{\prime}p^{\prime\rho}W^{(b)R}_{\rho\sigma}=0,\qquad
x^{\prime}p^{\prime\rho}W^{(c)R}_{\rho\sigma}=0.$ (108)

We thus have

$\displaystyle x^{\prime}p^{\prime\rho}W^{({\rm
i})}_{\rho\sigma}=x^{\prime}p^{\prime\rho}W^{(a)}_{\rho\sigma}-x^{\prime}p^{\prime\rho}W^{(c)L}_{\rho\sigma}.$
(109)

The first term in the RHS of (109) reads

$\displaystyle x^{\prime}p^{\prime\rho}W^{(a)}_{\rho\sigma}$ $\displaystyle=$
$\displaystyle\frac{F^{ade}}{(x^{\prime}p^{\prime}-P_{h}/z)^{2}}d^{\alpha\rho}(P_{h}/z)\times{\rm
Tr}\left[\hbox to0.0pt{/\hss}{\mkern-1.0muP_{\it X}}\gamma_{\rho}\,\,\hbox
to0.0pt{/\hss}{\mkern-1.0mup}\,T^{e}M^{\star,ad}_{\sigma,\beta}\right]$ (110)
$\displaystyle\times$ $\displaystyle\left[-w^{\alpha}\epsilon^{\beta
P_{h}wS_{\perp}}-w^{\beta}\epsilon^{\alpha
P_{h}wS_{\perp}}\right]\left.{\Delta\hat{G}_{3\bar{T}}(z)}\right|_{\rm
3-gluon}$ $\displaystyle=$ $\displaystyle\frac{1}{z}\epsilon^{\beta
P_{h}wS_{\perp}}\frac{F^{ade}}{(x^{\prime}p^{\prime}-P_{h}/z)^{2}}\times{\rm
Tr}\left[\hbox to0.0pt{/\hss}{\mkern-1.0muP_{\it X}}\hbox
to0.0pt{/\hss}{\mkern-1.0muP_{h}}\,\,\hbox
to0.0pt{/\hss}{\mkern-1.0mup}\,T^{e}M^{\star,ad}_{\sigma,\beta}\right]\left.\frac{\Delta\hat{G}_{3\bar{T}}(z)}{z}\right|_{\rm
3-gluon},$

where $P_{X}=xp+x^{\prime}p^{\prime}-P_{h}/z$, $F^{abc}\equiv-if^{abc}$ and
$d^{\mu\sigma}(k)\equiv k^{2}g^{\mu\sigma}-k^{\mu}k^{\sigma}$ and we have
denoted the $\widehat{N}_{i}$-contribution to $\Delta\hat{G}_{3\bar{T}}(z)$ in
the RHS of (9) by $\Delta\hat{G}_{3\bar{T}}(z)|_{\rm 3-gluon}$. This result is
shown in Fig. 22.

Figure 22: Diagrammatic representation for
$x^{\prime}p^{\prime\rho}W^{(a)}_{\rho\sigma}$.

Next we consider the second term in the RHS of (109). Since its hard part is
proportional to $f^{abc}$, the contribution from
$\hat{\Gamma}_{FS}^{\alpha\beta\gamma}$ drops. The contribution from
$\hat{\Gamma}_{AS}^{\alpha\beta\gamma}$ can be written as

$\displaystyle x^{\prime}p^{\prime\rho}W^{(c)L}_{\rho\sigma}$
$\displaystyle\sim-\frac{1}{2}z\epsilon^{\beta
P_{h}wS_{\perp}}\frac{F^{ade}}{(x^{\prime}p^{\prime}-P_{h}/z)^{2}}\int
d\left(\frac{1}{z^{\prime}}\right)\frac{z^{\prime}}{z^{2}}\frac{1}{1-z/z^{\prime}}$
$\displaystyle\times\left(1-2\frac{z}{z^{\prime}}\right){\rm Tr}\left[\hbox
to0.0pt{/\hss}{\mkern-1.0muP_{\it X}}\hbox
to0.0pt{/\hss}{\mkern-1.0muP_{h}}\,\,\hbox
to0.0pt{/\hss}{\mkern-1.0mup}\,T^{e}M^{\star,ad}_{\sigma,\beta}\right]\times\left[-\widehat{N}_{2}\left(\frac{1}{z}-\frac{1}{z^{\prime}},\frac{1}{z}\right)+\widehat{N}_{2}\left(\frac{1}{z^{\prime}},\frac{1}{z}\right)+2\widehat{N}_{1}\left(\frac{1}{z^{\prime}},\frac{1}{z}\right)\right]$
$\displaystyle=-\frac{1}{2}z\epsilon^{\beta
P_{h}wS_{\perp}}\frac{F^{ade}}{(x^{\prime}p^{\prime}-P_{h}/z)^{2}}{\rm
Tr}\left[\hbox to0.0pt{/\hss}{\mkern-1.0muP_{\it X}}\hbox
to0.0pt{/\hss}{\mkern-1.0muP_{h}}\,\,\hbox
to0.0pt{/\hss}{\mkern-1.0mup}\,T^{e}M^{\star,ad}_{\sigma,\beta}\right]$
$\displaystyle\times\int
d\left(\frac{1}{z^{\prime}}\right)\frac{1}{z}\left\\{-\frac{1}{1-z/z^{\prime}}+\frac{z^{\prime}}{z}\right\\}\left[-\widehat{N}_{2}\left(\frac{1}{z}-\frac{1}{z^{\prime}},\frac{1}{z}\right)+\widehat{N}_{2}\left(\frac{1}{z^{\prime}},\frac{1}{z}\right)+2\widehat{N}_{1}\left(\frac{1}{z^{\prime}},\frac{1}{z}\right)\right]$
$\displaystyle=-\frac{1}{2}z\epsilon^{\beta
P_{h}wS_{\perp}}\frac{F^{ade}}{(x^{\prime}p^{\prime}-P_{h}/z)^{2}}{\rm
Tr}\left[\hbox to0.0pt{/\hss}{\mkern-1.0muP_{\it X}}\hbox
to0.0pt{/\hss}{\mkern-1.0muP_{h}}\,\,\hbox
to0.0pt{/\hss}{\mkern-1.0mup}\,T^{e}M^{\star,ad}_{\sigma,\beta}\right]$
$\displaystyle\times\int
d\left(\frac{1}{z^{\prime}}\right)\frac{1}{z}\left(-2\times\frac{1}{1-z/z^{\prime}}\right)\left[-\widehat{N}_{2}\left(\frac{1}{z}-\frac{1}{z^{\prime}},\frac{1}{z}\right)+\widehat{N}_{2}\left(\frac{1}{z^{\prime}},\frac{1}{z}\right)+2\widehat{N}_{1}\left(\frac{1}{z^{\prime}},\frac{1}{z}\right)\right]$
$\displaystyle=\frac{1}{z}\epsilon^{\beta
P_{h}wS_{\perp}}\frac{F^{ade}}{(x^{\prime}p^{\prime}-P_{h}/z)^{2}}{\rm
Tr}\left[\hbox to0.0pt{/\hss}{\mkern-1.0muP_{\it X}}\hbox
to0.0pt{/\hss}{\mkern-1.0muP_{h}}\,\,\hbox
to0.0pt{/\hss}{\mkern-1.0mup}\,T^{e}M^{\star,ad}_{\sigma,\beta}\right]$
$\displaystyle\times\int
d\left(\frac{1}{z^{\prime}}\right)\frac{1}{1/z-1/z^{\prime}}\left[-\widehat{N}_{2}\left(\frac{1}{z}-\frac{1}{z^{\prime}},\frac{1}{z}\right)+\widehat{N}_{2}\left(\frac{1}{z^{\prime}},\frac{1}{z}\right)+2\widehat{N}_{1}\left(\frac{1}{z^{\prime}},\frac{1}{z}\right)\right],$
(111)

which can be diagrammatically written as Fig. 23. Using the EOM relation (9),
one finds (110) is equal to (111), which implies (109)=0. This completes the
proof for the relation $x^{\prime}p^{\prime\rho}W^{({\rm
i})}_{\rho\sigma}(xp,x^{\prime}p^{\prime},P_{h}/z)=0$.

Figure 23: Diagrammatic representation for
$x^{\prime}p^{\prime\rho}W^{(c)L}_{\rho\sigma}$ in the $qg\to gq$ channel.

###  B.2 Contribution from quark-antiquark-gluon FF: $W^{({\rm
ii})}_{\rho\sigma}$

Next we show $W^{({\rm ii})}_{\rho\sigma}$ shown in Fig. 24 satisfies (104).
Fig. 24 (b)’ is defined to include the contribution of the type Fig. 24 (c)’
which is not a part of the hard cross section for a quark-antiquark-gluon
contribution. Accordingly the corresponding hard cross section is written as

$\displaystyle W^{({\rm
iii})}_{\rho\sigma}=W^{(a)}_{\rho\sigma}+\left(W^{(b)^{\prime}L}_{\rho\sigma}-W^{(c)^{\prime}L}_{\rho\sigma}\right)+\left(W^{(b)^{\prime}R}_{\rho\sigma}-W^{(c)^{\prime}R}_{\rho\sigma}\right).$
(112)

Figure 24: Diagrams contributing to $W^{({\rm iii})}_{\rho\sigma}$.
$M^{d}_{\rho}$ represents the $gq\to q\bar{q}q$ scattering amplitude.

Taking into account of the relation (107), we have

$\displaystyle x^{\prime}p^{\prime\rho}W^{(b)^{\prime}L}_{\rho\sigma}=0,\qquad
x^{\prime}p^{\prime\rho}W^{(b)^{\prime}R}_{\rho\sigma}=0,\qquad
x^{\prime}p^{\prime\rho}W^{(c)^{\prime}R}_{\rho\sigma}=0.$ (113)

Therefore we obtain

$\displaystyle x^{\prime}p^{\prime\rho}W^{({\rm
iii})}_{\rho\sigma}=x^{\prime}p^{\prime\rho}W^{(a)}_{\rho\sigma}-x^{\prime}p^{\prime\rho}W^{(c)^{\prime}L}_{\rho\sigma}.$
(114)

The second term in the RHS of this equation can be written as

$\displaystyle x^{\prime}p^{\prime\rho}W^{(c)^{\prime}L}_{\rho\sigma}$
$\displaystyle\sim$ $\displaystyle-\frac{1}{z}\frac{2}{C_{F}}\epsilon^{\beta
P_{h}wS_{\perp}}\frac{F^{ade}}{(x^{\prime}p^{\prime}-P_{h}/z)^{2}}\times{\rm
Tr}\left[\hbox to0.0pt{/\hss}{\mkern-1.0muP_{\it X}}\hbox
to0.0pt{/\hss}{\mkern-1.0muP_{h}}\,\,\hbox
to0.0pt{/\hss}{\mkern-1.0mup}\,T^{e}M^{\star,ad}_{\sigma,\beta}\right]\widetilde{D}_{FT},$
(115)

which is diagrammatically written as Fig. 25. We again find the coefficient of
$(2/C_{F})\widetilde{D}_{FT}$ in (115) is equal to the coefficient of
$\Delta\hat{G}_{3\bar{T}}(z)/z|_{\rm 3-gluon}$ in (110), which shows (114)
vanishes owing to the EOM relation (9).

Figure 25: Diagrammatic representation of
$x^{\prime}p^{\prime\rho}W^{(c)^{\prime}L}_{\rho\sigma}$.

###  B.3 Contribution from kinematical FFs: $W^{({\rm iii})}_{\rho\sigma}$

Using the explicit forms of the intrinsic and kinematical FFs in (1) and (2),
$W^{({\rm iii})}_{\rho\sigma}(xp,x^{\prime}p^{\prime},P_{h}/z)$ can be written
as

$\displaystyle W^{({\rm
iii})}_{\rho\sigma}(xp,x^{\prime}p^{\prime},P_{h}/z)={M_{h}\over 2}{\rm
Tr}\left[\left(\epsilon^{P_{h}wS_{\perp}\mu}w^{\nu}+\epsilon^{P_{h}wS_{\perp}\nu}w^{\mu}\right)S_{\mu\nu,\rho\sigma}({P_{h}/z})\right]z\,\left(\widehat{G}_{T}^{(1)}(z)+\Delta\widehat{H}_{T}^{(1)}(z)\right)$
$\displaystyle\qquad\qquad-{M_{h}\over 2}{\rm
Tr}\left[g_{\perp}^{\mu\nu}\epsilon^{P_{h}wS_{\perp}\lambda}\left.{\partial
S_{\mu\nu,\rho\sigma}(k)\over\partial
k^{\lambda}}\right|_{c.l.}\right]\widehat{G}_{T}^{(1)}(z)$
$\displaystyle\qquad\qquad-{M_{h}\over 8}{\rm
Tr}\left[\left(\epsilon^{P_{h}wS_{\perp}\\{\mu}g_{\perp}^{\nu\\}\lambda}+\epsilon^{P_{h}w\lambda\\{\mu}S_{\perp}^{\nu\\}}\right)\left.{\partial
S_{\mu\nu,\rho\sigma}(k)\over\partial
k^{\lambda}}\right|_{c.l.}\right]\Delta\widehat{H}_{T}^{(1)}(z).$ (116)

with $g_{\perp}^{\mu\nu}=g^{\mu\nu}-P_{h}^{\mu}w^{\nu}-P_{h}^{\nu}w^{\mu}$. To
prove
$x^{\prime}p^{\prime\rho}W^{(iii)}_{\rho\sigma}(xp,x^{\prime}p^{\prime},P_{h}/z)=0$,
one needs to show the coefficients of $\widehat{G}_{T}^{(1)}(z)$ and
$\Delta\widehat{H}_{T}^{(1)}(z)$ in (116) satisfy this property. To this end
we first note that one can set $k$ on-shell,
$k=(k^{+},\,k^{-},\,\vec{k}_{\perp})=(k^{+},{\vec{k}_{\perp}^{2}\over
2k^{+}},\vec{k}_{\perp})$, in $S_{\mu\nu,\rho\sigma}(k)$ by regarding $k^{-}$
as a dependent variable, since we take the collinear limit $k\to P_{h}/z$
after taking the derivative. We also introduce the following tensors for an
on-shell $k$:

$\displaystyle
A_{1}^{\mu\nu\lambda}(k)=\epsilon^{P_{h}wS_{\perp}\lambda}g_{\perp}^{\mu\nu}(k),$
(117) $\displaystyle A_{2}^{\mu\nu\lambda}(k)={1\over k\cdot
w}\left(\epsilon^{kwS_{\perp}\mu}g_{\perp}^{\nu\lambda}(k)+\epsilon^{kwS_{\perp}\nu}g_{\perp}^{\mu\lambda}(k)+\epsilon^{kw\lambda\mu}S_{\perp}^{\nu}(k)+\epsilon^{kw\lambda\nu}S_{\perp}^{\mu}(k)\right),$
(118)

where

$\displaystyle
g_{\perp}^{\mu\nu}(k)=g^{\mu\nu}-{k^{\mu}w^{\nu}+k^{\nu}w^{\mu}\over k\cdot
w},\qquad S_{\perp}^{\mu}(k)=S_{\perp}^{\mu}-{k\cdot S_{\perp}\over k\cdot
w}w^{\mu}.$ (119)

Since $g_{\perp}^{\mu\nu}(k)k_{\mu}=0$ and $S_{\perp}^{\mu}(k)k_{\mu}=0$,
$A_{1,2}^{\mu\nu\lambda}(k)k_{\mu}=A_{1,2}^{\mu\nu\lambda}(k)k_{\nu}=0$.
$A_{1,2}^{\mu\nu\lambda}(k)$ also satisfy the following relations:

$\displaystyle
A_{1}^{\mu\nu\lambda}(P_{h}/z)=\epsilon^{P_{h}wS_{\perp}\lambda}g_{\perp}^{\mu\nu},$
(120) $\displaystyle
A_{2}^{\mu\nu\lambda}(P_{h}/z)=\epsilon^{P_{h}wS_{\perp}\\{\mu}g_{\perp}^{\nu\\}\lambda}+\epsilon^{P_{h}w\lambda\\{\mu}S_{\perp}^{\nu\\}},$
(121) $\displaystyle\left.{\partial A_{1}^{\mu\nu\lambda}(k)\over\partial
k^{\lambda}}\right|_{k=P_{h}/z}=-z\left(\epsilon^{P_{h}wS_{\perp}\mu}w^{\nu}+\epsilon^{P_{h}wS_{\perp}\nu}w^{\mu}\right),$
(122) $\displaystyle\left.{\partial A_{2}^{\mu\nu\lambda}(k)\over\partial
k^{\lambda}}\right|_{k=P_{h}/z}=-4z\left(\epsilon^{P_{h}wS_{\perp}\mu}w^{\nu}+\epsilon^{P_{h}wS_{\perp}\nu}w^{\mu}\right).$
(123)

From these relations, we have

$\displaystyle
A_{1,2}^{\mu\nu\lambda}(k)S_{\mu\nu,\rho\sigma}(k)x^{\prime}p^{\prime\rho}=0.$
(124)

This is because the Lorentz indices $\mu$ and $\nu$ provide physical
polarizations for the on-shell gluon with momentum $k$. By taking the
derivative of (124) with respect to $k^{\lambda}$ and setting $k\to P_{h}/z$,
it is easy to see $x^{\prime}p^{\prime\rho}W^{({\rm
iii})}_{\rho\sigma}(xp,x^{\prime}p^{\prime},P_{h}/z)=x^{\prime}p^{\prime\sigma}W^{({\rm
iii})}_{\rho\sigma}(xp,x^{\prime}p^{\prime},P_{h}/z)=0$, which completes the
proof.

##  Appendix C Separation of the 3-body partonic cross sections based on
$z^{\prime}$-dependence

Here we discuss separation of the 3-body cross section (52) based on
$z^{\prime}$-dependence. Inserting (2.1) into (51), we write the cross section
for $\Im\widehat{N}_{i}$ as

$\displaystyle
E_{h}\frac{d\sigma(p,p^{\prime},P_{h};S_{\perp})}{d^{3}P_{h}}\sim\int\left(\frac{1}{z}\right)z^{2}\int
d\left(\frac{1}{z^{\prime}}\right)\delta((xp+x^{\prime}p^{\prime}-P_{h}/z)^{2})$
$\displaystyle\qquad\qquad\qquad\times\frac{z^{\prime}}{z}P\left(\frac{1}{1/z-1/z^{\prime}}\right)\sum_{i=1}^{3}\Im\widehat{N}_{i}\hat{\sigma}(\frac{1}{z^{\prime}},\frac{1}{z}),$
(125)

where $\hat{\sigma}(\frac{1}{z^{\prime}},\frac{1}{z})$ is a partonic hard
cross section defined by

$\displaystyle\Re\left[-i{\Omega^{\mu}_{\alpha}}{\Omega^{\nu}_{\beta}}{\Omega^{\lambda}_{\gamma}}\frac{if^{abc}}{N}\hat{\Gamma}_{FA}^{\alpha\beta\gamma}\left(\frac{1}{z^{\prime}},\frac{1}{z}\right)S^{L}_{\mu\nu\lambda,abc}(z^{\prime},z))\right]$
$\displaystyle\qquad\qquad\qquad=\delta((xp+x^{\prime}p^{\prime}-P_{h}/z)^{2})\sum_{i=1}^{3}\Im\widehat{N}_{i}\hat{\sigma}(\frac{1}{z^{\prime}},\frac{1}{z}),$
(126)

to which diagrams shown in Fig. 26 contribute to in the $qg\to gq$ channel.

Figure 26: An example of diagrams which contribute to $qg\to gq$ channel.
Notation $pc\equiv P_{h}/z$ and $pc^{\prime}\equiv P_{h}/z^{\prime}$ are used.

The amplitude in Fig. 26 has two propagators with $z^{\prime}$-dependence
(\small1⃝ and \small2⃝). For these two propagators, $z^{\prime}$-dependence is
written as follows:

$\displaystyle\mbox{\footnotesize1⃝: }\frac{x\hbox
to0.0pt{/\hss}{\mkern-1.0mup}-\hbox
to0.0pt{/\hss}{\mkern-1.0muP_{h}}/z^{\prime}}{(xp-
P_{h}/z^{\prime})^{2}}=\underbrace{-\frac{\hbox
to0.0pt{/\hss}{\mkern-1.0muP_{h}}/z}{\hat{t}}}_{\mbox{\small1⃝}-{\rm
I}}+\frac{z^{\prime}}{z}\underbrace{\frac{x\hbox
to0.0pt{/\hss}{\mkern-1.0mup}}{\hat{t}}}_{\mbox{\small1⃝}-{\rm II}},$ (127)

and

$\displaystyle\mbox{\footnotesize2⃝: }\frac{x\hbox
to0.0pt{/\hss}{\mkern-1.0mup}+x^{\prime}\hbox
to0.0pt{/\hss}{\mkern-1.0mup}^{\prime}-\hbox
to0.0pt{/\hss}{\mkern-1.0muP_{h}}/z^{\prime}}{(xp+x^{\prime}p^{\prime}-P_{h}/z^{\prime})^{2}}=\underbrace{\frac{\hbox
to0.0pt{/\hss}{\mkern-1.0muP_{h}}/z}{\hat{s}}}_{\mbox{\small2⃝}-{\rm
I}}+\frac{1}{1-z/z^{\prime}}\underbrace{\frac{x\hbox
to0.0pt{/\hss}{\mkern-1.0mup}+x^{\prime}\hbox
to0.0pt{/\hss}{\mkern-1.0mup}^{\prime}-\hbox
to0.0pt{/\hss}{\mkern-1.0muP_{h}}/z}{\hat{s}}}_{\mbox{\small2⃝}-{\rm II}}.$
(128)

In these two equations, we call each term in the RHS \small1⃝-I, \small1⃝-II,
\small2⃝-I and\small2⃝-II as shown in the figure. Then, depending on the
combination of the propagators, one can separate the $z^{\prime}$-dependence
of the cross section as

$\displaystyle E_{h}\frac{d\sigma(p,p^{\prime},P_{h};S_{\perp})}{d^{3}P_{h}}$
$\displaystyle\sim$ $\displaystyle\int\left(\frac{1}{z}\right)z^{2}\int
d\left(\frac{1}{z^{\prime}}\right)\delta((xp+x^{\prime}p^{\prime}-P_{h}/z)^{2})$
(129) $\displaystyle\times$
$\displaystyle\frac{z^{\prime}}{z}P\left(\frac{1}{1/z-1/z^{\prime}}\right)\sum_{i=1}^{3}\Im\widehat{N}_{i}\left[\hat{\sigma}_{p1}+\frac{1}{1-z/z^{\prime}}\hat{\sigma}_{p2}+\frac{z^{\prime}}{z}\hat{\sigma}_{p3}\right],$

where each partonic cross section is defined as

$\displaystyle\hat{\sigma}_{p1}$ $\displaystyle\sim\cdots{\rm
Tr}[\cdots(\mbox{\small2⃝}-{\rm I})\gamma^{\mu}(\mbox{\small1⃝}-{\rm
I})\cdots]\cdots,$ (130) $\displaystyle\hat{\sigma}_{p2}$
$\displaystyle\sim\cdots{\rm Tr}[\cdots(\mbox{\small2⃝}-{\rm
II})\gamma^{\mu}(\mbox{\small1⃝}-{\rm I})\cdots]+\cdots{\rm
Tr}[\cdots(\mbox{\small2⃝}-{\rm II})\gamma^{\mu}(\mbox{\small1⃝}-{\rm
II})\cdots]\cdots,$ (131) $\displaystyle\hat{\sigma}_{p3}$
$\displaystyle\sim\cdots{\rm Tr}[\cdots(\mbox{\small2⃝}-{\rm
I})\gamma^{\mu}(\mbox{\small1⃝}-{\rm II})\cdots]+\cdots{\rm
Tr}[\cdots(\mbox{\small2⃝}-{\rm II})\gamma^{\mu}(\mbox{\small1⃝}-{\rm
II})\cdots],\cdots$ (132)

and we used the relation $z^{\prime}/z\times
1/(1-z/z^{\prime})=z^{\prime}/z+1/(1-z/z^{\prime})$.

Taking into account of the overall factor $z^{\prime}/z\times
1/(1/z-1/z^{\prime})$ and rearranging the $z^{\prime}$-dependence, one obtains
the cross section in the following form:

$\displaystyle E_{h}\frac{d\sigma(p,p^{\prime},P_{h};S_{\perp})}{d^{3}P_{h}}$
$\displaystyle\sim$ $\displaystyle\int\left(\frac{1}{z}\right)z^{2}\int
d\left(\frac{1}{z^{\prime}}\right)\delta((xp+x^{\prime}p^{\prime}-P_{h}/z)^{2})$
(133) $\displaystyle\times$
$\displaystyle\sum_{i=1}^{3}\Im\widehat{N}_{i}\left[\frac{1}{1/z-1/z^{\prime}}\hat{\sigma}_{1}+\frac{1}{z}\frac{1}{(1/z-1/z^{\prime})^{2}}\hat{\sigma}_{2}+{z^{\prime}}\hat{\sigma}_{3}+\frac{z^{\prime
2}}{z}\hat{\sigma}_{4}\right],$

where

$\displaystyle\hat{\sigma}_{1}$ $\displaystyle=$
$\displaystyle\hat{\sigma}_{p1}+\hat{\sigma}_{p2}+\hat{\sigma}_{p3},$ (134)
$\displaystyle\hat{\sigma}_{2}$ $\displaystyle=$
$\displaystyle\hat{\sigma}_{p2},$ (135) $\displaystyle\hat{\sigma}_{3}$
$\displaystyle=$
$\displaystyle\hat{\sigma}_{p1}+\hat{\sigma}_{p2}+\hat{\sigma}_{p3},$ (136)
$\displaystyle\hat{\sigma}_{4}$ $\displaystyle=$
$\displaystyle\hat{\sigma}_{p3}.$ (137)

This form is the expression used in (52). One can thus separate different
$z^{\prime}$-dependence of the cross section.

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