Measurement of form-factor independent observables in the decay $B^{0} \to K^{*0} \mu^+ \mu^-$

We present a measurement of form-factor independent angular observables in the decay \mbox{$B^0\to K^{*}(892)^{0}\mu^+ \mu^-$}. The analysis is based on a data sample corresponding to an integrated luminosity of 1.0fb$^{-1}$, collected by the LHCb experiment in $pp$ collisions at a center-of-mass energy of 7TeV. Four observables are measured in six bins of the dimuon invariant mass squared, $q^2$, in the range $0.1<q^2<19.0$GeV$^{2}$/c$^{4}$. Agreement with Standard Model predictions is found for 23 of the 24 measurements. A local discrepancy, corresponding to $3.7$ Gaussian standard deviations, is observed in one $q^2$ bin for one of the observables. Considering the 24 measurements as independent, the probability to observe such a discrepancy, or larger, in one is $0.5\%$.

The rare decay B 0 → K * 0 µ + µ − , where K * 0 indicates the K * (892) 0 → K + π − decay, is a flavor-changing neutral current process that proceeds via loop and box amplitudes in the Standard Model (SM). In extensions of the SM, contributions from new particles can enter in competing amplitudes and modify the angular distributions of the decay products. This decay has been widely studied from both theoretical [1][2][3] and experimental [4][5][6][7] perspectives. Its angular distribution is described by three angles (θ , θ K and φ) and the dimuon invariant mass squared, q 2 ; θ is the angle between the flight direction of the µ + (µ − ) and the B 0 (B 0 ) meson in the dimuon rest frame; θ K is the angle between the flight direction of the charged kaon and the B 0 (B 0 ) meson in the K * 0 (K * 0 ) rest frame; and φ is the angle between the decay planes of the K * 0 (K * 0 ) and the dimuon system in the B 0 (B 0 ) meson rest frame. A formal definition of the angles can be found in Ref. [7]. Using the definitions of Ref. [1] and summing over B 0 and B 0 mesons, the differential angular distribution can be written as − F L cos 2 θ K cos 2θ + S 3 sin 2 θ K sin 2 θ cos 2φ + S 4 sin 2θ K sin 2θ cos φ + S 5 sin 2θ K sin θ cos φ + S 6 sin 2 θ K cos θ + S 7 sin 2θ K sin θ sin φ + S 8 sin 2θ K sin 2θ sin φ + S 9 sin 2 θ K sin 2 θ sin 2φ , where the q 2 dependent observables F L and S i are bilinear combinations of the K * 0 decay amplitudes. These in turn are functions of the Wilson coefficients, which contain information about short distance effects and are sensitive to physics beyond the SM, and formfactors, which depend on long distance effects. Combinations of F L and S i with reduced formfactor uncertainties have been proposed independently by several authors [2,3,[8][9][10]. In particular, in the large recoil limit (low-q 2 ) the observables denoted as P 4 , P 5 , P 6 and P 8 [11] are largely free from form-factor uncertainties. These observables are defined as This Letter presents the measurement of the observables S j and the respective observables P i . This is the first measurement of these quantities by any experiment. Moreover, these observables provide complementary information about physics beyond the SM with respect to the angular observables previously measured in this decay [4][5][6][7]. The data sample analyzed corresponds to an integrated luminosity of 1.0 fb −1 of pp collisions at a center-of-mass energy of 7 TeV collected by the LHCb experiment in 2011. Charged conjugation is implied throughout this Letter, unless otherwise stated.
The LHCb detector [12] 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, a largearea silicon-strip detector located upstream of a dipole magnet with a bending power of approximately 4 Tm, and three stations of siliconstrip detectors and straw drift tubes placed downstream of the magnet. The combined tracking system provides a momentum measurement with relative uncertainty that varies from 0.4% at 5 GeV/c to 0.6% at 100 GeV/c, and impact parameter resolution of 20 µm for tracks with high transverse momentum (p T ). Charged hadrons are identified using two ringimaging Cherenkov detectors [13]. Muons are identified by a system composed of alternating layers of iron and multiwire proportional chambers [14].
The trigger [15] 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. Candidate events for this analysis are required to pass a hardware trigger, which selects muons with p T > 1.48 GeV/c. In the software trigger, at least one of the final state particles is required to have both p T > 1.0 GeV/c and impact parameter larger than 100 µm with respect to all of the primary pp interaction vertices in the event. Finally, the tracks of two or more of the final state particles are required to form a vertex that is significantly displaced from the primary vertex.
Simulated events are used in several stages of the analysis, pp collisions are generated using Pythia 6.4 [16] with a specific LHCb configuration [17]. Decays of hadronic particles are described by EvtGen [18], in which final state radiation is generated using Photos [19]. The interaction of the generated particles with the detector and its response are implemented using the Geant4 toolkit [20] as described in Ref. [21]. This analysis uses the same selection and acceptance correction technique as described in Ref. [7].
Signal candidates are required to pass a loose preselection: the B 0 vertex is required to be well separated from the primary pp interaction point; the impact parameter with respect to the primary pp interaction point is required to be small for the B 0 candidate and large for the final state particles; and the angle between the B 0 momentum and the vector from the primary vertex to the B 0 decay vertex is required to be small. Finally, the reconstructed invariant mass of the K * 0 candidate is required to be in the range 792 < m Kπ < 992 MeV/c 2 . To further reject combinatorial background events, a boosted decision tree (BDT) [22] using the AdaBoost algorithm [23] is applied. The BDT combines kinematic and geometrical properties of the event.
Several sources of peaking background have been considered. The decays B 0 → J/ψ K * 0 and B 0 → ψ(2S)K * 0 , where the charmonium resonances decay into a muon pair, are rejected by vetoing events for which the dimuon system has an invariant mass (m µµ ) in the range 2946− 3176 MeV/c 2 or 3586 − 3766 MeV/c 2 . Both vetoes are extended downwards by 150 MeV/c 2 for B 0 candidates with invariant mass (m Kπµµ ) in the range 5150 − 5230 MeV/c 2 to account for the radiative tails of the charmonium resonances. They are also extended upwards by 25 MeV/c 2 for candidates with 5370 < m Kπµµ < 5470 MeV/c 2 , to account for non-Gaussian reconstruction effects. Backgrounds from B 0 → J/ψ K * 0 decays with the kaon or pion from the K * 0 decay and one of the muons from the J/ψ meson being misidentified and swapped with each other, are rejected by assigning the muon mass hypothesis to the K + or π − and vetoing candidates for which the resulting invariant mass is in the range 3036 < m µµ < 3156 MeV/c 2 . Background from B 0 s → φ(→ K + K − )µ + µ − decays is removed by assigning the kaon mass hypothesis to the pion candidate and rejecting events for which the resulting invariant mass K + K − is consistent with the φ mass. A similar veto is applied to remove Λ 0 b → Λ(1520)(→ pK − )µ + µ − events. After these vetoes, the remaining peaking background is estimated to be negligibly small. It has been verified with the simulation that these vetos do not bias the angular observables. In total, 883 signal candidates are observed in the range 0.1 < q 2 < 19.0 GeV 2 /c 4 , with a signal over background ratio of about 5.
Detector acceptance effects are accounted for by weighting the candidates with the inverse of their efficiency. The efficiency is determined as a function of the three angles and q 2 by using a large sample of simulated events and assuming factorization in the three angles. Possible nonfactorizable acceptance effects are evaluated and included in the systematic uncertainties. Several control channels, in particular the decay B 0 → J/ψ K * 0 , which has the same final state as the signal, are used to verify the agreement between data and simulation.
Due to the limited number of signal candidates in this dataset, we do not fit the data to the full differential distribution of Eq. 1. In Ref. [7], the data were "folded" at φ = 0 (φ → φ + π for φ < 0) to reduce the number of parameters in the fit, while cancelling the terms containing sin φ and cos φ. Here, similar folding techniques are applied to specific regions of the three-dimensional angular space to exploit the (anti)-symmetries of the differential decay rate with respect to combinations of angular variables. This simplifies the differential decay rate without losing experimental sensitivity. This technique is discussed in more detail in Ref. [24]. The following sets of transformations are used to determine the observables of interest P 4 , S 4 : Each transformation preserves the first five terms and the corresponding S i term in Eq. 1, and cancels the other angular terms. Thus, the resulting angular distributions depend only on F L , S 3 and one of the observables S 4,5,7,8 . Four independent likelihood fits to the B 0 invariant mass and the transformed angular distributions are performed to extract the observables P i and S i . The signal invariant mass shape is parametrized with the sum of two Crystal Ball functions [25], where the parameters are extracted from the fit to B 0 → J/ψ K * 0 decays in data. The background invariant mass shape is parametrized with an exponential function, while its angular distribution is parametrized with the direct product of three second-order polynomials, dependent on φ, cos θ K and cos θ . The angular observables F L and S 3 are allowed to vary in the angular fit and are treated as nuisance parameters in this analysis. Their fit values agree with Ref. [7].
The presence of a K + π − system in an Swave configuration, due to a non-resonant contribution or to feed-down from K + π − scalar resonances, results in additional terms in the differential angular distribution. Denoting the right-hand side of Eq. 1 by W P , the differential decay rate takes the form where and W SP is given by S sin θ K sin θ sin φ +A (8) S sin θ K sin 2θ sin φ .
The factor F S is the fraction of the S-wave component in the K * 0 mass window, and W SP contains all the interference terms, A (i) S , of the S-wave with the K * 0 transversity amplitudes as defined in Ref. [26]. In Ref. [7], F S was measured to be less than 0.07 at 68% confidence level. The maximum value that the quantities A (i) S can assume is a function of F S and F L [11]. The S-wave contribution is neglected in the fit to data, but its effect is evaluated and assigned as a systematic uncertainty using pseudo-experiments. A large number of pseudo-experiments with F S = 0.07 and with the interference terms set to their maximum allowed values are generated. All other parameters, including the angular observables, are set to their measured values in data. The pseudoexperiments are fitted ignoring S-wave and interference contributions. The corresponding bias in the measurement of the angular observables is assigned as a systematic uncertainty. The results of the angular fits to the data are presented in Table 1. The statistical uncertainties are determined using the Feldman-Cousins method [27]. The systematic uncertainty takes into account the limited knowledge of the angular acceptance, uncertainties in the signal and background invariant mass models, the angular model for the background, and the impact of a possible S-wave amplitude. Effects due to B 0 /B 0 production asymmetry have been considered and found negligibly small. The comparison between the measurements and the theoretical predictions from Ref. [11] are shown in Fig. 1 for the observables P 4 and P 5 . The observables P 6 and P 8 (as well as S 7 and S 8 ) are suppressed by the small size of the strong phase difference between the decay amplitudes, and therefore are expected to be close to zero Table 1: Measurement of the observables P 4,5,6,8 and S 4,5,7,8 in the six q 2 bins of the analysis. For the observables P i the measurement in the q 2 -bin 1.0 < q 2 < 6.0 GeV 2 /c 4 , which is the theoretically preferred region at large recoil, is also reported. The first uncertainty is statistical and the second is systematic. across the whole q 2 region. In general, the measurements agree with SM expectations [11], apart from a sizeable discrepancy in the interval 4.30 < q 2 < 8.68 GeV 2 /c 4 for the observable P 5 . The p-value, calculated using pseudo-experiments, with respect to the upper bound of the theoretical predictions given in Ref. [11], for the observed deviation is 0.02%, corresponding to 3.7 Gaussian standard deviations (σ). If we consider the 24 measurements as independent, the probability that at least one varies from the expected value by 3.7 σ or more is approximately 0.5%. A discrepancy of 2.5 σ is observed integrating over the region 1.0 < q 2 < 6.0 GeV 2 /c 4 (see Ta-ble 1), which is considered the most robust region for theoretical predictions at large recoil. The discrepancy is also observed in the observable S 5 . The value of S 5 quantifies the asymmetry between decays with positive and negative value of cos θ K for |φ| < π/2, averaged with the opposite asymmetry of events with |φ| > π/2 [1]. As a cross check, this asymmetry was also determined from a counting analysis. The result is consistent with the value for S 5 determined from the fit. It is worth noting that the predictions for the first two q 2 -bins and for the region 1.0 < q 2 < 6.0 GeV 2 /c 4 are also calculated in Ref. [28], where power corrections to the QCD factorization framework and reso-nance contributions are considered. However, there is not yet in the literature unanimous consensus about the best approach to treat these power corrections. The technique used in Ref. [28] leads to a larger theoretical uncertainty with respect to Ref. [11].
In conclusion, we measure for the first time the angular observables S 4 , S 5 , S 7 , S 8 and the corresponding form-factor independent observables P 4 , P 5 , P 6 and P 8 in the decay B 0 → K * 0 µ + µ − . These measurements have been performed in six q 2 bins for each of the four observables. Agreement with SM predictions [11] is observed for 23 of the 24 measurements, while a local discrepancy of 3.7 σ is observed in the interval 4.30 < q 2 < 8.68 GeV 2 /c 4 for the observable P 5 . Integrating over the region 1.0 < q 2 < 6.0 GeV 2 /c 4 , the observed discrepancy in P 5 is 2.5 σ. The observed discrepancy in the angular observable P 5 could be caused by a smaller value of the Wilson coefficient C 9 with respect to the SM, as has been suggested to explain some other small inconsistencies between the B 0 → K * 0 µ + µ − data [7] and SM predictions [29]. Measurements with more data and further theoretical studies will be important to draw more definitive conclusions about this discrepancy.