EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-PH-EP-2014-191 LHCb-PAPER-2014-032 August 4, 2014

Measurement of $C\\!P$ asymmetries in the decays ${B^{0}\rightarrow
K^{*0}\mu^{+}\mu^{-}}$ and ${B^{+}\rightarrow K^{+}\mu^{+}\mu^{-}}$

The LHCb collaboration†††Authors are listed at the end of this paper.

The direct $C\\!P$ asymmetries of the decays ${B^{0}\rightarrow
K^{*0}\mu^{+}\mu^{-}}$ and ${B^{+}\rightarrow K^{+}\mu^{+}\mu^{-}}$ are
measured using $pp$ collision data corresponding to an integrated luminosity
of 3.0$\mbox{\,fb}^{-1}$ collected with the LHCb detector. The respective
control modes ${B^{0}\rightarrow J/\psi K^{*0}}$ and ${B^{+}\rightarrow J/\psi
K^{+}}$ are used to account for detection and production asymmetries. The
measurements are made in several intervals of $\mu^{+}\mu^{-}$ invariant mass
squared, with the $\phi(1020)$ and charmonium resonance regions excluded.
Under the hypothesis of zero $C\\!P$ asymmetry in the control modes, the
average values of the asymmetries are

$\displaystyle{\cal A}_{C\\!P}({B^{0}\rightarrow K^{*0}\mu^{+}\mu^{-}})$
$\displaystyle=-0.035\pm 0.024\pm 0.003,$ $\displaystyle{\cal
A}_{C\\!P}({B^{+}\rightarrow K^{+}\mu^{+}\mu^{-}})$
$\displaystyle=\phantom{-}0.012\pm 0.017\pm 0.001,$

where the first uncertainties are statistical and the second are due to
systematic effects. Both measurements are consistent with the Standard Model
prediction of small $C\\!P$ asymmetry in these decays.

Published in JHEP

© CERN on behalf of the LHCb collaboration, license CC-BY-4.0.

##  1 Introduction

The processes ${B^{0}\rightarrow K^{*0}(\rightarrow
K^{+}\pi^{-})\mu^{+}\mu^{-}}$ and ${B^{+}\rightarrow K^{+}\mu^{+}\mu^{-}}$ are
rare decays111The inclusion of charge conjugate modes is implied unless
explicitly stated. of $B$ mesons involving $b\rightarrow s$ quark-level
transitions, and have small branching fractions, measured as $(1.06\pm
0.10)\times 10^{-6}$ [1] and $(4.36\pm 0.23)\times 10^{-7}$ [2]. In the
Standard Model (SM) there are no tree-level Feynman diagrams for these
processes, which proceed via box or electroweak loop (penguin) diagrams. The
SM amplitudes are suppressed at loop order, increasing the sensitivity of
measurements in these decay channels to physics beyond the SM. Additionally,
leading form-factor uncertainties cancel in the measurement of asymmetries,
allowing for precise theoretical predictions. Examples include the isospin
asymmetry [3], the zero crossing point of the $\mu^{+}\mu^{-}$ forward-
backward asymmetry [4, 1], and the direct $C\\!P$ asymmetry, ${\cal
A}_{C\\!P}$.

This paper describes measurements of ${\cal A}_{C\\!P}$ in ${B^{0}\rightarrow
K^{*0}\mu^{+}\mu^{-}}$ and ${B^{+}\rightarrow K^{+}\mu^{+}\mu^{-}}$ decays
using data corresponding to 3.0$\mbox{\,fb}^{-1}$ of integrated luminosity
collected by LHCb in 2011 and 2012, at centre-of-mass energies of 7 and
8$\mathrm{\,Te\kern-1.00006ptV}$, respectively. The direct $C\\!P$ asymmetry
is defined as

${\cal A}_{C\\!P}\equiv\frac{\Gamma(\kern
1.79993pt\overline{\kern-1.79993ptB}{}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{(*)}\mu^{+}\mu^{-})-\Gamma({B\rightarrow
K^{(*)}\mu^{+}\mu^{-}})}{\Gamma(\kern
1.79993pt\overline{\kern-1.79993ptB}{}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{(*)}\mu^{+}\mu^{-})+\Gamma({B\rightarrow
K^{(*)}\mu^{+}\mu^{-}})},$ (1)

where $\Gamma$ is the decay width for the given mode. Non-SM physics
contributions could produce interfering diagrams, enhancing the magnitude of
${\cal A}_{C\\!P}$ in ${B^{0}\rightarrow K^{*0}\mu^{+}\mu^{-}}$ decays from
the SM prediction of $\mathcal{O}(10^{-3})$ [5] to values up to $\pm 0.15$
[6]. Measurements have already been obtained at LHCb using a data set
corresponding to an integrated luminosity of 1.0$\mbox{\,fb}^{-1}$, collected
in 2011, ${\cal A}_{C\\!P}({B^{0}\rightarrow K^{*0}\mu^{+}\mu^{-}})=-0.072\pm
0.040$ [7] and ${\cal A}_{C\\!P}({B^{+}\rightarrow
K^{+}\mu^{+}\mu^{-}})=0.000\pm 0.034$ [8], which are the dominant
contributions to the world-average values [9]. These are consistent both with
the SM predictions and with previous results from BaBar [10] and Belle [11].

##  2 Detector and simulation

The LHCb detector [12] is a single-arm forward spectrometer covering the
pseudorapidity range $2<\eta<5$, 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 (VELO) surrounding the
$pp$ interaction region, a large-area silicon-strip detector located upstream
of a dipole magnet with a bending power of about $4{\rm\,Tm}$, and three
stations of silicon-strip detectors and straw drift tubes [13] placed
downstream of the magnet. The polarity of the dipole magnet is reversed
periodically throughout data-taking. The tracking system provides a
measurement of momentum, $p$, with a relative uncertainty that varies from
0.4% at low momentum to 0.6% at 100${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The
minimum distance of a track to a primary $pp$ interaction vertex (PV), the
impact parameter (IP), is measured with a resolution of $(15+29/\mbox{$p_{\rm
T}$}){\,\upmu\rm m}$, where $p_{\rm T}$ is the component of $p$ transverse to
the beam, in ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. Different types of
charged hadrons are distinguished using information from two ring-imaging
Cherenkov (RICH) detectors [14]. 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 [15]. The trigger [16] 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 events are used in the process of selecting candidates, examining
background contributions, and in determining the efficiency of the selections.
In the simulation, $pp$ collisions are generated using Pythia [17,
*Sjostrand:2007gs] with a specific LHCb configuration [19]. Decays of hadronic
particles are described by EvtGen [20], in which final-state radiation is
generated using Photos [21]. The interaction of the generated particles with
the detector and its response are implemented using the Geant4 toolkit [22,
*Agostinelli:2002hh] as described in Ref. [24]. The simulated samples are
reweighted to model more accurately the data distributions in variables used
in the analysis. These include the $p_{\rm T}$ of the $B$ meson, the number of
tracks in the event, and the $\chi^{2}$ of the vertex fit to the final-state
tracks, which may differ due to misalignments of the detector and mismodelling
of the material description in the VELO region. In addition, information about
the IP and momentum resolution is used. The particle identification (PID)
performance is corrected to match the data using
$D^{*+}\rightarrow(D^{0}\rightarrow K^{-}\pi^{+})\pi^{+}$ and
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\rightarrow\mu^{+}\mu^{-}$
control channels. The ${B^{+}\rightarrow K^{+}\mu^{+}\mu^{-}}$ samples are
also reweighted for the $p_{\rm T}$ of the decay products.

##  3 Selection of events

Candidates are first required to pass the hardware trigger, which selects
muons with $\mbox{$p_{\rm T}$}>1.48{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. In
the subsequent software trigger, at least one of the final-state particles is
required to have both $\mbox{$p_{\rm
T}$}>0.8{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and IP $>100{\,\upmu\rm m}$ with
respect to all of the PVs in the event. Finally, the tracks of two or more of
the final-state particles are required to form a vertex that is significantly
displaced from the PVs.

All ${B^{0}\rightarrow K^{*0}\mu^{+}\mu^{-}}$ and ${B^{+}\rightarrow
K^{+}\mu^{+}\mu^{-}}$ candidates must pass the same initial selection
criteria. A requirement on the $B$ candidate vertex fit $\chi^{2}$ per degree
of freedom is applied to provide a good quality vertex fit. Additionally, the
angle between the momentum vector of the $B$ candidate and the vector between
the primary and $B$ candidate decay vertices must be less than 14 mrad, and
the $B$ candidates must be consistent with originating from the PV. The tracks
from the $B$ candidate decay products are required to be well separated from
the PV, helping to reject events where a final-state track does not come from
the decay vertex of the $B$ meson. The kaons, pions and muons must be
positively identified by PID information from the RICH detectors and muon
systems, combined using likelihood functions.

This initial selection is followed by a more stringent selection using
multivariate methods, based on a boosted decision tree (BDT) [25, 26]. For
${B^{+}\rightarrow K^{+}\mu^{+}\mu^{-}}$ decays, simulated signal decays are
used for the BDT training, along with data from the upper mass sideband,
$5700<m(K^{+}\mu^{+}\mu^{-})<6000{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$,
which is not used in the remainder of the analysis. The BDT uses a selection
of geometric and kinematic variables and has an efficiency of 90% for signal
while removing 95% of background. Following previous analyses [1], the
${B^{0}\rightarrow K^{*0}\mu^{+}\mu^{-}}$ BDT training uses a signal sample
containing background-subtracted data from the ${B^{0}\rightarrow
J/\psi(\rightarrow\mu^{+}\mu^{-})K^{*0}}$ control mode, and a background
sample from the upper mass sideband
$5350<m(K^{+}\pi^{-}\mu^{+}\mu^{-})<7000{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$.
This reduces combinatorial background to small levels.

The $C\\!P$ asymmetry can vary as a function of the $\mu^{+}\mu^{-}$ invariant
mass squared, $q^{2}$ [5], and hence the measurement is made in several
$q^{2}$ bins. The analysis is restricted to ${B^{0}\rightarrow
K^{*0}\mu^{+}\mu^{-}}$ candidates in the range
$0.1<q^{2}<19.0{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$, and
${B^{+}\rightarrow K^{+}\mu^{+}\mu^{-}}$ candidates satisfying
$0.1<q^{2}<22.0{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$. Three regions
are then removed from both samples, corresponding to the $\phi(1020)$
($0.98<q^{2}<1.10{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$),
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$
($8.0<q^{2}<11.0{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$), and
$\psi{(2S)}$ ($12.5<q^{2}<15.0{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$)
resonances. The remaining ${B\rightarrow K^{(*)}\mu^{+}\mu^{-}}$ candidates
are divided into ${17\,(14)}$ bins that are approximately
1${\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$ wide. The control decay modes,
${B\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{(*)}}$, are
selected from the range
$8.41<q^{2}<10.24{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$. The
${B^{0}\rightarrow K^{*0}\mu^{+}\mu^{-}}$ candidates are required to have a
$K^{+}\pi^{-}$ mass that lies within
$100{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the known $K^{*0}$ mass [9].

Tracks near the edge of the detector acceptance can be swept out by the
magnetic field, depending on their charge. This results in the observation of
highly asymmetric decay rates for such candidates, as fewer or no candidates
with the opposite flavour can be reconstructed. Therefore, fiducial criteria
are applied to remove candidates that are reconstructed near the edges of the
acceptance. These regions are removed by requiring that the kaon associated
with the ${B\rightarrow K^{(*)}\mu^{+}\mu^{-}}$ candidates has momentum which
satisfies $p_{z}>2500\,(2000){\mathrm{\,Me\kern-1.00006ptV\\!/}c}$ and
$|p_{x}|/(p_{z}-2500\,(2000))>0.33$, where $p_{z,x}$ are the components of the
momentum, measured in ${\mathrm{\,Me\kern-1.00006ptV\\!/}c}$, in the direction
of beam travel and in the bending plane, respectively.

There are several background contributions in the signal mass region that
require specific vetoes. For the ${B^{+}\rightarrow K^{+}\mu^{+}\mu^{-}}$
decays, there is a background from $B^{+}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}(\rightarrow K^{+}\pi^{-})\pi^{+}$
decays where the pions are misidentified as muons. These are removed by
computing the mass of the $K^{+}\mu^{-}$ pair under the $K^{+}\pi^{-}$ mass
hypothesis, and rejecting candidates that satisfy
$1850<m(K^{+}\pi^{-})<1880{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. Both
modes have backgrounds from
${B\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{(*)}}$ events in
which a muon from the decay of the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$ meson and a final-state hadron are misidentified as each other. These
events are vetoed if $m(h^{\pm}\mu^{\mp})$, calculated under the dimuon
hypothesis, lies within $60{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the
known ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ or $\psi{(2S)}$ mass, and
the hadron can be positively identified as a muon. Other backgrounds for the
${B^{0}\rightarrow K^{*0}\mu^{+}\mu^{-}}$ decay mode include
${B^{+}\rightarrow K^{+}\mu^{+}\mu^{-}}$ events combined with a random pion in
the event, $\mathchar 28931\relax^{0}_{b}\rightarrow pK^{-}\mu^{+}\mu^{-}$
decays where at least one hadron is misidentified, and
$B^{0}_{s}\rightarrow\phi\mu^{+}\mu^{-}$ events in which a kaon is
misidentified as a pion. These are suppressed using a combination of mass and
PID requirements similar to those used for the
${B\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{(*)}}$
background. The final peaking background comes from ${B^{0}\rightarrow
K^{*0}\mu^{+}\mu^{-}}$ events in which the kaon and pion are misidentified as
each other. Since the charge of the kaon identifies the produced meson as
either a $B^{0}$ or a $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$, a
misidentification can directly lead to an incorrect asymmetry being measured.

Therefore, the PID information is used to remove events in which the
likelihood functions indicate that the reconstructed pion has a higher
probability of being a true kaon than the reconstructed kaon. After the vetoes
are applied, all of these backgrounds are reduced to less than 1% of the level
of the signal, and are neglected for the rest of the analysis, as are the
singly Cabibbo-suppressed backgrounds $B^{0}\rightarrow\rho^{0}\mu^{+}\mu^{-}$
and $B^{+}\rightarrow\pi^{+}\mu^{+}\mu^{-}$.

##  4 Measurement of direct $C\\!P$ asymmetries

Asymmetries in production rate and detection efficiency may bias the
measurements and must be accounted for. To first order and for small
asymmetries, the raw asymmetry measured, $\mathcal{A}_{\rm raw}$, is related
to the $C\\!P$ asymmetry by

$\mathcal{A}_{\rm raw}({B\rightarrow K^{(*)}\mu^{+}\mu^{-}})={\cal
A}_{C\\!P}({B\rightarrow
K^{(*)}\mu^{+}\mu^{-}})+\mathcal{A}_{P}+\mathcal{A}_{D},$ (2)

where any terms from $B^{0}$ mixing are neglected, and the production,
$\mathcal{A}_{P}$, and detection, $\mathcal{A}_{D}$, asymmetries are given by

$\mathcal{A}_{P}\equiv\frac{\sigma(\overline{B})-\sigma(B)}{\sigma(\overline{B})+\sigma(B)}\qquad\mathrm{and}\qquad\mathcal{A}_{D}\equiv\frac{\epsilon(\overline{f})-\epsilon(f)}{\epsilon(\overline{f})+\epsilon(f)},$
(3)

where $\sigma$ represents the $B$ meson production cross-section in the LHCb
acceptance, and $\epsilon$ is the detection and reconstruction efficiency for
a given final state. The detection asymmetry has two components, one that
arises from the different interaction cross-sections of positive and negative
particles with the detector material, and another that is due to differences
between the left- and right-hand sides of the detector. The latter effect can
be reduced by using data collected with both polarities of the magnet, and
taking the average. To account for the remaining asymmetries, the control
modes ${B\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{(*)}}$ are
used. These modes have the same particles in the final state and similar
kinematic properties to the ${B\rightarrow K^{(*)}\mu^{+}\mu^{-}}$ modes, and
hence have similar production and detection asymmetries. Negligible direct CP
violation is expected for the control modes, as confirmed by measurements [9,
27]. Assuming that the control modes have zero $C\\!P$ asymmetry, ${\cal
A}_{C\\!P}$ can be calculated from

${\cal A}_{C\\!P}({B\rightarrow K^{(*)}\mu^{+}\mu^{-}})=\mathcal{A}_{\rm
raw}({B\rightarrow K^{(*)}\mu^{+}\mu^{-}})-\mathcal{A}_{\rm
raw}({B\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{(*)}}).$ (4)

Differences in the production and detection efficiencies of the control and
signal modes are considered as sources of systematic uncertainty.

Figure 1: Unbinned maximum-likelihood fits to the $K^{+}\pi^{-}\mu^{+}\mu^{-}$
mass distributions of the integrated data set for (a) ${B^{0}\rightarrow
K^{*0}\mu^{+}\mu^{-}}$ and (b) $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}\mu^{+}\mu^{-}$ decays for one
magnet polarity, and (c) ${B^{0}\rightarrow K^{*0}\mu^{+}\mu^{-}}$ and (d)
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}\mu^{+}\mu^{-}$ for the other. The
blue, solid line represents the total fit, the red, short-dashed line
represents the signal component and the grey, long-dashed line represents the
combinatorial background. Figure 2: Unbinned maximum-likelihood fits to the
$K^{+}\mu^{+}\mu^{-}$ mass distributions of the integrated data set for (a)
${B^{+}\rightarrow K^{+}\mu^{+}\mu^{-}}$ and (b) $B^{-}\rightarrow
K^{-}\mu^{+}\mu^{-}$ decays for one magnet polarity, and (c)
${B^{+}\rightarrow K^{+}\mu^{+}\mu^{-}}$ and (d) $B^{-}\rightarrow
K^{-}\mu^{+}\mu^{-}$ for the other. The blue, solid line represents the total
fit, the red, short-dashed line represents the signal component and the grey,
long-dashed line represents the combinatorial background.

The raw asymmetries are determined via unbinned maximum-likelihood fits to the
mass distributions of the candidates. The data set contains approximately
1,000,000 ${B^{+}\rightarrow J/\psi K^{+}}$, 320,000 ${B^{0}\rightarrow J/\psi
K^{*0}}$, 4600 ${B^{+}\rightarrow K^{+}\mu^{+}\mu^{-}}$, and 2200
${B^{0}\rightarrow K^{*0}\mu^{+}\mu^{-}}$ signal events in the $B$ mass range
$5170<m(K^{(*)}\mu^{+}\mu^{-})<5700{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$.
The fit shapes used are very similar for all four modes. The signal component
is the sum of two Crystal Ball functions [28], with common mean and tail
parameters, but different widths, and the combinatorial background is modelled
by an exponential function. The ${B^{0}\rightarrow J/\psi K^{*0}}$ mode has an
extra contribution arising from $B^{0}_{s}\rightarrow J/\psi\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ decays, which is modelled by the
same pair of Crystal Ball functions as the signal, but with the mean shifted
by the $B^{0}_{s}-B^{0}$ mass difference [9].

All four data sets are split by magnet polarity and charge of the kaon, and
the ${B\rightarrow K^{(*)}\mu^{+}\mu^{-}}$ data sets are also divided into the
${17\,(14)}$ $q^{2}$ bins. The fit is first performed on the four
${B\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{(*)}}$ data sets,
where the higher number of candidates allows a precise determination of the
fit shape parameters. Values for the combined $B$ yield and the raw asymmetry,
$\mathcal{A}_{\rm raw}({B\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{(*)}})$, for each magnet polarity are determined from the fit. The
raw asymmetries in the ${B\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{(*)}}$ modes are measured to be $-0.015\,(-0.012)$ for one magnet
polarity and $-0.013\,(-0.014)$ for the other. The signal shape parameters are
then fixed for the fit to the ${B\rightarrow K^{(*)}\mu^{+}\mu^{-}}$ mode in
each $q^{2}$ bin, and the values for the combined $B$ yield and
$\mathcal{A}_{\rm raw}({B\rightarrow K^{(*)}\mu^{+}\mu^{-}})$ are determined
from these fits in the same way. The fits performed on the ${B\rightarrow
K^{(*)}\mu^{+}\mu^{-}}$ data sets split by kaon charge and magnet polarity are
shown in Figs. 1 and 2.

The values for ${\cal A}_{C\\!P}({B\rightarrow K^{(*)}\mu^{+}\mu^{-}})$ are
determined according to Eq. 4 for each magnet polarity, and the arithmetic
mean of the resulting two values provides the final value for ${\cal
A}_{C\\!P}$ in each $q^{2}$ bin. To obtain an overall value of ${\cal
A}_{C\\!P}$ across all $q^{2}$ bins, an average, weighted by the signal yield
and efficiency in each bin, is calculated,

${\cal A}_{C\\!P}=\frac{\sum_{i}(N_{i}{\cal
A}_{C\\!P}^{i})/\epsilon_{i}}{\sum_{i}N_{i}/\epsilon_{i}},$ (5)

where $N_{i}$, $\epsilon_{i}$, and ${\cal A}_{C\\!P}^{i}$ are the signal
yield, signal efficiency, and the value of the $C\\!P$ asymmetry in the
$i\mathrm{th}$ $q^{2}$ bin.

Table 1: Summary of the sources of systematic uncertainty for the measurements of ${\cal A}_{C\\!P}({B\rightarrow K^{(*)}\mu^{+}\mu^{-}})$. The ranges shown in parentheses indicate the minimum and maximum values of the systematic uncertainties in different $q^{2}$ bins, while the numbers outside the parentheses are the values averaged over $q^{2}$. These may be outside the ranges as the uncertainties are determined by methods affected by statistical fluctuations. There is no systematic uncertainty due to duplicate candidates in the ${B^{+}\rightarrow K^{+}\mu^{+}\mu^{-}}$ decay.  Source | ${B^{0}\rightarrow K^{*0}\mu^{+}\mu^{-}}$ | ${B^{+}\rightarrow K^{+}\mu^{+}\mu^{-}}$
---|---|---
Kinematic differences | $0.0015\,(0.0025-0.0118)$ | $0.0007\,(0.0007-0.0040)$
Signal shape | $0.0018\,(0.0003-0.0057)$ | $0.0001\,(0.0001-0.0039)$
Background shape | $0.0015\,(0.0016-0.0205)$ | $0.0002\,(0.0001-0.0012)$
Duplicate candidates | $0.0015\,(0.0001-0.0061)$ | $-$
Total | $0.0032\,(0.0063-0.0215)$ | $0.0007\,(0.0011-0.0043)$

##  5 Systematic uncertainties

The systematic effects that require consideration are all of a similar
magnitude for the ${B^{0}\rightarrow K^{*0}\mu^{+}\mu^{-}}$ decays, and are
listed in order of importance for the ${B^{+}\rightarrow K^{+}\mu^{+}\mu^{-}}$
analysis.

In the construction of Eq. 4, an assumption is made that the kinematic
properties of the particles in the control and signal modes are identical, and
therefore the production and detection asymmetries are the same for both
modes. However, because the muons from the control mode must originate from
the decay of a ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ meson, there is
a slight difference in the kinematic properties. To estimate the effect that
this may have on the result, the data from the control mode are reweighted to
the signal mode data so that the distributions match in a chosen kinematic
variable. The raw asymmetry, which is approximately the sum of the production
and detection asymmetries, is then recalculated from a fit to the weighted
data. The difference between the values with the weighted and unweighted data
is taken as a contribution to the systematic uncertainty. This procedure is
repeated for eight kinematic variables including the momenta and
pseudorapidity of the particles and the decay time of the $B$ meson. The sum
in quadrature of the differences for each variable is assigned as the overall
systematic uncertainty in each $q^{2}$ bin.

In the mass fit, different functions are used to check if the shape used
affects the result. The fit is repeated, first replacing the signal component
with an Apollonios function, which is the exponential of a hyperbola combined
with a low-mass power-law tail [29], and a second time with a second-order
Chebychev polynomial modelling the combinatorial background. The differences
in the fit results with respect to the nominal fit are assigned as systematic
uncertainties.

For the ${B^{0}\rightarrow K^{*0}\mu^{+}\mu^{-}}$ channel, a further source of
systematic uncertainty arises from events that contain duplicate candidates,
one with the kaon and pion identified correctly, and one with them swapped.
The PID requirement described earlier removes one of each pair of these
candidates, but may occasionally select the incorrect candidate. The fit is
repeated with both candidates weighted by a factor of one-half, i.e. assuming
both are equally likely to be correct, rather than with one of them removed.
The difference in the fit result is taken as the systematic uncertainty
associated with the choice of selection of events with kaon-to-pion swaps.
Backgrounds from other decays that are not fully removed by the selection are
assumed to exhibit no $C\\!P$ asymmetry.

None of the systematic uncertainties is larger than 20% of the statistical
uncertainty in any $q^{2}$ bin, and the overall systematic uncertainty is less
than 7% of the statistical one. A summary of the systematic uncertainties,
indicating the range of each uncertainty across the $q^{2}$ bins along with
the values for the full data set, is given in Table 1.

Table 2: Values of ${\cal A}_{C\\!P}$ in ${B^{0}\rightarrow K^{*0}\mu^{+}\mu^{-}}$ decays in each of the 14 $q^{2}$ bins used in the analysis. The first uncertainties are statistical and the second are systematic. $q^{2}$ bin [$\mathrm{GeV}^{2}/{c}^{4}$] | Yield |  ${\cal A}_{C\\!P}$
---|---|---
0.10$-$ | 0.98 |  $304$$\,\pm\,$ | $18$ |  $-$0.087 $\,\pm\,$ | 0.060 $\,\pm\,$ | 0.006
1.10$-$ | 2.00 |  $105$$\,\pm\,$ | $11$ |  $-$0.176 $\,\pm\,$ | 0.106 $\,\pm\,$ | 0.009
2.00$-$ | 3.00 |  $120$$\,\pm\,$ | $13$ |  $-$0.146 $\,\pm\,$ | 0.102 $\,\pm\,$ | 0.008
3.00$-$ | 4.00 |  $101$$\,\pm\,$ | $12$ |  $-$0.013 $\,\pm\,$ | 0.113 $\,\pm\,$ | 0.014
4.00$-$ | 5.00 |  $120$$\,\pm\,$ | $13$ |  $-$0.076 $\,\pm\,$ | 0.106 $\,\pm\,$ | 0.012
5.00$-$ | 6.00 |  $143$$\,\pm\,$ | $13$ |  $-$0.030 $\,\pm\,$ | 0.097 $\,\pm\,$ | 0.009
6.00$-$ | 7.00 |  $144$$\,\pm\,$ | $14$ |  +0.020 $\,\pm\,$ | 0.095 $\,\pm\,$ | 0.008
7.00$-$ | 8.00 |  $177$$\,\pm\,$ | $15$ |  +0.099 $\,\pm\,$ | 0.087 $\,\pm\,$ | 0.006
11.0$-$ | 11.8 |  $144$$\,\pm\,$ | $14$ |  $-$0.021 $\,\pm\,$ | 0.093 $\,\pm\,$ | 0.007
11.8$-$ | 12.5 |  $147$$\,\pm\,$ | $14$ |  +0.031 $\,\pm\,$ | 0.093 $\,\pm\,$ | 0.022
15.0$-$ | 16.0 |  $205$$\,\pm\,$ | $16$ |  $-$0.125 $\,\pm\,$ | 0.075 $\,\pm\,$ | 0.009
16.0$-$ | 17.0 |  $216$$\,\pm\,$ | $16$ |  $-$0.002 $\,\pm\,$ | 0.074 $\,\pm\,$ | 0.010
17.0$-$ | 18.0 |  $169$$\,\pm\,$ | $14$ |  $-$0.059 $\,\pm\,$ | 0.085 $\,\pm\,$ | 0.009
18.0$-$ | 19.0 |  $105$$\,\pm\,$ | $11$ |  $-$0.054 $\,\pm\,$ | 0.108 $\,\pm\,$ | 0.016
Figure 3: Values of ${\cal A}_{C\\!P}$ for ${B^{0}\rightarrow K^{*0}\mu^{+}\mu^{-}}$ decays in each of the the 14 $q^{2}$ bins used in the analysis. The error bars are the sum of the statistical and systematic uncertainties in quadrature. The dashed line represents the weighted average value, and the grey band indicates $\pm 1\sigma$. The vertical red lines show the $\phi(1020)$, ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$, and $\psi{(2S)}$ regions, which are vetoed.  Table 3: Values of ${\cal A}_{C\\!P}$ in ${B^{+}\rightarrow K^{+}\mu^{+}\mu^{-}}$ decays in each of the 17 $q^{2}$ bins used in the analysis. The first uncertainties are statistical and the second are systematic. $q^{2}$ bin [$\mathrm{GeV}^{2}/{c}^{4}$] | Yield |  ${\cal A}_{C\\!P}$
---|---|---
0.10$-$ | 0.98 |  $387$$\,\pm\,$ | $22$ |  +0.088 $\,\pm\,$ | 0.057 $\,\pm\,$ | 0.001
1.10$-$ | 2.00 |  $277$$\,\pm\,$ | $19$ |  $-$0.004 $\,\pm\,$ | 0.068 $\,\pm\,$ | 0.002
2.00$-$ | 3.00 |  $367$$\,\pm\,$ | $22$ |  +0.042 $\,\pm\,$ | 0.059 $\,\pm\,$ | 0.001
3.00$-$ | 4.00 |  $334$$\,\pm\,$ | $21$ |  $-$0.034 $\,\pm\,$ | 0.063 $\,\pm\,$ | 0.001
4.00$-$ | 5.00 |  $307$$\,\pm\,$ | $20$ |  $-$0.021 $\,\pm\,$ | 0.064 $\,\pm\,$ | 0.001
5.00$-$ | 6.00 |  $332$$\,\pm\,$ | $21$ |  +0.031 $\,\pm\,$ | 0.062 $\,\pm\,$ | 0.002
6.00$-$ | 7.00 |  $355$$\,\pm\,$ | $22$ |  +0.026 $\,\pm\,$ | 0.060 $\,\pm\,$ | 0.001
7.00$-$ | 8.00 |  $371$$\,\pm\,$ | $22$ |  +0.041 $\,\pm\,$ | 0.059 $\,\pm\,$ | 0.002
11.0$-$ | 11.8 |  $232$$\,\pm\,$ | $18$ |  $-$0.047 $\,\pm\,$ | 0.076 $\,\pm\,$ | 0.002
11.8$-$ | 12.5 |  $247$$\,\pm\,$ | $17$ |  +0.018 $\,\pm\,$ | 0.070 $\,\pm\,$ | 0.002
15.0$-$ | 16.0 |  $287$$\,\pm\,$ | $19$ |  +0.120 $\,\pm\,$ | 0.065 $\,\pm\,$ | 0.004
16.0$-$ | 17.0 |  $287$$\,\pm\,$ | $19$ |  +0.028 $\,\pm\,$ | 0.066 $\,\pm\,$ | 0.001
17.0$-$ | 18.0 |  $349$$\,\pm\,$ | $21$ |  $-$0.030 $\,\pm\,$ | 0.058 $\,\pm\,$ | 0.001
18.0$-$ | 19.0 |  $222$$\,\pm\,$ | $17$ |  $-$0.061 $\,\pm\,$ | 0.074 $\,\pm\,$ | 0.003
19.0$-$ | 20.0 |  $121$$\,\pm\,$ | $13$ |  $-$0.048 $\,\pm\,$ | 0.105 $\,\pm\,$ | 0.003
20.0$-$ | 21.0 |  $95$$\,\pm\,$ | $12$ |  $-$0.012 $\,\pm\,$ | 0.120 $\,\pm\,$ | 0.003
21.0$-$ | 22.0 |  $50$$\,\pm\,$ | $8$ |  $-$0.290 $\,\pm\,$ | 0.161 $\,\pm\,$ | 0.004

##  6 Results

The results in each $q^{2}$ bin are shown in Table 2 and Fig. 3 for
${B^{0}\rightarrow K^{*0}\mu^{+}\mu^{-}}$, and Table 3 and Fig. 4 for
${B^{+}\rightarrow K^{+}\mu^{+}\mu^{-}}$. The values of the $C\\!P$
asymmetries in ${B\rightarrow K^{(*)}\mu^{+}\mu^{-}}$ decays are

$\displaystyle{\cal A}_{C\\!P}({B^{0}\rightarrow K^{*0}\mu^{+}\mu^{-}})$
$\displaystyle=-0.035\pm 0.024\pm 0.003,$ $\displaystyle{\cal
A}_{C\\!P}({B^{+}\rightarrow K^{+}\mu^{+}\mu^{-}})$
$\displaystyle=\phantom{-}0.012\pm 0.017\pm 0.001,$

where the first uncertainties are statistical and the second are due to
systematic effects. They are obtained under the hypothesis of zero $C\\!P$
asymmetry in the control modes, ${B^{0}\rightarrow J/\psi K^{*0}}$ and
${B^{+}\rightarrow J/\psi K^{+}}$. Both of these results, which supersede the
previous 1.0$\mbox{\,fb}^{-1}$ measurements [7, 8], are consistent with the SM
predictions, and the uncertainties on the measurements are almost a factor of
two smaller than the previous best values.

Figure 4: Values of ${\cal A}_{C\\!P}$ for ${B^{+}\rightarrow
K^{+}\mu^{+}\mu^{-}}$ decays in each of the the 17 $q^{2}$ bins used in the
analysis. The error bars are the sum of the statistical and systematic
uncertainties in quadrature. The dashed line represents the weighted average
value, and the grey band indicates $\pm 1\sigma$. The vertical red lines show
the $\phi(1020)$, ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$, and
$\psi{(2S)}$ regions, which are vetoed.

## Acknowledgements

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, FAPERJ and FINEP (Brazil); NSFC
(China); CNRS/IN2P3 (France); BMBF, DFG, HGF and MPG (Germany); SFI (Ireland);
INFN (Italy); FOM and NWO (The Netherlands); MNiSW and NCN (Poland); MEN/IFA
(Romania); MinES and FANO (Russia); MinECo (Spain); SNSF and SER
(Switzerland); NASU (Ukraine); STFC (United Kingdom); NSF (USA). The Tier1
computing centres are supported by IN2P3 (France), KIT and BMBF (Germany),
INFN (Italy), NWO and SURF (The Netherlands), PIC (Spain), GridPP (United
Kingdom). We are indebted to the communities behind the multiple open source
software packages on which we depend. We are also thankful for the computing
resources and the access to software R&D tools provided by Yandex LLC
(Russia). Individual groups or members have received support from EPLANET,
Marie Skłodowska-Curie Actions and ERC (European Union), Conseil général de
Haute-Savoie, Labex ENIGMASS and OCEVU, Région Auvergne (France), RFBR
(Russia), XuntaGal and GENCAT (Spain), Royal Society and Royal Commission for
the Exhibition of 1851 (United Kingdom).

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Sutcliffe53, K. Swientek27, S. Swientek9, V. Syropoulos42, M. Szczekowski28,
P. Szczypka39,38, D. Szilard2, T. Szumlak27, S. T’Jampens4, M. Teklishyn7, G.
Tellarini16,f, F. Teubert38, C. Thomas55, E. Thomas38, J. van Tilburg41, V.
Tisserand4, M. Tobin39, S. Tolk42, L. Tomassetti16,f, D. Tonelli38, S. Topp-
Joergensen55, N. Torr55, E. Tournefier4, S. Tourneur39, M.T. Tran39, M.
Tresch40, A. Tsaregorodtsev6, P. Tsopelas41, N. Tuning41, M. Ubeda Garcia38,
A. Ukleja28, A. Ustyuzhanin63, U. Uwer11, V. Vagnoni14, G. Valenti14, A.
Vallier7, R. Vazquez Gomez18, P. Vazquez Regueiro37, C. Vázquez Sierra37, S.
Vecchi16, J.J. Velthuis46, M. Veltri17,h, G. Veneziano39, M. Vesterinen11, B.
Viaud7, D. Vieira2, M. Vieites Diaz37, X. Vilasis-Cardona36,p, A. Vollhardt40,
D. Volyanskyy10, D. Voong46, A. Vorobyev30, V. Vorobyev34, C. Voß62, H.
Voss10, J.A. de Vries41, R. Waldi62, C. Wallace48, R. Wallace12, J. Walsh23,
S. Wandernoth11, J. Wang59, D.R. Ward47, N.K. Watson45, D. Websdale53, M.
Whitehead48, J. Wicht38, D. Wiedner11, G. Wilkinson55, M.P. Williams45, M.
Williams56, F.F. Wilson49, J. Wimberley58, J. Wishahi9, W. Wislicki28, M.
Witek26, G. Wormser7, S.A. Wotton47, S. Wright47, S. Wu3, K. Wyllie38, Y.
Xie61, Z. Xing59, Z. Xu39, Z. Yang3, X. Yuan3, O. Yushchenko35, M. Zangoli14,
M. Zavertyaev10,b, L. Zhang59, W.C. Zhang12, Y. Zhang3, A. Zhelezov11, A.
Zhokhov31, L. Zhong3, A. Zvyagin38.

1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Milano, Milano, Italy
22Sezione INFN di Padova, Padova, Italy
23Sezione INFN di Pisa, Pisa, Italy
24Sezione INFN di Roma Tor Vergata, Roma, Italy
25Sezione INFN di Roma La Sapienza, Roma, Italy
26Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
27AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
28National Center for Nuclear Research (NCBJ), Warsaw, Poland
29Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
30Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
31Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
32Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
33Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
34Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
35Institute for High Energy Physics (IHEP), Protvino, Russia
36Universitat de Barcelona, Barcelona, Spain
37Universidad de Santiago de Compostela, Santiago de Compostela, Spain
38European Organization for Nuclear Research (CERN), Geneva, Switzerland
39Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
40Physik-Institut, Universität Zürich, Zürich, Switzerland
41Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
42Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
43NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
44Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
45University of Birmingham, Birmingham, United Kingdom
46H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
47Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
48Department of Physics, University of Warwick, Coventry, United Kingdom
49STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
50School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
51School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
52Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
53Imperial College London, London, United Kingdom
54School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
55Department of Physics, University of Oxford, Oxford, United Kingdom
56Massachusetts Institute of Technology, Cambridge, MA, United States
57University of Cincinnati, Cincinnati, OH, United States
58University of Maryland, College Park, MD, United States
59Syracuse University, Syracuse, NY, United States
60Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
61Institute of Particle Physics, Central China Normal University, Wuhan,
Hubei, China, associated to 3
62Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11
63National Research Centre Kurchatov Institute, Moscow, Russia, associated to
31
64Instituto de Fisica Corpuscular (IFIC), Universitat de Valencia-CSIC,
Valencia, Spain, associated to 36
65KVI - University of Groningen, Groningen, The Netherlands, associated to 41
66Celal Bayar University, Manisa, Turkey, associated to 38

aUniversidade Federal do Triângulo Mineiro (UFTM), Uberaba-MG, Brazil
bP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
cUniversità di Bari, Bari, Italy
dUniversità di Bologna, Bologna, Italy
eUniversità di Cagliari, Cagliari, Italy
fUniversità di Ferrara, Ferrara, Italy
gUniversità di Firenze, Firenze, Italy
hUniversità di Urbino, Urbino, Italy
iUniversità di Modena e Reggio Emilia, Modena, Italy
jUniversità di Genova, Genova, Italy
kUniversità di Milano Bicocca, Milano, Italy
lUniversità di Roma Tor Vergata, Roma, Italy
mUniversità di Roma La Sapienza, Roma, Italy
nUniversità della Basilicata, Potenza, Italy
oAGH - University of Science and Technology, Faculty of Computer Science,
Electronics and Telecommunications, Kraków, Poland
pLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
qHanoi University of Science, Hanoi, Viet Nam
rUniversità di Padova, Padova, Italy
sUniversità di Pisa, Pisa, Italy
tScuola Normale Superiore, Pisa, Italy
uUniversità degli Studi di Milano, Milano, Italy