In radiative decays of the J/ψ meson, the BES and BESIII experiments have observed a family of pseudoscalar (JPC=0−+J^{PC}=0^{-+}) structures clustered at the proton–antiproton mass threshold (2mp≈1876.52m_p\approx 1876.5 MeV): the sub-threshold p p̄ enhancement in J/ψ→γppˉJ/\psi\to\gamma p\bar p, the X(1835) peak in J/ψ→γη′π+π−J/\psi\to\gamma\eta'\pi^+\pi^-, an anomalous distortion of that line shape at the p p̄ threshold, and — with the largest data sample — a new narrow resonance X(1880). Whether these signals manifest a genuine nucleon–antinucleon bound or quasi-bound state (baryonium), or are ordinary mesonic resonances distorted by strong N N̄ final-state interactions without any N N̄ pole, has remained unsettled for two decades despite convergent hints from potential models, Bethe–Salpeter analyses, and coupled-channel fits. The surviving open core is: determining the pole content and compositeness of the 0−+ amplitude near the p p̄ threshold, and deciding at quantified significance whether an N N̄ bound/quasi-bound state is required.

Background

The suggestion that some mesons are not elementary but bound states of a nucleon and an antinucleon goes back to Fermi and Yang in 1949; such states are called baryonium. They are expected to be difficult to observe: the nucleon–antinucleon (N N̄) interaction contains a strong absorptive part from annihilation into mesons, usually modeled as a complex (optical) potential, which makes any N N̄ bound state broad and its signature in a mass spectrum easily confused with a smooth threshold distortion produced by final-state interactions (FSI) — the rescattering of the produced particles after the primary reaction. A sharp test bed is provided by radiative decays of the J/ψ meson, the 1S1S charmonium bound state produced in huge samples at e+e− colliders; in decays J/ψ→γ+hadronsJ/\psi\to\gamma+\mathrm{hadrons} the photon is radiated from the initial charm quarks, so the hadronic system is produced gluon-rich and near threshold, and any sub-threshold enhancement stands out.

The BES collaboration observed an enhancement in the p p̄ invariant-mass spectrum at the p p̄ threshold (2m_p ≈ 1876.5 MeV) in J/ψ→γppˉJ/\psi\to\gamma p\bar p (Bai et al., 2003), and a pseudoscalar resonance X(1835) in J/ψ→γπ+π−η′J/\psi\to\gamma\pi^+\pi^-\eta' (Ablikim et al., 2005), where η′ is the heavier pseudoscalar mixing partner of the η meson; BESIII confirmed the X(1835) and found additional resonances X(2120) and X(2370) (2011). Partial wave analysis — a decomposition of the decay amplitude into angular-momentum eigenwaves whose quantum numbers JPCJ^{PC} (spin, parity, charge conjugation) label the underlying resonance — established JPC=0−+J^{PC}=0^{-+} for the X(1835) (BESIII 2015) and for the p p̄ threshold structure itself, with a pole mass slightly below threshold (BESIII 2012).

Competing mechanisms were proposed early. In the Watson–Migdal FSI framework with the Jülich N N̄ model, the threshold enhancement can be produced by S-wave rescattering in the isovector channel without a new particle (Sibirtsev et al., 2005). Conversely, the Paris N N̄ potential — a semi-phenomenological optical potential fitted to N N̄ scattering and antiprotonic-atom data — supports a broad quasi-bound state in the spin-singlet, isospin-singlet S wave (the 11S0{}^{11}S_0 channel); Dedonder, Loiseau, El-Bennich and Wycech (2009) showed that this quasi-bound state, interfering with a background amplitude, can describe both the J/ψ→γppˉJ/\psi\to\gamma p\bar p enhancement and the X(1835) peak. Bound-state calculations in the Bethe–Salpeter equation — the standard relativistic two-body bound-state integral equation — with phenomenological kernels were used by Wang (2010, 2011) to identify the X(1835) with a p p̄ bound state of mass 1833.7 MeV; that number is calibrated to the observed mass rather than predicted from it. Besides baryonium, the X(1835) has been interpreted as the second radial excitation of the η′ (Huang and Zhu, 2006), as a pseudoscalar glueball (a meson with no valence quarks, dominated by gluons), or as a mixture of these components; the 3P0 quark-pair-creation model applied to an assumed N N̄ bound state predicts dominant decays into ρρ, ωω and π a₀(1450) (Samart et al., 2012).

Two later measurements sharpened the question into a genuine pole-structure problem. Analyzing J/ψ→γη′π+π−J/\psi\to\gamma\eta'\pi^+\pi^-, BESIII (2016) observed an abrupt change of slope of the η′π+π− line shape exactly at the p p̄ threshold; two equally good descriptions — a Flatté line shape (a coupled-channel resonance form that incorporates the opening of a decay threshold) with strong coupling to p p̄, or the coherent sum of two resonances — each imply a moleculelike or bound p p̄ state with significance above 7σ, but the data could not distinguish them. With a sample of about 10 billion J/ψ, BESIII (2024) found that the analogous distortion in J/ψ→γ3(π+π−)J/\psi\to\gamma 3(\pi^+\pi^-) requires two overlapping resonances, X(1840) and a new narrow X(1880) with mass 1882.1±1.7±0.7 MeV and width 30.7±5.5±2.4 MeV (above 10σ), which the collaboration states indicates the existence of a p p̄ bound state.

Interpreting such structures requires the analytic continuation of the scattering amplitude to complex energies: a genuine bound state is a pole below threshold on the physical Riemann sheet of the amplitude; a virtual state is a pole on an unphysical sheet below threshold, which produces a threshold enhancement without a bound particle; a cusp is a threshold singularity from the opening of a channel; and the compositeness measures what fraction of the state's wavefunction is a given two-body cluster. Coupled-channel analyses respecting unitarity (conservation of probability across open channels) can in principle discriminate these alternatives, and the same N N̄ final-state interaction enters near-threshold e+e−→NNˉe^+e^-\to N\bar N cross sections, providing independent constraints. Recent work in exactly this spirit (Niu et al. 2024; Jia et al. 2025; Yang, Dai and Meißner; Ji and Meißner) still reaches mixed conclusions, which is why the question remains open.

Problem Statement

Determine the mechanism that produces the pseudoscalar (JPC=0−+J^{PC}=0^{-+}) structures observed at and near the proton–antiproton mass threshold in J/ψ radiative decays — the sub-threshold p p̄ enhancement in J/ψ→γppˉJ/\psi\to\gamma p\bar p, the X(1835)/X(1840) peak and its threshold distortion in J/ψ→γη′π+π−J/\psi\to\gamma\eta'\pi^+\pi^- and J/ψ→γ3(π+π−)J/\psi\to\gamma 3(\pi^+\pi^-), and the narrow X(1880), whose quantum numbers are not yet measured — and in particular decide whether a genuine nucleon–antinucleon bound or quasi-bound state (baryonium) exists in the 1.8–1.9 GeV region and contributes to these structures, or whether they are ordinary mesonic resonances (for example radial excitations of the η′ or glueball-mixed states) distorted by N N̄ final-state interactions without any N N̄ pole. All requested results jointly serve this single objective of identifying the nature of the near-threshold 0−+ amplitude. The objective is to determine: (i) the pole content of the 0−+0^{-+} amplitude in this mass region — how many poles, on which Riemann sheets, with which masses, widths, and couplings to the p p̄ (and n n̄) channels — as determined from the relevant invariant-mass spectra, consistently with two-body unitarity and with existing N N̄ scattering and antiprotonic-atom constraints; (ii) whether the data require an N N̄ composite component — a pole below threshold on the physical sheet, or a near-threshold pole with substantial N N̄ compositeness — or admit an equally good description with purely mesonic poles plus N N̄ rescattering (cusp or virtual-state type); (iii) a quantified statistical discrimination among the baryonium, η′-radial-excitation, glueball-mixture, and pure-final-state-interaction hypotheses across the available production and decay channels (J/ψ radiative decays, two-photon production, decay patterns such as π a₀(1450), ωω, K_S^0 K_S^0 η, and near-threshold e+e−→NNˉe^+e^-\to N\bar N cross sections).

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