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

TL;DR

The paper presents a nonperturbative quantum-bootstrap approach to heavy-quark bound states using the Cornell potential. By recasting the Schrödinger problem into moment recursions for $\\mu_n=\\langle x^n\\rangle$ and enforcing positivity of coupled Hankel matrices on the half-line, it derives energy bounds that converge exponentially with the bootstrap depth $K$. Applied to charmonium, bottomonium, and a hypothetical toponium system, the method reproduces spin-averaged $1S$ and $1P$ centroids with deviations below $0.5\%$ from PDG data and predicts a near-threshold $1S$ centroid around $344\ \text{GeV}$ for $t\\bar t$, in line with threshold enhancements reported by LHC experiments. The results demonstrate that spectra can be extracted from algebraic consistency and positivity without explicit wavefunctions, offering a complementary, efficient platform for nonperturbative spectroscopy with potential extensions to spin, relativistic effects, and field-theoretic contexts.

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 Schrödinger 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.

Figures (1)

• Figure 1: Algorithmic flow of the radial quantum bootstrap with SDP feasibility checks. The feasible energy set $\mathcal{S}_K$ contracts monotonically with depth $K$.