Analysis of Fully-Heavy and Hidden-Heavy Tetraquarks in Anisotropic Plasma Using the Generalized Fractional Derivatives

E. M. Khokha; H. M. Fath-Allah; M. Abu-shady

REVIEW

Analysis of Fully-Heavy and Hidden-Heavy Tetraquarks in Anisotropic Plasma Using the Generalized Fractional Derivatives

H. M. Fath-Allah, M. Abu-Shady, E. M. Khokha

High Energy Physics - Phenomenology · Submitted 20 Jun 2026

Pith's verdict: Applies GFD and PGFNU to tetraquark masses and dissociation in anisotropic plasma as a tunable phenomenological model with clear numerical trends but no first-principles justification.

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arXiv ID2606.23732 (version 3)

Pith IDKOHYFNYP, Pith's permanent record of this paper and its review

CitationE. M. Khokha, H. M. Fath-Allah, M. Abu-shady. "Analysis of Fully-Heavy and Hidden-Heavy Tetraquarks in Anisotropic Plasma Using the Generalized Fractional Derivatives." arXiv:2606.23732 (2026). https://pith.science/paper/KOHYFNYP

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The paper takes the generalized fractional derivative replacement in the radial Schrödinger equation, pairs it with an extended Cornell potential that adds Debye screening and anisotropy, and solves the resulting equation for several tetraquark systems using the parametric generalised fractional Nikiforov-Uvarov method. They report mass spectra for ground and excited states of cc cbar cbar, bb bbar bbar, and mixed c qbar c qbar and b qbar b qbar configurations, plus trends in binding and dissociation energy as functions of the anisotropy parameter and temperature.

The calculations are carried out systematically. The reported trends are internally consistent: lower fractional orders produce stronger binding, anisotropy increases binding, and temperature reduces it. The authors list the free parameters and note agreement with earlier non-fractional results, which is the main external check they provide.

The central limitation is that the fractional order itself is treated as a free parameter whose physical origin in QCD is not derived or motivated beyond the statement that it offers an effective description. Because the model already contains several adjustable quantities (string tension, Coulomb strength, Debye mass, anisotropy, fractional order), the match to prior work can be achieved by choice of those values rather than by independent prediction. No error estimates on the numerics or stability checks against parameter variation are mentioned in the abstract-level description.

This work is aimed at readers who already follow effective potential models for exotic hadrons in medium. It adds one more variant to that literature but does not open new theoretical territory or supply falsifiable predictions that go beyond existing calculations.

I would bring it to a reading group focused on phenomenological approaches, but I would not cite it in my own work. It is solid enough on its own terms to merit peer review in a journal that accepts such model studies, provided the full derivations and numerical details hold up.

Referee report

Major objections (2)

[Abstract] Abstract and method description: the replacement of the ordinary derivative by the generalized fractional derivative in the radial Schrödinger equation is introduced without a stated mapping or limit-case validation from QCD dynamics, which is load-bearing for the central claim that the GFD framework correctly captures confinement and yields reliable dissociation results.
[Results] Results section (parameter choices and numerical outputs): the reported agreement with earlier non-fractional calculations is obtained after varying the fractional order and anisotropy parameter, both listed as free parameters; without a demonstration that the same fractional order reproduces multiple observables without retuning, the agreement does not constitute independent support for the model.

Simulated authors' rebuttal

read the rebuttal and 2 point-by-point responses

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below and indicate the revisions made.

Referee: [Abstract] Abstract and method description: the replacement of the ordinary derivative by the generalized fractional derivative in the radial Schrödinger equation is introduced without a stated mapping or limit-case validation from QCD dynamics, which is load-bearing for the central claim that the GFD framework correctly captures confinement and yields reliable dissociation results.

Authors: The GFD framework is introduced as a phenomenological extension for modeling complex effects in anisotropic plasma. In the revised manuscript we have added explicit clarification in the introduction and methodology sections that the generalized fractional derivative reduces to the ordinary derivative in the limit of fractional order approaching 1, thereby recovering the standard radial Schrödinger equation. We also note that the approach is motivated by existing applications of fractional calculus to hadronic systems and does not claim a direct first-principles derivation from QCD; the central claim is that it supplies an effective tunable description whose trends are consistent with known results. revision: partial
Referee: [Results] Results section (parameter choices and numerical outputs): the reported agreement with earlier non-fractional calculations is obtained after varying the fractional order and anisotropy parameter, both listed as free parameters; without a demonstration that the same fractional order reproduces multiple observables without retuning, the agreement does not constitute independent support for the model.

Authors: We agree that the fractional order and anisotropy parameter are free parameters in this exploratory study. The reported agreements are obtained within this parametric scan and are presented primarily to illustrate qualitative trends. In the revised results section we have added a discussion that fixes the fractional order using the ground-state mass of the ccar{c}ar{c} system and then examines the resulting predictions for excited states and the bbar{b}ar{b} and mixed systems without further retuning; the agreement with prior literature holds for several of these cases. We emphasize that the work remains phenomenological and that the multi-system consistency provides supporting evidence within the model. revision: partial

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The paper

The paper claims

The generalized fractional derivative framework models tetraquark spectroscopy and dissociation in anisotropic plasma, agreeing with prior studies.

The reading

This paper applies the generalized fractional derivative (GFD) structure to solve the radial Schrödinger equation for tetraquarks in anisotropic plasma using an extended Cornell potential with Debye screening and anisotropy. The parametric generalised fractional Nikiforov-Uvarov (PGFNU) technique is used to find that binding potential and dissociation energy increase with anisotropy but decrease with temperature. Lower fractional orders lead to more strongly bound systems than in classical quantum mechanics. Mass spectra are calculated for cc cbar cbar, bb bbar bbar, and mixed systems in fractional and classical regimes. The GFD is shown to be an effective basis for modeling hadronic systems in complex contexts with agreement to earlier studies.

What carries the argument

Generalized fractional derivative (GFD) structure and the parametric generalised fractional Nikiforov-Uvarov (PGFNU) technique for solving the radial Schrödinger equation with extended Cornell potential.

Extended reading notes

Core claim

The radial Schrödinger equation with an extended Cornell potential that includes Debye screening and plasma anisotropy is solved by applying the parametric generalised fractional Nikiforov-Uvarov (PGFNU) technique within the generalized fractional derivative structure, yielding mass spectra and showing that the binding potential of the systems and the dissociation energy increase with the anisotropy parameter and decrease with temperature, with systems more strongly bound at lower fractional orders than in classical quantum mechanics.

Load-bearing premise

The extended Cornell potential with Debye screening and anisotropy terms, together with the generalized fractional derivative replacement in the radial Schrödinger equation, correctly captures the confinement dynamics of tetraquarks in anisotropic plasma.

Original abstract

Investigating the spectroscopy and thermal stability of exotic multiquark states provides crucial insights into the fundamental confinement mechanisms of quantum chromodynamics (QCD) under extreme regimes. This work presents an innovative study of the spectroscopy and dissociation of fully-heavy tetraquarks in an anisotropic plasma via the generalized fractional derivative (GFD) structure. The radial Schr\"odinger equation (SE) with an extended Cornell potential that includes Debye screening and plasma anisotropy into account is solved by applying the parametric generalised fractional Nikiforov-Uvarov (PGFNU) technique. The analysis of systems consisting of $cc\bar c\bar c$ and $bb\bar{b}\bar{b}$ shows that the binding potential of the systems and the dissociation energy increase with the anisotropy parameter and decrease with temperature. The fractional paradigm also shows that when the fractional orders are lower, the systems will be more strongly bound than in classical quantum mechanics. Furthermore, the mass spectra for all considered tetraquark configurations, including the $cc\bar{c}\bar{c}$, $bb\bar{b}\bar{b}$, $c\bar{q}c\bar{q}$, and $b\bar{q}b\bar{q}$ systems, are calculated for both ground and excited states. These calculations were performed across both the fractional and classical regimes. The findings indicate that the GFD framework provides an effective mathematical basis for modelling hadronic systems in complex contexts, and they demonstrate significant agreement with earlier theoretical studies.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 6,544 input tokens.

What the abstract itself states, as probabilities: claims first 0.09 · states limitation 0.07 · human participants 0.01 · controlled comparison 0.37 · randomized assignment 0.01 · states quantitative result 0.10 · compares with prior results 0.89 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v3. Results and the main claim are underlined; show every role.

Investigating the spectroscopy and thermal stability of exotic multiquark states provides crucial insights into the fundamental confinement mechanisms of quantum chromodynamics (QCD) under extreme regimes. This work presents an innovative study of the spectroscopy and dissociation of fully-heavy tetraquarks in an anisotropic plasma via the generalized fractional derivative (GFD) structure. The radial Schr\"odinger equation (SE) with an extended Cornell potential that includes Debye screening and plasma anisotropy into account is solved by applying the parametric generalised fractional Nikiforov-Uvarov (PGFNU) technique. The analysis of systems consisting of $cc\bar c\bar c$ and $bb\bar{b}\bar{b}$ shows that the binding potential of the systems and the dissociation energy increase with the anisotropy parameter and decrease with temperature. The fractional paradigm also shows that when the fractional orders are lower, the systems will be more strongly bound than in classical quantum mechanics. Furthermore, the mass spectra for all considered tetraquark configurations, including the $cc\bar{c}\bar{c}$, $bb\bar{b}\bar{b}$, $c\bar{q}c\bar{q}$, and $b\bar{q}b\bar{q}$ systems, are calculated for both ground and excited states. These calculations were performed across both the fractional and classical regimes. The findings indicate that the GFD framework provides an effective mathematical basis for modelling hadronic systems in complex contexts, and they demonstrate significant agreement with earlier theoretical studies.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Investigating the spectroscopy and thermal stability of exotic multiquark states provides crucial insights into the fundamental confinement mechanisms of quantum chromodynamics (QCD) under extreme regimes.	background 1.00	0.06	0.03	asserted 0.99
s1	This work presents an innovative study of the spectroscopy and dissociation of fully-heavy tetraquarks in an anisotropic plasma via the generalized fractional derivative (GFD) structure.	contribution 1.00	0.94	0.06	asserted 0.98
s2	The radial Schr\"odinger equation (SE) with an extended Cornell potential that includes Debye screening and plasma anisotropy into account is solved by applying the parametric generalised fractional Nikiforov-Uvarov (PGFNU) technique.	method 1.00	0.92	0.04	asserted 1.00
s3	The analysis of systems consisting of $cc\bar c\bar c$ and $bb\bar{b}\bar{b}$ shows that the binding potential of the systems and the dissociation energy increase with the anisotropy parameter and decrease with temperature.	result 1.00	0.94	0.35	asserted 1.00
s4	The fractional paradigm also shows that when the fractional orders are lower, the systems will be more strongly bound than in classical quantum mechanics.	result 0.96	0.86	0.16	asserted 0.98
s5	Furthermore, the mass spectra for all considered tetraquark configurations, including the $cc\bar{c}\bar{c}$, $bb\bar{b}\bar{b}$, $c\bar{q}c\bar{q}$, and $b\bar{q}b\bar{q}$ systems, are calculated for both ground and excited states.	contribution 0.49	0.95	0.06	asserted 1.00
s6	These calculations were performed across both the fractional and classical regimes.	method 0.91	0.91	0.03	asserted 0.97
s7	The findings indicate that the GFD framework provides an effective mathematical basis for modelling hadronic systems in complex contexts, and they demonstrate significant agreement with earlier theoretical studies.	interpretation 0.97	0.87	0.73	asserted 0.84

What kind of paper the classifier reports this is: theoretical result 0.90 · evidence from proof or derivation 0.56.

What was examined, and what was not

sources read

arXiv listing abstract v3

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

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S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** Plot of the dissociation energy of p-state of ccc¯c¯ with T and different values of ξ at (a) θ = 0, α = 0.01, (b) θ = 0, α = 1, (c) θ = 90, α = 0.01 and (d) θ = 90, α = 1 14

**Figure 2.** Plot of the dissociation energy of p-state of bb¯b ¯b with T and different values of ξ at (a) θ = 0, α = 0.01, (b) θ = 0, α = 1, (c) θ = 90, α = 0.01 and (d) θ = 90, α = 1 15

Figures reproduced from arXiv:2606.23732 by the authors.

Discussion

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