Effect of Rainbow Gravity, PDM, and Magnetic Field on Thermodynamics Properties of Charmonium and Bottomonium

Abstract

This study investigates the Klein-Gordon (KG) oscillator equation in the spacetime of a cosmic string under the influence of rainbow gravity, an external magnetic field, and a position-dependent mass (PDM). An analytical solution of the KG oscillator is derived, and the resulting energy equation is used to compute the numerical energy levels of charmonium and bottomonium for two types of rainbow functions. The computed values show good agreement with experimental data. Furthermore, the energy equation is used to evaluate the partition function, from which various thermodynamic properties of charmonium and bottomonium are derived. For rainbow function 1, increasing the inverse temperature parameter \(\:\beta\:\) results in a decrease in the partition function, internal energy, and entropy, while the free energy increases. The specific heat capacity exhibits a peak before declining. For rainbow function 2, the partition function and free energy increase with \(\:\beta\:\), whereas internal energy and entropy decrease. The specific heat capacity again displays a non-monotonic behavior, rising to a maximum and then decreasing as \(\:\beta\:\) increases. These results reveal how rainbow gravity, magnetic field, and PDM collectively influence the quantum and thermodynamic characteristics of relativistic quark systems.