EEC Quark Jet Function Overview

• The EEC quark jet function is the universal collinear kernel that governs quark-initiated radiation in small-angle energy-energy correlators and underpins SCET factorization.
• It is formulated in both momentum and position space, with perturbative calculations and nonperturbative modifications that capture the transition to hadronization.
• Phenomenological extensions, including multiplicity conditioning and flavor resolution, highlight its role in decoding angular energy distributions in pp, DIS, and heavy-ion collisions.

Searching arXiv for papers on the EEC quark jet function and closely related EEC jet-function literature. Fetching targeted arXiv results for collinear EEC factorization, nonperturbative EEC jet functions, multiplicity-conditioned EEC jet functions, and universal quark jet functions in SCET. The EEC quark jet function is the collinear, flavor-resolved object that encodes how a quark-initiated jet contributes to the energy-energy correlator (EEC) in factorized descriptions of angular energy flow. In the literature summarized here, the term appears in several closely related senses: as a formal SCET jet function governing the small-angle EEC in $e^+e^-$ annihilation, as a nonperturbative modification of that function in the transition to hadronization, as a multiplicity-conditioned generalization, and as a phenomenological decomposition of in-jet EEC distributions into pair counting, angular structure, and energy weighting in hadronic and heavy-ion collisions. Across these settings, the common role of the quark EEC jet function is to isolate universal or approximately universal collinear quark dynamics from hard production and from soft or medium-induced effects (Dixon et al., 2019).

1. Definition and scope

In the collinear limit of the $e^+e^-$ EEC, where $z=(1-\cos\chi)/2\to 0$, the cumulant factorizes as

$$\Sigma(z,\ln\frac{Q^2}{\mu^2},\mu) = \int_0^1 dx\, x^2\, \vec J\!\left(\ln\frac{z x^2 Q^2}{\mu^2},\mu\right)\cdot \vec H\!\left(x,\frac{Q^2}{\mu^2},\mu\right),$$

with $\vec J=\{J_q,J_g\}$ the EEC jet-function vector in flavor space and $\vec H$ the corresponding hard function (Dixon et al., 2019). In this formulation, the quark jet function $J_q$ is the universal object governing the collinear singular behavior of the EEC for quark-initiated radiation.

The same physics is also expressed in momentum space through

$$\Sigma(z,Q) = \int_0^1 dx\, x^2\, \mathbf{J}(x q_T,\mu)\cdot \mathbf{H}(x,Q,\mu),$$

with $q_T=\sqrt{z}\,Q$ and $\mathbf{J}=(J_q,J_g)$ (Herrmann et al., 23 Jul 2025). This suggests that the quark EEC jet function may be viewed either as a logarithmic-space object governing resummation or as a transverse-momentum-space object describing collinear energy flow.

A distinct but related usage appears in jet EEC studies in hadronic collisions, where the in-jet EEC is defined as

$e^+e^-$0

with

$e^+e^-$1

In that context, the papers do not define a formal perturbative quark jet function, but they rewrite the observable into factors that play a jet-function-like role phenomenologically (Shen et al., 2024).

2. Operator and factorization formulations

The formal SCET definition of the quark EEC jet function is given as a gauge-invariant matrix element with the EEC measurement inserted,

$e^+e^-$2

where $e^+e^-$3 is the collinear quark field (Dixon et al., 2019). In this formulation, $e^+e^-$4 is the probability density, weighted by energy, for a quark jet to contain two energy depositions separated by an angle smaller than $e^+e^-$5.

A later momentum-space treatment rewrites the resummed EEC in transverse position space,

$e^+e^-$6

where $e^+e^-$7 is the position-space differential jet function (Herrmann et al., 23 Jul 2025). This formulation is used to introduce a nonperturbative jet function multiplicatively in $e^+e^-$8-space.

The factorization logic is closely connected to timelike DGLAP evolution. The hard function obeys

$e^+e^-$9

while the jet function obeys

$z=(1-\cos\chi)/2\to 0$0

with the same timelike splitting matrix $z=(1-\cos\chi)/2\to 0$1 (Herrmann et al., 23 Jul 2025). This shared evolution is the central reason the EEC quark jet function is universal within the collinear limit.

A different nonperturbative factorization language appears in the treatment of EECs in $z=(1-\cos\chi)/2\to 0$2 and DIS through di-hadron fragmentation moments. There the integrated EEC jet function is defined as

$z=(1-\cos\chi)/2\to 0$3

and the unintegrated version as

$z=(1-\cos\chi)/2\to 0$4

These functions are explicitly presented as universal across $z=(1-\cos\chi)/2\to 0$5 and DIS (Guo et al., 17 Dec 2025).

3. Renormalization-group structure and perturbative content

The renormalization-group structure of the EEC quark jet function is governed by timelike splitting kernels. In the collinear-limit factorization, the jet function obeys

$z=(1-\cos\chi)/2\to 0$6

so the relevant anomalous dimensions are Mellin moments of the timelike splitting matrix at $z=(1-\cos\chi)/2\to 0$7 (Dixon et al., 2019). At leading logarithmic order, the convolution simplifies and the evolution becomes multiplicative.

The one-loop quark EEC jet function in that framework is

$z=(1-\cos\chi)/2\to 0$8

and the two-loop constant $z=(1-\cos\chi)/2\to 0$9 is extracted using the EEC sum rule and known endpoint information (Dixon et al., 2019). The same work emphasizes that the quark jet function is the essential ingredient in all-orders resummation of the EEC’s single-logarithmic collinear singularities.

The broader SCET quark jet function for invariant-mass observables is not EEC-specific, but it provides the universal collinear infrastructure underlying many factorization theorems. Its standard definition is

$$\Sigma(z,\ln\frac{Q^2}{\mu^2},\mu) = \int_0^1 dx\, x^2\, \vec J\!\left(\ln\frac{z x^2 Q^2}{\mu^2},\mu\right)\cdot \vec H\!\left(x,\frac{Q^2}{\mu^2},\mu\right),$$0

with $$\Sigma(z,\ln\frac{Q^2}{\mu^2},\mu) = \int_0^1 dx\, x^2\, \vec J\!\left(\ln\frac{z x^2 Q^2}{\mu^2},\mu\right)\cdot \vec H\!\left(x,\frac{Q^2}{\mu^2},\mu\right),$$1 (Brüser et al., 2018). That massless quark jet function has been computed to three loops, with perturbative coefficients organized as

$$\Sigma(z,\ln\frac{Q^2}{\mu^2},\mu) = \int_0^1 dx\, x^2\, \vec J\!\left(\ln\frac{z x^2 Q^2}{\mu^2},\mu\right)\cdot \vec H\!\left(x,\frac{Q^2}{\mu^2},\mu\right),$$2

and later extended to four-loop predictions in the threshold/SV+NSV framework (Brüser et al., 2018, Goyal et al., 29 Dec 2025). These results are not EEC-specific, and the papers explicitly avoid claiming a direct EEC factorization formula from them. A plausible implication is that they furnish perturbative building blocks for EEC-related factorizations when the observable-specific measurement reduces to a collinear invariant-mass constraint.

4. Nonperturbative quark EEC jet function

The transition from perturbative to nonperturbative dynamics in the collinear EEC is described by a nonperturbative modification of the jet function in the near-side region of $$\Sigma(z,\ln\frac{Q^2}{\mu^2},\mu) = \int_0^1 dx\, x^2\, \vec J\!\left(\ln\frac{z x^2 Q^2}{\mu^2},\mu\right)\cdot \vec H\!\left(x,\frac{Q^2}{\mu^2},\mu\right),$$3 annihilation (Herrmann et al., 23 Jul 2025). In that approach, the perturbative position-space jet function is modified at the initial jet scale by

$$\Sigma(z,\ln\frac{Q^2}{\mu^2},\mu) = \int_0^1 dx\, x^2\, \vec J\!\left(\ln\frac{z x^2 Q^2}{\mu^2},\mu\right)\cdot \vec H\!\left(x,\frac{Q^2}{\mu^2},\mu\right),$$4

with

$$\Sigma(z,\ln\frac{Q^2}{\mu^2},\mu) = \int_0^1 dx\, x^2\, \vec J\!\left(\ln\frac{z x^2 Q^2}{\mu^2},\mu\right)\cdot \vec H\!\left(x,\frac{Q^2}{\mu^2},\mu\right),$$5

The fitted parameters are

$$\Sigma(z,\ln\frac{Q^2}{\mu^2},\mu) = \int_0^1 dx\, x^2\, \vec J\!\left(\ln\frac{z x^2 Q^2}{\mu^2},\mu\right)\cdot \vec H\!\left(x,\frac{Q^2}{\mu^2},\mu\right),$$6

$$\Sigma(z,\ln\frac{Q^2}{\mu^2},\mu) = \int_0^1 dx\, x^2\, \vec J\!\left(\ln\frac{z x^2 Q^2}{\mu^2},\mu\right)\cdot \vec H\!\left(x,\frac{Q^2}{\mu^2},\mu\right),$$7

with fit quality

$$\Sigma(z,\ln\frac{Q^2}{\mu^2},\mu) = \int_0^1 dx\, x^2\, \vec J\!\left(\ln\frac{z x^2 Q^2}{\mu^2},\mu\right)\cdot \vec H\!\left(x,\frac{Q^2}{\mu^2},\mu\right),$$8

The extracted $$\Sigma(z,\ln\frac{Q^2}{\mu^2},\mu) = \int_0^1 dx\, x^2\, \vec J\!\left(\ln\frac{z x^2 Q^2}{\mu^2},\mu\right)\cdot \vec H\!\left(x,\frac{Q^2}{\mu^2},\mu\right),$$9 GeV is interpreted as the characteristic scale where the EEC transitions from perturbative jet evolution to non-perturbative hadronization (Herrmann et al., 23 Jul 2025).

That analysis also emphasizes flavor dependence. The $\vec J=\{J_q,J_g\}$0 data are quark-jet dominated, whereas earlier EEC-in-jet measurements in $\vec J=\{J_q,J_g\}$1 collisions were gluon-jet dominated. The paper reports $\vec J=\{J_q,J_g\}$2 GeV for quark-dominated $\vec J=\{J_q,J_g\}$3 data and $\vec J=\{J_q,J_g\}$4 GeV for gluon-dominated $\vec J=\{J_q,J_g\}$5 jets from earlier work, and interprets this as the first direct evidence for flavor dependence in the EEC non-perturbative region (Herrmann et al., 23 Jul 2025). The paper also states that the $\vec J=\{J_q,J_g\}$6 data are not sufficient to disentangle separate nonperturbative functions $\vec J=\{J_q,J_g\}$7 and $\vec J=\{J_q,J_g\}$8, so the fit uses $\vec J=\{J_q,J_g\}$9.

A complementary nonperturbative framework constructs EEC jet functions from moments of di-hadron fragmentation functions. There the integrated EEC jet function satisfies

$\vec H$0

and evolves through the $\vec H$1 timelike anomalous dimensions (Guo et al., 17 Dec 2025). This implies that nonperturbative EEC jet functions can be tied directly to fragmentation observables rather than only to fitted profile functions.

5. Generalizations: multiplicity conditioning and charge resolution

A recent extension introduces the multiplicity-conditioned EEC jet function as the EEC measured in jets selected by a fixed charged-particle multiplicity class (Duan et al., 1 Apr 2026). With normalized multiplicity

$\vec H$2

the conditioned EEC is

$\vec H$3

The key formal object is the multiplicity generating function

$\vec H$4

which modifies the EEC jet-function evolution.

The main perturbative result is that in the region

$\vec H$5

the conditioned EEC retains a power-law form,

$\vec H$6

but with a $\vec H$7-dependent exponent $\vec H$8 (Duan et al., 1 Apr 2026). When $\vec H$9, since $J_q$0, the multiplicity conditioning disappears and the evolution reduces to the standard quark or gluon EEC jet-function evolution. This suggests that the multiplicity-conditioned object is a genuine generalization rather than a different observable class.

A distinct generalization arises in DIS 1-Jettiness with simultaneous jet-charge measurement, where the standard jet function is replaced by a charged jet function

$J_q$1

defined by insertion of a jet-charge measurement operator into the SCET quark jet function (Chien et al., 4 Dec 2025). The factorization theorem is

$J_q$2

$J_q$3

Although this concerns jet charge rather than EEC, it establishes a general template: a universal quark jet function can be resolved with additional measurements while preserving factorized universality (Chien et al., 4 Dec 2025). A plausible implication is that EEC jet functions admit analogous measurement-resolved generalizations.

6. Phenomenological and heavy-ion realizations

In heavy-flavor-tagged and inclusive jet EEC studies in $J_q$4, $J_q$5+Pb, and Pb+Pb collisions, the observable is decomposed as

$J_q$6

where the three factors are the average number of particle pairs per jet, the normalized pair-angular distribution, and the average energy weight $J_q$7 (Shen et al., 2024). The authors use this decomposition to “unravel” what drives differences between pure quark jets, heavy-flavor-tagged jets, and medium-modified jets.

For $J_q$8 collisions, the reported average number of particle pairs per jet is

$J_q$9

After rescaling out the pair-count factor, the EEC peak positions show a mass hierarchy: $$\Sigma(z,Q) = \int_0^1 dx\, x^2\, \mathbf{J}(x q_T,\mu)\cdot \mathbf{H}(x,Q,\mu),$$0-tagged jets peak around $$\Sigma(z,Q) = \int_0^1 dx\, x^2\, \mathbf{J}(x q_T,\mu)\cdot \mathbf{H}(x,Q,\mu),$$1, $$\Sigma(z,Q) = \int_0^1 dx\, x^2\, \mathbf{J}(x q_T,\mu)\cdot \mathbf{H}(x,Q,\mu),$$2-tagged jets around $$\Sigma(z,Q) = \int_0^1 dx\, x^2\, \mathbf{J}(x q_T,\mu)\cdot \mathbf{H}(x,Q,\mu),$$3, and light quark jets around $$\Sigma(z,Q) = \int_0^1 dx\, x^2\, \mathbf{J}(x q_T,\mu)\cdot \mathbf{H}(x,Q,\mu),$$4 (Shen et al., 2024). The paper interprets this as a dead-cone-like broadening hierarchy.

In p+Pb collisions, the ratio

$$\Sigma(z,Q) = \int_0^1 dx\, x^2\, \mathbf{J}(x q_T,\mu)\cdot \mathbf{H}(x,Q,\mu),$$5

shows suppression at very small angles $$\Sigma(z,Q) = \int_0^1 dx\, x^2\, \mathbf{J}(x q_T,\mu)\cdot \mathbf{H}(x,Q,\mu),$$6, with the strongest suppression for $$\Sigma(z,Q) = \int_0^1 dx\, x^2\, \mathbf{J}(x q_T,\mu)\cdot \mathbf{H}(x,Q,\mu),$$7-tagged jets, down to about $$\Sigma(z,Q) = \int_0^1 dx\, x^2\, \mathbf{J}(x q_T,\mu)\cdot \mathbf{H}(x,Q,\mu),$$8, and nearly negligible modification for intermediate angles $$\Sigma(z,Q) = \int_0^1 dx\, x^2\, \mathbf{J}(x q_T,\mu)\cdot \mathbf{H}(x,Q,\mu),$$9 (Shen et al., 2024).

In Pb+Pb collisions, the same work finds a shift of the EEC toward larger $q_T=\sqrt{z}\,Q$0, with suppression at small $q_T=\sqrt{z}\,Q$1, enhancement at large $q_T=\sqrt{z}\,Q$2, and increased average pair counts,

$q_T=\sqrt{z}\,Q$3

The corresponding PbPb/pp ratios are $q_T=\sqrt{z}\,Q$4, $q_T=\sqrt{z}\,Q$5, $q_T=\sqrt{z}\,Q$6, and $q_T=\sqrt{z}\,Q$7 (Shen et al., 2024). The medium modification is attributed to the interplay of increased multiplicity, outward-shifted pair-angle distributions, and reduced average energy weight per pair.

A related heavy-ion study at $q_T=\sqrt{z}\,Q$8 TeV compares quark and gluon jets and states that pure quark jets exhibit strong EEC enhancement at large angular scales, while gluon jets show a bimodal enhancement pattern at both small and large scales (Chen et al., 2024). For quark jets in Pb+Pb, the EEC ratio shows suppression at small $q_T=\sqrt{z}\,Q$9 and enhancement at larger $\mathbf{J}=(J_q,J_g)$0, with the pair-count distribution shifted to large $\mathbf{J}=(J_q,J_g)$1 and the average weight $\mathbf{J}=(J_q,J_g)$2 globally suppressed (Chen et al., 2024). The paper explicitly notes that it does not define a formal perturbative EEC quark jet function, but it identifies this flavor-dependent EEC substructure as the closest phenomenological analogue.

7. Flavor dependence, misconceptions, and related directions

A recurrent result across the literature is that the quark-jet EEC is narrower than the gluon-jet EEC. A phenomenological study of flavor dependence states that the EEC distribution of a gluon jet is broader and that the transition from perturbative to non-perturbative regime occurs at a larger angle for a gluon jet than for a quark jet (Apolinário et al., 17 Feb 2025). The same work attributes this to standard QCD color and splitting physics, emphasizing $\mathbf{J}=(J_q,J_g)$3 versus $\mathbf{J}=(J_q,J_g)$4, and reports that a 10% change in quark fraction gives roughly a 10% change in the small-angle EEC (Apolinário et al., 17 Feb 2025).

This bears directly on a common misconception: observed modifications of inclusive-jet EECs do not isolate medium-induced changes of a fixed quark jet function unless flavor composition and selection effects are controlled. The heavy-ion studies explicitly stress that pre-quenching structural differences between quark and gluon jets materially affect the observed EEC ratios (Chen et al., 2024, Apolinário et al., 17 Feb 2025). This suggests that the “EEC quark jet function” in phenomenology is often an effective, sample-dependent quantity rather than a universal operator matrix element.

Another misconception is that the EEC quark jet function should be identified with the standard SCET invariant-mass jet function. The SCET jet function $\mathbf{J}=(J_q,J_g)$5 is indeed universal and central to many factorization theorems, and three-loop and four-loop results are available (Brüser et al., 2018, Goyal et al., 29 Dec 2025). However, the EEC-specific papers distinguish the EEC jet function as an observable-dependent collinear object whose measurement structure is not the same as a simple invariant-mass constraint (Dixon et al., 2019, Herrmann et al., 23 Jul 2025).

A further development concerns massive quarks in the two-jet limit. In the back-to-back EEC, a recent massive-case factorization analysis keeps the quark jet function inside the universal Sudakov factor while moving finite heavy-quark mass effects into a coefficient function that depends on $\mathbf{J}=(J_q,J_g)$6 (Aglietti et al., 24 Jun 2026). The paper argues that this improved scheme is required for a smooth massless limit. This suggests that heavy-quark mass effects do not invalidate the jet-function picture, but they do alter how finite terms are separated from the universal collinear sector.

Taken together, the literature presents the EEC quark jet function as a family of related objects rather than a single universally adopted definition. In formal small-angle factorization it is the universal collinear quark function evolved by timelike splitting kernels (Dixon et al., 2019). In global analyses of near-side EEC data it acquires a fitted nonperturbative profile (Herrmann et al., 23 Jul 2025). In fragmentation-based treatments it is a moment of di-hadron fragmentation functions universal across $\mathbf{J}=(J_q,J_g)$7 and DIS (Guo et al., 17 Dec 2025). In multiplicity-conditioned studies it becomes a generating-function-dressed observable with a $\mathbf{J}=(J_q,J_g)$8-dependent anomalous dimension (Duan et al., 1 Apr 2026). In jet and heavy-ion phenomenology it is often represented operationally by a decomposition into pair multiplicity, angular pair distribution, and energy weight (Shen et al., 2024, Chen et al., 2024). The convergence of these perspectives suggests that the EEC quark jet function is best understood as the quark-sector collinear kernel of EEC factorization, together with its experimentally relevant nonperturbative and environment-dependent resolutions.