This paper studies how simple models of space-time defects and a magnetic flux change the energy levels and masses of heavy quark systems. The authors solve the non‑relativistic Schrödinger equation for quark pairs (charmonium and bottomonium) and for fully‑heavy tetraquarks (states made only of charm or only of bottom quarks). They report analytic formulas for the bound‑state energies and compute mass spectra that they say agree with existing theory and experiment.

To do this they use the Analytical Exact Iteration Method (AEIM), a technique for finding exact eigenvalues and wave functions of the Schrödinger problem. The calculation is set in a simple curved background that models a cosmic string — a type of topological defect — and includes an Aharonov–Bohm magnetic flux (a magnetic field described by a flux number Φ). The quark interaction is modelled by a modified Cornell potential: a short‑range Coulomb term, a long‑range linear confining term, plus an extra harmonic term (proportional to r^2) and an inverse quadratic term (proportional to 1/r^2). In this setup the space‑time defect (called α in the paper) and the flux Φ change the effective centrifugal barrier and so shift the energy levels.

The authors derive closed analytic expressions for energies at arbitrary radial excitations and give explicit ground and excited state wave functions. They use these expressions to produce numerical mass spectra for c¯c and b¯b mesons and for fully‑charmed and fully‑bottom tetraquarks (cc¯c¯c and bb¯b¯b). The paper shows that the defect parameter α shifts the energy curves and that the total energy grows roughly linearly with α in their examples. They also state that their tetraquark masses agree with previous theoretical studies that treat tetraquarks as diquark–antidiquark systems.

Why this matters: fully‑heavy tetraquarks are a relatively clean laboratory for the strong force because they avoid complications from light quarks and pion exchange. Finding that a background geometric defect or a magnetic flux can change binding energies suggests that nontrivial geometry or external fields can influence hadron spectroscopy in model calculations. The analytic solutions also provide formulas that other theorists can use to explore parameter dependence without repeating heavy numerics.