Measurement of CP -Averaged Observables in the B 0 → K ∗ 0 µ + µ − Decay

An angular analysis of the B 0 → K (cid:1) 0 ð → K þ π − Þ μ þ μ − decay is presented using a dataset corresponding to an integrated luminosity of 4 . 7 fb − 1 of pp collision data collected with the LHCb experiment. The full set of CP -averaged observables are determined in bins of the invariant mass squared of the dimuon system. Contamination from decays with the K þ π − system in an S -wave configuration is taken into account. The tension seen between the previous LHCb results and the standard model predictions persists with the new data. The precise value of the significance of this tension depends on the choice of theory nuisance parameters.


Measurement of CP-Averaged
An angular analysis of the B 0 → K Ã0 ð→ K þ π − Þμ þ μ − decay is presented using a dataset corresponding to an integrated luminosity of 4.7 fb −1 of pp collision data collected with the LHCb experiment. The full set of CP-averaged observables are determined in bins of the invariant mass squared of the dimuon system. Contamination from decays with the K þ π − system in an S-wave configuration is taken into account. The tension seen between the previous LHCb results and the standard model predictions persists with the new data. The precise value of the significance of this tension depends on the choice of theory nuisance parameters. DOI: 10.1103/PhysRevLett.125.011802 Decays mediated by the quark-level transition b → sl þ l − , where l represents a lepton, have been the subject of intense recent study, as angular observables [1][2][3][4][5][6][7][8], branching fractions [8][9][10][11], and ratios of branching fractions between decays with different flavors of leptons [12][13][14][15][16] have been measured to be in tension with standard model (SM) predictions. Such decays are suppressed in the SM, as they proceed only through amplitudes that involve electroweak loop diagrams. The decays are sensitive to virtual contributions from new particles, which could have masses that are inaccessible to direct searches. The observed anomalies with respect to SM predictions can be explained consistently in new physics models that introduce an additional vector or axial-vector contribution [17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35]. However, there is still considerable debate about whether some of the observations might instead be explained by hadronic uncertainties associated with the transition form factors, or by other long-distance effects [36][37][38][39].
The LHCb Collaboration presented a measurement of the angular observables of the B 0 → K Ã0 μ þ μ − decay in Ref. [1] and found that the data could be explained by modifying the real part of the vector coupling strength of the decays, conventionally denoted ReðC 9 Þ. The analysis used the nuisance parameters from Ref. [40], implemented in the EOS software package described in Ref. [41], and found a 3.4 standard deviation (σ) tension with the SM value of ReðC 9 Þ. The tension observed depends on the values of various SM nuisance parameters, including form-factor parameters and subleading corrections used to account for long-distance QCD interference effects with the charmonium modes. Using the FLAVIO software package [42], with its default SM nuisance parameters, gives a tension of 3.0σ with respect to the SM value of ReðC 9 Þ when fitting the angular observables from Ref. [1]. The nuisance parameters include a recent treatment of the subleading corrections [43,44] that was not available at the time of the previous analysis.
This Letter presents the most precise measurements of the complete set of CP-averaged angular observables in the decay B 0 → K Ã0 μ þ μ − . The dataset corresponds to an integrated luminosity of 4.7 fb −1 of pp collisions collected with the LHCb experiment. The data were taken in the years 2011, 2012, and 2016, at center-of-mass energies of 7, 8, and 13 TeV, respectively. The analysis uses the same technique as the analysis described in Ref. [1] but the data sample contains approximately twice as many B 0 decays, owing to the addition of the 2016 data. The bb production cross section increases by roughly a factor of 2 between the Run 1 and 2016 datasets [45]. The same 2011 and 2012 (Run 1) data as in Ref. [1] are used in the present analysis. The results presented in this Letter supersede the previous LHCb publication. The combination of the Run 1 dataset with the 2016 dataset requires a simultaneous angular fit to account for efficiency and reconstruction differences between years. Throughout this Letter, K Ã0 is used to refer to the K Ã ð892Þ 0 resonance and the inclusion of charge-conjugate processes is implied. The K Ã0 meson is reconstructed through the decay The final state of the B 0 → K Ã0 μ þ μ − decay can be described by the invariant mass squared of the dimuon system q 2 , the invariant mass of the K þ π − system, and the three decay angles, ⃗ Ω ¼ðcos θ l ; cos θ K ; ϕÞ. The angle between the μ þ (μ − ) and the direction opposite to that of the B 0 (B 0 ) in the rest frame of the dimuon system is denoted θ l . The angle between the direction of the K þ (K − ) and the B 0 (B 0 ) in the rest frame of the K Ã0 (K Ã0 ) system is denoted θ K . The angle between the plane defined by the dimuon pair and the plane defined by the kaon and pion in the B 0 (B 0 ) rest frame is denoted ϕ. A full description of the angular basis is provided in Ref. [46].
Following the definitions given in Refs. [1,47], the CPaveraged angular distribution of the B 0 → K Ã0 μ þ μ − decay with the K þ π − system in a P-wave configuration can be written as where F L is the fraction of the longitudinal polarization of the K Ã0 meson, A FB is the forward-backward asymmetry of the dimuon system, and S i are other CP-averaged observables [1]. The K þ π − system can also be in an S-wave configuration, which modifies the angular distribution to where F S denotes the S-wave fraction and the coefficients S 11 , S 13 -S 17 arise from interference between the S-and Pwave amplitudes. Throughout this Letter, F S and the interference terms between the S and P wave are treated as nuisance parameters. Additional sets of observables, for which the leading B 0 → K Ã0 form-factor uncertainties cancel, can be built from F L , A FB , and S 3 -S 9 . Examples of such optimized observables include the P ð0Þ i series of observables [48]. The notation used in this Letter again follows Ref. [1], for example, P 0 The LHCb detector is a single-arm forward spectrometer covering the pseudorapidity range 2 < η < 5, described in detail in Refs. [49,50]. The detector includes a vertex detector surrounding the proton-proton interaction region, tracking stations on either side of a dipole magnet, ringimaging Cherenkov (RICH) detectors, electromagnetic and hadronic calorimeters, and muon chambers.
Simulated signal events are used in this analysis to determine the impact of the detector geometry, trigger, reconstruction, and candidate selection on the angular distribution of the signal. The simulation is produced using the software described in Refs. [51][52][53][54][55][56]. Corrections derived from the data are applied to the simulation to account for mismodeling of the charge multiplicity of the event, B 0 momentum spectrum, and B 0 vertex quality. Similarly, the simulated particle identification (PID) performance is corrected to match that determined from control samples selected from the data [57,58].
The online event selection is performed by a trigger, which comprises a hardware stage, based on information from the calorimeter and muon systems, followed by a software stage that applies a full event reconstruction [59]. Offline, signal candidates are formed from a pair of oppositely charged tracks that are identified as muons, combined with a K Ã0 candidate.
The distribution of the reconstructed K þ π − μ þ μ − invariant mass, mðK þ π − μ þ μ − Þ, is used to discriminate signal from background. This distribution is fitted simultaneously with the three decay angles. The distribution of the reconstructed K þ π − mass, mðK þ π − Þ, depends on the K þ π − angular-momentum configuration and is used to constrain the S-wave fraction. The analysis procedure is cross-checked by performing a fit of the b → ccs tree-level decay B 0 → J=ψK Ã0 , with J=ψ → μ þ μ − , which results in the same final-state particles. Hereafter, the B 0 → J=ψð→ μ þ μ − ÞK Ã0 decay and the equivalent decay via the ψð2SÞ resonance are denoted by B 0 → J=ψK Ã0 and B 0 → ψð2SÞK Ã0 , respectively. PHYSICAL REVIEW LETTERS 125, 011802 (2020) 011802-2 Two types of backgrounds are considered: combinatorial background, where the selected particles do not originate from a single b-hadron decay; and peaking backgrounds, where a single decay is selected but with some of the final-state particles misidentified. The combinatorial background is distributed smoothly in mðK þ π − μ þ μ − Þ, whereas the peaking backgrounds can accumulate in specific regions of the reconstructed mass. In addition, the decays B 0 → J=ψK Ã0 , B 0 → ψð2SÞK Ã0 , and B 0 → ϕð1020Þð→ μ þ μ − ÞK Ã0 are removed by rejecting events with q 2 in the ranges 8.0 <q 2 < 11.0 GeV 2 =c 4 , 12.5 <q 2 < 15.0 GeV 2 =c 4 ,o r0.98 <q 2 < 1.10 GeV 2 =c 4 .
The criteria used to select candidates from the Run 1 data are the same as those described in Ref. [1]. The selection of the 2016 data follows closely that of the Run 1 data.
Candidates are required to have 5170 <mðK þ π − μ þ μ − Þ < 5700 and 795.9 <mðK þ π − Þ < 995.9 MeV=c 2 . The four tracks of the final-state particles are required to have significant impact parameter (IP) with respect to all primary vertices (PVs) in the event. The tracks are fitted to a common vertex, which is required to be of good quality. The IP of the B 0 candidate is required to be small with respect to one of the PVs. The vertex of the B 0 candidate is required to be significantly displaced from the same PV. The angle between the reconstructed B 0 momentum and the vector connecting this PV to the reconstructed B 0 decay vertex, θ DIRA , is also required to be small. To avoid the same track being used to construct more than one of the final state particles, the opening angle between every pair of tracks is required to be larger than 1 mrad.
Combinatorial background is reduced further using a boosted decision tree (BDT) algorithm [60,61]. The BDT is trained entirely on data with B 0 → J=ψK Ã0 candidates used as a proxy for the signal and candidates from the uppermass sideband 5350 <mðK þ π − μ þ μ − Þ < 7000 MeV=c 2 used as a proxy for the background. The training uses a cross-validation technique [62] and is performed separately for the Run 1 and 2016 datasets. The input variables used are the reconstructed B 0 decay time and vertex-fit quality, the momentum and transverse momentum of the B 0 candidate θ DIRA , PID information from the RICH detectors and the muon system, and variables describing the isolation of the final-state tracks [63]. Variables are only used in the BDT if they do not have a strong correlation with the decay angles or q 2 . A requirement is placed on the BDT output to maximize the signal significance. This requirement rejects more than 97% of the remaining combinatorial background, while retaining more than 85% of the signal. The signal efficiency of the BDT is uniform in the mðK þ π − μ þ μ − Þ and mðK þ π − Þ distributions. Peaking where the latter constitutes a background if the kaon from theK Ã0 decay is misidentified as the pion and vice versa. In each case, at least one particle needs to be misidentified for the decay to be reconstructed as a signal candidate. Vetoes to reduce these peaking backgrounds are formed by placing requirements on the invariant mass of the candidates, recomputed with the relevant change in the particle mass hypotheses, and by using PID information. In addition, in order to avoid having a strongly peaking contribution to the cos θ K angular distribution in the upper mass sideband, B þ → K þ μ þ μ − candidates with K þ μ þ μ − invariant mass within 60 MeV=c 2 of the B þ mass are removed. The background from b-hadron decays with two hadrons misidentified as muons is negligible. The signal efficiency and residual peaking backgrounds are estimated using simulated events. The vetoes remove a negligible amount of signal. The largest residual backgrounds are at the level of 1% or less of the expected signal yield. This is sufficiently small such that these backgrounds are neglected in the angular analysis and are considered only as sources of systematic uncertainty.
For every q 2 bin, a fit is performed in both the standard and the optimized basis. For each basis, four datasets are fit simultaneously: the mðK þ π − μ þ μ − Þ and angular distributions of candidates in the Run 1 data; the equivalent distributions for the 2016 data; and the mðK þ π − Þ distributions of candidates in the Run 1 and the 2016 datasets. The signal fraction is shared between the two datasets from each data-taking period. The CP-averaged angular observables and the S-wave fraction are shared between all datasets.
The fitted probability density functions (PDFs) are of an identical form to those of Ref. [1], as are the q 2 bins used. In addition to the narrow q 2 bins, results are obtained for the wider bins 1.1 <q 2 < 6.0 and 15.0 <q 2 < 19.0 GeV 2 =c 4 .
The angular distribution of the signal is described using Eq. (1). The P ð0Þ i observables are determined by reparametrizing Eq. (1) using a basis comprising F L , P 1;2;3 , and P 0 4;5;6;8 . The angular distribution is multiplied by an acceptance model used to account for the effect of the reconstruction and candidate selection. The acceptance function is parametrized in four dimensions, according to εðcos θ l ; cos θ K ; ϕ;q 2 Þ ¼ X ijmn c ijmn L i ðcos θ l ÞL j ðcos θ K ÞL m ðϕÞL n ðq 2 Þ; ð3Þ where the terms L h ðxÞ denote Legendre polynomials of order h and the values of the angles and q 2 are rescaled to the range −1 <x<þ1 when evaluating the polynomials. For the cos θ l , cos θ K , and ϕ angles, the sum in Eq. (3) encompasses L h ðxÞ up to fourth, fifth, and sixth order, respectively. The q 2 parametrization comprises L h ðxÞ up to PHYSICAL REVIEW LETTERS 125, 011802 (2020) 011802-3 fifth order. Simulation indicates that the acceptance function can be assumed to be flat across mðK þ π − Þ. The coefficients c ijmn are determined using a principal moment analysis of simulated B 0 → K Ã0 μ þ μ − decays. As all of the relevant kinematic variables needed to describe the decay are used in this parametrization, the acceptance function does not depend on the decay model used in the simulation.
In the narrow q 2 bins, the acceptance is taken to be constant across each bin and is included in the fit by multiplying Eq. (2) by the acceptance function evaluated with the value of q 2 fixed at the bin center. In the wider q 2 bins, the shape of the acceptance can vary significantly across the bin. In the likelihood, candidates are therefore weighted by the inverse of the acceptance function and parameter uncertainties are obtained using a bootstrapping technique [64].
The background angular distribution is modeled with second-order polynomials in cos θ l , cos θ K , and ϕ, with the angular coefficients allowed to vary in the fit. This angular distribution is assumed to factorize in the three decay angles, which is confirmed to be the case for candidates in the upper mass sideband of the data.
The mðK þ π − μ þ μ − Þ distribution of the signal candidates is modeled using the sum of two Gaussian functions with a common mean, each with a power-law tail on the low mass side. The parameters describing the signal mass shape are determined from a fit to the B 0 → J=ψK Ã0 decay in the data and are subsequently fixed when fitting the B 0 → K Ã0 μ þ μ − candidates. For each of the q 2 bins, a scale factor that is determined from simulation is included to account for the difference in resolution between the B 0 → J=ψK Ã0 and B 0 → K Ã0 μ þ μ − decay modes. A component is included in the B 0 → J=ψK Ã0 fit to account forB 0 s → J=ψK Ã0 decays, which are at the level of ∼1% of the B 0 → J=ψK Ã0 signal yield. The background from the equivalent Cabibbo-suppressed penguin decay,B 0 s → K Ã0 μ þ μ − [65], is negligible and is ignored in the fit of the signal decay. The combinatorial background is described well by an exponential distribution in mðK þ π − μ þ μ − Þ.
The K Ã0 signal component in the mðK þ π − Þ distribution is modeled using a relativistic Breit-Wigner function for the P-wave component and the LASS parametrization [66] for the S-wave component. The combinatorial background is described by a linear function in mðK þ π − Þ.
The decay B 0 → J=ψK Ã0 is used to cross-check the analysis procedure in the region 8.0 <q 2 < 11.0 GeV 2 =c 4 . This decay is selected in the data with negligible background contamination. The angular structure has been determined by measurements made by the BABAR, Belle, and LHCb Collaborations [67][68][69]. The B 0 → J=ψK Ã0 angular observables obtained from the Run 1 and 2016 LHCb data, using the acceptance correction derived as described above, are in good agreement with these previous measurements. Figure 1 shows the projection of the fitted PDF on the K þ π − μ þ μ − mass distribution. The B 0 → K Ã0 μ þ μ − yield, integrated over the q 2 ranges 0.10 <q 2 < 0.98, 1.1 <q 2 < 8.0, 11.0 <q 2 < 12.5, and 15.0 <q 2 < 19.0 GeV 2 =c 4 , is determined to be 2398 AE 57 for the Run 1 data, and 2187 AE 53 for the 2016 data.
Pseudoexperiments, generated using the results of the best fit to data, are used to assess the bias and coverage of the fit. The majority of observables have a bias of less than 10% of their statistical uncertainty, with the largest bias being 17%, and all observables have an uncertainty estimate within 10% of the true uncertainty. The biases are driven by boundary effects in the observables. The largest effect comes from requiring that F S ≥ 0, which can bias F S to larger values. This can then result in a bias in the P-wave observables [see Eq.
(2)]. The statistical uncertainty is corrected to account for any under-or overcoverage and a systematic uncertainty equal to the size of the observed bias is assigned.
The size of other sources of systematic uncertainty is estimated using pseudoexperiments, in which one or more parameters are varied and the angular observables are determined with and without this variation. The systematic uncertainty is then taken as the difference between the two 011802-4 models. The pseudoexperiments are generated with signal yields many times larger than the data, in order to render statistical fluctuations negligible.
The size of the total systematic uncertainty varies depending on the angular observable and the q 2 bin. The majority of observables in both the S i and P ð0Þ i basis have a total systematic uncertainty between 5% and 25% of the statistical uncertainty. For F L , the systematic uncertainty tends to be larger, typically between 20% and 50%. The systematic uncertainties are given in Table 3 of Ref. [70].
The dominant systematic uncertainties arise from the peaking backgrounds that are neglected in the analysis, the bias correction, and, for the narrow q 2 bins, from the uncertainty associated with evaluating the acceptance at a fixed point in q 2 . For the peaking backgrounds, the systematic uncertainty is evaluated by injecting additional candidates, drawn from the angular distributions of the background modes, into the pseudoexperiment data. The systematic uncertainty for the bias correction is determined directly from the pseudoexperiments used to validate the fit. The systematic uncertainty from the variation of the acceptance with q 2 is determined by moving the point in q 2 at which the acceptance is evaluated to halfway between the bin center and the upper or the lower edge. The largest deviation is taken as the systematic uncertainty. Examples of further sources of systematic uncertainty investigated include the mðK þ π − Þ line shape for the S-wave contribution, the assumption that the acceptance function is flat across the mðK þ π − Þ mass, the effect of the B þ → K þ μ þ μ − veto on the angular distribution of the background and the order of polynomial used for the background parametrization. These sources make a negligible contribution to the total uncertainty. With respect to the analysis of Ref. [1], the systematic uncertainty from residual differences between data and simulation is significantly reduced, owing to an improved decay model for B 0 → J=ψK Ã0 decays [68].
The CP-averaged observables F L , A FB , S 5 , and P 0 5 that are obtained from the S i and P ð0Þ i fits are shown together with their respective SM predictions in Fig. 2. The results for all observables are given in Figs. 1 and 2 and Tables 1 and 2 of Ref. [70]. In addition, the statistical correlation between the observables is provided in Tables 4-23. The SM predictions are based on the prescription of Ref. [44], which combines light-cone sum rule calculations [43], valid in the low-q 2 region, with lattice determinations at high q 2 [71,72] to yield more precise determinations of the form factors over the full q 2 range. For the P ð0Þ i observables, predictions from Ref. [73] are shown using form factors  [43,44], with the exception of the P 0 5 distribution, which is compared to SM predictions based on Refs. [73,74]. from Ref. [74]. These predictions are restricted to the region q 2 < 8.0 GeV 2 =c 4 . The results from Run 1 and the 2016 data are in excellent agreement. A stand-alone fit to the Run 1 data reproduces exactly the central values of the observables obtained in Ref. [1].
Considering the observables individually, the results are largely in agreement with the SM predictions. The local discrepancy in the P 0 5 observable in the 4.0 <q 2 < 6.0 and 6.0 <q 2 < 8.0 GeV 2 =c 4 bins reduces from the 2.8 and 3.0σ observed in Ref. [1] to 2.5 and 2.9σ.H o w e v e r ,a s discussed below, the overall tension with the SM is observed to increase mildly.
Using the FLAVIO software package [42], a fit of the angular observables is performed varying the parameter ReðC 9 Þ. The default FLAVIO SM nuisance parameters are used, including form-factor parameters and subleading corrections to account for long-distance QCD interference effects with the charmonium decay modes [43,44]. The same q 2 bins as in Ref. [1] are included. The 3.0σ discrepancy with respect to the SM value of ReðC 9 Þ obtained with the Ref. [1] dataset changes to 3.3σ with the dataset used here. The best fit to the angular distribution is obtained with a shift in the SM value of ReðC 9 Þ by −0.99 þ0. 25 −0.21 . The tension observed in any such fit will depend on the effective coupling(s) varied, the handling of the SM nuisance parameters, and the q 2 bins that are included in the fit. For example, the 6.0 <q 2 < 8.0 GeV 2 =c 4 bin is known to be associated with larger theoretical uncertainties [47]. Neglecting this bin, a FLAVIO fit gives a tension of 2.4σ using the observables from Ref. [1] and 2.7σ tension with the measurements reported here.
In summary, using 4.7 fb −1 of pp collision data collected with the LHCb experiment during the years 2011, 2012, and 2016, a complete set of CP-averaged angular observables has been measured for the B 0 → K Ã0 μ þ μ − decay. These are the most precise measurements of these quantities to date.
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: CAPES, CNPq,