Polarized Fragmenting Jet Functions

• Polarized fragmenting jet functions are spin-dependent hadron-in-jet correlators that extend ordinary fragmentation functions by retaining detailed polarization information of the initiating parton and observed hadron.
• They match onto universal polarized fragmentation functions through perturbative coefficients in both exclusive and semi-inclusive frameworks, evolving via multiplicative renormalization or DGLAP-type equations.
• Applications include probing QCD spin dynamics and enabling 3D nucleon imaging via jet substructure, advancing the extraction of helicity, transversity, and polarizing fragmentation functions.

Searching arXiv for recent and foundational work on polarized fragmenting jet functions and related hadron-in-jet formalisms. Polarized fragmenting jet functions are spin-dependent hadron-in-jet correlators that generalize ordinary fragmentation functions to the environment of a reconstructed jet and generalize unpolarized fragmenting jet functions by retaining polarization information for the initiating parton, the observed hadron, or both. In the contemporary SCET formulation, they appear in both semi-inclusive and exclusive jet production, with collinear versions relevant when only the hadron longitudinal momentum fraction is measured and TMD versions relevant when the hadron transverse momentum relative to the standard jet axis is also resolved (Kang et al., 2023). The subject sits at the intersection of jet substructure, QCD factorization, polarized fragmentation, and spin-sensitive hadron tomography. Earlier unpolarized fragmenting jet function and semi-inclusive fragmenting jet function formalisms established the factorization architecture and renormalization structure (Procura et al., 2011, Kang et al., 2016), while recent work provided a complete polarized extension for inclusive and exclusive jet production (Kang et al., 2023). A parallel line of work in quarkonium physics showed that even unpolarized fragmenting jet functions can diagnose polarization-sensitive production mechanisms indirectly, especially through NRQCD channel discrimination in $J/\psi$-in-jet observables (Baumgart et al., 2014, Dai et al., 2017).

1. Definition and scope

The fragmenting jet function was introduced in SCET as an object $\mathcal{G}_i^h(s,z)$ describing a jet initiated by parton $i$, with invariant mass squared $s$, containing an identified hadron $h$ carrying large light-cone momentum fraction $z$ (Procura et al., 2011). Its defining role is to interpolate between the inclusive jet function $J_i(s)$, which resolves jet structure but not an identified hadron, and the ordinary fragmentation function $D_i^h(z)$, which resolves hadronization but not measured jet structure (Procura et al., 2011).

In exclusive jet production, the polarized generalization is formulated as a spin-dependent analogue of the standard fragmenting jet function, with matching relation

$$\Delta_{(T)} \mathscr G_i^h(\omega,R,z,\mu) = \sum_j \int_z^1 \frac{dx}{x}\, \Delta_{(T)}\mathscr J_{ij}(\omega,R,x,\mu)\, \Delta_{(T)}D_j^h\!\left(\frac{z}{x},\mu\right) +\mathcal O\!\left(\frac{\Lambda_{\rm QCD}^2}{\omega^2\tan^2(R/2)}\right),$$

where $\Delta$ denotes longitudinal polarization transfer and $\mathcal{G}_i^h(s,z)$0 transverse polarization transfer (Zhao, 2023, Kang et al., 2023).

In semi-inclusive jet production, the corresponding polarized objects are semi-inclusive polarized fragmenting jet functions, or in the terminology of the recent literature, polarized jet fragmentation functions. They depend on two longitudinal momentum fractions,

$\mathcal{G}_i^h(s,z)$1

where $\mathcal{G}_i^h(s,z)$2 is the fraction of the initiating parton momentum carried by the jet and $\mathcal{G}_i^h(s,z)$3 is the fraction of the jet momentum carried by the hadron (Kang et al., 2023, Kang et al., 2016). This distinction between exclusive and semi-inclusive kinematics is foundational: in exclusive production the jet takes the full initiating-parton momentum so $\mathcal{G}_i^h(s,z)$4, whereas in semi-inclusive production out-of-jet hard radiation makes $\mathcal{G}_i^h(s,z)$5 nontrivial (Kang et al., 2023).

The modern subject therefore comprises three closely related layers. First is the original exclusive FJF formalism for identified hadrons in measured jets (Procura et al., 2011). Second is the semi-inclusive FJF formalism adapted to inclusive collider jet measurements, with DGLAP-type evolution in $\mathcal{G}_i^h(s,z)$6 and explicit jet-radius dependence (Kang et al., 2016). Third is the polarized extension, in which spin labels are retained for hadrons and, where relevant, for fragmenting quarks or gluons, both in collinear and TMD regimes (Kang et al., 2023).

2. SCET operator structure and polarization decomposition

The SCET building blocks are the gauge-invariant collinear quark and gluon fields

$\mathcal{G}_i^h(s,z)$7

with collinear Wilson line

$\mathcal{G}_i^h(s,z)$8

and light-cone decomposition

$\mathcal{G}_i^h(s,z)$9

These definitions underpin both unpolarized and polarized formalisms (Kang et al., 2023).

For polarized hadrons inside jets, the hadron momentum and spin are parameterized as

$i$0

(Kang et al., 2023). In the semi-inclusive collinear case, the quark correlators are defined by

$i$1

corresponding respectively to unpolarized fragmentation, longitudinal spin transfer, and transverse spin transfer from a transversely polarized quark to a transversely polarized spin-$i$2 hadron (Kang et al., 2023). The gluon correlators admit unpolarized and longitudinally polarized channels,

$i$3

but there is no gluon transversity channel for spin-$i$4 hadrons (Kang et al., 2023).

When the hadron transverse momentum relative to the standard jet axis is measured, the TMD quark correlator is decomposed into the full leading-twist set

$i$5

inside the jet (Zhao, 2023, Kang et al., 2023). These are the direct in-jet analogues of ordinary TMD fragmentation functions, now promoted to jet-dependent objects (Kang et al., 2023). The gluon parent polarization structures are unpolarized, circularly polarized, and linearly polarized, with hadron polarization again taken as unpolarized, longitudinal, or transverse (Zhao, 2023).

A crucial structural point is that the polarized fragmenting jet function framework is not limited to one polarization channel. It includes unpolarized, helicity, and transversity transfer in the collinear sector, and all leading-twist quark and gluon TMD spin structures in the transverse-momentum-sensitive sector (Zhao, 2023, Kang et al., 2023).

3. Factorization, matching, and renormalization

The central factorization statement is that polarized fragmenting jet functions match onto the corresponding universal polarized fragmentation functions at the jet scale. For semi-inclusive collinear polarized FJFs,

$i$6

with perturbative matching coefficients $i$7 and $i$8 linking polarized jet functions to polarized collinear fragmentation functions (Zhao, 2023). The exclusive counterpart is

$i$9

(Zhao, 2023, Kang et al., 2023).

For semi-inclusive unpolarized FJFs, the corresponding matching formula is

$s$0

valid up to power corrections

$s$1

(Kang et al., 2016). This unpolarized relation is the direct precursor of the polarized versions.

Renormalization and evolution differ sharply between semi-inclusive and exclusive formulations. In the semi-inclusive case, renormalization is nonlocal in $s$2, and the RG equation is timelike DGLAP: $s$3 (Zhao, 2023). For the unpolarized semi-inclusive FJF the exact counterpart is

$s$4

(Kang et al., 2016). The two-variable structure is essential: renormalization acts in $s$5, while matching onto fragmentation functions acts in $s$6 (Kang et al., 2016).

In the exclusive case, by contrast, renormalization is multiplicative. The exclusive polarized FJF obeys

$s$7

with anomalous dimension

$s$8

(Zhao, 2023). For exclusive collinear polarized FJFs in the inclusive/exclusive unified treatment, the anomalous dimensions are polarization independent and equal to those of the corresponding exclusive jet functions, because the UV poles are proportional to $s$9 and arise from soft limits insensitive to hadron polarization (Kang et al., 2023).

4. Collinear and TMD polarized fragmenting jet functions

The collinear polarized FJF regime applies when only the hadron longitudinal momentum fraction in the jet is measured. In this regime the relevant nonperturbative inputs are collinear polarized fragmentation functions such as the helicity FF and transversity FF (Kang et al., 2023). The polarized splitting kernels entering the evolution are

$h$0

$h$1

$h$2

and for transversity

$h$3

(Kang et al., 2023). These govern semi-inclusive polarized DGLAP evolution in the jet variable $h$4.

The TMD polarized FJF regime applies when the hadron transverse momentum relative to the standard jet axis is resolved in the region

$h$5

(Kang et al., 2023, Zhao, 2023). In this case the standard jet axis introduces sensitivity to in-jet soft recoil, and the factorization acquires an explicit soft function. For the unpolarized channel,

$h$6

(Zhao, 2023). The same structural pattern extends to all polarized TMD FJFs (Zhao, 2023, Kang et al., 2023).

The in-jet TMD FF is defined by multiplying the unsubtracted TMD FF by the in-jet soft factor, and its natural Collins–Soper scale is modified by the jet radius,

$h$7

(Kang et al., 2023). This is one of the central conceptual results of the polarized formalism: the in-jet TMD FFs have the same evolution structure as ordinary TMD FFs, but with the natural Collins–Soper scale set by the jet scale rather than the full hard scale (Kang et al., 2023).

A practical implication is that polarized hadron-in-jet measurements probe universal polarized FFs and TMD FFs in a new perturbative environment. This suggests a jet-substructure route to extracting helicity, transversity, Collins-type, and polarizing fragmentation functions that is complementary to SIDIS and $h$8 annihilation (Kang et al., 2023, Zhao, 2023).

5. Observables, processes, and phenomenological applications

Polarized fragmenting jet functions enter cross sections for processes of the type

$h$9

(Zhao, 2023). The fully differential cross section admits a rich azimuthal decomposition with structure functions $z$0, $z$1, $z$2, $z$3, and others, each isolating a specific polarized jet-fragmentation structure (Zhao, 2023, Kang et al., 2023).

Two benchmark asymmetries illustrate the phenomenological role of polarized FJFs. In semi-inclusive polarized $z$4 collisions, the transversity-sensitive asymmetry

$z$5

probes the transversity PDF $z$6 and the transversity TMD FJF $z$7, hence the transversity fragmentation function of the hadron (Kang et al., 2023). In exclusive polarized $z$8 production, the asymmetry

$z$9

accesses the worm-gear TMD PDF $J_i(s)$0 and the longitudinally polarized exclusive TMD FJF $J_i(s)$1, hence the helicity FF $J_i(s)$2 (Kang et al., 2023).

Earlier semi-inclusive polarized jet-fragmentation studies concentrated on $J_i(s)$3 production inside jets. For longitudinally polarized $J_i(s)$4, the asymmetry

$J_i(s)$5

was proposed as a probe of the helicity FF $J_i(s)$6 through the jet-level helicity function $J_i(s)$7 (Kang et al., 2020). For transversely polarized $J_i(s)$8, the observable

$J_i(s)$9

probes the polarizing TMD fragmentation function $D_i^h(z)$0 through the in-jet function $D_i^h(z)$1 (Kang et al., 2020).

The thesis-level formulation broadened these observables considerably. It included inclusive hadron-in-jet in $D_i^h(z)$2 or $D_i^h(z)$3, longitudinal spin transfer to $D_i^h(z)$4 in jets, transverse $D_i^h(z)$5 polarization in jets, and back-to-back $D_i^h(z)$6jet at the EIC with hadron in jet (Zhao, 2023). A notable conceptual claim is that polarized jet fragmentation functions provide a new route to 3D nucleon imaging because the hadron-in-jet measurement separates the transverse momentum associated with the incoming parton side from the fragmentation side (Zhao, 2023).

6. Relation to quarkonium polarization and indirect polarization sensitivity

The phrase “polarized fragmenting jet functions” also has an indirect, conceptually distinct usage in quarkonium phenomenology. Two papers on quarkonium inside jets are highly relevant but do not define operator-level polarized FJFs (Baumgart et al., 2014, Dai et al., 2017). Instead, they use ordinary unpolarized FJFs as diagnostics for polarization-sensitive NRQCD channel composition.

In the 2014 analysis of quarkonium production mechanisms with jet substructure, the observable was a jet of fixed energy $D_i^h(z)$7, fixed cone size $D_i^h(z)$8, containing a $D_i^h(z)$9 with energy fraction $$\Delta_{(T)} \mathscr G_i^h(\omega,R,z,\mu) = \sum_j \int_z^1 \frac{dx}{x}\, \Delta_{(T)}\mathscr J_{ij}(\omega,R,x,\mu)\, \Delta_{(T)}D_j^h\!\left(\frac{z}{x},\mu\right) +\mathcal O\!\left(\frac{\Lambda_{\rm QCD}^2}{\omega^2\tan^2(R/2)}\right),$$0, described by the unpolarized FJF $$\Delta_{(T)} \mathscr G_i^h(\omega,R,z,\mu) = \sum_j \int_z^1 \frac{dx}{x}\, \Delta_{(T)}\mathscr J_{ij}(\omega,R,x,\mu)\, \Delta_{(T)}D_j^h\!\left(\frac{z}{x},\mu\right) +\mathcal O\!\left(\frac{\Lambda_{\rm QCD}^2}{\omega^2\tan^2(R/2)}\right),$$1 (Baumgart et al., 2014). The central factorization formula was

$$\Delta_{(T)} \mathscr G_i^h(\omega,R,z,\mu) = \sum_j \int_z^1 \frac{dx}{x}\, \Delta_{(T)}\mathscr J_{ij}(\omega,R,x,\mu)\, \Delta_{(T)}D_j^h\!\left(\frac{z}{x},\mu\right) +\mathcal O\!\left(\frac{\Lambda_{\rm QCD}^2}{\omega^2\tan^2(R/2)}\right),$$2

(Baumgart et al., 2014). By comparing NRQCD channels $$\Delta_{(T)} \mathscr G_i^h(\omega,R,z,\mu) = \sum_j \int_z^1 \frac{dx}{x}\, \Delta_{(T)}\mathscr J_{ij}(\omega,R,x,\mu)\, \Delta_{(T)}D_j^h\!\left(\frac{z}{x},\mu\right) +\mathcal O\!\left(\frac{\Lambda_{\rm QCD}^2}{\omega^2\tan^2(R/2)}\right),$$3, $$\Delta_{(T)} \mathscr G_i^h(\omega,R,z,\mu) = \sum_j \int_z^1 \frac{dx}{x}\, \Delta_{(T)}\mathscr J_{ij}(\omega,R,x,\mu)\, \Delta_{(T)}D_j^h\!\left(\frac{z}{x},\mu\right) +\mathcal O\!\left(\frac{\Lambda_{\rm QCD}^2}{\omega^2\tan^2(R/2)}\right),$$4, $$\Delta_{(T)} \mathscr G_i^h(\omega,R,z,\mu) = \sum_j \int_z^1 \frac{dx}{x}\, \Delta_{(T)}\mathscr J_{ij}(\omega,R,x,\mu)\, \Delta_{(T)}D_j^h\!\left(\frac{z}{x},\mu\right) +\mathcal O\!\left(\frac{\Lambda_{\rm QCD}^2}{\omega^2\tan^2(R/2)}\right),$$5, and $$\Delta_{(T)} \mathscr G_i^h(\omega,R,z,\mu) = \sum_j \int_z^1 \frac{dx}{x}\, \Delta_{(T)}\mathscr J_{ij}(\omega,R,x,\mu)\, \Delta_{(T)}D_j^h\!\left(\frac{z}{x},\mu\right) +\mathcal O\!\left(\frac{\Lambda_{\rm QCD}^2}{\omega^2\tan^2(R/2)}\right),$$6, the paper arrived at the robust prediction that if the depolarizing $$\Delta_{(T)} \mathscr G_i^h(\omega,R,z,\mu) = \sum_j \int_z^1 \frac{dx}{x}\, \Delta_{(T)}\mathscr J_{ij}(\omega,R,x,\mu)\, \Delta_{(T)}D_j^h\!\left(\frac{z}{x},\mu\right) +\mathcal O\!\left(\frac{\Lambda_{\rm QCD}^2}{\omega^2\tan^2(R/2)}\right),$$7 matrix element dominates, then the gluon FJF will diminish with increasing energy for fixed momentum fraction $$\Delta_{(T)} \mathscr G_i^h(\omega,R,z,\mu) = \sum_j \int_z^1 \frac{dx}{x}\, \Delta_{(T)}\mathscr J_{ij}(\omega,R,x,\mu)\, \Delta_{(T)}D_j^h\!\left(\frac{z}{x},\mu\right) +\mathcal O\!\left(\frac{\Lambda_{\rm QCD}^2}{\omega^2\tan^2(R/2)}\right),$$8, and $$\Delta_{(T)} \mathscr G_i^h(\omega,R,z,\mu) = \sum_j \int_z^1 \frac{dx}{x}\, \Delta_{(T)}\mathscr J_{ij}(\omega,R,x,\mu)\, \Delta_{(T)}D_j^h\!\left(\frac{z}{x},\mu\right) +\mathcal O\!\left(\frac{\Lambda_{\rm QCD}^2}{\omega^2\tan^2(R/2)}\right),$$9 (Baumgart et al., 2014).

The 2017 follow-up extended this idea from the FJF itself to measurable cross sections in $\Delta$0 dijets at $\Delta$1 TeV (Dai et al., 2017). The schematic factorization was

$\Delta$2

(Dai et al., 2017). Its central conclusion was that if a suitably normalized $\Delta$3-in-jet cross section decreases with jet energy $\Delta$4 at fixed $\Delta$5, especially for $\Delta$6, then the depolarizing $\Delta$7 must be the dominant channel, supporting an explanation for the observed lack of prompt $\Delta$8 polarization at high $\Delta$9 (Dai et al., 2017).

These works are significant for the encyclopedia topic because they clarify a potential misconception. They are not papers about explicitly polarized fragmenting jet functions in the operator sense. Rather, they show that ordinary unpolarized FJFs and FJF-based jet observables can be used to infer polarization-sensitive information through channel dominance in NRQCD (Dai et al., 2017).

7. Historical development and open directions

The subject developed through several distinct stages. The original SCET formulation introduced the fragmenting jet function $\mathcal{G}_i^h(s,z)$00 as the correct leading-power semi-inclusive object for an identified hadron inside a jet of measured invariant mass and established the replacement rule

$\mathcal{G}_i^h(s,z)$01

in factorization theorems (Procura et al., 2011). The semi-inclusive generalization then adapted the framework to inclusive collider jet measurements by introducing the semi-inclusive fragmenting jet function $\mathcal{G}_i^h(s,z)$02, showing that it obeys timelike DGLAP evolution and resums single logarithms of $\mathcal{G}_i^h(s,z)$03 up to NLL$\mathcal{G}_i^h(s,z)$04 (Kang et al., 2016).

The explicitly polarized extension first appeared under the terminology “polarized jet fragmentation functions” (Kang et al., 2020), and was subsequently developed into a broader SCET-based framework with operator definitions, collinear and TMD factorization, NLO matching coefficients, and phenomenology for $\mathcal{G}_i^h(s,z)$05, $\mathcal{G}_i^h(s,z)$06, RHIC, LHC, HERA, and especially the EIC (Zhao, 2023). The most complete one-loop treatment of polarized fragmenting jet functions in both inclusive and exclusive jet production, including longitudinal and transverse polarization, was then given in 2023 (Kang et al., 2023).

Several open directions are strongly suggested by the literature. One is the extraction of poorly known polarized FFs, especially helicity, transversity, and polarizing fragmentation functions, from hadron-in-jet observables (Kang et al., 2020, Kang et al., 2023). Another is the use of polarized jet fragmentation for 3D nucleon imaging, where simultaneous sensitivity to $\mathcal{G}_i^h(s,z)$07 and $\mathcal{G}_i^h(s,z)$08 provides cleaner separation of initial- and final-state transverse dynamics than in standard SIDIS (Zhao, 2023). A further plausible implication is that dihadron and multi-hadron extensions of polarized fragmenting jet functions may become particularly important, given the known role of interference DiFFs and recursive spin-dependent hadronization mechanisms in generating spin asymmetries and jet handedness (Radici, 2011, Kerbizi et al., 2018).

A final conceptual distinction remains important. The operator-level polarized FJF formalism treats spin-resolved hadrons inside reconstructed jets through SCET factorization and matching onto polarized FFs (Kang et al., 2023). Model studies of polarized hadronization, such as the NJL-jet treatment of the Collins function and recursive string-based models of polarized quark fragmentation, do not define polarized FJFs themselves, but they are highly informative about the nonperturbative spin-dependent fragmentation dynamics that such functions encode (Matevosyan et al., 2012, Kerbizi et al., 2018). This suggests that the mature subject of polarized fragmenting jet functions is both a factorization framework and a bridge between perturbative jet physics and nonperturbative spin-dependent hadronization.