Bayesian statistics maps the Bc meson spectrum and exposes confinement’s limits

The Bc meson, a bound state pairing a charm quark with a bottom antiquark, is the only known quarkonium built from two heavy quarks of different flavors, and it has now become the subject of the first fully Bayesian, uncertainty-propagated spectroscopic analysis. In a study published in The European Physical Journal C, A. Christas Mony and Rohit Dhir applied Markov chain Monte Carlo sampling to three confining potential models — the standard Cornell potential, a logarithmically modified extension, and a screened Cornell form — and propagated the correlated parameter uncertainties through the entire predicted spectrum up to the 6D multiplet. Their central result is a clear split in reliability: the low-lying spectrum is remarkably stable across all three confinement models, while the predictions for excited states become increasingly separated as the states probe larger distances, accompanied by uncertainties that grow steadily with excitation.

The Bc system occupies a unique position in heavy-quark physics. Because the quark and antiquark carry different flavors, the meson cannot annihilate directly into gluons, so every excited state below the lowest strong-decay threshold decays exclusively through radiative or weak transitions. This produces a sub-threshold spectrum that is both wider and populated by narrower states than in charmonium or bottomonium. The absence of charge conjugation symmetry has a further consequence: spin-singlet and spin-triplet states with the same orbital angular momentum can mix through the antisymmetric spin–orbit interaction, an effect entirely absent in equal-flavor quarkonia and directly sensitive to the interplay between the one-gluon-exchange and confinement components of the interquark potential. The characteristic dynamical scales of the Bc also lie between those of charmonium and bottomonium, placing it in an intermediate kinematic regime that must be calibrated on its own terms rather than borrowed from either neighbor.

Experimentally, the Bc family remains sparsely mapped. The complete Particle Data Group listing contains only the ground state Bc(1S) at 6274.47 ± 0.32 MeV, first observed by the CDF collaboration and confirmed by LHCb, CMS, and ATLAS, and the first radial excitation Bc(2S) at 6871.2 ± 1.0 MeV. LHCb has observed a wide structure in the Bc+γ mass spectrum consistent with the lowest excited P-wave states, while ATLAS has reported the first observation of the vector Bc*+ state. The individual 1P fine-structure components remain unresolved, however, and the low production rate — which requires simultaneous charm-anticharm and bottom-antibottom pair creation — means that the bulk of the predicted spectrum, including all D-wave states and radial excitations beyond 2S, still awaits discovery.

The theoretical framework is a non-relativistic potential model in which the heavy quark–antiquark interaction combines a short-distance Coulombic term from one-gluon exchange with a long-distance confining term. The authors solved the radial Schrödinger equation numerically using the Runge–Kutta method and added spin-dependent corrections — spin–spin, spin–orbit, and tensor interactions — perturbatively at order 1/m_Q². The logarithmic deformation modifies the intermediate-distance behavior through an effective string tension that decreases smoothly with separation while preserving asymptotic linearity, whereas the screened potential progressively suppresses the confining force at large distances altogether. Parameters for each potential were sampled with the affine-invariant ensemble sampler in the emcee package, using uniform priors within physically motivated bounds and a likelihood built from the two PDG masses, their mass difference, and the lattice QCD hyperfine splitting of 55 ± 3 MeV.

The Bayesian treatment matters because the Bc sector offers only three effectively independent experimental constraints while the models carry four to six free parameters, making deterministic chi-squared fits formally under-determined and susceptible to multiple acceptable solutions whose excited-state predictions diverge. Roughly 5000 posterior samples per potential, checked against convergence diagnostics including integrated autocorrelation times, effective sample sizes, and Gelman–Rubin statistics, yielded median predictions with asymmetric 1σ credible intervals. The ground state emerged with sub-MeV precision in all three models — 6274.48, 6274.47, and 6274.47 MeV — and the 1S hyperfine splittings of 55.71, 55.65, and 55.27 MeV agree with the lattice determinations of 55 ± 3 and 54 ± 3 MeV, validating the spin–contact parameterization.

A tension does appear with the newest data. ATLAS recently measured the 1S hyperfine splitting as 64.5 ± 1.4 MeV statistically, with a systematic uncertainty of −1.4 to +1.0 MeV, notably higher than the posterior medians, which were based on the lattice input available when the analysis was performed. To test the sensitivity of the inference to that constraint, the authors broadened the lattice uncertainty from 3 to 10 MeV through posterior importance reweighting using 300,000 uniformly thinned samples. The posterior median of the smearing parameter ρ shifted by roughly +0.21, +0.24, and +0.18 GeV for the three potentials while remaining within the original 68 percent credible intervals, indicating that the width of the ρ posterior is not an artifact of the assumed lattice precision and that the hyperfine splitting constrains a correlated combination of parameters rather than fixing any single one.

The picture changes sharply at higher excitation. The mass gap between the Cornell and logarithmically modified potentials accumulates at roughly 14 MeV per unit of radial quantum number, reaching about 70 MeV by the 6S state, with the modified potential clustering excited levels more tightly because of its shallower effective confining slope. The screened potential diverges even further: its 6S prediction lies about 260 MeV below the Cornell result, with an uncertainty of +150 to −241 MeV. Uncertainties widen monotonically with excitation — the Cornell 6³S₁ mass carries an interval of +39.46 to −80.35 MeV — reflecting the absence of experimental constraints on higher levels. The D-wave sector shows the largest inter-potential separation, about 90 MeV at 6D, because higher-l states are more spatially extended and therefore more sensitive to long-range potential differences, although that separation remains comparable to the credible interval, limiting the discriminating power to at most the 1σ level.

The flavor-asymmetry-driven mixing produces distinctive predictions. P-wave singlet–triplet mixing angles start at 4.82 and 6.62 degrees for the Cornell and modified potentials, with uncertainties of roughly ±9 to 10 degrees at 1P, and rise to about 19 to 20 degrees by 6P with much smaller uncertainties. D-wave mixing angles are large and negative throughout, near −52.5 degrees at 1D, with uncertainties of only about 1.5 degrees. The analysis also finds a nonstandard D-wave mass ordering — 1D₂ below 1³D₃ below 1³D₁ below 1D₂′ — persisting through 4D for the first two potentials, driven by a sign reversal in the symmetric spin–orbit contribution as scalar confinement overtakes one-gluon exchange. The LHCb P-wave structure, described at current resolution by a two-peak fit at 6704.8 and 6752.4 MeV, cannot yet be assigned to individual 1P states, but the predicted 1P masses populate the observed region, providing a qualitative consistency check.

Wave-function and velocity diagnostics trace where the model itself reaches its limits. The squared radial wave function at the origin, which governs leptonic decays and production rates, falls by about 59 percent from 1S to 6S under the Cornell potential, while root-mean-square radii grow from 1.71 to 8.47 GeV⁻¹. The squared charm-quark velocity climbs steadily with excitation, exceeding the diagnostic benchmark of 0.5 from approximately the 3S, 3P, and 3D states onward — or from the 4S, 4P, and 4D states under the screened potential — while the bottom quark remains non-relativistic throughout, with its squared velocity staying below 0.09. This means the higher-excitation predictions should be read as indicative rather than precision benchmarks, since higher-order relativistic corrections grow relatively more important even as the leading spin-dependent splittings shrink in absolute size.

Regge trajectories, which correlate squared masses with radial and orbital quantum numbers, add a complementary view. The radial trajectories of S-wave states show pronounced non-linearity under all three potentials, diminishing for P- and D-wave families as the centrifugal barrier pushes the wave functions outward toward the linearly confining region. The independently fitted curvature coefficients are not fixed to a universal value but shift systematically with the potential form, and the orbital slopes fall with increasing radial excitation. Model comparison using the widely applicable information criterion and Pareto-smoothed leave-one-out cross-validation found no statistically significant preference among the three confinement forms, so the authors retained the uniform-prior results as the baseline. They also caution that the credible intervals for states above the lowest open-flavor threshold are conditional on single-channel potentials and exclude coupled-channel effects, which existing calculations suggest can shift masses by tens to hundreds of MeV in a strongly state- and model-dependent way. Resolving the predicted fine structure and mixing pattern will require higher-resolution measurements at LHCb and other facilities, with access to the higher excitations depending on increased luminosity and dedicated reconstruction strategies.

Subject of Research: Bayesian MCMC analysis of the Bc meson spectrum using three confining potential models

Article Title: \({B_c}\) meson spectroscopy from Bayesian MCMC: probing confinement and state mixing

Article References: Mony, A. C., & Dhir, R. (2026). $${B_c}$$ meson spectroscopy from Bayesian MCMC: probing confinement and state mixing. The European Physical Journal C, 86(10), Article 1221. https://doi.org/10.1140/epjc/s10052-026-16315-6

Image Credits: AI Generated

DOI: 10.1140/epjc/s10052-026-16315-6

Keywords: Bc meson, spectroscopy, Bayesian inference, Markov chain Monte Carlo, Cornell potential, confinement, quarkonium, Regge trajectories, spin-orbit mixing, lattice QCD, LHCb, potential models