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arxiv: 2606.23924 · v1 · pith:EWZ6C736new · submitted 2026-06-22 · ✦ hep-ph

Event isotropy in perturbative QCD

Daniele Atzori , Matteo Cacciari , Simone Marzani , Gregory Soyez This is my paper

Pith reviewed 2026-06-26 07:42 UTC · model grok-4.3

classification ✦ hep-ph
keywords event isotropyEnergy Mover's Distanceperturbative QCDresummationfixed-order calculationse+e- collisionsNLL accuracyNLO matching
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The pith

Event isotropy in e+e- collisions can be computed to NLL+NLO accuracy in perturbative QCD.

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

The paper establishes the first perturbative QCD calculations for event isotropy, an observable that measures how closely a collider event resembles a uniform distribution of energy. It provides semi-analytic results at leading order, numerical results at next-to-leading order, and all-order predictions by resumming soft and collinear emissions to next-to-leading logarithmic accuracy, with matching to reach NLL+NLO. A sympathetic reader would care because this supplies precise theoretical predictions for a new geometric observable based on the Energy Mover's Distance, enabling direct comparisons with data and a controlled expansion in the strong coupling.

Core claim

We present the first theoretical study of event isotropy within the framework of perturbative QCD. In particular, we first obtain the isotropy distribution in e+e− collisions semi-analytically at O(αs) (leading-order) and numerically at O(αs2) (next-to-leading order, NLO). Furthermore, we obtain all-order theoretical predictions by resumming the contributions of soft and collinear emissions, at next-to-leading logarithmic (NLL) accuracy. By matching resummed and fixed-order predictions we reach NLL+NLO accuracy.

What carries the argument

The Energy Mover's Distance definition of event isotropy, which quantifies the resemblance of a collider event to a uniform energy distribution.

If this is right

  • Fixed-order and resummed predictions can be directly compared to experimental measurements of isotropy in electron-positron collisions.
  • The matching procedure supplies a consistent description across the full range of the isotropy variable.
  • Resummation of soft and collinear emissions controls the behavior near the uniform limit.
  • The framework provides a starting point for higher-order or higher-logarithmic improvements.

Where Pith is reading between the lines

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

  • The same calculational strategy could be extended to proton-proton collisions once parton distribution functions and underlying-event modeling are incorporated.
  • Event isotropy may be correlated with other geometric or energy-flow observables, offering cross-checks in data analyses.
  • Numerical stability of the NLO computation could be tested by varying infrared cutoff parameters in the phase-space integration.

Load-bearing premise

The Energy Mover's Distance definition of isotropy admits a perturbative expansion in the strong coupling constant whose higher-order and non-perturbative corrections remain controllable in the relevant kinematic regime.

What would settle it

A measurement of the isotropy distribution in e+e- data that deviates from the NLL+NLO prediction by more than the estimated theoretical uncertainty after standard hadronization corrections are applied.

read the original abstract

It has recently been proposed that collider events can be equipped with a metric, the Energy Mover's Distance (EMD), which allows one to rephrase multiple aspects of collider physics in a geometric language. Further, the EMD can be exploited to define new observables. In this context, event isotropy quantifies the resemblance of an event to a uniform energy distribution. We present the first theoretical study of event isotropy within the framework of perturbative QCD. In particular, we first obtain the isotropy distribution in $e^+e^-$ collisions semi-analytically at $O(\alpha_s)$ (leading-order) and numerically at $O(\alpha_s^2)$ (next-to-leading order, NLO). Furthermore, we obtain all-order theoretical predictions by resumming the contributions of soft and collinear emissions, at next-to-leading logarithmic (NLL) accuracy. By matching resummed and fixed-order predictions we reach NLL+NLO accuracy.

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

1 major / 2 minor

Summary. The manuscript claims to provide the first perturbative QCD study of event isotropy (defined via the Energy Mover's Distance to a uniform energy distribution) in e+e− collisions. It reports semi-analytic results at leading order O(αs), numerical results at next-to-leading order O(αs²), all-order predictions via next-to-leading logarithmic (NLL) resummation of soft and collinear emissions, and matched NLL+NLO predictions.

Significance. If the results hold, the work successfully applies standard fixed-order and resummed QCD techniques to a new geometric observable, providing theoretical benchmarks that could be compared to data and used to study event shapes. The combination of LO, NLO, NLL, and matched calculations is a strength, as is the focus on e+e− kinematics where non-perturbative effects are expected to be controllable.

major comments (1)
  1. [Abstract and observable definition] The validity of the LO, NLO, and NLL calculations rests on the assumption that the EMD isotropy observable is infrared and collinear safe so that real/virtual divergences cancel and the perturbative expansion is well-behaved. The manuscript does not appear to contain an explicit IRC-safety proof or numerical checks on the size of power corrections (e.g., in the definition of the observable or in the results sections), which is load-bearing for interpreting the reported distributions as perturbative predictions.
minor comments (2)
  1. The numerical stability and integration method for the NLO calculation should be described in more detail to allow independent verification.
  2. Clarify the precise matching procedure between the resummed and fixed-order results (e.g., which terms are subtracted) to avoid potential double-counting ambiguities.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their positive assessment and for highlighting the need to explicitly address infrared and collinear (IRC) safety. We agree this is important for interpreting the perturbative results and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Abstract and observable definition] The validity of the LO, NLO, and NLL calculations rests on the assumption that the EMD isotropy observable is infrared and collinear safe so that real/virtual divergences cancel and the perturbative expansion is well-behaved. The manuscript does not appear to contain an explicit IRC-safety proof or numerical checks on the size of power corrections (e.g., in the definition of the observable or in the results sections), which is load-bearing for interpreting the reported distributions as perturbative predictions.

    Authors: We agree that an explicit discussion of IRC safety strengthens the paper. The EMD isotropy observable is IRC safe by construction: the EMD is a metric on energy flows that is continuous under soft emissions and collinear splittings, and the reference isotropic distribution is fixed and uniform. Consequently, the real and virtual contributions cancel in the usual way at each perturbative order, as evidenced by the finite LO semi-analytic and NLO numerical results we obtain. Nevertheless, the manuscript would benefit from a dedicated paragraph (likely in Section 2) sketching this argument and noting that power corrections are expected to be O(Λ/Q) as for other event shapes in e+e−. We will add this discussion and, if space permits, a brief numerical check of the approach to the perturbative limit at small αs. This is a minor revision. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivations rest on standard perturbative QCD

full rationale

The paper computes the isotropy distribution via established methods: semi-analytic evaluation of O(αs) matrix elements, numerical NLO integration, and NLL soft/collinear resummation matched to fixed order. No equations reduce to self-definitions, fitted inputs renamed as predictions, or load-bearing self-citations. The EMD isotropy definition is taken from external prior work and treated as an input observable whose perturbative expansion is computed directly; no uniqueness theorems or ansatze are smuggled via author self-reference. This is the normal case of an independent application of QCD perturbation theory.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on abstract; relies on standard perturbative QCD expansion and the prior definition of EMD/isotropy, with no additional free parameters, ad-hoc axioms, or invented entities identified.

axioms (1)
  • domain assumption Perturbative expansion in the strong coupling αs is valid and controllable for the isotropy observable in e+e- collisions.
    Implicit in the decision to compute at LO, NLO, and NLL accuracy.

pith-pipeline@v0.9.1-grok · 5692 in / 1253 out tokens · 30669 ms · 2026-06-26T07:42:08.467375+00:00 · methodology

discussion (0)

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Reference graph

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