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

TL;DR

The paper tackles the spin-averaged $B_c$ spectrum using a minimal spin-independent Cornell potential, anchored to the experimental $1S$ centroid via $V_0$. It employs a two-stage VMC+GFMC projector approach to solve the radial Schrödinger problem and project energies for each $(n,\ell)$ channel, scanning $(\sigma,\kappa)$ to minimize the RMSE against experimental centroids. A low-RMSE valley in parameter space is identified, with a representative best point $\big(\sigma,\kappa,V_0\big) = (0.1625~{\rm GeV}^2, 0.6125, 0.901875~{\rm GeV})$, yielding spin-averaged masses within tens of MeV of data and showing consistency with canonical heavy-quarkonium parameter regions. The results establish a numerically controlled baseline for heavy quark spectroscopy that can be extended to spin-dependent splittings and relativistic corrections, linking with broader quarkonium phenomenology and lattice results.

Abstract

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 $(σ,κ)$, 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.

Figures (6)

• Figure 1: RMSE surface in the $(\sigma,\kappa)$ plane with $V_0(\sigma,\kappa)$ contours. The dark band indicates the low-RMSE valley, including the best point of Eq. \ref{['eq:best_point']}.
• Figure 2: Spin-averaged $B_c$ masses at the best valley parameters as functions of (a) the number of radial grid points $N_r$ at fixed $r_{\max}=15~\mathrm{GeV}^{-1}$ and (b) the radial cutoff $r_{\max}$ at fixed $N_r=10^3$.
• Figure 3: Spin-averaged $B_c$ masses versus time step $\Delta\tau$ at fixed $N_{steps}=10^4$.
• Figure 4: Spin-averaged $B_c$ masses versus number of GFMC steps $N_{steps}$ at fixed $\Delta\tau = 10^{-3}$. All states approach a common plateau for $N_{steps}\gtrsim 10^{4}$.
• Figure 5: Spin-averaged $B_c$ masses for each state in the tower. For every level, the box and whisker band summarises the spread of theoretical predictions and the experimental centroid taken from the literature.