Abstract
Observation of an anomalous line shape of the η ′ π + π − mass spectrum near the pp mass threshold in J/ψ → γη ′ π + π
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9 J/ψ events collected by the BESIII experiment in 2012, we study the J/ψ → γη ′ π + π − process and observe a significant abrupt change in the slope of the η ′ π + π − invariant mass distribution at the proton-antiproton (pp) mass threshold. We use two models to characterize the η ′ π + π − line shape around 1.85 GeV/c 2 : one which explicitly incorporates the opening of a decay threshold in the mass spectrum (Flatté formula) , and another which is the coherent sum of two resonant amplitudes. Both fits show almost equally good agreement with data, and suggest the existence of either a broad state around 1.85 GeV/c 2 with strong couplings to pp final states or a narrow state just below the pp mass threshold. Although we cannot distinguish between the fits, either one supports the existence of a pp molecule-like state or bound state with greater than 7σ significance. The state X(1835) was first observed by the BESII experiment as a peak in the η ′ π + π − invariant mass distribution in J/ψ → γη ′ π + π − decays [1] . This observation was later confirmed by BESIII studies of the same process [2] with mass and width measured to be M = 1836.5±3 S decays, where its spin-parity was determined to be J P = 0 − by a partial wave analysis (PWA) [3] . An anomalously strong enhancement at the proton-antiproton (pp) mass threshold, dubbed X(pp), was first observed by BESII in J/ψ → γpp decays [4] ; this observation was confirmed by BESIII [5] and CLEO [6] . This enhancement structure was subsequently determined to have spin-parity J P = 0 − by BESIII [7] . Among the various theoretical interpretations on the nature of the X(1835) and X(pp) [8] [9] [10] [11] [12] , a particularly intriguing one suggests that the two structures originate from a pp bound state [13] [14] [15] [16] [17] . If the X(1835) is really a pp bound state, it should have a strong coupling to 0 − pp systems, in which case the line shape of X(1835) at the pp mass threshold would be affected by the opening of the X(1835) → pp decay mode. A study of the η ′ π + π − line shape of X(1835) with high statistical precision therefore provides valuable information that helps clarify the nature of the X(1835) and X(pp).
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In this Letter, we report the observation of a significant abrupt change in slope of the X(1835) → η ′ π + π − line shape at the pp mass threshold in a sample of J/ψ → γη ′ π + π − events collected in the BESIII detector at the BEPCII e + e − storage ring. The η ′ is reconstructed in its two major decay modes:
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The data sample used in this analysis contains a total of 1.09 × 10 9 J/ψ decay events [18] accumulated by the BESIII experiment in 2012.
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The BESIII detector [19] is a magnetic spectrometer operating at BEPCII [20] , a double-ring e + e − collider with center of mass energies between 2.0 and 4.6 GeV. The cylindrical core of the BESIII detector consists of a helium-based main drift chamber (MDC), a plastic scintillator time-of-flight system (TOF), and a CsI(Tl) electromagnetic calorimeter (EMC) that are all enclosed in a superconducting solenoidal magnet providing a 0.9 T magnetic field. The solenoid is supported by an octagonal flux-return yoke with resistive plate counter muon identifier modules interleaved with steel. The acceptance of charged particles and photons is 93% of the 4π solid angle. The charged-particle momentum resolution at 1 GeV/c is 0.5%; the EMC measures 1 GeV photons with an energy resolution of 2.5% (5%) in the barrel (end cap) regions. A GEANT4-based [21] Monte Carlo (MC) simulation software package is used to optimize the event selection criteria, estimate backgrounds, and determine the detection efficiency. The KKMC [22] generator is used to simulate J/ψ production.
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Charged tracks are reconstructed using hits in the MDC. The point of closest approach of each charged track to the e + e − interaction point is required to be within 20 cm in the beam direction and 2 cm in the plane perpendicular to the beam direction. The reconstructed polar angle between the charged-track and beam direction is restricted to |cos θ| < 0.93. The TOF and energy loss (dE/dx) information are combined to form particle identification confidence levels for the π, K, and p hypotheses; each track is assigned to the particle type that corresponds to the hypothesis with the highest confidence level. Photon candidates are selected from showers in the EMC with energy deposited in the EMC barrel (|cos θ| < 0.8) or end-cap regions (0.86 < |cos θ| < 0.92) to be greater than 100 MeV. EMC cluster timing requirements are used to suppress electronic noise and energy deposits that are unrelated to the event.
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To select events from J/ψ → γη ′ π + π − , four charged tracks with zero net charge are required. Among the four tracks, at least three of them should be positively identified as pions. At least two photon candidates are required for the η ′ → γπ + π − mode; three photon candidates are required for the η 
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If two or more combinations pass these two criteria, the one with the smallest A five-constraint (5C) kinematic fit, energy-momentum conservation with an additional constraint on the η → γγ invariant mass to be equal to m η , is performed, and χ 2 5C < 40 is required. Then the η ′ candidate is formed from the ηπ + π − combination that best satisfies
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− invariant mass spectra of the surviving events are shown in Fig. 1 , where peaks corresponding to the X(1835), X(2120), X(2370), η c [2] , and a structure near 2.6 GeV/c 2 that has not been seen before are evident for both η ′ decays. Thanks to the high statistical precision, an abrupt change in slope of the X(1835) line shape at the pp mass threshold is evident in both event samples.
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An inclusive sample of 10 9 J/ψ decays events that are generated according to the Lund-Charm model [23] and Particle Data Group (PDG) [24] decay tables, is used to study potential background processes. These include events with no real η ′ s in the final state (non-η ′ ) and those from J/ψ → π 0 η ′ π + π − . We use η ′ mass sideband events to estimate the non-η
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determine the shape of their contribution to the selected η ′ π + π − mass spectrum and re-weight this shape by the ratio of MC-determined efficiencies for J/ψ → γη ′ π + π − and J/ψ → π 0 η ′ π + π − events; the total weight after re-weighting is the estimated number of J/ψ → π 0 η ′ π + π − background events. Our studies of background processes show that neither the four peaks mentioned above nor the abrupt change in the line shape at 2m p are caused by background processes.
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We perform simultaneous fits to the η ′ π + π − invariant mass distributions between 1. ′ π + π − contribution, the phase space for J/ψ → γη ′ π + π − is considered: according to the J P of f 1 (1510) and X(1835), J/ψ → γf 1 (1510) and J/ψ → γX(1835) are S-wave and P -wave processes, respectively; all other processes are assumed to be S-wave processes. Without explicit mention, all components are treated as incoherent contributions. In the simultaneous fits, the masses and widths of resonances, as well as the branching fraction for J/ψ radiative decays to η ′ π + π − final states (including resonances and non-resonant η ′ π + π − ) are constrained to be the same for both η ′ decay channels. The fit results are shown in Fig. 2 , where it is evident that using a simple Breit-Wigner function to describe the X(1835) line shape fails near the pp mass threshold. The log L (L is the combined likelihood of simultaneous fits) of this fit is 630503.3. Typically, there are two circumstances where an abrupt distortion of a resonance's line shape shows up: a threshold effect caused by the opening of an additional decay mode; or interference between two resonances. We tried to fit the data for both of these possibilities. The dashed dotted vertical line shows the position of pp mass threshold, the dots with error bars are data, the solid curves are total fit results, the dashed curves are the X(1835), the short-dashed curves the f1(1510), the dash-dot curves the X(2120), and the long-dashed curves are the non-resonant η ′ π + π − fit results; the shaded histograms are background events. The inset shows the data and the global fit between 1.8 GeV/c 2 and 1.95 GeV/c 2 .
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In the first model, we assume the state around 1.85 GeV/c 2 couples to pp. The line shape of η ′ π + π − above the pp threshold is therefore affected by the opening of the X(1835) → pp decay channel, similar to the distortion of the f 0 (980) → π + π − line shape at the KK threshold. To study this, the Flatté formula [25] is used for the X(1835) line shape:
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Here T is the decay amplitude, ρ out is the phase space for J/ψ → γη ′ π + π − , M is a parameter with the dimension of mass, s is the square of the η ′ π + π − system's mass, ρ k is the phase space for decay mode k, and g Approximately:
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where g 2 0 is the sum of g 2 of all decay modes other than X(1835) → pp, ρ 0 is the maximum two-body decay phase space volume [24] and g 2 pp /g 2 0 is the ratio between the coupling strength to the pp channel and the sum of all other channels.
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The fit results for this model are shown in Fig. 3 . The Flattè model fit has a log L = 630549.5 that is In the second model, we assume the existence of a narrow resonance near the pp threshold and that the interference between this resonance and the X(1835) produces the line shape distortion. Here we denote this narrow resonance as "X(1870)." For this case we represent the line shape in the vicinity on 1835 MeV by the square of T , where
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Here, ρ out and s have the same meaning as they had in Eq. 1; M 1 , Γ 1 , M 2 and Γ 2 represent the masses and widths of the X(1835) and X(1870) resonances respectively; and β and θ are the relative η ′ π + π − coupling strengths and the phase between the two resonances.
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The fit results for the second model are shown in Fig. 4 . The log L of this fit is 630540.3, which is improved by 37 with 4 additional parameters over that for the fit using one simple Breit-Wigner. The X(1835) mass is 1825.3 ± 2.4 MeV/c 2 and width is 245.2 ± 13.1 MeV/c 2 ; The X(1870) mass is 1870.2 ± 2.2 MeV/c 2 and width is 13.0 ± 6.1 MeV/c 2 , with a statistical significance that is 7.9σ. It is known that there are two non-trivial solutions in a fit using a coherent sum of two Breit-Wigner functions [27] . In the parameterization of Eq. 3, the two solutions share the same M 1 , Γ 1 , M 2 and Γ 2 , but have different values of β and θ, which means that the only observable difference between the solutions are branching fractions of the two Breit-Wigner functions. The product branching fractions with constructive interference are B(J/ψ → γX(1835))·B(X(1835
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and B(J/ψ → γX(1870))·B(X(1870) → η ′ π + π − ) = (2.03 ± 0.12) × 10 −7 , while the solution with destructive interference gives B(J/ψ → γX(1835))·B(X(1835)
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In this model, the X(1920) is not included in the fit because its significance is just 3.9σ. Considering systematic uncertainties (see below), the significance of X(1870) is larger than 7σ.
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The systematic uncertainties come from data-MC differences in the tracking, photon detection and particle identification efficiencies, the kinematic fit, requirements on the invariant mass distribution of γγ, signal selection of ρ 0 , η and η ′ , total number of J/ψ events, branching fractions for intermediate states decays, fit ranges, background descriptions, mass resolutions, and Fit results of using a coherent sum of two Breit-Wigner amplitudes. The dashed dotted vertical line shows the position of pp mass threshold, the dots with error bars are data, the solid curves are total fit results, the dashed curves are the sum of X(1835) and X(1870), the short-dashed curves are the f1(1510), the dash-dotted curves are the X(2120), the long-dashed curves are non-resonant η ′ π + π − fit results, the shaded histograms are background events. The inset shows the data and the global fit between 1.8 GeV/c 2 and 1.95 GeV/c 2 . Table II .
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In summary, the J/ψ → γη ′ π + π − process is studied with 1.09 × 10 9 J/ψ events collected at the BESIII experiment in 2012. We observed a significant distortion of the η ′ π + π − line shape near the pp mass threshold that cannot be accommodated by an ordinary Breit-Wigner resonance function. Two typical models for such a line shape are used to fit the data. The first model assumes the state around 1.85 GeV/c 2 couples with pp and the distortion reflects the opening of the pp decay channel. The fit result for this model yields a strong coupling between the broad structure and the pp of g 2 , with a statistical significance larger than 7σ. With current data, both models fit the data well with fit qualities, and both suggest the existence of a state, either a broad state with strong couplings to pp, or a narrow state just below the pp mass threshold. For the broad state above the pp mass threshold, its strong couplings to pp suggests the existence of a pp molecule-like state. For the narrow state just below pp mass threshold, its very narrow width suggests that it be an unconventional meson, most likely a pp bound state. So both fits support the existence of a pp molecule-like or bound state. With current statistics, more sophisticated models such as a mixture of above two models cannot be ruled out. In order to elucidate further the nature of the states around 1.85 GeV/c 2 , more data are needed to further study J/ψ → γη ′ π + π − process. Also, line shapes for other decay modes should be studied near the pp mass threshold, including further studies of J/ψ → γpp and J/ψ → γηK
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The BESIII collaboration thanks the staff of BEPCII and the IHEP computing center for their strong support. This work is supported in part by National Key

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