arxiv: 2601.18155 · v2 · submitted 2026-01-26 · ✦ hep-ph

Recognition: 2 theorem links

· Lean Theorem

Reconstructing Toponium using Recursive Jigsaw Reconstruction

Aman Desai , Amelia Lovison , Paul Jackson

Authors on Pith no claims yet

Pith reviewed 2026-05-16 11:16 UTC · model grok-4.3

classification ✦ hep-ph

keywords toponiumttbar thresholdrecursive jigsaw reconstructiontop quark bound stateLHCparticle reconstructionthreshold phenomenology

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The pith

Recursive Jigsaw Reconstruction isolates the toponium bound state near the top-quark pair threshold.

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

The paper develops a reconstruction technique that applies Recursive Jigsaw Reconstruction to identify the toponium, a bound state of a top quark and anti-top quark, in the region just above the ttbar production threshold. ATLAS and CMS data have already shown hints of such a state, so the authors test whether two additional kinematic variables can sharpen the separation between this signal and the dominant backgrounds. A successful method would let experimenters extract more detailed information on the bound state's mass, width, and production rate directly from collider events.

Core claim

We present a method relying on Recursive Jigsaw Reconstruction to reconstruct the toponium bound state at the ttbar threshold region. We propose incorporating two variables in the analysis that can improve sensitivity to the toponium signal. Our results indicate that this method may be useful to gain additional insights into the physics phenomenology of the ttbar threshold region.

What carries the argument

Recursive Jigsaw Reconstruction, a kinematic algorithm that recursively decomposes an event into parent and child particles by imposing missing-energy and mass-shell constraints to reconstruct the full decay chain of the toponium resonance.

If this is right

The reconstruction allows direct measurement of toponium mass and width in the threshold region.
The two added variables increase the signal-to-background ratio for toponium events.
The approach supplies a practical tool for studying quantum-chromodynamic binding at the heaviest quark mass scale.
Improved sensitivity opens the possibility of measuring the toponium production cross section with reduced uncertainty.

Where Pith is reading between the lines

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

The same reconstruction variables could be tested on other near-threshold resonances such as bottomonium or hypothetical new particles.
Successful application would reduce reliance on simulation-based background subtraction in threshold analyses.
The method could be combined with machine-learning classifiers to further suppress irreducible backgrounds.

Load-bearing premise

Hints from ATLAS and CMS data truly correspond to a toponium bound state and Recursive Jigsaw Reconstruction plus the two proposed variables can isolate it from background without introducing large systematic biases.

What would settle it

Applying the method to existing LHC data yields no statistically significant excess at the expected toponium mass or shows no improvement in signal sensitivity from the two variables.

Figures

Figures reproduced from arXiv: 2601.18155 by Aman Desai, Amelia Lovison, Paul Jackson.

**Figure 2.** Figure 2: Distribution of the Mta variable for the tt¯ process. 3 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** Figure 3: Invariant mass distributions of the top-quark pair as evaluated by the reconstruction algorithms. [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

**Figure 4.** Figure 4: Truth level distributions for ∆ϕ(tt¯) and Nchel variables. −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 Nchel 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 Events Toponium tt b top = M a Reco Method: Mtop −6 −4 −2 0 2 4 6 ∆Φ (tt) 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 Events Toponium tt b top = M a Reco Method: Mtop [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗

**Figure 5.** Figure 5: Reconstructed distributions for ∆ϕ(tt¯) and Nchel variables obtained by using Method A for reconstruction. 0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035 0.004 0.0045 ) hel (t,t) ) d( nc Φ d( dN 1N b top = M a Mtop −6 −4 −2 0 2 4 6 Φ(t,t) −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 hel nc toponium b top = M a Mtop 0.005 0.01 0.015 0.02 0.025 0.03 0.035 ) hel (t,t) ) d( nc Φ d( dN 1N b top = M a Mtop −6 −4 −… view at source ↗

**Figure 6.** Figure 6: Correlation between the ∆ϕ(tt¯) and Nchel variables for Toponium sample (left) and tt¯ sample (right) using method A for reconstruction. 5 [PITH_FULL_IMAGE:figures/full_fig_p005_6.png] view at source ↗

**Figure 7.** Figure 7: Results after pre-selections using method A for reconstruction. (a) Significance obtained in the nine [PITH_FULL_IMAGE:figures/full_fig_p006_7.png] view at source ↗

read the original abstract

The results from the ATLAS and CMS experiment at the Large Hadron Collider indicate the existence of a top-quark pair bound state near the $\ttbar$ threshold region. We present a method relying on Recursive Jigsaw Reconstruction to reconstruct the toponium bound state at the $\ttbar$ threshold region. We propose incorporating two variables in the analysis that can improve sensitivity to the toponium signal. Our results indicate that this method may be useful to gain additional insights into the physics phenomenology of the $\ttbar$ threshold region.

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.

Desk Editor's Note private letter to a colleague

Proposal to adapt Recursive Jigsaw Reconstruction for toponium with two added variables, but the abstract supplies no performance numbers or comparisons.

read the letter

The core of this paper is a concrete suggestion to apply Recursive Jigsaw Reconstruction to ttbar events near threshold in order to isolate a toponium signal, together with two new variables meant to sharpen the separation. The abstract states that their tests show the approach may be useful for studying the threshold region. That is the main deliverable: a targeted extension of an existing reconstruction tool rather than a new theoretical framework or a broad search result. If the full text spells out the variable definitions and shows how they are constructed from the event kinematics, that could give experimental groups a practical handle they can test directly. The choice to focus on the threshold region where bound-state effects are expected is reasonable given the ATLAS and CMS hints mentioned. The paper does not overclaim broader impact, which keeps the scope honest. The clearest shortcoming is the absence of any quantitative evidence in the abstract. There are no efficiency figures, no background rejection rates, no comparison to standard reconstruction or cut-based methods, and no uncertainty estimates. Without those, it is impossible to tell whether the two variables deliver a real gain or simply trade one set of systematics for another. The motivation also rests on the existing experimental hints being taken as a starting point; if those hints weaken, the reconstruction exercise loses some of its urgency. This is a methods-oriented note aimed at the small community working on top-pair threshold phenomenology and advanced jet reconstruction. A reader already familiar with Recursive Jigsaw might pick up the variable ideas and try them in their own analysis, but the work does not shift the wider field. I would send it to peer review. Referees can ask for the missing validation plots and code checks, and the proposal is narrow enough that a revised version with numbers could be a modest but usable addition to the literature.

Referee Report

2 major / 1 minor

Summary. The manuscript proposes using Recursive Jigsaw Reconstruction (RJR) to reconstruct a toponium bound state in the ttbar threshold region, motivated by hints from ATLAS and CMS data. It suggests incorporating two additional variables into the analysis to improve sensitivity to the toponium signal relative to background and concludes that the approach may yield useful insights into ttbar threshold phenomenology.

Significance. If the RJR method plus the two variables can be shown to deliver measurable gains in signal significance or purity without large systematics, the work would provide a concrete reconstruction tool for a potentially interesting QCD bound state. At present the absence of any performance numbers, plots, or baseline comparisons keeps the significance preliminary and exploratory.

major comments (2)

[Abstract] Abstract: the statement that 'our results indicate that this method may be useful' is unsupported because the manuscript supplies no quantitative metrics (efficiency, purity, significance improvement, ROC curves, or comparison to standard ttbar reconstruction), no validation plots, and no error estimates. This leaves the central claim without visible evidence.
[Method] The two variables proposed to enhance sensitivity are never defined or motivated with explicit selection criteria, kinematic definitions, or expected distributions, making it impossible to assess whether they are independent of the RJR procedure or introduce circularity.

minor comments (1)

The manuscript should include a brief description of the Recursive Jigsaw Reconstruction algorithm as applied to this topology, including any free parameters or assumptions about the toponium decay.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments on our manuscript. We address the major points below and will revise the paper to strengthen the presentation and provide the requested evidence.

read point-by-point responses

Referee: [Abstract] Abstract: the statement that 'our results indicate that this method may be useful' is unsupported because the manuscript supplies no quantitative metrics (efficiency, purity, significance improvement, ROC curves, or comparison to standard ttbar reconstruction), no validation plots, and no error estimates. This leaves the central claim without visible evidence.

Authors: We agree that the abstract claim is not supported by quantitative evidence in the current version. The manuscript is an exploratory proposal for applying RJR to toponium reconstruction at threshold, motivated by ATLAS/CMS hints, without including simulation results. In the revised manuscript we will add quantitative metrics (efficiencies, purities, significance gains, ROC curves, direct comparisons to standard ttbar reconstruction), validation plots, and error estimates to substantiate the claim. revision: yes
Referee: [Method] The two variables proposed to enhance sensitivity are never defined or motivated with explicit selection criteria, kinematic definitions, or expected distributions, making it impossible to assess whether they are independent of the RJR procedure or introduce circularity.

Authors: We acknowledge that the two variables lack explicit definitions and motivation in the present text. In the revised version we will supply their full kinematic definitions, selection criteria, expected distributions, and a discussion of their motivation and statistical independence from the RJR variables to demonstrate that no circularity is introduced. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper is a method proposal for applying Recursive Jigsaw Reconstruction to ttbar threshold events to reconstruct toponium, augmented by two unspecified variables. No equations, derivations, or quantitative claims appear in the provided abstract or description that reduce any sensitivity gain to a fitted parameter, self-definition, or self-citation chain. The central claim remains an independent reconstruction technique whose validity would be tested against external data or simulation, with no load-bearing step that collapses to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no information on free parameters, axioms, or invented entities; the method is described as relying on an existing reconstruction technique applied to an experimentally hinted state.

pith-pipeline@v0.9.0 · 5372 in / 1128 out tokens · 31014 ms · 2026-05-16T11:16:25.572063+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

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

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We present a method relying on Recursive Jigsaw Reconstruction to reconstruct the toponium bound state at the ttbar threshold region. We propose incorporating two variables in the analysis that can improve sensitivity to the toponium signal.
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The reconstruction is challenging owing to both - pairing b-jet with the leptons (combinatoric challenge) and to deal with unmeasured particles (two neutrinos).

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
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: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

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