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arxiv: 2603.27846 · v2 · submitted 2026-03-29 · ✦ hep-ph · nucl-th

Recognition: 2 theorem links

· Lean Theorem

Full energy fraction and angular dependence of medium-induced splittings in the large-N_c limit

Carlota Andres , Fabio Dominguez
Authors on Pith no claims yet

Pith reviewed 2026-05-14 21:18 UTC · model grok-4.3

classification ✦ hep-ph nucl-th
keywords medium-induced splittingslarge-Nc limitharmonic oscillator approximationjet quenchingquark-gluon plasmasemi-hard approximationparton energy fractionsplitting angle
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The pith

In the large-Nc limit and harmonic oscillator approximation, medium-induced parton splittings admit fully analytical path-integral evaluations with complete dependence on energy fraction z and angle θ.

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

Jets produced in heavy-ion collisions are modified by the quark-gluon plasma, so their substructure observables encode information about the medium only if the elementary splitting processes are understood in detail. The paper computes medium-induced 1 to 2 splittings double-differential in the energy fraction z carried by the emitted parton and the splitting angle θ, keeping the full resummation of multiple scatterings instead of restricting to soft emissions. In the large-Nc limit together with the harmonic oscillator approximation for the medium potential, every relevant path integral can be performed in closed form for gluon emission off quarks or gluons and for quark emission off gluons. The resulting semi-analytical expressions are computationally efficient and serve as benchmarks; they also show that the ordinary semi-hard approximation fails over most of phase space while an improved version that keeps leading inverse-energy corrections remains reliable whenever all partons are energetic.

Core claim

In the large-Nc limit and under the harmonic oscillator approximation, the medium-induced 1→2 splittings double-differential in z and θ admit fully analytical evaluations of all path integrals for any splitting channel, with complete resummation of multiple scatterings. This yields a computationally efficient semi-analytical framework. Direct comparison demonstrates that the standard semi-hard approximation is unreliable across most of phase space even for high-energy emitters, whereas the improved semi-hard approximation that retains leading corrections in inverse powers of the parton energies provides a robust description provided all participating partons remain sufficiently energetic.

What carries the argument

The large-Nc limit combined with the harmonic oscillator approximation to the medium-induced potential, which reduces the path integrals governing the splitting amplitudes to closed-form expressions.

If this is right

  • Jet substructure observables such as groomed jet mass or angularities can be computed with full z and θ dependence for direct comparison with heavy-ion data.
  • Multiple-scattering resummation is available in closed form, removing the need for Monte Carlo sampling of scatterings inside the medium.
  • The improved semi-hard approximation can be safely used in phenomenological codes when all partons exceed a modest energy threshold.
  • The analytic results supply reference curves against which more complete numerical or 1/Nc-corrected calculations can be validated.

Where Pith is reading between the lines

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

  • The closed-form expressions could be inserted directly into Monte Carlo event generators to generate medium-modified parton showers at far lower computational cost.
  • Comparison of the analytic curves to exact numerical solutions outside the harmonic oscillator regime would map the kinematic domain where the approximation remains valid.
  • The same framework could be adapted to study angular dependence of medium-induced radiation in proton-nucleus collisions or in cold nuclear matter.
  • Running-coupling corrections or finite-Nc effects could be added perturbatively on top of the present analytic baseline.

Load-bearing premise

The harmonic oscillator form of the medium-induced potential together with the large-Nc limit must remain accurate for the energies and angles that occur in realistic heavy-ion collisions.

What would settle it

A numerical evaluation of the exact path integrals for a chosen splitting channel at moderate parton energies, performed without imposing the harmonic oscillator form, would falsify the analytic result if the numerical and analytic curves disagree beyond controlled numerical error.

read the original abstract

Jets produced in relativistic heavy-ion collisions are modified by their interactions with the quark-gluon plasma (QGP), making jet substructure observables sensitive probes of QGP dynamics. A quantitative description of these modifications requires understanding how the medium affects elementary parton splittings with full dependence on both their energy fraction $z$ and splitting angle $\theta$, beyond the widely used soft emitted-gluon approximation. Here, we study medium-induced $1 \to 2$ splittings double-differential in $z$ and $\theta$, with full resummation of multiple scatterings, and show that in the large-$N_c$ limit and under the harmonic oscillator (HO) approximation, all path integrals can be evaluated analytically for any splitting channel, providing a computationally efficient semi-analytical result. We also revisit the semi-hard approximation (SHA), extending it to include leading corrections in inverse powers of the partons energies, which we denote the improved semi-hard approximation (ISHA), and assess its validity through a comparison with the large-$N_c$-HO results. Our analysis shows that while the SHA is found to be unreliable across most of phase space, even for high-energy emitters, the ISHA provides a robust approximation for splittings where all partons are sufficiently energetic.

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 / 1 minor

Summary. The manuscript claims that in the large-N_c limit and under the harmonic oscillator approximation for the medium-induced potential, all path integrals for 1→2 splittings can be evaluated analytically for arbitrary energy fraction z and splitting angle θ, yielding computationally efficient semi-analytical expressions. It further introduces an improved semi-hard approximation (ISHA) that includes leading 1/E corrections, compares it to the large-N_c HO results, and concludes that the standard semi-hard approximation is unreliable while ISHA remains robust when all partons are sufficiently energetic.

Significance. If the analytic tractability and domain of validity hold, the work supplies a significant technical advance by delivering double-differential (z,θ) medium-induced splitting kernels beyond the soft-gluon limit. These could be directly incorporated into Monte Carlo jet quenching codes, enabling more precise predictions for jet substructure observables and tighter constraints on QGP transport coefficients from heavy-ion data.

major comments (1)
  1. [Abstract] Abstract and results: the central claim of analytic path-integral evaluation for any splitting channel rests on the HO potential V(r)=(1/2)q̂r², yet no quantitative error estimate or comparison against the full non-quadratic kernel is supplied for finite-z kinematics where the transverse scale set by θ pushes typical r outside the regime controlled by the quadratic truncation.
minor comments (1)
  1. The manuscript would benefit from an explicit statement of the formation-time and transverse-momentum scales that delineate the HO regime, together with a brief numerical check against the exact kernel for a representative finite-z point.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive assessment of our work's significance and for the constructive feedback. We address the major comment below, providing clarification on the scope of the harmonic oscillator approximation while acknowledging the value of additional validation.

read point-by-point responses

  1. Referee: [Abstract] Abstract and results: the central claim of analytic path-integral evaluation for any splitting channel rests on the HO potential V(r)=(1/2)q̂r², yet no quantitative error estimate or comparison against the full non-quadratic kernel is supplied for finite-z kinematics where the transverse scale set by θ pushes typical r outside the regime controlled by the quadratic truncation.

    Authors: We agree that a quantitative error estimate comparing the HO results to the full non-quadratic kernel for finite-z kinematics would provide useful additional context. The HO potential is adopted because it permits the exact analytic evaluation of all relevant path integrals in the large-N_c limit, which constitutes the paper's primary technical contribution. This approximation is standard in the medium-induced radiation literature and is justified for the high-energy parton regime we target, where typical transverse separations remain small. We have added a dedicated paragraph in Section 3.2 discussing the expected validity range of the HO approximation, citing supporting analyses from prior works on soft emissions, and clarifying that deviations for larger angles are expected to be controlled by the high-energy limit. A full numerical benchmark against the complete kernel for arbitrary z is computationally intensive and outside the present scope, but we have updated the abstract and conclusions to better delineate the approximation's domain. This constitutes a partial revision. revision: partial

Circularity Check

0 steps flagged

No circularity: analytic results follow from standard path-integral formalism plus explicit large-Nc and HO approximations

full rationale

The derivation applies the large-Nc limit and harmonic-oscillator truncation to the medium-induced potential, allowing closed-form evaluation of the path integrals for 1→2 splittings. These are stated external approximations whose validity is discussed separately; the resulting semi-analytic expressions are not obtained by fitting parameters to the target observables, by self-definition, or by load-bearing self-citation. The abstract and claimed results contain no reduction of the final formulae to the input data or to prior results by the same authors. The ISHA comparison is an independent cross-check, not a circular step.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Central claim rests on two domain assumptions: the large-Nc limit simplifying color algebra and the harmonic oscillator model for the medium scattering potential.

axioms (2)
  • domain assumption large-Nc limit
    Invoked to simplify color factors and allow analytic evaluation of path integrals for all splitting channels.
  • domain assumption harmonic oscillator approximation for the medium
    Models the transverse momentum broadening from multiple scatterings in the QGP as a harmonic potential.

pith-pipeline@v0.9.0 · 5527 in / 1282 out tokens · 39957 ms · 2026-05-14T21:18:32.797459+00:00 · methodology

discussion (0)

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

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matches
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supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
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contradicts
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unclear
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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Quantum simulating multi-particle processes in high energy nuclear physics: dijet production and color (de)coherence

    hep-ph 2026-04 unverdicted novelty 6.0

    A framework is developed that encodes leading-order QCD antenna and dipole processes as quantum circuits, with benchmarks against analytic limits in simplified media.

Reference graph

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