Determination of the X(3872) meson quantum numbers

The quantum numbers of the X(3872) meson are determined to be JPC = 1++ based on angular correlations in B+ to X(3872) K+ decays, where X(3872) to pi+pi- J/psi and J/psi to \mu+\mu-. The data correspond to 1.0 fb-1 of pp collisions collected by the LHCb detector. The only alternative assignment allowed by previous measurements, JPC=2-+, is rejected with a confidence level equivalent to more than eight Gaussian standard deviations using the likelihood-ratio test in the full angular phase space. This result favors exotic explanations of the X(3872) state.

It has been almost ten years since the narrow Xð3872Þ state was discovered in B þ decays by the Belle experiment [1,2]. Subsequently, its existence has been confirmed by several other experiments [3][4][5]. Recently, its production has been studied at the LHC [6,7]. However, the nature of this state remains unclear. Among the open possibilities are conventional charmonium and exotic states such as D Ã0 " D 0 molecules [8], tetraquarks [9], or their mixtures [10]. Determination of the quantum numbers, total angular momentum J, parity P, and charge-conjugation C, is important to shed light on this ambiguity. The C parity of the state is positive since the Xð3872Þ ! J=c decay has been observed [11,12].
The CDF experiment analyzed three-dimensional (3D) angular correlations in a relatively high-background sample of 2292 AE 113 inclusively reconstructed Xð3872Þ ! þ À J=c , J=c ! þ À decays dominated by prompt production in p " p collisions. The unknown polarization of the Xð3872Þ mesons limited the sensitivity of the measurement of J PC [13]. A 2 fit of J PC hypotheses to the binned 3D distribution of the J=c and helicity angles ( J=c , ) [14][15][16] and the angle between their decay planes (Á J=c ; ¼ J=c À ) excluded all spinparity assignments except for 1 þþ or 2 Àþ . The Belle Collaboration observed 173 AE 16 B ! Xð3872ÞK (K ¼ K AE or K 0 S ), Xð3872Þ ! þ À J=c , J=c ! ' þ ' À decays [17]. The reconstruction of the full decay chain resulted in a small background and polarized Xð3872Þ mesons, making their helicity angle ( X ) and orientation of their decay plane ( X ) sensitive to J PC as well. By studying onedimensional distributions in three different angles without exploiting correlations, they concluded that their data were equally well described by the 1 þþ and 2 Àþ hypotheses.
In this Letter, we report the first analysis of the complete five-dimensional angular correlations of the B þ ! Xð3872ÞK þ , Xð3872Þ ! þ À J=c , J=c ! þ À decay chain using ffiffi ffi s p ¼ 7 TeV pp collision data corresponding to 1:0 fb À1 collected in 2011 by the LHCb experiment. The LHCb detector [19] is a single-arm forward spectrometer covering the pseudorapidity range 2 < < 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, 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 placed downstream. The combined tracking system has momentum resolution Áp=p that varies from 0.4% at 5 GeV to 0.6% at 100 GeV, and impact parameter resolution of 20 m for tracks with high transverse momentum (p T ) [20]. Charged hadrons are identified using two ring-imaging Cherenkov detectors. Photon, electron, and hadron candidates 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. The trigger [21] 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.
In the off-line analysis, J=c ! þ À candidates are selected with the following criteria: p T ðÞ > 0:9 GeV, p T ðJ=c Þ > 1:5 GeV, 2 per degree of freedom for the two muons to form a common vertex, 2 vtx ð þ À Þ=ndf < 9, and a mass consistent with the J=c meson. The separation of the J=c decay vertex from the nearest primary vertex (PV) must be at least 3 standard deviations.
Combinations of K þ À þ candidates that are consistent with originating from a common vertex with 2 vtx ðK þ À þ Þ=ndf < 9, with each charged hadron (h) separated from all PVs [ 2 IP ðhÞ > 9] and having p T ðhÞ > 0:25 GeV, are selected. The quantity 2 IP ðhÞ is defined as the difference between the 2 of the PV reconstructed with and without the considered particle. Kaon and pion candidates are required to satisfy ln½LðKÞ=LðÞ > 0 and <5, respectively, where L is the particle identification likelihood [22]. If both same-sign hadrons in this combination meet the kaon requirement, only the particle with higher p T is considered a kaon candidate. We combine J=c candidates with K þ À þ candidates to form B þ candidates, which must satisfy 2 vtx ðJ=c K þ À þ Þ=ndf < 9, p T ðB þ Þ > 2 GeV and have decay time greater than 0.25 ps. The J=c K þ À þ mass is calculated using the known J=c mass and the B vertex as constraints.
Four discriminating variables (x i ) are used in a likelihood ratio to improve the background suppression: the minimal 2 IP ðhÞ, 2 vtx ðJ=c K þ þ À Þ=ndf, 2 IP ðB þ Þ, and the cosine of the largest opening angle between the J=c and the charged-hadron transverse momenta. The latter peaks at positive values for the signal, as the B þ meson has a high transverse momentum. Background events in which particles are combined from two different B decays peak at negative values, while those due to random combinations of particles are more uniformly distributed. The four 1D signal probability density functions (PDFs) P sig ðx i Þ are obtained from a simulated sample of B þ ! c ð2SÞK þ , c ð2SÞ ! þ À J=c decays, which are kinematically similar to the signal decays. The data sample of B þ ! c ð2SÞK þ events is used as a control sample for P sig ðx i Þ and for systematic studies in the angular analysis. The background PDFs P bkg ðx i Þ are obtained from the data in the B þ mass sidebands (4.85-5.10 and 5.45-6.50 GeV). We require À2 P 4 i¼1 ln½P sig ðx i Þ=P bkg ðx i Þ < 1:0, which preserves about 94% of the Xð3872Þ signal events.
About 38000 candidates are selected in a AE2 mass range around the B þ peak in the MðJ=c þ À K þ Þ distribution, with a signal purity of 89%. The ÁM ¼ Mð þ À J=c Þ À MðJ=c Þ distribution is shown in Fig. 1. Fits to the c ð2SÞ and Xð3872Þ signals are shown in the insets. A Crystal Ball function [23] with symmetric tails is used for the signal shapes. The background is assumed to be linear. The c ð2SÞ fit is performed in the 539.2-639.2 MeV range leaving all parameters free to vary. It yields 5642 AE 76 signal (230 AE 21 background) candidates with a ÁM resolution of ÁM ¼ 3:99 AE 0:05 MeV, corresponding to a signal purity of 99.2% within a AE2:5 ÁM region. When fitting in the 723-823 MeV range, the signal tail parameters are fixed to the values obtained in the c ð2SÞ fit, which also describe well the simulated Xð3872Þ signal distribution. The fit yields 313 AE 26 B þ ! Xð3872ÞK þ candidates with a resolution of 5:5 AE 0:5 MeV. The number of background candidates is 568 AE 31 including the sideband regions. The signal purity is 68% within a AE2:5 ÁM signal region. The dominant source of background is from B þ ! J=c K 1 ð1270Þ þ , K 1 ð1270Þ þ ! K þ þ À decays, as found by studying the K þ þ À mass distribution.
The angular correlations in the B þ decay carry information about the Xð3872Þ quantum numbers. To discriminate between the 1 þþ and 2 Àþ assignments, we use a likelihood-ratio test, which in general provides the most powerful discrimination between two hypotheses [24]. The PDF for each J PC hypothesis J X is defined in the 5D angular space ðcos X ; cos ; Á X; ; cos J=c ; Á X;J=c Þ by the normalized product of the expected decay matrix element (M) squared and of the reconstruction efficiency (), P ðjJ X Þ ¼ jMðjJ X Þj 2 ðÞ=IðJ X Þ, where IðJ X Þ ¼ R jMðjJ X Þj 2 ðÞd. The efficiency is averaged over the þ À mass [MðÞ] using a simulation [25][26][27][28][29] that assumes the Xð3872Þ ! ð770ÞJ=c , ð770Þ ! þ À decay [7,17,30]. The observed MðÞ distribution is in good agreement with this simulation. The line shape of the ð770Þ resonance can change slightly depending on the spin hypothesis. The effect on ðÞ is found to be very small and is neglected. We follow the approach adopted in Ref. [13] to predict the matrix elements. The angular correlations are obtained using the helicity formalism, 222001-2 where are particle helicities and D J 1 ; 2 are Wigner functions [14][15][16]. The helicity couplings A J=c ; are expressed in terms of the LS couplings [31,32], B LS , where L is the orbital angular momentum between the system and the J=c meson, and S is the sum of their spins. Since the energy release in the Xð3872Þ ! ð770ÞJ=c decay is small, the lowest value of L is expected to dominate, especially because the next-to-minimal value is not allowed by parity conservation. The lowest value for the 1 þþ hypothesis is L ¼ 0, which implies S ¼ 1. With only one LS amplitude present, the angular distribution is completely determined without free parameters. For the 2 Àþ hypothesis, the lowest value is L ¼ 1, which implies S ¼ 1 or 2. As both LS combinations are possible, the 2 Àþ hypothesis implies two parameters, which are chosen to be the real and imaginary parts of B 11 =ðB 11 þ B 12 Þ. Since they are related to strong dynamics, they are difficult to predict theoretically and are treated as nuisance parameters.
We define a test statistic t ¼ À2 ln½Lð2 Àþ Þ=Lð1 þþ Þ, where the Lð2 Àþ Þ likelihood is maximized with respect to . The efficiency ðÞ is not determined on an eventby-event basis, since it cancels in the likelihood ratio except for the normalization integrals. A large sample of simulated events with uniform angular distributions passed through a full simulation of the detection and the data selection process is used to carry out the integration, IðJ X Þ / P N MC i¼1 jMð i jJ X Þj 2 , where N MC is the number of reconstructed simulated events. The background in the data is subtracted in the log likelihoods using the sPlot technique [33] by assigning to each candidate in the fitted ÁM range an event weight (sWeight) w i based on its ÁM value, À2 lnLðJ X Þ ¼ Às w 2 P N data i¼1 w i lnP ð i jJ X Þ. Here, s w is a constant scaling factor, s w ¼ P N data i¼1 w i = P N data i¼1 w 2 i , which accounts for statistical fluctuations in the background subtraction. Positive (negative) values of the test statistic for the data t data favor the 1 þþ (2 Àþ ) hypothesis. The analysis procedure has been extensively tested on simulated samples for the 1 þþ and 2 Àþ hypotheses with different values of generated using the EVTGEN package [27].
The value of that minimizes À2 lnLðJ X ¼ 2 Àþ ; Þ in the data is ¼ ð0:671 AE 0:046; 0:280 AE 0:046Þ. This is compatible with the value reported by Belle, (0.64,0.27) [17]. The value of the test statistic observed in the data is t data ¼ þ99, thus favoring the 1 þþ hypothesis. Furthermore, is consistent with the value of obtained from fitting a large background-free sample of simulated 1 þþ events, (0:650 AE 0:011, 0:294 AE 0:012). The value of t data is compared with the distribution of t in the simulated experiments to determine a p value for the 2 Àþ hypothesis via the fraction of simulated experiments yielding a value of t > t data . We simulate 2 million experiments with the value of , and the number of signal and background events, as observed in the data. The background is assumed to be saturated by the B þ ! J=c K 1 ð1270Þ þ decay, which provides a good description of its angular correlations. None of the values of t from the simulated experiments even approach t data , indicating a p value smaller than 1=ð2 Â 10 6 Þ, which corresponds to a rejection of the 2 Àþ hypothesis with greater than 5 significance. As shown in Fig. 2, the distribution of t is reasonably well approximated by a Gaussian function. Based on the mean and rms spread of the t distribution for the 2 Àþ experiments, this hypothesis is rejected with a significance of 8:4. The deviations of the t distribution from the Gaussian function suggest this is a plausible estimate. Using phase-space B þ ! J=c K þ þ À decays as a model for the background events, we obtain a consistent result. The value of t data falls into the region where the probability density for the 1 þþ simulated experiments is high. Integrating the 1 þþ distribution from À1 to t data gives C:L:ð1 þþ Þ ¼ 34%.
The value of t is the sum of the single-event likelihood ratios ln½P ð i j2 Àþ ;Þ=P ð i j1 þþ Þ over the analyzed data sample and is therefore proportional to its average value. Even though this is the most effective way to discriminate between the two hypotheses, the agreement with the 1 þþ hypothesis might have been coincidental if the data were inconsistent with both tested hypotheses. However, the full shape of the single-event likelihood-ratio distribution also shows good consistency between the data and the distribution expected for the 1 þþ case, as illustrated in Fig. 3.
We vary the data selection criteria to probe for possible biases from the background subtraction and the efficiency corrections. The nominal selection does not bias the MðÞ distribution. By requiring Q ¼ MðJ=c Þ À MðJ=c Þ À MðÞ < 0:1 GeV, we reduce the background level by a factor of 4, while losing only 21% of the signal. The significance of the 2 Àþ rejection changes very little, in agreement with the simulations. By tightening the  2 (color online). Distribution of the test statistic t for the simulated experiments with J PC ¼ 2 Àþ and ¼ (black circles on the left) and with J PC ¼ 1 þþ (red triangles on the right). A Gaussian fit to the 2 Àþ distribution is overlaid (blue solid line). The value of the test statistic for the data t data is shown by the solid vertical line. PRL 110, 222001 (2013) P H Y S I C A L R E V I E W L E T T E R S week ending 31 MAY 2013 222001-3 requirements on the p T of , K, and candidates, we decrease the signal efficiency by about 50% with similar reduction in the background level. In all cases, the significance of the 2 Àþ rejection is reduced by a factor consistent with the simulations.
In the analysis we use simulations to calculate the IðJ X Þ integrals. In an alternative approach to the efficiency estimates, we use the B þ ! c ð2SÞK þ events observed in the data weighted by the inverse of 1 ÀÀ matrix element squared. We obtain a value of t data that corresponds to 8:2 rejection of the 2 Àþ hypothesis.
As an additional goodness-of-fit test for the 1 þþ hypothesis, we project the data onto five 1D and ten 2D binned distributions in all five angles and their combinations. They are all consistent with the distributions expected for the 1 þþ hypothesis. Some of them are inconsistent with the distributions expected for the (2 Àþ ,) hypothesis. The most significant inconsistency is observed for the 2D projections onto cos X vs cos . The separation between the 1 þþ and 2 Àþ hypotheses increases when using correlations between these two angles, as illustrated in Fig. 4.
In summary, we unambiguously establish that the values of total angular momentum, parity, and charge-conjugation eigenvalues of the Xð3872Þ state are 1 þþ . This is achieved through the first analysis of the full five-dimensional angular correlations between final state particles in B þ ! Xð3872ÞK þ , Xð3872Þ ! þ À J=c , J=c ! þ À decays using a likelihood-ratio test. The 2 Àþ hypothesis is excluded with a significance of more than 8 Gaussian standard deviations. This result rules out the explanation of the Xð3872Þ meson as a conventional c2 ð1 1 D 2 Þ state.
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 following national agencies: CAPES, CNPq,  3 (color online). Distribution of À ln½P ð i j2 Àþ ;Þ= P ð i j1 þþ Þ for the data (points with error bars) compared to the distributions for the simulated experiments with J PC ¼ 1 þþ (red solid histogram) and with J PC ¼ 2 Àþ , ¼ (blue dashed histogram) after the background subtraction using sWeights. The simulated distributions are normalized to the number of signal candidates observed in the data. Bin contents and uncertainties are divided by bin width because of unequal bin sizes. and GridPP (United Kingdom). We are thankful for the computing resources put at our disposal by Yandex LLC (Russia), as well as to the communities behind the multiple open source software packages that we depend on.