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

Christas Mony A., Rohit Dhir

Abstract

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.

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

Abstract

We present a comprehensive Bayesian study of the 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 -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 states with uncertainties, serving as benchmarks for ongoing and future experiments.

Paper Structure

This paper contains 14 sections, 38 equations, 8 figures, 10 tables.

Table of Contents

  1. Introduction
  2. Methodology
  3. Deterministic Parameter Fits
  4. MCMC Exploration of Parameter Space
  5. Wave Function at the Origin and Related Quantities
  6. Regge Trajectories
  7. Numerical Results and Discussion
  8. Mass Spectra
  9. Wave Function at the Origin and Related Quantities
  10. Effects of Modification in the Cornell Potential
  11. Regge Trajectories
  12. Summary and Conclusions
  13. Limitations of Deterministic Parameter Fits
  14. MCMC Configuration and Convergence

Figures (8)

  • Figure 1: Comparison of the Cornell (Potential I, Eq. \ref{['eqn:1_cornell']}) and Modified Cornell (Potential II, Eq. \ref{['eqn:corplusln']}) potentials, using median parameter values Table \ref{['tab:Bc_parameters']}. Shaded bands denote credible regions based on $\sim 5000$ MCMC samples.
  • Figure 2: Corner plot for Cornell Potential (Potential I).
  • Figure 3: Corner plot for Modified Cornell Potential (Potential II).
  • Figure 4: The $B_c$ mass spectrum (in MeV) computed using Cornell (Potential I) and modified Cornell (Potential II) potentials in blue (left) and red (right), respectively. The black diamonds are experimental values ParticleDataGroup:2024cfk. The dotted lines indicate the thresholds for open-flavor $B_{(s)}^{(*)} D_{(s)}^{(*)}$ channels.
  • Figure 5: Radial wave functions of the $B_c$ system obtained using Cornell (Potential I, Eq. \ref{['eqn:1_cornell']}, left) and Modified Cornell (Potential II, Eq. \ref{['eqn:corplusln']}) potentials, right). Shaded bands represent the propagated uncertainties from the MCMC analysis ($\sim 5000$ samples).
  • ...and 3 more figures