Differential branching fraction and angular analysis of the decay $B_s^0\to\phi\mu^{+}\mu^{-}$

The determination of the differential branching fraction and the first angular analysis of the decay $B_s^0\to\phi\mu^{+}\mu^{-}$ are presented using data, corresponding to an integrated luminosity of $1.0\,{\rm fb}^{-1}$, collected by the LHCb experiment at $\sqrt{s}=7\,{\rm TeV}$. The differential branching fraction is determined in bins of $q^{2}$, the invariant dimuon mass squared. Integration over the full $q^{2}$ range yields a total branching fraction of ${\cal B}(B_s^0\to\phi\mu^{+}\mu^{-}) = (7.07\,^{+0.64}_{-0.59}\pm 0.17 \pm 0.71)\times 10^{-7}$, where the first uncertainty is statistical, the second systematic, and the third originates from the branching fraction of the normalisation channel. An angular analysis is performed to determine the angular observables $F_{\rm L}$, $S_3$, $A_6$, and $A_9$. The observables are consistent with Standard Model expectations.


Introduction
The B 0 s → φµ + µ − (φ → K + K − ) decay 1 involves a b → s quark transition and therefore constitutes a flavour changing neutral current (FCNC) process. Since FCNC processes are forbidden at tree level in the Standard Model (SM), the decay is mediated by higher order (box and penguin) diagrams. In scenarios beyond the SM new particles can affect both the branching fraction of the decay and the angular distributions of the decay products.
The angular configuration of the K + K − µ + µ − system is defined by the decay angles θ K , θ , and Φ. Here, θ K (θ ) denotes the angle of the K − (µ − ) with respect to the direction of flight of the B 0 s meson in the K + K − (µ + µ − ) centre-of-mass frame, and Φ denotes the relative angle of the µ + µ − and the K + K − decay planes in the B 0 s meson centre-of-mass frame [1]. In contrast to the decay B 0 → K * 0 µ + µ − , the final state of the decay B 0 s → φµ + µ − is not flavour specific. The differential decay rate, depending on the decay angles and the invariant mass squared of the dimuon system is given by where equal numbers of produced B 0 s and B 0 s mesons are assumed [2]. The q 2 -dependent angular observables S i and A i correspond to CP averages and CP asymmetries, respectively. Integrating Eq. 1 over two angles, under the assumption of massless leptons, results in three distributions, each depending on one decay angle which retain sensitivity to the angular observables F L (= S c 1 = −S c 2 ), S 3 , A 6 , and A 9 . Of particular interest is the T -odd asymmetry A 9 where possible large CP -violating phases from contributions beyond the SM would not be suppressed by small strong phases [1]. This paper presents a measurement of the differential branching fraction and the angular observables F L , S 3 , A 6 , and A 9 in six bins of q 2 . In addition, the total branching fraction is determined. The data used in the analysis were recorded by the LHCb experiment in 2011 in pp collisions at √ s = 7 TeV and correspond to an integrated luminosity of 1.0 fb −1 .
1 The inclusion of charge conjugated processes is implied throughout this paper.

2 The LHCb detector
The LHCb detector [3] 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 placed downstream. 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. Charged hadrons are identified using two ring-imaging Cherenkov detectors. Photon, electron and hadron candidates 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. The LHCb trigger system [4] 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. Simulated signal event samples are generated to determine the trigger, reconstruction and selection efficiencies. Exclusive samples are analysed to estimate possible backgrounds. The simulation generates pp collisions using Pythia 6.4 [5] with a specific LHCb configuration [6]. Decays of hadronic particles are described by EvtGen [7] in which final state radiation is generated using Photos [8]. The interaction of the generated particles with the detector and its response are implemented using the Geant4 toolkit [9] as described in Ref. [10]. Data driven corrections are applied to the simulated events to account for differences between data and simulation. These include the impact parameter resolution, tracking efficiency, and particle identification performance. In addition, simulated events are reweighted depending on the transverse momentum (p T ) of the B 0 s meson, the vertex fit quality, and the track multiplicity to match distributions of control samples from data.

Selection of signal candidates
Signal candidates are accepted if they are triggered by particles of the B 0 s → φµ + µ − (φ → K + K − ) final state. The hardware trigger requires either a high transverse momentum muon or muon pair, or a high transverse energy (E T ) hadron. The first stage of the software trigger selects events containing a muon (or hadron) with p T > 0.8 GeV/c (E T > 1.5 GeV/c) and a minimum IP with respect to all primary interaction vertices in the event of 80 µm (125 µm). In the second stage of the software trigger the tracks of two or more final state particles are required to form a vertex that is significantly displaced from all primary vertices (PVs) in the event.
Candidates are selected if they pass a loose preselection that requires the kaon and muon tracks to have a large χ 2 IP with respect to the PV. The χ 2 IP is defined as the difference between the χ 2 of the PV reconstructed with and without the considered track. The four tracks forming a B 0 s candidate are fit to a common vertex, which is required to be of good quality and well separated from the PV. The angle between the B 0 s momentum vector and the vector connecting the PV with the B 0 s decay vertex is required to be small. Furthermore, B 0 s candidates are required to have a small IP with respect to the PV. The invariant mass of the K + K − system is required to be within 12 MeV/c 2 of the known φ mass [11].
To further reject combinatorial background events, a boosted decision tree (BDT) [12] using the AdaBoost algorithm [13] is applied. The BDT training uses B 0 s → J/ψ φ (J/ψ → µ + µ − ) candidates as proxy for the signal, and candidates in the B 0 s → φµ + µ − mass sidebands (5100 < m(K + K − µ + µ − ) < 5166 MeV/c 2 and 5566 < m(K + K − µ + µ − ) < 5800 MeV/c 2 ) as background. The input variables of the BDT are the χ 2 IP of all final state tracks and of the B 0 s candidate, the angle between the B 0 s momentum vector and the vector between PV and B 0 s decay vertex, the vertex fit χ 2 , the flight distance significance and transverse momentum of the B 0 s candidate, and particle identification information of the muons and kaons in the final state.
Several types of b-hadron decays can mimic the final state of the signal decay and constitute potential sources of peaking background. The resonant decays In the region 5416 < m(B 0 s ) < 5566 MeV/c 2 the vetoes are extended by 50 MeV/c 2 to higher m(µ + µ − ) to reject a small fraction of J/ψ and ψ(2S) decays that are misreconstructed at higher masses. The decay B 0 → K * 0 µ + µ − (K * 0 → K + π − ) can be reconstructed as signal if the pion is misidentified as a kaon. This background is strongly suppressed by particle identification criteria. In the narrow φ mass window, 2.4 ± 0.5 misidentified B 0 → K * 0 µ + µ − candidates are expected within ±50 MeV/c 2 of the known B 0 s mass of 5366 MeV/c 2 [11]. The resonant decay B 0 s → J/ψ φ can also constitute a source of peaking background if the K + (K − ) is misidentified as µ + (µ − ) and vice versa. Similarly, the decay B 0 → J/ψ K * 0 (K * 0 → K + π − ) where the π − (µ − ) is misidentified as µ − (K − ) can mimic the signal decay. These backgrounds are rejected by requiring that the invariant mass of the K + µ − (K − µ + ) system, with kaons reconstructed under the muon mass hypothesis, is not within ±50 MeV/c 2 around the known J/ψ mass of 3096 MeV/c 2 [11], unless both the kaon and the muon pass stringent particle identification criteria. The expected number of background events from double misidentification in the B 0 s signal mass region is 0.9 ± 0.5. All other backgrounds studied, , and the decay Λ 0 b → Λ(1520) µ + µ − , have been found to be negligible. 4 Differential branching fraction Figure 1 shows the µ + µ − versus the K + K − µ + µ − invariant mass of the selected candidates. The signal decay B 0 s → φµ + µ − is clearly visible in the B 0 s signal region. The determination of the differential branching fraction is performed in six bins of q 2 , corresponding to the binning chosen for the analysis of the decay B 0 → K * 0 µ + µ − [14]. Figure 2 shows the K + K − µ + µ − mass distribution in the six q 2 bins. The signal yields are determined by extended unbinned maximum likelihood fits to the reconstructed B 0 s mass distributions. The signal component is modeled by a double Gaussian function. The resolution parameters are obtained from the resonant B 0 s → J/ψ φ decay. A q 2 -dependent scaling factor, determined with simulated B 0 s → φµ + µ − events, is introduced to account for the observed q 2 dependence of the mass resolution. The combinatorial background is described by a single exponential function. The veto of the radiative tails of the charmonium resonances is accounted for by using a scale factor. The resulting signal yields are given in Table 1. Fitting for the signal yield over the full q 2 region, 174 ± 15 signal candidates are found. A fit of the normalisation mode B 0 s → J/ψ φ yields (20.36 ± 0.14) × 10 3 candidates. The differential branching fraction of the signal decay in the q 2 interval spanning from where N sig and N J/ψ φ denote the yields of the B 0 s → φµ + µ − and B 0 s → J/ψ φ candidates and φµ + µ − and J/ψ φ denote their respective efficiencies. Since the reconstruction and selection efficiency of the signal decay depends on q 2 , a separate efficiency ratio J/ψ φ / φµ + µ − is determined for every q 2 bin. The branching fractions used in Eq. 5 are given by B(B 0 s → J/ψ φ) = (10.50 ± 1.05)×10 −4 [15] and B(J/ψ → µ + µ − ) = (5.93 ± 0.06)×10 −2 [11]. The resulting q 2 -dependent differential branching fraction dB(B 0 s → φµ + µ − )/dq 2 is shown in Fig. 3. Possible contributions from B 0 s decays to K + K − µ + µ − , with the K + K − pair in an S-wave configuration, are neglected in this analysis. The S-wave fraction is expected to Table 1: Signal yield and differential branching fraction dB(B 0 s → φµ + µ − )/dq 2 in six bins of q 2 . Results are also quoted for the region 1 < q 2 < 6 GeV/c 2 where theoretical predictions are most reliable. The first uncertainty is statistical, the second systematic, and the third from the branching fraction of the normalisation channel.  [15] for the K + K − mass window used in this analysis.
The total branching fraction is determined by summing the differential branching fractions in the six q 2 bins. Using the form factor calculations described in Ref. [16] the signal fraction rejected by the charmonium vetoes is determined to be 17.7%. This number is confirmed by a different form factor calculation detailed in Ref. [17]. No uncertainty is assigned to the vetoed signal fraction. Correcting for the charmonium vetoes, the branching fraction ratio B (B 0 The systematic uncertainties will be discussed in detail in Sec. 4.1. Using the known branching fraction of the normalisation channel the total branching fraction is B(B 0 s → φµ + µ − ) = 7.07 +0.64 −0.59 ± 0.17 ± 0.71 × 10 −7 , where the first uncertainty is statistical, the second systematic and the third from the uncertainty on the branching fraction of the normalisation channel.

Systematic uncertainties on the differential branching fraction
The dominant source of systematic uncertainty on the differential branching fraction arises from the uncertainty on the branching fraction of the normalisation channel B 0 s → J/ψ φ (J/ψ → µ + µ − ), which is known to an accuracy of 10% [15]. This uncertainty is fully correlated between all q 2 bins. resonances. The solid curve shows the leading order SM prediction, scaled to the fitted total branching fraction. The prediction uses the SM Wilson coefficients and leading order amplitudes given in Ref. [2], as well as the form factor calculations in Ref. [16]. B 0 s mixing is included as described in Ref. [1]. The dashed curve denotes the leading order prediction scaled to a total branching fraction of 16 × 10 −7 [18].
Many of the systematic uncertainties affect the relative efficiencies J/ψ φ / φµ + µ − that are determined using simulation. The limited size of the simulated samples causes an uncertainty of ∼ 1% on the ratio in each bin. Simulated events are corrected for known discrepancies between simulation and data. The systematic uncertainties associated with these corrections (e.g. tracking efficiency and performance of the particle identification) are typically of the order of 1-2%. The correction procedure for the impact parameter resolution has an effect of up to 5%. Averaging the relative efficiency within the q 2 bins leads to a systematic uncertainty of 1-2%. Other systematic uncertainties of the same magnitude include the trigger efficiency and the uncertainties of the angular distributions of the signal decay B 0 s → φµ + µ − . The influence of the signal mass shape is found to be 0.5%. The background shape has an effect of up to 5%, which is evaluated by using a linear function to describe the mass distribution of the background instead of the nominal exponential shape. Peaking backgrounds cause a systematic uncertainty of 1-2% on the differential branching fraction. The size of the systematics uncertainties on the differential branching fraction, added in quadrature, ranges from 4-6%. This is small compared to the dominant systematic uncertainty of 10% due to the branching fraction of the normalisation channel, which is given separately in Table 1, and the statistical uncertainty.

Angular analysis
The angular observables F L , S 3 , A 6 , and A 9 are determined using unbinned maximum likelihood fits to the distributions of cos θ K , cos θ , Φ, and the invariant mass of the K + K − µ + µ − system. The detector acceptance and the reconstruction and selection of the signal decay distort the angular distributions given in Eqs. 2-4. To account for this angular acceptance effect, an angle-dependent efficiency is introduced that factorises in cos θ K and cos θ , and is independent of the angle Φ, i.e. (cos θ K , cos θ , Φ) = K (cos θ K ) · (cos θ ). The factors K (cos θ K ) and (cos θ ) are determined from fits to simulated events. Even Chebyshev polynomial functions of up to fourth order are used to parametrise K (cos θ K ) and (cos θ ) for each bin of q 2 . The point-to-point dissimilarity method described in Ref. [19] confirms that the angular acceptance effect is well described by the acceptance model.
Taking the acceptances into account and integrating Eq. 1 over two angles, results in The terms ξ i are correction factors with respect to Eqs. 2-4 and are given by the angular integrals (1 + cos 2 θ ) (cos θ )d cos θ , Three two-dimensional maximum likelihood fits in the decay angles and the reconstructed B 0 s mass are performed for each q 2 bin to determine the angular observables. The observable F L is determined in the fit to the cos θ K distribution described by Eq. 6. The cos θ distribution given by Eq. 7 is used to determine A 6 . Both S 3 and A 9 are measured Error bars include statistical and systematic uncertainties added in quadrature. The solid curves are the leading order SM predictions, using the Wilson coefficients and leading order amplitudes given in Ref. [2], as well as the form factor calculations in Ref. [16]. B 0 s mixing is included as described in Ref. [1].
from the Φ distribution, as described by Eq. 8. In the fit of the Φ distribution a Gaussian constraint is applied to the parameter F L using the value of F L determined from the cos θ K distribution. The constraint on F L has negligible influence on the values of S 3 and A 9 . The angular distribution of the background events is fit using Chebyshev polynomial functions of second order. The mass shapes of the signal and background are described by the sum of two Gaussian distributions with a common mean, and an exponential function, respectively.
The measured angular observables are presented in Fig. 4 and Table 2. The 68% confidence intervals are determined using the Feldman-Cousins method [20] and the nuisance parameters are included using the plug-in method [21]. Table 2: Results for the angular observables F L , S 3 , A 6 , and A 9 in bins of q 2 . The first uncertainty is statistical, the second systematic.

Systematic uncertainties on the angular observables
The dominant systematic uncertainty on the angular observables is due to the angular acceptance model. Using the point-to-point dissimilarity method detailed in Ref. [19], the acceptance model is shown to describe the angular acceptance effect for simulated events at the level of 10%. A cross-check of the angular acceptance using the normalisation channel B 0 s → J/ψ φ shows good agreement of the angular observables with the values determined in Refs. [22] and [23]. For the determination of the systematic uncertainty due to the angular acceptance model, variations of the acceptance curves are used that have the largest impact on the angular observables. The resulting systematic uncertainty is of the order of 0.05-0.10, depending on the q 2 bin.
The limited amount of simulated events accounts for a systematic uncertainty of up to 0.02. The simulation correction procedure (for tracking efficiency, impact parameter resolution, and particle identification performance) has negligible effect on the angular observables. The description of the signal mass shape leads to a negligible systematic uncertainty. The background mass model causes an uncertainty of less than 0.02. The model of the angular distribution of the background can have a large effect since the statistical precision of the background sample is limited. To estimate the effect, the parameters describing the background angular distribution are determined in the high B 0 s mass sideband (5416 < m(K + K − µ + µ − ) < 5566 MeV/c 2 ) using a relaxed requirement on the φ mass. The effect is typically 0.05-0.10. Peaking backgrounds cause systematic deviations of the order of 0.01-0.02. The total systematic uncertainties, evaluated by adding all components in quadrature, are small compared to the statistical uncertainties.
In addition, the first angular analysis of the decay B 0 s → φµ + µ − has been performed. The angular observables F L , S 3 , A 6 , and A 9 are determined in bins of q 2 , using the distributions of cos θ K , cos θ , and Φ. The results are summarised in Fig. 4, and the numerical values are given in Table 2. All measured angular observables are consistent with the leading order SM expectation.