Differential branching fraction and angular moments analysis of the decay B0 → K+π−μ+μ− in the K0,2∗(1430)0 region

Measurements of the differential branching fraction and angular moments of the decay B0 → K+π−μ+μ− in the K+π− invariant mass range 1330 < m(K+π−) < 1530 MeV/c2 are presented. Proton-proton collision data are used, corresponding to an integrated luminosity of 3 fb−1 collected by the LHCb experiment. Differential branching fraction measurements are reported in five bins of the invariant mass squared of the dimuon system, q2, between 0.1 and 8.0 GeV2/c4. For the first time, an angular analysis sensitive to the S-, P- and D-wave contributions of this rare decay is performed. The set of 40 normalised angular moments describing the decay is presented for the q2 range 1.1-6.0 GeV2/c4.

of the theory uncertainty on hadronic effects or if it requires a New Physics explanation.
Since short-distance effects should be universal in all b → sµµ transitions, measuring other such transitions can shed light on this situation. Recently, the S-wave contribution to B 0 → K + π − µ + µ − decays has been measured in the 644 < m(K + π − ) < 1200 MeV/c 2 region [9].
Since the dominant structures in the K + π − invariant mass spectrum of B 0 → K + π − µ + µ − above the P-wave K * (892) 0 are resonances in the 1430 MeV/c 2 region, this is a natural region to study. The relevant K * 0 states above the K * (892) 0 mass range are listed in table 1. Throughout this paper, the symbol K * 0 denotes any neutral strange meson in an excited state that decays to a K + π − final state. In the 1430 MeV/c 2 region, contributions are expected from the S-wave K * 0 (1430) 0 , P-wave K * (1410) 0 and D-wave K * 2 (1430) 0 states, as well as the broad P-wave K * (1680) 0 state. The mass region of the higher K * J resonances was studied in ref. [11] with model-dependent theoretical predictions based on QCD form-factors. However, since the form-factors for broad resonances remain poorly known, a more model-independent prescription was provided in ref. [12], which is used in this analysis.
The m(K + π − ) distribution for B 0 → K + π − µ + µ − decays in the range 1.1 < q 2 < 6.0 GeV 2 /c 4 and 630 < m(K + π − ) < 1630 MeV/c 2 is shown in figure 1, where q 2 ≡ m 2 (µ + µ − ). The candidates are obtained using the selection described in section 4 and the background component is subtracted using the sPlot technique [13]. The main structures are observed around the mass of the K * (892) 0 resonance and in the 1430 MeV/c 2 region. This paper presents the first measurements of the differential branching fraction and angular moments of B 0 → K + π − µ + µ − in the region 1330 < m(K + π − ) < 1530 MeV/c 2 . The values of the differential branching fraction are reported in five bins of q 2 between 0.1 and 8.0 GeV 2 /c 4 , and in the range 1.1 < q 2 < 6.0 GeV 2 /c 4 for which the angular moments are also measured. The measurements are based on samples of pp collisions collected by the LHCb experiment in Run 1, corresponding to integrated luminosities of 1.0 fb −1 at a centre-of-mass energy of 7 TeV and 2.0 fb −1 at 8 TeV.  Figure 1. Background-subtracted m(K + π − ) distribution for B 0 → K + π − µ + µ − decays in the range 1.1 < q 2 < 6.0 GeV 2 /c 4 . The region 1330 < m(K + π − ) < 1530 MeV/c 2 is indicated by the blue, hatched area.  Angle conventions for (a) B 0 → K − π + µ − µ + and (b) B 0 → K + π − µ + µ − , as described in ref. [12]. The leptonic and hadronic frames are back-to-back with a commonŷ axis. For the dihedral angle φ between the leptonic and hadronic decay planes, there is an additional sign flip φ → −φ compared to previous LHCb analyses [1][2][3][4].

Angular distribution
The final state of the decay B 0 → K + π − µ + µ − is fully described by five kinematic variables: three decay angles (θ , θ K , φ), m(K + π − ), and q 2 . Figure 2a shows the angle conventions for the B 0 decay (containing a b quark): the back-to-back leptonic and hadronic systems share a commonŷ axis and have oppositex andẑ axes. The negatively charged lepton is used to define the leptonic helicity angle θ for the B 0 . The quadrant of the dihedral angle φ between the dimuon and the K * 0 → K − π + decay planes is determined by requiring the azimuthal angle of the µ − to be zero in the leptonic helicity frame. The azimuthal angle of the K − in the hadronic helicity frame is then equal to φ. Compared to the dihedral angle used in the previous LHCb analyses [1][2][3][4], there is a sign flip, φ → −φ, in the convention used here. For the B 0 decay (containing a b quark), the charge conjugation is performed explicitly, and the angles are shown in figure 2b, where for the B 0 , the µ + -3 -

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and K + directions are used to define the angles. An additional minus sign is added to the dihedral angle when performing the CP conjugation, in order to keep the measured angular observables the same between B 0 and B 0 in the absence of direct CP violation.
In the limit where q 2 is large compared to the square of the muon mass, the CPaveraged differential decay rate of B 0 → K + π − µ + µ − with the K + π − system in a S-, P-, or D-wave configuration can be expanded in an orthonormal basis of angular functions f i (Ω) as where dΩ = dcos θ dcos θ K dφ, and L and R denote the (left-and right-handed) chirality of the lepton system [12]. The sign η L→R i = ±1 depends on whether f i changes sign under θ → π + θ .
The orthonormal angular basis is constructed out of spherical harmonics, Y m l ≡ Y m l (θ , φ), and reduced spherical harmonics, P m l ≡ √ 2πY m l (θ K , 0). The transversity-basis moments of the 41 orthonormal angular functions are given in appendix A. The convention is that the amplitudes correspond to the B 0 decay, with the corresponding amplitudes for the The measured angular observables are averaged over the range 1330 < m(K + π − ) < 1530 MeV/c 2 and 1.1 < q 2 < 6.0 GeV 2 /c 4 . This q 2 range is part of the large-recoil regime where the recoiling K * 0 has a relatively large energy, E K * 0 , as measured in the rest frame of the parent B meson. In the limit Λ QCD /E K * 0 → 0, the uncertainties arising from hadronic effects in the relevant form-factors are reduced at leading order, resulting in more reliable theory predictions [5]. The high-q 2 region above the ψ(2S) resonance is polluted by broad charmonium resonances and is also phase-space suppressed for higher m(K + π − ) masses. Therefore, that region is not considered in this study.
In the present analysis, the first moment, Γ 1 (q 2 ), corresponds to the total decay rate. From this, 40 normalised moments for i ∈ {2, . . . , 41} are defined as These form the set of observables that are measured in the angular moments analysis described in section 8.

Detector and simulation
The LHCb detector [14,15] 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 of the magnet. The tracking system provides a measurement of momentum 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 (PV), the impact parameter (IP), 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. Different types of charged hadrons are distinguished using information from two ring-imaging Cherenkov detectors. 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. The online event selection is performed by a trigger, 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. Simulated signal events are used to determine the effect of the detector geometry, trigger, reconstruction and selection on the angular distributions of the signal and of the B 0 → J/ψ K * (892) 0 mode, which is used for normalisation. Additional simulated samples are used to estimate the contribution from specific background processes. In the simulation, pp collisions are generated using Pythia [16,17] with a specific LHCb configuration [18]. Decays of hadronic particles are described by EvtGen [19], in which final-state radiation is generated using Photos [20]. The interaction of the generated particles with the detector, and its response, are implemented using the Geant4 toolkit [21] as described in ref. [22]. Data-driven techniques are used to correct the simulation for mismodelling of the detector occupancy, the B 0 meson momentum and vertex quality distributions, and particle identification performance.

Selection of signal candidates
The B 0 → K + π − µ + µ − signal candidates are first required to pass the hardware trigger, which selects events containing at least one muon with transverse momentum p T > 1.48 GeV/c in the 7 TeV data or p T > 1.76 GeV/c in the 8 TeV data. In the subsequent software trigger, at least one of the final-state particles is required to have both p T > 1.0 GeV/c and an impact parameter larger than 100 µm with respect to all PVs in the event. Finally, the tracks of two or more of the final-state particles are required to form a vertex significantly displaced from all PVs.
Signal candidates are formed from a pair of oppositely charged tracks identified as muons, combined with two oppositely charged tracks identified as a kaon and a pion. These signal candidates are then required to pass a set of loose preselection requirements, identical to those described in ref. [4] with the exception that the K + π − system is permitted to be in the wider mass range 630 < m(K + π − ) < 1630 MeV/c 2 . This allows the decay B 0 → J/ψ K * (892) 0 to be used as a normalisation mode for the branching fraction measurement. Candidates are required to have good quality vertex and track fits, and a reconstructed B 0 invariant mass in the range 5170 < m(K + π − µ + µ − ) < 5700 MeV/c 2 . From this point onwards, the normalisation mode is selected in the range 796 < m(K + π − ) < 996 MeV/c 2 and the signal in the range 1330 < m(K + π − ) < 1530 MeV/c 2 .

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The backgrounds from combining unrelated particles, mainly from different b and c hadron decays, are referred to as combinatorial. Such backgrounds are suppressed with the use of a Boosted Decision Tree (BDT) [23,24]. The BDT used for the present analysis is identical to that described in ref. [4] and the same working point is used.
Exclusive background processes can mimic the signal if their final states are misidentified or misreconstructed. For the present analysis, the requirements of ref. [4] for the K * (892) 0 region are applied to a wider m(K + π − ) invariant mass window. However, to reduce the expected contamination from peaking background to the level of 2% of the signal yield, it is necessary to modify two of them. First, the requirement to remove contributions from B 0 → J/ψ K * (892) 0 candidates, where the π − (K + ) is misidentified as a µ − (µ + ) and the µ − (µ + ) is misidentified as a π − (K + ), is tightened by extending the invariant mass window of the µ + π − (K + µ − ) system and requiring stricter muon identification criteria. Second, the requirement to remove the contributions from genuine B 0 → K + π − µ + µ − decays where the two hadron hypotheses are interchanged is tightened by requiring stricter hadron identification criteria.

Acceptance correction
The triggering, reconstruction and selection of candidates distorts their kinematic distributions. The dominant acceptance effects are due to the requirements on track momentum and impact parameter.
The method for obtaining the acceptance correction, described in ref. [4], is extended to include the m(K + π − ) dimension. The efficiency is parameterised in terms of Legendre polynomials of order n, L n (x), as As the polynomials are defined over the domain x ∈ [−1, 1], the variables q 2 , φ and m (K + π − ) are used, which are obtained by linearly transforming q 2 , φ and m(K + π − ) to lie in this range. The sum in eq. (5.1) encompasses L n (x) up to fourth order in cos θ and m (K + π − ), sixth order in φ and q 2 , and eighth order in cos θ K . The coefficients c hijkl are determined using a moment analysis of simulated B 0 → K + π − µ + µ − decays, generated according to a phase space distribution. The angular acceptance as a function of cos θ , cos θ K and φ in the region 1.1 < q 2 < 6.0 GeV 2 /c 4 and 1330 < m( The invariant mass m(K + π − µ + µ − ) is used to discriminate between signal and background. The signal distribution is modelled as 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 shape of the mass distribution of the signal are determined from a fit to the Relative efficiency 0 0.5 1 LHCb simulation Figure 3. Relative efficiency in cos θ , cos θ K and φ in the region 1.1 < q 2 < 6.0 GeV 2 /c 4 and 1330 < m(K + π − ) < 1530 MeV/c 2 as determined from a moment analysis of simulated B 0 → K + π − µ + µ − decays, shown as a histogram. The efficiency function is shown by the blue, dashed line.
B 0 → J/ψ K * (892) 0 control mode, as shown in figure 4, and are subsequently fixed when fitting the B 0 → K + π − µ + µ − candidates. An additional component is included in the fit to the control mode to model the contribution from B 0 s → J/ψ K * 0 decays. A single scaling factor is used to correct the width of the Gaussian functions to account for variations in the shape of the mass distribution of the signal observed in simulation, due to the different regions of m(K + π − ) and q 2 between the control mode and signal mode. The combinatorial background is modelled using an exponential function. The fit to B 0 → K + π − µ + µ − candidates in the range 1.1 < q 2 < 6.0 GeV 2 /c 4 is shown in figure 4. The signal yield in the range 1.1 < q 2 < 6.0 GeV 2 /c 4 is 229 ± 21. The fits to B 0 → K + π − µ + µ − candidates in each of the q 2 bins used for the differential branching fraction measurement are shown in appendix B.
In order to obtain the acceptance-corrected yield, the efficiency function described in section 5 is used to evaluate an acceptance weight for each candidate. An average acceptance weight is determined for both the B 0 → J/ψ K * 0 candidates and the signal candidates in each q 2 bin. The acceptance-corrected yield is then equal to the measured yield multiplied by the average weight. The average weight is calculated within the ±50 MeV/c 2 signal window around the mean B 0 mass and also in the background region taken from the upper mass sideband in the range 5350 < m(K + π − µ + µ − ) < 5700 MeV/c 2 . The latter is subsequently used to subtract the background contribution from the average weight obtained in the ±50 MeV/c 2 window, taking into account the extrapolated background yield in this window. This method avoids making any assumption about the unknown angular distribution of the B 0 → K + π − µ + µ − decay.
The results for the differential branching fraction are given in figure 5. The uncertainties shown are the sums in quadrature of the statistical and systematic uncertainties. The results are also presented in table 2. The various sources of the systematic uncertainties are described in section 9.   Table 2. Differential branching fraction of B 0 → K + π − µ + µ − in bins of q 2 for the range 1330 < m(K + π − ) < 1530 MeV/c 2 . The first uncertainty is statistical, the second systematic and the third due to the uncertainty on the B 0 → J/ψ K * (892) 0 and J/ψ → µ + µ − branching fractions.

Angular moments analysis
The angular observables defined in section 2 are determined using a moments analysis of the angular distribution, as outlined in ref. [12]. This approach has the advantage of producing stable measurements with well-defined uncertainties even for small data samples. Similar methods using angular moments are described in refs. [27,28]. The 41 background-subtracted and acceptance-corrected moments are estimated as and the corresponding covariance matrix is estimated as

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Here n sig and n bkg correspond to the candidates in the signal and background regions, respectively. The signal region is defined within ±50 MeV/c 2 of the mean B 0 mass, and the background region in the range 5350 < m(K + π − µ + µ − ) < 5700 MeV/c 2 . The scale factor x is the ratio of the estimated number of background candidates in the signal region over the number of candidates in the background region and is used to normalise the background subtraction. It has been checked in data that the angular distribution of the background is independent of m(K + π − µ + µ − ) within the precision of this measurement, and that the uncertainty on x has negligible impact on the results. The weights, w k , are the reciprocals of the candidates' efficiencies and account for the acceptance, described in section 5. The covariance matrix describing the statistical uncertainties on the 40 normalised moments is computed as The results for the normalised moments, Γ i , are given in figure 6. The uncertainties shown are the sums in quadrature of the statistical and systematic uncertainties. The results are also presented in table 3. The various sources of the systematic uncertainties are described in section 9. The complete set of numerical values for the measured moments and the covariance matrix is provided in ref. [29].
The distributions of each of the decay angles within the signal region are shown in figure 7. The estimated signal distribution is derived from the moments model by evaluating the sum in eq. (2.1), which is found to provide a good representation of the data for each of the decay angles.
The D-wave fraction, F D , is estimated from the moments Γ 5 and Γ 10 as Naively, one would expect a large D-wave contribution in this region, as was seen in the amplitude analysis of B 0 → J/ψ K + π − [26]. However, in B 0 → K + π − µ + µ − no significant D-wave contribution is seen and, with the limited statistics currently available, it is only possible to set an upper limit of F D < 0.29 at 95% confidence level using the approach in ref. [30]. This might be an indication of a large breaking of QCD factorisation due to non-factorizable diagrams where additional gluons are exchanged between the K + π − and the cc, before the J/ψ decays into µ + µ − . For electroweak penguins, similar effects could occur due to charm loops [8]. Additionally, the values of the moments Γ 2 and Γ 3 imply the presence of large interference effects between the S-and P-or D-wave contributions.

Systematic uncertainties
The main sources of systematic uncertainty for the measurements of the differential branching fraction and angular moments are described in detail below and summarised in table 4. They are significantly smaller than the statistical uncertainties. The differential branching fraction and angular moments analysis share several common systematic effects: the statistical uncertainty on the acceptance function due to the    Table 3. Measurement of the normalised moments, Γ i , of the decay B 0 → K + π − µ + µ − in the range 1.1 < q 2 < 6.0 GeV 2 /c 4 and 1330 < m(K + π − ) < 1530 MeV/c 2 . The first uncertainty is statistical and the second systematic.  Table 4. Summary of the main sources of systematic uncertainty for the differential branching fraction and the angular moments analysis. Typical ranges are quoted for the different q 2 bins used in the differential branching fraction measurement, and for the moments measured in the angular analysis. The systematic uncertainties are significantly smaller than the statistical ones.

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size of the simulated sample from which it is determined, differences between data and the simulated decays used to determine the acceptance function and contributions from residual peaking background candidates. The differential branching fraction has, in addition, a systematic uncertainty due to the uncertainty on the branching fraction of the decay B 0 → J/ψ K * (892) 0 , which is dominant and is shown separately in table 2.
The size of the systematic uncertainties associated with the determination of the acceptance correction and residual peaking background contributions are evaluated using pseudoexperiments, in which samples are generated varying one or more parameters. The differential branching fraction and each of the moments are evaluated using both the nominal model and the systematically varied models. In general, the systematic uncertainty is taken as the average of the difference between the nominal and varied models over a large number of pseudoexperiments. The exception to this is the statistical uncertainty of the acceptance function, due to the limited size of the simulated samples, for which the standard deviation is used instead. For this, pseudoexperiments are generated where the acceptance is varied according to the covariance matrix of the moments of the acceptance function.
The effect of differences between the data candidates and the simulated candidates is evaluated using pseudoexperiments, where candidates are generated with an acceptance determined from simulated candidates without applying the corrections for the differences between data and simulation described in section 3.
The effect of residual peaking background contributions is evaluated using pseudoexperiments, where peaking background components are generated in addition to the signal and the combinatorial background. The angular distributions of the peaking backgrounds are taken from data by isolating the decays using dedicated selections.
All other sources of systematic uncertainties investigated, such as the choice of the m(K + π − µ + µ − ) signal model and the resolution in the angular variables, are found to have a negligible impact.

Conclusions
This paper presents measurements of the differential branching fraction and angular moments of the decay B 0 → K + π − µ + µ − in the K + π − invariant mass range 1330 < m(K + π − ) < 1530 MeV/c 2 . The data sample corresponds to an integrated luminosity of 3 fb −1 of pp collision data collected by the LHCb experiment. The differential branching fraction is reported in five narrow q 2 bins between 0.1 and 8.0 GeV 2 /c 4 and in the range 1.1 < q 2 < 6.0 GeV 2 /c 4 , where an angular moments analysis is also performed.
The measured values of the angular observables Γ 2 and Γ 3 point towards the presence of large interference effects between the S-and P-or D-wave contributions. Using only Γ 5 and Γ 10 it is possible to estimate the D-wave fraction, F D , yielding an upper limit of F D < 0.29 at 95% confidence level. This value is lower than naively expected from amplitude analyses of B 0 → J/ψ K + π − decays [26].
The underlying Wilson coefficients may be extracted from the normalised moments and covariance matrix presented in this analysis, when combined with a prediction for the form factors. While first estimates for the form factors are given in ref. [11], no interpretation of the results in terms of the Wilson coefficients is made at this time. With additional -13 -

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input from theory, these results could provide further contributions to understanding the pattern of deviations with respect to SM predictions that has been observed in other b → sµµ transitions.

Acknowledgments
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: It should be noted that in addition to dependence on the amplitudes there is an overall kinematic factor of kq 2 , where k is the B 0 break-up momentum given by and m B is the B 0 mass. Figure 8 shows the fits to the m(K + π − µ + µ − ) distribution in each of the q 2 bins used for the differential branching fraction measurement.  Figure 8. Invariant mass m(K + π − µ + µ − ) distributions of the signal decay B 0 → K + π − µ + µ − in each of the q 2 bins used for the differential branching fraction measurement. The solid black line represents the total fitted function. The individual components of the signal (blue, shaded area) and combinatorial background (red, hatched area) are also shown.

B Mass distributions
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