Small-Angle EEC: Collinear QCD Dynamics

• Small-Angle EEC is the near-side limit of energy-weighted correlations that probes collinear QCD dynamics in e+e- annihilation.
• It captures single-logarithmic enhancements from perturbative splittings, enabling precision resummation versus double-Sudakov behavior.
• Its factorization framework, involving universal jet functions and process-dependent hard functions, provides insights into QCD evolution and hadronization.

The small-angle Energy-Energy Correlator (EEC) is the collinear, or near-side, limit of the energy-weighted angular correlation of final-state radiation in $e^+e^-$ annihilation. In standard notation, the observable measures the distribution of hadron pairs as a function of their opening angle $\chi$, and is most naturally studied with $z=\frac12(1-\cos\chi)$, so that the small-angle regime is $z\to0$, equivalently $\chi\to0$. In modern precision QCD, this limit is important because it isolates universal collinear dynamics rather than dijet recoil, and its singular terms form a single-logarithmic series, in sharp contrast with the double-Sudakov structure of the back-to-back region $z\to1$ (Neill et al., 2022).

1. Definition and kinematic regime

The EEC in $e^+e^-$ collisions is the normalized energy-weighted angular distribution of all final-state pairs. A standard form is

$$\frac{d\Sigma}{d\chi} = \sum_{i,j}\int d\sigma_{ij}\,\frac{E_iE_j}{Q^2}\,\delta\!\bigl(\chi-\theta_{ij}\bigr),$$

with $Q$ the center-of-mass energy and $\theta_{ij}$ the opening angle between hadrons $\chi$0 and $\chi$1. The angular variable used for endpoint analysis is

$\chi$2

so that, for $\chi$3,

$\chi$4

The scale $\chi$5 is the physically relevant transverse scale governing the approach to the nonperturbative region (Herrmann et al., 23 Jul 2025).

The distinction between the two endpoint regions is structural, not merely kinematic. The small-angle limit $\chi$6 is the near-side collinear regime, in which the two measured energy deposits are nearly aligned and typically belong to the same jet. The back-to-back limit $\chi$7, by contrast, is a recoil-sensitive dijet configuration. The modern literature treats these as different asymptotic problems; the detailed TMD/SCET machinery used for $\chi$8 should not be transplanted to $\chi$9 (Neill et al., 2022, Kang et al., 2024).

2. Collinear singularity and logarithmic structure

The enhancement of the small-angle EEC is generated by collinear radiation. Because the observable sums over all pairs with energy weights, perturbative splittings that produce energetic daughters at small relative angle contribute strongly. In this limit, the EEC probes the same universal splitting dynamics that govern timelike evolution and fragmentation. A particularly important structural statement is that the small-angle region is single logarithmic, whereas the back-to-back region carries double Sudakov logarithms (Neill et al., 2022).

Schematic singular terms in the collinear limit therefore have the form

$z=\frac12(1-\cos\chi)$0

rather than the stronger $z=\frac12(1-\cos\chi)$1 hierarchy. In distributional language, the leading-power endpoint expansion can be organized as

$z=\frac12(1-\cos\chi)$2

with

$z=\frac12(1-\cos\chi)$3

The total normalization,

$z=\frac12(1-\cos\chi)$4

links the small-angle and back-to-back regions and later becomes important for determining jet-function boundary data (Dixon et al., 2019).

Fixed-order information in this regime is unusually explicit. The first analytic NLO QCD result gives the small-$z=\frac12(1-\cos\chi)$5 expansion through next-to-leading power, including the complete coefficients of $z=\frac12(1-\cos\chi)$6, $z=\frac12(1-\cos\chi)$7, $z=\frac12(1-\cos\chi)$8, and constant terms, with full color dependence. The leading $z=\frac12(1-\cos\chi)$9 coefficient agrees with the older jet-calculus prediction, while the subleading terms expose more intricate collinear dynamics, including triple-collinear structure in specific channels (Dixon et al., 2018). The review literature also emphasizes a useful physical simplification of EEC-type observables: direct soft contributions are suppressed by the energy weighting, so soft radiation enters primarily through recoil of energetic collinear particles rather than as an independent leading source of small-angle enhancement (Neill et al., 2022).

3. Factorization and resummation

The leading-power collinear limit admits a factorization theorem most naturally written for the cumulant

$z\to0$0

Its factorized form is

$z\to0$1

where $z\to0$2 is a universal EEC jet function and $z\to0$3 is a process-dependent hard function in flavor space (Dixon et al., 2019). The natural jet scale is $z\to0$4, while the hard function is naturally evaluated at $z\to0$5.

The renormalization-group structure is governed by timelike evolution. The hard function obeys

$z\to0$6

and RG invariance implies

$z\to0$7

At leading logarithmic accuracy the evolution is controlled by the $z\to0$8 Mellin moment of the timelike splitting matrix, while beyond LL it depends also on Mellin derivatives around $z\to0$9. This yields an all-orders single-logarithmic resummation, and the collinear EEC was pushed to NNLL accuracy in QCD and $\chi\to0$0 SYM in the 2019 factorization analysis (Dixon et al., 2019).

A later formulation recasts the collinear factorization theorem in transverse-position space, where nonperturbative broadening can be inserted multiplicatively. In that representation,

$\chi\to0$1

and the hard and jet functions satisfy timelike DGLAP-type equations. A notable feature of this formulation is that there is no separate perturbative soft function in the factorized collinear expression; the resummation is organized through the hard function and the EEC jet function evolving under timelike splitting (Herrmann et al., 23 Jul 2025).

4. Perturbative-to-nonperturbative transition

A unified description of the full near-side region $\chi\to0$2 was obtained at NNLO+NNLL by supplementing the perturbative collinear factorization theorem with a nonperturbative jet function in $\chi\to0$3-space (Herrmann et al., 23 Jul 2025). The construction introduces

$\chi\to0$4

with $\chi\to0$5 regulating the Landau pole and $\chi\to0$6. In the fit, quark and gluon nonperturbative functions were taken equal because the $\chi\to0$7 EEC is quark dominated.

The global analysis used eight historical EEC datasets spanning

$\chi\to0$8

with the fit restricted to the near-side region and excluding the first point of each dataset. The fitted parameters were:

Quantity	Value
$\chi\to0$9	$z\to1$0
$z\to1$1	$z\to1$2
$z\to1$3	$z\to1$4
$z\to1$5	$z\to1$6
$z\to1$7	$z\to1$8

The extracted $z\to1$9 was interpreted as the characteristic transition scale between perturbative and nonperturbative jet dynamics in quark-dominated $e^+e^-$0 EEC. The same analysis notes that plotting the EEC against $e^+e^-$1 reveals a transition peak around $e^+e^-$2, while around $e^+e^-$3 the distribution enters a “free-hadron region” and flattens toward the geometric scaling

$e^+e^-$4

By comparison with a gluon-dominated inclusive EEC-in-jet study in $e^+e^-$5 collisions, which found $e^+e^-$6, this was presented as first evidence for flavor dependence in the nonperturbative EEC transition (Herrmann et al., 23 Jul 2025).

A complementary all-angle renormalon analysis is also relevant for the endpoint structure. In the bubble-sum approximation it identifies a leading $e^+e^-$7 renormalon and a corresponding $e^+e^-$8 power correction

$e^+e^-$9

or, equivalently,

$$\frac{d\Sigma}{d\chi} = \sum_{i,j}\int d\sigma_{ij}\,\frac{E_iE_j}{Q^2}\,\delta\!\bigl(\chi-\theta_{ij}\bigr),$$0

Near $$\frac{d\Sigma}{d\chi} = \sum_{i,j}\int d\sigma_{ij}\,\frac{E_iE_j}{Q^2}\,\delta\!\bigl(\chi-\theta_{ij}\bigr),$$1, this behaves as $$\frac{d\Sigma}{d\chi} = \sum_{i,j}\int d\sigma_{ij}\,\frac{E_iE_j}{Q^2}\,\delta\!\bigl(\chi-\theta_{ij}\bigr),$$2, so the leading nonperturbative correction is more singular in angle than the fixed-order $$\frac{d\Sigma}{d\chi} = \sum_{i,j}\int d\sigma_{ij}\,\frac{E_iE_j}{Q^2}\,\delta\!\bigl(\chi-\theta_{ij}\bigr),$$3 behavior. The same work develops an $$\frac{d\Sigma}{d\chi} = \sum_{i,j}\int d\sigma_{ij}\,\frac{E_iE_j}{Q^2}\,\delta\!\bigl(\chi-\theta_{ij}\bigr),$$4-scheme subtraction of the leading renormalon to stabilize the perturbative series (Schindler et al., 2023).

5. Light-ray OPE, conformal theory, and post-confinement scaling

In conformal theories the small-angle EEC has an especially sharp operator interpretation. The light-ray/OPE analysis in $$\frac{d\Sigma}{d\chi} = \sum_{i,j}\int d\sigma_{ij}\,\frac{E_iE_j}{Q^2}\,\delta\!\bigl(\chi-\theta_{ij}\bigr),$$5 SYM shows that the leading endpoint behavior is controlled by the twist-two tower exchanged in the short-distance product of the two calorimeters. The resulting all-orders form is

$$\frac{d\Sigma}{d\chi} = \sum_{i,j}\int d\sigma_{ij}\,\frac{E_iE_j}{Q^2}\,\delta\!\bigl(\chi-\theta_{ij}\bigr),$$6

equivalently a power law determined by analytically continued twist-two data. In that setting the small-angle EEC is governed by the same spin-three twist-two anomalous dimension that appears in spacelike language, providing a direct bridge between timelike energy flow and light-ray OPE methods (Korchemsky, 2019).

An independent NNLO computation of the EEC in $$\frac{d\Sigma}{d\chi} = \sum_{i,j}\int d\sigma_{ij}\,\frac{E_iE_j}{Q^2}\,\delta\!\bigl(\chi-\theta_{ij}\bigr),$$7 SYM, based on an infrared-finite triple-discontinuity representation of a four-point correlator, makes the small-angle structure explicit. For the rescaled quantity $$\frac{d\Sigma}{d\chi} = \sum_{i,j}\int d\sigma_{ij}\,\frac{E_iE_j}{Q^2}\,\delta\!\bigl(\chi-\theta_{ij}\bigr),$$8, with $$\frac{d\Sigma}{d\chi} = \sum_{i,j}\int d\sigma_{ij}\,\frac{E_iE_j}{Q^2}\,\delta\!\bigl(\chi-\theta_{ij}\bigr),$$9, the three-loop asymptotics are

$Q$0

That analysis is notable because the elliptic sector survives at leading power in the collinear limit, and the result agrees with the light-ray OPE prediction (Henn et al., 2019).

More recently, the light-ray perspective has been extended into the post-confinement regime. A 2025 study argues that the small-angle EEC plateau continues to exhibit predictive $Q$1-scaling after confinement sets in, and that the leading post-confinement scaling is governed by the time-like DGLAP anomalous dimension at Mellin spin $Q$2. In the same framework, the nonperturbative OPE coefficients are identified with moments of the dihadron fragmentation function, establishing a direct correspondence between the hadronic light-ray OPE and DFF moments (Chang et al., 21 Jul 2025).

6. Measurements, variants, and scope

Experimentally, the small-angle EEC remains less fully mapped than the back-to-back region, but the near-side regime is no longer only a formal asymptotic limit. A charged-particle-only ALEPH analysis at the $Q$3 pole reports a high-resolution measurement of the projected two-point correlator with logarithmic binning down to $Q$4 internally and $Q$5 in the publicly reported spectrum. Because the analysis excludes self-correlations, avoids double counting, and uses only charged particles normalized by the fixed collision energy, it is not identical to the historical inclusive LEP EEC convention; its value lies in improved small-angle resolution and a detector correction strategy tailored to the collinear tail (Bossi et al., 17 May 2025).

Semi-inclusive generalizations make the collinear structure more explicit. In hadron-tagged $Q$6 energy correlators, the small-angle regime factorizes in terms of semi-inclusive energy correlators (SIECs), and the ordinary inclusive small-angle EEC is recovered by summing over hadron species and taking the third Mellin moment in the hadron energy fraction. In that sense, the standard near-side EEC can be viewed as a special Mellin projection of a broader semi-inclusive fragmentation problem (Zhu, 1 Sep 2025).

A final source of confusion concerns the relation to in-jet correlators at hadron colliders. The jet-internal two-point EECs measured by STAR and ALICE use pair distance

$Q$7

inside reconstructed jets and are therefore distinct observables, even though they probe analogous near-side physics. Those measurements resolve perturbative, transition, and hadronization-dominated regions and report characteristic scales around $Q$8, but they are not the global $Q$9 small-angle EEC (Collaboration, 21 Feb 2025, Collaboration, 16 Jun 2026). Within the $\theta_{ij}$0 context itself, the modern consensus is more specific: the small-angle EEC is a precision probe of universal collinear QCD, governed by timelike evolution and resum-mable single logarithms, yet sensitive enough to hadronization that a quantitatively accurate treatment now requires an explicit description of the perturbative-to-nonperturbative transition (Neill et al., 2022, Herrmann et al., 23 Jul 2025).