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arxiv: 2511.10986 · v2 · submitted 2025-11-14 · ✦ hep-ph

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

Tarik Akan This is my paper

Pith reviewed 2026-05-17 22:54 UTC · model grok-4.3

classification ✦ hep-ph
keywords B_c mesonCornell potentialVariational Monte CarloGreen's function Monte Carlospin-averaged spectrumheavy quarkoniumMonte Carlo methods
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The pith

A Cornell potential calibrated to the B_c ground state and evolved with VMC plus GFMC reproduces spin-averaged masses within tens of MeV of experiment.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper computes the spin-averaged B_c meson masses in a nonrelativistic Cornell potential by treating the system as spin-independent. Parameters are fixed by anchoring the 1S centroid to experiment and scanning a grid in the string tension and Coulomb coefficient, with the constant term set by the measured ground-state mass. Variational Monte Carlo supplies optimized trial wave functions carrying the correct nodal structure for each radial and angular channel. Green's function Monte Carlo then projects the ground-state energies while systematic errors are controlled by locating plateau regions in time step, projection length, walker number, and grid spacing. At the representative low-error point the calculated centroids lie within a few tens of MeV of the measured values and the fitted parameters remain consistent with those used for ordinary heavy quarkonia.

Core claim

The spin-averaged B_c spectrum is obtained 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 V0 fixed at each point by the experimental ground state mass. The spectrum is obtained with a two-stage Monte Carlo approach. Variational Monte Carlo provides optimized radial trial states with the desired nodal pattern. Fixed-node Green's function Monte Carlo then projects the corresponding ground-state energies for each (n, ℓ) channel. Controlled scans identify plateau regions.

What carries the argument

Two-stage Monte Carlo procedure in which Variational Monte Carlo supplies optimized radial trial states carrying the required nodal pattern and fixed-node Green's function Monte Carlo projects the ground-state energies for each (n, ℓ) channel.

If this is right

  • The same calibration and projection procedure yields energies for all low-lying (n, ℓ) channels once the potential parameters are chosen.
  • Quantitative control of discretization and projection errors is achieved by identifying plateau regions in time-step, projection-time, walker-population, and radial-grid scans.
  • The fitted values of σ and κ lie inside the range obtained from canonical heavy-quarkonium analyses.
  • The method supplies a controlled baseline against which spin-dependent corrections can later be added.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same anchoring-plus-scan procedure could be repeated for any other heavy or exotic meson for which a few experimental centroids are known.
  • Because the low-RMSE valley is two-dimensional rather than a single point, the calculation naturally supplies a range of acceptable potentials rather than a unique set of parameters.
  • Adding relativistic or finite-mass corrections on top of this nonrelativistic baseline would test how far the simple Cornell form remains reliable for the B_c system.

Load-bearing premise

The Cornell parameters can be 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 V0 fixed at each point by the experimental ground state mass.

What would settle it

A new experimental measurement of any higher spin-averaged B_c centroid (2S or 3S) lying more than roughly 50 MeV outside the low-RMSE valley found in the (σ, κ) scan.

Figures

Figures reproduced from arXiv: 2511.10986 by Tarik Akan.

Figure 1
Figure 1. Figure 1: RMSE surface in the (σ, κ) plane with V0(σ, κ) contours. The dark band indicates the low-RMSE valley, including the best point of Eq. (23). Because the calibration relies directly on GFMC energies, it is essential to understand how the Monte Carlo control parameters and the radial discretisation affect the results. The most important numerical inputs are the imaginary time step ∆τ , the number of time step… view at source ↗
Figure 2
Figure 2. Figure 2: Spin-averaged Bc masses at the best valley parameters as functions of (a) the number of radial grid points Nr at fixed rmax = 15 GeV−1 and (b) the radial cutoff rmax at fixed Nr = 103 . The mixed estimator of the energy is then measured at several time steps ∆τi while Nsteps and the other control parameters are held fixed. The total projection time is τtot = Nsteps ∆τ. (24) If ∆τ is chosen too small at fix… view at source ↗
Figure 3
Figure 3. Figure 3: Spin-averaged Bc masses versus time step ∆τ at fixed Nsteps = 104 . A complementary test of projection quality is obtained by varying Nsteps at fixed ∆τ = 10−3 , chosen from the central plateau in [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Spin-averaged Bc masses versus number of GFMC steps Nsteps at fixed ∆τ = 10−3 . All states approach a common plateau for Nsteps ≳ 104 . Within the low-RMSE valley we further discriminate between nearby points by looking at the residuals for each state in the Bc tower. Points where the deviations from experiment drift systematically with excitation (for example, progressively overshooting the higher radial … view at source ↗
Figure 5
Figure 5. Figure 5: Spin-averaged Bc 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 [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Same as Fig. 5, shown as splittings relative to the 1 [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗
read the original 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 $(\sigma,\kappa)$, 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript computes the spin-averaged B_c meson spectrum in a spin-independent Cornell potential using a two-stage Monte Carlo procedure. Variational Monte Carlo (VMC) generates optimized radial trial wave functions with specified nodal structure, after which fixed-node Green's function Monte Carlo (GFMC) projects the ground-state energies in each (n, ℓ) channel. Potential parameters are calibrated by anchoring the additive constant V0 to the experimental 1S centroid and scanning a grid in string tension σ and Coulomb strength κ to minimize root-mean-square error against the full set of measured spin-averaged centroids; at a representative low-RMSE point the computed masses lie within tens of MeV of experiment and the fitted parameters are stated to be consistent with canonical heavy-quarkonium analyses.

Significance. The explicit scans over GFMC time step, projection time, walker population, and radial discretization provide quantitative control of numerical systematics, which is a clear methodological strength. The work supplies a concrete demonstration that VMC/GFMC can be applied to the B_c system once the potential is calibrated. However, because the calibration is performed directly against the same experimental tower later used for comparison, the reported agreement is the minimized residual of the fit rather than an independent prediction; this limits the strength of the validation claim.

major comments (2)
  1. [Abstract] Abstract and parameter-calibration description: V0 is fixed to the experimental 1S mass and (σ, κ) are chosen by minimizing RMSE to the entire experimental spin-averaged tower. Consequently the quoted agreement 'at the level of a few tens of MeV' is the post-fit residual, not an a-priori prediction. The central claim therefore requires explicit qualification that the comparison is internal to the calibration procedure.
  2. [Results / parameter determination] Consistency statement (abstract and results section): the assertion that the fitted Cornell parameters are 'consistent with canonical heavy quarkonium analyses' is not supported by a direct benchmark. No comparison is shown of the achieved RMSE (or of the resulting σ and κ values) against the residuals obtained when the identical Cornell form and fitting protocol are applied to the charmonium or bottomonium towers.
minor comments (1)
  1. [Method] Notation: the precise functional form of the trial radial functions (including how nodes are imposed for excited states) should be stated explicitly, together with the definition of the radial grid used in the VMC optimization.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive comments. We appreciate the positive assessment of the numerical controls and the applicability of the VMC-GFMC method to the B_c system. We address each major comment below, indicating the revisions that will be incorporated.

read point-by-point responses

  1. Referee: [Abstract] Abstract and parameter-calibration description: V0 is fixed to the experimental 1S mass and (σ, κ) are chosen by minimizing RMSE to the entire experimental spin-averaged tower. Consequently the quoted agreement 'at the level of a few tens of MeV' is the post-fit residual, not an a-priori prediction. The central claim therefore requires explicit qualification that the comparison is internal to the calibration procedure.

    Authors: We agree that the procedure constitutes a calibration of the Cornell parameters directly to the experimental B_c spin-averaged centroids, with V0 anchored to the 1S state and (σ, κ) chosen to minimize the RMSE over the full tower. The reported agreement is therefore the post-fit residual rather than an independent prediction. In the revised manuscript we will explicitly qualify this in the abstract and in the parameter-determination section, stating that the masses are reproduced to within tens of MeV inside the calibration procedure. revision: yes

  2. Referee: [Results / parameter determination] Consistency statement (abstract and results section): the assertion that the fitted Cornell parameters are 'consistent with canonical heavy quarkonium analyses' is not supported by a direct benchmark. No comparison is shown of the achieved RMSE (or of the resulting σ and κ values) against the residuals obtained when the identical Cornell form and fitting protocol are applied to the charmonium or bottomonium towers.

    Authors: We acknowledge that a direct side-by-side benchmark applying the identical VMC-GFMC fitting protocol to the charmonium and bottomonium towers is not presented. The consistency claim rests on the fitted σ and κ values lying within the ranges commonly used in the literature for successful Cornell-potential descriptions of heavy quarkonia. A complete parallel re-analysis of those systems lies outside the scope of the present focused study. To address the concern we will add a clarifying sentence in the results section that references the typical literature ranges and qualifies the basis of the consistency statement. revision: partial

Circularity Check

1 steps flagged

Fitted Cornell parameters to B_c centroids make reported mass agreement the minimized fit residual

specific steps
  1. fitted input called prediction [Abstract]
    "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 V0 fixed at each point by the experimental ground state mass. ... 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"

    Parameters (σ, κ, V0) are chosen specifically to minimize RMSE to the experimental tower; the 'predicted' masses at the selected best point are therefore the fitted values whose deviation from experiment has already been minimized by construction, rendering the quoted agreement a report of the fit quality rather than an out-of-sample validation.

full rationale

The paper explicitly calibrates the potential parameters by anchoring to the experimental 1S mass and scanning (σ, κ) to minimize RMSE against the full set of measured spin-averaged centroids; the subsequent claim of agreement 'at the level of a few tens of MeV' at the best-fit point is therefore the residual of that same fit rather than an independent prediction from first principles. The VMC/GFMC numerics themselves are not at issue, but the validation loop is internal to the calibration procedure described in the abstract.

Axiom & Free-Parameter Ledger

3 free parameters · 2 axioms · 0 invented entities

The central claim rests on three fitted parameters (σ, κ, V0) whose values are chosen to reproduce the experimental 1S mass and to minimize mismatch with the rest of the tower; the non-relativistic spin-independent treatment is taken as given without independent justification in the abstract.

free parameters (3)
  • string tension σ
    Scanned on a grid and selected by RMSE minimization against the experimental spin-averaged tower after anchoring the 1S state.
  • Coulomb strength κ
    Scanned on a grid and selected by RMSE minimization against the experimental spin-averaged tower after anchoring the 1S state.
  • additive constant V0
    Fixed at each (σ, κ) point so that the calculated 1S mass exactly equals the experimental ground-state mass.
axioms (2)
  • domain assumption The B_c meson can be treated as a non-relativistic two-body system in a spin-independent potential.
    Stated explicitly in the abstract as the modeling framework.
  • domain assumption The Cornell form (Coulomb plus linear) is an adequate effective potential for the spin-averaged spectrum.
    Used without derivation; parameters are calibrated rather than derived from QCD.

pith-pipeline@v0.9.0 · 5495 in / 1714 out tokens · 39480 ms · 2026-05-17T22:54:49.008823+00:00 · methodology

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

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    hep-ph 2026-04 unverdicted novelty 6.0

    Bayesian MCMC sampling of Cornell and log-modified Cornell potentials reproduces known B_c states and supplies mass predictions for higher excitations with propagated uncertainties.

  2. Relativistic effects in heavy mesons

    hep-ph 2026-01 unverdicted novelty 4.0

    A relativistic potential model with few parameters qualitatively reproduces heavy meson masses and radiative widths, remaining finite at zero light quark mass.

Reference graph

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