EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-EP-2016-141 LHCb-PAPER-2016-012 15 June 2016

Measurements of the S-wave fraction in $B^{0}\rightarrow
K^{+}\pi^{-}\mu^{+}\mu^{-}$ decays and the $B^{0}\rightarrow
K^{\ast}(892)^{0}\mu^{+}\mu^{-}$ differential branching fraction

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

A measurement of the differential branching fraction of the decay
${B^{0}\rightarrow K^{\ast}(892)^{0}\mu^{+}\mu^{-}}$ is presented together
with a determination of the S-wave fraction of the $K^{+}\pi^{-}$ system in
the decay $B^{0}\rightarrow K^{+}\pi^{-}\mu^{+}\mu^{-}$. The analysis is based
on $pp$-collision data corresponding to an integrated luminosity of 3 fb-1
collected with the LHCb experiment. The measurements are made in bins of the
invariant mass squared of the dimuon system, $q^{2}$. Precise theoretical
predictions for the differential branching fraction of $B^{0}\rightarrow
K^{\ast}(892)^{0}\mu^{+}\mu^{-}$ decays are available for the $q^{2}$ region
$1.1<q^{2}<6.0\,{\rm GeV}^{2}/c^{4}$. In this $q^{2}$ region, for the
$K^{+}\pi^{-}$ invariant mass range $796<m_{K\pi}<996\,{\rm MeV}/c^{2}$, the
S-wave fraction of the $K^{+}\pi^{-}$ system in $B^{0}\rightarrow
K^{+}\pi^{-}\mu^{+}\mu^{-}$ decays is found to be

$F_{\rm S}=0.101\pm 0.017({\rm stat})\pm 0.009({\rm syst}),$

and the differential branching fraction of $B^{0}\rightarrow
K^{\ast}(892)^{0}\mu^{+}\mu^{-}$ decays is determined to be

${\rm d}\mathcal{B}/{\rm d}q^{2}=(0.342_{\,-0.017}^{\,+0.017}({\rm
stat})\pm{0.009}({\rm syst})\pm 0.023({\rm norm}))\times 10^{-7}c^{4}/{\rm
GeV}^{2}.$

The differential branching fraction measurements presented are the most
precise to date and are found to be in agreement with Standard Model
predictions.

Published in JHEP 11 (2016) 047

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

##  1 Introduction

The decay ${{B}^{0}}\\!\rightarrow{{K}^{*0}}{\mu^{+}\mu^{-}}$ proceeds via a
${b}\\!\rightarrow{s}\,{\ell^{+}}{\ell^{-}}$ flavour-changing neutral-current
transition. In the Standard Model (SM), this transition is forbidden at tree
level and must therefore occur via a loop-level process. Extensions to the SM
predict new particles that can contribute to the
${b}\\!\rightarrow{s}\,{\ell^{+}}{\ell^{-}}$ process and affect the rate and
angular distribution of the decay. Recently, global analyses of measurements
involving ${b}\\!\rightarrow{s}\,{\ell^{+}}{\ell^{-}}$ processes have reported
significant deviations from SM predictions [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11,
12, 13, 14, 15]. These deviations could be explained either by new particles
[3, 4, 10, 11, 14, 15, 16] or by unexpectedly large hadronic effects [17, 13,
9].

In this paper, the symbol ${{K}^{*0}}$ denotes any neutral strange meson in an
excited state that decays to a $K^{+}$ and a $\pi^{-}$.111Inclusion of charge
conjugate processes is implied throughout this paper unless otherwise noted.
For invariant masses of the $K^{+}\pi^{-}$ system in the range considered in
this analysis, the ${{K}^{*0}}$ decay products are predominantly found in a P-
or S-wave state. The fractional size of the scalar (S-wave) component of the
${{K}^{+}}{{\pi}^{-}}$ system ($F_{\rm S}$) depends on the squared invariant
mass of the dimuon system (${q^{2}}$). This dependence is expected to be
similar to that of the longitudinal polarisation fraction ($F_{\rm L}$) of the
${K}^{*}(892)^{0}$ meson [18, 19, 20].

The S-wave fraction is predicted to be maximal in the ${q^{2}}$ range
$1.0<{q^{2}}<6.0{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$ [18, 19, 20]. A
previous analysis by the LHCb collaboration set an upper limit of $F_{\rm
S}<0.07$ at 68% confidence level for invariant masses of the $K^{+}\pi^{-}$
system in the range $792<m_{K\pi}<992{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$
[21]. The measurement was performed by exploiting the phase shift of the
${K}^{*}(892)^{0}$ Breit–Wigner function around the corresponding pole mass.

In all previous determinations of the differential branching fraction of
${{B}^{0}}\\!\rightarrow{K}^{*}(892)^{0}{\mu^{+}\mu^{-}}$ decays [21, 22, 23,
24, 25], the ${K}^{*}(892)^{0}$ was selected by requiring a window of size
80–380${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ around the known
${K}^{*}(892)^{0}$ mass, but no correction was made for the scalar fraction.
This fraction was assumed to be small and was treated as a systematic
uncertainty. The measurements of the differential branching fraction of
${{B}^{0}}\\!\rightarrow{K}^{*}(892)^{0}{\mu^{+}\mu^{-}}$ decays are included
in global analyses of ${b}\\!\rightarrow{s}\,{\ell^{+}}{\ell^{-}}$ processes.
As these analyses make use of theory predictions which are made purely for the
resonant P-wave part of the $K^{+}\pi^{-}$ system, an accurate assessment of
the S-wave component in ${{B}^{0}}\\!\rightarrow{{K}^{*0}}{\mu^{+}}{\mu^{-}}$
decays is critical.

In this paper, the first measurement of $F_{\rm S}$ in
${{B}^{0}}\\!\rightarrow{{K}^{*0}}{\mu^{+}\mu^{-}}$ decays is presented. The
measurement is performed through a fit to the kaon helicity angle [21, 26],
$\theta_{K}$, and the $m_{K\pi}$ spectrum, in the range
$644<m_{K\pi}<1200{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. Motivated by
previous estimates of the S-wave fraction [21, 18, 19, 20], $F_{\rm S}$ is
also determined in a narrower window of
$796<m_{K\pi}<996{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. The values of
$F_{\rm S}$ are reported in eight bins of ${q^{2}}$ of approximately
$2{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$ width, and in two larger bins
$1.1<{q^{2}}<6.0{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$ and
$15.0<{q^{2}}<19.0{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$. The choice of
${q^{2}}$ bins is identical to that of Ref. [27].

The measurements of $F_{\rm S}$ allow the determination of the differential
branching fraction of the
${{{B}^{0}}\\!\rightarrow{K}^{*}(892)^{0}{\mu^{+}}{\mu^{-}}}$ decay. The
differential branching fraction is determined by normalising the
${{{B}^{0}}\\!\rightarrow{K}^{*}(892)^{0}{\mu^{+}}{\mu^{-}}}$ yield in each
${q^{2}}$ bin to the total event yield of the
${{{B}^{0}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{*0}}}$ control channel, where the
$J/\psi\rightarrow\mu^{+}\mu^{-}$ decay mode is used. The measurements are
made using a $pp$-collision data sample recorded by the LHCb experiment in Run
1, corresponding to an integrated luminosity of 3$\mbox{\,fb}^{-1}$. These
data were collected at centre-of-mass energies of $7$ and
$8\mathrm{\,Te\kern-1.00006ptV}$ during 2011 and 2012 respectively. The
differential branching fraction measurement is complementary to the angular
analysis presented in Ref. [27], and supersedes that of Ref. [21]. The latter
analysis was performed on a 1$\mbox{\,fb}^{-1}$ subset of the Run 1 data
sample.

This paper is organised as follows. Section 2 describes the angular and
$m_{K\pi}$ distributions of
${{B}^{0}}\\!\rightarrow{{K}^{*0}}{\mu^{+}}{\mu^{-}}$ decays with the
$K^{+}\pi^{-}$ system in a P- or S-wave state. Section 3 describes the LHCb
detector and the procedure used to generate simulated data. The reconstruction
and selection of
${{{B}^{0}}\\!\rightarrow{{K}^{+}}{{\pi}^{-}}{\mu^{+}\mu^{-}}}$ candidates are
described in Sec. 4. Section 5 describes the parameterisation of the mass
distributions and Sec. 6 describes the determination of $F_{\rm S}$, including
the method used to correct for the detection and selection biases. The
measurement of the differential branching fraction of
${{B}^{0}}\\!\rightarrow{K}^{*}(892)^{0}{\mu^{+}\mu^{-}}$ decays is presented
in Sec. 7. The systematic uncertainties affecting the measurements are
discussed in Sec. 8. Finally, the conclusions are presented in Sec. 9.

##  2 The angular distribution and $F_{\rm S}$

The final state of the ${{B}^{0}}\\!\rightarrow{{K}^{*0}}{\mu^{+}}{\mu^{-}}$
decay is completely described by ${q^{2}}$, and the three decay angles,
$\vec{\Omega}\equiv(\cos{\theta_{K}},\,\cos{\theta_{\ell}},\,\phi)$ [21]. The
angle between the $\mu^{+}$ ($\mu^{-}$) and the direction opposite to that of
the ${{B}^{0}}$ (${{\kern 1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}}$)
meson in the rest frame of the dimuon system is denoted by $\theta_{\ell}$.
The angle between the direction of the $K^{+}$ ($K^{-}$) and the ${{B}^{0}}$
(${{\kern 1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}}$) meson in the rest
frame of the ${{K}^{*0}}$ (${{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}^{*0}}$) is denoted by $\theta_{K}$.
The angle between the plane defined by the dimuon pair and the plane defined
by the kaon and pion in the ${{B}^{0}}$ (${{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}}$) rest frame is denoted by
$\phi$.

In the limit that the dimuon mass is large compared to the mass of the muons
(${q^{2}}\gg 4m_{\mu}^{2}$), this choice of the angular basis allows the
differential decay rates of
${{B}^{0}}\\!\rightarrow{{K}^{*0}}{\mu^{+}\mu^{-}}$ and ${{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}}\\!\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}^{*0}}{\mu^{+}\mu^{-}}$ decays to be
written as

$\begin{split}\\!\\!\\!\\!\\!\\!\\!\frac{\mathrm{d}^{5}(\Gamma+\overline{\Gamma})}{\mathrm{d}m_{K\pi}\mathrm{d}{q^{2}}\,\mathrm{d}\vec{\Omega}}=\frac{9}{32\pi}\left[\frac{}{}\right.&\\!\\!\\!\\!\left.\frac{}{}{(I_{1}^{s}+\bar{I}_{1}^{s})}\sin^{2}\theta_{K}(1+3\cos
2\theta_{\ell})+{(I_{1}^{c}+\bar{I}_{1}^{c})}\cos^{2}\theta_{K}(1-\cos
2\theta_{\ell})~{}+\right.\\\
&\\!\\!\\!\\!\left.\frac{}{}{(I_{3}+\bar{I}_{3})}\sin^{2}\theta_{K}\sin^{2}\theta_{\ell}\cos
2\phi+{{(I_{4}+\bar{I}_{4})\sin 2\theta_{K}\sin
2\theta_{\ell}\cos\phi}}~{}+\right.\\\
&\\!\\!\\!\\!\frac{}{}\left.{{{(I_{5}+\bar{I}_{5})}\sin
2\theta_{K}\sin\theta_{\ell}\cos\phi}}+(I_{6s}+\bar{I}_{6s})\sin^{2}\theta_{K}\cos\theta_{\ell}~{}+\right.\\\
&\\!\\!\\!\\!\frac{}{}\left.{{{(I_{7}+\bar{I}_{7})}\sin
2\theta_{K}\sin\theta_{\ell}\sin\phi}}+{{{(I_{8}+\bar{I}_{8})}\sin
2\theta_{K}\sin 2\theta_{\ell}\sin\phi}}~{}+\right.\\\
&\\!\\!\\!\\!\frac{}{}\left.(I_{9}+\bar{I}_{9})\sin^{2}\theta_{K}\sin^{2}\theta_{\ell}\sin
2\phi+(I_{10}+\bar{I}_{10})(1-\cos 2\theta_{\ell})~{}+\right.\\\
&\\!\\!\\!\\!\frac{}{}\left.(I_{11}+\bar{I}_{11})\cos\theta_{K}(1-\cos
2\theta_{\ell})~{}+\right.\\\
&\\!\\!\\!\\!\frac{}{}\left.(I_{14}+\bar{I}_{14})\sin\theta_{K}\sin
2\theta_{\ell}\cos\phi+(I_{15}+\bar{I}_{15})\sin\theta_{K}\sin\theta_{\ell}\cos\phi~{}+\right.\\\
&\\!\\!\\!\\!\frac{}{}\left.(I_{16}+\bar{I}_{16})\sin\theta_{K}\sin\theta_{\ell}\sin\phi+(I_{17}+\bar{I}_{17})\sin\theta_{K}\sin
2\theta_{\ell}\sin\phi\frac{}{}\right],\\\ \end{split}$ (1)

where $\Gamma$ and $\overline{\Gamma}$ denote the decay rates of the
${{B}^{0}}$ and ${{\kern 1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}}$
respectively. The 15 coefficients $I_{j}$ ($\bar{I}_{j}$) are bilinear
combinations of the ${{K}^{*0}}$ (${{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}^{*0}}$) decay amplitudes and vary
with ${q^{2}}$ and $m_{K\pi}$. The numbering of the coefficients follows the
convention used in Ref. [27]. Coefficients $I_{j}$ with $j\leq 9$ involve
P-wave amplitudes only, coefficient $I_{10}$ involves S-wave amplitudes only
and coefficients with $11\leq j\leq 17$ describe the interference between P-
and S-wave amplitudes [28].

The polarity of the LHCb dipole magnet, discussed in Sec. 3, is reversed
periodically. Coupled with the fact that ${{B}^{0}}$ and ${{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}}$ decays are studied
simultaneously, this results in a symmetric detection efficiency in $\phi$.
Therefore, the angular distribution is simplified by performing a
transformation of the $\phi$ angle such that

$\phi^{\prime}=\begin{cases}\phi+\pi&\text{~{}if~{}}\phi<0\\\
\phi&\text{~{}otherwise},\end{cases}$ (2)

which results in the cancellation of terms in Eq. 1 that have a $\sin\phi$ or
$\cos\phi$ dependence.

The remaining $I_{j}$ and $\bar{I}_{j}$ coefficients can be written in terms
of the decay amplitudes given in Ref. [27]. Defining
$\vec{\Omega}^{\prime}\equiv(\cos{\theta_{K}},\,\cos{\theta_{\ell}},\,\phi^{\prime})$,
the resulting differential decay rate has the form

$\begin{split}\frac{\mathrm{d}^{5}(\Gamma+\overline{\Gamma})}{\mathrm{d}m_{K\pi}\mathrm{d}{q^{2}}\,\mathrm{d}\vec{\Omega}^{\prime}}=&\,\frac{1}{4\pi}G_{\rm
S}\left|f_{\rm LASS}(m_{K\pi})\right|^{2}(1-\cos{2\theta_{\ell}})~{}+\\\
&\frac{3}{4\pi}G_{\rm P}^{0}\left|f_{\rm
BW}(m_{K\pi})\right|^{2}\cos^{2}\theta_{K}(1-\cos{2\theta_{\ell}})~{}+\\\
&\frac{\sqrt{3}}{2\pi}{\rm Re}\left[\left(G_{\rm SP}^{\rm Re}+iG_{\rm SP}^{\rm
Im}\right)f_{\rm LASS}(m_{K\pi})f^{\ast}_{\rm
BW}(m_{K\pi})\right]\cos{\theta_{K}}(1-\cos{2\theta_{\ell}})+~{}\\\
&\frac{9}{16\pi}G_{\rm P}^{\perp\parallel}\left|f_{\rm
BW}(m_{K\pi})\right|^{2}\sin^{2}\theta_{K}\left(1+\frac{1}{3}\cos{2\theta_{\ell}}\right)+~{}\\\
&\frac{3}{8\pi}S_{3}(G_{\rm P}^{0}+G_{\rm P}^{\perp\parallel})\left|f_{\rm
BW}(m_{K\pi})\right|^{2}\sin^{2}\theta_{K}\sin^{2}\theta_{\ell}\cos{2\phi^{\prime}}+~{}\\\
&\frac{3}{2\pi}A_{\rm FB}(G_{\rm P}^{0}+G_{\rm
P}^{\perp\parallel})\left|f_{\rm
BW}(m_{K\pi})\right|^{2}\sin^{2}\theta_{K}\cos{\theta_{\ell}}+~{}\\\
&\frac{3}{4\pi}S_{9}(G_{\rm P}^{0}+G_{\rm P}^{\perp\parallel})\left|f_{\rm
BW}(m_{K\pi})\right|^{2}\sin^{2}\theta_{K}\sin^{2}\theta_{\ell}\sin{2\phi^{\prime}},\end{split}$
(3)

where $f_{\rm BW}(m_{K\pi})$ denotes the $m_{K\pi}$ dependence of the resonant
P-wave component, which is modelled using a relativistic Breit–Wigner
function. The S-wave component is modelled using the LASS parameterisation
[29], $f_{\rm LASS}(m_{K\pi})$. The exact definitions of the P- and S-wave
line shapes are given in Appendix A. The real-valued coefficients $G_{\rm
S}^{\phantom{R}}$, $G_{\rm SP}^{\rm Re}$, $G_{\rm SP}^{\rm Im}$, $G_{\rm
P}^{0}$ and $G_{\rm P}^{\perp\parallel}$ are bilinear combinations of the
$q^{2}$-dependent parts of the ${{K}^{*0}}$ (${{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}^{*0}}$) helicity amplitudes
$A_{i}^{L,R}({q^{2}})$ ($\overline{A}_{i}^{L,R}({q^{2}})$) and are given by

$\displaystyle G_{\rm S}^{\phantom{R}}=$
$\displaystyle\,|A_{S}^{L}({q^{2}})|^{2}+|A_{S}^{R}({q^{2}})|^{2}+|\overline{A}_{S}^{L}({q^{2}})|^{2}+|\overline{A}_{S}^{R}({q^{2}})|^{2},$
$\displaystyle G_{\rm SP}^{\rm Re}+iG_{\rm SP}^{\rm Im}=$
$\displaystyle\,A_{S}^{L}A_{0}^{L\ast}+A_{S}^{R}A_{0}^{R\ast}+\overline{A}_{S}^{L}\overline{A}_{0}^{L\ast}+\overline{A}_{S}^{R}\overline{A}_{0}^{R\ast},$
$\displaystyle G_{\rm P}^{0}=$
$\displaystyle\,|A_{0}^{L}({q^{2}})|^{2}+|A_{0}^{R}({q^{2}})|^{2}+|\overline{A}_{0}^{L}({q^{2}})|^{2}+|\overline{A}_{0}^{R}({q^{2}})|^{2},$
(4) $\displaystyle G_{\rm P}^{\perp\parallel}=$
$\displaystyle\displaystyle\sum\limits_{i=\perp,\parallel}|A_{i}^{L}({q^{2}})|^{2}+|A_{i}^{R}({q^{2}})|^{2}+|\overline{A}_{i}^{L}({q^{2}})|^{2}+|\overline{A}_{i}^{R}({q^{2}})|^{2},$

where $L$ and $R$ denote the (left- and right-handed) chiralities of the
dimuon system. These coefficients are determined through the extended maximum
likelihood fit described in Sec. 6.2. The coefficients $S_{3}$, $A_{\rm FB}$
and $S_{9}$ are $C\\!P$-averaged observables that are defined in Ref. [27].
The integral of Eq. 3 with respect to $\cos{\theta_{\ell}}$ and
$\phi^{\prime}$ is independent of these observables. However, detection
effects that are either asymmetric or non-uniform in $\cos{\theta_{\ell}}$ and
$\phi^{\prime}$ introduce a residual dependence on these observables. In this
analysis, $S_{3}$, $A_{\rm FB}$ and $S_{9}$ are set to their measured values
[27]. The systematic uncertainty associated with this choice is negligible.

Using the definitions of Eq. 2, the S-wave fraction $F_{\rm S}$ in the range
$a<m_{K\pi}<b$ can be determined from the coefficients $G_{\rm
S}^{\phantom{R}}$ and $G_{\rm P}^{0,\perp\parallel}$, through

$F_{\rm S}|_{a}^{b}=\frac{G_{\rm
S}^{\phantom{R}}\int_{a}^{b}{\textrm{d}m_{K\pi}\left|f_{\rm
LASS}(m_{K\pi})\right|^{2}}}{G_{\rm
S}\int_{a}^{b}\textrm{d}m_{K\pi}\left|f_{\rm
LASS}(m_{K\pi})\right|^{2}+\left(G_{\rm P}^{0}+G_{\rm
P}^{\perp\parallel}\right)\int_{a}^{b}\textrm{d}m_{K\pi}\left|f_{\rm
BW}(m_{K\pi})\right|^{2}}.$ (5)

##  3 Detector and simulation

The LHCb detector [30, 31] 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 divided into three sub-systems: a silicon-strip vertex detector
surrounding the $pp$ interaction region, a large-area silicon-strip detector
that is located upstream of a dipole magnet with a bending power of about
$4{\mathrm{\,Tm}}$, and three stations of silicon-strip detectors and straw
drift tubes situated downstream of the magnet. The tracking system provides a
measurement of the momentum, $p$, of charged particles with a relative
uncertainty that varies from 0.5% at low momentum to 1.0% at
200${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The minimum distance of a track to
a primary vertex (PV), the impact parameter, is measured with a resolution of
$(15+29/\mbox{$p_{\mathrm{T}}$}){\,\upmu\mathrm{m}}$, where $p_{\mathrm{T}}$
is the component of the momentum 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. 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 [32], 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.

A large sample of simulated events is used to determine the effect of the
detector geometry, trigger, and the selection criteria on the angular
distribution of the signal, and to determine the ratio of efficiencies between
the signal and the
${{{B}^{0}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{*0}}}$ normalisation mode. In the simulation, $pp$ collisions are
generated using Pythia [33, *Sjostrand:2007gs] with a specific LHCb
configuration [35]. The decay of the ${B}^{0}$ meson is described by EvtGen
[36], which generates final-state radiation using Photos [37]. As described in
Ref. [38], the Geant4 toolkit [39, *Agostinelli:2002hh] is used to implement
the interaction of the generated particles with the detector and the detector
response. Data-driven corrections are applied to the simulation following the
procedure of Ref. [27]. These corrections account for the small level of
mismodelling of the detector occupancy, the ${{B}^{0}}$ momentum and vertex
quality, and the particle identification (PID) performance.

##  4 Selection of signal candidates

The ${{B}^{0}}\\!\rightarrow{{K}^{*0}}{\mu^{+}\mu^{-}}$ signal candidates are
first required to pass the hardware trigger, which selects events containing
at least one muon with transverse momentum
$\mbox{$p_{\mathrm{T}}$}>1.48{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ in the
7$\mathrm{\,Te\kern-1.00006ptV}$ data or
$\mbox{$p_{\mathrm{T}}$}>1.76{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ in the
8$\mathrm{\,Te\kern-1.00006ptV}$ data. In the subsequent software trigger, at
least one of the final-state particles is required to have
$\mbox{$p_{\mathrm{T}}$}>1.7{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ in the
7$\mathrm{\,Te\kern-1.00006ptV}$ data or
$\mbox{$p_{\mathrm{T}}$}>1.6{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ in the
8$\mathrm{\,Te\kern-1.00006ptV}$ data, unless the particle is identified as a
muon in which case
$\mbox{$p_{\mathrm{T}}$}>1.0{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ is required.
The final-state particles that satisfy these transverse momentum criteria are
also required to have an impact parameter larger than $100{\,\upmu\mathrm{m}}$
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 that is significantly
displaced from the PVs.

Signal candidates are formed from a pair of oppositely charged tracks that are
identified as muons, combined with a ${{K}^{*0}}$ meson candidate. The
${{K}^{*0}}$ candidate is formed from two oppositely charged tracks that are
identified as a kaon and a pion. These signal candidates are required to pass
a set of loose preselection requirements, which are identical to those
described in Ref. [27], with the exception that the ${{K}^{*0}}$ candidate is
required to have an invariant mass in the wider
$644<m_{K\pi}<1200{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ range. The
preselection requirements exploit the decay topology of
${{B}^{0}}\\!\rightarrow{{K}^{*0}}{\mu^{+}}{\mu^{-}}$ transitions and restrict
the data sample to candidates with good quality vertex and track fits.
Candidates are required to have a reconstructed ${{B}^{0}}$ invariant mass
($m_{K\pi\mu\mu}$) in the range
$5170<m_{K\pi\mu\mu}<5780{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$.

The backgrounds formed by combining particles 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) [41, 42]. The BDT
used for the present analysis is identical to that described in Ref. [27] and
the same working point is used. The BDT selection has a signal efficiency of
90% while removing 95% of the combinatorial background surviving the
preselection. The efficiency of the BDT is uniform with respect to
$m_{K\pi\mu\mu}$ in the above mass range.

Specific background processes can mimic the signal if their final states are
misidentified or misreconstructed. The requirements of Ref. [27] are
reassessed and found to reduce the sum of all backgrounds from such decay
processes to a level of less than 2% of the expected signal yield. The only
requirement that is modified in the present analysis is that responsible for
removing genuine ${{B}^{0}}\\!\rightarrow{{K}^{*0}}{\mu^{+}}{\mu^{-}}$ decays,
where the track of the genuine pion is reconstructed with the kaon hypothesis
and vice versa. These misidentified signal candidates occur more often in the
wider $m_{K\pi}$ window used for the present analysis, and are reduced by
tightening the requirements made on the kaon and pion PID information provided
by the RICH detectors. After the application of all the selection criteria,
this specific background process is reduced to less than 1% of the level of
the signal.

##  5 The $K^{+}\pi^{-}\mu^{+}\mu^{-}$ and $K^{+}\pi^{-}$ mass distributions

The $K^{+}\pi^{-}\mu^{+}\mu^{-}$ invariant mass is used to discriminate
between signal and background. The distribution of the signal candidates is
modelled using the sum of two Gaussian functions with a common mean, each with
a power law tail on the lower side. The parameters describing this model are
determined from fits to
${{B}^{0}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{*0}}$ data in a ${q^{2}}$ range
$9.22<{q^{2}}<9.96{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$ and with an
$m_{K\pi}$ range of
$644<m_{K\pi}<1200{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, shown in Fig. 1.
These parameters are fixed for the subsequent fits to the
${{B}^{0}}\\!\rightarrow{{K}^{*0}}{\mu^{+}}{\mu^{-}}$ candidates in the same
$m_{K\pi}$ range. In samples of simulated
${{B}^{0}}\\!\rightarrow{{K}^{*0}}{\mu^{+}}{\mu^{-}}$ decays, the
$m_{K\pi\mu\mu}$ resolution is observed to differ from that in
${{B}^{0}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{*0}}$ decays by 2 to 8% depending on ${q^{2}}$. A correction
factor is therefore derived from the simulation and is applied to the widths
of the Gaussian functions in the different ${q^{2}}$ bins. In the fits to
${{B}^{0}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{*0}}$ decays, an additional component is included to account for
the ${{{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{*0}}}$ process. The size of this additional component is taken to
be 0.8% of the ${{B}^{0}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{*0}}$ signal [43]. The fit to the
${{B}^{0}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{*0}}$ mode gives $389\,577\pm 649$ decays. In the fits to
${{B}^{0}}\\!\rightarrow{{K}^{*0}}{\mu^{+}}{\mu^{-}}$ decays, the ${{{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}}\\!\rightarrow{{K}^{*0}}{\mu^{+}\mu^{-}}}$
contribution is neglected. The systematic uncertainty related to ignoring this
background process is negligible. For both
${{B}^{0}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{*0}}$ and ${{B}^{0}}\\!\rightarrow{{K}^{*0}}{\mu^{+}}{\mu^{-}}$
decays, the combinatorial background in the $K^{+}\pi^{-}\mu^{+}\mu^{-}$
invariant mass spectrum is described by an exponential function. The
${{B}^{0}}\\!\rightarrow{{K}^{*0}}{\mu^{+}}{\mu^{-}}$ yield integrated over
the ${q^{2}}$ ranges
$0.1<{q^{2}}<8.0{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$,
$11.0<{q^{2}}<12.5{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$ and
$15.0<{q^{2}}<19.0{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$ is determined
to be $2593\pm 60$. The ${q^{2}}$ regions
$8.0<{q^{2}}<11.0{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$ and
$12.5<{q^{2}}<15.0{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$ are dominated
by the contributions from
${{B}^{0}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{*0}}$ and ${{{B}^{0}}\\!\rightarrow{\psi{(2S)}}{{K}^{*0}}}$
decays respectively and are therefore excluded in the fits to the signal
${{B}^{0}}\\!\rightarrow{{K}^{*0}}{\mu^{+}}{\mu^{-}}$ decays.

Figure 1:  Invariant mass $m_{K\pi\mu\mu}$ of (left) the
${{B}^{0}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{*0}}$ decay and (right) the signal decay
${{B}^{0}}\\!\rightarrow{{K}^{*0}}{\mu^{+}}{\mu^{-}}$ integrated over the
${q^{2}}$ regions described in the text. The individual signal (blue shaded
area) and background (red hatched area) components are shown. The solid line
denotes the total fitted distribution.

As discussed in Sec. 2, the $K^{+}\pi^{-}$ invariant mass distribution of the
signal candidates is modelled with two distributions. A relativistic
Breit–Wigner function is used for the P-wave component and the LASS
parameterisation for the S-wave component. The parameters of these functions
are fixed to the values determined in
${{B}^{0}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{*0}}$ decays using the model described in Ref. [44]. A systematic
uncertainty is assigned for this choice.

The $K^{+}\pi^{-}$ invariant mass distribution of the combinatorial background
is modelled using an empirical threshold function of the form

$f_{\rm bkg}(m_{K\pi})=(m_{K\pi}-m_{\rm thr})^{1/\alpha},$ (6)

where $m_{\rm thr}=634{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ is given by
the sum of the pion and kaon masses [45], and $\alpha$ is a parameter
determined from fits to the data. This model has been validated on data from
the upper $m_{K\pi\mu\mu}$ sideband, defined as
$5350<m_{K\pi\mu\mu}<5780{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, where no
resonant structure in the $m_{K\pi}$ spectrum is observed.

##  6 Determination of the S-wave fraction

###  6.1 Efficiency correction

The trigger, selection, and detector geometry bias the distributions of the
decay angles $\cos{\theta_{K}}$, $\cos{\theta_{\ell}}$, $\phi^{\prime}$, as
well as the ${q^{2}}$ and $m_{K\pi}$ distributions. The dominant sources of
bias are the geometrical acceptance of the detector and the requirements on
the track momentum, the impact parameter, and the PID of the hadrons.

The method for obtaining the efficiency correction, described in Ref [27], is
extended to also include the $m_{K\pi}$ dimension. The detection efficiency is
expressed in terms of orthonormal Legendre polynomials of order $n$,
$P_{n}(x)$, as

$\epsilon({q^{2}},m_{K\pi},\vec{\Omega}^{\prime})=\sum\limits_{g,h,i,j,k}c_{ghijk}P_{g}(m_{K\pi})P_{h}(\cos{\theta_{\ell}})P_{i}(\cos{\theta_{K}})P_{j}(\phi^{\prime})P_{k}({{q^{2}}}).$
(7)

As the polynomials are orthonormal over the domain $x\in[-1,1]$, the
observables $m_{K\pi}$, $\phi^{\prime}$, and ${q^{2}}$ are linearly
transformed to lie within this domain when evaluating the efficiency. The sum
in Eq. 7 runs up to 5th order for $\cos{\theta_{K}}$ and $\phi^{\prime}$, and
up to 8th, 7th and 6th order for $\cos{\theta_{\ell}}$, ${q^{2}}$ and
$m_{K\pi}$ respectively. The coefficients $c_{ghijk}$ are determined using a
principal moment analysis of simulated four-body
${{{B}^{0}}\\!\rightarrow{{K}^{+}}{{\pi}^{-}}{\mu^{+}\mu^{-}}}$ phase-space
decays. Two-dimensional projections of the detection efficiency as a function
of $\cos{\theta_{K}}$–${q^{2}}$ and $m_{K\pi}$–${q^{2}}$ are shown in Fig. 2.

Figure 2:  Two-dimensional projections of the efficiency (left) in the
$\cos{\theta_{K}}$–${q^{2}}$ plane and (right) in the $m_{K\pi}$–${q^{2}}$
plane, determined from a principal moments analysis of simulated four-body
${{{B}^{0}}\\!\rightarrow{{K}^{+}}{{\pi}^{-}}{\mu^{+}\mu^{-}}}$ phase-space
decays. The colour scale denotes the efficiency in arbitrary units. The lack
of entries in the top right corner of the $m_{K\pi}$–${q^{2}}$ distribution is
due to the limited phase space available in the decay of the ${{B}^{0}}$
meson.

###  6.2 Fit to the mass and angular distributions

An extended maximum likelihood fit to $m_{K\pi\mu\mu}$, $m_{K\pi}$ and
$\cos{\theta_{K}}$ is performed in each bin of ${q^{2}}$ in order to determine
the coefficients $G_{\rm S}^{\phantom{R}}$, $G_{\rm SP}^{\rm Re}$, $G_{\rm
SP}^{\rm Im}$ and $G_{\rm P}^{\perp\parallel}$ averaged over the ${q^{2}}$
bin. Given these coefficients, the S-wave fraction $F_{\rm S}$ is extracted
using Eq. 5. The angular distribution of the signal is described by Eq. 3
multiplied by the efficiency model evaluated at the centre of the ${q^{2}}$
bin ($q^{2}_{\rm bc}$). Integrating over $\cos{\theta_{\ell}}$ and
$\phi^{\prime}$ simplifies the fit, while retaining the sensitivity to the
parameters related to $F_{\rm S}$. The resulting angular and $m_{K\pi}$
distribution of the signal, $P_{\rm sig}$, within a bin $q^{2}_{\rm
min}<{q^{2}}<q^{2}_{\rm max}$, is given by

$P_{\rm sig}(m_{K\pi},\cos{\theta_{K}})=\int_{q^{2}_{\rm min}}^{q^{2}_{\rm
max}}\int_{0}^{\pi}\int_{-1}^{1}\mathrm{d}\\!\cos{\theta_{\ell}}{\rm
d}\phi^{\prime}\textrm{d}{q^{2}}\left[\frac{\mathrm{d}^{5}(\Gamma+\overline{\Gamma})}{\mathrm{d}m_{K\pi}\mathrm{d}{q^{2}}\,\mathrm{d}\vec{\Omega}^{\prime}}\times\epsilon(q^{2}_{\rm
bc},m_{K\pi},\vec{\Omega}^{\prime})\right],$ (8)

The overall scale of $P_{\rm sig}$ is set by fixing the parameter $G_{\rm
P}^{0}$ to an arbitrary value. The $m_{K\pi\mu\mu}$ distribution of the signal
is assumed to factorise with $P_{\rm sig}(m_{K\pi},\cos{\theta_{K}})$. This
assumption is validated using simulated events.

The $\cos{\theta_{K}}$ distribution of the combinatorial background is
modelled with a second-order polynomial where all parameters are allowed to
vary in the fit. The $m_{K\pi}$, $m_{K\pi\mu\mu}$ and $\cos{\theta_{K}}$
distributions of the combinatorial background are assumed to factorise. This
assumption has been validated on data from the upper $m_{K\pi\mu\mu}$
sideband. Figure 3 shows the projections of the probability distribution
function on the angular and mass distributions for the ${q^{2}}$ bin
$1.1<{q^{2}}<6.0{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$. Projections of
other ${q^{2}}$ bins are provided in Appendix B.

Figure 3: Angular and mass distributions for the ${q^{2}}$ bin
$1.1<{q^{2}}<6.0{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$. The
distributions of $\cos{\theta_{K}}$ and $m_{K\pi}$ are shown for candidates in
the signal $m_{K\pi\mu\mu}$ window of $\pm
50{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ around the known ${{B}^{0}}$ mass.
The solid line denotes the total fitted distribution. The individual
components, signal (blue shaded area) and background (red hatched area), are
also shown.

###  6.3 Result for $F_{\rm S}$

Using Eq. 5, $F_{\rm S}$ is determined in the full $m_{K\pi}$ region of the
fit, $\left.F_{\rm S}\right|_{644}^{1200}$, and in the narrow $m_{K\pi}$
region, $\left.F_{\rm S}\right|_{796}^{996}$. The statistical uncertainty on
$F_{\rm S}$ is determined using the following procedure. Values of the
parameters of the fit are generated according to a multi-dimensional
bifurcated Gaussian distribution. This distribution is constructed out of the
correlation matrix of the fit and the asymmetric uncertainties obtained from a
profile likelihood. For each generated set of parameters of the fit, a value
of $F_{\rm S}$ is computed. The 68% confidence interval is defined by taking
the $16^{\rm th}$–$84^{\rm th}$ percentiles of the resulting distribution of
$F_{\rm S}$. The correct coverage of this method is validated using
pseudoexperiments generated with a wide range of $F_{\rm S}$ values.

Figure 4 shows the values of $\left.F_{\rm S}\right|_{644}^{1200}$ and
$\left.F_{\rm S}\right|_{796}^{996}$ in each ${q^{2}}$ bin. The uncertainties
given are a quadratic sum of statistical and systematic uncertainties. The
results are also reported in Table 1. The sources of systematic uncertainty
are detailed in Sec. 8. As expected, the shape of the measured $F_{\rm S}$
distribution is found to be compatible with the smoothly varying distribution
of $F_{\rm L}$ measured in Ref. [27].

Figure 4:  Results for the S-wave fraction ($F_{\rm S}$) in bins of $q^{2}$ in the range (left) $644<m_{K\pi}<1200{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and (right) $796<m_{K\pi}<996{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. The uncertainties shown are the quadratic sum of the statistical and systematic uncertainties. The shape of $F_{\rm S}$ is found to be compatible with the smoothly varying distribution of $F_{\rm L}$, as measured in Ref. [27].  Table 1:  S-wave fraction ($F_{\rm S}$) in bins of ${q^{2}}$ for two $m_{K\pi}$ regions. The first uncertainty is statistical and the second systematic.  $q^{2}$ bin $(\\!{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}})$ | $\left.F_{\rm S}\right|_{796}^{996}$ | $\left.F_{\rm S}\right|_{644}^{1200}$
---|---|---
$0.10<{q^{2}}<0.98$ | $\phantom{+}0.021_{\,-0.011}^{\,+0.015}\pm{0.009}$ | $\phantom{+}0.052_{\,-0.027}^{\,+0.035}\pm{0.013}$
$1.1<{q^{2}}<2.5$ | $\phantom{+}0.144_{\,-0.030}^{\,+0.035}\pm{0.010}$ | $\phantom{+}0.304_{\,-0.053}^{\,+0.058}\pm{0.013}$
$2.5<{q^{2}}<4.0$ | $\phantom{+}0.029_{\,-0.020}^{\,+0.031}\pm{0.010}$ | $\phantom{+}0.071_{\,-0.049}^{\,+0.069}\pm{0.015}$
$4.0<{q^{2}}<6.0$ | $\phantom{+}0.117_{\,-0.023}^{\,+0.027}\pm{0.008}$ | $\phantom{+}0.254_{\,-0.044}^{\,+0.048}\pm{0.012}$
$6.0<{q^{2}}<8.0$ | $\phantom{+}0.033_{\,-0.019}^{\,+0.022}\pm{0.009}$ | $\phantom{+}0.082_{\,-0.045}^{\,+0.049}\pm{0.016}$
$11.0<{q^{2}}<12.5$ | $\phantom{+}0.021_{\,-0.016}^{\,+0.021}\pm{0.007}$ | $\phantom{+}0.049_{\,-0.039}^{\,+0.048}\pm{0.014}$
$15.0<{q^{2}}<17.0$ | $-0.008_{\,-0.014}^{\,+0.033}\pm{0.006}$ | $-0.016_{\,-0.030}^{\,+0.069}\pm{0.012}$
$17.0<{q^{2}}<19.0$ | $\phantom{+}0.018_{\,-0.017}^{\,+0.013}\pm{0.009}$ | $\phantom{+}0.034_{\,-0.032}^{\,+0.024}\pm{0.019}$
$1.1<{q^{2}}<6.0$ | $\phantom{+}0.101_{\,-0.017}^{\,+0.017}\pm{0.009}$ | $\phantom{+}0.224_{\,-0.033}^{\,+0.032}\pm{0.013}$
$15.0<{q^{2}}<19.0$ | $\phantom{+}0.010_{\,-0.014}^{\,+0.017}\pm{0.007}$ | $\phantom{+}0.019_{\,-0.025}^{\,+0.030}\pm{0.015}$

The presence of a nonresonant P-wave component in the $K^{+}\pi^{-}$ system
has been suggested in Refs. [46, 47]. However, no evidence for such a
component was found in the current data sample. The effect of neglecting a
nonresonant P-wave contribution with a relative phase and magnitude varied
within the statistical uncertainties determined in this analysis, was found to
be negligible.

##  7 Differential branching fraction of the decay
${{{B}^{0}}\\!\rightarrow{K}^{*}(892)^{0}{\mu^{+}}{\mu^{-}}}$

The differential branching fraction of the decay
${{{B}^{0}}\\!\rightarrow{K}^{*}(892)^{0}{\mu^{+}}{\mu^{-}}}$ is estimated by
normalising the signal yield, $n_{{{K}^{*0}}\mu^{+}\mu^{-}}$, obtained from
the fit described in Sec. 6.2, to the total event yield of the decay
${{{B}^{0}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{*0}}}$, $n_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{*0}}}$. The number of
${{{B}^{0}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{*0}}}$ events is obtained from a fit to the $m_{K\pi\mu\mu}$
spectrum using the same ${q^{2}}$ range as for the fit to determine the
$m_{K\pi\mu\mu}$ mass shape parameters (Sec. 5), but for an $m_{K\pi}$ range
$796<m_{K\pi}<996~{}{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. This yield has
to be corrected for the S-wave fraction within the narrow $m_{K\pi}$ window of
${{{B}^{0}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{*0}}}$ decays, $F_{\rm S}^{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{*0}}}$. The value of $F_{\rm
S}^{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}{{K}^{*0}}}$ is obtained
from Ref. [48] and is adjusted to the $m_{K\pi}$ range
$796<m_{K\pi}<996~{}{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. The ratio of
${{B}^{0}}\\!\rightarrow{{K}^{*0}}{\mu^{+}}{\mu^{-}}$ and
${{{B}^{0}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{*0}}}$ events is corrected for the relative efficiency between
the two decays,
$R_{\epsilon}=\epsilon_{J/\psi{{K}^{*0}}}/\epsilon_{{{K}^{*0}}\mu^{+}\mu^{-}}$.
This ratio is determined using simulated samples of
${{{B}^{0}}\\!\rightarrow{K}^{*}(892)^{0}{\mu^{+}}{\mu^{-}}}$ and
${{{B}^{0}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{K}^{*}(892)^{0}}$ decays. The angular distributions of these samples
are corrected to account for the presence of P- and S-wave components with a
relative abundance given by the measurements of Sec. 6.3 and Ref. [48]. The
systematic uncertainty associated with this correction is determined by
varying the components within the uncertainties of the measured values and
recalculating $R_{\epsilon}$. The resulting uncertainty on $R_{\epsilon}$ is
negligible.

The differential branching fraction of
${{{B}^{0}}\\!\rightarrow{K}^{*}(892)^{0}{\mu^{+}}{\mu^{-}}}$ decays in a
${q^{2}}$ bin of width $(q^{2}_{\rm max}-q^{2}_{\rm min})$ is given by

$\begin{split}\frac{\textrm{d}\mathcal{B}}{\textrm{d}{q^{2}}}=&\frac{R_{\epsilon}}{(q^{2}_{\rm
max}-q^{2}_{\rm min})}\frac{(1-\left.F_{\rm
S}\right|_{644}^{1200})n_{{{K}^{*0}}\mu^{+}\mu^{-}}}{(1-F_{\rm
S}^{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{*0}}})n_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{*0}}}}\mathcal{B}({{{B}^{0}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{*0}}})\mathcal{B}({{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\rightarrow\mu^{+}\mu^{-}),\end{split}$ (9)

where $\left.F_{\rm S}\right|_{644}^{1200}$, $R_{\epsilon}$ and
$n_{{{K}^{*0}}\mu^{+}\mu^{-}}$ correspond to quantities measured within the
relevant ${q^{2}}$ bin. The branching fraction
$\mathcal{B}({{B}^{0}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{K}^{*}(892)^{0})$ obtained from Ref. [49] is

$\mathcal{B}({{B}^{0}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{K}^{*}(892)^{0})=(1.19\pm 0.01\pm 0.08)\times 10^{-3},$

where the first uncertainty is statistical and the second systematic. The
branching fraction for ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\rightarrow\mu^{+}\mu^{-}$ decays is taken from Ref. [45]. The
resulting differential branching fraction is shown in Fig. 5. The
uncertainties given are a quadratic sum of statistical and systematic
uncertainties and the bands shown indicate the SM prediction from Refs. [50,
51]. The results are also reported in Table 2. The various sources of
systematic uncertainties are described in Sec. 8.

Figure 5:  Differential branching fraction of ${{{B}^{0}}\\!\rightarrow{K}^{*}(892)^{0}{\mu^{+}}{\mu^{-}}}$ decays as a function of ${q^{2}}$. The data are overlaid with the SM prediction from Refs. [50, 51]. No SM prediction is included in the region close to the narrow $c\bar{c}$ resonances. The result in the wider $q^{2}$ bin $15.0<{q^{2}}<19.0{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$ is also presented. The uncertainties shown are the quadratic sum of the statistical and systematic uncertainties, and include the uncertainty on the ${{{B}^{0}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}{{K}^{*0}}}$ and ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\rightarrow\mu^{+}\mu^{-}$ branching fractions.  Table 2:  Differential branching fraction of ${{{B}^{0}}\\!\rightarrow{K}^{*}(892)^{0}{\mu^{+}}{\mu^{-}}}$ decays in bins of ${q^{2}}$. The first uncertainty is statistical, the second systematic and the third due to the uncertainty on the ${{{B}^{0}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}{{K}^{*0}}}$ and ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\rightarrow\mu^{+}\mu^{-}$ branching fractions.  $q^{2}$ bin $(\\!{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}})$ |  $\textrm{d}\mathcal{B}/\textrm{d}{q^{2}}\times 10^{-7}$ $(c^{4}/\mathrm{\,Ge\kern-1.00006ptV}^{2})$
---|---
$0.10<{q^{2}}<0.98$ | $1.016_{\,-0.073}^{\,+0.067}\pm{0.029}\pm 0.069$
$1.1<{q^{2}}<2.5$ | $0.326_{\,-0.031}^{\,+0.032}\pm{0.010}\pm 0.022$
$2.5<{q^{2}}<4.0$ | $0.334_{\,-0.033}^{\,+0.031}\pm{0.009}\pm 0.023$
$4.0<{q^{2}}<6.0$ | $0.354_{\,-0.026}^{\,+0.027}\pm{0.009}\pm 0.024$
$6.0<{q^{2}}<8.0$ | $0.429_{\,-0.027}^{\,+0.028}\pm{0.010}\pm 0.029$
$11.0<{q^{2}}<12.5$ | $0.487_{\,-0.032}^{\,+0.031}\pm{0.012}\pm 0.033$
$15.0<{q^{2}}<17.0$ | $0.534_{\,-0.037}^{\,+0.027}\pm{0.020}\pm 0.036$
$17.0<{q^{2}}<19.0$ | $0.355_{\,-0.022}^{\,+0.027}\pm{0.017}\pm 0.024$
$1.1<{q^{2}}<6.0$ | $0.342_{\,-0.017}^{\,+0.017}\pm{0.009}\pm 0.023$
$15.0<{q^{2}}<19.0$ | $0.436_{\,-0.019}^{\,+0.018}\pm{0.007}\pm 0.030$

The total branching fraction of the
${{{B}^{0}}\\!\rightarrow{K}^{*}(892)^{0}{\mu^{+}}{\mu^{-}}}$ decay is
obtained from the sum over the eight ${q^{2}}$ bins. To account for the
fraction of signal events in the vetoed ${q^{2}}$ regions, a correction factor
of $1.532\pm 0.001({\rm stat})\pm 0.010({\rm syst})$ is applied. This factor
is determined using the calculation in Ref. [52] and form factors from Ref.
[53]. The systematic uncertainty is determined by recalculating the
extrapolation factor using the form factors from Ref. [54] and taking the
difference to the nominal value. The resulting total branching fraction is

$\mathcal{B}({{{B}^{0}}\\!\rightarrow{K}^{*}(892)^{0}{\mu^{+}}{\mu^{-}}})=(0.904^{\,+0.016}_{\,-0.015}\pm
0.010\pm 0.006\pm 0.061)\times 10^{-6},$

where the uncertainties, from left to right, are statistical, systematic, from
the extrapolation to the full ${q^{2}}$ region and due to the uncertainty of
the branching fraction of the normalisation mode.

##  8 Systematic uncertainties

The sources of systematic uncertainty considered can alter the angular and
mass distributions, as well as the ratio of efficiencies between the signal
and control channels. In general, the systematic uncertainties are
significantly smaller than the statistical uncertainties. The various sources
of systematic uncertainty are discussed in detail below and are summarised in
Table 3. Motivated by Eq. 9, the systematic uncertainty for $F_{\rm S}$ is
presented for the $m_{K\pi}$ region
$644<m_{K\pi}<1200{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. Typical ranges
are quoted in order to summarise the effect the systematic uncertainties have
across the various ${q^{2}}$ bins. Sources of systematic uncertainty that can
affect both $F_{\rm S}$ and the differential branching fraction are treated as
100% correlated.

Table 3: Summary of the main sources of systematic uncertainty on
$\left.F_{\rm S}\right|_{644}^{1200}$ and
$\textrm{d}\mathcal{B}/\textrm{d}{q^{2}}$. Typical ranges are quoted in order
to summarise the effect the systematic uncertainties have across the various
${q^{2}}$ bins.

Source | $\left.F_{\rm S}\right|_{644}^{1200}$ |  $\textrm{d}\mathcal{B}/\textrm{d}{q^{2}}$ $\times 10^{-7}(c^{4}/\rm{GeV}^{2})$
---|---|---
Data-simulation differences |  $0.008$–$0.013$ |  $0.004$–$0.021$
Efficiency model |  $0.001$–$0.010$ |  $0.001$–$0.012$
S-wave $m_{K\pi}$ model |  $0.001$–$0.017$ |  $0.001$–$0.015$
${{B}^{0}}\rightarrow{K}^{*}(892)^{0}$ form factors | – |  $0.003$–$0.017$
$\mathcal{B}({{B}^{0}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}(\rightarrow\mu^{+}\mu^{-}){{K}^{*0}})$ | – |  $0.025$–$0.079$

###  8.1 Systematic uncertainties on the S-wave fraction

The impact of each source of systematic uncertainty on $F_{\rm S}$ is
estimated using pseudoexperiments, where samples are generated varying one or
more parameters. The value of $F_{\rm S}$ is determined using both the nominal
model and the alternative model. For every pseudoexperiment, the difference
between the two values of $F_{\rm S}$ is computed. In general, the systematic
uncertainty is then taken as the average of this difference over a large
number of pseudoexperiments. The exception to this is the statistical
uncertainty of the efficiency correction. In order to account for this
statistical variation, the standard deviation of the difference between the
two values of $F_{\rm S}$ from each pseudoexperiment is used instead. The
systematic uncertainty is evaluated in each ${q^{2}}$ bin separately. The
pseudodata are generated with signal and background yields many times larger
than those of the data, rendering statistical effects negligible. The main
systematic uncertainties on $F_{\rm S}$ originate from the efficiency
correction function and the choice of model used to describe the S-wave
component of the $m_{K\pi}$ distribution of the signal.

There are two main systematic uncertainties associated with the efficiency
correction function used for determining $F_{\rm S}$. Firstly, an uncertainty
arises from residual data-simulation differences. After all corrections to the
simulation are applied, a difference at the level of 10% remains in the
momentum spectrum of the pions between simulated and genuine
${{B}^{0}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{*0}}$ decays. A new efficiency correction is derived after
weighting the simulated phase-space sample to account for this difference. The
second main systematic uncertainty associated with the efficiency correction
is due to the order of the polynomials used to describe the efficiency
function. To evaluate this uncertainty, a new efficiency correction is derived
in which the polynomial order in $q^{2}$ is increased by two. This change is
motivated by a small residual difference between the ${q^{2}}$ dependence of
the nominal efficiency correction and the simulated phase-space sample, near
the upper kinematic edge of the ${q^{2}}$ range. Uncertainties due to the
limited size of the simulation sample used to derive the efficiency
correction, as well as due to the evaluation of the efficiency correction at
the centre of the ${q^{2}}$ bin are also assessed and are found to be
negligible.

To assess the modelling of the S-wave component in the $m_{K\pi}$
distribution, pseudoexperiments are produced where the LASS line shape is
exchanged for the sum of resonant $K^{*}_{0}(800)^{0}$ (also known as the
$\kappa$ resonance) and $K^{*}_{0}(1430)^{0}$ contributions. An additional
variation is considered where the parameters of the LASS distribution,
determined in $B^{0}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{*0}}$ decays using the model described in Ref. [44], are
exchanged for those measured by the LASS collaboration [29]. The largest of
the two variations is taken as the systematic uncertainty on the S-wave model.
Systematic uncertainties associated with the modelling of the P-wave
$m_{K\pi}$ distribution of the signal are found to be negligible.

Integrating the differential decay rate given in Eq. 3 over
$\cos{\theta_{\ell}}$ and $\phi^{\prime}$ results in the cancellation of terms
involving the angular observables $S_{3}$, $A_{\rm FB}$ and $S_{9}$. However
the integral of the product of the differential decay rate with the efficiency
correction, given in Eq. 8, results in a residual dependence of the signal
distribution on these angular observables. By generating pseudoexperiments
with observables $S_{3}$, $A_{\rm FB}$ and $S_{9}$ either set to zero or
varied within the uncertainties measured in Ref. [27], the systematic
uncertainty on $F_{\rm S}$ is assessed. Even considering the largest variation
observed, the resulting systematic uncertainty is negligible.

All other sources of systematic uncertainties described in Ref. [27], such as
the modelling of the $m_{K\pi\mu\mu}$ distribution of the signal and
background, the choice of the $m_{K\pi}$ and $\cos\theta_{K}$ background
models and the effect of residual specific backgrounds, are found to be sub-
dominant. The effect of neglecting a possible D-wave $K^{+}\pi^{-}$ component,
arising from the tail of the $K^{*}_{2}(1430)^{0}$, is also assessed and found
to be negligible.

###  8.2 Systematic uncertainties on the differential branching fraction

Systematic uncertainties affecting the differential branching fraction
predominantly arise through: the knowledge of $R_{\epsilon}$, the ratio of the
reconstruction and selection efficiencies described in Sec. 7; the uncertainty
of the branching fraction of the decay
${{B}^{0}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{*0}}$, which is shown as a separate systematic uncertainty in
Table 2; and systematic uncertainties related to the determination of $F_{\rm
S}$, which are propagated to the differential branching fraction measurement.

The imperfect knowledge of the $B\rightarrow K^{*}$ form-factor model used in
the generation of the ${{B}^{0}}\\!\rightarrow{{K}^{*0}}{\mu^{+}}{\mu^{-}}$
simulated sample affects the determination of the ratio of efficiencies
$R_{\epsilon}$. A systematic uncertainty is therefore assessed by weighting
simulated events to account for the variations between the models described in
Refs. [50] and [54].

As described in Sec. 8.1, after all corrections to the simulation are applied,
a small difference remains in the momentum spectrum of the pions between
simulated and genuine
${{B}^{0}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{*0}}$ decays. The ratio $R_{\epsilon}$, and consequently
$\textrm{d}\mathcal{B}/\textrm{d}{q^{2}}$, is therefore calculated by
weighting the simulated
${{{B}^{0}}\\!\rightarrow{K}^{*}(892)^{0}{\mu^{+}}{\mu^{-}}}$ and
${{{B}^{0}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{K}^{*}(892)^{0}}$ decays to account for the observed differences.

Other sources of systematic uncertainties affecting the determination of the
signal yield, such as the choice of model to describe the $m_{K\pi\mu\mu}$
distribution of the signal and the background components, the choice of the
$m_{K\pi}$ and $\cos\theta_{K}$ models to describe the background, and the
effect of residual specific backgrounds, are found to be negligible.

##  9 Conclusions

This paper presents the first measurement of the S-wave fraction in the
$K^{+}\pi^{-}$ system of ${{B}^{0}}\\!\rightarrow{{K}^{*0}}{\mu^{+}}{\mu^{-}}$
decays using a data sample corresponding to an integrated luminosity of
$3\mbox{\,fb}^{-1}$ collected at the LHCb experiment. Accounting for the
measured S-wave fraction in the wide $m_{K\pi}$ region, the first measurement
of the P-wave component of the differential branching fraction of
${{{B}^{0}}\\!\rightarrow{K}^{*}(892)^{0}{\mu^{+}}{\mu^{-}}}$ decays is
reported in bins of ${q^{2}}$. All previous measurements of the differential
branching fraction have compared the combination of S- and P-wave components
to the theory prediction, which is made purely for the resonant P-wave part of
the $K^{+}\pi^{-}$ system. The measurements of the S-wave fraction presented
in this paper are compatible with theory predictions [18, 19, 20] and support
previous estimates [21]. In the absence of any previous measurement, such
estimates have been used to assign a systematic uncertainty for a possible
S-wave component [21]. The measurements of the S-wave fraction presented in
this paper allow these estimates to be replaced with an accurate assessment of
the scalar component in ${{B}^{0}}\\!\rightarrow{{K}^{*0}}{\mu^{+}}{\mu^{-}}$
decays. The resulting measurements of the differential branching fraction of
${{{B}^{0}}\\!\rightarrow{K}^{*}(892)^{0}{\mu^{+}}{\mu^{-}}}$ decays are the
most precise to date and are in good agreement with the SM predictions.

## Acknowledgements

We would like to thank Gudrun Hiller and Martin Jung for useful discussions
regarding the treatment of the $K^{+}\pi^{-}$ system. 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 and MPG (Germany); 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). We acknowledge the computing resources that are
provided by CERN, IN2P3 (France), KIT and DESY (Germany), INFN (Italy), SURF
(The Netherlands), PIC (Spain), GridPP (United Kingdom), RRCKI and Yandex LLC
(Russia), CSCS (Switzerland), IFIN-HH (Romania), CBPF (Brazil), PL-GRID
(Poland) and OSC (USA). We are indebted to the communities behind the multiple
open source software packages on which we depend. Individual groups or members
have received support from AvH Foundation (Germany), EPLANET, Marie
Skłodowska-Curie Actions and ERC (European Union), Conseil Général de Haute-
Savoie, Labex ENIGMASS and OCEVU, Région Auvergne (France), RFBR and Yandex
LLC (Russia), GVA, XuntaGal and GENCAT (Spain), Herchel Smith Fund, The Royal
Society, Royal Commission for the Exhibition of 1851 and the Leverhulme Trust
(United Kingdom).

Appendices

##  Appendix A The $m_{K\pi}$ distribution of the signal

The $K^{+}\pi^{-}$ invariant mass distribution of the signal candidates is
modelled by two distributions. For the P-wave component, a relativistic Breit-
Wigner function is used, given by

$\begin{split}f_{\rm
BW}(m_{K\pi})=&\sqrt{kp}\left(\frac{k}{k_{892}}\right)\frac{B^{\prime}_{1}(k,k_{892},d)B^{\prime}_{0}(p,p_{892},d)}{m_{K\pi}^{2}-m_{892}^{2}-im_{892}\Gamma_{892}(m_{K\pi})},\end{split}$
(10)

where $\sqrt{kp}$ is the phase-space factor, $\Gamma_{892}(m_{K\pi})$ is given
by

$\Gamma_{892}(m_{K\pi})=\Gamma_{892}B^{\prime\,2}_{1}(k,k_{892},d)\left(\frac{k}{k_{892}}\right)^{3}\left(\frac{m_{892}}{m_{K\pi}}\right),$
(11)

and $B^{\prime}$ are Blatt–Weisskopf barrier factors as defined in Ref. [45].
The parameter $d$ is the meson radius parameter and is set to
1.6$\mathrm{\,Ge\kern-1.00006ptV}^{-1}c$ [44]. The systematic uncertainty
associated with the choice of this value is negligible. The parameters
$m_{892}$ and $\Gamma_{892}$ are the pole mass and width of the
${K}^{*}(892)^{0}$ resonance, and $k$ ($p$) is the momentum of the $K^{+}$
(${{K}^{*0}}$) in the rest frame of the ${{K}^{*0}}$ (${{B}^{0}}$) evaluated
at a given $m_{K\pi}$. The parameters $k_{892}$ and $p_{892}$ are the values
of $k$ and $p$ evaluated at the pole mass of the ${K}^{*}(892)^{0}$ resonance.
In Eq. 10, the orbital angular momentum between the ${K}^{*}(892)^{0}$ and the
dimuon system is considered to be zero. The inclusion of a higher orbital
angular momentum component has a negligible effect on the measurements.

The S-wave component of the signal is modelled using the LASS parameterisation
[29], given by

$f_{\rm
LASS}(m_{K\pi})=\sqrt{kp}B^{\prime}_{1}(k,k_{1430},d)\left(\frac{k}{k_{1430}}\right)\left(\frac{1}{\cot{\delta_{B}}-i}+e^{2i\delta_{B}}\frac{1}{\cot{\delta_{R}}-i}\right),$
(12)

where $k_{1430}$ is the momentum of the ${{K}^{*0}}$ in the ${{B}^{0}}$ rest
frame, evaluated at the pole mass of the ${K}_{0}^{*}(1430)^{0}$ resonance.
The terms $\cot{\delta_{B}}$ and $\cot{\delta_{R}}$ are given by

$\cot{\delta_{B}}=\frac{1}{ak}+\frac{rk}{2}$ (13)

and

$\cot{\delta_{R}}=\frac{m_{1430}^{2}-m_{K\pi}^{2}}{m_{1430}\Gamma_{1430}(m_{K\pi})},$
(14)

with the running width $\Gamma_{1430}(m_{K\pi})$ in turn given by

$\Gamma_{1430}(m_{K\pi})=\Gamma_{1430}\frac{k}{k_{1430}}\frac{m_{1430}}{m_{K\pi}}\,.$
(15)

The parameters $m_{1430}$ and $\Gamma_{1430}$ are the pole mass and width of
the ${K}_{0}^{*}(1430)^{0}$ resonance, and $k_{1430}$ is the momentum of the
kaon in the ${{K}^{*0}}$ rest frame, evaluated at the pole mass of the
${K}_{0}^{*}(1430)^{0}$ resonance. The second term of Eq. 12 is equivalent to
a Breit–Wigner function for the ${K}_{0}^{*}(1430)^{0}$. The first term of Eq.
12 contains two empirical parameters $\left\\{a,r\right\\}$. These parameters
are fixed to the values $a=3.83\,{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}^{-1}$
and $r=2.86{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}^{-1}$, determined in
${{B}^{0}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{*0}}$ decays using the model described in Ref. [44].

In order to assess the systematic effect of this choice, these parameters are
also fixed to values from the LASS experiment,
$a=1.94{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}^{-1}$ and
$r=1.76{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}^{-1}$. The resulting systematic
uncertainty is found to be negligible.

##  Appendix B Likelihood fit projections

Figures 6–9 show the projections of the fitted probability density function on
$m_{K\pi\mu\mu}$, $m_{K\pi}$ and $\cos{\theta_{K}}$. Figure 6 shows the wider
${q^{2}}$ bins of
$1.1<{q^{2}}<6.0{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$ and
$15.0<{q^{2}}<19.0{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$, Figs. 7–9
show the $m_{K\pi\mu\mu}$, $m_{K\pi}$ and $\cos{\theta_{K}}$ projections
respectively for the finer ${q^{2}}$ bins. In all figures, the solid line
denotes the total fitted distribution. The individual components, signal (blue
shaded area) and background (red hatched area), are also shown.

Figure 6:  Angular and mass distributions for the ${q^{2}}$ bins
$1.1<{q^{2}}<6.0{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$ (left) and
$15.0<{q^{2}}<19.0{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$ (right). The
distributions of $\cos{\theta_{K}}$ and $m_{K\pi}$ are shown for candidates in
the signal $m_{K\pi\mu\mu}$ window of $\pm
50{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ around the known ${{B}^{0}}$ mass.

Figure 7:  The $K^{+}\pi^{-}\mu^{+}\mu^{-}$ invariant mass distributions for
the fine ${q^{2}}$ bins.

Figure 8:  The ${{K}^{+}}{{\pi}^{-}}$ invariant mass distributions for the
fine ${q^{2}}$ bins for candidates in the signal $m_{K\pi\mu\mu}$ window of
$\pm 50{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ around the known ${{B}^{0}}$
mass.

Figure 9:  The $\cos{\theta_{K}}$ angular distributions for the fine ${q^{2}}$
bins for candidates in the signal $m_{K\pi\mu\mu}$ window of $\pm
50{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ around the known ${{B}^{0}}$ mass.

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LHCb collaboration

R. Aaij39, B. Adeva38, M. Adinolfi47, Z. Ajaltouni5, S. Akar6, J. Albrecht10,
F. Alessio39, M. Alexander52, S. Ali42, G. Alkhazov31, P. Alvarez Cartelle54,
A.A. Alves Jr58, S. Amato2, S. Amerio23, Y. Amhis7, L. An40, L. Anderlini18,
G. Andreassi40, M. Andreotti17,g, J.E. Andrews59, R.B. Appleby55, O. Aquines
Gutierrez11, F. Archilli1, P. d’Argent12, A. Artamonov36, M. Artuso60, E.
Aslanides6, G. Auriemma26,s, M. Baalouch5, S. Bachmann12, J.J. Back49, A.
Badalov37, C. Baesso61, W. Baldini17, R.J. Barlow55, C. Barschel39, S.
Barsuk7, W. Barter39, V. Batozskaya29, V. Battista40, A. Bay40, L. Beaucourt4,
J. Beddow52, F. Bedeschi24, I. Bediaga1, L.J. Bel42, V. Bellee40, N.
Belloli21,i, K. Belous36, I. Belyaev32, E. Ben-Haim8, G. Bencivenni19, S.
Benson39, J. Benton47, A. Berezhnoy33, R. Bernet41, A. Bertolin23, M.-O.
Bettler39, M. van Beuzekom42, S. Bifani46, P. Billoir8, T. Bird55, A.
Birnkraut10, A. Bitadze55, A. Bizzeti18,u, T. Blake49, F. Blanc40, J. Blouw11,
S. Blusk60, V. Bocci26, T. Boettcher57, A. Bondar35, N. Bondar31,39, W.
Bonivento16, S. Borghi55, M. Borisyak67, M. Borsato38, F. Bossu7, M. Boubdir9,
T.J.V. Bowcock53, E. Bowen41, C. Bozzi17,39, S. Braun12, M. Britsch12, T.
Britton60, J. Brodzicka55, E. Buchanan47, C. Burr55, A. Bursche2, J.
Buytaert39, S. Cadeddu16, R. Calabrese17,g, M. Calvi21,i, M. Calvo Gomez37,m,
P. Campana19, D. Campora Perez39, L. Capriotti55, A. Carbone15,e, G.
Carboni25,j, R. Cardinale20,h, A. Cardini16, P. Carniti21,i, L. Carson51, K.
Carvalho Akiba2, G. Casse53, L. Cassina21,i, L. Castillo Garcia40, M.
Cattaneo39, Ch. Cauet10, G. Cavallero20, R. Cenci24,t, M. Charles8, Ph.
Charpentier39, G. Chatzikonstantinidis46, M. Chefdeville4, S. Chen55, S.-F.
Cheung56, V. Chobanova38, M. Chrzaszcz41,27, X. Cid Vidal38, G. Ciezarek42,
P.E.L. Clarke51, M. Clemencic39, H.V. Cliff48, J. Closier39, V. Coco58, J.
Cogan6, E. Cogneras5, V. Cogoni16,f, L. Cojocariu30, G. Collazuol23,o, P.
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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 Savoie Mont-Blanc, 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
9I. Physikalisches Institut, RWTH Aachen University, Aachen, Germany
10Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
11Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
12Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
13School of Physics, University College Dublin, Dublin, Ireland
14Sezione INFN di Bari, Bari, Italy
15Sezione INFN di Bologna, Bologna, Italy
16Sezione INFN di Cagliari, Cagliari, Italy
17Sezione INFN di Ferrara, Ferrara, Italy
18Sezione INFN di Firenze, Firenze, Italy
19Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
20Sezione INFN di Genova, Genova, Italy
21Sezione INFN di Milano Bicocca, Milano, Italy
22Sezione INFN di Milano, Milano, Italy
23Sezione INFN di Padova, Padova, Italy
24Sezione INFN di Pisa, Pisa, Italy
25Sezione INFN di Roma Tor Vergata, Roma, Italy
26Sezione INFN di Roma La Sapienza, Roma, Italy
27Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
28AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
29National Center for Nuclear Research (NCBJ), Warsaw, Poland
30Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
31Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
32Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
33Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
34Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
35Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
36Institute for High Energy Physics (IHEP), Protvino, Russia
37Universitat de Barcelona, Barcelona, Spain
38Universidad de Santiago de Compostela, Santiago de Compostela, Spain
39European Organization for Nuclear Research (CERN), Geneva, Switzerland
40Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
41Physik-Institut, Universität Zürich, Zürich, Switzerland
42Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
43Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
44NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
45Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
46University of Birmingham, Birmingham, United Kingdom
47H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
48Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
49Department of Physics, University of Warwick, Coventry, United Kingdom
50STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
51School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
52School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
53Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
54Imperial College London, London, United Kingdom
55School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
56Department of Physics, University of Oxford, Oxford, United Kingdom
57Massachusetts Institute of Technology, Cambridge, MA, United States
58University of Cincinnati, Cincinnati, OH, United States
59University of Maryland, College Park, MD, United States
60Syracuse University, Syracuse, NY, United States
61Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
62University of Chinese Academy of Sciences, Beijing, China, associated to 3
63Institute of Particle Physics, Central China Normal University, Wuhan,
Hubei, China, associated to 3
64Departamento de Fisica , Universidad Nacional de Colombia, Bogota, Colombia,
associated to 8
65Institut für Physik, Universität Rostock, Rostock, Germany, associated to 12
66National Research Centre Kurchatov Institute, Moscow, Russia, associated to
32
67Yandex School of Data Analysis, Moscow, Russia, associated to 32
68Instituto de Fisica Corpuscular (IFIC), Universitat de Valencia-CSIC,
Valencia, Spain, associated to 37
69Van Swinderen Institute, University of Groningen, Groningen, The
Netherlands, associated to 42

aUniversidade Federal do Triângulo Mineiro (UFTM), Uberaba-MG, Brazil
bLaboratoire Leprince-Ringuet, Palaiseau, France
cP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
dUniversità di Bari, Bari, Italy
eUniversità di Bologna, Bologna, Italy
fUniversità di Cagliari, Cagliari, Italy
gUniversità di Ferrara, Ferrara, Italy
hUniversità di Genova, Genova, Italy
iUniversità di Milano Bicocca, Milano, Italy
jUniversità di Roma Tor Vergata, Roma, Italy
kUniversità di Roma La Sapienza, Roma, Italy
lAGH - University of Science and Technology, Faculty of Computer Science,
Electronics and Telecommunications, Kraków, Poland
mLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
nHanoi University of Science, Hanoi, Viet Nam
oUniversità di Padova, Padova, Italy
pUniversità di Pisa, Pisa, Italy
qUniversità degli Studi di Milano, Milano, Italy
rUniversità di Urbino, Urbino, Italy
sUniversità della Basilicata, Potenza, Italy
tScuola Normale Superiore, Pisa, Italy
uUniversità di Modena e Reggio Emilia, Modena, Italy
vIligan Institute of Technology (IIT), Iligan, Philippines
†Deceased