Fragmentation Energy Correlators (FECs)

• Fragmentation Energy Correlators (FECs) are energy-weighted angular observables defined through fragmentation functions like TMD FFs, DiFFs, and track functions, providing insights into hadronization and confinement.
• They encompass various constructions—including azimuthal EECs, one-point in-jet correlators, semi-inclusive, near-side dihadron, and heavy-flavor correlators—each tailored to specific processes and spin dynamics.
• FECs enable precise studies of jet substructure and nucleon spin via robust factorization frameworks, connecting perturbative calculations with nonperturbative fragmentation and providing a bridge to the confinement regime.

Fragmentation Energy Correlators (FECs) are energy-weighted angular observables whose defining or factorized structure is controlled by fragmentation. In contemporary QCD usage, the label spans several closely related constructions: azimuthal generalizations of the Energy-Energy Correlator (EEC) in $e^+e^-$ and SIDIS, one-point energy correlators inside jets, semi-inclusive energy correlators anchored on an identified hadron, near-side dihadron correlators, track-weighted correlators, and heavy-flavor tagged correlators. What unifies them is that the observable is built from, or factorizes through, TMD fragmentation functions, dihadron fragmentation functions (DiFFs), track functions, EEC jet functions, or heavy-flavor fragmentation matrix elements, so that the correlator becomes a probe of hadronization, confinement, and spin-dependent fragmentation (Kang et al., 2023, Mi et al., 29 Jul 2025, Kang et al., 23 Jul 2025).

1. Terminological scope and taxonomy

The recent literature does not use a single universally standardized name for this class of observables. Instead, closely related objects appear under designations such as “fragmenting energy correlator,” “Semi-Inclusive Energy Correlator,” “one-point energy correlator inside jets,” “EEC-DiFF,” “energy correlators on tracks,” and heavy-flavor energy correlators. This suggests that “Fragmentation Energy Correlator” functions primarily as an umbrella term for energy correlators that are explicitly conditioned on fragmentation, rather than as the name of one unique observable (Liu et al., 2024, Zhu, 1 Sep 2025, Jaarsma et al., 2023).

A recurrent misconception is that fragmentation sensitivity arises only after one abandons inclusive EECs. The cited work instead shows several mechanisms by which fragmentation becomes explicit: by tagging a hadron and measuring energy flow around it; by restricting the hadronic subset, such as charged tracks or heavy-flavor hadrons; by focusing on near-side hadron pairs within a jet; or by introducing azimuthal dependence that projects out Collins-type and related fragmentation structures. Another misconception is that all such observables are back-to-back constructions. Semi-inclusive energy correlators were introduced precisely to derive transverse-momentum moments of TMD distributions and fragmentation functions without enforcing the back-to-back kinematics (Liu et al., 2024).

Realization	Typical regime or process	Fragmentation content
Azimuthal EEC	$e^+e^-$, SIDIS	TMD FFs $D_1$, $H_1^\perp$
One-point EC inside jets	$e^+e^-$, $pp$ jets	siTMDFJF, TMDFFs, in-jet soft function
Near-side dihadron EEC	$e^+e^-$, SIDIS, $pp$	DiFFs, EEC-DiFFs
Semi-inclusive EC	identified-hadron $e^+e^-$, DIS	TMD FFs, SIECs
Energy correlators on tracks	jets at hadron colliders	track functions
Heavy-flavor energy correlator	heavy-hadron pairs in jets	heavy-flavor fragmentation, HQET matrix elements

2. Observable definitions and kinematics

The simplest explicit in-jet realization is the one-point energy correlator introduced for reconstructed jets. In $e^+e^-$, it is defined by

$e^+e^-$0

where $e^+e^-$1 is the hadron energy fraction inside the jet and $e^+e^-$2 is the angle between the hadron and the jet axis. The observable therefore measures the radial energy profile inside the jet and directly probes the transverse-momentum structure of fragmentation in the small-angle regime $e^+e^-$3 (Mi et al., 29 Jul 2025).

A second archetype is the azimuthal-angle-dependent EEC in $e^+e^-$4, which supplements the usual polar-angle variable with an azimuth $e^+e^-$5. In the back-to-back limit,

$e^+e^-$6

and the factorized correlator takes the form

$e^+e^-$7

The $e^+e^-$8 term is the defining Collins-type modulation, while the angle-independent part measures the unpolarized energy flow baseline (Kang et al., 2023).

Semi-inclusive constructions anchor the correlator on a tagged hadron rather than on two freely summed energy deposits. In the fragmenting energy correlator formulation, one defines operator-valued weighted cross sections with energy-flow insertions around the observed hadron, and in semi-inclusive $e^+e^-$9 annihilation one may equivalently work with the cumulative operator

$D_1$0

which measures the fraction of energy carried by radiation within angle $D_1$1 around the tagged hadron (Liu et al., 2024, Zhu, 1 Sep 2025).

Near-side dihadron observables replace the jet axis or external reference direction by the opening angle of a hadron pair. The standard variable is

$D_1$2

and in the dihadron framework one writes

$D_1$3

so that the energy correlator constrains the relative transverse momentum $D_1$4 of the pair for fixed $D_1$5 and energy fractions (Kang et al., 23 Jul 2025).

3. Factorization and nonperturbative building blocks

The defining property of FECs is that their factorization theorem is governed by fragmentation objects. In the Collins-type azimuthal EEC, the fundamental building blocks are the unpolarized EEC jet function $D_1$6 and the Collins-type EEC jet function $D_1$7. These are related, respectively, to the unpolarized TMD fragmentation function $D_1$8 and the Collins fragmentation function $D_1$9, with perturbative and nonperturbative Sudakov factors and rapidity evolution incorporated in impact-parameter space. The resulting decomposition into $H_1^\perp$0 and $H_1^\perp$1 shows that the azimuthal modulation directly isolates fragmentation dynamics (Kang et al., 2023).

Inside jets, the one-point EC is encoded in an EC jet function constructed from the semi-inclusive TMD fragmenting jet function: $H_1^\perp$2 The underlying siTMDFJF factorizes into a hard inside-jet function, a TMDFF $H_1^\perp$3, and an in-jet soft function $H_1^\perp$4. This makes the one-point correlator a fragmentation-weighted angular observable whose $H_1^\perp$5-dependence is governed by collinear fragmentation and soft recoil inside the jet (Mi et al., 29 Jul 2025).

Near-side dihadron formulations introduce a distinct nonperturbative object, the EEC-DiFF: $H_1^\perp$6 This quantity is an angularly weighted integral of the DiFF over internal dihadron phase space and is explicitly designed to enter the near-side EEC factorization formula (Kang et al., 23 Jul 2025).

A closely related construction appears in the nonperturbative proof of collinear EEC factorization, where the EEC jet function is written directly in terms of transverse-momentum-sensitive DiFFs and single-hadron fragmentation functions. In that framework, the jet function $H_1^\perp$7 is a weighted integral over $H_1^\perp$8 with the EEC angular constraint imposed by a $H_1^\perp$9-function. This identifies the collinear EEC jet function as a universal nonperturbative fragmentation object, valid not only in the OPE region but also in the confinement transition region (Lee et al., 15 Jul 2025).

Track-based realizations add a different nonperturbative layer. Energy correlators on tracks are computed by replacing calorimetric energy weights with charged-energy weights, which introduces track functions $e^+e^-$0 and $e^+e^-$1 and their moments $e^+e^-$2. In this formalism, the bulk of a correlator is reweighted by first moments such as $e^+e^-$3, while contact terms involve higher moments such as $e^+e^-$4. This is the canonical example in which a correlator remains analytically tractable but ceases to be purely perturbative because the charged-hadron selection is fragmentation-sensitive (Jaarsma et al., 2023).

4. Collinear, near-side, and confinement regimes

The small-angle regime is central to the modern theory of FECs. For projected $e^+e^-$5-point energy correlators in $e^+e^-$6, the relevant effective scale is

$e^+e^-$7

where $e^+e^-$8 is the projected largest-angle variable. As $e^+e^-$9, the perturbative partonic regime gives way to the confinement transition and then to the free hadron regime. In that setting, the leading nonperturbative correction is governed by a universal matrix element $pp$0, and the leading nonperturbative corrections for projected $pp$1-point energy correlators are described by the same universal parameter for any $pp$2 (Lee et al., 2024).

The light-ray OPE refines this picture by showing that the leading power corrections obey a classical scaling behavior that is violated at the quantum level. The leading nonperturbative contribution to projected correlators scales as $pp$3, while the dependence on the hard scale $pp$4 is calculable in perturbation theory through anomalous dimensions of twist-2 light-ray operators. The analysis identifies two nonperturbative quantities describing the fragmentation of quarks and gluons, which function as the quark and gluon sectors of the leading power correction (Chen et al., 2024).

Near-side dihadron formulations partition the small-angle domain more explicitly into a free hadron region, a transition region, and a quark/gluon region. In the free hadron and transition regions, the near-side EEC is expressed directly in terms of the EEC-DiFF. At large relative transverse momentum $pp$5, the EEC-DiFF reproduces the $pp$6 perturbative EEC jet function. This supplies an explicit bridge between hadron-level and parton-level descriptions and implies that a formal matching procedure can interpolate continuously between them (Kang et al., 23 Jul 2025).

The nonperturbative proof of collinear EEC factorization sharpens the interpretation of the confinement transition. In that formulation, the collinear EEC is written as

$pp$7

where the same universal jet function $pp$8 governs both the perturbative OPE region and the nonperturbative region $pp$9. This gives a rigorous description of the confinement transition region in terms of DiFFs and their weighted moments (Lee et al., 15 Jul 2025).

These results support a broad interpretation of FECs as observables that not only resolve perturbative jet substructure, but also provide a controlled interpolation to the hadronization regime. That interpretation is especially strong for near-side and projected correlators, where the governing scale is directly tied to intra-jet transverse momentum rather than to the full hard scale.

5. Spin, flavor, and nucleon structure

Spin dependence is one of the most developed applications of FECs. In the azimuthal EEC of $e^+e^-$0 and SIDIS, the unpolarized piece probes $e^+e^-$1, while the $e^+e^-$2 Collins-type term probes $e^+e^-$3. In SIDIS, the corresponding structure functions include

$e^+e^-$4

so that unpolarized, transversity-Collins, and Sivers-type information is all accessible within a single energy-correlator framework (Kang et al., 2023).

The first measurement of one- and two-point energy correlators within jets in transversely polarized proton-proton collisions at $e^+e^-$5 GeV was reported by STAR. Those observables quantify the energy-weighted angular distribution of single hadrons and hadron pairs within jets, respectively, and sizable spin-dependent asymmetries were observed for $e^+e^-$6, $e^+e^-$7, and $e^+e^-$8 pairs. The analysis emphasized that projecting the fragmentation dynamics onto Mellin moments provides sensitivity to nucleon transversity while minimizing uncertainties from nonperturbative fragmentation functions. This places within-jet FECs in direct correspondence with transversity phenomenology and three-dimensional nucleon tomography (Collaboration, 16 Apr 2026).

A collinear-factorization implementation of near-side EECs in the dihadron fragmentation framework further simplifies the transversity program. In that approach, one defines EEC-weighted DiFFs $e^+e^-$9 and $pp$0, and obtains SIDIS and $pp$1 observables structurally analogous to ordinary collinear factorization expressions for PDFs and single-hadron fragmentation functions. The paper argues that this removes complications from intrinsic transverse momentum modeling or resonance modeling in the dihadron invariant mass distribution, while preserving direct sensitivity to $pp$2 through $pp$3 (Kang et al., 30 Apr 2026).

The same dihadron strategy has now been extended to transversely polarized proton-proton collisions. There the near-side EEC for two hadrons inside a jet factorizes into unpolarized PDFs, transversity PDFs, hard functions, and EEC-weighted DiFFs $pp$4 and $pp$5. Numerical predictions show remarkable agreement with the recent STAR measurement and indicate that data at large jet transverse momentum have a slight preference for extractions of transversity that are consistent with lattice QCD computations of the nucleon tensor charges (Kang et al., 2 Jul 2026).

Heavy-flavor FECs provide a complementary flavor-tagged direction. The heavy-flavor energy correlator

$pp$6

restricts the energy-flow operators to heavy-flavor hadrons and factorizes into a hard function vector and a heavy-flavor jet function vector. In the region $pp$7, the angular distribution is controlled by the massive splitting function $pp$8, while the normalization becomes proportional to products of fragmentation fractions $pp$9. This separates partonic heavy-pair production from subsequent heavy-hadron formation and provides a flavor-specific FEC in both vacuum and medium (Barata et al., 26 Aug 2025).

Longitudinal-spin energy correlators extend the program further. In polarized DIS, the spin-dependent energy correlator in the current fragmentation region is connected with longitudinally polarized TMDs, whereas in the target fragmentation region it is connected with helicity nucleon energy correlators (NECs). This establishes a direct correspondence between spin-dependent energy correlation patterns and both TMD and NEC descriptions of proton spin, with joint $e^+e^-$0LL/NNLL predictions for current and target fragmentation regions (Gao et al., 22 Sep 2025).

6. Phenomenology, measurements, and open directions

Several phenomenological advantages recur across the literature. First, normalized correlators can substantially reduce scale dependence. For the one-point EC inside jets, normalization by the inclusive jet cross section cancels jet-radius logarithms $e^+e^-$1, leaving residual scale sensitivity mainly through the jet scale $e^+e^-$2. The resulting observable is primarily sensitive to the jet scale and to TMD fragmentation dynamics, especially gluon TMDFFs in $e^+e^-$3 jets (Mi et al., 29 Jul 2025).

Second, Breit-frame DIS EECs and their fragmentation-sensitive generalizations are robust against laboratory-frame pseudorapidity cuts in the back-to-back limit. The Breit-frame construction correlates final-state hadrons with the incoming proton direction using the weight

$e^+e^-$4

and in the back-to-back region the observable is sensitive to universal TMD PDFs and fragmentation functions within standard TMD factorization. This makes DIS EECs and related FECs attractive for precision studies at future lepton-hadron facilities (Li et al., 2021).

Third, semi-inclusive $e^+e^-$5 correlators provide a unified hadronization program across all polar angles. In the large-angle regime they are controlled by TMD FFs and EEC jet functions; in the small-angle jet fragmentation region they are controlled by SIECs with DGLAP-type evolution. Joint $e^+e^-$6LL/NNLL predictions have been obtained for these complementary regimes, and the resulting observables provide direct, theoretically controlled access to the QCD dynamics underlying hadronization (Zhu, 1 Sep 2025).

Experimental coverage is now broad. FEC-like observables have been analyzed or proposed in $e^+e^-$7, SIDIS, $e^+e^-$8, and heavy-ion-relevant heavy-flavor contexts. They have been measured within jets in transversely polarized proton-proton collisions at RHIC, compared with PYTHIA 8 in $e^+e^-$9 and $e^+e^-$0 jet studies, and developed as part of future EIC programs in both unpolarized and spin-dependent settings (Collaboration, 16 Apr 2026, Mi et al., 29 Jul 2025).

Several open problems follow directly from the cited work. Formal matching between EEC-DiFFs and perturbative jet functions beyond $e^+e^-$1, systematic treatment of inhomogeneous evolution terms in the dihadron sector, higher-order spin-dependent calculations, and broader universality tests across $e^+e^-$2, SIDIS, and $e^+e^-$3 remain active directions. The same is true for extensions to higher-point correlators, other hadron species, and more differential flavor- and spin-tagged observables. This suggests that FECs are best viewed not as a single observable, but as a developing framework in which energy correlator technology and fragmentation theory are being merged into a common precision language for hadronization, confinement, and QCD spin structure.