Energy Correlator Jet Function

• Energy Correlator Jet Function is a measured collinear object that captures the angular-energy distribution within jets across various observables.
• It underpins factorization theorems in SCET by isolating collinear emissions and enabling precise renormalization-group evolution without Sudakov double logarithms.
• The function adapts to different regimes, including heavy-quark jets and medium-modified scenarios, revealing insights into nonperturbative and nuclear dynamics.

The energy correlator jet function is the collinear object that encodes how a jet contributes to an energy correlator at fixed angular resolution. In modern usage, the term does not denote a single universal formula; rather, it refers to a family of measured collinear functions appearing in factorization theorems for projected $N$-point correlators, near-side energy-energy correlators (EECs), heavy-flavor jets, and medium-modified jet observables. Across these settings, the common role is to isolate the angular-energy structure of final-state collinear radiation while hard production, and in some cases soft or medium dynamics, factorize separately (Craft et al., 2022, Guo et al., 17 Dec 2025).

1. Scope and terminology

In the literature, “energy correlator jet function” is used for several closely related constructions. Some are standard SCET matrix elements for projected correlators, some are fragmentation-based EEC jet functions differential in a transverse scale, and some are medium-modified analogues or kernels that play the same structural role without being universal vacuum jet functions in the strict SCET sense (Craft et al., 2022, Guo et al., 17 Dec 2025, Fu et al., 2024, Singh et al., 2024).

Context	Representative object	Defining feature
Projected correlators	$\vec J^{[\nu]}$, $J_Q^{[N]}$	Collinear function for projected $N$-point correlators
Near-side EEC	$\Gamma_i(\mu)$, $\Gamma_i(\mu,q_T)$	Energy-weighted di-hadron fragmentation object
Medium-modified EEC	$\mathcal K(\theta)$, $J_i(xQ,\chi,\mu)$	Angular kernel or jet function with medium dependence
Other realizations	$J_{\rm EEC}^f(b_\perp)$, $J_c(z_c,\chi,\omega_JR,\mu_J)$	Observable-specific measured collinear functions

This diversity is not merely terminological. In heavy-flavor jet substructure, the jet function localizes the heavy-quark mass dependence. In near-side EEC factorization, the jet function is built from di-hadron fragmentation functions and carries the nonperturbative collinear physics of energy-weighted hadron pairs. In cold-matter and heavy-ion applications, the corresponding object acquires explicit dependence on $\vec J^{[\nu]}$0, path length, Glauber exchange, or universal medium correlators. This suggests that the phrase is best understood as denoting a measured collinear sector tailored to a specific energy-correlator observable, rather than a single observable-independent function.

2. Vacuum factorization and renormalization-group structure

A broad SCET formulation was given for projected $\vec J^{[\nu]}$1-point energy correlators and their analytic continuation to non-integer $\vec J^{[\nu]}$2. In the collinear limit $\vec J^{[\nu]}$3, the cumulant obeys

$\vec J^{[\nu]}$4

with jet-function evolution

$\vec J^{[\nu]}$5

The relevant anomalous dimensions are timelike twist-2 moments evaluated at spin $\vec J^{[\nu]}$6, and the one-loop jet functions are analytic in $\vec J^{[\nu]}$7, with explicit dependence on $\vec J^{[\nu]}$8 and $\vec J^{[\nu]}$9 (Chen et al., 2020).

A more direct EEC-specific formulation was later given in terms of integrated and unintegrated EEC jet functions. The integrated object is

$J_Q^{[N]}$0

while the unintegrated function differential in relative transverse momentum is

$J_Q^{[N]}$1

For the near-side EEC in $J_Q^{[N]}$2, the small-angle factorization is written as

$J_Q^{[N]}$3

with an analogous DIS factorization. A central point is that for any fixed nonzero angle the EEC is IR safe, whereas the angle-integrated EEC is IR sensitive and depends on di-hadron fragmentation physics through $J_Q^{[N]}$4. The corresponding $J_Q^{[N]}$5-space evolution is DGLAP-like; unlike standard TMDs, the paper emphasizes that there are no Sudakov double logarithms and that only single logarithms need to be resummed (Guo et al., 17 Dec 2025).

These vacuum formulations establish the canonical role of the energy correlator jet function: it is the process-independent collinear factor, whereas process dependence enters through hard functions, PDFs in DIS, or analogous short-distance coefficients.

3. Heavy-quark energy-correlator jet functions

For identified heavy-flavor jets, the jet function becomes the locus of heavy-quark mass dependence. The projected $J_Q^{[N]}$6-point correlator factorization theorem is

$J_Q^{[N]}$7

with flavor vectors

$J_Q^{[N]}$8

The heavy-quark jet function is defined by the SCET matrix element

$J_Q^{[N]}$9

where $N$0 is the gauge-invariant collinear SCET field for a massive quark (Craft et al., 2022).

The natural mass variable is

$N$1

so the characteristic transition occurs at

$N$2

The regimes are $N$3 (massless-like), $N$4 (mass-sensitive), and $N$5 (dead-cone). At one loop, the heavy-quark jet function has an analytic structure described as unusually simple: for integer $N$6, the result reduces to transcendental functions of weight 1 only, with alphabet $N$7. The ultraviolet poles are identical to those of the massless quark jet function, so the anomalous dimension is unchanged; mass effects enter through finite infrared-sensitive terms. The massless limit $N$8 reproduces the known one-loop massless jet functions, while $N$9 yields the perturbative dead-cone turnover (Craft et al., 2022).

This heavy-flavor formulation made possible what the paper describes as the first full next-to-leading-logarithmic calculation of a heavy-quark jet substructure observable at the LHC. In that sense, the energy correlator jet function is both a formal factorization ingredient and the mechanism by which heavy-quark mass effects become visible in projected correlators.

4. Nonperturbative structure and flavor dependence

Near-side EEC analyses extend the jet function into the nonperturbative region by working in impact-parameter space. One formulation introduces a two-component jet function $\Gamma_i(\mu)$0 and its differential version $\Gamma_i(\mu)$1, with momentum-space factorization

$\Gamma_i(\mu)$2

and $\Gamma_i(\mu)$3-space representation

$\Gamma_i(\mu)$4

The nonperturbative modification is imposed multiplicatively at the initial jet scale,

$\Gamma_i(\mu)$5

The fit to eight $\Gamma_i(\mu)$6 datasets with $\Gamma_i(\mu)$7 gave

$\Gamma_i(\mu)$8

with

$\Gamma_i(\mu)$9

The paper identifies $\Gamma_i(\mu,q_T)$0 with a quark-jet transition scale around $\Gamma_i(\mu,q_T)$1 GeV and contrasts it with a gluon-dominated $\Gamma_i(\mu,q_T)$2 EEC-in-jet scale of $\Gamma_i(\mu,q_T)$3 GeV, interpreting the difference as evidence for flavor dependence of the EEC jet function’s nonperturbative sector (Herrmann et al., 23 Jul 2025).

A different near-side formulation does not introduce an explicit perturbative SCET jet function, but instead uses a universal nonperturbative TMD fragmentation distribution as the effective small-angle governing object: $\Gamma_i(\mu,q_T)$4 with

$\Gamma_i(\mu,q_T)$5

The near-side EEC is then modeled as

$\Gamma_i(\mu,q_T)$6

In this framework the TMD fragmentation distribution acts as a phenomenological proxy for a near-side energy-correlator jet function, with the small-$\Gamma_i(\mu,q_T)$7 behavior $\Gamma_i(\mu,q_T)$8 associated with the free-hadron region and the scale dependence governed by a nonperturbative Collins–Soper kernel (Liu et al., 2024).

Together, these approaches show that the energy correlator jet function has a genuinely nonperturbative sector. In one formulation that sector is fitted directly as a multiplicative $\Gamma_i(\mu,q_T)$9-space factor on a perturbative jet function; in another, the near-side object itself is modeled as a universal TMD fragmentation distribution.

5. Medium-modified and nuclear-matter generalizations

In cold nuclear matter, the object closest to an energy correlator jet function is the angular kernel $\mathcal K(\theta)$0 that governs the EEC-sensitive part of the cross section. In DIS,

$\mathcal K(\theta)$1

with

$\mathcal K(\theta)$2

The vacuum-like term is

$\mathcal K(\theta)$3

while the medium correction is

$\mathcal K(\theta)$4

In the small-phase regime, $\mathcal K(\theta)$5 gives $\mathcal K(\theta)$6, in contrast to the vacuum scaling $\mathcal K(\theta)$7. This contrast underlies the large-angle enhancement of the EEC within the jet cone. The same paper treats $\mathcal K(\theta)$8 as the closest analogue of an energy correlator jet function, while stressing that it is not a universal SCET jet function because it contains explicit medium structure through $\mathcal K(\theta)$9 and $J_i(xQ,\chi,\mu)$0 (Fu et al., 10 Oct 2025, Fu et al., 2024).

In heavy-ion SCET, the jet function is promoted to a gauge-invariant operator-valued object evolved with collinear, soft-medium, and Glauber interactions. In the regime $J_i(xQ,\chi,\mu)$1, the factorization theorem is

$J_i(xQ,\chi,\mu)$2

and the quark jet function is

$J_i(xQ,\chi,\mu)$3

with $J_i(xQ,\chi,\mu)$4. At single scattering,

$J_i(xQ,\chi,\mu)$5

which factorizes the observable-dependent jet function from a universal medium correlator $J_i(xQ,\chi,\mu)$6. In this EFT framework, radiative corrections generate both DGLAP evolution in $J_i(xQ,\chi,\mu)$7 and BFKL evolution in rapidity $J_i(xQ,\chi,\mu)$8, and the one-loop single-scattering medium-induced jet function reproduces GLV, while the refactorized collinear-soft sector connects to BDMPS-Z in the appropriate regime (Singh et al., 2024).

These medium generalizations shift the meaning of “jet function” in a precise way. In cold matter, the jet-function role may be played by an angular kernel; in heavy-ion EFT, it becomes an operator-defined medium-modified jet function with coupled $J_i(xQ,\chi,\mu)$9- and $J_{\rm EEC}^f(b_\perp)$0-evolution and explicit factorization from universal medium correlators.

6. Alternative realizations and extensions

The same concept appears in several newer observable classes. For boosted heavy-boson jets, the most explicit object called an EEC jet function is

$J_{\rm EEC}^f(b_\perp)$1

which arises after weighting TMD fragmentation functions with the EEC energy factors and using the momentum sum rule. The corresponding Sudakov-region factorization takes the form hard $J_{\rm EEC}^f(b_\perp)$2. In that setting the angular peak at

$J_{\rm EEC}^f(b_\perp)$3

is interpreted not as a Breit–Wigner structure but as the boosted image of the back-to-back Sudakov region of the $J_{\rm EEC}^f(b_\perp)$4-pole EEC (Holguin et al., 28 Jan 2026).

A different extension concerns the observable basis itself. A new parametrization of projected correlators replaces the largest pairwise angle $J_{\rm EEC}^f(b_\perp)$5 by a radius $J_{\rm EEC}^f(b_\perp)$6 measured from a “special” particle and gives the cumulative form

$J_{\rm EEC}^f(b_\perp)$7

The factorization formula of traditional projected correlators still applies, with the important caveat that the jet function differs only at $J_{\rm EEC}^f(b_\perp)$8, i.e. at NNLL. The paper therefore presents the new variables as a measurement basis in which the same leading collinear dynamics survive through NLL, while the actual jet-function calculation should simplify at NNLL and beyond (Alipour-fard et al., 2024).

A one-point generalization inside jets introduces a measured, TMD-sensitive jet function

$J_{\rm EEC}^f(b_\perp)$9

built from the semi-inclusive TMD fragmenting jet function. Its normalized version is designed so that the dependence on the factorization scale is significantly reduced, leaving the dominant sensitivity to the jet scale $J_c(z_c,\chi,\omega_JR,\mu_J)$0. The paper presents numerical calculations at NNLL accuracy for global logarithms and LL accuracy for non-global logarithms (Mi et al., 29 Jul 2025).

Taken together, these developments show that the energy correlator jet function is now a broad organizing concept spanning projected correlators, near-side EECs, heavy-flavor jets, nuclear matter, boosted color-singlet decays, and TMD-sensitive one-point observables. The common structure is always a measured collinear function that resolves energy flow in angle space, but its precise definition, operator content, and evolution depend on the observable and on whether the relevant long-distance physics is hadronization, heavy-quark mass, or medium dynamics.