Differential branching fraction and angular analysis of the decay $\boldsymbol{B^{0} \rightarrow K^{*0} \mu^+ \mu^-}$

The angular distributions and the partial branching fraction of the decay $B^{0} \rightarrow K^{*0} \mu^+ \mu^-$ are studied using an integrated luminosity of $0.37\,fb^{-1}$ of data collected with the LHCb detector. The forward-backward asymmetry of the muons, $A_{\mathrm{FB}}$, the fraction of longitudinal polarisation, $F_{\mathrm{L}}$, and the partial branching fraction, $\mathrm{d}{\mathcal B}/\mathrm{d}q^{2}$, are determined as a function of the dimuon invariant mass. The measurements are in good agreement with the Standard Model predictions and are the most precise to date. In the dimuon invariant mass squared range $1.00-6.00{\mathrm{\,Ge\kern -0.1em V}}^2/c^4$, the results are $A_{\mathrm{FB}}=-0.06\,^{+0.13}_{-0.14} \pm 0.04$, $F_{\mathrm{L}}=0.55\pm 0.10\pm 0.03$ and $\mathrm{d}{\mathcal B}/\mathrm{d}q^{2}=(0.42 \pm 0.06\pm 0.03) \times 10^{-7}c^4/{\mathrm{\,Ge\kern -0.1em V}}^2$. In each case, the first error is statistical and the second systematic.

The process B 0 → K * 0 µ + µ − is a flavour changing neutral current decay. In the Standard Model (SM) such decays are suppressed, as they can only proceed via loop processes involving electroweak penguin or box diagrams.
As-yet undiscovered particles could give additional contributions with comparable amplitudes, and the decay is therefore a sensitive probe of new phenomena. A number of angular observables in B 0 → K * 0 µ + µ − decays can be theoretically predicted with good control of the relevant form factor uncertainties. These include the forward-backward asymmetry of the muons, A FB , and the fraction of longitudinal polarisation, F L , as functions of the dimuon invariant mass squared, q 2 [1]. These observables have previously been measured by the BaBar, Belle, and CDF experiments [2]. A more precise determination of A FB is of particular interest as, in the 1.00 < q 2 < 6.00 GeV 2 /c 4 region, previous measurements favour an asymmetry with the opposite sign to that expected in the SM. If confirmed, this would be an unequivocal sign of phenomena not described by the SM. This letter presents the most precise measurements of A FB , F L and the partial branching fraction, dB/dq 2 , to date. The data used for this analysis were taken with the LHCb detector at CERN during 2011 and correspond to an integrated luminosity of 0.37 fb −1 . The K * 0 is reconstructed through its decay into the K + π − final state. The LHCb detector [3] is a single-arm spectrometer designed to study b-hadron decays. A silicon strip vertex detector positioned around the interaction region is used to measure the trajectory of charged particles and allows the reconstruction of the primary proton-proton interactions and the displaced secondary vertices characteristic of B-meson decays. A dipole magnetic field and further charged particle tracking stations allow momenta in the range 5 < p < 100 GeV/c to be determined with a precision of δp/p = 0.4-0.6%. The experiment has an acceptance for charged particles with pseudorapidity between 2 and 5. Two ring imaging Cherenkov (RICH) detectors allow kaons to be separated from pions or muons over a momentum range 2 < p < 100 GeV/c. Muons are identified on the basis of the number of hits in detectors interleaved with an iron muon filter.
The B 0 → K * 0 µ + µ − angular distribution is governed by six q 2 -dependent transversity amplitudes. The decay can be described by q 2 and the three angles θ l , θ K , φ. For the B 0 (B 0 ), θ l is the angle between the µ + (µ − ) and the opposite of the B 0 (B 0 ) direction in the dimuon rest frame, θ K the angle between the kaon and the direction opposite to the B meson in the K * 0 rest frame, and φ the angle between the µ + µ − and K + π − decay planes in the B rest frame. The inclusion of charge conjugate modes is implied throughout this letter. At a given q 2 , neglecting the muon mass, the normalised partial differential width integrated over θ K and φ is and integrated over θ l and φ it is These expressions do not include any broad S-wave contribution to the B 0 → K + π − µ + µ − decay and any contribution from low mass tails of higher K * 0 resonances. These contributions are assumed to be small and are neglected in the rest of the analysis. Signal candidates are isolated from the background using a set of selection criteria which are detailed below. An event-by-event weight is then used to correct for the bias induced by the reconstruction, trigger and selection criteria. In order to extract A FB and F L , simultaneous fits are made to the K + π − µ + µ − invariant mass distribution and the angular distributions. The partial branching fraction is measured by comparing the efficiency corrected yield of B 0 → K * 0 µ + µ − decays to the yield of B 0 → J/ψ K * 0 , where J/ψ → µ + µ − .
Candidate B 0 → K * 0 µ + µ − events are first required to pass a hardware trigger which selects muons with a transverse momentum, p T > 1.48 GeV/c. In the subsequent software trigger, at least one of the final state particles is required to have both p T > 0.8 GeV/c and impact parameter > 100 µm with respect to all of the primary protonproton interaction vertices in the event [4]. Finally, the tracks of two or more of the final state particles are required to form a vertex which is significantly displaced from the primary vertices in the event [5].
The combinatorial background, which is smoothly distributed in the reconstructed K + π − µ + µ − invariant mass, is reduced using a Boosted Decision Tree (BDT).
The BDT uses information about the event kinematics, vertex and track quality, impact parameter and particle identification information from the RICH and muon detectors. The variables that are used in the BDT are chosen so as to induce the minimum possible distortion in the angular and q 2 distributions. For example, no additional requirement is made on the p T of both of the muons as, at low q 2 , this would remove a large proportion of events with | cos θ l | ∼ 1. The BDT is trained entirely on data, using samples that are independent of that which is used to make the measurements: triggered and fully reconstructed B 0 → J/ψ K * 0 events are used as a proxy for the signal decay, and events from the upper B 0 → K * 0 µ + µ − mass sideband (5350 < m K + π − µ + µ − < 5600 MeV/c 2 ) are used as a background sample. The lower mass sideband is not used, as it contains background events formed from partially reconstructed B decays. These events make a negligible contribution in the signal region and have properties different from the combinatorial background which is the dominant background in this region.
A cut is made on the BDT output in order to optimise the sensitivity to A FB averaged over all q 2 . The selected sample has a signal-to-background ratio of three to one.
Peaking backgrounds from B 0 considered and reduced with a set of vetoes. In each case, for the decay to be a potential signal candidate, at least one particle needs to be misidentified. For example, B 0 → J/ψ K * 0 events where a kaon or pion is swapped for one of the muons, peak around the nominal B 0 mass and evade the J/ψ veto described above. Vetoes for each of these backgrounds are formed by changing the relevant particle mass hypotheses and recomputing the invariant masses, and by making use of the particle identification information. In order to avoid having a strongly peaking contribution to the cos θ K angular distribution in the upper mass sideband, B + → K + µ + µ − candidates are removed. Events with K + µ + µ − invariant mass within 60 MeV/c 2 of the nominal B + mass are rejected. The vetoes for all of these peaking backgrounds remove a negligible amount of signal.
After the application of the BDT cut and the above vetoes, a fit is made to the K + π − µ + µ − invariant mass distribution in the entire accepted mass range (see Fig. 1). A double-Gaussian distribution is used for the signal mass shape and an exponential function for the background. The signal shape is fixed from data using a fit to the B 0 → J/ψ K * 0 mass peak. In the full q 2 range, in a signal mass window of ±50 MeV/c 2 (±2.5σ) around the measured B 0 mass, the fit gives an estimate of 337 ± 21 signal events with a background of 97 ± 6 events.
The residual peaking background is estimated using simulated events. As detailed below, the accuracy of the simulation is verified by comparing the particle (mis-) identification probabilities with those derived from control channels selected from the data. The residual peak- ing backgrounds are reduced to a level of 6.1 events, i.e. 1.8% of the 337 observed signal events. The backgrounds from B 0 s → φµ + µ − and B 0 → J/ψ K * 0 decays do not give rise to any forward-backward asymmetry and are ignored. However, in addition to the above backgrounds, B 0 → K * 0 µ + µ − decays with the kaon and pion swapped give rise to a 0.7% contribution. The change in the sign of the particle which is taken to be the kaon results in a B 0 (B 0 ) being reconstructed as a B 0 (B 0 ), therefore changing the sign of A FB for the candidate. This misidentification is accounted for in the fit for the angular observables.
The selected B 0 → K * 0 µ + µ − candidates are weighted in order to correct for the effects of the reconstruction, trigger and selection. The weights are derived from simulated B 0 → K * 0 µ + µ − events and are normalised such that the average weight is one. In order to be independent of the physics model used in the simulation, the weights are computed based on cos θ K , cos θ l and q 2 on an event-by-event basis. The variation of detector efficiency with the φ angle is small and ignoring this variation does not bias the measurements. Only events with 0.10 < q 2 < 19.00 GeV 2 /c 4 are analysed.
Owing to the relatively unbiased selection, 89% of events have weights between 0.7 and 1.3, and only 3% of events have a weight above 2. The distortions in the distributions of cos θ K , cos θ l and q 2 that are induced originate from two main sources. Firstly, in order to pass through the iron muon filter and give hits in the muon stations, tracks must have at least 3 GeV/c momentum. At low q 2 this removes events with | cos θ l | ∼ 1. This effect stems from the geometry of the LHCb detector and is therefore relatively easy to model. Secondly, events with cos θ K ∼ 1, and hence a slow pion, are removed both by the pion reconstruction and by the impact parameter requirements used in the trigger and BDT selection.
A number of control samples are used to verify the simulation quality and to correct for differences with respect to the data. The reproduction of the B 0 momentum and pseudorapidity distributions is verified using B 0 → J/ψ K * 0 decays. These decays are also used to check that the simulation reproduces the measured properties of selected events. The hadron and muon (mis-)identification probabilities are adjusted using decays where the tested particle type can be determined without the use of the particle identification algorithms. A tag and probe approach with J/ψ → µ + µ − decays is used to isolate a clean sample of genuine muons. The decay D * + → D 0 π + , where D 0 → K − π + , is used to give an unambiguous source of kaons and pions. The statistical precision with which it is possible to make the data/simulation comparison gives rise to a systematic uncertainty in the weights which is evaluated below.
The observables A FB and F L are extracted in bins of q 2 . In each bin, a simultaneous fit to the K + π − µ + µ − invariant mass distribution and the cos θ K and cos θ l distributions is performed. The angular distributions are fitted in both the signal mass window and in the upper mass sideband which determines the background parameters. The angular distributions for the signal are given by Eqs. 1 and 2 and a second order polynomial in cos θ K and in cos θ l is used for the background.
In order to obtain a positive probability density function over the entire angular range, Eqs. 1 and 2 imply that the conditions |A FB | ≤ 3 4 (1 − F L ) and 0 < F L < 1 must be satisfied. To account for this, the maximum likelihood values for A FB and F L are extracted by performing a profile-likelihood scan over the allowed range. The uncertainty on the central value of A FB and F L is calculated by integrating the probability density extracted from the likelihood, assuming a flat prior in A FB and F L , inside the allowed range. This gives an (asymmetric) 68% confidence interval.
The partial branching fraction is measured in each of the q 2 bins from a fit to the efficiency corrected K + π − µ + µ − mass spectrum. The efficiencies are determined relative to the B 0 → J/ψ K * 0 decay which is used as a normalisation mode.
The event weighting and fitting procedure is validated by fitting the angular distribution of B 0 → J/ψ K * 0 events, where the physics parameters are known from previous measurements [6].
The product of the B 0 → J/ψ K * 0 and J/ψ → µ + µ − branching fractions is ∼ 75 times larger than the branching fraction of B 0 → K * 0 µ + µ − , allowing a precise test of the procedure to be made. Fitting the B 0 → J/ψ K * 0 angular distribution, weighted according to the event-by-event procedure described above, yields values for F L and A FB in good agreement with those found previously.
For B 0 → K * 0 µ + µ − , the fit results for A FB , F L and dB/dq 2 are shown in Fig. 2   with the signal and background yields in Table I. The fit projections are given in the appendix. Signal candidates are observed in each q 2 bin with more than 5σ significance. The compatibility of the fits and the data are assessed using a binned χ 2 test and all fits are found to be of good quality. The measurements in all three quantities are more precise than those of previous experiments and are in good agreement with the SM predictions. The predictions are taken from Ref. [7]. In the low q 2 region they rely on the factorisation approach [8], which loses accuracy when approaching the J/ψ resonance; in the high q 2 region, an operator product expansion in the inverse b-quark mass, 1/m b , and in 1/ q 2 is used [9], which is only valid above the open charm threshold. In both regions the form factor calculations are taken from Ref. [10] and a dimensional estimate is made on the uncertainty from expansion corrections [11].
For the determination of A FB and F L , the dominant systematic uncertainties arise from the event-by-event weights which are extracted from simulated events, and from the model used to describe the angular distribution of the background. The uncertainty on the event-byevent weights is evaluated by fluctuating these weights within their statistical uncertainties and repeating the fitting procedure. The uncertainty from the background model which is used is estimated by changing this model to one which uses binned templates from the upper mass sideband rather than a polynomial parameterisation.
The dominant systematic errors for the determination of dB/dq 2 arise from the uncertainties on the particle identification and track reconstruction efficiencies. These efficiencies are extracted from control channels and are limited by the relevant sample sizes. The systematic uncertainty is estimated by fluctuating the efficiencies within the relevant uncertainties and repeating the fitting procedure. An additional systematic uncertainty of ∼ 4% arises from the uncertainty in the B 0 → J/ψ K * 0 and J/ψ → µ + µ − branching fractions [12].
The total systematic error on each of A FB and F L (dB/dq 2 ) is typically ∼ 30% (50%) of the statistical error, and hence adds ∼ 4% (∼ 11%) to the total uncertainty.
In summary, using 0.37 fb −1 of data taken with the LHCb detector during 2011, A FB , F L and dB/dq 2 have been determined for the decay B 0 → K * 0 µ + µ − . These are the most precise measurements of these quantities to-date. All three observables show good agreement with the SM predictions.  I. Central values with statistical and systematic uncertainties for AFB, FL and dB/dq 2 as a function of q 2 . The B 0 → K * 0 µ + µ − signal and background yields in the ±50 MeV/c 2 signal mass window with their statistical uncertainties are also indicated, together with the statistical significance of the signal peak that is observed.