Learning holographic QCD with unflavoured meson spectra

TL;DR

The paper addresses extracting the holographic QCD background from hadron spectra by training neural networks to learn the 5D warp factor, dilaton, and chiral-symmetry breaking potential directly from unflavored meson masses. It treats the bulk-mode equations as discretized Schrödinger-like problems within a deconstructed 5D framework, enabling efficient computation of spectra and background fields via finite-difference eigenproblems. Key findings include a positive dilaton profile intermediate between linear and quadratic forms, a bulk scalar potential with Tuneable cubic and quartic terms, and accurate predictions for the ρ, a1, a2, f0 spectra as well as reasonable π masses, all achieved with a repository of trained models. The approach demonstrates a scalable, physics-informed deep-learning pathway to constrain bottom-up AdS/QCD backgrounds and underscores the utility of lattice-like discretization in holographic model fitting.

Abstract

We develop a neural network framework to predict the five-dimensional background geometry, dilaton potential, and chiral symmetry breaking scalar potential of holographic QCD from unflavored meson mass spectra. The model was trained in a discretized form of the Schrödinger-like equation, which resembles a linear moose in ``deconstructed" 5 dimensions with Dirichlet boundary conditions, in contrast to the AdS/DL with ``emergent" space-time. Using the $ρ$, $a_1$, $a_2$, and $f_0$ unflavored mesons and their excitations as training data, the model learns confining effective potentials and computes a dilaton profile that satisfies the null energy condition. The network predicts the IR behavior of dilaton to be in-between linear and quadratic forms. Moreover, the symmetry-breaking bulk potential of the scalar field, $V(X)= k_1 X^3+k_2 X^4$, was computed, and the parameters $k_1$ and $k_2$ predicted to be $\sim - \ 8$ and $\sim 17$ respectively. The deep-learned parameters, metric, and the dilaton profile were then used to predict the pion mass and its spectrum with good accuracy. A Python code, along with the trained models, is provided to facilitate further studies

Figures (6)

• Figure 1: The learned functions $v(z)$ (left) and $A(z)$ (right) after training. The shaded regions represent the standard deviation over different runs, and the solid lines represent the mean value.
• Figure 2: The plot of $d\phi/dz$ (left) and $\phi(z)$ (right). The solid line represents the best fit corresponding to the minimum value of $\mathcal{L}_{\text{mass}}^{(\rho, a_1, a_2, f_0)}$. Unlike Figs. \ref{['fig:v_A_plots']} and \ref{['fig:phi_plot']}, where the spread was symmetric and shown as mean $\pm$ standard deviation, the $d\phi/dz$ here exhibit a skewed variation, which is why we only plot the $d\phi/dz$ corresponding to the best run. However, the $\phi(z)$ (right panel) obtained shows a symmetric spread, which is represented by the mean. The dotted line and the dashed line in the right panel represent a linear and quadratic fit to the mean value of $\phi(z)$, respectively.
• Figure 3: Effective potentials for the $\rho$, $a_1$, $a_2$, and $f_0$ mesons. The solid line represents the best fit corresponding to the minimum value of $\mathcal{L}_{\text{mass}}^{(\rho, a_1, a_2, f_0)}$.
• Figure 4: Mass spectra of the $\rho$, $f_0$, $a_1$ and $a_2$ mesons ParticleDataGroup:2024cfk. The predicted masses correspond to the mean, and the error bars show the standard deviation across multiple runs.
• Figure 5: Mass spectrum of the predicted $\pi$ meson as shown in Table \ref{['tab:pi_masses']}. The error bars represent the standard deviation over different runs.