EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-PH-EP-2015-107 LHCb-PAPER-2015-017 3 August 2015

Amplitude analysis of ${{B}^{0}}\rightarrow{{\kern
4.14793pt\overline{\kern-4.14793ptD}{}}{}^{0}}{{K}^{+}}{{\pi}^{-}}$ decays

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

The Dalitz plot distribution of ${{B}^{0}}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{K}^{+}}{{\pi}^{-}}$ decays is
studied using a data sample corresponding to $3.0\mbox{\,fb}^{-1}$ of $pp$
collision data recorded by the LHCb experiment during 2011 and 2012. The data
are described by an amplitude model that contains contributions from
intermediate ${{K}^{*}}(892)^{0}$, ${{K}^{*}}(1410)^{0}$,
$K^{*}_{2}(1430)^{0}$ and $D^{*}_{2}(2460)^{-}$ resonances. The model also
contains components to describe broad structures, including the
$K^{*}_{0}(1430)^{0}$ and $D^{*}_{0}(2400)^{-}$ resonances, in the $K\pi$
S-wave and the $D\pi$ S- and P-waves. The masses and widths of the
$D^{*}_{0}(2400)^{-}$ and $D^{*}_{2}(2460)^{-}$ resonances are measured, as
are the complex amplitudes and fit fractions for all components included in
the amplitude model. The model obtained will be an integral part of a future
determination of the angle $\gamma$ of the CKM quark mixing matrix using
${{B}^{0}}\rightarrow{D}{{K}^{+}}{{\pi}^{-}}$ decays.

Published in Phys. Rev. D

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

##  1 Introduction

Dalitz plot (DP) analysis of ${{B}^{0}}\rightarrow{D}{{K}^{+}}{{\pi}^{-}}$
decays has been proposed as a way to measure the unitarity triangle angle
$\gamma$ [1, 2]. The sensitivity to
$\gamma\equiv\arg\left[-V_{ud}V_{ub}^{*}/(V_{cd}V_{cb}^{*})\right]$, where
$V_{xy}$ are elements of the Cabibbo-Kobayashi-Maskawa (CKM) quark mixing
matrix [3, 4], originates from the interference of
$\bar{b}\rightarrow\bar{c}u\bar{s}$ and $\bar{b}\rightarrow\bar{u}c\bar{s}$
amplitudes. Such interference occurs when the neutral $D$ meson is
reconstructed in a final state that is accessible to both ${\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$ and ${D}^{0}$ decays and
therefore corresponds to an admixture of the two states [5, 6]. One of the
largest components of the ${{B}^{0}}\rightarrow{D}{{K}^{+}}{{\pi}^{-}}$ final
state is ${{B}^{0}}\rightarrow D{{K}^{*}}(892)^{0}$, for which both
$\bar{b}\rightarrow\bar{c}u\bar{s}$ and $\bar{b}\rightarrow\bar{u}c\bar{s}$
amplitudes are colour suppressed, making them comparable in magnitude and
potentially enhancing ${C\\!P}$ violation effects [7]. Decay diagrams for the
quasi-two-body contributions from ${{B}^{0}}\rightarrow D{{K}^{*}}(892)^{0}$
and ${{B}^{0}}\rightarrow D^{*}_{2}(2460)^{-}{{K}^{+}}$ decays are shown in
Fig. 1. Observables sensitive to $\gamma$ have been measured by LHCb with a
quasi-two-body approach [8], but a DP analysis is expected to be more
sensitive because interference between resonances provides the possibility to
resolve ambiguities in the determination of $\gamma$.

Figure 1: Decay diagrams for the quasi-two-body contributions to
${{B}^{0}}\rightarrow{D}{{K}^{+}}{{\pi}^{-}}$ from (a) ${{B}^{0}}\rightarrow
D{{K}^{*}}(892)^{0}$ and (b) ${{B}^{0}}\rightarrow
D^{*}_{2}(2460)^{-}{{K}^{+}}$ decays.

In order to determine $\gamma$ with this method, it is necessary to have an
amplitude model of the ${{B}^{0}}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{K}^{+}}{{\pi}^{-}}$ decay that
proceeds through the favoured $\bar{b}\rightarrow\bar{c}u\bar{s}$ transition.
This can be achieved by reconstructing the ${{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}$ meson through the
${{K}^{+}}{{\pi}^{-}}$ decay; in this way the contribution from the
$\bar{b}\rightarrow\bar{c}u\bar{s}$ amplitude is dominant and effects due to
$\bar{b}\rightarrow\bar{u}c\bar{s}$ amplitudes can be neglected. While the
analysis of this decay chain is not itself sensitive to $\gamma$, its outcome
will be an integral part of a future analysis using, for example,
${D}\rightarrow{{K}^{+}}{{K}^{-}}$ decays where $C\\!P$ violation effects are
expected as the final state is common to both ${{D}^{0}}$ and ${{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}$ decays.

Dalitz plot analyses of $B$ meson decays to final states containing a charmed
meson and two charged particles (either pions or kaons) also provide
opportunities for studies of the spectroscopy of charmed mesons. Results in
this area have recently been obtained from DP analyses of
${{B}^{0}_{s}}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{K}^{-}}{{\pi}^{+}}$ [9, 10],
${{{B}^{+}}}\rightarrow{{D}^{-}}{{K}^{+}}{{\pi}^{+}}$ [11] and
${{B}^{0}}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{\pi}^{+}}{{\pi}^{-}}$ [12]
decays, all from LHCb. As the branching fraction for
${{B}^{0}}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{K}^{+}}{{\pi}^{-}}$ decays is
smaller than that for ${{B}^{0}}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{\pi}^{+}}{{\pi}^{-}}$ decays,
the analysis presented in this paper is not as sensitive to the parameters of
charm resonances as that of Ref. [12]. However, the much larger sample of $B$
mesons available at LHCb compared to that used in the only other published DP
analysis of ${{B}^{0}}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{\pi}^{+}}{{\pi}^{-}}$ decays
from the Belle collaboration [13] allows useful results on excited charm
mesons to be obtained. Moreover, results on charm meson spectroscopy obtained
from DP analysis of ${{B}^{0}}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{K}^{+}}{{\pi}^{-}}$ decays
provide important independent cross-checks of results from studies of the
${{B}^{0}}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{\pi}^{+}}{{\pi}^{-}}$ DP, as
possible biases due to other structures in the Dalitz plots are different
between the two modes.

In this paper an amplitude analysis of the ${{B}^{0}}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{K}^{+}}{{\pi}^{-}}$ decay is
reported. The inclusion of charge-conjugate processes is implied throughout
the paper. The analysis is based on a data sample corresponding to an
integrated luminosity of $3.0\,{\rm fb}^{-1}$ of $pp$ collision data collected
with the LHCb detector in 2011 and 2012, when the collision centre-of-mass
energy was $\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$ ($1.0\,{\rm fb}^{-1}$)
and $\sqrt{s}=8\mathrm{\,Te\kern-1.00006ptV}$ ($2.0\,{\rm fb}^{-1}$),
respectively. Previously, the branching fraction for the three-body decay has
been measured [14, 15], and the ${{B}^{0}}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{K}^{*}}(892)^{0}$ [16, 17] and
${{B}^{0}}\rightarrow D^{*}_{2}(2460)^{-}{{K}^{+}}$ [14] contributions have
been measured using quasi-two-body approaches; however this is the first DP
analysis of the ${{B}^{0}}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{K}^{+}}{{\pi}^{-}}$ decay.

The paper is organised as follows. A description of the LHCb detector,
reconstruction and simulation software is given in Sec. 2. The selection of
signal candidates is described in Sec. 3, and the determination of signal and
background yields is presented in Sec. 4. An overview of the Dalitz plot
analysis formalism is given in Sec. 5, and details of the implementation of
the amplitude analysis are presented in Sec. 6. The evaluation of systematic
uncertainties is described in Sec. 7, with results and a brief summary given
in Sec. 8.

##  2 LHCb detector and software

The LHCb detector [18, 19] 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 [20] surrounding the $pp$
interaction region, a large-area silicon-strip detector located upstream of a
dipole magnet with a bending power of about $4{\rm\,Tm}$, and three stations
of silicon-strip detectors and straw drift tubes [21] placed downstream of the
magnet. The polarity of the dipole magnet is reversed periodically throughout
data-taking. The tracking system provides a measurement of momentum, $p$, of
charged particles with a relative uncertainty that varies from 0.5% at low
momentum to 1.0% at 200$\mathrm{\,Ge\kern-1.00006ptV}$ (natural units with
$c=\hbar=1$ are used). The minimum distance of a track to a primary vertex,
the impact parameter (IP), is measured with a resolution of
$(15+29/\mbox{$p_{\rm T}$}){\,\upmu\rm m}$, where $p_{\rm T}$ is the component
of the momentum transverse to the beam, in $\mathrm{\,Ge\kern-1.00006ptV}$.
Different types of charged hadrons are distinguished using information from
two ring-imaging Cherenkov detectors [22]. Photon, electron and hadron
candidates are identified by a calorimeter system consisting of scintillating-
pad and preshower detectors, an electromagnetic calorimeter and a hadronic
calorimeter. Muons are identified by a system composed of alternating layers
of iron and multiwire proportional chambers [23].

The trigger [24] consists of a hardware stage, based on information from the
calorimeter and muon systems, followed by a software stage, in which all
tracks with $\mbox{$p_{\rm T}$}>500~{}(300)\mathrm{\,Me\kern-1.00006ptV}$ are
reconstructed for data collected in 2011 (2012). The software trigger used in
this analysis requires a two-, three- or four-track secondary vertex with
significant displacement from any primary $pp$ interaction vertex (PV). At
least one charged particle must have $\mbox{$p_{\rm
T}$}>1.7\mathrm{\,Ge\kern-1.00006ptV}$ and be inconsistent with originating
from a PV. A multivariate algorithm [25] is used for the identification of
secondary vertices consistent with the decay of a $b$ hadron.

In the offline selection, the objects that prompted a positive trigger
decision are associated with reconstructed particles. Selection requirements
can therefore be made not only on whether the hardware trigger decision was
due to a signature in the calorimeters or in the muon system, but on whether
the decision was due to the signal candidate, other particles produced in the
$pp$ collision, or a combination of both. Signal candidates are accepted
offline if at least one of the final-state particles created a cluster in the
hadronic calorimeter with sufficient transverse energy to fire the hardware
trigger. Events that are triggered at the hardware level by another particle
in the event are also retained. After all selection requirements are imposed,
67 % of events in the sample were triggered by the decay products of the
signal candidate, while the remainder were triggered only by another particle
in the event.

Simulated events are used to characterise the detector response to signal and
certain types of background events. In the simulation, $pp$ collisions are
generated using Pythia [26, *Sjostrand:2007gs] with a specific LHCb
configuration [28]. Decays of hadronic particles are described by EvtGen [29],
in which final-state radiation is generated using Photos [30]. The interaction
of the generated particles with the detector, and its response, are
implemented using the Geant4 toolkit [31, *Agostinelli:2002hh] as described in
Ref. [33].

##  3 Selection requirements

The selection requirements follow closely those used in Ref. [10]. The more
copious ${{B}^{0}}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{\pi}^{+}}{{\pi}^{-}}$ decay is
topologically and kinematically similar to the ${{B}^{0}}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{K}^{+}}{{\pi}^{-}}$ channel,
allowing it to be used as a control mode to optimise the selection
requirements. Loose initial requirements are used to obtain a visible signal
peak of ${{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{\pi}^{+}}{{\pi}^{-}}$
candidates. The tracks are required to be of good quality and must be above
thresholds in $p$, $p_{\rm T}$ and $\chi^{2}_{\rm IP}$, where $\chi^{2}_{\rm
IP}$ is defined as the difference in $\chi^{2}$ of a given PV reconstructed
with and without the considered particle. The ${{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}\rightarrow{{K}^{+}}{{\pi}^{-}}$
candidate must satisfy criteria on its vertex quality ($\chi^{2}_{\rm vtx}$)
and flight distance from any PV and from the $B$ candidate vertex, and must
have invariant mass $m({{K}^{+}}{{\pi}^{-}})$ in the range
$1814$–$1914\mathrm{\,Me\kern-1.00006ptV}$. A requirement on the output of a
boosted decision tree that identifies ${{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}\rightarrow{{K}^{+}}{{\pi}^{-}}$
decays originating from $b$ hadron decays (${{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}$ BDT) [34, 35] is also applied.
Candidate $B$ mesons are selected with requirements on invariant mass,
$\chi^{2}_{\rm IP}$ and on the cosine of the angle between the $B$ momentum
vector and the line from the PV under consideration to the ${B}$ vertex
($\cos\theta_{\rm dir}$). A requirement is placed on the $\chi^{2}$ of a
kinematic fit [36] to the candidate’s decay chain in which the ${{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}$ mass is constrained to its
nominal value. The four final-state tracks must satisfy pion and kaon particle
identification (PID) criteria.

A neural network [37] is used to discriminate between signal decays and
combinatorial background. The sPlot technique [38], with the $B$ candidate
mass as the discriminating variable, is used to separate statistically
${{B}^{0}}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{\pi}^{+}}{{\pi}^{-}}$ decays
from background. The signal and background weights returned by this method are
applied to the candidates, which are then used to train the network. The
network is trained using 16 variables. They include the $\chi^{2}_{\rm IP}$ of
the four final-state tracks and the following variables associated to the
${{\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}$ candidate:
$\chi^{2}_{\rm IP}$, $\chi^{2}_{\rm vtx}$, the square of the flight distance
from the PV divided by its uncertainty squared ($\chi^{2}_{\rm flight}$),
$\cos\theta_{\rm dir}$, and the output of the ${{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}$ BDT. In addition, the
following variables associated to the ${B}$ candidate are included: $p_{\rm
T}$, $\chi^{2}_{\rm IP}$, $\chi^{2}_{\rm vtx}$, $\chi^{2}_{\rm flight}$, and
$\cos\theta_{\rm dir}$. The $p_{\rm T}$ asymmetry [10] and track multiplicity
in a cone with half-angle of 1.5 units of the plane of pseudorapidity and
azimuthal angle (measured in radians) around the $B$ candidate flight
direction, which contain information about the isolation of the $B$ candidate
from the rest of the event, are also used. The input quantities to the neural
network depend only weakly on the position of the candidate in the ${B}$ decay
Dalitz plot and therefore any requirement on the network output cannot
appreciably bias the DP distribution. A requirement imposed on the network
output reduces the combinatorial background remaining after the initial
selection by a factor of five while retaining more than $90\,\%$ of the
signal.

The ${{B}^{0}}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{K}^{+}}{{\pi}^{-}}$ candidates
must satisfy the same selection as the ${{B}^{0}}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{\pi}^{+}}{{\pi}^{-}}$ decays,
except for the PID requirement on the positively charged track from the $B$
decay vertex, which is imposed to preferentially select kaons rather than
pions. The PID efficiency is evaluated from calibration samples of
${{D}^{0}}\rightarrow{{K}^{-}}{{\pi}^{+}}$ decays from the
${{D}^{*+}}\rightarrow{{D}^{0}}{{\pi}^{+}}$ decay chain. The kinematics of
this decay chain can be exploited to obtain clean samples without using the
PID information [22]. The PID efficiency of the requirements on the four
tracks in the final state is around $50\,\%$ and varies depending on the
kinematics of the tracks, as described in detail in Sec. 6.1.

Candidates are vetoed when the difference between
$m({{K}^{+}}{{\pi}^{-}}{{\pi}^{-}})$ and $m({{K}^{+}}{{\pi}^{-}})$ lies within
$\pm 2.5\mathrm{\,Me\kern-1.00006ptV}$ of the known
${{D}^{*}}(2010)^{-}$–${{\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}$
mass difference [39] to remove background containing
${{D}^{*-}}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{\pi}^{-}}$ decays. Candidates
are also rejected if a similar mass difference calculated with the pion mass
hypothesis applied to the bachelor kaon satisfies the same criterion. To
reject backgrounds from ${{B}^{0}}\rightarrow{{D}^{-}}{{K}^{+}}$,
${{D}^{-}}\rightarrow{{K}^{+}}{{\pi}^{-}}{{\pi}^{-}}$ decays, it is required
that the combination of the pion from the ${{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}$ candidate together with the
two bachelor particles does not have an invariant mass in the range
$1850$–$1890\mathrm{\,Me\kern-1.00006ptV}$. Additionally, candidates are
removed if the pion and kaon originating directly from the ${B}^{0}$ decay
combine to give an invariant mass consistent with that of the ${{D}^{0}}$
meson ($1835$–$1880\mathrm{\,Me\kern-1.00006ptV}$). This removes candidates
with the ${{\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}$ wrongly
reconstructed as well as potential background from
${{B}^{0}}\rightarrow{{D}^{0}}{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}$ decays. Other incorrectly
reconstructed candidates are removed by vetoing candidates where the pion from
the ${{\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}$ decay and the
kaon originating directly from the ${B}^{0}$ decay give an invariant mass in
the range $1850$–$1885\mathrm{\,Me\kern-1.00006ptV}$. At least one of the pion
candidates is required to have no associated hits in the muon system to remove
potential background from
${{B}^{0}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}{{K}^{*0}}$
decays. Charmless decays of $b$ hadrons are suppressed by the use of the
${{\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}$ BDT and further
reduced to a negligible level by requiring that the ${{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}$ candidate vertex is separated
from the ${{B}^{0}}$ decay vertex by at least $1\rm\,mm$.

Signal candidates are retained for further analysis if they have invariant
mass in the range $5100$–$5900\mathrm{\,Me\kern-1.00006ptV}$. After all
selection requirements are applied, fewer than 1 % of selected events also
contain a second candidate. Such multiple candidates are retained and treated
in the same manner as other candidates; the associated systematic uncertainty
is negligible.

##  4 Determination of signal and background yields

The signal and background yields are determined from an extended maximum
likelihood fit to the $B$ candidate invariant mass distribution. The fit
allows for signal decays, combinatorial background and contributions from
other $b$ hadron decays. The decay chain ${{B}^{0}}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{*0}}{{K}^{+}}{{\pi}^{-}}$ with
${{\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{*0}}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}\gamma$ or ${{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{\pi}^{0}}$ forms a partially
reconstructed background that peaks at low $B$ candidate mass as the neutral
particle is not included in the candidate. Misidentified
${{B}^{0}}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{(*)0}}{{\pi}^{+}}{{\pi}^{-}}$,
${{\kern 1.00006pt\overline{\kern-1.00006pt\mathchar
28931\relax}}{}^{0}_{b}}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{(*)0}}{{K}^{+}}{\overline{p}}$ and
${B}^{0}_{({s})}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{(*)0}}{{K}^{+}}{{K}^{-}}$ decays
are found to contribute to the background. A possible contribution from the
highly suppressed ${{B}^{0}_{s}}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{K}^{+}}{{\pi}^{-}}$ decay mode
is also included in the fit.

The signal peak is modelled with the sum of two Crystal Ball [40] functions,
which have tails on opposite sides and which share a common mean. The tail
parameters are fixed to the values found in fits to simulated signal decays.
The relative normalisation and the ratio of widths of the two functions are
constrained, within uncertainties, to the values determined in a fit to
${{B}^{0}}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{\pi}^{+}}{{\pi}^{-}}$ data.
The ${{B}^{0}_{s}}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{K}^{+}}{{\pi}^{-}}$ shape is
modelled identically to the signal peak, with the difference between the
${{B}^{0}}$ and ${{B}^{0}_{s}}$ mass peak positions fixed to its known value
[39]. An exponential shape is used to model the combinatorial background.

Misidentified and partially reconstructed backgrounds are modelled with
smoothed non-parametric functions. Simulated samples are used to obtain the
shape for ${{B}^{0}}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{*0}}{{K}^{+}}{{\pi}^{-}}$ decays,
with the ${{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{*0}}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}\gamma$ and ${{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{\pi}^{0}}$ contributions
generated in the correct proportions [39]. To account for differences between
simulation and data, for example in the polarisation of the ${{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{*0}}$ meson, the $B$ candidate
invariant mass distribution for partially reconstructed
${{B}^{0}}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{*0}}{{K}^{+}}{{\pi}^{-}}$ decays is
allowed to be shifted by an offset that is a free parameter of the fit. The
misidentified background shapes are also obtained from simulated samples. For
${{B}^{0}}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{(*)0}}{{\pi}^{+}}{{\pi}^{-}}$
decays the ${{\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}$ and
${{\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{*0}}$ contributions are
combined according to their relative branching fractions [39]. The backgrounds
from ${{\kern 1.00006pt\overline{\kern-1.00006pt\mathchar
28931\relax}}{}^{0}_{b}}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{(*)0}}{{K}^{+}}{\overline{p}}$ and
${B}^{0}_{({s})}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{(*)0}}{{K}^{+}}{{K}^{-}}$ decays
are assumed to have equal branching fractions for the decays with ${{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}$ and ${{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{*0}}$ mesons in the final state, as
the decays involving ${{\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{*0}}$
mesons have not yet been measured. The simulated samples are reweighted
according to the relevant particle identification and misidentification
probabilities and to match known DP distributions [12, 41, 42, 43]. The
(mis)identification probabilities take account of track kinematics and are
calculated using calibration samples of
${{D}^{*+}}\rightarrow{{D}^{0}}{{\pi}^{+}}$,
${{D}^{0}}\rightarrow{{K}^{-}}{{\pi}^{+}}$ and ${\mathchar
28931\relax}\rightarrow p{{\pi}^{-}}$ decays [22, 19]. The yields of the
misidentified backgrounds relative to the signal are constrained, within
uncertainties, to their expected values based on their known branching
fractions [12, 41, 42] and misidentification probabilities.

There are 14 free parameters in the fit model: the mean and width of the
signal shape, the relative normalisation and relative width of the two Crystal
Ball functions, the slope of the exponential function, the offset of the
${{B}^{0}}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{*0}}{{K}^{+}}{{\pi}^{-}}$ shape and
the yields of the eight contributions. The result of the fit is shown in Fig.
2 and the yields are summarised in Table 1. The yields are also reported
within the signal region used in the Dalitz plot fit, corresponding to $\pm
2.5$ widths of the ${B}^{0}$ signal shape
($5248.55$–$5309.05\mathrm{\,Me\kern-1.00006ptV}$). The distribution of
candidates in the signal region over the Dalitz plot is shown in Fig. 3(a).
Note that a ${{B}^{0}}$ mass constraint is applied to calculate the variables
that are used to describe the Dalitz plot [44], improving the resolution of
those variables and giving a unique kinematic boundary.

Table 1: Yields from the fit to the ${{\kern 1.79997pt\overline{\kern-1.79997ptD}{}}{}^{0}}{{K}^{+}}{{\pi}^{-}}$ data sample. The full mass range is $5100$–$5900\mathrm{\,Me\kern-0.90005ptV}$ and the signal region is $5248.55$–$5309.05\mathrm{\,Me\kern-0.90005ptV}$.  Component | Full mass range | Signal region
---|---|---
${{B}^{0}}\rightarrow{{\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{K}^{+}}{{\pi}^{-}}$ | $2576\pm 72\phantom{1}$ | $2344\pm 66$
${{B}^{0}_{s}}\rightarrow{{\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{K}^{+}}{{\pi}^{-}}$ | $\phantom{11}55\pm 27\phantom{1}$ | $\phantom{111}1\pm 1\phantom{1}$
comb. bkgd. | $5540\pm 187$ | $\phantom{1}684\pm 23$
${{B}^{0}}\rightarrow{{\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{*0}}{{K}^{+}}{{\pi}^{-}}$ | $1750\pm 99\phantom{1}$ | $\phantom{111}6\pm 1\phantom{1}$
${{B}^{0}}\rightarrow{{\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{(*)0}}{{\pi}^{+}}{{\pi}^{-}}$ | $\phantom{1}485\pm 47\phantom{1}$ | $\phantom{11}51\pm 5\phantom{1}$
${{\kern 1.00006pt\overline{\kern-1.00006pt\mathchar 28931\relax}}{}^{0}_{b}}\rightarrow{{\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{(*)0}}{{K}^{+}}{\overline{p}}$ | $\phantom{11}95\pm 26\phantom{1}$ | $\phantom{11}18\pm 5\phantom{1}$
${{B}^{0}}\rightarrow{{\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{(*)0}}{{K}^{+}}{{K}^{-}}$ | $\phantom{1}127\pm 27\phantom{1}$ | $\phantom{11}10\pm 2\phantom{1}$
${{B}^{0}_{s}}\rightarrow{{\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{(*)0}}{{K}^{+}}{{K}^{-}}$ | $\phantom{11}54\pm 18\phantom{1}$ | $\phantom{11}14\pm 5\phantom{1}$

Figure 2: Results of the fit to the $B$ candidate invariant mass distribution
with (a) linear and (b) logarithmic $y$-axis scales. The components are as
described in the legend.

Figure 3: Distribution of ${{B}^{0}}\rightarrow{{\kern
1.79997pt\overline{\kern-1.79997ptD}{}}{}^{0}}{{K}^{+}}{{\pi}^{-}}$ candidates
in the signal region over (a) the Dalitz plot and (b) the square Dalitz plot.
The definition of the square Dalitz plot is given in Sec. 6.1.

##  5 Dalitz plot analysis formalism

In ${{B}^{0}}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{K}^{+}}{{\pi}^{-}}$ decays,
resonances are expected in the $m^{2}({{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{\pi}^{-}})$ and
$m^{2}({{K}^{+}}{{\pi}^{-}})$ combinations, so the Dalitz plot shown in Fig.
3(a) is defined in terms of these two invariant mass squared terms. For a
fixed ${{B}^{0}}$ mass, these two invariant mass squared combinations can be
used to calculate all other relevant kinematic quantities.

The isobar model [45, 46, 47] is used to describe the complex decay amplitude.
The total amplitude is given by the coherent sum of amplitudes from resonant
and nonresonant intermediate contributions and is given by

${\cal A}\left(m^{2}({{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{\pi}^{-}}),m^{2}({{K}^{+}}{{\pi}^{-}})\right)=\sum_{j=1}^{N}c_{j}F_{j}\left(m^{2}({{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{\pi}^{-}}),m^{2}({{K}^{+}}{{\pi}^{-}})\right)\,,$
(1)

where $c_{j}$ are complex coefficients describing the relative contribution
for each intermediate process. The resonant dynamics are contained in the
$F_{j}\left(m^{2}({{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{\pi}^{-}}),m^{2}({{K}^{+}}{{\pi}^{-}})\right)$
terms that are normalised such that the integral of the squared magnitude over
the DP is unity for each term. For a ${{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{\pi}^{-}}$ resonance
$F_{j}\left(m^{2}({{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{\pi}^{-}}),m^{2}({{K}^{+}}{{\pi}^{-}})\right)$
is given by

$F\left(m^{2}({{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{\pi}^{-}}),m^{2}({{K}^{+}}{{\pi}^{-}})\right)=R\left(m({{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{\pi}^{-}})\right)\times
X(|\vec{p}\,|\,r_{\rm BW})\times X(|\vec{q}\,|\,r_{\rm BW})\times
T(\vec{p},\vec{q}\,)\,,$ (2)

where $\vec{p}$ is the bachelor particle momentum and $\vec{q}$ is the
momentum of one of the resonance daughters, both evaluated in the ${{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{\pi}^{-}}$ rest frame. The
functions $R$, $X$ and $T$ are described below.

The $X(z)$ terms are Blatt–Weisskopf barrier factors [48], where
$z=|\vec{q}\,|\,r_{\rm BW}$ or $|\vec{p}\,|\,r_{\rm BW}$ and $r_{\rm BW}$ is
the barrier radius which is set to
$4.0\mathrm{\,Ge\kern-1.00006ptV}^{-1}\approx 0.8\rm\,fm$ [10] for all
resonances. The barrier factors are angular momentum dependent and are given
by

$\begin{array}[]{rcl}L=0\ :\ X(z)&=&1\,,\\\ L=1\ :\
X(z)&=&\sqrt{\frac{1+z_{0}^{2}}{1+z^{2}}}\,,\\\ L=2\ :\
X(z)&=&\sqrt{\frac{z_{0}^{4}+3z_{0}^{2}+9}{z^{4}+3z^{2}+9}}\,,\\\ L=3\ :\
X(z)&=&\sqrt{\frac{z_{0}^{6}+6z_{0}^{4}+45z_{0}^{2}+225}{z^{6}+6z^{4}+45z^{2}+225}}\,,\end{array}$
(3)

where $z_{0}$ is the value of $z$ at the pole mass of the resonance and $L$ is
the orbital angular momentum between the resonance and the bachelor particle.
Since the parent and daughter particles all have zero spin, $L$ is also the
spin of the resonance.

The $T(\vec{p},\vec{q})$ terms describe the angular distributions in the
Zemach tensor formalism [49, 50] and are given by

$\begin{array}[]{rcl}L=0\ :\ T(\vec{p},\vec{q})&=&1\,,\\\ L=1\ :\
T(\vec{p},\vec{q})&=&-\,2\,\vec{p}\cdot\vec{q}\,,\\\ L=2\ :\
T(\vec{p},\vec{q})&=&\frac{4}{3}\left[3(\vec{p}\cdot\vec{q}\,)^{2}-(|\vec{p}\,||\vec{q}\,|)^{2}\right]\,,\\\
L=3\ :\
T(\vec{p},\vec{q})&=&-\,\frac{24}{15}\left[5(\vec{p}\cdot\vec{q}\,)^{3}-3(\vec{p}\cdot\vec{q}\,)(|\vec{p}\,||\vec{q}\,|)^{2}\right]\,.\end{array}$
(4)

These expressions are proportional to the Legendre polynomials, $P_{L}(x)$,
where $x$ is the cosine of the angle between $\vec{p}$ and $\vec{q}$ (referred
to as the helicity angle).

The $R\left(m({{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{\pi}^{-}})\right)$ functions
are the mass lineshapes. Resonant contributions are typically described by the
relativistic Breit–Wigner (RBW) function

$R(m)=\frac{1}{(m_{0}^{2}-m^{2})-i\,m_{0}\Gamma(m)}\,,$ (5)

where the mass-dependent decay width is

$\Gamma(m)=\Gamma_{0}\left(\frac{q}{q_{0}}\right)^{2L+1}\left(\frac{m_{0}}{m}\right)X^{2}(q\,r_{\rm
BW})\,,$ (6)

where $q_{0}$ is the value of $q=|\vec{q}\,|$ when the invariant mass is equal
to the pole mass of the resonance, $m_{0}$.

The large phase space available in $B$ decays allows for the presence of
nonresonant amplitudes (i.e. contributions that do not proceed via a known
resonance) that vary across the Dalitz plot. An exponential form factor (EFF)
has been found to describe nonresonant contributions well in several DP
analyses of $B$ decays [51],

$R(m)=\exp\left[-\alpha m^{2}\right]\,,$ (7)

where $\alpha$ is a shape parameter that must be determined from the data and
$m$ is a two-body invariant mass ($m({{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{\pi}^{-}})$ in this example).

The RBW function is a good model for narrow resonances that are well separated
from any other resonant or nonresonant contribution of the same spin. This
approach is known to break down in the $K\pi$ S-wave because the ${{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}^{*}_{0}}(1430)$ resonance interferes
strongly with a slowly varying nonresonant term, as described in Ref. [52].
The LASS lineshape [53] has been developed to combine these two contributions,

$\displaystyle R(m)$ $\displaystyle=$
$\displaystyle\frac{m}{q\cot{\delta_{B}}-iq}+\exp\left[2i\delta_{B}\right]\frac{m_{0}\Gamma_{0}\frac{m_{0}}{q_{0}}}{(m_{0}^{2}-m^{2})-im_{0}\Gamma_{0}\frac{q}{m}\frac{m_{0}}{q_{0}}}\,,$
(8) $\displaystyle{\rm where}\ \cot{\delta_{B}}$ $\displaystyle=$
$\displaystyle\frac{1}{aq}+\frac{1}{2}rq\,,$ (9)

and where $m_{0}$ and $\Gamma_{0}$ are the pole mass and width of the ${{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}^{*}_{0}}(1430)$ state, and $a$ and
$r$ are shape parameters.

The $D\pi$ S-wave nonresonant contribution can be described by the “dabba”
lineshape [54], defined as

$R(m)=\frac{B^{\prime}(m^{2})(m^{2}-s_{A})\rho}{1-\beta(m^{2}-m^{2}_{\rm
min})-iB^{\prime}(m^{2})(m^{2}-s_{A})\rho}\,,$ (10)

where

$B^{\prime}(m^{2})=b\exp\left[-\alpha(m^{2}-m^{2}_{\rm min})\right]\,.$ (11)

Here $m_{\rm min}$ is the invariant mass at threshold,
$s_{A}=m^{2}_{D}-0.5\,m^{2}_{\pi}$ is the Adler zero, $\rho$ is a phase-space
factor and $b$, $\alpha$ and $\beta$ are parameters with values fixed to
$24.49\mathrm{\,Ge\kern-1.00006ptV}^{-2}$,
$0.1\mathrm{\,Ge\kern-1.00006ptV}^{-2}$ and
$0.1\mathrm{\,Ge\kern-1.00006ptV}^{-2}$, respectively, according to Ref. [54].

Ignoring reconstruction and selection effects, the DP probability density
function would be

${\cal{P}}_{\rm phys}\left(m^{2}({{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{\pi}^{-}}),m^{2}({{K}^{+}}{{\pi}^{-}})\right)=\frac{|{\cal
A}\left(m^{2}({{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{\pi}^{-}}),m^{2}({{K}^{+}}{{\pi}^{-}})\right)|^{2}}{\int\\!\\!\int_{\rm
DP}~{}{|{\cal A}|^{2}}~{}dm^{2}({{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{\pi}^{-}})\,dm^{2}({{K}^{+}}{{\pi}^{-}})}\,,$
(12)

where the dependence of ${\cal A}$ on the DP position has been suppressed in
the denominator for brevity. The primary results of most Dalitz plot analyses
are the complex coefficients, as defined in Eq. (1). These, however, depend on
the choice of phase convention, amplitude formalism and normalisation used in
each analysis. The convention-independent quantities of fit fractions and
interference fit fractions provide a way to reliably compare results between
different analyses. Fit fractions are defined as the integral of the amplitude
for each intermediate component squared, divided by that of the coherent
matrix element squared for all intermediate contributions,

${\it FF}_{j}=\frac{\int\\!\\!\int_{\rm DP}\left|c_{j}F_{j}\left(m^{2}({{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{\pi}^{-}}),m^{2}({{K}^{+}}{{\pi}^{-}})\right)\right|^{2}~{}dm^{2}({{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{\pi}^{-}})\,dm^{2}({{K}^{+}}{{\pi}^{-}})}{\int\\!\\!\int_{\rm
DP}\left|{\cal A}\right|^{2}~{}dm^{2}({{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{\pi}^{-}})\,dm^{2}({{K}^{+}}{{\pi}^{-}})}\,.$
(13)

The fit fractions need not sum to unity due to possible net constructive or
destructive interference, described by interference fit fractions defined (for
$i<j$ only) by

${\it FF}_{ij}=\frac{\int\\!\\!\int_{\rm
DP}2\,\mathcal{R}e\left[c_{i}c_{j}^{*}F_{i}F_{j}^{*}\right]~{}dm^{2}({{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{\pi}^{-}})\,dm^{2}({{K}^{+}}{{\pi}^{-}})}{\int\\!\\!\int_{\rm
DP}\left|{\cal A}\right|^{2}~{}dm^{2}({{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{\pi}^{-}})\,dm^{2}({{K}^{+}}{{\pi}^{-}})}\,,$
(14)

where the dependence of $F_{i}^{(*)}$ and ${\cal A}$ on the DP position has
been omitted.

##  6 Dalitz plot fit

###  6.1 Signal efficiency

The variation of the efficiency across the phase space is studied in terms of
the square Dalitz plot (SDP). The SDP is defined by variables $m^{\prime}$ and
$\theta^{\prime}$ that range between 0 and 1 and are given for the ${{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{K}^{+}}{{\pi}^{-}}$ case by

$\mbox{$m^{\prime}$}\equiv\frac{1}{\pi}\arccos\left(2\frac{m({{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{\pi}^{-}})-m^{\rm
min}_{{{\kern
1.39998pt\overline{\kern-1.39998ptD}{}}{}^{0}}{{\pi}^{-}}}}{m^{\rm
max}_{{{\kern
1.39998pt\overline{\kern-1.39998ptD}{}}{}^{0}}{{\pi}^{-}}}-m^{\rm
min}_{{{\kern
1.39998pt\overline{\kern-1.39998ptD}{}}{}^{0}}{{\pi}^{-}}}}-1\right)\hskip
28.45274pt{\rm and}\hskip
28.45274pt\mbox{$\theta^{\prime}$}\equiv\frac{1}{\pi}\theta({{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{\pi}^{-}})\,,$ (15)

where $m^{\rm max}_{{{\kern
1.39998pt\overline{\kern-1.39998ptD}{}}{}^{0}}{{\pi}^{-}}}=m_{{{B}^{0}}}-m_{{{K}^{+}}}$
and $m^{\rm min}_{{{\kern
1.39998pt\overline{\kern-1.39998ptD}{}}{}^{0}}{{\pi}^{-}}}=m_{{{\kern
1.39998pt\overline{\kern-1.39998ptD}{}}{}^{0}}}+m_{{{\pi}^{-}}}$ are the
kinematic boundaries of $m({{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{\pi}^{-}})$ allowed in the
${{B}^{0}}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{K}^{+}}{{\pi}^{-}}$ decay and
$\theta({{\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{\pi}^{-}})$ is
the helicity angle of the ${{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{\pi}^{-}}$ system (the angle
between the ${{K}^{+}}$ and the ${{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}$ meson in the ${{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{\pi}^{-}}$ rest frame). The
distribution of selected events across the SDP is shown in Fig. 3(b).

Efficiency variation across the SDP is caused by the detector acceptance and
by trigger, selection and PID requirements. The efficiency is evaluated with
simulated samples generated uniformly over the SDP. Data-driven corrections
are applied to correct for known differences between data and simulation in
the tracking, trigger and PID efficiencies, using identical methods to those
described in Ref. [10]. The efficiency functions are obtained by fitting the
corrected simulation with two-dimensional cubic splines. Figure 4 shows the
histograms used to model the variation of the efficiency over the SDP for
candidates triggered by (a) signal decays and (b) by the rest of the event.

Figure 4: Efficiency variation as a function of SDP position for candidates
triggered by (a) signal decay products and (b) by the rest of the event. The
vertical white stripe is due to the ${D}^{*}$ veto and the curved white band
is due to the ${\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$ veto.

###  6.2 Background studies

Among the background yields in the signal region, given in Table 1, the only
sizeable components are due to combinatorial background and
${{B}^{0}}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{(*)0}}{{\pi}^{+}}{{\pi}^{-}}$
decays. The SDP distributions of these backgrounds are described by the
histograms shown in Fig. 5.

Figure 5: SDP distributions of the background contributions from (a)
combinatorial background and (b) ${{B}^{0}}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{(*)0}}{{\pi}^{+}}{{\pi}^{-}}$
decays.

Combinatorial background contributes $22\,\%$ of candidates in the signal
region. The shape of this contribution over the SDP is obtained from the high
$B$ mass sideband ($5400$–$5900\mathrm{\,Me\kern-1.00006ptV}$). As seen in
Fig. 2, this sideband is dominated by combinatorial background, with a small
contribution from ${{B}^{0}}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{(*)0}}{{\pi}^{+}}{{\pi}^{-}}$
decays. The SDP distribution of the ${{B}^{0}}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{(*)0}}{{\pi}^{+}}{{\pi}^{-}}$
contribution in the sideband region is modelled as described below for the
signal region and is subtracted from the sideband sample. To check that the
SDP shape of the combinatorial background does not vary with the $B$ candidate
mass, a sample of doubly charged ${{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{K}^{\pm}}{{\pi}^{\pm}}$
candidates is investigated. This confirms that the SDP distribution from the
sideband is a reliable model for combinatorial background in the signal
region.

The ${{B}^{0}}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{(*)0}}{{\pi}^{+}}{{\pi}^{-}}$
contribution accounts for $1.6\,\%$ of the candidates in the signal region.
Its SDP distribution is obtained from a simulated sample, with the
${{B}^{0}}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{\pi}^{+}}{{\pi}^{-}}$ and
${{B}^{0}}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{*0}}{{\pi}^{+}}{{\pi}^{-}}$
contributions combined and reweighted as described in Sec. 4. The
${{B}^{0}}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{\pi}^{+}}{{\pi}^{-}}$ decay
mode is the dominant component of this background in the signal region.

Due to resolution effects, ${{B}^{0}}\rightarrow{{D}^{*}}(2010)^{-}{{K}^{+}}$,
${{D}^{*}}(2010)^{-}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{\pi}^{-}}$ decays are not
entirely removed by the ${D}^{*}$ veto. Although this component corresponds to
real ${{B}^{0}}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{\pi}^{-}}{{K}^{+}}$ decays, it
is treated as incoherent (i.e. as background) since its interference with the
remainder of the signal is negligible. Its SDP distribution is modelled as a
Gaussian peak in $m({{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{\pi}^{-}})$, with mean fixed
to the known ${{D}^{*}}(2010)^{-}$ mass and width and yield determined in the
fit to data.

###  6.3 Amplitude model for ${{B}^{0}}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{K}^{+}}{{\pi}^{-}}$ decays

The Dalitz plot fit is performed using the Laura++ [55] package. The
likelihood function is given by

${\cal L}=\prod_{i}^{N_{c}}\Bigg{[}\sum_{k}N_{k}{\cal
P}_{k}\left(m^{2}_{i}({{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{\pi}^{-}}),m^{2}_{i}({{K}^{+}}{{\pi}^{-}})\right)\Bigg{]}\,,$
(16)

where the index $i$ runs over $N_{c}$ candidates, $k$ runs over the signal and
background components and $N_{k}$ is the yield in each component. The
probability density function for the signal component, ${\cal P}_{\rm sig}$,
is given by Eq. (12) with the $|{\cal A}\left(m^{2}({{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{\pi}^{-}}),m^{2}({{K}^{+}}{{\pi}^{-}})\right)|^{2}$
terms multiplied by the efficiency function described in Sec. 6.1. As it is
possible for the minimisation procedure to find a local minimum of the
negative log likelihood (NLL) profile, the fit is repeated many times with
randomised initial values to ensure the global minimum is found.

The nominal Dalitz plot fit model for ${{B}^{0}}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{K}^{+}}{{\pi}^{-}}$ decays is
composed of several resonant and nonresonant amplitudes. Only those amplitudes
that provide significant contributions or that aid the fit stability are
included. Unnatural spin-parity states are not considered as these do not
decay to two pseudoscalars. The eight amplitudes included in this model are
listed in Table 2. The width of the narrowest signal contribution to the
Dalitz plot ($\sim 50\mathrm{\,Me\kern-1.00006ptV}$) is far larger than the
mass resolution ($\sim 2.4\mathrm{\,Me\kern-1.00006ptV}$); therefore,
resolution effects are neglected.

Table 2: Signal contributions to the fit model, where parameters and
uncertainties are taken from Ref. [39]. The models are described in Sec. 5.

Resonance | Spin | DP axis | Model | Parameters
---|---|---|---|---
${{K}^{*}}(892)^{0}$ | 1 | $m^{2}({{K}^{+}}{{\pi}^{-}})$ | RBW |  $m_{0}=895.81\pm 0.19\mathrm{\,Me\kern-1.00006ptV}$, $\Gamma_{0}=47.4\pm 0.6\mathrm{\,Me\kern-1.00006ptV}$
${{K}^{*}}(1410)^{0}$ | 1 | $m^{2}({{K}^{+}}{{\pi}^{-}})$ | RBW |  $m_{0}=1414\pm 15\mathrm{\,Me\kern-1.00006ptV}$, $\Gamma_{0}=232\pm 21\mathrm{\,Me\kern-1.00006ptV}$
$K^{*}_{0}(1430)^{0}$ | 0 | $m^{2}({{K}^{+}}{{\pi}^{-}})$ | LASS | Determined from data (see text)
$K^{*}_{2}(1430)^{0}$ | 2 | $m^{2}({{K}^{+}}{{\pi}^{-}})$ | RBW |  $m_{0}=1432.4\pm 1.3\mathrm{\,Me\kern-1.00006ptV}$, $\Gamma_{0}=109\pm 5\mathrm{\,Me\kern-1.00006ptV}$
$D^{*}_{0}(2400)^{-}$ | 0 | $m^{2}({{\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{\pi}^{-}})$ | RBW | Determined from data (see Table 3)
$D^{*}_{2}(2460)^{-}$ | 2 | $m^{2}({{\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{\pi}^{-}})$ | RBW
Nonresonant | 0 | $m^{2}({{\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{\pi}^{-}})$ | dabba | Fixed (see text)
Nonresonant | 1 | $m^{2}({{\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{\pi}^{-}})$ | EFF | Determined from data (see text)

The real and imaginary parts of the complex coefficients $c_{j}$ defined in
Eq. (1) are free parameters of the fit except for the coefficient of the
$D^{*}_{2}(2460)^{-}$ component, which is fixed to 1 as a reference. The
phases and magnitudes of the complex coefficients, as well as fit fractions
and interference fit fractions are derived from these free parameters. In
addition, the masses and widths of the $D^{*}_{0}(2400)^{-}$ and
$D^{*}_{2}(2460)^{-}$ resonances are determined from the fit to data and are
reported in Table 3. The statistical uncertainties on all parameters of
interest are calculated using large samples of simulated pseudoexperiments.
These pseudoexperiments are also used to determine the correlations between
the statistical uncertainties on the parameters, which are given in Appendix
A. The LASS parameters are determined to be $m_{0}=1450\pm
80\mathrm{\,Me\kern-1.00006ptV}$, $\Gamma_{0}=400\pm
230\mathrm{\,Me\kern-1.00006ptV}$, $a=3.2\pm 1.8\mathrm{\,Ge\kern-1.00006ptV}$
and $r=0.9\pm 1.1\mathrm{\,Ge\kern-1.00006ptV}$, while the parameter of the
EFF lineshape of the $D\pi$ P-wave nonresonant amplitude is determined to be
$\alpha=0.88\pm 0.10\mathrm{\,Ge\kern-1.00006ptV}^{-2}$.

Table 3: Masses and widths $(\mathrm{Me\kern-0.90005ptV})$ determined in the fit to data, with statistical uncertainties only. Resonance | Mass | Width
---|---|---
$D^{*}_{0}(2400)^{-}$ | $2360\pm 15$ | $255\pm 26$
$D^{*}_{2}(2460)^{-}$ | $2465.6\pm 1.8\phantom{1}$ | $46.0\pm 3.4$

The values of the fit fractions and complex coefficients obtained from the fit
are shown in Table 4, while the values of the interference fit fractions are
given in Appendix B. The sum of the fit fractions is seen to exceed unity,
mostly due to interference within the $D\pi$ and $K\pi$ S-waves. Note that in
Table 4, and all results for fit fractions and derived quantities, values are
reported separately for both the ${K}^{*}_{0}(1430)^{0}$ and nonresonant
components of the LASS lineshape, as well as their coherent sum.

Table 4: Complex coefficients and fit fractions determined from the Dalitz
plot fit. Uncertainties are statistical only. Note that the fit fractions,
magnitudes and phases are derived quantities.

Resonance | Fit fraction | Isobar model coefficients
---|---|---
| (%) | Real part | Imaginary part | Magnitude | Phase
${{K}^{*}}(892)^{0}$ | $37.4\pm 1.5$ | $-0.00\pm 0.15$ | $-1.27\pm 0.06$ | $1.27\pm 0.06$ | $-1.57\pm 0.11$
${{K}^{*}}(1410)^{0}$ | $\phantom{1}0.7\pm 0.3$ | $\phantom{-}0.15\pm 0.06$ | $-0.09\pm 0.09$ | $0.18\pm 0.07$ | $-0.54\pm 0.21$
${K}^{*}_{0}(1430)^{0}$ | $\phantom{1}5.1\pm 2.0$ | $\phantom{-}0.14\pm 0.38$ | $\phantom{-}0.45\pm 0.15$ | $0.47\pm 0.09$ | $\phantom{-}1.27\pm 0.95$
LASS nonresonant | $\phantom{1}4.8\pm 3.8$ | $-0.10\pm 0.24$ | $\phantom{-}0.44\pm 0.14$ | $0.46\pm 0.14$ | $\phantom{-}1.79\pm 0.65$
LASS total | $\phantom{1}6.7\pm 2.7$ |  |  |  |
${K}^{*}_{2}(1430)^{0}$ | $\phantom{1}7.4\pm 1.7$ | $-0.32\pm 0.09$ | $-0.47\pm 0.07$ | $0.57\pm 0.05$ | $-2.16\pm 0.19$
$D^{*}_{0}(2400)^{-}$ | $19.3\pm 2.8$ | $-0.80\pm 0.08$ | $-0.44\pm 0.14$ | $0.91\pm 0.07$ | $-2.64\pm 0.15$
$D^{*}_{2}(2460)^{-}$ | $23.1\pm 1.2$ | $\phantom{-}1.00$ | $\phantom{-}0.00$ | $1.00$ | $\phantom{-}0.00$
$D\pi$ S-wave (dabba) | $\phantom{1}6.6\pm 1.4$ | $-0.39\pm 0.09$ | $\phantom{-}0.36\pm 0.17$ | $0.53\pm 0.07$ | $\phantom{-}2.40\pm 0.27$
$D\pi$ P-wave (EFF) | $\phantom{1}8.9\pm 1.6$ | $-0.62\pm 0.06$ | $-0.03\pm 0.06$ | $0.62\pm 0.06$ | $-3.09\pm 0.10$
Total fit fraction | 113.4 |  |  |  |

Projections of the data and the nominal fit model onto
$m({{K}^{+}}{{\pi}^{-}})$, $m({{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{\pi}^{-}})$ and $m({{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{K}^{+}})$ are shown in Fig. 6.
Zooms are provided in the regions of the main resonant contributions in Fig.
7. Projections onto the cosine of the helicity angle of the ${{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{\pi}^{-}}$ and
${{K}^{+}}{{\pi}^{-}}$ systems are shown in Figs. 8 and 9, respectively. These
projections all show good agreement between data and the fit model.

Angular moments provide a useful method to investigate the helicity structure
of the decays. The angular moments in $m({{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{\pi}^{-}})$ and
$m({{K}^{+}}{{\pi}^{-}})$, obtained as in Refs. [10, 11], are shown in Figs.
10 and 11, respectively. The contributions due to the ${{K}^{*}}(892)^{0}$ and
$D_{2}^{*}(2460)^{-}$ resonances are seen as peaks in moments up to order 2
and order 4, respectively, as expected for spin-1 and spin-2 resonances.
Reflections make the interpretation of moments at higher masses more
difficult. However, the good agreement seen between data and the fit model
provides further support for the fit model being a good description of the
data.

Figure 6: Projections of the data and amplitude fit results onto (a)
$m({{\kern 1.79997pt\overline{\kern-1.79997ptD}{}}{}^{0}}{{\pi}^{-}})$, (c)
$m({{K}^{+}}{{\pi}^{-}})$ and (e) $m({{\kern
1.79997pt\overline{\kern-1.79997ptD}{}}{}^{0}}{{K}^{+}})$, with the same
projections shown in (b), (d) and (f) with a logarithmic $y$-axis scale.
Components are described in the legend.

Figure 7: Projections of the data and amplitude fit results onto (a)
$m({{\kern 1.79997pt\overline{\kern-1.79997ptD}{}}{}^{0}}{{\pi}^{-}})$ in the
$D^{*}_{2}(2460)^{-}$ region and (b) the low $m({{K}^{+}}{{\pi}^{-}})$ region.
Components are as shown in Fig. 6.

Figure 8: Projections of the data and amplitude fit results onto
$\cos\theta({{\kern
1.79997pt\overline{\kern-1.79997ptD}{}}{}^{0}}{{\pi}^{-}})$ in the mass ranges
(a) $2.04<m({{\kern
1.79997pt\overline{\kern-1.79997ptD}{}}{}^{0}}{{\pi}^{-}})<2.35\mathrm{\,Ge\kern-0.90005ptV}$
and (b) $2.35<m({{\kern
1.79997pt\overline{\kern-1.79997ptD}{}}{}^{0}}{{\pi}^{-}})<2.55\mathrm{\,Ge\kern-0.90005ptV}$.
Components are as shown in Fig. 6.

Figure 9: Projections of the data and amplitude fit results onto
$\cos\theta({{K}^{+}}{{\pi}^{-}})$ in the mass ranges (a)
$m({{K}^{+}}{{\pi}^{-}})<0.8\mathrm{\,Ge\kern-0.90005ptV}$, (b)
$0.8<m({{K}^{+}}{{\pi}^{-}})<1.0\mathrm{\,Ge\kern-0.90005ptV}$, (c)
$1.0<m({{K}^{+}}{{\pi}^{-}})<1.3\mathrm{\,Ge\kern-0.90005ptV}$ and (d)
$1.4<m({{K}^{+}}{{\pi}^{-}})<1.5\mathrm{\,Ge\kern-0.90005ptV}$. Components are
as shown in Fig. 6.

Figure 10: Background-subtracted and efficiency-corrected Legendre moments up
to order 7 calculated as a function of $m({{\kern
1.79997pt\overline{\kern-1.79997ptD}{}}{}^{0}}{{\pi}^{-}})$ for data (black
data points) and the fit result (solid blue curve).

Figure 11: Background-subtracted and efficiency-corrected Legendre moments up
to order 7 calculated as a function of $m({{K}^{+}}{{\pi}^{-}})$ for data
(black data points) and the fit result (solid blue curve).

The quality of the fit is evaluated by determining a two-dimensional
$\chi^{2}$ value, comparing the data and fit model in 144 bins across the SDP
that are defined adaptively to ensure they are approximately equally
populated. The pull in each of these bins, defined as the difference between
the data and the fit model divided by the uncertainty, is shown in Fig. 12.
The effective number of degrees of freedom of the $\chi^{2}$ is between
$N_{\rm bins}-N_{\rm pars}-1$ and $N_{\rm bins}-1$, where $N_{\rm pars}$ is
the number of free parameters in the fit, yielding a reduced $\chi^{2}$ in the
range 0.99 – 1.22. Additional unbinned tests of fit quality [56] also show
acceptable agreement between the data and the fit model.

Figure 12: Differences between the data SDP distribution and the fit model
across the SDP, in terms of the per-bin pull.

##  7 Systematic uncertainties

Systematic uncertainties are considered from sources that can be divided into
two categories, experimental and model uncertainties. Experimental
uncertainties arise due to the signal and background yields in the signal
region, the distributions of the background components across the SDP, the
variation of efficiency across the SDP, and possible bias due to the fitting
procedure. Model uncertainties are considered due to the fixed parameters in
the signal model, the addition or removal of marginal amplitudes, and the
choice of models for the $K\pi$ S-wave and the $D\pi$ S- and P-waves. The
uncertainties due to all of these sources are combined in quadrature.

The signal and background yields are determined from the fit to the $B$
candidate invariant mass distribution. Systematic uncertainties are considered
on the total yields due to both statistical and systematic variations of these
yields, as evaluated in Ref. [15]. The yields in the signal region are varied
appropriately and the effects of these variations on the parameters of
interest are assigned as systematic uncertainties.

The histograms describing the background distributions are each varied within
statistical uncertainties to establish the uncertainty due to the imperfect
knowledge of the distribution of these backgrounds across the SDP. The
histograms describing the efficiency variation across the SDP are also varied
within their uncertainties prior to the spline fit. The efficiency histograms
are also varied with local correlations to account for the effects of any
unknown correlations between neighbouring bins. For each parameter the larger
uncertainty is considered from the correlated and uncorrelated variations. In
addition, the binning scheme of the control sample used to evaluate PID
performance is varied.

An ensemble of pseudoexperiments is used to establish systematic uncertainties
related to any potential fit bias. Pseudoexperiments are generated using the
parameters determined in the fit to data, and mean fitted values of the
parameters are determined from Gaussian fits to the distributions of results
obtained in the ensemble. The differences between the generated and mean
fitted values are found to be small. Uncertainties are determined as the sum
in quadrature of the difference between the generated and mean fitted values,
and the uncertainty on the mean fitted value from the Gaussian fit.

Systematic uncertainties due to fixed parameters in the fit model are
determined by repeating the fit with these parameters varied. The fixed masses
and widths are varied within their uncertainties as shown in Table 2, while
the Blatt–Weisskopf barrier radii, $r_{\rm BW}$, are varied independently for
${{\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{\pi}^{-}}$ and
${{K}^{+}}{{\pi}^{-}}$ resonances between 3 and 5
$\mathrm{\,Ge\kern-1.00006ptV}^{-1}$ [10].

The ${{K}^{*}}(1410)^{0}$ amplitude does not contribute significantly to the
nominal fit model so the fit is repeated with this component removed and the
change in results is assigned as the associated systematic uncertainty. In
addition, uncertainties are assigned from the changes in the results when an
amplitude due to the ${{K}^{*}}(1680)^{0}$ resonance or from a virtual
$B^{*}(5325)^{+}$ component, as described in Ref. [10], is included in the
model.

A Dalitz plot analysis of ${{B}^{0}_{s}}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{K}^{-}}{{\pi}^{+}}$ observed
both spin-1 and spin-3 resonances at $m({{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{K}^{-}})\sim
2.86\mathrm{\,Ge\kern-1.00006ptV}$ [9, 10]. A spin-3 resonance has also been
seen at $m({{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{\pi}^{-}})\sim
2.76\mathrm{\,Ge\kern-1.00006ptV}$ in ${{B}^{0}}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{\pi}^{+}}{{\pi}^{-}}$ decays
[12], while an analysis of
${{{B}^{+}}}\rightarrow{{D}^{-}}{{K}^{+}}{{\pi}^{+}}$ shows a spin-1 resonance
at similar $m({{D}^{+}}{{\pi}^{-}})$ values [11]. Uncertainties are assigned
corresponding to the changes in results when either a spin-1 or spin-3 state
at $m({{\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{\pi}^{-}})\sim
2.76\mathrm{\,Ge\kern-1.00006ptV}$ is added to the nominal model.

The LASS model used to describe the $K\pi$ S-wave is replaced with a Flatté
shape [57], which accounts for the ${K}\eta^{\prime}$ threshold near the
${K}^{*}_{0}(1430)^{0}$ mass, and a resonant term with a modified mass
dependent width [54] for the $\kappa$ (also known as ${K}^{*}_{0}(800)$)
state. This model gives an NLL that is worse by 4.4 units. A model-independent
description of the $K\pi$ S-wave is also used to fit the data, with the
uncertainty on each parameter taken as the larger of the two differences. This
alternative improves the NLL by 8.8 units but introduces an additional four
free parameters into the fit. The dabba lineshape used to describe the
${{\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{\pi}^{-}}$ S-wave is
replaced with an EFF lineshape, while a power-law model is introduced as an
alternative to the EFF lineshape that describes the ${{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{\pi}^{-}}$ P-wave. These
alterations worsen the NLL by 3.0 units and improve it by 0.8 units
respectively.

The total systematic uncertainties due to experimental and model effects for
fit fractions and complex coefficients are given together with the results in
Sec. 8. The contributions from different sources to the systematic
uncertainties on the fit fractions, masses and widths are given in Tables 5
and 6. The dominant experimental systematic uncertainty on most parameters is
due to either the efficiency variation or background distributions across the
SDP, while the model uncertainties are generally dominated by the effects of
alternative descriptions of the $K\pi$ S-wave and the $D\pi$ S- and P-waves.
For the parameters of broad components, the model uncertainties dominate;
however for other parameters the statistical, experimental and model
uncertainties are all of a similar magnitude.

Table 5: Experimental systematic uncertainties on the fit fractions (%) and
masses and widths $(\mathrm{Me\kern-0.90005ptV})$. Uncertainties given on the
central values are statistical only.

Resonance | Central value | S/B fraction | Efficiency | Background SDP | Fit bias | Total
---|---|---|---|---|---|---
${{K}^{*}}(892)^{0}$ | $\phantom{1}37.4\pm 1.5$ | 0.60 | 0.83 | 0.50 | 0.31 | 1.17
${{K}^{*}}(1410)^{0}$ | $\phantom{11}0.7\pm 0.3$ | 0.06 | 0.39 | 0.69 | 0.05 | 0.80
${K}^{*}_{0}(1430)^{0}$ | $\phantom{11}5.1\pm 2.0$ | 0.28 | 1.48 | 1.85 | 0.33 | 2.41
LASS nonresonant | $\phantom{11}4.8\pm 3.8$ | 0.51 | 2.25 | 2.86 | 0.86 | 3.77
LASS total | $\phantom{11}6.7\pm 2.7$ | 0.26 | 1.86 | 1.60 | 1.02 | 2.67
${K}^{*}_{2}(1430)^{0}$ | $\phantom{11}7.4\pm 1.7$ | 0.23 | 0.72 | 0.53 | 0.54 | 1.07
$D^{*}_{0}(2400)^{-}$ | $\phantom{1}19.3\pm 2.8$ | 0.21 | 1.39 | 1.43 | 0.40 | 2.04
$D^{*}_{2}(2460)^{-}$ | $\phantom{1}23.1\pm 1.2$ | 0.70 | 0.70 | 0.49 | 0.15 | 1.11
$D\pi$ S-wave (dabba) | $\phantom{11}6.6\pm 1.4$ | 0.03 | 0.81 | 0.59 | 0.57 | 1.15
$D\pi$ P-wave (EFF) | $\phantom{11}8.9\pm 1.6$ | 0.86 | 1.91 | 0.52 | 0.38 | 2.19
$m\left(D^{*}_{0}(2400)^{-}\right)$ | $2360\pm 15$ | 4.6 | 8.1 | 7.0 | 3.7 | 12.2
$m\left(D^{*}_{2}(2460)^{-}\right)$ | $2465.6\pm 1.8\phantom{1}$ | 0.01 | 0.37 | 0.22 | 0.29 | 0.51
$\Gamma\left(D^{*}_{0}(2400)^{-}\right)$ | $\phantom{1}255\pm 26$ | 2.8 | 13.1 | 13.9 | 4.8 | 19.9
$\Gamma\left(D^{*}_{2}(2460)^{-}\right)$ |  1$46.0\pm 3.4$ | 0.5 | 0.9 | 0.9 | 0.5 | 1.4

Table 6: Model uncertainties on the fit fractions (%) and masses and widths
$(\mathrm{Me\kern-0.90005ptV})$. Uncertainties given on the central values are
statistical only.

Resonance | Central value | Fixed parameters | Add/remove | Alternative models | Total
---|---|---|---|---|---
${{K}^{*}}(892)^{0}$ | $\phantom{1}37.4\pm 1.5$ | 0.75 | 1.14 | 1.09 | 1.74
${{K}^{*}}(1410)^{0}$ | $\phantom{11}0.7\pm 0.3$ | 0.18 | 0.70 | 0.22 | 0.76
${K}^{*}_{0}(1430)^{0}$ | $\phantom{11}5.1\pm 2.0$ | 0.79 | 3.30 | 0.23 | 3.40
LASS nonresonant | $\phantom{11}4.8\pm 3.8$ | 1.10 | 3.99 | 5.20 | 6.65
LASS total | $\phantom{11}6.7\pm 2.7$ | 0.53 | 1.42 | 5.21 | 5.43
${K}^{*}_{2}(1430)^{0}$ | $\phantom{11}7.4\pm 1.7$ | 0.36 | 1.87 | 0.56 | 1.98
$D^{*}_{0}(2400)^{-}$ | $\phantom{1}19.3\pm 2.8$ | 0.55 | 1.95 | 7.11 | 7.40
$D^{*}_{2}(2460)^{-}$ | $\phantom{1}23.1\pm 1.2$ | 0.18 | 0.73 | 0.99 | 1.24
$D\pi$ S-wave (dabba) | $\phantom{11}6.6\pm 1.4$ | 0.27 | 1.40 | 3.46 | 3.74
$D\pi$ P-wave (EFF) | $\phantom{11}8.9\pm 1.6$ | 0.31 | 1.99 | 2.15 | 2.95
$m\left(D^{*}_{0}(2400)^{-}\right)$ | $2360\pm 15$ | $6.1$ | $9.3$ | $25.6$ | $27.9$
$m\left(D^{*}_{2}(2460)^{-}\right)$ | $2465.6\pm 1.8\phantom{1}$ | $0.09$ | $1.05$ | $0.48$ | $1.15$
$\Gamma\left(D^{*}_{0}(2400)^{-}\right)$ | $\phantom{1}255\pm 26$ | $4.0$ | $18.0$ | $43.5$ | $47.2$
$\Gamma\left(D^{*}_{2}(2460)^{-}\right)$ | $\phantom{1}46.0\pm 3.4$ | $1.4$ | $0.5$ | $2.4$ | $2.9$

The stability of the fit results is confirmed by several cross-checks. The
data sample is divided into subsamples according to the flavour of the $B$
candidate, the trigger decision, the polarity of the magnet and the year of
data taking. Each subsample is fitted separately and all are seen to be
consistent with the nominal fit results. The fit is also repeated with
additional resonance components to test the fit model. Resonances up to spin-3
were considered for all three pairs of daughters with the mass and width
allowed to vary in the fit. No additional significant contributions were
observed.

##  8 Results and summary

The results for the complex coefficients are reported in Table 7, with results
for the fit fractions given in Table 8. The results for the interference fit
fractions are given in Appendix B. Upper limits are determined on the fit
fractions of the $K^{*}(1410)^{0}$ and $D_{3}^{*}(2760)^{-}$ components, which
do not give significant contributions, using the method described in Ref.
[10].

The fit fractions for resonant contributions are converted into quasi-two-body
product branching fractions by multiplying by
$\mathcal{B}({{B}^{0}}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{K}^{+}}{{\pi}^{-}})=(9.2\pm
0.6\pm 0.7\pm 0.6)\times 10^{-5}$, which is obtained from the ratio of
branching fractions $\mathcal{B}({{B}^{0}}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{K}^{+}}{{\pi}^{-}})/\mathcal{B}({{B}^{0}}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{\pi}^{+}}{{\pi}^{-}})$ [15]
multiplied by the latest result for $\mathcal{B}({{B}^{0}}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{\pi}^{+}}{{\pi}^{-}})$ [12],
accounting for the different ${{D}^{*}}(2010)^{-}$ veto windows used in the
analyses and for the 3.7 % residual background due to
${{B}^{0}}\rightarrow{{D}^{*}}(2010)^{-}{{K}^{+}}$ decays. The statistical
correlation between the results of Ref. [15] and this analysis is found to be
negligible. The results for the product branching fractions are shown in Table
9. For the ${{K}^{+}}{{\pi}^{-}}$ resonances, where the branching fractions to
$K\pi$ are known, the product branching fractions are converted to the $B$
decay branching fractions, shown in Table 10. The results for
$\mathcal{B}({{B}^{0}}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{K}^{*}}(892)^{0})$ and
$\mathcal{B}({{B}^{0}}\rightarrow
D^{*}_{2}(2460)^{-}{{K}^{+}})\times\mathcal{B}(D^{*}_{2}(2460)^{-}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{\pi}^{-}})$ are consistent
with, and more precise than, previous measurements [14, 16, 17].

Table 7: Results for the complex amplitudes and their uncertainties presented
(top) in terms of real and imaginary parts and (bottom) in terms and
magnitudes and phases. The three quoted errors are statistical, experimental
systematic and model uncertainties, respectively.

Resonance | Real part | Imaginary part
---|---|---
${{K}^{*}}(892)^{0}$ |  $-0.00$ $\>\pm\>$ |  $0.15$ $\>\pm\>$ | 0.24 $\>\pm\>$ | 0.34 |  $-1.27$ $\>\pm\>$ |  $0.06$ $\>\pm\>$ | 0.03 $\>\pm\>$ | 0.06
${{K}^{*}}(1410)^{0}$ |  $\phantom{-}0.15$ $\>\pm\>$ |  $0.06$ $\>\pm\>$ | 0.04 $\>\pm\>$ | 0.09 |  $-0.09$ $\>\pm\>$ |  $0.09$ $\>\pm\>$ | 0.18 $\>\pm\>$ | 0.18
${K}^{*}_{0}(1430)^{0}$ |  $\phantom{-}0.14$ $\>\pm\>$ |  $0.38$ $\>\pm\>$ | 0.48 $\>\pm\>$ | 0.38 |  $\phantom{-}0.45$ $\>\pm\>$ |  $0.15$ $\>\pm\>$ | 0.37 $\>\pm\>$ | 0.17
LASS nonresonant |  $-0.10$ $\>\pm\>$ |  $0.24$ $\>\pm\>$ | 0.16 $\>\pm\>$ | 0.42 |  $\phantom{-}0.44$ $\>\pm\>$ |  $0.14$ $\>\pm\>$ | 0.17 $\>\pm\>$ | 0.23
${K}^{*}_{2}(1430)^{0}$ |  $-0.32$ $\>\pm\>$ |  $0.09$ $\>\pm\>$ | 0.15 $\>\pm\>$ | 0.23 |  $-0.47$ $\>\pm\>$ |  $0.07$ $\>\pm\>$ | 0.14 $\>\pm\>$ | 0.15
$D^{*}_{0}(2400)^{-}$ |  $-0.80$ $\>\pm\>$ |  $0.08$ $\>\pm\>$ | 0.07 $\>\pm\>$ | 0.22 |  $-0.44$ $\>\pm\>$ |  $0.14$ $\>\pm\>$ | 0.12 $\>\pm\>$ | 0.18
$D^{*}_{2}(2460)^{-}$ | 1.00 | 0.00
$D\pi$ S-wave (dabba) |  $-0.39$ $\>\pm\>$ |  $0.09$ $\>\pm\>$ | 0.09 $\>\pm\>$ | 0.14 |  $\phantom{-}0.36$ $\>\pm\>$ |  $0.17$ $\>\pm\>$ | 0.14 $\>\pm\>$ | 0.23
$D\pi$ P-wave (EFF) |  $-0.62$ $\>\pm\>$ |  $0.06$ $\>\pm\>$ | 0.03 $\>\pm\>$ | 0.11 |  $-0.03$ $\>\pm\>$ |  $0.06$ $\>\pm\>$ | 0.05 $\>\pm\>$ | 0.10

Resonance | Magnitude | Phase
---|---|---
${{K}^{*}}(892)^{0}$ |  $\phantom{-}1.27$ $\>\pm\>$ | 0.06 $\>\pm\>$ | 0.03 $\>\pm\>$ | 0.05 |  $-1.57$ $\>\pm\>$ | 0.11 $\>\pm\>$ | 0.16 $\>\pm\>$ | 0.27
${{K}^{*}}(1410)^{0}$ |  $\phantom{-}0.18$ $\>\pm\>$ | 0.07 $\>\pm\>$ | 0.10 $\>\pm\>$ | 0.11 |  $-0.54$ $\>\pm\>$ | 0.21 $\>\pm\>$ | 0.55 $\>\pm\>$ | 1.04
${K}^{*}_{0}(1430)^{0}$ |  $\phantom{-}0.47$ $\>\pm\>$ | 0.09 $\>\pm\>$ | 0.10 $\>\pm\>$ | 0.14 |  $\phantom{-}1.27$ $\>\pm\>$ | 0.95 $\>\pm\>$ | 1.04 $\>\pm\>$ | 0.81
LASS nonresonant |  $\phantom{-}0.46$ $\>\pm\>$ | 0.14 $\>\pm\>$ | 0.16 $\>\pm\>$ | 0.29 |  $\phantom{-}1.79$ $\>\pm\>$ | 0.65 $\>\pm\>$ | 0.35 $\>\pm\>$ | 0.69
${K}^{*}_{2}(1430)^{0}$ |  $\phantom{-}0.57$ $\>\pm\>$ | 0.05 $\>\pm\>$ | 0.04 $\>\pm\>$ | 0.08 |  $-2.16$ $\>\pm\>$ | 0.19 $\>\pm\>$ | 0.43 $\>\pm\>$ | 0.43
$D^{*}_{0}(2400)^{-}$ |  $\phantom{-}0.91$ $\>\pm\>$ | 0.07 $\>\pm\>$ | 0.06 $\>\pm\>$ | 0.17 |  $-2.64$ $\>\pm\>$ | 0.15 $\>\pm\>$ | 0.14 $\>\pm\>$ | 0.23
$D^{*}_{2}(2460)^{-}$ | 1.00 | 0.00
$D\pi$ S-wave (dabba) |  $\phantom{-}0.53$ $\>\pm\>$ | 0.07 $\>\pm\>$ | 0.04 $\>\pm\>$ | 0.14 |  $\phantom{-}2.40$ $\>\pm\>$ | 0.27 $\>\pm\>$ | 0.24 $\>\pm\>$ | 0.44
$D\pi$ P-wave (EFF) |  $\phantom{-}0.62$ $\>\pm\>$ | 0.06 $\>\pm\>$ | 0.04 $\>\pm\>$ | 0.11 |  $-3.09$ $\>\pm\>$ | 0.10 $\>\pm\>$ | 0.07 $\>\pm\>$ | 0.17

Table 8: Results for the fit fractions and their uncertainties (%). The three quoted errors are statistical, experimental systematic and model uncertainties, respectively. Upper limits are given at 90 % (95 %) confidence level.  Resonance | Fit fraction | Upper limit
---|---|---
${{K}^{*}}(892)^{0}$ | 37.4$\>\pm\>$ | 1.5 $\>\pm\>$ | 1.2 $\>\pm\>$ | 1.7 |
${{K}^{*}}(1410)^{0}$ | 0.7$\>\pm\>$ | 0.3 $\>\pm\>$ | 0.8 $\>\pm\>$ | 0.8 |  $<$ 3.2 (3.7)
${K}^{*}_{0}(1430)^{0}$ | 5.1$\>\pm\>$ | 2.0 $\>\pm\>$ | 2.4 $\>\pm\>$ | 3.4 |
LASS nonresonant | 4.8$\>\pm\>$ | 3.8 $\>\pm\>$ | 3.8 $\>\pm\>$ | 6.7 |
LASS total | 6.7$\>\pm\>$ | 2.7 $\>\pm\>$ | 2.7 $\>\pm\>$ | 5.4 |
${K}^{*}_{2}(1430)^{0}$ | 7.4$\>\pm\>$ | 1.7 $\>\pm\>$ | 1.1 $\>\pm\>$ | 2.0 |
$D^{*}_{0}(2400)^{-}$ | 19.3$\>\pm\>$ | 2.8 $\>\pm\>$ | 2.0 $\>\pm\>$ | 7.4 |
$D^{*}_{2}(2460)^{-}$ | 23.1$\>\pm\>$ | 1.2 $\>\pm\>$ | 1.1 $\>\pm\>$ | 1.2 |
$D_{3}^{*}(2760)^{-}$ |  |  $<$ 1.0 (1.1)
$D\pi$ S-wave (dabba) | 6.6$\>\pm\>$ | 1.4 $\>\pm\>$ | 1.2 $\>\pm\>$ | 3.7 |
$D\pi$ P-wave (EFF) | 8.9$\>\pm\>$ | 1.6 $\>\pm\>$ | 2.2 $\>\pm\>$ | 3.0 |
Table 9: Results for the product branching fractions. The four quoted errors are statistical, experimental systematic, model and PDG uncertainties, respectively. Upper limits are given at 90 % (95 %) confidence level.  Resonance | Product branching fraction ($10^{-5}$) | Upper limit ($10^{-5}$)
---|---|---
${{K}^{*}}(892)^{0}$ | 3.42 $\>\pm\>$ | 0.13 $\>\pm\>$ | 0.10 $\>\pm\>$ | 0.16 $\>\pm\>$ | 0.40 |
${{K}^{*}}(1410)^{0}$ | 0.07 $\>\pm\>$ | 0.03 $\>\pm\>$ | 0.08 $\>\pm\>$ | 0.07 $\>\pm\>$ | 0.01 |  $<$ 0.29 (0.34)
${K}^{*}_{0}(1430)^{0}$ | 0.47 $\>\pm\>$ | 0.18 $\>\pm\>$ | 0.22 $\>\pm\>$ | 0.31 $\>\pm\>$ | 0.05 |
LASS nonresonant | 0.44 $\>\pm\>$ | 0.34 $\>\pm\>$ | 0.34 $\>\pm\>$ | 0.61 $\>\pm\>$ | 0.05 |
LASS total | 0.61 $\>\pm\>$ | 0.25 $\>\pm\>$ | 0.25 $\>\pm\>$ | 0.49 $\>\pm\>$ | 0.07 |
${K}^{*}_{2}(1430)^{0}$ | 0.68 $\>\pm\>$ | 0.15 $\>\pm\>$ | 0.10 $\>\pm\>$ | 0.18 $\>\pm\>$ | 0.08 |
$D^{*}_{0}(2400)^{-}$ | 1.77 $\>\pm\>$ | 0.26 $\>\pm\>$ | 0.19 $\>\pm\>$ | 0.67 $\>\pm\>$ | 0.20 |
$D^{*}_{2}(2460)^{-}$ | 2.12 $\>\pm\>$ | 0.10 $\>\pm\>$ | 0.11 $\>\pm\>$ | 0.11 $\>\pm\>$ | 0.25 |
$D_{3}^{*}(2760)^{-}$ |  |  $<$ 0.10 (0.11)
$D\pi$ S-wave (dabba) | 0.60 $\>\pm\>$ | 0.13 $\>\pm\>$ | 0.11 $\>\pm\>$ | 0.34 $\>\pm\>$ | 0.07 |
$D\pi$ P-wave (EFF) | 0.81 $\>\pm\>$ | 0.15 $\>\pm\>$ | 0.20 $\>\pm\>$ | 0.27 $\>\pm\>$ | 0.09 |
Table 10: Results for the branching fractions. The four quoted errors are statistical, experimental systematic, model and PDG uncertainties, respectively. Upper limits are given at 90 % (95 %) confidence level.  Resonance | Branching fraction ($10^{-5}$) | Upper limit ($10^{-5}$)
---|---|---
${{K}^{*}}(892)^{0}$ | 5.13 $\>\pm\>$ | 0.20 $\>\pm\>$ | 0.15 $\>\pm\>$ | 0.24 $\>\pm\>$ | 0.60 |
${{K}^{*}}(1410)^{0}$ | 1.59 $\>\pm\>$ | 0.68 $\>\pm\>$ | 1.81 $\>\pm\>$ | 1.59 $\>\pm\>$ | 0.36 |  $<$ 6.7 (7.8)
${K}^{*}_{0}(1430)^{0}$ | 0.71 $\>\pm\>$ | 0.27 $\>\pm\>$ | 0.33 $\>\pm\>$ | 0.47 $\>\pm\>$ | 0.08 |
LASS nonresonant | 0.66 $\>\pm\>$ | 0.51 $\>\pm\>$ | 0.51 $\>\pm\>$ | 0.92 $\>\pm\>$ | 0.08 |
LASS total | 0.92 $\>\pm\>$ | 0.38 $\>\pm\>$ | 0.38 $\>\pm\>$ | 0.74 $\>\pm\>$ | 0.11 |
${K}^{*}_{2}(1430)^{0}$ | 2.04 $\>\pm\>$ | 0.45 $\>\pm\>$ | 0.30 $\>\pm\>$ | 0.54 $\>\pm\>$ | 0.25 |

The masses and widths of the $D^{*}_{0}(2400)^{-}$ and $D^{*}_{2}(2460)^{-}$
states are found to be

$\displaystyle m(D^{*}_{0}(2400)^{-})$ $\displaystyle=$ $\displaystyle(2360\pm
15\pm 12\pm 28)\mathrm{\,Me\kern-1.00006ptV}$
$\displaystyle\Gamma(D^{*}_{0}(2400)^{-})$ $\displaystyle=$
$\displaystyle(255\pm 26\pm 20\pm 47)\mathrm{\,Me\kern-1.00006ptV}$
$\displaystyle m(D^{*}_{2}(2460)^{-})$ $\displaystyle=$
$\displaystyle(2465.6\pm 1.8\pm 0.5\pm 1.2)\mathrm{\,Me\kern-1.00006ptV}$
$\displaystyle\Gamma(D^{*}_{2}(2460)^{-})$ $\displaystyle=$
$\displaystyle(46.0\pm 3.4\pm 1.4\pm 2.9)\mathrm{\,Me\kern-1.00006ptV}\,,$

where the three uncertainties are statistical, experimental systematic and
model systematic, respectively. These are consistent with, though less precise
than, recent results from a DP analysis of ${{B}^{0}}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{\pi}^{+}}{{\pi}^{-}}$ decays
[12]. They also show good agreement with, and have similar precision to,
earlier measurements of these quantities [13, 39].

In summary, the first amplitude analysis of ${{B}^{0}}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{K}^{+}}{{\pi}^{-}}$ decays has
been presented, using a data sample corresponding to 3.0$\mbox{\,fb}^{-1}$ of
$pp$ collision data collected by the LHCb experiment. A good description of
the data is obtained with a model containing contributions from intermediate
${{K}^{*}}(892)^{0}$, ${{K}^{*}}(1410)^{0}$, $K^{*}_{2}(1430)^{0}$ and
$D^{*}_{2}(2460)^{-}$ resonances, with additional components to describe broad
structures in the $K\pi$ S-wave and the $D\pi$ S- and P-waves. The masses and
widths of the $D^{*}_{0}(2400)^{-}$ and $D^{*}_{2}(2460)^{-}$ resonances are
measured, as are the complex amplitudes and fit fractions for all components
included in the amplitude model. The results can be used in conjunction with
an analysis of ${{B}^{0}}\rightarrow{D}{{K}^{+}}{{\pi}^{-}}$ decays, where the
neutral $D$ meson is reconstructed in final states such as
${{K}^{+}}{{K}^{-}}$, to measure the CKM unitarity triangle parameter $\gamma$
[1, 2].

## Acknowledgements

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

Appendices

##  Appendix A Correlation matrices

The statistical correlations between the complex coefficients, masses and
widths determined from the fit to data are given in terms of real and
imaginary parts, and in terms of magnitudes and phases in Tables 11, and 12.
The correlations are determined using the same sample of simulated
pseudoexperiments used to calculate the statistical uncertainties on fit
parameters.

Table 11: Statistical correlations between the real ($x$) and imaginary ($y$)
parts of the complex coefficients that are free parameters of the fit.
Correlations with the masses ($m$) and widths ($\Gamma$) that are determined
from the fit are also included. The correlations are determined from the same
sample of simulated pseudoexperiments used to evaluate systematic
uncertainties. The labels correspond to: (0) ${{K}^{*}}(892)^{0}$, (1)
${{K}^{*}}(1410)^{0}$, (2) ${K}^{*}_{0}(1430)^{0}$, (3) LASS nonresonant, (4)
${K}^{*}_{2}(1430)^{0}$, (5) $D^{*}_{0}(2400)^{-}$, (6) $D^{*}_{2}(2460)^{-}$,
(7) $D\pi$ S-wave (dabba), (8) $D\pi$ P-wave (EFF).

| $x_{0}$ | $y_{0}$ | $x_{1}$ | $y_{1}$ | $x_{2}$ | $y_{2}$ | $x_{3}$ | $y_{3}$ | $x_{4}$ | $y_{4}$ | $x_{5}$ | $y_{5}$ | $x_{7}$ | $y_{7}$ | $x_{8}$ | $y_{8}$ | $m_{6}$ | $\Gamma_{6}$ | $m_{5}$ | $\Gamma_{5}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$x_{0}$ |  1.00 |  0.14 |  0.02 |  0.17 | $-0.18$ |  0.15 | $-0.17$ |  0.03 |  0.39 | $-0.37$ |  0.16 | $-0.09$ | $-0.18$ | $-0.29$ |  0.07 | $-0.37$ | $-0.19$ |  0.02 |  0.04 |  0.12
$y_{0}$ |  |  1.00 | $-0.06$ | $-0.08$ |  0.09 | $-0.11$ | $-0.00$ |  0.14 |  0.03 |  0.03 |  0.12 |  0.13 | $-0.12$ | $-0.24$ |  0.35 | $-0.07$ |  0.09 |  0.12 |  0.01 | $-0.04$
$x_{1}$ |  |  |  1.00 | $-0.09$ | $-0.08$ | $-0.14$ | $-0.04$ |  0.29 | $-0.01$ |  0.03 | $-0.20$ |  0.18 | $-0.21$ | $-0.31$ | $-0.17$ | $-0.36$ | $-0.08$ | $-0.17$ | $-0.23$ |  0.02
$y_{1}$ |  |  |  |  1.00 |  0.05 |  0.34 |  0.09 | $-0.01$ |  0.60 | $-0.39$ |  0.02 | $-0.03$ | $-0.03$ |  0.07 | $-0.02$ | $-0.04$ | $-0.16$ | $-0.02$ |  0.07 |  0.02
$x_{2}$ |  |  |  |  |  1.00 | $-0.13$ |  0.81 |  0.09 |  0.14 | $-0.16$ | $-0.15$ |  0.21 |  0.10 |  0.13 |  0.04 | $-0.15$ | $-0.07$ |  0.01 | $-0.25$ |  0.01
$y_{2}$ |  |  |  |  |  |  1.00 |  0.08 |  0.28 |  0.24 | $-0.36$ | $-0.13$ |  0.08 | $-0.04$ | $-0.01$ | $-0.04$ |  0.04 | $-0.02$ | $-0.01$ | $-0.09$ | $-0.10$
$x_{3}$ |  |  |  |  |  |  |  1.00 |  0.20 |  0.13 | $-0.34$ | $-0.10$ |  0.33 | $-0.08$ |  0.08 | $-0.04$ | $-0.05$ | $-0.07$ | $-0.03$ | $-0.33$ | $-0.09$
$y_{3}$ |  |  |  |  |  |  |  |  1.00 | $-0.12$ | $-0.12$ | $-0.51$ |  0.29 | $-0.34$ | $-0.53$ |  0.09 | $-0.12$ |  0.09 | $-0.14$ | $-0.36$ |  0.01
$x_{4}$ |  |  |  |  |  |  |  |  |  1.00 | $-0.53$ |  0.32 | $-0.10$ | $-0.20$ |  0.03 |  0.02 | $-0.27$ | $-0.17$ | $-0.06$ |  0.10 | $-0.00$
$y_{4}$ |  |  |  |  |  |  |  |  |  |  1.00 |  0.06 | $-0.09$ |  0.40 |  0.10 |  0.14 |  0.16 |  0.11 | $-0.05$ |  0.07 | $-0.03$
$x_{5}$ |  |  |  |  |  |  |  |  |  |  |  1.00 | $-0.29$ | $-0.03$ |  0.22 |  0.02 |  0.04 | $-0.08$ |  0.02 |  0.45 | $-0.06$
$y_{5}$ |  |  |  |  |  |  |  |  |  |  |  |  1.00 | $-0.31$ | $-0.34$ |  0.06 |  0.03 |  0.24 | $-0.14$ | $-0.72$ | $-0.33$
$x_{7}$ |  |  |  |  |  |  |  |  |  |  |  |  |  1.00 |  0.46 |  0.05 |  0.10 | $-0.11$ | $-0.01$ |  0.27 |  0.17
$y_{7}$ |  |  |  |  |  |  |  |  |  |  |  |  |  |  1.00 | $-0.16$ |  0.36 | $-0.01$ |  0.17 |  0.32 |  0.08
$x_{8}$ |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  1.00 | $-0.03$ |  0.01 | $-0.06$ | $-0.09$ | $-0.14$
$y_{8}$ |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  1.00 |  0.29 |  0.02 |  0.15 | $-0.02$
$m_{6}$ |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  1.00 |  0.09 |  0.01 | $-0.04$
$\Gamma_{6}$ |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  1.00 | $-0.03$ |  0.00
$m_{5}$ |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  1.00 |  0.27
$\Gamma_{5}$ |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  1.00

Table 12: Statistical correlations between the magnitudes ($a$) and phases
($\Delta$) of the complex coefficients that are free parameters of the fit.
Correlations with the masses ($m$) and widths ($\Gamma$) that are determined
from the fit are also included. The correlations are determined from the same
sample of simulated pseudoexperiments used to evaluate systematic
uncertainties. The labels correspond to: (0) ${{K}^{*}}(892)^{0}$, (1)
${{K}^{*}}(1410)^{0}$, (2) ${K}^{*}_{0}(1430)^{0}$, (3) LASS nonresonant, (4)
${K}^{*}_{2}(1430)^{0}$, (5) $D^{*}_{0}(2400)^{-}$, (6) $D^{*}_{2}(2460)^{-}$,
(7) $D\pi$ S-wave (dabba), (8) $D\pi$ P-wave (EFF).

| $a_{0}$ | $\Delta_{0}$ | $a_{1}$ | $\Delta_{1}$ | $a_{2}$ | $\Delta_{2}$ | $a_{3}$ | $\Delta_{3}$ | $a_{4}$ | $\Delta_{4}$ | $a_{5}$ | $\Delta_{5}$ | $a_{7}$ | $\Delta_{7}$ | $a_{8}$ | $\Delta_{8}$ | $m_{6}$ | $\Gamma_{6}$ | $m_{5}$ | $\Gamma_{5}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$a_{0}$ |  1.00 | $-0.11$ | $-0.08$ |  0.05 |  0.24 |  0.03 | $-0.14$ |  0.04 |  0.04 |  0.01 |  0.21 |  0.02 |  0.22 | $-0.22$ |  0.38 | $-0.00$ | $-0.04$ | $-0.09$ | $-0.01$ |  0.04
$\Delta_{0}$ |  |  1.00 | $-0.10$ |  0.12 |  0.03 |  0.20 |  0.14 |  0.14 | $-0.01$ |  0.45 | $-0.05$ |  0.10 | $-0.08$ |  0.35 | $-0.07$ |  0.05 | $-0.22$ |  0.06 |  0.07 |  0.09
$a_{1}$ |  |  |  1.00 | $-0.35$ | $-0.25$ |  0.06 |  0.32 |  0.01 | $-0.01$ | $-0.38$ |  0.04 | $-0.22$ | $-0.06$ |  0.31 |  0.02 |  0.24 |  0.07 | $-0.18$ | $-0.20$ |  0.01
$\Delta_{1}$ |  |  |  |  1.00 |  0.24 |  0.15 |  0.16 |  0.00 |  0.14 |  0.38 | $-0.18$ | $-0.13$ |  0.16 |  0.26 |  0.22 | $-0.06$ | $-0.09$ | $-0.04$ | $-0.12$ |  0.05
$a_{2}$ |  |  |  |  |  1.00 |  0.04 |  0.11 | $-0.28$ | $-0.07$ |  0.39 | $-0.09$ |  0.07 |  0.27 |  0.10 |  0.01 |  0.01 | $-0.15$ |  0.04 | $-0.03$ | $-0.13$
$\Delta_{2}$ |  |  |  |  |  |  1.00 |  0.15 |  0.77 | $-0.20$ | $-0.10$ | $-0.01$ |  0.24 |  0.09 |  0.13 |  0.03 | $-0.09$ |  0.05 |  0.01 |  0.23 | $-0.10$
$a_{3}$ |  |  |  |  |  |  |  1.00 | $-0.06$ |  0.00 | $-0.11$ |  0.16 | $-0.30$ | $-0.12$ |  0.56 |  0.01 |  0.12 |  0.02 | $-0.07$ | $-0.27$ | $-0.04$
$\Delta_{3}$ |  |  |  |  |  |  |  |  1.00 | $-0.28$ | $-0.19$ |  0.15 |  0.41 | $-0.24$ | $-0.08$ | $-0.00$ | $-0.13$ |  0.04 |  0.02 |  0.41 |  0.08
$a_{4}$ |  |  |  |  |  |  |  |  |  1.00 |  0.13 |  0.05 | $-0.17$ |  0.14 |  0.18 | $-0.00$ |  0.06 |  0.00 |  0.14 | $-0.19$ | $-0.09$
$\Delta_{4}$ |  |  |  |  |  |  |  |  |  |  1.00 | $-0.16$ |  0.16 |  0.17 |  0.12 |  0.00 | $-0.02$ | $-0.19$ | $-0.03$ |  0.03 |  0.01
$a_{5}$ |  |  |  |  |  |  |  |  |  |  |  1.00 |  0.24 | $-0.26$ | $-0.06$ |  0.11 |  0.05 | $-0.07$ |  0.01 |  0.19 |  0.39
$\Delta_{5}$ |  |  |  |  |  |  |  |  |  |  |  |  1.00 | $-0.00$ | $-0.28$ | $-0.01$ | $-0.20$ | $-0.20$ |  0.15 |  0.76 |  0.27
$a_{7}$ |  |  |  |  |  |  |  |  |  |  |  |  |  1.00 | $-0.01$ |  0.17 | $-0.06$ |  0.04 |  0.13 | $-0.05$ | $-0.24$
$\Delta_{7}$ |  |  |  |  |  |  |  |  |  |  |  |  |  |  1.00 | $-0.12$ |  0.37 |  0.03 | $-0.00$ | $-0.27$ | $-0.12$
$a_{8}$ |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  1.00 | $-0.04$ | $-0.01$ |  0.04 |  0.06 |  0.15
$\Delta_{8}$ |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  1.00 | $-0.37$ | $-0.02$ | $-0.33$ | $-0.01$
$m_{6}$ |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  1.00 |  0.09 |  0.01 | $-0.04$
$\Gamma_{6}$ |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  1.00 | $-0.03$ |  0.00
$m_{5}$ |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  1.00 |  0.27
$\Gamma_{5}$ |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  1.00

##  Appendix B Results for interference fit fractions

The central values of the interference fit fractions are given in Table 13.
The statistical, experimental systematic and model uncertainties on these
quantities are given in Tables 14, 15 and 16.

Table 13: Interference fit fractions (%) from the nominal DP fit. The amplitudes are all pairwise products involving: ($A_{0}$) ${{K}^{*}}(892)^{0}$, ($A_{1}$) ${{K}^{*}}(1410)^{0}$, ($A_{2}$) ${K}^{*}_{0}(1430)^{0}$, ($A_{3}$) LASS nonresonant, ($A_{4}$) ${K}^{*}_{2}(1430)^{0}$, ($A_{5}$) $D^{*}_{0}(2400)^{-}$, ($A_{6}$) $D^{*}_{2}(2460)^{-}$, ($A_{7}$) $D\pi$ S-wave (dabba), ($A_{8}$) $D\pi$ P-wave (EFF). The diagonal elements are the same as the conventional fit fractions.  | $\phantom{-}A_{0}$ | $\phantom{-}A_{1}$ | $\phantom{-}A_{2}$ | $\phantom{-}A_{3}$ | $\phantom{-}A_{4}$ | $\phantom{-}A_{5}$ | $\phantom{-}A_{6}$ | $\phantom{-}A_{7}$ | $\phantom{-}A_{8}$
---|---|---|---|---|---|---|---|---|---
$A_{0}$ | $\phantom{-}37.4$ | $\phantom{-}1.8$ | $-0.0$ | $\phantom{-}0.0$ | $\phantom{-}0.0$ | $-2.4$ | $-0.2$ | $-2.6$ | $-5.0$
$A_{1}$ |  | $\phantom{-}0.7$ | $-0.0$ | $\phantom{-}0.0$ | $\phantom{-}0.0$ | $-1.1$ | $-0.3$ | $-0.6$ | $-1.1$
$A_{2}$ |  |  | $\phantom{-}5.1$ | $-3.2$ | $-0.0$ | $\phantom{-}2.9$ | $\phantom{-}1.5$ | $\phantom{-}1.6$ | $\phantom{-}1.9$
$A_{3}$ |  |  |  | $\phantom{-}4.8$ | $\phantom{-}0.0$ | $-8.4$ | $\phantom{-}0.1$ | $\phantom{-}2.1$ | $\phantom{-}0.2$
$A_{4}$ |  |  |  |  | $\phantom{-}7.4$ | $-0.3$ | $-1.0$ | $-1.5$ | $-0.4$
$A_{5}$ |  |  |  |  |  | $\phantom{-}19.3$ | $-0.0$ | $\phantom{-}2.8$ | $\phantom{-}0.0$
$A_{6}$ |  |  |  |  |  |  | $\phantom{-}23.1$ | $-0.0$ | $\phantom{-}0.0$
$A_{7}$ |  |  |  |  |  |  |  | $\phantom{-}6.6$ | $-0.0$
$A_{8}$ |  |  |  |  |  |  |  |  | $\phantom{-}8.9$
Table 14: Statistical uncertainties on the interference fit fractions (%). The amplitudes are all pairwise products involving: ($A_{0}$) ${{K}^{*}}(892)^{0}$, ($A_{1}$) ${{K}^{*}}(1410)^{0}$, ($A_{2}$) ${K}^{*}_{0}(1430)^{0}$, ($A_{3}$) LASS nonresonant, ($A_{4}$) ${K}^{*}_{2}(1430)^{0}$, ($A_{5}$) $D^{*}_{0}(2400)^{-}$, ($A_{6}$) $D^{*}_{2}(2460)^{-}$, ($A_{7}$) $D\pi$ S-wave (dabba), ($A_{8}$) $D\pi$ P-wave (EFF). The diagonal elements are the same as the conventional fit fractions.  | $A_{0}$ | $A_{1}$ | $A_{2}$ | $A_{3}$ | $A_{4}$ | $A_{5}$ | $A_{6}$ | $A_{7}$ | $A_{8}$
---|---|---|---|---|---|---|---|---|---
$A_{0}$ | 1.5 | 0.7 | 0.0 | 0.0 | 0.0 | 0.9 | 0.4 | 0.4 | 0.5
$A_{1}$ |  | 0.3 | 0.0 | 0.0 | 0.0 | 0.7 | 0.3 | 0.4 | 0.5
$A_{2}$ |  |  | 2.0 | 2.6 | 0.0 | 1.1 | 0.5 | 1.5 | 0.5
$A_{3}$ |  |  |  | 3.8 | 0.0 | 3.2 | 0.4 | 3.4 | 0.8
$A_{4}$ |  |  |  |  | 1.7 | 0.6 | 0.1 | 0.6 | 0.5
$A_{5}$ |  |  |  |  |  | 2.8 | 0.0 | 3.1 | 0.0
$A_{6}$ |  |  |  |  |  |  | 1.2 | 0.0 | 0.0
$A_{7}$ |  |  |  |  |  |  |  | 1.4 | 0.0
$A_{8}$ |  |  |  |  |  |  |  |  | 1.6
Table 15: Experimental systematic uncertainties on the interference fit fractions (%). The amplitudes are all pairwise products involving: ($A_{0}$) ${{K}^{*}}(892)^{0}$, ($A_{1}$) ${{K}^{*}}(1410)^{0}$, ($A_{2}$) ${K}^{*}_{0}(1430)^{0}$, ($A_{3}$) LASS nonresonant, ($A_{4}$) ${K}^{*}_{2}(1430)^{0}$, ($A_{5}$) $D^{*}_{0}(2400)^{-}$, ($A_{6}$) $D^{*}_{2}(2460)^{-}$, ($A_{7}$) $D\pi$ S-wave (dabba), ($A_{8}$) $D\pi$ P-wave (EFF). The diagonal elements are the same as the conventional fit fractions.  | $A_{0}$ | $A_{1}$ | $A_{2}$ | $A_{3}$ | $A_{4}$ | $A_{5}$ | $A_{6}$ | $A_{7}$ | $A_{8}$
---|---|---|---|---|---|---|---|---|---
$A_{0}$ | 1.2 | 0.5 | 0.0 | 0.0 | 0.0 | 2.3 | 0.2 | 0.4 | 0.5
$A_{1}$ |  | 0.8 | 0.0 | 0.0 | 0.0 | 0.4 | 0.8 | 0.7 | 0.9
$A_{2}$ |  |  | 2.4 | 3.1 | 0.0 | 1.4 | 2.0 | 2.6 | 1.1
$A_{3}$ |  |  |  | 3.8 | 0.0 | 3.6 | 0.2 | 2.6 | 0.3
$A_{4}$ |  |  |  |  | 1.1 | 2.2 | 0.2 | 0.5 | 1.0
$A_{5}$ |  |  |  |  |  | 2.0 | 0.0 | 4.0 | 0.0
$A_{6}$ |  |  |  |  |  |  | 1.1 | 0.0 | 0.0
$A_{7}$ |  |  |  |  |  |  |  | 1.2 | 0.0
$A_{8}$ |  |  |  |  |  |  |  |  | 2.2
Table 16: Model uncertainties on the interference fit fractions (%). The amplitudes are all pairwise products involving: ($A_{0}$) ${{K}^{*}}(892)^{0}$, ($A_{1}$) ${{K}^{*}}(1410)^{0}$, ($A_{2}$) ${K}^{*}_{0}(1430)^{0}$, ($A_{3}$) LASS nonresonant, ($A_{4}$) ${K}^{*}_{2}(1430)^{0}$, ($A_{5}$) $D^{*}_{0}(2400)^{-}$, ($A_{6}$) $D^{*}_{2}(2460)^{-}$, ($A_{7}$) $D\pi$ S-wave (dabba), ($A_{8}$) $D\pi$ P-wave (EFF). The diagonal elements are the same as the conventional fit fractions.  | $A_{0}$ | $A_{1}$ | $A_{2}$ | $A_{3}$ | $A_{4}$ | $A_{5}$ | $A_{6}$ | $A_{7}$ | $A_{8}$
---|---|---|---|---|---|---|---|---|---
$A_{0}$ | 1.7 | 0.8 | 0.0 | 0.0 | 0.0 | 2.6 | 0.7 | 0.8 | 1.4
$A_{1}$ |  | 0.8 | 0.0 | 0.0 | 0.0 | 0.8 | 0.6 | 0.8 | 1.1
$A_{2}$ |  |  | 3.4 | 2.9 | 0.0 | 5.8 | 1.7 | 3.6 | 2.7
$A_{3}$ |  |  |  | 6.6 | 0.0 | 8.3 | 0.3 | 3.5 | 0.7
$A_{4}$ |  |  |  |  | 2.0 | 1.7 | 0.2 | 1.2 | 1.0
$A_{5}$ |  |  |  |  |  | 7.4 | 0.0 | 4.5 | 0.0
$A_{6}$ |  |  |  |  |  |  | 1.2 | 0.0 | 0.0
$A_{7}$ |  |  |  |  |  |  |  | 3.7 | 0.0
$A_{8}$ |  |  |  |  |  |  |  |  | 2.9

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LHCb collaboration

R. Aaij38, B. Adeva37, M. Adinolfi46, A. Affolder52, Z. Ajaltouni5, S. Akar6,
J. Albrecht9, F. Alessio38, M. Alexander51, S. Ali41, G. Alkhazov30, P.
Alvarez Cartelle53, A.A. Alves Jr57, S. Amato2, S. Amerio22, Y. Amhis7, L.
An3, L. Anderlini17,g, J. Anderson40, M. Andreotti16,f, J.E. Andrews58, R.B.
Appleby54, O. Aquines Gutierrez10, F. Archilli38, P. d’Argent11, A.
Artamonov35, M. Artuso59, E. Aslanides6, G. Auriemma25,n, M. Baalouch5, S.
Bachmann11, J.J. Back48, A. Badalov36, C. Baesso60, W. Baldini16,38, R.J.
Barlow54, C. Barschel38, S. Barsuk7, W. Barter38, V. Batozskaya28, V.
Battista39, A. Bay39, L. Beaucourt4, J. Beddow51, F. Bedeschi23, I. Bediaga1,
L.J. Bel41, I. Belyaev31, E. Ben-Haim8, G. Bencivenni18, S. Benson38, J.
Benton46, A. Berezhnoy32, R. Bernet40, A. Bertolin22, M.-O. Bettler38, M. van
Beuzekom41, A. Bien11, S. Bifani45, T. Bird54, A. Birnkraut9, A. Bizzeti17,i,
T. Blake48, F. Blanc39, J. Blouw10, S. Blusk59, V. Bocci25, A. Bondar34, N.
Bondar30,38, W. Bonivento15, S. Borghi54, M. Borsato7, T.J.V. Bowcock52, E.
Bowen40, C. Bozzi16, S. Braun11, D. Brett54, M. Britsch10, T. Britton59, J.
Brodzicka54, N.H. Brook46, A. Bursche40, J. Buytaert38, S. Cadeddu15, R.
Calabrese16,f, M. Calvi20,k, M. Calvo Gomez36,p, P. Campana18, D. Campora
Perez38, L. Capriotti54, A. Carbone14,d, G. Carboni24,l, R. Cardinale19,j, A.
Cardini15, P. Carniti20, L. Carson50, K. Carvalho Akiba2,38, R. Casanova
Mohr36, G. Casse52, L. Cassina20,k, L. Castillo Garcia38, M. Cattaneo38, Ch.
Cauet9, G. Cavallero19, R. Cenci23,t, M. Charles8, Ph. Charpentier38, M.
Chefdeville4, S. Chen54, S.-F. Cheung55, N. Chiapolini40, M. Chrzaszcz40, X.
Cid Vidal38, G. Ciezarek41, P.E.L. Clarke50, M. Clemencic38, H.V. Cliff47, J.
Closier38, V. Coco38, J. Cogan6, E. Cogneras5, V. Cogoni15,e, L. Cojocariu29,
G. Collazuol22, P. Collins38, A. Comerma-Montells11, A. Contu15,38, A. Cook46,
M. Coombes46, S. Coquereau8, G. Corti38, M. Corvo16,f, B. Couturier38, G.A.
Cowan50, D.C. Craik48, A. Crocombe48, M. Cruz Torres60, S. Cunliffe53, R.
Currie53, C. D’Ambrosio38, J. Dalseno46, P.N.Y. David41, A. Davis57, K. De
Bruyn41, S. De Capua54, M. De Cian11, J.M. De Miranda1, L. De Paula2, W. De
Silva57, P. De Simone18, C.-T. Dean51, D. Decamp4, M. Deckenhoff9, L. Del
Buono8, N. Déléage4, D. Derkach55, O. Deschamps5, F. Dettori38, B. Dey40, A.
Di Canto38, F. Di Ruscio24, H. Dijkstra38, S. Donleavy52, F. Dordei11, M.
Dorigo39, A. Dosil Suárez37, D. Dossett48, A. Dovbnya43, K. Dreimanis52, L.
Dufour41, G. Dujany54, F. Dupertuis39, P. Durante38, R. Dzhelyadin35, A.
Dziurda26, A. Dzyuba30, S. Easo49,38, U. Egede53, V. Egorychev31, S.
Eidelman34, S. Eisenhardt50, U. Eitschberger9, R. Ekelhof9, L. Eklund51, I. El
Rifai5, Ch. Elsasser40, S. Ely59, S. Esen11, H.M. Evans47, T. Evans55, A.
Falabella14, C. Färber11, C. Farinelli41, N. Farley45, S. Farry52, R. Fay52,
D. Ferguson50, V. Fernandez Albor37, F. Ferrari14, F. Ferreira Rodrigues1, M.
Ferro-Luzzi38, S. Filippov33, M. Fiore16,38,f, M. Fiorini16,f, M. Firlej27, C.
Fitzpatrick39, T. Fiutowski27, K. Fohl38, P. Fol53, M. Fontana10, F.
Fontanelli19,j, R. Forty38, O. Francisco2, M. Frank38, C. Frei38, M.
Frosini17, J. Fu21, E. Furfaro24,l, A. Gallas Torreira37, D. Galli14,d, S.
Gallorini22,38, S. Gambetta50, M. Gandelman2, P. Gandini55, Y. Gao3, J. García
Pardiñas37, J. Garofoli59, J. Garra Tico47, L. Garrido36, D. Gascon36, C.
Gaspar38, U. Gastaldi16, R. Gauld55, L. Gavardi9, G. Gazzoni5, A. Geraci21,v,
D. Gerick11, E. Gersabeck11, M. Gersabeck54, T. Gershon48, Ph. Ghez4, A.
Gianelle22, S. Gianì39, V. Gibson47, O. G. Girard39, L. Giubega29, V.V.
Gligorov38, C. Göbel60, D. Golubkov31, A. Golutvin53,31,38, A. Gomes1,a, C.
Gotti20,k, M. Grabalosa Gándara5, R. Graciani Diaz36, L.A. Granado Cardoso38,
E. Graugés36, E. Graverini40, G. Graziani17, A. Grecu29, E. Greening55, S.
Gregson47, P. Griffith45, L. Grillo11, O. Grünberg63, B. Gui59, E. Gushchin33,
Yu. Guz35,38, T. Gys38, C. Hadjivasiliou59, G. Haefeli39, C. Haen38, S.C.
Haines47, S. Hall53, B. Hamilton58, T. Hampson46, X. Han11, S. Hansmann-
Menzemer11, N. Harnew55, S.T. Harnew46, J. Harrison54, J. He38, T. Head39, V.
Heijne41, K. Hennessy52, P. Henrard5, L. Henry8, J.A. Hernando Morata37, E.
van Herwijnen38, M. Heß63, A. Hicheur2, D. Hill55, M. Hoballah5, C. Hombach54,
W. Hulsbergen41, T. Humair53, N. Hussain55, D. Hutchcroft52, D. Hynds51, M.
Idzik27, P. Ilten56, R. Jacobsson38, A. Jaeger11, J. Jalocha55, E. Jans41, A.
Jawahery58, F. Jing3, M. John55, D. Johnson38, C.R. Jones47, C. Joram38, B.
Jost38, N. Jurik59, S. Kandybei43, W. Kanso6, M. Karacson38, T.M. Karbach38,†,
S. Karodia51, M. Kelsey59, I.R. Kenyon45, M. Kenzie38, T. Ketel42, B.
Khanji20,38,k, C. Khurewathanakul39, S. Klaver54, K. Klimaszewski28, O.
Kochebina7, M. Kolpin11, I. Komarov39, R.F. Koopman42, P. Koppenburg41,38, M.
Korolev32, L. Kravchuk33, K. Kreplin11, M. Kreps48, G. Krocker11, P.
Krokovny34, F. Kruse9, W. Kucewicz26,o, M. Kucharczyk26, V. Kudryavtsev34, A.
K. Kuonen39, K. Kurek28, T. Kvaratskheliya31, V.N. La Thi39, D. Lacarrere38,
G. Lafferty54, A. Lai15, D. Lambert50, R.W. Lambert42, G. Lanfranchi18, C.
Langenbruch48, B. Langhans38, T. Latham48, C. Lazzeroni45, R. Le Gac6, J. van
Leerdam41, J.-P. Lees4, R. Lefèvre5, A. Leflat32,38, J. Lefrançois7, O.
Leroy6, T. Lesiak26, B. Leverington11, Y. Li7, T. Likhomanenko65,64, M.
Liles52, R. Lindner38, C. Linn38, F. Lionetto40, B. Liu15, X. Liu3, S. Lohn38,
I. Longstaff51, J.H. Lopes2, D. Lucchesi22,r, M. Lucio Martinez37, H. Luo50,
A. Lupato22, E. Luppi16,f, O. Lupton55, F. Machefert7, F. Maciuc29, O. Maev30,
K. Maguire54, S. Malde55, A. Malinin64, G. Manca7, G. Mancinelli6, P.
Manning59, A. Mapelli38, J. Maratas5, J.F. Marchand4, U. Marconi14, C. Marin
Benito36, P. Marino23,38,t, R. Märki39, J. Marks11, G. Martellotti25, M.
Martinelli39, D. Martinez Santos42, F. Martinez Vidal66, D. Martins Tostes2,
A. Massafferri1, R. Matev38, A. Mathad48, Z. Mathe38, C. Matteuzzi20, K.
Matthieu11, A. Mauri40, B. Maurin39, A. Mazurov45, M. McCann53, J. McCarthy45,
A. McNab54, R. McNulty12, B. Meadows57, F. Meier9, M. Meissner11, M. Merk41,
D.A. Milanes62, M.-N. Minard4, D.S. Mitzel11, J. Molina Rodriguez60, S.
Monteil5, M. Morandin22, P. Morawski27, A. Mordà6, M.J. Morello23,t, J.
Moron27, A.B. Morris50, R. Mountain59, F. Muheim50, J. Müller9, K. Müller40,
V. Müller9, M. Mussini14, B. Muster39, P. Naik46, T. Nakada39, R.
Nandakumar49, I. Nasteva2, M. Needham50, N. Neri21, S. Neubert11, N.
Neufeld38, M. Neuner11, A.D. Nguyen39, T.D. Nguyen39, C. Nguyen-Mau39,q, V.
Niess5, R. Niet9, N. Nikitin32, T. Nikodem11, D. Ninci23, A. Novoselov35, D.P.
O’Hanlon48, A. Oblakowska-Mucha27, V. Obraztsov35, S. Ogilvy51, O.
Okhrimenko44, R. Oldeman15,e, C.J.G. Onderwater67, B. Osorio Rodrigues1, J.M.
Otalora Goicochea2, A. Otto38, P. Owen53, A. Oyanguren66, A. Palano13,c, F.
Palombo21,u, M. Palutan18, J. Panman38, A. Papanestis49, M. Pappagallo51, L.L.
Pappalardo16,f, C. Parkes54, G. Passaleva17, G.D. Patel52, M. Patel53, C.
Patrignani19,j, A. Pearce54,49, A. Pellegrino41, G. Penso25,m, M. Pepe
Altarelli38, S. Perazzini14,d, P. Perret5, L. Pescatore45, K. Petridis46, A.
Petrolini19,j, M. Petruzzo21, E. Picatoste Olloqui36, B. Pietrzyk4, T.
Pilař48, D. Pinci25, A. Pistone19, A. Piucci11, S. Playfer50, M. Plo
Casasus37, T. Poikela38, F. Polci8, A. Poluektov48,34, I. Polyakov31, E.
Polycarpo2, A. Popov35, D. Popov10,38, B. Popovici29, C. Potterat2, E.
Price46, J.D. Price52, J. Prisciandaro39, A. Pritchard52, C. Prouve46, V.
Pugatch44, A. Puig Navarro39, G. Punzi23,s, W. Qian4, R. Quagliani7,46, B.
Rachwal26, J.H. Rademacker46, B. Rakotomiaramanana39, M. Rama23, M.S. Rangel2,
I. Raniuk43, N. Rauschmayr38, G. Raven42, F. Redi53, S. Reichert54, M.M.
Reid48, A.C. dos Reis1, S. Ricciardi49, S. Richards46, M. Rihl38, K.
Rinnert52, V. Rives Molina36, P. Robbe7,38, A.B. Rodrigues1, E. Rodrigues54,
J.A. Rodriguez Lopez62, P. Rodriguez Perez54, S. Roiser38, V. Romanovsky35, A.
Romero Vidal37, M. Rotondo22, J. Rouvinet39, T. Ruf38, H. Ruiz36, P. Ruiz
Valls66, J.J. Saborido Silva37, N. Sagidova30, P. Sail51, B. Saitta15,e, V.
Salustino Guimaraes2, C. Sanchez Mayordomo66, B. Sanmartin Sedes37, R.
Santacesaria25, C. Santamarina Rios37, M. Santimaria18, E. Santovetti24,l, A.
Sarti18,m, C. Satriano25,n, A. Satta24, D.M. Saunders46, D. Savrina31,32, M.
Schiller38, H. Schindler38, M. Schlupp9, M. Schmelling10, T. Schmelzer9, B.
Schmidt38, O. Schneider39, A. Schopper38, M. Schubiger39, M.-H. Schune7, R.
Schwemmer38, B. Sciascia18, A. Sciubba25,m, A. Semennikov31, I. Sepp53, N.
Serra40, J. Serrano6, L. Sestini22, P. Seyfert11, M. Shapkin35, I.
Shapoval16,43,f, Y. Shcheglov30, T. Shears52, L. Shekhtman34, V. Shevchenko64,
A. Shires9, R. Silva Coutinho48, G. Simi22, M. Sirendi47, N. Skidmore46, I.
Skillicorn51, T. Skwarnicki59, E. Smith55,49, E. Smith53, I. T. Smith50, J.
Smith47, M. Smith54, H. Snoek41, M.D. Sokoloff57,38, F.J.P. Soler51, F.
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Zavertyaev10,b, L. Zhang3, Y. Zhang3, A. Zhelezov11, A. Zhokhov31, L. Zhong3.

1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université Savoie Mont-Blanc, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Milano, Milano, Italy
22Sezione INFN di Padova, Padova, Italy
23Sezione INFN di Pisa, Pisa, Italy
24Sezione INFN di Roma Tor Vergata, Roma, Italy
25Sezione INFN di Roma La Sapienza, Roma, Italy
26Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
27AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
28National Center for Nuclear Research (NCBJ), Warsaw, Poland
29Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
30Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
31Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
32Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
33Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
34Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
35Institute for High Energy Physics (IHEP), Protvino, Russia
36Universitat de Barcelona, Barcelona, Spain
37Universidad de Santiago de Compostela, Santiago de Compostela, Spain
38European Organization for Nuclear Research (CERN), Geneva, Switzerland
39Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
40Physik-Institut, Universität Zürich, Zürich, Switzerland
41Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
42Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
43NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
44Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
45University of Birmingham, Birmingham, United Kingdom
46H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
47Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
48Department of Physics, University of Warwick, Coventry, United Kingdom
49STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
50School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
51School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
52Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
53Imperial College London, London, United Kingdom
54School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
55Department of Physics, University of Oxford, Oxford, United Kingdom
56Massachusetts Institute of Technology, Cambridge, MA, United States
57University of Cincinnati, Cincinnati, OH, United States
58University of Maryland, College Park, MD, United States
59Syracuse University, Syracuse, NY, United States
60Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
61Institute of Particle Physics, Central China Normal University, Wuhan,
Hubei, China, associated to 3
62Departamento de Fisica , Universidad Nacional de Colombia, Bogota, Colombia,
associated to 8
63Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11
64National Research Centre Kurchatov Institute, Moscow, Russia, associated to
31
65Yandex School of Data Analysis, Moscow, Russia, associated to 31
66Instituto de Fisica Corpuscular (IFIC), Universitat de Valencia-CSIC,
Valencia, Spain, associated to 36
67Van Swinderen Institute, University of Groningen, Groningen, The
Netherlands, associated to 41

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