Spin-averaged $B_c$ Spectrum in a Cornell-type Potential Using VMC Baseline and GFMC Evolution

In this work, the spin-averaged $B_c$ spectrum is computed in a naive Cornell framework, treating the meson as a nonrelativistic system in a spin-independent potential. The Cornell parameters are calibrated directly to the spin-averaged $B_c$ tower by anchoring the $1S$ centroid and scanning a grid in $(\sigma,\kappa)$, with the additive constant $V_0$ fixed at each point by the experimental ground state mass. The spectrum is obtained with a two stage Monte Carlo approach. Variational Monte Carlo (VMC) provides optimized radial trial states with the desired nodal pattern. Fixed node Green's function Monte Carlo (GFMC) then projects the corresponding ground state energies for each $(n,\ell)$ channel. Controlled scans over the GFMC time step, projection time, walker population, and radial grid identify plateau regions where discretization and projection systematics are quantitatively under control. At a representative best point in the low-RMSE valley, the predicted spin-averaged masses agree with the experimental centroids at the level of a few tens of MeV, and the fitted Cornell parameters are consistent with canonical heavy quarkonium analyses.