EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-PH-EP-2015-051 LHCb-PAPER-2015-007 21 June 2016

First observation and amplitude analysis of the
${{{B}^{-}}}\rightarrow{{D}^{+}}{{K}^{-}}{{\pi}^{-}}$ decay

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

The ${{{B}^{-}}}\rightarrow{{D}^{+}}{{K}^{-}}{{\pi}^{-}}$ decay is observed in
a data sample corresponding to $3.0\mbox{\,fb}^{-1}$ of $pp$ collision data
recorded by the LHCb experiment during 2011 and 2012. Its branching fraction
is measured to be ${\cal
B}({{{B}^{-}}}\rightarrow{{D}^{+}}{{K}^{-}}{{\pi}^{-}})=(7.31\pm 0.19\pm
0.22\pm 0.39)\times 10^{-5}$ where the uncertainties are statistical,
systematic and from the branching fraction of the normalisation channel
${{{B}^{-}}}\rightarrow{{D}^{+}}{{\pi}^{-}}{{\pi}^{-}}$, respectively. An
amplitude analysis of the resonant structure of the
${{{B}^{-}}}\rightarrow{{D}^{+}}{{K}^{-}}{{\pi}^{-}}$ decay is used to measure
the contributions from quasi-two-body
${{{B}^{-}}}\rightarrow{D}_{0}^{*}(2400)^{0}{{K}^{-}}$,
${{{B}^{-}}}\rightarrow{D}_{2}^{*}(2460)^{0}{{K}^{-}}$, and
${{{B}^{-}}}\rightarrow{D}_{J}^{*}(2760)^{0}{{K}^{-}}$ decays, as well as from
nonresonant sources. The ${D}_{J}^{*}(2760)^{0}$ resonance is determined to
have spin 1.

Submitted to Phys. Rev. D

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

##  1 Introduction

Excited charmed mesons are of great theoretical and experimental interest as
they allow detailed studies of QCD in an interesting energy regime. Good
progress has been achieved in identifying and measuring the parameters of the
orbitally excited states, notably from Dalitz plot (DP) analyses of three-body
$B$ decays. Relevant examples include the studies of
${{{B}^{-}}}\rightarrow{{D}^{+}}{{\pi}^{-}}{{\pi}^{-}}$ [1, 2] and ${{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}}\rightarrow{{D}^{0}}{{\pi}^{+}}{{\pi}^{-}}$
[3] decays, which provide information on excited neutral and charged charmed
mesons (collectively referred to as $D^{**}$ states), respectively. First
results on excited charm-strange mesons have also recently been obtained with
the DP analysis technique [4, 5, 6]. Studies of prompt charm resonance
production in $e^{+}e^{-}$ and $pp$ collisions [7, 8] have revealed a number
of additional high mass states. Most of these higher mass states are not yet
confirmed by independent analyses, and their spectroscopic identification is
unclear. Analyses of resonances produced directly from $e^{+}e^{-}$ and $pp$
collisions do not allow determination of the quantum numbers of the produced
states, but can distinguish whether or not they have natural spin parity (i.e.
$J^{P}$ in the series $0^{+},1^{-},2^{+},...$). The current experimental
knowledge of the neutral $D^{**}$ states is summarised in Table 1 (here and
throughout the paper, natural units with $\hbar=c=1$ are used). The
$D^{*}_{0}(2400)^{0}$, $D_{1}(2420)^{0}$, $D_{1}^{\prime}(2430)^{0}$ and
$D^{*}_{2}(2460)^{0}$ mesons are generally understood to be the four orbitally
excited (1P) states. The experimental situation as well as the spectroscopic
identification of the heavier states is less clear.

Table 1: Measured properties of neutral $D^{**}$ states. Where more than one uncertainty is given, the first is statistical and the others systematic.  Resonance | Mass | Width | $J^{P}$ | Ref.
---|---|---|---|---
| $(\mathrm{Me\kern-1.00006ptV})$ | $(\mathrm{Me\kern-1.00006ptV})$ |  |
$D^{*}_{0}(2400)^{0}$ | $2318\pm 29$ | $267\pm 40$ | $0^{+}$ | [9]
$D_{1}(2420)^{0}$ | $2421.4\pm 0.6$ | $27.4\pm 2.5$ | $1^{+}$ | [9]
$D_{1}^{\prime}(2430)^{0}$ | $2427\pm 26\pm 20\pm 15$ | $384\,^{+107}_{-75}\pm 24\pm 70$ | $1^{+}$ | [1]
$D^{*}_{2}(2460)^{0}$ | $2462.6\pm 0.6$ | $49.0\pm 1.3$ | $2^{+}$ | [9]
$D^{*}(2600)$ | $2608.7\pm 2.4\pm 2.5$ | $93\pm 6\pm 13$ | natural | [7]
$D^{*}(2650)$ | $2649.2\pm 3.5\pm 3.5$ | $140\pm 17\pm 19$ | natural | [8]
$D^{*}(2760)$ | $2763.3\pm 2.3\pm 2.3$ | $60.9\pm 5.1\pm 3.6$ | natural | [7]
$D^{*}(2760)$ | $2760.1\pm 1.1\pm 3.7$ | $74.4\pm 3.4\pm 19.1$ | natural | [8]

The ${{{B}^{-}}}\rightarrow{{D}^{+}}{{K}^{-}}{{\pi}^{-}}$ decay can be used to
study neutral $D^{**}$ states. The ${{D}^{+}}{{K}^{-}}{{\pi}^{-}}$ final state
is expected to exhibit resonant structure only in the ${{D}^{+}}{{\pi}^{-}}$
channel, and unlike the Cabibbo-favoured ${{D}^{+}}{{\pi}^{-}}{{\pi}^{-}}$
final state does not contain any pair of identical particles. This simplifies
the analysis of the contributing excited charm states, since partial wave
analysis can be used to help determine the resonances that contribute.

One further motivation to study
${{{B}^{-}}}\rightarrow{{D}^{+}}{{K}^{-}}{{\pi}^{-}}$ decays is related to the
measurement of the angle $\gamma$ of the unitarity triangle defined as
$\gamma\equiv\arg\left[-V_{ud}V_{ub}^{*}/(V_{cd}V_{cb}^{*})\right]$, where
$V_{xy}$ are elements of the Cabibbo-Kobayashi-Maskawa (CKM) quark mixing
matrix [10, 11]. One of the most powerful methods to determine $\gamma$ uses
${{{B}^{-}}}\rightarrow D{{K}^{-}}$ decays, with the neutral $D$ meson
decaying to $C\\!P$ eigenstates [12, 13]. The sensitivity to $\gamma$ arises
due to the interference of amplitudes proportional to the CKM matrix elements
$V_{ub}$ and $V_{cb}$, associated with ${{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}$ and ${{D}^{0}}$ production
respectively. However, a challenge for such methods is to determine the ratio
of magnitudes of the two amplitudes, $r_{B}$, that must be known to extract
$\gamma$. This is usually handled by including $D$ meson decays to additional
final states in the analysis. By contrast, in ${{{B}^{-}}}\rightarrow
D^{**}{{K}^{-}}$ decays the efficiency-corrected ratio of yields of
${{{B}^{-}}}\rightarrow
D^{**}{{K}^{-}}\rightarrow{{D}^{-}}{{\pi}^{+}}{{K}^{-}}$ and
${{{B}^{-}}}\rightarrow
D^{**}{{K}^{-}}\rightarrow{{D}^{+}}{{\pi}^{-}}{{K}^{-}}$ decays gives
$r_{B}^{2}$ directly [14]. The decay ${{{B}^{-}}}\rightarrow
D^{**}{{K}^{-}}\rightarrow D\pi^{0}{{K}^{-}}$ where the $D$ meson is
reconstructed in $C\\!P$ eigenstates can be used to search for $C\\!P$
violation driven by $\gamma$. Measurement of the first two of these processes
would therefore provide knowledge of $r_{B}$ in ${{{B}^{-}}}\rightarrow
D^{**}{{K}^{-}}$ decays, indicating whether or not a competitive measurement
of $\gamma$ can be made with this approach.

In this paper, the ${{{B}^{-}}}\rightarrow{{D}^{+}}{{K}^{-}}{{\pi}^{-}}$ decay
is studied for the first time, with the ${{D}^{+}}$ meson reconstructed
through the ${{K}^{-}}{{\pi}^{+}}{{\pi}^{+}}$ decay mode. The inclusion of
charge conjugate processes is implied. The topologically similar
${{{B}^{-}}}\rightarrow{{D}^{+}}{{\pi}^{-}}{{\pi}^{-}}$ decay is used as a
control channel and for normalisation of the branching fraction measurement. A
large ${{{B}^{-}}}\rightarrow{{D}^{+}}{{K}^{-}}{{\pi}^{-}}$ signal yield is
found, corresponding to a clear first observation of the decay, and allowing
investigation of the DP structure of the decay. The amplitude analysis allows
studies of known resonances, searches for higher mass states and measurement
of the properties, including the quantum numbers, of any resonances that are
observed. 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, approximately one third of which was collected during
2011 when the collision centre-of-mass energy was
$\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$ and the rest during 2012 with
$\sqrt{s}=8\mathrm{\,Te\kern-1.00006ptV}$.

The paper is organised as follows. A brief description of the LHCb detector as
well as reconstruction and simulation software is given in Sec. 2. The
selection of signal candidates is described in Sec. 3, and the branching
fraction measurement is presented in Sec. 4. Studies of the backgrounds and
the fit to the $B$ candidate invariant mass distribution are in Sec. 4.1, with
studies of the signal efficiency and a definition of the square Dalitz plot
(SDP) in Sec. 4.2. Systematic uncertainties on, and the results for, the
branching fraction are discussed in Secs. 4.3 and 4.4 respectively. A study of
the angular moments of ${{{B}^{-}}}\rightarrow{{D}^{+}}{{K}^{-}}{{\pi}^{-}}$
decays is given in Sec. 5, with results used to guide the Dalitz plot analysis
that follows. An overview of the Dalitz plot analysis formalism is given in
Sec. 6, and details of the implementation of the amplitude analysis are
presented in Sec. 7. The evaluation of systematic uncertainties is described
in Sec. 8. The results and a summary are given in Sec. 9.

##  2 LHCb detector

The LHCb detector [15, 16] 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 [17] 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 [18] 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$\mathrm{\,Ge\kern-1.00006ptV}$. 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
[19]. 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 [20].

The trigger [21] 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 line
used in the analysis reported in this paper requires a two-, three- or four-
track secondary vertex with significant displacement from the primary $pp$
interaction vertices (PVs). At least one charged particle must have
$\mbox{$p_{\rm T}$}>1.7\mathrm{\,Ge\kern-1.00006ptV}$ and be inconsistent with
originating from the PV. A multivariate algorithm [22] is used for the
identification of secondary vertices consistent with the decay of a $b$
hadron.

In the offline selection, the objects that fired the trigger are associated
with reconstructed particles. Selection requirements can therefore be made not
only on the trigger line that fired, 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 one of the
final state particles created a cluster in the hadronic calorimeter with
sufficient transverse energy to fire the hardware trigger. These candidates
are referred to as “triggered on signal” or TOS. Events that are triggered at
the hardware level by another particle in the event, referred to as “triggered
independent of signal” or TIS, are also retained. After all selection
requirements are imposed, 57 % of events in the sample were triggered by the
decay products of the signal candidate (TOS), while the remainder were
triggered only by another particle in the event (TIS-only).

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 [23, *Sjostrand:2007gs] with a specific LHCb
configuration [25]. Decays of hadronic particles are described by EvtGen [26],
in which final state radiation is generated using Photos [27]. The interaction
of the generated particles with the detector and its response are implemented
using the Geant4 toolkit [28, *Agostinelli:2002hh] as described in Ref. [30].

##  3 Selection requirements

Most selection requirements are optimised using the
${{{B}^{-}}}\rightarrow{{D}^{+}}{{\pi}^{-}}{{\pi}^{-}}$ control channel. Loose
initial selection requirements on the quality of the tracks combined to form
the $B$ candidate, as well as on their $p$, $p_{\rm T}$ and
$\chi^{2}_{\rm{IP}}$, are applied to obtain a visible peak in the invariant
mass distribution. The $\chi^{2}_{\rm{IP}}$ is the difference between the
$\chi^{2}$ of the PV reconstruction with and without the considered particle.
Only candidates with invariant mass in the range
$1770<m({{K}^{-}}{{\pi}^{+}}{{\pi}^{+}})<1968\mathrm{\,Me\kern-1.00006ptV}$
are retained. Further requirements are imposed on the vertex quality
($\chi^{2}_{\rm{vtx}}$) and flight distance from the associated PV of the $B$
and $D$ candidates. The $B$ candidate must also satisfy requirements on its
invariant mass and on the cosine of the angle between the momentum vector and
the line joining the PV under consideration to the ${B}$ vertex
($\cos\theta_{\rm{dir}}$). The initial selection requirements are found to be
about $90\,\%$ efficient on simulated signal decays.

Two neural networks [31] are used to further separate signal from background.
The first is designed to separate candidates that contain real
${{D}^{+}}\rightarrow{{K}^{-}}{{\pi}^{+}}{{\pi}^{+}}$ decays from those that
do not; the second separates
${{{B}^{-}}}\rightarrow{{D}^{+}}{{\pi}^{-}}{{\pi}^{-}}$ signal decays from
background combinations. Both networks are trained using the
${{D}^{+}}{{\pi}^{-}}{{\pi}^{-}}$ control channel, where the sPlot technique
[32] is used to statistically separate
${{{B}^{-}}}\rightarrow{{D}^{+}}{{\pi}^{-}}{{\pi}^{-}}$ signal decays from
background combinations using the $D$ ($B$) candidate mass as the
discriminating variable for the first (second) network. The first network
takes as input properties of the $D$ candidate and its daughter tracks,
including information about kinematics, track and vertex quality. The second
uses a total of 27 input variables. They include the $\chi^{2}_{\rm{IP}}$ of
the two “bachelor” pions (i.e. pions that originate directly from the $B$
decay) and properties of the $D$ candidate including its $\chi^{2}_{\rm{IP}}$,
$\chi^{2}_{\rm{vtx}}$, $\cos\theta_{\rm{dir}}$, the output of the $D$ neural
network and the square of the flight distance divided by its uncertainty
squared ($\chi^{2}_{\rm{flight}}$). Variables associated to the $B$ candidate
are also used, including $p_{\rm T}$, $\chi^{2}_{\rm{IP}}$,
$\chi^{2}_{\rm{vtx}}$, $\chi^{2}_{\rm{flight}}$ and $\cos\theta_{\rm{dir}}$.
The $p_{\rm T}$ asymmetry 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 [33], which contain
information about the isolation of the $B$ candidate from the rest of the
event, are also used in the network. The neural network input quantities
depend only weakly on the kinematics of the ${B}$ decay. A requirement is
imposed on the second neural network output that reduces the combinatorial
background by an order of magnitude while retaining about $75\,\%$ of the
signal.

The selection criteria for the
${{{B}^{-}}}\rightarrow{{D}^{+}}{{K}^{-}}{{\pi}^{-}}$ and
${{{B}^{-}}}\rightarrow{{D}^{+}}{{\pi}^{-}}{{\pi}^{-}}$ candidates are
identical except for the particle identification (PID) requirement on the
bachelor track that differs between the two modes. All five final state
particles for each decay mode have PID criteria applied to preferentially
select either pions or kaons. Tight requirements are placed on the higher
momentum pion from the ${{D}^{+}}$ decay and on the bachelor kaon in
${{{B}^{-}}}\rightarrow{{D}^{+}}{{K}^{-}}{{\pi}^{-}}$ to suppress backgrounds
from ${{D}^{+}_{s}}\rightarrow{{K}^{-}}{{K}^{+}}{{\pi}^{+}}$ and
${{{B}^{-}}}\rightarrow{{D}^{+}}{{\pi}^{-}}{{\pi}^{-}}$ decays, respectively.
The combined efficiency of the PID requirements on the five final state tracks
is around $70\,\%$ for ${{{B}^{-}}}\rightarrow{{D}^{+}}{{\pi}^{-}}{{\pi}^{-}}$
decays and around $40\,\%$ for
${{{B}^{-}}}\rightarrow{{D}^{+}}{{K}^{-}}{{\pi}^{-}}$ decays. The PID
efficiency depends on the kinematics of the tracks, as described in detail in
Sec. 4.2, and is determined using samples of
$D^{0}\rightarrow{{K}^{-}}{{\pi}^{+}}$ decays selected in data by exploiting
the kinematics of the $D^{*+}\rightarrow D^{0}{{\pi}^{+}}$ decay chain to
obtain clean samples without using the PID information.

To improve the $B$ candidate invariant mass resolution, track momenta are
scaled [34, 35] with calibration parameters determined by matching the
measured peak of the $J/\psi\rightarrow\mu^{+}\mu^{-}$ decay to the known
$J/\psi$ mass [9]. Furthermore, a fit to the kinematics and topology of the
decay chain [36] is used to adjust the four-momenta of the tracks from the $D$
candidate so that their combined invariant mass matches the world average
value for the ${{D}^{+}}$ meson [9]. An additional $B$ mass constraint is
applied in the calculation of the variables that are used in the Dalitz plot
fit.

To remove potential background from misreconstructed ${{\mathchar
28931\relax}^{+}_{c}}$ decays, candidates are rejected if the invariant mass
of the $D$ candidate lies in the range
$2280$–$2300\mathrm{\,Me\kern-1.00006ptV}$ when the proton mass hypothesis is
applied to the low momentum pion track. Possible backgrounds from
${{{B}^{-}}}$ meson decays without an intermediate charm meson are suppressed
by the requirement on the output value from the first neural network, and any
surviving background of this type is removed by requiring that the $D$
candidate vertex is displaced by at least $1\rm\,mm$ from the $B$ decay
vertex. The efficiency of this requirement is about $85\,\%$.

Signal candidates are retained for further analysis if they have an invariant
mass in the range $5100$–$5800\mathrm{\,Me\kern-1.00006ptV}$. After all
selection requirements are applied, fewer than $1\,\%$ of events with one
candidate 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 Branching fraction determination

The ratio of branching fractions is calculated from the signal yields with
event-by-event efficiency corrections applied as a function of square Dalitz
plot position. The calculation is

$\frac{\mathcal{B}({{{B}^{-}}}\rightarrow{{D}^{+}}{{K}^{-}}{{\pi}^{-}})}{\mathcal{B}({{{B}^{-}}}\rightarrow{{D}^{+}}{{\pi}^{-}}{{\pi}^{-}})}=\frac{N^{\rm
corr}({{{B}^{-}}}\rightarrow{{D}^{+}}{{K}^{-}}{{\pi}^{-}})}{N^{\rm
corr}({{{B}^{-}}}\rightarrow{{D}^{+}}{{\pi}^{-}}{{\pi}^{-}})}\,,$ (1)

where $N^{\rm corr}=\sum_{i}W_{i}/\epsilon_{i}$ is the efficiency-corrected
yield. The index $i$ sums over all candidates in the data sample and $W_{i}$
is the signal weight for each candidate, which is determined from the fits
described in Sec. 4.1 and shown in Figs. 1 and 2, using the sPlot technique
[32]. Each fit is performed simultaneously to decays in the TOS and TIS-only
categories. The efficiency of candidate $i$, $\epsilon_{i}$, is obtained
separately for each trigger subsample as described in Sec. 4.2.

###  4.1 Determination of signal and background yields

The candidates that survive the selection requirements are comprised of signal
decays and various categories of background. Combinatorial background arises
from random combinations of tracks (possibly including a real
${{D}^{+}}\rightarrow{{K}^{-}}{{\pi}^{+}}{{\pi}^{+}}$ decay). Partially
reconstructed backgrounds originate from $b$ hadron decays with additional
particles that are not part of the reconstructed decay chain. Misidentified
decays also originate from $b$ hadron decays, but where one of the final state
particles has been incorrectly identified (e.g. a pion as a kaon). The signal
(normalisation channel) and background yields are obtained from unbinned
maximum likelihood fits to the ${{D}^{+}}{{K}^{-}}{{\pi}^{-}}$
(${{D}^{+}}{{\pi}^{-}}{{\pi}^{-}}$) invariant mass distributions.

Both the ${{{B}^{-}}}\rightarrow{{D}^{+}}{{K}^{-}}{{\pi}^{-}}$ and
${{{B}^{-}}}\rightarrow{{D}^{+}}{{\pi}^{-}}{{\pi}^{-}}$ signal shapes are
modelled by the sum of two Crystal Ball (CB) functions [37] with a common mean
and tails on opposite sides, where the high-mass tail accounts for non-
Gaussian reconstruction effects. The ratio of widths of the CB shapes and the
relative normalisation of the narrower CB shape are constrained within their
uncertainties to the values found in fits to simulated signal samples. The
tail parameters of the CB shapes are also fixed to those found in simulation.

The combinatorial backgrounds in both ${{D}^{+}}{{K}^{-}}{{\pi}^{-}}$ and
${{D}^{+}}{{\pi}^{-}}{{\pi}^{-}}$ samples are modelled with linear functions;
the slope of this function is allowed to differ between the two trigger
subsamples. The decay ${{{B}^{-}}}\rightarrow{{D}^{*+}}{{K}^{-}}{{\pi}^{-}}$
is a partially reconstructed background for ${{D}^{+}}{{K}^{-}}{{\pi}^{-}}$
candidates, where the ${{D}^{*+}}$ decays to either ${{D}^{+}}\gamma$ or
${{D}^{+}}{{\pi}^{0}}$ and the neutral particle is not reconstructed.
Similarly the decay ${{{B}^{-}}}\rightarrow{{D}^{*+}}{{\pi}^{-}}{{\pi}^{-}}$
forms a partially reconstructed background to the
${{D}^{+}}{{\pi}^{-}}{{\pi}^{-}}$ final state. These are modelled with non-
parametric shapes determined from simulated samples. The shapes are
characterised by a sharp edge around $100\mathrm{\,Me\kern-1.00006ptV}$ below
the $B$ peak, where the exact position of the edge depends on properties of
the decay including the ${{D}^{*+}}$ polarisation. The fit quality improves
when the shape is allowed to be offset by a small shift that is determined
from the data.

Most potential sources of misidentified backgrounds have broad ${B}$ candidate
invariant mass distributions, and hence are absorbed in the combinatorial
background component in the fit. The decays ${{{B}^{-}}}\rightarrow
D^{(*)+}{{\pi}^{-}}{{\pi}^{-}}$ and
${{{B}^{-}}}\rightarrow{{D}^{+}_{s}}{{K}^{-}}{{\pi}^{-}}$, however, give
distinctive shapes in the mass distribution of ${{D}^{+}}{{K}^{-}}{{\pi}^{-}}$
candidates. For ${{D}^{+}}{{\pi}^{-}}{{\pi}^{-}}$ candidates the only
significant misidentified background contribution is from
${{{B}^{-}}}\rightarrow D^{(*)+}{{K}^{-}}{{\pi}^{-}}$ decays. The
misidentified background shapes are also modelled with non-parametric shapes
determined from simulated samples.

The simulated samples used to obtain signal and background shapes are
generated with flat distributions in the phase space of their SDPs. For
${{{B}^{-}}}\rightarrow{{D}^{+}}{{\pi}^{-}}{{\pi}^{-}}$ and
${{{B}^{-}}}\rightarrow{{D}^{*+}}{{\pi}^{-}}{{\pi}^{-}}$ decays, accurate
models of the distributions across the SDP are known [1, 2], so the simulated
samples are reweighted using the
${{{B}^{-}}}\rightarrow{{D}^{+}}{{\pi}^{-}}{{\pi}^{-}}$ data sample; this
affects the shape of the misidentified background component in the fit to the
${{D}^{+}}{{K}^{-}}{{\pi}^{+}}$ sample. Additionally, the ${{D}^{+}}$ and
${{D}^{*+}}$ portions of this background are combined according to their known
branching fractions. All of the shapes, except for that of the combinatorial
background, are common between the two trigger subsamples in each fit, but the
signal and background yields in the subsamples are independent. In total there
are 15 free parameters in the fit to the ${{D}^{+}}{{\pi}^{-}}{{\pi}^{-}}$
sample: yields in each subsample for signal, combinatorial,
${{{B}^{-}}}\rightarrow D^{(*)+}{{K}^{-}}{{\pi}^{-}}$ and
${{{B}^{-}}}\rightarrow{{D}^{*+}}{{\pi}^{-}}{{\pi}^{-}}$ backgrounds; the
combinatorial slope in each subsample; the double CB peak position, the width
of the narrower CB, the ratio of CB widths and the fraction of entries in the
narrower CB shape; and the shift parameter of the partially reconstructed
background. The result of the ${{D}^{+}}{{\pi}^{-}}{{\pi}^{-}}$ fit is shown
in Fig. 1 for both trigger subsamples and gives a combined signal yield of
approximately 49 000 decays. Component yields are given in Table 2.

Figure 1: Results of the fit to the ${{{B}^{-}}}\rightarrow{{D}^{+}}{{\pi}^{-}}{{\pi}^{-}}$ candidate invariant mass distribution for the (left) TOS and (right) TIS-only subsamples. Data points are shown in black, the full fitted model as solid blue lines and the components as shown in the legend. Table 2: Yields of the various components in the fit to ${{{B}^{-}}}\rightarrow{{D}^{+}}{{\pi}^{-}}{{\pi}^{-}}$ candidate invariant mass distribution. Component | TOS | TIS-only
---|---|---
$N({{{B}^{-}}}\rightarrow{{D}^{+}}{{\pi}^{-}}{{\pi}^{-}})$ | $29\,190\pm 204$ | $19\,416\pm 159$
$N({{{B}^{-}}}\rightarrow D^{(*)+}{{K}^{-}}{{\pi}^{-}})$ | $\phantom{29\,}807\pm 123$ | $\phantom{19\,}401\pm 84\phantom{9}$
$N({{{B}^{-}}}\rightarrow{{D}^{*+}}{{\pi}^{-}}{{\pi}^{-}})$ | $12\,120\pm 115$ | $\phantom{1\,}8551\pm 96\phantom{9}$
$N(\rm{comb.\ bkgd.})$ | $\phantom{29\,}784\pm 54\phantom{4}$ | $\phantom{9\,4}746\pm 47\phantom{9}$

There are a total of 17 free parameters in the fit to the
${{D}^{+}}{{K}^{-}}{{\pi}^{-}}$ sample: yields in each subsample for signal,
combinatorial, ${{{B}^{-}}}\rightarrow{{D}^{*+}}{{K}^{-}}{{\pi}^{-}}$,
${{{B}^{-}}}\rightarrow{{D}^{+}_{s}}{{K}^{-}}{{\pi}^{-}}$ and
${{{B}^{-}}}\rightarrow D^{(*)+}{{\pi}^{-}}{{\pi}^{-}}$ backgrounds; the
combinatorial slope in each subsample; the same signal shape parameters as for
the ${{D}^{+}}{{\pi}^{-}}{{\pi}^{-}}$ fit; and the shift parameter of the
partially reconstructed background. Figure 2 shows the result of the
${{D}^{+}}{{K}^{-}}{{\pi}^{-}}$ fit for the two trigger subsamples that yield
a total of approximately 2000
${{{B}^{-}}}\rightarrow{{D}^{+}}{{K}^{-}}{{\pi}^{-}}$ decays. The yields of
all fit components are shown in Table 3. The statistical signal significance,
estimated in the conventional way from the change in negative log-likelihood
from the fit when the signal component is removed, is in excess of $60$
standard deviations ($\sigma$).

Figure 2: Results of the fit to the ${{{B}^{-}}}\rightarrow{{D}^{+}}{{K}^{-}}{{\pi}^{-}}$ candidate invariant mass distribution for the (left) TOS and (right) TIS-only subsamples. Data points are shown in black, the full fitted model as solid blue lines and the components as shown in the legend. Table 3: Yields of the various components in the fit to ${{{B}^{-}}}\rightarrow{{D}^{+}}{{K}^{-}}{{\pi}^{-}}$ candidate invariant mass distribution. Component | TOS | TIS-only
---|---|---
$N({{{B}^{-}}}\rightarrow{{D}^{+}}{{K}^{-}}{{\pi}^{-}})$ | $1112\pm 37$ | $891\pm 32$
$N({{{B}^{-}}}\rightarrow D^{(*)+}{{\pi}^{-}}{{\pi}^{-}})$ | $\phantom{1}114\pm 34$ | $\phantom{8}23\pm 27$
$N({{{B}^{-}}}\rightarrow{{D}^{+}_{s}}{{K}^{-}}{{\pi}^{-}})$ | $\phantom{11}69\pm 17$ | $\phantom{8}40\pm 15$
$N({{{B}^{-}}}\rightarrow{{D}^{*+}}{{K}^{-}}{{\pi}^{-}})$ | $\phantom{1}518\pm 26$ | $361\pm 21$
$N(\rm{comb.\ bkgd.})$ | $\phantom{1}238\pm 38$ | $253\pm 36$

###  4.2 Signal efficiency

Since both ${{{B}^{-}}}\rightarrow{{D}^{+}}{{K}^{-}}{{\pi}^{-}}$ and
${{{B}^{-}}}\rightarrow{{D}^{+}}{{\pi}^{-}}{{\pi}^{-}}$ decays have non-
trivial DP distributions, it is necessary to understand the variation of the
efficiency across the phase space. Since, moreover, the efficiency variation
tends to be strongest close to the kinematic boundaries of the conventional
Dalitz plot, it is convenient to model these effects in terms of the SDP
defined by variables $m^{\prime}$ and $\theta^{\prime}$ which are valid in the
range 0 to 1 and are given for the ${{D}^{+}}{{K}^{-}}{{\pi}^{-}}$ case by

$\mbox{$m^{\prime}$}\equiv\frac{1}{\pi}\arccos\left(2\frac{m({{D}^{+}}{{\pi}^{-}})-m^{\rm
min}_{{{D}^{+}}{{\pi}^{-}}}}{m^{\rm max}_{{{D}^{+}}{{\pi}^{-}}}-m^{\rm
min}_{{{D}^{+}}{{\pi}^{-}}}}-1\right)\hskip 28.45274pt{\rm and}\hskip
28.45274pt\mbox{$\theta^{\prime}$}\equiv\frac{1}{\pi}\theta({{D}^{+}}{{\pi}^{-}})\,,$
(2)

where $m^{\rm max}_{{{D}^{+}}{{\pi}^{-}}}=m_{{{{B}^{-}}}}-m_{{{K}^{-}}}$ and
$m^{\rm min}_{{{D}^{+}}{{\pi}^{-}}}=m_{{{D}^{+}}}+m_{{{\pi}^{-}}}$ are the
kinematic boundaries of $m({{D}^{+}}{{\pi}^{-}})$ allowed in the
${{{B}^{-}}}\rightarrow{{D}^{+}}{{K}^{-}}{{\pi}^{-}}$ decay and
$\theta({{D}^{+}}{{\pi}^{-}})$ is the helicity angle of the
${{D}^{+}}{{\pi}^{-}}$ system (the angle between the ${{K}^{-}}$ and the
${{D}^{+}}$ meson momenta in the ${{D}^{+}}{{\pi}^{-}}$ rest frame). For the
${{D}^{+}}{{\pi}^{-}}{{\pi}^{-}}$ case, $m^{\prime}$ and $\theta^{\prime}$ are
defined in terms of the ${{\pi}^{-}}{{\pi}^{-}}$ mass and helicity angle,
respectively, since with this choice only the region of the SDP with
$\mbox{$\theta^{\prime}$}({{\pi}^{-}}{{\pi}^{-}})<0.5$ is populated due to the
symmetry of the two pions in the final state.

Efficiency variation across the SDP is caused by the detector acceptance and
by trigger, selection and PID requirements. The efficiency variation is
evaluated for both ${{D}^{+}}{{K}^{-}}{{\pi}^{-}}$ and
${{D}^{+}}{{\pi}^{-}}{{\pi}^{-}}$ final states 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. [5]. The efficiency functions are fitted with two-dimensional cubic
splines to smooth out statistical fluctuations due to limited sample size.

The efficiency is studied separately for the TOS and TIS-only categories. The
efficiency maps for each trigger subsample are shown for
${{{B}^{-}}}\rightarrow{{D}^{+}}{{K}^{-}}{{\pi}^{-}}$ decays in Fig. 3.
Regions of relatively high efficiency are seen where all decay products have
comparable momentum in the $B$ rest frame; the efficiency drops sharply in
regions with a low momentum bachelor track due to geometrical effects. The
efficiency maps are used to calculate the ratio of branching fractions and
also as inputs to the ${{D}^{+}}{{K}^{-}}{{\pi}^{-}}$ Dalitz plot fit.

Figure 3: Signal efficiency across the SDP for (left) TOS and (right) TIS-only
${{{B}^{-}}}\rightarrow{{D}^{+}}{{K}^{-}}{{\pi}^{-}}$ decays. The relative
uncertainty at each point is typically $5\,\%$.

###  4.3 Systematic uncertainties

Table 4 summarises the systematic uncertainties on the measurement of the
ratio of branching fractions. Selection effects cancel in the ratio of
branching fractions, except for inefficiency due to the ${{\mathchar
28931\relax}^{+}_{c}}$ veto. The invariant mass fits are repeated both with a
wider veto ($2270$–$2310\mathrm{\,Me\kern-1.00006ptV}$) and with no veto, and
changes in the yields are used to assign a relative systematic uncertainty of
$0.2\,\%$.

To estimate the uncertainty arising from the choice of invariant mass fit
model, the ${{D}^{+}}{{K}^{-}}{{\pi}^{-}}$ mass fit is varied by replacing the
signal shape with the sum of two bifurcated Gaussian functions, removing the
smoothing of the non-parametric functions, using exponential and second-order
polynomial functions to describe the combinatorial background, varying fixed
parameters within their uncertainties and varying the binning of histograms
used to reweight the simulated background samples. For the
${{D}^{+}}{{\pi}^{-}}{{\pi}^{-}}$ fit the same variations are made. The
relative changes in the yields are summed in quadrature to give a relative
systematic uncertainty on the ratio of branching fractions of $2.0\,\%$.

The systematic uncertainty due to PID is estimated by accounting for three
sources: the intrinsic uncertainty of the calibration ($1.0\,\%$); possible
differences in the kinematics of tracks in simulated samples, used to reweight
the calibration data samples, to those in the data ($1.7\,\%$); the
granularity of the binning in the reweighting procedure ($0.7\,\%$). Combining
these in quadrature, the total relative systematic uncertainty from PID is
$2.1\,\%$.

The bins of the efficiency maps are varied within uncertainties to make 100
new efficiency maps, for both ${{D}^{+}}{{K}^{-}}{{\pi}^{-}}$ and
${{D}^{+}}{{\pi}^{-}}{{\pi}^{-}}$ modes. The efficiency-corrected yields are
evaluated for each new map and their distributions are fitted with Gaussian
functions. The widths of these are used to assign a relative systematic
uncertainty on the ratio of branching fractions of $0.8\,\%$.

Table 4: Relative systematic uncertainties on the measurement of the ratio of branching fractions for ${{{B}^{-}}}\rightarrow{{D}^{+}}{{K}^{-}}{{\pi}^{-}}$ and ${{{B}^{-}}}\rightarrow{{D}^{+}}{{\pi}^{-}}{{\pi}^{-}}$ decays.  Source | Uncertainty (%)
---|---
${{\mathchar 28931\relax}^{+}_{c}}$ veto | 0.2
Fit model | 2.0
Particle identification | 2.1
Efficiency modelling | 0.8
Total | 3.0

A number of additional cross-checks are performed to test the branching
fraction result. The neural network and PID requirements are both tightened
and loosened. The data sample is divided by dipole magnet polarity and year of
data taking. The branching fraction is also calculated separately for TOS and
TIS-only events. All cross-checks give consistent results.

###  4.4 Results

The ratio of branching fractions is found to be

$\frac{\mathcal{B}({{{B}^{-}}}\rightarrow{{D}^{+}}{{K}^{-}}{{\pi}^{-}})}{\mathcal{B}({{{B}^{-}}}\rightarrow{{D}^{+}}{{\pi}^{-}}{{\pi}^{-}})}=0.0720\pm
0.0019\pm 0.0021\,,$

where the first uncertainty is statistical and the second systematic. The
statistical uncertainty includes contributions from the event weighting used
in Eq. (1) and from the shape parameters that are allowed to vary in the fit
[38]. The world average value of
$\mathcal{B}({{{B}^{-}}}\rightarrow{{D}^{+}}{{\pi}^{-}}{{\pi}^{-}})=(1.07\pm
0.05)\times 10^{-3}$ [9] assumes that ${{{B}^{+}}}{{{B}^{-}}}$ and
${{B}^{0}}{{\kern 1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}}$ are produced
equally in the decay of the $\Upsilon(4S)$ resonance. Using
$\Gamma(\Upsilon(4S)\rightarrow{{{B}^{+}}}{{{B}^{-}}})/\Gamma(\Upsilon(4S)\rightarrow{{B}^{0}}{{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}})=1.055\pm 0.025$ [9] gives a
corrected value of
$\mathcal{B}({{{B}^{-}}}\rightarrow{{D}^{+}}{{\pi}^{-}}{{\pi}^{-}})=(1.01\pm
0.05)\times 10^{-3}$. This allows the branching fraction of
${{{B}^{-}}}\rightarrow{{D}^{+}}{{K}^{-}}{{\pi}^{-}}$ decays to be determined
as

$\mathcal{B}({{{B}^{-}}}\rightarrow{{D}^{+}}{{K}^{-}}{{\pi}^{-}})=(7.31\pm
0.19\pm 0.22\pm 0.39)\times 10^{-5}\,,$

where the third uncertainty is from
$\mathcal{B}({{{B}^{-}}}\rightarrow{{D}^{+}}{{\pi}^{-}}{{\pi}^{-}})$. This
measurement represents the first observation of the
${{{B}^{-}}}\rightarrow{{D}^{+}}{{K}^{-}}{{\pi}^{-}}$ decay.

##  5 Study of angular moments

To investigate which amplitudes should be included in the DP analysis of
${{{B}^{-}}}\rightarrow{{D}^{+}}{{K}^{-}}{{\pi}^{-}}$ decays, a study of its
angular moments is performed. Such an analysis is particularly useful for
${{{B}^{-}}}\rightarrow{{D}^{+}}{{K}^{-}}{{\pi}^{-}}$ decays because resonant
contributions are only expected to appear in the ${{D}^{+}}{{\pi}^{-}}$
combination, and therefore the distributions should be free of effects from
reflections that make them more difficult to interpret.

The analysis is performed by calculating moments from the Legendre polynomials
$P_{L}$ of order up to $2J_{\rm max}$, where $J_{\rm max}$ is the maximum spin
of the resonances considered. Each candidate is weighted according to its
value of $P_{L}\left(\cos\theta({{D}^{+}}{{\pi}^{-}})\right)$ with an
efficiency correction applied, and background contributions subtracted. The
results for $J_{\rm max}=3$ are shown in Fig. 4 for the ${{D}^{+}}{{\pi}^{-}}$
invariant mass range $2.0$–$3.0\mathrm{\,Ge\kern-1.00006ptV}$. The
distributions of $\left\langle P_{5}\right\rangle$ and $\left\langle
P_{6}\right\rangle$ are compatible with being flat, which implies that there
are no significant spin 3 contributions. Considering only contributions up to
spin 2, the following expressions are used to interpret Fig. 4

$\displaystyle\left\langle P_{0}\right\rangle\propto\,$
$\displaystyle\left|h_{0}\right|{}^{2}+\left|h_{1}\right|{}^{2}+\left|h_{2}\right|{}^{2}\,,$
(3) $\displaystyle\left\langle P_{1}\right\rangle\propto\,$
$\displaystyle\frac{2}{\sqrt{3}}\left|h_{0}\right|\left|h_{1}\right|\cos\left(\delta_{0}-\delta_{1}\right)+\frac{4}{\sqrt{15}}\left|h_{1}\right|\left|h_{2}\right|\cos\left(\delta_{1}-\delta_{2}\right)\,,$
(4) $\displaystyle\left\langle P_{2}\right\rangle\propto\,$
$\displaystyle\frac{2}{\sqrt{5}}\left|h_{0}\right|\left|h_{2}\right|\cos\left(\delta_{0}-\delta_{2}\right)+\frac{2}{5}\left|h_{1}\right|{}^{2}+\frac{2}{7}\left|h_{2}\right|{}^{2}\,,$
(5) $\displaystyle\left\langle P_{3}\right\rangle\propto\,$
$\displaystyle\frac{6}{7}\sqrt{\frac{3}{5}}\left|h_{1}\right|\left|h_{2}\right|\cos\left(\delta_{1}-\delta_{2}\right)\,,$
(6) $\displaystyle\left\langle P_{4}\right\rangle\propto\,$
$\displaystyle\frac{2}{7}\left|h_{2}\right|{}^{2}\,,$ (7)

where S-, P- and D-wave contributions are denoted by amplitudes
$h_{j}e^{i\delta_{j}}$ ($j=0,1,2$ respectively). The $D^{*}_{2}(2460)^{0}$
resonance is clearly seen in the $\left\langle P_{4}\right\rangle$
distribution of Fig. 4(e). The distribution of $\left\langle
P_{3}\right\rangle$ shows interference between spin 1 and 2 contributions,
indicating the presence of a broad, possibly nonresonant, spin 1 contribution
at low $m({{D}^{+}}{{\pi}^{-}})$. The difference in shape between
$\left\langle P_{1}\right\rangle$ and $\left\langle P_{3}\right\rangle$ shows
interference between spin 1 and 0 indicating that a broad spin 0 component is
similarly needed.

Figure 4: The first seven Legendre-polynomial weighted moments for background-
subtracted and efficiency-corrected
${{{B}^{-}}}\rightarrow{{D}^{+}}{{K}^{-}}{{\pi}^{-}}$ data (black points) as a
function of $m({{D}^{+}}{{\pi}^{-}})$ in the range
$2.0$–$3.0\mathrm{\,Ge\kern-0.90005ptV}$. Candidates from both TOS and TIS-
only subsamples are included. The blue line shows the result of the DP fit
described in Sec. 7.

##  6 Dalitz plot analysis formalism

A Dalitz plot [39] is a representation of the phase-space for a three-body
decay in terms of two of the three possible two-body invariant mass squared
combinations. In ${{{B}^{-}}}\rightarrow{{D}^{+}}{{K}^{-}}{{\pi}^{-}}$ decays,
resonances are expected in the $m^{2}({{D}^{+}}{{\pi}^{-}})$ combination,
therefore this and $m^{2}({{D}^{+}}{{K}^{-}})$ are chosen to define the DP
axes. For a fixed ${{{B}^{-}}}$ mass, all other relevant kinematic quantities
can be calculated from these two invariant mass squared combinations.

The complex decay amplitude is described using the isobar approach [40, 41,
42], where the total amplitude is calculated as a coherent sum of amplitudes
from resonant and nonresonant intermediate processes. The total amplitude is
then given by

${\cal
A}\left(m^{2}({{D}^{+}}{{\pi}^{-}}),m^{2}({{D}^{+}}{{K}^{-}})\right)=\sum_{j=1}^{N}c_{j}F_{j}\left(m^{2}({{D}^{+}}{{\pi}^{-}}),m^{2}({{D}^{+}}{{K}^{-}})\right)\,,$
(8)

where $c_{j}$ are complex coefficients giving the relative contribution of
each intermediate process. The
$F_{j}\left(m^{2}({{D}^{+}}{{\pi}^{-}}),m^{2}({{D}^{+}}{{K}^{-}})\right)$
terms contain the resonance dynamics, which are composed of several terms and
are normalised such that the integral of the squared magnitude over the DP is
unity for each term. For a ${{D}^{+}}{{\pi}^{-}}$ resonance

$F\left(m^{2}({{D}^{+}}{{\pi}^{-}}),m^{2}({{D}^{+}}{{K}^{-}})\right)=R\left(m({{D}^{+}}{{\pi}^{-}})\right)\times
X(|\vec{p}\,|\,r_{\rm BW})\times X(|\vec{q}\,|\,r_{\rm BW})\times
T(\vec{p},\vec{q}\,)\,,$ (9)

where the functions $R$, $X$ and $T$ are described below, and $\vec{p}$ and
$\vec{q}$ are the bachelor particle momentum and the momentum of one of the
resonance daughters, respectively, both evaluated in the
${{D}^{+}}{{\pi}^{-}}$ rest frame.

The $X(z)$ terms, where $z=|\vec{q}\,|\,r_{\rm BW}$ or $|\vec{p}\,|\,r_{\rm
BW}$, are Blatt–Weisskopf barrier factors [43] with barrier radius $r_{\rm
BW}$, and are given by

$\begin{array}[]{rcl}L=0\ :\ X(z)&=&1\,,\\\ L=1\ :\
X(z)&=&\sqrt{\frac{1+z_{0}^{2}}{1+z^{2}}}\,,\\\ L=2\ :\
X(z)&=&\sqrt{\frac{z_{0}^{4}+3z_{0}^{2}+9}{z^{4}+3z^{2}+9}}\,,\\\ L=3\ :\
X(z)&=&\sqrt{\frac{z_{0}^{6}+6z_{0}^{4}+45z_{0}^{2}+225}{z^{6}+6z^{4}+45z^{2}+225}}\,,\end{array}$
(10)

where $z_{0}$ is the value of $z$ when the invariant mass is equal to the pole
mass of the resonance and $L$ is the spin of the resonance. For a
${{D}^{+}}{{\pi}^{-}}$ resonance, since the ${{B}^{-}}$ meson has zero spin,
$L$ is also the orbital angular momentum between the resonance and the kaon.
The barrier radius, $r_{\rm BW}$, is taken to be
$4.0\mathrm{\,Ge\kern-1.00006ptV}^{-1}\approx 0.8\rm\,fm$ [44, 5] for all
resonances.

The terms $T(\vec{p},\vec{q})$ describe the angular probability distribution
and are given in the Zemach tensor formalism [45, 46] by

$\begin{array}[]{rcl}L=0\ :\ T(\vec{p},\vec{q})&=&1\,,\\\ L=1\ :\
T(\vec{p},\vec{q})&=&-\,2\,\vec{p}\cdot\vec{q}\,,\\\ L=2\ :\
T(\vec{p},\vec{q})&=&\frac{4}{3}\left[3(\vec{p}\cdot\vec{q}\,)^{2}-(|\vec{p}\,||\vec{q}\,|)^{2}\right]\,,\\\
L=3\ :\
T(\vec{p},\vec{q})&=&-\,\frac{24}{15}\left[5(\vec{p}\cdot\vec{q}\,)^{3}-3(\vec{p}\cdot\vec{q}\,)(|\vec{p}\,||\vec{q}\,|)^{2}\right]\,,\end{array}$
(11)

which 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 function $R\left(m({{D}^{+}}{{\pi}^{-}})\right)$ of Eq. (9) is the mass
lineshape. The resonant contributions considered in the DP model are described
by the relativistic Breit–Wigner (RBW) function

$R(m)=\frac{1}{(m_{0}^{2}-m^{2})-i\,m_{0}\Gamma(m)}\,,$ (12)

where the mass-dependent decay width is

$\Gamma(m)=\Gamma_{0}\left(\frac{q}{q_{0}}\right)^{2L+1}\left(\frac{m_{0}}{m}\right)X^{2}(q\,r_{\rm
BW})\,,$ (13)

where $q_{0}$ is the value of $q=|\vec{q}\,|$ for $m=m_{0}$. Virtual
contributions, from resonances with pole masses outside the kinematically
accessible region of the phase space, can also be modelled by this shape with
one modification: the pole mass $m_{0}$ is replaced with $m_{0}^{\rm{eff}}$, a
mass in the kinematically allowed region, in the calculation of the parameter
$q_{0}$. This effective mass is defined by the ad hoc formula [5]

$m_{0}^{\rm{eff}}(m_{0})=m^{\rm{min}}+(m^{\rm{max}}-m^{\rm{min}})\left(1+\tanh\left(\frac{m_{0}-\frac{m^{\rm{min}}+m^{\rm{max}}}{2}}{m^{\rm{max}}-m^{\rm{min}}}\right)\right)\,,$
(14)

where $m^{\rm{max}}$ and $m^{\rm{min}}$ are the upper and lower limits of the
kinematically allowed range, respectively. For virtual contributions, only the
tail of the RBW function enters the Dalitz plot.

Given the large available phase-space in the $B$ decay, it is possible to have
nonresonant amplitudes (i.e. contributions that are not from any known
resonance, including virtual states) that vary across the Dalitz plot. A model
that has been found to describe well nonresonant contributions in several $B$
decay DP analyses is an exponential form factor (EFF) [47],

$R(m)=e^{-\alpha m^{2}}\,,$ (15)

where $m$ is a two-body (in this case $D\pi$) invariant mass and $\alpha$ is a
shape parameter that must be determined from the data.

Neglecting reconstruction effects, the DP probability density function would
be

${\cal{P}}_{\rm
phys}\left(m^{2}({{D}^{+}}{{\pi}^{-}}),m^{2}({{D}^{+}}{{K}^{-}})\right)=\frac{|{\cal
A}\left(m^{2}({{D}^{+}}{{\pi}^{-}}),m^{2}({{D}^{+}}{{K}^{-}})\right)|^{2}}{\int\\!\\!\int_{\rm
DP}~{}{|{\cal
A}|^{2}}~{}dm^{2}({{D}^{+}}{{\pi}^{-}})\,dm^{2}({{D}^{+}}{{K}^{-}})}\,,$ (16)

where the dependence of ${\cal A}$ on the DP position has been suppressed in
the denominator for brevity. The complex coefficients, given by $c_{j}$ in Eq.
(8), are the primary results of most Dalitz plot analyses. However, these
depend on the choice of normalisation, phase convention and amplitude
formalism in each analysis. Fit fractions and interference fit fractions are
also reported as these provide a convention-independent method to allow
meaningful comparisons of results. The fit fraction is defined as the integral
of the amplitude for a single component squared divided by that of the
coherent matrix element squared for the complete Dalitz plot,

${\it FF}_{j}=\frac{\int\\!\\!\int_{\rm
DP}\left|c_{j}F_{j}\left(m^{2}({{D}^{+}}{{\pi}^{-}}),m^{2}({{D}^{+}}{{K}^{-}})\right)\right|^{2}~{}dm^{2}({{D}^{+}}{{\pi}^{-}})\,dm^{2}({{D}^{+}}{{K}^{-}})}{\int\\!\\!\int_{\rm
DP}\left|{\cal
A}\right|^{2}~{}dm^{2}({{D}^{+}}{{\pi}^{-}})\,dm^{2}({{D}^{+}}{{K}^{-}})}\,.$
(17)

The fit fractions do not necessarily sum to unity due to the potential
presence of net constructive or destructive interference, described by
interference fit fractions defined for $i<j$ only by

${\it FF}_{ij}=\frac{\int\\!\\!\int_{\rm
DP}2\,\mathcal{R}e\left[c_{i}c_{j}^{*}F_{i}F_{j}^{*}\right]~{}dm^{2}({{D}^{+}}{{\pi}^{-}})\,dm^{2}({{D}^{+}}{{K}^{-}})}{\int\\!\\!\int_{\rm
DP}\left|{\cal
A}\right|^{2}~{}dm^{2}({{D}^{+}}{{\pi}^{-}})\,dm^{2}({{D}^{+}}{{K}^{-}})}\,,$
(18)

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

##  7 Dalitz plot fit

The Laura++ [48] package is used to perform the Dalitz plot fit, with the two
trigger subsamples fitted simultaneously using the Jfit method [49]. The two
subsamples have separate signal and background yields, efficiency maps and
background SDP distributions, but all parameters of the signal model are
common. The likelihood function that is used is

${\cal L}=\prod_{i}^{N_{c}}\Bigg{[}\sum_{k}N_{k}{\cal
P}_{k}\left(m^{2}_{i}({{D}^{+}}{{\pi}^{-}}),m^{2}_{i}({{D}^{+}}{{K}^{-}})\right)\Bigg{]}\,,$
(19)

where the index $i$ runs over $N_{c}$ candidates, while $k$ distinguishes the
signal and background components with $N_{k}$ the yield in each component. The
probability density function for signal events, ${\cal P}_{\rm sig}$, is given
by Eq. (16) where the $|{\cal
A}\left(m^{2}({{D}^{+}}{{\pi}^{-}}),m^{2}({{D}^{+}}{{K}^{-}})\right)|^{2}$
terms are multiplied by the efficiency function described in Sec. 4.2. The
mass resolution is approximately $2.4\mathrm{\,Me\kern-1.00006ptV}$, which is
much lower than the width of the narrowest contribution to the Dalitz plot
($\sim 50\mathrm{\,Me\kern-1.00006ptV}$); therefore, this has negligible
effect on the likelihood and is not considered further.

The signal and background yields that enter the Dalitz plot fit are taken from
the mass fit described in Sec. 4.1. Only candidates in the signal region,
defined as $\pm 2.5\sigma$ around the $B$ signal peak, where $\sigma$ is the
width of the peak, are used in the Dalitz plot fit. Within this region, in the
TOS subsample the result of the $B$ candidate invariant mass fit corresponds
to yields of $1060\pm 35$, $37\pm 6$, $26\pm 8$ and $16\pm 4$ in the signal,
combinatorial background, $D^{(*)+}{{\pi}^{-}}{{\pi}^{-}}$ and
${{D}^{+}_{s}}{{K}^{-}}{{\pi}^{-}}$ components, respectively. The equivalent
yields in the TIS-only subsample are $849\pm 30$, $39\pm 6$, $5\pm 5$ and
$9\pm 3$ candidates. The contribution from ${{D}^{*+}}{{K}^{-}}{{\pi}^{-}}$
decays is negligible in the signal window. The distributions of the candidates
in the signal region over the DP and SDP are shown in Fig. 5.

Figure 5: Distribution of
${{{B}^{-}}}\rightarrow{{D}^{+}}{{K}^{-}}{{\pi}^{-}}$ candidates in the signal
region over (left) the DP and (right) the SDP. Candidates from both TOS and
TIS-only subsamples are included.

The SDP distributions of the $D^{(*)+}{{\pi}^{-}}{{\pi}^{-}}$ and
${{D}^{+}_{s}}{{K}^{-}}{{\pi}^{-}}$ background sources are obtained from
simulated samples using the same procedures as described for their invariant
mass distributions in Sec. 4.1. The distribution of combinatorial background
events is modelled by considering ${{D}^{+}}{{K}^{-}}{{\pi}^{-}}$ candidates
in the sideband high-mass range $5500$–$5800\mathrm{\,Me\kern-1.00006ptV}$,
with contributions from $D^{(*)+}{{\pi}^{-}}{{\pi}^{-}}$ in this region
subtracted. The dependence of the SDP distribution on $B$ candidate mass was
investigated and found to be negligible. The SDP distributions of these
backgrounds are shown in Fig. 6. These histograms are used to model the
background contributions in the Dalitz plot fit.

Figure 6: Square Dalitz plot distributions used in the Dalitz plot fit for
(top) combinatorial background, (middle) ${{{B}^{-}}}\rightarrow
D^{(*)+}{{\pi}^{-}}{{\pi}^{-}}$ decays and (bottom)
${{{B}^{-}}}\rightarrow{{D}^{+}_{s}}{{K}^{-}}{{\pi}^{-}}$ decays. Candidates
the TOS (TIS-only) subsamples are shown in the left (right) column.

Using the results of the moments analysis of Sec. 5 as a guide, the nominal
Dalitz plot fit model for
${{{B}^{-}}}\rightarrow{{D}^{+}}{{K}^{-}}{{\pi}^{-}}$ decays is determined by
considering several resonant, nonresonant and virtual amplitudes. Those that
do not contribute significantly and that do not aid the stability of the fit
are removed. Only natural spin-parity intermediate states are considered, as
unnatural spin-parity states do not decay to two pseudoscalars. The resulting
signal model, referred to below as the nominal DP model, consists of the seven
amplitudes shown in Table 5: three resonances, two virtual resonances and two
nonresonant terms. Parts of the model are known to be approximations. In
particular both S- and P-waves in the $D\pi$ system are modelled with
overlapping broad structures. The nominal model gives a better description of
the data than any of the alternative models considered; alternative models are
used to assign systematic uncertainties as discussed in Sec. 8.

Table 5: Signal contributions to the fit model, where parameters and uncertainties are taken from Ref. [9]. States labelled with subscript $v$ are virtual contributions.  Resonance | Spin | DP axis | Model | Parameters
---|---|---|---|---
$D^{*}_{0}(2400)^{0}$ | 0 | $m^{2}(D\pi)$ | RBW |  $m=2318\pm 29\mathrm{\,Me\kern-1.00006ptV}$, $\Gamma=267\pm 40\mathrm{\,Me\kern-1.00006ptV}$
$D^{*}_{2}(2460)^{0}$ | 2 | $m^{2}(D\pi)$ | RBW | Determined from data (see Table 6)
$D^{*}_{J}(2760)^{0}$ | 1 | $m^{2}(D\pi)$ | RBW
Nonresonant | 0 | $m^{2}(D\pi)$ | EFF | Determined from data (see text)
Nonresonant | 1 | $m^{2}(D\pi)$ | EFF
$D^{*}_{v}(2007)^{0}$ | 1 | $m^{2}(D\pi)$ | RBW |  $m=2006.98\pm 0.15\mathrm{\,Me\kern-1.00006ptV}$, $\Gamma=2.1\mathrm{\,Me\kern-1.00006ptV}$
$B^{*0}_{v}$ | 1 | $m^{2}(DK)$ | RBW |  $m=5325.2\pm 0.4\mathrm{\,Me\kern-1.00006ptV}$, $\Gamma=0.0\mathrm{\,Me\kern-1.00006ptV}$

The free parameters in the fit are the $c_{j}$ terms introduced in Eq. (8),
with the real and imaginary parts of these complex coefficients determined for
each amplitude in the fit model. The $D^{*}_{2}(2460)^{0}$ component, as the
reference amplitude, is the exception with real and imaginary parts fixed to 1
and 0, respectively. Fit fractions and interference fit fractions are derived
from these free parameters, as are the magnitudes and phases of the complex
coefficients. Statistical uncertainties for the derived parameters are
calculated using large samples of simulated pseudoexperiments to ensure that
non-trivial correlations are accounted for. Several other parameters are also
determined from the fit as described below.

In Dalitz plot fits it is common for the minimisation procedure to find local
minima of the likelihood function. To find the global minimum, the fit is
performed many times using randomised starting values for the complex
coefficients. In addition to the global minimum of the likelihood,
corresponding to the results reported below, several additional minima are
found. Two of these have negative log-likelihood (NLL) values close to that of
global minimum. The main differences between secondary minima and the global
minimum are the interference patterns in the $D\pi$ S- and P-waves, as shown
in App. A.

The shape parameters, defined in Eq. (15), for the nonresonant components are
determined from the fit to data to be $0.36\pm
0.03\mathrm{\,Ge\kern-1.00006ptV}^{-2}$ and $0.36\pm
0.04\mathrm{\,Ge\kern-1.00006ptV}^{-2}$ for the S-wave and P-wave,
respectively, where the uncertainties are statistical only. The mass and width
of the $D^{*}_{2}(2460)^{0}$ resonance are determined from the fit to improve
the fit quality. Since the mass and width of the $D^{*}_{J}(2760)^{0}$ state
have not been precisely determined by previous experiments, these parameters
are also allowed to vary in the fit. The masses and widths of the
$D^{*}_{2}(2460)^{0}$ and $D^{*}_{J}(2760)^{0}$ are reported in Table 6.

The spin of the $D^{*}_{J}(2760)^{0}$ state has not been determined
previously. Fits are performed with all values up to 3, and spin 1 is found to
be preferred with changes relative to the spin 0, 2 and 3 hypotheses of
$2\Delta{\rm NLL}=37.3,49.5$ and $48.2$ units, respectively. For comparison,
the value of $2\Delta{\rm NLL}$ obtained from a fit with the
$D^{*}_{1}(2760)^{0}$ state excluded is 75.0 units. The alternative models
discussed in Sec. 8 give very similar values and therefore do not affect the
conclusion that the $D^{*}_{J}(2760)^{0}$ state has spin 1.

Table 6: Masses and widths determined in the fit to data, with statistical uncertainties only. Resonance | Mass $(\mathrm{Me\kern-1.00006ptV})$ | Width $(\mathrm{Me\kern-1.00006ptV})$
---|---|---
$D^{*}_{2}(2460)^{0}$ | $2464.0\pm 1.4\phantom{1}$ | $43.8\pm 2.9$
$D^{*}_{J}(2760)^{0}$ | $2781\pm 18$ | $177\pm 32$

The values of the complex coefficients and fit fractions returned by the fit
are shown in Table 7. Results for the interference fit fractions are given in
App. B. The total fit fraction exceeds unity mostly due to interference
between the $D^{*}_{0}(2400)^{0}$ and S-wave nonresonant contributions.

Table 7: Complex coefficients and fit fractions determined from the Dalitz
plot fit. Uncertainties are statistical only.

|  | Isobar model coefficients
---|---|---
Resonance | Fit fraction (%) | Real part | Imaginary part | Magnitude | Phase
$D^{*}_{0}(2400)^{0}$ | $\phantom{1}8.3\pm 2.6$ | $-0.04\pm 0.07$ | $-0.51\pm 0.07$ | $0.51\pm 0.09$ | $-1.65\pm 0.16$
$D^{*}_{2}(2460)^{0}$ | $31.8\pm 1.5$ | $\phantom{-}1.00$ | $\phantom{-}0.00$ | $1.00$ | $\phantom{-}0.00$
$D^{*}_{1}(2760)^{0}$ | $\phantom{1}4.9\pm 1.2$ | $-0.32\pm 0.06$ | $-0.23\pm 0.07$ | $0.39\pm 0.05$ | $-2.53\pm 0.24$
S-wave nonresonant | $38.0\pm 7.4$ | $\phantom{-}0.93\pm 0.09$ | $-0.58\pm 0.08$ | $1.09\pm 0.09$ | $-0.56\pm 0.09$
P-wave nonresonant | $23.8\pm 5.6$ | $-0.43\pm 0.09$ | $\phantom{-}0.75\pm 0.09$ | $0.87\pm 0.09$ | $\phantom{-}2.09\pm 0.15$
$D^{*}_{v}(2007)^{0}$ | $\phantom{1}7.6\pm 2.3$ | $\phantom{-}0.16\pm 0.08$ | $\phantom{-}0.46\pm 0.09$ | $0.49\pm 0.07$ | $\phantom{-}1.24\pm 0.17$
$B^{*}_{v}$ | $\phantom{1}3.6\pm 1.9$ | $-0.07\pm 0.08$ | $\phantom{-}0.33\pm 0.07$ | $0.34\pm 0.06$ | $\phantom{-}1.78\pm 0.23$
Total fit fraction | $118.1$ |  |  |  |

The consistency of the fit model and the data is evaluated in several ways.
Numerous one-dimensional projections (including several shown below and those
shown in Sec. 5) show good agreement. A two-dimensional $\chi^{2}$ value is
determined by comparing the data and the fit model in $100$ equally populated
bins across the SDP. The pull, i.e. the difference between the data and fit
model divided by the uncertainty, is shown with this SDP binning in Fig. 7.
The $\chi^{2}$ value obtained is found to be within the bulk of the
distribution expected from simulated pseudoexperiments. Other unbinned fit
quality tests [50] also show acceptable agreement between the data and the fit
model.

Figure 7: Differences between the data SDP distribution and the fit model
across the SDP, in terms of the per-bin pull.

Figure 8 shows projections of the nominal fit model and the data onto
$m(D\pi)$, $m(DK)$ and $m(K\pi)$. Zooms are provided around the resonant
structures on $m(D\pi)$ in Fig. 9. Projections of the cosine of the helicity
angle of the $D\pi$ system are shown in Fig. 10. Good agreement is seen
between the data and the fit model.

Figure 8: Projections of the data and amplitude fit onto (a) $m(D\pi)$, (c)
$m(DK)$ and (e) $m(K\pi)$, with the same projections shown in (b), (d) and (f)
with a logarithmic $y$-axis scale. Components are described in the legend.

Figure 9: Projections of the data and amplitude fit onto $m(D\pi)$ in (a) the
threshold region, (b) the $D^{*}_{2}(2460)^{0}$ region and (c) the
$D^{*}_{1}(2760)^{0}$ region. Components are as shown in Fig. 8.

Figure 10: Projections of the data and amplitude fit onto the cosine of the
helicity angle for the $D\pi$ system in (a) the threshold region, (b) the
$D^{*}_{2}(2460)^{0}$ region and (c) the $D^{*}_{1}(2760)^{0}$ region.
Components are as shown in Fig. 8.

##  8 Systematic uncertainties

Sources of systematic uncertainty are divided into two categories:
experimental and model uncertainties. The sources of experimental systematic
uncertainty are: the signal and background yields in the signal region; the
SDP distributions of the background components; the efficiency variation
across the SDP; possible fit bias. The considered model uncertainties are: the
fixed parameters in the amplitude model; the addition or removal of marginal
amplitudes; the choice of models for the nonresonant contributions. The
systematic uncertainties from each source are combined in quadrature.

The signal and background yields in the signal region are determined from the
fit to the $B$ candidate invariant mass distribution, as described in Sec.
4.1. The uncertainty on each yield (including systematic uncertainty evaluated
as in Sec. 4.3) is calculated, and the yields varied accordingly in the DP
fit. The deviations from the nominal DP fit result are assigned as systematic
uncertainties.

The effect of imperfect knowledge of the background distributions over the SDP
is tested by varying the histograms used to model the shapes within their
statistical uncertainties. For $D^{(*)+}{{\pi}^{-}}{{\pi}^{-}}$ decays the
ratio of the ${{D}^{*+}}$ and ${{D}^{+}}$ contributions is varied. Where
applicable, the reweighting of the SDP distribution of the simulated samples
is removed.

The uncertainty related to the knowledge of the variation of efficiency across
the SDP is determined by varying the efficiency histograms before the spline
fit is performed. The central bin in each cell of $3\times 3$ bins is varied
by its statistical uncertainty and the surrounding bins in the cell are varied
by interpolation. This procedure accounts for possible correlations between
the bins, since a systematic effect on a given bin is likely also to affect
neighbouring bins. The effects on the DP fit results are assigned as
systematic uncertainties. An additional systematic uncertainty is assigned by
varying the binning scheme of the control sample used to determine the PID
efficiencies.

Systematic uncertainties related to possible intrinsic fit bias are
investigated using an ensemble of pseudoexperiments. Differences between the
input and fitted values from the ensemble for the fit parameters are found to
be small. Systematic uncertainties are assigned as the sum in quadrature of
the difference between the input and output values and the uncertainty on the
mean of the output value determined from a fit to the ensemble.

Systematic uncertainties due to fixed parameters in the fit model are
determined by varying the parameters within their uncertainties and repeating
the fit. The fixed parameters considered are the mass and width of the
$D^{*}_{0}(2400)^{0}$ resonance and the Blatt–Weisskopf barrier radius,
$r_{\rm BW}$. The mass and width are varied by the uncertainties shown in
Table 5 and the barrier radius is varied between 3 and 5
$\mathrm{\,Ge\kern-1.00006ptV}^{-1}$ [5]. For each fit parameter, the
difference compared to the nominal fit model is assigned as a systematic
uncertainty for each source.

The marginal $B^{*0}_{v}$ component is removed from the model and the changes
in the other parameters are assigned as the systematic uncertainties. Dalitz
plot analysis of ${{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}}\rightarrow{{D}^{0}}{{K}^{+}}{{\pi}^{-}}$
revealed that a structure at $m({{D}^{0}}{{K}^{+}})\sim
2.86\mathrm{\,Ge\kern-1.00006ptV}$ has both spin 1 and spin 3 components [4,
5]. Although there is no evidence for a spin 3 resonance in this analysis, the
excess at $m({{D}^{+}}{{\pi}^{-}})\sim 2.76\mathrm{\,Ge\kern-1.00006ptV}$
could have a similar composition. A putative $D^{*}_{3}(2760)$ resonance is
added to the fit model, and the effect on the other parameters is used to
assign systematic uncertainties.

The EFF lineshapes used to model the nonresonant S- and P-wave contributions
are replaced by a power-law model and the change in the fit parameters used as
a systematic uncertainty. The dependence of the results on the effective pole
mass description of Eq. (14) that is used for the virtual resonance
contributions is found by using a fixed width in Eq. (12), removing the
dependency on $m^{\rm eff}_{0}$.

The total experimental and model systematic uncertainties for fit fractions
and complex coefficients are summarised in Tables 8 and 9, respectively. The
contributions for the fit fractions, masses and widths are broken down in
Tables 10 and 11. Similar tables summarising the systematic uncertainties on
the interference fit fractions are given in App. B. The largest source of
experimental systematic uncertainty on the fit fractions is due to the
efficiency variation. For the model uncertainty on the fit fractions, the
addition and removal of marginal components and variation of fixed parameters
dominate. In general, the model uncertainties are larger than the experimental
systematic uncertainties for the fit fractions and the masses and widths.

Table 8: Experimental systematic uncertainties on the fit fractions and
complex amplitudes.

|  | Isobar model coefficients
---|---|---
Resonance | Fit fraction (%) | Real part | Imaginary part | Magnitude | Phase
$D^{*}_{0}(2400)^{0}$ | 0.6 | 0.03 | 0.02 | 0.02 | 0.06
$D^{*}_{2}(2460)^{0}$ | 0.9 | – | – | – | –
$D^{*}_{1}(2760)^{0}$ | 0.4 | 0.03 | 0.03 | 0.01 | 0.08
S-wave nonresonant | 1.5 | 0.03 | 0.03 | 0.02 | 0.04
P-wave nonresonant | 2.1 | 0.03 | 0.05 | 0.03 | 0.05
$D^{*}_{v}(2007)^{0}$ | 1.3 | 0.03 | 0.04 | 0.04 | 0.07
$B^{*}_{v}$ | 0.9 | 0.22 | 0.02 | 0.03 | 0.11

Table 9: Model uncertainties on the fit fractions and complex amplitudes.

|  | Isobar model coefficients
---|---|---
Resonance | Fit fraction (%) | Real part | Imaginary part | Magnitude | Phase
$D^{*}_{0}(2400)^{0}$ | 1.9 | 0.28 | 0.13 | 0.15 | 0.51
$D^{*}_{2}(2460)^{0}$ | 1.4 | – | – | – | –
$D^{*}_{1}(2760)^{0}$ | 0.9 | 0.03 | 0.03 | 0.03 | 0.08
S-wave nonresonant | 10.8 | 0.17 | 0.15 | 0.20 | 0.11
P-wave nonresonant | 3.7 | 0.34 | 0.68 | 0.12 | 0.95
$D^{*}_{v}(2007)^{0}$ | 1.5 | 0.56 | 0.77 | 0.05 | 0.60
$B^{*}_{v}$ | 1.6 | 0.09 | 0.08 | 0.07 | 0.27

Table 10: Breakdown of experimental systematic uncertainties on the fit fractions (%) and masses $(\mathrm{Me\kern-0.90005ptV})$ and widths $(\mathrm{Me\kern-0.90005ptV})$. | Nominal | S/B frac. | Eff. | Bkg. | Fit bias | Total
---|---|---|---|---|---|---
$D^{*}_{0}(2400)^{0}$ | $8.3\pm 2.6$ | 0.2 | 0.5 | 0.1 | 0.3 | 0.6
$D^{*}_{2}(2460)^{0}$ | $31.8\pm 1.5$ | 0.2 | 0.8 | 0.0 | 0.2 | 0.9
$D^{*}_{1}(2760)^{0}$ | $4.9\pm 1.2$ | 0.2 | 0.2 | 0.1 | 0.2 | 0.3
S-wave nonresonant | $38.0\pm 7.4$ | 0.7 | 0.5 | 0.4 | 1.2 | 1.5
P-wave nonresonant | $23.8\pm 5.6$ | 1.0 | 1.6 | 0.7 | 0.5 | 2.1
$D^{*}_{v}(2007)^{0}$ | $7.6\pm 2.3$ | 0.7 | 1.0 | 0.3 | 0.3 | 1.3
$B^{*}_{v}$ | $3.6\pm 1.9$ | 0.3 | 0.3 | 0.2 | 0.8 | 0.9
$m\left(D^{*}_{2}(2460)^{0}\right)$ | $2464.0\pm 1.4$ | 0.1 | 0.1 | 0.0 | 0.2 | 0.2
$\Gamma\left(D^{*}_{2}(2460)^{0}\right)$ | $43.8\pm 2.9$ | 0.3 | 0.3 | 0.0 | 0.4 | 0.6
$m\left(D^{*}_{1}(2760)^{0}\right)$ | $2781\pm 18\phantom{.}$ | 1 | 4 | 0 | 2 | 6
$\Gamma\left(D^{*}_{1}(2760)^{0}\right)$ | $177\pm 32\phantom{.}$ | 3 | 1 | 2 | 5 | 7
Table 11: Breakdown of model uncertainties on the fit fractions (%) and masses $(\mathrm{Me\kern-0.90005ptV})$ and widths $(\mathrm{Me\kern-0.90005ptV})$. | Nominal | Add/rem | Alt. models | Fixed params | Total
---|---|---|---|---|---
$D^{*}_{0}(2400)^{0}$ | $8.3\pm 2.6$ | 2.0 | 0.1 | 0.2 | 2.0
$D^{*}_{2}(2460)^{0}$ | $31.8\pm 1.5$ | 1.3 | 0.2 | 0.4 | 1.4
$D^{*}_{1}(2760)^{0}$ | $4.9\pm 1.2$ | 0.8 | 0.1 | 0.3 | 0.9
S-wave nonresonant | $38.0\pm 7.4$ | 4.8 | 4.5 | 5.4 | 10.8
P-wave nonresonant | $23.8\pm 5.6$ | 2.6 | 2.1 | 3.0 | 3.7
$D^{*}_{v}(2007)^{0}$ | $7.6\pm 2.3$ | 0.6 | 0.1 | 1.4 | 1.5
$B^{*}_{v}$ | $3.6\pm 1.9$ | 0.7 | 1.0 | 1.1 | 1.6
$m\left(D^{*}_{2}(2460)^{0}\right)$ | $2464.0\pm 1.4$ | 0.5 | 0.1 | 0.1 | 0.5
$\Gamma\left(D^{*}_{2}(2460)^{0}\right)$ | $43.8\pm 2.9$ | 0.8 | 1.4 | 0.6 | 1.7
$m\left(D^{*}_{1}(2760)^{0}\right)$ | $2781\pm 18\phantom{.}$ | 6 | 6 | 1 | 11
$\Gamma\left(D^{*}_{1}(2760)^{0}\right)$ | $177\pm 32\phantom{.}$ | 16 | 9 | 1 | 20

Several cross-checks are performed to confirm the stability of the results.
The data sample is divided into two parts depending on the charge of the $B$
candidate, the polarity of the magnet and the year of data taking. Selection
effects are also checked by varying the requirement on the neural network
output variable and the PID criteria applied to the bachelor kaon. A fit is
performed for each of the subsamples individually and each is seen to be
consistent with the default fit results, although in some cases one of the
secondary minima described in App. A becomes the preferred solution. To cross-
check the amplitude model, the fit is repeated many times with an extra
resonance with fixed mass, width and spin included in the model. All possible
mass and width values, and spin up to 3, were considered. None of the
additional resonances are found to contribute significantly.

##  9 Results and summary

The results for the complex coefficients are reported in Tables 12 and 13 in
terms of real and imaginary parts and of magnitudes and phases, respectively.
The results for the fit fractions are given in Table 14 and the results for
the interference fit fractions are given in App. B. The fit fractions for
resonant contributions are converted into quasi-two-body product branching
fractions by multiplying by
$\mathcal{B}({{{B}^{-}}}\rightarrow{{D}^{+}}{{K}^{-}}{{\pi}^{-}})=(7.31\pm
0.19\pm 0.22\pm 0.39)\times 10^{-5}$, as determined in Sec. 4.4. These product
branching fractions are shown in Table 15; they cannot be converted into
absolute branching fractions because the branching fractions for the resonance
decays to ${{D}^{+}}{{\pi}^{-}}$ are unknown.

Table 12: Results for the complex amplitudes and their uncertainties. The three quoted errors are statistical, experimental systematic and model uncertainties, respectively. | Isobar model coefficients
---|---
Resonance | Real part | Imaginary part
$D^{*}_{0}(2400)^{0}$ | $-0.04\pm 0.07\pm 0.03\pm 0.28$ | $-0.51\pm 0.07\pm 0.02\pm 0.13$
$D^{*}_{2}(2460)^{0}$ | $1.00$ | $0.00$
$D^{*}_{1}(2760)^{0}$ | $-0.32\pm 0.06\pm 0.03\pm 0.03$ | $-0.23\pm 0.07\pm 0.03\pm 0.03$
S-wave nonresonant | $\phantom{-}0.93\pm 0.09\pm 0.03\pm 0.17$ | $-0.58\pm 0.08\pm 0.03\pm 0.15$
P-wave nonresonant | $-0.43\pm 0.09\pm 0.03\pm 0.34$ | $\phantom{-}0.75\pm 0.09\pm 0.05\pm 0.68$
$D^{*}_{v}(2007)^{0}$ | $\phantom{-}0.16\pm 0.08\pm 0.03\pm 0.56$ | $\phantom{-}0.46\pm 0.09\pm 0.04\pm 0.77$
$B^{*}_{v}$ | $-0.07\pm 0.08\pm 0.22\pm 0.09$ | $\phantom{-}0.33\pm 0.07\pm 0.02\pm 0.08$
Table 13: Results for the complex amplitudes and their uncertainties. The three quoted errors are statistical, experimental systematic and model uncertainties, respectively. | Isobar model coefficients
---|---
Resonance | Magnitude | Phase
$D^{*}_{0}(2400)^{0}$ | $0.51\pm 0.09\pm 0.02\pm 0.15$ | $-1.65\pm 0.16\pm 0.06\pm 0.50$
$D^{*}_{2}(2460)^{0}$ | $1.00$ | $0.00$
$D^{*}_{1}(2760)^{0}$ | $0.39\pm 0.05\pm 0.01\pm 0.03$ | $-2.53\pm 0.24\pm 0.08\pm 0.08$
S-wave nonresonant | $1.09\pm 0.09\pm 0.02\pm 0.20$ | $-0.56\pm 0.09\pm 0.04\pm 0.11$
P-wave nonresonant | $0.87\pm 0.09\pm 0.03\pm 0.11$ | $\phantom{-}2.09\pm 0.15\pm 0.05\pm 0.95$
$D^{*}_{v}(2007)^{0}$ | $0.49\pm 0.07\pm 0.04\pm 0.05$ | $\phantom{-}1.24\pm 0.17\pm 0.07\pm 0.60$
$B^{*}_{v}$ | $0.34\pm 0.06\pm 0.03\pm 0.07$ | $\phantom{-}1.78\pm 0.23\pm 0.11\pm 0.27$
Table 14: Results for the fit fractions and their uncertainties (%). The three quoted errors are statistical, experimental systematic and model uncertainties, respectively. Resonance | Fit fraction
---|---
$D^{*}_{0}(2400)^{0}$ | $\phantom{2}8.3\pm 2.6\pm 0.6\pm\phantom{2}1.9$
$D^{*}_{2}(2460)^{0}$ | $31.8\pm 1.5\pm 0.9\pm\phantom{2}1.4$
$D^{*}_{1}(2760)^{0}$ | $\phantom{2}4.9\pm 1.2\pm 0.3\pm\phantom{2}0.9$
S-wave nonresonant | $38.0\pm 7.4\pm 1.5\pm 10.8$
P-wave nonresonant | $23.8\pm 5.6\pm 2.1\pm\phantom{2}3.7$
$D^{*}_{v}(2007)^{0}$ | $\phantom{2}7.6\pm 2.3\pm 1.3\pm\phantom{2}1.5$
$B^{*}_{v}$ | $\phantom{2}3.6\pm 1.9\pm 0.9\pm\phantom{2}1.6$
Table 15: Results for the product branching fractions ${\cal B}({{{B}^{-}}}\rightarrow R{{K}^{-}})\times{\cal B}(R\rightarrow{{D}^{+}}{{\pi}^{-}})$ ($10^{-6}$). The four quoted errors are statistical, experimental systematic, model and inclusive branching fraction uncertainties, respectively. Resonance | Branching fraction
---|---
$D^{*}_{0}(2400)^{0}$ | $\phantom{2}6.1\pm 1.9\pm 0.5\pm 1.4\pm 0.4$
$D^{*}_{2}(2460)^{0}$ | $23.2\pm 1.1\pm 0.6\pm 1.0\pm 1.6$
$D^{*}_{1}(2760)^{0}$ | $\phantom{2}3.6\pm 0.9\pm 0.3\pm 0.7\pm 0.2$
S-wave nonresonant | $27.8\pm 5.4\pm 1.1\pm 7.9\pm 1.9$
P-wave nonresonant | $17.4\pm 4.1\pm 1.5\pm 2.7\pm 1.2$
$D^{*}_{v}(2007)^{0}$ | $\phantom{2}5.6\pm 1.7\pm 1.0\pm 1.1\pm 0.4$
$B^{*}_{v}$ | $\phantom{2}2.6\pm 1.4\pm 0.6\pm 1.2\pm 0.2$

The masses and widths of the $D^{*}_{2}(2460)^{0}$ and $D^{*}_{1}(2760)^{0}$
are determined to be

$\displaystyle m(D^{*}_{2}(2460)^{0})$ $\displaystyle=$
$\displaystyle(2464.0\pm 1.4\pm 0.5\pm 0.2)\mathrm{\,Me\kern-1.00006ptV}\,,$
$\displaystyle\Gamma(D^{*}_{2}(2460)^{0})$ $\displaystyle=$
$\displaystyle\phantom{24}(43.8\pm 2.9\pm 1.7\pm
0.6)\mathrm{\,Me\kern-1.00006ptV}\,,$ $\displaystyle m(D^{*}_{1}(2760)^{0})$
$\displaystyle=$
$\displaystyle\phantom{.0}(2781\pm\phantom{.}18\pm\phantom{.}11\pm\phantom{.0}6)\mathrm{\,Me\kern-1.00006ptV}\,,$
$\displaystyle\Gamma(D^{*}_{1}(2760)^{0})$ $\displaystyle=$
$\displaystyle\phantom{.02}(177\pm\phantom{.}32\pm\phantom{.}20\pm\phantom{.0}7)\mathrm{\,Me\kern-1.00006ptV}\,,$

where the three quoted errors are statistical, experimental systematic and
model uncertainties, respectively. The results for the $D^{*}_{2}(2460)^{0}$
are within $2\,\sigma$ of the world average values [9]. The mass of the
$D^{*}_{1}(2760)^{0}$ resonance is similarly consistent with previous
measurements. The measured width of this state is larger than previous
measurements by 2 to 3 times the uncertainties. Future studies based on much
larger data samples will be required to better understand these states.

The measurement of
$\mathcal{B}({{{B}^{-}}}\rightarrow{{D}^{+}}{{K}^{-}}{{\pi}^{-}})$ corresponds
to the first observation of this decay mode. Therefore, the resonant
contributions to the decay are also first observations. The significance of
the ${{{B}^{-}}}\rightarrow D^{*}_{1}(2760)^{0}{{K}^{-}}$ observation is
investigated by removing the corresponding resonance from the DP model. A fit
without the $D^{*}_{1}(2760)^{0}$ component increases the value of
$2\Delta{\rm NLL}$ by $75.0$ units, corresponding to a high statistical
significance. Only the systematic effects due to uncertainties in the DP model
could in principle significantly change the conclusion regarding the need for
this resonance. However, in alternative DP models where a $D\pi$ resonance
with spin 3 is added and where the $B^{*}_{v}$ contribution is removed, the
shift in $2\Delta{\rm NLL}$ remains above 50 units. The alternative models
also do not significantly impact the level at which the $D^{*}_{1}(2760)^{0}$
state is preferred to be spin 1. Therefore, these results represent the first
observation of the ${{{B}^{-}}}\rightarrow D^{*}_{1}(2760)^{0}{{K}^{-}}$ and
the measurement of the spin of the $D^{*}_{1}(2760)^{0}$ resonance.

In summary, the ${{{B}^{-}}}\rightarrow{{D}^{+}}{{K}^{-}}{{\pi}^{-}}$ decay
has been observed in a data sample corresponding to $3.0\mbox{\,fb}^{-1}$ of
$pp$ collision data recorded by the LHCb experiment. An amplitude analysis of
its Dalitz plot distribution has been performed, in which a model containing
resonant contributions from the $D^{*}_{0}(2400)^{0}$, $D^{*}_{2}(2460)^{0}$
and $D^{*}_{1}(2760)^{0}$ states in addition to both S-wave and P-wave
nonresonant amplitudes and components due to virtual $D^{*}_{v}(2007)^{0}$ and
$B^{*0}_{v}$ resonances was found to give a good description of the data. The
${{{B}^{-}}}\rightarrow D^{*}_{2}(2460)^{0}{{K}^{-}}$ decay may in future be
used to determine the angle $\gamma$ of the CKM unitarity triangle. The
results provide insight into the spectroscopy of charm mesons, and demonstrate
that further progress may be obtained with Dalitz plot analyses of larger data
samples.

## 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).

## References

  * [1] Belle collaboration, K. Abe et al., _Study of ${{{B}^{-}}}\rightarrow D^{**0}{{\pi}^{-}}\ (D^{**0}\rightarrow D^{(*)+}{{\pi}^{-}})$ decays_, Phys. Rev. D69 (2004) 112002, arXiv:hep-ex/0307021
  * [2] BaBar collaboration, B. Aubert et al., _Dalitz plot analysis of ${{{B}^{-}}}\rightarrow{{D}^{+}}{{\pi}^{-}}{{\pi}^{-}}$_, Phys. Rev. D79 (2009) 112004, arXiv:0901.1291
  * [3] Belle collaboration, A. Kuzmin et al., _Study of $\bar{B}^{0}\rightarrow D^{0}\pi^{+}\pi^{-}$ decays_, Phys. Rev. D76 (2007) 012006, arXiv:hep-ex/0611054
  * [4] LHCb collaboration, R. Aaij et al., _Observation of overlapping spin- $1$ and spin-$3$ $\bar{D}^{0}K^{-}$ resonances at mass $2.86$ GeV/$c^{2}$_, Phys. Rev. Lett. 113 (2014) 162001, arXiv:1407.7574
  * [5] LHCb collaboration, R. Aaij et al., _Dalitz plot analysis of $B^{0}_{s}\rightarrow\bar{D}^{0}K^{-}\pi^{+}$ decays_, Phys. Rev. D90 (2014) 072003, arXiv:1407.7712
  * [6] BaBar collaboration, J. P. Lees et al., _Dalitz plot analyses of $B^{0}\rightarrow D^{-}D^{0}K^{+}$ and $B^{+}\rightarrow\overline{D}^{0}D^{0}K^{+}$ decays_, Phys. Rev. D91 (2015) 052002, arXiv:1412.6751
  * [7] BaBar collaboration, P. del Amo Sanchez et al., _Observation of new resonances decaying to $D\pi$ and $D^{*}\pi$ in inclusive $e^{+}e^{-}$ collisions near $\sqrt{s}=$10.58 GeV_, Phys. Rev. D82 (2010) 111101, arXiv:1009.2076
  * [8] LHCb collaboration, R. Aaij et al., _Study of $D_{J}$ meson decays to $D^{+}\pi^{-}$, $D^{0}\pi^{+}$ and $D^{*+}\pi^{-}$ final states in $pp$ collisions_, JHEP 09 (2013) 145, arXiv:1307.4556
  * [9] Particle Data Group, K. A. Olive et al., _Review of particle physics_ , Chin. Phys. C38 (2014) 090001
  * [10] N. Cabibbo, _Unitary symmetry and leptonic decays_ , Phys. Rev. Lett. 10 (1963) 531
  * [11] M. Kobayashi and T. Maskawa, _$C\\!P$ -violation in the renormalizable theory of weak interaction_, Progress of Theoretical Physics 49 (1973) 652
  * [12] M. Gronau and D. London, _How to determine all the angles of the unitarity triangle from ${{B}^{0}}\rightarrow D{{K}^{0}_{\rm\scriptscriptstyle S}}$ and ${{B}^{0}_{s}}\rightarrow D\phi$ _, Phys. Lett. B253 (1991) 483
  * [13] M. Gronau and D. Wyler, _On determining a weak phase from charged $B$ decay asymmetries_, Phys. Lett. B265 (1991) 172
  * [14] N. Sinha, _Determining $\gamma$ using $B\rightarrow D^{**}K$_, Phys. Rev. D70 (2004) 097501, arXiv:hep-ph/0405061
  * [15] LHCb collaboration, A. A. Alves Jr. et al., _The LHCb detector at the LHC_, JINST 3 (2008) S08005
  * [16] LHCb collaboration, R. Aaij et al., _LHCb detector performance_ , arXiv:1412.6352, to appear in Int. J. Mod. Phys. A
  * [17] R. Aaij et al., _Performance of the LHCb Vertex Locator_ , JINST 9 (2014) P09007, arXiv:1405.7808
  * [18] R. Arink et al., _Performance of the LHCb Outer Tracker_ , JINST 9 (2014) P01002, arXiv:1311.3893
  * [19] M. Adinolfi et al., _Performance of the LHCb RICH detector at the LHC_, Eur. Phys. J. C73 (2013) 2431, arXiv:1211.6759
  * [20] A. A. Alves Jr. et al., _Performance of the LHCb muon system_ , JINST 8 (2013) P02022, arXiv:1211.1346
  * [21] R. Aaij et al., _The LHCb trigger and its performance in 2011_, JINST 8 (2013) P04022, arXiv:1211.3055
  * [22] V. V. Gligorov and M. Williams, _Efficient, reliable and fast high-level triggering using a bonsai boosted decision tree_ , JINST 8 (2013) P02013, arXiv:1210.6861
  * [23] T. Sjöstrand, S. Mrenna, and P. Skands, _PYTHIA 6.4 physics and manual_ , JHEP 05 (2006) 026, arXiv:hep-ph/0603175
  * [24] T. Sjöstrand, S. Mrenna, and P. Skands, _A brief introduction to PYTHIA 8.1_ , Comput. Phys. Commun. 178 (2008) 852, arXiv:0710.3820
  * [25] I. Belyaev et al., _Handling of the generation of primary events in Gauss, the LHCb simulation framework_ , J. Phys. Conf. Ser. 331 (2011) 032047
  * [26] D. J. Lange, _The EvtGen particle decay simulation package_ , Nucl. Instrum. Meth. A462 (2001) 152
  * [27] P. Golonka and Z. Was, _PHOTOS Monte Carlo: A precision tool for QED corrections in $Z$ and $W$ decays_, Eur. Phys. J. C45 (2006) 97, arXiv:hep-ph/0506026
  * [28] Geant4 collaboration, J. Allison et al., _Geant4 developments and applications_ , IEEE Trans. Nucl. Sci. 53 (2006) 270
  * [29] Geant4 collaboration, S. Agostinelli et al., _Geant4: a simulation toolkit_ , Nucl. Instrum. Meth. A506 (2003) 250
  * [30] M. Clemencic et al., _The LHCb simulation application, Gauss: Design, evolution and experience_, J. Phys. Conf. Ser. 331 (2011) 032023
  * [31] M. Feindt and U. Kerzel, _The NeuroBayes neural network package_ , Nucl. Instrum. Meth. A 559 (2006) 190
  * [32] M. Pivk and F. R. Le Diberder, _sPlot: A statistical tool to unfold data distributions_ , Nucl. Instrum. Meth. A555 (2005) 356, arXiv:physics/0402083
  * [33] LHCb collaboration, R. Aaij et al., _Observation of $CP$ violation in $B^{\pm}\rightarrow DK^{\pm}$ decays_, Phys. Lett. B712 (2012) 203, Erratum ibid. B713 (2012) 351, arXiv:1203.3662
  * [34] LHCb collaboration, R. Aaij et al., _Measurements of the $\Lambda_{b}^{0}$, $\Xi_{b}^{-}$, and $\Omega_{b}^{-}$ baryon masses_, Phys. Rev. Lett. 110 (2013) 182001, arXiv:1302.1072
  * [35] LHCb collaboration, R. Aaij et al., _Precision measurement of $D$ meson mass differences_, JHEP 06 (2013) 065, arXiv:1304.6865
  * [36] W. D. Hulsbergen, _Decay chain fitting with a Kalman filter_ , Nucl. Instrum. Meth. A552 (2005) 566, arXiv:physics/0503191
  * [37] T. Skwarnicki, A study of the radiative cascade transitions between the Upsilon-prime and Upsilon resonances, PhD thesis, Institute of Nuclear Physics, Krakow, 1986, DESY-F31-86-02
  * [38] LHCb collaboration, R. Aaij et al., _Observation of $B^{0}\rightarrow\bar{D}^{0}K^{+}K^{-}$ and evidence for $B^{0}_{s}\rightarrow\bar{D}^{0}K^{+}K^{-}$_, Phys. Rev. Lett. 109 (2012) 131801, arXiv:1207.5991
  * [39] R. H. Dalitz, _On the analysis of tau-meson data and the nature of the tau-meson_ , Phil. Mag. 44 (1953) 1068
  * [40] G. N. Fleming, _Recoupling effects in the isobar model. 1. General formalism for three-pion scattering_ , Phys. Rev. 135 (1964) B551
  * [41] D. Morgan, _Phenomenological analysis of $I=\frac{1}{2}$ single-pion production processes in the energy range 500 to 700 MeV_, Phys. Rev. 166 (1968) 1731
  * [42] D. Herndon, P. Soding, and R. J. Cashmore, _A generalised isobar model formalism_ , Phys. Rev. D11 (1975) 3165
  * [43] J. Blatt and V. E. Weisskopf, Theoretical nuclear physics, J. Wiley (New York), 1952
  * [44] BaBar collaboration, B. Aubert et al., _Dalitz-plot analysis of the decays $B^{\pm}\rightarrow K^{\pm}\pi^{\mp}\pi^{\pm}$_, Phys. Rev. D72 (2005) 072003, Erratum ibid. D74 (2006) 099903, arXiv:hep-ex/0507004
  * [45] C. Zemach, _Three pion decays of unstable particles_ , Phys. Rev. 133 (1964) B1201
  * [46] C. Zemach, _Use of angular-momentum tensors_ , Phys. Rev. 140 (1965) B97
  * [47] Belle collaboration, A. Garmash et al., _Dalitz analysis of the three-body charmless decays ${{{B}^{+}}}\rightarrow{{K}^{+}}{{\pi}^{+}}{{\pi}^{-}}$ and ${{{B}^{+}}}\rightarrow{{K}^{+}}{{K}^{+}}{{K}^{-}}$_, Phys. Rev. D71 (2005) 092003, arXiv:hep-ex/0412066
  * [48] Laura++ Dalitz plot fitting package, http://laura.hepforge.org
  * [49] E. Ben-Haim, R. Brun, B. Echenard, and T. E. Latham, _JFIT: a framework to obtain combined experimental results through joint fits_ , arXiv:1409.5080
  * [50] M. Williams, _How good are your fits? Unbinned multivariate goodness-of-fit tests in high energy physics_ , JINST 5 (2010) P09004, arXiv:1006.3019

##  Appendix A Secondary minima

The results, in terms of fit fractions and complex coefficients, corresponding
to the two secondary minima discussed in Sec. 7 are compared to those of the
global minimum in Table 16. The main difference between the global and
secondary minima is in the interference pattern in the $D\pi$ P-waves, while
the third minimum exhibits a different interference pattern in the $D\pi$
S-wave than the global minimum and has a very large total fit fraction due to
strong destructive interference.

Table 16: Results for the fit fractions and complex coefficients for the
secondary minima with $2{\rm NLL}$ values 2.8 and 3.3 units greater than that
of the global minimum of the NLL function.

Resonance | Fit fraction (%) | Real part | Imaginary part | Magnitude | Phase
---|---|---|---|---|---
$2\Delta{\rm NLL}$ | 0 | 2.8 | 3.3 | 0 | 2.8 | 3.3 | 0 | 2.8 | 3.3 | 0 | 2.8 | 3.3 | 0 | 2.8 | 3.3
$D^{*}_{0}(2400)^{0}$ | $8.3$ | $9.6$ | $84.4$ | $-0.04$ | $-0.03$ | $-1.38$ | $-0.51$ | $-0.55$ | $-0.72$ | $0.51$ | $0.55$ | $1.56$ | $-1.65$ | $-1.62$ | $-2.66$
$D^{*}_{2}(2460)^{0}$ | $31.8$ | $31.5$ | $34.9$ | $1.00$ | $0.00$ | $1.00$ | $0.00$
$D^{*}_{1}(2760)^{0}$ | $4.9$ | $4.6$ | $5.5$ | $-0.32$ | $-0.30$ | $-0.30$ | $-0.23$ | $-0.24$ | $-0.26$ | $0.39$ | $0.38$ | $0.40$ | $-2.53$ | $-2.46$ | $-2.42$
S-wave nonresonant | $38.0$ | $36.2$ | $4.6$ | $0.93$ | $0.89$ | $-0.33$ | $-0.58$ | $-0.60$ | $0.15$ | $1.09$ | $1.07$ | $-0.36$ | $-0.56$ | $-0.59$ | $2.71$
P-wave nonresonant | $23.8$ | $22.6$ | $-31.9$ | $-0.43$ | $0.83$ | $-0.84$ | $0.75$ | $0.15$ | $0.45$ | $0.87$ | $0.85$ | $0.96$ | $2.09$ | $2.96$ | $2.65$
$D^{*}_{v}(2007)^{0}$ | $7.6$ | $7.1$ | $11.9$ | $0.16$ | $-0.38$ | $-0.28$ | $0.46$ | $-0.29$ | $-0.51$ | $0.49$ | $0.48$ | $0.58$ | $1.24$ | $-2.49$ | $-2.07$
$B^{*}_{v}$ | $3.6$ | $1.0$ | $25.0$ | $-0.07$ | $-0.16$ | $-0.31$ | $0.33$ | $0.09$ | $0.79$ | $0.34$ | $0.18$ | $0.85$ | $1.78$ | $2.61$ | $1.94$
Total fit fraction | 118.1 | 112.6 | 198.3 |  |  |  |  |  |  |  |  |  |  |  |

##  Appendix B Results for interference fit fractions

The central values and statistical errors for the interference fit fractions
are shown in Table 17. The experimental systematic and model uncertainties are
given in Tables 18 and 19. The interference fit fractions are common to both
trigger subsamples.

Table 17: Interference fit fractions (%) and statistical uncertainties. The
amplitudes are: ($A_{0}$) $D^{*}_{v}(2007)^{0}$, ($A_{1}$)
$D^{*}_{0}(2400)^{0}$, ($A_{2}$) $D^{*}_{2}(2460)^{0}$, ($A_{3}$)
$D^{*}_{1}(2760)^{0}$, ($A_{4}$) $B^{*}_{v}$, ($A_{5}$) nonresonant S-wave,
($A_{6}$) nonresonant P-wave. The diagonal elements are the same as the
conventional fit fractions.

| $\phantom{-}A_{0}$ | $\phantom{-}A_{1}$ | $\phantom{-}A_{2}$ | $\phantom{-}A_{3}$ | $\phantom{-}A_{4}$ | $\phantom{-}A_{5}$ | $\phantom{-}A_{6}$
---|---|---|---|---|---|---|---
$A_{0}$ | $\phantom{-}7.6\pm 2.3$ | $\phantom{-}0.0\pm 0.0$ | $\phantom{-1}0.0\pm 0.0$ | $\phantom{-}2.4\pm 0.9$ | $\phantom{-}4.8\pm 1.3$ | $\phantom{-1}0.0\pm 0.0$ | $-14.2\pm 5.3$
$A_{1}$ |  | $\phantom{-}8.3\pm 2.6$ | $\phantom{-1}0.0\pm 0.0$ | $\phantom{-}0.0\pm 0.0$ | $-1.6\pm 0.7$ | $\phantom{-}18.1\pm 2.6$ | $\phantom{-1}0.0\pm 0.0$
$A_{2}$ |  |  | $\phantom{-}31.8\pm 1.5$ | $\phantom{-}0.0\pm 0.0$ | $-2.3\pm 0.6$ | $\phantom{-1}0.0\pm 0.0$ | $\phantom{-1}0.0\pm 0.0$
$A_{3}$ |  |  |  | $\phantom{-}4.9\pm 1.2$ | $\phantom{-}2.0\pm 0.8$ | $\phantom{-1}0.0\pm 0.0$ |  1$-9.6\pm 2.9$
$A_{4}$ |  |  |  |  | $\phantom{-}3.6\pm 1.9$ |  1$-6.7\pm 2.3$ | $-11.1\pm 3.6$
$A_{5}$ |  |  |  |  |  | $\phantom{-}38.0\pm 7.4$ | $\phantom{-1}0.0\pm 0.0$
$A_{6}$ |  |  |  |  |  |  | $\phantom{-}23.8\pm 5.6$

Table 18: Experimental systematic uncertainies on the interference fit fractions (%). The amplitudes are: ($A_{0}$) $D^{*}_{v}(2007)^{0}$, ($A_{1}$) $D^{*}_{0}(2400)^{0}$, ($A_{2}$) $D^{*}_{2}(2460)^{0}$, ($A_{3}$) $D^{*}_{1}(2760)^{0}$, ($A_{4}$) $B^{*}_{v}$, ($A_{5}$) nonresonant S-wave, ($A_{6}$) nonresonant P-wave. The diagonal elements are the same as the conventional fit fractions. | $A_{0}$ | $A_{1}$ | $A_{2}$ | $A_{3}$ | $A_{4}$ | $A_{5}$ | $A_{6}$
---|---|---|---|---|---|---|---
$A_{0}$ | 1.3 | 0.0 | 0.0 | 0.4 | 0.6 | 0.0 | 2.6
$A_{1}$ |  | 0.6 | 0.0 | 0.0 | 0.4 | 0.6 | 0.0
$A_{2}$ |  |  | 0.9 | 0.0 | 0.3 | 0.0 | 0.0
$A_{3}$ |  |  |  | 0.4 | 0.2 | 0.0 | 0.7
$A_{4}$ |  |  |  |  | 0.9 | 1.1 | 1.2
$A_{5}$ |  |  |  |  |  | 1.5 | 0.0
$A_{6}$ |  |  |  |  |  |  | 2.1
Table 19: Model systematic uncertainies on the interference fit fractions (%). The amplitudes are: ($A_{0}$) $D^{*}_{v}(2007)^{0}$, ($A_{1}$) $D^{*}_{0}(2400)^{0}$, ($A_{2}$) $D^{*}_{2}(2460)^{0}$, ($A_{3}$) $D^{*}_{1}(2760)^{0}$, ($A_{4}$) $B^{*}_{v}$, ($A_{5}$) nonresonant S-wave, ($A_{6}$) nonresonant P-wave. The diagonal elements are the same as the conventional fit fractions. |  1$A_{0}$1 |  1$A_{1}$1 |  1$A_{2}$1 |  1$A_{3}$1 |  1$A_{4}$1 |  1$A_{5}$ |  1$A_{6}$1
---|---|---|---|---|---|---|---
$A_{0}$ | 1.5 | 0.0 | 0.0 | 0.1 | 1.4 |  10.0 | 1.1
$A_{1}$ |  | 1.9 | 0.0 | 0.0 | 1.7 |  14.8 | 0.0
$A_{2}$ |  |  | 1.4 | 0.0 | 0.5 |  10.0 | 0.0
$A_{3}$ |  |  |  | 0.9 | 0.6 |  10.0 | 3.4
$A_{4}$ |  |  |  |  | 1.6 |  12.8 | 0.4
$A_{5}$ |  |  |  |  |  |  110.8 | 0.0
$A_{6}$ |  |  |  |  |  |  | 3.67

LHCb collaboration

R. Aaij41, 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, 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. Bizzeti17,i, T. Blake48, F. Blanc39, J. Blouw10, S. Blusk59, V.
Bocci25, A. Bondar34, N. Bondar30,38, W. Bonivento15, S. Borghi54, A.
Borgia59, 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. Chrzaszcz40,26, 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, I. Counts56, B.
Couturier38, G.A. Cowan50, D.C. Craik48, A.C. 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, 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. Ferreira Rodrigues1, M.
Ferro-Luzzi38, S. Filippov33, M. Fiore16,38,f, M. Fiorini16,f, M. Firlej27, C.
Fitzpatrick39, T. Fiutowski27, P. Fol53, M. Fontana10, F. Fontanelli19,j, R.
Forty38, O. Francisco2, M. Frank38, C. Frei38, M. Frosini17, J. Fu21,38, E.
Furfaro24,l, A. Gallas Torreira37, D. Galli14,d, S. Gallorini22,38, S.
Gambetta19,j, 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, E.
Gersabeck11, M. Gersabeck54, T. Gershon48, Ph. Ghez4, A. Gianelle22, S.
Gianì39, V. Gibson47, 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, 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, J. Lefrançois7, O. Leroy6, T. Lesiak26, B.
Leverington11, Y. Li7, T. Likhomanenko64, M. Liles52, R. Lindner38, C. Linn38,
F. Lionetto40, B. Liu15, S. Lohn38, I. Longstaff51, J.H. Lopes2, P. Lowdon40,
D. Lucchesi22,r, H. Luo50, A. Lupato22, E. Luppi16,f, O. Lupton55, F.
Machefert7, I.V. Machikhiliyan31, F. Maciuc29, O. Maev30, S. Malde55, A.
Malinin64, G. Manca15,e, 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, Z. Mathe38, C. Matteuzzi20, A. Mauri40, B. Maurin39, A. Mazurov45, M.
McCann53, J. McCarthy45, A. McNab54, R. McNulty12, B. McSkelly52, B.
Meadows57, F. Meier9, M. Meissner11, M. Merk41, D.A. Milanes62, M.-N. Minard4,
J. Molina Rodriguez60, S. Monteil5, M. Morandin22, P. Morawski27, A. Mordà6,
M.J. Morello23,t, J. Moron27, A.-B. Morris50, R. Mountain59, F. Muheim50, K.
Müller40, 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, 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, E. Picatoste
Olloqui36, B. Pietrzyk4, T. Pilař48, D. Pinci25, A. Pistone19, S. Playfer50,
M. Plo Casasus37, T. Poikela38, F. Polci8, A. Poluektov48,34, I. Polyakov31,
E. Polycarpo2, A. Popov35, D. Popov10, 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, 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, B. Schmidt38, O. Schneider39, A.
Schopper38, 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, N.A. Smith52, E. Smith55,49, E.
Smith53, J. Smith47, M. Smith54, H. Snoek41, M.D. Sokoloff57,38, F.J.P.
Soler51, F. Soomro39, D. Souza46, B. Souza De Paula2, B. Spaan9, P.
Spradlin51, S. Sridharan38, F. Stagni38, M. Stahl11, S. Stahl38, O.
Steinkamp40, O. Stenyakin35, F. Sterpka59, S. Stevenson55, S. Stoica29, S.
Stone59, B. Storaci40, S. Stracka23,t, M. Straticiuc29, U. Straumann40, R.
Stroili22, L. Sun57, W. Sutcliffe53, K. Swientek27, S. Swientek9, V.
Syropoulos42, M. Szczekowski28, P. Szczypka39,38, T. Szumlak27, S. T’Jampens4,
M. Teklishyn7, G. Tellarini16,f, F. Teubert38, C. Thomas55, E. Thomas38, J.
van Tilburg41, V. Tisserand4, M. Tobin39, J. Todd57, S. Tolk42, L.
Tomassetti16,f, D. Tonelli38, S. Topp-Joergensen55, N. Torr55, E. Tournefier4,
S. Tourneur39, K. Trabelsi39, M.T. Tran39, M. Tresch40, A. Trisovic38, A.
Tsaregorodtsev6, P. Tsopelas41, N. Tuning41,38, M. Ubeda Garcia38, A.
Ukleja28, A. Ustyuzhanin65, U. Uwer11, C. Vacca15,e, V. Vagnoni14, G.
Valenti14, A. Vallier7, R. Vazquez Gomez18, P. Vazquez Regueiro37, C. Vázquez
Sierra37, S. Vecchi16, J.J. Velthuis46, M. Veltri17,h, G. Veneziano39, M.
Vesterinen11, J.V. Viana Barbosa38, B. Viaud7, D. Vieira2, M. Vieites Diaz37,
X. Vilasis-Cardona36,p, A. Vollhardt40, D. Volyanskyy10, D. Voong46, A.
Vorobyev30, V. Vorobyev34, C. Voß63, J.A. de Vries41, R. Waldi63, C.
Wallace48, R. Wallace12, J. Walsh23, S. Wandernoth11, J. Wang59, D.R. Ward47,
N.K. Watson45, D. Websdale53, A. Weiden40, M. Whitehead48, D. Wiedner11, G.
Wilkinson55,38, M. Wilkinson59, M. Williams38, M.P. Williams45, M. Williams56,
H.W. Wilschut67, F.F. Wilson49, J. Wimberley58, J. Wishahi9, W. Wislicki28, M.
Witek26, G. Wormser7, S.A. Wotton47, S. Wright47, K. Wyllie38, Y. Xie61, Z.
Xu39, Z. Yang3, X. Yuan34, O. Yushchenko35, M. Zangoli14, M. 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