---
title: Polarized Fragmenting Jet Functions
url: https://www.emergentmind.com/topics/polarized-fragmenting-jet-functions
type: topic
---

# Polarized Fragmenting Jet 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 [2311.00672]. 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 [1102.0489], [1606.07063], while recent work provided a complete polarized extension for inclusive and exclusive jet production [2311.00672]. 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 [1406.2295], [1707.08629].

## 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 \(z\) [1102.0489]. 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 [1102.0489].

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 \(\Delta_T\) transverse polarization transfer [2309.10838], [2311.00672].

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,
\[
z=\frac{\omega_J}{\omega},\qquad z_h=\frac{\omega_h}{\omega_J},
\]
where \(z\) is the fraction of the initiating parton momentum carried by the jet and \(z_h\) is the fraction of the jet momentum carried by the hadron [2311.00672], [1606.07063]. This distinction between exclusive and semi-inclusive kinematics is foundational: in exclusive production the jet takes the full initiating-parton momentum so \(z=1\), whereas in semi-inclusive production out-of-jet hard radiation makes \(z\) nontrivial [2311.00672].

The modern subject therefore comprises three closely related layers. First is the original exclusive FJF formalism for identified hadrons in measured jets [1102.0489]. Second is the semi-inclusive FJF formalism adapted to inclusive collider jet measurements, with DGLAP-type evolution in \(z\) and explicit jet-radius dependence [1606.07063]. 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 [2311.00672].

## 2. SCET operator structure and polarization decomposition

The SCET building blocks are the gauge-invariant collinear quark and gluon fields
\[
\chi_n = W_n^\dagger \xi_n \, , \qquad \mathcal{B}_{n\perp}^\mu = \frac{1}{g}\bigl[W_n^\dagger i D_{n\perp}^\mu W_n\bigr] \, ,
\]
with collinear Wilson line
\[
W_n(x) = \sum_{\textrm{perms} \exp\!\left(-g \frac{1}{\overline{n}\cdot \mathcal{P} \overline{n}\cdot A_n(x)\right)
\]
and light-cone decomposition
\[
p^\mu = p^- \frac{n^\mu}{2} + p^+ \frac{\overline{n}^\mu}{2} + p_\perp^\mu \, .
\]
These definitions underpin both unpolarized and polarized formalisms [2311.00672].

For polarized hadrons inside jets, the hadron momentum and spin are parameterized as
\[
p_h = \left[\frac{M_h^2}{p_h^-},\, p_h^-,\, \boldsymbol{0}\right], \qquad S_h = \left[- \lambda_h \frac{M_h}{p_h^-},\, \lambda_h \frac{p_h^-}{M_h},\, \boldsymbol{S}_{h\perp}\right]
\]
[2311.00672]. In the semi-inclusive collinear case, the quark correlators are defined by
\[
\mathcal{G}_q^h(z,z_h,p_TR,\mu), \qquad \Delta \mathcal{G}_q^h(z,z_h,p_TR,\mu), \qquad \Delta_T \mathcal{G}_q^h(z,z_h,p_TR,\mu),
\]
corresponding respectively to unpolarized fragmentation, longitudinal spin transfer, and transverse spin transfer from a transversely polarized quark to a transversely polarized spin-\(\tfrac12\) hadron [2311.00672]. The gluon correlators admit unpolarized and longitudinally polarized channels,
\[
\mathcal{G}_g^h(z,z_h,p_TR,\mu), \qquad \Delta \mathcal{G}_g^h(z,z_h,p_TR,\mu),
\]
but there is no gluon transversity channel for spin-\(\tfrac12\) hadrons [2311.00672].

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
\[
\mathcal D_1,\ \mathcal D_{1T}^{\perp},\ \mathcal G_{1L},\ \mathcal G_{1T},\ \mathcal H_1,\ \mathcal H_1^\perp,\ \mathcal H_{1L}^\perp,\ \mathcal H_{1T}^\perp
\]
inside the jet [2309.10838], [2311.00672]. These are the direct in-jet analogues of ordinary TMD fragmentation functions, now promoted to jet-dependent objects [2311.00672]. The gluon parent polarization structures are unpolarized, circularly polarized, and linearly polarized, with hadron polarization again taken as unpolarized, longitudinal, or transverse [2309.10838].

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 [2309.10838], [2311.00672].

## 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,
\[
\Delta_{(T)} \mathcal{G}_i^h (z, z_h, \omega_J,\mu) = \sum_j \int_{z_h}^1 \frac{dz_h'}{z_h'} \Delta_{(T)}\mathcal J_{ij}(z,z_h',\omega_J,\mu)\, \Delta_{(T)}D^{h/j}\!\left(\frac{z_h}{z_h'},\mu\right),
\]
with perturbative matching coefficients \(\Delta\mathcal J_{ij}\) and \(\Delta_T\mathcal J_{ij}\) linking polarized jet functions to polarized collinear fragmentation functions [2309.10838]. The exclusive counterpart is
\[
\Delta_{(T)} \mathscr G_i^h(\omega,R,z,\mu) = \sum_j \int_z^1 \frac{dx}{x}\, \Delta_{(T)}\mathscr J_{ij}(\omega,R,x,\mu)\, \Delta_{(T)}D_j^h\!\left(\frac{z}{x},\mu\right) +\mathcal O\!\left(\frac{\Lambda_{\rm QCD}^2}{\omega^2\tan^2(R/2)}\right)
\]
[2309.10838], [2311.00672].

For semi-inclusive unpolarized FJFs, the corresponding matching formula is
\[
{\cal G}_i^h(z,z_h,\omega_J,\mu) = \sum_j \int_{z_h}^1 \frac{dz_h'}{z_h'} {\mathcal J}_{ij}\left(z,z_h',\omega_J,\mu\right) D_j^h\left(\frac{z_h}{z_h'},\mu\right),
\]
valid up to power corrections
\[
{\mathcal O}\!\left(\frac{\Lambda^2_{\rm QCD}{\omega^2\tan^2(R/2)}\right)
\]
[1606.07063]. 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 \(z\), and the RG equation is timelike DGLAP:
\[
\mu \frac{d}{d\mu}\, \Delta_{(T)} \mathcal G_i^h(z,z_h,\omega_J,\mu) = \frac{\alpha_s(\mu)}{\pi} \sum_k \int_z^1 \frac{dz'}{z'}\, \Delta_{(T)}P_{ji}\!\left(\frac{z}{z'}\right) \Delta_{(T)} \mathcal G_i^h(z',z_h,\omega_J,\mu)
\]
[2309.10838]. For the unpolarized semi-inclusive FJF the exact counterpart is
\[
\mu \frac{d}{d\mu} {\cal G}_i^h(z,z_h, \omega_J, \mu) = \frac{\alpha_s(\mu)}{\pi} \sum_k \int_z^1  \frac{dz'}{z'} P_{ki}\left(\frac{z}{z'} \right) {\cal G}_k^h(z',z_h, \omega_J,\mu)
\]
[1606.07063]. The two-variable structure is essential: renormalization acts in \(z\), while matching onto fragmentation functions acts in \(z_h\) [1606.07063].

In the exclusive case, by contrast, renormalization is multiplicative. The exclusive polarized FJF obeys
\[
\mu \frac{d}{d\mu}\Delta_{(T)}\mathscr G_i^h(\omega,R,z,\mu) = \gamma_{\mathscr G^i}(\mu)\, \Delta_{(T)}\mathscr G_i^h(\omega,R,z,\mu),
\]
with anomalous dimension
\[
\gamma_{\mathscr G^i}(\mu) = \Gamma_{\rm cusp}^i(\alpha_s)\ln\!\frac{\mu^2}{\omega^2\tan^2(R/2)} +\gamma^i(\alpha_s)
\]
[2309.10838]. 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 \(\delta(1-z_h)\) and arise from soft limits insensitive to hadron polarization [2311.00672].

## 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 [2311.00672]. The polarized splitting kernels entering the evolution are
\[
\Delta P_{qq}(z)= C_F\left[\frac{1+z^2}{(1-z)_+}+\frac{3}{2}\delta(1-z)\right],
\]
\[
\Delta P_{gq}(z)=C_F(2-z), \qquad \Delta P_{qg}(z)=T_F(2z-1),
\]
\[
\Delta P_{gg}(z) = 2C_A\left[\frac{1}{(1-z)_+}-2z+1\right]+\frac{\beta_0}{2}\delta(1-z),
\]
and for transversity
\[
\Delta_T P_{qq}(z) = C_F\left[\frac{2z}{(1-z)_+}+\frac{3}{2}\delta(1-z)\right]
\]
[2311.00672]. These govern semi-inclusive polarized DGLAP evolution in the jet variable \(z\).

The TMD polarized FJF regime applies when the hadron transverse momentum relative to the standard jet axis is resolved in the region
\[
\Lambda_{\rm QCD}\lesssim j_\perp \ll p_T R
\]
[2311.00672], [2309.10838]. 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,
\[
\mathcal D_1^{h/c}(z,z_h,\omega_JR,\boldsymbol j_\perp,\mu) = \hat H_{c\to i}^U(z,\omega_JR,\mu) \int d^2\boldsymbol k_\perp\, d^2\boldsymbol\lambda_\perp\, \delta^{(2)}(z_h\boldsymbol\lambda_\perp+\boldsymbol k_\perp-\boldsymbol j_\perp) D_1^{h/i}(z_h,\boldsymbol k_\perp,\mu,\nu)\, S_i(\boldsymbol\lambda_\perp,\mu,\nu R)
\]
[2309.10838]. The same structural pattern extends to all polarized TMD FJFs [2309.10838], [2311.00672].

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,
\[
\sqrt{\zeta_J}=p_TR
\]
[2311.00672]. 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 [2311.00672].

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 \(e^+e^-\) annihilation [2311.00672], [2309.10838].

## 5. Observables, processes, and phenomenological applications

Polarized fragmenting jet functions enter cross sections for processes of the type
\[
p(p_A,S_A)+\big(p(p_B)/e(p_\ell)\big)\to (\text{jet}(\eta_J,p_{JT},R)\,h(z_h,\boldsymbol j_\perp,S_h))+X
\]
[2309.10838]. The fully differential cross section admits a rich azimuthal decomposition with structure functions \(F_{UU,U}\), \(F_{LU,L}\), \(F_{UU,T}^{\sin(\hat\phi_h-\hat\phi_{S_h})}\), \(F_{TU,T}^{\cos(\phi_S-\hat\phi_{S_h})}\), and others, each isolating a specific polarized jet-fragmentation structure [2309.10838], [2311.00672].

Two benchmark asymmetries illustrate the phenomenological role of polarized FJFs. In semi-inclusive polarized \(pp\) collisions, the transversity-sensitive asymmetry
\[
A_{TU,T}^{\cos(\phi_S-\hat\phi_{S_h})} \equiv \frac{F_{TU,T}^{\cos(\phi_S-\hat\phi_{S_h})}{F_{UU,U}}
\]
probes the transversity PDF \(h_1\) and the transversity TMD FJF \(\mathcal H_1^{h/q}\), hence the transversity fragmentation function of the hadron [2311.00672]. In exclusive polarized \(ep\) production, the asymmetry
\[
A_{TU,L}^{\cos(\phi_q-\phi_S)} \equiv \frac{F_{TU,L}^{\cos(\phi_q-\phi_S)}{F_{UU,U}}
\]
accesses the worm-gear TMD PDF \(g_{1T}\) and the longitudinally polarized exclusive TMD FJF \(\mathscr G_{1L}^{h/q}\), hence the helicity FF \(G_{1L}^{h/q}\) [2311.00672].

Earlier semi-inclusive polarized jet-fragmentation studies concentrated on \(\Lambda\) production inside jets. For longitudinally polarized \(\Lambda\), the asymmetry
\[
D_{LL}^{\text{jet}\Lambda} = \frac{F_{LU,L}}{F_{UU,U}}
\]
was proposed as a probe of the helicity FF \(G_1^{\Lambda/c}\) through the jet-level helicity function \(\mathcal G_1^{\Lambda/c}\) [2005.02398]. For transversely polarized \(\Lambda\), the observable
\[
P_\Lambda= \frac{F_{UU,T}^{\sin(\hat\phi_h-\hat\phi_{S_h})}}{F_{UU,U}}
\]
probes the polarizing TMD fragmentation function \(D_{1T}^{\perp\,\Lambda/c}\) through the in-jet function \(\mathcal D_{1T}^{\perp\,\Lambda/c}\) [2005.02398].

The thesis-level formulation broadened these observables considerably. It included inclusive hadron-in-jet in \(pp\) or \(ep\), longitudinal spin transfer to \(\Lambda\) in jets, transverse \(\Lambda\) polarization in jets, and back-to-back \(e+\)jet at the EIC with hadron in jet [2309.10838]. 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 [2309.10838].

## 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 [1406.2295], [1707.08629]. 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 \(E\), fixed cone size \(R\), containing a \(J/\psi\) with energy fraction \(z=E_\psi/E_{\rm jet}\), described by the unpolarized FJF \({\cal G}_i^\psi(E,R,z,\mu)\) [1406.2295]. The central factorization formula was
\[
{\cal G}_i^\psi(E, R, z,\mu) = \sum_j \int_z^1 \frac{dy}{y} {\cal J}_{ij}(E, R, y, \mu) D_{j\to\psi}\left(\frac{z}{y},\mu\right)\times\left[1 + {\cal O}\left(\frac{m_\psi^2}{4E^2\tan^2(R/2)}\right)\right]
\]
[1406.2295]. By comparing NRQCD channels \({}^3S_1^{(1)}\), \({}^3S_1^{(8)}\), \({}^1S_0^{(8)}\), and \({}^3P_J^{(8)}\), the paper arrived at the robust prediction that if the depolarizing \({}^1S_0^{(8)}\) matrix element dominates, then the gluon FJF will diminish with increasing energy for fixed momentum fraction \(z\), and \(z>0.5\) [1406.2295].

The 2017 follow-up extended this idea from the FJF itself to measurable cross sections in \(pp\to\) dijets at \(\sqrt s=13\) TeV [1707.08629]. The schematic factorization was
\[
\frac{d\sigma}{dEdz}=\sum_{a,b,i,j} H_{ab\rightarrow ij}\otimes f_{a/p}\otimes f_{b/p} \otimes J_{j} \otimes S \otimes \mathcal{G}^{\psi}_{i}(E,R,z,\mu)
\]
[1707.08629]. Its central conclusion was that if a suitably normalized \(J/\psi\)-in-jet cross section decreases with jet energy \(E\) at fixed \(z\), especially for \(z>0.5\), then the depolarizing \(^{1}S_{0}^{[8]}\) must be the dominant channel, supporting an explanation for the observed lack of prompt \(J/\psi\) polarization at high \(p_T\) [1707.08629].

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 [1707.08629].

## 7. Historical development and open directions

The subject developed through several distinct stages. The original SCET formulation introduced the fragmenting jet function \(\mathcal G_i^h(s,z)\) as the correct leading-power semi-inclusive object for an identified hadron inside a jet of measured invariant mass and established the replacement rule
\[
J_i(s,\mu) \;\to\; \frac{1}{2(2\pi)^3}\,\mathcal G_i^h(s,z,\mu)\,dz
\]
in factorization theorems [1102.0489]. The semi-inclusive generalization then adapted the framework to inclusive collider jet measurements by introducing the semi-inclusive fragmenting jet function \({\mathcal G}_i^h(z,z_h,\omega_J,R,\mu)\), showing that it obeys timelike DGLAP evolution and resums single logarithms of \(R\) up to NLL\(_R\) [1606.07063].

The explicitly polarized extension first appeared under the terminology “polarized jet fragmentation functions” [2005.02398], and was subsequently developed into a broader SCET-based framework with operator definitions, collinear and TMD factorization, NLO matching coefficients, and phenomenology for \(pp\), \(ep\), RHIC, LHC, HERA, and especially the EIC [2309.10838]. 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 [2311.00672].

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 [2005.02398], [2311.00672]. Another is the use of polarized jet fragmentation for 3D nucleon imaging, where simultaneous sensitivity to \(q_T\) and \(j_\perp\) provides cleaner separation of initial- and final-state transverse dynamics than in standard SIDIS [2309.10838]. 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 [1111.3383], [1802.00962].

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 [2311.00672]. 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 [1205.5813], [1802.00962]. 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.

Source: https://www.emergentmind.com/topics/polarized-fragmenting-jet-functions