Observation of the Anomalous Shape of $X(1840)$ in $J/\psi \rightarrow \gamma 3(\pi^+ \pi^-)$ Indicating a Second Resonance Near $p\bar{p}$ Threshold

Using a sample of $(10087\pm44)\times 10^6$ $J/\psi$ events, which is about 45 times larger than that was previously analyzed, a further investigation on the $J/\psi\rightarrow \gamma 3(\pi^+\pi^-)$ decay is performed. A significant distortion at 1.84 GeV/$c^2$ in the line-shape of the $3(\pi^+\pi^-)$ invariant mass spectrum is observed for the first time, which could be resolved by two overlapping resonant structures, $X(1840)$ and $X(1880)$. The new state $X(1880)$ is observed with a statistical significance larger than $10\sigma$. The mass and width of $X(1880)$ are determined to be $1882.1\pm1.7\pm0.7$ MeV/$c^2$ and $30.7\pm5.5 \pm2.4$ MeV, respectively, which indicates the existence of a $p\bar{p}$ bound state.

a Also at the Moscow Institute of Physics and Technology, Moscow 141700, Russia b Also at the Novosibirsk State University, Novosibirsk, 630090, Russia c Also at the NRC "Kurchatov Institute", PNPI, 188300, Gatchina, Russia Using a sample of (10087 ± 44) × 10 6 J/ψ events, which is about 45 times larger than that was previously analyzed, a further investigation on the J/ψ → γ3(π + π − ) decay is performed.A significant distortion at 1.84 GeV/c 2 in the line-shape of the 3(π + π − ) invariant mass spectrum is observed for the first time, which could be resolved by two overlapping resonant structures, X(1840) and X(1880).The new state X(1880) is observed with a statistical significance larger than 10σ.The mass and width of X(1880) are determined to be 1882.1 ± 1.7 ± 0.7 MeV/c 2 and 30.7 ± 5.5 ± 2.4 MeV, respectively, which indicates the existence of a pp bound state.
A distinct resonance, X(1835) [1], in the π + π − η ′ invariant mass spectrum and a dramatic pp mass threshold enhancement [2] in J/ψ → γpp were first observed by BESII, which stimulated both theoretical and experimental interests in their nature.Some theoretical models are proposed to interpret their internal structures, e.g. a pp bound state [3][4][5][6], a pseudoscalar glueball [7][8][9], or a radial excitation of the η ′ meson [10].Subsequently these resonances were confirmed by BESIII [11,12] and CLEO [13] experiments and found to have the same J P C of 0 −+ [14,15].Meanwhile, a prominent structure, X(1840), was observed in the 3(π MeV/c 2 and a width of 83 ± 14 ± 11 MeV [16].It was interpreted as a new decay mode of X(1835), although its width is substantially narrower than that of X(1835) [12].Of interest is that an updated analysis of J/ψ → γπ + π − η ′ observed a significant abrupt change in slope of the X(1835) → π + π − η ′ line-shape at the pp mass threshold, which could be originated from the opening of an additional pp decay channel (threshold effect) or the interference between two different resonance contributions [17].To understand whether a similar phenomenon to that of J/ψ → γπ + π − η ′ exists around the pp mass threshold in the M (6π) spectrum, it is worth a more detailed investigation on the X(1840) line-shape in J/ψ → γ3(π + π − ) with higher precision.In this letter we report an anomalous line-shape of X(1840) in the M (6π) spectrum in J/ψ → γ3(π + π − ) with a sample of (10087 ± 44) × 10 6 J/ψ events [18] collected with the BE-SIII detector.The size of the sample is about 45 times greater than that used in Ref. [16].
The BESIII detector records symmetric e + e − collisions provided by the BEPCII storage ring [19] in the center-of-mass energy range from 2.0 to 4.95 GeV, which is described in detail in [20][21][22][23].Simulated data samples produced with a geant4-based [24] Monte Carlo (MC) package, which includes the geometric description of the BESIII detector [25] and the detector response, are used to determine detection efficiencies and to estimate backgrounds.The simulation models the beam energy spread and initial state radiation (ISR) in the e + e − annihilation with the generator kkmc [26].All particle decays are modelled with evtgen [27,28] using branching fractions either taken from the Particle Data Group [29], when available, or otherwise estimated with lundcharm [30,31].Final state radiation (FSR) from charged final state particles is incorporated using the photos package [32].
Charged tracks detected in the main drift chamber (MDC) are required to be within a polar angle (θ) range of |cosθ| < 0.93, where θ is defined with respect to the z-axis, the symmetry axis of the MDC.The distance of closest approach to the interaction point must be less than 10 cm along the z-axis, and less than 1 cm in the transverse plane.Photon candidates are reconstructed using clusters of energy deposited in the electromagnetic calorimeter (EMC), where a minimum energy of 25 MeV for the barrel region (|cos θ| < 0.8) and 50 MeV for the endcap region (0.86 < |cos θ| < 0.92) is required.To suppress electronic noise and showers unrelated to the event, the difference between the EMC time and the event start time is required to be within [0, 700] ns.
Candidates for the signal are required to have six charged tracks with zero net charge and at least one photon.All the charged tracks are assumed to be pions.A four-momentum-constraint (4C) kinematic fit is performed under the hypothesis of J/ψ → γ3(π + π − ), and the χ 2 4C of this kinematic fit is required to be less than 30.For events with more than one photon candidate, the γ3(π + π − ) combination with the minimum χ 2 4C is retained.To suppress the backgrounds with a final state of γγ3(π + π − ), the χ 2 4C is required to be less than that for the kinematically similar γγ3(π + π − ) hypothesis.Furthermore, for the events containing at least two photons, the γγ invariant mass is required to be outside the π 0 mass window of |M γγ − m π 0 | < 0.01 GeV/c 2 to veto the backgrounds with π 0 in their final states.
The J/ψ → γK 0 S K 0 S π + π − process with a subsequent decay of K 0 S to π + π − has the same final state as the signal decay.To suppress this background, the K 0 S candidates are reconstructed from secondary vertex fit (SVF) to all π + π − pairs.The K 0 S candidates are tagged by passing the SVF successfully and requiring the π + π − invariant mass in a range of |M π + π − − m K 0 S | < 0.005 GeV/c 2 , where m K 0 S is the K 0 S known mass.Given the existence of mis-reconstructed K 0 S for the signal, events with the number of K 0 S candidates less than 2 are retained for further analysis.
After applying the above requirements, the M (6π) spectrum is shown in Figure 1, where, in addition to the well established η c peak and the peak around 3.07 GeV/c 2 from J/ψ → 3(π + π − ) background channel, a distinct structure around 1.84 GeV/c 2 is apparent, and an anomalous line-shape near the pp mass threshold is clearly observed, as shown in the inset plot.
With exactly the same processes of simulated inclusive J/ψ events as for the data, no peaking background contribution around 1.84 GeV/c 2 is found.The remaining background is mainly from J/ψ → π 0 3(π + π − ), for which, we use a one-dimensional data-driven method to determine its contribution.We select the J/ψ → π 0 3(π + π − ) events from data firstly and then implement the signal selection criteria on these events.The M (6π) spectrum extracted based on these surviving events is further reweighted by the ratio of MC-determined efficiencies for J/ψ → γ3(π + π − ) to J/ψ → π 0 3(π + π − ) events.To ensure that the anomalous line-shape in data is not caused by the distortion of the detection efficiency due to event selection bias, we studied the phase space MC events of J/ψ → γ3(π + π − ).As a result, neither the 1.84 GeV/c 2  peaking structure nor the abrupt change in the line-shape near the pp mass threshold is caused by the background processes or the distortion of the the event selection efficiency.
We perform an unbinned maximum likelihood fit to the M (6π) spectrum between 1.55 and 2.07 GeV/c 2 with the X(1840) peak represented by the efficiency corrected Breit-Wigner (BW) function convolved with a Gaussian function to account for the mass resolution, which is determined to be 4 MeV/c 2 from the MC simulation.The dominant background to the X(1840) peak is from the non-resonant contribution of J/ψ → γ3(π + π − ), whose shape is obtained through MC simulation and the fraction is free in the fit.The J/ψ → 3(π + π − )π 0 background contributions are estimated with the data-driven approach as described above.The remaining background is described by a free second-order polynomial function.Without explicit mention, all components are treated as incoherent contributions.The fit quality is significantly poor, as shown in Fig. 2. The goodness of fit is studied using a χ 2 test and the χ 2 value per number of degrees of freedom (ndof ) is found to be χ 2 /ndof = 399.0/45.This implies that a simple resonant structure fails to describe the M (6π) spectrum.
To resolve the discrepancy from data, two different models for the line shape of the structure around 1.84 GeV/c 2 are applied to investigate the resonances in the M (6π) spectrum.With an assumption of the line-shape of 3(π + π − ) above the pp mass threshold affected by the opening of the X(1840) → pp decay (model I), we try to describe the anomalous shape with a Flatté formula [33], where M is a parameter with the dimension of mass, s is the mass square of the 3(π + π − ) combination, ρ j is the phase space for the decay mode j, and g 2 j is the corre- The dots with error bars are data, the solid curve in red is the total fit result, the dashed line in blue is the X(1840) signal, the dash-dotted line in green is the background events from J/ψ → π 0 3(π + π − ), and the dotted line in magenta is the sum of background.
sponding coupling strength.The j g 2 j ρ j term describes how the decay width varies with s.Approximately, where g 2 0 is the sum of g 2 of all decay modes other than X(1840) → pp, ρ 0 is the maximum two-body decay phase space volume [29] and g 2 p p/g 2 0 is the ratio between the coupling strength to the pp channel and the sum of all other channels.This fit, as illustrated in Fig. 3(a), yields M = 1.818±0.009GeV/c 2 , g 2 0 = 18.0±2.8GeV 2 /c 4 , and g 2 p p = 51.4 ± 14.8 GeV 2 /c 4 .This model fit has a log L that is improved over the simple Breit-Wigner one by 42.8.The significance of g 2 p p/g 2 0 being non-zero is 9.2σ.The goodness of the fit χ 2 /ndof = 317.9/44,yet not enough to be acceptable for a good description of data.
A comparison between the fit result of model I and the data reveals a tension around the pp mass threshold.To obtain a better description on data, another model allows for interference between two resonant components (model II) and the coherent sum of them is defined as where M 1 , Γ 1 , M 2 and Γ 2 represent the masses and widths of the two resonant structures, denoted as X(1840) and X(1880), respectively.β is a complex parameter accounting for the contribution of X(1880) relative to the X(1840) as well as the phase between them.The fit with model II improves the fit quality significantly (χ 2 /ndof = 155.6/41), in particular for the region around the pp mass threshold, which is illustrated   1880) is found to be larger than 10σ, which is determined by comparing the log-likelihood value and the number of degrees of freedom between model II and model I using Wilks' theorem [34].As discussed in Ref. [35], a fit using a coherent sum of two BW functions may result in two nontrivial solutions with the FIG. 4. The acceptance-corrected angular distribution with respect to |cos θ| for the X(1840) and X(1880), respectively.The curves represent the fit results with the function of 1 + cos 2 θ to the above two components: the solid curve in blue is for the X(1840); the dashed one in red is for the X(1880).same resonant parameters.We make an extensive investigation on the fit and find the second solution with the same fit quality, which yields N X(1840) = 36506 ± 8740 and N X(1880) = 22097 ± 5794, corresponding to the destructive interference as described in Ref. [35].The figure of second solution is given in the Supplemental Material [36].
Since the X(1835) is known as a pseudoscalar meson, the X(1840) is supposed to be a pseudoscalar particle as well considering the similar behaviours with those of the X(1835).For the radiative J/ψ decay to a pseudoscalar meson, the polar angle of the photon in the J/ψ rest frame, denoted as θ, is expected to follow a 1 + cos 2 θ distribution.The |cos θ| is divided into nine bins in a region of [0, 0.9] to investigate the angular distribution.The number of signal events corresponding to the constructive interference solution in each bin is obtained with the same fit procedure as mentioned above.The result is shown in Fig. 4. As a result, the angular distributions of X(1840) and X(1880) both agree with 1+cos 2 θ and support the interpretation of the pseudoscalar mesons.With the hypotheses of pseudoscalar mesons, the detection efficiencies for J/ψ → γX(1840) and J/ψ → γX(1880), 17.4% and 18.4%, are obtained from the MC simulation.The product branching fractions corresponding to that two solutions are summarized in Table I.
Sources of systematic uncertainties and their corresponding contributions to the measurement of the branching fractions are summarized in Table II.The uncertainties come from data-MC differences (tracking, photon detection, 4C kinematic fit etc.), total number of J/ψ events, and background uncertainty from the change of fit range, MC model, MC statistics.For the MC model uncertainties due to the unknown spin-parity of the structures, we use the difference between phase space and a pseudoscalar meson hypothesis.In accordance with the previous publication [37], we keep the efficiency with the track helix correction as the nominal value in this work, and take the difference between the efficiencies with and without correction as the systematic uncertainty from the 4C kinematic fit.The main contribution of systematic uncertainty comes from the uncertainty in the background estimation which is accessed by changing fit ranges.The uncertainty caused by the contribution above 2.07 GeV of the M (6π) spectrum has a considerable effect on the parameterization of the remaining background, which results in a large uncertainty of the branching fractions.Meanwhile, the impact of this uncertainty on the statistical significance of X( 1880) is considered, and the smallest statistical significance is chosen as the final result.The total systematic uncertainty is obtained by adding all of the mentioned ones in quadrature under the assumption that they are independent.The total systematic uncertainties on mass and width are estimated from the background uncertainty due to fit range and background description, and found to be ± 2.5 MeV/c 2 and ± 7.7 MeV for the X(1840), ± 0.7 MeV/c 2 and ± 2.4 MeV for the X(1880), respectively.Since the mass resolution of 4 MeV/c 2 is much smaller than the width of these structures, the uncertainty from the detector resolution is found to be negligible.The final results given in Table I are obtained considering all the systematic uncertainties reported in Table II.In summary, a study of the radiative decay J/ψ → γ3(π + π − ) is performed with a sample of (10087 ± 44) × 10 6 J/ψ events accumulated at the BESIII detector.A significant distortion of the M (6π) distribution near the pp mass threshold is observed for the first time, which is analogous to the distortion observed in the π + π − η ′ invariant mass spectrum in J/ψ → γπ + π − η ′ [17].
To understand this anomalous line-shape, a few interpretations including a single structure described by a single BW or with threshold effect (model I) and a coherent sum of two structures (model II) are tested.We find that neither a simple BW nor a Flatté function could provide a reasonable description of data.The scheme of a coherent sum of two structures gives a much better description on the anomalous line-shape in the M (6π) spectrum.According to the fit results, the narrow structure, X(1880), has a mass of M = 1882.1±1.7±0.7 MeV/c 2 and a width of Γ = 30.7 ± 5.5 ± 2.4 MeV.The significance of X( 1880) is larger than 10σ compared to the fit result with model I considering any effect associated to the systematic uncertainties.The mass and width of X(1840) are measured to be M = 1832.5±3.1±2.5 MeV/c 2 and Γ = 80.7±5.2±7.7 MeV, which are in agreement with the previous work [16].Two solutions with the same fit quality and the identical resonant parameters but different branching fractions due to the constructive or destructive interference are summarized in Table I.
Compared with the two structures observed in the M (π + π − η ′ ) spectrum [17], the X(1840) has a consistent mass with that X(1835) but much narrower width.The mass and width of the X(1880) obtained in this work are in reasonable agreement with those reported in Ref. [17], which are 1870.2± 2.2 +2.3 −0.7 MeV/c 2 and 13.0 ± 6.1 +2.1 −3.8 MeV, respectively.This further supports the existence of a pp bound state just spanning the pp mass threshold.At present, more sophisticated parameterizations such as a mixture of above two models cannot be ruled out.The observed anomalous line-shape in the M (6π) spectrum in J/ψ → γ3(π + π − ) and the π + π − η ′ invariant mass spectrum in J/ψ → γπ + π − η ′ reveal complex resonant structures near the pp mass threshold.To establish the relationship between different resonances in the mass region of [1.8, 1.9] GeV/c 2 and determine the nature of the underlying resonant structures, more data along with additional measurements including the determination of the spin-parity quantum numbers and the coupled channel amplitude analysis are highly desirable.
The BESIII Collaboration thanks the staff of BEPCII and the IHEP computing center for their strong support.

FIG. 1 .
FIG.1.M (6π) distribution from J/ψ → γ3(π + π − ) events.The dots with error bars are data.The inset shows the data between 1.75 and 1.95 GeV/c 2 and the vertical dotted line represents the pp mass threshold.

FIG. 3 .
FIG. 3. Fit result of the M (6π) distribution with model I (a), and solution I for the model II (b).The dashed line in blue is the X(1840) signal for (a), and the sum of X(1840) and X(1880) for (b).

d
Also at Goethe University Frankfurt, 60323 Frankfurt am Main, Germany e Also at Key Laboratory for Particle Physics, Astrophysics and Cosmology, Ministry of Education; Shanghai Key Laboratory for Particle Physics and Cosmology; Institute of Nuclear and Particle Physics, Shanghai 200240, People's Republic of China f Also at Key Laboratory of Nuclear Physics and Ion-beam Application (MOE) and Institute of Modern Physics, Fudan University, Shanghai 200443, People's Republic of China g Also at State Key Laboratory of Nuclear Physics and Technology, Peking University, Beijing 100871, People's Republic of China h Also at School of Physics and Electronics, Hunan University, Changsha 410082, China i Also at Guangdong Provincial Key Laboratory of Nuclear Science, Institute of Quantum Matter, South China Normal University, Guangzhou 510006, China j Also at Frontiers Science Center for Rare Isotopes, Lanzhou University, Lanzhou 730000, People's Republic of China k Also at Lanzhou Center for Theoretical Physics, Lanzhou University, Lanzhou 730000, People's Republic of China l Also at the Department of Mathematical Sciences, IBA, Karachi 75270, Pakistan

TABLE I .
The fitted parameters of the two coherent resonant structures and the corresponding product branching fractions.Solution I and II refer to the solutions characterizing model II as discussed in the text.

TABLE II .
Relative systematic uncertainties in the product branching fractions (in percent).