University of Birmingham Model-independent confirmation of the Z (4430)-state

The decay B 0 → ψ ð 2 S Þ K þ π − is analyzed using 3 fb − 1 of pp collision data collected with the LHCb detector. A model-independent description of the ψ ð 2 S Þ π mass spectrum is obtained, using as input the K π mass spectrum and angular distribution derived directly from data, without requiring a theoretical description of resonance shapes or their interference. The hypothesis that the ψ ð 2 S Þ π mass spectrum can be described in terms of K π reflections alone is rejected with more than 8 σ significance. This provides confirmation, in a model-independent way, of the need for an additional resonant component in the mass region of the Z ð 4430 Þ − exotic state.

The decay B 0 → ψð2SÞK þ π − is analyzed using 3 fb −1 of pp collision data collected with the LHCb detector. A model-independent description of the ψð2SÞπ mass spectrum is obtained, using as input the Kπ mass spectrum and angular distribution derived directly from data, without requiring a theoretical description of resonance shapes or their interference. The hypothesis that the ψð2SÞπ mass spectrum can be described in terms of Kπ reflections alone is rejected with more than 8σ significance. This provides confirmation, in a model-independent way, of the need for an additional resonant component in the mass region of the Zð4430Þ − exotic state. DOI

I. INTRODUCTION
Almost all known mesons and baryons can be described in the quark model with combinations of two or three quarks, although the existence of higher multiplicity configurations, as well as additional gluonic components, is, in principle, not excluded [1]. For many years, significant effort has been devoted to the search for such exotic configurations. In the baryon sector, resonances with a fivequark content have been searched for extensively [2][3][4][5].
Recently, LHCb has observed a resonance in the J=ψp channel, compatible with being a pentaquark-charmonium state [6]. In the meson sector, several charmoniumlike states, that could be interpreted as four-quark states [7,8], have been reported by a number of experiments, but not all of them have been confirmed.
The BABAR Collaboration [25] searched for the Zð4430Þ − state in a data sample statistically comparable to Belle's. They used a model-independent approach to test whether an interpretation of the experimental data is possible in terms of the known resonances in the Kπ system. The Kπ mass and angular distributions were determined from data and used to predict the observed ψð2SÞπ mass spectrum. It was found that the observed ψð2SÞπ mass spectrum was compatible with being described by reflections of the Kπ system. Therefore, no clear evidence for a Zð4430Þ − was established, although BABAR's analysis did not exclude the observation by Belle.
The present article describes the details of an LHCb analysis that was briefly reported in Ref. [12]. Adopting a model-independent approach, along the lines of BABAR's strategy, the structures observed in the ψð2SÞπ mass spectrum are predicted in terms of the reflections of the Kπ system mass and angular composition, without introducing any modelling of the resonance line shapes and their interference patterns. The compatibility of these predictions with data is quantified.

II. LHCb DETECTOR
The LHCb detector [26,27] is a single-arm forward spectrometer covering the pseudorapidity range 2 < η < 5, designed for the study of particles containing b or c quarks. The detector includes a high-precision tracking system consisting of a silicon-strip vertex detector surrounding the pp interaction region [28], a large-area silicon-strip detector located upstream of a dipole magnet with a bending power of about 4 Tm, and three stations of silicon-strip detectors and straw drift tubes [29] placed downstream of the magnet. The tracking system provides a measurement of momentum, p, of charged particles with a relative uncertainty that varies from 0.5% at low momentum to 1.0% at 200 GeV=c. The minimum distance of a track to a primary vertex, the impact parameter, is measured with a resolution of ð15 þ 29=p T Þ μm, where p T is the component of the momentum transverse to the beam, in GeV=c. Different types of charged hadrons are distinguished using information from two ring-imaging Cherenkov detectors [30]. Photons, electrons, and hadrons are identified by a calorimeter system consisting of scintillating-pad and preshower detectors, an electromagnetic calorimeter, and a hadronic calorimeter. Muons are identified by a system composed of alternating layers of iron and multiwire proportional chambers [31]. The online event selection is performed by a trigger [32], which consists of a hardware stage, based on information from the calorimeter and muon systems, followed by a software stage, which applies a full event reconstruction.

III. DATA SAMPLES AND CANDIDATE SELECTION
The results presented in this paper are based on data from pp collisions collected by the LHCb experiment, corresponding to integrated luminosities of 1 and 2 fb −1 at center-of-mass energies of 7 TeV in 2011 and 8 TeV in 2012, respectively.
In the simulation, pp collisions are generated using PYTHIA [33] with a specific LHCb configuration [34]. The decays of hadronic particles are described by EvtGen [35], in which final-state radiation is generated using PHOTOS [36]. The interaction of the generated particles with the detector, and its response, are implemented using the GEANT4 toolkit [37] as described in Ref. [38]. Samples of simulated events, generated with both 2011 and 2012 conditions, are produced for the decay B 0 → ψð2SÞK þ π − with a uniform three-body phase-space distribution and the ψð2SÞ decaying into two muons. These simulated events are used to tune the event selection and for efficiency and resolution studies.
The selection is similar to that used in Ref. [12] and consists of a cut-based preselection followed by a multivariate analysis. Track-fit quality and particle identification requirements are applied to all charged tracks. The B 0 candidate reconstruction starts by requiring two well-identified muons, with opposite charges, having p T > 2 GeV=c and forming a good quality vertex. The dimuon invariant mass has to lie in the window 3630-3734 MeV=c 2 , around the ψð2SÞ mass. To obtain a B 0 candidate, each dimuon pair is required to form a good vertex with a kaon and a pion candidate, with opposite charges. Pions and kaons are required to be inconsistent with coming from any primary vertex (PV) and to have tranverse momenta greater than 200 MeV=c.
The B 0 candidate has to have p T > 2 GeV=c, a reconstructed decay time exceeding 0.25 ps, and an invariant mass in the window 5200-5380 MeV=c 2 around the nominal B 0 mass. Contributions from ϕ → K þ K − decay, where one of the kaons is misidentified as a pion, are removed by vetoing the region 1010-1030 MeV=c 2 of the dihadron invariant mass calculated assuming that the π − candidate has the K − mass.
To reduce the combinatorial background, a requirement is imposed on the output of a multivariate discriminator based on the likelihood ratio [39].
The four variables used as input are the smaller χ 2 IP of the kaon and the pion, where χ 2 IP is the difference in the PV fit χ 2 with and without the track under consideration; the μ þ μ − Kπ vertex-fit quality; the B 0 candidate impact parameter significance with respect to the PV; and the cosine of the largest opening angle between the ψð2SÞ and each of the charged hadrons in the plane transverse to the beam. After the multivariate selection, the B 0 candidate invariantmass distribution appears as shown in Fig. 1 with a fitted curve superimposed. The fit model consists of a Hypatia distribution [40] to describe the signal and an exponential function to describe the background. Table I provides the fit results and the signal and background yields in the signal region. The width of the distribution, σ B 0 , is defined as half the symmetric interval around M B 0 containing 68.7% of the total signal. The signal region is defined by the AE2σ B 0 interval around M B 0 .
Sideband subtraction is used to remove the background which is dominated by combinations of ψð2SÞ mesons from b-hadron decays with random kaons and pions. Sidebands are identified by the intervals ½M B 0 − 80; A weight, W signal , is attributed to each candidate: unit  weight is assigned to candidates in the signal region; the ratio of the background yield in the signal region and in the sidebands, with a negative sign, is the weight assigned to sideband candidates; and zero weight is assigned to candidates in the remaining regions.

IV. EFFICIENCY AND RESOLUTION
The reconstruction and selection efficiency has been evaluated using simulated samples. The efficiency is calculated as a function of four variables: the Kπ system invariant mass, m Kπ ; the cosine of the K Ã0 helicity angle, cos θ K Ã0 ; the cosine of the ψð2SÞ helicity angle, cos θ ψð2SÞ ; and the angle between the Kπ and the μ þ μ − planes calculated in the B 0 rest frame, Δϕ Kπ;μμ (this variable is called ϕ in Ref. [12]). The helicity angle of the K Ã0 [ψð2SÞ] is defined as the angle between the K þ (μ þ ) direction and the B 0 direction in the K Ã0 [ψð2SÞ] rest frame. This fourdimensional (4D) space is subdivided in 24, 25, 5, and 4 bins of the respective variables. The value of the efficiency, at each point of the 4D space, is evaluated as a multilinear interpolation of the values at the 16 bins' centers surrounding it. To the points falling in a border 4D bin, where interpolation is not possible, the value of the efficiency at the bin center is assigned.
To visualize the behavior, 2D efficiency plots are shown, as functions of all the possible variable pairs, in Fig. 2. Table II lists the resolutions (average uncertainty) of the reconstructed event variables as evaluated on simulated events. They are found to be very small compared to the width of any possible structure searched for in this analysis; therefore, no resolution corrections are applied. In addition, the smooth behavior of the m ψð2SÞπ resolution, shown in Fig. 3, demonstrates that structures in the m ψð2SÞπ spectrum could not be caused by resolution effects.

V. K Ã RESONANCES
A number of K Ã0 resonances with masses up to slightly above the kinematic limit of 1593 MeV=c 2 can decay to the Kπ final state and contribute to the B 0 → ψð2SÞK þ π − decay. Table III lists these K Ã0 states as well as resonances just above the kinematic limit.   LHCb Simulation  The m Kπ spectrum of candidate events, shown in the left plot of Fig. 4, is dominated by the K Ã ð892Þ 0 meson. A structure in the K Ã ð1410Þ 0 , K Ã 0 ð1430Þ 0 , and K Ã 2 ð1430Þ 0 mass regions is also clearly visible. In addition, a nonresonant component is evident. A contribution from the low-mass tail of excited states above the kinematic limit is expected, in particular from the spin-1 K Ã ð1680Þ 0 and the spin-3 K Ã 3 ð1780Þ 0 due to their large widths. The right plot of Fig. 4 shows the cos θ K Ã0 distribution which highlights the rich angular structure of the Kπ system. The resonant structures of the Kπ system can be also seen in the 2D distributions shown in Fig. 5. The plot on the right illustrates how the structures present in the Kπ system considerably influence the ψð2SÞπ system.

VI. EXTRACTION OF THE MOMENTS OF THE Kπ SYSTEM
Background-subtracted and efficiency-corrected data are subdivided in m Kπ bins of width 30 MeV=c 2 , which is suitable for observing the Kπ resonance structures. For each m Kπ bin, the cos θ K Ã0 distribution can be expressed as an expansion in terms of Legendre polynomials. The coefficients of this expansion contain all of the information on the angular structure of the system and characterize the spin of the contributing resonances. The angular distribution, after integration over the ψð2SÞ decay angles, can be written as where l max depends on the maximum orbital angular momentum necessary to describe the Kπ system, (1) are called unnormalized moments (moments, in the following) and can be calculated as integrals of the product of the corresponding Legendre polynomial and the cos θ K Ã0 distribution. Resonances of the Kπ system with spin s can contribute to the moments up to hP U 2s i. Interference between resonances with spin s 1 and s 2 can contribute to moments up to hP U s 1 þs 2 i. For large samples, the moments are determined from the data as where N reco is the number of reconstructed and selected candidates in the m Kπ bin. The superscript i labels the   candidate, W i signal is the weight which implements the sideband background subtraction, and ϵ i ¼ ϵðm i Kπ ; cos θ i K Ã0 ; cos θ i ψð2SÞ ; Δϕ i Kπ;μμ Þ is the efficiency correction, obtained as described in Sec. III.
The dependence of the first six moments on m Kπ is shown in Fig. 6. Together with moment hP U 0 i, represented in the left plot of Fig. 4, moments hP U 2 i and hP U 4 i show the S, P, and D wave amplitudes in the mass regions of the K Ã ð892Þ 0 , K Ã ð1410Þ 0 , K Ã 0 ð1430Þ 0 , and K Ã 2 ð1430Þ 0 resonances. The behavior of the moment hP U 6 i, generated by an F wave, shows that any contribution from K Ã 3 ð1780Þ 0 is small. A resonant ψð2SÞπ state would, in general, contribute to all Kπ moments.
A detailed discussion of these moments, together with the expressions relating moments to the amplitudes, can be found in Ref. [25] and references therein.

VII. ANALYSIS OF THE m ψð2SÞπ SPECTRUM
The reflection of the mass and angular structure of the Kπ system into the ψð2SÞπ invariant mass spectrum is investigated to establish whether it is sufficient to explain the data distribution. This is achieved by comparing the experimental m ψð2SÞπ spectrum to that of a simulated sample which accounts for the measured mass spectrum and the angular distribution of the Kπ system by means of appropriate weights. The comparison is performed in three  configurations of the Kπ spin contributions. The simplest configuration corresponds to including the contributions of S, P, and D waves, which account for all resonances with mass below the kinematic limit and the K Ã ð1680Þ 0 meson, just above it (see Table III). In the second configuration, the K Ã 3 ð1780Þ 0 meson is also allowed to contribute. This represents a rather unlikely assumption since it implies a sizeable presence of spin-3 resonances at low m Kπ . This configuration can be considered as an extreme case that provides a valuable test for the robustness of the method. In the third configuration, a more realistic choice is made by limiting the maximum spin as a function of m Kπ .
For each of the three configurations, 50 million simulated events are generated according to the B 0 → ψð2SÞK þ π − phase-space decay. The simulation does not include detector effects because it will be compared to efficiency-corrected data. The simulated m Kπ distribution is forced to reproduce the Kπ spectrum in the data (left plot of Fig. 4) by attributing to each event a weight proportional to the ratio between the real and simulated m Kπ spectra in the appropriate bin. Finally, the angular structure of the Kπ system is modified in the simulated sample by applying an additional weight to each event computed as where hP N j i ¼ 2hP U j i=N corr are the normalized moments, derived from the moments hP U j i of Eq. (2), and N corr is the background-subtracted and efficiency-corrected yield of the m Kπ bin where the event lies. The behavior of the first six normalized moments is shown in Fig. 7. The value and the uncertainty of these moments, at a given m Kπ value, are estimated by linearly interpolating adjacent points and their AE1σ values, respectively, as shown by the shaded (yellow) bands in the figures. The experimental distribution of the ψð2SÞπ system invariant mass, m ψð2SÞπ , is shown by the black points in the left plot of Fig. 8. 1 The dotted (black) line represents the pure phase-space simulation; the dash-dotted (red) line shows the effect of the m Kπ modulation; while in the continuous (blue) line, the angular structure of the Kπ system has been taken into account by allowing S, P, and D waves to contribute, which corresponds to setting l max ¼ 4 in Eq. This plot uses an improved parametrization of the B 0 mass spectrum with respect to Fig. 1 in Ref. [12]. uncertainty in the simulation but only as an indicative measure of the limited data sample used to compute moments. Since the band and the error bars on the black points are related to the same statistical uncertainty on the data, they should not be combined when estimating the statistical significance of deviations of the data from the prediction.
When spin-3 Kπ states are included, by setting l max ¼ 6, the predicted m ψð2SÞπ spectrum is modified as shown on the right plot of Fig. 8. Even though the l max ¼ 6 solution apparently provides a better description of the data, it is shown in the following that it is largely incompatible with the data.
In Fig. 9, the maximum Legendre polynomial order is limited as a function of m Kπ , according to m Kπ > 1000 MeV=c 2 : ð4Þ Figure 9 demonstrates that with this better-motivated l max assignment, the simulation cannot reproduce adequately the m ψð2SÞπ distribution. The disagreement is more evident when looking at the same spectra in different intervals of m Kπ , as shown in Fig. 10. Here, the candidates are subdivided according to the m Kπ intervals defined in Eq. (4). The last interval is further split into 1000 MeV=c 2 < m Kπ < 1390 MeV=c 2 and m Kπ > 1390 MeV=c 2 . Except for the mass region around 4430 MeV=c 2 , all slices exhibit good agreement between the data and the simulation. The peaking structure is particularly evident in the region 1000 MeV=c 2 < m Kπ < 1390 MeV=c 2 , between the K Ã ð892Þ 0 and the resonances above 1400 MeV=c 2 .  to the pure phase-space case; in the dash-dotted (red) line, the m Kπ spectrum is weighted to reproduce the experimental distribution; in the continuous (blue) line, the angular structure of the Kπ system is incorporated using Legendre polynomials with index l max variable according to m Kπ as described in Eq. (4), reaching up to l max ¼ 4. The shaded (yellow) bands are related to the uncertainty on normalized moments, which is due to the statistical uncertainty that comes from the data. Therefore, the two uncertainties should not be combined when comparing data and Monte Carlo predictions. See the text for further details.

VIII. STATISTICAL SIGNIFICANCE OF THE RESULT
The fact that the m ψð2SÞπ spectrum cannot be explained as a reflection of the angular structure of the Kπ system has been illustrated qualitatively. In this section, the disagreement is quantified via a hypothesis-testing procedure using a likelihood-ratio estimator. The compatibility between the expected m ψð2SÞπ distribution, accounting for the reflections of the Kπ angular structure, and that observed experimentally is tested for the three l max assignments described in the previous section, with three sets of about 1000 pseudoexperiments each. For each pseudoexperiment, data and simulated samples involved in the analysis chain are reproduced as pseudosamples, generated at the same statistical level as in the real case. The signal candidate pseudosamples are extracted from a ðm Kπ ; cos θ K Ã0 ; cos θ ψð2SÞ ; Δϕ Kπ;μμ Þ distribution obtained by an independent EvtGen [35] phase-space sample of B 0 → ψð2SÞK þ π − events. The distribution is generated, for each of the three l max cases previously discussed, in order to reproduce the (m Kπ ,cos θ K Ã0 ) behavior. The background pseudosamples are simulated according to the ðm Kπ ; cos θ K Ã0 ; cos θ ψð2SÞ ; Δϕ Kπ;μμ Þ distribution of the candidates in the B 0 invariant mass sidebands. Finally, to mimic the calculation of the efficiency correction factors, two additional samples are generated by extracting events from two distributions, in the same 4D space, obtained from the simulation with full detector effects, before and after the application of the analysis chain. The sum of the signal and background samples is then subject to background subtraction and efficiency correction, exactly as for the real data, and moments are calculated. In the pseudoexperiments, events are simulated with equal amounts of each ψð2SÞ polarization state. Effects related to ψð2SÞ polarization are only included in the pseudosample via their correlation with the Kπ mass and cos θ K Ã0 distributions, which are derived from the data. It has been checked that this does not significantly influence the results although the validity of such approximate treatment of ψð2SÞ polarization is, in general, analysis dependent and not necessarily appropriate in other experimental situations.
The Monte Carlo method described in Sec. VII is used for each pseudoexperiment to produce an m ψð2SÞπ probability density function, F l max , for each of the three l max configurations. To test for the presence of possible contributions from the ψð2SÞπ dynamics, which are expected to be present in moments of all orders, a fourth configuration is introduced by setting l max to the unphysically large value of 30. By including moments up to l max ¼ 30, most of the features of the m ψð2SÞπ spectrum in data are well described, as can be seen in Fig. 11. The logarithm of the likelihood ratio is used to define the test statistic The dash-dotted (red) line is obtained by modifying the m Kπ spectrum of the phase-space simulation according to the m Kπ experimental spectrum. In the continuous (blue) line, the angular structure of the Kπ system is incorporated using Legendre polynomials with variable index l max according to Eq. (4). The shaded (yellow) bands are related to the uncertainty on normalized moments, which is due to the statistical uncertainty that comes from the data. Therefore, the two uncertainties should not be combined when comparing data and Monte Carlo predictions. See the text for further details.
where the sum runs over the events in the pseudo-or real experiments. An exotic state in the ψð2SÞπ system would give contributions to all Kπ Legendre polynomial moments, whereas the conventional Kπ resonances contribute only to moments corresponding to their spin and their interferences. If, for instance, the B 0 → ψð2SÞK þ π − decay proceeds through S, P, and D Kπ resonances, then only moments with l max ≤ 4 would exhibit significant activity. Therefore, activity in moments of order l max > 4 would suggest the presence of other resonant states contributing to the decay. Lower-order Kπ Legendre polynomial moments, determined from data and used to build the prediction, although strongly dominated by the conventional Kπ resonances, could also contain a contribution from the exotic state. As a consequence, a relatively small ψð2SÞπ resonant contribution could be accommodated by the Monte Carlo prediction. Conversely, a significant disagreement would imply that the ψð2SÞπ invariant mass spectrum cannot be explained as a reflection of the activity of known resonances in the Kπ system and would therefore constitute strong evidence for the presence of exotic states in the The ΔNLL l max distributions of the pseudoexperiments are shown in Fig. 12 (points with error bars) for each of the three l max settings. They are consistent with Gaussian distributions. The statistical significance, S, to rule out the different hypotheses is the distance, in units of standard deviations, between the mean value of the ΔNLL l max (dashed red arrow in Fig. 12) and the observed value of the real experiment (continuous black arrow in Fig. 12). This ranges from 8 to 15 standard deviations, as listed in Table IV.  The table also gives the statistical significance obtained by restricting the analysis to the region 1000 MeV=c 2 < m Kπ < 1390 MeV=c 2 , where the presence of the structure around the Zð4430Þ − mass is most evident, as shown in Fig. 13. Thus, the hypothesis that the data can be explained solely in terms of plausible Kπ degrees of freedom can be  IV. Significance, S, in units of standard deviations, at which the hypothesis that m ψð2SÞπ data can be described as a reflection of the Kπ system angular structure is excluded, for different configurations of the Kπ system angular contributions. In the second column, the whole m Kπ spectrum has been analyzed, while in the third one, the specified m Kπ cut is applied. ruled out without making any assumption on the exact shapes of the Kπ resonances present and their interference patterns.

IX. SUMMARY AND CONCLUSIONS
A satisfactory description of the ψð2SÞπ mass spectrum in the decay B 0 → ψð2SÞK þ π − cannot be obtained solely from the reflections of the angular structure of the Kπ system. In particular, a clear peaking structure in the 4430 MeV=c 2 mass region remains unexplained. Through a hypothesis-testing procedure based on the likelihood-ratio estimator, compatibility between the data and predictions taking into account the reflections of Kπ states up to spin 3, is excluded with a significance exceeding 8σ. The most plausible configuration, which allows Kπ states with spin values depending on the Kπ mass, is excluded with a significance of more than 15σ.
This work represents an alternative and modelindependent confirmation of the existence of a ψð2SÞπ resonance in the same mass region in which previous model-dependent amplitude analyses have found signals [9,11,12].