arxiv: 2604.09198 · v2 · submitted 2026-04-10 · ⚛️ nucl-th

Recognition: 2 theorem links

· Lean Theorem

Unified Extraction of In-Medium Heavy Quark Potentials from RHIC to LHC Energies via Deep Learning

Jiamin Liu , Kai Zhou , Baoyi Chen

Authors on Pith no claims yet

Pith reviewed 2026-05-10 17:19 UTC · model grok-4.3

classification ⚛️ nucl-th

keywords heavy quark potentialbottomonium suppressiondeep learningBayesian inferenceheavy ion collisionsquark gluon plasmanuclear modification factorin-medium potential

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The pith

Deep learning applied to bottomonium data from RHIC and LHC extracts an in-medium heavy quark potential whose real part stays near the vacuum Cornell form while the imaginary part drives most suppression.

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

The paper seeks to determine the temperature- and distance-dependent potential felt by heavy quarks inside the quark-gluon plasma by matching a parameterized model to measured bottomonium nuclear modification factors across several collision systems and energies. It evolves bottomonium wave functions with a time-dependent Schrödinger equation inside hydrodynamic backgrounds and uses convolutional neural networks trained on many simulated datasets to learn the mapping from potential parameters to observable suppression. A Bayesian inversion with stochastic gradient Langevin dynamics then finds the parameters that best reproduce the combined RHIC and LHC data. A reader would care because this potential controls how heavy quarks bind or dissociate in the medium and therefore shapes predictions for all heavy-flavor observables. If the result holds, the same potential form can be used consistently from the lower densities at RHIC to the higher densities at the LHC.

Core claim

The central claim is that a simultaneous Bayesian extraction from bottomonium R_AA measured in Pb-Pb collisions at 5.02 TeV and 2.76 TeV and in Au-Au collisions at 200 GeV yields an in-medium heavy quark potential whose real part remains close to the vacuum Cornell form, implying a relatively weak screened Debye mass, while the imaginary part is more tightly constrained by the data and supplies the dominant contribution to suppression over the full energy range.

What carries the argument

The in-medium heavy quark potential V(T,r) with separate real and imaginary parts, inserted into a time-dependent Schrödinger equation for b b-bar dipole evolution inside a hydrodynamically expanding medium; parameters are learned by convolutional neural networks trained on simulated R_AA and inverted via stochastic gradient Langevin dynamics to obtain posterior distributions.

If this is right

The real part's closeness to vacuum implies that color screening remains modest at the temperatures reached in these collisions.
Dissociation driven by the imaginary part, rather than reduction in binding energy, is the main mechanism for the observed suppression.
A single potential parametrization works across the density range spanned by RHIC and LHC data.
The method provides a unified, data-driven route to the potential instead of separate fits at each energy.
The extracted imaginary part can be directly inserted into calculations of other quarkonium states or heavy-flavor transport.

Where Pith is reading between the lines

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

Models of in-medium quarkonium should emphasize medium-induced width over adjustments to the real binding potential.
The weak screening result could guide targeted lattice QCD studies of the real part at finite temperature.
The same extraction pipeline can be applied to charmonium data or to future higher-precision measurements to test consistency.
If the hydrodynamic background is varied, the stability of the extracted imaginary part would indicate how much the conclusion depends on the medium evolution model.

Load-bearing premise

The true in-medium potential can be captured by the chosen functional form and that the hydrodynamic background used for the evolution is sufficiently accurate; if either is wrong the extracted parameters will not reflect the actual medium effects.

What would settle it

A new lattice QCD calculation of the heavy quark potential at the relevant temperatures that shows substantially stronger Debye screening than the weak value extracted here, or a new set of bottomonium R_AA measurements at an intermediate energy that cannot be reproduced by the same potential parameters.

read the original abstract

We use deep learning under Bayesian perspective to quantitatively extract the in-medium heavy quark (HQ) potential from bottomonium nuclear modification factors ($R_{AA}$) measured across multiple heavy ion collision systems at the Large Hadron Collider (LHC) and the Relativistic Heavy-Ion Collider (RHIC). The in-medium HQ potential, comprising both a real and imaginary part, is parameterized and incorporated into a time-dependent Schr\"odinger equation to model the wave function evolution of $b\bar{b}$ dipoles within a hydrodynamically evolving hot QCD medium. We construct Convolutional Neural Networks (CNNs) to capture the non-linear correspondence between the heavy quark potential $V(T,r)$ and the bottomonium $R_{AA}$ for Pb-Pb collisions at 5.02 TeV and 2.76 TeV, and Au-Au collisions at 200 GeV. Training datasets are generated by sampling the potential parameters and are further augmented using Principal Component Analysis (PCA) and Gaussian Process Regression (GPR). After validating the stability and correctness of the CNNs, we employ Stochastic Gradient Langevin Dynamics (SGLD) to perform a simultaneous Bayesian inverse extraction of the optimal potential parameters and their posterior distributions using experimental data of bottomonium $R_{AA}$ in both LHC and RHIC energies. Our joint multi-energy extraction suggests that, within the present parametrization and hydrodynamic background, the real part of the in-medium potential remains close to the vacuum Cornell form, corresponding to a relatively weak screened Debye mass across RHIC to LHC energies. By contrast, the imaginary part is more strongly constrained by the data and provides the dominant contribution to bottomonium suppression from RHIC to LHC energies.

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

They train CNNs on Schrödinger evolutions to invert bottomonium R_AA from RHIC and LHC into one parameterized potential, but the result sits inside a single fixed hydrodynamic background.

read the letter

The paper's core move is to generate training data by solving the time-dependent Schrödinger equation for bottomonium wave functions in a hydrodynamically evolving medium, then train CNNs to map parameterized V(T,r) to R_AA. They augment the samples with PCA and GPR, validate the networks, and run SGLD to obtain posteriors on the real and imaginary parts using data from Au-Au at 200 GeV plus Pb-Pb at 2.76 and 5.02 TeV. That joint multi-energy extraction is new for this observable set. They also report that, within their parametrization, the real part stays close to the vacuum Cornell form while the imaginary part supplies most of the suppression. The pipeline is built from established pieces and the data combination is a clear step beyond single-energy fits. The soft spot is the fixed hydrodynamic background. Temperature and flow profiles are not varied, so any mismatch in initial conditions, viscosity, equation of state, or lifetime is absorbed into the extracted potential parameters. The headline claim about weak Debye screening therefore inherits the accuracy of that particular hydro setup across the full energy range. The abstract gives no numbers on validation against synthetic data with known ground truth, though the full text may contain them. This work is for people who run heavy-quark transport codes and need a data-constrained potential to insert. A reader who already works with bottomonium suppression will see the most value. It deserves peer review because the method is coherent on its own terms and the multi-energy data use is worth referee scrutiny, even if the hydro systematics need explicit tests.

Referee Report

2 major / 2 minor

Summary. The paper proposes a Bayesian deep learning framework to extract parameters of a parameterized in-medium heavy quark potential (real and imaginary parts) from bottomonium R_AA data across RHIC (Au-Au at 200 GeV) and LHC (Pb-Pb at 2.76 and 5.02 TeV) energies. Training data are generated by solving the time-dependent Schrödinger equation for b b-bar wave functions in a hydrodynamically evolving medium with sampled potential parameters; CNNs learn the forward map from V(T,r) to R_AA, augmented via PCA and GPR, and SGLD performs the inverse extraction to obtain posterior distributions. The central result is that, within the chosen parametrization and background, Re(V) remains close to the vacuum Cornell form with weak Debye screening while Im(V) dominates suppression.

Significance. If robust, the work supplies a unified multi-energy, data-driven extraction of the in-medium potential, clarifying the relative roles of real and imaginary components in bottomonium suppression. It combines established components (TDSE evolution, hydrodynamics) with CNN regression and SGLD sampling for Bayesian inversion, providing a reproducible template for similar inverse problems in heavy-ion physics. The joint analysis across energies is a clear strength.

major comments (2)

[Training data generation (methods)] The hydrodynamic background (temperature and flow profiles) is held fixed when generating training data for the CNNs. No marginalization or propagation of uncertainties arising from initial conditions, equation of state, shear viscosity, or medium lifetime is performed. Because any mismatch between the assumed evolution and reality is absorbed into the inferred potential parameters, the posteriors on Re(V) and Im(V) do not incorporate this systematic; this directly underpins the headline claim of weak screening in the real part.
[CNN construction and validation] The validation of the CNNs is stated to confirm stability and correctness, yet no quantitative metrics are supplied (e.g., mean-squared error on held-out synthetic data, recovery accuracy for known input potentials, or calibration of predicted R_AA uncertainties). Without these, the reliability of the learned forward map prior to SGLD inversion cannot be assessed at the level required for the central extraction.

minor comments (2)

[Abstract] The abstract refers to augmentation with PCA and GPR but does not detail the procedure, the resulting training-set size, or any checks that the augmentation preserves the physical mapping.
[Potential parametrization] The explicit functional form and all free parameters of V(T,r) (real and imaginary) should be written out with equation numbers so that the dimensionality of the SGLD sampling is immediately clear.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and constructive feedback on our manuscript. We address each major comment below and have prepared revisions to strengthen the presentation of our methods and results.

read point-by-point responses

Referee: The hydrodynamic background (temperature and flow profiles) is held fixed when generating training data for the CNNs. No marginalization or propagation of uncertainties arising from initial conditions, equation of state, shear viscosity, or medium lifetime is performed. Because any mismatch between the assumed evolution and reality is absorbed into the inferred potential parameters, the posteriors on Re(V) and Im(V) do not incorporate this systematic; this directly underpins the headline claim of weak screening in the real part.

Authors: We agree that fixing the hydrodynamic background means the extracted posteriors are conditional on this choice and do not marginalize over uncertainties in initial conditions, EOS, viscosity, or lifetime. This is a standard limitation in such inverse problems given the prohibitive cost of regenerating large training datasets for varied hydro evolutions. The headline claim of weak screening in Re(V) is therefore presented as holding within the employed hydrodynamic model. In the revised manuscript we will add an explicit discussion of this systematic uncertainty, including a note that future extensions could incorporate marginalization over hydro parameters. revision: partial
Referee: The validation of the CNNs is stated to confirm stability and correctness, yet no quantitative metrics are supplied (e.g., mean-squared error on held-out synthetic data, recovery accuracy for known input potentials, or calibration of predicted R_AA uncertainties). Without these, the reliability of the learned forward map prior to SGLD inversion cannot be assessed at the level required for the central extraction.

Authors: We acknowledge that the manuscript describes CNN validation only qualitatively. In the revised version we will add quantitative metrics in the CNN construction section, including mean-squared error on a held-out test set of synthetic R_AA values, recovery accuracy when inverting known input potentials, and calibration checks on the predicted uncertainties. These additions will allow readers to assess the forward map reliability before the SGLD inversion step. revision: yes

Circularity Check

0 steps flagged

No significant circularity: standard Bayesian extraction from external data

full rationale

The paper parameterizes V(T,r), solves the TDSE in a fixed hydrodynamic background to generate training R_AA, trains CNN surrogates, and uses SGLD to infer posterior parameters from measured bottomonium R_AA at RHIC and LHC. The headline conclusions about Re(V) remaining close to the vacuum Cornell form and Im(V) dominating suppression are direct outputs of this fit to independent experimental data. No derivation step reduces the result to the inputs by construction, no self-citation is load-bearing for the central claim, and the hydro background plus parametrization are stated as explicit modeling choices rather than smuggled assumptions. The analysis is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on a specific parameterization of V(T,r) whose parameters are fitted to data, plus the assumption that the time-dependent Schrödinger equation plus hydrodynamic background faithfully maps potential to R_AA.

free parameters (1)

potential parameters
Parameters controlling the real (screening) and imaginary (absorption) parts of V(T,r) are sampled during training and optimized against experimental R_AA.

axioms (2)

domain assumption The evolution of b bbar dipoles is accurately described by the time-dependent Schrödinger equation inside a hydrodynamically evolving medium.
This is the forward model used both to generate training data and to interpret the extracted potential.
domain assumption The hydrodynamic background provides a reliable temperature and flow profile for the medium.
The background is an external input that enters the Schrödinger evolution.

pith-pipeline@v0.9.0 · 5616 in / 1562 out tokens · 61156 ms · 2026-05-10T17:19:01.215932+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

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

VI =−i T a0 (a1r+a2 ra3), VR =−α (μD + e−μD r / r) + 2σ/μD − σ e−μD r (2 + μD r)/μD with μD = a4 T … ; Tsw switching to vacuum Cornell
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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

time-dependent Schrödinger equation … coupled with the hydrodynamic model … CNNs … SGLD … joint multi-energy extraction

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.

Forward citations

Cited by 1 Pith paper

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

Effects of event-by-event hydrodynamic fluctuations on bottomonium dynamics in Pb--Pb collisions at $\sqrt{s_{NN}} = 5.02$ TeV
nucl-th 2026-05 unverdicted novelty 4.0

Event-by-event hydrodynamic fluctuations have marginal effects on bottomonium R_AA and v2 in 5.02 TeV Pb-Pb collisions.

Reference graph

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