Measurement of the $C\P$ asymmetry in $B^+ \rightarrow K^+ \mu^+ \mu^-$ decays

A measurement of the $C\P$ asymmetry in $B^+ \rightarrow K^+ \mu^+ \mu^-$ decays is presented using $pp$ collision data, corresponding to an integrated luminosity of 1.0${\,fb}^{-1}$, recorded by the LHCb experiment during 2011 at a centre-of-mass energy of 7 TeV. The measurement is performed in seven bins of $\mu^+ \mu^-$ invariant mass squared in the range ${0.05<q^{2}<22.00{\mathrm{GeV^2/}c^4}}$, excluding the ${J/\psi}$ and $\psi{(2S)}$ resonance regions. Production and detection asymmetries are corrected for using the $B^+ \rightarrow J/\psi K^+$ decay as a control mode. Averaged over all the bins, the $C\P$ asymmetry is found to be ${{\cal A}_{C\P} = 0.000\pm 0.033{(stat.)} \pm0.005 {(syst.)} \pm 0.007{}(J/\psi K^+)}$, where the third uncertainty is due to the $C\P$ asymmetry of the control mode. This is consistent with the Standard Model prediction.

The rare decay B + → K + µ + µ − is a flavour-changing neutral current process mediated by electroweak loop (penguin) and box diagrams. The absence of tree-level diagrams for the decay results in a small value of the Standard Model (SM) prediction for the branching fraction, which is supported by a measurement of (4.36 ± 0.23) × 10 −7 [1]. Physics processes beyond the SM that may enter via the loop and box diagrams could have large effects on observables of the decay. Examples include the decay rate, the µ + µ − forward-backward asymmetry [1][2][3], and the CP asymmetry [2,4], as functions of the µ + µ − invariant mass squared (q 2 ).
The CP asymmetry is defined as where Γ is the decay rate of the mode. This asymmetry is predicted to be of order 10 −4 in the SM [5], but can be significantly enhanced in models beyond the SM [6]. Current measurements including the dielectron mode, A CP (B → K + + − ), from BaBar and Belle give −0.03 ± 0.14 and 0.04 ± 0.10, respectively [2,4], and are consistent with the SM.
The CP asymmetry has already been measured at LHCb in B 0 → K * 0 µ + µ − decays [7], A CP = −0.072 ± 0.040. Assuming that contributions beyond the SM are independent of the flavour of the spectator quark, A CP should be similar for both B + → K + µ + µ − and B 0 → K * 0 µ + µ − decays.
In this Letter, a measurement of A CP in B + → K + µ + µ − decays is presented using pp collision data, corresponding to an integrated luminosity of 1.0 fb −1 , recorded at a centre-of-mass energy of 7 TeV at LHCb in 2011. The inclusion of charge conjugate modes is implied throughout unless explicitly stated.
The LHCb detector [8] 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 (IP) resolution of 20 µm for tracks with high transverse momentum (p T ). Charged hadrons are identified using two ring-imaging Cherenkov detectors [9]. Muons are identified by a system composed of alternating layers of iron and multiwire proportional chambers [10].
Samples of simulated events are used to determine the efficiency of selecting B + → K + µ + µ − signal events and to study certain backgrounds. In the simulation, pp collisions are generated using Pythia 6.4 [11] with a specific LHCb configuration [12]. Decays of hadronic particles are described by EvtGen [13], in which final-state radiation is generated using Photos [14]. The interaction of the generated particles with the detector and its response are implemented using the Geant4 toolkit [15] as described in Ref. [16]. The simulated samples are corrected to reproduce the data distributions of the B + meson p T and vertex χ 2 , the track χ 2 of the kaon, as well as the detector IP resolution, particle identification and momentum resolution.
Candidate events are first required to pass a hardware trigger, which selects muons with p T > 1.48 GeV/c [17]. In the subsequent software trigger, at least one of the final-state particles is required to have p T > 1.0 GeV/c and IP > 100 µm with respect to all primary pp interaction vertices (PVs) in the event. Finally, the tracks of two or more of the final-state particles are required to form a vertex that is displaced from the PVs.
An initial selection is applied to the B + → K + µ + µ − candidates to enhance signal decays and suppress combinatorial background. Candidate B + mesons must satisfy requirements on their direction and flight distance, to ensure consistency with originating from the PV. The decay products must pass criteria regarding the χ 2 IP , where χ 2 IP is defined as the difference in χ 2 of a given PV reconstructed with and without the considered particle. There is also a requirement on the vertex χ 2 of the µ + µ − pair. All the tracks are required to have p T > 250 MeV/c.
Additional background rejection is achieved by using a boosted decision tree (BDT) [18] that implements the AdaBoost algorithm [19]. The BDT uses the p T and χ 2 IP of the muons and the B + meson candidate, as well as the decay time, vertex χ 2 , and flight direction of the B + candidate and the χ 2 IP of the kaon. Data, corresponding to an integrated luminosity of 0.1 fb −1 , are used to optimise this selection, leaving 0.9 fb −1 for the determination of A CP .
Following the multivariate selection, candidate events pass several requirements to remove specific sources of background. Particle identification (PID) criteria are applied to kaon candidates to reduce the number of pions incorrectly identified as kaons. Candidates with µ + µ − invariant mass in the ranges 2.95 < m µµ < 3.18 GeV/c 2 and 3.59 < m µµ < 3.77 GeV/c 2 are removed to reject backgrounds from tree level B + → J/ψ (→ µ + µ − )K + and B + → ψ(2S)(→ µ + µ − )K + decays. Those in the first range are selected as B + → J/ψ K + decays, which are used as a control sample. If m Kµµ < 5.22 GeV/c 2 , the vetoes are extended downwards by 0.25 and 0.19 GeV/c 2 , respectively, to remove the radiative tails of the resonant decays. If 5.35 < m Kµµ < 5.50 GeV/c 2 the vetoes are extended upwards by 0.05 GeV/c 2 to remove misreconstructed resonant candidates that appear at large m µµ and m Kµµ . Further vetoes are applied to remove B + → J/ψ K + events in which the kaon and a muon have been swapped, and contributions from decays involving charm mesons such as B + → D 0 (→ K + π − )π + where both pions are misidentified as muons. After these selection requirements have been applied, there are two sources of background that are difficult to distinguish from the signal. These are B + → K + π + π − and B + → π + µ + µ − decays, which both contribute at the level of 1% of the signal yield. These peaking backgrounds are accounted for during the analysis.
In order to perform a measurement of A CP , the production and detection asymmetries associated with the measurement must be considered. The raw measured asymmetry is, to first order, where the production and detection asymmetries are defined as where R and represent the B meson production rate and kaon detection efficiency, respectively. The detection asymmetry has two components: one due to the different interactions of positive and negative kaons with the detector material, and a left-right asymmetry due to particles of different charges being deflected to opposite sides of the detector by the magnet. The component of the detection asymmetry from muon reconstruction is small and neglected. Since the LHCb experiment reverses the magnetic field, about half of the data used in the analysis is taken with each polarity. Therefore, an average of the measurements with the two polarities is used to suppress significantly the second effect. To account for both the detection and production asymmetries, the decay B + → J/ψ K + is used, which has the same final-state particles as B + → K + µ + µ − and very similar kinematic properties. The CP asymmetry in B + → J/ψ K + decays has been measured as (1 ± 7) × 10 −3 [20,21]. Neglecting the difference in the final-state kinematic properties of the kaon, the production and detection asymmetries are the same for both modes, and the value of the CP asymmetry can be obtained via Differences in the kinematic properties are accounted for by a systematic uncertainty.
In the data set, approximately 1330 B + → K + µ + µ − and 218,000 B + → J/ψ K + signal decays are reconstructed. To measure any variation in A CP as a function of q 2 , which improves the sensitivity of the measurement to physics beyond the SM, the B + → K + µ + µ − dataset is divided into the seven q 2 bins used in Ref. [1]. The measurement is also made in a bin of 1 < q 2 < 6 GeV 2 /c 4 , which is of particular theoretical interest. To determine the number of B + decays in each bin, a simultaneous unbinned maximum likelihood fit is performed to the invariant mass distributions of the B + → K + µ + µ − and B + → J/ψ K + candidates in the range 5.10 < m Kµµ < 5.60 GeV/c 2 . The signal shape is parameterised by a Cruijff function [22], and the combinatorial background is described by an exponential function. All parameters of the signal and combinatorial background are allowed to vary freely in the fit. Additionally, there is background from partially-reconstructed decays such as B 0 → K * 0 (→ K + π − )µ + µ − or B 0 → J/ψ K * 0 (→ K + π − ) where the pion is undetected. For the B + → K + µ + µ − distribution, these decays are fitted by an ARGUS function [23] convolved with a Gaussian function to account for detector resolution. For the B + → J/ψ K + decays the partially-reconstructed background is modelled by another Cruijff function. The shapes of the peaking backgrounds, due to B + → K + π + π − and B + → π + µ + µ − decays, are taken from fits to simulated events.
In each q 2 bin, the B + → J/ψ K + and B + → K + µ + µ − data sets are divided according to the charge of the B + meson and magnet polarity, providing eight distinct subsets. These are fitted simultaneously with the parameters of the signal Cruijff function common for all eight subsets. For each subset, the only independent fitting parameters are the combined yield of the B + and B − decays and the values of A RAW for the signal, control and background modes for each magnet polarity. The fits to the invariant mass distributions of the B + → K + µ + µ − candidates in the full q 2 range are shown in Fig. 1.
The value of A CP for each magnet polarity is determined from Eq. 5, and an average with equal weights is taken to obtain a single value for the q 2 bin. To obtain the final value of A CP for the full dataset, an average is taken of the values in each q 2 bin, weighted according to the signal efficiency and the number of B + → K + µ + µ − decays in the bin, where N i , i , and A i CP are the signal yield, signal efficiency, and the fitted value of the CP asymmetry in the ith q 2 bin.
Several assumptions are made about the backgrounds. The partially-reconstructed Table 1: Systematic uncertainties on A CP from non-cancelling asymmetries arising from kinematic differences between B + → J/ψ K + and B + → K + µ + µ − decays, and fit uncertainties arising from the choice of signal shape, mass fit range and combinatorial background shape, and from the treatment of the asymmetries in the B + → π + µ + µ − and partially-reconstructed (PR) backgrounds. The total is the sum in quadrature of each component. background is assumed to exhibit no CP asymmetry. For B + → π + µ + µ − , A CP is also assumed to be zero [24]. For the B + → K + π + π − decay, A CP in each q 2 bin is taken from a recent LHCb measurement [25]. The effect of these assumptions on the result is investigated as a systematic uncertainty. Various sources of systematic uncertainty are considered. The analysis relies on the assumption that the B + → K + µ + µ − and B + → J/ψ K + decays have the same final-state kinematic distributions, so that the relation in Eq. 5 is exact. To estimate the bias associated with this assumption, the kinematic distributions of B + → J/ψ K + decays are reweighted to match those of B + → K + µ + µ − , and the value of A RAW is recalculated. The variables used are the momentum, p T and pseudorapidity of the B + and K + mesons, as well as the B + decay time and the position of the kaon in the detector. The difference between the two values of A RAW for each variable is taken as the systematic uncertainty. The total systematic uncertainty associated to the different kinematic behaviour of the two decays in each q 2 bin is calculated by adding each individual contribution in quadrature.
The choice of fit model also introduces systematic uncertainties. The fit is repeated using a different signal model, replacing the Cruijff function with the sum of two Crystal Ball functions [26] that have the same mean and tail parameters, but different Gaussian widths. The difference in the value of A CP using these two fits is assigned as the uncertainty. The fit is also repeated using a reduced mass range of 5.17 < m Kµµ < 5.60 GeV/c 2 to investigate the effect of excluding the partially-reconstructed background. The difference in results obtained by modelling the combinatorial background using a second-order polynomial, rather than an exponential function, produces a small systematic uncertainty.
Uncertainties also arise from the assumptions made about the asymmetries in background events. Phenomena beyond the SM could cause the CP asymmetry Table 2: Values of A CP and the signal yields in the seven q 2 bins, the weighted average, and their associated uncertainties.

Stat.
Syst. q 2 bin ( GeV 2 /c 4 ) in B + → π + µ + µ − decays to be large [24], and so the analysis is performed again for values of A CP (B + → π + µ + µ − ) = ±0.5, with the larger of the two deviations in A CP (B + → K + µ + µ − ) taken as the systematic uncertainty. As the partially-reconstructed background can arise from B 0 → K * 0 µ + µ − decays, the value of A CP for this source background is taken to be −0.072 [7], the value from the LHCb measurement, neglecting any further CP violation in angular distributions. The difference in the fit result compared to the zero A CP hypothesis is taken as the systematic uncertainty. Variations in A CP (B + → K + π + π − ) have a negligible effect on the final result. A summary of the systematic uncertainties is shown in Table 1. The value of A CP calculated by performing the fits on the data set integrated over q 2 is consistent with that from the weighted average of the q 2 bins. The results for A CP in each q 2 bin and the weighted average are displayed in Table 2, as well as in Fig. 2. The value of the raw asymmetry in B + → J/ψ K + determined from the fit is −0.016 ± 0.002. The CP asymmetry in B + → K + µ + µ − decays is measured to be A CP = 0.000 ± 0.033 (stat.) ± 0.005 (syst.) ± 0.007 (J/ψ K + ), where the third uncertainty is due to the uncertainty on the known value of A CP (B + → J/ψ K + ). This compares with the current world average of −0.05 ± 0.13 [20], and previous measurements including the dielectron final-state [2,4]. This result is consistent with the SM, as well as the B 0 → K * 0 µ + µ − decay mode, and improves the precision of the current world average for the dimuon mode by a factor of four. With the recent observation of resonant structure in the low-recoil region above the ψ(2S) resonance [27], care should be taken when interpreting the result in this region. Interesting effects due to physics beyond the SM are possible through interference with this resonant structure, and could be investigated in a future update of the measurement of A CP .