On quarkonium masses in 3D non-commutative space

TL;DR

This paper investigates how non-commutative geometry at the quantum scale affects quarkonium mass calculations, providing theoretical corrections and setting bounds on the fundamental length scale of space.

Contribution

It introduces a modified approach to calculating quarkonium masses in non-commutative space using WKB and Pekeris approximations, deriving corrections and bounds on the fundamental length.

Findings

Relative correction at Planck scale is about 10^{-39}

Upper bound on fundamental length is approximately 10^{-18} meters

Provides theoretical constraints consistent with current experimental data

Abstract

We modify the calculation of quarkonium masses using the radial WKB and Pekeris-type approximations for the case of three-dimensional, rotationally invariant non-commutative space. We obtain corrections to the charmonium ($\text{c}\bar{\text{c}}$), bottomonium ($\text{b}\bar{\text{b}}$) and bottom-charmed meson ($\text{c}\bar{\text{b}}$) masses due to the discrete short distance structure of the space introduced by the space non-commutativity. For the fundamental length at the Planck scale we obtain relative correction at the order of $10^{-39}$, and taking into account the current experimental data, we obtain the upper bound at the order of ${10^{-18}\textrm{ m}}$ for the fundamental length scale of space.