Observation of the decay B 0 s → ψ (2 S ) K + π −

The decay B 0 s → ψ( 2 S ) K + π − is observed using a data set corresponding to an integrated luminosity of 3 . 0 fb − 1 collected by the LHCb experiment in pp collisions at centre-of-mass energies of 7 and 8 TeV. The branching fraction relative to the B 0 → ψ( 2 S ) K + π − decay mode is measured to be where f s / f d indicates the uncertainty due to the ratio of probabilities for a b quark to hadronise into a B 0 s or B 0 meson. Using an amplitude analysis, the fraction of decays proceeding via an intermediate K ∗ ( 892 ) 0 meson is measured to be 0 . 645 ± 0 . 049 ( stat ) ± 0 . 049 ( syst ) and its longitudinal polarisation fraction is 0 . 524 ± 0 . 056 ( stat ) ± 0 . 029 ( syst ) . The relative branching fraction for this component is determined to be syst 32 % . In addition, the mass splitting between the B 0 s and B 0 mesons is measured as


B(B
.38 ± 0.36 (stat) ± 0.22 (syst) ± 0.31 (f s /f d ) %, where f s /f d indicates the uncertainty due to the ratio of probabilities for a b quark to hadronise into a B 0 s or B 0 meson.Using an amplitude analysis, the fraction of decays proceeding via an intermediate K * (892) 0 meson is measured to be 0.645 ± 0.049 (stat) ± 0.049 (syst) and its longitudinal polarisation fraction is 0.524 ± 0.056 (stat) ± 0.029 (syst).The relative branching fraction for this component is determined to be = 5.58 ± 0.57 (stat) ± 0.40 (syst) ± 0.32 (f s /f d ) %.

Introduction
The large data set collected by the LHCb experiment has allowed precision measurements of time-dependent CP violation in the B 0 s → J/ψφ and B 0 s → J/ψf 0 (980) decay modes [1,2]. 1  The results are interpreted assuming that these decays are dominated by colour-suppressed tree-level amplitudes (Fig. 1).Higher-order penguin amplitudes, which are difficult to calculate in QCD, also contribute (Fig. 1).Reference [3] suggests that the effect of these processes can be determined from decay modes such as B 0 s → J/ψK * (892) 0 where they dominate.The B 0 s → J/ψK * (892) 0 decay mode was first observed by the CDF collaboration [4] and subsequently studied in detail by the LHCb collaboration [5].Recently, interest in b-hadron decays to final states containing charmonia has been generated by the observation of the Z(4430) − state in the B 0 → ψ(2S)K + π − decay chain by the Belle [6][7][8] and LHCb collaborations [9].As this state is charged and has a minimal quark content of ccdu, it is interpreted as evidence for the existence of non-qq mesons [10].Evidence for similar exotic structures in B 0 → χ c1,c2 K + π − and B 0 → J/ψK + π − decays has been reported by the Belle collaboration [11,12].If these structures correspond to real particles they should be visible in other decay modes.
This letter reports the first observation of the decay B 0 s → ψ(2S)K + π − and presents measurements of the inclusive branching fraction and the fraction of decays that proceed via an intermediate K * (892) 0 resonance, as determined from an amplitude analysis of the final state.The amplitude analysis also allows the determination of the longitudinal polarisation fraction of the K * (892) 0 meson.Additionally the mass difference between B 0 s and B 0 mesons is reported.

Detector and simulation
The LHCb detector [13,14] 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 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 [15] 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 (15 + 29/p T ) µm, where p T is the component of the momentum transverse to the beam, in GeV/c.Large samples of B + → J/ψ K + and J/ψ → µ + µ − decays, collected concurrently with the data set used here, were used to calibrate the momentum scale of the spectrometer to a precision of 0.03 % [16].
Different types of charged hadrons are distinguished using information from two ringimaging Cherenkov detectors [17].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 [18].The online event selection is performed by a trigger [19], 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.In this analysis candidates are first required to pass the hardware trigger, which selects muons and dimuon pairs based on the transverse momentum.At the subsequent software stage, events are triggered by a ψ(2S) → µ + µ − candidate where the ψ(2S) is required to be consistent with coming from the decay of a b hadron by either using impact parameter requirements on daughter tracks or detachment of the ψ(2S) candidate from the primary vertex.
The analysis is performed using data corresponding to an integrated luminosity of 1.0 fb −1 collected in pp collisions at a centre-of-mass energy of 7 TeV and 2.0 fb −1 collected at 8 TeV.In the simulation, pp collisions are generated using Pythia [20] with a specific LHCb configuration [21].Decays of hadronic particles are described by EvtGen [22], in which final state radiation is generated using Photos [23].The interaction of the generated particles with the detector and its response are implemented using the Geant4 toolkit [24] as described in Ref. [25].

Event selection
The selection of candidates is divided into two parts.First, a loose selection is performed that retains the majority of signal events whilst reducing the background substantially.After this the B 0 → ψ(2S)K + π − peak is clearly visible.Subsequently, a multivariate method is used to further improve the signal-to-background ratio and to allow the observation of the B 0 s → ψ(2S)K + π − decay.The selection starts by reconstructing the dimuon decay of the ψ(2S) meson.Pairs of oppositely charged particles identified as muons with p T > 550 MeV/c are combined to form ψ(2S) candidates.The invariant mass of the dimuon pair is required to be within 60 MeV/c 2 of the known ψ(2S) mass [26].To form B 0 (s) candidates, the selected ψ(2S) mesons are combined with oppositely charged kaon and pion candidates.Good track quality for these hadrons is ensured by requiring that they have p T > 250 MeV/c.Fake tracks created by the reconstruction due to random hits in the detector are suppressed by selecting on the output of a neural network trained to discriminate between these and genuine tracks associated to particles.Combinatorial background from hadrons originating in the primary vertex (PV) is suppressed by requiring that both hadrons are significantly displaced from any PV.Well-identified hadrons are selected using the information provided by the Cherenkov detectors.This is combined with kinematic information using a neural network to provide a probability that a particle is a kaon (P K ), pion (P π ) or proton (P p ).
It is required that P K is larger than 0.1 for the K + candidate and that P π is larger than 0.2 for the π − candidate.
A kinematical vertex fit is applied to the B 0 (s) candidates [27].To improve the invariant mass resolution, the fit is performed with the requirement that the B 0 (s) candidate points to the PV and the ψ(2S) is mass constrained to the known value [26].A good quality of the vertex fit χ 2 , χ 2 DTF , is required.To ensure good separation between the B 0 and B 0 s signals, the uncertainty on the reconstructed mass returned by the fit must be less than 11 MeV/c 2 .Combinatorial background from particles produced in the primary vertex is further reduced by requiring the decay time of the B 0 (s) meson to exceed 0.3 ps.Four criteria are applied to reduce background from specific b-hadron decay modes.First, the candidate is rejected if the invariant mass of the hadron pair calculated assuming that both particles are kaons is within 10 MeV/c 2 of the known φ meson mass [26], suppressing B 0 s → ψ(2S)φ decays where one of the kaons is misidentified as a pion.Second, to suppress B 0 → ψ(2S)π + π − events where one of the pions is incorrectly identified as a kaon, it is required that P K > P π for the kaon candidate.This rejects 80 % of the background from this source whilst retaining 90 % of B 0 (s) signal candidates.Third, to suppress background from Λ 0 b → ψ(2S)pπ − decays where the proton is misidentifed as a kaon, candidates with P p > 0.3 and an invariant mass within 15 MeV/c 2 of the known Λ 0 b mass [26] are discarded.Finally, to reduce background from a B + → ψ(2S)K + decay combined with a random pion, candidates where the reconstructed ψ(2S)K + invariant mass is within 16 MeV/c 2 of the known B + mass [26] are rejected.Background from the decay Λ 0 b → ψ(2S)pK − with misidentified hadrons does not peak at the B 0 s mass and is modelled in the fit.
To further improve the signal-to-background ratio, a multivariate analysis based on a neural network is used.This is trained using simulated B 0 signal events together with candidates from data with a mass between 5500 and 5600 MeV/c 2 that are not used for subsequent analysis.Eight variables that give good separation between signal and background are used: the number of clusters in the large-area silicon tracker upstream of the magnet, P K for the kaon candidate, P π for the pion candidate, the transverse momentum of the B 0 (s) , the minimum impact parameter to any primary vertex for each of the two hadrons, χ 2 DTF and the flight distance of the B 0 (s) candidate divided by its uncertainty.The ratio N S / √ N S + N B is used as a figure of merit, where N S (N B ) is the number of signal (background) events determined from the invariant mass fit (see Sect. 4).
The maximum value of this ratio is found for a threshold on the neural network that rejects 98% of the background and retains 81% of the signal for subsequent analysis.

Invariant mass fit
A maximum likelihood fit is made to the unbinned ψ(2S)K + π − invariant mass distribution, m(ψ(2S)K + π − ), to extract the B 0 and B 0 s signal yields.The B 0 signal component is modelled by the sum of two Crystal Ball functions [28] with common tail parameters and an additional Gaussian component, all with a common mean.All parameters are fixed to values determined from the simulation apart from the common mean and an overall resolution scale factor.The simulation is tuned to match the invariant mass resolution seen in data for the B + → J/ψK + and B 0 → J/ψ K + π − decay modes.Consequently, the resolution scale factor is consistent with unity in the fit to data.The B 0 s component is modelled with the same function, with the mean value of the B 0 s meson mass left free in the fit.The resolution parameters in this case are multiplied by a factor of 1.06, determined from simulation, which accounts for the additional energy release in this decay.
Combinatorial background is modelled by an exponential function.A significant component from B 0 s → ψ(2S)φ decays is visible at lower masses than the B 0 peak.This is modelled in the fit by a bifurcated Gaussian function where the shape parameters are constrained to the values obtained in the simulation and the yield to the value determined in data under the hypothesis that both hadrons are kaons.Additional components from B 0 (s) → ψ(2S)π + π − and Λ 0 b → ψ(2S)pK − decays are modelled by bifurcated Gaussian functions.The shapes of these components are fixed using the simulation and the yields are determined by normalising the simulation samples to the number of candidates for each modes found in data using dedicated selections.Contributions from partially reconstructed decays are accounted for in the combinatorial background.In total, the fit has ten free parameters.Variations of this fit model are considered as systematic uncertainties.
Figure 2 shows the invariant mass distribution observed in the data together with the result of a fit to the model described above.Binning the data, a χ 2 -probability of 0.30 is found.The moderate mismodelling of the B 0 peak is accounted for in the systematic uncertainties.The fit determines that there are 329 ± 22 B 0 s decays and 24207 ± 160 B 0 decays.The B 0 s → ψ(2S)K + π − mode is observed with high significance.The precision of the momentum scale calibration of 0.03% translates to an uncertainty on the B 0 and B 0 s meson masses of 0.3 MeV/c 2 .Therefore, it is chosen to quote only the mass difference in which this uncertainty largely cancels, The systematic uncertainty on the mass difference is due to the momentum scale and mass fit model.Varying the momentum scale by 0.03% leads to an uncertainty of 0.04 MeV/c 2 .The effect of the fit model is evaluated by considering several variations: the relative fraction of the two Crystal Ball functions is left free; the slope of the combinatorial background is constrained using candidates where the kaon and pion have the same charge; the Gaussian constraints on the background from the B 0 s → ψ(2S)φ mode are removed; and the tail parameters of the Crystal Ball functions are left free.The largest variation in the mass splitting is 0.06 MeV/c 2 .The total systematic uncertainty is given by summing the individual components in quadrature.

Amplitude analysis
Figure 3 shows the Dalitz plot of the selected There is a clear enhancement around the known K * (892) 0 mass [26] and no other significant enhancements elsewhere.To determine the fraction of decays that proceed via the K * (892) 0 resonance, an amplitude analysis is performed, similar to that used in Ref. [9] for the analysis of the B 0 → ψ(2S)K + π − mode.In the helicity basis, defined in Fig. 4, the final-state particles are described using three angles Ω ≡ (cos θ K , cos θ µ , φ) and the invariant K + π − mass, m Kπ ≡ m(K + π − ).The total amplitude is S(m Kπ , Ω)ε(m Kπ , Ω) + B(m Kπ , Ω), where S(m Kπ , Ω) represents  the coherent sum over the helicity amplitudes for each considered K + π − resonance or non-resonant component.The detection efficiency, ε(m Kπ , cos θ K , cos θ µ , φ), is evaluated using simulation and parameterised using a combination of Legendre polynomials and spherical harmonic moments, given by The coefficients of the efficiency parameterisation are computed by summing over the N MC events simulated uniformly in the phase-space as  where g i = p i q i , with p i (q i ) being the momentum of the K * 0 (K + ) meson in the B 0 (K * 0 ) rest frame and m min(max) Kπ is the minimum (maximum) value allowed for m Kπ in the available phase space of the decay.This approach provides a description of the multidimensional correlations without assuming factorisation.In practice, the sum is over a finite number of moments (a ≤ 2, b ≤ 2, c ≤ 2 and d ≤ 2) and only coefficients with a statistical significance larger than five standard deviations from zero are retained.The one-dimensional projections of the parameterised efficiency are shown in Fig. 5, superimposed on the simulated event distributions.
The background probability density function, B(m Kπ , Ω), is determined using a similar method as for the efficiency parameterisation.In this case the sum in Eq. ( 2) is over the selected events with m(ψ(2S)K + π − ) > 5390 MeV/c 2 and g i ≡ 1.Only moments with a ≤ 2, b = 0, c = 0 and d ≤ 2 and a statistical significance larger than five standard deviations from zero are retained.The one-dimensional projections of the parameterised background distribution are shown in Fig. 6, superimposed on the sideband data.As a consistency check, the (m Kπ , Ω) distributions for events with m(ψ(2S)K + π − ) > 5390 MeV/c 2 are found to be compatible with the same distributions obtained from a like-sign (ψ(2S)K ± π ± ) sample in the same mass range.The default amplitude model is constructed using contributions from the K * (892) 0 resonance and a K + π − S-wave modelled using the LASS parameterisation [29].The magnitudes and phases of all components are measured relative to those of the zero helicity state of the K * (892) 0 meson and the masses and widths of the resonances are fixed to their known values [26].The remaining eight free parameters are determined using a maximum likelihood fit of the amplitude to the data in the signal window.The background fraction is fixed to 0.28, as determined from the fit described in Sect. 4. The fit fraction for any resonance R is defined in the full phase space, as f R = S R dm Kπ dΩ / S dm Kπ dΩ, where S R is the signal amplitude with all amplitude terms set to zero except those for R. The fractions of each component determined by the fit are f K * (892) 0 = 0.645 ± 0.049, and f S−wave = 0.339 ± 0.052, where the uncertainty is statistical only.Variations of the S-wave description and default mixture of K + π − resonances, including the introduction of the spin-2 K * 2 (1430) 0 meson or an exotic Z − c meson, are considered but found to give larger values of the Poisson likelihood χ 2 [30] per degree of freedom or lead to components with fit fractions that are consistent with zero.For each model the number of degrees of freedom is calibrated using simulated experiments.The variations in amplitude model are considered as sources of systematic uncertainty.The longitudinal polarisation fraction of the K * (892) 0 meson is defined as , where H 0,+,− are the magnitudes of the K * (892) 0 helicity amplitudes.This is measured to be f L = 0.524 ± 0.056, where the uncertainty is statistical.The projections of the default fit for the helicity angles and invariant K + π − mass are shown in Fig. 7.

Systematic uncertainties of amplitude analysis
A summary of possible sources of systematic uncertainties that affect the amplitude analysis is reported in Table 1.The size of each contribution is determined using a set of simulated experiments, of the same size as the data, generated under the hypothesis of an alternative amplitude model.These are fitted once with the default model and again with the alternative model.The experiment-by-experiment difference in the measured fit fractions and f L is then computed and the sum in quadrature of the mean and standard deviation is assigned as a systematic uncertainty to the corresponding parameter.The systematic dependence on the K + π − amplitude model is determined using the above procedure, where the alternative model also contains a spin-2 K * 2 (1430) component.This leads to the dominant systematic uncertainty on the K * (892) 0 fit fraction and f L .The systematic dependence on the K + π − S-wave model is determined using simulated experiments where a combination of a non-resonant term and a K * 0 (1430) contribution is used in place of the LASS parameterisation.In addition, the amplitude model contains parameters that are fixed in the default fit such as the Blatt-Weisskopf radius and the masses and widths of the resonances.The radius controls the effective hadron size and is set to 1.6 GeV −1 by default.An alternative model is considered where this is changed to 3.0 GeV −1 .
A large source of systematic uncertainty comes from the choice of convention for the mass, m, in the (p/m) L R terms of the amplitude.The default amplitude model follows the convention in Ref. [26] by using the resonance mass.This is different to that in Ref. [9] where the running resonance mass (m Kπ ) is used in the denominator.This choice is motivated by the improved fit quality obtained when using the resonance mass.
The systematic uncertainty related to the combinatorial background parameterisation is determined using an amplitude model with an alternative background description that allows for higher moment contributions (a ≤ 2, b ≤ 2, c ≤ 2 and d ≤ 2).The combinatorial background normalisation is determined from the high B 0 s -mass sidebands and is fixed in the amplitude fit.The systematic uncertainty related to the level of the background is estimated by using an amplitude model with the background fraction modified by ±10%.
The efficiency parameterisation is tested by re-evaluating the coefficients, allowing for higher order moments (a ≤ 4, b ≤ 4, c ≤ 4 and d ≤ 4).Similarly, to test the dependence of the efficiency model on the neural network requirement, an alternative model is used with the efficiency parameterisation determined from the simulated events that are selected without applying the requirement.

Branching fraction results
Two ratios of branching fractions are calculated, B(B ).These are determined from the signal yields given in Sect. 4 correcting for the relative detector acceptance using simulation.The simulated B 0 s samples are reweighted with the results of the angular analysis presented in Sect. 5. Similarly, the B 0 simulated data are reweighted to match the results given in Ref. [9].For the inclusive branching ratio, the relative efficiency between the two modes is found to be 0.975 ± 0.014 whilst for the K * (892) 0 component it is 1.027 ± 0.021.The uncertainty on these values is propagated to the systematic uncertainty.
Since the same final state is considered in the signal and normalisation mode, most sources of systematic uncertainty cancel in the ratio.The remaining sources are discussed in the following.Several variations of the invariant mass fit model are considered.The largest change in the ratio of yields observed in these tests is 3.7%, which is assigned as a systematic uncertainty.Differences in the p T spectra of the B 0 meson are seen comparing data and the reweighted simulation.If the p T spectrum in the simulation is further reweighted to match the data, the efficiency ratio changes by 0.7%, which is assigned as a systematic uncertainty.
To test the impact of the chosen K + π − amplitude model for the B 0 s channel, the simulated events are reweighted using a model consisting of the K * (892) 0 resonance, the LASS [29] description of the S-wave and the K * 2 (1430) resonance.This changes the efficiency ratio by 0.6%, which is assigned as a systematic uncertainty.To calculate the K * (892) 0 branching ratio, the fraction of candidates from this source is needed.For the B 0 s channel this is determined from the amplitude analysis to be 0.645 ± 0.069.The corresponding fraction for the B 0 channel is 0.591 ± 0.009 [9].The uncertainties discussed above are summed in quadrature and are summarised in Table 2.The statistical (systematic) component of these uncertainties is propagated to the statistical (systematic) uncertainty on the final ratio of branching fractions.The limited knowledge of the fragmentation fractions, f s /f d = 0.259 ± 0.015 [31][32][33], results in an uncertainty of 5.8% which is quoted separately from the others.

Summary
Using a data set corresponding to an integrated luminosity of 3.0 fb −1 collected in pp collisions at centre-of-mass energies of 7 and 8 TeV, the decay B  This is consistent with, though less precise than, the value 87.21 ± 0.31 MeV/c 2 obtained by averaging the results in Refs.[34,35].The ratio of branching fractions between the B 0 s and B 0 modes is measured to be The fraction of decays proceeding via an intermediate K * (892) 0 meson is measured with an amplitude analysis to be 0.645 ± 0.049 (stat) ± 0.049 (syst).No significant structure is seen in the distribution of m(ψ(2S)π − ).The longitudinal polarisation fraction, f L , of the K * (892) 0 meson is determined as 0.524 ± 0.056 (stat) ± 0.029 (syst).This is consistent with the value measured in the corresponding decay that proceeds through an intermediate J/ψ meson, f L = 0.50 ± 0.08 (stat) ± 0.02 (syst) [5].The present data set does not allow a test of the prediction given in Ref. [36] that f L should be lower for decays closer to the kinematic endpoint.
Using the K * (892) 0 fraction determined in this analysis for the B 0 s component and the corresponding number for the B 0 mode from Ref. [9], the following ratio of branching fractions is measured

Figure 2 :
Figure2: Invariant mass distribution for selected ψ(2S)K + π − candidates in the data.A fit to the model described in the text is superimposed.The full fit model is shown by the solid (red) line, the combinatorial background by the solid (yellow) and the sum of background from the exclusive b → ψ(2S)X modes considered in the text by the shaded (blue) area.The maximum of the y-scale is restricted so as to be able to see more clearly the B 0 s → ψ(2S)K + π − signal.The lower plot shows the differences between the fit and measured values divided by the corresponding uncertainty of the measured value, the so-called pull distribution.

Figure 4 :
Figure 4: Definition of the helicity angles.

Table 1 :
Summary of systematic uncertainties on the K * (892) 0 fit fraction and f L .Rows marked with (*) refer to uncorrelated sources of uncertainty between the B 0 s and B 0 modes for the computation of the ratio of branching fractions.

Table 2 :
Systematic uncertainties on the ratio of branching fractions.
Università di Bologna, Bologna, Italy e Università di Cagliari, Cagliari, Italy f Università di Ferrara, Ferrara, Italy g Università di Firenze, Firenze, Italy h Università di Urbino, Urbino, Italy i Università di Modena e Reggio Emilia, Modena, Italy j Università di Genova, Genova, Italy k Università di Milano Bicocca, Milano, Italy l Università di Roma Tor Vergata, Roma, Italy m Università di Roma La Sapienza, Roma, Italy n Università della Basilicata, Potenza, Italy o AGH -University of Science and Technology, Faculty of Computer Science, Electronics and Telecommunications, Kraków, Poland p LIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain q Hanoi University of Science, Hanoi, Viet Nam r Università di Padova, Padova, Italy s Università di Pisa, Pisa, Italy t Scuola Normale Superiore, Pisa, Italy u Università degli Studi di Milano, Milano, Italy v Politecnico di Milano, Milano, Italy d