Quantum Bootstrap Approach to a Non-Relativistic Potential for Quarkonium systems

TL;DR

This paper applies the quantum bootstrap method to non-relativistic quarkonium systems, accurately calculating known spectra and predicting a toponium state consistent with recent experimental hints, demonstrating the method's effectiveness.

Contribution

It introduces a quantum bootstrap approach for quarkonium spectra, providing precise calculations for known systems and predicting a toponium state aligned with experimental observations.

Findings

Accurately reproduces charmonium and bottomonium spectra with less than 0.5% deviation.

Predicts a toponium 1S mass of approximately 344.3 GeV.

Supports the interpretation of recent experimental signals as toponium formation.

Abstract

The quantum bootstrap method is applied to determine the bound-state spectrum of Quarkonium systems using a non-relativistic potential approximation. The method translates the Schr\"odinger equation into a set of algebraic recursion relations for radial moments $\langle r^m \rangle$, which are constrained by the positive semidefiniteness of their corresponding Hankel matrices. The numerical implementation is first validated by calculating the $1S$ and $1P$ mass centroids for both charmonium ($c\bar{c}$) and bottomonium ($b\bar{b}$) systems, finding deviations of less than 0.5\% from experimental data from the Particle Data Group (PDG). This analysis is then extended to the hypothetical toponium ($t\bar{t}$) system, predicting a $1S$ ground state mass of $M \approx 344.3 \text{ GeV}$. This theoretical mass is in agreement with the energy of the recently observed resonance-like enhancement in the $t\bar{t}$ cross-section by the ATLAS and CMS collaborations. This result provides theoretical support for the interpretation of this experimental phenomenon as the formation of a quasi-bound toponium state and highlights the predictive power of the non-relativistic potential approach for systems of two massive quarks.