Polarizing Fragmentation Function (PFF)

• PFF is the leading-twist TMD fragmentation function that describes how unpolarized partons produce transversely polarized spin-½ hadrons through T-odd spin-momentum correlations.
• Its correlator structure distinctively separates it from the Collins function, with a vanishing collinear integration that connects its first moment to twist-3 fragmentation dynamics.
• Phenomenological studies, including Belle and hadronic collision analyses, reveal measurable flavor asymmetries and highlight the growing importance of the gluon PFF in polarized hyperon production.

Searching arXiv for recent and foundational papers on the polarizing fragmentation function, especially $D_{1T}^{\perp}$, $\Lambda$ polarization, TMD classification, and twist-3 matching. The polarizing fragmentation function (PFF), conventionally denoted $D_{1T}^{\perp}$, is the leading-twist transverse-momentum-dependent (TMD) fragmentation function that describes the production of a transversely polarized spin-$\tfrac12$ hadron from an unpolarized parton through a correlation between the hadron transverse spin and the transverse momentum generated in fragmentation (Metz et al., 2016, Chen et al., 2015). In practice it is most closely associated with spontaneous transverse polarization of $\Lambda$ hyperons in $e^+e^-$ annihilation, semi-inclusive deep-inelastic scattering (SIDIS), and hadronic collisions, and it has become the central nonperturbative quantity in modern analyses of polarized hyperon production (Zaccheddu et al., 2022).

1. Definition, nomenclature, and conceptual placement

The defining physical process is

$q \to h^\uparrow + X,$

with the parent quark unpolarized and the observed hadron transversely polarized. The PFF is therefore distinct from the Collins function, which instead describes fragmentation of a transversely polarized quark into an unpolarized hadron. In the standard classification of leading-twist spin-$\tfrac12$ TMD fragmentation functions, $D_{1T}^{\perp}$ occupies the “unpolarized quark $\to$ transversely polarized hadron” slot, whereas $\Lambda$0 occupies the “transversely polarized quark $\Lambda$1 unpolarized hadron” slot (Metz et al., 2016, D'Alesio et al., 2022).

This distinction is central to the interpretation of hyperon polarization. A common misconception is to treat the PFF as a variant of the Collins mechanism. It is not. The spin degree of freedom resides on the hadron side rather than the fragmenting quark side. Another common misconception is to regard it as a purely collinear fragmentation effect. At leading twist, the PFF is intrinsically TMD: its characteristic spin-momentum correlation vanishes after unweighted transverse-momentum integration (Chen et al., 2015).

Broader reviews of fragmentation functions did not always discuss the PFF explicitly by name. For example, one review of unpolarized and polarized fragmentation functions emphasized $\Lambda$2, the Collins function $\Lambda$3, and polarized dihadron fragmentation functions, while placing the PFF only indirectly within the broader TMD taxonomy (Radici, 2011).

Notation	Usage
$\Lambda$4	Standard TMD notation for the polarizing fragmentation function
$\Lambda$5	Older left-right asymmetry notation used in phenomenology
$\Lambda$6	Compact notation used in the quark-quark correlator classification of three-dimensional FFs
$\Lambda$7	Helicity-formalism notation for the same object

2. Correlator structure and formal classification

The formal definition of the PFF arises from the quark-quark fragmentation correlator. In the Amsterdam-style TMD decomposition, the quark correlator projected with $\Lambda$8 contains

$\Lambda$9

This makes explicit that the PFF multiplies the antisymmetric transverse structure $D_{1T}^{\perp}$0, which is the hallmark of a T-odd transverse-spin–transverse-momentum correlation (Metz et al., 2016).

The same review also gives the gluon analogue,

$D_{1T}^{\perp}$1

so the PFF exists for both quark and gluon fragmentation at the TMD level (Metz et al., 2016).

A systematic correlator classification written directly in terms of the quark-quark correlator organizes fragmentation functions into spin-independent, vector-polarization-dependent, and tensor-polarization-dependent sectors. For spin-$D_{1T}^{\perp}$2 hadrons only the spin-independent and vector-polarization-dependent sectors occur, and the PFF belongs to the vector-polarization-dependent part $D_{1T}^{\perp}$3, not to the tensor-polarization sector (Chen et al., 2015). In that notation the relevant term is

$D_{1T}^{\perp}$4

where the paper’s $D_{1T}^{\perp}$5 is the modern $D_{1T}^{\perp}$6 (Chen et al., 2015).

This formal placement has two important consequences. First, the PFF is a genuine single-hadron polarized TMD FF, not a dihadron object. Second, for spin-1 hadrons the ordinary PFF remains in the vector-polarization sector; spin-1-specific tensor-polarized structures are additional, not replacements for $D_{1T}^{\perp}$7 (Chen et al., 2015).

3. Symmetry properties, collinear moments, and twist-3 matching

In the complete TMD classification, the PFF is explicitly identified as leading twist, chiral-even, and naive-$D_{1T}^{\perp}$8-odd (Chen et al., 2015). The chiral-even character follows from its placement in the vector Dirac projection, whereas its naive-$D_{1T}^{\perp}$9-odd character reflects the fact that fragmentation can support nonzero T-odd correlations through final-state phases rather than being forbidden by time reversal. A review of fragmentation functions states explicitly that $\tfrac12$0 and $\tfrac12$1 are naive-T-odd, and also stresses that time reversal does not forbid such FFs because the fragmentation correlator involves out-states (Metz et al., 2016).

The PFF is also subject to a positivity bound,

$\tfrac12$2

which has been used in phenomenological fits (Metz et al., 2016).

A crucial structural fact is that the unweighted collinear integral vanishes. In the three-dimensional classification, the PFF appears in the leading-twist “quark polarization $\tfrac12$3, hadron polarization $\tfrac12$4” sector, and its entry in the integrated column is $\tfrac12$5, i.e. there is no surviving collinear one-dimensional FF after $\tfrac12$6 integration (Chen et al., 2015). This does not mean that the PFF is disconnected from collinear factorization. Rather, its first transverse-momentum moment defines a twist-3 fragmentation correlator. In one formulation,

$\tfrac12$7

and it is this quantity whose QCD evolution is studied in the twist-3 framework (Kang, 2010).

The corresponding leading-order evolution is not closed. It contains a diagonal term proportional to $\tfrac12$8 itself and an off-diagonal mixing term involving the two-variable F-type twist-3 correlator $\tfrac12$9. The diagonal kernel is

$\Lambda$0

which is the same diagonal kernel as for the ordinary unpolarized fragmentation function, while the full evolution remains more complicated because of the off-diagonal twist-3 mixing (Kang, 2010). This is one of the central clarifications of the twist-3 PFF formalism: the PFF moment does not evolve autonomously, but its self-evolution is unpolarized-like.

A second clarification comes from explicit matching between TMD and collinear twist-3 descriptions of polarized hyperon production in SIDIS. In the overlap region

$\Lambda$1

the perturbative large-$\Lambda$2 tail of $\Lambda$3 can be expressed in terms of the same twist-3 fragmentation functions that enter the collinear twist-3 cross section, including purely gluonic correlators (Ikarashi et al., 16 Dec 2025). In that analysis, the PFF tail takes the form

$\Lambda$4

with $\Lambda$5 built from quark twist-3 FFs and $\Lambda$6 from gluon twist-3 FFs (Ikarashi et al., 16 Dec 2025). This establishes that the TMD PFF and the collinear twist-3 fragmentation mechanism are two descriptions of the same QCD effect in different kinematic limits.

4. Measured observables and factorization regimes

The cleanest observables for the PFF are those in which a transversely polarized $\Lambda$7 is produced in an otherwise unpolarized process. In near back-to-back two-hadron production,

$\Lambda$8

the polarization-sensitive structure function is

$\Lambda$9

while the denominator is

$e^+e^-$0

The experimentally used normal polarization is therefore

$e^+e^-$1

or, in $e^+e^-$2-space,

$e^+e^-$3

In this channel the PFF enters linearly in the numerator and the unpolarized FF enters in the denominator, which makes associated production a particularly direct probe (Zaccheddu et al., 2022, D'Alesio et al., 2022).

Single-inclusive $e^+e^-$4 production with thrust-axis reconstruction,

$e^+e^-$5

is also a TMD observable, but it is theoretically more delicate because the thrust measurement produces a non-global observable. The measured transverse polarization is

$e^+e^-$6

and in a TMD factorization treatment the polarized numerator is written in terms of $e^+e^-$7, a Bessel $e^+e^-$8 kernel, a perturbative Sudakov factor, and a non-global logarithm factor $e^+e^-$9 (Zaccheddu et al., 2022). The thrust-axis formalism developed for this process shows that the PFF is the leading-power mechanism when the $q \to h^\uparrow + X,$0 transverse momentum is measured with respect to the thrust axis (Gamberg et al., 2021).

A particularly important conceptual point is that the axis definition matters. If the same single-inclusive $q \to h^\uparrow + X,$1 final state is analyzed using the $q \to h^\uparrow + X,$2 transverse momentum relative to the thrust axis, the relevant leading-power mechanism is the TMD PFF $q \to h^\uparrow + X,$3. If instead the $q \to h^\uparrow + X,$4 transverse momentum is defined relative to the incoming lepton axis in the CM frame, the relevant description is collinear twist-3, and the polarization is generated by the intrinsic twist-3 fragmentation function $q \to h^\uparrow + X,$5 rather than by $q \to h^\uparrow + X,$6 (Gamberg et al., 2021). The difference is not merely kinematic bookkeeping; it is a change of factorization regime.

In SIDIS,

$q \to h^\uparrow + X,$7

the spontaneous transverse $q \to h^\uparrow + X,$8 polarization isolates the PFF through

$q \to h^\uparrow + X,$9

with

$\tfrac12$0

The corresponding electron–jet process at the EIC,

$\tfrac12$1

admits an analogous ratio

$\tfrac12$2

but with the TMD PDF sector and the fragmentation sector deconvolved more cleanly than in ordinary SIDIS (Kang et al., 2021).

5. Phenomenology and extractions

Before direct extractions became possible, the PFF was used chiefly as a phenomenological mechanism for longstanding hyperon polarization puzzles. A single-inclusive $\tfrac12$3 analysis made this mechanism explicit by showing how two fragmentation geometries with opposite spin correlation are weighted differently by the elementary $\tfrac12$4 angular dependence, leading to a net polarization proportional to

$\tfrac12$5

in a simplified treatment (Anselmino et al., 2019). That study did not extract the PFF, but it identified the $\tfrac12$6-differential signature needed for a clean test.

A dedicated extraction of TMD PFFs from back-to-back Belle $\tfrac12$7 data used a Gaussian model and obtained

$\tfrac12$8

with a flavor pattern in which the $\tfrac12$9-quark PFF is positive, while the $D_{1T}^{\perp}$0-, $D_{1T}^{\perp}$1-, and sea-quark PFFs are negative (Callos et al., 2020). In that analysis the associated hadron serves as a flavor analyzer: $D_{1T}^{\perp}$2 primarily weights the $D_{1T}^{\perp}$3 sector, $D_{1T}^{\perp}$4 the $D_{1T}^{\perp}$5 sector, and $D_{1T}^{\perp}$6 the $D_{1T}^{\perp}$7 sector. This established that Belle data already contained nontrivial flavor information.

A subsequent Belle extraction treated both associated production and inclusive $D_{1T}^{\perp}$8-in-jet data within a Gaussian TMD framework and explicitly parameterized four flavor structures, $D_{1T}^{\perp}$9 and sea (D'Alesio et al., 2021). The preferred fit retained

$\to$0

plus the polarized width, giving 8 free parameters. The best-fit values were

$\to$1

$\to$2

$\to$3

$\to$4

$\to$5

The fit quality was

$\to$6

for associated production only and

$\to$7

for the full dataset (D'Alesio et al., 2021). The extracted qualitative picture was positive $\to$8, negative $\to$9, sizeable negative $\Lambda$00, and negative sea.

A renewed analysis implemented a fuller TMD-factorization scheme, combining CSS evolution for the double-hadron channel with SCET-based factorization for the single-inclusive thrust-axis observable, and worked directly with the first $\Lambda$01-space moment $\Lambda$02 (Zaccheddu et al., 2022). In the double-hadron fit, the preferred configurations gave

$\Lambda$03

and

$\Lambda$04

for two different nonperturbative choices, while other combinations still satisfied

$\Lambda$05

For the combined associated-plus-single-inclusive fit the quoted quality was

$\Lambda$06

The cuts

$\Lambda$07

were used to remain in a safer TMD region (Zaccheddu et al., 2022). The main conclusion was that Belle data support a nonzero $\Lambda$08 PFF, but the extraction retains visible model dependence in the nonperturbative transverse sector.

Model calculations have also been informative by failure. A TMD-factorization study using a spectator-diquark model for $\Lambda$09 found agreement in sign and rough size for $\Lambda$10 and $\Lambda$11, but strong disagreement for $\Lambda$12 and $\Lambda$13 (Li et al., 2020). That analysis concluded that Belle data require strong flavor asymmetry, likely opposite signs for $\Lambda$14 and $\Lambda$15, and non-negligible sea fragmentation contributions.

6. Universality, hadronic measurements, and present frontiers

Universality has been a recurring theme in the PFF literature. Fragmentation reviews emphasize that T-odd TMD FFs such as $\Lambda$16 are expected to be universal across $\Lambda$17 and SIDIS because the relevant fragmentation correlators carry future-pointing Wilson lines in both cases (Metz et al., 2016). Modern phenomenology has therefore used Belle-based PFF extractions to make SIDIS and EIC predictions rather than treating the function as process specific.

A dedicated EIC study of transversely polarized $\Lambda$18 production concluded that future SIDIS pseudo-data with a proton beam can significantly reduce the uncertainties of the $\Lambda$19-quark and sea PFFs, while leaving the strange sector much less constrained (Kang et al., 2021). In that framework the PFF enters the spontaneous-polarization observable

$\Lambda$20

and the electron–jet channel provides a particularly attractive environment because the TMD PDF and TMD FF sectors are more cleanly separated than in ordinary SIDIS (Kang et al., 2021). This suggests that the most stringent future universality tests will likely combine Belle/Belle II, EIC SIDIS, and EIC jet-based measurements.

Hadronic collisions add a qualitatively new component: sensitivity to the gluon PFF. The first measurement of transverse polarization of $\Lambda$21 and $\Lambda$22 hyperons inside jets in unpolarized $\Lambda$23 collisions at

$\Lambda$24

explicitly interpreted the observable as generated by the PFF (Collaboration, 22 Sep 2025). For $\Lambda$25, the polarization shows a clear jet-$\Lambda$26 dependence, with a fitted slope

$\Lambda$27

and a sign change from negative to positive from low to high jet $\Lambda$28. For $\Lambda$29, the polarization mostly remains negative, with average value

$\Lambda$30

and slope

$\Lambda$31

Because gluon-initiated subprocesses dominate jet production in part of the measured $\Lambda$32 phase space, these data provide the first experimental leverage on the gluon PFF, which $\Lambda$33 data do not constrain (Collaboration, 22 Sep 2025).

The present frontier is therefore defined by three linked issues. The first is the gluon sector: both the 2025 jet measurement and the 2025 twist-3/TMD matching study indicate that purely gluonic fragmentation correlators are not peripheral but structurally necessary in a complete description (Collaboration, 22 Sep 2025, Ikarashi et al., 16 Dec 2025). The second is nonperturbative modeling: Belle-based extractions remain sensitive to the choice of $\Lambda$34, $\Lambda$35, and related large-$\Lambda$36 ingredients (Zaccheddu et al., 2022). The third is channel compatibility: joint descriptions of associated and single-inclusive $\Lambda$37 data are feasible within a unified TMD framework, but they are not yet free of residual tension or model dependence (D'Alesio et al., 2022).

In that sense, the polarizing fragmentation function is no longer a purely hypothetical explanation for hyperon polarization. It is now a formally classified leading-twist TMD FF, extracted phenomenologically from Belle data, linked to twist-3 fragmentation through evolution and matching, and experimentally probed in both lepton- and hadron-initiated processes. The unresolved questions concern precision, flavor separation, gluon fragmentation, nonperturbative transverse structure, and the quantitative realization of universality across processes.