Quantum numbers of the $X(3872)$ state and orbital angular momentum in its $\rho^0 J\psi$ decay

Angular correlations in $B^+\to X(3872) K^+$ decays, with $X(3872)\to \rho^0 J/\psi$, $\rho^0\to\pi^+\pi^-$ and $J/\psi \to\mu^+\mu^-$, are used to measure orbital angular momentum contributions and to determine the $J^{PC}$ value of the $X(3872)$ meson. The data correspond to an integrated luminosity of 3.0 fb$^{-1}$ of proton-proton collisions collected with the LHCb detector. This determination, for the first time performed without assuming a value for the orbital angular momentum, confirms the quantum numbers to be $J^{PC}=1^{++}$. The $X(3872)$ is found to decay predominantly through S wave and an upper limit of $4\%$ at $95\%$ C.L. is set on the fraction of D wave.

The X(3872) state was discovered in B +,0 → X(3872)K +,0 , X(3872) → π + π − J/ψ , J/ψ → + − decays by the Belle experiment [1] and subsequently confirmed by other experiments [2][3][4]. 1 Its production was also studied at the LHC [5,6]. However, the nature of this state remains unclear. The X(3872) state is narrow, has a mass very close to the D 0 D * 0 threshold and decays to ρ 0 J/ψ and ωJ/ψ final states with comparable branching fractions [7], thus violating isospin symmetry. This suggests that the X(3872) particle may not be a simple cc state, and exotic states such as D 0 D * 0 molecules [8], tetraquarks [9] or mixtures of states [10] have been proposed to explain its composition. The X(3872) quantum numbers, such as total angular momentum J, parity P , and charge-conjugation C, impose constraints on the theoretical models of this state. The orbital angular momentum L in the X(3872) decay may also provide information on its internal structure.
The choices for J P C were narrowed down to two possibilities, 1 ++ or 2 −+ , by the CDF collaboration, via an analysis of the angular correlations in inclusively reconstructed X(3872) → π + π − J/ψ and J/ψ → µ + µ − decays, dominated by prompt production in pp collisions [16]. Using 1.0 fb −1 of pp collision data collected by LHCb, J P C = 2 −+ was ruled out in favor of the 1 ++ assignment, using the angular correlations in the same decay chain, with the X(3872) state produced in B + → X(3872)K + decays [17]. Both angular analyses assumed that the lowest orbital angular momentum between the X(3872) decay products (L min ) dominated the matrix element. Significant contributions from L min + 2 amplitudes could invalidate the 1 ++ assignment. Since the phase-space limit on M (π + π − ) is close to the ρ 0 pole (775.3 ± 0.3 MeV [7]), the energy release in the X(3872) decay, Q ≡ M (J/ψ π + π − ) − M (J/ψ ) − M (π + π − ), is a small fraction of the X(3872) mass, making the orbital angular momentum barrier effective. 3 However, an exotic component in X(3872) could induce contributions from higher orbital angular momentum for models in which the size of the X(3872) state is substantially larger than the compact sizes of the charmonium states. Therefore, it is important to probe the X(3872) spin-parity without any assumptions about L. A determination of the magnitude of contributions from L min + 2 amplitudes for the correct J P C is also of interest, since a substantial value would suggest an anomalously large size of the X(3872) state. In this article, we extend our previous analysis [17] of five-dimensional angular correlations in B + → X(3872)K + , X(3872) → ρ 0 J/ψ , ρ 0 → π + π − , J/ψ → µ + µ − decays to accomplish these goals. The integrated luminosity of the data sample has been tripled by adding 8 TeV pp collision data collected in 2012.
1 The inclusion of charge-conjugate states is implied in this article. 2 We use mass and momentum units in which c = 1. 3 Dimuon candidates are constrained to the known J/ψ mass [7]. The LHCb detector is a single-arm forward spectrometer covering the pseudorapidity range 2 < η < 5, described in detail in Ref. [18,19]. The X(3872) candidate selection, which is based on reconstructing B + → (J/ψ → µ + µ − )π + π − K + candidates using particle identification information, transverse momentum (p T ) thresholds and requiring separation of tracks and the B + vertex from the primary pp interaction vertex, is improved relative to that of Ref. [17]. The signal efficiency is increased by lowering requirements on p T for muons from 0.90 to 0.55 GeV and for hadrons from 0.25 to 0.20 GeV. The background is further suppressed without significant loss of signal by requiring Q < 250 MeV. The X(3872) mass resolution (σ ∆M ) is improved from about 5.5 MeV to 2.8 MeV by constraining the B + candidate to its known mass and requiring its momentum to point to a pp collision vertex in the kinematic fit of its decay. The distribution of ∆M ≡ M (π + π − J/ψ ) − M (J/ψ ) is shown in Fig. 1. A Crystal Ball function [20] with symmetric tails is used to model the signal shape, while the background is assumed to be linear. An unbinned maximum likelihood fit yields 1011 ± 38 B + → X(3872)K + decays and 1468 ± 44 background entries in the 725 < ∆M < 825 MeV range used in the angular analysis. The signal purity is 80% within 2.5 σ ∆M from the signal peak. From studying the K + π + π − mass distribution, the dominant source of background is found to be B + → J/ψ K 1 (1270) + , Angular correlations in the B + decay chain are analyzed using an unbinned maximumlikelihood fit to determine the X(3872) quantum numbers and orbital angular momentum in its decay. The probability density function (P) for each J P C hypothesis, J X , is defined in the five-dimensional angular space Ω ≡ (cos θ X , cos θ ρ , ∆φ X,ρ , cos θ J/ψ , ∆φ X,J/ψ ), where θ X , θ ρ and θ J/ψ are the helicity angles [21][22][23] in X(3872), ρ 0 and J/ψ decays, respectively, and ∆φ X,ρ , ∆φ X,J/ψ are the angles between the decay planes of the X(3872) particle and of its decay products. The quantity P is the normalized product of the expected decay matrix element (M) squared and of the reconstruction efficiency ( ), P(Ω|J X ) = |M(Ω|J X )| 2 (Ω)/I(J X ), where I(J X ) = |M(Ω|J X )| 2 (Ω)d Ω. The efficiency is averaged over the π + π − mass using a simulation [24][25][26][27][28] of the X(3872) → ρ 0 J/ψ , ρ 0 → π + π − decay. The lineshape of the ρ 0 resonance can change slightly depending on the X(3872) spin hypothesis. The effect on (Ω) is very small and is neglected. The angular correlations are obtained using the helicity formalism [16], where λ are particle helicities, ∆λ µ = λ µ + − λ µ − , and D J λ 1 , λ 2 are Wigner functions [21][22][23]. The helicity couplings, A λ J/ψ ,λρ , are expressed in terms of the LS couplings, B LS , with help of Clebsch-Gordan coefficients, where L is the orbital angular momentum between the ρ 0 and the J/ψ mesons, and S is the sum of their spins, Possible values of L are constrained by parity conservation, P X = P J/ψ P ρ (−1) L = (−1) L .
In the previous analyses [14,16,17], only the minimal value of the angular momentum, L min , was allowed. Thus, for the preferred J P C = 1 ++ hypothesis, the D wave was neglected allowing only S-wave decays. In this work all L values are allowed in Eq.
(2). The corresponding B LS amplitudes are listed in Table 1. Values of J X up to four are analyzed.
Since the orbital angular momentum in the B + decay equals J X , high values are suppressed by the angular momentum barrier. In fact, the highest observed spin of any resonance produced in B decays is three [29,30]. Since P is insensitive to the overall normalization of the B LS couplings and to the phase of the matrix element, the B LS amplitude with the lowest L and S is set to the arbitrary reference value (1, 0). The set of other possible complex B LS amplitudes, which are free parameters in the fit, is denoted as α.
is the unbinned likelihood, and N data is the number of selected candidates. The background is subtracted using the sPlot technique [31] by assigning a weight, w i , to each candidate based on its ∆M value (see Fig. 1). No correlations between ∆M and Ω are observed. Prompt production of X(3872) in pp collisions gives negligible contribution to the selected sample. Statistical fluctuations in the background subtraction are taken into Table 1: Parity-allowed LS couplings in the X(3872) → ρ 0 J/ψ decay. For comparison, we also list a subset of these couplings corresponding to the lowest L, used in the previous determinations [14,16,17] of the X(3872) quantum numbers.
The efficiency (Ω) is not determined on an event-by-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, where N MC is the number of reconstructed simulated events. The negative log-likelihood is minimized for each J X value with respect to free B LS couplings, yielding their estimated set of valuesα. Hereinafter, L(J X ) ≡ L(J X ,α).
The 1 ++ hypothesis gives the highest likelihood value. From angular momentum and parity conservation, there are two possible values of orbital angular momentum in the X(3872) decay for this J P C value, L = 0 or 2. For the S-wave decay, the total spin of the ρ 0 and J/ψ mesons must be S = 1; thus B 01 is the only possible LS amplitude. For the D-wave decay, two values are possible, S = 1 or 2, corresponding to the amplitudes B 21 and B 22 , respectively. The squared magnitudes of both of these D-wave amplitudes are consistent with zero, as demonstrated by the ratios |B 21 | 2 /|B 01 | 2 = 0.002 ± 0.004 and |B 22 | 2 /|B 01 | 2 = 0.007 ± 0.008. Overall, the D-wave significance is only 0. integration of the matrix element squared. The distribution of f D values weighted by the likelihood values is shown in Fig. 2. It peaks at 0.4% with a non-Gaussian tail at higher values. An upper limit of f D < 4% at 95% C.L. is determined using a Bayesian approach. The likelihood ratio t ≡ −2 ln[L(J alt X )/L(1 ++ )] is used as a test variable to discriminate between the 1 ++ and alternative spin hypotheses considered (J alt X ). The values of t in the data (t data ) are positive, favoring the 1 ++ assignment. They are incompatible with the distributions of t observed in experiments simulated under various J alt X hypotheses, as illustrated in Fig. 3. To quantify these disagreements we calculate the approximate significance of rejection (p-value) of J alt X as (t data − t )/σ(t), where t and σ(t) are the mean and rms deviations of the t distribution under the J alt X hypothesis. In all spin configurations tested, we exclude the alternative spin hypothesis with a significance of more than 16 standard deviations. Values of t in data are consistent with those expected in the 1 ++ case as shown in Fig. 3, with fractions of simulated 1 ++ experiments with t < t data in the 25%-91% range. Projections of the data and of the fit P onto individual angles show good consistency with the 1 ++ assignment. Inconsistency with the other assignments is apparent when correlations between various angles are exploited. For example, the data projection onto cos θ X is consistent only with the 1 ++ fit projection after requiring | cos θ ρ | > 0.6 (see Fig. 4), while inconsistency with the other quantum number assignments is less clear without the cos θ ρ requirement. The selection criteria are varied to probe for possible biases from the background subtraction and the efficiency corrections. By requiring Q < 0.1 GeV, the background level is reduced by more than a factor of two, while losing only 20% of the signal. By tightening the requirements on the p T of π, K and µ candidates, we decrease the signal efficiency by around 75% with similar reduction in the background level. In all cases, the significance of the rejection of the disfavored hypotheses is compatible with that expected from the simulation. Likewise, the best fit f D values determined for these subsamples of data change within the expected statistical fluctuations and remain consistent with the upper limit we have set.
In summary, the analysis of the angular correlations in B + → X(3872)K + , X(3872) → π + π − J/ψ , J/ψ → µ + µ − decays, performed for the first time without any assumption about the orbital angular momentum in the X(3872) decay, confirms that the eigenvalues of total angular momentum, parity and charge-conjugation of the X(3872) state are 1 ++ . These quantum numbers are consistent with those predicted by the molecular or tetraquark models and with the χ c1 (2 3 P 1 ) charmonium state [32]. Other charmonium states are excluded. No significant D-wave fraction is found, with an upper limit of 4% at 95% C.L. The S-wave dominance is expected in the charmonium or tetraquark models, in which the X(3872) state has a compact size. An extended size, as that predicted by the molecular model, implies more favorable conditions for the D wave. However, conclusive discrimination among models is difficult because quantitative predictions are not available.
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:    Figure 4: Background-subtracted distribution of cos θ X for candidates with | cos θ ρ | > 0.6 for the data (points with error bars) compared to the expected distributions for various X(3872) J P C assignments (solid histograms) with the B LS amplitudes obtained by the fit to the data in the five-dimensional angular space. The fit displays are normalized to the observed number of the signal events in the full angular phase space. arXiv:hep-ex/0312021.