The baryonium interpretation of the X(1835): establishing a near-threshold nucleon-antinucleon ${}^{11}S_0$ bound state and its decay pattern

The X(1835), a broad pseudoscalar resonance discovered by BESII in $J/\psi\to\gamma\pi^+\pi^-\eta'$ just below the proton-antiproton threshold, is the longest-standing candidate for a nucleon-antinucleon bound state (baryonium). Samart, Yan, Gutsche and Faessler (Phys. Rev. D 85, 114033 (2012)) computed its relative partial decay widths in the ${}^3P_0$ quark model assuming a ${}^{11}S_0$ $N\bar N$ bound state, finding dominant $\rho\rho$, $\omega\omega$ and $\pi a_0(1450)$ modes, but left open whether microscopic $N\bar N$ dynamics produces the required near-threshold bound state at all, what its wave function - and hence the absolute widths - is, and whether the discriminating channels $\pi a_0(1450)$ (sizeable in the baryonium picture), $K K^*$ and $K^*K^*$ (large only in the competing $\eta'$ radial-excitation picture) show observable signals. BESIII has since fixed the spin-parity to $J^{PC}=0^{-+}$ and uncovered an anomalous two-pole line shape near the $p\bar p$ threshold (X(1840) and the new X(1880)); modern $N\bar N$ dynamics (chiral effective theory, constituent quark models, QCD sum rules) finds near-threshold structures but no consensus microscopic bound-state wave function; and no dedicated experimental search for the discriminating decay channels exists.

Background

The X(1835) is a broad resonance with mass near 1.83-1.84 GeV and width of order 100-200 MeV (channel- and fit-dependent), discovered by the BESII experiment in 2005 in the invariant mass of the $\pi^+\pi^-\eta'$ system produced in the radiative decay $J/\psi\to\gamma\pi^+\pi^-\eta'$ and confirmed by BESIII in 2011 with more than 20 sigma significance. It lies just below the proton-antiproton threshold ($2m_p$ is about 1876 MeV). Its observation revived interest in baryonium, a bound state of a nucleon and an antinucleon, an idea going back to Fermi and Yang. The direct experimental trigger was the strong near-threshold enhancement in the $p\bar p$ invariant-mass spectrum of $J/\psi\to\gamma p\bar p$ observed by BES in 2003: reanalyses including final-state interactions gave a mass consistent with the X(1835), suggesting that the two observations involve the same state. Because the neutron-antiproton threshold lies within a few MeV of the proton one, an isospin-singlet $N\bar N$ bound state couples to both.

In the spectroscopic notation ${}^{2I+1\,2S+1}L_J$ for baryon-antibaryon states, the configuration compatible with a pseudoscalar meson is ${}^{11}S_0$: isospin singlet, spin singlet, S-wave, J = 0, with overall quantum numbers $I^G(J^{PC}) = 0^+(0^{-+})$. At the time of the source paper this assignment was favored but not experimentally established. A competing interpretation treats the X(1835) as the second radial excitation of the $\eta'$ meson, i.e. an ordinary quark-antiquark meson.

The source paper (Samart, Yan, Gutsche and Faessler, 2012) assumed the X(1835) to be a ${}^{11}S_0$ $N\bar N$ bound state and computed its partial decay widths to two mesons in the ${}^3P_0$ (quark-pair-creation) model, in which a hadron decays through creation of a quark-antiquark pair with vacuum quantum numbers ${}^3P_0$ and effective coupling $\lambda$; the amplitude is an overlap of final-state meson wave functions with the bound-state wave function $\Phi_X(\vec p\,)$, and the widths depend on baryon and meson size parameters $a$ and $b$. Because no reliable $N\bar N$ potential existed, only relative widths were computed - in the low-momentum approximation these are insensitive to $\Phi_X$, $\lambda$, $a$ and $b$. The dominant modes were found to be $\rho\rho$, $\omega\omega$ and $\pi a_0(1450)$, where $a_0(1450)$ is the isovector scalar quark-antiquark meson, and the $\pi a_0(1450)$ channel was proposed as the most promising search channel. The calculation left several issues open. First, microscopically derived $N\bar N$ potentials had not produced the required near-threshold ${}^{11}S_0$ bound state, although the semi-phenomenological Paris-type analysis of Dedonder, Loiseau, El-Bennich and Wycech described both $J/\psi\to\gamma p\bar p$ and $\eta'\pi\pi$ spectra with a broad ${}^{11}S_0$ pole. Second, without a microscopic wave function $\Phi_X$ and a calibrated coupling $\lambda$, absolute widths and branching fractions cannot be predicted, and corrections beyond the low-momentum approximation remained uncomputed; the authors also noted that the widths to $\eta f_0(600)$, $\eta' f_0(600)$ and $\pi a_0(980)$ are difficult to calculate in any quark model because the nature of the $f_0(600)$ and $a_0(980)$ mesons is not clear and they may have large non-$q\bar q$ components. Third, the channels $K K^*$ and $K^*K^*$ discriminate between interpretations: $N\bar N$ bound states annihilate weakly into strange final states, so these modes are suppressed for baryonium, whereas the $\eta'$ radial-excitation interpretation predicts large $K K^*$ and $K^*K^*$ partial widths; observing or excluding them would therefore separate the two pictures.

Later experiments sharpened the threshold physics without settling it. BESIII determined the spin-parity of the X(1835) to be $J^{PC}=0^{-+}$ (2015) and found an anomalous line shape of the $\eta'\pi^+\pi^-$ spectrum at the $p\bar p$ threshold (2016) described about equally well by a Flatte-type threshold effect (a Flatte line shape describes a resonance coupled to a nearby open threshold) and by two overlapping resonances, either fit supporting a $p\bar p$ moleculelike or bound state with significance above 7 sigma. A companion structure X(1840) in $3(\pi^+\pi^-)$ (2013) was resolved in 2024 into X(1840) plus a new resonance X(1880) just above the $p\bar p$ threshold. Antiprotonic-atom level shifts and widths, $N\bar N$ scattering data, and these line shapes are the empirical constraints any microscopic $N\bar N$ dynamics must satisfy.

Problem Statement

Determine whether the X(1835) - together with the near-threshold structures observed alongside it - is a nucleon-antinucleon bound state (baryonium). The three components below are jointly one test of this single hypothesis, following the program left open by Samart et al. (2012); an answer may use any sound route, provided it settles the identification itself rather than merely completing one component in isolation.

(1) Establish, from microscopic $N\bar N$ dynamics - for example constituent-quark-model or chiral effective-field-theory interactions, treated in a coupled-channel framework constrained by $N\bar N$ scattering data, antiprotonic-atom level shifts and widths, and the measured $J/\psi$ radiative-decay invariant-mass spectra - whether a near-threshold ${}^{11}S_0$ (isospin-singlet, spin-singlet S-wave) bound-state pole with $I^G(J^{PC}) = 0^+(0^{-+})$ exists; determine whether it is to be identified with the broad X(1835) itself or with a companion pole near the $p\bar p$ threshold (as suggested by the two-pole line-shape interpretations of the X(1840)/X(1880) and $\eta'\pi\pi$ data); and determine its momentum-space wave function $\Phi_X(\vec p\,)$.

(2) With that wave function, compute the absolute partial decay widths of the bound state in a ${}^3P_0$-type quark-pair-creation framework: the two-meson channels $\rho\rho$, $\omega\omega$, $\pi a_0(1450)$, $\eta f_0(1370)$ and $\eta' f_0(1370)$, with the momentum dependence beyond the low-momentum approximation treated explicitly and the sensitivities to the pair-creation coupling $\lambda$ and the hadron size parameters $a$ and $b$ quantified.

(3) Test the interpretation experimentally through its discriminating predictions: the product branching fractions $B(J/\psi\to\gamma X)\,B(X\to\pi a_0(1450))$, predicted to be sizeable in the baryonium picture, and $B(J/\psi\to\gamma X)\,B(X\to K K^*)$ and $B(J/\psi\to\gamma X)\,B(X\to K^*K^*)$, predicted small for baryonium but large if X(1835) is the second radial excitation of $\eta'$ - so that the $N\bar N$ bound-state assignment can be separated from the $\eta'$ radial-excitation alternative.

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