$\boldsymbol{B_c}$ Meson Spectroscopy from Bayesian MCMC: Probing Confinement and State Mixing

We present a comprehensive Bayesian study of the $B_c$ meson spectrum using non-relativistic Cornell and logarithmically modified Cornell potentials, introducing the logarithmic term as the minimal deformation that preserves short-range Coulombic and long-range linear confinement while adding controlled flexibility at intermediate distances to probe the sensitivity of higher excited states to the confining form. Model parameters are sampled via Markov chain Monte Carlo (MCMC), enabling rigorous propagation of correlated uncertainties to all predictions. Spin-dependent interactions are treated perturbatively, with unequal heavy-quark masses accounted for consistently. Both potentials reproduce the known states within uncertainties, with small errors for low-lying states that grow for higher radial and orbital excitations. Analyzing radial and orbital Regge trajectories using linear and nonlinear parametrizations, we observe pronounced nonlinearity for low $S$-waves trending toward linearity at higher excitations. The modified potential yields modest, systematic shifts in higher excited states, reflecting the logarithmic correction's impact. We provide updated theoretical predictions for excited $B_c$ states with uncertainties, serving as benchmarks for ongoing and future experiments.