# Inclusive and semi-inclusive production of spin-3/2 hadrons in $e^{+}e^{-}$
annihilation

Jing Zhao  Zhe Zhang  Zuo-tang Liang  Tianbo Liu<EMAIL_ADDRESS>Ya-jin Zhou
<EMAIL_ADDRESS>
###### Abstract

We investigate the inclusive and semi-inclusive productions of spin-3/2
hadrons, such as $\Omega$, in unpolarized $e^{+}e^{-}$ annihilation. The
general differential cross sections are expressed in terms of structure
functions in accordance to the polarization of the hadron and the azimuthal
modulations. We derive a complete definition of quark transverse momentum
dependent (TMD) fragmentation functions (FFs) to spin-3/2 hadrons for the
first time from the decomposition of the quark-quark correlation matrix at
leading twist, 14 of which are newly defined corresponding to rank-3 tensor
polarized hadron. The collinear FFs are obtained from the $k_{T}$-integrated
correlation matrix, and only two TMD FFs with rank-3 tensor polarization have
nonvanishing collinear counterparts. Then we perform a leading order
calculation of the unpolarized differential cross sections. In the single-
hadron inclusive production, only two structure functions are found nonzero
and none of the rank-3 tensor polarized FFs contributes. For the nearly back-
to-back two-hadron production, half of the 48 structure functions are found
nonzero even if the spin of the second hadron is not analyzed, and ten of the
rank-3 tensor polarized TMD FFs contribute. Therefore, one can study the
rank-3 tensor polarized FFs via the production of a spin-3/2 hadron and an
unpolarized hadron in unpolarized $e^{+}e^{-}$ collision experiments. These
newly defined FFs can be further applied in semi-inclusive deep inelastic
scattering processes for the study of nucleon structures.

##  I Introduction

The quantum chromodynamics (QCD) is nowadays known as the fundamental theory
of strong interactions within the framework of the Yang-Mills gauge field
theory. The color confinement, as a feature of QCD, prevents the direct
measurement of its fundamental degrees of freedom, quarks and gluons, by any
modern detectors. To understand the emergence of color neutral hadrons from
colored quarks and gluons produced at high-energy scatterings has become one
of the main issues in nuclear and particle physics [1, 2, 3, 4, 5, 6, 7, 8,
9]. Due to the nonperturbative nature of QCD at hadronic scales, it is still
challenging to calculate the hadronization from first principles, and most
analyses of hadron involved high energy reactions rely on the QCD
factorization [10]. In the factorized formalism, the cross section is
approximately expressed as a convolution of the short-distance partonic
scattering cross section, which can be calculated order by order in
perturbation theory, and some universal process-independent long-distance
functions with overall corrections suppressed by inverse powers of some hard
scale.

The fragmentation function (FF), e.g., $D_{f\to h}(z)$, was first introduced
in QCD to describe inclusive production of a desired hadron in $e^{+}e^{-}$
annihilation [11], although the concept arose right after the parton model
[12]. It describes the probability density of observing the hadron $h$
carrying a fraction $z$ of the longitudinal momentum from a parton with flavor
$f$. If an additional hadron is identified in the final state, the reaction
will become a double scale process when the transverse momentum imbalance,
$|{\bm{P}}_{1T}+{\bm{P}}_{2T}|$, of the two hadrons is much smaller than the
hard scale $Q=\sqrt{s}$, the center-of-mass energy of the reaction. One needs
to apply the transverse momentum dependent (TMD) factorization [11, 13, 14]
and the cross section is expressed as the convolution of partonic hard part
and two TMD FFs $D_{f\to h}(z,k_{T}^{2})$, where $k_{T}$ is the transverse
momentum of the parton with respect to the motion of the observed hadron.
Taking the quark spin degree of freedom into account, one can define polarized
TMD FFs, which encode the correlation between quark transverse momentum and
its spin during the hadronization. They lead to rich phenomena in high energy
reactions, such as the azimuthal asymmetries measured by Belle [15, 16, 17],
BaBar [18], and BESIII [19] in inclusive production of two hadrons in
unpolarized $e^{+}e^{-}$ annihilation. In recent years, a factorization scheme
is also developed to access the TMD FF in the single hadron production process
in $e^{+}e^{-}$ annihilation by identifying the thrust axis, which provides
the sensitivity to the transverse momentum of the detected hadron [20, 21, 22,
23]. Corresponding phenomenological studies based on the data from Belle [24,
25] have been carried out [26, 27, 28, 29]. While this process provides us new
opportunities to learn TMD FFs, it is beyond the scope of this study. In the
rest of this paper, we only focus on inclusive and semi-inclusive processes as
described above without identifying the thrust axis.

The TMD FFs extracted from $e^{+}e^{-}$ annihilation also play an important
role in the study of nucleon structures through the semi-inclusive deep
inelastic scattering (SIDIS) process. For instance, the Collins fragmentation
function $H_{1}^{\perp}(z,k_{T}^{2})$ [30], which is naively interpreted as
the probability density of transversely polarized quark fragmenting to
unpolarized hadron, can lead to a target single spin asymmetry by convoluting
with quark transversity distribution, which describes the net density of a
transversely polarized quark in a transversely polarized nucleon. This
azimuthal asymmetry, usually referred to as the Collins asymmetry [31], is
currently the main observable for phenomenological extraction of quark
transversity distributions of the nucleon. Besides, by selecting different
type of hadrons, such as pions and kaons, in the SIDIS process, one may have
flavor separation in the extraction of the partonic structures of the nucleon
[32].

The spin, as a fundamental property of all particles, is proven a powerful
quantity to test theories and models, especially in hadron physics. Although
the spin effect in the hadronization process exists even in the fragmentation
to spin-0 or unpolarized hadrons, such as the Collins effect [30, 33], much
richer information can be obtained from the measurement of the spin state of
the hadron [34, 35, 36, 37, 38, 39, 40, 41, 42, 43]. The recent measurement of
the spontaneous polarization of $\Lambda$ from unpolarized $e^{+}e^{-}$
collisions by Belle has received great interest [24]. This observation is
suggested to be explained by a naively time-reversal-odd (T-odd) FF
$D_{1T}^{\perp}(z,k_{T}^{2})$ and several parametrizations have been proposed
by fitting the Belle data [44, 45, 46]. The polarized FFs to $\Lambda$ also
provide opportunities to study nucleon spin structures via the spin transfer
in SIDIS process [47, 48, 49, 50, 51, 52] with particular sensitivity to the
spin of strange quarks. Extending to vector mesons, such as $\rho$ and
$K^{*}$, there are five additional spin states, referred to as rank-2 tensor
polarization [53, 54, 55, 56], from the decomposition of the spin density
matrix. The measurements of the rank-2 tensor polarization of vector mesons
produced in $e^{+}e^{-}$ [57, 58, 59, 59, 60] and $pp$ [61, 64, 62, 63, 65,
64, 66, 67, 68, 69] collisions have been carried out and the results incite
extensive theoretical studies [55, 70, 71, 72, 80, 73, 76, 74, 75, 77, 78, 79,
81, 82, 83, 84]. A systematic study of the Lorentz decomposition of the quark-
quark correlator suggests that the spin alignment of vector mesons is solely
determined by a rank-2 tensor polarized FF $D_{1LL}(z)$ independent of the
spin of the fragmenting quark [85] and some parametrizations have been
obtained by fitting the LEP data [57, 58, 59].

In this paper, we present a systematic study of the TMD FFs for spin-3/2
baryons at leading twist. Apart from the three components of the vector
polarization and the five components of the rank-2 tensor polarization, we
define the seven components of the rank-3 tensor polarization from the
decomposition of the spin density matrix, while keeping the orthogonal
relation among different spin projectors. Then we derive the complete set of
TMD FFs at leading twist via Lorentz decompositions of the quark-quark
correlator and 14 of them are related to the rank-3 tensor polarization. The
corresponding collinear FFs are obtained from the $k_{T}$-integrated quark-
quark correlator and only two of the rank-3 tensor polarized FFs have
collinear counter parts. As an ideal process to investigate FFs, we study the
production of spin-3/2 hadrons in unpolarized $e^{+}e-$ annihilation. For the
inclusive production, we derive the differential cross section in terms of
nine structure functions in accordance to different angular dependence and
polarization of the hadron. As a single scale process, we apply the collinear
factorization formalism. Expressing the structure functions in terms of FFs,
we find that none of the rank-3 tensor polarized FFs contributes at leading
order.

To access the TMD FFs, it is necessary to have a second scale, which can be
achieved by measuring one additional hadron. Since the measurement of the spin
states of the two hadrons is very challenging, we consider the inclusive
production of a spin-3/2 hadron and an unpolarized hadron in unpolarized
$e^{+}e^{-}$ annihilation. The differential cross section can be generally
expressed in terms of 48 structure functions. Within the kinematic region
where the two hadrons are nearly back to back, we apply the TMD factorization
formalism and perform a leading order calculation. At leading twist, 24 of the
structure functions are found nonzero as convolutions of two TMD FFs and ten
of the structure functions have contributions from the rank-3 tensor polarized
TMD FFs. Similar to the vector polarization of the baryon, the rank-3 tensor
polarization of spin-3/2 hadrons can, in principle, be measured via weak
decays, and hence these newly defined rank-3 tensor polarized TMD FFs can be
explored via $e^{+}e^{-}\to\Omega hX$ at Belle, BaBar, and BESIII experiments.

The paper is organized as follows. In Sec. II, we provide a decomposition of
the spin density matrix of spin-3/2 hadrons. In Sec. III, we derive the TMD
FFs and their collinear counter parts at leading twist from the quark-quark
correlation function. The general kinematic analysis of the differential cross
section and the calculation of the structure functions in terms of the FFs in
the parton model are presented in Sec. IV.1 for the inclusive production
process and in Sec. IV.2 for the semi-inclusive production process. A summary
and outlook is given in Sec. V.

##  II Spin density matrix of spin-3/2 particles

The complete information of a quantum system is given by the density matrix,
which is capable of describing both pure states and mixed states. Focusing on
the spin degree of freedom, one can use the spin density matrix to
characterize the spin state of a system. Since a massive spin-$s$ particle
belongs to the $(2s+1)$-dimensional representation of the $SO(3)$ group, the
corresponding spin density matrix $\rho$ is a $(2s+1)\times(2s+1)$ matrix.

We start from the spin density matrix for spin-3/2 particles, which has been
derived in the study of nucleon resonance [86]. According to the Hermiticity
condition $\rho=\rho^{\dagger}$, one can decompose $\rho$ on a basis of 16
independent Hermitian matrices,

$\displaystyle\rho=\frac{1}{4}\left(\bm{1}+\frac{4}{5}S^{i}\Sigma^{i}+\frac{2}{3}T^{ij}\Sigma^{ij}+\frac{8}{9}R^{ijk}\Sigma^{ijk}\right).$
(1)

Here the $(\Sigma^{a})$s with $a=i$, $ij$, and $ijk$ are traceless matrices,
corresponding to vector polarization, rank-2 tensor polarization, and rank-3
tensor polarization states, respectively. When summing over all spin states,
only the scalar term $\bm{1}$ contributes and the normalization condition
${\rm Tr[\rho]=1}$ is satisfied. On the $s_{z}$ basis, the three
$(\Sigma^{i})$s are represented as

$\Sigma^{x}=\begin{pmatrix}0&\frac{\sqrt{3}}{2}&0&0\\\
\frac{\sqrt{3}}{2}&0&1&0\\\ 0&1&0&\frac{\sqrt{3}}{2}\\\
0&0&\frac{\sqrt{3}}{2}&0\\\
\end{pmatrix},\quad\Sigma^{y}=\begin{pmatrix}0&-\frac{i\sqrt{3}}{2}&0&0\\\
\frac{i\sqrt{3}}{2}&0&-i&0\\\ 0&i&0&-\frac{i\sqrt{3}}{2}\\\
0&0&\frac{i\sqrt{3}}{2}&0\\\
\end{pmatrix},\quad\Sigma^{z}=\begin{pmatrix}\frac{3}{2}&0&0&0\\\
0&\frac{1}{2}&0&0\\\ 0&0&-\frac{1}{2}&0\\\ 0&0&0&-\frac{3}{2}\\\
\end{pmatrix}.$ (2)

They form a representation of the angular momentum algebra,
$[\Sigma^{i},\Sigma^{j}]=i\epsilon^{ijk}\Sigma^{k}$, and the Casimir operator
is given by
$\Sigma^{x}\Sigma^{x}+\Sigma^{y}\Sigma^{y}+\Sigma^{z}\Sigma^{z}=\frac{15}{4}{\bm{1}}$.
The five rank-2 tensor polarization basis $(\Sigma^{ij})$s can be constructed
from $(\Sigma^{i})$s,

$\displaystyle\Sigma^{ij}$
$\displaystyle=\frac{1}{2}\left(\Sigma^{i}\Sigma^{j}+\Sigma^{j}\Sigma^{i}\right)-\frac{5}{4}\delta^{ij}{\bm{1}},$
(3)

where the second term subtracts the scalar component from the product in the
first term and one has the relation

$\displaystyle\Sigma^{xx}+\Sigma^{yy}+\Sigma^{zz}$ $\displaystyle=0.$ (4)

Similarly, we construct the seven rank-3 tensor polarization basis
$(\Sigma^{ijk})$s as

$\displaystyle\Sigma^{ijk}$
$\displaystyle=\frac{1}{6}\Sigma^{\\{i}\Sigma^{j}\Sigma^{k\\}}-\frac{41}{60}\left(\delta^{ij}\Sigma^{k}+\delta^{jk}\Sigma^{i}+\delta^{ki}\Sigma^{j}\right)$
$\displaystyle=\frac{1}{3}\left(\Sigma^{ij}\Sigma^{k}+\Sigma^{jk}\Sigma^{i}+\Sigma^{ki}\Sigma^{j}\right)-\frac{4}{15}\left(\delta^{ij}\Sigma^{k}+\delta^{jk}\Sigma^{i}+\delta^{ki}\Sigma^{j}\right),$
(5)

where the symbol $\\{\cdots\\}$ stands for a symmetrization of the indices,
i.e., the sum over all permutations. The second term substracts the vector
components from the product in the first term and one can derive the relations

$\displaystyle\Sigma^{xxz}+\Sigma^{yyz}+\Sigma^{zzz}$ $\displaystyle=0,$ (6)
$\displaystyle\Sigma^{xxy}+\Sigma^{yyy}+\Sigma^{zzy}$ $\displaystyle=0,$ (7)
$\displaystyle\Sigma^{xxx}+\Sigma^{yyx}+\Sigma^{zzx}$ $\displaystyle=0.$ (8)

The basis matrices constructed via the above procedure automatically satisfy
the orthogonal relation

$\displaystyle{\rm Tr}[\Sigma^{i}\Sigma^{jk}]={\rm
Tr}[\Sigma^{i}\Sigma^{jkl}]={\rm Tr}[\Sigma^{ij}\Sigma^{klm}]=0,$ (9)

while one still has the freedom for the choice of basis matrices within each
subspace. For convenience, we impose additional orthogonal conditions for the
choice of the basis, requiring ${\rm Tr}[\Sigma^{a}\Sigma^{b}]=0$ if
$\Sigma^{a}\neq\Sigma^{b}$, where $\Sigma^{a}={\bm{1}}$, $\Sigma^{i}$,
$\Sigma^{ij}$, and $\Sigma^{ijk}$. Then the 16 $(\Sigma^{a})$s can serve as
spin projectors and the coefficients $S^{i}$, $T^{ij}$, and $R^{ijk}$ in Eq.
(1) can be calculated from $\langle\Sigma^{a}\rangle\equiv{\rm
Tr}[\rho\Sigma^{a}]$ with corresponding $\Sigma^{a}$. While the spin density
matrices for spin-1/2 and spin-1 are well known and even available in some
textbooks [53, 54, 55, 56], the one for spin-3/2 particles is not widely
utilized. Our choice of the decomposition basis matrices, Eqs. (3) and (5),
are consistent with those in Refs. [86, 87, 88] up to a normalization
constant, but different from the form in Ref. [89]. We comment that the basis
matrices in Ref. [89] do not satisfy the orthogonal condition and hence should
not be directly used as projectors to extract corresponding coefficients.

Following the common convention, we define the spin vector $S^{i}$ in the
hadron rest frame as

$\displaystyle S^{i}$ $\displaystyle=\left(S_{T}^{x},S_{T}^{y},S_{L}\right),$
(10)

where the components are given by

$\displaystyle S_{L}=\langle\Sigma^{z}\rangle,\quad
S_{T}^{x}=\langle\Sigma^{x}\rangle,\quad S_{T}^{y}=\langle\Sigma^{y}\rangle.$
(11)

Similarly, we define the rank-2 spin tensor $T^{ij}$ as

$\displaystyle T^{ij}$
$\displaystyle=\frac{1}{2}\begin{pmatrix}-S_{LL}+S_{TT}^{xx}&S_{TT}^{xy}&S_{LT}^{x}\\\
S_{TT}^{xy}&-S_{LL}-S_{TT}^{xx}&S_{LT}^{y}\\\
S_{LT}^{x}&S_{LT}^{y}&2S_{LL}\end{pmatrix},$ (12)

where the components are given by

$\displaystyle S_{LL}$ $\displaystyle=\langle\Sigma^{zz}\rangle,\quad
S_{LT}^{x}=2\langle\Sigma^{xz}\rangle,\quad
S_{LT}^{y}=2\langle\Sigma^{yz}\rangle,$ $\displaystyle S_{TT}^{xy}$
$\displaystyle=2\langle\Sigma^{xy}\rangle,\quad
S_{TT}^{xx}=\langle\Sigma^{xx}-\Sigma^{yy}\rangle.$ (13)

The rank-3 spin tensor $R^{ijk}$ is defined as

$R^{ijk}=\frac{1}{4}\left[\begin{aligned}
&\left(\begin{array}[]{ccc}-3S_{LLT}^{x}+S_{TTT}^{xxx}&-S_{LLT}^{y}+S_{TTT}^{yxx}&-2S_{LLL}+S_{LTT}^{xx}\\\
-S_{LLT}^{y}+S_{TTT}^{yxx}&-S_{LLT}^{x}-S_{TTT}^{xxx}&S_{LTT}^{xy}\\\
-2S_{LLL}+S_{LTT}^{xx}&S_{LTT}^{xy}&4S_{LLT}^{x}\end{array}\right)\\\
&\left(\begin{array}[]{ccc}-S_{LLT}^{y}+S_{TTT}^{yxx}&-S_{LLT}^{x}-S_{TTT}^{xxx}&S_{LTT}^{xy}\\\
-S_{LLT}^{x}-S_{TTT}^{xxx}&-3S_{LLT}^{y}-S_{TTT}^{yxx}&-2S_{LLL}-S_{LTT}^{xx}\\\
S_{LTT}^{xy}&-2S_{LLL}-S_{LTT}^{xx}&4S_{LLT}^{y}\end{array}\right)\\\
&\left(\begin{array}[]{ccc}-2S_{LLL}+S_{LTT}^{xx}&S_{LTT}^{xy}&4S_{LLT}^{x}\\\
S_{LTT}^{xy}&-2S_{LLL}-S_{LTT}^{xx}&4S_{LLT}^{y}\\\
4S_{LLT}^{x}&4S_{LLT}^{y}&4S_{LLL}\end{array}\right)\end{aligned}\right],$
(14)

where the components are given by

$\displaystyle S_{LLL}$ $\displaystyle=\langle\Sigma^{zzz}\rangle,\quad
S_{LLT}^{x}=\langle\Sigma^{xzz}\rangle,\quad
S_{LLT}^{y}=\langle\Sigma^{yzz}\rangle,\quad
S_{LTT}^{xy}=4\langle\Sigma^{xyz}\rangle,$ $\displaystyle S_{LTT}^{xx}$
$\displaystyle=2\langle\Sigma^{xxz}-\Sigma^{yyz}\rangle,\quad
S_{TTT}^{xxx}=\langle\Sigma^{xxx}-3\Sigma^{xyy}\rangle,\quad
S_{TTT}^{yxx}=\langle 3\Sigma^{yxx}-\Sigma^{yyy}\rangle.$ (15)

We should note that one may have different expressions for $S^{i}$, $T^{ij}$,
and $R^{ijk}$ due to the convention of the value range of the components. The
value ranges of the spin components defined above and their physical
interpretations are provided in Appendix A.

One can further define the total degree of polarization from the spin density
matrix,

$\displaystyle d=\frac{1}{\sqrt{2s}}\sqrt{(2s+1){\rm Tr}[\rho^{2}]-1}.$ (16)

According to Eq. (1), the total degree of polarization of a spin-3/2 particles
is

$\displaystyle d$
$\displaystyle=\frac{1}{3}\left\\{\frac{12}{3}S^{i}S_{i}+2T^{ij}T_{ij}+\frac{8}{3}R^{ijk}R_{ijk}\right\\}^{\frac{1}{2}}$
$\displaystyle=\frac{1}{3}\bigg{\\{}\frac{12}{5}\Big{[}(S_{L})^{2}+(S_{T}^{x})^{2}+(S_{T}^{y})^{2}\Big{]}$
$\displaystyle\quad+\Big{[}3(S_{LL})^{2}+(S_{LT}^{x})^{2}+(S_{LT}^{y})^{2}+(S_{TT}^{xx})^{2}+(S_{TT}^{xy})^{2}\Big{]}$
$\displaystyle\quad+\frac{1}{3}\Big{[}20(S_{LLL})^{2}+30\left((S_{LLT}^{x})^{2}+(S_{LLT}^{y})^{2}\right)$
$\displaystyle\quad\quad+3\left((S_{LTT}^{xx})^{2}+(S_{LTT}^{xy})^{2}\right)+2\left((S_{TTT}^{xxx})^{2}+(S_{TTT}^{yxx})^{2}\right)\Big{]}\bigg{\\}}^{\frac{1}{2}}.$
(17)

Its value ranges between 0 and 1.

The spin vector and tensors for a moving hadron can be obtained via a Lorentz
boost from those defined in the rest frame. For convenience, we use the light-
cone coordinates of a vector $v^{\mu}=(v^{+},v^{-},{\bm{v}}_{\perp})$ with
$v^{\pm}=(v^{0}\pm v^{3})/\sqrt{2}$ and introduce two null vectors
$n^{\mu}=(0,1,{\bm{0}}_{\perp})$ and $\bar{n}^{\mu}=(1,0,{\bm{0}}_{\perp})$.
Following the convention in Ref. [37], we take the large component of the
hadron momentum along the “$-$” direction and decompose the hadron momentum as
$P^{\mu}=n^{\mu}(P\cdot\bar{n})+\bar{n}^{\mu}\,M^{2}/(2P\cdot\bar{n})$, where
$M$ is the mass of the hadron. Then, we can express the spin vector and
tensors in Lorentz covariant forms as

$\displaystyle S^{\mu}$
$\displaystyle=S_{L}\left(\frac{M}{2P\cdot\bar{n}}\bar{n}^{\mu}-\frac{P\cdot\bar{n}}{M}n^{\mu}\right)+{S}_{T}^{\mu},$
(18) $\displaystyle T^{\mu\nu}$
$\displaystyle=\frac{1}{2}\Bigg{\\{}S_{LL}\Bigg{[}\frac{1}{2}\left(\frac{M}{P\cdot\bar{n}}\right)^{2}\bar{n}^{\mu}\bar{n}^{\nu}+2\left(\frac{P\cdot\bar{n}}{M}\right)^{2}n^{\mu}n^{\nu}-\bar{n}^{\\{\mu}n^{\nu\\}}+g_{T}^{\mu\nu}\Bigg{]}$
$\displaystyle\quad+\frac{1}{2}\left(\frac{M}{P\cdot\bar{n}}\right)\bar{n}^{\\{\mu}S_{LT}^{\nu\\}}-\left(\frac{P\cdot\bar{n}}{M}\right)n^{\\{\mu}S_{LT}^{\nu\\}}+S_{TT}^{\mu\nu}\Bigg{\\}},$
(19) $\displaystyle R^{\mu\nu\rho}$
$\displaystyle=\frac{1}{4}\Bigg{\\{}S_{LLL}\Bigg{[}\frac{1}{2}\left(\frac{M}{P\cdot\bar{n}}\right)^{3}\bar{n}^{\mu}\bar{n}^{\nu}\bar{n}^{\rho}-\frac{1}{2}\left(\frac{M}{P\cdot\bar{n}}\right)\left(\bar{n}^{\\{\mu}\bar{n}^{\nu}n^{\rho\\}}-\bar{n}^{\\{\mu}g_{T}^{\nu\rho\\}}\right)$
$\displaystyle\quad+\left(\frac{P\cdot\bar{n}}{M}\right)\left(\bar{n}^{\\{\mu}n^{\nu}n^{\rho\\}}-n^{\\{\mu}g_{T}^{\nu\rho\\}}\right)-4\left(\frac{P\cdot\bar{n}}{M}\right)^{3}n^{\mu}n^{\nu}n^{\rho}\Bigg{]}$
$\displaystyle\quad+\frac{1}{2}\left(\frac{M}{P\cdot\bar{n}}\right)^{2}\bar{n}^{\\{\mu}\bar{n}^{\nu}S_{LLT}^{\rho\\}}+2\left(\frac{P\cdot\bar{n}}{M}\right)^{2}n^{\\{\mu}n^{\nu}S_{LLT}^{\rho\\}}-2\bar{n}^{\\{\mu}n^{\nu}S_{LLT}^{\rho\\}}+\frac{1}{2}S_{LLT}^{\\{\mu}g_{T}^{\nu\rho\\}}$
$\displaystyle\quad+\frac{1}{4}\left(\frac{M}{P\cdot\bar{n}}\right)\bar{n}^{\\{\mu}S_{LTT}^{\nu\rho\\}}-\frac{1}{2}\left(\frac{P\cdot\bar{n}}{M}\right)n^{\\{\mu}S_{LTT}^{\nu\rho\\}}+S_{TTT}^{\mu\nu\rho}\Bigg{\\}},$
(20)

where the transverse components of $S_{T}^{\mu}$, $S_{LT}^{\mu}$,
$S_{TT}^{\mu\nu}$, $S_{LLT}^{\mu}$, $S_{LTT}^{\mu\nu}$, and
$S_{TTT}^{\mu\nu\rho}$ are given by

$\displaystyle S_{T}^{i}$ $\displaystyle=(S_{T}^{x},S_{T}^{y}),\quad
S_{LT}^{i}=(S_{LT}^{x},S_{LT}^{y}),\quad
S_{LLT}^{i}=(S_{LLT}^{x},S_{LLT}^{y}),$ $\displaystyle S_{TT}^{ij}$
$\displaystyle=\begin{pmatrix}S_{TT}^{xx}&S_{TT}^{xy}\\\
S_{TT}^{xy}&-S_{TT}^{xx}\end{pmatrix},\quad
S_{LTT}^{ij}=\begin{pmatrix}S_{LTT}^{xx}&S_{LTT}^{xy}\\\
S_{LTT}^{xy}&-S_{LTT}^{xx}\end{pmatrix},$ $\displaystyle S_{TTT}^{ijk}$
$\displaystyle=\left[\begin{pmatrix}S_{TTT}^{xxx}&S_{TTT}^{yxx}\\\
S_{TTT}^{yxx}&-S_{TTT}^{xxx}\end{pmatrix},\begin{pmatrix}S_{TTT}^{yxx}&-S_{TTT}^{xxx}\\\
-S_{TTT}^{xxx}&-S_{TTT}^{yxx}\end{pmatrix}\right],$ (21)

and their longitudinal components are zeros. The transverse metric
$g_{T}^{\mu\nu}$ is defined as

$\displaystyle g_{T}^{\mu\nu}$
$\displaystyle=g^{\mu\nu}-\bar{n}^{\mu}n^{\nu}-n^{\mu}\bar{n}^{\nu}.$ (22)

One can also find that the spin tensors $S$, $T$, and $R$ are orthogonal to
the momentum $P$ of the hadron,

$\displaystyle P_{\mu}S^{\mu}=0,\quad P_{\mu}T^{\mu\nu}=0,\quad
P_{\mu}R^{\mu\nu\rho}=0,$ (23)

because these relations are not altered by the Lorentz boost.

##  III TMD Fragmentation functions

In this section, we derive quark FFs at leading twist from the decomposition
of quark-quark correlation function.

The matrix element that describes the emergence of a spin-3/2 hadron of
momentum $P$ characterized by the spin tensors $S$, $T$, and $R$ from a quark
is $\langle P,S,T,R,X|\bar{\psi}(0)|0\rangle$, where $\psi$ is the quark field
operator and $X$ stands for undetected hadrons. Then one can define the
unintegrated quark-quark correlation function,

$\displaystyle\Delta_{\alpha\beta}\left(k,P,S,T,R\right)=$
$\displaystyle\sum_{X}\int\frac{\mathrm{d}^{4}\xi}{(2\pi)^{4}}e^{ik\cdot\xi}\langle
0|\mathcal{L}(\infty,\xi)\psi_{\alpha}(\xi)|P,S,T,R,X\rangle$
$\displaystyle\times\langle
P,S,T,R,X|\bar{\psi}_{\beta}(0)\mathcal{L}^{\dagger}(\infty,0)|0\rangle,$ (24)

where $k$ is the momentum of the quark, $\alpha$ and $\beta$ are Dirac
indices, and the symbol $\displaystyle{\sum_{X}}$ also implies the integration
over the momenta of the undetected hadrons labeled by $X$. The gauge link
$\mathcal{L}(y_{2},y_{1})$, which ensures the gauge invariance of the
correlation function, is defined as a path integral

$\displaystyle\mathcal{L}(y_{2},y_{1})$ $\displaystyle={\cal
P}\exp\left[-ig\int_{y_{1}}^{y_{2}}dy\cdot A(y)\right],$ (25)

where $g$ is the strong coupling constant and $A(y)$ is the gluon field
operator. The correlation function defined in Eq. (24) fulfills the
Hermiticity condition,

$\displaystyle\Delta(k,P,S,T,R)$
$\displaystyle=\gamma^{0}\Delta^{\dagger}(k,P,S,T,R)\gamma^{0},$ (26)

and the parity invariance,

$\displaystyle\Delta(k,P,S,T,R)$
$\displaystyle=\gamma^{0}\Delta(\bar{k},\bar{P},-\bar{S},\bar{T},-\bar{R})\gamma^{0},$
(27)

where $\bar{k}^{\mu}$, $\bar{P}^{\mu}$, $\bar{T}^{\mu\nu}$, and
$\bar{R}^{\mu\nu\rho}$ represent a sign flip of all space components. The
time-reversal invariance has no additional constraints on the correlation
function [90, 91, 92] because of the final-state interaction or formally the
gauge link.

From the explicit Dirac indices in Eq. (24), one can observe that the
correlation function is a $4\times 4$ matrix in the Dirac space. It can be
expanded on a basis of 16 Dirac $\gamma$ matrices, which we choose as $\bf 1$,
$\gamma_{5}$, $\gamma^{\mu}$, $\gamma^{\mu}\gamma_{5}$, and
$i\sigma^{\mu\nu}\gamma_{5}$. The coefficients are given by the combinations
of the momenta $k$, $P$, and the spin tensors $S$, $T$, $R$ from the spin
density matrix. After imposing the Hermiticity (26) and the parity invariance
(27), we obtain the general decomposition of the quark-quark correlation
function (24) for a spin-3/2 hadron as

$\displaystyle\Delta(k,P,S,T,R)$
$\displaystyle=B_{1}M\mathbf{1}+B_{2}\not{P}+B_{3}\not{k}+\left(\frac{B_{4}}{M}\sigma_{\mu\nu}P^{\mu}k^{\nu}\right)+\left(iB_{5}k\cdot
S\gamma_{5}\right)$
$\displaystyle\quad+B_{6}M\not{S}\gamma_{5}+B_{7}\frac{k\cdot
S}{M}\not{P}\gamma_{5}+B_{8}\frac{k\cdot
S}{M}\not{k}\gamma_{5}+iB_{9}\sigma_{\mu\nu}\gamma_{5}S^{\mu}P^{\nu}$
$\displaystyle\quad+iB_{10}\sigma_{\mu\nu}\gamma_{5}S^{\mu}k^{\nu}+iB_{11}\frac{k\cdot
S}{M^{2}}\sigma_{\mu\nu}\gamma_{5}P^{\mu}k^{\nu}+\left(B_{12}\frac{\epsilon_{\mu\nu\rho\sigma}\gamma^{\mu}P^{\nu}k^{\rho}S^{\sigma}}{M}\right)$
$\displaystyle\quad+\frac{B_{13}}{M}k_{\mu}k_{\nu}T^{\mu\nu}{\bf
1}+\frac{B_{14}}{M^{2}}k_{\mu}k_{\nu}T^{\mu\nu}\not{P}+\frac{B_{15}}{M^{2}}k_{\mu}k_{\nu}T^{\mu\nu}\not{k}$
$\displaystyle\quad+\left(\frac{B_{16}}{M^{3}}k_{\mu}k_{\nu}T^{\mu\nu}\sigma_{\rho\sigma}P^{\rho}k^{\sigma}\right)+B_{17}k_{\mu}T^{\mu\nu}\gamma_{\nu}+\left(\frac{B_{18}}{M}\sigma_{\nu\rho}P^{\rho}k_{\mu}T^{\mu\nu}\right)$
$\displaystyle\quad+\left(\frac{B_{19}}{M}\sigma_{\nu\rho}k^{\rho}k_{\mu}T^{\mu\nu}\right)+\left(\frac{B_{20}}{M^{2}}\epsilon_{\mu\nu\rho\sigma}\gamma^{\mu}\gamma_{5}P^{\nu}k^{\rho}k_{\tau}T^{\tau\sigma}\right)$
$\displaystyle\quad+\left(i\frac{B_{21}}{M^{2}}k_{\mu}k_{\nu}k_{\rho}R^{\mu\nu\rho}\gamma_{5}\right)+\frac{B_{22}}{M^{3}}k_{\mu}k_{\nu}k_{\rho}R^{\mu\nu\rho}\not{k}\gamma_{5}+\frac{B_{23}}{M^{3}}k_{\mu}k_{\nu}k_{\rho}R^{\mu\nu\rho}\not{P}\gamma_{5}$
$\displaystyle\quad+i\frac{B_{24}}{M^{4}}k_{\mu}k_{\nu}k_{\rho}R^{\mu\nu\rho}\sigma_{\tau\lambda}\gamma_{5}k^{\tau}P^{\lambda}+\frac{B_{25}}{M}k_{\mu}k_{\nu}R^{\mu\nu\rho}\gamma_{\rho}\gamma_{5}$
$\displaystyle\quad+i\frac{B_{26}}{M^{2}}\sigma_{\rho\tau}\gamma_{5}k^{\tau}k_{\mu}k_{\nu}R^{\mu\nu\rho}+i\frac{B_{27}}{M^{2}}\sigma_{\rho\tau}\gamma_{5}P^{\tau}k_{\mu}k_{\nu}R^{\mu\nu\rho}$
$\displaystyle\quad+\left(\frac{B_{28}}{M^{3}}\epsilon_{\mu\nu\rho\sigma}\gamma^{\mu}k^{\nu}P^{\rho}k_{\tau}k_{\lambda}R^{\tau\lambda\sigma}\right),$
(28)

where the Dirac indices are suppressed and $\epsilon_{\mu\nu\rho\sigma}$ is
the totally antisymmetric tensor with $\epsilon_{+-12}=1$. The $B_{i}$’s are
Lorentz scalar functions of $k\cdot P$ and $k^{2}$, and the mass factor $M$ is
introduced to balance the dimension. The terms in parentheses are often
referred to as naively T-odd functions. The terms in Eq. (28) can be divided
into four categories, including the unpolarized ones which do not contain $S$,
$T$, or $R$, the vector polarized ones which contain the spin vector $S$, the
rank-2 tensor polarized ones which contain the spin tensor $T$, and the rank-3
tensor polarized ones which contain the spin tensor $R$. The rank-3 tensor
polarized terms, $B_{21}$ – $B_{28}$, are newly defined for spin-3/2 hadrons,
while the unpolarized, the vector polarized, and the rank-2 tensor polarized
ones also exist in the decomposition of the correlation function of spin-1
hadrons [55].

To deal with quark transverse momentum ${\bm{k}}_{T}$ in the fragmentation
process, we choose the momentum of the hadron along the longitudinal direction
with $P^{-}$ to be the large component, and $P^{+}=M^{2}/(2P^{-})$ is fixed by
the hadron mass. Correspondingly, we perform a Sudakov decomposition of the
quark momentum,

$\displaystyle k^{\mu}$
$\displaystyle=\frac{z\left(k^{2}+\bm{k}_{T}^{2}\right)}{2P^{-}}\bar{n}^{\mu}+\frac{P^{-}}{z}n^{\mu}+{k}_{T}^{\mu},$
(29)

where $z$ is the longitudinal momentum fraction carried by the hadron from the
fragmenting quark and $k_{T}^{\mu}$ is the transverse part of the full four-
momentum vector $k^{\mu}$ with $k_{T}^{2}=-{\bm{k}}_{T}^{2}$. By integrating
Eq. (24) over $k^{+}$, or, equivalently $k^{2}$, we obtain the quark-quark
correlation function,

$\Delta\left(z,k_{T}\right)=\left.\frac{1}{4z}\int\mathrm{d}k^{+}\Delta\left(k,P,S,T,R\right)\right|_{k^{-}=\frac{P^{-}}{z}},$
(30)

which leads to the definition of quark TMD FFs after the Dirac decomposition
as shown in Eq. (28).

Here, we consider $P^{-}$ as a large momentum component and collect the
leading twist, i.e., twist-two, TMD FFs, which can be projected out from the
correlator $\Delta(z,k_{T})$ by the Dirac matrices $\gamma^{-}$,
$\gamma^{-}\gamma_{5}$, and $i\sigma^{i-}\gamma_{5}$. Then a complete set of
quark TMD FFs for spin-3/2 hadrons at leading twist are defined as

$\displaystyle\Delta_{U}\left(z,k_{T}\right)=$
$\displaystyle\frac{1}{4}\left\\{D_{1}\left(z,k_{T}^{2}\right)\not{n}+\left(H_{1}^{\perp}\left(z,k_{T}^{2}\right)\sigma_{\mu\nu}\frac{k_{T}^{\mu}}{M}n^{\nu}\right)\right\\},$
(31) $\displaystyle\Delta_{L}\left(z,k_{T}\right)=$
$\displaystyle\frac{1}{4}\left\\{G_{1L}\left(z,k_{T}^{2}\right)S_{L}\gamma_{5}\not{n}+H_{1L}^{\perp}\left(z,k_{T}^{2}\right)S_{L}i\sigma_{\mu\nu}\gamma_{5}n^{\mu}\frac{k_{T}^{\nu}}{M}\right\\},$
(32) $\displaystyle\Delta_{T}\left(z,k_{T}\right)=$
$\displaystyle\frac{1}{4}\left\\{G_{1T}^{\perp}\left(z,k_{T}^{2}\right)\frac{\bm{S}_{T}\cdot\bm{k}_{T}}{M}\gamma_{5}\not{n}+H_{1T}\left(z,k_{T}^{2}\right)i\sigma_{\mu\nu}\gamma_{5}n^{\mu}S_{T}^{\nu}\right.$
$\displaystyle\left.-H_{1T}^{\perp}\left(z,k_{T}^{2}\right)i\sigma_{\mu\nu}\gamma_{5}n^{\mu}\frac{k_{T}^{\nu\rho}}{M^{2}}S_{T\rho}+\left(D_{1T}^{\perp}\left(z,k_{T}^{2}\right)\epsilon_{T}^{\mu\nu}\frac{k_{T\mu}}{M}S_{T\nu}\not{n}\right)\right\\},$
(33) $\displaystyle\Delta_{LL}\left(z,k_{T}\right)=$
$\displaystyle\frac{1}{4}\left\\{D_{1LL}\left(z,k_{T}^{2}\right)S_{LL}\not{n}-\left(H_{1LL}^{\perp}\left(z,k_{T}^{2}\right)S_{LL}\sigma_{\mu\nu}\frac{k_{T}^{\mu}}{M}n^{\nu}\right)\right\\},$
(34) $\displaystyle\Delta_{LT}\left(z,k_{T}\right)=$
$\displaystyle\frac{1}{4}\left\\{D_{1LT}^{\perp}\left(z,k_{T}^{2}\right)\frac{\bm{S}_{LT}\cdot\bm{k}_{T}}{M}\not{n}-\left(G_{1LT}^{\perp}\left(z,k_{T}^{2}\right)\epsilon_{T}^{\mu\nu}\frac{k_{T\mu}}{M}S_{LT\nu}\gamma_{5}\not{n}\right)\right.$
$\displaystyle\left.-\left(H_{1LT}\left(z,k_{T}^{2}\right)\sigma_{\mu\nu}S_{LT}^{\mu}n^{\nu}\right)+\left(H_{1LT}^{\perp}\left(z,k_{T}^{2}\right)\sigma_{\mu\nu}\frac{k_{T}^{\mu\rho}}{M^{2}}S_{LT\rho}n^{\nu}\right)\right\\},$
(35) $\displaystyle\Delta_{TT}\left(z,k_{T}\right)=$
$\displaystyle\frac{1}{4}\left\\{D_{1TT}^{\perp}\left(z,k_{T}^{2}\right)S_{TT\mu\nu}\frac{k_{T}^{\mu\nu}}{M^{2}}\not{n}+\left(G_{1TT}^{\perp}\left(z,k_{T}^{2}\right)\epsilon_{T\nu}^{\mu}\frac{k_{T\mu\rho}}{M_{h}^{2}}S_{TT}^{\nu\rho}\gamma_{5}\not{n}\right)\right.$
$\displaystyle\left.-\left(H_{1TT}^{\perp}\left(z,k_{T}^{2}\right)\sigma_{\mu\nu}S_{TT}^{\mu\rho}\frac{k_{T\rho}}{M_{h}}n^{\nu}\right)-\left(H_{1TT}^{\perp\perp}\left(z,k_{T}^{2}\right)\sigma_{\mu\nu}\frac{k_{T}^{\mu\rho\sigma}}{M^{3}}S_{TT\rho\sigma}n^{\nu}\right)\right\\},$
(36) $\displaystyle\Delta_{LLL}(z,k_{T})=$
$\displaystyle\frac{1}{4}\left\\{G_{1LLL}\left(z,k_{T}^{2}\right)S_{LLL}\gamma_{5}\not{n}+H_{1LLL}^{\perp}\left(z,k_{T}^{2}\right)S_{LLL}i\sigma_{\mu\nu}\gamma_{5}n^{\mu}\frac{k_{T}^{\nu}}{M}\right\\},$
(37) $\displaystyle\Delta_{LLT}(z,k_{T})=$
$\displaystyle\frac{1}{4}\left\\{G_{1LLT}^{\perp}\left(z,k_{T}^{2}\right)\frac{\bm{S}_{LLT}\cdot\bm{k}_{T}}{M}\gamma_{5}\not{n}-H_{1LLT}^{\perp}\left(z,k_{T}^{2}\right)i\sigma_{\mu\nu}\gamma_{5}n^{\mu}\frac{k_{T}^{\nu\rho}}{M^{2}}S_{LLT\rho}\right.$
$\displaystyle\left.+H_{1LLT}\left(z,k_{T}^{2}\right)i\sigma_{\mu\nu}\gamma_{5}n^{\mu}S_{LLT}^{\nu}+\left(D_{1LLT}^{\perp}\left(z,k_{T}^{2}\right)\epsilon_{T}^{\mu\nu}\frac{k_{T\mu}}{M}S_{LLT\nu}\not{n}\right)\right\\},$
(38) $\displaystyle\Delta_{LTT}(z,k_{T})=$
$\displaystyle\frac{1}{4}\left\\{G_{1LTT}^{\perp}\left(z,k_{T}^{2}\right)S_{LTT\mu\nu}\frac{k_{T}^{\mu\nu}}{M^{2}}\gamma_{5}\not{n}+H_{1LTT}^{\perp\perp}\left(z,k_{T}^{2}\right)i\sigma_{\mu\nu}\gamma_{5}n^{\mu}\frac{k_{T}^{\nu\rho\sigma}}{M^{3}}S_{LTT\rho\sigma}\right.$
$\displaystyle\left.+H_{1LTT}^{\perp}\left(z,k_{T}^{2}\right)i\sigma_{\mu\nu}\gamma_{5}n^{\mu}S_{LTT}^{\nu\rho}\frac{k_{T\rho}}{M}+\left(D_{1LTT}^{\perp}\left(z,k_{T}^{2}\right)\epsilon_{T\nu}^{\mu}\frac{k_{T\mu\rho}}{M^{2}}S_{LTT}^{\nu\rho}\not{n}\right)\right\\},$
(39) $\displaystyle\Delta_{TTT}(z,k_{T})=$
$\displaystyle\frac{1}{4}\left\\{G_{1TTT}^{\perp}\left(z,k_{T}^{2}\right)S_{TTT\mu\nu\rho}\frac{k_{T}^{\mu\nu\rho}}{M^{3}}\gamma_{5}\not{n}+H_{1TTT}^{\perp\perp}\left(z,k_{T}^{2}\right)i\sigma_{\mu\nu}\gamma_{5}n^{\mu}\frac{k_{T}^{\nu\rho\sigma\tau}}{M^{4}}S_{TTT\rho\sigma\tau}\right.$
$\displaystyle\left.+H_{1TTT}^{\perp}\left(z,k_{T}^{2}\right)i\sigma_{\mu\nu}\gamma_{5}n^{\mu}S_{TTT}^{\nu\rho\sigma}\frac{k_{T\rho\sigma}}{M^{2}}+\left(D_{1TTT}^{\perp}\left(z,k_{T}^{2}\right)\epsilon_{T\nu}^{\mu}\frac{k_{T\mu\rho\sigma}}{M^{3}}S_{TTT}^{\nu\rho\sigma}\not{n}\right)\right\\},$
(40)

where $\bm{S}_{T}\cdot\bm{k}_{T}=-S_{T}^{\mu}k_{T\mu}$,
$\bm{S}_{LT}\cdot\bm{k}_{T}=-S_{LT}^{\mu}k_{T\mu}$, and
$\bm{S}_{LLT}\cdot\bm{k}_{T}=-S_{LLT}^{\mu}k_{T\mu}$. The terms corresponding
to different spin components of the hadron have been separated and labeled by
$U$ for unpolarized hadron, by $L$ and $T$ for vector polarized hadron as
defined in Eq. (11), by $LL$, $LT$, and $TT$ for rank-2 tensor polarized
hadron as defined in Eq. (13), and by $LLL$, $LLT$, $LTT$, and $TTT$ for
rank-3 tensor polarized hadron as defined in Eq. (15). The transverse spin
tensors $S_{T}$, $S_{LT}$, $S_{TT}$, $S_{LLT}$, $S_{LTT}$, and $S_{TTT}$ are
explicitly written in Eq. (21). The transverse antisymmetric tensor is defined
as
$\epsilon_{T}^{\mu\nu}=\epsilon^{\mu\nu\rho\sigma}\bar{n}_{\rho}n_{\sigma}$.
We name the TMD FFs following the common convention. The $D$, $G$, and $H$ are
used to represent unpolarized, longitudinally polarized, and transversely
polarized quarks, corresponding to the Dirac matrices $\gamma^{-}$,
$\gamma^{-}\gamma_{5}$, and $i\sigma^{i-}\gamma_{5}$, respectively. The
subscript “1” indicates that the TMD FFs are at leading twist. A superscript
“$\perp$” is assigned if there is inhomogeneous dependence on the quark
transverse momentum, i.e., $k_{T}^{\mu}$, $k_{T}^{\mu\nu}$,
$k_{T}^{\mu\nu\rho}$, and $k_{T}^{\mu\nu\rho\sigma}$, which are completely
symmetric and traceless tensors [80] as defined in Eqs. (230)–(232). The
superscript “$\perp\perp$” in $H_{1TT}^{\perp\perp}$, $H_{1LTT}^{\perp\perp}$,
and $H_{1TTT}^{\perp\perp}$ is introduced to differentiate them from
$H_{1TT}^{\perp}$, $H_{1LTT}^{\perp}$, and $H_{1TTT}^{\perp}$.

Among the 32 TMD FFs at leading twist, two are for the unpolarized hadron
state, six are for the vector polarized hadron state, ten are for the rank-2
tensor polarized hadron state, and 14 are for the rank-3 tensor polarized
hadron state. While the unpolarized, the vector polarized, and the rank-2
tensor polarized ones have been defined in the study of spin-1 hadrons, the
rank-3 tensor polarized TMD FFs are newly defined and only exist for hadrons
with $s\geq 3/2$. The procedure presented above can be extended to the
analysis of hadrons with higher spins. For a rank-$n$ tensor polarized hadron
state, one can define $4n+2$ TMD FFs at leading twist.

The collinear FFs are defined from the $k_{T}$-integrated correlation
function,

$\Delta(z)=\left.\frac{z}{4}\int\mathrm{d}^{2}\bm{k}_{T}\mathrm{~{}d}k^{+}\Delta\left(k,P,S,T,R\right)\right|_{k^{-}=\frac{P_{h}^{-}}{z}}.$
(41)

The terms with inhomogeneous $k_{T}$ dependence in Eqs. (31)–(40) all vanish
and a complete set of collinear FFs are defined as

$\displaystyle\Delta_{U}(z)$
$\displaystyle=\frac{1}{4}D_{1}\left(z\right)\not{n},$ (42)
$\displaystyle\Delta_{L}(z)$
$\displaystyle=\frac{1}{4}G_{1L}\left(z\right)S_{L}\gamma_{5}\not{n},$ (43)
$\displaystyle\Delta_{T}(z)$
$\displaystyle=\frac{1}{4}H_{1T}\left(z\right)i\sigma_{\mu\nu}\gamma_{5}n^{\mu}S_{T}^{\nu},$
(44) $\displaystyle\Delta_{LL}(z)$
$\displaystyle=\frac{1}{4}D_{1LL}\left(z\right)S_{LL}\not{n},$ (45)
$\displaystyle\Delta_{LT}(z)$
$\displaystyle=\frac{1}{4}H_{1LT}\left(z\right)\sigma_{\mu\nu}S_{LT}^{\mu}n^{\nu},$
(46) $\displaystyle\Delta_{LLL}(z)$
$\displaystyle=\frac{1}{4}G_{1LLL}\left(z\right)S_{LLL}\gamma_{5}\not{n},$
(47) $\displaystyle\Delta_{LLT}(z)$
$\displaystyle=\frac{1}{4}H_{1LLT}\left(z\right)i\sigma_{\mu\nu}\gamma_{5}n^{\mu}S_{LLT}^{\nu},$
(48)

where we keep the same symbols as their TMD counterparts. Among the seven
collinear FFs, $G_{1LLL}(z)$ and $H_{1LLT}(z)$ are newly defined for the
rank-3 tensor polarized hadron state and only exist for hadrons with $s\geq
3/2$, while the other ones have the same definitions as those for a spin-1
hadron [55].

We should note that recovering the collinear FFs from TMD FFs is actually a
nontrivial issue in the factorization framework. In the Collins-Soper-Sterman
(CSS) formalism [13, 11, 93, 94], the differential cross section is expressed
into a W term and a Y term, also referred to as the “W+Y” prescription. The W
term dominates the small transverse momentum region where the TMD
factorization works. The Y term contains the fixed-order term which dominates
the large transverse momentum region within the collinear factorization and
the asymptotic term that matches the regions. Since the TMD factorization is
not suitable for large transverse momentum, the direct integration of the W
term over the transverse momentum to infinity will vanish, and thus the
collinear FFs are not directly obtained by integrating the TMD FFs. Various
methods have been proposed in literature to deal with this problem and related
issues [94, 95, 96, 97, 98, 99, 100, 101, 102]. To suppress the W term at
large transverse momentum, one may directly impose a cutoff at some transverse
momentum value [94, 100]. Alternatively, one can work in the Fourier conjugate
space, the $b$ space where the resummation is usually implemented, and
introduce a cutoff at some small $b$ value [95, 96, 97, 98, 99]. Besides, it
is also useful to make the W term short circuited above some large transverse
momentum for the computing efficiency [101, 102]. On the other hand, an
effective cutoff of the Y term in the small transverse momentum region need to
be introduced accordingly [93, 103]. Therefore, the integration over
transverse momentum is a nontrivial issue for a consistent factorization
framework and appropriate regularization is required.

##  IV Production of spin-3/2 hadrons in $e^{+}e^{-}$ annihilation

The cleanest process to study FFs is the inclusive hadron production from
$e^{+}e^{-}$ annihilation at a large center-of-mass energy $\sqrt{s}=Q$, which
serves as a hard probe allowing us to use the parton language. Within the QCD
factorization, the cross section can be approximately expressed as a
convolution of the production of quarks at short distance and quark FFs to the
desired hadron in a certain spin state and overall corrections are suppressed
by inverse powers of the hard scale. The inclusive production of a single
spin-3/2 hadron, $e^{+}e^{-}\to hX$, is characterized by a single hard scale
$Q$ and thus not sensitive to the confined motion of quarks during the
fragmentation. To have the sensitivity to quark transverse momentum, we can
consider the inclusive production of two nearly back to back hadrons,
$e^{+}e^{-}\to h_{1}h_{2}X$, where the transverse momentum imbalance of the
two hadrons provides a soft scale in addition to the hard scale. As a double-
scale process, one should apply the TMD factorization in which the cross
section is expressed as the convolution of the production of quarks at short
distance and two quark TMD FFs to the detected hadrons $h_{1}$ and $h_{2}$.

The spin states of the produced hadron can be measured by analyzing the
spatial distribution of the decayed particles. While the strong decays can
only be utilized to measure the rank-2 tensor polarization because of the
parity conservation, one may extract the vector and rank-3 tensor polarization
through weak decays, which have been widely applied in the analysis of
$\Lambda$ hyperon polarization. Therefore, to access the rank-3 tensor
polarization of the produced hadron, one can choose the identified hadron as
the $\Omega$, which only has weak decays.

In this section, we consider both inclusive and semi-inclusive productions of
the spin-$3/2$ hadron in $e^{+}e^{-}$ annihilation processes. Considering the
existing electron-positron colliders, PEP-II, KEKB, and BEPC, the calculation
is restricted to unpolarized lepton beams. We first derive the general cross
sections in terms of structure functions, which correspond to different
angular distributions and spin states of the hadron. At leading twist, we then
express the structure functions in terms of the FFs as defined in Sec. III.

###  IV.1 Inclusive production of the $\Omega$ in unpolarized $e^{+}e^{-}$
collisions

Figure 1: Left: The illustration of the inclusive production of $\Omega$ in
$e^{+}e^{-}$ annihilation. Right: The diagram of the process
$e^{+}e^{-}\to\Omega X$ under the one-photon-exchange approximation.

We consider the process

$\displaystyle e^{-}(l_{1})+e^{+}(l_{2})\to\Omega(P)+X(P_{X}),$ (49)

where the variables in parentheses label the four momenta of the corresponding
particles. With one-photon-exchange approximation as illustrated in Fig. 1,
one can write the differential cross section as

$\displaystyle
P^{0}\frac{d\sigma}{d^{3}\bm{P}}=\frac{\alpha^{2}}{2Q^{6}}L_{\mu\nu}W^{\mu\nu},$
(50)

where $\alpha$ is the electromagnetic fine structure constant,

$\displaystyle L_{\mu\nu}\left(l_{1},l_{2}\right)$
$\displaystyle=2[l_{1\mu}l_{2\nu}+l_{1\nu}l_{2\mu}-g_{\mu\nu}(l_{1}\cdot
l_{2})]$ (51)

is the leptonic tensor, and

$\displaystyle W^{\mu\nu}\left(q;P,S,T,R\right)=$
$\displaystyle\frac{1}{(2\pi)}\sum_{X}\left\langle
0\left|J^{\mu}(0)\right|P_{X};P,S,T,R\right\rangle\left\langle
P_{X};P,S,T,R\left|J^{\nu}(0)\right|0\right\rangle$
$\displaystyle\times(2\pi)^{4}\delta^{(4)}\left(q-P_{X}-P\right)$ (52)

is the hadronic tensor. Here $q=l_{1}+l_{2}$ is the momentum of the virtual
photon with $q^{2}=Q^{2}$, $J^{\mu}$ is the electromagnetic current operator,
and the symbol $\displaystyle{\sum_{X}}$ represents the sum over the hadronic
states $X$ implying the momentum integrals.

For the unpolarized process, we only need the symmetric part of the hadronic
tensor $W^{\mu\nu}$, which satisfies the Hermiticity and the parity invariance
relations,

$\displaystyle
W^{*\mu\nu}\left(q;P,S,T,R\right)=W^{\nu\mu}\left(q;P,S,T,R\right),$ (53)
$\displaystyle
W^{\mu\nu}\left(q;P,S,T,R\right)=W_{\mu\nu}\left(q;\bar{P},-\bar{S},\bar{T},-\bar{R}\right).$
(54)

Besides, the gauge invariance requires
$q_{\mu}W^{\mu\nu}=W^{\mu\nu}q_{\nu}=0$. To impose this condition, it is
convenient to introduce the so-called conserved vectors and tensors,

$\displaystyle\widetilde{g}^{\mu\nu}$
$\displaystyle=g^{\mu\nu}-\frac{q^{\mu}q^{\nu}}{q^{2}},$ (55)
$\displaystyle\widetilde{P}^{\mu}$ $\displaystyle=P^{\mu}-\frac{P\cdot
q}{q^{2}}q^{\mu},$ (56) $\displaystyle\widetilde{T}^{\mu q}$
$\displaystyle=T^{\mu q}-\frac{T^{qq}}{q^{2}}q^{\mu},$ (57)
$\displaystyle\widetilde{T}^{\mu\nu}$ $\displaystyle=T^{\mu\nu}-\frac{T^{\mu
q}q^{\nu}}{q^{2}}-\frac{T^{\nu
q}q^{\mu}}{q^{2}}+\frac{T^{qq}q^{\mu}q^{\nu}}{q^{4}},$ (58)
$\displaystyle\widetilde{R}^{\mu\nu q}$ $\displaystyle=R^{\mu\nu
q}-\frac{R^{\mu qq}q^{\nu}}{q^{2}}-\frac{R^{\nu
qq}q^{\mu}}{q^{2}}+\frac{R^{qqq}q^{\mu}q^{\nu}}{q^{4}},$ (59)

which vanish when contract with $q$. Here, we have used the short-handed
notation $A^{\mu q}\equiv A^{\mu\nu}q_{\nu}$ for conciseness. Then, we can
expand the hadronic tensor $W^{\mu\nu}$ as

$\displaystyle
W^{\mu\nu}=\sum_{i=1}^{2}V_{U,i}t_{U,i}^{\mu\nu}+\sum_{i=1}^{1}V_{V,i}t_{V,i}^{\mu\nu}+\sum_{i=1}^{4}V_{T,i}t_{T,i}^{\mu\nu}+\sum_{i=1}^{2}V_{R,i}t_{R,i}^{\mu\nu},$
(60)

where the nine basis tensors are

$\displaystyle t_{U}^{\mu\nu}$
$\displaystyle=\left\\{\widetilde{g}^{\mu\nu},\widetilde{P}^{\mu}\widetilde{P}^{\nu}\right\\},$
(61) $\displaystyle t_{V}^{\mu\nu}$
$\displaystyle=\epsilon^{SPq\\{\mu}\widetilde{P}^{\nu\\}},$ (62)
$\displaystyle t_{T}^{\mu\nu}$
$\displaystyle=\left\\{T^{qq}\widetilde{g}^{\mu\nu},T^{qq}\widetilde{P}^{\mu}\widetilde{P}^{\nu},\widetilde{T}^{q\\{\mu}\widetilde{P}^{\nu\\}},\widetilde{T}^{\mu\nu}\right\\},$
(63) $\displaystyle t_{R}^{\mu\nu}$
$\displaystyle=\left\\{\epsilon^{R^{qq}Pq\\{\mu}\widetilde{P}^{\nu\\}},R_{\rho}^{~{}q\\{\mu}\epsilon^{\nu\\}\rho
Pq}\right\\},$ (64)

with the subscripts $U$, $V$, $T$, and $R$ indicating the polarization of the
hadron $\Omega$. The coefficients $V_{U,i}$, $V_{V,i}$, $V_{T,i}$, and
$V_{R,i}$ are scalar functions of $P\cdot q$ and $q^{2}$, often referred to as
structure functions. Although the final definition of structure functions are
linear combinations of these coefficients, the number of independent structure
functions is completely determined by the number of basis tensor from this
kinematic analysis.

After contracting the leptonic tensor and the hadronic tensor, we can obtain
the general differential cross section in Eq. (50). Whereas, it is practically
convenient to specify a reference frame and to separate the contributions from
each structure function according to the angular distribution of the observed
hadron as well as its spin states. As a natural choice, we consider the
differential cross section in the center-of-mass frame of leptons and define
the temporal basis vector

$\displaystyle\hat{t}^{\mu}$ $\displaystyle=\frac{q^{\mu}}{Q}.$ (65)

The spatial vector $\hat{z}$ is chosen opposite to the direction of the hadron
momentum and can be written in covariant form as

$\displaystyle\hat{z}^{\mu}$
$\displaystyle=\frac{z_{h}q^{\mu}-2P^{\mu}}{z_{h}Q\sqrt{1-\gamma_{h}^{2}}},$
(66)

where

$\displaystyle z_{h}=\frac{2P\cdot
q}{Q^{2}},\quad\gamma_{h}=\frac{2M}{z_{h}Q}.$ (67)

We keep the $\gamma_{h}$ here since the mass of the $\Omega$ may not be
negligible. Following the definitions of the light-cone basis vectors

$\displaystyle\bar{n}^{\mu}=\frac{1}{\sqrt{2}}(\hat{t}^{\mu}+\hat{z}^{\mu}),\quad
n^{\mu}=\frac{1}{\sqrt{2}}(\hat{t}^{\mu}-\hat{z}^{\mu}),$ (68)

we can express the transverse metric $g_{T}^{\mu\nu}$ and the transverse
antisymmetric tensor $\epsilon_{T}^{\mu\nu}$ with external momenta as

$\displaystyle g_{T}^{\mu\nu}$
$\displaystyle=g^{\mu\nu}-\frac{P^{\mu}q^{\nu}+P^{\nu}q^{\mu}}{P\cdot
q\left(1-\gamma_{h}^{2}\right)}+\frac{\gamma_{h}^{2}}{1-\gamma_{h}^{2}}\left(\frac{q^{\mu}q^{\nu}}{Q^{2}}+\frac{P^{\mu}P^{\nu}}{M^{2}}\right),$
(69) $\displaystyle\epsilon_{T}^{\mu\nu}$ $\displaystyle=\frac{1}{P\cdot
q\sqrt{1-\gamma_{h}^{2}}}\epsilon^{\mu\nu\rho\sigma}q_{\rho}P_{\sigma}.$ (70)

Furthermore, we set the lepton momentum in the $\hat{x}-\hat{z}$ plane and
define the transverse spatial basis vectors as

$\displaystyle\hat{x}^{\mu}=\frac{g_{T}^{\mu\nu}l_{1\nu}}{\sqrt{-g_{T}^{\mu\nu}l_{1\mu}l_{1\nu}}},\quad\hat{y}^{\mu}=\epsilon_{T}^{\mu\nu}\hat{x}_{\nu}.$
(71)

In the coordinate system given by $\\{\hat{t},\hat{x},\hat{y},\hat{z}\\}$, one
can explicitly write down the momenta as

$\displaystyle l_{1}^{\mu}$
$\displaystyle=\frac{Q}{2}\left(1,\sin{\theta},0,\cos{\theta}\right),$ (72)
$\displaystyle l_{2}^{\mu}$
$\displaystyle=\frac{Q}{2}\left(1,-\sin{\theta},0,-\cos{\theta}\right),$ (73)
$\displaystyle P^{\mu}$
$\displaystyle=\frac{z_{h}Q}{2}\left(1,0,0,-\sqrt{1-\gamma_{h}^{2}}\right),$
(74) $\displaystyle q^{\mu}$ $\displaystyle=(Q,0,0,0),$ (75)

where $\theta$ is the polar angle of the electron momentum $l_{1}$. It can be
defined in Lorentz invariant form as

$\displaystyle\cos{\theta}=\frac{2y-1}{\sqrt{1-\gamma_{h}^{2}}},\quad\sin{\theta}=\sqrt{\frac{4y-4y^{2}-\gamma_{h}^{2}}{1-\gamma_{h}^{2}}},$
(76)

where $y=(P\cdot l_{1})/(P\cdot q)$.

By construction, the hadron momentum $P$ has zero transverse component and the
large component is $P^{-}$. So the longitudinal and transverse components of
the spin tensors in Eqs. (18)–(20) have the same meanings in this reference
frame and coordinate system. In addition to $\hat{x}$ and $\hat{y}$, which can
be used to project out the transverse components, we define a longitudinal
basis vector,

$\displaystyle\hat{L}^{\mu}=-\frac{M}{2P\cdot\bar{n}}\bar{n}^{\mu}+\frac{P\cdot\bar{n}}{M}n^{\mu},$
(77)

which satisfies the orthogonal relation $P\cdot\hat{L}=0$. Then we redefine
the spin components in Eqs. (11), (13), and (15) as

$\displaystyle S_{L}$ $\displaystyle=S^{\mu}\hat{L}_{\mu},$ (78)
$\displaystyle S_{T}^{x}$
$\displaystyle=|S_{T}|\cos\phi_{T}=-S^{\mu}\hat{x}_{\mu},$ (79) $\displaystyle
S_{T}^{y}$ $\displaystyle=|S_{T}|\sin\phi_{T}=-S^{\mu}\hat{y}_{\mu},$ (80)
$\displaystyle S_{LL}$ $\displaystyle=T^{\mu\nu}\hat{L}_{\mu}\hat{L}_{\nu},$
(81) $\displaystyle S_{LT}^{x}$
$\displaystyle=|S_{LT}|\cos\phi_{LT}=-2T^{\mu\nu}\hat{L}_{\mu}\hat{x}_{\nu},$
(82) $\displaystyle S_{LT}^{y}$
$\displaystyle=|S_{LT}|\sin\phi_{LT}=-2T^{\mu\nu}\hat{L}_{\mu}\hat{y}_{\nu},$
(83) $\displaystyle S_{TT}^{xx}$ $\displaystyle=|S_{TT}|\cos
2\phi_{TT}=2T^{\mu\nu}\hat{x}_{\mu}\hat{x}_{\nu}+T^{\mu\nu}\hat{L}_{\mu}\hat{L}_{\nu},$
(84) $\displaystyle S_{TT}^{xy}$ $\displaystyle=|S_{TT}|\sin
2\phi_{TT}=2T^{\mu\nu}\hat{x}_{\mu}\hat{y}_{\nu},$ (85) $\displaystyle
S_{LLL}$
$\displaystyle=R^{\mu\nu\rho}\hat{L}_{\mu}\hat{L}_{\nu}\hat{L}_{\rho},$ (86)
$\displaystyle S_{LLT}^{x}$
$\displaystyle=|S_{LLT}|\cos\phi_{LLT}=-R^{\mu\nu\rho}\hat{L}_{\mu}\hat{L}_{\nu}\hat{x}_{\rho},$
(87) $\displaystyle S_{LLT}^{y}$
$\displaystyle=|S_{LLT}|\sin\phi_{LLT}=-R^{\mu\nu\rho}\hat{L}_{\mu}\hat{L}_{\nu}\hat{y}_{\rho},$
(88) $\displaystyle S_{LTT}^{xx}$ $\displaystyle=|S_{LTT}|\cos
2\phi_{LTT}=4R^{\mu\nu\rho}\hat{L}_{\mu}\hat{x}_{\nu}\hat{x}_{\rho}+2R^{\mu\nu\rho}\hat{L}_{\mu}\hat{L}_{\nu}\hat{L}_{\rho},$
(89) $\displaystyle S_{LTT}^{xy}$ $\displaystyle=|S_{LTT}|\sin
2\phi_{LTT}=4R^{\mu\nu\rho}\hat{L}_{\mu}\hat{x}_{\nu}\hat{y}_{\rho},$ (90)
$\displaystyle S_{TTT}^{xxx}$ $\displaystyle=|S_{TTT}|\cos
3\phi_{TTT}=-4R^{\mu\nu\rho}\hat{x}_{\mu}\hat{x}_{\nu}\hat{x}_{\rho}-3R^{\mu\nu\rho}\hat{L}_{\mu}\hat{L}_{\nu}\hat{x}_{\rho},$
(91) $\displaystyle S_{TTT}^{yxx}$ $\displaystyle=|S_{TTT}|\sin
3\phi_{TTT}=-4R^{\mu\nu\rho}\hat{y}_{\mu}\hat{x}_{\nu}\hat{x}_{\rho}-R^{\mu\nu\rho}\hat{L}_{\mu}\hat{L}_{\nu}\hat{y}_{\rho}.$
(92)

Since $\hat{x}$, $\hat{y}$, and $\hat{L}$ are constructed with external
momenta of the reaction, the quantities $S_{L}$, $|S_{T}|$, $\phi_{T}$,
$S_{LL}$, $|S_{LT}|$, $\phi_{LT}$, $|S_{TT}|$, $\phi_{TT}$, $S_{LLL}$,
$|S_{LLT}|$, $\phi_{LLT}$, $|S_{LTT}|$, $\phi_{LTT}$, $|S_{TTT}|$, and
$\phi_{TTT}$ are Lorentz scalars, although the meaning of transverse or
longitudinal direction is more clearly understood in this particular frame.
The differential cross section can be expressed as

$\displaystyle P^{0}\frac{d\sigma}{d^{3}\bm{P}}=$
$\displaystyle\frac{\alpha^{2}}{Q^{4}}\left\\{\left(1+\cos^{2}{\theta}\right)F_{U}^{T}+\left(1-\cos^{2}{\theta}\right)F_{U}^{L}+|S_{T}|\sin{\phi_{T}}\sin{2\theta}F_{T}^{\sin{\phi_{T}}}\right.$
$\displaystyle+S_{LL}\left[\left(1+\cos^{2}{\theta}\right)F_{LL}^{T}+\left(1-\cos^{2}{\theta}\right)F_{LL}^{L}\right]$
$\displaystyle+|S_{LT}|\cos{\phi_{LT}}\sin{2\theta}F_{LT}^{\cos{\phi_{LT}}}+|S_{TT}|\cos{2\phi_{TT}}\sin^{2}{\theta}F_{TT}^{\cos{2\phi_{TT}}}$
$\displaystyle\left.+|S_{LLT}|\sin{\phi_{LLT}}\sin{2\theta}F_{LLT}^{\sin{\phi_{LLT}}}+|S_{LTT}|\sin{2\phi_{LTT}}\sin^{2}{\theta}F_{LTT}^{\sin{2\phi_{LTT}}}\right\\},$
(93)

where the $F$’s, as functions of $z_{h}$ and $Q^{2}$, are structure functions,
which are linear combinations of the $V$’s in Eq. (60). The nine structure
functions are defined according to their contributions to certain hadron
polarization states as labeled by the subscripts and the azimuthal modulations
as labeled by the superscripts. For those contributing flat distributions in
the azimuthal angles, we add superscripts $T$ and $L$ to represent the
transverse and longitudinal polarization states of the virtual photon, which
lead to $\left(1+\cos^{2}{\theta}\right)$ and
$\left(1-\cos^{2}{\theta}\right)$ distributions in the polar angle,
respectively. The two structure functions, $F_{LLT}^{\sin\phi_{LLT}}$ and
$F_{LTT}^{\sin 2\phi_{LTT}}$, are newly introduced for the production of
rank-3 tensor polarized hadrons from unpolarized $e^{+}e^{-}$ collisions,
while the other ones also exist in the production of a spin-1 hadron [79].

Now we calculate the structure functions in the parton model. According to the
factorization, we write the hadronic tensor,

$\displaystyle W^{\mu\nu}(q;P,S,T,R)=\frac{1}{2\pi}\int
d^{4}k\operatorname{Tr}[\Delta(k;P,S,T,R)H^{\mu\nu}(k,q)],$ (94)

as a convolution between the correlation function, in Eq. (24), and the hard
factor

$\displaystyle H^{\mu\nu}(k,q)$
$\displaystyle=\gamma^{\mu}(\not{q}-\not{k})\gamma^{\nu}(2\pi)\delta_{+}((q-k)^{2}),$
(95)

where the subscript “+” of $\delta$ function indicates picking the positive
energy solution. With the collinear approximation, one can derive the the
hadronic tensor at leading twist as

$\displaystyle W^{\mu\nu}(q;P,S,T,R)$
$\displaystyle=\frac{2}{z}\mathrm{Tr}\left[\Delta(z)\gamma^{\mu}\not{\bar{n}}\gamma^{\nu}\right],$
(96)

where the $k_{T}$-integrated correlation function $\Delta(z)$ is defined in
Eq. (41). As has been derived in Eqs. (42)–(48), it is parametrized in terms
of collinear FFs,

$\displaystyle\Delta(z)=$
$\displaystyle\frac{1}{4}\Big{\\{}D_{1}\left(z\right)\not{n}+G_{1L}\left(z\right)S_{L}\gamma_{5}\not{n}+H_{1T}\left(z\right)i\sigma_{\mu\nu}\gamma_{5}n^{\mu}S_{T}^{\nu}$
$\displaystyle+D_{1LL}\left(z\right)S_{LL}\not{n}+H_{1LT}\left(z\right)\sigma_{\mu\nu}S_{LT}^{\mu}n^{\nu}$
$\displaystyle+G_{1LLL}\left(z\right)S_{LLL}\gamma_{5}\not{n}+H_{1LLT}\left(z\right)i\sigma_{\mu\nu}\gamma_{5}n^{\mu}S_{LLT}^{\nu}\Big{\\}}.$
(97)

In this process, the collinear FF to $\Omega$ does not couple to other long-
distance functions. As a consequence of the helicity conservation, the chiral-
odd FFs decouple from the cross section in the limit of massless quarks, and
only the chiral-even FFs contribute. Besides, for the unpolarized collision,
the leptonic tensor $L_{\mu\nu}$ is symmetric under the one-photon-exchange
approximation, and thus we only need to consider the symmetric part of the
hadronic tensor $W^{\mu\nu}$. After contracting the leptonic tensor and the
hadronic tensor, we obtain the differential cross section in terms of
collinear FFs as

$\displaystyle
P^{0}\frac{d\sigma}{d^{3}\bm{P}}=N_{c}\sum_{q}e_{q}^{2}\frac{\alpha^{2}}{Q^{4}}\frac{1}{z_{h}}\left(1+\cos^{2}{\theta}\right)\left[D_{1,q}(z_{h})+D_{1LL,q}(z_{h})S_{LL}\right],$
(98)

where the sum of “$q$” runs over all active quark and antiquark flavors with
$e_{q}$ being the charge in unit of the positron charge and a subscript “$q$”
is explicitly added to the FFs. As expected from the above analysis, only two
terms, $D_{1}$ and $D_{1LL}$, contribute to the cross section. Comparing with
general expression from kinematic analysis (93), we can write the structure
functions in terms of collinear FFs at leading twist as

$\displaystyle F_{U}^{T}(z_{h},Q^{2})$
$\displaystyle=\frac{N_{c}}{z_{h}}\sum_{q}e_{q}^{2}D_{1,q}(z_{h},Q^{2}),$ (99)
$\displaystyle F_{LL}^{T}(z_{h},Q^{2})$
$\displaystyle=\frac{N_{c}}{z_{h}}\sum_{q}e_{q}^{2}D_{1LL,q}(z_{h},Q^{2}),$
(100)

where $N_{c}=3$ is the number of colors. Since none of the newly defined
rank-3 tensor polarized FFs contributes, the unpolarized differential cross
section of the inclusive production of a spin-3/2 hadron shares the same form
as that of the inclusive production of a spin-1 hadron. Therefore, one cannot
extract rank-3 tensor polarized FFs from the measurement of inclusive
production of a spin-3/2 hadron in unpolarized $e^{+}e^{-}$ annihilation, or
in other words, any nonzero result of the measurement of a spin-3/2 hadron in
a rank-3 tensor polarization state will indicate high-twist or high-order
effects.

###  IV.2 Semi-inclusive production of the $\Omega$ in unpolarized
$e^{+}e^{-}$ collisions

Figure 2: Left: The illustration of the semi-inclusive production of $\Omega$
in $e^{+}e^{-}$ annihilation with the second observed hadron $h$ nearly back
to the $\Omega$. Right: The diagram of the process $e^{+}e^{-}\to\Omega hX$
under the one-photon-exchange approximation.

To access TMD FFs, we consider the process

$\displaystyle e^{-}(l_{1})+e^{+}(l_{2})\to\Omega(P_{1})+h(P_{2})+X(P_{X}),$
(101)

as illustrated in Fig. 2, where the variables in parentheses are the four
momenta of the corresponding particles and for simplicity the spin of the
second hadron $h$ is not taken into account. With one-photon-exchange
approximation, the differential cross section can be expressed as

$\displaystyle\frac{P_{1}^{0}P_{2}^{0}d\sigma}{d^{3}\bm{P}_{1}d^{3}\bm{P}_{2}}=\frac{\alpha^{2}}{4Q^{6}}L_{\mu\nu}W^{\mu\nu},$
(102)

where the leptonic tensor $L_{\mu\nu}$ has the same definition in Eq. (51) and
the hadronic tensor $W^{\mu\nu}$ is given by

$\displaystyle W^{\mu\nu}\left(q;P_{1},S,T,R;P_{2}\right)$
$\displaystyle=\frac{1}{(2\pi)^{4}}\sum_{X}(2\pi)^{4}\delta^{4}\left(q-P_{X}-P_{1}-P_{2}\right)$
$\displaystyle\quad\times\left\langle
0\left|J^{\mu}(0)\right|P_{X};P_{1},S,T,R;P_{2}\right\rangle\left\langle
P_{X};P_{1},S,T,R;P_{2}\left|J^{\nu}(0)\right|0\right\rangle,$ (103)

where $S$, $T$, and $R$ represent the spin states of the first hadron, while
the second hadron is unpolarized. The hadronic tensor satisfies the
Hermiticity and the parity invariance relations,

$\displaystyle W^{*\mu\nu}\left(q;P_{1},S,T,R;P_{2}\right)$
$\displaystyle=W^{\nu\mu}\left(q;P_{1},S,T,R;P_{2}\right),$ (104)
$\displaystyle W^{\mu\nu}\left(q;P_{1},S,T,R;P_{2}\right)$
$\displaystyle=W_{\mu\nu}\left(q;\bar{P}_{1},-\bar{S},\bar{T},-\bar{R};\bar{P}_{2}\right),$
(105)

and the gauge invariance requirement $q_{\mu}W^{\mu\nu}=W^{\mu\nu}q_{\nu}=0$.

Following the same procedure in Sec. IV.1, one can, in principle, construct
the basis tensors from the conserved vectors and tensors by exhausting all
possible combinations. However, it is not efficient with increasing number of
measured momenta. Instead of directly constructing all basis tensors, we have
a systematic procedure for the semi-inclusive process, where the hadronic
tensor depends on three external momenta. Because the four vectors
$P_{1}^{\mu}$, $P_{2}^{\mu}$, $q^{\mu}$, and $\epsilon^{\mu qP_{1}P_{2}}$ form
complete basis of the spacetime, we can first construct the basic Lorentz
tensors,

$\displaystyle t_{U}^{\mu\nu}$
$\displaystyle=\left\\{\widetilde{g}^{\mu\nu},\widetilde{P}_{1}^{\mu}\widetilde{P}_{1}^{\nu},\widetilde{P}_{2}^{\mu}\widetilde{P}_{2}^{\nu},\widetilde{P}_{1}^{\\{\mu}\widetilde{P}_{2}^{\nu\\}}\right\\},$
(106) $\displaystyle t_{U}^{\mathcal{P},\mu\nu}$
$\displaystyle=\left\\{\widetilde{P}_{1}^{\\{\mu}\epsilon^{\nu\\}qP_{1}P_{2}},\widetilde{P}_{2}^{\\{\mu}\epsilon^{\nu\\}qP_{1}P_{2}}\right\\},$
(107)

which are grouped into two sets $t_{U}^{\mu\nu}$ and $t_{U}^{{\cal
P},\mu\nu}$, corresponding to parity conserving and parity nonconserving ones,
respectively. The conserved vectors $\widetilde{P}_{1}$ and
$\widetilde{P}_{2}$ are defined in a similar way to Eq. (56). The unpolarized
basis tensors are directly given by those in $t_{U}^{\mu\nu}$. One may also
include those in $t_{U}^{{\cal P}\mu\nu}$ if parity-violating channels, e.g.,
the exchange of a $Z$ boson, are taken into account. Since all the spin
components can essentially be expressed into scalars or pseudoscalars, as
explicitly written in Eqs. (78)–(92), the polarized basis tensors can be
obtained by multiplying the basic Lorentz tensors by a spin-dependent scalar
or pseudoscalar. Considering the parity conserving case, we multiply the basic
Lorentz tensors in $t_{U}^{\mu\nu}$ by a spin-dependent scalar and multiply
those in $t_{U}^{{\cal P},\mu\nu}$ by a spin-dependent pseudoscalar. Then the
eight vector polarized basis tensors, 16 rank-2 tensor polarized basis
tensors, and 20 rank-3 tensor polarized basis tensors are constructed from
those in $t_{U}^{\mu\nu}$ and $t_{U}^{{\cal P},\mu\nu}$ as

$\displaystyle t_{V}^{\mu\nu}$
$\displaystyle=\left\\{\epsilon^{SqP_{1}P_{2}}\right\\}t_{U}^{\mu\nu},\left\\{S\cdot
q,S\cdot P_{2}\right\\}t_{U}^{\mathcal{P},\mu\nu},$ (108) $\displaystyle
t_{T}^{\mu\nu}$
$\displaystyle=\left\\{T^{P_{2}P_{2}},T^{P_{2}q},T^{qq}\right\\}t_{U}^{\mu\nu},\left\\{\epsilon^{T^{P_{2}}P_{1}P_{2}q},\epsilon^{T^{q}P_{1}P_{2}q}\right\\}t_{U}^{\mathcal{P},\mu\nu},$
(109) $\displaystyle t_{R}^{\mu\nu}$
$\displaystyle=\left\\{\epsilon^{R^{P_{2}P_{2}}P_{1}P_{2}q},\epsilon^{R^{P_{2}q}P_{1}P_{2}q},\epsilon^{R^{qq}P_{1}P_{2}q}\right\\}t_{U}^{\mu\nu},\left\\{R^{P_{2}P_{2}P_{2}},R^{qqq},R^{P_{2}P_{2}q},R^{P_{2}qq}\right\\}t_{U}^{\mathcal{P},\mu\nu}.$
(110)

With the 48 basis tensors above, we can expand the hadronic tensor
$W^{\mu\nu}$ as

$\displaystyle
W^{\mu\nu}=\sum_{i=1}^{4}V_{U,i}t_{U,i}^{\mu\nu}+\sum_{i=1}^{8}V_{V,i}t_{V,i}^{\mu\nu}+\sum_{i=1}^{16}V_{T,i}t_{T,i}^{\mu\nu}+\sum_{i=1}^{20}V_{R,i}t_{R,i}^{\mu\nu},$
(111)

where the coefficients $V$’s are scalar functions of $q^{2}$, $P_{1}\cdot q$,
$P_{2}\cdot q$, and $P_{1}\cdot P_{2}$. We should note that this procedure is
invalid for the inclusive process because the number of momenta in the
hadronic tensor is not enough to form a complete basis of the spacetime.

Contracting the hadronic tensor with the leptonic tensor, one can derive the
differential cross section in Eq. (102), and it is common to introduce some
dimensionless variables to replace the scalar products of momenta,

$\displaystyle z_{h1}$ $\displaystyle=\frac{2P_{1}\cdot q}{Q^{2}},\quad
z_{h2}=\frac{2P_{2}\cdot q}{Q^{2}},\quad\xi=\frac{2P_{1}\cdot P_{2}}{Q^{2}},$
$\displaystyle y_{1}$ $\displaystyle=\frac{l_{1}\cdot P_{1}}{q\cdot
P_{1}},\quad y_{2}=\frac{l_{1}\cdot P_{2}}{q\cdot
P_{2}},\quad\gamma_{h1}=\frac{2M_{1}}{z_{h1}Q},\quad\gamma_{h2}=\frac{2M_{2}}{z_{h2}Q},$
(112)

where $M_{1}$ and $M_{2}$ are the masses of the two hadrons, respectively.

Figure 3: The Collins-Soper frame for $e^{+}e^{-}\rightarrow\Omega hX$.

To separate contributions from different structure functions, it is in
practice convenient to specify some reference frame and coordinate system to
define angular distributions. For this process, there are two commonly used
frames. One is the Collins-Soper (CS) frame [104], illustrated in Fig. 3. It
is defined as the center-of-mass frame of the leptons with the longitudinal
spatial axis chosen as the exterior angle bisector of the momenta of the two
hadrons in the final state. In this frame, one can write down the components
of the momenta as

$\displaystyle l_{1}^{\mu}$
$\displaystyle=\frac{Q}{2}(1,\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta),$
(113) $\displaystyle l_{2}^{\mu}$
$\displaystyle=\frac{Q}{2}(1,-\sin\theta\cos\phi,-\sin\theta\sin\phi,-\cos\theta),$
(114) $\displaystyle P_{1}^{\mu}$
$\displaystyle=\frac{z_{h1}Q}{2}\left(1,\sqrt{1-\gamma_{h1}^{2}}\sin\beta,0,-\sqrt{1-\gamma_{h1}^{2}}\cos\beta\right),$
(115) $\displaystyle P_{2}^{\mu}$
$\displaystyle=\frac{z_{h2}Q}{2}\left(1,\sqrt{1-\gamma_{h2}^{2}}\sin\beta,0,\sqrt{1-\gamma_{h2}^{2}}\cos\beta\right),$
(116) $\displaystyle q^{\mu}$ $\displaystyle=(Q,0,0,0),$ (117)

where the angles $\beta$, $\theta$, and $\phi$, as illustrated in Fig. 3, can
also be defined in Lorentz invariant forms,

$\displaystyle\cos{2\beta}$ $\displaystyle=\frac{2\xi-
z_{h1}z_{h2}}{z_{h1}z_{h2}\sqrt{\left(1-\gamma_{h1}^{2}\right)}\sqrt{\left(1-\gamma_{h2}^{2}\right)}}\approx\frac{2\xi-
z_{h1}z_{h2}}{z_{h1}z_{h2}},$ (118) $\displaystyle\cos{\theta}$
$\displaystyle=\frac{1-2y_{2}}{2\sqrt{1-\gamma_{h2}^{2}}\cos{\beta}}-\frac{1-2y_{1}}{2\sqrt{1-\gamma_{h1}^{2}}\cos{\beta}}\approx\frac{y_{1}-y_{2}}{\cos{\beta}},$
(119) $\displaystyle\cos{\phi}$
$\displaystyle=\frac{1-2y_{2}}{2\sqrt{1-\gamma_{h2}^{2}}\sin{\beta}\sin{\theta}}+\frac{1-2y_{1}}{2\sqrt{1-\gamma_{h1}^{2}}\sin{\beta}\sin{\theta}}\approx\frac{1-y_{1}-y_{2}}{\sin{\beta\sin{\theta}}},$
(120)

where “$\approx$” represents the approximation of neglecting the hadron masses
although the masses are kept in this calculation.

Figure 4: The two-hadron center-of-mass frame for $e^{+}e^{-}\rightarrow\Omega
hX$.

The other frame is the two-hadron back-to-back frame, which can be chosen as
the center-of-mass frame of the two hadrons in the final state, referred to as
the c.m. frame. As illustrated in Fig. 4, we construct the coordinate system
by choosing the temporal basis vector as

$\displaystyle\hat{t}^{\mu}$
$\displaystyle=\frac{P_{1}^{\mu}+P_{2}^{\mu}}{\sqrt{M_{1}^{2}+M_{2}^{2}+2P_{1}\cdot
P_{2}}},$ (121)

and the longitudinal spatial basis vector as

$\displaystyle\hat{z}^{\mu}$
$\displaystyle=\frac{P_{2}^{\mu}\left(M_{1}^{2}+P_{1}\cdot
P_{2}\right)-P_{1}^{\mu}\left(M_{2}^{2}+P_{1}\cdot
P_{2}\right)}{\sqrt{\left(P_{1}\cdot
P_{2}\right)^{2}-M_{1}^{2}M_{2}^{2}}\sqrt{M_{1}^{2}+M_{2}^{2}+2P_{1}\cdot
P_{2}}}.$ (122)

The light-cone basis vectors are given by
$\bar{n}^{\mu}=(\hat{t}^{\mu}+\hat{z}^{\mu})/\sqrt{2}$ and
${n}^{\mu}=(\hat{t}^{\mu}-\hat{z}^{\mu})/\sqrt{2}$, which are different from
those in Eq. (68). Correspondingly, the transverse metric tensor
$g_{T}^{\mu\nu}$ and the transverse antisymmetric tensor
$\epsilon_{T}^{\mu\nu}$ are given by

$\displaystyle g^{\mu\nu}_{T}$
$\displaystyle=g^{\mu\nu}-\frac{\left(P_{1}\cdot
P_{2}\right)\left(P_{1}^{\mu}P_{2}^{\nu}+P_{1}^{\nu}P_{2}^{\mu}\right)}{\left(P_{1}\cdot
P_{2}\right)^{2}-M_{1}^{2}M_{2}^{2}}+\frac{M_{1}^{2}P_{2}^{\mu}P_{2}^{\nu}+M_{2}^{2}P_{1}^{\mu}P_{1}^{\nu}}{\left(P_{1}\cdot
P_{2}\right)^{2}-M_{1}^{2}M_{2}^{2}}$ $\displaystyle\approx
g^{\mu\nu}-\frac{P_{1}^{\mu}P_{2}^{\nu}+P_{1}^{\nu}P_{2}^{\mu}}{P_{1}\cdot
P_{2}},$ (123) $\displaystyle\epsilon_{T}^{\mu\nu}$
$\displaystyle=\epsilon^{\mu\nu\rho\sigma}\frac{P_{2\rho}P_{1\sigma}}{\sqrt{\left(P_{1}\cdot
P_{2}\right)^{2}-M_{1}^{2}M_{2}^{2}}}$
$\displaystyle\approx\epsilon^{\mu\nu\rho\sigma}\frac{P_{2\rho}P_{1\sigma}}{P_{1}\cdot
P_{2}}.$ (124)

By construction, the hadrons have zero transverse momenta and the large
momentum components are $P_{1}^{-}$ and $P_{2}^{+}$ according to this choice
of $\hat{t}$ and $\hat{z}$. Whereas, the virtual photon has transverse
momentum in this frame. We choose the transverse spatial basis vectors as

$\displaystyle\hat{x}^{\mu}$
$\displaystyle=\frac{g_{T}^{\mu\nu}q_{\nu}}{\sqrt{-g_{T}^{\mu\nu}q_{\mu}q_{\nu}}},$
(125) $\displaystyle\hat{y}^{\mu}$
$\displaystyle=\epsilon_{T}^{\mu\nu}\hat{x}_{\nu},$ (126)

and thus the virtual photon is in the $\hat{x}-\hat{z}$ plane.

In this calculation, we find the CS frame is more convenient to describe the
angular distributions of the produced hadrons, but the spin components are
easier to be defined in the c.m. frame. Therefore, taking advantage of the
both frames, we will use $\theta$ and $\phi$ introduced in the CS frame and
the spin components in the c.m. frame, $S_{L}$, $|S_{T}|$, $\phi_{T}$,
$S_{LL}$, $|S_{LT}|$, $\phi_{LT}$, $|S_{TT}|$, $\phi_{TT}$, $S_{LLL}$,
$|S_{LLT}|$, $\phi_{LLT}$, $|S_{LTT}|$, $\phi_{LTT}$, $|S_{TTT}|$, and
$\phi_{TTT}$, which have the same expressions as Eqs. (78)–(92) but with the
basis vectors replaced by Eqs. (125) and (126). Since all these variables are
defined in Lorentz invariant forms, one can in principle evaluate them in
arbitrary frame, although the meanings of the variables are clearer in certain
frames.

After contracting the hadronic tensor and the leptonic tensor, we can express
the differential cross section in terms of 48 structure functions according to
the angular distributions and the spin states of the produced hadrons,

$\displaystyle\frac{P_{1}^{0}P_{2}^{0}d\sigma}{d^{3}\bm{P}_{1}d^{3}\bm{P}_{2}}=\frac{\alpha^{2}}{4Q^{4}}\times$
$\displaystyle\Bigg{\\{}\left[\left(1+\cos^{2}\theta\right)F_{U,U}^{T}+\left(1-\cos^{2}\theta\right)F_{U,U}^{L}+(\sin
2\theta\cos\phi)F_{U,U}^{\cos\phi}+\left(\sin^{2}\theta\cos
2\phi\right)F_{U,U}^{\cos 2\phi}\right]$
$\displaystyle+S_{L}\left[\left(\sin^{2}\theta\sin 2\phi\right)F_{L,U}^{\sin
2\phi}+(\sin 2\theta\sin\phi)F_{L,U}^{\sin\phi}\right]$
$\displaystyle+\left|{S}_{T}\right|\left[\operatorname{sin}\phi_{T}\left(\left(1+\cos^{2}\theta\right)F_{T,U}^{T}+\left(1-\cos^{2}\theta\right)F_{T,U}^{L}+(\sin
2\theta\cos\phi)F_{T,U}^{\cos\phi}\right.\right.$
$\displaystyle\left.\left.+\left(\sin^{2}\theta\cos 2\phi\right)F_{T,U}^{\cos
2\phi}\right)+\cos\phi_{T}\left(\left(\sin^{2}\theta\sin
2\phi\right)F_{T,U}^{\sin 2\phi}+(\sin
2\theta\sin\phi)F_{T,U}^{\sin\phi}\right)\right]$
$\displaystyle+S_{LL}\left[\left(1+\cos^{2}\theta\right)F_{LL,U}^{T}+\left(1-\cos^{2}\theta\right)F_{LL,U}^{L}+(\sin
2\theta\cos\phi)F_{LL,U}^{\cos\phi}+\left(\sin^{2}\theta\cos
2\phi\right)F_{LL,U}^{\cos 2\phi}\right]$
$\displaystyle+\left|S_{LT}\right|\left[\operatorname{cos}\phi_{LT}\left(\left(1+\cos^{2}\theta\right)F_{LT,U}^{T}+\left(1-\cos^{2}\theta\right)F_{LT,U}^{L}+(\sin
2\theta\cos\phi)F_{LT,U}^{\cos\phi}\right.\right.$
$\displaystyle\left.\left.+\left(\sin^{2}\theta\cos 2\phi\right)F_{LT,U}^{\cos
2\phi}\right)+\sin\phi_{LT}\left(\left(\sin^{2}\theta\sin
2\phi\right)F_{LT,U}^{\sin 2\phi}+(\sin
2\theta\sin\phi)F_{LT,U}^{\sin\phi}\right)\right]$
$\displaystyle+\left|S_{TT}\right|\left[\operatorname{cos}2\phi_{TT}\left(\left(1+\cos^{2}\theta\right)F_{TT,U}^{T}+\left(1-\cos^{2}\theta\right)F_{TT,U}^{L}+(\sin
2\theta\cos\phi)F_{TT,U}^{\cos\phi}\right.\right.$
$\displaystyle\left.\left.+\left(\sin^{2}\theta\cos 2\phi\right)F_{TT,U}^{\cos
2\phi}\right)+\sin 2\phi_{TT}\left(\left(\sin^{2}\theta\sin
2\phi\right)F_{TT,U}^{\sin 2\phi}+(\sin
2\theta\sin\phi)F_{TT,U}^{\sin\phi}\right)\right]$
$\displaystyle+S_{LLL}\left[\left(\sin^{2}\theta\sin
2\phi\right)F_{LLL,U}^{\sin 2\phi}+(\sin
2\theta\sin\phi)F_{LLL,U}^{\sin\phi}\right]$
$\displaystyle+\left|S_{LLT}\right|\left[\operatorname{sin}\phi_{LLT}\left(\left(1+\cos^{2}\theta\right)F_{LLT,U}^{T}+\left(1-\cos^{2}\theta\right)F_{LLT,U}^{L}+(\sin
2\theta\cos\phi)F_{LLT,U}^{\cos\phi}\right.\right.$
$\displaystyle\left.\left.+\left(\sin^{2}\theta\cos
2\phi\right)F_{LLT,U}^{\cos
2\phi}\right)+\cos\phi_{LLT}\left(\left(\sin^{2}\theta\sin
2\phi\right)F_{LLT,U}^{\sin 2\phi}+(\sin
2\theta\sin\phi)F_{LLT,U}^{\sin\phi}\right)\right]$
$\displaystyle+\left|S_{LTT}\right|\left[\operatorname{sin}2\phi_{LTT}\left(\left(1+\cos^{2}\theta\right)F_{LTT,U}^{T}+\left(1-\cos^{2}\theta\right)F_{LTT,U}^{L}+(\sin
2\theta\cos\phi)F_{LTT,U}^{\cos\phi}\right.\right.$
$\displaystyle\left.\left.+\left(\sin^{2}\theta\cos
2\phi\right)F_{LTT,U}^{\cos 2\phi}\right)+\cos
2\phi_{LTT}\left(\left(\sin^{2}\theta\sin 2\phi\right)F_{LTT,U}^{\sin
2\phi}+(\sin 2\theta\sin\phi)F_{LTT,U}^{\sin\phi}\right)\right]$
$\displaystyle+\left|S_{TTT}\right|\left[\operatorname{sin}3\phi_{TTT}\left(\left(1+\cos^{2}\theta\right)F_{TTT,U}^{T}+\left(1-\cos^{2}\theta\right)F_{TTT,U}^{L}+(\sin
2\theta\cos\phi)F_{TTT,U}^{\cos\phi}\right.\right.$
$\displaystyle\left.\left.+\left(\sin^{2}\theta\cos
2\phi\right)F_{TTT,U}^{\cos 2\phi}\right)+\cos
3\phi_{TTT}\left(\left(\sin^{2}\theta\sin 2\phi\right)F_{TTT,U}^{\sin
2\phi}+(\sin 2\theta\sin\phi)F_{TTT,U}^{\sin\phi}\right)\right]\Bigg{\\}},$
(127)

where the two subscripts for each structure function represent the
polarization states of the two hadrons, $\Omega$ and $h$, respectively, though
the spin of the second hadron is not taken into account and thus effectively
unpolarized. The superscripts have the same meanings as those in the single
hadron inclusive production process, either labeling the azimuthal modulation
or representing the virtual photon polarization.

The $F$’s, as scalar functions of $z_{h1}$, $z_{h2}$, $Q^{2}$, and
${\bm{q}}_{T}^{2}\equiv-g_{T}^{\mu\nu}q_{\mu}q_{\nu}$, are linear combinations
of the $V$’s in Eq. (111). Among them, the 20 rank-3 tensor polarized ones are
newly defined and only exist when one of the detected two hadrons has spin
$s\geq 3/2$, while the four unpolarized ones, the eight vector polarized ones,
and the 16 rank-2 tensor polarized ones also exist if one of the observed
hadrons is spin-1.

Now we calculate the structure functions in the parton model and only focus on
the kinematic region in which the produced two hadrons are nearly back to
back. It corresponds to the region ${\bm{q}}_{T}^{2}\ll Q^{2}$, where one can
apply the TMD factorization. At the leading order, we express the hadronic
tensor $W^{\mu\nu}$ as the production of a quark-antiquark pair convoluting
with two correlation functions,

$\displaystyle W^{\mu\nu}=N_{c}z_{1}z_{2}\sum_{q}e_{q}^{2}\int
d^{2}\bm{k}_{1T}d^{2}\bm{k}_{2T}\delta^{(2)}(\bm{k}_{1T}+\bm{k}_{2T}-\bm{q}_{T}){\rm
Tr}[\Delta^{\Omega/q}\gamma^{\mu}\Delta^{h/\bar{q}}\gamma^{\nu}],$ (128)

where the transverse momenta ${\bm{k}}_{1T}$, ${\bm{k}}_{2T}$, and
${\bm{q}}_{T}$ are defined in the c.m. frame. Since the two hadrons have zero
transverse momentum components in this frame, the ${\bm{k}}_{1T}$ and
${\bm{k}}_{2T}$ directly correspond to the quark transverse momentum in the
definition of the correlation function, while the longitudinal direction is
flipped for the second hadron in comparison to Eq. (30). Hence, the light-cone
longitudinal momentum fractions are given by $z_{1}=k_{1}^{-}/P_{1}^{-}$ and
$z_{2}=k_{2}^{+}/P_{2}^{+}$. The superscripts “$\Omega/q$” and “$h/\bar{q}$”
of the $\Delta$’s indicate the relevant quark and hadron, and will be
suppressed in the rest of the paper if no confusion is caused. The sum on
“$q$” runs over all active flavors of quark and antiquark.

Alternative to the complete parametrization in Eqs. (31)–(40), we write the
$\Delta$’s in a concise form,

$\displaystyle\Delta(z_{1},k_{1T})$
$\displaystyle=\frac{1}{4}\left\\{\Delta^{\left[\gamma^{-}\right]}(z_{1},k_{1T})\gamma^{+}-\Delta^{\left[\gamma^{-}\gamma_{5}\right]}(z_{1},k_{1T})\gamma^{+}\gamma_{5}+\Delta^{\left[i\sigma^{i-}\gamma_{5}\right]}(z_{1},k_{1T})i\sigma_{-i}\gamma_{5}\right\\},$
(129) $\displaystyle\Delta(z_{2},k_{2T})$
$\displaystyle=\frac{1}{4}\left\\{\Delta^{\left[\gamma^{+}\right]}(z_{2},k_{2T})\gamma^{-}-\Delta^{\left[\gamma^{+}\gamma_{5}\right]}(z_{2},k_{2T})\gamma^{-}\gamma_{5}+\Delta^{\left[i\sigma^{j+}\gamma_{5}\right]}(z_{2},k_{2T})i\sigma_{+j}\gamma_{5}\right\\},$
(130)

where we use the notation $\Delta^{[\Gamma]}\left(z,k_{T}\right)\equiv{\rm
Tr}\left[\Delta\left(z,k_{T}\right)\Gamma\right]$ with $\Gamma=\gamma^{-}$,
$\gamma^{-}\gamma_{5}$, and $i\sigma^{i-}\gamma_{5}$ for the hadron $\Omega$
and $\Gamma=\gamma^{+}$, $\gamma^{+}\gamma_{5}$, and $i\sigma^{i+}\gamma_{5}$
for the hadron $h$. A complete parametrization of
$\Delta^{[\Gamma]}(z,k_{T})$’s in terms of TMD FFs are provided in Appendix B.
Since $h$ is unpolarized, there are only two leading-twist TMD FFs, which are
defined as

$\displaystyle\Delta^{[\gamma^{+}]}(z_{2},k_{2T})$
$\displaystyle=D_{1}(z_{2},k_{2T}^{2}),$ (131)
$\displaystyle\Delta^{[i\sigma^{i+}\gamma_{5}]}(z_{2},k_{2T})$
$\displaystyle=-\frac{\epsilon_{T}^{ij}k_{2Tj}}{M_{2}}H_{1}^{\perp}(z_{2},k_{2T}^{2}).$
(132)

Substituting the FFs in Eq. (128) with Eqs. (129) and (130), one can easily
find that only two terms

$\displaystyle{\rm
Tr}\Big{[}\Delta^{\left[\gamma^{-}\right]}(z_{1},k_{1T})\gamma^{+}\gamma^{\mu}\Delta^{\left[\gamma^{+}\right]}(z_{2},k_{2T})\gamma^{-}\gamma^{\nu}\Big{]},$
$\displaystyle{\rm
Tr}\Big{[}\Delta^{\left[i\sigma^{i-}\gamma_{5}\right]}(z_{1},k_{1T})i\sigma_{-i}\gamma_{5}\gamma^{\mu}\Delta^{\left[i\sigma^{j+}\gamma_{5}\right]}(z_{2},k_{2T})i\sigma_{+j}\gamma_{5}\gamma^{\nu}\Big{]},$

contribute to the symmetric part of $W^{\mu\nu}$, because the chiral-odd TMD
FFs in $\Delta^{[i\sigma^{i-}\gamma_{5}]}$ must couple to another chiral-odd
function for the helicity conservation of massless quarks. Then the
differential cross section at the leading-twist approximation is

$\displaystyle\frac{P_{1}^{0}P_{2}^{0}d\sigma}{d^{3}\bm{P}_{1}d^{3}\bm{P}_{2}}$
$\displaystyle=\frac{\alpha^{2}}{4Q^{4}}\frac{1}{4}N_{c}z_{1}z_{2}\sum_{q}e_{q}^{2}\int
d^{2}\bm{k}_{1T}d^{2}\bm{k}_{2T}\delta^{(2)}(\bm{k}_{1T}+\bm{k}_{2T}-\bm{q}_{T})$
$\displaystyle\times\left\\{\left(1+\cos^{2}\theta\right)\Delta^{\left[\gamma^{-}\right]}(z_{1},k_{1T})\Delta^{\left[\gamma^{+}\right]}(z_{2},k_{2T})+\left[\sin^{2}\theta\cos
2\phi\left(\delta^{i1}\delta^{j1}-\delta^{i2}\delta^{j2}\right)\right.\right.$
$\displaystyle\left.\left.+\sin^{2}\theta\sin
2\phi\left(\delta^{i1}\delta^{j2}+\delta^{i2}\delta^{j1}\right)\right]\Delta^{\left[i\sigma^{i-}\gamma_{5}\right]}(z_{1},k_{1T})\Delta^{\left[i\sigma^{j+}\gamma_{5}\right]}(z_{2},k_{2T})\right\\}.$
(133)

Comparing Eq. (133) with Eq. (127), we can obtain the structure functions,
which are convolutions of two TMD FFs. For conciseness, we introduce the
transverse momentum convolution notation

$\displaystyle{\cal
C}\left[w_{a}(k_{1T},k_{2T})D(z_{1},k_{1T}^{2})D(z_{2},k_{2T}^{2})\right]$
$\displaystyle\equiv\frac{1}{4}N_{c}z_{1}z_{2}\sum_{q}e_{q}^{2}\int
d^{2}\bm{k}_{1T}d^{2}\bm{k}_{2T}\delta^{(2)}(\bm{k}_{1T}+\bm{k}_{2T}-\bm{q}_{T})w_{a}(k_{1T},k_{2T})D_{q}(z_{1},k_{1T}^{2})D_{\bar{q}}(z_{2},k_{2T}^{2}),$
(134)

where $D_{q}(z_{1},k_{1T}^{2})$ is a TMD FF for the first hadron $\Omega$,
$D_{\bar{q}}(z_{2},k_{2T}^{2})$ is a TMD FF for the second hadron $h$, and
$w_{a}(k_{1T},k_{2T})$, with $a=1,\cdots,10$, is one of the dimensionless
scalar functions as provided below,

$\displaystyle w_{1}=-\frac{\hat{q}_{\scriptscriptstyle T}\cdot
k_{\scriptscriptstyle 1T}}{M_{1}},\quad
w_{2}=-\frac{\hat{q}_{\scriptscriptstyle T}\cdot k_{\scriptscriptstyle
2T}}{M_{2}},\quad w_{3}=\frac{2(\hat{q}_{\scriptscriptstyle T}\cdot
k_{\scriptscriptstyle 1T})(\hat{q}_{\scriptscriptstyle T}\cdot
k_{\scriptscriptstyle 2T})+k_{\scriptscriptstyle 1T}\cdot
k_{\scriptscriptstyle 2T}}{M_{1}M_{2}},$ $\displaystyle
w_{4}=\frac{k_{\scriptscriptstyle 1T}^{ij}\hat{q}_{\scriptscriptstyle
Ti}k_{{\scriptscriptstyle 2Tj}}+2k_{\scriptscriptstyle
1T}^{ij}\hat{q}_{\scriptscriptstyle Ti}\hat{q}_{\scriptscriptstyle
Tj}(\hat{q}_{\scriptscriptstyle T}\cdot k_{\scriptscriptstyle
2T})}{M_{1}^{2}M_{2}},\quad w_{5}=\frac{2k_{\scriptscriptstyle
1T}^{ij}\hat{q}_{\scriptscriptstyle Ti}\hat{q}_{\scriptscriptstyle
Tj}}{M_{1}^{2}},$ $\displaystyle w_{6}=\frac{2\left[k_{\scriptscriptstyle
1T}^{ijl}\hat{q}_{\scriptscriptstyle Ti}\hat{q}_{\scriptscriptstyle
Tj}k_{\scriptscriptstyle 2Tl}+2k_{\scriptscriptstyle
1T}^{ijl}\hat{q}_{\scriptscriptstyle Ti}\hat{q}_{\scriptscriptstyle
Tj}\hat{q}_{\scriptscriptstyle Tl}(k_{\scriptscriptstyle
2T}\cdot\hat{q}_{\scriptscriptstyle T})\right]}{M_{1}^{3}M_{2}},\quad
w_{7}=-\frac{k_{\scriptscriptstyle 1T}\cdot k_{\scriptscriptstyle
2T}}{M_{1}M_{2}},\quad w_{8}=\frac{4k_{\scriptscriptstyle
1T}^{ijl}\hat{q}_{\scriptscriptstyle Ti}\hat{q}_{\scriptscriptstyle
Tj}\hat{q}_{\scriptscriptstyle Tl}}{M_{1}^{3}},$ $\displaystyle
w_{9}=\frac{4\left[k_{\scriptscriptstyle 1T}^{ijlm}\hat{q}_{\scriptscriptstyle
Ti}\hat{q}_{\scriptscriptstyle Tj}\hat{q}_{\scriptscriptstyle
Tl}k_{\scriptscriptstyle 2Tm}+2k_{\scriptscriptstyle
1T}^{ijlm}\hat{q}_{\scriptscriptstyle Ti}\hat{q}_{\scriptscriptstyle
Tj}\hat{q}_{\scriptscriptstyle Tl}\hat{q}_{\scriptscriptstyle
Tm}(k_{\scriptscriptstyle 2T}\cdot\hat{q}_{\scriptscriptstyle
T})\right]}{M_{1}^{4}M_{2}},\quad w_{10}=\frac{2k_{\scriptscriptstyle
1T}^{ij}\hat{q}_{\scriptscriptstyle Ti}k_{\scriptscriptstyle
2Tj}}{M_{1}^{2}M_{2}},$ (135)

where $\hat{q}_{T}^{\mu}\equiv g_{T}^{\mu\nu}q_{\nu}/\sqrt{{\bm{q}}_{T}^{2}}$
is the direction of the virtual photon transverse momentum in the c.m. frame.

At leading twist, 24 structure functions have nontrivial expressions. Two of
the nonvanishing structure functions are for unpolarized hadron state,

$\displaystyle F_{U,U}^{T}$ $\displaystyle={\cal
C}\left[D_{1}(z_{1},k_{1T}^{2})D_{1}(z_{2},k_{2T}^{2})\right],$ (136)
$\displaystyle F_{U,U}^{\cos 2\phi}$ $\displaystyle={\cal
C}\left[w_{3}(k_{1T},k_{2T})H_{1}^{\perp}(z_{1},k_{1T}^{2})H_{1}^{\perp}(z_{2},k_{2T}^{2})\right],$
(137)

where the $\cos{2\phi}$ azimuthal modulation term $F_{U,U}^{\cos 2\phi}$ is
widely used to extract the Collins function $H_{1}^{\perp}(z,k_{T}^{2})$ [15,
16, 17, 18]. Four of the nonvanishing structure functions are for vector
polarized hadron states, including one for $S_{L}$ dependence,

$\displaystyle F_{L,U}^{\sin 2\phi}$ $\displaystyle=-{\cal
C}\left[w_{3}(k_{1T},k_{2T})H_{1L}^{\perp}(z_{1},k_{1T}^{2})H_{1}^{\perp}(z_{2},k_{2T}^{2})\right],$
(138)

and three for $S_{T}$ dependence,

$\displaystyle F_{T,U}^{T}$ $\displaystyle={\cal
C}\left[w_{1}(k_{1T},k_{2T})D_{1T}^{\perp}(z_{1},k_{1T}^{2})D_{1}(z_{2},k_{2T}^{2})\right],$
(139) $\displaystyle F_{T,U}^{\sin\left(2\phi+\phi_{T}\right)}$
$\displaystyle=\frac{1}{2}\left(F_{T,U}^{\cos 2\phi}+F_{T,U}^{\sin
2\phi}\right)={\cal
C}\left[w_{4}(k_{1T},k_{2T})H_{1T}^{\perp}(z_{1},k_{1T}^{2})H_{1}^{\perp}(z_{2},k_{2T}^{2})\right],$
(140) $\displaystyle F_{T,U}^{\sin\left(2\phi-\phi_{T}\right)}$
$\displaystyle=\frac{1}{2}\left(F_{T,U}^{\sin 2\phi}-F_{T,U}^{\cos
2\phi}\right)=-{\cal
C}\left[w_{2}(k_{1T},k_{2T})H_{1T}(z_{1},k_{1T}^{2})H_{1}^{\perp}(z_{2},k_{2T}^{2})\right].$
(141)

Eight of the nonvanishing structure functions are for rank-2 tensor polarized
hadron states, including two for $S_{LL}$ dependence,

$\displaystyle F_{LL,U}^{T}$ $\displaystyle={\cal
C}\left[D_{1LL}(z_{1},k_{1T}^{2})D_{1}(z_{2},k_{2T}^{2})\right],$ (142)
$\displaystyle F_{LL,U}^{\cos{2\phi}}$ $\displaystyle=-{\cal
C}[w_{3}(k_{1T},k_{2T})H_{1LL}^{\perp}(z_{1},k_{1T}^{2})H_{1}^{\perp}(z_{2},k_{2T}^{2})],$
(143)

three for $S_{LT}$ dependence,

$\displaystyle F_{LT,U}^{T}$ $\displaystyle={\cal
C}\left[w_{1}(k_{1T},k_{2T})D_{1LT}^{\perp}(z_{1},k_{1T}^{2})D_{1}(z_{2},k_{2T}^{2})\right],$
(144) $\displaystyle F_{LT,U}^{\sin\left(2\phi+\phi_{LT}\right)}$
$\displaystyle=\frac{1}{2}\left(F_{LT,U}^{\cos 2\phi}+F_{LT,U}^{\sin
2\phi}\right)=-{\cal
C}\left[w_{2}(k_{1T},k_{2T})H_{1LT}(z_{1},k_{1T}^{2})H_{1}^{\perp}(z_{2},k_{2T}^{2})\right],$
(145) $\displaystyle F_{LT,U}^{\sin\left(2\phi-\phi_{LT}\right)}$
$\displaystyle=\frac{1}{2}\left(F_{LT,U}^{\sin 2\phi}-F_{LT,U}^{\cos
2\phi}\right)=-{\cal
C}\left[{w}_{4}(k_{1T},k_{2T})H_{1LT}^{\perp}(z_{1},k_{1T}^{2})H_{1}^{\perp}(z_{2},k_{2T}^{2})\right],$
(146)

and three for $S_{TT}$ dependence,

$\displaystyle F_{TT,U}^{T}$ $\displaystyle={\cal
C}\left[w_{5}(k_{1T},k_{2T})D_{1TT}^{\perp}(z_{1},k_{1T}^{2})D_{1}(z_{2},k_{2T}^{2})\right],$
(147) $\displaystyle F_{TT,U}^{\sin\left(2\phi+2\phi_{TT}\right)}$
$\displaystyle=\frac{1}{2}\left(F_{TT,U}^{\cos 2\phi}+F_{TT,U}^{\sin
2\phi}\right)={\cal
C}\left[{w}_{7}(k_{1T},k_{2T})H_{1TT}^{\perp}(z_{1},k_{1T}^{2})H_{1}^{\perp}(z_{2},k_{2T}^{2})\right],$
(148) $\displaystyle F_{TT,U}^{\sin\left(2\phi-2\phi_{TT}\right)}$
$\displaystyle=\frac{1}{2}\left(F_{TT,U}^{\sin 2\phi}-F_{TT,U}^{\cos
2\phi}\right)={\cal
C}\left[w_{6}(k_{1T},k_{2T})H_{1TT}^{\perp\perp}(z_{1},k_{1T}^{2})H_{1}^{\perp}(z_{2},k_{2T}^{2})\right].$
(149)

Ten of the nonvanishing structure functions are for rank-3 tensor polarized
hadron states, including one for $S_{LLL}$ dependence,

$\displaystyle F_{LLL,U}^{\sin 2\phi}=-{\cal
C}\left[w_{3}(k_{1T},k_{2T})H_{1LLL}^{\perp}(z_{1},k_{1T}^{2})H_{1}^{\perp}(z_{2},k_{2T}^{2})\right],$
(150)

three for $S_{LLT}$ dependence,

$\displaystyle F_{LLT,U}^{T}$ $\displaystyle={\cal
C}\left[w_{1}(k_{1T},k_{2T})D_{1LLT}^{\perp}(z_{1},k_{1T}^{2})D_{1}(z_{2},k_{2T}^{2})\right],$
(151) $\displaystyle F_{LLT,U}^{\sin\left(2\phi+\phi_{LLT}\right)}$
$\displaystyle=\frac{1}{2}\left(F_{LLT,U}^{\cos 2\phi}+F_{LLT,U}^{\sin
2\phi}\right)={\cal
C}\left[{w}_{4}(k_{1T},k_{2T})H_{1LLT}^{\perp}(z_{1},k_{1T}^{2})H_{1}^{\perp}(z_{2},k_{2T}^{2})\right],$
(152) $\displaystyle F_{LLT,U}^{\sin\left(2\phi-\phi_{LLT}\right)}$
$\displaystyle=\frac{1}{2}\left(F_{LLT,U}^{\sin 2\phi}-F_{LLT,U}^{\cos
2\phi}\right)=-{\cal
C}\left[w_{2}(k_{1T},k_{2T})H_{1LLT}(z_{1},k_{1T}^{2})H_{1}^{\perp}(z_{2},k_{2T}^{2})\right],$
(153)

three for $S_{LTT}$ dependence,

$\displaystyle F_{LTT,U}^{T}$ $\displaystyle=-{\cal
C}\left[w_{5}(k_{1T},k_{2T})D_{1LTT}^{\perp}(z_{1},k_{1T}^{2})D_{1}(z_{2},k_{2T}^{2})\right],$
(154) $\displaystyle F_{LTT,U}^{\sin\left(2\phi+2\phi_{LTT}\right)}$
$\displaystyle=\frac{1}{2}\left(F_{LTT,U}^{\cos 2\phi}+F_{LTT,U}^{\sin
2\phi}\right)=-{\cal
C}\left[w_{6}(k_{1T},k_{2T})H_{1LTT}^{\perp\perp}(z_{1},k_{1T}^{2})H_{1}^{\perp}(z_{2},k_{2T}^{2})\right],$
(155) $\displaystyle F_{LTT,U}^{\sin\left(2\phi-2\phi_{LTT}\right)}$
$\displaystyle=\frac{1}{2}\left(F_{LTT,U}^{\sin 2\phi}-F_{LTT,U}^{\cos
2\phi}\right)={\cal
C}\left[w_{7}(k_{1T},k_{2T})H_{1LTT}^{\perp}(z_{1},k_{1T}^{2})H_{1}^{\perp}(z_{2},k_{2T}^{2})\right],$
(156)

and three for $S_{TTT}$ dependence,

$\displaystyle F_{TTT,U}^{T}$ $\displaystyle=-{\cal
C}\left[w_{8}(k_{1T},k_{2T})D_{1TTT}^{\perp}(z_{1},k_{1T}^{2})D_{1}(z_{2},k_{2T}^{2})\right],$
(157) $\displaystyle F_{TTT,U}^{\sin\left(2\phi+3\phi_{TTT}\right)}$
$\displaystyle=\frac{1}{2}\left(F_{TTT,U}^{\cos 2\phi}+F_{TTT,U}^{\sin
2\phi}\right)=-{\cal
C}\left[w_{9}(k_{1T},k_{2T})H_{1TTT}^{\perp\perp}(z_{1},k_{1T}^{2})H_{1}^{\perp}(z_{2},k_{2T}^{2})\right],$
(158) $\displaystyle F_{TTT,U}^{\sin\left(2\phi-3\phi_{TTT}\right)}$
$\displaystyle=\frac{1}{2}\left(F_{TTT,U}^{\sin 2\phi}-F_{TTT,U}^{\cos
2\phi}\right)=-{\cal
C}\left[w_{10}(k_{1T},k_{2T})H_{1TTT}^{\perp}(z_{1},k_{1T}^{2})H_{1}^{\perp}(z_{2},k_{2T}^{2})\right].$
(159)

The measurement of these 24 structure functions, particularly the ten for
rank-3 tensor polarized states, can be utilized to study the TMD FFs for a
spin-3/2 hadron, although the other 24 structure functions in the unpolarized
differential cross section only arise at high twist or high order.

##  V Summary and outlook

In this paper, we have studied the inclusive and semi-inclusive production of
spin-3/2 hadrons in $e^{+}e^{-}$ annihilation.

We describe the spin state of a hadron with the spin density matrix $\rho$.
For a spin-3/2 hadron, its spin density matrix contains 16 independent
components, including one for unpolarized state, three for vector polarized
states, five for rank-2 tensor polarized states, and seven for rank-3 tensor
polarized states. By imposing orthogonal relations, we are able to use the
basis matrices as spin projectors to extract the spin components of the hadron
state.

Considering existing $e^{+}e^{-}$ collision experiments, such as Belle II,
BaBar, and BESIII, we only focus on unpolarized collisions and the parity
violating effect via a $Z$-boson exchange is neglected. Through kinematic
analyses, we derive the differential cross section of inclusive production of
a spin-3/2 hadron in unpolarized $e^{+}e^{-}$ collisions in terms of nine
structure functions, among which $F_{LLT}^{\sin\phi_{LLT}}$ and $F_{LTT}^{\sin
2\phi_{LTT}}$ are newly defined and only exist when spin of the produced
hadron no less than $3/2$. For the process with two identified hadrons, we
only consider the spin of one observed hadron. Taking advantage of one
additional observed momentum, we can simplify the kinematic analysis of the
hadronic tensor with a systematic procedure, in which all basis tensors are
constructed from spin-independent basic Lorentz tensors multiplied by a spin-
dependent scalar or pseudoscalar. Then the differential cross section are
expressed in terms of 48 structure functions in accordance to the angular
distributions and spin states of the produced hadrons. Among the 48 structure
functions, 20 are newly defined, corresponding to the production of rank-3
tensor polarized hadron state, while the other 28 also exist for the
production of a spin-1 hadron and an unpolarized hadron.

Applying the factorized formalism, we express the cross sections of these
reactions as convolutions between the production of quarks at a hard scale and
some long-distance functions describing the hadronization process. At leading
twist, we provide a complete parametrization of the long-distance function,
i.e., the quark-quark correlation function, in terms of TMD FFs, including two
for the unpolarized hadron state, six for the vector polarized hadron states,
ten for rank-2 tensor polarized hadron states, and 14 for rank-3 tensor
polarized states. At the collinear limit only seven of them are nonvanishing,
including $D_{1}(z)$ for the unpolarized hadron state, $G_{1L}(z)$ and
$H_{1T}(z)$ for the vector polarized hadron states, $D_{1LL}(z)$ and
$H_{1LT}(z)$ for rank-2 tensor polarized hadron states, and $G_{1LLL}(z)$ and
$H_{1LLT}(z)$ for rank-3 tensor polarized hadron states.

For the inclusive process, the reaction is not sensitive to quark transverse
momentum and one applies the collinear factorization. We perform a leading
order calculation in the parton model and find that none of the FFs for rank-3
tensor polarized hadron states contributes to the cross section. Therefore,
any observation of a rank-3 tensor polarized hadron state, as well as vector
polarized hadron state, from such unpolarized collision will imply high-twist
or high-order effects.

For the semi-inclusive process, we consider the kinematic region
${\bm{q}}_{T}^{2}\ll Q^{2}$ and apply the TMD factorization formalism. Through
a leading order calculation, we find half of the structure functions vanish at
leading twist, but ten of the leading-twist nontrivial structure functions are
contributions from rank-3 tensor polarized hadron states, and therefore can be
utilized to study the rank-3 tensor polarized TMD FFs for spin-3/2 hadrons.

The hadron production in high energy $e^{+}e^{-}$ annihilation, as the
cleanest process to study FFs, has been widely used to extract unpolarized FF
$D_{1}(z)$ and its TMD counterpart $D_{1}(z,k_{T}^{2})$. The azimuthal
asymmetry data of two hadron productions from Belle, BaBar, and BESIII have
been combined with SIDIS data from HERMES, COMPASS, and JLab to extract the
Collins function $H_{1}^{\perp}(x,k_{T}^{2})$. Furthermore, the recent
measurement of $\Lambda$ polarization by Belle has stimulated extensive
studies of FFs to polarized hadron states. These results encourage us to
explore more possibilities for the study of FFs and the role of spin in the
hadronization process at $e^{+}e^{-}$ machines. The FFs to $\Omega$, a
spin-3/2 baryon, may provide an opportunity, not only because it has more spin
states and thus more observables, but also because it has unique sensitivity
to strange quarks.

Recently, the BESIII Collaboration measured the polarization of the $\Omega$
from $\psi(3686)\rightarrow\Omega^{-}\bar{\Omega}^{+}$ via a chain of weak
decays $\Omega^{-}\rightarrow K^{-}\Lambda$ and $\Lambda\rightarrow p\pi^{-}$
[105]. Although these data cannot be directly utilized to study FFs, the
developed techniques to analyze the spin state of the produced $\Omega$ can be
applied in other processes with $\Omega$ produced from parton fragmentation.
The Belle II experiment with 40 times higher luminosity than the Belle
experiment is expected to produce enough $\Omega$ events for the spin state
analysis. Therefore, the analysis of the spin state of $\Omega$ and the
extraction of the rank-3 tensor polarized FFs are in principle feasible with
future data from the Belle II experiment.

One can also consider the spin of both hadrons in the semi-inclusive process,
which will require much more statistics in experiment for the analysis of both
spin states, which may be possible with the Belle II experiment. The FFs to
$\Omega$ obtained in $e^{+}e^{-}$ annihilation can also be applied in the
SIDIS process and have advantages to study the strange sea distributions in
the nucleon.

Therefore, the study of TMD FFs to spin-3/2 hadrons will provide new
opportunities for understanding the role of spin in the hadronization process.
The semi-inclusive data of the $\Omega$ production at existing $e^{+}e^{-}$
collision experiments can be utilized to explore the new FFs and more efforts
are expected from both theoretical and experimental sides.

## Acknowledgements

We thank Zhiqing Liu and Jinlong Zhang for useful discussions. This work was
supported by the National Natural Science Foundation of China (Approvals No.
12175117, No. 11890713, and No. 11890700) and Shandong Province Natural
Science Foundation Grant No. ZR2020MA098.

##  Appendix A PROBABILITY INTERPRETATIONS OF THE SPIN COMPONENTS

A spin-3/2 hadron has four independent spin states and the corresponding spin
density matrix $\rho$ is a $4\times 4$ matrix, which in the $s_{z}$
representation can be explicitly written as

$\rho=\frac{1}{4}\left(\begin{array}[]{cccc}\rho_{\frac{3}{2}\frac{3}{2}}&\rho_{\frac{3}{2}\frac{1}{2}}&\rho_{\frac{3}{2}-\frac{1}{2}}&\rho_{\frac{3}{2}-\frac{3}{2}}\\\
\rho_{\frac{1}{2}\frac{3}{2}}&\rho_{\frac{1}{2}\frac{1}{2}}&\rho_{\frac{1}{2}-\frac{1}{2}}&\rho_{\frac{1}{2}-\frac{3}{2}}\\\
\rho_{-\frac{1}{2}\frac{3}{2}}&\rho_{-\frac{1}{2}\frac{1}{2}}&\rho_{-\frac{1}{2}-\frac{1}{2}}&\rho_{-\frac{1}{2}-\frac{3}{2}}\\\
\rho_{-\frac{3}{2}\frac{3}{2}}&\rho_{-\frac{3}{2}\frac{1}{2}}&\rho_{-\frac{3}{2}-\frac{1}{2}}&\rho_{-\frac{3}{2}-\frac{3}{2}}\end{array}\right).$
(160)

Comparing with the decomposition in Eq. (1) and the spin vector and tensors,
(10), (12), and (14), one can match the matrix elements with spin components
as

$\displaystyle\rho_{\frac{3}{2}\frac{3}{2}}$
$\displaystyle=1+\frac{6}{5}S_{L}+S_{LL}+\frac{2}{3}S_{LLL},$ (161)
$\displaystyle\rho_{\frac{1}{2}\frac{1}{2}}$
$\displaystyle=1+\frac{2}{5}S_{L}-S_{LL}-2S_{LLL},$ (162)
$\displaystyle\rho_{-\frac{1}{2}-\frac{1}{2}}$
$\displaystyle=1-\frac{2}{5}S_{L}-S_{LL}+2S_{LLL},$ (163)
$\displaystyle\rho_{-\frac{3}{2}-\frac{3}{2}}$
$\displaystyle=1-\frac{6}{5}S_{L}+S_{LL}-\frac{2}{3}S_{LLL},$ (164)
$\displaystyle\rho_{\frac{3}{2}\frac{1}{2}}$
$\displaystyle=\frac{2\sqrt{3}}{5}(S_{T}^{x}-iS_{T}^{y})+\frac{\sqrt{3}}{3}(S_{LT}^{x}-iS_{LT}^{y})+\frac{2\sqrt{3}}{3}(S_{LLT}^{x}-iS_{LLT}^{y}),$
(165) $\displaystyle\rho_{\frac{1}{2}\frac{3}{2}}$
$\displaystyle=\frac{2\sqrt{3}}{5}(S_{T}^{x}+iS_{T}^{y})+\frac{\sqrt{3}}{3}(S_{LT}^{x}+iS_{LT}^{y})+\frac{2\sqrt{3}}{3}(S_{LLT}^{x}+iS_{LLT}^{y}),$
(166) $\displaystyle\rho_{\frac{1}{2}-\frac{1}{2}}$
$\displaystyle=\frac{4}{5}(S_{T}^{x}-iS_{T}^{y})-2(S_{LLT}^{x}-iS_{LLT}^{y}),$
(167) $\displaystyle\rho_{-\frac{1}{2}\frac{1}{2}}$
$\displaystyle=\frac{4}{5}(S_{T}^{x}+iS_{T}^{y})-2(S_{LLT}^{x}+iS_{LLT}^{y}),$
(168) $\displaystyle\rho_{-\frac{1}{2}-\frac{3}{2}}$
$\displaystyle=\frac{2\sqrt{3}}{5}(S_{T}^{x}-iS_{T}^{y})-\frac{\sqrt{3}}{3}(S_{LT}^{x}-iS_{LT}^{y})+\frac{2\sqrt{3}}{3}(S_{LLT}^{x}-iS_{LLT}^{y}),$
(169) $\displaystyle\rho_{-\frac{3}{2}-\frac{1}{2}}$
$\displaystyle=\frac{2\sqrt{3}}{5}(S_{T}^{x}+iS_{T}^{y})-\frac{\sqrt{3}}{3}(S_{LT}^{x}+iS_{LT}^{y})+\frac{2\sqrt{3}}{3}(S_{LLT}^{x}+iS_{LLT}^{y}),$
(170) $\displaystyle\rho_{\frac{3}{2}-\frac{1}{2}}$
$\displaystyle=\frac{\sqrt{3}}{3}(S_{TT}^{xx}-iS_{TT}^{xy})+\frac{\sqrt{3}}{3}(S_{LTT}^{xx}-iS_{LTT}^{xy}),$
(171) $\displaystyle\rho_{-\frac{1}{2}\frac{3}{2}}$
$\displaystyle=\frac{\sqrt{3}}{3}(S_{TT}^{xx}+iS_{TT}^{xy})+\frac{\sqrt{3}}{3}(S_{LTT}^{xx}+iS_{LTT}^{xy}),$
(172) $\displaystyle\rho_{\frac{1}{2}-\frac{3}{2}}$
$\displaystyle=\frac{\sqrt{3}}{3}(S_{TT}^{xx}-iS_{TT}^{xy})-\frac{\sqrt{3}}{3}(S_{LTT}^{xx}-iS_{LTT}^{xy}),$
(173) $\displaystyle\rho_{\frac{1}{2}-\frac{3}{2}}$
$\displaystyle=\frac{\sqrt{3}}{3}(S_{TT}^{xx}+iS_{TT}^{xy})-\frac{\sqrt{3}}{3}(S_{LTT}^{xx}+iS_{LTT}^{xy}),$
(174) $\displaystyle\rho_{\frac{3}{2}-\frac{3}{2}}$
$\displaystyle=\frac{2}{3}(S_{TTT}^{xxx}+iS_{TTT}^{yxx}),$ (175)
$\displaystyle\rho_{-\frac{3}{2}\frac{3}{2}}$
$\displaystyle=\frac{2}{3}(S_{TTT}^{xxx}-iS_{TTT}^{yxx}).$ (176)

To analyze the probability interpretation of each spin component, it is
convenient to choose the hadron rest frame, while the general situation for a
moving hadron can be obtained via a Lorentz boost. By measuring the spin along
a particular direction
$\hat{\bm{n}}=(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)$ specified by
the polar angle $\theta$ and the azimuthal angle $\phi$, one will obtain an
eigenstate of the spin operator,

$\displaystyle\Sigma^{i}\hat{n}_{i}=\Sigma^{x}\sin\theta\cos\phi+\Sigma^{y}\sin\theta\sin\phi+\Sigma^{z}\cos\theta.$
(177)

Following the common convention, we represent the eigenstate by
$\left|m_{(\theta,\phi)}\right>$, where $m=-3/2$, $-1/2$, $1/2$, and $3/2$ is
the magnetic quantum number along the direction $(\theta,\phi)$. The
probability of finding the specified eigenstate can be evaluated from the spin
density matrix $\rho$ as

$\displaystyle
P\left(m_{(\theta,\phi)}\right)=\operatorname{Tr}\left[\rho|m_{(\theta,\phi)}\rangle\langle
m_{(\theta,\phi)}|\right].$ (178)

Then all the spin components can be expressed in terms of the
$P\left(m_{(\theta,\phi)}\right)$’s. The three spin components of the spin
vector $S^{i}$, defined in Eq. (11), are given by

$\displaystyle S_{L}=$
$\displaystyle\frac{3}{2}\left[P\left(\frac{3}{2}_{(0,0)}\right)-P\left(-\frac{3}{2}_{(0,0)}\right)\right]+\frac{1}{2}\left[P\left(\frac{1}{2}_{(0,0)}\right)-P\left(-\frac{1}{2}_{(0,0)}\right)\right],$
(179) $\displaystyle S_{T}^{x}=$
$\displaystyle\frac{3}{2}\left[P\left(\frac{3}{2}_{(\frac{\pi}{2},0)}\right)-P\left(-\frac{3}{2}_{(\frac{\pi}{2},0)}\right)\right]+\frac{1}{2}\left[P\left(\frac{1}{2}_{(\frac{\pi}{2},0)}\right)-P\left(-\frac{1}{2}_{(\frac{\pi}{2},0)}\right)\right],$
(180) $\displaystyle S_{T}^{y}=$
$\displaystyle\frac{3}{2}\left[P\left(\frac{3}{2}_{(\frac{\pi}{2},\frac{\pi}{2})}\right)-P\left(-\frac{3}{2}_{(\frac{\pi}{2},\frac{\pi}{2})}\right)\right]+\frac{1}{2}\left[P\left(\frac{1}{2}_{(\frac{\pi}{2},\frac{\pi}{2})}\right)-P\left(-\frac{1}{2}_{(\frac{\pi}{2},\frac{\pi}{2})}\right)\right].$
(181)

The five spin components of the rank-2 spin tensor $T^{ij}$, defined in Eq.
(13), are given by

$\displaystyle S_{LL}=$
$\displaystyle\left[P\left(\frac{3}{2}_{(0,0)}\right)+P\left(-\frac{3}{2}_{(0,0)}\right)\right]-\left[P\left(\frac{1}{2}_{(0,0)}\right)+P\left(-\frac{1}{2}_{(0,0)}\right)\right],$
(182) $\displaystyle S_{LT}^{x}=$ $\displaystyle
2\left\\{\left[P\left(\frac{3}{2}_{(\frac{\pi}{4},0)}\right)+P\left(-\frac{3}{2}_{(\frac{\pi}{4},0)}\right)\right]-\left[P\left(\frac{3}{2}_{(-\frac{\pi}{4},0)}\right)+P\left(-\frac{3}{2}_{(-\frac{\pi}{4},0)}\right)\right]\right\\},$
(183) $\displaystyle S_{LT}^{y}=$ $\displaystyle
2\left\\{\left[P\left(\frac{3}{2}_{(\frac{\pi}{4},\frac{\pi}{2})}\right)+P\left(-\frac{3}{2}_{(\frac{\pi}{4},\frac{\pi}{2})}\right)\right]-\left[P\left(\frac{3}{2}_{(-\frac{\pi}{4},\frac{\pi}{2})}\right)+P\left(-\frac{3}{2}_{(-\frac{\pi}{4},\frac{\pi}{2})}\right)\right]\right\\},$
(184) $\displaystyle S_{TT}^{xx}=$ $\displaystyle
2\left\\{\left[P\left(\frac{3}{2}_{(\frac{\pi}{2},0)}\right)+P\left(-\frac{3}{2}_{(\frac{\pi}{2},0)}\right)\right]-\left[P\left(\frac{3}{2}_{(\frac{\pi}{2},\frac{\pi}{2})}\right)+P\left(-\frac{3}{2}_{(\frac{\pi}{2},\frac{\pi}{2})}\right)\right]\right\\},$
(185) $\displaystyle S_{TT}^{xy}=$ $\displaystyle
2\left\\{\left[P\left(\frac{3}{2}_{(\frac{\pi}{2},\frac{\pi}{4})}\right)+P\left(-\frac{3}{2}_{(\frac{\pi}{2},\frac{\pi}{4})}\right)\right]-\left[P\left(\frac{3}{2}_{(\frac{\pi}{2},-\frac{\pi}{4})}\right)+P\left(-\frac{3}{2}_{(\frac{\pi}{2},-\frac{\pi}{4})}\right)\right]\right\\}.$
(186)

The seven spin components of the rank-3 spin tensor $R^{ijk}$, defined in Eq.
(15), are given by

$\displaystyle S_{LLL}=$
$\displaystyle\frac{3}{10}\left[P\left(\frac{3}{2}_{(0,0)}\right)-P\left(-\frac{3}{2}_{(0,0)}\right)\right]-\frac{9}{10}\left[P\left(\frac{1}{2}_{(0,0)}\right)-P\left(-\frac{1}{2}_{(0,0)}\right)\right],$
(187) $\displaystyle S_{LLT}^{x}=$
$\displaystyle-\frac{1}{60}\left\\{129\left[P\left(\frac{3}{2}_{(\frac{\pi}{2},0)}\right)-P\left(-\frac{3}{2}_{(\frac{\pi}{2},0)}\right)\right]+23\left[P\left(\frac{1}{2}_{(\frac{\pi}{2},0)}\right)-P\left(-\frac{1}{2}_{(\frac{\pi}{2},0)}\right)\right]\right\\}$
$\displaystyle+\frac{\sqrt{2}}{24}\left\\{27\left[P\left(\frac{3}{2}_{(\frac{\pi}{4},0)}\right)-P\left(-\frac{3}{2}_{(\frac{\pi}{4},0)}\right)\right]+\left[P\left(\frac{1}{2}_{(\frac{\pi}{4},0)}\right)-P\left(-\frac{1}{2}_{(\frac{\pi}{4},0)}\right)\right]\right\\}$
$\displaystyle+\frac{\sqrt{2}}{24}\left\\{27\left[P\left(\frac{3}{2}_{(-\frac{\pi}{4},0)}\right)-P\left(-\frac{3}{2}_{(-\frac{\pi}{4},0)}\right)\right]+\left[P\left(\frac{1}{2}_{(-\frac{\pi}{4},0)}\right)-P\left(-\frac{1}{2}_{(-\frac{\pi}{4},0)}\right)\right]\right\\},$
(188) $\displaystyle S_{LLT}^{y}=$
$\displaystyle-\frac{1}{60}\left\\{129\left[P\left(\frac{3}{2}_{(\frac{\pi}{2},\frac{\pi}{2})}\right)-P\left(-\frac{3}{2}_{(\frac{\pi}{2},\frac{\pi}{2})}\right)\right]+23\left[P\left(\frac{1}{2}_{(\frac{\pi}{2},\frac{\pi}{2})}\right)-P\left(-\frac{1}{2}_{(\frac{\pi}{2},\frac{\pi}{2})}\right)\right]\right\\}$
$\displaystyle+\frac{\sqrt{2}}{24}\left\\{27\left[P\left(\frac{3}{2}_{(\frac{\pi}{4},\frac{\pi}{2})}\right)-P\left(-\frac{3}{2}_{(\frac{\pi}{4},\frac{\pi}{2})}\right)\right]+\left[P\left(\frac{1}{2}_{(\frac{\pi}{4},\frac{\pi}{2})}\right)-P\left(-\frac{1}{2}_{(\frac{\pi}{4},\frac{\pi}{2})}\right)\right]\right\\}$
$\displaystyle+\frac{\sqrt{2}}{24}\left\\{27\left[P\left(\frac{3}{2}_{(-\frac{\pi}{4},\frac{\pi}{2})}\right)-P\left(-\frac{3}{2}_{(-\frac{\pi}{4},\frac{\pi}{2})}\right)\right]+\left[P\left(\frac{1}{2}_{(-\frac{\pi}{4},\frac{\pi}{2})}\right)-P\left(-\frac{1}{2}_{(-\frac{\pi}{4},\frac{\pi}{2})}\right)\right]\right\\},$
(189) $\displaystyle S_{LTT}^{xx}=$
$\displaystyle\frac{\sqrt{2}}{12}\left\\{27\left[P\left(\frac{3}{2}_{(\frac{\pi}{4},0)}\right)-P\left(-\frac{3}{2}_{(\frac{\pi}{4},0)}\right)\right]+\left[P\left(\frac{1}{2}_{(\frac{\pi}{4},0)}\right)-P\left(-\frac{1}{2}_{(\frac{\pi}{4},0)}\right)\right]\right\\}$
$\displaystyle-\frac{\sqrt{2}}{12}\left\\{27\left[P\left(\frac{3}{2}_{(-\frac{\pi}{4},0)}\right)-P\left(-\frac{3}{2}_{(-\frac{\pi}{4},0)}\right)\right]+\left[P\left(\frac{1}{2}_{(-\frac{\pi}{4},0)}\right)-P\left(-\frac{1}{2}_{(-\frac{\pi}{4},0)}\right)\right]\right\\}$
$\displaystyle-\frac{\sqrt{2}}{12}\left\\{27\left[P\left(\frac{3}{2}_{(\frac{\pi}{4},\frac{\pi}{2})}\right)-P\left(-\frac{3}{2}_{(\frac{\pi}{4},\frac{\pi}{2})}\right)\right]+\left[P\left(\frac{1}{2}_{(\frac{\pi}{4},\frac{\pi}{2})}\right)-P\left(-\frac{1}{2}_{(\frac{\pi}{4},\frac{\pi}{2})}\right)\right]\right\\}$
$\displaystyle+\frac{\sqrt{2}}{12}\left\\{27\left[P\left(\frac{3}{2}_{(-\frac{\pi}{4},\frac{\pi}{2})}\right)-P\left(-\frac{3}{2}_{(-\frac{\pi}{4},\frac{\pi}{2})}\right)\right]+\left[P\left(\frac{1}{2}_{(-\frac{\pi}{4},\frac{\pi}{2})}\right)-P\left(-\frac{1}{2}_{(-\frac{\pi}{4},\frac{\pi}{2})}\right)\right]\right\\},$
(190) $\displaystyle S_{LTT}^{xy}=$
$\displaystyle\frac{1}{12}\left\\{27\left[P\left(\frac{3}{2}(0,0)\right)-P\left(-\frac{3}{2}(0,0)\right)\right]+\left[P\left(\frac{1}{2}(0,0)\right)-P\left(-\frac{1}{2}(0,0)\right)\right]\right\\}$
$\displaystyle+\frac{1}{12}\left\\{27\left[P\left(\frac{3}{2}_{(\frac{\pi}{2},0)}\right)-P\left(-\frac{3}{2}_{(\frac{\pi}{2},0)}\right)\right]+\left[P\left(\frac{1}{2}_{(\frac{\pi}{2},0)}\right)-P\left(-\frac{1}{2}_{(\frac{\pi}{2},0)}\right)\right]\right\\}$
$\displaystyle+\frac{1}{12}\left\\{27\left[P\left(\frac{3}{2}_{(\frac{\pi}{2},\frac{\pi}{2})}\right)-P\left(-\frac{3}{2}_{(\frac{\pi}{2},\frac{\pi}{2})}\right)\right]+\left[P\left(\frac{1}{2}_{(\frac{\pi}{2},\frac{\pi}{2})}\right)-P\left(-\frac{1}{2}_{(\frac{\pi}{2},\frac{\pi}{2})}\right)\right]\right\\}$
$\displaystyle-\frac{\sqrt{2}}{6}\left\\{27\left[P\left(\frac{3}{2}_{(\frac{\pi}{4},0)}\right)-P\left(-\frac{3}{2}_{(\frac{\pi}{4},0)}\right)\right]+\left[P\left(\frac{1}{2}_{(\frac{\pi}{4},0)}\right)-P\left(-\frac{1}{2}_{(\frac{\pi}{4},0)}\right)\right]\right\\}$
$\displaystyle-\frac{\sqrt{2}}{6}\left\\{27\left[P\left(\frac{3}{2}_{(\frac{\pi}{4},\frac{\pi}{2})}\right)-P\left(-\frac{3}{2}_{(\frac{\pi}{4},\frac{\pi}{2})}\right)\right]+\left[P\left(\frac{1}{2}_{(\frac{\pi}{4},\frac{\pi}{2})}\right)-P\left(-\frac{1}{2}_{(\frac{\pi}{4},\frac{\pi}{2})}\right)\right]\right\\}$
$\displaystyle-\frac{\sqrt{2}}{6}\left\\{27\left[P\left(\frac{3}{2}_{(\frac{\pi}{2},\frac{\pi}{4})}\right)-P\left(-\frac{3}{2}_{(\frac{\pi}{2},\frac{\pi}{4})}\right)\right]+\left[P\left(\frac{1}{2}_{(\frac{\pi}{2},\frac{\pi}{4})}\right)-P\left(-\frac{1}{2}_{(\frac{\pi}{2},\frac{\pi}{4})}\right)\right]\right\\}$
$\displaystyle+\frac{\sqrt{3}}{4}\left\\{27\left[P\left(\frac{3}{2}_{(\theta_{xyz},\frac{\pi}{4})}\right)-P\left(-\frac{3}{2}_{(\theta_{xyz},\frac{\pi}{4})}\right)\right]+\left[P\left(\frac{1}{2}_{(\theta_{xyz},-\frac{\pi}{4})}\right)-\left(-\frac{1}{2}_{(\theta_{xyz},-\frac{\pi}{4})}\right)\right]\right\\},$
(191) $\displaystyle S_{TTT}^{xxx}=$
$\displaystyle\frac{1}{4}\left\\{27\left[P\left(\frac{3}{2}_{(\frac{\pi}{2},0)}\right)-P\left(-\frac{3}{2}_{(\frac{\pi}{2},0)}\right)\right]+\left[P\left(\frac{1}{2}_{(\frac{\pi}{2},0)}\right)-P\left(-\frac{1}{2}_{(\frac{\pi}{2},0)}\right)\right]\right\\}$
$\displaystyle-\frac{\sqrt{2}}{8}\left\\{27\left[P\left(\frac{3}{2}_{(\frac{\pi}{2},\frac{\pi}{4})}\right)-P\left(-\frac{3}{2}_{(\frac{\pi}{2},\frac{\pi}{4})}\right)\right]+\left[P\left(\frac{1}{2}_{(\frac{\pi}{2},\frac{\pi}{4})}\right)-P\left(-\frac{1}{2}_{(\frac{\pi}{2},\frac{\pi}{4})}\right)\right]\right\\}$
$\displaystyle-\frac{\sqrt{2}}{8}\left\\{27\left[P\left(\frac{3}{2}_{(\frac{\pi}{2},-\frac{\pi}{4})}\right)-P\left(-\frac{3}{2}_{(\frac{\pi}{2},-\frac{\pi}{4})}\right)\right]+\left[P\left(\frac{1}{2}_{(\frac{\pi}{2},-\frac{\pi}{4})}\right)-P\left(-\frac{1}{2}_{(\frac{\pi}{2},-\frac{\pi}{4})}\right)\right]\right\\},$
(192) $\displaystyle S_{TTT}^{yxx}=$
$\displaystyle-\frac{1}{4}\left\\{27\left[P\left(\frac{3}{2}_{(\frac{\pi}{2},\frac{\pi}{2})}\right)-P\left(-\frac{3}{2}_{(\frac{\pi}{2},\frac{\pi}{2})}\right)\right]+\left[P\left(\frac{1}{2}_{(\frac{\pi}{2},\frac{\pi}{2})}\right)-P\left(-\frac{1}{2}_{(\frac{\pi}{2},\frac{\pi}{2})}\right)\right]\right\\}$
$\displaystyle+\frac{\sqrt{2}}{8}\left\\{27\left[P\left(\frac{3}{2}_{(\frac{\pi}{2},\frac{\pi}{4})}\right)-P\left(-\frac{3}{2}_{(\frac{\pi}{2},\frac{\pi}{4})}\right)\right]+\left[P\left(\frac{1}{2}_{(\frac{\pi}{2},\frac{\pi}{4})}\right)-P\left(-\frac{1}{2}_{(\frac{\pi}{2},\frac{\pi}{4})}\right)\right]\right\\}$
$\displaystyle-\frac{\sqrt{2}}{8}\left\\{27\left[P\left(\frac{3}{2}_{(\frac{\pi}{2},-\frac{\pi}{4})}\right)-P\left(-\frac{3}{2}_{(\frac{\pi}{2},-\frac{\pi}{4})}\right)\right]+\left[P\left(\frac{1}{2}_{(\frac{\pi}{2},-\frac{\pi}{4})}\right)-P\left(-\frac{1}{2}_{(\frac{\pi}{2},-\frac{\pi}{4})}\right)\right]\right\\},$
(193)

where $\theta_{xyz}$ indicates the angle $\arctan(\sqrt{2})$.

The domains of the spin components can be obtained through the probabilistic
interpretations

$\displaystyle S_{L},S_{T}^{x},S_{T}^{y}\in[-\frac{3}{2},\frac{3}{2}],$ (194)
$\displaystyle S_{LL}\in[-1,1],\quad
S_{LT}^{x},S_{LT}^{y},S_{TT}^{xx},S_{TT}^{xy}\in[-\sqrt{3},\sqrt{3}],$ (195)
$\displaystyle S_{LLL}\in[-\frac{9}{10},\frac{9}{10}],\quad
S_{LLT}^{x},S_{LLT}^{y}\in[-\frac{3+\sqrt{21}}{10},\frac{3+\sqrt{21}}{10}],$
(196) $\displaystyle S_{LTT}^{xx},S_{LTT}^{xy}\in[-\sqrt{3},\sqrt{3}],\quad
S_{TTT}^{xxx},S_{TTT}^{yxx}\in[-3,3].$ (197)

We note that the ranges may change in accordance with different normalization
conventions.

##  Appendix B FRAGMENTATION FUNCTIONS AND SYMMETRIC TRACELESS TENSORS

The TMD FFs defined in Eqs. (31)–(40) can be separated through the projection
by the Dirac matrix,

$\displaystyle\Delta^{[\Gamma]}\left(z,k_{T}\right)$
$\displaystyle=\operatorname{Tr}\left[\Delta\left(z,k_{T}\right)\Gamma\right],$
(198) $\displaystyle\Delta^{[\Gamma]}(z)$
$\displaystyle=\operatorname{Tr}[\Delta(z)\Gamma].$ (199)

Taking $\Gamma=\gamma^{-}$, $\gamma^{-}\gamma_{5}$, and
$i\sigma^{i-}\gamma_{5}$, one can obtain all leading-twist TMD FFs,

$\displaystyle\Delta_{U}^{\left[\gamma^{-}\right]}\left(z,k_{T}\right)$
$\displaystyle=D_{1},$ (200)
$\displaystyle\Delta_{L}^{\left[\gamma^{-}\right]}\left(z,k_{T}\right)$
$\displaystyle=0,$ (201)
$\displaystyle\Delta_{T}^{\left[\gamma^{-}\right]}\left(z,k_{T}\right)$
$\displaystyle=\left(\epsilon_{T}^{\mu\nu}S_{T\nu}\frac{k_{T\mu}}{M}D_{1T}^{\perp}\right),$
(202) $\displaystyle\Delta_{LL}^{\left[\gamma^{-}\right]}\left(z,k_{T}\right)$
$\displaystyle=S_{LL}D_{1LL},$ (203)
$\displaystyle\Delta_{LT}^{\left[\gamma^{-}\right]}\left(z,k_{T}\right)$
$\displaystyle=\frac{\bm{S}_{hLT}\cdot\bm{k}_{T}}{M}D_{1LT}^{\perp},$ (204)
$\displaystyle\Delta_{TT}^{\left[\gamma^{-}\right]}\left(z,k_{T}\right)$
$\displaystyle=S_{TT\mu\nu}\frac{k_{T}^{\mu\nu}}{M^{2}}D_{1TT}^{\perp},$ (205)
$\displaystyle\Delta_{LLL}^{\left[\gamma^{-}\right]}\left(z,k_{T}\right)$
$\displaystyle=0,$ (206)
$\displaystyle\Delta_{LLT}^{\left[\gamma^{-}\right]}\left(z,k_{T}\right)$
$\displaystyle=\left(\epsilon_{T}^{\mu\nu}S_{LLT\nu}\frac{k_{T\mu}}{M}D_{1LLT}^{\perp}\right),$
(207)
$\displaystyle\Delta_{LTT}^{\left[\gamma^{-}\right]}\left(z,k_{T}\right)$
$\displaystyle=\left(\epsilon_{T\nu}^{\mu}S_{LTT}^{\nu\rho}\frac{k_{T\mu\rho}}{M^{2}}D_{1LTT}^{\perp}\right),$
(208)
$\displaystyle\Delta_{TTT}^{\left[\gamma^{-}\right]}\left(z,k_{T}\right)$
$\displaystyle=\left(\epsilon_{T\nu}^{\mu}S_{TTT}^{\nu\rho\sigma}\frac{k_{T\mu\rho\sigma}}{M^{3}}D_{1TTT}^{\perp}\right),$
(209)
$\displaystyle\Delta_{U}^{\left[\gamma^{-}\gamma_{5}\right]}\left(z,k_{T}\right)$
$\displaystyle=0,$ (210)
$\displaystyle\Delta_{L}^{\left[\gamma^{-}\gamma_{5}\right]}\left(z,k_{T}\right)$
$\displaystyle=S_{L}G_{1L},$ (211)
$\displaystyle\Delta_{T}^{\left[\gamma^{-}\gamma_{5}\right]}\left(z,k_{T}\right)$
$\displaystyle=\frac{\bm{S}_{hT}\cdot\bm{k}_{T}}{M}G_{1T}^{\perp},$ (212)
$\displaystyle\Delta_{LL}^{\left[\gamma^{-}\gamma_{5}\right]}\left(z,k_{T}\right)$
$\displaystyle=0,$ (213)
$\displaystyle\Delta_{LT}^{\left[\gamma^{-}\gamma_{5}\right]}\left(z,k_{T}\right)$
$\displaystyle=-\left(\epsilon_{T}^{\mu\nu}S_{LT\nu}\frac{k_{T\mu}}{M}G_{1LT}^{\perp}\right),$
(214)
$\displaystyle{\Delta}_{TT}^{\left[\gamma^{-}\gamma_{5}\right]}\left(z,k_{T}\right)$
$\displaystyle=\left(\epsilon_{T\nu}^{\mu}S_{TT}^{\nu\rho}\frac{k_{T\mu\rho}}{M^{2}}G_{1TT}^{\perp}\right),$
(215)
$\displaystyle\Delta_{LLL}^{\left[\gamma^{-}\gamma_{5}\right]}\left(z,k_{T}\right)$
$\displaystyle=S_{LLL}G_{1LLL},$ (216)
$\displaystyle\Delta_{LLT}^{\left[\gamma^{-}\gamma_{5}\right]}\left(z,k_{T}\right)$
$\displaystyle=\frac{\bm{S}_{hLLT}\cdot\bm{k}_{T}}{M}G_{1LLT}^{\perp},$ (217)
$\displaystyle\Delta_{LTT}^{\left[\gamma^{-}\gamma_{5}\right]}\left(z,k_{T}\right)$
$\displaystyle={S}_{hLTT\mu\nu}\frac{{k}_{T}^{\mu\nu}}{M^{2}}G_{1LTT}^{\perp},$
(218)
$\displaystyle\Delta_{TTT}^{\left[\gamma^{-}\gamma_{5}\right]}\left(z,k_{T}\right)$
$\displaystyle=S_{TTT\mu\nu\rho}\frac{k_{T}^{\mu\nu\rho}}{M^{3}}G_{1TTT}^{\perp},$
(219)
$\displaystyle\Delta_{U}^{\left[i\sigma^{i-}\gamma_{5}\right]}\left(z,k_{T}\right)$
$\displaystyle=\left(\frac{\epsilon_{T}^{ij}k_{Tj}}{M}H_{1}^{\perp}\right),$
(220)
$\displaystyle\Delta_{L}^{\left[i\sigma^{i-}\gamma_{5}\right]}\left(z,k_{T}\right)$
$\displaystyle=S_{L}\frac{k_{T}^{i}}{M}H_{1L}^{\perp},$ (221)
$\displaystyle\Delta_{T}^{\left[i\sigma^{i-}\gamma_{5}\right]}\left(z,k_{T}\right)$
$\displaystyle=S_{T}^{i}H_{1T}-\frac{k_{T}^{ij}}{M}S_{Tj}H_{1T}^{\perp},$
(222)
$\displaystyle\Delta_{LL}^{\left[i\sigma^{i-}\gamma_{5}\right]}\left(z,k_{T}\right)$
$\displaystyle=-\left(S_{LL}\frac{\epsilon_{T}^{ij}k_{Tj}}{M}H_{1LL}^{\perp}\right),$
(223)
$\displaystyle\Delta_{LT}^{\left[i\sigma^{i-}\gamma_{5}\right]}\left(z,k_{T}\right)$
$\displaystyle=-\left(\epsilon_{T}^{ij}S_{LTj}H_{1LT}\right)+\left(\epsilon_{Tj}^{i}\frac{k_{T}^{jl}}{M^{2}}S_{LTl}H_{1LT}^{\perp}\right),$
(224)
$\displaystyle\Delta_{TT}^{\left[i\sigma^{i-}\gamma_{5}\right]}\left(z,k_{T}\right)$
$\displaystyle=-\left(\epsilon_{T}^{ij}S_{TTjl}\frac{k_{T}^{l}}{M}H_{1TT}^{\perp}\right)-\left(\epsilon^{i}_{Tj}\frac{k_{T}^{jlm}}{M^{3}}S_{TTlm}H_{1TT}^{\perp\perp}\right),$
(225)
$\displaystyle\Delta_{LLL}^{\left[i\sigma^{i-}\gamma_{5}\right]}\left(z,k_{T}\right)$
$\displaystyle=S_{LLL}\frac{k_{T}^{i}}{M}H_{1LLL}^{\perp},$ (226)
$\displaystyle\Delta_{LLT}^{\left[i\sigma^{i-}\gamma_{5}\right]}\left(z,k_{T}\right)$
$\displaystyle=S_{LLT}^{i}H_{1LLT}-\frac{k_{T}^{ij}}{M^{2}}S_{LLTj}H_{1LLT}^{\perp},$
(227)
$\displaystyle\Delta_{LTT}^{\left[i\sigma^{i-}\gamma_{5}\right]}\left(z,k_{T}\right)$
$\displaystyle=S_{LTT}^{ij}\frac{k_{Tj}}{M}H_{1LTT}^{{}^{\perp}}+\frac{k_{T}^{ijl}}{M^{3}}S_{LTTjl}H_{1LTT}^{\perp\perp},$
(228)
$\displaystyle\Delta_{TTT}^{\left[i\sigma^{i-}\gamma_{5}\right]}\left(z,k_{T}\right)$
$\displaystyle=S_{TTT}^{ijl}\frac{k_{Tjl}}{M^{2}}H_{1TTT}^{{}^{\perp}}+\frac{k_{T}^{ijlm}}{M^{3}}S_{TTTjlm}H_{1TTT}^{\perp\perp},$
(229)

where the terms in parentheses are often referred to as naively T-odd terms
and the indices $i$, $j$, $k$, and $l$ indicate the transverse components. The
TMD FFs are defined as functions of $z$ and $k_{T}^{2}=-\bm{k}_{T}^{2}$, with
inhomogeneous $k_{T}$ dependence factored out. The completely symmetric and
traceless tensors $k_{T}^{i_{1}\cdots i_{n}}$ are defined as [80]

$\displaystyle k_{T}^{ij}$
$\displaystyle=k_{T}^{i}k_{T}^{j}-\frac{1}{2}k_{T}^{2}g_{T}^{ij},$ (230)
$\displaystyle k_{T}^{ijk}$
$\displaystyle=k_{T}^{i}k_{T}^{j}k_{T}^{k}-\frac{1}{4}k_{T}^{2}\left(g_{T}^{ij}k_{T}^{k}+g_{T}^{ik}k_{T}^{j}+g_{T}^{jk}k_{T}^{i}\right),$
(231) $\displaystyle k_{T}^{ijkl}$
$\displaystyle=k_{T}^{i}k_{T}^{j}k_{T}^{k}k_{T}^{l}$
$\displaystyle\quad-\frac{1}{6}k_{T}^{2}\left(g_{T}^{ij}k_{T}^{kl}+g_{T}^{ik}k_{T}^{jl}+g_{T}^{il}k_{T}^{jk}+g_{T}^{jk}k_{T}^{il}+g_{T}^{jl}k_{T}^{ik}+g_{T}^{kl}k_{T}^{ij}\right)$
$\displaystyle\quad-\frac{1}{8}\left(k_{T}^{2}\right)^{2}\left(g_{T}^{ij}g_{T}^{kl}+g_{T}^{ik}g_{T}^{jl}+g_{T}^{il}g_{T}^{jk}\right),$
(232)

which satisfy

$\displaystyle g_{Tij}k_{T}^{ij}=g_{Tij}k_{T}^{ijk}=g_{Tij}k_{T}^{ijkl}=0.$
(233)

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