aainstitutetext: Department of Physics and Astronomy, University of
California, Los Angeles, California 90095, USAbbinstitutetext: Mani L. Bhaumik
Institute for Theoretical Physics, University of California, Los Angeles,
California 90095, USAccinstitutetext: Center for Frontiers in Nuclear Science,
Stony Brook University, Stony Brook, New York 11794, USAddinstitutetext: Key
Laboratory of Atomic and Subatomic Structure and Quantum Control (MOE),
Guangdong Basic Research Center of Excellence for Structure and Fundamental
Interactions of Matter, Institute of Quantum Matter, South China Normal
University, Guangzhou 510006, Chinaeeinstitutetext: Guangdong-Hong Kong Joint
Laboratory of Quantum Matter, Guangdong Provincial Key Laboratory of Nuclear
Science, Southern Nuclear Science Computing Center, South China Normal
University, Guangzhou 510006, Chinaffinstitutetext: Southern Center for
Nuclear-Science Theory (SCNT), Institute of Modern Physics, Chinese Academy of
Sciences, Huizhou 516000, Chinagginstitutetext: Center for Theoretical
Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

# Polarized fragmenting jet functions in Inclusive and Exclusive Jet
Production

Zhong-Bo Kang d,e,f , Hongxi Xing g , Fanyi Zhao d,e,a and Yiyu Zhou
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In this work, we present a complete theoretical framework for analyzing the
distribution of polarized hadrons within jets, with and without measuring the
transverse momentum relative to the standard jet axis. Using soft-collinear
effective theory (SCET), we derive the factorization and provide the
theoretical calculation of both semi-inclusive and exclusive fragmenting jet
functions (FJFs) under longitudinal and transverse polarization. With the
polarized FJFs, one gains access to a variety of new observables that can be
used for extracting both collinear and transverse momentum dependent parton
distribution functions (PDFs) and fragmentation functions (FFs). As examples,
we provide numerical results for the spin asymmetry
$A_{TU,T}^{\cos(\phi_{S}-\hat{\phi}_{S_{h}})}$ from polarized semi-inclusive
hadron-in-jet production in polarized $pp$ collisions at RHIC kinematics,
where a transversely polarized quark would lead to the transverse spin of the
final-state hadron inside the jet and is thus sensitive to the transversity
fragmentation functions. Similarly, another spin asymmetry,
$A_{TU,L}^{\cos(\phi_{q}-\phi_{S})}$ from polarized exclusive hadron-in-jet
production in polarized $ep$ collisions at EIC kinematics would allow us to
access the helicity fragmentation functions. These observables demonstrate
promising potential in investigating transverse momentum dependent PDFs and
FFs and are worthwhile for further measurements.

MIT-CTP/5633

##  1 Introduction

Jet substructure, originally developed to exploit the flow of energy within
jets of particles at the Large Hadron Collider (LHC) to enhance new physics
searches Butterworth:2008iy , has since emerged as a primary tool for studying
Quantum Chromodynamics (QCD). Not only has jet substructure been vastly
studied in ongoing experiments at the Relativistic Heavy Ion Collider (RHIC)
and the LHC Newman:2013ada ; Kogler:2018hem ; Larkoski:2017jix ;
Connors:2017ptx ; Cunqueiro:2021wls ; Belmont:2023fau , but also in future
facilities such as the Electron-Ion Collider (EIC) AbdulKhalek:2021gbh ;
AbdulKhalek:2022hcn . For example, at the LHC, the jet substructure has been
successfully used to tag the origin of jets in precision measurements and
searches for new physics. At both the RHIC and the LHC, the jet substructure
has been an important tool for studying the properties of the hot and dense
quark-gluon plasma. On the other hand, at the EIC, as a good proxy of parton-
level dynamics, jet allows for a clean factorization between the target and
current-fragmentation regions, which is generally difficult in hadron
productions Aschenauer:2019kzf ; Wang:2019bvb ; Whitehill:2022mpq  in deep
inelastic scatterings (DIS).

Most recently, one particular set of jet substructure observables, the
distribution of hadrons inside the jet, has received a lot of attention
Boussarie:2023izj ; LHCb:2022rky ; ATLAS:2017pgl ; ATLAS:2019dsv ;
ATLAS:2011myc ; CMS:2014jjt ; ALICE:2023jgm . Studying hadron distribution
inside the jet enables us to pinpoint fragmentation functions of various kinds
and to explore the elusive hadronization process. They are complementary to
the more standard processes, such as semi-inclusive hadron production in deep
inelastic scattering and hadron production in $e^{+}e^{-}$ collisions, and
allow us to study QCD factorization and to test the universality properties of
the associated fragmentation functions in different processes Collins:2011zzd
. For example, one can measure the distribution of hadrons inside the jet as a
function of the hadron momentum fraction $z_{h}$, where $z_{h}$ is the ratio
of the transverse momentum of the hadron ($p_{hT}$) to that of the jet
($p_{JT}$), both relative to the beam axis. Such a $z_{h}$ dependence can be
used for extracting the $z_{h}$-dependence of the collinear fragmentation
functions (FFs) Kang:2016ehg ; Anderle:2017cgl . On the other hand, if one
measures both $z_{h}$ and the transverse momentum $\bm{j}_{\bot}$ distribution
of the hadrons with respect to the jet axis, one would be able to study the
transverse momentum-dependent fragmentation functions (TMD FFs) Kang:2017glf .

The theoretical object which describes the momentum distribution of hadrons
inside a fully reconstructed jet is called fragmenting jet functions (FJFs).
For the $z_{h}$-dependence while integrating over the hadron’s transverse
momentum with respect to the jet axis, we would have collinear FJFs, which can
be matched onto the standard collinear FFs. These collinear FJFs have been
studied in the so-called semi-inclusive jet production, as well as the
exclusive jet production. For semi-inclusive jet production, one measures the
signal jet while completely inclusive on other particles in the final state,
see the studies in e.g. Arleo:2013tya ; Kaufmann:2015hma ; Kang:2016ehg ;
Dai:2016hzf ; Kang:2019ahe ; Zhang:2022bhq ; Liu:2023fsq . On the other hand,
in the context being discussed, “exclusive” jet production refers to
situations where a fixed number of jets are produced in the final state but
one vetoes additional jets. For example, in the back-to-back electron-jet
production in electron-proton collisions, our factorization would depend on
the exclusive FJFs, where we have an electron and a single signal jet in the
final state. See the studies along this line Procura:2009vm ; Jain:2011xz ;
Jain:2011iu ; Chien:2015ctp ; Kang:2020xyq . On the other hand, measuring both
$z_{h}$ and $\bm{j}_{\bot}$ distribution of the hadrons with respect to the
standard jet axis would give us the transverse momentum dependent fragmenting
jet functions (TMD FJFs). There are close relationships Bain:2016rrv ;
Kang:2017glf ; Makris:2017arq ; Neill:2018wtk ; Kang:2019ahe  between these
TMD FJFs and the standard TMD FFs Bacchetta:2000jk ; Mulders:2000sh ;
Metz:2016swz . More recently, studies have been advanced to the polarized
sector, including the polarization of both the parton initiating the jet and
the hadron produced within the jet Kang:2020xyq ; Kang:2021ffh ; Kang:2021kpt
; Zhao:2023lav . For a recent study exploring dihadron fragmentation functions
(or two hadrons) inside the jet, see Bacchetta:2023njc ; Chien:2021yol . For
the TMD study of the hadron with respect to the Winner-Take-All jet axis, see
Neill:2016vbi ; Neill:2018wtk ; Gutierrez-Reyes:2019vbx ; Makris:2021drz . As
for the TMD study inside the groomed jet, see Makris:2017arq ; Makris:2018npl
; Gutierrez-Reyes:2019msa . More recently, exploring the connections between
TMD physics and the energy-energy correlators has seen a lot of progress
Gao:2019ojf ; Li:2020bub ; Kang:2023big .

For the polarized collinear FJFs and TMD FJFs, we have performed studies in
previous publications, mainly in Kang:2020xyq ; Kang:2021ffh  for semi-
inclusive and exclusive jet productions. Even though we provided theoretical
formalism and illustrated some phenomenological studies there, for the
computations of FJFs, only partial results were available in those previous
publications, but not the complete full results. This is the goal of our
current paper. We will perform all the relevant computations and give all
associated matching coefficients that allow us to connect these FJFs to the
standard FFs. In doing so, we also update our formalism using the more recent
approach outlined in the TMD Handbook Boussarie:2023izj . We provide
additional phenomenological studies at the RHIC and the future EIC.

The rest of the paper is organized as follows: In Section 2, we establish the
theoretical foundation for collinear FJFs, elucidate a number of correlations
between the polarization of the hadrons and fragmenting partons, and
illuminate their physical significance and connections to standard
fragmentation functions. In Section 3, we introduce the semi-inclusive and
exclusive TMD FJFs. We discuss their factorization and evolution properties,
as well as their relations to the standard TMD FFs. In Section 4, we employ
this framework to predict the spin asymmetry
$A_{TU,T}^{\cos(\phi_{S}-\hat{\phi}_{S_{h}})}$ for the semi-inclusive
production of transversely polarized $\Lambda$ baryons inside jet in
transversely polarized $pp$ collisions at RHIC kinematics, and
$A_{TU,L}^{\cos(\phi_{q}-\phi_{S})}$ for the exclusive production of
longitudinally polarized $\Lambda/\overline{\Lambda}$ inside jet in polarized
$ep$ collisions at EIC kinematics. Finally, we provide a summary of our
findings in Section 5. We collect a few additional details in the Appendix.

##  2 The collinear fragmenting jet functions (FJFs)

In this section, we introduce the definition of the semi-inclusive and
exclusive fragmenting jet functions (FJFs) in soft-collinear effective theory
(SCET) Bauer:2000ew ; Bauer:2000yr ; Bauer:2001ct ; Bauer:2001yt ;
Bauer:2002nz  with both unpolarized and polarized fragmenting hadron. Such
FJFs are used to describe the longitudinal momentum fraction distribution of
hadrons within jets. We first provide the operator definitions of FJFs, then
perform the calculation to NLO, and lastly derive and solve the RG evolution
equations.

###  2.1 Collinear FJFs in semi-inclusive jet productions

Figure 1: Illustration for the distribution of hadrons $h$ inside a jet that
is initiated by a parton $q$. The hadron has a momentum $p_{h}$ and spin
$S_{h}$, while the jet has a momentum $p_{J}$.

We can construct the semi-inclusive fragmenting quark and gluon jet functions
using the corresponding gauge invariant quark and gluon fields in SCET which
are given by:

$\chi_{n}=W_{n}^{\dagger}\xi_{n}\,,\quad\mathcal{B}_{n\perp}^{\mu}=\frac{1}{g}\bqty{W_{n}^{\dagger}iD_{n\perp}^{\mu}W_{n}}\,,$
(1)

where the subscript $n$ indicates that the field is along the $n$-collinear
direction, and $n^{\mu}=\pqty{1,0,0,1}$ is the light-cone vector whose spatial
component is aligned with the jet axis. In Eq. 1, the covariant derivative is
$iD_{n\perp}^{\mu}=\mathcal{P}_{n\perp}^{\mu}+gA_{n\perp}^{\mu}$, where
$\mathcal{P}^{\mu}$ is the label momentum operator. Moreover, $W_{n}$ is the
Wilson line of collinear gluons:

$W_{n}\pqty{x}=\sum_{\textrm{perms}}\exp(-g\frac{1}{\bar{n}\cdot\mathcal{P}}\bar{n}\cdot
A_{n}\pqty{x})\,.$ (2)

It is also useful to define the conjugate light-cone vector
$\bar{n}^{\mu}=(1,0,0,-1)$, such that $n^{2}=\bar{n}^{2}=0$ and
$n\cdot\bar{n}=2$. Thus, any four-vector $p^{\mu}$ can be decomposed as
$p^{\mu}=[p^{+},p^{-},\bm{p}_{\perp}]$111We use square bracket for vectors
written in light cone coordinates., where $p^{+}=n\cdot p$ and
$p^{-}=\bar{n}\cdot p$, namely:

$p^{\mu}=p^{-}\frac{n^{\mu}}{2}+p^{+}\frac{\bar{n}^{\mu}}{2}+p_{\perp}^{\mu}\,.$
(3)

For a hadron in the reference frame where it has no transverse momentum and
moves along the $+z$-direction, it will have a large $p_{h}^{-}$ component and
a small $p_{h}^{+}$ component, i.e., $p_{h}^{+}\ll p_{h}^{-}$. Thus the
momentum $p_{h}$ and spin $S_{h}$ of the hadron inside the jet can be
parameterized as:

$\displaystyle
p_{h}=\bqty{\frac{M_{h}^{2}}{p_{h}^{-}},p_{h}^{-},\bm{0}}\,,\quad
S_{h}=\bqty{-\lambda_{h}\frac{M_{h}}{p_{h}^{-}},\lambda_{h}\frac{p_{h}^{-}}{M_{h}},\bm{S}_{h\perp}}\,.$
(4)

Here $\lambda_{h}$ is the helicity of the hadron with mass $M_{h}$, and
$\bm{S}_{h\perp}$ is the transverse polarization of the hadron inside the jet.

Having these collinear quark and gluon fields readily available, we can
formulate the correlator definitions for the quark and gluon semi-inclusive
FJFs, each with different polarizations Procura:2009vm ; Jain:2011iu ;
Kang:2016mcy ; Kang:2016ehg :

(5) (6)

where $N_{c}$ is the number of colors for quarks and $(d-2)$ is the number of
polarizations for gluons in $d$ dimensional space-time. Note that we only
consider massless quark flavors. For studies about heavy meson, see e.g. Refs.
vonKuk:2023jfd ; Makris:2018npl ; Dai:2018ywt ; Anderle:2017cgl . The state
$\ket{\pqty{Jh}X}$ denotes the final state, encompassing both unobserved
particles labeled as $X$, and the observed jet $J$ containing an identified
hadron $h$, which is collectively referred to as $\pqty{Jh}$. Additionally,
$\omega$ represents the large light-cone component for the momentum of the
quark or gluon initiating the jet, whereas $\omega_{J}=p_{J}^{-}$ and
$\omega_{h}=p_{h}^{-}$ correspond respectively to the large light-cone
component for the momentum of the jet itself and the identified hadron within
the jet, as shown in Fig. 1. In our convention and the perturbative
calculations performed below, we choose a frame where the jet has no
transverse momentum and $\omega_{J}\simeq 2E_{J}$ with $E_{J}$ the jet energy.
However, in the actual experimental measurements at the collider experiments
such as proton-proton collisions at the LHC, the jet has a transverse momentum
$p_{T}$ with a jet radius $R$ Cacciari:2008gp  in the lab frame. Since we will
perform phenomenological studies for the relevant collider experiments, we
include $p_{T}R$ as an argument in our collinear FJFs in
LABEL:e.collinear_FJFs_q, LABEL:e.collinear_Delta_FJFs_q,
LABEL:e.collinear_Delta_T_FJFs_q, 5 and 6.

At the same time, the energy fractions $z$ and $z_{h}$ are defined as:

$z=\frac{\omega_{J}}{\omega}\,,\quad z_{h}=\frac{\omega_{h}}{\omega_{J}}\,.$
(7)

As given in LABEL:e.collinear_FJFs_q, LABEL:e.collinear_Delta_FJFs_q and
LABEL:e.collinear_Delta_T_FJFs_q, to obtain the helicity and transversity
distributions of the hadron in quark FJFs, namely $\Delta\mathcal{G}_{q}^{h}$
and $\Delta_{T}\mathcal{G}_{q}^{h}$, we replace the $\not{\bar{n}}$ in
LABEL:e.collinear_FJFs_q by $\not{\bar{n}}\gamma_{5}$ and
$\not{\bar{n}}\gamma_{\perp}^{i}\gamma_{5}$. The unpolarized collinear FJF
$\mathcal{G}_{q,g}^{h}$ represents the situation where an unpolarized quark or
gluon initiates a jet that carries a momentum fraction $z$ of the parent
parton, and we further observe an unpolarized hadron $h$ inside the jet which
carries the momentum fraction $z_{h}$ of the jet. Similarly,
$\Delta\mathcal{G}_{q,g}^{h}$ stands for the case where a longitudinally
polarized parton initiates a jet in which a longitudinally polarized hadron
$h$ is observed. Finally, $\Delta_{T}\mathcal{G}_{q}^{h}$ is the case for a
transversely polarized quark going into a jet in which a transversely
polarized hadron is observed. Note that we do not have the corresponding
$\Delta_{T}\mathcal{G}_{g}^{h}$ for gluons when hadron $h$ is a spin-1/2
particle, due to the helicity-conservation constraint. This is the same reason
why the gluon transversity does not exist for the spin-1/2 nucleon
Barone:2003fy .

Figure 2:  Contributions required for studying semi-inclusive FJFs. In (A),
both the quark and the gluon are inside the jet, while in (B) and (C) only the
quark or gluon is inside the jet.

####  2.1.1 NLO calculation

Although the semi-inclusive FJFs
$\mathcal{G}_{i}^{h}\pqty{z,z_{h},p_{T}R,\mu}$ are not perturbatively
calculable, they can be matched onto the standard collinear fragmentation
functions as long as $p_{T}R\gg\Lambda_{\rm QCD}$ Kang:2016ehg . The matching
coefficients can be computed by replacing the hadron $h$ by a parton $j$ and
then computing $\mathcal{G}_{i}^{j}\pqty{z,z_{h},p_{T}R,\mu}$ perturbatively.
Prior to describing the process of computing the semi-inclusive FJFs, it is
important to emphasize that the outcomes are influenced by the choice of jet
algorithms. To streamline our explanation, however, we solely concentrate on
the anti-$k_{T}$ algorithm Cacciari:2008gp  in this study.

We will now compute the bare partonic semi-inclusive FJFs
$\mathcal{G}_{i}^{j}\pqty{z,z_{h},p_{T}R}$ in order to determine the UV
counter terms. At the leading order (LO), one has a single parton. It is this
parton $i$ that forms the jet as well as presenting the parton $j$ inside the
jet. Thus we have $z=1$ and $z_{h}=1$ at LO and bare polarized semi-inclusive
FJFs are given by:

$\Delta_{(T)}\mathcal{G}_{i}^{j,(0)}(z,z_{h},p_{T}R)=\delta_{ij}\delta(1-z)\delta(1-z_{h})\,.$
(8)

Here and throughout the rest of the paper, we use the symbol $\Delta_{(T)}$ to
collectively represent both the longitudinally ($\Delta$) and transversely
($T$) polarized cases.

At the next-to-leading order (NLO), we have to consider the $1\to 2$ splitting
process in the real emission diagrams as shown in Fig. 2, as well as virtual
diagrams. In dimensional regularization, since virtual diagrams become
scaleless integrals and vanish, we thus only consider the real diagrams.
Following the discussion in Kang:2016ehg , we consider the cases of both
partons in the jet and only one parton in the jet as shown in Fig. 2. They are
given by the same integrals over the transverse momentum $q_{\bot}$ of the
parton $j$, but are constrained by different Heaviside functions. Working in
pure dimensional regularization with $d=4-2\epsilon$ dimensions, we have:

$\displaystyle\Delta_{(T)}\mathcal{G}_{i}^{j,(1)}(z,z_{h},\omega_{J})$
$\displaystyle=\frac{\alpha_{s}}{2\pi}\frac{(e^{\gamma_{E}}\mu^{2})^{\epsilon}}{\Gamma(1-\epsilon)}\bigg{(}\delta(1-z)\Delta_{(T)}\hat{P}_{ji}(z_{h},\epsilon)\int\frac{\differential{q_{\perp}^{2}}}{(q_{\perp}^{2})^{1+\epsilon}}\Theta^{\text{anti-}k_{T}}_{\mathrm{both}}$
$\displaystyle\qquad\qquad\qquad\quad+\delta(1-z_{h})\Delta_{(T)}\hat{P}_{ji}(z,\epsilon)\int\frac{\differential{q_{\perp}^{2}}}{(q_{\perp}^{2})^{1+\epsilon}}\Theta^{\text{anti-}k_{T}}_{j}\bigg{)}\,,$
(9)

where the $\Theta^{\text{anti-}k_{T}}_{\mathrm{both}}$ and
$\Theta^{\text{anti-}k_{T}}_{j}$ are the jet algorithm constraints with both
partons in jet (Fig. 2A) and only one parton in jet (Fig. 2B, 2C),
respectively, and the superscript “anti-$k_{T}$” indicates that anti-$k_{T}$
algorithm is used. The form of $\Theta^{\text{anti-}k_{T}}_{\mathrm{both}}$
and $\Theta^{\text{anti-}k_{T}}_{j}$ are given by the Heaviside functions
Ellis:2010rwa ; Kang:2016ehg ; Kang:2017mda :

$\displaystyle\Theta^{\text{anti-}k_{T}}_{\textrm{both}}$
$\displaystyle=\theta\pqty{z_{h}(1-z_{h})\omega_{J}\tan{\frac{\mathcal{R}}{2}}-q_{\perp}}\,,$
(10) $\displaystyle\Theta^{\text{anti-}k_{T}}_{j}$
$\displaystyle=\theta\pqty{q_{\perp}-(1-z)\omega_{J}\tan\frac{\mathcal{R}}{2}}\,,$
(11)

where $\mathcal{R}$ is the angular separation between three vector momenta
inside the jet Ellis:2010rwa . This is to be compared with the jet radius $R$
in the hadron collider that specifies the separation in pseudo-rapidity and
azimuthal angle as in $\sqrt{(\Delta\eta)^{2}+(\Delta\phi)^{2}}$. For jets
with a small jet radius $R\ll 1$, we have

$\displaystyle\mathcal{R}=\frac{R}{\cosh(\eta)}\,,$ (12)

where $\eta$ is the jet pseudo-rapidity. Note that

$\displaystyle\omega_{J}\tan\frac{\mathcal{R}}{2}\approx
2E_{J}\,\frac{\mathcal{R}}{2}=p_{T}\cosh(\eta)\frac{R}{\cosh(\eta)}=p_{T}R\,,$
(13)

and we will thus include the argument $p_{T}R$ in the FJFs.

The longitudinally polarized functions $\Delta\hat{P}_{ji}(z,\epsilon)$ in
Section 2.1.1 are given in Vogelsang:1996im :

$\displaystyle\Delta\hat{P}_{qq}(z,\epsilon)$
$\displaystyle=C_{F}\bqty{\frac{1+z^{2}}{1-z}-\epsilon(1-z)}\,,$ (14)
$\displaystyle\Delta\hat{P}_{gq}(z,\epsilon)$
$\displaystyle=C_{F}\bqty{2-z+2\epsilon(1-z)}\,,$ (15)
$\displaystyle\Delta\hat{P}_{qg}(z,\epsilon)$
$\displaystyle=T_{F}\bqty{2z-1-2\epsilon(1-z)}\,,$ (16)
$\displaystyle\Delta\hat{P}_{gg}(z,\epsilon)$
$\displaystyle=2C_{A}\bqty{\frac{1}{1-z}-2z+1+2\epsilon(1-z)}\,.$ (17)

The transversely polarized splitting functions
$\Delta_{T}\hat{P}_{ji}(z,\epsilon)$ only exist for $\Delta_{T}\hat{P}_{qq}$
because there is no gluon transversity fragmentation function for spin-1/2
hadron as we mentioned above. We have the following expression for
$\Delta_{T}\hat{P}_{qq}(z,\epsilon)$ Vogelsang:1997ak

$\Delta_{T}\hat{P}_{qq}(z,\epsilon)=C_{F}\left[\frac{2z}{1-z}\right]\,.$ (18)

By inserting the $\Theta$ functions for anti-$k_{T}$ algorithm and carrying
out the integration in Section 2.1.1, we obtain the bare results222More
details on the $q_{\perp}$ integral and relevant expansions as $\epsilon\to 0$
can be found in Appendix A. for longitudinally polarized semi-inclusive FJFs
$\Delta\mathcal{G}^{j}_{i,\mathrm{bare}}(z,z_{h},p_{T}R)$ with
$i,j\in\Bqty{q,g}$:

$\displaystyle\Delta\mathcal{G}^{q}_{q,\mathrm{bare}}$
$\displaystyle(z,z_{h},p_{T}R)=\delta(1-z)\delta(1-z_{h})$
$\displaystyle+\frac{\alpha_{s}}{2\pi}\bqty{\pqty{\frac{1}{\epsilon}+L}\Delta
P_{qq}(z)\delta(1-z_{h})-\pqty{\frac{1}{\epsilon}+L}\Delta
P_{qq}(z_{h})\delta(1-z)}$
$\displaystyle+\delta(1-z)\frac{\alpha_{s}}{2\pi}\bqty{2C_{F}(1+z_{h}^{2})\pqty{\frac{\ln(1-z_{h})}{1-z_{h}}}_{+}+C_{F}(1-z_{h})+2\Delta
P_{qq}(z_{h})\ln(z_{h})}$
$\displaystyle-\delta(1-z_{h})\frac{\alpha_{s}}{2\pi}\bqty{2C_{F}(1+z^{2})\pqty{\frac{\ln(1-z)}{1-z}}_{+}+C_{F}(1-z)}\,,$
(19) $\displaystyle\Delta\mathcal{G}^{g}_{q,\mathrm{bare}}$
$\displaystyle(z,z_{h},p_{T}R)=\frac{\alpha_{s}}{2\pi}\bqty{\pqty{\frac{1}{\epsilon}+L}\Delta
P_{gq}(z)\delta(1-z_{h})-\pqty{\frac{1}{\epsilon}+L}\Delta
P_{gq}(z_{h})\delta(1-z)}$
$\displaystyle+\delta(1-z)\frac{\alpha_{s}}{2\pi}\bigg{[}2\Delta
P_{gq}(z_{h})\ln(z_{h}(1-z_{h}))-2C_{F}(1-z_{h})\bigg{]}$
$\displaystyle-\delta(1-z_{h})\frac{\alpha_{s}}{2\pi}\bigg{[}2\Delta
P_{gq}(z)\ln(1-z)-2C_{F}(1-z)\bigg{]}\,,$ (20)
$\displaystyle\Delta\mathcal{G}^{q}_{g,\mathrm{bare}}$
$\displaystyle(z,z_{h},p_{T}R)=\frac{\alpha_{s}}{2\pi}\bqty{\pqty{\frac{1}{\epsilon}+L}\Delta
P_{qg}(z)\delta(1-z_{h})-\pqty{\frac{1}{\epsilon}+L}\Delta
P_{qg}(z_{h})\delta(1-z)}$
$\displaystyle+\delta(1-z)\frac{\alpha_{s}}{2\pi}\bigg{[}2\Delta
P_{qg}(z_{h})\ln(z_{h}(1-z_{h}))+2T_{F}(1-z_{h})\bigg{]}$
$\displaystyle-\delta(1-z_{h})\frac{\alpha_{s}}{2\pi}\bigg{[}2\Delta
P_{qg}(z)\ln(1-z)+2T_{F}(1-z)\bigg{]}\,,$ (21)
$\displaystyle\Delta\mathcal{G}^{g}_{g,\mathrm{bare}}$
$\displaystyle(z,z_{h},p_{T}R)=\delta(1-z)\delta(1-z_{h})$
$\displaystyle+\frac{\alpha_{s}}{2\pi}\bqty{\pqty{\frac{1}{\epsilon}+L}\Delta
P_{gg}(z)\delta(1-z_{h})-\pqty{\frac{1}{\epsilon}+L}\Delta
P_{gg}(z_{h})\delta(1-z)}$
$\displaystyle+\delta(1-z)\frac{\alpha_{s}}{2\pi}\bqty{4C_{A}\pqty{2(1-z_{h})^{2}+z_{h}}\pqty{\frac{\ln(1-z_{h})}{1-z_{h}}}_{+}}$
$\displaystyle+\delta(1-z)\frac{\alpha_{s}}{2\pi}\bigg{[}2\Delta
P_{gg}(z_{h})\ln(z_{h})-4C_{A}(1-z_{h})\bigg{]}$
$\displaystyle-\delta(1-z_{h})\frac{\alpha_{s}}{2\pi}\bqty{4C_{A}\pqty{2(1-z)^{2}+z}\pqty{\frac{\ln(1-z)}{1-z}}_{+}-4C_{A}(1-z)}\,,$
(22)

where the logarithm

$\displaystyle L\equiv\ln(\frac{\mu^{2}}{\left(p_{T}R\right)^{2}})\,.$ (23)

This indicates that the natural scale for the collinear FJFs is given by
$\mu_{\mathscr{G}}\sim p_{T}R$. As for the transversely polarized semi-
inclusive FJFs, we obtain:

$\displaystyle\Delta_{T}\mathcal{G}^{q}_{q,\mathrm{bare}}$
$\displaystyle(z,z_{h},p_{T}R)=\delta(1-z)\delta(1-z_{h})$
$\displaystyle+\frac{\alpha_{s}}{2\pi}\bqty{\pqty{\frac{1}{\epsilon}+L}\Delta_{T}P_{qq}(z)\delta(1-z_{h})-\pqty{\frac{1}{\epsilon}+L}\Delta_{T}P_{qq}(z_{h})\delta(1-z)}$
$\displaystyle+\delta(1-z)\frac{\alpha_{s}}{2\pi}\bqty{4C_{F}z_{h}\pqty{\frac{\ln(1-z_{h})}{1-z_{h}}}_{+}+2\Delta_{T}P_{qq}(z_{h})\ln(z_{h})}$
$\displaystyle-\delta(1-z_{h})\frac{\alpha_{s}}{2\pi}\bqty{4C_{F}z\pqty{\frac{\ln(1-z)}{1-z}}_{+}}\,.$
(24)

Here the functions $\Delta_{(T)}P_{ji}(z)$ are the longitudinally
(transversely) polarized Altarelli-Parisi splitting kernels:

$\displaystyle\Delta P_{qq}(z)$
$\displaystyle=C_{F}\bqty{\frac{1+z^{2}}{(1-z)_{+}}+\frac{3}{2}\delta(1-z)}\,,$
(25) $\displaystyle\Delta P_{gq}(z)$ $\displaystyle=C_{F}\bqty{2-z}\,,$ (26)
$\displaystyle\Delta P_{qg}(z)$ $\displaystyle=T_{F}\bqty{2z-1}\,,$ (27)
$\displaystyle\Delta P_{gg}(z)$
$\displaystyle=2C_{A}\bqty{\frac{1}{(1-z)_{+}}-2z+1}+\frac{\beta_{0}}{2}\delta(1-z)\,,$
(28) $\displaystyle\Delta_{T}P_{qq}(z)$
$\displaystyle=C_{F}\bqty{\frac{2z}{(1-z)_{+}}+\frac{3}{2}\delta(1-z)}\,,$
(29)

where $\beta_{0}\equiv\frac{11}{3}C_{A}-\frac{4}{3}T_{F}n_{f}$ and $n_{f}$ is
number of flavors. The “plus” distributions are defined as usual by:

$\int_{0}^{1}\differential{z}f\pqty{z}\bqty{g\pqty{z}}_{+}=\int_{0}^{1}\differential{z}\Big{(}f\pqty{z}-f\pqty{1}\Big{)}g\pqty{z}\,.$
(30)

####  2.1.2 Renormalization and matching onto collinear FFs

In this section, we will renormalize the bare semi-inclusive FJFs obtained in
the previous section, and then match the results to the standard collinear
FFs. In doing so, it is important to point out that the $1/\epsilon$ poles
with a factor of $\Delta_{\pqty{T}}P_{ij}\pqty{z_{h}}\delta\pqty{1-z}$ in Eqs.
19, 20, 21, 22 and 24 are the infrared (IR) poles, that would be matched onto
the standard longitudinally (transversely) polarized collinear FFs. On the
other hand, the poles with a factor of
$\Delta_{\pqty{T}}P_{ij}\pqty{z}\delta\pqty{1-z_{h}}$ are the ultraviolet (UV)
poles which will be taken care of by renormalization. Since the UV poles do
not involve the variable $z_{h}$, we should expect that $z_{h}$ is merely a
parameter in renormalization. This is exactly the same as the unpolarized
situation studied in Kang:2016ehg . With this in mind, the relationship
between the bare and renormalized semi-inclusive FJFs is given by

$\Delta_{\pqty{T}}\mathcal{G}_{i,\mathrm{bare}}^{j}\pqty{z,z_{h},p_{T}R}=\sum_{k}\int_{z}^{1}\frac{\differential{z^{\prime}}}{z^{\prime}}\Delta_{\pqty{T}}Z_{ik}\pqty{\frac{z}{z^{\prime}},\mu}\Delta_{\pqty{T}}\mathcal{G}_{k}^{j}\pqty{z^{\prime},z_{h},p_{T}R,\mu}\,,$
(31)

where $\Delta_{\pqty{T}}Z_{ik}\pqty{\frac{z}{z^{\prime}},\mu}$ is the
renormalization matrix and
$\Delta_{\pqty{T}}\mathcal{G}_{k}^{j}\pqty{z^{\prime},z_{h},p_{T}R,\mu}$ are
the renormalized semi-inclusive FJFs. For the aforementioned reason, the
convolution only involves the variable $z$. The renormalized semi-inclusive
FJFs satisfy the following RG evolution equations:

$\mu\frac{\differential{}}{\differential{\mu}}\Delta_{\pqty{T}}\mathcal{G}_{i}^{j}\pqty{z,z_{h},p_{T}R,\mu}=\sum_{k}\int_{z}^{1}\frac{\differential{z^{\prime}}}{z^{\prime}}\Delta_{\pqty{T}}\gamma_{ik}^{\mathcal{G}}\pqty{\frac{z}{z^{\prime}},\mu}\Delta_{\pqty{T}}\mathcal{G}_{k}^{j}\pqty{z^{\prime},z_{h},p_{T}R,\mu}\,,$
(32)

with the anomalous dimension matrix given by:

$\Delta_{\pqty{T}}\gamma_{ik}^{\mathcal{G}}\pqty{\frac{z}{z^{\prime}},\mu}=-\sum_{k}\int_{z}^{1}\frac{\differential{z^{\prime}}}{z^{\prime}}\pqty{\Delta_{\pqty{T}}Z}_{il}^{-1}\pqty{\frac{z}{z^{\prime}},\mu}\mu\frac{\differential{}}{\differential{\mu}}\Delta_{\pqty{T}}Z_{lk}\pqty{z^{\prime},\mu}\,,$
(33)

and $\pqty{\Delta_{\pqty{T}}Z}_{il}^{-1}$ is the inverse of the
renormalization matrix that is defined such that:

$\sum_{k}\int_{z}^{1}\frac{\differential{z^{\prime}}}{z^{\prime}}\pqty{\Delta_{\pqty{T}}Z}_{il}^{-1}\pqty{\frac{z}{z^{\prime}},\mu}\Delta_{\pqty{T}}Z_{lj}\pqty{z^{\prime},\mu}=\delta_{ij}\delta\pqty{1-z}\,.$
(34)

Up to $\order{\alpha_{s}}$, the renormalization matrix is

$\Delta_{\pqty{T}}Z_{ij}\pqty{z,\mu}=\delta_{ij}\delta\pqty{1-z}+\frac{\alpha_{s}\pqty{\mu}}{2\pi}\frac{1}{\epsilon}\Delta_{\pqty{T}}P_{ji}\pqty{z}\,,$
(35)

and therefore the anomalous dimension matrix is given by:

$\Delta_{\pqty{T}}\gamma_{ij}^{\mathcal{G}}\pqty{z,\mu}=\frac{\alpha_{s}\pqty{\mu}}{\pi}\Delta_{\pqty{T}}P_{ji}\pqty{z}\,.$
(36)

We therefore observe that the evolution of the renormalized polarized semi-
inclusive FJFs conforms to the time-like DGLAP equation for collinear
polarized FFs Altarelli:1977zs :

$\mu\frac{\differential{}}{\differential{\mu}}\Delta_{\pqty{T}}\mathcal{G}_{i}^{h}\pqty{z,z_{h},p_{T}R,\mu}=\frac{\alpha_{s}\pqty{\mu}}{\pi}\sum_{k}\int_{z}^{1}\frac{\differential{z^{\prime}}}{z^{\prime}}\Delta_{\pqty{T}}P_{ji}\pqty{\frac{z}{z^{\prime}}}\Delta_{\pqty{T}}\mathcal{G}_{j}^{h}\pqty{z^{\prime},z_{h},p_{T}R,\mu}\,,$
(37)

where the leading order splitting kernels are given in Eqs. 25, 26, 27, 28 and
29. Notice that the hadronic FJFs have been reinstated since the hadronic and
partonic FJFs would contain the same UV counter terms and thus follow the same
RG evolution equations. As we mentioned in the previous section that the
natural scale for the collinear FJFs is given by $\mu_{\mathscr{G}}=p_{T}R$,
thus one would use the above renormalization group equations to evolve these
FJFs from their natural scale $p_{T}R$ to the hard scale $p_{T}$ for the jet
production. The effect of this would be to resum the logarithms of the jet
radius $R$ for narrow jets $R\ll 1$. Analogous behaviors were observed in the
context of the semi-inclusive unpolarized FJFs in Kang:2016ehg .

With the UV poles eliminated by renormalization, we can proceed to dealing
with the IR poles, which can be addressed by matching onto the collinear
polarized FFs. Such matching can be done at a scale
$\mu\gg\Lambda_{\mathrm{QCD}}$ as follows:

$\Delta_{\pqty{T}}\mathcal{G}_{i}^{h}\pqty{z,z_{h},p_{T}R,\mu}=\sum_{j}\int_{z_{h}}^{1}\frac{\differential{z_{h}^{\prime}}}{z_{h}^{\prime}}\Delta_{\pqty{T}}\mathcal{J}_{ij}\pqty{z,z_{h}^{\prime},p_{T}R,\mu}\Delta_{\pqty{T}}D_{h/j}\pqty{\frac{z_{h}}{z_{h}^{\prime}},\mu}\,,$
(38)

where again, the hadronic semi-inclusive FJFs and collinear FFs are
reinstated. Here, $\Delta_{(T)}D_{h/j}(z_{h},\mu)$ is the collinear
fragmentation function characterizing a parton $j$ fragmenting into a hadron
$h$. Specifically, $\Delta D_{h/j}$ ($\Delta_{T}D_{h/j}$) is the helicity
(transversity) fragmentation function describing a longitudinally
(transversely) polarized parton fragmenting into a longitudinally
(transversely) polarized hadron. And the matching relation above holds, with a
power correction of order
$\order{\Lambda_{\mathrm{QCD}}^{2}/\pqty{p_{T}R)^{2}}}$ Jain:2011xz ;
Chien:2015ctp ; Kang:2016ehg . In this case, however, different from the
renormalization procedure, $z_{h}$ is being convoluted and $z$ is merely a
parameter. This process is similar to the approach used for unpolarized FJFs
described in Kang:2016ehg , except that instead of the unpolarized FFs,
collinear polarized FFs are used in this case. The perturbative results at the
parton level for the polarized collinear FFs are given by:

$\Delta_{\pqty{T}}D_{j/i}\pqty{z_{h},\mu}=\delta_{ij}\delta\pqty{1-z_{h}}+\frac{\alpha_{s}}{2\pi}\Delta_{\pqty{T}}P_{ji}\pqty{z_{h}}\pqty{-\frac{1}{\epsilon}}\,.$
(39)

Finally, the matching coefficients $\Delta_{\pqty{T}}\mathcal{J}_{ij}$ for
anti-$k_{T}$ jet algorithm are as follows

$\displaystyle\Delta\mathcal{J}_{qq}(z,z_{h},p_{T}R,\mu)$
$\displaystyle=\delta(1-z)\delta(1-z_{h})+\frac{\alpha_{s}}{2\pi}\bigg{\\{}L\bigg{[}\Delta
P_{qq}(z)\delta(1-z_{h})-\Delta P_{qq}(z_{h})\delta(1-z)\bigg{]}$
$\displaystyle\quad+\delta(1-z)\bqty{2C_{F}(1+z_{h}^{2})\pqty{\frac{\ln(1-z_{h})}{1-z_{h}}}_{+}+C_{F}(1-z_{h})+\Delta\mathcal{I}_{qq}^{\text{anti-}k_{T}}(z_{h})}$
$\displaystyle\quad-\delta(1-z_{h})\bqty{2C_{F}(1+z^{2})\pqty{\frac{\ln(1-z)}{1-z}}_{+}+C_{F}(1-z)}\bigg{\\}}\,,$
(40) $\displaystyle\Delta\mathcal{J}_{qg}(z,z_{h},p_{T}R,\mu)$
$\displaystyle=\frac{\alpha_{s}}{2\pi}\bigg{\\{}L\bigg{[}\Delta
P_{gq}(z)\delta(1-z_{h})-\Delta P_{gq}(z_{h})\delta(1-z)\bigg{]}$
$\displaystyle\quad+\delta(1-z)\bigg{[}2\Delta
P_{gq}(z_{h})\ln(1-z_{h})-2C_{F}(1-z_{h})+\Delta\mathcal{I}_{gq}^{\text{anti-}k_{T}}(z_{h})\bigg{]}$
$\displaystyle\quad-\delta(1-z_{h})\bigg{[}2\Delta
P_{gq}(z)\ln(1-z)-2C_{F}(1-z)\bigg{]}\bigg{\\}}\,,$ (41)
$\displaystyle\Delta\mathcal{J}_{gq}(z,z_{h},p_{T}R,\mu)$
$\displaystyle=\frac{\alpha_{s}}{2\pi}\bigg{\\{}L\bigg{[}\Delta
P_{qg}(z)\delta(1-z_{h})-\Delta P_{qg}(z_{h})\delta(1-z)\bigg{]}$
$\displaystyle\quad+\delta(1-z)\bigg{[}2\Delta
P_{qg}(z_{h})\ln(1-z_{h})+2T_{F}(1-z_{h})+\Delta\mathcal{I}_{qg}^{\text{anti-}k_{T}}(z_{h})\bigg{]}$
$\displaystyle\quad-\delta(1-z_{h})\bigg{[}2\Delta
P_{qg}(z)\ln(1-z)+2T_{F}(1-z)\bigg{]}\bigg{\\}}\,,$ (42)
$\displaystyle\Delta\mathcal{J}_{gg}(z,z_{h},p_{T}R,\mu)$
$\displaystyle=\delta(1-z)\delta(1-z_{h})+\frac{\alpha_{s}}{2\pi}\bigg{\\{}L\bigg{[}\Delta
P_{gg}(z)\delta(1-z_{h})-\Delta P_{gg}(z_{h})\delta(1-z)\bigg{]}$
$\displaystyle\quad+\delta(1-z)\bigg{[}4C_{A}\pqty{2(1-z_{h})^{2}+z_{h}}\pqty{\frac{\ln(1-z_{h})}{1-z_{h}}}_{+}$
$\displaystyle\qquad\qquad\qquad-4C_{A}(1-z_{h})+\Delta\mathcal{I}_{gg}^{\text{anti-}k_{T}}(z_{h})\bigg{]}$
$\displaystyle\quad-\delta(1-z_{h})\bqty{4C_{A}\pqty{2(1-z)^{2}+z}\pqty{\frac{\ln(1-z)}{1-z}}_{+}-4C_{A}(1-z)}\bigg{\\}}\,,$
(43) $\displaystyle\Delta_{T}\mathcal{J}_{qq}(z,z_{h},p_{T}R,\mu)$
$\displaystyle=\delta(1-z)\delta(1-z_{h})+\frac{\alpha_{s}}{2\pi}\bigg{\\{}L\bigg{[}\Delta_{T}P_{qq}(z)\delta(1-z_{h})-\Delta_{T}P_{qq}(z_{h})\delta(1-z)\bigg{]}$
$\displaystyle\quad+\delta(1-z)\bqty{4C_{F}z_{h}\pqty{\frac{\ln(1-z_{h})}{1-z_{h}}}_{+}+\Delta_{T}\mathcal{I}_{qq}^{\text{anti-}k_{T}}(z_{h})}$
$\displaystyle\quad-\delta(1-z_{h})\bqty{4C_{F}z\pqty{\frac{\ln(1-z)}{1-z}}_{+}}\bigg{\\}}\,,$
(44)

where we have defined $\Delta_{(T)}\mathcal{I}_{ij}^{\text{anti-}k_{T}}$ as:

$\begin{split}\Delta\mathcal{I}_{ij}^{\text{anti-}k_{T}}\pqty{z_{h}}&=2\Delta
P_{ji}\pqty{z_{h}}\ln(z_{h})\,,\\\
\Delta_{T}\mathcal{I}_{ij}^{\text{anti-}k_{T}}\pqty{z_{h}}&=2\Delta_{T}P_{ji}\pqty{z_{h}}\ln(z_{h})\,.\\\
\end{split}$ (45)

The matching coefficient functions $\Delta\mathcal{J}_{ij}$ for the
longitudinally polarized case have been given in the previous publication
Kang:2020xyq  with a different notation $\mathcal{J}^{L}_{ij}$. Here, we
provide the detailed steps of how such matching coefficients are derived. At
the same time, the matching coefficient for the transversely polarized case
$\Delta_{T}\mathcal{J}_{qq}$ is presented here for the first time.

###  2.2 Collinear FJFs in exclusive jet productions

Exclusive jet production, such as back-to-back dijet production in proton-
proton collisions Kang:2020xez ; Gao:2023ulg  or back-to-back electron-jet
production in electron-proton collisions Liu:2018trl , can provide valuable
insight into the understanding of the fundamental dynamics of hadron structure
and interactions. However, in such situations, typically the radiation outside
the jet is restricted. For example, in back-to-back dijet production, one
usually restricts the transverse momentum imbalance $q_{T}$ to be much smaller
than the average transverse momentum of the jets. In this case, the
hard/collinear gluon emission outside the jet would move the imbalance $q_{T}$
away from the back-to-back region. Because of that, the collinear radiation
can only happen inside the jet. Consequently, for jet function calculations in
exclusive jet production, we only have to consider Fig. 2A while B and C would
not contribute. We refer to the corresponding FJFs as exclusive FJFs below,
which are key ingredients in the factorization formalism for these processes.
They describe the probability distribution for a parton in the jet to fragment
into a hadron with a given collinear momentum fraction in exclusive jet
production.

The perturbative computations for exclusive FJFs
$\Delta_{(T)}\mathscr{G}_{i}^{j}(z_{h},p_{T}R,\mu)$ are identical to those for
the semi-inclusive FJFs, except that we keep only the contributions from Fig.
2A. Since both partons are always inside the jet, we always have $z=1$, i.e.
the jet carries all the energy of the parton that initiates the jet in this
case. Because of this, we no longer keep the variable $z$ in the FJFs and thus
the exclusive FJFs only depend on $z_{h}$. The RG equations for exclusive FJFs
can also be derived in the same manner as above, and we find that they satisfy
the following equation

$\mu\frac{\differential{}}{\differential{\mu}}\Delta_{(T)}\mathscr{G}_{i}^{h}(z_{h},p_{T}R,\mu)=\gamma_{\mathscr{G}}^{i}(\mu)\Delta_{(T)}\mathscr{G}_{i}^{h}(z_{h},p_{T}R,\mu)\,,$
(46)

where the index $i$ is not summed over, and the anomalous dimensions
$\gamma_{\mathscr{G}}^{i}(\mu)$ with $i=q,g$ at this order are given

$\displaystyle\gamma_{\mathscr{G}}^{q}(\mu)=$
$\displaystyle\frac{\alpha_{s}(\mu)}{\pi}\left(C_{F}L+\frac{3}{2}C_{F}\right)\,,$
(47) $\displaystyle\gamma_{\mathscr{G}}^{g}(\mu)=$
$\displaystyle\frac{\alpha_{s}(\mu)}{\pi}\left(C_{A}L+\frac{\beta_{0}}{2}\right)\,.$
(48)

A few comments are in order. First of all, it is important to emphasize that
the RG equation for exclusive FJFs is a multiplicative renormalization,
instead of a convolution over $z$ as in the semi-inclusive FJFs given in the
previous section. Secondly, the anomalous dimension
$\gamma_{\mathscr{G}}^{i}(\mu)$ is the same as that of the exclusive jet
functions in Ellis:2010rwa  and exclusive unpolarized FJFs Kang:2019ahe ;
Chien:2015ctp ; Waalewijn:2012sv . Since in the factorization formalism, one
simply replaces the exclusive jet function with the exclusive FJFs, thus they
should have the same anomalous dimensions simply because of the RG
consistency. At the same time, we find that the UV divergent terms are all
proportional to $\delta(1-z_{h})$ where the radiated gluon becomes soft, and
thus the result should be independent of the polarization of the partons. This
explains the anomalous dimensions for both longitudinally and transversely
polarized FJFs are the same as the unpolarized FJFs. The solution to the RG
equation for the exclusive FJFs is then:

$\Delta_{(T)}\mathscr{G}_{i}^{h}(z_{h},p_{T}R,\mu)=\Delta_{(T)}\mathscr{G}_{i}^{h}\pqty{z_{h},p_{T}R,\mu_{\mathscr{G}}}\exp\bqty{\int_{\mu_{\mathscr{G}}}^{\mu}\frac{\differential{\mu^{\prime}}}{\mu^{\prime}}\gamma_{\mathscr{G}}^{i}\pqty{\mu^{\prime}}}\,,$
(49)

where the scale $\mu_{\mathscr{G}}$ is the characteristic scale that
eliminates the large logarithms in the fixed-order perturbative calculations.
The logarithm $L$ indicates that the natural scale for the exclusive FJFs is
also given by $\mu_{\mathscr{G}}=p_{T}R$, the same as for the semi-inclusive
FJFs. One would thus evolve the FJFs from the natural scale
$\mu_{\mathscr{G}}\sim p_{T}R$ to the hard scale of the process $\mu\sim
p_{T}$ and effectively resum the logarithm of jet radius, $\ln R$.

Finishing the discussion on the RG equation for the exclusive FJFs, let us now
turn to their matching to the standard collinear FFs. Similar to the semi-
inclusive FJFs, the exclusive fragmenting jet functions
$\Delta_{(T)}\mathscr{G}_{i}^{h}(z_{h},p_{T}R,\mu)$ are also closely related
to the fragmentation functions $\Delta_{(T)}D_{h/j}$ and in this work we label
the corresponding matching coefficients as $\Delta_{(T)}\mathscr{J}_{ij}$.
Following Jain:2011xz ; Waalewijn:2012sv ; Chien:2015ctp ; Kang:2019ahe , one
has the following expansion for exclusive FJFs:

$\displaystyle\Delta_{(T)}\mathscr{G}_{i}^{h}(z_{h},p_{T}R,\mu)=\sum_{j}\int_{z_{h}}^{1}\frac{\differential{z^{\prime}_{h}}}{z^{\prime}_{h}}\Delta_{(T)}\mathscr{J}_{ij}(z^{\prime}_{h},p_{T}R,\mu)\Delta_{(T)}D_{h/j}\pqty{\frac{z_{h}}{z^{\prime}_{h}},\mu}\,.$
(50)

Again this equation is valid up to power corrections of order
$\order{\Lambda_{\mathrm{QCD}}^{2}/\pqty{p_{T}R)^{2}}}$ for light hadrons. For
heavy meson fragmenting jet functions, $\Lambda_{\rm QCD}$ should be replaced
by the heavy quark mass $m_{Q}$ in the above equation Chien:2015ctp ;
Baumgart:2014upa . We provide the NLO results of matching coefficients
$\Delta_{(T)}\mathscr{J}_{ij}$, which depend on the jet algorithm. The results
for unpolarized FJFs with cone algorithm were given in Procura:2011aq , while
those for anti-$k_{T}$ jets were first written down in Waalewijn:2012sv .
While the results for the unpolarized FJFs are available in the literature,
the coefficients for the polarized FJFs are given for the first time in our
current paper. We provide the detailed derivations of exclusive FJFs with
polarizations for anti-$k_{T}$ jets in Appendix B. Here we only list the final
results:

$\displaystyle\Delta\mathscr{J}_{qq}(z_{h},p_{T}R,\mu)$
$\displaystyle=\delta(1-z_{h})+\frac{\alpha_{s}C_{F}}{2\pi}\bigg{[}\delta(1-z_{h})\pqty{\frac{L^{2}}{2}-\frac{\pi^{2}}{12}}-\Delta
P_{qq}(z_{h})L$
$\displaystyle\qquad\qquad\qquad\qquad\quad+1-z_{h}+\Delta\hat{\mathscr{I}}_{qq}^{\text{anti-}k_{T}}(z_{h})\bigg{]}\,,$
(51) $\displaystyle\Delta\mathscr{J}_{qg}(z_{h},p_{T}R,\mu)$
$\displaystyle=\frac{\alpha_{s}C_{F}}{2\pi}\bigg{[}-\Delta
P_{gq}(z_{h})L-2(1-z_{h})+\Delta\hat{\mathscr{I}}_{qg}^{\text{anti-}k_{T}}(z_{h})\bigg{]}\,,$
(52) $\displaystyle\Delta\mathscr{J}_{gq}(z_{h},p_{T}R,\mu)$
$\displaystyle=\frac{\alpha_{s}T_{F}}{2\pi}\bigg{[}-\Delta
P_{qg}(z_{h})L+2(1-z_{h})+\Delta\hat{\mathscr{I}}_{gq}^{\text{anti-}k_{T}}(z_{h})\bigg{]}\,,$
(53) $\displaystyle\Delta\mathscr{J}_{gg}(z_{h},p_{T}R,\mu)$
$\displaystyle=\delta(1-z_{h})+\frac{\alpha_{s}C_{A}}{2\pi}\bigg{[}\delta(1-z_{h})\pqty{\frac{L^{2}}{2}-\frac{\pi^{2}}{12}}-\Delta
P_{gg}(z_{h})L$
$\displaystyle\qquad\qquad\qquad\qquad\quad-4(1-z_{h})+\Delta\hat{\mathscr{I}}_{gg}^{\text{anti-}k_{T}}(z_{h})\bigg{]}\,,$
(54) $\displaystyle\Delta_{T}\mathscr{J}_{qq}(z_{h},p_{T}R,\mu)$
$\displaystyle=\delta(1-z_{h})+\frac{\alpha_{s}C_{F}}{2\pi}\bigg{[}\delta(1-z_{h})\pqty{\frac{L^{2}}{2}-\frac{\pi^{2}}{12}}$
$\displaystyle\qquad\qquad\qquad\qquad\quad-\Delta_{T}P_{qq}(z_{h})L+\Delta_{T}\hat{\mathscr{I}}_{q}^{\text{anti-}k_{T}}(z_{h})\bigg{]}\,,$
(55)

where for anti-$k_{T}$ jets, the jet-algorithm dependent pieces,
$\Delta_{(T)}\hat{\mathscr{I}}_{ij}^{\text{anti-}k_{T}}(z_{h})$, are given by

$\displaystyle\Delta\hat{\mathscr{I}}_{qq}^{\text{anti-}k_{T}}\pqty{z_{h}}$
$\displaystyle=2\Delta
P_{qq}(z_{h})\ln(z_{h})+2\pqty{1+z_{h}^{2}}\pqty{\frac{\ln(1-z_{h})}{1-z_{h}}}_{+}\,,$
(56)
$\displaystyle\Delta\hat{\mathscr{I}}_{qg}^{\text{anti-}k_{T}}\pqty{z_{h}}$
$\displaystyle=2\Delta P_{gq}(z_{h})\ln{\Big{(}z_{h}(1-z_{h})\Big{)}}\,,$ (57)
$\displaystyle\Delta\hat{\mathscr{I}}_{gq}^{\text{anti-}k_{T}}\pqty{z_{h}}$
$\displaystyle=2\Delta P_{qg}(z_{h})\ln{\Big{(}z_{h}(1-z_{h})\Big{)}}\,,$ (58)
$\displaystyle\Delta\hat{\mathscr{I}}_{gg}^{\text{anti-}k_{T}}\pqty{z_{h}}$
$\displaystyle=2\Delta
P_{gg}(z_{h})\ln(z_{h})+4\Big{(}2(1-z_{h})^{2}+z_{h}\Big{)}\pqty{\frac{\ln(1-z_{h})}{1-z_{h}}}_{+}\,,$
(59)
$\displaystyle\Delta_{T}\hat{\mathscr{I}}_{qq}^{\text{anti-}k_{T}}\pqty{z_{h}}$
$\displaystyle=2\Delta_{T}P_{qq}(z_{h})\ln(z_{h})+4z_{h}\pqty{\frac{\ln(1-z_{h})}{1-z_{h}}}_{+}\,,$
(60)

where the splitting kernels $\Delta_{(T)}{P}_{ji}(z_{h})$ Vogelsang:1996im ;
Vogelsang:1997ak  have been introduced in Eqs. 25, 26, 27, 28 and 29.

##  3 The transverse momentum dependent FJFs (TMD FJFs)

In this section, we study TMD FJFs in both the semi-inclusive jet production
and the exclusive jet production, including polarizations of the initiating
parton and the final observed hadron inside the jet. They are termed as
polarized TMD FJFs Kang:2020xyq . Some partial results are available in the
previous publications Kang:2020xyq ; Kang:2021ffh , but never the complete
results. For example, the hard matching functions for both longitudinally and
transversely polarized cases are presented for the first time. Below, we
provide the complete results for all the relevant polarized semi-inclusive and
exclusive TMD FJFs. The operator definitions of these functions in SCET are
presented, followed by their factorization formalism, which involves hard
functions, soft functions, and TMD FFs.

The kinematics is set up as in Fig. 3, a hadron $h$ is observed inside the
jet, carrying a fraction $z_{h}$ of the jet longitudinal momentum (not labeled
in the figure), a transverse momentum $\bm{j}_{\perp}$ with azimuthal angle
$\hat{\phi}_{h}$, and a transverse spin $\bm{S}_{h\bot}$ with azimuthal angle
$\hat{\phi}_{S_{h}}$. Both $\bm{j}_{\perp}$ and $\bm{S}_{h\bot}$ are measured
with respect to the jet axis $z_{J}$, while both azimuthal angles
$\hat{\phi}_{h}$ and $\hat{\phi}_{S_{h}}$ are measured within the jet
transverse plane and with respect to $x_{J}$, the $x$-axis that is rotated
into jet coordinates333Please see Eqs. 155 and 156 for the parameterization of
$\bm{j}_{\perp}$ and $\bm{S}_{h\bot}$..

Figure 3: Illustration for the distribution of hadrons inside a jet with
transverse momentum $\bm{j}_{\perp}$ and azimuthal angle $\hat{\phi}_{h}$, and
transverse spin $\bm{S}_{h\bot}$ and azimuthal angle $\hat{\phi}_{S_{h}}$.
Both azimuthal angles are measured with respect to the jet coordinate $x_{J}$
and within the jet transverse plane.

###  3.1 TMD FJFs in semi-inclusive jet productions

Again we start with the semi-inclusive TMD FJFs, which is an essential
ingredient for studying the transverse momentum distribution of hadrons inside
a jet in e.g. single inclusive jet production in proton-proton collisions
($p+p\to{\rm jet}(h)+X$) or in electron-proton collisions ($e+p\to{\rm
jet}(h)+X$). Following the previous publication Kang:2020xyq , we define the
general correlators for TMD FJFs initiated by quark or gluon as:

$\displaystyle\Delta^{h/q}\pqty{z,z_{h},\bm{j}_{\perp},S_{h}}$
$\displaystyle=\frac{z}{2N_{c}}\delta\pqty{z_{h}-\frac{\omega_{h}}{\omega_{J}}}\matrixelement{0}{\delta\pqty{\omega-\bar{n}\cdot\mathcal{P}}\delta^{2}\pqty{\mathcal{P}_{\perp}/z_{h}+\bm{j}_{\perp}}\chi_{n}(0)}{(Jh)X}$
(61)
$\displaystyle\quad\times\matrixelement{(Jh)X}{\overline{\chi}_{n}(0)}{0},$
$\displaystyle\Delta^{h/g,\mu\nu}\pqty{z,z_{h},\bm{j}_{\perp},S_{h}}$
$\displaystyle=\frac{z\omega}{(d-2)\pqty{N_{c}^{2}-1}}\delta\pqty{z_{h}-\frac{\omega_{h}}{\omega_{J}}}$
(62)
$\displaystyle\quad\times\matrixelement{0}{\delta\pqty{\omega-\bar{n}\cdot\mathcal{P}}\delta^{2}\pqty{\mathcal{P}_{\perp}/z_{h}+\bm{j}_{\perp}}\mathcal{B}_{n\perp}^{\mu}(0)}{(Jh)X}\matrixelement{(Jh)X}{\mathcal{B}_{n\perp}^{\nu}(0)}{0}\,,$

where we have the energy fractions $z$ and $z_{h}$ defined the same as in Eq.
7. One can parameterize these correlators at the leading power Metz:2016swz ;
Kang:2020xyq :

$\displaystyle\Delta^{h/q}(z,z_{h},\bm{j}_{\perp},S_{h})$
$\displaystyle=\Delta^{h/q\,[\not{n}]}\frac{\not{\bar{n}}}{2}-\Delta^{h/q\,[\not{n}\gamma_{5}]}\frac{\not{\bar{n}}\gamma_{5}}{2}+\Delta^{h/q\,[in_{\nu}\sigma^{k\nu}\gamma_{5}]}\frac{i\bar{n}_{\mu}\sigma^{k\mu}\gamma_{5}}{2}\,,$
(63) $\displaystyle\Delta^{h/g,kl}(z,z_{h},\bm{j}_{\perp},S_{h})$
$\displaystyle=\frac{1}{2}\delta_{T}^{kl}\pqty{\delta_{T}^{mn}\Delta^{h/g,mn}}-\frac{i}{2}\epsilon_{T}^{kl}\pqty{i\epsilon_{T}^{mn}\Delta^{h/g,mn}}+\hat{S}\Delta^{h/g,kl}\,.$
(64)

Here we have defined
$\Delta^{h/q[\Gamma]}\equiv\frac{1}{4}\Tr(\Delta^{h/q}\Gamma)$, and we have
$\hat{S}O^{kl}=\frac{1}{2}\pqty{O^{kl}+O^{lk}-\delta_{T}^{kl}O^{\rho\rho}}$
and $\delta_{T}^{kl}=-g_{T}^{kl}$. The three terms on the right-hand side of
Eq. 63 correspond (in order) to the TMD FJFs with unpolarized, longitudinally
polarized, and transversely polarized initial quarks. On the other hand, the
three terms on the right-hand side of Eq. 64 correspond to unpolarized,
circularly polarized, and linearly polarized gluons. More details have been
presented in the previous work Kang:2020xyq . For completeness, here we
provide the parameterization of quark TMD FJFs:

$\displaystyle\Delta^{h/q[\not{n}]}$
$\displaystyle=\mathcal{D}_{1}^{h/q}(z,z_{h},\bm{j}_{\perp})-\frac{\epsilon^{kl}_{T}j_{\perp}^{k}S_{h\perp}^{l}}{z_{h}M_{h}}\mathcal{D}_{1T}^{\perp
h/q}(z,z_{h},\bm{j}_{\perp})\,,$ (65)
$\displaystyle\Delta^{h/q[\not{n}\gamma_{5}]}$
$\displaystyle=\lambda_{h}\mathcal{G}_{1L}^{h/q}(z,z_{h},\bm{j}_{\perp})-\frac{\bm{j}_{\perp}\cdot\bm{S}_{h\perp}}{z_{h}M_{h}}\mathcal{G}_{1T}^{h/q}(z,z_{h},\bm{j}_{\perp})\,,$
(66) $\displaystyle\Delta^{h/q[in_{\nu}\sigma^{k\nu}\gamma_{5}]}$
$\displaystyle=S_{h\perp}^{k}\mathcal{H}_{1}^{h/q}(z,z_{h},\bm{j}_{\perp})-\frac{\epsilon^{kl}_{T}j_{\perp}^{l}}{z_{h}M_{h}}\mathcal{H}_{1}^{\perp
h/q}(z,z_{h},\bm{j}_{\perp})-\frac{j_{\perp}^{k}}{z_{h}M_{h}}\lambda_{h}{\mathcal{H}_{1L}^{\perp
h/q}(z,z_{h},\bm{j}_{\perp})}$
$\displaystyle\quad+\frac{j_{\perp}^{k}\bm{j}_{\perp}\cdot\bm{S}_{h\perp}-\frac{1}{2}\bm{j}_{\perp}^{2}S_{h\perp}^{k}}{z_{h}^{2}M_{h}^{2}}\mathcal{H}_{1T}^{\perp
h/q}(z,z_{h},\bm{j}_{\perp})\,,$ (67)

The functions $\mathcal{D}$, $\mathcal{G}$ and $\mathcal{H}$ on the right-hand
side of Eqs. 65, 66 and 67 represent the TMD FJFs initiated by unpolarized,
longitudinally polarized, and transversely polarized initial quark
respectively. The parameterization for gluon TMD FJFs is given by

$\displaystyle\delta_{T}^{kl}\Delta^{h/g,kl}$
$\displaystyle=\mathcal{D}_{1}^{h/g}(z,z_{h},\bm{j}_{\perp})-\frac{\epsilon^{ij}_{T}j_{\perp}^{i}S_{h\perp}^{j}}{z_{h}M_{h}}\mathcal{D}_{1}^{\perp
h/g}(z,z_{h},\bm{j}_{\perp})\,,$ (68) $\displaystyle
i\epsilon_{T}^{kl}\Delta^{h/g,kl}$
$\displaystyle=\lambda_{h}\mathcal{G}_{1L}^{h/g}(z,z_{h},\bm{j}_{\perp})-\frac{\bm{j}_{\perp}\cdot\bm{S}_{h\perp}}{z_{h}M_{h}}\mathcal{G}_{1T}^{h/g}(z,z_{h},\bm{j}_{\perp})\,,$
(69) $\displaystyle\hat{S}\Delta^{h/g,kl}$
$\displaystyle=\frac{j_{\perp}^{k}j_{\perp}^{l}}{2z_{h}^{2}M_{h}^{2}}\mathcal{H}_{1}^{\perp
h/g}(z,z_{h},\bm{j}_{\perp})-\frac{\epsilon_{T}^{j_{\perp}\\{k}S_{h\perp}^{l\\}}+\epsilon_{T}^{S_{h\perp}\\{k}j_{\perp}^{l\\}}}{8z_{h}M_{h}}\mathcal{H}_{1T}^{h/g}(z,z_{h},\bm{j}_{\perp})$
$\displaystyle\quad+\frac{\epsilon_{T}^{j_{\perp}\\{k}j_{\perp}^{l\\}}}{4z_{h}^{2}M_{h}^{2}}\pqty{\lambda_{h}\mathcal{H}_{1L}^{\perp,h/g}(z,z_{h},\bm{j}_{\perp})-\frac{\bm{j}_{\perp}\cdot\bm{S}_{h\perp}}{z_{h}M_{h}}\mathcal{H}_{1T}^{\perp
h/g}(z,z_{h},\bm{j}_{\perp})}\,,$ (70)

where the functions $\mathcal{D}$, $\mathcal{G}$ and $\mathcal{H}$ represent
the TMD FJFs initiated by unpolarized, circularly polarized, and linearly
polarized gluons, respectively. The two $\mathcal{G}$ functions with
circularly polarized gluons are anti-symmetric of indices $k,\,l=1,2$, and the
four $\mathcal{H}$ functions related to linearly polarized gluons are
symmetric and traceless combinations of $k,\,l=1,2$. We have adopted the
notation $v_{T}^{\\{k}w_{T}^{l\\}}=v_{T}^{k}w_{T}^{l}+v_{T}^{l}w_{T}^{k}$ as
in Mulders:2000sh ; Boussarie:2023izj .

Since TMD FJFs represent the hadron fragmentation inside a fully reconstructed
jet, their physical meaning is similar to that of standard TMD FFs as reviewed
in Metz:2016swz . Naturally, we adopt the calligraphic font of the letters
used by the corresponding TMD FFs as the notations of TMD FJFs. For example,
$\mathcal{H}_{1}^{h/q}$ in Eq. 67 is the so-called quark transversity TMD
FJFs, i.e. a transversely polarized quark initiates a jet, in which we further
observe a transversely polarized hadron. On the other hand,
$\mathcal{G}_{1L}^{h/g}$ in Eq. 69 is a helicity gluon TMD FJFs, i.e. a
longitudinally polarized gluon initiates a jet, in which we further observe a
longitudinally polarized hadron.

In this study, we focus on the kinematic region
$\Lambda_{\mathrm{QCD}}\lesssim j_{\perp}\ll p_{T}R$, necessitating the
adoption of the TMD factorization Collins:2011zzd . In the previous work
Kang:2017glf , the TMD factorization for transverse momentum distribution of
unpolarized hadrons inside the jet was derived. Over there, the evolution
equations are based on the separate rapidity renormalization group equations
for the so-called “unsubtracted” TMD FFs and soft functions. Below, in the
process of reviewing the TMD factorization, we update the formalism with the
more recent language outlined in the TMD Handbook Boussarie:2023izj . This new
formalism makes the evolution of the in-jet TMD FJFs simpler and more
transparent than before. We then present the hard matching functions, where
the expressions for longitudinally and transversely polarized cases are
provided here for the first time.

####  3.1.1 TMD Factorization

In the kinematic region under consideration, the radiation relevant at leading
power is restricted to collinear radiation within the jet, characterized by
the momentum that scales as $p_{c}=\bqty{p_{c}^{+},p_{c}^{-},p_{c\perp}}\sim
p_{c}^{-}\bqty{\lambda^{2},1,\lambda}$, where the power counting parameter
$\lambda\sim j_{\perp}/p_{T}$. Additionally, soft radiation of order
$j_{\perp}$ is also relevant. It is worth noting that harder emissions are
only permitted outside the jet cone and will thus only impact the
determination of the jet axis. Consequently, the hadron transverse momentum
$j_{\perp}$, which is defined with respect to the jet axis, remains intact
from the radiations external to the jet. Taking the unpolarized case as an
example, a factorized formalism for the unpolarized TMD FJFs within SCET can
be formulated Kang:2017glf :

$\displaystyle\mathcal{D}_{1}^{h/c}\pqty{z,z_{h},\bm{j}_{\perp},p_{T}R,\mu,\zeta_{J}}$
$\displaystyle=\hat{H}^{U}_{c\to
i}\pqty{z,p_{T}R,\mu}\int\differential[2]{\bm{k}_{\perp}}\differential[2]{\bm{\lambda}_{\perp}}\delta^{2}\pqty{z_{h}\bm{\lambda}_{\perp}+\bm{k}_{\perp}-\bm{j}_{\perp}}$
$\displaystyle\quad\times
D_{1}^{h/i(u)}\pqty{z_{h},\bm{k}_{\perp},\mu,\zeta/\nu^{2}}S_{i}\pqty{\bm{\lambda}_{\perp},\mu,\nu\mathcal{R}/2}\,,$
(71)

where $S_{i}\pqty{\bm{\lambda}_{\perp},\mu,\nu\mathcal{R}/2}$ is the in-jet
soft function, and
$D_{1}^{h/i(u)}\pqty{z_{h},\bm{k}_{\perp},\mu,\zeta/\nu^{2}}$ is the
“unsubtracted” TMD FFs with the superscript “$(u)$” to emphasize this fact.
Here $\mu$ is the standard renormalization scale as above while $\nu$ is a
rapidity scale associated with the rapidity divergence Chiu:2011qc ;
Chiu:2012ir , and $\zeta$ is the so-called Collins-Soper scale Collins:2011zzd
. We also have a scale $\zeta_{J}$ to be defined below. The $\delta$ function
establishes a relationship between the hadron transverse momentum
$\bm{j}_{\perp}$, relative to the jet axis, and two other momenta: the
transverse momentum $\bm{\lambda}_{\perp}$ of soft radiation, and the hadron
transverse momentum $\bm{k}_{\perp}$ with respect to the fragmenting parton.
Notice that $\bm{\lambda}_{\perp}$ is multiplied by $z_{h}$ to adjust for the
dissimilarity between the fragmenting parton and the observed hadron. The
relationship among the three aforementioned transverse momenta is illustrated
in Fig. 4.

Figure 4: Illustration of the soft radiation (purple) and fragmentation
process (red). The initial fragmenting quark carries a momentum $p_{f}$ along
the jet axis $z_{J}$. It undergoes a soft radiation of $\bm{\lambda}_{\bot}$,
and then fragments into a hadron with momentum $p_{h}$. The momentum $p_{h}$
has a transverse component of $\bm{k}_{\bot}$ with respect to the quark after
soft radiation, or a transverse component of $\bm{j}_{\bot}$ with respect to
the jet axis.

As is common practice in TMD physics, we convert the aforementioned expression
from the transverse momentum space to the coordinate $b$-space with the
following transformation:

$\displaystyle\mathcal{D}_{1}^{h/c}\pqty{z,z_{h},\bm{j}_{\perp},p_{T}R,\mu,\zeta_{J}}=$
$\displaystyle\hat{H}^{U}_{c\to i}\pqty{z,p_{T}R,\mu}$
$\displaystyle\times\int\frac{\differential[2]{\bm{b}}}{(2\pi)^{2}}e^{i\bm{j}_{\perp}\cdot\bm{b}/z_{h}}\widetilde{D}_{1}^{h/i(u)}\pqty{z_{h},\bm{b},\mu,\zeta/\nu^{2}}\widetilde{S}_{i}(\bm{b},\mu,\nu\mathcal{R}/2)\,,$
(72)

where we have defined the Fourier transform for both
$\widetilde{D}_{1}^{h/i(u)}\pqty{z_{h},\bm{b},\mu,\zeta/\nu^{2}}$ and
$\widetilde{S}_{i}(\bm{b},\mu,\nu\mathcal{R}/2)$ with $i=q,\,g$ as:

$\displaystyle\widetilde{D}_{1}^{h/i(u)}\pqty{z_{h},\bm{b},\mu,\zeta/\nu^{2}}$
$\displaystyle=\frac{1}{z_{h}^{2}}\int\differential[2]{\bm{k}_{\perp}}e^{-i\bm{k}_{\perp}\cdot\bm{b}/z_{h}}D_{1}^{h/i(u)}\pqty{z_{h},\bm{k}_{\perp},\mu,\zeta/\nu^{2}}$
$\displaystyle=\frac{1}{z_{h}^{2}}\int\differential{k_{\perp}}k_{\perp}2\pi
J_{0}\pqty{\frac{bk_{\perp}}{z_{h}}}D_{1}^{h/i(u)}\pqty{z_{h},\bm{k}_{\perp},\mu,\zeta/\nu^{2}}\,,$
(73) $\displaystyle\widetilde{S}_{i}(\bm{b},\mu,\nu\mathcal{R}/2)$
$\displaystyle=\int\differential[2]{\bm{\lambda_{\perp}}}e^{-i\bm{\lambda}_{\perp}\cdot\bm{b}}S_{i}\pqty{\bm{\lambda}_{\perp},\mu,\nu\mathcal{R}/2}\,.$
(74)

The $\mu$\- and $\nu$-renormalization group equations for the “unsubstracted”
TMD FFs $D_{1}^{h/i(u)}$ are well known:

$\displaystyle\mu\frac{d}{d\mu}\ln\widetilde{D}_{1}^{h/i(u)}\pqty{z_{h},\bm{b},\mu,\zeta/\nu^{2}}$
$\displaystyle=\gamma_{\mu,i}^{D}(\mu,\zeta/\nu^{2}),$ (75)
$\displaystyle\nu\frac{d}{d\nu}\ln\widetilde{D}_{1}^{h/i(u)}\pqty{z_{h},\bm{b},\mu,\zeta/\nu^{2}}$
$\displaystyle=\gamma_{\nu,i}^{D}(b,\mu),$ (76)

where the leading order $\mu$\- and $\nu$-anomalous dimensions are given by

$\displaystyle\gamma_{\mu,q}^{D}(\mu,\zeta/\nu^{2})=$
$\displaystyle\frac{\alpha_{s}}{\pi}C_{F}\left(\ln\left(\frac{\nu^{2}}{\zeta}\right)+\frac{3}{2}\right),$
(77) $\displaystyle\gamma_{\mu,g}^{D}(\mu,\zeta/\nu^{2})=$
$\displaystyle\frac{\alpha_{s}}{\pi}C_{A}\left(\ln\left(\frac{\nu^{2}}{\zeta}\right)+\frac{\beta_{0}}{2C_{A}}\right),$
(78) $\displaystyle\gamma_{\nu,i}^{D}(b,\mu)=$
$\displaystyle\frac{\alpha_{s}}{\pi}C_{i}\ln\left(\frac{\mu^{2}}{\mu_{b}^{2}}\right)\,,$
(79)

with $C_{q}=C_{F}$, $C_{g}=C_{A}$, and $\mu_{b}\equiv 2e^{-\gamma_{E}}/b$.

On the other hand, the in-jet soft function up to the NLO is given by
Kang:2017glf ; Buffing:2018ggv ; Kang:2021ffh

$\widetilde{S}_{i}\pqty{\bm{b},\mu,\nu\mathcal{R}/2}=1+\frac{\alpha_{s}}{2\pi}C_{i}\bqty{-\frac{1}{2}\ln[2](\frac{\mu^{2}}{\mu_{b}^{2}})-\ln(\frac{\mu^{2}}{\mu_{b}^{2}})\ln(\frac{\nu^{2}\tan[2](\mathcal{R}/2)}{\mu^{2}})-\frac{\pi^{2}}{12}}\,.$
(80)

The corresponding $\mu$\- and $\nu$-renormalization group equations are given
by

$\displaystyle\mu\frac{d}{d\mu}\ln\widetilde{S}_{i}\pqty{\bm{b},\mu,\nu\mathcal{R}/2}=\,$
$\displaystyle\gamma_{\mu,i}^{S}\pqty{b,\mu,\nu\mathcal{R}/2}\,,$ (81)
$\displaystyle\nu\frac{d}{d\nu}\ln\widetilde{S}_{i}\pqty{\bm{b},\mu,\nu\mathcal{R}/2}=\,$
$\displaystyle\gamma_{\nu,i}^{S}\pqty{b,\mu}\,,$ (82)

where the anomalous dimensions are given by

$\displaystyle\gamma_{\mu,i}^{S}\pqty{b,\mu,\nu\mathcal{R}/2}$
$\displaystyle=-\frac{\alpha_{s}}{\pi}C_{i}\ln(\frac{\nu^{2}\tan[2](\mathcal{R}/2)}{\mu^{2}})\,,$
(83) $\displaystyle\gamma_{\nu,i}^{S}\pqty{b,\mu}$
$\displaystyle=-\frac{\alpha_{s}}{\pi}C_{i}\ln(\frac{\mu^{2}}{\mu_{b}^{2}})\,.$
(84)

It is instructive to compare this in-jet soft function $\widetilde{S}_{i}$
with the standard soft function
$\widetilde{S}_{n\bar{n},i}\pqty{\bm{b},\mu,\nu}$, which arises in the TMD
factorization for the semi-inclusive deep inelastic scattering (SIDIS) and
Drell-Yan production (involving quarks) or for the Higgs production in proton-
proton collisions (involving gluons), whose expression is given by

$\displaystyle\widetilde{S}_{n\bar{n},i}\pqty{\bm{b},\mu,\nu}=1+\frac{\alpha_{s}}{\pi}C_{i}\bqty{-\frac{1}{2}\ln[2](\frac{\mu^{2}}{\mu_{b}^{2}})-\ln(\frac{\mu^{2}}{\mu_{b}^{2}})\ln(\frac{\nu^{2}}{\mu^{2}})-\frac{\pi^{2}}{12}}\,.$
(85)

After replacing $\nu\to\nu\tan(\mathcal{R}/2)$, the in-jet soft function
$\widetilde{S}_{i}$ is exactly equal to $\sqrt{\widetilde{S}_{n\bar{n},i}}$ at
the NLO:

$\displaystyle\widetilde{S}_{i}\pqty{\bm{b},\mu,\nu\mathcal{R}/2}=\left.\sqrt{\widetilde{S}_{n\bar{n},i}\pqty{\bm{b},\mu,\nu}}\right|_{\nu\to\nu\tan(\mathcal{R}/2)}\,.$
(86)

Now since the standard “subtracted” TMD FFs are defined as a combination of
“unsubtracted” TMD FFs $\widetilde{D}_{1}^{h/i(u)}$ and $\widetilde{S}_{i}$
Boussarie:2023izj , we would define the “subtracted” in-jet TMD FFs as a
product of $\widetilde{D}_{1}^{h/i(u)}$ and $\widetilde{S}_{i}$ as follows:

$\displaystyle\widetilde{D}_{1}^{h/i}\pqty{z_{h},\bm{b},\mu,\zeta_{J}}\equiv\widetilde{D}_{1}^{h/i(u)}\pqty{z_{h},\bm{b},\mu,\zeta/\nu^{2}}\widetilde{S}_{i}(\bm{b},\mu,\nu\mathcal{R}/2)\,,$
(87)

where the rapidity divergences cancel out between $D_{1}^{h/i(u)}$ and
$\widetilde{S}_{i}$ due to the sign difference shown in Eqs. (79) and (84). At
the same time, we find that a slightly different Collins-Soper scale
$\zeta_{J}$ arises in the in-jet “subtracted” TMD FFs, which is related to the
Collins-Soper scale $\zeta$ for the standard TMD FFs,

$\displaystyle\sqrt{\zeta_{J}}\equiv\sqrt{\zeta}\,\mathcal{R}/2=p_{T}R\,,$
(88)

where we have used $\tan(\mathcal{R}/2)\approx\mathcal{R}/2$ for narrow jets
$\mathcal{R}\ll 1$. Note that in our regularization scheme, the natural
Collins-Soper scale for TMD FFs is
$\sqrt{\zeta}=\omega_{J}=2E_{J}=2p_{T}\cosh\eta$ and using
$\mathcal{R}=R/\cosh\eta$, we obtain the natural scale $\sqrt{\zeta_{J}}$ for
the unpolarized TMD FJFs, which is simply $p_{T}R$. With this newly defined
Collins-Soper scale $\zeta_{J}$, following Boussarie:2023izj ; Ebert:2019okf ,
we can thus convert the $\nu$-RG equations above as

$\displaystyle\frac{d}{d\ln\sqrt{\zeta_{J}}}\ln\widetilde{D}_{1}^{h/i}\pqty{z_{h},\bm{b},\mu,\zeta_{J}}=\widetilde{K}(b,\mu)\,,$
(89)

where $\widetilde{K}(b,\mu)=-\gamma_{\nu,i}^{D}(b,\mu)$ in Eq. (79) is the so-
called Collins-Soper evolution kernel. In the small-$b$ region where
$1/b\gg\Lambda_{\rm QCD}$, it can be computed perturbatively and the four-loop
expressions are available in Moult:2022xzt ; Duhr:2022yyp . Notice that
$\widetilde{K}(b,\mu)$ would become non-perturbative in the large-$b$ region,
see recent numerical computations in lattice QCD Shanahan:2021tst ;
LPC:2022ibr ; LatticePartonLPC:2023pdv ; Shu:2023cot ; Avkhadiev:2023poz . On
the other hand, the standard $\mu$-RG equation is given by

$\displaystyle\frac{d}{d\ln\mu}\ln\widetilde{D}_{1}^{h/i}\pqty{z_{h},\bm{b},\mu,\zeta_{J}}=\gamma_{\mu}^{i}[\alpha_{s}(\mu),\zeta_{J}/\mu^{2}]\,,$
(90)

where the anomalous dimension
$\gamma_{\mu}^{i}[\alpha_{s}(\mu),\zeta_{J}/\mu^{2}]$ are given by

$\displaystyle\gamma_{\mu}^{i}[\alpha_{s}(\mu),\zeta_{J}/\mu^{2}]=-\Gamma_{\rm
cusp}^{i}[\alpha_{s}(\mu)]\,\ln\left(\frac{\zeta_{J}}{\mu^{2}}\right)+\gamma_{\mu}^{i}[\alpha_{s}(\mu)]\,,$
(91)

where we have written the more general form Boussarie:2023izj  with
$\Gamma_{\rm cusp}^{i}$ and $\gamma_{\mu}^{i}$ the cusp and non-cusp anomalous
dimensions. They have the perturbative expansion $\Gamma_{\rm
cusp}^{i}[\alpha_{s}]=\sum_{n=1}\Gamma_{n-1}^{i}\left(\alpha_{s}/4\pi\right)^{n}$
and
$\gamma_{\mu}^{i}[\alpha_{s}]=\sum_{n=1}\gamma_{n}^{i}\left(\alpha_{s}/4\pi\right)^{n}$
and at the next-to-leading logarithmic (NLL) order, we have

$\displaystyle\Gamma_{0}^{q}=\,$ $\displaystyle
4\,C_{F},\qquad\Gamma_{1}^{q}=4\,C_{F}\left[\left(\frac{67}{9}-\frac{\pi^{2}}{3}\right)C_{A}-\frac{20}{9}T_{F}n_{f}\right],\qquad\gamma_{0}^{q}=6\,C_{F},$
(92) $\displaystyle\Gamma_{0,1}^{g}=\,$
$\displaystyle\frac{C_{A}}{C_{F}}\Gamma_{0,1}^{q},\qquad\gamma_{0}^{g}=2\beta_{0}\,.$
(93)

One can solve the $\zeta_{J}$\- and $\mu$-RG evolution equation to obtain the
“subtracted” in-jet TMD FFs

$\displaystyle\widetilde{D}_{1}^{h/i}\pqty{z_{h},\bm{b},\mu,\zeta_{J}}=\widetilde{D}_{1}^{h/i}\pqty{z_{h},\bm{b},\mu_{0},\zeta_{0}}$
$\displaystyle\exp\left[\int_{\mu_{0}}^{\mu}\frac{d\mu^{\prime}}{\mu^{\prime}}\gamma_{\mu}^{i}[\alpha_{s}(\mu),\zeta_{J}/\mu^{2}]\right]$
$\displaystyle\times$
$\displaystyle\exp\left[\widetilde{K}(b,\mu_{0})\ln\frac{\sqrt{\zeta_{J}}}{\sqrt{\zeta_{0}}}\right]\,.$
(94)

One typically evolves the in-jet TMD FFs $\widetilde{D}_{1}^{h/i}$ from the
initial scale $\mu_{0}=\sqrt{\zeta_{0}}=\mu_{b}$ to the final scale $\mu\sim
p_{T}$ of the jet and $\sqrt{\zeta_{J}}=p_{T}R$ as mentioned above. On the
other hand, at the initial scale $\mu_{0}=\sqrt{\zeta_{0}}=\mu_{b}$, one can
expand $\widetilde{D}_{1}^{h/i}\pqty{z_{h},\bm{b},\mu_{0},\zeta_{0}}$ in terms
of the corresponding collinear FFs when $\mu_{b}\gg\Lambda_{\rm QCD}$.

To generalize the above TMD factorization to the polarized TMD FJFs, let us
define the following operator $\mathscr{C}$:

$\displaystyle\mathscr{C}[\widetilde{D}^{h/i,(n)}]\equiv\int\frac{b^{n+1}\differential{b}}{2\pi
n!}\pqty{\frac{z_{h}^{2}M_{h}^{2}}{j_{\perp}}}^{n}J_{n}\pqty{\frac{j_{\perp}b}{z_{h}}}\widetilde{D}^{h/i,(n)}\pqty{z_{h},\bm{b},\mu,\zeta_{J}}\,,$
(95)

where $\widetilde{D}^{h/i,(n)}$ is $n$-th moment of the TMD FFs in the Fourier
$b$-space:

$\widetilde{D}^{h/i,(n)}\pqty{z_{h},\bm{b},\mu,\zeta_{J}}=\frac{1}{z_{h}^{2}}\frac{2\pi
n!}{\pqty{z_{h}^{2}M_{h}^{2}}^{n}}\int\differential{k_{\perp}}k_{\perp}\pqty{\frac{k_{\perp}}{b}}^{n}J_{n}\pqty{\frac{bk_{\perp}}{z_{h}}}D^{h/i}\pqty{z_{h},\bm{k}_{\perp},\mu,\zeta_{J}}\,.$
(96)

In addition, we suppress the superscript $(0)$ when $n=0$. One can easily
verify that with $n=0$ and $D=D_{1}$, the combination of Eqs. (73) and (87)
would lead to Eq. 96. It is important to realize that each in-jet TMD FF
$\widetilde{D}^{h/i,(n)}\pqty{z_{h},\bm{b},\mu,\zeta_{J}}$ would follow the
same Collins-Soper evolution equation as in Eq. 89, as well as the same
$\mu$-RG equation as in Eq. 90. They are identical to the standard TMD FFs
with the replacement $\zeta$ by $\zeta_{J}$. Thus, their evolved results would
be given exactly by Eq. 94, except one replaces
$\widetilde{D}_{1}^{h/i}\pqty{z_{h},\bm{b},\mu,\zeta_{J}}$ by the
corresponding TMD FFs
$\widetilde{D}^{h/i,(n)}\pqty{z_{h},\bm{b},\mu,\zeta_{J}}$ defined above and
given below for specific TMD FFs. Note that the evolved in-jet unpolarized TMD
FFs $\widetilde{D}_{1}^{h/i}\pqty{z_{h},\bm{b},\mu,\zeta_{J}}$ is given by Eq.
94, which depends on the TMD FFs
$\widetilde{D}_{1}^{h/i}\pqty{z_{h},\bm{b},\mu_{0},\zeta_{0}}$ at the initial
scale $\mu_{0}$ and $\zeta_{0}$. Often, one further matches the TMD FFs at the
initial scales onto collinear FFs. For example, the matching coefficients for
the unpolarized TMD FFs are known up to ${\rm N^{3}LO}$ Luo:2020epw ;
Ebert:2020qef . On the other hand, the matching coefficients for
$\widetilde{H}_{1}^{h/q}$ are known up to NNLO Gutierrez-Reyes:2018iod . We
can now write down the factorization for all the TMD FJFs as:

$\displaystyle\mathcal{D}_{1}^{h/c}\pqty{z,z_{h},\bm{j}_{\perp},p_{T}R,\mu,\zeta_{J}}$
$\displaystyle=\hat{H}_{c\to
i}^{U}(z,p_{T}R,\mu)\mathscr{C}\bqty{\widetilde{D}_{1}^{h/i}}\,,$ (97)
$\displaystyle\mathcal{D}_{1T}^{\perp,h/c}\pqty{z,z_{h},\bm{j}_{\perp},p_{T}R,\mu,\zeta_{J}}$
$\displaystyle=\hat{H}_{c\to
i}^{U}(z,p_{T}R,\mu)\mathscr{C}\bqty{\widetilde{D}_{1T}^{\perp,h/i,(1)}}\,,$
(98)
$\displaystyle\mathcal{G}_{1L}^{h/c}\pqty{z,z_{h},\bm{j}_{\perp},p_{T}R,\mu,\zeta_{J}}$
$\displaystyle=\hat{H}_{c\to
i}^{L}(z,p_{T}R,\mu)\mathscr{C}\bqty{\widetilde{G}_{1L}^{h/i}}\,,$ (99)
$\displaystyle\mathcal{G}_{1T}^{h/c}\pqty{z,z_{h},\bm{j}_{\perp},p_{T}R,\mu,\zeta_{J}}$
$\displaystyle=\hat{H}_{c\to
i}^{L}(z,p_{T}R,\mu)\mathscr{C}\bqty{\widetilde{G}_{1T}^{h/i,(1)}}\,,$ (100)
$\displaystyle\mathcal{H}_{1}^{h/c}\pqty{z,z_{h},\bm{j}_{\perp},p_{T}R,\mu,\zeta_{J}}$
$\displaystyle=\hat{H}_{c\to
i}^{T}(z,p_{T}R,\mu)\mathscr{C}\bqty{\widetilde{H}_{1}^{h/i}}\,,$ (101)
$\displaystyle\mathcal{H}_{1}^{\perp,h/c}\pqty{z,z_{h},\bm{j}_{\perp},p_{T}R,\mu,\zeta_{J}}$
$\displaystyle=\hat{H}_{c\to
i}^{T}(z,p_{T}R,\mu)\mathscr{C}\bqty{\widetilde{H}_{1}^{\perp,h/i,(1)}}\,,$
(102)
$\displaystyle\mathcal{H}_{1L}^{\perp,h/c}\pqty{z,z_{h},\bm{j}_{\perp},p_{T}R,\mu,\zeta_{J}}$
$\displaystyle=\hat{H}_{c\to
i}^{T}(z,p_{T}R,\mu)\mathscr{C}\bqty{\widetilde{H}_{1L}^{\perp,h/i,(1)}}\,,$
(103)
$\displaystyle\mathcal{H}_{1T}^{\perp,h/c}\pqty{z,z_{h},\bm{j}_{\perp},p_{T}R,\mu,\zeta_{J}}$
$\displaystyle=\hat{H}_{c\to
i}^{T}(z,p_{T}R,\mu)\mathscr{C}\bqty{\widetilde{H}_{1T}^{\perp,h/i,(2)}}\,.$
(104)

Here the superscripts $U$, $L$ and $T$ of $\hat{H}_{c\to i}$ represent
unpolarized, longitudinally polarized and transversely polarized hard matching
functions, respectively. The hard matching functions will be provided in the
next subsection. The above equations also show how various TMD FJFs are
matched onto their corresponding TMD FFs, with which the matching of the
scenarios listed in Table 1 can be performed.

|  | Quark polarization
---|---|---
|  | U | L | T
| U | $\mathcal{D}_{1}=\includegraphics[height=15.00002pt,valign={c}]{./figures/FJFs-table/1-1.pdf}$ |  | $\mathcal{H}_{1}^{\perp}=\includegraphics[height=15.00002pt,valign={c}]{./figures/FJFs-table/1-3-1.pdf}-\includegraphics[height=15.00002pt,valign={c}]{./figures/FJFs-table/1-3-2.pdf}$
| L |  | $\mathcal{G}_{1L}=\includegraphics[height=15.00002pt,valign={c}]{./figures/FJFs-table/2-2-1.pdf}-\includegraphics[height=15.00002pt,valign={c}]{./figures/FJFs-table/2-2-2.pdf}$ | $\mathcal{H}_{1L}^{\perp}=\includegraphics[height=15.00002pt,valign={c}]{./figures/FJFs-table/3-2-1.pdf}-\includegraphics[height=15.00002pt,valign={c}]{./figures/FJFs-table/3-2-2.pdf}$
|  |  |  | $\mathcal{H}_{1}=\includegraphics[height=15.00002pt,valign={c}]{./figures/FJFs-table/3-3-1.pdf}-\includegraphics[height=15.00002pt,valign={c}]{./figures/FJFs-table/3-3-2.pdf}$
Hadron polarization | T | $\mathcal{D}_{1T}^{\perp}=\includegraphics[height=15.00002pt,valign={c}]{./figures/FJFs-table/3-1-1.pdf}-\includegraphics[height=15.00002pt,valign={c}]{./figures/FJFs-table/3-1-2.pdf}$ | $\mathcal{G}_{1T}=\includegraphics[height=15.00002pt,valign={c}]{./figures/FJFs-table/3-2-1.pdf}-\includegraphics[height=15.00002pt,valign={c}]{./figures/FJFs-table/3-2-2.pdf}$ | $\mathcal{H}_{1T}^{\perp}=\includegraphics[height=15.00002pt,valign={c}]{./figures/FJFs-table/3-3-3.pdf}-\includegraphics[height=15.00002pt,valign={c}]{./figures/FJFs-table/3-3-4.pdf}$
Table 1:  Summary of the semi-inclusive TMD FJFs. The header row represents
the polarization of the quark (indicated by the blue line) that initiates the
jet, while the header column indicates the corresponding polarizations of
produced hadrons (indicated by the red arrow from the red dot). Shown here is
for quark TMD FJFs, with $U,\,L,\,T$ representing the unpolarized case, and
longitudinal and transverse polarization. For gluon TMD FJFs, one would
interpret $L,\,T,$ as circular and linear polarization Metz:2016swz .

####  3.1.2 Hard matching functions

The hard matching functions $\hat{H}_{c\to i}^{U,L,T}$ describe the out-of-jet
radiation during which an energetic parton $c$, generated in a hard scattering
event, undergoes a splitting into a parton $i$, which subsequently initiates a
jet with energy $\omega_{J}$ and radius $R$. The hard matching functions for
the unpolarized case $\hat{H}_{c\to i}^{U}$ are available previously
Kang:2016ehg ; Kang:2016mcy ; Kang:2017mda ; Cal:2019hjc . They describe the
splitting from an unpolarized initial parton $c$ to an unpolarized parton $i$,
irrespective of the polarization of the final-state hadron inside the jet.
Because of that, we would have the same hard matching functions for both
unpolarized TMD FJFs $\mathcal{D}_{1}^{h/c}$ and the polarized TMD FJFs
$\mathcal{D}_{1T}^{\perp,h/c}$, as indicated in Eqs. (97) and (98). We remind
the reader that $\mathcal{D}_{1T}^{\perp,h/c}$ stands for the situation where
an unpolarized parton $c$ splits into an unpolarized parton $i$ that fragments
into a transversely polarized hadron $h$. This arises from the correlation
between the hadron’s transverse momentum with respect to the jet axis and the
transverse spin of the hadron itself. For the same reasons, we have the same
longitudinally polarized hard matching functions $\hat{H}_{c\to i}^{L}$ for
the TMD FJFs $\mathcal{G}_{1L}^{h/c}$ and $\mathcal{G}_{1T}^{h/c}$, as
indicated in Eqs. (99) and (100). Likewise, we have the same transversely
polarized hard matching functions $\hat{H}_{c\to i}^{T}$ for the TMD FJFs
$\mathcal{H}_{1}^{h/c}$, $\mathcal{H}_{1}^{\perp,h/c}$,
$\mathcal{H}_{1L}^{\perp,h/c}$, and $\mathcal{H}_{1T}^{\perp,h/c}$, as
indicated in Eqs. 101, 102, 103 and 104. Looking at Table 1, the FJFs listed
in the same column possess identical hard matching functions, due to the fact
that the parton polarization is the same in the same column.

For completeness, we list the unpolarized results $\hat{H}_{c\to i}^{U}$ here.
We then provide in addition the results $\hat{H}_{c\to i}^{L,T}$ for the
polarized cases.

$\displaystyle\hat{H}_{q\to q^{\prime}}^{U}(z,p_{T}R,\mu)$
$\displaystyle=\delta_{qq^{\prime}}\delta(1-z)+\delta_{qq^{\prime}}\frac{\alpha_{s}}{2\pi}\bigg{[}C_{F}\delta(1-z)\pqty{-\frac{L^{2}}{2}-\frac{3}{2}L+\frac{\pi^{2}}{12}}$
$\displaystyle\quad+P_{qq}(z)L-2C_{F}(1+z^{2})\pqty{\frac{\ln(1-z)}{1-z}}_{+}-C_{F}(1-z)\bigg{]}\,,$
(111) $\displaystyle\hat{H}_{q\to g}^{U}(z,p_{T}R,\mu)$
$\displaystyle=\frac{\alpha_{s}}{2\pi}\bigg{[}\Big{(}L-2\ln(1-z)\Big{)}P_{gq}(z)-C_{F}z\bigg{]}\,,$
(112) $\displaystyle\hat{H}_{g\to g}^{U}(z,p_{T}R,\mu)$
$\displaystyle=\delta(1-z)+\frac{\alpha_{s}}{2\pi}\bigg{[}\delta(1-z)\pqty{-C_{A}\frac{L^{2}}{2}-\frac{\beta_{0}}{2}L+C_{A}\frac{\pi^{2}}{12}}$
$\displaystyle\quad+P_{gg}(z)L-\frac{4C_{A}\pqty{1-z+z^{2}}^{2}}{z}\pqty{\frac{\ln(1-z)}{1-z}}_{+}\bigg{]}\,,$
(113) $\displaystyle\hat{H}_{g\to q}^{U}(z,p_{T}R,\mu)$
$\displaystyle=\frac{\alpha_{s}}{2\pi}\bigg{[}\Big{(}L-2\ln(1-z)\Big{)}P_{qg}(z)-2T_{F}z(1-z)\bigg{]}\,,$
(114)

where the leading splitting kernels are given by:

$\displaystyle P_{qq}(z)$
$\displaystyle=C_{F}\bqty{\frac{1+z^{2}}{(1-z)_{+}}+\frac{3}{2}\delta(1-z)}\,,$
(115) $\displaystyle P_{gq}(z)$
$\displaystyle=C_{F}\frac{1+\pqty{1-z}^{2}}{z}\,,$ (116) $\displaystyle
P_{qg}(z)$ $\displaystyle=T_{F}\bqty{z^{2}+\pqty{1-z}^{2}}\,,$ (117)
$\displaystyle P_{gg}(z)$
$\displaystyle=2C_{A}\bqty{\frac{z}{(1-z)_{+}}+\frac{1-z}{z}+z\pqty{1-z}}+\frac{\beta_{0}}{2}\delta(1-z)\,.$
(118)

The hard matching functions for the longitudinally polarized parton case are
given here for the first time:

$\displaystyle\hat{H}^{L}_{q\to q^{\prime}}(z,p_{T}R,\mu)$
$\displaystyle=\delta_{qq^{\prime}}\delta(1-z)+\delta_{qq^{\prime}}\frac{\alpha_{s}}{2\pi}\bigg{[}C_{F}\delta(1-z)\pqty{-\frac{L^{2}}{2}-\frac{3}{2}L+\frac{\pi^{2}}{12}}$
$\displaystyle\quad+\Delta
P_{qq}(z)L-2C_{F}(1+z^{2})\pqty{\frac{\ln(1-z)}{1-z}}_{+}-C_{F}(1-z)\bigg{]}\,,$
(119) $\displaystyle\hat{H}^{L}_{q\to g}(z,p_{T}R,\mu)$
$\displaystyle=\frac{\alpha_{s}}{2\pi}\bigg{[}\Big{(}L-2\ln(1-z)\Big{)}\Delta
P_{gq}(z)+2C_{F}(1-z)\bigg{]}\,,$ (120) $\displaystyle\hat{H}^{L}_{g\to
g}(z,p_{T}R,\mu)$
$\displaystyle=\delta(1-z)+\frac{\alpha_{s}}{2\pi}\bigg{[}\delta(1-z)\pqty{-C_{A}\frac{L^{2}}{2}-\frac{\beta_{0}}{2}L+C_{A}\frac{\pi^{2}}{12}}$
$\displaystyle\quad+\Delta
P_{gg}(z)L+4C_{A}(1-z)-4C_{A}(2(1-z)^{2}+z)\pqty{\frac{\ln(1-z)}{1-z}}_{+}\bigg{]}\,,$
(121) $\displaystyle\hat{H}^{L}_{g\to q}(z,p_{T}R,\mu)$
$\displaystyle=\frac{\alpha_{s}}{2\pi}\bigg{[}\Big{(}L-2\ln(1-z)\Big{)}\Delta
P_{qg}(z)-2T_{F}(1-z)\bigg{]}\,,$ (122)

Finally, for the transversely polarized case, we only have contributions from
the $q\to q$ channel, for the same reason mentioned before, i.e. gluon
transversity FF does not exist for spin-1/2 hadron. The transversely polarized
hard matching function is then given by

$\displaystyle\hat{H}^{T}_{q\to q^{\prime}}(z,p_{T}R,\mu)$
$\displaystyle=\delta_{qq^{\prime}}\delta(1-z)+\delta_{qq^{\prime}}\frac{\alpha_{s}}{2\pi}\bigg{\\{}\Delta_{T}P_{qq}(z)L+C_{F}\bigg{[}-4z\pqty{\frac{\ln(1-z)}{1-z}}_{+}$
$\displaystyle\quad+\pqty{-\frac{3}{2}L-\frac{L^{2}}{2}+\frac{\pi^{2}}{12}}\delta(1-z)\bigg{]}\bigg{\\}}\,.$
(123)

Note that the leading order splitting kernels for the polarized cases are
given in Eqs. 25, 26, 27, 28 and 29.

The hard matching functions $\hat{H}_{i\to j}^{U,L,T}\pqty{z,p_{T}R,\mu}$
follow the RG equations below:

$\displaystyle\mu\frac{\differential}{\differential{\mu}}\hat{H}_{i\to
j}^{U}\pqty{z,p_{T}R,\mu}$
$\displaystyle=\sum_{k}\int_{z}^{1}\frac{\differential{z^{\prime}}}{z^{\prime}}\gamma_{ik}^{U}\pqty{\frac{z}{z^{\prime}},p_{T}R,\mu}\hat{H}_{k\to
j}^{U}\pqty{z^{\prime},p_{T}R,\mu}\,,$ (124)
$\displaystyle\mu\frac{\differential}{\differential{\mu}}\hat{H}_{i\to
j}^{L}\pqty{z,p_{T}R,\mu}$
$\displaystyle=\sum_{k}\int_{z}^{1}\frac{\differential{z^{\prime}}}{z^{\prime}}\gamma_{ik}^{L}\pqty{\frac{z}{z^{\prime}},p_{T}R,\mu}\hat{H}_{k\to
j}^{L}\pqty{z^{\prime},p_{T}R,\mu}\,,$ (125)
$\displaystyle\mu\frac{\differential}{\differential{\mu}}\hat{H}_{q\to
q^{\prime}}^{T}\pqty{z,p_{T}R,\mu}$
$\displaystyle=\int_{z}^{1}\frac{\differential{z^{\prime}}}{z^{\prime}}\gamma_{qq}^{T}\pqty{\frac{z}{z^{\prime}},p_{T}R,\mu}\hat{H}_{q\to
q^{\prime}}^{T}\pqty{z^{\prime},p_{T}R,\mu}\,,$ (126)

where indices $i$, $j$ and $k$ all represent partons $q$ or $g$. The anomalous
dimensions $\gamma_{ij}^{U}\pqty{z,p_{T}R,\mu}$ for the unpolarized hard
matching functions are given by:

$\displaystyle\gamma_{qq}^{U}\pqty{z,p_{T}R,\mu}$
$\displaystyle=\frac{\alpha_{s}}{\pi}\pqty{P_{qq}\pqty{z}-C_{F}L\delta\pqty{1-z}-\frac{3C_{F}}{2}\delta\pqty{1-z}}\,,$
(127) $\displaystyle\gamma_{qg}^{U}\pqty{z,p_{T}R,\mu}$
$\displaystyle=\frac{\alpha_{s}}{\pi}P_{gq}\pqty{z}\,,$ (128)
$\displaystyle\gamma_{gg}^{U}\pqty{z,p_{T}R,\mu}$
$\displaystyle=\frac{\alpha_{s}}{\pi}\pqty{P_{gg}\pqty{z}-C_{A}L\delta\pqty{1-z}-\frac{\beta_{0}}{2}\delta\pqty{1-z}}\,,$
(129) $\displaystyle\gamma_{gq}^{U}\pqty{z,p_{T}R,\mu}$
$\displaystyle=\frac{\alpha_{s}}{\pi}P_{qg}\pqty{z}\,.$ (130)

On the other hand, the anomalous dimensions for the polarized cases,
$\gamma_{ij}^{L\pqty{T}}\pqty{z,p_{T}R,\mu}$ are given by:

$\displaystyle\gamma_{qq}^{L}\pqty{z,p_{T}R,\mu}$
$\displaystyle=\frac{\alpha_{s}}{\pi}\pqty{\Delta
P_{qq}\pqty{z}-C_{F}L\delta\pqty{1-z}-\frac{3C_{F}}{2}\delta\pqty{1-z}}\,,$
(131) $\displaystyle\gamma_{qg}^{L}\pqty{z,p_{T}R,\mu}$
$\displaystyle=\frac{\alpha_{s}}{\pi}\Delta P_{gq}\pqty{z}\,,$ (132)
$\displaystyle\gamma_{gg}^{L}\pqty{z,p_{T}R,\mu}$
$\displaystyle=\frac{\alpha_{s}}{\pi}\pqty{\Delta
P_{gg}\pqty{z}-C_{A}L\delta\pqty{1-z}-\frac{\beta_{0}}{2}\delta\pqty{1-z}}\,,$
(133) $\displaystyle\gamma_{gq}^{L}\pqty{z,p_{T}R,\mu}$
$\displaystyle=\frac{\alpha_{s}}{\pi}\Delta P_{qg}\pqty{z}\,,$ (134)
$\displaystyle\gamma_{qq}^{T}\pqty{z,p_{T}R,\mu}$
$\displaystyle=\frac{\alpha_{s}}{\pi}\pqty{\Delta_{T}P_{qq}\pqty{z}-C_{F}L\delta\pqty{1-z}-\frac{3C_{F}}{2}\delta\pqty{1-z}}\,.$
(135)

Obviously, the natural scale for the hard matching functions is $\mu\sim
p_{T}R$ as indicated in the logarithm $L$ in the perturbative results.
Therefore, we can resum the large logarithms of jet radius $\ln R$ by evolving
the hard matching functions from scale $\mu\sim p_{T}R$ to the hard scattering
scale $\mu\sim p_{T}$ with the RG equations. We notice that similar to the
unpolarized case studied in Kang:2017glf :

$\gamma_{ij}^{U}\pqty{z,p_{T}R,\mu}=\delta_{ij}\delta(1-z)\Gamma_{i}\pqty{p_{T}R,\mu}+\frac{\alpha_{s}}{\pi}P_{ji}(z)\,,$
(136)

where $\Gamma_{i}$ are given by:

$\Gamma_{q}\pqty{p_{T}R,\mu}=\frac{\alpha_{s}}{\pi}C_{F}\pqty{-L-\frac{3}{2}}\,,\quad\Gamma_{g}\pqty{p_{T}R,\mu}=\frac{\alpha_{s}}{\pi}C_{A}\pqty{-L-\frac{\beta_{0}}{2C_{A}}}\,.$
(137)

Following the same exercise, we can express the anomalous dimensions
$\gamma_{ij}^{L(T)}\pqty{z,p_{T}R,\mu}$ in Eqs. 131, 132, 133, 134 and 135
into a general form:

$\gamma_{ij}^{L(T)}\pqty{z,p_{T}R,\mu}=\delta_{ij}\delta(1-z)\Gamma_{i}\pqty{p_{T}R,\mu}+\frac{\alpha_{s}}{\pi}\Delta_{(T)}P_{ji}(z)\,,$
(138)

where the functions $\Gamma_{i}\pqty{p_{T}R,\mu}$ in the first term are
independent of polarization and are therefore the same as the unpolarized
$\Gamma_{i}$ in Eq. 137.

###  3.2 TMD FJFs in exclusive jet production

For the exclusive jet production, exclusive TMD FJFs
$\widetilde{\mathscr{D}}_{1}^{h/i}\pqty{z_{h},\bm{j}_{\perp},p_{T}R,\mu,\zeta_{J}}$
arises in the factorization formalism. Since one cannot have out-of-jet hard
radiation as explained in Section 2.2, the factorization formalism in the
$j_{\perp}\ll p_{T}R$ region would be given by

$\displaystyle\widetilde{\mathscr{D}}_{1}^{h/i}\pqty{z_{h},\bm{j}_{\perp},p_{T}R,\mu,\zeta_{J}}=$
$\displaystyle\int\differential[2]{\bm{k}_{\perp}}\differential[2]{\bm{\lambda}_{\perp}}\delta^{2}\pqty{z_{h}\bm{\lambda}_{\perp}+\bm{k}_{\perp}-\bm{j}_{\perp}}D_{1}^{h/i(u)}\pqty{z_{h},\bm{k}_{\perp},\mu,\zeta/\nu^{2}}S_{i}\pqty{\bm{\lambda}_{\perp},\mu,\nu\mathcal{R}/2}$
$\displaystyle=$
$\displaystyle\int\frac{\differential[2]{\bm{b}}}{(2\pi)^{2}}e^{i\bm{j}_{\perp}\cdot\bm{b}/z_{h}}\widetilde{D}_{1}^{h/i}\pqty{z_{h},\bm{b},\mu,\zeta_{J}}\,,$
(139)

where we have used Eq. 87. Again for exclusive jet production, we no longer
have the dependence on $z$ as in the semi-inclusive TMD FJFs. Following the
same procedure as before, we can generalize this to the polarized exclusive
TMD FJFs. With the operator $\mathscr{C}$ defined in Eq. 95, the unpolarized
exclusive TMD FJFs are given by:

$\widetilde{\mathscr{D}}_{1}^{h/i}\pqty{z_{h},p_{T}{R},\bm{j}_{\perp},\mu,\zeta_{J}}=\mathscr{C}\bqty{\widetilde{D}_{1}^{h/i}}\,,$
(140)

Similarly for exclusive TMD FJFs with different polarizations, we have:

$\displaystyle\widetilde{\mathscr{D}}_{1T}^{\perp,h/i}\pqty{z_{h},\bm{j}_{\perp},p_{T}R,\mu,\zeta_{J}}$
$\displaystyle=\mathscr{C}\bqty{\widetilde{D}_{1T}^{\perp,h/i,(1)}}\,,$ (141)
$\displaystyle\widetilde{\mathscr{G}}_{1L}^{h/i}\pqty{z_{h},\bm{j}_{\perp},p_{T}R,\mu,\zeta_{J}}$
$\displaystyle=\mathscr{C}\bqty{\widetilde{G}_{1L}^{h/i}}\,,$ (142)
$\displaystyle\widetilde{\mathscr{G}}_{1T}^{\perp,h/i}\pqty{z_{h},\bm{j}_{\perp},p_{T}R,\mu,\zeta_{J}}$
$\displaystyle=\mathscr{C}\bqty{\widetilde{G}_{1T}^{\perp,h/i,(1)}}\,,$ (143)
$\displaystyle\widetilde{\mathscr{H}}_{1}^{h/i}\pqty{z_{h},\bm{j}_{\perp},p_{T}R,\mu,\zeta_{J}}$
$\displaystyle=\mathscr{C}\bqty{\widetilde{H}_{1}^{h/i}}\,,$ (144)
$\displaystyle\widetilde{\mathscr{H}}_{1}^{\perp,h/i}\pqty{z_{h},\bm{j}_{\perp},p_{T}R,\mu,\zeta_{J}}$
$\displaystyle=\mathscr{C}\bqty{\widetilde{H}_{1}^{\perp,h/i,(1)}}\,,$ (145)
$\displaystyle\widetilde{\mathscr{H}}_{1L}^{\perp,h/i}\pqty{z_{h},\bm{j}_{\perp},p_{T}R,\mu,\zeta_{J}}$
$\displaystyle=\mathscr{C}\bqty{\widetilde{H}_{1L}^{\perp,h/i,(1)}}\,,$ (146)
$\displaystyle\widetilde{\mathscr{H}}_{1T}^{\perp,h/i}\pqty{z_{h},\bm{j}_{\perp},p_{T}R,\mu,\zeta_{J}}$
$\displaystyle=\mathscr{C}\bqty{\widetilde{H}_{1T}^{\perp,h/i,(2)}}\,.$ (147)

##  4 Phenomenology

There has been growing phenomenological work for studying hadron distribution
inside the jet, in particular in connection with the 3D imaging of the
hadrons. For example, for single inclusive jet production in proton-proton
collisions as well as electron-proton scatterings, unpolarized and polarized
collinear hadron distribution inside the jet have been studied
Kaufmann:2015hma ; Kang:2016ehg ; Kang:2020xyq ; Zhao:2023lav . The TMD
distribution of hadrons inside the jet produced in transversely polarized
proton-proton collisions is sensitive to the Collins TMD FFs and has been
studied in Kang:2017btw . For exclusive jet production, such as back-to-back
$Z$+jet production in proton-proton collisions and back-to-back lepton+jet
production in DIS, TMD hadron distribution inside jets has been studied
Kang:2019ahe ; Kang:2021kpt ; Kang:2021ffh ; Arratia:2022oxd ; Arratia:2020nxw
.

Before we present the phenomenology, we first provide more details on TMD FFs.
In previous sections, we have discussed the evolution of FJFs in the
perturbative region, i.e., $1/b\gg\Lambda_{\mathrm{QCD}}$. However, to do
calculations for phenomenology, we must take care of the non-perturbative
evolution of TMDs at the large-$b$ region. In order to do this, we adopt the
$b_{*}$-prescription Collins:1984kg . Alternative approaches can be found in
Kulesza:2002rh ; Qiu:2000hf ; Catani:2015vma ; Ebert:2016gcn ; Monni:2016ktx .
The $b_{*}$ is defined as:

$b_{*}\equiv\frac{b}{\sqrt{1+b^{2}/b_{\max}^{2}}}\,,$ (148)

where we choose $b_{\max}=1.5~{}\mathrm{GeV}^{-1}$. With this definition, the
magnitude of $b_{*}$ approaches $b$ when $b\ll b_{\max}$, and approaches
$b_{\max}$ in the large-$b$ region. For the perturbative contribution, we work
at the NLL order, where we keep $\Gamma_{0,1}^{i}$ and $\gamma_{0}^{i}$ in
Eqs. 94 and 91. For the non-perturbative contribution, we include the
following non-perturbative Sudakov factor Sun:2014dqm ; Echevarria:2020hpy :

$S_{\mathrm{NP}}^{q}(b,Q_{0},\sqrt{\zeta_{J}})=\frac{g_{2}}{2}\ln(\frac{b}{b_{*}})\ln(\frac{\sqrt{\zeta_{J}}}{Q_{0}})+\frac{g_{h}}{z_{h}^{2}}b^{2}\,,$
(149)

where $Q_{0}^{2}=2.4~{}\mathrm{GeV}^{2}$, $g_{2}=0.84$ and $g_{h}=0.042$, and
$\sqrt{\zeta_{J}}=p_{T}R$ following Eq. 88. For gluon TMD FFs, we use
Kang:2017glf

$S_{\mathrm{NP}}^{g}(b,Q_{0},\sqrt{\zeta_{J}})=\frac{C_{A}}{C_{F}}\frac{g_{2}}{2}\ln(\frac{b}{b_{*}})\ln(\frac{\sqrt{\zeta_{J}}}{Q_{0}})+\frac{g_{h}}{z_{h}^{2}}b^{2}\,.$
(150)

Now that we have properly addressed the non-perturbative evolution, we can
proceed with predicting relevant observables. In this section, we provide two
additional phenomenological examples. For the case of single inclusive jet
production, we compute the transverse momentum distribution for hadrons inside
the jet at RHIC kinematics, where a transversely polarized proton collides
with an unpolarized proton, producing a jet with a transversely polarized
$\Lambda$-baryon inside. This process will be sensitive to both transversity
distributions $h_{1}$ and transversity TMD FJFs $\mathcal{H}_{1}$. For the
exclusive jet production, we study back-to-back lepton jet production in
lepton-proton collisions, where the incoming proton is transversely polarized
and the final hadron inside the jet is longitudinally polarized. This
observable is sensitive to the “worm-gear” TMD PDFs $g_{1T}$ and
longitudinally polarized TMD FJFs $\mathcal{G}_{1L}$.

###  4.1 Transverse spin transfer to $\Lambda$ in jet in polarized $pp$
collisions

For semi-inclusive process, we consider the $pp$ collision in its center-of-
mass frame:

$p\pqty{P_{A},S_{A}}+p\pqty{P_{B}}\to\bqty{\mathrm{jet}(p_{J})h\pqty{z_{h},\bm{j}_{\perp},S_{h}}}+X\,,$
(151)

where $P_{A}$ and $P_{B}$ are the momenta of the incident protons, $S_{A}$ is
the polarization of the proton $A$. They are defined as:

$\displaystyle P_{A}$ $\displaystyle=\frac{\sqrt{s}}{2}\pqty{1,0,0,1},\quad
P_{B}=\frac{\sqrt{s}}{2}\pqty{1,0,0,-1}\,,$ (152) $\displaystyle S_{A}$
$\displaystyle=\bqty{-\lambda_{A}\frac{M_{A}}{p_{A}^{-}},\lambda_{A}\frac{p_{A}^{-}}{M_{A}},\bm{S}_{T}}\,,$
(153)

where $s$ is the squared proton center of mass energy. Notice that $S_{A}$ is
written in light-cone coordinate defined in Eq. 3, and $\lambda_{A}$ is the
helicity of the incoming proton with mass $M_{A}$, $p_{A}^{-}$ is the large
light-cone momentum of the proton $A$, $\phi_{S}$ is the azimuthal angle
between the transverse spin $\bm{S}_{T}$ of the initial proton and the
reaction plane, with the reaction plane defined by the incoming beam direction
and jet axis direction (painted yellow in Fig. 5). For the jet production,
$p_{J}$ is the jet momentum, with $p_{T}$ being the magnitude of its
transverse component:

$p_{J}=\pqty{p_{T}\cosh{\eta},p_{T},0,p_{T}\sinh{\eta}}\,.$ (154)

Note here the $y$-component of $p_{J}$ is zero because we put the jet within
the reaction plane ($xz$-plane). The hadron inside the jet has a transverse
momentum $\bm{j}_{\perp}$ with respect to the jet axis, and collinear momentum
fraction $z_{h}$ of the jet momentum. Together with the hadron’s transverse
spin $\bm{S}_{h\bot}$, they are given by:

$\displaystyle\bm{j}_{\perp}$
$\displaystyle=\absolutevalue{\bm{j}_{\perp}}\pqty{\cos{\hat{\phi}_{h}}\cos{\theta_{J}},\sin{\hat{\phi}_{h}},-\cos{\hat{\phi}_{h}}\sin{\theta_{J}}}\,,$
(155) $\displaystyle\bm{S}_{h\bot}$
$\displaystyle=\absolutevalue{\bm{S}_{h\bot}}\pqty{\cos{\hat{\phi}_{S_{h}}}\cos{\theta_{J}},\sin{\hat{\phi}_{S_{h}}},-\cos{\hat{\phi}_{S_{h}}}\sin{\theta_{J}}}\,,$
(156)

where $\theta_{J}$ is the angle between jet axis and $z$-axis,
$\hat{\phi}_{h}$ is the azimuthal angle between the momentum $p_{h}$ of the
produced hadron and the reaction plane, and $\hat{\phi}_{S_{h}}$ is the
azimuthal angle between the transverse spin $\bm{S}_{h\bot}$ and the reaction
plane. The “hat” that decorates $\hat{\phi}_{h}$ and $\hat{\phi}_{S_{h}}$
indicates that these angles are measured in the jet coordinates.

Figure 5: Illustration for the distribution of hadrons inside jets in the
collisions of a polarized proton and an unpolarized proton or lepton. The
reaction plane is painted yellow. For detailed illustration of jet cone and
hadron kinematics, please see Fig. 3.

For single inclusive jet production, the most general azimuthal
dependence444Here we consider the situation where only one of the incoming
hadrons could be polarized and the final-state hadron can be polarized. for
the hadron $z_{h}$ and $\bm{j}_{\perp}$ distributions inside the jet is given
by Kang:2020xyq :

$\displaystyle\frac{\differential{\sigma^{p\pqty{S_{A}}+p\to\mathrm{jet}h\pqty{S_{h}}+X}}}{\differential{\eta}\differential[2]{\bm{p}_{T}}\differential{z_{h}}\differential[2]{\bm{j}_{\perp}}}$
$\displaystyle=F_{UU,U}+\absolutevalue{\bm{S}_{T}}\sin(\phi_{S}-\hat{\phi}_{h})F^{\sin(\phi_{S}-\hat{\phi}_{h})}_{TU,U}$
$\displaystyle\quad+\lambda_{h}\bqty{\lambda_{A}F_{LU,L}+\absolutevalue{\bm{S}_{T}}\cos(\phi_{S}-\hat{\phi}_{h})F^{\cos(\phi_{S}-\hat{\phi}_{h})}_{TU,L}}$
$\displaystyle\quad+\absolutevalue{\bm{S}_{h_{T}}}\bigg{[}\sin(\hat{\phi}_{h}-\hat{\phi}_{S_{h}})F^{\sin(\hat{\phi}_{h}-\hat{\phi}_{S_{h}})}_{UU,T}+\lambda_{A}\cos(\hat{\phi}_{h}-\hat{\phi}_{S_{h}})F^{\cos(\hat{\phi}_{h}-\hat{\phi}_{S_{h}})}_{LU,T}$
$\displaystyle\qquad\qquad\quad+\absolutevalue{\bm{S}_{T}}\bigg{(}\cos(\phi_{S}-\hat{\phi}_{S_{h}})F^{\cos(\phi_{S}-\hat{\phi}_{S_{h}})}_{TU,T}$
$\displaystyle\qquad\qquad\qquad\quad+\cos(2\hat{\phi}_{h}-\phi_{S}-\hat{\phi}_{S_{h}})F^{\cos(2\hat{\phi}_{h}-\phi_{S}-\hat{\phi}_{S_{h}})}_{TU,T}\bigg{)}\bigg{]}\,,$
(157)

where $F_{AB,C}$ denotes the spin-dependent structure functions, with $A$, $B$
and $C$ indicating the polarization of incoming proton $A$, incoming proton
$B$ (or electron), and the fragmented hadron inside the jet. $\lambda_{A}$ and
$\absolutevalue{\bm{S}_{\perp}}$ are the longitudinal and transverse spin of
the initial polarized proton $A$, while $\lambda_{h}$ and
$\absolutevalue{\bm{S}_{h_{T}}}$ are the longitudinal and transverse
polarization of the hadron inside jet measured in the fragmenting parton
helicity frame.

The structure function $F_{UU,U}(z_{h},j_{\perp})$ is defined by:

$\displaystyle F_{UU,U}(z_{h},j_{\perp})$
$\displaystyle=\frac{\alpha_{s}^{2}}{s}\sum_{a,b,c}\int_{x_{1}^{\min}}^{1}\frac{\differential{x_{1}}}{x_{1}}f_{1}^{a/A}(x_{1},\mu)\int_{x_{2}^{\min}}^{1}\frac{\differential{x_{2}}}{x_{2}}f_{2}^{b/B}(x_{2},\mu)$
$\displaystyle\quad\times\int_{z^{\min}}^{1}\frac{\differential{z}}{z^{2}}\hat{\sigma}_{ab}^{c}\pqty{\hat{s},\hat{p}_{T},\hat{\eta},\mu}\mathcal{D}_{1}^{h/c}\pqty{z,z_{h},\bm{j}_{\perp},p_{T}R,\mu,\zeta_{J}}$
$\displaystyle\equiv\mathcal{C}\bqty{ff\mathcal{D}_{1}\hat{\sigma}}\,,$ (158)

where $f_{1}^{a/A}(x_{1},\mu)$ and $f_{2}^{b/B}(x_{2},\mu)$ are the collinear
unpolarized PDFs in the proton with corresponding momentum fractions $x_{1}$
and $x_{2}$, and $\hat{\sigma}_{ab}^{c}$ is the hard function for unpolarized
parton $a$, $b$ to unpolarized parton $c$. On the other hand,
$\mathcal{D}_{1}^{h/c}\pqty{z,z_{h},\bm{j}_{\perp},p_{T}R,\mu,\zeta_{J}}$ are
the unpolarized TMD FJFs in Section 3.1.1 and we have studied extensively in
the previous section. The lower integration limits $x_{1}^{\min}$,
$x_{2}^{\min}$ and $z^{\min}$ can be found in Kaufmann:2015hma ; Kang:2016ehg
. The variables $\hat{s}$, $\hat{p}_{T}$ and $\hat{\eta}$ are the squared
parton center of mass energy, transverse momentum and rapidity of parton $c$,
respectively, and are related to their hadron analogues as:

$\hat{s}=x_{1}x_{2}s\,,\quad\hat{p}_{T}=p_{T}/z\,,\quad\hat{\eta}=\eta-\frac{1}{2}\ln(\frac{x_{1}}{x_{2}})\,.$
(159)

In Eq. 158, we also defined the notation
$\mathcal{C}\bqty{ff\mathcal{D}_{1}\hat{\sigma}}$, where parton flavors are
summed for PDFs and FJFs along with their corresponding unpolarized hard
functions. Using this notation, we can write down the expressions for all the
polarized structure functions:

$\displaystyle F^{\sin(\phi_{S}-\hat{\phi}_{h})}_{TU,U}(z_{h},j_{\perp})$
$\displaystyle=\mathcal{C}\bqty{\frac{j_{\perp}}{z_{h}M_{h}}h_{1}f_{1}\mathcal{H}_{1}^{\perp}\Delta_{T}\hat{\sigma}}\,,$
(160) $\displaystyle F_{LU,L}(z_{h},j_{\perp})$
$\displaystyle=\mathcal{C}\Big{[}g_{1L}f_{1}\mathcal{G}_{1L}\Delta\hat{\sigma}\Big{]}\,,$
(161) $\displaystyle
F^{\cos(\phi_{S}-\hat{\phi}_{h})}_{TU,L}(z_{h},j_{\perp})$
$\displaystyle=-\mathcal{C}\bqty{\frac{j_{\perp}}{z_{h}M_{h}}h_{1}f_{1}\mathcal{H}_{1L}^{\perp}\Delta_{T}\hat{\sigma}}\,,$
(162) $\displaystyle
F^{\sin(\hat{\phi}_{h}-\hat{\phi}_{S_{h}})}_{UU,T}(z_{h},j_{\perp})$
$\displaystyle=-\mathcal{C}\bqty{\frac{j_{\perp}}{z_{h}M_{h}}f_{1}f_{1}\mathcal{D}_{1T}^{\perp}\hat{\sigma}}\,,$
(163) $\displaystyle
F^{\cos(\hat{\phi}_{h}-\hat{\phi}_{S_{h}})}_{LU,T}(z_{h},j_{\perp})$
$\displaystyle=-\mathcal{C}\bqty{\frac{j_{\perp}}{z_{h}M_{h}}g_{1L}f_{1}\mathcal{G}_{1T}\Delta\hat{\sigma}}\,,$
(164) $\displaystyle
F^{\cos(\phi_{S}-\hat{\phi}_{S_{h}})}_{TU,T}(z_{h},j_{\perp})$
$\displaystyle=\mathcal{C}\Big{[}h_{1}f_{1}\mathcal{H}_{1}\Delta_{T}\hat{\sigma}\Big{]}\,,$
(165) $\displaystyle
F^{\cos(2\hat{\phi}_{h}-\phi_{S}-\hat{\phi}_{S_{h}})}_{TU,T}(z_{h},j_{\perp})$
$\displaystyle=-\mathcal{C}\bqty{\frac{j^{2}_{\perp}}{2z_{h}^{2}M_{h}^{2}}h_{1}f_{1}\mathcal{H}_{1T}^{\perp}\Delta_{T}\hat{\sigma}}\,.$
(166)

The corresponding polarizations for hard functions $\hat{\sigma}_{ab}^{c}$,
$\Delta\hat{\sigma}_{ab}^{c}$ and $\Delta_{T}\hat{\sigma}_{ab}^{c}$ are given
in Table 2. The expressions for $F_{UU,U}$ and
$F^{\sin(\hat{\phi}_{h}-\hat{\phi}_{S_{h}})}_{UU,T}$ were given in the
previous publication Kang:2020xyq  while Kang:2017btw  performed a
phenomenological study for $F^{\sin(\phi_{S}-\hat{\phi}_{h})}_{TU,U}$, that is
related to the Collins TMD FFs. The detailed expressions for all the other
structure functions are written down here for the first time.

$c$ $a$ | U | L | T
---|---|---|---
U | $\hat{\sigma}_{ab}^{c}$ |  |
L |  | $\Delta\hat{\sigma}_{ab}^{c}$ |
T |  |  | $\Delta_{T}\hat{\sigma}_{ab}^{c}$
Table 2: Hard functions for parton $a$, $b$ to parton $c$. The header row
represents the polarization of the parton $a$ while the header column
indicates the polarizations of the parton $c$. Parton $b$ is always
unpolarized.

Relevant measurements have been performed for the $\pi^{\pm}$-in-jet
production in the case of Collins FFs STAR:2022hqg . More relevantly, a
measurement of transverse spin transfer to $\Lambda/\overline{\Lambda}$
hyperons in polarized $pp$ collisions (no jet is involved) was performed by
the STAR collaboration at RHIC STAR:2018fqv . This suggests that measuring the
transverse polarization of $\Lambda/\overline{\Lambda}$ hyperons inside the
jet in transversely polarized $pp$ collisions could be achieved at RHIC. This
process would correspond to the structure function
$F^{\cos(\phi_{S}-\hat{\phi}_{S_{h}})}_{TU,T}$, which is sensitive to both
transversity PDFs $h_{1}$ and transversity TMD FJFs $\mathcal{H}_{1}^{h/q}$.
Below, we focus on this structure function and define the following spin
asymmetry for $\Lambda$-in-jet productions:

$\displaystyle A_{TU,T}^{\cos(\phi_{S}-\hat{\phi}_{S_{h}})}\equiv
F_{TU,T}^{\cos(\phi_{S}-\hat{\phi}_{S_{h}})}\bigg{/}F_{UU,U}\,.$ (171)

In order to compute $F_{TU,T}^{\cos(\hat{\phi}_{h}-\hat{\phi}_{S_{h}})}$, we
will need to parameterize the collinear transversity PDF $h_{1}(x,\mu)$, which
appears in the initial state of $F_{TU,T}^{\cos(\phi_{S}-\hat{\phi}_{S_{h}})}$
in Eq. 165. For this, we follow the parameterization in Kang:2015msa  and
write the quark transversity as:

$h_{1}^{q}\pqty{x,Q_{0}}=N_{q}^{h}x^{a_{q}}\pqty{1-x}^{b_{q}}\frac{\pqty{a_{q}+b_{q}}^{a_{q}+b_{q}}}{a_{q}^{a_{q}}b_{q}^{b_{q}}}\frac{1}{2}\pqty{f_{1}^{q}\pqty{x,Q_{0}}+g_{1}^{q}\pqty{x,Q_{0}}}\,,$
(172)

at initial scale $Q_{0}=1.27~{}\mathrm{GeV}$, for up and down quarks $q=u$ and
$d$ only. The $f_{1}^{q}$ and $g_{1}^{q}$ are the collinear unpolarized and
helicity parton distributions respectively, for which we use the parameters
from Cocuzza:2022jye . Lastly, in the generation of TMD FFs grids, we also
need the transversity FFs. In this work we use the parameterization from
Kang:2021kpt , which assumes a simple normalization of the collinear
unpolarized FFs:

$H_{\Lambda/q}\pqty{z,Q}=N_{q}^{H}D_{\Lambda/q}\pqty{z,Q}\,.$ (173)

Here $H_{\Lambda/q}$ and $D_{\Lambda/q}$ are the transversity and unpolarized
FFs of the $\Lambda$ baryon, and $N_{q}^{H}$ is the fitted parameter. We also
follow the assumptions made in Callos:2020qtu ; Kang:2021kpt  and demand:

$D_{\Lambda/q}=D_{\overline{\Lambda}/q}=\frac{1}{2}D_{\Lambda/\overline{\Lambda}\leftarrow
q}\,,$ (174)

as for the unpolarized $\Lambda$ baryon FFs that appear in the denominator of
$A_{TU,T}^{\cos(\phi_{S}-\hat{\phi}_{S_{h}})}$, we use the AKK parametrization
Albino:2005mv . For both the transversity PDFs and transversity FFs, we also
vary their fitted parameters according to the extracted uncertainties,
weighted by Gaussian random number with a mean of 0 and variance of 1.

Using the above setup, we now present the first prediction for the spin
asymmetry $A_{TU,T}^{\cos(\phi_{S}-\hat{\phi}_{S_{h}})}$. We choose the jet
kinematics consistent with the available data from STAR STAR:2022hqg , with
the center of mass energy at $\sqrt{s}=200~{}\mathrm{GeV}$, rapidity ranges
$\eta\in\pqty{0,0.9}$ and $\eta\in\pqty{-0.9,0}$. The jets are reconstructed
using the anti-$k_{T}$ algorithm with radius $R=0.6$. The numerical
implementation is very similar to that used in the longitudinal momentum
distribution of hadrons inside jets Kang:2016ehg , with the numerical DGLAP
evolution tool developed in Pegasus Vogt:2004ns . The RG evolution of the
various parts of the cross sections is performed as outlined in Kang:2017glf .

Figure 6:  Hadron $j_{\perp}$-distribution within jets in $pp$ collisions at
$\sqrt{s}=200~{}\mathrm{GeV}$. Jets are detected in the rapidity interval
$\eta\in\pqty{0,0.9}$ (i.e., the jet is scattered forward relative to the
polarized beam), and are reconstructed using the anti-$k_{T}$ algorithm with
$R=0.6$. We vary the jet transverse momentum from 5 to 15 GeV, and choose
average $z_{h}$ at 0.2 (top row) and 0.4 (bottom row), which is within the
typical kinematic region according to STAR:2022hqg . The blue band is obtained
by varying the parameters for transversity PDFs and transversity FFs within
their fitted uncertainties. While the yellow band is obtained by keeping the
fitted parameters of transversity FFs fixed at their central values and
varying only the parameters of transversity PDFs. The red lines show the
theory obtained from the central of the fitted parameters.

In Fig. 6, the prediction to $A_{TU,T}^{\cos(\phi_{S}-\hat{\phi}_{S_{h}})}$
for the jet$(\Lambda)$ production at RHIC kinematics is presented. The blue
band is obtained by varying the fitted parameters of both transversity PDFs
and transversity FFs, while the yellow band is obtained by varying the
parameters of transversity PDFs only (with parameters for transversity FFs
fixed at central values). We can therefore deduce that the uncertainty of
theoretical prediction for the observable
$A_{TU,T}^{\cos(\phi_{S}-\hat{\phi}_{S_{h}})}$ comes largely from the
transversity FFs, i.e., variation of $N_{q}^{H}$ in Eq. 173. Given the large
uncertainty resulting from the currently available extraction of the
transversity FFs for $\Lambda$ baryon, this observable, when measured in the
future, can be used to constrain the uncertainty on the $\Lambda$ transversity
FFs.

Figure 7: Same plot as Fig. 6, but with jet scattered backward relative to the
polarized beam.

In Fig. 7, the same predictions are presented but with a negative rapidity
range. Similar patterns occur as in Fig. 6, but with a much smaller magnitude,
indicating a much smaller probability of producing polarized final particles
in the opposite direction of the polarized beam.

Overall, in proton-proton collisions for $\Lambda$-in-jet production, where
one of the proton beams is transversely polarized, and the resulting Lambda
particles in the jet also exhibit transverse polarization, a distinct
observation emerges. Specifically, in the forward region (positive rapidity),
a moderate spin asymmetry $A_{TU,T}^{\cos(\phi_{S}-\hat{\phi}_{S_{h}})}\sim
1.5\%$ can be observed at RHIC kinematics, while in the backward region
(negative rapidity), the spin asymmetry
$A_{TU,T}^{\cos(\phi_{S}-\hat{\phi}_{S_{h}})}$ appears to be very small. This
is largely due to the probing $x$ range for the transversity PDF
$h_{1}(x,\mu)$: in the forward rapidity region $x\sim 0.2$ at $p_{T}\sim 15$
GeV where $h_{1}(x,\mu)$ is sizable Kang:2015msa ; in the backward rapidity
region $x\sim 0.05$ for $p_{T}\sim 15$ GeV where $h_{1}(x,\mu)$ is very small.

###  4.2 Back-to-back electron-jet production with longitudinally polarized
$\Lambda$ in jet

For exclusive jet production, we choose to study back-to-back electron-jet
production in deep inelastic $ep$ scattering:

$e\pqty{\ell,\lambda_{e}}+p\pqty{P,S}\to
e\pqty{\ell^{\prime}}+\bqty{\mathrm{jet}(p_{J})h\pqty{z_{h},\bm{j}_{\perp},S_{h}}}+X\,,$
(175)

where $\ell$ and $\ell^{\prime}$ are the momenta of the incident and scattered
lepton respectively. The incident electron could carry a helicity
$\lambda_{e}$ and the scattered lepton is assumed to be unpolarized but with a
transverse momentum $\bm{\ell}^{\prime}_{T}$. Here we use $P$ and $S$ to
represent the momentum and polarization of the incoming proton, and the jet
has a momentum $p_{J}$ with transverse component $\bm{p}_{T}$. The hadron
inside the jet has a transverse momentum $\bm{j}_{\perp}$ with respect to the
jet axis and collinear momentum fraction $z_{h}$ of the jet, as well as a
polarization $\bm{S}_{h}$ with respect to the jet axis. The kinematics is
shown in Fig. 8. In the back-to-back region where
$\absolutevalue{\bm{q}_{T}}\ll\absolutevalue{\bm{p}_{T}}$ with the transverse
momentum imbalance $\bm{q}_{T}\equiv\bm{\ell}^{\prime}_{T}+\bm{p}_{T}$, TMD
factorization formalism for the full differential cross section has been given
in the previous publication Kang:2021ffh , where various structure functions
were defined and studied. For example, the structure function
$F_{UU,U}^{\cos(\phi_{q}-\hat{\phi}_{h})}$ for $\pi^{\pm}$-in-jet production
is sensitive to Boer-Mulder function and Collins function, while
$F_{UU,T}^{\cos(\hat{\phi}_{h}-\hat{\phi}_{S_{h}})}$ from $\Lambda$-in-jet
production has the sensitivity to the polarizing TMD FFs $D_{1T}^{\bot}$. At
the same time, in Arratia:2020nxw , the structure function
$F_{TU,U}^{\,\sin(\phi_{S}-\hat{\phi}_{h})}$ is computed for $\pi^{+/-}$-in-
jet production, and is found to be able to further constrain the Collins FFs.

Figure 8: Illustration for the distribution of hadrons inside jets in the
collisions of a polarized proton and an unpolarized lepton. For a detailed
illustration of jet and hadron kinematics, please see Fig. 3.

In this section, we studied yet another observable - the spin asymmetry
$A_{TU,L}^{\cos(\phi_{q}-\phi_{S})}$ defined as

$\displaystyle
A_{TU,L}^{\cos(\phi_{q}-\phi_{S})}\equiv\frac{F_{TU,L}^{\cos(\phi_{q}-\phi_{S})}}{F_{UU,U}}\,,$
(176)

where the structure $F_{TU,L}^{\cos(\phi_{q}-\phi_{S})}$ in the numerator
denotes the spin-dependent structure functions, with $T$, $U$ and $L$ being
the polarization of incoming proton, incoming electron, and the final-state
hadron inside the jet, respectively. Here, $\phi_{S}$ is the angle between the
transverse spin of the incoming proton $\bm{S}_{T}$ and the $x$-axis, while
$\phi_{q}$ is the azimuthal angle between the transverse momentum imbalance
$\bm{q}_{T}$ and the $x$-axis, see Fig. 8. Following Kang:2021ffh , this
structure function can be factorized as follows:

$\displaystyle F_{TU,L}^{\cos(\phi_{q}-\phi_{S})}(z_{h},j_{\perp})$
$\displaystyle=\hat{\sigma}_{0}H\pqty{Q,\mu}\sum_{q}e_{q}^{2}\mathscr{G}_{1L}^{h/q}\pqty{z_{h},\bm{j}_{\perp},p_{T}R,\mu,\zeta_{J}}$
$\displaystyle\times\int\frac{b^{2}\differential{b}}{2\pi}J_{1}\pqty{q_{T}b}xM\,\widetilde{g}_{1T}^{(1)q}\pqty{x,\bm{b},\mu,\zeta}\overline{S}_{\mathrm{global}}\pqty{b^{2},\mu}\overline{S}_{\mathrm{cs}}\pqty{b^{2},R,\mu}\,,$
(177)

where the global soft function $\overline{S}_{\mathrm{global}}$ and the
collinear-soft function $\overline{S}_{\mathrm{cs}}$ are given in Kang:2021ffh
. The bar in $\overline{S}_{\mathrm{global}/\mathrm{cs}}$ indicates that their
dependence on azimuthal angle is averaged. $\hat{\sigma}_{0}$ is the partonic
cross section of electron-quark scattering:

$\hat{\sigma}_{0}=\frac{\alpha_{\mathrm{em}}\alpha_{s}}{\hat{s}Q^{2}}\frac{2\pqty{\hat{u}^{2}+\hat{s}^{2}}}{\hat{t}^{2}}\,,$
(178)

with the Mandelstam variables $\hat{s}\equiv\pqty{xP+\ell}^{2}$,
$\hat{t}\equiv\pqty{\ell-\ell^{\prime}}^{2}$ and
$\hat{u}\equiv\pqty{xP-\ell^{\prime}}^{2}$, where $x$ is the fraction of
collinear momentum carried by the parton inside the incoming proton. The
electromagnetic and strong coupling constants are denoted as
$\alpha_{\mathrm{em}}$ and $\alpha_{s}$, and the virtuality of the exchanged
photon is defined as $Q^{2}\equiv-\pqty{\ell-\ell^{\prime}}^{2}=-\hat{t}$.

It is evident from Eq. 177 that this spin asymmetry
$A_{TU,L}^{\cos(\phi_{q}-\phi_{S})}$ has sensitivity on both “worm-gear” TMD
PDFs $\widetilde{g}_{1T}^{q}\pqty{x,\bm{b},\mu,\zeta}$, as well as the
longitudinally polarized exclusive TMD FJFs
$\mathscr{G}_{1L}\pqty{z_{h},\bm{j}_{\perp},p_{T}R,\mu,\zeta_{J}}$ in Eq. 142,
which is connected to the helicity TMD FFs $G_{1L}^{h/q}$. The observable
describes the following scattering process: a longitudinally polarized quark
inside the transversely polarized proton (as denoted by the TMD PDFs
$g_{1T}^{q}$) scatters with an unpolarized electron and then initiates a jet
in which a longitudinally polarized hadron is observed. To generate a
prediction, for the first moment of “worm-gear” distribution
$\widetilde{g}_{1T}^{(1)q}$, we adopt its parameterization in
$\bm{k}_{\perp}$-space from Bhattacharya:2021twu , and Fourier transform it to
the conjugate $\bm{b}$-space:

$\widetilde{g}_{1T}^{(1)q}\pqty{x,\bm{b},\mu,\zeta}=\frac{2\pi}{M^{2}}\int\differential{k}_{\perp}\frac{{k}_{\perp}^{2}}{b}J_{1}({k}_{\perp}b)g_{1T}^{q}(x,\bm{k}_{\perp}^{2})=g_{1T}^{\pqty{1}q}\pqty{x}\exp(-\frac{1}{4}b^{2}\langle
k_{\perp}^{2}\rangle\big{|}_{g_{1T}^{q}})\,,$ (179)

where $M\approx 938.272~{}\mathrm{MeV}$ is the proton mass and
$g_{1T}^{\pqty{1}q}\pqty{x}$ is the collinear function that is parameterized
at $Q_{0}=2~{}\mathrm{GeV}$ Bhattacharya:2021twu . Note that this is a simple
parton model extraction without TMD evolution, we thus also do not take into
account TMD evolution for $\widetilde{g}_{1T}^{q}$. Since CT10 PDFs Lai:2010vv
are used in the extraction of $\widetilde{g}_{1T}^{q}$, we will use them in
the computation of $F_{UU,U}$ as well. For the final-state hadron, we observe
the longitudinally polarized $\Lambda/\overline{\Lambda}$ production. The
corresponding helicity TMD FFs is given by

$\displaystyle\widetilde{G}_{1L}^{\Lambda/q}\pqty{z_{h},\bm{b},\mu,\zeta_{J}}={G}_{1L}^{\Lambda/q}\pqty{z_{h},\mu_{b_{*}}}$
$\displaystyle\exp\left[\int_{\mu_{b_{*}}}^{\mu}\frac{d\mu^{\prime}}{\mu^{\prime}}\gamma_{\mu}^{i}[\alpha_{s}(\mu),\zeta_{J}/\mu^{2}]\right]S_{NP}^{q}(b,Q_{0},\sqrt{\zeta_{J}})\,,$
(180)

where we use the TMD evolution in Eq. 94 and we assume the non-perturbative
Sudakov is the same as the unpolarized case in Eq. 149 Koike:2006fn . For the
collinear helicity FFs for $\Lambda/\overline{\Lambda}$, we take the
parameterizations from deFlorian:1997zj , in which three scenarios are
provided, each with a different assumption for the source of contribution to
$\Lambda$ longitudinal polarization. Specifically,

  * •

Scenario 1: only polarized $s$ quark contributes;

  * •

Scenario 2: $u$ and $d$ quarks have equal contributions, while the
distribution of $s$ quark has an opposite sign relative to $u$ and $d$ quarks;
$u$ and $d$ helicity FFs are negative

  * •

Scenario 3: $u$, $d$, and $s$ quarks have the same distribution functions,
which are all positive.

Figure 9: Longitudinally polarized hadron $j_{\perp}$-distribution (left) and
$q_{T}$-distribution (right) within jets in $ep$ collisions at
$\sqrt{s}=105~{}\mathrm{GeV}$. The event inelasticity $y$ is constrained
within $\pqty{0.1,0.9}$. Jets are detected with transverse momentum
$p_{T}\in\pqty{10,15}~{}\mathrm{GeV}$, and are reconstructed using the
anti-$k_{T}$ algorithm with $R=0.6$. For the $j_{\bot}$-distribution, $q_{T}$
is chosen to be 0.5 GeV, while for the $q_{T}$-distribution, $j_{\bot}$ is
chosen to be 0.5 GeV. The uncertainty band is obtained by computing the
1-$\sigma$ band of the fitted parameters for the “worm-gear” PDFs $g_{1T}$,
and the three scenarios for $\Lambda/\overline{\Lambda}$ are shown by blue
(scenario 1), red (scenario 2) and green (scenario 3). We collectively denote
$\Lambda/\overline{\Lambda}$ as $\Lambda$. Scenario 1 is consistent with zero
because the current extraction of the strange quark worm-gear function is
zero.

With the above setup, we are able to make predictions for the exclusive
process $ep\to e+\mathrm{jet}\pqty{\Lambda/\overline{\Lambda}}+X$, in which we
measure the longitudinal polarization of the final state
$\Lambda/\overline{\Lambda}$ inside the jet. In Fig. 9, we plot the spin
asymmetry $A_{TU,L}^{\cos(\phi_{q}-\phi_{S})}$ defined in Eq. 176 with EIC
kinematics, where $\sqrt{s}=105~{}\mathrm{GeV}$, jet transverse momentum
$p_{T}\in\pqty{10,15}~{}\mathrm{GeV}$, and jet radius parameter $R=0.6$. We
also introduce the quantity event inelasticity
$y=1-\frac{\ell_{T}^{\prime}}{\sqrt{s}}e^{-y_{e}}\in\pqty{0.1,0.9}$, where
$\ell_{T}^{\prime}$ and $y_{e}$ are the measured transverse momentum and
rapidity of the scattered electron. Constraining the event inelasticity
ensures that our calculation stays in the region with reasonable resolution on
$x$ and $Q^{2}$, as well as avoiding the phase where QED radiative corrections
are important Arratia:2020nxw . On the left panel, we plot
$A_{TU,L}^{\cos(\phi_{q}-\phi_{S})}$ as a function of hadron transverse
momentum $j_{\bot}$, while on the right panel, we plot the same spin asymmetry
as a function of the electron-jet transverse momentum imbalance $q_{T}$. For
the $j_{\bot}$-distribution (left), $q_{T}$ is chosen to be 0.5 GeV, while for
the $q_{T}$-distribution (right), $j_{\bot}$ is chosen to be 0.5 GeV. The
uncertainty band is obtained by computing the 1-$\sigma$ band of the fitted
parameters for the “worm-gear” PDFs $g_{1T}$, and the three scenarios for the
helicity fragmentation functions of $\Lambda/\overline{\Lambda}$ are shown by
blue (scenario 1), red (scenario 2) and green (scenario 3). We collectively
denote $\Lambda/\overline{\Lambda}$ as $\Lambda$.

As can be inferred from Fig. 9, the different scenarios for helicity FFs
provided in deFlorian:1997zj  can result in drastically different values in
$A_{TU,L}^{\cos(\phi_{q}-\phi_{S})}$. Notably, Scenario 1 is consistent with
zero because only the $s$ quark helicity FFs contribute in this scenario.
However, the current extraction of the “worm-gear” function
${g}_{1T}^{(1)q}(x)$ of the incident proton receives contributions solely from
$u$ and $d$ quarks, and the $s$ quark worm-gear function vanish
Bhattacharya:2021twu . As for Scenarios 2 and 3, distinct measurable
asymmetries emerge with different signs, simply because in Scenario 2, both
$u$ and $d$ helicity FFs are both negative, while they are positive in
Scenario 3. This observable thus enables the discrimination of previously
indistinguishable helicity fragmentation functions for
$\Lambda/\overline{\Lambda}$ from different quark flavors.

##  5 Conclusion

The study of fragmenting jet functions (FJFs) has become a topic of paramount
importance in high-energy physics, particularly at the LHC and RHIC, where
measurements have been made for a broad range of identified particles within
jets. These studies not only provide crucial information at the LHC and RHIC,
but also hold the potential to yield novel insights at the upcoming Electron-
Ion Collider (EIC).

It is thus important to study both longitudinal and transverse momentum
distribution of hadrons inside the jet, which are characterized by the so-
called semi-inclusive fragmenting jet functions and exclusive fragmenting jet
functions. They are critical ingredients in the factorization theorems. The
semi-inclusive fragmenting jet functions arise in the situation e.g. where
single inclusive jet production in proton-proton or electron-proton
collisions, while the exclusive fragmenting jet functions appear in the
situation e.g. where back-to-back electron-jet production in electron-proton
collisions. There have been partial results available in previous
publications, but never the complete full results, which we set up to do in
this paper. In this work, we have studied the hadron longitudinal momentum
fraction $z_{h}$ distribution, as well as the hadron transverse momentum
$j_{\perp}$ distribution within jets, for both semi-inclusive and exclusive
fragmenting jet functions. We set up factorization formalism within SCET
framework that allows for the systematic determination of these distributions.
We first calculated all the components of the factorization theorem up to NLO,
then we resummed all the associated large logarithms $\ln{R}$ and
$\ln(p_{T}R/j_{\perp})$.

In the phenomenology for semi-inclusive process, we study the transverse
polarization of Lambda hyperons inside the jet produced in transversely
polarized proton-proton collisions, which is sensitive to the elusive TMD
transversity fragmentation functions. Our numerical estimate shows that this
observable is promising at the RHIC. For the exclusive jet production process,
we study the longitudinal polarization of Lambda hyperon inside the jet, in
the back-to-back electron-jet production in electron-proton collisions where
the proton is transversely polarized. We find that this observable is
sensitive to the “worm-gear” TMD PDF $g_{1T}$ and the TMD helicity
fragmentation functions and its measurement is promising at the future EIC.
With the various applications for probing the 3D QCD structures, the future
for fragmenting jet observables is promising with all the planned measurements
at the RHIC and the EIC.

###### Acknowledgements.

We thank Y. He, H. Wang and S. Yang for useful discussions. H.X. and Y.Z. are
supported by the Guangdong Major Project of Basic and Applied Basic Research
No. 2020B0301030008, No. 2022A1515010683, the National Natural Science
Foundation of China under Grants No. 12022512, No. 12035007. Z.K. is supported
by the National Science Foundation under Grant No. PHY-1945471. F.Z. is
supported by the U.S. Department of Energy, Office of Science, Office of
Nuclear Physics under grant Contract Number DE-SC0011090 and Co-design Center
for Quantum Advantage (C2QA) under Contract Number DE-SC0012704.

##  Appendix A Perturbative NLO semi-inclusive FJFs

As defined in Section 2.1.1, we have the leading order and one-loop bare semi-
inclusive FJFs for longitudinally and transversely polarized quarks and
gluons. To get the final expression as given in Eqs. 19, 20, 21, 22 and 24, we
first perform the $q_{\perp}$ integral:

$\displaystyle\int\frac{\differential{q_{\perp}^{2}}}{\pqty{q_{\perp}^{2}}^{1+\epsilon}}\Theta^{\text{anti-}k_{T}}_{\textrm{both}}\pqty{q_{\perp}}$
$\displaystyle=\frac{-1}{\epsilon}\pqty{\omega_{J}^{2}\tan[2](\frac{\mathcal{R}}{2})}^{-\epsilon}z_{h}^{-2\epsilon}\pqty{1-z_{h}}^{-2\epsilon}\,,$
(181)
$\displaystyle\int\frac{\differential{q_{\perp}^{2}}}{\pqty{q_{\perp}^{2}}^{1+\epsilon}}\Theta^{\text{anti-}k_{T}}_{j}\pqty{q_{\perp}}$
$\displaystyle=\frac{1}{\epsilon}\pqty{\omega_{J}^{2}\tan[2](\frac{\mathcal{R}}{2})}^{-\epsilon}\pqty{1-z}^{-2\epsilon}\,,$
(182)

then apply the following expansions as $\epsilon\to 0$:

$\displaystyle\frac{1}{\pqty{1-z}^{1+2\epsilon}}$
$\displaystyle=\frac{-1}{2\epsilon}\delta\pqty{1-z}+\frac{1}{\pqty{1-z}_{+}}-2\epsilon\pqty{\frac{\ln(1-z)}{\pqty{1-z}}}_{+}+\order{\epsilon^{2}}\,,$
(183) $\displaystyle\frac{1}{z^{2\epsilon}}$
$\displaystyle=1-2\epsilon\ln(z)+\order{\epsilon^{2}}\,,$ (184)
$\displaystyle\pqty{\frac{e^{\gamma_{E}}\mu^{2}}{\omega_{J}^{2}\tan[2](\frac{\mathcal{R}}{2})}}^{\epsilon}\frac{1}{\Gamma\pqty{1-\epsilon}}$
$\displaystyle=1+L\epsilon+\frac{1}{12}\pqty{6L^{2}-\pi^{2}}\epsilon^{2}+\order{\epsilon^{3}}\,,$
(185)

where $\displaystyle
L\equiv\ln(\frac{\mu^{2}}{\omega_{J}^{2}\tan[2](\frac{\mathcal{R}}{2})})=\ln(\frac{\mu^{2}}{(p_{T}R)^{2}})$.

##  Appendix B Calculation of collinear exclusive FJFs

In this section, we provide the derivation for the matching coefficients
$\Delta_{(T)}\mathscr{J}_{ij}$ of the exclusive fragmenting jet functions
$\Delta_{(T)}\mathscr{G}_{i}^{h}(z_{h},p_{T}R,\mu)$ to the collinear
fragmentation function $\Delta_{(T)}D_{i}^{h}(z_{h},\mu)$ for anti-${k}_{T}$
jets. The unpolarized results were first written down in the appendix of
Waalewijn:2012sv . We start by specifying the phase space constraint from the
jet algorithm, which was nicely outlined in Ellis:2010rwa . Consider a parton
splitting process, $i(\ell)\rightarrow j(q)+k(\ell-q)$, where an incoming
parton $i$ with momentum $\ell$ splits into a parton $j$ with momentum $q$ and
a parton $k$ with momentum $\ell-q$. The four-vector $\ell^{\mu}$ can be
decomposed in light-cone coordinates as
$\ell^{\mu}=\pqty{\ell^{+},\ell^{-}=\omega,0_{\perp}}$ where
$\ell^{\pm}=\ell^{0}\mp\ell^{z}$. The constraint for anti-$k_{T}$ algorithm
with radius $R$ is given by:

$\displaystyle\Theta^{\text{anti-}k_{T}}$
$\displaystyle=\theta\pqty{\tan[2](\frac{\mathcal{R}}{2})-\frac{q^{+}\omega^{2}}{q^{-}\pqty{\omega-\ell^{-}}^{2}}}\,.$
(186)

For fragmenting jet functions, the above constraint leads to a constraint on
the jet invariant mass $m_{J}^{2}=\omega\ell^{+}$ Procura:2011aq , which is
derived and listed as follows:

$\displaystyle\delta_{\text{anti-}k_{T}}$
$\displaystyle=\theta\pqty{z_{h}(1-z_{h})\omega^{2}\tan[2](\frac{\mathcal{R}}{2})-m_{J}^{2}}\theta\pqty{m_{J}^{2}}\,,$
(187)

where $z_{h}=\omega_{h}/\omega_{J}=\omega_{h}/\omega$, $\omega_{h}$ is the
large light-cone momentum of the final hadron, and $\omega=\omega_{J}$ since
both partons are in the jet (see Fig. 2 (A)).

|  | Quark polarization
---|---|---
|  | U | L | T
| U | $\mathscr{G}_{q}^{h}=\includegraphics[height=15.00002pt,valign={c}]{./figures/FJFs-table/1-1.pdf}$ |  |
| L |  | $\Delta\mathscr{G}_{q}^{h}=\includegraphics[height=15.00002pt,valign={c}]{./figures/FJFs-table/2-2-1.pdf}-\includegraphics[height=15.00002pt,valign={c}]{./figures/FJFs-table/2-2-2.pdf}$ |
Hadron pol. | T |  |  | $\Delta_{T}\mathscr{G}_{q}^{h}=\includegraphics[height=15.00002pt,valign={c}]{./figures/FJFs-table/3-3-1.pdf}-\includegraphics[height=15.00002pt,valign={c}]{./figures/FJFs-table/3-3-2.pdf}$
Table 3:  Summary of the exclusive collinear FJFs. The header row represents
the polarization of the quark (indicated by the blue line) that initiates the
jet, while the header column indicates the corresponding polarizations of
produced hadrons (indicated by the red arrow from the red dot). Shown here is
for quark FJFs, with $U,\,L,\,T$ representing the unpolarized case, and
longitudinal and transverse polarization. For gluon FJFs, one would interpret
$L,\,T,$ as circular and linear polarization.

The exclusive FJFs $\Delta_{(T)}\mathscr{G}_{i}^{h}(z_{h},p_{T}R,\mu)$ can be
matched onto corresponding fragmentation functions
$\Delta_{(T)}D_{i}^{h}(z_{h},\mu)$ as listed in Table 3:

$\displaystyle\Delta_{(T)}\mathscr{G}_{i}^{h}(z_{h},p_{T}R,\mu)$
$\displaystyle=\sum_{j}\int_{z_{h}}^{1}\frac{\differential{z^{\prime}_{h}}}{z^{\prime}_{h}}\Delta_{(T)}\mathscr{J}_{ij}(z^{\prime}_{h},p_{T}R,\mu)\Delta_{(T)}D_{h/j}\pqty{\frac{z_{h}}{z^{\prime}_{h}},\mu}$
$\displaystyle\quad+\order{\frac{\Lambda_{\mathrm{QCD}}^{2}}{(p_{T}R)^{2}}}\,,$
(193)

where $\Delta_{(T)}\mathscr{J}_{ij}$ are the matching coefficients. The
exclusive FJFs $\Delta_{(T)}\mathscr{G}_{i}^{j}\pqty{z_{h},m_{J}^{2},\mu}$
with $i,j\in\Bqty{q,g}$ has been extensively studied in Jain:2011xz ;
Ritzmann:2014mka . Using pure dimensional regularization with $4-2\epsilon$
dimensions in the $\overline{\mathrm{MS}}$ scheme, the bare results at
$\order{\alpha_{s}}$ can be written in the following compact form Giele:1991vf
; Ritzmann:2014mka ; Chien:2015ctp :

$\Delta_{(T)}\mathscr{G}_{i,\mathrm{bare}}^{j}\pqty{z_{h},m_{J}^{2}}=\frac{\alpha_{s}}{2\pi}\frac{\pqty{e^{\gamma_{E}}\mu^{2}}^{\epsilon}}{\Gamma(1-\epsilon)}\Delta_{(T)}\hat{P}_{ji}(z_{h},\epsilon)z_{h}^{-\epsilon}(1-z_{h})^{-\epsilon}\pqty{m_{J}^{2}}^{-1-\epsilon}\,.$
(194)

Generalized from Chien:2015ctp ,
$\Delta_{(T)}\mathscr{G}_{i,\mathrm{bare}}^{j}\pqty{z_{h},m_{J}^{2},\mu}$ is
related to $\Delta_{(T)}\mathscr{G}_{i,\mathrm{bare}}^{h}(z_{h},p_{T}R,\mu)$
by:

$\Delta_{(T)}\mathscr{G}_{i,\mathrm{bare}}^{h}(z_{h},p_{T}R)=\int\differential{m_{J}^{2}}\Delta_{(T)}\mathscr{G}_{i,\mathrm{bare}}^{h}\pqty{z_{h},m_{J}^{2}}\delta_{\text{anti-}k_{T}}\,,$
(195)

notice that we reinstated hadronic FJFs. The splitting functions
$\Delta_{(T)}\hat{P}_{ji}(z_{h},\epsilon)$ are given in Eqs. 14, 15, 16, 17
and 18. By performing the integration over $m_{J}^{2}$ with the constraints
imposed by the jet algorithm $\delta_{\text{anti-}k_{T}}$, one obtains the
bare exclusive FJFs
$\Delta_{(T)}\mathscr{G}_{i,\mathrm{bare}}^{j}(z_{h},p_{T}R)$. We present the
results for anti-$k_{T}$ jets here, as their explicit expressions are not
available in the literature:

$\displaystyle\Delta\mathscr{G}_{q,\text{bare}}^{q}(z_{h},p_{T}R)$
$\displaystyle=\frac{\alpha_{s}}{2\pi}\bigg{\\{}C_{F}\pqty{\frac{1}{\epsilon^{2}}+\frac{3}{2\epsilon}+\frac{L}{\epsilon}}\delta(1-z_{h})-\frac{1}{\epsilon}\Delta
P_{qq}(z_{h})+C_{F}(1-z_{h})$
$\displaystyle\quad+C_{F}\delta(1-z_{h})\pqty{\frac{L^{2}}{2}-\frac{\pi^{2}}{12}}+\Delta
P_{qq}(z_{h})(2\ln(z_{h})-L)$
$\displaystyle\quad+\frac{3C_{F}L}{2}\delta\pqty{1-z_{h}}+2C_{F}\pqty{1+z_{h}^{2}}\pqty{\frac{\ln(1-z_{h})}{1-z_{h}}}_{+}\bigg{\\}}\,,$
(196) $\displaystyle\Delta\mathscr{G}_{q,\text{bare}}^{g}(z_{h},p_{T}R)$
$\displaystyle=\frac{\alpha_{s}}{2\pi}\bigg{\\{}\frac{-1}{\epsilon}\Delta
P_{gq}(z_{h})+\Delta
P_{gq}(z_{h})\Big{[}2\ln(z_{h}(1-z_{h}))-L\Big{]}-2C_{F}(1-z_{h})\bigg{\\}}\,,$
(197) $\displaystyle\Delta\mathscr{G}_{g,\text{bare}}^{q}(z_{h},p_{T}R)$
$\displaystyle=\frac{\alpha_{s}}{2\pi}\bigg{\\{}\frac{-1}{\epsilon}\Delta
P_{qg}(z_{h})+\Delta
P_{qg}(z_{h})\Big{[}2\ln(z_{h}(1-z_{h}))-L\Big{]}+2T_{F}(1-z_{h})\bigg{\\}}\,,$
(198) $\displaystyle\Delta\mathscr{G}_{g,\text{bare}}^{g}(z_{h},p_{T}R)$
$\displaystyle=\frac{\alpha_{s}}{2\pi}\bigg{\\{}C_{A}\pqty{\frac{1}{\epsilon^{2}}+\frac{1}{\epsilon}\frac{\beta_{0}}{2C_{A}}+\frac{L}{\epsilon}}\delta(1-z_{h})-\frac{1}{\epsilon}\Delta
P_{gg}(z_{h})$
$\displaystyle\quad+C_{A}\delta(1-z_{h})\pqty{\frac{L^{2}}{2}-\frac{\pi^{2}}{12}}+\Delta
P_{gg}(z_{h})(2\ln(z_{h})-L)+\frac{\beta_{0}L}{2}\delta(1-z_{h})$
$\displaystyle\quad-4C_{A}(1-z_{h})+4C_{A}\Big{[}2(1-z_{h})^{2}+z_{h}\Big{]}\pqty{\frac{\ln(1-z_{h})}{1-z_{h}}}_{+}\bigg{\\}}\,,$
(199) $\displaystyle\Delta_{T}\mathscr{G}_{q,\text{bare}}^{q}(z_{h},p_{T}R)$
$\displaystyle=\frac{\alpha_{s}}{2\pi}\bigg{\\{}C_{F}\pqty{\frac{1}{\epsilon^{2}}+\frac{3}{2\epsilon}+\frac{L}{\epsilon}}\delta(1-z_{h})-\frac{1}{\epsilon}\Delta_{T}P_{qq}(z_{h})$
$\displaystyle\quad+C_{F}\delta(1-z_{h})\pqty{\frac{L^{2}}{2}-\frac{\pi^{2}}{12}}+\Delta_{T}P_{qq}(z_{h})(2\ln(z_{h})-L)$
$\displaystyle\quad+\frac{3C_{F}L}{2}\delta\pqty{1-z_{h}}+4C_{F}z_{h}\pqty{\frac{\ln(1-z_{h})}{1-z_{h}}}_{+}\bigg{\\}}\,,$
(200)

where, as given in the main text, $\beta_{0}$ is defined as:

$\beta_{0}\equiv\frac{11}{3}C_{A}-\frac{4}{3}T_{F}n_{f}\,$ (201)

and $\Delta_{(T)}{P}_{ji}(z_{h})$ are given in Eqs. 25, 26, 27, 28 and 29. It
is instructive to point out that the $\epsilon$ poles in the first term of
Eqs. 196, 199 and 200 correspond to ultraviolet (UV) divergences, and they are
related to the renormalization of the FJFs
$\Delta_{(T)}\mathscr{G}_{i,\text{bare}}^{j}(z_{h},p_{T}R)$. All the remaining
$\epsilon$ poles in Eqs. 196, 197, 198, 199 and 200 are infrared (IR) poles,
and they match exactly with those in the fragmentation functions
$\Delta_{(T)}D_{j/i}(z_{h},\mu)$, which we will show below.
$\Delta_{(T)}\mathscr{G}_{i,\text{bare}}^{h}(z_{h},p_{T}R)$ is renormalized
by:

$\Delta_{(T)}\mathscr{G}_{i,\text{bare}}^{h}(z_{h},p_{T}R)=\mathcal{Z}_{\mathscr{G}}^{i}(\mu)\Delta_{(T)}\mathscr{G}_{i}^{h}(z_{h},p_{T}R,\mu)\,,$
(202)

where $i$ is not summed over in the above equation. The corresponding
renormalization group (RG) equation is:

$\mu\frac{\differential{}}{\differential{\mu}}\Delta_{(T)}\mathscr{G}_{i}^{h}(z_{h},p_{T}R,\mu)=\gamma_{\mathscr{G}}^{i}(\mu)\Delta_{(T)}\mathscr{G}_{i}^{h}(z_{h},p_{T}R,\mu)\,,$
(203)

where the anomalous dimension $\gamma_{\mathscr{G}}^{i}(\mu)$ is:

$\gamma_{\mathscr{G}}^{i}(\mu)=-\pqty{\mathcal{Z}_{\mathscr{G}}^{i}(\mu)}^{-1}\mu\frac{\differential{}}{\differential{\mu}}\mathcal{Z}_{\mathscr{G}}^{i}(\mu)\,.$
(204)

The solution to Eq. 204 is then:

$\Delta_{(T)}\mathscr{G}_{i}^{h}(z_{h},p_{T}R,\mu)=\Delta_{(T)}\mathscr{G}_{i}^{h}\pqty{z_{h},p_{T}R,\mu_{\mathscr{G}}}\exp(\int_{\mu_{\mathscr{G}}}^{\mu}\frac{\differential{\mu^{\prime}}}{\mu^{\prime}}\gamma_{\mathscr{G}}^{i}\pqty{\mu^{\prime}})\,,$
(205)

where the scale $\mu_{\mathscr{G}}$ should be the characteristic scale chosen
such that large logarithms in the fixed-order calculation vanish. The counter
terms $\mathcal{Z}_{\mathscr{G}}^{i}(\mu)$ are given by555Note here the
counter terms for polarized quark and gluon FJFs are the same as those of the
unpolarized ones as shown in Chien:2015ctp . For explanation, see the main
text in Section 2.2.:

$\displaystyle\mathcal{Z}_{\mathscr{G}}^{q}(\mu)$
$\displaystyle=1+\frac{\alpha_{s}}{2\pi}C_{F}\bqty{\frac{1}{\epsilon^{2}}+\frac{3}{2\epsilon}+\frac{L}{\epsilon}},$
(206) $\displaystyle\mathcal{Z}_{\mathscr{G}}^{g}(\mu)$
$\displaystyle=1+\frac{\alpha_{s}}{2\pi}C_{A}\bqty{\frac{1}{\epsilon^{2}}+\frac{1}{\epsilon}\frac{\beta_{0}}{2C_{A}}+\frac{L}{\epsilon}}\,.$
(207)

From these results we obtain the anomalous dimension
$\gamma_{\mathscr{G}}^{i}(\mu)$ with the following form:

$\displaystyle\gamma_{\mathscr{G}}^{i}(\mu)=\Gamma_{\text{cusp}}^{i}\pqty{\alpha_{s}}\ln(\frac{\mu^{2}}{(p_{T}R)^{2}})+\gamma^{i}\pqty{\alpha_{s}}\,,$
(208)

where
$\Gamma_{\text{cusp}}^{i}=\sum_{n}\Gamma_{n-1}^{i}\pqty{\frac{\alpha_{s}}{4\pi}}^{n}$
and $\gamma^{i}=\sum_{n}\gamma_{n-1}^{i}\pqty{\frac{\alpha_{s}}{4\pi}}^{n}$.
The lowest-order coefficients can be extracted from the above calculations:

$\displaystyle\Gamma_{0}^{q}=4C_{F},$
$\displaystyle\quad\gamma_{0}^{q}=6C_{F},$ (209)
$\displaystyle\Gamma_{0}^{g}=4C_{A},$
$\displaystyle\quad\gamma_{0}^{g}=2\beta_{0}\,,$ (210)

and higher-order results can be found in Jain:2011xz ; Becher:2006mr ;
Becher:2009th ; Echevarria:2012pw ; Moch:2004pa . After the subtraction of the
UV counter terms specified in Eqs. 206 and 207, the renormalized FJFs
$\Delta_{(T)}\mathscr{G}_{i}^{j}(\omega,R,z_{h},\mu)$ are given by:

$\displaystyle\Delta\mathscr{G}_{q}^{q}(z_{h},p_{T}R,\mu)$
$\displaystyle=\frac{\alpha_{s}}{2\pi}\bigg{\\{}\frac{-1}{\epsilon}\Delta
P_{qq}(z_{h})+C_{F}\delta(1-z_{h})\pqty{\frac{L^{2}}{2}-\frac{\pi^{2}}{12}}+C_{F}(1-z_{h})$
$\displaystyle\quad+\Delta
P_{qq}(z_{h})(2\ln(z_{h})-L)+\frac{3C_{F}L}{2}\delta\pqty{1-z_{h}}$