EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-EP-2017-317 LHCb-PAPER-2017-033 22 December 2017

Amplitude analysis of the decay ${{\kern
3.73305pt\overline{\kern-3.73305ptB}{}}{}^{0}}\rightarrow
K_{\mathrm{\scriptscriptstyle S}}^{0}\pi^{+}\pi^{-}$ and first observation of
the $C\\!P$ asymmetry in ${{\kern
3.73305pt\overline{\kern-3.73305ptB}{}}{}^{0}}\rightarrow
K^{*}(892)^{-}\pi^{+}$

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

The time-integrated Dalitz plot of the three-body hadronic charmless decay
${{{\kern 1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}}\rightarrow
K_{\mathrm{\scriptscriptstyle S}}^{0}\pi^{+}\pi^{-}}$ is studied using a $pp$
collision data sample recorded with the LHCb detector, corresponding to an
integrated luminosity of $3.0\mbox{\,fb}^{-1}$. The decay amplitude is
described with an isobar model. Relative contributions of the isobar
amplitudes to the ${{{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}}\rightarrow
K_{\mathrm{\scriptscriptstyle S}}^{0}\pi^{+}\pi^{-}}$ decay branching fraction
and $C\\!P$ asymmetries of the flavour-specific amplitudes are measured. The
$C\\!P$ asymmetry between the conjugate ${{{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}}\rightarrow
K^{*}(892)^{-}\pi^{+}}$ and ${{{B}^{0}}\rightarrow K^{*}(892)^{+}\pi^{-}}$
decay rates is determined to be $-0.308\pm 0.062$.

Submitted to Phys. Rev. Lett.

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

The breaking of the invariance of the weak interaction under the combined
action of the charge conjugation ($C$) and parity ($P$) transformations is
firmly established in the $K$\- and $B$-meson systems [1, 2, 3]. In
particular, significant $C\\!P$ asymmetries at the level of 10% or more have
been measured in the decays of $B$ mesons into two light pseudoscalars. The
$C\\!P$ asymmetries in the decays of ${{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}}\rightarrow K^{-}\pi^{+}$ and
$B^{-}\rightarrow K^{-}\pi^{0}$ are observed to be different [4], while, in
predictions based on the QCD factorisation approach, the two asymmetries are
expected to be similar [5]. This apparent discrepancy is often referred to in
the literature as the $K\pi$ puzzle [6, 7, 8, 9]. The study of the flavour-
specific quasi-two-body amplitudes which contribute to the decay ${{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}}\\!\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}{{\pi}^{+}}{{\pi}^{-}}$ offers the possibility to measure $C\\!P$
asymmetries. In particular, the decays with a vector and a pseudoscalar in the
final state, such as ${\overline{B}}^{0}\rightarrow K^{*}(892)^{-}\pi^{+}$,
may help to shed light on the $K\pi$ puzzle.

In the Standard Model (SM) [10, 11], the mixing-induced $C\\!P$ asymmetries in
the quark-level transitions ${b}\\!\rightarrow{q}{\overline{q}}{s}$
($q=u,d,s$) which govern the decay ${{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}}\\!\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}{{\pi}^{+}}{{\pi}^{-}}$ are predicted to be approximately equal to those
in ${b}\\!\rightarrow{c}{\overline{c}}{s}$ transitions, such as
${{B}^{0}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}$. The existence of new
particles in extensions of the SM could introduce additional weak phases that
contribute along with the SM mixing phase [12, 13, 14, 15]. In general, for
each of the studied $C\\!P$ eigenstates, the current experimental measurements
of ${b}\\!\rightarrow{q}{\overline{q}}{s}$ decays [4] show good agreement with
the results from ${b}\\!\rightarrow{c}{\overline{c}}{s}$ decays [4]. There is
nonetheless room for contributions from physics beyond the SM and, hence, the
need for precision measurements of these weak mixing phases.

The mixing-induced $C\\!P$-violating phase can be measured by means of a
decay-time-dependent analysis of the Dalitz plot (DP) [16] of the decay
${{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}}\\!\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}{{\pi}^{+}}{{\pi}^{-}}$ [17, 18, 19, 20]. Such an analysis requires the
initial flavour of the ${\kern 1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}$
meson to be determined or “tagged”. A recent study of the yields of the
charmless three-body decays ${{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}}\\!\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}{{\pi}^{+}}{{\pi}^{-}}$ has been reported in Ref. [21]. The ${{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}}\\!\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}{{\pi}^{+}}{{\pi}^{-}}$ yields are comparable to those obtained at the
BaBar and Belle experiments but the lower tagging efficiency at LHCb does not
yet allow a precise flavour-tagged analysis to be performed. The decay-time-
integrated untagged DP of this mode is studied in this Letter. The DP of the
decay ${{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}}\\!\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}{{\pi}^{+}}{{\pi}^{-}}$ is modelled by a sum of quasi-two-body amplitudes
(the isobar parameterisation) and the model is fit to the LHCb data to measure
the relative branching fractions and the $C\\!P$ asymmetries of flavour-
specific final states.

The analysis reported in this Letter is performed using $pp$ collision data
recorded with the LHCb detector, corresponding to integrated luminosities of
1.0$\mbox{\,fb}^{-1}$ at a centre-of-mass energy of
7$\mathrm{\,Te\kern-1.00006ptV}$ in 2011 and to 2.0$\mbox{\,fb}^{-1}$ at a
centre-of-mass energy of 8$\mathrm{\,Te\kern-1.00006ptV}$ in 2012. The LHCb
detector [22, 23] is a single-arm forward spectrometer covering the
pseudorapidity range $2<\eta<5$, designed for the study of particles
containing $b$ or $c$ quarks. Signal candidates are accepted if one of the
final-state particles from the signal decay deposits sufficient energy
transverse to the beamline in the hadronic calorimeter to pass the hardware
trigger. Events that are triggered at the hardware level by another particle
in the event are also retained. In a second step, a software trigger requires
a two-, three- or four-track secondary vertex with a significant displacement
from any primary $pp$ interaction vertex (PV). At least one charged particle
must have a large transverse momentum and be inconsistent with originating
from a PV. A multivariate algorithm [24] is used for the identification of
secondary vertices consistent with the decay of a $b$ hadron.

The selection procedure is described in detail in Ref. [21]. Decays of
${{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{-}}$ are reconstructed in two different
categories: the first involving ${K}^{0}_{\mathrm{\scriptscriptstyle S}}$
mesons that decay early enough for the resulting pions to be reconstructed in
the vertex detector; and the second containing those
${K}^{0}_{\mathrm{\scriptscriptstyle S}}$ mesons that decay later, such that
track segments of the pions cannot be formed in the vertex detector. These
categories are referred to as Long and Downstream, respectively. Downstream
${K}^{0}_{\mathrm{\scriptscriptstyle S}}$ were not reconstructed in the
software trigger in 2011, but they were reconstructed and used for triggering
in 2012. Furthermore, an improved software trigger with larger $b$-hadron
efficiency, in particular in the Downstream category, was used for the second
part of the 2012 data taking. To take into account the different levels of
trigger efficiency, the data sample is divided into 2011, 2012a, and 2012b
data-taking periods, and each period is further divided according to the
${K}^{0}_{\mathrm{\scriptscriptstyle S}}$ reconstruction category, giving a
total of six subsamples. The 2012b sample is the largest, corresponding to an
integrated luminosity 1.4$\mbox{\,fb}^{-1}$, and has the highest trigger
efficiency.

The events passing the trigger requirements are then filtered in two stages.
Initial requirements are applied to further reduce the size of the data sample
and increase the signal purity, before a multivariate classifier, based mostly
on topological variables derived from the vertexing of the candidates, is
implemented [21]. The selection requirement placed on the output of the
multivariate classifier is defined for each data subsample to yield a signal
purity close to 90%. Particle identification (PID) requirements are applied in
order to reduce backgrounds from decays where either a proton, kaon or muon is
misidentified as a pion. These criteria are optimised to reduce the cross-feed
background coming from the decays
${{B}^{0}_{s}}\\!\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}{{K}^{\pm}}{{\pi}^{\mp}}$, where the kaon is misidentified as a pion. The
same invariant-mass vetoes on charmed and charmonium resonances as in Ref.
[21] are used in this analysis. The invariant-mass distribution of signal
candidates from the six aforementioned subsamples is displayed in Fig. 1 with
the result of a simultaneous fit. The candidates selected for the subsequent
DP analysis are those in the ${K}^{0}_{\mathrm{\scriptscriptstyle S}}$
${\pi}^{+}$ ${\pi}^{-}$ mass range [5227,5343]
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$.

Figure 1: Invariant mass distributions of
${{K}^{0}_{\mathrm{\scriptscriptstyle S}}}{{\pi}^{+}}{{\pi}^{-}}$ candidates,
summing the two years of data taking and the two
${K}^{0}_{\mathrm{\scriptscriptstyle S}}$ reconstruction categories. The sum
of the partially reconstructed contributions from $B$ to open charm decays,
charmless hadronic decays, ${{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}}\\!\rightarrow\eta^{\prime}{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}$ and charmless radiative decays are denoted ${\kern
1.79993pt\overline{\kern-1.79993ptB}{}}^{0}_{(s)}\\!\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}{{\pi}^{+}}{{\pi}^{-}}(X)$.

The DP analysis technique [16] is employed to study the dynamics of the three-
body decay ${{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}}\\!\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}{{\pi}^{+}}{{\pi}^{-}}$. A decay-time-integrated untagged probability
density function (p.d.f.) is built to describe the phase space of the decay as
a function of the DP kinematical variables. In this case, the p.d.f. becomes
an incoherent sum of the $\mathcal{A}(s_{+},s_{-})$ and $\kern
1.79993pt\overline{\kern-1.79993pt\mathcal{A}}(s_{+},s_{-})$ Lorentz-invariant
transition amplitudes of the decays
${{B}^{0}}\\!\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}{{\pi}^{+}}{{\pi}^{-}}$ and ${{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}}\\!\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}{{\pi}^{+}}{{\pi}^{-}}$, respectively,

$\mathcal{P}(s_{+},s_{-})=\frac{|\mathcal{A}(s_{+},s_{-})|^{2}+|\kern
1.79993pt\overline{\kern-1.79993pt\mathcal{A}}(s_{+},s_{-})|^{2}}{\iint_{\rm
DP}\left(|\mathcal{A}(s_{+},s_{-})|^{2}+|\kern
1.79993pt\overline{\kern-1.79993pt\mathcal{A}}(s_{+},s_{-})|^{2}\right){\rm
d}s_{+}{\rm d}s_{-}}\,,\\\ $ (1)

where the kinematical variables $s_{\pm}$ denote the mass squared,
$m_{{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}\pi^{\pm}}^{2}$.

The total amplitude $\overline{\mathcal{A}}(s_{+},s_{-})$ of the decay
${{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}}\\!\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}{{\pi}^{+}}{{\pi}^{-}}$ is described as a coherent sum of the amplitudes
of possible intermediate resonances and nonresonant contributions. The decay
amplitudes for ${B}^{0}$ and ${\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}$ are given by

$\mathcal{A}=\sum_{j=1}^{N}c_{j}F_{j}(s_{+},s_{-})\,,\,\,\overline{\mathcal{A}}=\sum_{j=1}^{N}\overline{c}_{j}\kern
1.79993pt\overline{\kern-1.79993ptF}_{j}(s_{+},s_{-})\,,$ (2)

where $F_{j}$ and $\kern 1.79993pt\overline{\kern-1.79993ptF}_{j}$ are the DP
spin-dependent dynamical functions for the resonance $j$ and $c_{j}$ are
complex coefficients that account for the relative magnitudes and phases of
the $N$ intermediate (resonant and nonresonant) components. The spin-dependent
functions $F_{j}(s_{+},s_{-})$, embedding the resonance lineshape and the
angular distributions, are constructed in the Zemach tensor formalism [25].
The weak-phase dependence is included in the $c_{j}$ coefficients. The results
obtained for each isobar amplitude are expressed in this paper as a magnitude
and a phase.

The analysis method consists of a simultaneous DP fit to the six data
subsamples defined above, with the shared isobar parameters determined using
an unbinned maximum likelihood fit. The DP model is built starting from the
most significant amplitudes as determined in previous studies [17, 18, 19,
20]. An algorithm to select the relevant additional amplitudes is defined
before examining the data. A resonant amplitude is retained in the DP model if
at least one of the following requirements is met: (1) a goodness-of-fit
estimator based on the point-to-point dissimilarity test [26] decreases when
the component is removed from the fit, (2) the likelihood ratio of the two
hypotheses (component in and out) decreases, or (3) the significance of the
magnitude of the component is at least three statistical standard deviations,
neglecting systematic uncertainties. In particular, the components of the
isobar DP model, $f_{0}(1500){{K}^{0}_{\mathrm{\scriptscriptstyle S}}}$ and
$K^{*}(1680)^{-}\pi^{+}$, which were not considered in previous studies, meet
all three criteria. By contrast, the amplitude
$f_{2}(1270){{K}^{0}_{\mathrm{\scriptscriptstyle S}}}$ is not retained.

Table 1:  Components of the DP model used in the fit. The individual amplitudes are referred to by the resonance they contain. The parameter values are given in ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ for the masses and $\mathrm{\,Me\kern-1.00006ptV}$ for the widths, except for $f_{0}(980)$ resonance. The parameter $m_{0}$ is the pole mass of the resonance and $\Gamma_{0}$ its natural width. The mass-dependent lineshapes employed to model the resonances are indicated in the third column. Relativistic Breit-Wigner and Gounaris-Sakurai lineshapes are denoted RBW and GS, respectively. EFKLLM is a parameterisation of the ${{K}^{0}_{\mathrm{\scriptscriptstyle S}}}\pi^{-}$ S-wave lineshape, $(K\pi)^{-}_{0}$.  Resonance | Parameters | Lineshape | Value references
---|---|---|---
$K^{*}(892)^{-}$ |  | $m_{0}=891.66\pm 0.26$
---
$\Gamma_{0}=50.8\pm 0.9$
RBW | [27]
$(K\pi)^{-}_{0}$ |  | $\mathcal{R}e(\lambda_{0})=0.204\pm 0.103$
---
$\mathcal{I}m(\lambda_{0})=0$
$\mathcal{R}e(\lambda_{1})=1$
$\mathcal{I}m(\lambda_{1})=0$
EFKLLM [28] | [28]
$K^{*}_{2}(1430)^{-}$ |  | $m_{0}=1425.6\pm 1.5$
---
$\Gamma_{0}=98.5\pm 2.7$
RBW | [27]
$K^{*}(1680)^{-}$ |  | $m_{0}=1717\pm 27$
---
$\Gamma_{0}=332\pm 110$
Flatté [29] | [27]
$f_{0}(500)$ |  | $\mathrm{\,m}_{0}=513\pm 32$
---
$\Gamma_{0}=335\pm 67$
RBW | [30]
$\rho(770)^{0}$ |  | $m_{0}=775.26\pm 0.25$
---
$\Gamma_{0}=149.8\pm 0.8$
GS [31] | [27]
$f_{0}(980)$ |  | $m_{0}=965\pm 10$
---
$g_{\pi}=0.165\pm 0.025$ GeV
$g_{K}=0.695\pm 0.119$ GeV
Flatté | [32]
$f_{0}(1500)$ |  | $m_{0}=1505\pm 6$
---
$\Gamma_{0}=109\pm 7$
RBW | [27]
$\chi_{c0}$ |  | $m_{0}=3414.75\pm 0.31$
---
$\Gamma_{0}=10.5\pm 0.6$
RBW | [27]
Nonresonant (NR) |  | Phase space |

The signal DP model p.d.f. $\mathcal{S}(s_{+},s_{-})$ is built from the
coherent sum of the amplitudes listed in Table 1, normalising each isobar
coefficient to the $K^{*}(892)^{+}\pi^{-}$ reference amplitude. The choice of
the $K^{*}(892)^{\pm}\pi^{\mp}$ amplitudes as a reference provides the most
stable DP fit. The phases of the reference amplitude and its conjugate are
fixed to zero and the magnitude of the reference amplitude is arbitrarily
fixed at 2.

Two dominant backgrounds contaminate the ${{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}}\\!\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}{{\pi}^{+}}{{\pi}^{-}}$ candidate samples: a combinatorial background and
a cross-feed background from the decay ${{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}}\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}K^{\pm}\pi^{\mp}$. The fractions of these backgrounds are measured from
the invariant-mass fits performed in Ref. [21] and their DP distributions are
determined from the data. The combinatorial background DP model is built from
the DP histogram of the ${{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}}\\!\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}{{\pi}^{+}}{{\pi}^{-}}$ candidates with an invariant mass larger than 5450
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. The DP model of the cross-feed
background is measured from ${{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}}\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}K^{\pm}\pi^{\mp}$ candidates, where the $K^{\pm}$ is reconstructed under
the $\pi^{\pm}$ hypothesis [33]. The signal fraction depends on the
reconstruction category; it is determined from the fit to the invariant-mass
distribution and ranges from 85% (Downstream) to 95% (Long). The p.d.f. in Eq.
1 is modified to account for the background components and the signal
reconstruction efficiency across the DP, as determined from simulated events.

Two additional observables are formed from the isobar complex coefficients and
are measured in the simultaneous DP fit. The asymmetry observables
$\mathcal{A}_{\rm raw}$ are derived from the measured isobar parameters of an
amplitude $j$, $c_{j}$ and $\overline{c}_{j}$

$\displaystyle\mathcal{A}_{\rm raw}$ $\displaystyle=$
$\displaystyle\frac{|\overline{c}_{j}|^{2}-|c_{j}|^{2}}{|\overline{c}_{j}|^{2}+|c_{j}|^{2}}.$
(3)

These observables are directly measured for flavour-specific final states. By
contrast, the asymmetry of the mode $\overline{B}^{0}\rightarrow
f_{0}(980){{K}^{0}_{\mathrm{\scriptscriptstyle S}}}$ is determined using the
patterns of its interference with flavour-specific amplitudes. The $C\\!P$
asymmetry is related to the raw asymmetry by
$\mathcal{A}_{{C\\!P}}=\mathcal{A}_{\rm raw}-\mathcal{A}_{\Delta}$. The
correction asymmetry is defined at first order as
$\mathcal{A}_{\Delta}=A_{P}({{B}^{0}})+A_{D}(\pi)$, where $A_{P}({{B}^{0}})$
is the production asymmetry between the ${B}^{0}$ and ${\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}$ mesons and $A_{D}(\pi)$ is the
detection asymmetry between ${\pi}^{+}$ and ${\pi}^{-}$ mesons. The production
asymmetry $A_{P}({{B}^{0}})$ has been determined to be
$A_{P}({{B}^{0}})=(-0.35\pm 0.81)\%$ [34]. Using $D_{s}^{+}$ decay modes [35],
the pion detection asymmetry is measured to be consistent with zero with a
0.25% uncertainty. The difference in the nuclear cross-sections for ${K}^{0}$
and ${\kern 1.99997pt\overline{\kern-1.99997ptK}{}}{}^{0}$ interactions in
material results in a negligible bias [36]. The uncertainty due to the
correction asymmetries and the experimental systematic uncertainty are added
in quadrature.

The rate of a single process is proportional to the square of the relevant
matrix element (see Eq. 1). This involves the ensemble of its interferences
with other components. It is convenient to define the fit fraction of the
process $i$, ${\cal F}_{i}$, as

${\cal F}_{i}=\frac{\iint_{\rm DP}\left|c_{i}F_{i}(s_{+},s_{-})\right|^{2}{\rm
d}s_{+}{\rm d}s_{-}}{\iint_{\rm
DP}\left|\sum_{j}c_{j}F_{j}(s_{+},s_{-})\right|^{2}{\rm d}s_{+}{\rm
d}s_{-}}\;.$ (4)

Simulation is used to determine the selection efficiency of the signal. The
simulation does not perfectly reproduce the detector response and these
imperfections are corrected for in several respects. Firstly, the particle
identification and misidentification efficiencies are determined from a
calibration sample using reconstructed ${{D}^{*+}}\rightarrow{{D}^{0}}\pi^{+}$
decays, where the ${D}^{0}$ meson decays to the Cabibbo-favoured
${{K}^{-}}{{\pi}^{+}}$ final state. The variation of the PID performance with
the track kinematics is included in the procedure. The calibration is
performed using samples from the same data-taking period, accounting for the
variation in the performance of the hadron identification detectors over time.
Secondly, inaccuracies of the tracking simulation are mitigated by a weighting
of the simulated tracking efficiency to match that measured in a calibration
sample [37]. Analogous corrections are applied to the
${K}^{0}_{\mathrm{\scriptscriptstyle S}}$ decay-products tracking and
vertexing efficiencies. Finally, a control sample of
${{D}^{*+}}\rightarrow{{D}^{0}}(\rightarrow{{K}^{-}}{{\pi}^{+}})\pi^{+}$
decays is used to quantify the differences of the hardware trigger response in
data and simulation for pions and kaons, separated by positive and negative
hadron charges, as a function of their transverse momentum [38]. The
uncertainties assigned to these corrections are taken as a source of
systematic uncertainties.

Two categories of systematic uncertainties are considered: experimental and
related to the DP model. The former category comprises the uncertainties on
the fraction of signal, the fit biases, the variation of the signal efficiency
across the DP (including the choice of the efficiency binning) and the
background DP models. The DP model uncertainties arise from the limited
knowledge of the fixed parameters of the resonance-lineshape models, the
marginal components neglected in the amplitude fit model and the modelling of
the ${{K}^{0}_{\mathrm{\scriptscriptstyle S}}}\pi^{-}$ and $\pi^{+}\pi^{-}$
S-wave components.

All experimental uncertainties are estimated by means of pseudoexperiments, in
which samples for each reconstruction category are simulated and fitted
exactly as for the data sample. For each pseudoexperiment, a single parameter
governing a systematic effect (e.g. the signal fraction) is varied according
to its uncertainty. The standard deviation of the distribution of the fit
results in an ensemble of 500 pseudoexperiments is taken as the corresponding
systematic error estimate. The largest biases observed are at the few percent
level. The final result is corrected for any observed bias where it is
significant. The dominant contribution to the experimental uncertainty is the
efficiency determination.

The mass and the width of each resonance given in Table 1 are varied
individually and symmetrically by one standard deviation to evaluate the
impact of the fixed parameters of the isobar resonance lineshapes. The Blatt-
Weisskopf radius parameter, fixed at $4\mathrm{\,Ge\kern-1.00006ptV}^{-1}$, is
varied by $\pm 1\mathrm{\,Ge\kern-1.00006ptV}^{-1}$.

To evaluate the systematic uncertainties related to the marginal components of
the DP model, the effect of adding the resonance $f_{2}(1270)$ (which is not
retained by the previous criteria) and removing of the $f_{0}(500)$ component
(the least significant contribution in the nominal model) is considered by
repeating the fit with and without these components. Based upon this new
model, a pseudoexperiment with a signal yield much larger than that of the
data is then generated and fit back with the nominal model. The related
systematic uncertainty estimate is taken as the difference between the
generated and fitted values.

A critical part of the isobar model design is the description of
$K_{S}^{0}\pi^{\pm}$ S-wave components. Two parameterisations of these
contributions have been studied: LASS [39] and EFKLLM [28]. The latter
provides the best fit to the data. The log-likelihood difference between the
two model hypotheses is $-2\Delta\ln{\cal L}=85$. Given this large difference,
no systematic uncertainty is then assigned to the choice of the EFKLLM
parameterisation. All model uncertainties are combined in quadrature to form
the total model systematic uncertainty

The Dalitz plot projections are shown in Fig. 2 with the result of the fit
superimposed. The $C\\!P$-averaged fit fractions related to the quasi two-body
and nonresonant amplitudes are derived from the isobar coefficients with Eq. 4

Figure 2:  Projections of the sum of all data categories (black points) and
the nominal fit function onto the DP variables (left) $m_{K_{\rm
S}^{0}\pi^{+}}^{2}$, (right) $m_{K_{\rm S}^{0}\pi^{-}}^{2}$ and (bottom)
$m_{\pi^{+}\pi^{-}}^{2}$, restricted to the two-body low invariant-mass
regions. The full fit is shown by the solid blue line and the signal model by
the dashed red line. The observed difference is due to the (green)
combinatorial and (light red) cross-feed background contributions, barely
visible in these projections.

$\displaystyle\mathcal{F}(K^{*}(892)^{-}\pi^{+})$ $\displaystyle=$
$\displaystyle\quad 9.43\pm 0.40\pm 0.33\pm 0.34\;\%\,,$
$\displaystyle\mathcal{F}((K\pi)^{-}_{0}\pi^{+})$ $\displaystyle=$
$\displaystyle\quad 32.7\pm\,\,\,1.4\pm\,\,\,1.5\pm\,\,\,1.1\;\%\,,$
$\displaystyle\mathcal{F}(K^{*}_{2}(1430)^{-}\pi^{+})$ $\displaystyle=$
$\displaystyle\quad 2.45\,\,_{-\;\;\;\;0.08}^{+\;\;\;\;0.10}\pm 0.14\pm
0.12\;\%\,,$ $\displaystyle\mathcal{F}(K^{*}(1680)^{-}\pi^{+})$
$\displaystyle=$ $\displaystyle\quad 7.34\pm 0.30\pm 0.31\pm 0.06\;\%\,,$
$\displaystyle\mathcal{F}(f_{0}(980){{K}^{0}_{\mathrm{\scriptscriptstyle
S}}})$ $\displaystyle=$ $\displaystyle\quad
18.6\pm\,\,\,0.8\pm\,\,\,0.7\pm\,\,\,1.2\;\%\,,$
$\displaystyle\mathcal{F}(\rho(770)^{0}{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}})$ $\displaystyle=$
$\displaystyle\quad\,\,\,3.8\,\,_{-\;\;\;\;\;\;1.6}^{+\;\;\;\;\;\;1.1}\pm\,\,\,0.7\pm\,\,\,0.4\;\%\,,$
$\displaystyle\mathcal{F}(f_{0}(500){{K}^{0}_{\mathrm{\scriptscriptstyle
S}}})$ $\displaystyle=$ $\displaystyle\quad
0.32\,\,_{-\;\;\;\;0.08}^{+\;\;\;\;0.40}\pm 0.19\pm 0.23\;\%\,,$
$\displaystyle\mathcal{F}(f_{0}(1500){{K}^{0}_{\mathrm{\scriptscriptstyle
S}}})$ $\displaystyle=$ $\displaystyle\quad 2.60\pm 0.54\pm 1.28\pm
0.60\;\%\;,$
$\displaystyle\mathcal{F}(\chi_{c0}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}})$
$\displaystyle=$ $\displaystyle\quad
2.23\,\,_{-\;\;\;\;0.32}^{+\;\;\;\;0.40}\pm 0.22\pm 0.13\;\%\,,$
$\displaystyle\mathcal{F}({{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}{{\pi}^{+}}{{\pi}^{-}})^{\rm NR}$ $\displaystyle=$ $\displaystyle\quad
24.3\pm\,\,\,1.3\pm\,\,\,3.7\pm\,\,\,4.5\;\%\,,$

where the statistical, experimental systematic and model uncertainties are
split accordingly in that order. The results are in agreement with the
measurements obtained by the BaBar and Belle collaborations with decay-time-
dependent flavour-tagged analyses [17, 18], insofar as the DP model components
can be compared.

The measurements of the $C\\!P$ asymmetries are

$\displaystyle\mathcal{A}_{{C\\!P}}(K^{*}(892)^{-}\pi^{+})$ $\displaystyle=$
$\displaystyle-0.308\pm 0.060\pm 0.011\pm 0.012\,,$
$\displaystyle\mathcal{A}_{{C\\!P}}((K\pi)^{-}_{0}\pi^{+})$ $\displaystyle=$
$\displaystyle-0.032\pm 0.047\pm 0.016\pm 0.027\,,$
$\displaystyle\mathcal{A}_{{C\\!P}}(K^{*}_{2}(1430)^{-}\pi^{+})$
$\displaystyle=$
$\displaystyle-0.29\phantom{2}\pm\,\,\,0.22\pm\,\,\,0.09\pm\,\,\,0.03\,,$
$\displaystyle\mathcal{A}_{{C\\!P}}(K^{*}(1680)^{-}\pi^{+})$ $\displaystyle=$
$\displaystyle-0.07\phantom{2}\pm\,\,\,0.13\pm\,\,\,0.02\pm\,\,\,0.03\,,$
$\displaystyle\mathcal{A}_{{C\\!P}}(f_{0}(980){{K}^{0}_{\mathrm{\scriptscriptstyle
S}}})$ $\displaystyle=$
$\displaystyle\;\;\;\;\,0.28\phantom{2}\pm\,\,\,0.27\pm\,\,\,0.05\pm\,\,\,0.14\,,$

where the uncertainties are statistical, experimental systematic and from the
model. The statistical significance of having observed a nonvanishing $C\\!P$
asymmetry in the decay $\overline{B}^{0}\rightarrow K^{*}(892)^{-}\pi^{+}$,
built from the likelihood ratio for the null hypothesis, is 6.7 standard
deviations and reduces to about 6 standard deviations taking into account the
systematic uncertainties. This measurement constitutes the first observation
of a $C\\!P$-violating asymmetry in the decay $\overline{B}^{0}\rightarrow
K^{*}(892)^{-}\pi^{+}$. The measured value is in good agreement with the world
average $\mathcal{A}_{{C\\!P}}(K^{*}(892)^{-}\pi^{+})=-0.23\pm 0.06$ with a
similar precision.

## 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 (The Netherlands); MNiSW and NCN (Poland); MEN/IFA (Romania);
MinES and FASO (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),
ANR, Labex P2IO, ENIGMASS and OCEVU, and Région Auvergne-Rhône-Alpes (France),
RFBR and Yandex LLC (Russia), GVA, XuntaGal and GENCAT (Spain), Herchel Smith
Fund, the Royal Society, the English-Speaking Union and the Leverhulme Trust
(United Kingdom).

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Stagni40, M. Stahl12, S. Stahl40, P. Stefko41, S. Stefkova55, O. Steinkamp42,
S. Stemmle12, O. Stenyakin37, M. Stepanova31, H. Stevens10, S. Stone61, B.
Storaci42, S. Stracka24,p, M.E. Stramaglia41, M. Straticiuc30, U. Straumann42,
J. Sun3, L. Sun64, W. Sutcliffe55, K. Swientek28, V. Syropoulos44, T.
Szumlak28, M. Szymanski63, S. T’Jampens4, A. Tayduganov6, T. Tekampe10, G.
Tellarini17,g, F. Teubert40, E. Thomas40, J. van Tilburg43, M.J. Tilley55, V.
Tisserand4, M. Tobin41, S. Tolk49, L. Tomassetti17,g, D. Tonelli24, F.
Toriello61, R. Tourinho Jadallah Aoude1, E. Tournefier4, M. Traill53, M.T.
Tran41, M. Tresch42, A. Trisovic40, A. Tsaregorodtsev6, P. Tsopelas43, A.
Tully49, N. Tuning43,40, A. Ukleja29, A. Usachov7, A. Ustyuzhanin35, U.
Uwer12, C. Vacca16,f, A. Vagner69, V. Vagnoni15,40, A. Valassi40, S. Valat40,
G. Valenti15, R. Vazquez Gomez40, P. Vazquez Regueiro39, S. Vecchi17, M. van
Veghel43, J.J. Velthuis48, M. Veltri18,r, G. Veneziano57, A. Venkateswaran61,
T.A. Verlage9, M. Vernet5, M. Vesterinen57, J.V. Viana Barbosa40, B. Viaud7,
D. Vieira63, M. Vieites Diaz39, H. Viemann67, X. Vilasis-Cardona38,m, M.
Vitti49, V. Volkov33, A. Vollhardt42, B. Voneki40, A. Vorobyev31, V.
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Watson47, D. Websdale55, A. Weiden42, C. Weisser58, M. Whitehead40, J.
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Williams58, T. Williams47, F.F. Wilson51,40, J. Wimberley60, M. Winn7, J.
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Wraight53, K. Wyllie40, Y. Xie65, M. Xu65, Z. Xu4, Z. Yang3, Z. Yang60, Y.
Yao61, H. Yin65, J. Yu65, X. Yuan61, O. Yushchenko37, K.A. Zarebski47, M.
Zavertyaev11,c, L. Zhang3, Y. Zhang7, A. Zhelezov12, Y. Zheng63, X. Zhu3, V.
Zhukov33, J.B. Zonneveld52, S. Zucchelli15.

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
6Aix Marseille Univ, CNRS/IN2P3, CPPM, 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
17Universita e INFN, Ferrara, Ferrara, Italy
18Sezione INFN di Firenze, Firenze, Italy
19Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
20Sezione INFN di Genova, Genova, Italy
21Universita & INFN, Milano-Bicocca, Milano, Italy
22Sezione 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
35Yandex School of Data Analysis, Moscow, Russia
36Budker Institute of Nuclear Physics (SB RAS), Novosibirsk, Russia
37Institute for High Energy Physics (IHEP), Protvino, Russia
38ICCUB, Universitat de Barcelona, Barcelona, Spain
39Universidad de Santiago de Compostela, Santiago de Compostela, Spain
40European Organization for Nuclear Research (CERN), Geneva, Switzerland
41Institute of Physics, Ecole Polytechnique Fédérale de Lausanne (EPFL),
Lausanne, Switzerland
42Physik-Institut, Universität Zürich, Zürich, Switzerland
43Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
44Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
45NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
46Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
47University of Birmingham, Birmingham, United Kingdom
48H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
49Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
50Department of Physics, University of Warwick, Coventry, United Kingdom
51STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
52School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
53School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
54Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
55Imperial College London, London, United Kingdom
56School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
57Department of Physics, University of Oxford, Oxford, United Kingdom
58Massachusetts Institute of Technology, Cambridge, MA, United States
59University of Cincinnati, Cincinnati, OH, United States
60University of Maryland, College Park, MD, United States
61Syracuse University, Syracuse, NY, United States
62Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
63University of Chinese Academy of Sciences, Beijing, China, associated to 3
64School of Physics and Technology, Wuhan University, Wuhan, China, associated
to 3
65Institute of Particle Physics, Central China Normal University, Wuhan,
Hubei, China, associated to 3
66Departamento de Fisica , Universidad Nacional de Colombia, Bogota, Colombia,
associated to 8
67Institut für Physik, Universität Rostock, Rostock, Germany, associated to 12
68National Research Centre Kurchatov Institute, Moscow, Russia, associated to
32
69National Research Tomsk Polytechnic University, Tomsk, Russia, associated to
32
70Instituto de Fisica Corpuscular, Centro Mixto Universidad de Valencia -
CSIC, Valencia, Spain, associated to 38
71Van Swinderen Institute, University of Groningen, Groningen, The
Netherlands, associated to 43

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
wNovosibirsk State University, Novosibirsk, Russia
†Deceased