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arxiv: 2606.02714 · v1 · pith:ERS5BQADnew · submitted 2026-06-01 · ✦ hep-ph · hep-ex· nucl-th

Projected Energy Correlators: Two-Loop Jet Functions and NNLL Resummation

Kyle Lee , Yibei Li , Zhen Xu , Xiaoyuan Zhang This is my paper

Pith reviewed 2026-06-28 13:25 UTC · model grok-4.3

classification ✦ hep-ph hep-exnucl-th
keywords energy correlatorsjet functionsNNLL resummationcollinear factorizationQCDnonperturbative correctionsalpha_s extraction
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0 comments X

The pith

Two-loop jet functions enable NNLL resummation of projected N-point energy correlators up to N=6.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes NNLL collinear resummation for projected N-point energy correlators up to N=6 by computing the required two-loop jet functions semi-analytically. This is done for both electron-positron annihilation and Higgs decay to gluons, with matching to NLO fixed-order results and inclusion of leading non-perturbative corrections via universal soft matrix elements. A sympathetic reader would care as this brings higher-point observables under quantitative perturbative control, potentially enabling alpha_s extractions with different systematics from lower-point ones.

Core claim

The key new ingredient is the two-loop jet function for N=4,5,6 computed using Integration-by-Parts and differential equations, which permits the NNLL resummation of projected N-point energy correlators up to N=6 matched to fixed-order NLO predictions, including leading non-perturbative corrections described by two universal soft matrix elements.

What carries the argument

Two-loop jet function for N=4,5,6 computed semi-analytically via Integration-by-Parts and differential equations.

If this is right

  • The matched NNLL distributions for ENCs up to N=6 can be compared with parton-shower simulations.
  • Sensitivity of the spectra and their ratios to alpha_s and the soft matrix elements can be analyzed.
  • Higher-point projected energy correlators achieve quantitative control at NNLL accuracy.
  • This opens the possibility for future alpha_s extractions with complementary systematics.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The anomalous dimensions for (N-1)-point correlators governing the evolution of the soft matrix elements could be used to relate corrections across different N.
  • Ratios of higher N correlators to the two-point one might reduce experimental and theoretical uncertainties in measurements.
  • Applying similar methods to other processes or observables could broaden the use of energy correlators in QCD studies.

Load-bearing premise

The leading non-perturbative corrections are described by two universal soft matrix elements of order Lambda_QCD whose evolution is governed by anomalous dimensions for (N-1)-point correlators.

What would settle it

A direct computation of the jet function at three loops or a comparison with experimental data showing significant deviation from the NNLL prediction after accounting for the included power corrections would falsify the claimed accuracy.

read the original abstract

We present the next-to-next-to-leading logarithmic (NNLL) collinear resummation of projected $N$-point energy correlators (ENCs) up to $N=6$, matched to fixed-order predictions at NLO, in both electron-positron annihilation and Higgs decay to gluons. The key new ingredient is the two-loop jet function for $N=4,5,6$, which we compute semi-analytically using Integration-by-Parts and differential equations. We further include the leading non-perturbative corrections for ENCs, described by two universal soft matrix elements $\overline{\Omega}_{1q},\overline{\Omega}_{1g}$ of order $\Lambda_{\rm QCD}$, whose evolution is governed by anomalous dimensions for $(N-1)$-point correlators. The matched distributions are compared with parton-shower simulations from Pythia8 and Herwig7, and we study the sensitivity of both the absolute spectra and their ratios to the two-point energy correlator under variations of $\alpha_s$ and $\overline{\Omega}_{1q,1g}$. Our results show that higher-point projected energy correlators are now under quantitative control at NNLL accuracy, opening the door to future $\alpha_s$ extractions with complementary systematics.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 1 minor

Summary. The paper presents NNLL collinear resummation of projected N-point energy correlators (ENCs) up to N=6 in e+e- annihilation and Higgs decay to gluons, matched to NLO fixed order. The central technical advance is the semi-analytic computation of the two-loop jet functions for N=4,5,6 via integration-by-parts reduction and differential equations. Leading non-perturbative power corrections are modeled by two universal soft matrix elements Ω̅1q and Ω̅1g whose evolution follows (N-1)-point anomalous dimensions. Results are compared to Pythia8 and Herwig7 parton showers, with studies of sensitivity to αs and the soft parameters in both absolute spectra and ratios to the two-point correlator.

Significance. If the two-loop jet-function results hold, the work brings projected ENCs under quantitative NNLL control, providing a new set of observables for αs extractions whose systematics are complementary to event shapes. The explicit IBP+DE computation of the N=4,5,6 jet functions and the consistent matching plus power-correction framework constitute a clear technical contribution to the precision QCD literature.

minor comments (1)
  1. The abstract and introduction would benefit from a brief statement of the kinematic cuts or fiducial phase-space definitions used for the N-point correlators when comparing to parton showers.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, recognition of the technical advances in the two-loop jet functions, and recommendation to accept. There are no major comments to address.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper computes two-loop jet functions for N=4,5,6 directly via Integration-by-Parts reduction and differential equations, then performs standard NNLL collinear resummation matched to NLO fixed order. Non-perturbative power corrections are introduced explicitly as an assumption using two universal soft matrix elements whose evolution follows from known anomalous dimensions; these are not derived from or fitted to the perturbative results within the paper. No quoted equation or step reduces the claimed predictions to inputs by construction, and no load-bearing self-citation chain is exhibited. The derivation chain is therefore self-contained and employs standard field methods without internal reduction.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard perturbative QCD factorization assumptions plus two universal non-perturbative matrix elements whose values and evolution are taken as inputs; no new entities are postulated.

free parameters (2)
  • Ω̅1q
    Universal soft matrix element for quarks of order ΛQCD used to parametrize leading non-perturbative corrections.
  • Ω̅1g
    Universal soft matrix element for gluons of order ΛQCD used to parametrize leading non-perturbative corrections.
axioms (2)
  • domain assumption Collinear factorization and all-order resummation in perturbative QCD
    Invoked to justify the NNLL resummation of the projected correlators.
  • domain assumption Existence and universality of the two soft matrix elements governing power corrections
    Invoked to include leading non-perturbative effects via evolution with (N-1)-point anomalous dimensions.

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Forward citations

Cited by 1 Pith paper

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  1. Putting Jet Substructure on Track(s)

    hep-ph 2026-06 unverdicted novelty 7.0

    First complete NLL calculations of projected energy correlators (up to 4-point) on tracks via factorization theorems and RG evolution, extending prior full-jet results.

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