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Heavy Quarkonium Spectrum and Decay Constants from a Neural-Network-Based Holographic Model

Yu Zhang, Xun Chen, Miguel Angel Martin Contreras

TL;DR

The paper tackles the challenge of simultaneously describing heavy quarkonium spectra and leptonic decay constants within a holographic QCD framework. It introduces a neural-network-based inverse method that reconstructs the dilaton profile $Φ(z)$ by learning $Φ'(z)$ with a constraint $Φ(0)=0$, from which the holographic potential $U(z)$ and the bulk equations yield the spectrum and decay constants. The method achieves RMS deviations of $δ_{ ext{RMS}} = 1.26\%$ for charmonium and $3.32\%$ for bottomonium, and reveals a non-quadratic UV behavior with rapid IR growth that improves agreement with experimental data over traditional soft-wall or WKB approaches. This data-driven holographic modeling provides a flexible, extensible platform for incorporating lattice results, finite-temperature effects, or additional observables to further illuminate non-perturbative QCD dynamics.

Abstract

We present a data-driven inverse construction of the dilaton field in a bottom-up AdS/QCD description of heavy vector quarkonia. Instead of adopting an \emph{ad hoc} analytic ansatz, we use a multilayer perceptron to learn \(Φ'(z)\) as a smooth function of the holographic coordinate, with \(Φ(0)=0\) imposed to ensure ultraviolet consistency. The dilaton and its derivatives obtained by automatic differentiation generate the holographic potential \(U(z)\), and the associated Schrödinger-like equation is discretized and diagonalized to extract the low-lying eigenmodes. Masses and decay constants are then evaluated from the eigenvalues and the near-boundary behavior of the bulk-to-boundary modes. Training on PDG data for charmonium and bottomonium yields a non-quadratic dilaton profile that resolves the longstanding difficulty of simultaneously reproducing both the heavy-quarkonium spectrum and the monotonic suppression of leptonic decay constants with radial excitation. The combined fit achieves RMS deviations of \(1.26\%\) (charmonium) and \(3.32\%\) (bottomonium). This work establishes neural-network reconstruction as a flexible tool for holographic modeling and provides a basis for future extensions incorporating additional channels, lattice constraints, or finite-temperature backgrounds.

Heavy Quarkonium Spectrum and Decay Constants from a Neural-Network-Based Holographic Model

TL;DR

The paper tackles the challenge of simultaneously describing heavy quarkonium spectra and leptonic decay constants within a holographic QCD framework. It introduces a neural-network-based inverse method that reconstructs the dilaton profile by learning with a constraint , from which the holographic potential and the bulk equations yield the spectrum and decay constants. The method achieves RMS deviations of for charmonium and for bottomonium, and reveals a non-quadratic UV behavior with rapid IR growth that improves agreement with experimental data over traditional soft-wall or WKB approaches. This data-driven holographic modeling provides a flexible, extensible platform for incorporating lattice results, finite-temperature effects, or additional observables to further illuminate non-perturbative QCD dynamics.

Abstract

We present a data-driven inverse construction of the dilaton field in a bottom-up AdS/QCD description of heavy vector quarkonia. Instead of adopting an \emph{ad hoc} analytic ansatz, we use a multilayer perceptron to learn \(Φ'(z)\) as a smooth function of the holographic coordinate, with \(Φ(0)=0\) imposed to ensure ultraviolet consistency. The dilaton and its derivatives obtained by automatic differentiation generate the holographic potential \(U(z)\), and the associated Schrödinger-like equation is discretized and diagonalized to extract the low-lying eigenmodes. Masses and decay constants are then evaluated from the eigenvalues and the near-boundary behavior of the bulk-to-boundary modes. Training on PDG data for charmonium and bottomonium yields a non-quadratic dilaton profile that resolves the longstanding difficulty of simultaneously reproducing both the heavy-quarkonium spectrum and the monotonic suppression of leptonic decay constants with radial excitation. The combined fit achieves RMS deviations of (charmonium) and (bottomonium). This work establishes neural-network reconstruction as a flexible tool for holographic modeling and provides a basis for future extensions incorporating additional channels, lattice constraints, or finite-temperature backgrounds.
Paper Structure (4 sections, 17 equations, 3 figures, 2 tables)

This paper contains 4 sections, 17 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Holographic Model
  3. Modeling Holographic QCD with MLP
  4. SUMMARY

Figures (3)

  • Figure 1: A neural network flowchart for reconstructing the dilaton field $\Phi(z)$.
  • Figure 2: Dilaton fields $\Phi(z)$ for charmonium (left) and bottomonium (right) obtained from different approaches: Non-linear approach MCVD MartinContreras:2021bis (blue), Machine Learning Perceptron (MLP) in orange, soft-Wall model Karch:2006pv, and WKB approach MartinContreras:2023eft.
  • Figure 3: Potentials $U(z)$ for charmonium (left) and bottomonium (right) obtained from different approaches: Non-linear approach MCVD MartinContreras:2021bis (blue), Machine Learning Perceptron (MLP) in orange, soft-wall model Karch:2006pv, and WKB approach MartinContreras:2023eft.