A proposed molecular signature in three exotic Pc states becomes much less certain once the analysis allows their coupling phases to vary. In a new preprint, fits with real couplings produced negative real effective ranges for Pc (4312)+, Pc (4440)+ and Pc (4457)+, a pattern the authors regard as consistent with a molecular interpretation. But after coupling phases were marginalised, the central values clustered near zero and the uncertainty widened enough to remove certainty about the sign.
The finding makes the interpretation of the states depend heavily on how the fit handles information the spectrum does not pin down. The study is an indirect, model-based inference from a binned invariant-mass spectrum. It does not establish that the states are molecular or rule out compact-pentaquark and kinematic alternatives.
A single fit to the full spectrum
The authors analysed the publicly available LHCb cos θPc-weighted J/ψ p invariant-mass histogram from HEPData record 89271. It contains 200 bins, each 2 MeV wide, spanning 4200 to 4600 MeV. These are spectrum bins rather than event-level data.
They used a global two-channel Flatté model, meaning one line-shape description for all three candidates, with fully complex couplings and derived complex effective-range-expansion parameters. The complete spectrum was represented as an incoherent sum of the three signal amplitudes, plus a sixth-order polynomial background and LHCb mass-resolution smearing.
The objective combined the data χ² with an explicit prior and was optimized locally in a 24-parameter space. The researchers used eight independent Nelder-Mead restarts for each fit variant and treated the spread between restarts as a convergence check. Uncertainty was propagated with 500 non-parametric bootstrap samples.
What the restricted fits suggested
The analysis converted the fitted parameters into two scattering quantities, a scattering length and an effective range, using an effective-range expansion, or ERE. Under the real-coupling assumption, the negative sign of the real effective range was the pattern treated as compatible with a molecular interpretation.
For Pc (4312)+, the real part of the scattering length was 0.96 ± 0.09 femtometres in Fit I and 0.64 ± 0.06 femtometres in Fit II. Its real effective range was −1.86 ± 0.09 and −1.87 ± 0.08 femtometres, respectively.
Pc (4440)+ showed a real scattering length of 0.42 ± 0.05 femtometres in Fit I and 0.32 ± 0.05 femtometres in Fit II. The corresponding real effective ranges were −1.20 ± 0.06 and −1.19 ± 0.06 femtometres.
For Pc (4457)+, the real scattering length was 2.05 ± 0.17 femtometres in Fit I and 0.87 ± 0.12 femtometres in Fit II. The real effective range was −1.19 ± 0.06 and −1.19 ± 0.07 femtometres. The negative effective-range result was therefore stable across the two real-coupling variants, even though the scattering-length estimates shifted.
The assumption that changed the story
The headline results fixed both coupling phases at zero for the real-coupling treatment. A separate phase-marginalised ensemble instead drew each of the two phases uniformly from 0 to 2π for every bootstrap sample.
That broader treatment substantially reduced the apparent constraint. Across the states and threshold fits, the real-coupling estimates of the real effective range ran from −1.87 to −1.19 femtometres. The phase-marginalised central values ran from −0.05 to 0.12 femtometres, with reported uncertainties from 0.83 to 1.34 femtometres. The authors say this removes sign certainty for the molecular signature.
The result highlights a limitation of the inclusive J/ψ p spectrum: the coupling phases are not constrained by it in this analysis. The paper therefore treats the molecular interpretation as dependent on the phase assumption rather than as a conclusion that survives every allowed fit.
The two closest states may not be cleanly separable
The fit found substantial line-shape overlap between Pc (4440)+ and Pc (4457)+. Their overlap integral was 0.57, while the overlap involving Pc (4312)+ and either of the other states was below 0.1. The close pair is separated by approximately 17 MeV.
That makes the use of an incoherent sum questionable for the Pc (4440)+ and Pc (4457)+ pair. The analysis quantifies the overlap and points to the possible importance of omitted coherent cross-terms, but it does not directly fit or establish physical interference.
Thresholds and optimisation still matter
The choice of the second channel's threshold was identified as the dominant systematic affecting the effective-range extraction. Between the two fits, it shifted the study's EBW parameter by 4 to 6 MeV per state and changed the absolute real scattering-length estimates by about 20% to 30%.
Other diagnostics also point to a fit that is sensitive to its setup. Restart-to-restart variation exposed local minima in the high-dimensional optimization, so a unique global minimum was not guaranteed. The EBW and |g2| parameters remained degenerate and were regularized with a soft prior rather than fully resolved.
The fitted visible-yield allocation varied as well. Pc (4457)+ was the largest visible-yield component in both variants, while the second fit assigned no joint non-negative least-squares amplitude to Pc (4440)+. That allocation depends on the global fit configuration.
A result bounded by its model
The authors describe the work as an indirect inference from a binned invariant-mass spectrum. The conclusions are tied to the selected Flatté parametrization, threshold configurations, background, resolution, prior and phase treatment. The sixth-order polynomial background is approximate, alternative background forms were not systematically scanned, and the non-relativistic T-matrix approximation was not systematically varied.
The bootstrap samples around the best-fit solution rather than fully refitting every resampled dataset. The analysis also does not resolve the coupling phases or the EBW-|g2| degeneracy, and it does not provide a complete systematic-uncertainty budget. The authors point to further constraints, broader background scans and more exhaustive optimization as ways to test how much the remaining uncertainty could change the inference.
The work is a preprint on arXiv dated August 25, 2026, with no journal or DOI listed in the supplied metadata. The authors report support from the Post-Doctoral Program of the Swiss Government Excellence Scholarship, grant 2025.0419, and computational resources from the University of Geneva.
Paper data and sources
Original title: Statistical Inference of Scattering Parameters for Exotic Hadronic States in the $J/ψ\,p$ Spectrum
Authors: André Aimé Atangana Likéné, P. Pradyun Hebbar, Alexis Franck Rothen et al.
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-25
DOI: Not available
Original paper · Full text