Semi-Inclusive Energy Correlators (SIECs)

• SIECs are energy-flow observables that condition on a trigger (hadron, jet, etc.) while remaining inclusive over unobserved radiation, providing a clear probe of QCD dynamics.
• They are applied across various processes—from SIDIS and e⁺e⁻ annihilation to heavy-ion and proton collisions—using TMD factorization and fragmentation operators.
• SIECs offer actionable insights into hadronization, spin-dependent fragmentation, and jet-medium interactions by linking measured angular energy flow to underlying QCD structure.

Semi-Inclusive Energy Correlators (SIECs) are a family of energy-flow observables in which one conditions on a specified hadron, jet, or hard trigger and measures energy-weighted angular correlations while remaining inclusive over the rest of the final state. In current usage, the term covers several closely related constructions: nucleon and fragmenting energy correlators in semi-inclusive processes, hadron-tagged and dihadron near-side correlators in $e^+e^-$ annihilation and SIDIS, per-jet correlators in proton and heavy-ion collisions, and bias-minimized jet correlators designed to remove trigger-induced distortions. Across these settings, SIECs serve as probes of hadronization, TMD structure, spin-dependent fragmentation, and jet-medium interactions (Liu et al., 2024, Zhu, 1 Sep 2025, Andres et al., 2024).

1. Terminology, scope, and conceptual structure

The defining feature of a SIEC is semi-inclusiveness: one identifies or conditions on part of the event, but sums inclusively over unobserved radiation. In the formulation that introduced the term explicitly, Semi-Inclusive Energy Correlators “gauge the correlation between the examined hadron and the surrounding radiations,” with the examined object taken either to be the incoming nucleon in the target fragmentation region or an identified final-state hadron in the current fragmentation region. This construction was proposed as a way to probe the multidimensional structure of hadrons and to relate angular energy flow directly to transverse-momentum moments of TMD parton distributions and fragmentation functions without enforcing back-to-back kinematics (Liu et al., 2024).

A second, explicitly semi-inclusive realization appears in single-hadron production in $e^+e^-$ annihilation. There the tagged hadron is inserted at operator level through $a_h^\dagger a_h$, and the energy correlator is taken between that hadron and the rest of the event. The same framework distinguishes two complementary limits: the back-to-back Sudakov regime, where the observable is described by TMD fragmentation functions, and the small-angle jet fragmentation region, where new fragmentation-like SIEC operators govern the angular energy pattern around the tagged hadron (Zhu, 1 Sep 2025).

The label is also used more broadly than its original hadron-centric definition. In heavy-ion jet substructure, the CMS two-point correlator measured inside inclusive jets is operationally a per-jet correlator conditioned on reconstructed jet $p_T$ and centrality; although the companion note does not use the term “SIEC” explicitly, it states that this structure “fits naturally into a semi-inclusive framework,” since the correlator is measured conditional on the presence of a jet in a specified kinematic bin and is inclusive over the rest of the event (Andres et al., 2024).

This breadth of usage underlies a common misconception: SIECs are not a single universally fixed observable. Rather, they form a class of conditional energy correlators whose precise definition depends on the trigger, the measured subset of final-state radiation, and the angular variable retained. What unifies them is the semi-inclusive logic and the use of energy-flow weighting to retain direct sensitivity to QCD radiation patterns (Liu et al., 2024, Andres et al., 2024).

2. Operator definitions and basic observables

At the most general level, SIECs are built from the energy-flow operator. In the hadron-structure formulation, the energy flow operator ${\cal E}(\Omega)$ measures the energy flowing to null infinity around direction $\Omega$, and acting on a final state $|X\rangle$ gives

$${\cal E}(\Omega)\,|X\rangle = E(\Omega)\,|X\rangle, \qquad E(\Omega) = \sum_{h\in X} E_h\,\delta(\Omega - \Omega_h)\,|X\rangle.$$

The corresponding $n$-point nucleon and fragmenting correlators insert one or more such operators into matrix elements of gauge-invariant quark fields, producing nonperturbative objects $f_{q,n}(x,\{\Omega_i\})$ and $e^+e^-$0 that encode energy flow around the incoming nucleon or an identified hadron, respectively (Liu et al., 2024).

In semi-inclusive $e^+e^-$1 annihilation with a tagged hadron $e^+e^-$2, the core operator-level object is

$e^+e^-$3

which is semi-inclusive because it is exclusive in the tagged hadron but inclusive over all other hadrons. The phenomenological energy-flow operator is often written in cumulant form as

$e^+e^-$4

so the observable measures the total energy within angle $e^+e^-$5 around the tagged hadron, normalized by the hadron energy (Zhu, 1 Sep 2025).

A closely related small-angle construction is the near-side dihadron energy-energy correlator. For two observed hadrons $e^+e^-$6, the relevant variable is

$e^+e^-$7

which in the near-side limit becomes the correlator variable $e^+e^-$8. The associated nonperturbative objects are the EEC-weighted dihadron fragmentation functions

$e^+e^-$9

constructed from $a_h^\dagger a_h$0 and $a_h^\dagger a_h$1 with an energy weight $a_h^\dagger a_h$2 and a delta constraint fixing $a_h^\dagger a_h$3. After the constrained integration, they depend effectively on the single combination $a_h^\dagger a_h$4 (Kang et al., 30 Apr 2026).

Jet-based versions fit the same pattern. The one-point energy correlator inside jets,

$a_h^\dagger a_h$5

measures energy deposition at angle $a_h^\dagger a_h$6 relative to the jet axis with weight $a_h^\dagger a_h$7. Although it is a one-point object rather than a pair correlator, it is explicitly described as a prototype of a Semi-Inclusive Energy Correlator because it is inclusive over hadron species and longitudinal fractions but differential in jet kinematics and in-jet transverse structure (Mi et al., 29 Jul 2025).

3. Factorization, evolution, and nonperturbative structure

One of the central developments in the SIEC program is the identification of distinct factorization regimes. In semi-inclusive $a_h^\dagger a_h$8 annihilation with a tagged hadron, the back-to-back limit $a_h^\dagger a_h$9 is controlled by TMD factorization, while the small-angle jet fragmentation region $p_T$0 is governed by collinear factorization in terms of SIEC operators $p_T$1 and $p_T$2. In the Sudakov region, the semi-inclusive correlator factorizes into a hard function, an EEC TMD jet function, and a physical TMD fragmentation function; in the jet fragmentation region, the cross section is written directly in terms of singlet and non-singlet SIECs with the same coefficient functions as in standard single-inclusive annihilation up to a factor $p_T$3. The same work derives Collins–Soper evolution on the TMD side and a modified DGLAP equation for SIECs in Mellin space, and reports joint N$p_T$4LL/NNLL predictions (Zhu, 1 Sep 2025).

Near-side dihadron SIECs realize a different simplification. Their leading-order factorization formulas in SIDIS and $p_T$5 have the same schematic structure as conventional collinear extractions of PDFs and fragmentation functions, but with $p_T$6 and $p_T$7 replacing $p_T$8 and $p_T$9. In SIDIS,

${\cal E}(\Omega)$0

while the corresponding ${\cal E}(\Omega)$1 observable factorizes into ${\cal E}(\Omega)$2 and ${\cal E}(\Omega)$3 pieces. The formal gain is that the nonperturbative input becomes one-variable, avoiding both intrinsic-${\cal E}(\Omega)$4 modeling and the complicated resonance structure of ${\cal E}(\Omega)$5-dependent DiFF fits (Kang et al., 30 Apr 2026).

Azimuthal-dependent energy correlators in SIDIS and ${\cal E}(\Omega)$6 provide another factorization channel that is often interpreted as semi-inclusive. In the back-to-back region, the relevant building blocks are the unpolarized EEC jet function ${\cal E}(\Omega)$7 and the Collins-type EEC jet function ${\cal E}(\Omega)$8, which are constructed from TMD fragmentation functions ${\cal E}(\Omega)$9 and Collins functions $\Omega$0. In SIDIS they enter structure functions such as $\Omega$1, $\Omega$2, and $\Omega$3, thereby turning the azimuthal dependence of the EEC into a probe of TMD PDFs and spin-dependent fragmentation (Kang et al., 2023).

A more recent fragmentation-based formulation defines boost-invariant Fragmentation Energy Correlators (FECs) as nonperturbative generalizations of fragmentation functions with explicit angular-energy information. The Collins-type quark FEC is singled out as the chiral-odd component describing the azimuthal asymmetry in fragmentation of a transversely polarized quark, and the corresponding semi-inclusive DIS hard coefficient has been computed at next-to-leading order for the quark non-singlet sector. This places Collins-type semi-inclusive correlators on the same footing as collinear fragmentation observables with a systematically improvable perturbative expansion (Cao et al., 23 Sep 2025).

4. Jet-based SIECs in proton and heavy-ion collisions

Jet-based correlators provide the most direct bridge between semi-inclusive energy-flow theory and collider measurements. ALICE defines the $\Omega$4-point energy correlator inside reconstructed jets as a per-jet observable,

$\Omega$5

with the two-point case reducing to the EEC and the three-point case defined through the largest pairwise angle in a triplet. Because the normalization is per selected jet, the measurement is already semi-inclusive in the experimental sense. ALICE reports EECs in inclusive jets and D-tagged jets in pp, inclusive jets in p–Pb, and discusses E3C/EEC as a probe of anomalous dimensions and $\Omega$6. In p–Pb the ratio $\Omega$7 is approximately unity at high jet $\Omega$8, while in the lowest jet-$\Omega$9 bin it shows about $|X\rangle$0 suppression at small angles and about $|X\rangle$1 enhancement at large angles; in D-tagged jets the EEC is suppressed relative to inclusive jets, consistently with a dead-cone interpretation (Liang-Gilman, 28 Jun 2025).

In heavy-ion collisions, the main conceptual issue is selection bias. Semi-analytic studies of the EEC in a quark-gluon plasma show that inclusive heavy-ion jet samples at fixed reconstructed $|X\rangle$2 are biased because the initial jet energy $|X\rangle$3 exceeds $|X\rangle$4 by the lost energy $|X\rangle$5. In the formal description,

$|X\rangle$6

so even universal hadronization produces an apparent shift of the small-angle transition because the relevant hard scale differs between pp and A–A. This work also emphasizes that medium-induced radiation enhances the correlator at intermediate angles, while energy loss and quenching weights suppress the large-angle signal; it further argues that semi-inclusive triggers such as $|X\rangle$7-tagged jets are the natural way to reduce this bias (Andres et al., 2024, Barata et al., 2023).

The most explicit heavy-ion SIEC construction is the bias-minimized ratio introduced for the CMS Pb–Pb measurement. The observable is

$|X\rangle$8

where $|X\rangle$9 is an unbiasing function derived from the measured E2C spectrum and the steeply falling jet spectrum. The derivation uses the approximate relation

$${\cal E}(\Omega)\,|X\rangle = E(\Omega)\,|X\rangle, \qquad E(\Omega) = \sum_{h\in X} E_h\,\delta(\Omega - \Omega_h)\,|X\rangle.$$0

evaluated with $${\cal E}(\Omega)\,|X\rangle = E(\Omega)\,|X\rangle, \qquad E(\Omega) = \sum_{h\in X} E_h\,\delta(\Omega - \Omega_h)\,|X\rangle.$$1, and the accompanying Letter reports that Pythia and Herwig reduce the impact of selection bias in the E2C by an order of magnitude while retaining sensitivity to other medium modifications (Andres et al., 2024).

Applied to CMS Pb–Pb data, the companion note shows that the raw E2C in 0–10% centrality and $${\cal E}(\Omega)\,|X\rangle = E(\Omega)\,|X\rangle, \qquad E(\Omega) = \sum_{h\in X} E_h\,\delta(\Omega - \Omega_h)\,|X\rangle.$$2 GeV is shifted to smaller $${\cal E}(\Omega)\,|X\rangle = E(\Omega)\,|X\rangle, \qquad E(\Omega) = \sum_{h\in X} E_h\,\delta(\Omega - \Omega_h)\,|X\rangle.$$3 than the pp baseline, with suppression at $${\cal E}(\Omega)\,|X\rangle = E(\Omega)\,|X\rangle, \qquad E(\Omega) = \sum_{h\in X} E_h\,\delta(\Omega - \Omega_h)\,|X\rangle.$$4 and apparent differences in the deep hadronization region $${\cal E}(\Omega)\,|X\rangle = E(\Omega)\,|X\rangle, \qquad E(\Omega) = \sum_{h\in X} E_h\,\delta(\Omega - \Omega_h)\,|X\rangle.$$5. After dividing by $${\cal E}(\Omega)\,|X\rangle = E(\Omega)\,|X\rangle, \qquad E(\Omega) = \sum_{h\in X} E_h\,\delta(\Omega - \Omega_h)\,|X\rangle.$$6, the intermediate-angle suppression is significantly reduced, the ratio becomes flat for $${\cal E}(\Omega)\,|X\rangle = E(\Omega)\,|X\rangle, \qquad E(\Omega) = \sum_{h\in X} E_h\,\delta(\Omega - \Omega_h)\,|X\rangle.$$7, and the large-angle enhancement for $${\cal E}(\Omega)\,|X\rangle = E(\Omega)\,|X\rangle, \qquad E(\Omega) = \sum_{h\in X} E_h\,\delta(\Omega - \Omega_h)\,|X\rangle.$$8 becomes more pronounced. The same note reports that the wide-angle enhancement decreases with decreasing centrality, that the 50–90% class becomes completely flat after applying $${\cal E}(\Omega)\,|X\rangle = E(\Omega)\,|X\rangle, \qquad E(\Omega) = \sum_{h\in X} E_h\,\delta(\Omega - \Omega_h)\,|X\rangle.$$9, and that in central events the enhancement decreases and shifts to slightly smaller $n$0 as jet $n$1 increases (Andres et al., 2024).

5. Spin, flavor, and nucleon-structure applications

One of the most active SIEC applications is transversity extraction. Near-side dihadron energy-energy correlators in SIDIS and $n$2 were proposed as a simplified, collinear-factorization-based strategy for accessing the transversity PDF $n$3. The key observables are the SIDIS asymmetry

$n$4

and the $n$5 asymmetry

$n$6

The reported phenomenology gives $n$7 at $n$8 and $n$9 GeV, $f_{q,n}(x,\{\Omega_i\})$0 in typical COMPASS/HERMES kinematics, and $f_{q,n}(x,\{\Omega_i\})$1 at Belle for $f_{q,n}(x,\{\Omega_i\})$2, while the unpolarized near-side $f_{q,n}(x,\{\Omega_i\})$3 correlator shows no visible resonance structure despite originating from $f_{q,n}(x,\{\Omega_i\})$4-dependent DiFFs (Kang et al., 30 Apr 2026).

Azimuthal EECs in SIDIS provide a parallel SIEC route to three-dimensional nucleon structure. The corresponding cross section contains modulations $f_{q,n}(x,\{\Omega_i\})$5, $f_{q,n}(x,\{\Omega_i\})$6, and others, with structure functions proportional to combinations such as $f_{q,n}(x,\{\Omega_i\})$7 and $f_{q,n}(x,\{\Omega_i\})$8. The resulting Collins and Sivers asymmetries were studied at EIC kinematics, and the Sivers asymmetry is reported to reach a few percent at small $f_{q,n}(x,\{\Omega_i\})$9, while Collins-type asymmetries become sizable when hadron species are separated rather than summed over $e^+e^-$00 (Kang et al., 2023).

Quantum-number-resolved and azimuthally dependent correlators extend this logic. By restricting the sums over hadrons to subsets $e^+e^-$01, one obtains correlators $e^+e^-$02 and corresponding EEC jet functions $e^+e^-$03 and $e^+e^-$04. In the back-to-back limit, the nonperturbative information in the OPE region collapses to a small set of moments $e^+e^-$05 and $e^+e^-$06, while the angular dependence remains perturbatively controlled. This is why charge-tagged or flavor-tagged semi-inclusive correlators can expose Collins, Sivers, Boer–Mulders, and worm-gear structures more clearly than fully inclusive EECs (Kang et al., 2023).

A broader implication is that SIECs naturally encode both unpolarized and spin-dependent tomography. That conclusion is explicit in the original SIEC proposal, which relates angular moments of nucleon and fragmenting correlators to transverse-momentum moments of TMD PDFs and TMD FFs, including Sivers and Collins effects, and emphasizes that the extraction involves only one TMD at a time rather than a convolution of two TMDs (Liu et al., 2024).

6. Generalizations, computation, and open issues

The SIEC program is no longer confined to two-point observables. The one-point EC inside jets is formulated through semi-inclusive TMD fragmenting jet functions,

$e^+e^-$07

with $e^+e^-$08 factorized into a hard function, a TMD fragmentation function, and an in-jet soft function. The resulting normalized EC jet function is computed at NNLL for global logarithms, LL for non-global logarithms, includes nonperturbative modeling through $e^+e^-$09, and is argued to be especially sensitive to gluon TMDFFs in pp collisions (Mi et al., 29 Jul 2025).

High-point correlators and their parametrization are also being reworked to make SIECs computationally feasible in high-multiplicity environments. A new parametrization of projected $e^+e^-$10-point energy correlators replaces the traditional largest pairwise angle $e^+e^-$11 by a radius $e^+e^-$12 measured from a “special” particle, yielding the cumulative form

$e^+e^-$13

where $e^+e^-$14 is the energy fraction inside a disk of radius $e^+e^-$15 around $e^+e^-$16. This reduces the computational scaling from $e^+e^-$17 for a jet with $e^+e^-$18 particles to $e^+e^-$19, independently of $e^+e^-$20, and preserves orientation information in resolved correlators. The authors state that the theoretical difference from traditional parametrizations first enters at NNLL through the jet function, while the simpler phase space is expected to ease calculations. That is particularly relevant for heavy-ion and semi-inclusive studies, where multiplicities are large and higher-point correlators are otherwise prohibitively expensive (Alipour-fard et al., 2024).

Several limitations remain central to the field. In the heavy-ion unbiasing analysis, $e^+e^-$21 is derived from the same E2C data, so uncertainties in E2C and $e^+e^-$22 are highly correlated; the note explicitly states that these correlations are not yet propagated and that the displayed error bands are likely overestimated. The same analysis also notes the absence of released bin-to-bin correlations from CMS and the restriction to charged particles within $e^+e^-$23 jets (Andres et al., 2024). In QGP theory, medium response is omitted in some semi-analytic EEC calculations, and the available medium-induced kernels are still limited by approximations such as harmonic-oscillator multiple scattering or first-order opacity (Andres et al., 2024, Barata et al., 2023).

A second misconception is terminological rather than technical. Because the same semi-inclusive logic appears in tagged-hadron fragmentation, near-side dihadron observables, per-jet correlators, one-point jet energy flow, and bias-minimized heavy-ion ratios, “SIEC” should be read as a framework rather than a single formula. What is shared across these constructions is a conditional energy correlator, a normalization that preserves semi-inclusive meaning, and a factorization structure that ties measured angular energy flow to hard scattering, fragmentation, and medium dynamics. This suggests that future progress will come from unifying these presently parallel lines of work rather than reducing them to one canonical definition (Liu et al., 2024, Zhu, 1 Sep 2025, Andres et al., 2024).