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arxiv: 2605.01539 · v1 · submitted 2026-05-02 · ✦ hep-ph · hep-ex· hep-th· nucl-ex· nucl-th

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Multimodal Fragmentation of All-Heavy Pentaquarks: Uncertainty-Aware Predictions for Hadron Colliders

Francesco Giovanni Celiberto
Authors on Pith no claims yet

Pith reviewed 2026-05-09 14:15 UTC · model grok-4.3

classification ✦ hep-ph hep-exhep-thnucl-exnucl-th
keywords all-charm pentaquarksfragmentation functionsuncertainty quantificationhadron collidersexotic hadronsheavy flavor QCDsemi-inclusive productionmultimodal fragmentation
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The pith

A multimodal set of fragmentation functions gives uncertainty-aware predictions for all-charm pentaquark production at hadron colliders.

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

The paper constructs a multimodal set of collinear fragmentation functions for S-wave all-charm pentaquark states. These functions combine estimates of missing higher-order perturbative effects with controlled variations in the nonperturbative wave function. The initial-scale inputs for charm fragmentation are adjusted to cover both compact multiquark and diquark-driven mechanisms. The framework is then applied to compute semi-inclusive pentaquark-plus-jet production at the HL-LHC and FCC using an NLL/NLO+ accuracy setup. A sympathetic reader would care because it supplies a practical way to forecast rare exotic-hadron signals while keeping track of theoretical uncertainties.

Core claim

The central claim is that leading-power fragmentation for all-charm pentaquarks can be described by the multimodal collinear fragmentation functions PQ5Q1.1, which incorporate perturbative uncertainties through missing higher-order variations and nonperturbative uncertainties through controlled modifications of the transverse-momentum structure of the wave function, all combined in a replica-like framework with refined initial-scale inputs that accommodate both compact multiquark and diquark-driven charm fragmentation mechanisms.

What carries the argument

The multimodal collinear fragmentation functions PQ5Q1.1, which integrate perturbative and nonperturbative uncertainty estimates to model the fragmentation of all-charm pentaquarks.

If this is right

  • NLL/NLO+ predictions for semi-inclusive pentaquark-plus-jet production become available at the HL-LHC and FCC with both perturbative and nonperturbative uncertainties quantified.
  • The same framework supplies a flexible tool for uncertainty-controlled studies that link exotic-hadron structure to heavy-flavor fragmentation and high-energy QCD.
  • The bottom sector is set aside for later work because it shows greater sensitivity to nonperturbative modeling choices.
  • The replica-like combination of uncertainty sources yields a practical way to generate families of fragmentation functions for collider phenomenology.

Where Pith is reading between the lines

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

  • The method could be extended to other multiquark states once analogous fragmentation functions are constructed for them.
  • Comparison with early LHC data on pentaquark candidates might help test whether the compact versus diquark mechanisms dominate in real production.
  • If the uncertainty bands prove reliable, the approach would support more targeted experimental searches for heavy exotic hadrons at future colliders.

Load-bearing premise

The transverse-momentum structure of the nonperturbative wave function can be modified in a controlled way to describe both compact multiquark and diquark-driven production mechanisms at the same time.

What would settle it

A measurement of the rate for pentaquark-plus-jet production at the HL-LHC that falls well outside the uncertainty bands obtained from the PQ5Q1.1 functions would show that the uncertainty modeling or the initial-scale inputs require revision.

Figures

Figures reproduced from arXiv: 2605.01539 by Francesco Giovanni Celiberto.

Figure 1
Figure 1. Figure 1: Main leading-order diagrams for the collinear fragmentation of a constituent heavy view at source ↗
Figure 2
Figure 2. Figure 2: Main leading-order diagrams for the collinear fragmentation of a constituent heavy view at source ↗
Figure 3
Figure 3. Figure 3: Momentum-fraction dependence of PQ5Q1.1 functions describing the collinear frag￾mentation of (anti)charm quarks into P5c pentaquarks, for direct (left) and scalar-diquark (right) initial-scale inputs. Shaded bands in the main panels denote the total uncertainty, ob￾tained by combining F-MHOU and F-NPWF variations. The first lower panel highlights the effect of F-MHOUs, while the second isolates F-NPWF unce… view at source ↗
Figure 4
Figure 4. Figure 4: Momentum-fraction dependence of PQ5Q1.1 functions describing the collinear frag￾mentation of gluons into P5c pentaquarks, for direct (left) and scalar-diquark (right) initial-scale inputs. Shaded bands in the main panels denote the total uncertainty, obtained by combining F-MHOU and F-NPWF variations. The first lower panel highlights the effect of F-MHOUs, while the second isolates F-NPWF uncertainties. Fo… view at source ↗
Figure 5
Figure 5. Figure 5: Energy dependence of PQ5Q1.1 functions describing the collinear fragmentation of all parton species to P5c pentaquarks within direct (left) or scalar-diquark (right) initial-scale inputs, at z = 0.5 ≃ ⟨z⟩. Central-value replicas are shown, corresponding to the default configurations for each mode (replica 0 for the direct case and replica 9 for the diquark one), without including F-MHOU or F-NPWF variation… view at source ↗
Figure 6
Figure 6. Figure 6: Energy dependence of PQ5Q1.1 functions describing the collinear fragmentation of all parton species to P5b pentaquarks within direct (left) or scalar-diquark (right) initial-scale inputs, at z = 0.5 ≃ ⟨z⟩. Central-value replicas are shown, corresponding to the default configurations for each mode (replica 0 for the direct case and replica 9 for the diquark one), without including F-MHOU or F-NPWF variation… view at source ↗
Figure 7
Figure 7. Figure 7: Schematic representation of the hybrid collinear and high-energy factorization for the view at source ↗
Figure 8
Figure 8. Figure 8: ∆Y distributions for semi-inclusive P5c plus jet production at √ s = 13 TeV (HL￾LHC), in the direct fragmentation mode (left) and in the scalar-diquark mode (right). Shaded bands in the main panels represent the total uncertainty, obtained by combining H-MHOU, F-MHOU, F-NPWF, and phase-space integration effects. The ancillary panels display: (i) the ratios of LL/LO and HE-NLO+ predictions to the NLL/NLO+ b… view at source ↗
Figure 9
Figure 9. Figure 9: ∆Y distributions for semi-inclusive P5c plus jet production at √ s = 100 TeV (FCC, nominal), in the direct fragmentation mode (left) and in the scalar-diquark mode (right). Shaded bands in the main panels represent the total uncertainty, obtained by combining H￾MHOU, F-MHOU, F-NPWF, and phase-space integration effects. The ancillary panels display: (i) the ratios of LL/LO and HE-NLO+ predictions to the NLL… view at source ↗
read the original abstract

We present an uncertainty-aware description of leading-power fragmentation for all-charm pentaquark states ($S$-wave $|cccc\bar{c}\rangle$) at hadron colliders. We construct a multimodal set of collinear fragmentation functions, PQ5Q1.1, incorporating both perturbative and nonperturbative uncertainties. Perturbative effects are estimated via missing higher-order variations (F-MHOUs), while the nonperturbative wave function is modeled through controlled modifications of its transverse-momentum structure (F-NPWF), consistently combined within a replica-like framework. The initial-scale input for constituent charm fragmentation is refined to describe both compact multiquark and diquark-driven production mechanisms. We employ the (sym)JETHAD interface to study NLL/NLO$^+$ semi-inclusive pentaquark-plus-jet production at the HL-LHC and future FCC. The bottom sector is left to future dedicated studies due to its enhanced sensitivity to nonperturbative modeling. Our results provide a flexible framework for uncertainty-controlled predictions, bridging exotic-hadron structure, heavy-flavor fragmentation, and high-energy QCD.

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

2 major / 2 minor

Summary. The manuscript constructs a multimodal set of collinear fragmentation functions (PQ5Q1.1) for S-wave all-charm pentaquarks (|cccc c-bar>) at hadron colliders. It incorporates perturbative missing-higher-order uncertainties (F-MHOU) and nonperturbative wave-function uncertainties (F-NPWF) combined in a replica-like framework. The initial-scale input for constituent charm fragmentation is refined to simultaneously describe compact multiquark and diquark-driven mechanisms. Semi-inclusive pentaquark-plus-jet cross sections are computed at NLL/NLO+ accuracy for HL-LHC and FCC using the (sym)JETHAD interface; the bottom sector is deferred.

Significance. If the central modeling choices are placed on firmer ground, the work supplies a practical uncertainty-aware framework for leading-power fragmentation of exotic all-heavy states. The controlled transverse-momentum modifications and replica combination constitute a concrete strength for propagating both perturbative and nonperturbative effects into collider predictions.

major comments (2)
  1. [Abstract, §2] Abstract and §2 (construction of PQ5Q1.1): the statement that the initial-scale input for constituent charm fragmentation is 'refined to describe both compact multiquark and diquark-driven production mechanisms' is load-bearing for the multimodal claim, yet the manuscript provides no explicit matching to the standard heavy-quark fragmentation function or first-principles wave-function calculation that would justify the refinement as more than a phenomenological adjustment. Without such a derivation or consistency check, the subsequent F-MHOU/F-NPWF envelope cannot be shown to bound the true theoretical uncertainty on the leading-power fragmentation function.
  2. [§3] §3 (uncertainty combination): the replica-like merging of F-MHOU and F-NPWF variations assumes a specific correlation structure between perturbative and nonperturbative sources that is not derived from the factorization theorem or validated against an external benchmark; this directly affects the width of the uncertainty bands reported for HL-LHC and FCC predictions.
minor comments (2)
  1. [§2] Notation for the PQ5Q1.1 set and the F-MHOU/F-NPWF labels should be defined at first use with a brief reminder of their relation to standard DGLAP evolution.
  2. [§4] Figure captions for the pentaquark-plus-jet distributions should explicitly state the kinematic cuts and the value of the factorization scale used in the NLL/NLO+ calculation.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We appreciate the referee's thorough review and constructive feedback on our manuscript. Below, we provide point-by-point responses to the major comments, indicating the revisions planned for the next version.

read point-by-point responses

  1. Referee: [Abstract, §2] Abstract and §2 (construction of PQ5Q1.1): the statement that the initial-scale input for constituent charm fragmentation is 'refined to describe both compact multiquark and diquark-driven production mechanisms' is load-bearing for the multimodal claim, yet the manuscript provides no explicit matching to the standard heavy-quark fragmentation function or first-principles wave-function calculation that would justify the refinement as more than a phenomenological adjustment. Without such a derivation or consistency check, the subsequent F-MHOU/F-NPWF envelope cannot be shown to bound the true theoretical uncertainty on the leading-power fragmentation function.

    Authors: We thank the referee for highlighting this point. The refinement of the initial-scale input is indeed a phenomenological adjustment aimed at accommodating both compact multiquark and diquark-driven mechanisms in the multimodal PQ5Q1.1 set. In the revised manuscript, we will add an explicit matching to the standard heavy-quark fragmentation function in the appropriate limit, as well as a brief discussion of the parameter choices that allow interpolation between the two mechanisms. This will serve as a consistency check and better support the claim that the F-MHOU/F-NPWF envelope provides a reasonable bound on the theoretical uncertainty. A complete first-principles derivation from wave-function calculations lies outside the present scope but will be addressed in follow-up work. revision: partial

  2. Referee: [§3] §3 (uncertainty combination): the replica-like merging of F-MHOU and F-NPWF variations assumes a specific correlation structure between perturbative and nonperturbative sources that is not derived from the factorization theorem or validated against an external benchmark; this directly affects the width of the uncertainty bands reported for HL-LHC and FCC predictions.

    Authors: The replica-like framework for merging F-MHOU and F-NPWF is a practical method to propagate both types of uncertainties into the predictions, assuming independence between perturbative and nonperturbative variations to obtain a conservative envelope. We acknowledge that this correlation structure is not derived from the factorization theorem. In the updated version, we will clarify this assumption in §3 and discuss its impact on the uncertainty bands for the HL-LHC and FCC cross sections. Validation against external benchmarks is currently limited by the absence of suitable calculations for all-charm pentaquarks, which we will note as a direction for future research. revision: yes

Circularity Check

1 steps flagged

Refined initial-scale input for charm fragmentation is adjusted to encode both production mechanisms, making subsequent multimodal predictions dependent on that modeling choice by construction

specific steps
  1. fitted input called prediction [Abstract]
    "The initial-scale input for constituent charm fragmentation is refined to describe both compact multiquark and diquark-driven production mechanisms. We construct a multimodal set of collinear fragmentation functions, PQ5Q1.1, incorporating both perturbative and nonperturbative uncertainties. Perturbative effects are estimated via missing higher-order variations (F-MHOUs), while the nonperturbative wave function is modeled through controlled modifications of its transverse-momentum structure (F-NPWF), consistently combined within a replica-like framework."

    The refinement is performed expressly to encode both production mechanisms inside a single initial-scale input; this input then directly supplies the PQ5Q1.1 set whose F-MHOU/F-NPWF variations generate the uncertainty-aware predictions. The collider observables are therefore not independent results but re-expressions of the same phenomenological adjustment, with no cited matching condition or external calculation that would break the dependence.

full rationale

The paper's central construction begins by refining the initial-scale input for constituent charm fragmentation specifically to describe both compact multiquark and diquark-driven mechanisms, then builds the PQ5Q1.1 multimodal fragmentation functions from this input together with F-MHOU and F-NPWF variations combined in a replica-like framework. These functions are subsequently used to generate predictions for NLL/NLO+ semi-inclusive pentaquark-plus-jet production. Because the refinement is presented as a phenomenological adjustment rather than a derivation from leading-power factorization, heavy-quark fragmentation matching, or an independent wave-function calculation, the uncertainty envelope and collider predictions reduce to variations around the chosen input. This matches the fitted-input-called-prediction pattern: the output is statistically and structurally forced by the modeling decision that was introduced to cover the desired mechanisms. The decision to defer the bottom sector due to nonperturbative sensitivity further underscores that the charm treatment remains under-constrained by external benchmarks.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 1 invented entities

Central claim rests on leading-power collinear factorization, NLL/NLO+ perturbative accuracy, and specific modeling of initial-scale inputs plus transverse-momentum modifications for the pentaquark wave function.

free parameters (2)
  • initial-scale input for constituent charm fragmentation
    Refined to describe both compact multiquark and diquark-driven production mechanisms.
  • transverse-momentum structure modifications
    Controlled modifications used to model nonperturbative uncertainties in the wave function.
axioms (2)
  • domain assumption Leading-power collinear factorization holds for pentaquark fragmentation
    Invoked for the construction of collinear fragmentation functions in semi-inclusive production.
  • standard math NLL/NLO+ accuracy in perturbative QCD calculations
    Employed via the (sym)JETHAD interface for predictions.
invented entities (1)
  • PQ5Q1.1 multimodal fragmentation functions no independent evidence
    purpose: Set of functions incorporating F-MHOUs and F-NPWF uncertainties for all-charm pentaquarks
    Newly constructed set whose validity depends on the modeling assumptions.

pith-pipeline@v0.9.0 · 5502 in / 1559 out tokens · 52710 ms · 2026-05-09T14:15:15.425985+00:00 · methodology

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