EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-EP-2016-155 LHCb-PAPER-2016-018 29 September 2016
Observation of structures consistent with exotic states from amplitude analysis of decays
The LHCb collaboration†††Authors are listed at the end of this paper.
The first full amplitude analysis of with , decays is performed with a data sample of 3 fb of collision data collected at and TeV with the LHCb detector. The data cannot be described by a model that contains only excited kaon states decaying into , and four structures are observed, each with significance over standard deviations. The quantum numbers of these structures are determined with significance of at least standard deviations. The lightest has mass consistent with, but width much larger than, previous measurements of the claimed state.
Published in Physical Review Letters 118, 022003 (2017).
© CERN on behalf of the LHCb collaboration, license CC-BY-4.0.
There has been a great deal of experimental and theoretical interest in mass structures in decays111Inclusion of charge-conjugate processes is implied. since the CDF collaboration presented evidence for a near-threshold mass peak, with width .222Units with are used. Much larger widths are expected for charmonium states at this mass because of open flavor decay channels , which should also make the kinematically suppressed decays undetectable. Therefore, it has been suggested that the peak could be a molecular state [3, 4, 5, 6, 7, 8, 9], a tetraquark state [10, 11, 12, 13, 14], a hybrid state [15, 16] or a rescattering effect [17, 18]. Subsequent measurements resulted in the confusing experimental situation summarized in Table 1. Searches for the narrow in decays were negative in the Belle [19, 20] (unpublished), LHCb  (0.37 fb) and BaBar  experiments. The structure was, however, observed by the CMS  and D0 [24, 25] collaborations.
In an unpublished update to their analysis , the CDF collaboration presented evidence for a second relatively narrow mass peak near . A second peak was also observed by the CMS collaboration at a mass which is higher by standard deviations, but its statistical significance was not determined . The Belle collaboration obtained evidence for a narrow () peak at in two-photon collisions, which implies or , and found no signal for .
The and states are the only known candidates for four-quark systems that contain neither of the light and quarks. Their confirmation, and determination of their quantum numbers, would allow new insights into the binding mechanisms present in multi-quark systems, and help improve understanding of QCD in the non-perturbative regime.
The data sample used in this work corresponds to an integrated luminosity of fb collected with the LHCb detector in collisions at center-of-mass energies 7 and 8 TeV. The LHCb detector is a single-arm forward spectrometer covering the pseudorapidity range , described in detail in Refs. [28, 29]. Thanks to the larger signal yield, corresponding to reconstructed decays, the roughly uniform efficiency and the relatively low background across the entire mass range, this data sample offers the best sensitivity to date, not only to probe for the previously claimed structures, but also to inspect the high mass region for the first time. All previous analyses were based on naive mass () fits, with Breit–Wigner (BW) signal peaks on top of incoherent background described by ad-hoc functional shapes (e.g. the three-body phase space distribution in decays). While the distribution has been observed to be smooth, several resonant contributions from kaon excitations (denoted generically as ) are expected. It is important to prove that any peaks are not merely reflections of states. If genuine states are present, it is crucial to determine their quantum numbers to aid their theoretical interpretation. Both of these tasks call for a proper amplitude analysis of decays, in which the observed and masses are analyzed simultaneously with the distributions of decay angles, without which the resolution of different resonant contributions is difficult, if not impossible.
In this Letter, results with a focus on mass structures are presented from the first amplitude analysis of decays. A detailed description of the analysis with more extensive discussion of the results on kaon spectroscopy can be found in Ref. . The data selection is similar to that described in Ref. , with modifications  that increase the signal yield per unit luminosity by about 50% at the expense of larger background. A pair with mass within of the known mass  is accepted as a candidate. To avoid reconstruction ambiguities, we require that there is exactly one candidate per combination, which reduces the yield by . A fit to the mass distribution of candidates yields events, with a background fraction () of in the region used in the amplitude analysis (twice the mass resolution on each side of its peak). The non- background is small () and neglected in the amplitude model, but considered as a source of systematic uncertainty.
We first try to describe the data with kaon excitations alone. We construct an amplitude model () using the helicity formalism [32, 33, 34] in which the six independent variables fully describing the , , , decay chain are , , , , and , where denotes helicity angles, and angles between decay planes. The set of angles is denoted by . The matrix element for a single resonance () with mass and width is assumed to factorize, , where is a complex BW function and describes the angular correlations, with being a set of complex helicity couplings which are determined from the data (1–4 independent couplings depending on ), where , and denotes the helicity. The total matrix element sums coherently over all possible resonances: . Detailed definitions of and of are given in Ref. . The free parameters are determined from the data by minimizing the unbinned six-dimensional (6D) negative log-likelihood (), where the probability density function (PDF) is proportional to , multiplied by the detection efficiency, plus a background term. The signal PDF is normalized by summing over events generated [35, 36] uniformly in decay phase space, followed by detector simulation  and data selection. This procedure accounts for the 6D efficiency in the reconstruction of the signal decays . We use mass sidebands to obtain a 6D parameterization of the background PDF .
Past experiments on states decaying to [38, 39, 40] had limited precision, gave somewhat inconsistent results, and provided evidence for only a few of the states expected from the quark model in the – range probed in our data. We have used the predictions of the relativistic potential model by Godfrey–Isgur  (horizontal black lines in Fig. 2) as a guide to the quantum numbers of the states to be included in the amplitude model. The masses and widths of all states are left free; thus our fits do not depend on details of the predictions, nor on previous measurements. We also include a constant nonresonant amplitude with , since such contributions can be produced, and can decay, in S-wave. Allowing the magnitude of the nonresonant amplitude to vary with does not improve fit qualities. While it is possible to describe the and distributions well with contributions alone, the fit projections onto do not provide an acceptable description of the data. For illustration we show in Fig. 1 the projection of a fit with the following composition: a nonresonant term plus candidates for two , two , and one of each of , , , , , , and states, labeled here with their intrinsic quantum numbers ( is the radial quantum number, the total spin of the valence quarks, the orbital angular momentum between quarks, and the total angular momentum of the bound state). The fit contains 104 free parameters. The value (144.9/68 bins) between the fit projection and the observed distribution corresponds to a p-value below . Adding even more resonances does not change the conclusion that non- contributions are needed.
The matrix element for , decays can be parameterized using and the , , , , angles. The angles and are not the same as in the decay chain since and are produced in decays of different particles. For the same reason, the muon helicity states are different between the two decay chains, and an azimuthal rotation by an angle is needed to align them [42, 30]. The parameters needed to characterize the decay chain, including , do not constitute new degrees of freedom since they can all be derived from and . We also consider possible contributions from , decays, which can be parameterized in a similar way . The total matrix element is obtained by summing all possible (), () and () contributions: .
We have explored adding and contributions of various quantum numbers to the fit models. Only contributions lead to significant improvements in the description of the data. The default resonance model is summarized in Table 2. It contains seven states (Fig. 2), four states, and and nonresonant components. There are 98 free parameters in this fit. Additional , or states are not significant. Projections of the fit onto the mass variables are displayed in Fig. 3. The value (71.5/68 bins) between the fit projection and the observed distribution corresponds to a p-value of 22%, where the effective number of degrees of freedom has been obtained with simulations of pseudoexperiments generated from the default amplitude model. Projections onto angular variables, and onto masses in different regions of the Dalitz plot, can be found in Ref. .
The systematic uncertainties  are obtained from the sum in quadrature of the changes observed in the fit results when: the and models are varied (the dominant errors); the BW amplitude parameterization is modified; only the left or right mass peak sidebands are used for the background parameterization; the mass selection is changed; the signal and background shapes are varied in the fit to which determines ; the weights assigned to simulated events, in order to improve agreement with the data on production characteristics and detector efficiency, are removed.
The significance of each (non)resonant contribution is calculated from the change in log-likelihood between fits with and without the contribution included. The distribution of between the two hypothesis should follow a distribution with number of degrees of freedom equal to the number of free parameters in its parameterization (doubled when and are free parameters). The validity of this assumption has been verified using simulated pseudoexperiments. The significances of the contributions are given after accounting for systematic uncertainties.
The composition of our amplitude model is in good agreement with the expectations for the states , and also in agreement with previous experimental results on states in this mass range  as illustrated in Fig. 2 and in Table 2. Effects of adding extra insignificant resonances of various , as well as of removing the least significant contributions, are included among the systematic variations of the fit amplitude. More detailed discussion of our results for kaon excitations can be found in Ref. .
A near-threshold structure in our data is the most significant () exotic contribution to our model. We determine its quantum numbers to be at significance from the change in relative to other assignments  including systematic variations. When fitted as a resonance, its mass () is in excellent agreement with previous measurements for the state, although the width () is substantially larger. The upper limit previously set for production of a narrow () state based on a small subset of our present data  does not apply to such a broad resonance, thus the present results are consistent with our previous analysis. The statistical power of the present data sample is not sufficient to study its phase motion . A model-dependent study discussed in Ref.  suggests that the structure may be affected by the nearby coupled-channel threshold. However, larger data samples will be required to resolve this issue.
We establish the existence of the structure with statistical significance of , at a mass of and a width of . Its quantum numbers are determined to be at significance. Due to interference effects, the data peak above the pole mass, underlining the importance of proper amplitude analysis.
The high mass region also shows structures that cannot be described in a model containing only states. These features are best described in our model by two resonances, () and (), with parameters given in Table 2. The resonances interfere with a nonresonant contribution that is also significant (). The significances of the quantum number determinations for the high mass states are and , respectively.
In summary, we have performed the first amplitude analysis of decays. We have obtained a good description of the data in the 6D phase space composed of invariant masses and decay angles. The amplitude model extracted from our data is consistent with expectations from the quark model and from the previous experimental results on such resonances. We determine the quantum numbers of the structure to be . This has a large impact on its possible interpretations, in particular ruling out the or molecular models [3, 4, 5, 6, 7, 8]. The width is substantially larger than previously determined. The below--threshold cusp [18, 9] may have an impact on the structure, but more data will be required to address this issue, as discussed in more detail in the companion article . The existence of the structure is established and its quantum numbers are determined to be . Molecular bound-states or cusps cannot account for these values. A hybrid charmonium state would have [15, 16]. Some tetraquark models expected ,  or ,  state(s) in this mass range. A tetraquark model implemented by Stancu  not only correctly assigned to , but also predicted a second state at mass not much higher than the mass. Calculations by Anisovich et al.  based on the diquark tetraquark model predicted only one state at a somewhat higher mass. Lebed–Polosa  predicted the peak to be a tetraquark, although they expected the peak to be a state in the same model. A lattice QCD calculation with diquark operators found no evidence for a tetraquark below .
The high mass region is investigated for the first time with good sensitivity and shows very significant structures, which can be described as two resonances: and . The work of Wang et al.  predicted a virtual state at . None of the observed states is consistent with the state seen in two-photon collisions by the Belle collaboration .
We express our gratitude to our colleagues in the CERN accelerator departments for the excellent performance of the LHC. We thank the technical and administrative staff at the LHCb institutes. We acknowledge support from CERN and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); NSFC (China); CNRS/IN2P3 (France); BMBF, DFG and MPG (Germany); INFN (Italy); FOM and NWO (The Netherlands); MNiSW and NCN (Poland); MEN/IFA (Romania); MinES and FASO (Russia); MinECo (Spain); SNSF and SER (Switzerland); NASU (Ukraine); STFC (United Kingdom); NSF (USA). We acknowledge the computing resources that are provided by CERN, IN2P3 (France), KIT and DESY (Germany), INFN (Italy), SURF (The Netherlands), PIC (Spain), GridPP (United Kingdom), RRCKI and Yandex LLC (Russia), CSCS (Switzerland), IFIN-HH (Romania), CBPF (Brazil), PL-GRID (Poland) and OSC (USA). We are indebted to the communities behind the multiple open source software packages on which we depend. Individual groups or members have received support from AvH Foundation (Germany), EPLANET, Marie Skłodowska-Curie Actions and ERC (European Union), Conseil Général de Haute-Savoie, Labex ENIGMASS and OCEVU, Région Auvergne (France), RFBR and Yandex LLC (Russia), GVA, XuntaGal and GENCAT (Spain), Herchel Smith Fund, The Royal Society, Royal Commission for the Exhibition of 1851 and the Leverhulme Trust (United Kingdom).
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