Constraining the annihilation optical potential and pole content of quark-model coupled-channel $p\bar p$ scattering: do the predicted near-threshold poles correspond to the observed $X(1835)$-family states?

A constituent-quark-model (CQM) coupled-channels calculation of nucleon-antinucleon scattering predicts a dense spectrum of $T$-matrix poles (resonances and virtual states) near the $p\bar p$ and $n\bar n$ thresholds, but the absorptive part of the interaction — a purely phenomenological Gaussian optical potential with parameters frozen at 2006 values — controls where the poles sit and how wide they are, so no assignment to the experimentally observed $X(1835)$, $X(1840)$, $X(1870)$, $X(1880)$ structures can yet be made. The open problem is to determine whether the predicted poles correspond to these observed near-threshold enhancements: this requires deriving or empirically constraining the annihilation term from modern data, quantifying the systematic uncertainties of the pole positions, and, on that basis, establishing or ruling out specific pole-to-state assignments. Neither the source group nor any later work has performed any of these steps within the quark-model framework, and even the effective-field-theory analyses that constrain the $N\bar N$ absorption from data leave the near-threshold pole content ambiguous.

Background

Experiments at the BESIII detector (Beijing Spectrometer III, operating at the BEPC-II collider) have observed a family of enhancements just below the proton-antiproton ($p\bar p$) threshold (about 1876 MeV): the $X(1835)$ in $J/\psi\to\gamma\pi^+\pi^-\eta'$, whose line shape shows an abrupt change at the $p\bar p$ threshold, and most recently an anomalous shape of the $X(1840)$ in $J/\psi\to\gamma 3(\pi^+\pi^-)$ indicating a second resonance, dubbed $X(1880)$, near threshold. Whether these structures are genuine baryonium states — meson-like bound states of a baryon and an antibaryon, a concept dating back to the 1940s — or dynamical threshold effects generated by the strong nucleon-antinucleon ($N\bar N$) final-state interaction is a long-standing open question of light-hadron spectroscopy.

One theoretical route is the constituent quark model (CQM), in which hadrons are described as systems of massive constituent quarks interacting through one-gluon exchange and meson-exchange forces. The Salamanca group (Ortega, Entem, Fernandez, Segovia) constructed an $N\bar N$ potential by applying the $G$-parity transformation (the operation that converts a potential between a nucleon and another nucleon into the corresponding nucleon-antinucleon one, flipping the sign of the meson-exchange terms) to a quark-model $NN$ interaction that accurately reproduces the deuteron and $NN$ phase shifts; the two-baryon dynamics is solved with the resonating group method (RGM), a technique that projects the quark-level Hamiltonian onto baryon-baryon channels, and the coupled $p\bar p$-$n\bar n$ channels are treated exactly. Because $N\bar N$ annihilation into mesons cannot be computed from the quark model, its flux-removing effect is modeled by a complex optical potential: its real part comes from one-pion and one-gluon exchange annihilation diagrams, while its imaginary (absorptive) part is taken as an energy-independent Gaussian, $V^{\rm Ann}_{q\bar q}(\vec q)=i\kappa e^{-\varsigma^2 q^2}$, with strength $\kappa=-0.74$ GeV$^{-2}$ and range $\varsigma=0.49$ fm fixed in 2006 by a fit to low-energy $N\bar N$ cross sections and never updated. The scattering singularities are located as poles of the $T$-matrix (the scattering amplitude) on the unphysical Riemann sheets of the complex energy plane — virtual states correspond to poles at real energies below a threshold on the second sheet, while resonances appear as complex-energy poles on the unphysical sheets; these pole positions, not the Breit-Wigner parameters used in experimental fits, are the model-independent definition of a resonance, though the mapping between the two is itself nontrivial.

With this machinery the 2024/2025 calculation found that without annihilation the interaction binds $N\bar N$ in the $J^{PC}=0^{++}$ and $1^{--}$ sectors, but switching on the optical potential dissolves all bound states into a richer spectrum of poles on non-physical sheets between roughly 1861 and 1896 MeV, whose positions and widths move continuously as $\kappa$ is varied — demonstrating that the pole spectrum is controlled by the two unconstrained annihilation parameters. The authors therefore declined to assign their poles to the observed $X$ states. In parallel, effective-field-theory (EFT) analyses — chiral EFT potentials constrained by $N\bar N$ scattering and antiprotonic-hydrogen data (Kang, Haidenbauer, Meißner 2014), and more recent nonrelativistic EFT fits of the BESIII/SND $e^+e^-\to N\bar N$ cross sections (Jia et al. 2025) — constrain the absorptive interaction empirically from data, and the 2025-2026 EFT work increasingly interprets the near-threshold structures as $N\bar N$ final-state-interaction effects rather than new narrow resonances. But no analysis has connected the quark-model pole spectrum to the data at this level of rigor: the CQM annihilation term remains unconstrained by modern data, no uncertainties are propagated to the pole positions, and no production amplitudes or line shapes have been computed from the CQM.

Problem Statement

The problem, as left open by Ortega, Entem, Fernández and Segovia and still untouched in August 2026, has one overarching objective: determine whether the near-threshold $T$-matrix poles predicted by the quark-model coupled-channels $N\bar N$ calculation correspond to the experimentally observed structures clustered at the $p\bar p$ threshold — the $X(1835)$, $X(1840)$, $X(1870)$ and $X(1880)$ — or whether the pole spectrum is an artifact of an unconstrained annihilation interaction. Answering this question requires: (1) replacing the phenomenological imaginary annihilation potential $i\kappa e^{-\varsigma^2 q^2}$ (with $\kappa,\varsigma$ frozen at 2006 values) by either a microscopic quark-level derivation or an empirical constraint from modern data — e.g. fits to inelastic $N\bar N$ cross sections, antiprotonic-hydrogen level shifts and widths, or multi-meson final states — thereby fixing the quantitative map from the annihilation parameters to pole positions, widths, and Riemann-sheet assignments; (2) propagating the systematic uncertainties of the constituent-quark-model parameter determination into error bands on the pole positions and widths; and (3) on that basis, establishing or ruling out specific pole-to-state assignments, with the Breit-Wigner-versus-pole-position comparison handled consistently. These three items state what a resolution of the single assignment question has to settle — an unconstrained absorptive interaction, the absence of error estimates on the poles, and the delicacy of the Breit-Wigner-versus-pole comparison are exactly the obstacles the source paper cites for declaring its own assignments premature — and not a prescribed sequence of computational steps. A direct confrontation with measured line shapes, which would additionally require modeling production amplitudes and final-state interactions in the relevant radiative $J/\psi$ decays and in $e^+e^-\to N\bar N$, is one possible route toward the assignment conclusion rather than a required component of it. Answers formulated within a different but microscopic-coupled-channel framework are admissible provided they confront the same quark-model pole spectrum and the same set of observed near-threshold structures and settle the same objective — an annihilation interaction constrained from modern data or from microscopics, quantified uncertainties on the pole positions and widths, and a resulting assignment or ruling-out; item (2) then applies to the corresponding parameter determination within that framework.

References

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