EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-EP-2019-063 LHCb-PAPER-2019-004 July 16, 2019

Amplitude analysis of the ${{B}^{0}_{({s})}}\rightarrow{{K}^{*0}}{{\kern
3.73305pt\overline{\kern-3.73305ptK}}{}^{*0}}$ decays and measurement of the
branching fraction of the ${{B}^{0}}\rightarrow{{K}^{*0}}{{\kern
3.73305pt\overline{\kern-3.73305ptK}}{}^{*0}}$ decay

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

The $B^{0}\rightarrow{{K}^{*0}}{{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}$ and
$B^{0}_{s}\rightarrow{{K}^{*0}}{{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}$ decays are studied using
proton-proton collision data corresponding to an integrated luminosity of
3$\text{\,fb}^{-1}$. An untagged and time-integrated amplitude analysis of
${{B}^{0}_{({s})}}\rightarrow({{K}^{+}}{{\pi}^{-}})({{K}^{-}}{{\pi}^{+}})$
decays in two-body invariant mass regions of
150$\text{\,Me\kern-1.00006ptV\\!/}c^{2}$ around the ${K}^{*0}$ mass is
performed. A stronger longitudinal polarisation fraction in the
${B^{0}\rightarrow{{K}^{*0}}{{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}}$ decay, ${f_{L}=0.724\pm
0.051\,({\rm stat})\pm 0.016\,({\rm syst})}$, is observed as compared to
${f_{L}=0.240\pm 0.031\,({\rm stat})\pm 0.025\,({\rm syst})}$ in the
${B^{0}_{s}\rightarrow{{K}^{*0}}{{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}}$ decay. The ratio of branching
fractions of the two decays is measured and used to determine
${\mathcal{B}}(\mbox{${{B}^{0}}\\!\rightarrow{{K}^{*0}}{{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}$})=(8.0\pm 0.9\,({\rm stat})\pm
0.4\,({\rm syst}))\times 10^{-7}$.

Published in JHEP 07 (2019) 032

© 2024 CERN for the benefit of the LHCb collaboration. CC-BY-4.0 licence.

##  1 Introduction

The ${{B}^{0}}\\!\rightarrow{{K}^{*0}}{{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}$ decay is a Flavour-Changing
Neutral Current (FCNC) process.111Throughout the text charge conjugation is
implied, $(K\pi)$ indicates either a $({{K}^{+}}{{\pi}^{-}})$ or a
$({{K}^{-}}{{\pi}^{+}})$ pair, ${B}^{0}_{({s})}$ indicates either a ${B}^{0}$
or a ${B}^{0}_{s}$ meson and ${K}^{*0}$ refers to the ${{K}^{*}}(892)^{0}$
resonance, unless otherwise stated. In the Standard Model (SM) this type of
processes is forbidden at tree level and occurs at first order through loop
penguin diagrams. Hence, FCNC processes are considered to be excellent probes
for physics beyond the SM, since contributions mediated by heavy particles,
contemplated in these theories, may produce effects measurable with the
current sensitivity.

Evidence of the ${{B}^{0}}\\!\rightarrow{{K}^{*0}}{{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}$ decay has been found by the
BaBar collaboration [1] with a measured yield of ${33.5}^{+9.1}_{-8.1}$
decays. An untagged time-integrated analysis was presented finding a branching
fraction of ${\mathcal{B}}=(1.28^{+0.35}_{-0.30}\pm 0.11)\times{10}^{-6}$ and
a longitudinal polarisation fraction of $f_{L}={0.80}^{+0.11}_{-0.12}\pm
0.06$. In untagged time-integrated analyses the distributions for ${B}^{0}$
and ${\kern 1.79993pt\overline{\kern-1.79993ptB}}{}^{0}$ decays are assumed to
be identical and summed, so that they can be fitted with a single amplitude.
However, if $C\\!P$-violation effects are present, the distribution is given
by the incoherent sum of the two contributions. The Belle collaboration also
searched for this decay [2] and a branching fraction of
${\mathcal{B}}=(0.26^{+0.33+0.10}_{-0.29-0.07})\times 10^{-6}$ was measured,
disregarding S–wave contributions. There is a $2.2$ standard-deviations
difference between the branching fraction measured by the two experiments. The
predictions of factorised QCD (QCDF) are
${\mathcal{B}}=(0.6^{+0.1+0.5}_{-0.1-0.3})\times 10^{-6}$ and
$f_{L}=0.69^{+0.01+0.34}_{-0.01-0.27}$ [3]. Perturbative QCD predicts
${\mathcal{B}}=(0.64^{+0.24}_{-0.23})\times 10^{-6}$ [4].222This reference
considers two scenarios for its predictions, both giving compatible results.
Only the first scenario considered therein is quoted here. These theoretical
predictions agree with the experimental results within the large
uncertainties. The measurement of $f_{L}$ agrees with the naïve hypothesis,
based on the quark helicity conservation and the $V$$-$$A$ nature of the weak
interaction, that charmless decays into pairs of vector mesons ($VV$) should
be strongly longitudinally polarised. See, for example, the Polarization in B
Decays review in Ref. [5].

The ${{B}^{0}_{s}}\\!\rightarrow{{K}^{*0}}{{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}$ decay was first observed by the
LHCb experiment with early LHC data [4]. A later untagged time-integrated
study, with data corresponding to $1\text{\,fb}^{-1}$ of integrated
luminosity, measured ${{\mathcal{B}}=(10.8\pm 2.1\pm 1.5)\times{10}^{-6}}$ and
${f_{L}=0.201\pm 0.057\pm 0.040}$ [6]. More recently, a complete
$C\\!P$-sensitive time-dependent analysis of
${{B}^{0}_{s}}\\!\rightarrow({{K}^{+}}{{\pi}^{-}})({{K}^{-}}{{\pi}^{+}})$
decays in the $(K\pi)$ mass range from 750 to
$1600\text{\,Me\kern-1.00006ptV\\!/}c^{2}$ has been published by LHCb [7],
with data corresponding to $3\text{\,fb}^{-1}$ of integrated luminosity. A
determination of ${f_{L}=0.208\pm 0.032\pm 0.046}$ was performed as well as
the first measurements of the mixing-induced $C\\!P$-violating phase
$\phi_{s}^{{d}{\overline{d}}}$ and of the direct $C\\!P$ asymmetry parameter
$|\lambda|$. These LHCb analyses of
${{B}^{0}_{s}}\\!\rightarrow({{K}^{+}}{{\pi}^{-}})({{K}^{-}}{{\pi}^{+}})$
decays lead to three conclusions: firstly, within their uncertainties, the
measured observables are compatible with the absence of $C\\!P$ violation;
secondly, a low polarisation fraction is found; finally, a large S–wave
contribution, as much as $60\%$, is measured in the
$150\text{\,Me\kern-1.00006ptV\\!/}c^{2}$ window around the ${K}^{*0}$ mass.
The low longitudinal polarisation fraction shows a tension with the prediction
of QCDF ($f_{L}=0.63^{+0.42}_{-0.29}$ [3]) and disfavours the hypothesis of
strongly longitudinally polarised $VV$ decays. Theoretical studies try to
explain the small longitudinal polarisation with mechanisms such as
contributions from annihilation processes [8, 3]. It is intriguing that the
two channels ${{B}^{0}}\\!\rightarrow{{K}^{*0}}{{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}$ and
${{B}^{0}_{s}}\\!\rightarrow{{K}^{*0}}{{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}$, which are related by U-spin
symmetry, implying the exchange of $d$ and $s$ quarks as displayed in Fig. 1,
show such different polarisations. A comprehensive theory review on
polarisation of charmless $VV$ neutral $B$-meson decays can be found in Ref.
[9].

Figure 1: Leading order Feynman diagrams for the
${{B}^{0}}\\!\rightarrow{{K}^{*0}}{{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}$ and
${{B}^{0}_{s}}\\!\rightarrow{{K}^{*0}}{{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}$ decays. Both modes are
dominated by a gluonic-penguin diagram.

Some authors consider the ${{B}^{0}_{s}}\\!\rightarrow{{K}^{*0}}{{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}$ decay as a golden channel for a
precision test of the CKM phase $\beta_{s}$ [10]. High-precision analyses of
this channel, dominated by the gluonic-penguin diagram, will require to
account for subleading amplitudes [11, 9]. The study of the
${{B}^{0}}\\!\rightarrow{{K}^{*0}}{{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}$ decay allows to control higher-
order SM contributions to the ${{B}^{0}_{s}}\\!\rightarrow{{K}^{*0}}{{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}$ channel employing U-spin
symmetry [10, 12]. In Refs. [12, 13] more precise QCDF predictions, involving
the relation between longitudinal branching fractions of the two channels, are
made.

In this work, an untagged and time-integrated amplitude analysis of the
${{B}^{0}}\\!\rightarrow({{K}^{+}}{{\pi}^{-}})({{K}^{-}}{{\pi}^{+}})$ and
${{B}^{0}_{s}}\\!\rightarrow({{K}^{+}}{{\pi}^{-}})({{K}^{-}}{{\pi}^{+}})$
decays in the two-body invariant mass regions of
150$\text{\,Me\kern-1.00006ptV\\!/}c^{2}$ around the ${K}^{*0}$ mass is
presented, as well as the determination of the
${{B}^{0}}\\!\rightarrow{{K}^{*0}}{{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}$ decay branching fraction. The
analysis uses data recorded in 2011 and 2012 at centre-of-mass energies of
$\sqrt{s}=7$ and $\sqrt{s}=8\text{\,Te\kern-1.00006ptV}$, respectively,
corresponding to an integrated luminosity of $3\text{\,fb}^{-1}$.

This paper is organised as follows. In Sect. 2 the formalism of the decay
amplitudes is presented. In Sect. 3 a brief description of the LHCb detector,
online selection algorithms and simulation software is given. The selection of
${{B}^{0}}\\!\rightarrow({{K}^{+}}{{\pi}^{-}})({{K}^{-}}{{\pi}^{+}})$ and
${{B}^{0}_{s}}\\!\rightarrow({{K}^{+}}{{\pi}^{-}})({{K}^{-}}{{\pi}^{+}})$
candidates is presented in Sect. 4. Sect. 5 describes the maximum-likelihood
fit to the four-body invariant-mass spectra and its results. The amplitude
analysis and its results are discussed in Sect. 6. The estimation of
systematic uncertainties is described in Sect. 7, and the determination of the
${{B}^{0}}\\!\rightarrow{{K}^{*0}}{{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}$ decay branching fraction
relative to the ${{B}^{0}_{s}}\\!\rightarrow{{K}^{*0}}{{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}$ mode in Sect. 8. Finally, the
results are summarised and conclusions are drawn in Sect. 9.

##  2 Amplitude analysis formalism

The ${{B}^{0}}\\!\rightarrow{{K}^{*0}}{{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}$ and
${{B}^{0}_{s}}\\!\rightarrow{{K}^{*0}}{{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}$ modes are weak decays of a
pseudoescalar particle into two vector mesons ($P\\!\rightarrow VV$). The
$B$-meson decays are followed by subsequent
${{K}^{*0}}\\!\rightarrow{{K}^{+}}{{\pi}^{-}}$ and ${{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$
decays. The study of the angular distribution employs the helicity angles
shown in Fig. 2: $\theta_{1(2)}$, defined as the angle between the direction
of the ${K}^{+(-)}$ meson and the direction opposite to the $B$-meson momentum
in the rest frame of the ${K}^{*0}$ (${\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}$) resonance, and $\phi$, the
angle between the decay planes of the two vector mesons in the $B$-meson rest
frame. From angular momentum conservation, three relative polarisations of the
final state are possible for $VV$ final states that correspond to longitudinal
($0$ or $L$), or transverse to the direction of motion and parallel
($\parallel$) or perpendicular ($\perp$) to each other. For the two-body
invariant mass of the $({{K}^{+}}{{\pi}^{-}})$ and $({{K}^{-}}{{\pi}^{+}})$
pairs, noted as $m_{1}\equiv M({{K}^{+}}{{\pi}^{-}})$ and $m_{2}\equiv
M({{K}^{-}}{{\pi}^{+}})$, a range of
$150$$\text{\,Me\kern-1.00006ptV\\!/}c^{2}$ around the known ${K}^{*0}$ mass
[5] is considered. Therefore, $(K\pi)$ pairs may not only originate from the
spin-1 ${K}^{*0}$ meson, but also from other spin states. This justifies that,
besides the helicity angles, a phenomenological description of the two-body
invariant mass spectra, employing the isobar model, is adopted in the analytic
model. In the isobar approach, the decay amplitude is modelled as a linear
superposition of quasi-two-body amplitudes [14, *Morgan:1968zza,
*Herndon:1973yn].

Figure 2: Definition of the helicity angles, employed in the angular analysis
of the ${{B}^{0}_{({s})}}\\!\rightarrow{{K}^{*0}}{{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}$ decays. Each angle is defined
in the rest frame of the decaying particle.

For the S–wave ($J=0$), the $K^{*}_{0}(1430)^{0}$ resonance, the possible
$K^{*}_{0}(700)^{0}$ (or $\kappa$) and a non-resonant component, $(K\pi)_{0}$,
need to be accounted for. This is done using the LASS parameterisation [17],
which is an effective-range elastic scattering amplitude, interfering with the
$K^{*}_{0}(1430)^{0}$ meson,

$\mathcal{M}_{0}(m)\propto\frac{m}{q}\left(\frac{1}{\cot\delta_{\beta}-i}+e^{2i\delta_{\beta}}\frac{M_{0}\Gamma_{0}(m)}{M_{0}^{2}-m^{2}-iM_{0}\Gamma_{0}(m)}\right),$
(1)

where

$\Gamma_{0}(m)=\Gamma_{0}\frac{M_{0}}{m}\left(\frac{q}{q_{0}}\right)$ (2)

represents the $K^{*}_{0}(1430)^{0}$ width. In Eq. (1) and Eq. (2) $q$ is the
$(K\pi)$ centre-of-mass decay momentum, and $M_{0}$, $\Gamma_{0}$ and $q_{0}$
are the $K^{*}_{0}(1430)^{0}$ mass, width and centre-of-mass decay momentum at
the pole, respectively. The effective-range elastic scattering amplitude
component depends on

$\cot\delta_{\beta}=\frac{1}{aq}+\frac{1}{2}bq,$

where $a$ is the scattering length and $b$ the effective range.

For the P–wave ($J=1$), only the ${K}^{*}(892)^{0}$ resonance is considered.
Other P–wave resonances, such as ${K}^{*}(1410)^{0}$ or ${K}^{*}(1680)^{0}$,
with pole masses much above the fit region, are neglected. Resonances with
higher spin, for instance the D–wave ${K}^{*}_{2}(1430)^{0}$ meson, are
negligible in the considered two-body mass range [7] and are also disregarded.
The ${K}^{*0}$ amplitude is parameterised with a spin-1 relativistic
Breit–Wigner amplitude,

$\mathcal{M}_{1}(m)\propto\frac{m}{q}\frac{M_{1}\Gamma_{1}(m)}{(M_{1}^{2}-m^{2})-iM_{1}\Gamma_{1}(m)}.$
(3)

The mass-dependent width is given by

$\Gamma_{1}(m)=\Gamma_{1}\frac{M_{1}}{m}\frac{1+r^{2}q_{1}^{2}}{1+r^{2}q^{2}}\left(\frac{q}{q_{1}}\right)^{3},$
(4)

where $M_{1}$ and $\Gamma_{1}$ are the ${K}^{*0}$ mass and width, $r$ is the
interaction radius parameterising the centrifugal barrier penetration factor,
and $q_{1}$ corresponds to the centre-of-mass decay momentum at the resonance
pole. The values of the mass propagator parameters are summarised in Table 1.

Table 1: Parameters of the mass propagators employed in the amplitude analysis. |  | $(K\pi)_{0}$ | ${K}^{*0}$
---|---|---|---
|  |  $J=0$ [17, 18] |  $J=1$ [5]
$M_{J}$ | [MeV/$c^{2}$],, | $1435\pm\phantom{2}7$ | $895.81\pm 0.19$
$\Gamma_{J}$ | [MeV] | $\phantom{1}279\pm 22$ | $\phantom{0}47.4\phantom{1}\pm 0.6\phantom{0}$
$r$ | [$c$/GeV] | $\phantom{00}-$ | $\phantom{00}3.0\phantom{1}\pm 0.5\phantom{0}$
$a$ | [$c$/GeV] | $\phantom{00}1.95\pm 0.11$ | $\phantom{00}-$
$b$ | [$c$/GeV] | $\phantom{00}1.76\pm 0.76$ | $\phantom{00}-$

The differential decay rate for ${B}^{0}_{({s})}$ mesons333Charge conjugation
is not implied in the rest of this section. For the charge-conjugated mode,
${{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}_{({s})}}\\!\rightarrow({{K}^{+}}{{\pi}^{-}})({{K}^{-}}{{\pi}^{+}})$,
the decay rate is obtained applying the transformation
$A_{i}\rightarrow\eta_{i}\bar{A}_{i}$ in Eq. (5) where the corresponding
$C\\!P$ eigenvalues, $\eta_{i}$, are given in Table 2. at production is given
by [19, 6],

$\displaystyle\frac{\mathrm{d}^{5}\Gamma}{\mathrm{d}{\rm
cos}\,\theta_{1}\mathrm{d}{\rm
cos}\,\theta_{2}\mathrm{d}\phi\mathrm{d}m_{1}\mathrm{d}m_{2}}$
$\displaystyle=\frac{9}{8\pi}\Phi_{4}(m_{1},m_{2})\left|\sum_{i=1}^{6}A_{i}g_{i}(m_{1},m_{2},\theta_{1},\theta_{2},\phi)\right|^{2}$
$\displaystyle=\sum_{i=1}^{6}\sum_{j\geq
i}^{6}\mathcal{R}e[A_{i}A^{*}_{j}F_{ij}],$ (5)

where $\Phi_{4}$ is the four-body phase space factor. The index $i$ runs over
the first column of Table 2 where the different decay amplitudes,
$A_{i}\equiv|A_{i}|e^{i\delta_{i}}$, and the angular-mass functions, $g_{i}$,
are listed. The angular dependence of these functions is obtained from
spherical harmonics as explained in Ref. [19]. For $C\\!P$-studies, the
$C\\!P$-odd, $A_{S}^{+}$, and $C\\!P$-even, $A_{S}^{-}$, eigenstates of the
S–wave polarisation amplitudes are preferred to the vector-scalar ($VS$) and
scalar-vector ($SV$) helicity amplitudes, to which they are related by

$A_{S}^{+}=\frac{A_{VS}+A_{SV}}{\sqrt{2}}\quad\text{and}\quad
A_{S}^{-}=\frac{A_{VS}-A_{SV}}{\sqrt{2}}.$

The remaining amplitudes, except for $A_{\perp}$, correspond to $C\\!P$-even
eigenstates. The contributions can be quantified by the terms $F_{ij}$,
defined as

$F_{ij}=\frac{9}{8\pi}\Phi_{4}(m_{1},m_{2})g_{i}(m_{1},m_{2},\theta_{1},\theta_{2},\phi)g_{j}^{*}(m_{1},m_{2},\theta_{1},\theta_{2},\phi)(2-\delta_{ij}),$
(6)

which are normalised according to

$\int F_{ij}\mathrm{d}m_{1}\mathrm{d}m_{2}\mathrm{d}{\rm
cos}\,\theta_{1}\mathrm{d}{\rm cos}\,\theta_{1}\mathrm{d}\phi=\delta_{ij}.$

This condition ensures that $\sum_{i=1}^{6}|A_{i}|^{2}=1$.

Table 2: Amplitudes, $A_{i}$, and angle-mass functions, $g_{i}(m_{1},m_{2},\theta_{1},\theta_{2},\phi)$, of the differential decay rate of Eq. (5). In particular, $A_{0}$, $A_{\parallel}$ and $A_{\perp}$ are the longitudinal, parallel and transverse helicity amplitudes of the P–wave whereas $A_{S}^{+}$ and $A_{S}^{-}$ are the combinations of $C\\!P$ eigenstate amplitudes of the $SV$ and $VS$ states and $A_{SS}$ is the double S–wave amplitude. The table indicates the corresponding $C\\!P$ eigenvalue, $\eta_{i}$. The mass propagators, $\mathcal{M}_{0,1}(m)$, are discussed in the text. $i$ | $A_{i}$ | $\phantom{-}\eta_{i}$ | $g_{i}(m_{1},m_{2},\theta_{1},\theta_{2},\phi)$
---|---|---|---
1 | $A_{0}$ | $\phantom{-}1$ | $\cos\theta_{1}\cos\theta_{2}{\cal M}_{1}(m_{1}){\cal M}_{1}(m_{2})$
2 | $A_{\parallel}$ | $\phantom{-}1$ | $\frac{1}{\sqrt{2}}\sin\theta_{1}\sin\theta_{2}\cos\phi{\cal M}_{1}(m_{1}){\cal M}_{1}(m_{2})$
3 | $A_{\perp}$ | $-1$ | $\frac{i}{\sqrt{2}}\sin\theta_{1}\sin\theta_{2}\sin\phi{\cal M}_{1}(m_{1}){\cal M}_{1}(m_{2})$
4 | $A_{S}^{+}$ | $-1$ | $-\frac{1}{\sqrt{6}}(\cos\theta_{1}{\cal M}_{1}(m_{1}){\cal M}_{0}(m_{2})-\cos\theta_{2}{\cal M}_{0}(m_{1}){\cal M}_{1}(m_{2}))$
5 | $A_{S}^{-}$ | $\phantom{-}1$ | $-\frac{1}{\sqrt{6}}(\cos\theta_{1}{\cal M}_{1}(m_{1}){\cal M}_{0}(m_{2})+\cos\theta_{2}{\cal M}_{0}(m_{1}){\cal M}_{1}(m_{2}))$
6 | $A_{SS}$ | $\phantom{-}1$ | $-\frac{1}{3}{\cal M}_{0}(m_{1}){\cal M}_{0}(m_{2})$

The polarisation fractions of the $VV$ amplitudes are defined as

$f_{L,\parallel,\perp}=\frac{|A_{0,\parallel,\perp}|^{2}}{|A_{0}|^{2}+|A_{\parallel}|^{2}+|A_{\perp}|^{2}},$

where $A_{0}$, $A_{\parallel}$ and $A_{\perp}$ are the longitudinal, parallel
and transverse amplitudes of the P–wave. Therefore, $f_{L}$ is the fraction of
${{B}^{0}_{({s})}}\\!\rightarrow{{K}^{*0}}{{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}$ longitudinally polarised
decays. The polarisation fractions are preferred to the amplitude moduli since
they are independent of the considered $(K\pi)$ mass range. The P–wave
amplitudes moduli can always be recovered as

$|A_{0,\parallel,\perp}|^{2}=(1-|A_{S}^{+}|^{2}-|A_{S}^{-}|^{2}-|A_{SS}|^{2})\,f_{L,\parallel,\perp}.$

The phase of all propagators is set to be zero at the ${K}^{*0}$ mass. In
addition, a global phase can be factorised without affecting the decay rate
setting $\delta_{0}\equiv 0$. The last two requirements establish the
definition of the amplitude phases ($\delta_{\parallel}$, $\delta_{\perp}$,
$\delta_{S}^{-}$, $\delta_{S}^{+}$ and $\delta_{SS}$) as the phase relative to
that of the longitudinal P–wave amplitude at the ${K}^{*0}$ mass.

Since ${B}^{0}_{({s})}$ mesons oscillate, the decay rate evolves with time.
The time-dependent amplitudes are obtained replacing
$A_{i}\rightarrow\mathcal{A}_{i}(t)$ and
$\bar{A}_{i}\rightarrow\bar{\mathcal{A}}_{i}(t)$ in Eq. (5) being

$\mathcal{A}_{i}(t)=\left[g_{+}(t)A_{i}+\eta_{i}\frac{{\rm q}}{{\rm
p}}g_{-}(t)\bar{A}_{i}\right]\quad\text{and}\quad\bar{\mathcal{A}}_{i}(t)=\left[\frac{{\rm
p}}{{\rm q}}g_{-}(t)A_{i}+\eta_{i}g_{+}(t)\bar{A}_{i}\right],$

with

$g_{+}(t)=\frac{1}{2}\left(e^{-\left(iM_{\rm
L}+\frac{{\Gamma_{\mathrm{L}}}}{2}\right)t}+e^{-\left(iM_{\rm
H}+\frac{{\Gamma_{\mathrm{H}}}}{2}\right)t}\right)\;\text{and}\;g_{-}(t)=\frac{1}{2}\left(e^{-\left(iM_{\rm
L}+\frac{{\Gamma_{\mathrm{L}}}}{2}\right)t}-e^{-\left(iM_{\rm
H}+\frac{{\Gamma_{\mathrm{H}}}}{2}\right)t}\right),$

where ${\Gamma_{\mathrm{L}}}$ and ${\Gamma_{\mathrm{H}}}$ are the widths of
the light and heavy mass eigenstates of the ${{B}^{0}_{({s})}}-{{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}_{({s})}}$ system and $M_{\rm L}$
and $M_{\rm H}$ are their masses. The coefficients p and q are the mixing
terms that relate the flavour and mass eigenstates,

${{B}^{0}_{({s})}}_{\rm H}={\rm p}{{B}^{0}_{({s})}}+{\rm q}{{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}_{({s})}}\quad\text{and}\quad{{B}^{0}_{({s})}}_{\rm
L}={\rm p}{{B}^{0}_{({s})}}-{\rm q}{{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}_{({s})}}.$

Masses and widths are often considered in their averages and differences,
${M=(M_{\rm L}+M_{\rm H})/2}$, $\Delta M=M_{\rm L}-M_{\rm H}$,
$\Gamma=({\Gamma_{\mathrm{L}}}+{\Gamma_{\mathrm{H}}})/2$ and
$\Delta\Gamma={\Gamma_{\mathrm{L}}}-{\Gamma_{\mathrm{H}}}$, in particular in
their relation with the mixing phase,

$\tan\phi_{(s)}=2\frac{\Delta M}{\Delta\Gamma}\left(1-\frac{|{\rm q}|}{|{\rm
p}|}\right).$ (7)

In this analysis, no attempt is made to identify the flavour of the initial
${B}^{0}_{({s})}$ meson and time-integrated spectra are considered.
Consequently, the selected candidates correspond to untagged and time-
integrated decay rates and there is no sensitivity to direct and mixing-
induced $C\\!P$ violation. Moreover, since the origin of phases is set in a
$C\\!P$-even eigenstate ($\delta_{0}=0$), for the $C\\!P$-odd eigenstates, the
untagged time-integrated decay is only sensitive to the phase difference
$\delta_{\perp}-\delta_{S}^{+}$. The present experimental knowledge is
compatible with small $C\\!P$ violation in mixing [20] and with the absence of
direct $C\\!P$ violation in the
${{B}^{0}_{s}}\\!\rightarrow({{K}^{+}}{{\pi}^{-}})({{K}^{-}}{{\pi}^{+}})$
system [7].

The dependence of the decay rate in an untagged and time-integrated analysis
of a ${B}^{0}_{({s})}$ meson can be expressed as

$\frac{\mathrm{d}^{5}(\Gamma+\overline{\Gamma})}{\mathrm{d}{\rm
cos}\,\theta_{1}\,\mathrm{d}{\rm
cos}\,\theta_{2}\,\mathrm{d}\phi\,\mathrm{d}m_{1}\,\mathrm{d}m_{2}}=N\sum_{i=1}^{6}\sum_{j\geq
i}^{6}\mathcal{R}e\left[A_{i}A_{j}^{*}\left(\frac{1-\eta_{i}}{{\Gamma_{\mathrm{H}}}}+\frac{1+\eta_{i}}{{\Gamma_{\mathrm{L}}}}\right)F_{ij}\delta_{\eta_{i}\eta_{j}}\right],$
(8)

where the $A_{i}$ amplitudes account for the the average of ${B}^{0}_{({s})}$
and ${\kern 1.79993pt\overline{\kern-1.79993ptB}}{}^{0}_{({s})}$ decays and
$N$ is a normalisation constant. For the ${B}^{0}$ meson, a further
simplification of the decay rate is considered, since
$\Delta\Gamma/\Gamma=-0.002\pm 0.010$ [20] the light and heavy mass eigenstate
widths can be assumed to be equal,

$\left(\frac{1-\eta_{i}}{{\Gamma_{\mathrm{H}}}}+\frac{1+\eta_{i}}{{\Gamma_{\mathrm{L}}}}\right)\approx\frac{2}{\Gamma},$

and this factor can be extracted as part of the normalisation constant in Eq.
(8). For the ${B}^{0}_{s}$ meson the central values
${\Gamma_{\mathrm{H}}}=0.618$$\text{\,ps}^{-1}$ and
${\Gamma_{\mathrm{L}}}=0.708$$\text{\,ps}^{-1}$ [20] are considered.

##  3 Detector and simulation

The LHCb detector [21, 22] 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 surrounding the $pp$
interaction region, a large-area silicon-strip detector 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 placed 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$\text{\,Ge\kern-1.00006ptV\\!/}c$. The minimum
distance of a track to a primary vertex (PV), the impact parameter (IP), is
measured with a resolution of $(15+29/p_{\mathrm{T}})\,\upmu\text{m}$, where
$p_{\mathrm{T}}$ is the component of the momentum transverse to the beam, in
$\text{\,Ge\kern-1.00006ptV\\!/}c$. Different types of charged hadrons are
distinguished using information from two ring-imaging Cherenkov detectors.
Photons, electrons and hadrons are identified by a calorimeter system
consisting of scintillating-pad and preshower detectors, an electromagnetic
and a hadronic calorimeter. Muons are identified by a system composed of
alternating layers of iron and multiwire proportional chambers.

The magnetic field deflects oppositely charged particles in opposite
directions and this can lead to detection asymmetries. Periodically reversing
the magnetic field polarity throughout the data-taking almost cancels the
effect. The configuration with the magnetic field pointing upwards
(downwards), MagUp (MagDown), bends positively (negatively) charged particles
in the horizontal plane towards the centre of the LHC ring.

The online event selection is performed by a trigger [23], 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. In
the offline selection, trigger signatures are associated with reconstructed
particles. Since the trigger system uses the $p_{\mathrm{T}}$ of the charged
particles, the phase-space and time acceptance is different for events where
signal tracks were involved in the trigger decision (called trigger-on-signal
or TOS throughout) and those where the trigger decision was made using
information from the rest of the event only (noTOS).

Simulated samples of the ${{B}^{0}}\\!\rightarrow{{K}^{*0}}{{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}$ and
${{B}^{0}_{s}}\\!\rightarrow{{K}^{*0}}{{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}$ decays with longitudinal
polarisation fractions of 0.81 and 0.64, respectively, are primarily employed
in these analyses, particularly for the acceptance description as explained in
Sect. 6. Simulated samples of the main peaking background contributions,
${{B}^{0}}\\!\rightarrow{{K}^{*0}}\phi({{K}^{+}}{{K}^{-}})$,
${{B}^{0}}\\!\rightarrow{{\rho}^{0}}{{K}^{*0}}$ and ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{{K}^{*0}}{p}{{\pi}^{-}}$, are also
considered. In the simulation, $pp$ collisions are generated using Pythia [24]
with a specific LHCb configuration [25]. Decays of hadronic particles are
described by EvtGen [26], in which final-state radiation is generated using
Photos [27]. The interaction of the generated particles with the detector, and
its response, are implemented using the Geant4 toolkit [28, 29] as described
in Ref. [30].

##  4 Signal selection

Both data and simulation are filtered with a preliminary selection. Events
containing four good quality tracks with
$p_{\mathrm{T}}>500$$\text{\,Me\kern-1.00006ptV\\!/}c$ are retained. In events
that contain more than one PV, the ${B}^{0}_{({s})}$ candidate constructed
with these four tracks is associated with the PV that has the smallest
$\chi^{2}_{\text{IP}}$, where $\chi^{2}_{\text{IP}}$ is defined as the
difference in the vertex-fit $\chi^{2}$ of the PV reconstructed with and
without the track or tracks in question. Each of the four tracks must fulfil
$\chi^{2}_{\text{IP}}>9$ with respect to the PV and originate from a common
vertex of good quality (${\chi^{2}/\mathrm{ndf}<15}$, where ndf is the number
of degrees of freedom of the vertex). To identify kaons and pions, a selection
in the difference of the log-likelihoods of the kaon and pion hypothesis
($\mathrm{DLL}_{{K}{\pi}}$) is applied. This selection is complemented with
fiducial constraints that optimise the particle identification determination:
the pion and kaon candidates are required to have
$3<p<100$$\text{\,Ge\kern-1.00006ptV\\!/}c$ and $1.5<\eta<4.5$ and be
inconsistent with muon hypothesis. The final state opposite charge $(K\pi)$
pairs are combined into ${K}^{*0}$ and ${\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}$ candidates with a mass within
$150$$\text{\,Me\kern-1.00006ptV\\!/}c^{2}$ of the ${K}^{*0}$ mass. The
${K}^{*0}$ and ${\kern 1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}$
candidates must have $p_{\mathrm{T}}>900$$\text{\,Me\kern-1.00006ptV\\!/}c$
and vertex $\chi^{2}/\mathrm{ndf}<9$. The intermediate resonances must combine
into ${B}^{0}_{({s})}$ candidates within
$500$$\text{\,Me\kern-1.00006ptV\\!/}c^{2}$ of the ${B}^{0}_{s}$ mass, with a
distance of closest approach between their trajectories of less than $0.3$ mm.
To guarantee that the ${B}^{0}_{({s})}$ candidate originates in the
interaction point, the cosine of the angle between the ${B}^{0}_{({s})}$
momentum and the direction of flight from the PV to the decay vertex is
required to be larger than 0.99 and the $\chi^{2}_{\text{IP}}$ with respect to
the PV has to be smaller than 25.

A multivariate selection based on a Boosted Decision Tree with Gradient Boost
[31, *Roe] (BDTG) is employed. It relies on the aforementioned variables and
on the ${B}^{0}_{({s})}$ candidate flight distance with respect to the PV and
its $p_{\mathrm{T}}$. Simulated ${{B}^{0}}\\!\rightarrow{{K}^{*0}}{{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}$ decays with tracks matched to
the generator particles and filtered with the preliminary selection are used
as signal sample, whereas the four-body invariant-mass sideband
$5600<M({{K}^{+}}{{\pi}^{-}}{{K}^{-}}{{\pi}^{+}})<5800\text{\,Me\kern-1.00006ptV\\!/}c^{2}$,
composed of purely combinatorial
$({{K}^{+}}{{\pi}^{-}})({{K}^{-}}{{\pi}^{+}})$ combinations, is used as
background sample for the BDTG training. The number of events in the signal
training sample of the BDTG is determined using the ratio between the
${B}^{0}_{s}$ and the ${B}^{0}$ yields from Ref. [6] and the ${B}^{0}_{s}$
yield obtained with a four-body mass fit to the data sample after the
preliminary selection. The number of events in the background training sample
of the BDTG is estimated by extrapolating the background yield in the sideband
into the $\pm 30$$\text{\,Me\kern-1.00006ptV\\!/}c^{2}$ window around the
${B}^{0}$ mass. The requirement on the BDTG output is chosen to maximise the
figure of merit $N_{S}/\sqrt{N_{S}+N_{B}}$, where $N_{S}$ and $N_{B}$ are the
expected output signal and background yields, respectively. Different BDTGs
are implemented for 2011 and 2012 data.

A comprehensive search for peaking backgrounds, mainly involving intermediate
charm particles, is performed. Decays of $B$ mesons sharing the same final
state with the signal,444The branching fractions in this section are taken
from Ref. [5]. such as
${{B}^{0}}\\!\rightarrow{{D}^{0}}({{K}^{+}}{{K}^{-}}){{\pi}^{+}}{{\pi}^{-}}$
(${\mathcal{B}}\sim 3\times 10^{-6}$) and
${{B}^{0}}\\!\rightarrow{{D}^{0}}({{\pi}^{+}}{{\pi}^{-}}){{K}^{+}}{{K}^{-}}$
(${\mathcal{B}}\sim 6\times 10^{-8}$) decays, are strongly suppressed by the
requirement in the $(K\pi)$ mass. Resonances in three-body combinations
(${K}^{+}$ ${K}^{-}$ ${\pi}^{+}$) and (${K}^{+}$ ${\pi}^{+}$ ${\pi}^{-}$) are
also explored. In the case of the former, the three-body invariant mass in the
data sample is above all known charm resonances. For the latter, no evidence
of candidates originated in ${{B}^{0}}\\!\rightarrow{{D}^{\mp}}{{K}^{\pm}}$
and ${{B}^{0}}\\!\rightarrow{{D}^{\mp}_{s}}{{K}^{\pm}}$ decays
(${\mathcal{B}}\sim 10^{-7}$) or in
${{B}^{0}_{s}}\\!\rightarrow{{D}^{\mp}_{s}}{{K}^{\pm}}$ decays
(${\mathcal{B}}\sim 10^{-6}$) is found. Three-body combinations with a pion
misidentified as a kaon are reconstructed, mainly searching for
${{B}^{0}}\\!\rightarrow{{D}^{-}}({{\pi}^{-}}{{\pi}^{-}}{{K}^{+}}){{\pi}^{+}}$
decays (${\mathcal{B}}=2.45\times 10^{-4}$), but also for
${{B}^{0}_{s}}\\!\rightarrow{{D}^{\pm}_{s}}{{K}^{\mp}}$ (${\mathcal{B}}\sim
10^{-5}$), ${{B}^{0}}\\!\rightarrow{{D}^{-}}{{K}^{+}}$ and
${{B}^{0}}\\!\rightarrow{{D}^{-}_{s}}{{K}^{+}}$ (${\mathcal{B}}\sim 10^{-6}$)
decays. All of them are suppressed to a negligible level by the applied
selection. A search of three-body combinations with a proton misidentified as
a kaon is performed, finding no relevant contribution from decays involving a
${\mathchar 28931\relax}^{+}_{c}$ baryon. Decays into five final-state
particles are also investigated. Contributions of the
${{B}^{0}}\\!\rightarrow\eta^{\prime}(\gamma{{\pi}^{+}}{{\pi}^{-}}){{K}^{*0}}$
decay can be neglected due to the small misidentification probability and the
four-body mass distribution whereas the
${{B}^{0}_{s}}\\!\rightarrow\phi(\pi^{0}\pi^{+}\pi^{-})\phi(K^{+}K^{-})$ decay
is negligible due to the requirement on the $(K\pi)$ mass.

##  5 Four-body mass spectrum

The signal and background yields are determined by means of a simultaneous
extended maximum-likelihood fit to the invariant-mass spectra of the four
final-state particles in the 2011 and 2012 data samples. The
${{B}^{0}_{({s})}}\\!\rightarrow({{K}^{+}}{{\pi}^{-}})({{K}^{-}}{{\pi}^{+}})$
signal decays are parameterised with double-sided Hypatia distributions [33]
with the same parameters except for their means that are shifted by the
difference between the ${B}^{0}$ and ${B}^{0}_{s}$ masses,
$87.13\text{\,Me\kern-1.00006ptV\\!/}c^{2}$ [5]. Misidentified
${{B}^{0}}\\!\rightarrow({{K}^{+}}{{\pi}^{-}})({{K}^{-}}{{K}^{+}})$ (including
${{B}^{0}}\\!\rightarrow{{K}^{*0}}\phi$ decays), ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow({p}{{\pi}^{-}})({{K}^{+}}{{\pi}^{-}})$ and
${{B}^{0}}\\!\rightarrow{{\rho}^{0}}{{K}^{*0}}$ decays are also considered in
the fit. Both the
${{B}^{0}}\\!\rightarrow({{K}^{+}}{{\pi}^{-}})({{K}^{-}}{{K}^{+}})$ and
${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow({p}{{\pi}^{-}})({{K}^{-}}{{\pi}^{+}})$
contributions are described with the sum of a Crystal Ball [34] function and a
Gaussian distribution which shares mean with the Crystal Ball core. The
parameters of these distributions are obtained from simulation, apart from the
mean and resolution values which are free to vary in the fit. Whereas the
distribution mean values are constrained to be the same in the 2011 and 2012
data, the resolution is allowed to have different values for the two samples.
The small contributions from ${{B}^{0}}\\!\rightarrow{{\rho}^{0}}{{K}^{*0}}$
and ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow({p}{{\pi}^{-}})({{K}^{-}}{{\pi}^{+}})$
decays have a broad distribution in the four-body mass and are the object of
specific treatment. The contribution from
${{B}^{0}}\\!\rightarrow{{\rho}^{0}}{{K}^{*0}}$ decays has an expected yield
of $3.5\pm 1.3$ ($6.6\pm 2.3$) in the 2011 (2012) sample. It is estimated from
the detection and selection efficiency measured with simulation, the collected
luminosities, the cross section for ${b}{\overline{b}}$ production, the
hadronisation fractions of ${B}^{0}$ and ${B}^{0}_{s}$ mesons and the known
branching fraction of the mode. Simulated events containing this decay mode
are added with negative weights to the final data sample to subtract its
contribution. The contribution of ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow({p}{{\pi}^{-}})({{K}^{-}}{{\pi}^{+}})$
decays in the 2011 (2012) sample is determined to be $36\pm 16$ ($120\pm 28$)
from a fit to the $({p}{{\pi}^{-}}{{K}^{-}}{{\pi}^{+}})$ four-body mass
spectrum of the selected data. In this study the four-body invariant mass is
recomputed assigning the proton mass to the kaon with the largest
$\mathrm{DLL}_{{p}{K}}$ value. In these fits the ${\mathchar
28931\relax}^{0}_{b}$ component is described with a Gaussian distribution and
the dominant
${{B}^{0}_{s}}\\!\rightarrow({{K}^{+}}{{\pi}^{-}})({{K}^{-}}{{\pi}^{+}})$
background is described with a Crystal Ball function. The parameters of both
lineshapes are obtained from simulation. The remaining contributions, mainly
${{B}^{0}}\\!\rightarrow({{K}^{+}}{{\pi}^{-}})({{K}^{-}}{{K}^{+}})$ and
partially reconstructed events, are parameterised with a decreasing
exponential with a free decay constant. The ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow({p}{{\pi}^{-}})({{K}^{-}}{{\pi}^{+}})$
decay angular distribution is currently unknown and its contribution can not
be subtracted with negatively weighted simulated events. Its subtraction is
commented further below.

Finally, contributions from partially reconstructed $b$-hadron decays and
combinatorial background are also considered. The former is composed of $B$\-
and ${B}^{0}_{s}$-meson decays containing neutral particles that are not
reconstructed. Because of the missing particle, the measured four-body
invariant mass of these candidates lies in the lower sideband of the spectrum.
All contributions to this background are jointly parameterised with an ARGUS
function [35] convolved with a Gaussian resolution function, with the same
width as the signal. The endpoint of the distribution is also fixed to the
${B}^{0}_{s}$ mass minus the ${\pi}^{0}$ mass. The combinatorial background is
composed of charged tracks that are not originating from the signal decay
chain. It is modelled with a linear distribution, with a free slope parameter,
separate for 2011 and 2012 data samples.

The results of the fit to the four-body mass spectrum are shown in Fig. 3 and
the yields are reported in Table 3. In total, about three hundred
${{B}^{0}}\\!\rightarrow({{K}^{+}}{{\pi}^{-}})({{K}^{-}}{{\pi}^{+}})$ signal
candidates are found, a factor seven larger than previous analyses [1, 2]. To
perform a background-subtracted amplitude analysis, the sPlot technique [36,
37] is applied to isolate either the
${{B}^{0}}\\!\rightarrow({{K}^{+}}{{\pi}^{-}})({{K}^{-}}{{\pi}^{+}})$ or the
${{B}^{0}_{s}}\\!\rightarrow({{K}^{+}}{{\pi}^{-}})({{K}^{-}}{{\pi}^{+}})$
decays. The contribution from ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow({p}{{\pi}^{-}})({{K}^{-}}{{\pi}^{+}})$,
for which the yield is fixed, is treated using extended weights according to
Appendix B.2 of Ref. [36]. The sPlot method suppresses the background
contributions using their relative abundance in the four-body invariant mass
spectrum and, therefore, no assumption is required for their phase-space
distribution.

Figure 3: Aggregated four-body invariant-mass fit result of the 2011 and 2012 data. The solid red distribution corresponds to the ${{B}^{0}_{s}}\\!\rightarrow({{K}^{+}}{{\pi}^{-}})({{K}^{-}}{{\pi}^{+}})$ decay, the solid cyan distribution to ${{B}^{0}}\\!\rightarrow({{K}^{+}}{{\pi}^{-}})({{K}^{-}}{{\pi}^{+}})$, the dotted dark blue line to ${{\mathchar 28931\relax}^{0}_{b}}\\!\rightarrow({p}{{\pi}^{-}})({{K}^{-}}{{\pi}^{+}})$, the dotted yellow line to ${{B}^{0}}\\!\rightarrow({{K}^{+}}{{\pi}^{-}})({{K}^{-}}{{K}^{+}})$ and the dotted cyan line represents the partially reconstructed background. The tiny combinatorial background contribution is not represented. The black points with error bars correspond to data to which the ${{B}^{0}}\\!\rightarrow{{\rho}^{0}}{{K}^{*0}}$ contribution has been subtracted with negatively weighted simulation, and the overall fit is represented by the thick blue line. Table 3: Signal and background yields for the 2011 and 2012 data samples, obtained from the fit to the four-body mass spectrum of the selected candidates. Statistical and systematic uncertainties are reported, the latter are estimated as explained in Sect. 8. Yield | 2011 sample | 2012 sample
---|---|---
${{B}^{0}}\\!\rightarrow({{K}^{+}}{{\pi}^{-}})({{K}^{-}}{{\pi}^{+}})$ | $\phantom{0}99\pm 12\pm\phantom{0}3$ | $\phantom{0}249\pm 19\pm\phantom{0}5$
${{B}^{0}_{s}}\\!\rightarrow({{K}^{+}}{{\pi}^{-}})({{K}^{-}}{{\pi}^{+}})$ | $617\pm 26\pm\phantom{0}8$ | $1337\pm 39\pm 12$
Misidentified ${{B}^{0}}\\!\rightarrow({{K}^{+}}{{\pi}^{-}})({{K}^{-}}{{K}^{+}})$ | $145\pm 17\pm\phantom{0}2$ | $\phantom{0}266\pm 27\pm\phantom{0}8$
Partially reconstructed background | $100\pm 15\pm\phantom{0}4$ | $\phantom{0}230\pm 25\pm\phantom{0}6$
Combinatorial background | $\phantom{00}7\pm\phantom{0}5\pm 11$ | $\phantom{00}48\pm 25\pm 25$

##  6 Amplitude analysis

Each of the background-subtracted samples of
${{B}^{0}}\\!\rightarrow({{K}^{+}}{{\pi}^{-}})({{K}^{-}}{{\pi}^{+}})$ and
${{B}^{0}_{s}}\\!\rightarrow({{K}^{+}}{{\pi}^{-}})({{K}^{-}}{{\pi}^{+}})$
decays is the object of a separate amplitude analysis based on the model
described in Sect. 2. As a first step, the effect of a non-uniform efficiency,
depending on the helicity angles and the two-body invariant masses, is
examined. For this purpose, four categories are defined according to the
hardware trigger decisions (TOS or noTOS) and data-taking period (2011 and
2012). The efficiency is accounted for through the complex integrals [38]

$\omega^{k}_{ij}=\int\varepsilon(m_{1},m_{2},\theta_{1},\theta_{2},\phi)F_{ij}\mathrm{d}m_{1}\mathrm{d}m_{2}\mathrm{d}{\rm
cos}\,\theta_{1}\mathrm{d}{\rm cos}\,\theta_{2}\mathrm{d}\phi,$ (9)

where $\varepsilon$ is the total phase-space dependent efficiency, $k$ is the
sample category and $F_{ij}$ are defined in Eq. (6). The integrals of Eq. (9)
are determined using simulated signal samples of each of the four categories,
selected with the same criteria applied to data. A single set of integrals is
used for both the ${B}^{0}_{s}$ and the ${B}^{0}$ amplitude analyses. A
probability density function (PDF) for each category is built

$S^{k}(m_{1},m_{2},\theta_{1},\theta_{2},\phi)=\frac{\sum\limits_{i=1}^{6}\sum\limits_{j\geq
i}^{6}\mathcal{R}e\left[A_{i}A_{j}^{*}\left(\frac{1-\eta_{i}}{{\Gamma_{\mathrm{H}}}}+\frac{1+\eta_{i}}{{\Gamma_{\mathrm{L}}}}\right)F_{ij}\delta_{\eta_{i}\eta_{j}}\right]}{\sum\limits_{i=1}^{6}\sum\limits_{j\geq
i}^{6}\mathcal{R}e\left[A_{i}A_{j}^{*}\left(\frac{1-\eta_{i}}{{\Gamma_{\mathrm{H}}}}+\frac{1+\eta_{i}}{{\Gamma_{\mathrm{L}}}}\right)\omega_{ij}^{k}\delta_{\eta_{i}\eta_{j}}\right]},$
(10)

where $A_{i}$ and $\eta_{i}$ are given in Table 2.

Candidates from all categories are processed in a simultaneous unbinned
maximum-likelihood fit, separately for each signal decay mode, using the PDFs
in Eq. (10). To avoid nonphysical values of the parameters during the
minimisation, some of them are redefined as

$\displaystyle\begin{split}f_{\parallel}&=x_{f_{\parallel}}(1-f_{L}),\\\
f_{\perp}&=(1-x_{f_{\parallel}})(1-f_{L}),\\\
|A_{S}^{+}|^{2}&=x_{|A_{S}^{+}|^{2}}(1-|A_{S}^{-}|^{2}),\\\
|A_{SS}|^{2}&=x_{|A_{SS}|^{2}}(1-|A_{S}^{-}|^{2}-|A_{S}^{+}|^{2}),\end{split}$

where $x_{f_{\parallel}}$, $x_{|A_{S}^{+}|^{2}}$ and $x_{|A_{SS}|^{2}}$ are
used in the fit, together with $f_{L}$,
$|A_{S}^{-}|^{2}$,$\delta_{\parallel}$, $\delta_{\perp}-\delta_{S}^{+}$,
$\delta_{S}^{-}$ and $\delta_{SS}$. The former three variables are free to
vary within the range $[0,1]$, ensuring that the sum of all the squared
amplitudes is never greater than $1$. The fit results are corrected for a
small reducible bias, originated in discrepancies between data and simulation,
as explained in Sect. 7. The final results are shown in Table 4.

Table 4: Results of the amplitude analysis of ${{B}^{0}}\\!\rightarrow({{K}^{+}}{{\pi}^{-}})({{K}^{-}}{{\pi}^{+}})$ and ${{B}^{0}_{s}}\\!\rightarrow({{K}^{+}}{{\pi}^{-}})({{K}^{-}}{{\pi}^{+}})$ decays. The observables above the line are directly obtained from the maximum-likelihood fit whereas those below are obtained from the former, as explained in the text, with correlations accounted for in their estimated uncertainties. For each result, the first quoted uncertainty is statistical and the second systematic. The estimation of the latter is described in Sect. 7. Parameter | ${{B}^{0}}\\!\rightarrow{{K}^{*0}}{{\kern 1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}$ | ${{B}^{0}_{s}}\\!\rightarrow{{K}^{*0}}{{\kern 1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}$
---|---|---
$f_{L}$ | $0.724\pm 0.051\pm 0.016$ | $0.240\pm 0.031\pm 0.025$
$x_{f_{\parallel}}$ | $\phantom{0}0.42\pm\phantom{0}0.10\pm\phantom{0}0.03$ | $0.307\pm 0.031\pm 0.010$
$|A_{S}^{-}|^{2}$ | $0.377\pm 0.052\pm 0.024$ | $0.558\pm 0.021\pm 0.014$
$x_{|A_{S}^{+}|^{2}}$ | $0.013\pm 0.027\pm 0.011$ | $0.109\pm 0.028\pm 0.024$
$x_{|A_{SS}|^{2}}$ | $0.038\pm 0.022\pm 0.006$ | $0.222\pm 0.025\pm 0.031$
$\delta_{\parallel}$ | $\phantom{0}2.51\pm\phantom{0}0.22\pm\phantom{0}0.06$ | $\phantom{0}2.37\pm\phantom{0}0.12\pm\phantom{0}0.06$
$\delta_{\perp}-\delta_{S}^{+}$ | $\phantom{0}5.44\pm\phantom{0}0.86\pm\phantom{0}0.22$ | $\phantom{0}4.40\pm\phantom{0}0.17\pm\phantom{0}0.07$
$\delta_{S}^{-}$ | $\phantom{0}5.11\pm\phantom{0}0.13\pm\phantom{0}0.04$ | $\phantom{0}1.80\pm\phantom{0}0.10\pm\phantom{0}0.06$
$\delta_{SS}$ | $\phantom{0}2.88\pm\phantom{0}0.35\pm\phantom{0}0.13$ | $\phantom{0}0.99\pm\phantom{0}0.13\pm\phantom{0}0.06$
$f_{\parallel}$ | $0.116\pm 0.033\pm 0.012$ | $0.234\pm 0.025\pm 0.010$
$f_{\perp}$ | $0.160\pm 0.044\pm 0.012$ | $0.526\pm 0.032\pm 0.019$
$|A_{S}^{+}|^{2}$ | $0.008\pm 0.013\pm 0.007$ | $0.048\pm 0.014\pm 0.011$
$|A_{SS}|^{2}$ | $0.023\pm 0.014\pm 0.004$ | $0.087\pm 0.011\pm 0.011$
S–wave fraction | $0.408\pm 0.050\pm 0.017$ | $0.694\pm 0.016\pm 0.010$

Figs. 4 and 5 show the one-dimensional projections of the amplitude fit to the
${{B}^{0}}\\!\rightarrow({{K}^{+}}{{\pi}^{-}})({{K}^{-}}{{\pi}^{+}})$ and
${{B}^{0}_{s}}\\!\rightarrow({{K}^{+}}{{\pi}^{-}})({{K}^{-}}{{\pi}^{+}})$
signal samples in which the background is statistically subtracted by means of
the sPlot technique. Three contributions are shown: $VV$, produced by
$({{K}^{+}}{{\pi}^{-}})$ $({{K}^{-}}{{\pi}^{+}})$ pairs originating in a
${K}^{*0}$ ${\kern 1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}$ decay; $VS$,
accounting for amplitudes in which only one of the $(K\pi)$ pairs originates
in a ${K}^{*0}$ decay; and $SS$, where none of the two $(K\pi)$ pairs
originate in a ${K}^{*0}$ decay.

The fraction of $VV$ decays, or purity at production, of the
${{B}^{0}}\\!\rightarrow{{K}^{*0}}{{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}$ signal, $f^{P}_{{{B}^{0}}}$, is
estimated from the amplitude analysis and found to be

$f^{P}_{{{B}^{0}}}\equiv
1-|A_{SS}|^{2}-|A_{S}^{+}|^{2}-|A_{S}^{-}|^{2}=0.592\pm
0.050\text{\,(stat)}\pm 0.017\text{\,(syst)}.$

The significance of this magnitude, computed as its value over the sum in
quadrature of the statistical and systematic uncertainty, is found to be 10.8
standard deviations. This significance corresponds to the presence of
${{B}^{0}}\\!\rightarrow{{K}^{*0}}{{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}$ $VV$ decays in the data sample.
The S–wave fraction of the decay is equal to $0.408=1-f^{P}_{{{B}^{0}}}$. For
the ${{B}^{0}_{s}}\\!\rightarrow{{K}^{*0}}{{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}$ mode the S–wave fraction is
found to be ${0.694\pm 0.016\text{\,(stat)}\pm 0.010\text{\,(syst)}}$.

Figure 4: Projections of the amplitude fit results for the
${{B}^{0}}\\!\rightarrow{{K}^{*0}}{{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}$ decay mode on the helicity
angles (top row: $\cos\theta_{1}$ left, $\cos\theta_{2}$ centre and $\phi$
right) and on the two-body invariant masses (bottom row:
$M({{K}^{+}}{{\pi}^{-}})$ left and $M({{K}^{-}}{{\pi}^{+}})$ centre). The
contributing partial waves: $VV$ (dashed red), $VS$ (dashed green) and $SS$
(dotted grey) are shown with lines. The black points correspond to data and
the overall fit is represented by the blue line. Figure 5: Projections of the
amplitude fit results for the ${{B}^{0}_{s}}\\!\rightarrow{{K}^{*0}}{{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}$ decay mode on the helicity
angles (top row: $\cos\theta_{1}$ left, $\cos\theta_{2}$ centre and $\phi$
right) and on the two-body invariant masses (bottom row:
$M({{K}^{+}}{{\pi}^{-}})$ left and $M({{K}^{-}}{{\pi}^{+}})$ centre). The
contributing partial waves: $VV$ (dashed red), $VS$ (dashed green) and $SS$
(dotted grey) are shown with lines. The black points correspond to data and
the overall fit is represented by the blue line.

##  7 Systematic uncertainties of the amplitude analysis

Several sources of systematic uncertainty that affect the results of the
amplitude analysis are considered and discussed in the following.

Fit method. Biases induced by the fitting method are evaluated with a large
ensemble of pseudoexperiments. For each signal decay, samples with the same
yield of signal observed in data (see Table 3) are generated according to the
PDF of Eq. (8) with inputs set to the results summarised in Table 4. The use
of the weights defined in Eq. (9) to account the detector acceptance would
require a full simulation and, instead, a parametric efficiency is considered.
For each observable, the mean deviation of the result from the input value is
assigned as a systematic uncertainty.

Description of the kinematic acceptance. The uncertainty on the signal
efficiency relies on the coefficients of Eq. (9) that are estimated with
simulation. To evaluate its impact on the amplitude analysis results, the fit
to data is repeated several times with alternative coefficients varied
according to their covariance matrix. The standard deviation of the
distribution of the fit results for each observable is assigned as a
systematic uncertainty.

Resolution. The fit performed assumes a perfect resolution on the phase-space
variables. The impact of the detector resolution on these variables is
estimated with sets of pseudoexperiments adding per-event random deviations
according to the resolution estimated from simulation. For each observable,
the mean deviation of the result from the measured value is assigned as a
systematic uncertainty.

P–wave mass model. The amplitude analysis is repeated with alternative values
of the parameters that define the P–wave mass propagator, detailed in Table 1,
randomly sampled from their known values [5]. The standard deviation of the
distribution of the amplitude fit results for each observable is assigned as a
systematic uncertainty.

S–wave mass model. In addition to the default S–wave propagator, described in
Sect. 2, two alternative models are used: the LASS lineshape with the
parameters of Table 5, obtained with ${{B}^{0}}\\!\rightarrow
J/\psi{{K}^{+}}{{\pi}^{-}}$ decays within the analysis of Ref. [39], and the
propagator proposed in Ref. [40]. The amplitude fit is performed with these
two alternatives and, for each observable, the largest deviation from the
baseline result is assigned as a systematic uncertainty.

Table 5: Alternative parameters of the LASS mass propagator used in the S–wave systematic uncertainty estimation. |  | $({{K}^{+}}{{\pi}^{-}})_{0}$
---|---|---
$M_{0}$ | [MeV/$c^{2}$],, | $\,1456.7\,$ | $\pm$ | $\,3.9$
$\Gamma_{0}$ | [MeV] | $323\,$ | $\pm$ | $\,11$
$a$ | [$c$/GeV] | $3.83\,$ | $\pm$ | $\,0.11$
$b$ | [$c$/GeV] | $2.86\,$ | $\pm$ | $\,0.22$

Differences between data and simulation. An iterative method [41], is used to
weight the simulated events and improve the description of the track
multiplicity and ${B}^{0}_{({s})}$-meson momentum distributions. The procedure
is repeated multiple times and, for each observable, the mean bias of the
amplitude fit result is corrected for in the results of Table 4 while its
standard deviation is assigned as a systematic uncertainty.

Background subtraction. The data set used in the amplitude analysis is
background subtracted using the sPlot method [36, 37] that relies in the
lineshapes of the four-body mass fit discussed in Sect. 5. The uncertainty
related to the determination of the signal weights is evaluated repeating the
amplitude analysis fits with weights obtained fitting the four-body invariant-
mass with two alternative models. In the first case, the model describing the
signal is defined by the sum of two Crystal Ball functions [34] with a common,
free, peak value and the resolution parameter fixed from simulation. In the
second case, the model describing the combinatorial background is assumed to
be an exponential function. The amplitude fit is performed with the sPlot-
weights obtained with the two alternatives and, for each observable, the
largest deviation from the baseline result is assigned as a systematic
uncertainty. This procedure is also used when addressing the systematic
uncertainties in the measured yields of the different subsamples, as discussed
in Sect. 8.

Peaking backgrounds. The uncertainty related to the fluctuations in the yields
of the ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow(p{{\pi}^{-}})({{K}^{-}}{{\pi}^{+}})$ and
${{B}^{0}}\\!\rightarrow{{\rho}^{0}}{{K}^{*0}}$ background contributions are
estimated repeating the amplitude-analysis fit with the yield values varied by
their uncertainties reported in Sect. 5. For each observable, the largest
deviation from the default result is assigned as a systematic uncertainty.
This procedure is also used when addressing the systematic uncertainties of
the four-body invariant mass yields in Sect. 8.

Time acceptance. The amplitude analysis does not account for possible decay-
time dependency of the efficiency, however, the trigger and the offline
selections may have an impact on it. This effect only affects
${B}^{0}_{s}$-meson decays and is accounted for by estimating effective
shifts: ${\Gamma_{\mathrm{H}}}=0.618\rightarrow 0.598$$\text{\,ps}^{-1}$ and
${\Gamma_{\mathrm{L}}}=0.708\rightarrow 0.732$$\text{\,ps}^{-1}$, which are
obtained with simulation. For each observable, the variation of the result of
the fit after introducing these values in the amplitude analysis is considered
as a systematic uncertainty.

The resulting systematic uncertainties and the corrected biases, originated in
the differences between data and simulation, are detailed in Table 6 for the
parameters of the amplitude-analysis fit. The corresponding values for the
derived observables are summarised in Table 7. The total systematic
uncertainty is computed as the sum in quadrature of the different
contributions.

Table 6: Systematic uncertainties for the parameters of the amplitude-analysis
fit of the
${{B}^{0}_{({s})}}\\!\rightarrow({{K}^{+}}{{\pi}^{-}})({{K}^{-}}{{\pi}^{+}})$
decay. The bias related to differences between data and simulation is included
in the results shown in Table 4.

Decay mode | ${{B}^{0}}\\!\rightarrow({{K}^{+}}{{\pi}^{-}})({{K}^{-}}{{\pi}^{+}})$
---|---
Parameter | $f_{L}$ | $x_{f_{\parallel}}$ | $|A_{S}^{-}|^{2}$ | $\phantom{-}x_{|A_{S}^{+}|^{2}}$ | $x_{|A_{SS}|^{2}}$ | $\delta_{\parallel}$ | $\delta_{\perp}-\delta_{S}^{+}$ | $\delta_{S}^{-}$ | $\delta_{SS}$
Bias data-simulation | 0.001 | 0.00 | 0.006 | $-0.001$ | 0.004 | 0.01 | $-0.01$ | 0.00 | 0.01
Fit method | 0.007 | 0.01 | 0.011 | $\phantom{-}0.009$ | 0.001 | 0.00 | $\phantom{-}0.01$ | 0.00 | 0.02
Kinematic acceptance | 0.005 | 0.01 | 0.006 | $\phantom{-}0.004$ | 0.002 | 0.03 | $\phantom{-}0.12$ | 0.01 | 0.04
Resolution | 0.007 | 0.00 | 0.005 | $\phantom{-}0.001$ | 0.002 | 0.00 | $\phantom{-}0.16$ | 0.00 | 0.02
P–wave mass model | 0.001 | 0.00 | 0.004 | $\phantom{-}0.001$ | 0.002 | 0.00 | $\phantom{-}0.01$ | 0.00 | 0.02
S–wave mass model | 0.007 | 0.01 | 0.016 | $\phantom{-}0.003$ | 0.002 | 0.03 | $\phantom{-}0.03$ | 0.03 | 0.02
Differences data-simulation | 0.004 | 0.00 | 0.002 | $\phantom{-}0.001$ | 0.001 | 0.01 | $\phantom{-}0.01$ | 0.01 | 0.01
Background subtraction | 0.002 | 0.01 | 0.006 | $\phantom{-}0.001$ | 0.002 | 0.01 | $\phantom{-}0.06$ | 0.01 | 0.09
Peaking backgrounds | 0.009 | 0.02 | 0.009 | $\phantom{-}0.003$ | 0.003 | 0.04 | $\phantom{-}0.06$ | 0.01 | 0.08
1-10 Total systematic unc. | 0.016 | 0.03 | 0.024 | $\phantom{-}0.011$ | 0.006 | 0.06 | $\phantom{-}0.22$ | 0.04 | 0.13

Decay mode | ${{B}^{0}_{s}}\\!\rightarrow({{K}^{+}}{{\pi}^{-}})({{K}^{-}}{{\pi}^{+}})$
---|---
Parameter | $f_{L}$ | $x_{f_{\parallel}}$ | $|A_{S}^{-}|^{2}$ | $\phantom{-}x_{|A_{S}^{+}|^{2}}$ | $x_{|A_{SS}|^{2}}$ | $\delta_{\parallel}$ | $\delta_{\perp}-\delta_{S}^{+}$ | $\delta_{S}^{-}$ | $\delta_{SS}$
Bias data-simulation | 0.004 | 0.003 | 0.007 | $-0.003$ | 0.021 | 0.05 | $\phantom{-}0.00$ | 0.05 | 0.07
Fit method | 0.001 | 0.000 | 0.001 | $\phantom{-}0.000$ | 0.000 | 0.00 | $\phantom{-}0.00$ | 0.00 | 0.00
Kinematic acceptance | 0.011 | 0.006 | 0.011 | $\phantom{-}0.021$ | 0.009 | 0.05 | $\phantom{-}0.07$ | 0.05 | 0.05
Resolution | 0.002 | 0.001 | 0.000 | $\phantom{-}0.002$ | 0.000 | 0.00 | $\phantom{-}0.00$ | 0.00 | 0.00
P–wave mass model | 0.001 | 0.000 | 0.001 | $\phantom{-}0.002$ | 0.009 | 0.00 | $\phantom{-}0.01$ | 0.00 | 0.01
S–wave mass model | 0.021 | 0.001 | 0.007 | $\phantom{-}0.011$ | 0.028 | 0.03 | $\phantom{-}0.02$ | 0.03 | 0.02
Differences data-simulation | 0.002 | 0.000 | 0.001 | $\phantom{-}0.001$ | 0.001 | 0.01 | $\phantom{-}0.00$ | 0.01 | 0.01
Background subtraction | 0.000 | 0.001 | 0.001 | $\phantom{-}0.001$ | 0.004 | 0.01 | $\phantom{-}0.01$ | 0.01 | 0.01
Peaking backgrounds | 0.003 | 0.008 | 0.002 | $\phantom{-}0.002$ | 0.002 | 0.02 | $\phantom{-}0.01$ | 0.02 | 0.01
Time acceptance | 0.008 | 0.014 | 0.008 | $\phantom{-}0.004$ | 0.005 | 0.00 | $\phantom{-}0.00$ | 0.00 | 0.00
1-10 Total systematic unc. | 0.025 | 0.010 | 0.014 | $\phantom{-}0.024$ | 0.031 | 0.06 | $\phantom{-}0.07$ | 0.06 | 0.05

Table 7: Systematic uncertainties for the derived observables of the
amplitude-analysis fit of the
${{B}^{0}_{({s})}}\\!\rightarrow({{K}^{+}}{{\pi}^{-}})({{K}^{-}}{{\pi}^{+}})$
decay. The bias related to differences between data and simulation is included
in the results shown in Table 4.

Decay mode | ${{B}^{0}}\\!\rightarrow({{K}^{+}}{{\pi}^{-}})({{K}^{-}}{{\pi}^{+}})$
---|---
Observable | $f_{\parallel}$ | $f_{\perp}$ | $|A_{S}^{+}|^{2}$ | $|A_{SS}|^{2}$ |  S–wave fraction
Bias data-simulation | 0.001 | $-0.001$ | $-0.001$ | 0.002 | 0.007
Fit method | 0.000 | $\phantom{-}0.007$ | $\phantom{-}0.005$ | 0.000 | 0.006
Kinematic acceptance | 0.003 | $\phantom{-}0.004$ | $\phantom{-}0.001$ | 0.003 | 0.006
Resolution | 0.001 | $\phantom{-}0.003$ | $\phantom{-}0.000$ | 0.001 | 0.006
P–wave mass model | 0.000 | $\phantom{-}0.001$ | $\phantom{-}0.000$ | 0.001 | 0.005
S–wave mass model | 0.000 | $\phantom{-}0.007$ | $\phantom{-}0.002$ | 0.002 | 0.008
Differences data-simulation | 0.001 | $\phantom{-}0.003$ | $\phantom{-}0.000$ | 0.001 | 0.002
Background subtraction | 0.005 | $\phantom{-}0.003$ | $\phantom{-}0.001$ | 0.001 | 0.002
Peaking backgrounds | 0.010 | $\phantom{-}0.003$ | $\phantom{-}0.002$ | 0.002 | 0.009
1-10 Total systematic unc. | 0.012 | $\phantom{-}0.012$ | $\phantom{-}0.007$ | 0.004 | 0.017

Decay mode | ${{B}^{0}_{s}}\\!\rightarrow({{K}^{+}}{{\pi}^{-}})({{K}^{-}}{{\pi}^{+}})$
---|---
Observable | $f_{\parallel}$ | $f_{\perp}$ | $|A_{S}^{+}|^{2}$ | $|A_{SS}|^{2}$ |  S–wave fraction
Bias data-simulation | 0.001 | $-0.005$ | $-0.002$ | 0.007 | 0.012
Fit method | 0.001 | $\phantom{-}0.001$ | $\phantom{-}0.000$ | 0.001 | 0.001
Kinematic acceptance | 0.005 | $\phantom{-}0.009$ | $\phantom{-}0.010$ | 0.004 | 0.004
Resolution | 0.000 | $\phantom{-}0.002$ | $\phantom{-}0.000$ | 0.001 | 0.002
P–wave mass model | 0.000 | $\phantom{-}0.001$ | $\phantom{-}0.001$ | 0.003 | 0.005
S–wave mass model | 0.006 | $\phantom{-}0.016$ | $\phantom{-}0.004$ | 0.009 | 0.006
Differences data-simulation | 0.001 | $\phantom{-}0.001$ | $\phantom{-}0.000$ | 0.001 | 0.001
Background subtraction | 0.001 | $\phantom{-}0.001$ | $\phantom{-}0.001$ | 0.002 | 0.002
Peaking backgrounds | 0.007 | $\phantom{-}0.005$ | $\phantom{-}0.001$ | 0.001 | 0.001
Time acceptance | 0.008 | $\phantom{-}0.016$ | $\phantom{-}0.003$ | 0.001 | 0.007
1-10 Total systematic unc. | 0.010 | $\phantom{-}0.019$ | $\phantom{-}0.011$ | 0.011 | 0.010

##  8 Determination of the ratio of branching fractions

In this analysis, the ${{B}^{0}}\\!\rightarrow{{K}^{*0}}{{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}$ branching fraction is measured
relative to that of ${{B}^{0}_{s}}\\!\rightarrow{{K}^{*0}}{{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}$ decays. Since both decays are
selected in the same data sample and share a common final state most
systematic effects cancel. However, some efficiency corrections, eg. those
originated from the difference in phase-space distributions of events of the
two modes, need to be accounted for. The amplitude fit provides the relevant
information to tackle the differences between the two decays.

This branching-fraction ratio is obtained as

$\frac{{\mathcal{B}}(\mbox{${{B}^{0}}\\!\rightarrow{{K}^{*0}}{{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}$})}{{\mathcal{B}}(\mbox{${{B}^{0}_{s}}\\!\rightarrow{{K}^{*0}}{{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}$})}=\frac{\varepsilon_{{{B}^{0}_{s}}}}{\varepsilon_{{{B}^{0}}}}\times\frac{\lambda^{f_{L}}_{{{B}^{0}_{s}}}}{\lambda^{f_{L}}_{{{B}^{0}}}}\times\frac{N_{{B}^{0}}\times
f^{D}_{{{B}^{0}}}}{N_{{B}^{0}_{s}}\times
f^{D}_{{{B}^{0}_{s}}}}\times\frac{f_{s}}{f_{d}},$ (11)

where, for each channel, $\varepsilon_{{{B}^{0}_{({s})}}}$ is the detection
efficiency, $\lambda^{f_{L}}_{{{B}^{0}_{({s})}}}$ is a polarisation-dependent
correction of the efficiency, originated in differences between the measured
polarisation and that assumed in simulation, $N_{{B}^{0}_{({s})}}$ is the
measured number of
${{B}^{0}_{({s})}}\\!\rightarrow({{K}^{+}}{{\pi}^{-}})({{K}^{-}}{{\pi}^{+}})$
candidates and $f^{D}_{{{B}^{0}_{({s})}}}$ represents the $VV$ signal purity
at detection. In this way $N_{{B}^{0}_{({s})}}\times
f^{D}_{{{B}^{0}_{({s})}}}$ represents the
${{B}^{0}_{({s})}}\\!\rightarrow{{K}^{*0}}{{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}$ yield. Finally, $f_{d}$ and
$f_{s}$ are the hadronisation fractions of a ${b}$-quark into a ${B}^{0}$ and
${B}^{0}_{s}$ meson, respectively.

The purity at detection and the $\lambda^{f_{L}}$ factor ratios,
$\kappa^{k}_{{{B}^{0}_{({s})}}}$, are obtained for each decay mode as

$\kappa^{k}_{{{B}^{0}_{({s})}}}\equiv\frac{\lambda^{f_{L}}_{{{B}^{0}_{({s})}}}}{f^{D}_{{{B}^{0}_{({s})}}}}=\frac{\sum\limits_{i=1}^{6}\sum\limits_{j\geq
i}^{6}\mathcal{R}e[A_{i}A_{j}^{*}\left(\frac{1-\eta_{i}}{{\Gamma_{\mathrm{H}}}}+\frac{1+\eta_{i}}{{\Gamma_{\mathrm{L}}}}\right)\omega^{k}_{ij}]}{(1-|A_{S}^{-}|^{2}-|A_{S}^{+}|^{2}-|A_{SS}|^{2})\sum\limits_{i=1}^{3}\sum\limits_{j\geq
i}^{3}\mathcal{R}e[A_{i}^{\rm sim}A_{j}^{{\rm
sim}*}\left(\frac{1-\eta_{i}}{{\Gamma_{\mathrm{H}}}}+\frac{1+\eta_{i}}{{\Gamma_{\mathrm{L}}}}\right)\omega^{k}_{ij}]},$
(12)

where the $\omega^{k}_{ij}$ coefficients are defined in Eq. (9), $A_{i}^{\rm
sim}$ are the amplitudes used to generate signal samples, and the $\eta_{i}$
values are given in Table 2. Also in this case, for the
${{B}^{0}}\\!\rightarrow{{K}^{*0}}{{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}$ decay, the
${\Gamma_{\mathrm{H}}}={\Gamma_{\mathrm{L}}}$ approximation is adopted.

The detection efficiency is determined from simulation for each channel
separately for the different categories discussed in Sect. 6: year of data
taking, trigger type and, in addition, the LHCb magnet polarity. An exception
is applied to the particle-identification selection whose efficiency is
determined from large control samples of
${{D}^{*+}}\rightarrow{{D}^{0}}{{\pi}^{+}}$,
${{D}^{0}}\rightarrow{{K}^{-}}{{\pi}^{+}}$ decays. Differences in kinematics
and detector occupancy between the control samples and the signal data are
accounted for in this particle-identification efficiency study [42, 43].

The different sources of systematic uncertainty in the branching fraction
determination are discussed below.

Systematic uncertainties in the factor $\boldsymbol{\kappa}$. The
uncertainties on the parameters of the amplitude analysis fit described in
Sect. 7 affect the determination of the factors $\kappa$ defined in Eq. (12)
as summarised in Table 8.

Systematic uncertainties in the signal yields. As discussed in Sect. 7
uncertainties on the signal yields arise from the model used to fit the four-
body invariant mass. The uncertainties from the different proposed alternative
signal and background lineshapes are summed in quadrature to compute the final
systematic uncertainty.

Systematic uncertainty in the efficiencies. A dedicated data method is
employed to estimate the uncertainty in the signal efficiency originated in
the PID selection.

Table 8: Systematic uncertainties in the factor $\kappa$ defined in Eq. (12)
split in categories. The bias originated in differences between data and
simulation is corrected for in the $\kappa$ results shown in Table 9.

Decay mode | ${{B}^{0}}\\!\rightarrow({{K}^{+}}{{\pi}^{-}})({{K}^{-}}{{\pi}^{+}})$ | ${{B}^{0}_{s}}\\!\rightarrow({{K}^{+}}{{\pi}^{-}})({{K}^{-}}{{\pi}^{+}})$
---|---|---
Year | 2011 | 2012 | 2011 | 2012
Trigger | TOS | noTOS | TOS | noTOS | TOS | noTOS | TOS | noTOS
Bias data-simulation | 0.01 | 0.03 | 0.02 | 0.01 | 0.04 | 0.03 | 0.02 | 0.02
Fit method | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00
Kinematic acceptance | 0.03 | 0.04 | 0.02 | 0.02 | 0.06 | 0.06 | 0.06 | 0.06
Resolution | 0.02 | 0.02 | 0.02 | 0.02 | 0.00 | 0.00 | 0.00 | 0.00
P–wave mass model | 0.02 | 0.02 | 0.02 | 0.02 | 0.05 | 0.04 | 0.05 | 0.04
S–wave mass model | 0.03 | 0.03 | 0.03 | 0.03 | 0.17 | 0.17 | 0.16 | 0.17
Differences data-simulation | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01
Background subtraction | 0.03 | 0.03 | 0.03 | 0.03 | 0.02 | 0.01 | 0.02 | 0.01
Peaking backgrounds | 0.03 | 0.04 | 0.03 | 0.04 | 0.01 | 0.01 | 0.01 | 0.01
Time acceptance | $-$ | $-$ | $-$ | $-$ | 0.08 | 0.07 | 0.08 | 0.07
1-9 Total systematic unc. | 0.06 | 0.08 | 0.06 | 0.07 | 0.19 | 0.19 | 0.17 | 0.18

The inputs employed for measuring the relative branching fraction are
summarised in Table 9. The factor $\kappa$ is different for the two decay
modes because of two main reasons: firstly, the discrepancy between the
polarisation assumed in simulation and its measurement is larger for the
${{B}^{0}_{s}}\\!\rightarrow{{K}^{*0}}{{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}$ than for the
${{B}^{0}}\\!\rightarrow{{K}^{*0}}{{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}$ decay. Secondly, the different
S–wave fraction of the decays. Also, the efficiency ratio of the two modes
deviating from one is explained upon the different polarisation of the
simulation samples. The LHCb detector is less efficient for values of
$\cos\theta_{1}$ ($\cos\theta_{2}$) close to unity because of slow pions
emitted in ${K}^{*0}$ (${\kern 1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}$)
decays and these are more frequent the larger is the longitudinal
polarisation.

The final result of the branching-fraction ratio is obtained as the weighted
mean of the per-category result obtained with Eq. (11) for the eight
categories of Table 9, and found to be

$\frac{{\mathcal{B}}(\mbox{${{B}^{0}}\\!\rightarrow{{K}^{*0}}{{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}$})}{{\mathcal{B}}(\mbox{${{B}^{0}_{s}}\\!\rightarrow{{K}^{*0}}{{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}$})}=0.0758\pm
0.0057\text{\,(stat)}\pm 0.0025\text{\,(syst)}\pm
0.0016\,\left(\frac{f_{s}}{f_{d}}\right).$ (13)

Considering that

${\mathcal{B}}(\mbox{${{B}^{0}_{s}}\\!\rightarrow{{K}^{*0}}{{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}$})=(1.11\pm
0.22\text{\,(stat)}\pm 0.12\text{\,(syst)})\times 10^{-5},$

from Ref. [5], the absolute branching fraction for the
${{B}^{0}}\\!\rightarrow{{K}^{*0}}{{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}$ mode is found to be

${\mathcal{B}}(\mbox{${{B}^{0}}\\!\rightarrow{{K}^{*0}}{{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}$})=(8.0\pm 0.9\text{\,(stat)}\pm
0.4\text{\,(syst)})\times 10^{-7}.$

It is worth noticing that, since the
${{B}^{0}_{s}}\\!\rightarrow{{K}^{*0}}{{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}$ branching fraction was
determined with the ${{B}^{0}}\\!\rightarrow{{K}^{*0}}\phi$ decay as a
reference [6], the uncertainty on $f_{s}/f_{d}$, which appears in the ratio of
Eq. (13), does not contribute to the absolute branching fraction measurement.

Table 9: Parameters used to determine
${\mathcal{B}}(\mbox{${{B}^{0}}\\!\rightarrow{{K}^{*0}}{{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}$})/{\mathcal{B}}(\mbox{${{B}^{0}_{s}}\\!\rightarrow{{K}^{*0}}{{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}$})$. When two uncertainties are
quoted, the first is statistical and the second systematic. The value of
$f_{s}/f_{d}$ is taken from Ref. [44].

Parameter | 2011 TOS MagUp | 2011 TOS MagDown | 2011 noTOS MagUp | 2011 noTOS MagDown
---|---|---|---|---
$N_{{{B}^{0}}}$ | $\phantom{1}21.8\pm\phantom{1}4.8\pm 1.2$ | $\phantom{1}33.7\pm\phantom{1}5.5\pm 1.4$ | $\phantom{1}10.8\pm\phantom{1}3.6\pm 0.9$ | $\phantom{1}33.5\pm\phantom{1}5.4\pm 1.4$
$N_{{{B}^{0}_{s}}}$ | $145.0\pm 10.9\pm 3.3$ | $177.3\pm 11.6\pm 3.5$ | $131.9\pm 10.5\pm 3.2$ | $162.5\pm 11.3\pm 3.4$
$\varepsilon_{{{B}^{0}_{s}}}/\varepsilon_{{{B}^{0}}}$ | $1.127\pm 0.018\pm 0.022$ | $1.074\pm 0.017\pm 0.030$ | $1.102\pm 0.029\pm 0.029$ | $1.144\pm 0.030\pm 0.026$
$\kappa_{{{B}^{0}}}$ | $1.88\pm 0.17\pm 0.06$ | $2.11\pm 0.21\pm 0.08$
$\kappa_{{{B}^{0}_{s}}}$ | $3.25\pm 0.16\pm 0.19$ | $3.27\pm 0.16\pm 0.19$

Parameter | 2012 TOS MagUp | 2012 TOS MagDown | 2012 noTOS MagUp | 2012 noTOS MagDown
---|---|---|---|---
$N_{{{B}^{0}}}$ | $73.0\pm 8.7\pm 2.3$ | $58.7\pm 8.1\pm 2.1$ | $64.1\pm 8.4\pm 2.2$ | $53.7\pm 7.9\pm 2.1$
$N_{{{B}^{0}_{s}}}$ | $311\phantom{.}\pm 16\phantom{.}\pm 5\phantom{.0}$ | $344\phantom{.}\pm 17\phantom{.}\pm 5\phantom{.0}$ | $346\phantom{.}\pm 17\phantom{.}\pm 5\phantom{.0}$ | $336\phantom{.}\pm 17\phantom{.}\pm 5\phantom{.0}$
$\varepsilon_{{{B}^{0}_{s}}}/\varepsilon_{{{B}^{0}}}$ | $1.102\pm 0.014\pm 0.053$ | $1.100\pm 0.014\pm 0.048$ | $1.180\pm 0.022\pm 0.065$ | $1.108\pm 0.021\pm 0.060$
$\kappa_{{{B}^{0}}}$ | $1.92\pm 0.18\pm 0.06$ | $2.07\pm 0.21\pm 0.07$
$\kappa_{{{B}^{0}_{s}}}$ | $3.27\pm 0.16\pm 0.17$ | $3.14\pm 0.15\pm 0.18$
$f_{s}/f_{d}$ | $0.259\pm 0.015$

##  9 Summary and final considerations

The first study of
${{B}^{0}}\\!\rightarrow({{K}^{+}}{{\pi}^{-}})({{K}^{-}}{{\pi}^{+}})$ decays
is performed with a data set recorded by the LHCb detector, corresponding to
an integrated luminosity of 3.0$\text{\,fb}^{-1}$ at centre-of-mass energies
of 7 and 8 TeV. The ${{B}^{0}}\\!\rightarrow{{K}^{*0}}{{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}$ mode is observed with $10.8$
standard deviations. An untagged and time-integrated amplitude analysis is
performed, taking into account the three helicity angles and the
$({{K}^{+}}{{\pi}^{-}})$ and $({{K}^{-}}{{\pi}^{+}})$ invariant masses in a
150$\text{\,Me\kern-1.00006ptV\\!/}c^{2}$ window around the ${K}^{*0}$ and
${\kern 1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}$ masses. Six
contributions are included in the fit: three correspond to the
${{B}^{0}}\\!\rightarrow{{K}^{*0}}{{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}$ P–wave, and three to the
S–wave, along with their interferences. A large longitudinal polarisation of
the ${{B}^{0}}\\!\rightarrow{{K}^{*0}}{{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}$ decay, ${f_{L}=0.724\pm
0.051\text{\,(stat)}\pm 0.016\text{\,(syst)}}$, is measured. The S–wave
fraction is found to be ${0.408\pm 0.050\text{\,(stat)}\pm
0.023\text{\,(syst)}}$.

A parallel study of the
${{B}^{0}_{s}}\\!\rightarrow({{K}^{+}}{{\pi}^{-}})({{K}^{-}}{{\pi}^{+}})$ mode
within $150\text{\,Me\kern-1.00006ptV\\!/}c^{2}$ of the ${K}^{*0}$ mass is
performed, superseding a previous LHCb analysis [6]. A small longitudinal
polarisation, ${f_{L}=0.240\pm 0.031\text{\,(stat)}\pm 0.025\text{\,(syst)}}$
and a large S–wave contribution of ${0.694\pm 0.016\text{\,(stat)}\pm
0.012\text{\,(syst)}}$ are measured for the
${{B}^{0}_{s}}\\!\rightarrow{{K}^{*0}}{{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}$ decay, confirming the previous
LHCb results of the time-dependent analysis of the same data [7].

The ratio of branching fractions

${\frac{{\mathcal{B}}(\mbox{${{B}^{0}}\\!\rightarrow{{K}^{*0}}{{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}$})}{{\mathcal{B}}(\mbox{${{B}^{0}_{s}}\\!\rightarrow{{K}^{*0}}{{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}$})}=0.0758\pm
0.0057\text{\,(stat)}\pm 0.0025\text{\,(syst)}\pm
0.0016\,\left(\frac{f_{s}}{f_{d}}\right)},$

is determined. With this ratio the ${{B}^{0}}\\!\rightarrow{{K}^{*0}}{{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}$ branching fraction is found to
be

${\mathcal{B}}(\mbox{${{B}^{0}}\\!\rightarrow{{K}^{*0}}{{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}$})=(8.0\pm 0.9\text{\,(stat)}\pm
0.4\text{\,(syst)})\times 10^{-7}.$

This value is smaller than the measurement from the BaBar collaboration [1],
due to the S–wave contribution. The measurement is compatible with the QCDF
prediction of Ref. [3]: ${(6^{\,+1+5}_{\,-1-3})\times 10^{-7}}$.

Using the ${B}^{0}_{s}$-meson averages [20] for
$y\equiv\Delta\Gamma/(2\Gamma)=0.064\pm 0.005$ and the mixing phase, defined
in Eq. (7), $\phi_{s}=-0.021\pm 0.031$, the ratio

$R_{sd}=\frac{{\mathcal{B}}(\mbox{${{B}^{0}_{s}}\\!\rightarrow{{K}^{*0}}{{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}$})f_{L}(\mbox{${{B}^{0}_{s}}\\!\rightarrow{{K}^{*0}}{{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}$})}{{\mathcal{B}}(\mbox{${{B}^{0}}\\!\rightarrow{{K}^{*0}}{{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}$})f_{L}(\mbox{${{B}^{0}}\\!\rightarrow{{K}^{*0}}{{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}$})}\frac{1-y^{2}}{1+y\,\cos\phi_{s}},$
(14)

is found to be

${{R_{sd}=3.48\pm 0.32\text{\,(stat)}\pm 0.19\text{\,(syst)}\pm
0.08\,(f_{d}/f_{s})\,\pm 0.02\,(y,\phi_{s})}=3.48\pm 0.38}.$

This result is inconsistent with the prediction of $R_{sd}=16.4\pm 5.2$ [13].
Within models such as QCDF or the soft-collinear effective theory, based on
the heavy-quark limit the predictions, longitudinal observables, such as the
one in Eq. (14), have reduced theoretical uncertainties as compared to
parallel and perpendicular ones. The heavy-quark limit also implies the
polarisation hierarchy $f_{L}\gg f_{\parallel,\perp}$. The measured value for
$R_{sd}$ and the $f_{L}$ result of the
${{B}^{0}_{s}}\\!\rightarrow{{K}^{*0}}{{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}$ decay put in question this
hierarchy. The picture is even more intriguing since, contrary to its U-spin
partner, the ${{B}^{0}}\\!\rightarrow{{K}^{*0}}{{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}$ decay is confirmed to be
strongly polarised.

## 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); MOST
and NSFC (China); CNRS/IN2P3 (France); BMBF, DFG and MPG (Germany); INFN
(Italy); NWO (Netherlands); MNiSW and NCN (Poland); MEN/IFA (Romania); MSHE
(Russia); MinECo (Spain); SNSF and SER (Switzerland); NASU (Ukraine); STFC
(United Kingdom); DOE NP and NSF (USA). We acknowledge the computing resources
that are provided by CERN, IN2P3 (France), KIT and DESY (Germany), INFN
(Italy), SURF (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); ANR, Labex P2IO and OCEVU,
and Région Auvergne-Rhône-Alpes (France); Key Research Program of Frontier
Sciences of CAS, CAS PIFI, and the Thousand Talents Program (China); RFBR, RSF
and Yandex LLC (Russia); GVA, XuntaGal and GENCAT (Spain); the Royal Society
and the Leverhulme Trust (United Kingdom).

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Quintana7, N.V. Raab15, B. Rachwal32, J.H. Rademacker50, M. Rama26, M. Ramos
Pernas43, M.S. Rangel2, F. Ratnikov39,74, G. Raven30, M. Ravonel Salzgeber44,
M. Reboud6, F. Redi45, S. Reichert12, F. Reiss10, C. Remon Alepuz76, Z. Ren3,
V. Renaudin59, S. Ricciardi53, S. Richards50, K. Rinnert56, P. Robbe9, A.
Robert10, A.B. Rodrigues45, E. Rodrigues61, J.A. Rodriguez Lopez69, M.
Roehrken44, S. Roiser44, A. Rollings59, V. Romanovskiy41, A. Romero Vidal43,
J.D. Roth77, M. Rotondo20, M.S. Rudolph63, T. Ruf44, J. Ruiz Vidal76, J.J.
Saborido Silva43, N. Sagidova35, B. Saitta24,f, V. Salustino Guimaraes65, C.
Sanchez Gras29, C. Sanchez Mayordomo76, B. Sanmartin Sedes43, R.
Santacesaria28, C. Santamarina Rios43, M. Santimaria20,44, E. Santovetti27,j,
G. Sarpis58, A. Sarti20,k, C. Satriano28,s, A. Satta27, M. Saur4, D.
Savrina36,37, S. Schael11, M. Schellenberg12, M. Schiller55, H. Schindler44,
M. Schmelling13, T. Schmelzer12, B. Schmidt44, O. Schneider45, A. Schopper44,
H.F. Schreiner61, M. Schubiger45, S. Schulte45, M.H. Schune9, R. Schwemmer44,
B. Sciascia20, A. Sciubba28,k, A. Semennikov36, E.S. Sepulveda10, A.
Sergi49,44, N. Serra46, J. Serrano8, L. Sestini25, A. Seuthe12, P. Seyfert44,
M. Shapkin41, T. Shears56, L. Shekhtman40,x, V. Shevchenko72, E. Shmanin73,
B.G. Siddi18, R. Silva Coutinho46, L. Silva de Oliveira2, G. Simi25,o, S.
Simone16,d, I. Skiba18, N. Skidmore14, T. Skwarnicki63, M.W. Slater49, J.G.
Smeaton51, E. Smith11, I.T. Smith54, M. Smith57, M. Soares17, l. Soares
Lavra1, M.D. Sokoloff61, F.J.P. Soler55, B. Souza De Paula2, B. Spaan12, E.
Spadaro Norella23,q, P. Spradlin55, F. Stagni44, M. Stahl14, S. Stahl44, P.
Stefko45, S. Stefkova57, O. Steinkamp46, S. Stemmle14, O. Stenyakin41, M.
Stepanova35, H. Stevens12, A. Stocchi9, S. Stone63, S. Stracka26, M.E.
Stramaglia45, M. Straticiuc34, U. Straumann46, S. Strokov75, J. Sun3, L.
Sun67, Y. Sun62, K. Swientek32, A. Szabelski33, T. Szumlak32, M. Szymanski4,
Z. Tang3, T. Tekampe12, G. Tellarini18, F. Teubert44, E. Thomas44, M.J.
Tilley57, V. Tisserand7, S. T’Jampens6, M. Tobin5, S. Tolk44, L.
Tomassetti18,g, D. Tonelli26, D.Y. Tou10, R. Tourinho Jadallah Aoude1, E.
Tournefier6, M. Traill55, M.T. Tran45, A. Trisovic51, A. Tsaregorodtsev8, G.
Tuci26,44,p, A. Tully51, N. Tuning29, A. Ukleja33, A. Usachov9, A.
Ustyuzhanin39,74, U. Uwer14, A. Vagner75, V. Vagnoni17, A. Valassi44, S.
Valat44, G. Valenti17, M. van Beuzekom29, H. Van Hecke78, E. van Herwijnen44,
C.B. Van Hulse15, J. van Tilburg29, M. van Veghel29, R. Vazquez Gomez44, P.
Vazquez Regueiro43, C. Vázquez Sierra29, S. Vecchi18, J.J. Velthuis50, M.
Veltri19,r, A. Venkateswaran63, M. Vernet7, M. Veronesi29, M. Vesterinen52,
J.V. Viana Barbosa44, D. Vieira4, M. Vieites Diaz43, H. Viemann70, X. Vilasis-
Cardona42,m, A. Vitkovskiy29, M. Vitti51, V. Volkov37, A. Vollhardt46, D. Vom
Bruch10, B. Voneki44, A. Vorobyev35, V. Vorobyev40,x, N. Voropaev35, R.
Waldi70, J. Walsh26, J. Wang5, M. Wang3, Y. Wang68, Z. Wang46, D.R. Ward51,
H.M. Wark56, N.K. Watson49, D. Websdale57, A. Weiden46, C. Weisser60, M.
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M.R.J. Williams58, T. Williams49, F.F. Wilson53, M. Winn9, W. Wislicki33, M.
Witek31, G. Wormser9, S.A. Wotton51, K. Wyllie44, D. Xiao68, Y. Xie68, H.
Xing66, A. Xu3, M. Xu68, Q. Xu4, Z. Xu6, Z. Xu3, Z. Yang3, Z. Yang62, Y.
Yao63, L.E. Yeomans56, H. Yin68, J. Yu68,aa, X. Yuan63, O. Yushchenko41, K.A.
Zarebski49, M. Zavertyaev13,c, M. Zeng3, D. Zhang68, L. Zhang3, W.C. Zhang3,z,
Y. Zhang44, A. Zhelezov14, Y. Zheng4, X. Zhu3, V. Zhukov11,37, J.B.
Zonneveld54, S. Zucchelli17,e.

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
4University of Chinese Academy of Sciences, Beijing, China
5Institute Of High Energy Physics (ihep), Beijing, China
6Univ. Grenoble Alpes, Univ. Savoie Mont Blanc, CNRS, IN2P3-LAPP, Annecy,
France
7Université Clermont Auvergne, CNRS/IN2P3, LPC, Clermont-Ferrand, France
8Aix Marseille Univ, CNRS/IN2P3, CPPM, Marseille, France
9LAL, Univ. Paris-Sud, CNRS/IN2P3, Université Paris-Saclay, Orsay, France
10LPNHE, Sorbonne Université, Paris Diderot Sorbonne Paris Cité, CNRS/IN2P3,
Paris, France
11I. Physikalisches Institut, RWTH Aachen University, Aachen, Germany
12Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
13Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
14Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
15School of Physics, University College Dublin, Dublin, Ireland
16INFN Sezione di Bari, Bari, Italy
17INFN Sezione di Bologna, Bologna, Italy
18INFN Sezione di Ferrara, Ferrara, Italy
19INFN Sezione di Firenze, Firenze, Italy
20INFN Laboratori Nazionali di Frascati, Frascati, Italy
21INFN Sezione di Genova, Genova, Italy
22INFN Sezione di Milano-Bicocca, Milano, Italy
23INFN Sezione di Milano, Milano, Italy
24INFN Sezione di Cagliari, Monserrato, Italy
25INFN Sezione di Padova, Padova, Italy
26INFN Sezione di Pisa, Pisa, Italy
27INFN Sezione di Roma Tor Vergata, Roma, Italy
28INFN Sezione di Roma La Sapienza, Roma, Italy
29Nikhef National Institute for Subatomic Physics, Amsterdam, Netherlands
30Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, Netherlands
31Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
32AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
33National Center for Nuclear Research (NCBJ), Warsaw, Poland
34Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
35Petersburg Nuclear Physics Institute NRC Kurchatov Institute (PNPI NRC KI),
Gatchina, Russia
36Institute of Theoretical and Experimental Physics NRC Kurchatov Institute
(ITEP NRC KI), Moscow, Russia, Moscow, Russia
37Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
38Institute for Nuclear Research of the Russian Academy of Sciences (INR RAS),
Moscow, Russia
39Yandex School of Data Analysis, Moscow, Russia
40Budker Institute of Nuclear Physics (SB RAS), Novosibirsk, Russia
41Institute for High Energy Physics NRC Kurchatov Institute (IHEP NRC KI),
Protvino, Russia, Protvino, Russia
42ICCUB, Universitat de Barcelona, Barcelona, Spain
43Instituto Galego de Física de Altas Enerxías (IGFAE), Universidade de
Santiago de Compostela, Santiago de Compostela, Spain
44European Organization for Nuclear Research (CERN), Geneva, Switzerland
45Institute of Physics, Ecole Polytechnique Fédérale de Lausanne (EPFL),
Lausanne, Switzerland
46Physik-Institut, Universität Zürich, Zürich, Switzerland
47NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
48Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
49University of Birmingham, Birmingham, United Kingdom
50H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
51Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
52Department of Physics, University of Warwick, Coventry, United Kingdom
53STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
54School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
55School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
56Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
57Imperial College London, London, United Kingdom
58School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
59Department of Physics, University of Oxford, Oxford, United Kingdom
60Massachusetts Institute of Technology, Cambridge, MA, United States
61University of Cincinnati, Cincinnati, OH, United States
62University of Maryland, College Park, MD, United States
63Syracuse University, Syracuse, NY, United States
64Laboratory of Mathematical and Subatomic Physics , Constantine, Algeria,
associated to 2
65Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
66South China Normal University, Guangzhou, China, associated to 3
67School of Physics and Technology, Wuhan University, Wuhan, China, associated
to 3
68Institute of Particle Physics, Central China Normal University, Wuhan,
Hubei, China, associated to 3
69Departamento de Fisica , Universidad Nacional de Colombia, Bogota, Colombia,
associated to 10
70Institut für Physik, Universität Rostock, Rostock, Germany, associated to 14
71Van Swinderen Institute, University of Groningen, Groningen, Netherlands,
associated to 29
72National Research Centre Kurchatov Institute, Moscow, Russia, associated to
36
73National University of Science and Technology “MISIS”, Moscow, Russia,
associated to 36
74National Research University Higher School of Economics, Moscow, Russia,
associated to 39
75National Research Tomsk Polytechnic University, Tomsk, Russia, associated to
36
76Instituto de Fisica Corpuscular, Centro Mixto Universidad de Valencia -
CSIC, Valencia, Spain, associated to 42
77University of Michigan, Ann Arbor, United States, associated to 63
78Los Alamos National Laboratory (LANL), Los Alamos, United States, associated
to 63

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, Vietnam
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
vH.H. Wills Physics Laboratory, University of Bristol, Bristol, United Kingdom
wMSU - Iligan Institute of Technology (MSU-IIT), Iligan, Philippines
xNovosibirsk State University, Novosibirsk, Russia
ySezione INFN di Trieste, Trieste, Italy
zSchool of Physics and Information Technology, Shaanxi Normal University
(SNNU), Xi’an, China
aaPhysics and Micro Electronic College, Hunan University, Changsha City, China
abLanzhou University, Lanzhou, China
†Deceased