EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

CERN-EP-2016-184 LHCb-PAPER-2016-026 August 3, 2016

Amplitude analysis of ${{{B}^{-}}}\rightarrow{{D}^{+}}{{\pi}^{-}}{{\pi}^{-}}$
decays

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

The Dalitz plot analysis technique is used to study the resonant substructures
of ${{{B}^{-}}}\rightarrow{{D}^{+}}{{\pi}^{-}}{{\pi}^{-}}$ decays in a data
sample corresponding to $3.0\mbox{\,fb}^{-1}$ of $pp$ collision data recorded
by the LHCb experiment during 2011 and 2012. A model-independent analysis of
the angular moments demonstrates the presence of resonances with spins 1, 2
and 3 at high ${{D}^{+}}{{\pi}^{-}}$ mass. The data are fitted with an
amplitude model composed of a quasi-model-independent function to describe the
${{D}^{+}}{{\pi}^{-}}$ S-wave together with virtual contributions from the
${{D}^{*}}(2007)^{0}$ and ${B}^{*0}$ states, and components corresponding to
the ${D}^{*}_{2}(2460)^{0}$, ${D}^{*}_{1}(2680)^{0}$, ${D}^{*}_{3}(2760)^{0}$
and ${D}^{*}_{2}(3000)^{0}$ resonances. The masses and widths of these
resonances are determined together with the branching fractions for their
production in ${{{B}^{-}}}\rightarrow{{D}^{+}}{{\pi}^{-}}{{\pi}^{-}}$ decays.
The ${{D}^{+}}{{\pi}^{-}}$ S-wave has phase motion consistent with that
expected due to the presence of the ${D}^{*}_{0}(2400)^{0}$ state. These
results constitute the first observations of the ${D}^{*}_{3}(2760)^{0}$ and
${D}^{*}_{2}(3000)^{0}$ resonances, with significances of $10\,\sigma$ and
$6.6\,\sigma$, respectively.

Submitted to Phys. Rev. D

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

##  1 Introduction

There is strong theoretical and experimental interest in charm meson
spectroscopy because it provides opportunities to study QCD predictions within
the context of different models [1, 2, 3, 4, 5]. Experimental knowledge of the
masses, widths and spins of the charged and neutral orbitally-excited (1P)
charm meson states has been gained through analyses of both prompt production
[6, 7] and three-body decays of $B$ mesons [8, 9, 10, 11, 12, 13]. Progress
has been equally strong for excited charm-strange ($c\bar{s}$) mesons [14, 15,
16, 17, 18]. These studies have in addition revealed several new states at
higher masses, most of which have not yet been confirmed by analyses of
independent data samples. Moreover, quantum numbers are only known for states
studied in amplitude analyses of multibody $B$ meson decays, since analyses of
promptly produced excited charm states only determine whether the spin-parity
is natural (i.e. $J^{P}=0^{+},1^{-},2^{+},...$) or unnatural (i.e.
$J^{P}=0^{-},1^{+},2^{-},...$), not the resonance spin. The experimental
status of the neutral excited charm 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 1P
states. The spectroscopic identification for heavier states is not clear.

Table 1: Measured properties of neutral excited charm states. World averages are given for the 1P resonances (top part), while all measurements are listed for the heavier states (bottom part). Where two uncertainties are given, the first is statistical and second systematic; where a third is given, it is due to model uncertainty. The uncertainties on the averages for the ${D}^{*}_{0}(2400)^{0}$ mass and the $D_{1}(2420)^{0}$ and ${D}^{*}_{2}(2460)^{0}$ masses and widths are inflated by scale factors to account for inconsistencies between measurements. The quoted ${D}^{*}_{2}(2460)^{0}$ averages do not include the recent result from Ref. [12].  Resonance | Mass $(\mathrm{Me\kern-1.00006ptV})$ | Width $(\mathrm{Me\kern-1.00006ptV})$ | $J^{P}$ | Ref.
---|---|---|---|---
${D}^{*}_{0}(2400)^{0}$ | $\phantom{.0}2318\pm 29\phantom{.}$ | $267\pm 40$ | $0^{+}$ | [19]
$D_{1}(2420)^{0}$ | $2421.4\pm 0.6$ | $27.4\pm 2.5$ | $1^{+}$ | [19]
$D_{1}^{\prime}(2430)^{0}$ | $\phantom{.0}2427\pm 40\phantom{.}$ | $384\,^{+130}_{-110}$ | $1^{+}$ | [19]
${D}^{*}_{2}(2460)^{0}$ | $2462.6\pm 0.6$ | $49.0\pm 1.3$ | $2^{+}$ | [19]
$D^{*}(2600)$ | $2608.7\pm 2.4\pm 2.5$ | $\phantom{1.}93\pm\phantom{1.}6\pm\phantom{1.}13$ | natural | [6]
$D^{*}(2650)$ | $2649.2\pm 3.5\pm 3.5$ | $\phantom{.}140\pm\phantom{.}17\pm\phantom{1.}19$ | natural | [7]
$D^{*}(2760)$ | $2763.3\pm 2.3\pm 2.3$ | $60.9\pm 5.1\pm\phantom{1}3.6$ | natural | [6]
$D^{*}(2760)$ | $2760.1\pm 1.1\pm 3.7$ | $74.4\pm 3.4\pm 19.1$ | natural | [7]
$D_{1}^{*}(2760)^{0}$ | $2781\pm 18\pm 11\pm 6$ | $177\pm 32\pm 20\pm 7$ | $1^{-}$ | [12]

The ${{{B}^{-}}}\rightarrow{{D}^{+}}{{\pi}^{-}}{{\pi}^{-}}$ decay mode has
been previously studied in Refs. [8, 9]. The inclusion of charge-conjugate
processes is implied throughout the paper. The Dalitz plot (DP) models that
were used contained components for two excited charm states, the
${D}^{*}_{0}(2400)^{0}$ and ${D}^{*}_{2}(2460)^{0}$ resonances, together with
nonresonant amplitudes. More recently, a DP analysis of
${{{B}^{-}}}\rightarrow{{D}^{+}}{{K}^{-}}{{\pi}^{-}}$ decays [12] included, in
addition, a contribution from the $D_{1}^{*}(2760)^{0}$ state. The properties
of this state indicate that it belongs to the 1D family [20, 21]. The
$D_{1}^{*}(2760)^{0}$ width is found to be larger than in previous
measurements based on prompt production, which may be due to a contribution
from an additional resonance, as would be expected if both 2S and 1D states
with spin-parity $J^{P}=1^{-}$ are present in this region. There should also
be a 1D state with $J^{P}=3^{-}$ at similar mass, as seen in the charm-strange
system [15, 16]. As yet there is no evidence for such a neutral charm state,
but a DP analysis of ${{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}}\rightarrow{{D}^{0}}{{\pi}^{+}}{{\pi}^{-}}$
decays [11] led to the first observation of the $D_{3}^{*}(2760)^{+}$ state.

One challenge for DP analyses with large data samples is the modelling of
broad resonances that interfere with nonresonant amplitudes in the same
partial wave. Inclusion of both contributions in an amplitude fit can violate
unitarity in the decay matrix element, and also gives results that are
difficult to interpret due to large interference effects. In the case of
${{{B}^{-}}}\rightarrow{{D}^{+}}{{\pi}^{-}}{{\pi}^{-}}$ decays this is
particularly relevant for the ${{D}^{+}}{{\pi}^{-}}$ S-wave, where both the
${D}^{*}_{0}(2400)^{0}$ resonance and a nonresonant contribution are expected.
In the ${{\pi}^{+}}{{\pi}^{-}}$ and ${{K}^{+}}{{\pi}^{-}}$ systems such
effects can be handled with a K-matrix approach or specific models such as the
LASS function [22] inspired by low-energy scattering data, respectively. In
the absence of any ${{D}^{+}}{{\pi}^{-}}$ scattering data, a viable
alternative approach is to use a quasi-model-independent description, in which
the partial wave is fitted using splines to describe the magnitude and phase
as a function of $m({{D}^{+}}{{\pi}^{-}})$. Determination of the phase depends
on interference of the S-wave with another partial wave, so that some model
dependence remains due to the description of the other amplitudes in the
decay. This approach was first applied to the $K\pi$ S-wave using
${{D}^{+}}\rightarrow{{K}^{-}}{{\pi}^{+}}{{\pi}^{+}}$ decays [23]. Subsequent
uses include further studies of the $K\pi$ S-wave [24, 25, 26, 27] as well as
the ${{K}^{+}}{{K}^{-}}$ [28] and ${{\pi}^{+}}{{\pi}^{-}}$ [29] S-waves, in
various processes. Similar methods have been used to determine the phase
motion of exotic hadron candidates [30, 31]. Quasi-model-independent
information on the ${{D}^{+}}{{\pi}^{-}}$ S-wave could be used to develop
better models of the dynamics in the ${{D}^{+}}{{\pi}^{-}}$ system [32, 33,
34, 35].

In this paper, the DP analysis technique is employed to study the contributing
amplitudes in ${{{B}^{-}}}\rightarrow{{D}^{+}}{{\pi}^{-}}{{\pi}^{-}}$ decays,
where the charm meson is reconstructed through
${{D}^{+}}\rightarrow{{K}^{-}}{{\pi}^{+}}{{\pi}^{+}}$ decays. The analysis is
based on a data sample corresponding to an integrated luminosity of
$3.0\mbox{\,fb}^{-1}$ of data collected with the LHCb detector during 2011
when the $pp$ collision centre-of-mass energy was
$\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$, and 2012 with
$\sqrt{s}=8\mathrm{\,Te\kern-1.00006ptV}$.

The paper is organised as follows. Section 2 provides a brief description of
the LHCb detector and the event reconstruction and simulation software. The
selection of signal candidates is described in Sec. 3 and the determination of
signal and background yields is presented in Sec. 4. The angular moments of
${{{B}^{-}}}\rightarrow{{D}^{+}}{{\pi}^{-}}{{\pi}^{-}}$ decays are studied in
Sec. 5 and are used to guide the amplitude analysis. The DP analysis formalism
is reviewed briefly in Sec. 6, and implementation of the amplitude fit is
given in Sec. 7. Experimental and model-dependent systematic uncertainties are
evaluated in Sec. 8, and the results and a summary are presented in Sec. 9.

##  2 LHCb detector

The LHCb detector [36, 37] is a single-arm forward spectrometer covering the
pseudorapidity range $2<\eta<5$, designed for the study of particles
containing $b$ or $c$ quarks. The detector includes a high-precision tracking
system consisting of a silicon-strip vertex detector surrounding the $pp$
interaction region, a large-area silicon-strip detector located upstream of a
dipole magnet with a bending power of about $4{\rm\,Tm}$, and three stations
of silicon-strip detectors and straw drift tubes 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 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.
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.

The trigger 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 [38] 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.

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 [39, *Sjostrand:2007gs] with a specific LHCb
configuration [41]. Decays of hadronic particles are described by EvtGen [42],
in which final state radiation is generated using Photos [43]. The interaction
of the generated particles with the detector and its response are implemented
using the Geant4 toolkit [44, *Agostinelli:2002hh] as described in Ref. [46].

##  3 Selection requirements

The selection criteria are the same as those used in Ref. [12], where a
detailed description is given, with the exception that only candidates that
are triggered by at least one of the signal tracks are retained in order to
minimise the uncertainty on the efficiency. First, loose requirements are
applied in order to obtain a visible peak in the $B$ candidate invariant mass
distribution. These criteria are found to be $91\,\%$ efficient on simulated
signal decays. The remaining data are then used to train two artificial neural
networks [47] that separate signal from different categories of background.
The first is designed to distinguish candidates that contain real
${{D}^{+}}\rightarrow{{K}^{-}}{{\pi}^{+}}{{\pi}^{+}}$ decays from those that
do not; the second separates signal
${{{B}^{-}}}\rightarrow{{D}^{+}}{{\pi}^{-}}{{\pi}^{-}}$ decays from background
combinations. The sPlot technique [48] is used to statistically separate
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 decay product
tracks, including information about kinematics, track and vertex quality. The
second uses a total of 27 input variables, including the output of the first
network, as described in Ref. [12]. The neural network input quantities depend
only weakly on the position in the DP, so that training the networks with the
same data sample used for the analysis does not bias the results. A
requirement that reduces the combinatorial background by an order of
magnitude, while retaining about $75\,\%$ of the signal, is imposed on the
second neural network output.

Particle identification (PID) requirements are applied to all five final state
tracks to select pions or kaons as necessary. Background from
${{D}^{+}_{s}}\rightarrow{{K}^{-}}{{K}^{+}}{{\pi}^{+}}$ decays, where the
${K}^{+}$ is misidentified as a ${\pi}^{+}$ meson, are suppressed using a
tight PID criterion on the higher momentum ${\pi}^{+}$ from the ${D}^{+}$
decay. The combined efficiency of the PID requirements on the five final state
tracks is determined using $D^{*+}\rightarrow D^{0}{{\pi}^{+}}$,
$D^{0}\rightarrow{{K}^{-}}{{\pi}^{+}}$ calibration data [49] and found to be
around $70\,\%$.

Potential background from ${{\mathchar
28931\relax}^{+}_{c}}\rightarrow{p}{{K}^{-}}{{\pi}^{+}}$ decays,
misreconstructed as ${{D}^{+}}$ candidates, is removed if the invariant mass
lies in the range $2280$–$2300\mathrm{\,Me\kern-1.00006ptV}$ when the proton
mass hypothesis is applied to the low momentum pion track. Decays of
${{B}^{-}}$ mesons to the
${{K}^{-}}{{\pi}^{+}}{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{-}}$ final state that do
not proceed via an intermediate charm state are removed by requiring that the
$D$ and $B$ candidate decay vertices are separated by at least $1\rm\,mm$. The
signal efficiency of this requirement is approximately $85\,\%$.

To improve mass resolution, the momenta of the final state tracks are rescaled
[50, 51] using weights obtained from a sample of
$J/\psi\rightarrow\mu^{+}\mu^{-}$ decays where the measured mass peak is
matched to the known value [19]. Additionally, a kinematic fit [52] is
performed to candidates in which the invariant mass of the $D$ decay products
is constrained to equal the world average $D$ mass [19]. A $B$ mass constraint
is added in the calculation of the variables that are used in the DP fit.

Candidate $B$ mesons with invariant mass in the range
$5100$–$5800\mathrm{\,Me\kern-1.00006ptV}$ are retained for further analysis.
Following all selection requirements, multiple candidates are found in
approximately $0.4\,\%$ of events. All candidates are retained and treated in
the same way.

##  4 Determination of signal and background yields

The signal and background yields are measured using an extended unbinned
maximum likelihood fit to the ${D}^{+}$ ${\pi}^{-}$ ${\pi}^{-}$ invariant mass
distribution. The candidates are comprised of true signal decays and several
sources of background. Partially reconstructed backgrounds come from $b$
hadron decays where one or more final state particles are not reconstructed.
Combinatorial background originates from random combinations of tracks,
potentially including a real
${{D}^{+}}\rightarrow{{K}^{-}}{{\pi}^{+}}{{\pi}^{+}}$ decay. Misidentified
background arises from $b$ hadron decays in which one of the final state
particles is not correctly identified. Potential residual background from
charmless $B$ decays is reduced to a negligible level by the requirement that
the flight distance of the $D$ candidate be greater than $1\rm\,mm$.

Signal candidates are modelled by the sum of two Crystal Ball (CB) functions
[53] with a common peak position of the Gaussian core and tails on opposite
sides. The relative normalisation of the narrower CB shape and the ratio of
widths of the CB functions are constrained, by including a Gaussian penalty
term in the likelihood, to the values found in fits to simulated samples. The
tail parameters of the CB shapes are fixed to those found in simulation.

The main source of partially reconstructed background is the
${{{B}^{-}}}\rightarrow{{D}^{*+}}{{\pi}^{-}}{{\pi}^{-}}$ channel with
subsequent ${{D}^{*+}}\rightarrow{{D}^{+}}\gamma$ or
${{D}^{*+}}\rightarrow{{D}^{+}}{{\pi}^{0}}$ decay, where the neutral particle
is not reconstructed. A non-parametric shape derived from simulation is used
to model this contribution. The shape is characterised by an 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. As in previous studies of similar processes [12,
54], the fit quality improves when the shape is allowed to be offset by a
small shift ($\approx 3.5\mathrm{\,Me\kern-1.00006ptV}$) that is determined
from the data.

The combinatorial background is modelled with a linear function, where the
slope is free to vary. Many sources of misidentified background have broad
${{D}^{+}}{{\pi}^{-}}{{\pi}^{-}}$ invariant mass distributions that can be
absorbed into the combinatorial background component. The exceptions are
${{{B}^{-}}}\rightarrow D^{(*)+}{{K}^{-}}{{\pi}^{-}}$ decays that produce
distinctive shapes in the ${B}$ candidate invariant mass distribution. These
backgrounds are combined into a single non-parametric shape determined from
simulated samples that are weighted to account for the known DP distribution
for ${{{B}^{-}}}\rightarrow{{D}^{+}}{{K}^{-}}{{\pi}^{-}}$ decays [12]. The
ratio of ${{D}^{+}}$ and ${{D}^{*+}}$ components in the
${{{B}^{-}}}\rightarrow D^{(*)+}{{K}^{-}}{{\pi}^{-}}$ background shape is
fixed from the measured values of the
${{{B}^{-}}}\rightarrow{{D}^{+}}{{\pi}^{-}}{{\pi}^{-}}$ and
${{{B}^{-}}}\rightarrow{{D}^{*+}}{{\pi}^{-}}{{\pi}^{-}}$ branching fractions
[8, 19] since
$\mathcal{B}({{{B}^{-}}}\rightarrow{{D}^{*+}}{{K}^{-}}{{\pi}^{-}})$ is
unknown.

Figure 1: Results of the fit to the $B$ candidate invariant mass distribution shown with (left) linear and (right) logarithmic $y$-axis scales. Contributions are as described in the legend. Table 2: Yields of the various components in the fit to ${{{B}^{-}}}\rightarrow{{D}^{+}}{{\pi}^{-}}{{\pi}^{-}}$ candidate invariant mass distribution. Note that the yields in the signal region are scaled from the full mass range.  Component | Full mass range | Signal region
---|---|---
$N({{{B}^{-}}}\rightarrow{{D}^{+}}{{\pi}^{-}}{{\pi}^{-}})$ | $29\,190\pm 204$ | $27\,956\pm 195$
$N({{{B}^{-}}}\rightarrow D^{(*)+}{{K}^{-}}{{\pi}^{-}})$ | $\phantom{29\,}807\pm 123$ | $\phantom{27\,}243\pm 37\phantom{1}$
$N({{{B}^{-}}}\rightarrow{{D}^{*+}}{{\pi}^{-}}{{\pi}^{-}})$ | $12\,120\pm 115$ | $\phantom{27\,9}70\pm 1\phantom{11}$
$N(\rm{comb.\ bkg.})$ | $\phantom{29\,}784\pm 54\phantom{4}$ | $\phantom{27\,}103\pm 7\phantom{11}$

There are 10 parameters in the fit that are free to vary: the yields for
signal and combinatorial, ${{{B}^{-}}}\rightarrow
D^{(*)+}{{K}^{-}}{{\pi}^{-}}$ and
${{{B}^{-}}}\rightarrow{{D}^{*+}}{{\pi}^{-}}{{\pi}^{-}}$ backgrounds; the
combinatorial background slope; the shared mean of the double CB shape, the
width and relative normalisation of the narrower CB and the ratio of CB
widths; and the shift parameter of the
${{{B}^{-}}}\rightarrow{{D}^{*+}}{{\pi}^{-}}{{\pi}^{-}}$ shape. The result of
the fit is shown in Fig. 1 and gives a signal yield of approximately 29 000
decays. The $\chi^{2}$ per degree of freedom for this projection of the fit is
$1.16$, calculated with statistical uncertainties only. Component yields are
shown in Table 2 for both the full fit range and the signal region defined as
$\pm 2.5\,\sigma$ around the $B$ peak, where $\sigma$ is the width parameter
of the dominant CB function in the signal shape; this corresponds to
$5235.3<m({{D}^{+}}{{\pi}^{-}}{{\pi}^{-}})<5320.8\mathrm{\,Me\kern-1.00006ptV}$.

A Dalitz plot [55] is a two-dimensional 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
${{{B}^{-}}}\rightarrow{{D}^{+}}{{\pi}^{-}}{{\pi}^{-}}$ decays there are two
indistinguishable pions in the final state, so the two
$m^{2}({{D}^{+}}{{\pi}^{-}})$ combinations are ordered by value and the DP
axes are defined as $m^{2}({{D}^{+}}{{\pi}^{-}})_{\rm min}$ and
$m^{2}({{D}^{+}}{{\pi}^{-}})_{\rm max}$. The ordering causes a “folding” of
the DP from the minimum value of $m^{2}({{D}^{+}}{{\pi}^{-}})_{\rm max}$,
which is $m_{{{B}^{-}}}m_{{D}^{+}}+m_{{\pi}^{-}}^{2}$, to the maximum value of
$m^{2}({{D}^{+}}{{\pi}^{-}})_{\rm min}$ at
$\left(m_{{{B}^{-}}}^{2}+m_{{D}^{+}}^{2}-2m_{{\pi}^{-}}^{2}\right)/2$. The DP
distribution of the candidates in the signal region that are used in the DP
fit is shown in Fig. 2 (left). The same data are shown in the square Dalitz
plot (SDP) in Fig. 2 (right). The SDP is defined by the variables $m^{\prime}$
and $\theta^{\prime}$, which are given by

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

where $m^{\rm max}_{{{\pi}^{-}}{{\pi}^{-}}}=m_{{{{B}^{-}}}}-m_{{{D}^{+}}}$ and
$m^{\rm min}_{{{\pi}^{-}}{{\pi}^{-}}}=2m_{{{\pi}^{-}}}$ are the kinematic
boundaries of $m({{\pi}^{-}}{{\pi}^{-}})$ and $\theta({{\pi}^{-}}{{\pi}^{-}})$
is the helicity angle of the ${{\pi}^{-}}{{\pi}^{-}}$ system (the angle
between the momenta of the ${D}$ meson and one of the pions, evaluated in the
${{\pi}^{-}}{{\pi}^{-}}$ rest frame). With $m^{\prime}$ and $\theta^{\prime}$
defined in terms of the ${{\pi}^{-}}{{\pi}^{-}}$ mass and helicity angle in
this way, only the region of the SDP with $\mbox{$\theta^{\prime}$}\leq 0.5$
is populated due to the symmetry of the two pions in the final state. The SDP
is used to describe the signal efficiency variation and distribution of
background candidates, as described in Sec. 7.

Figure 2: Distribution of
${{{B}^{-}}}\rightarrow{{D}^{+}}{{\pi}^{-}}{{\pi}^{-}}$ candidates in the
signal region over (left) the DP and (right) the SDP.

##  5 Study of angular moments

The angular moments of the
${{{B}^{-}}}\rightarrow{{D}^{+}}{{\pi}^{-}}{{\pi}^{-}}$ decays are studied to
investigate which amplitudes to include in the DP fit model. Angular moments
are determined by weighting the data by the Legendre polynomial
$P_{L}\left(\cos\theta({{D}^{+}}{{\pi}^{-}})\right)$, where
$\theta({{D}^{+}}{{\pi}^{-}})$ is the helicity angle of the
${{D}^{+}}{{\pi}^{-}}$ system, i.e. the angle between the momenta of the pion
in the ${{D}^{+}}{{\pi}^{-}}$ system and the other pion from the ${{B}^{-}}$
decay, evaluated in the ${{D}^{+}}{{\pi}^{-}}$ rest frame. The moment
$\left\langle P_{L}\right\rangle$ is the sum of the weighted data in a bin of
${{D}^{+}}{{\pi}^{-}}$ mass with background contributions subtracted using
sideband data and efficiency corrections, determined as in Sec. 7.1, applied.
Each of the moments contains contributions from certain partial waves and
interference terms. For the S-, P-, D- and F-wave amplitudes denoted by
$h_{j}e^{i\delta_{j}}$ ($j=0,1,2,3$ respectively),

$\displaystyle\left\langle P_{0}\right\rangle\propto\,$
$\displaystyle\left|h_{0}\right|{}^{2}+\left|h_{1}\right|{}^{2}+\left|h_{2}\right|{}^{2}+\left|h_{3}\right|{}^{2}\,,$
(2) $\displaystyle\left\langle P_{1}\right\rangle\propto\,$
$\displaystyle\frac{2}{\sqrt{3}}\left|h_{0}\right|\left|h_{1}\right|\cos\left(\delta_{0}-\delta_{1}\right)+\frac{4}{\sqrt{15}}\left|h_{1}\right|\left|h_{2}\right|\cos\left(\delta_{1}-\delta_{2}\right)+$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\frac{6}{\sqrt{35}}\left|h_{2}\right|\left|h_{3}\right|\cos\left(\delta_{2}-\delta_{3}\right)\,,$
(3) $\displaystyle\left\langle P_{2}\right\rangle\propto\,$
$\displaystyle\frac{6}{5}\sqrt{\frac{3}{7}}\left|h_{1}\right|\left|h_{3}\right|\cos\left(\delta_{1}-\delta_{3}\right)+\frac{2\left|h_{0}\right|\left|h_{2}\right|\cos\left(\delta_{0}-\delta_{2}\right)}{\sqrt{5}}+$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\frac{2\left|h_{1}\right|{}^{2}}{5}+\frac{2\left|h_{2}\right|{}^{2}}{7}+\frac{4\left|h_{3}\right|{}^{2}}{15}\,,$
(4) $\displaystyle\left\langle P_{3}\right\rangle\propto\,$
$\displaystyle\frac{6}{7}\sqrt{\frac{3}{5}}\left|h_{1}\right|\left|h_{2}\right|\cos\left(\delta_{1}-\delta_{2}\right)+\frac{2\left|h_{0}\right|\left|h_{3}\right|\cos\left(\delta_{0}-\delta_{3}\right)}{\sqrt{7}}+$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\frac{8\left|h_{2}\right|\left|h_{3}\right|\cos\left(\delta_{2}-\delta_{3}\right)}{3\sqrt{35}}\,,$
(5) $\displaystyle\left\langle P_{4}\right\rangle\propto\,$
$\displaystyle\frac{8\left|h_{1}\right|\left|h_{3}\right|\cos\left(\delta_{1}-\delta_{3}\right)}{3\sqrt{21}}+\frac{2\left|h_{2}\right|{}^{2}}{7}+\frac{2\left|h_{3}\right|{}^{2}}{11}\,,$
(6) $\displaystyle\left\langle P_{5}\right\rangle\propto\,$
$\displaystyle\frac{20}{33}\sqrt{\frac{5}{7}}\left|h_{2}\right|\left|h_{3}\right|\cos\left(\delta_{2}-\delta_{3}\right)\,,$
(7) $\displaystyle\left\langle P_{6}\right\rangle\propto\,$
$\displaystyle\frac{100\left|h_{3}\right|{}^{2}}{429}\,.$ (8)

These expressions assume that there are no contributions from partial waves
higher than F-wave. Thus, they are valid only in regions of the DP unaffected
by the folding, i.e. for $m({{D}^{+}}{{\pi}^{-}})\lesssim
3.2\mathrm{\,Ge\kern-1.00006ptV}$, where the full range of the
${{D}^{+}}{{\pi}^{-}}$ helicity angle distribution is available. Above this
mass, the orthogonality of the Legendre polynomials does not hold and a
straightforward interpretation of the angular moments in terms of the
contributing partial waves is not possible. Nevertheless, the angular moments
provide a useful way to judge the agreement of the fit result with the data,
complementary to the projections onto the invariant masses.

The unnormalised angular moments $\left\langle
P_{0}\right\rangle$–$\left\langle P_{6}\right\rangle$ are shown in Fig. 3 for
the ${{D}^{+}}{{\pi}^{-}}$ invariant mass range
$2.0$–$4.0\mathrm{\,Ge\kern-1.00006ptV}$. The ${D}^{*}_{2}(2460)^{0}$
resonance is clearly seen in the $\left\langle P_{4}\right\rangle$
distribution of Fig. 3(e). From Eqs. (3) and (5) it can be inferred that the
structures in the distributions of $\left\langle P_{1}\right\rangle$ and
$\left\langle P_{3}\right\rangle$ below $3\mathrm{\,Ge\kern-1.00006ptV}$
suggest that there is interference both between S- and P-wave amplitudes and
between P- and D-wave amplitudes. Therefore broad spin 0 and spin 1 components
are required in the DP model. In addition, structure in $\left\langle
P_{2}\right\rangle$ around $2.76\mathrm{\,Ge\kern-1.00006ptV}$ implies the
possible presence of a spin 1 resonance in that region. The angular moments
$\left\langle P_{7}\right\rangle$ and $\left\langle P_{8}\right\rangle$, shown
in Fig. 4, show no structure, consistent with the assumption that
contributions from higher partial waves and from the isospin-2 dipion channel
are small.

Figure 3: The first seven unnormalised angular moments for background-
subtracted and efficiency-corrected data (black points) as a function of
$m({{D}^{+}}{{\pi}^{-}})$ in the range
$2.0$–$4.0\mathrm{\,Ge\kern-0.90005ptV}$. The blue line shows the result of
the DP fit described in Sec. 7.

Figure 4: Unnormalised angular moments $\left\langle P_{7}\right\rangle$ and
$\left\langle P_{8}\right\rangle$ for background-subtracted and efficiency-
corrected data (black points) as a function of $m({{D}^{+}}{{\pi}^{-}})$ in
the range $2.0$–$4.0\mathrm{\,Ge\kern-0.90005ptV}$. The blue line shows the
result of the DP fit described in Sec. 7.

Figure 5: Zoomed views of the fourth and sixth unnormalised angular moments
for background-subtracted and efficiency-corrected data (black points) as a
function of $m({{D}^{+}}{{\pi}^{-}})$. The blue line shows the result of the
DP fit described in Sec. 7.

Zoomed views of the fourth and sixth moments in the region around
$m({{D}^{+}}{{\pi}^{-}})=3\mathrm{\,Ge\kern-1.00006ptV}$ are shown in Fig. 5.
A wide bump is visible in the distribution of $\left\langle
P_{4}\right\rangle$ at $m({{D}^{+}}{{\pi}^{-}})\approx
3\mathrm{\,Ge\kern-1.00006ptV}$. Although close to the point where the DP
folding affects the interpretation of the moments, this enhancement suggests
that an additional spin 2 resonance could be contributing in this region. A
peak is also seen at $m({{D}^{+}}{{\pi}^{-}})\approx
2.76\mathrm{\,Ge\kern-1.00006ptV}$ in the $\left\langle P_{6}\right\rangle$
distribution, suggesting that a spin 3 resonance should be included in the DP
model. As discussed in Sec. 1, other recent analyses [6, 7, 15, 16, 11, 12]
suggest that both spin 1 and spin 3 states could be expected in this region.

##  6 Dalitz plot analysis formalism

The isobar approach [56, 57, 58] is used to describe the complex decay
amplitude as the coherent sum of amplitudes for intermediate resonant and
nonresonant decays. The total amplitude is given by

${\cal A}\left(s,t\right)=\sum_{j=1}^{N}c_{j}F_{j}\left(s,t\right)\,,$ (9)

where the complex coefficients $c_{j}$ describe the relative contribution of
each intermediate process. Here, and for the remainder of this section,
$m^{2}({{D}^{+}}{{\pi}^{-}})_{\rm min}$ and $m^{2}({{D}^{+}}{{\pi}^{-}})_{\rm
max}$ are referred to as $s$ and $t$, respectively.

The resonant dynamics are encoded in the $F_{j}\left(s,t\right)$ terms, each
of which is normalised such that the integral of the magnitude squared across
the DP is unity. The amplitude is explicitly symmetrised to take account of
the Bose symmetry of the final state due to the identical pions, i.e.

${\cal A}\left(s,t\right)\mapsto{\cal A}\left(s,t\right)+{\cal
A}\left(t,s\right)\,.$ (10)

This substitution is implied throughout this section.

For a ${{D}^{+}}{{\pi}^{-}}$ resonance

$F\left(s,t\right)=R\left(s\right)\times X(|\vec{p}\,|\,r_{\rm BW})\times
X(|\vec{q}\,|\,r_{\rm BW})\times T(\vec{p},\vec{q}\,)\,,$ (11)

where $\vec{p}$ and $\vec{q}$ are the momenta, calculated in the
${{D}^{+}}{{\pi}^{-}}$ rest frame, of the particle not involved in the
resonance and one of the resonance decay products, respectively. The functions
$X$, $T$ and $R$ are described below.

The $X(z)$ terms are Blatt–Weisskopf barrier factors [59], where
$z=|\vec{q}\,|\,r_{\rm BW}$ or $|\vec{p}\,|\,r_{\rm BW}$ and $r_{\rm BW}$ is
the barrier radius, and are given by

$\begin{array}[]{rcl}L=0\ :\ X(z)&=&\displaystyle{1}\,,\\\ L=1\ :\
X(z)&=&\displaystyle{\sqrt{\frac{1+z_{0}^{2}}{1+z^{2}}}}\,,\\\ L=2\ :\
X(z)&=&\displaystyle{\sqrt{\frac{z_{0}^{4}+3z_{0}^{2}+9}{z^{4}+3z^{2}+9}}}\,,\\\
L=3\ :\
X(z)&=&\displaystyle{\sqrt{\frac{z_{0}^{6}+6z_{0}^{4}+45z_{0}^{2}+225}{z^{6}+6z^{4}+45z^{2}+225}}}\,,\end{array}$
(12)

where $L$ is the spin of the resonance and $z_{0}$ is defined as the value of
$z$ where the invariant mass is equal to the mass of the resonance. Since the
${{B}^{-}}$ meson has zero spin, $L$ is also the orbital angular momentum
between the resonance and the other pion. The barrier radius, $r_{\rm BW}$, is
taken to be $4.0\mathrm{\,Ge\kern-1.00006ptV}^{-1}\approx 0.8\rm\,fm$ [60, 16]
for all resonances.

The $T(\vec{p},\vec{q})$ functions describe the angular distribution and are
given in the Zemach tensor formalism [61, 62],

$\begin{array}[]{rcl}L=0\ :\ T(\vec{p},\vec{q})&=&\displaystyle{1}\,,\\\ L=1\
:\ T(\vec{p},\vec{q})&=&\displaystyle{-\,2\,\vec{p}\cdot\vec{q}}\,,\\\ L=2\ :\
T(\vec{p},\vec{q})&=&\displaystyle{\frac{4}{3}\left[3(\vec{p}\cdot\vec{q}\,)^{2}-(|\vec{p}\,||\vec{q}\,|)^{2}\right]}\,,\\\
L=3\ :\
T(\vec{p},\vec{q})&=&\displaystyle{-\,\frac{24}{15}\left[5(\vec{p}\cdot\vec{q}\,)^{3}-3(\vec{p}\cdot\vec{q}\,)(|\vec{p}\,||\vec{q}\,|)^{2}\right]}\,.\end{array}$
(13)

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

The function $R\left(s\right)$ of Eq. (11) describes the resonance lineshape.
Resonant contributions to the total amplitude are modelled by relativistic
Breit–Wigner (RBW) functions, given by

$R(s)=\frac{1}{(m_{0}^{2}-s)-i\,m_{0}\Gamma(\sqrt{s})}\,,$ (14)

with a mass-dependent decay width defined as

$\Gamma(m)=\Gamma_{0}\left(\frac{q}{q_{0}}\right)^{2L+1}\left(\frac{m_{0}}{m}\right)X^{2}(q\,r_{\rm
BW})\,,$ (15)

where $q_{0}$ is the value of $q\equiv|\vec{q}\,|$ when $m=m_{0}$ and
$\Gamma_{0}$ is the full width. Virtual contributions, from resonances with
pole masses outside the kinematically allowed region, can be described by RBW
functions with one modification: the pole mass $m_{0}$ is replaced with an
effective mass, $m_{0}^{\rm{eff}}$, in the allowed region of $s$, when the
parameter $q_{0}$ is calculated. The term $m_{0}^{\rm{eff}}$ is given by the
ad hoc formula [16]

$m_{0}^{\rm{eff}}(m_{0})=m^{\rm{min}}+(m^{\rm{max}}-m^{\rm{min}})\left(1+\tanh\left(\frac{m_{0}-\frac{m^{\rm{min}}+m^{\rm{max}}}{2}}{m^{\rm{max}}-m^{\rm{min}}}\right)\right)\,,$
(16)

where $m^{\rm{max}}$ and $m^{\rm{min}}$ are the upper and lower thresholds of
$s$. Note that $m_{0}^{\rm{eff}}$ is only used in the calculation of $q_{0}$,
so only the tail of such virtual contributions enters the DP.

A quasi-model-independent approach is used to describe the entire
${{D}^{+}}{{\pi}^{-}}$ spin 0 partial wave. The total ${{D}^{+}}{{\pi}^{-}}$
S-wave is fitted using cubic splines to describe the magnitude and phase
variation of the spin 0 amplitude. Knots are defined at fixed values of
$m({{D}^{+}}{{\pi}^{-}})$ and splines give a smooth interpolation of the
magnitude and phase of the S-wave between these points. The S-wave magnitude
and phase are both fixed to zero at the highest mass knot in order to ensure
sensible behaviour at the kinematic limit. For the knot at
$m({{D}^{+}}{{\pi}^{-}})=2.4$$\mathrm{\,Ge\kern-1.00006ptV}$, close to the
peak of the ${D}^{*}_{0}(2400)^{0}$ resonance, the magnitude and phase values
are fixed to 0.5 and 0, respectively, as a reference. The magnitude and phase
values at every other knot position are determined from the fit.

The folding of the Dalitz plot has implications for the choice of knot
positions. Since the S-wave amplitude varies with $m({{D}^{+}}{{\pi}^{-}})$,
its reflection onto the other DP axis gives a helicity angle distribution that
corresponds to higher partial waves. Equally, if knots are included at high
$m({{D}^{+}}{{\pi}^{-}})$, the quasi-model-independent ${{D}^{+}}{{\pi}^{-}}$
S-wave amplitude can absorb resonant contributions with non-zero spin due to
their reflections. To avoid this problem, only a single knot with floated
parameters is used above the minimum value of
$m^{2}({{D}^{+}}{{\pi}^{-}})_{\rm max}$, specifically at
$4.1\mathrm{\,Ge\kern-1.00006ptV}$ (as mentioned above, the amplitude is fixed
to zero at the highest mass knot at $5.1\mathrm{\,Ge\kern-1.00006ptV}$). At
lower $m({{D}^{+}}{{\pi}^{-}})$, knots are spaced every
$0.1\mathrm{\,Ge\kern-1.00006ptV}$ from $2.0\mathrm{\,Ge\kern-1.00006ptV}$ up
to $3.1\mathrm{\,Ge\kern-1.00006ptV}$, except that the knot at
$3.0\mathrm{\,Ge\kern-1.00006ptV}$ is removed in order to stabilise the fit.

Neglecting reconstruction effects, the DP probability density function would
be

${\cal{P}}_{\rm phys}\left(s,t\right)=\frac{|{\cal
A}\left(s,t\right)|^{2}}{\int\\!\\!\int_{\rm DP}~{}{|{\cal
A}\left(s,t\right)|^{2}}~{}ds\,dt}\,.$ (17)

The effects of nonuniform signal efficiency and of background contributions
are accounted for as described in Sec. 7. The probability density function
depends on the complex coefficients, introduced in Eq. (9), as well as the
masses and widths of the resonant contributions and the parameters describing
the ${{D}^{+}}{{\pi}^{-}}$ S-wave. These parameters are allowed to vary freely
in the fit. Results for the complex coefficients are dependent on the
amplitude formalism, normalisation and phase convention, and consequently may
be difficult to compare between different analyses. It is therefore useful to
define fit fractions and interference fit fractions to provide convention-
independent results. Fit fractions are defined as the integral over the DP for
a single contributing amplitude squared divided by that of the total amplitude
squared,

${\it FF}_{j}=\frac{\int\\!\\!\int_{\rm
DP}\left|c_{j}F_{j}\left(s,t\right)\right|^{2}~{}ds\,dt}{\int\\!\\!\int_{\rm
DP}\left|{\cal A}\left(s,t\right)\right|^{2}~{}ds\,dt}\,.$ (18)

The sum of fit fractions is not required to be unity due to the potential
presence of net constructive or destructive interference. Interference fit
fractions are defined, for $i<j$ only, as

${\it FF}_{ij}=\frac{\int\\!\\!\int_{\rm
DP}2\,\mathcal{R}e\left[c_{i}c_{j}^{*}F_{i}\left(s,t\right)F_{j}^{*}\left(s,t\right)\right]~{}ds\,dt}{\int\\!\\!\int_{\rm
DP}\left|{\cal A}\left(s,t\right)\right|^{2}~{}ds\,dt}\,.$ (19)

##  7 Dalitz plot fit

###  7.1 Signal efficiency

Variation of the efficiency across the phase space of
${{{B}^{-}}}\rightarrow{{D}^{+}}{{\pi}^{-}}{{\pi}^{-}}$ decays is studied in
terms of the SDP, since the efficiency variation is typically greatest close
to the kinematic boundaries of the conventional DP. The causes of efficiency
variation across the SDP are the detector acceptance and trigger, selection
and PID requirements. Simulated samples, generated uniformly over the SDP, are
used to evaluate the efficiency variation. Data-driven corrections are applied
to correct the simulation for known discrepancies with data, for the tracking,
trigger and PID efficiencies, using identical methods to those described in
Ref. [16]. The efficiency distributions are fitted with two-dimensional cubic
splines to smooth out statistical fluctuations due to limited sample size.
Figure 6 shows the efficiency variation over the SDP.

Figure 6: Signal efficiency across the SDP for
${{{B}^{-}}}\rightarrow{{D}^{+}}{{\pi}^{-}}{{\pi}^{-}}$ decays. The relative
uncertainty at each point is typically $5\,\%$.

###  7.2 Background studies

The yields presented in Table 2 show that the important background components
in the signal region are from combinatorial background and
${{{B}^{-}}}\rightarrow D^{(*)+}{{K}^{-}}{{\pi}^{-}}$ decays. The SDP
distribution of ${{{B}^{-}}}\rightarrow D^{(*)+}{{K}^{-}}{{\pi}^{-}}$ decays
is obtained from simulated samples using the same procedures as described in
Sec. 4 to apply weights and combine the ${{D}^{+}}$ and $D^{*+}$
contributions. The distribution of combinatorial background events is obtained
from ${{D}^{+}}{{\pi}^{-}}{{\pi}^{-}}$ candidates in the high-mass sideband,
defined to be $5500$–$5800\mathrm{\,Me\kern-1.00006ptV}$. Figure 7 shows the
SDP distributions of these backgrounds, which are used in the Dalitz plot fit.

Figure 7: Square Dalitz plot distributions for (left) combinatorial background
and (right) ${{{B}^{-}}}\rightarrow D^{(*)+}{{K}^{-}}{{\pi}^{-}}$ decays.

###  7.3 Amplitude model for
${{{B}^{-}}}\rightarrow{{D}^{+}}{{\pi}^{-}}{{\pi}^{-}}$ decays

The DP fit is performed using the Laura++ [63] package, and the likelihood
function is given by

${\cal L}=\prod_{i}^{n_{c}}\Bigg{[}\sum_{k}N_{k}{\cal
P}_{k}\left(s_{i},t_{i}\right)\Bigg{]}\,,$ (20)

where the index $i$ runs over $n_{c}$ candidates, while $k$ sums over the
probability density functions ${\cal P}_{k}$ with a yield of $N_{k}$
candidates in each component. For signal events ${\cal P}_{k}\equiv{\cal
P}_{\rm sig}$ is similar to Eq. (17), but is modified such that the $|{\cal
A}\left(s,t\right)|^{2}$ terms are multiplied by the efficiency function
described in Sec. 7.1. The mass resolution is approximately
$2.4\mathrm{\,Me\kern-1.00006ptV}$, which is much less 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. Its effect on the measurement of masses and widths of
resonances is, however, considered as a systematic uncertainty.

Using the results of the moments analysis presented in Sec. 5 as a guide, a
${{{B}^{-}}}\rightarrow{{D}^{+}}{{\pi}^{-}}{{\pi}^{-}}$ DP model is
constructed by including various resonant, nonresonant and virtual amplitudes.
Only intermediate states with natural spin-parity are included because
unnatural spin-parity states do not decay to two pseudoscalars. Amplitudes
that do not contribute significantly and cause the fit to become unstable are
discarded. Alternative and additional contributions that have been considered
include: an isobar description of the ${{D}^{+}}{{\pi}^{-}}$ S-wave including
the ${D}^{*}_{0}(2400)^{0}$ resonance and a nonresonant amplitude; a
nonresonant P-wave component; an isospin-2 $\pi\pi$ interaction described by a
unitary model as in Refs. [64, 24] (see also Refs. [65, 66, 67]); quasi-model-
independent descriptions of partial waves other than the
${{D}^{+}}{{\pi}^{-}}$ S-wave.

The resulting baseline signal model consists of the seven components listed in
Table 3: four resonances, two virtual resonances and a quasi-model-independent
description of the ${{D}^{+}}{{\pi}^{-}}$ S-wave. There are 42 free parameters
in this model. The broad P-wave structure indicated by the angular moments is
adequately described by the virtual $D^{*}(2007)^{0}$ and $B^{*0}$ amplitudes.
The peaks seen in various moments are described by the
${D}^{*}_{2}(2460)^{0}$, ${D}^{*}_{1}(2680)^{0}$, ${D}^{*}_{3}(2760)^{0}$ and
${D}^{*}_{2}(3000)^{0}$ resonances. Here, and throughout the paper, these
states are labelled as such since it is not clear if the
${D}^{*}_{1}(2680)^{0}$ state corresponds to one of the previously observed
peaks (see Table 1), while the parameters of the ${D}^{*}_{3}(2760)^{0}$
resonance seem to be consistent with earlier measurements. An excess at
$m({{D}^{+}}{{\pi}^{-}})\approx 3000\mathrm{\,Me\kern-1.00006ptV}$ was
reported in Ref. [7], but the parameters of this state were not reported with
systematic uncertainties. The baseline model provides a better quality fit
than the alternative models that are discussed in Sec. 8. The inclusion of all
components of the model is necessary to obtain a good description of the data,
as described in Sec. 9.

Table 3: Signal contributions to the fit model, where parameters and uncertainties are taken from Ref. [19]. States labelled with subscript $v$ are virtual contributions. The model “MIPW” refers to the quasi-model-independent partial wave approach.  Resonance | Spin | Model | Parameters
---|---|---|---
${D}^{*}_{2}(2460)^{0}$ | 2 | RBW | Determined from data (see Table 4)
${D}^{*}_{1}(2680)^{0}$ | 1 | RBW
${D}^{*}_{3}(2760)^{0}$ | 3 | RBW
${D}^{*}_{2}(3000)^{0}$ | 2 | RBW
${D}^{*}_{v}(2007)^{0}$ | 1 | RBW |  $m=2006.98\pm 0.15\mathrm{\,Me\kern-1.00006ptV}$, $\Gamma=2.1\mathrm{\,Me\kern-1.00006ptV}$
${B}^{*0}_{v}$ | 1 | RBW |  $m=5325.2\pm 0.4\mathrm{\,Me\kern-1.00006ptV}$, $\Gamma=0.0\mathrm{\,Me\kern-1.00006ptV}$
Total S-wave | 0 | MIPW | See text

The real and imaginary parts of the complex coefficients for each of the
components are free parameters of the fit, except for the
${D}^{*}_{2}(2460)^{0}$ contribution that is taken to be a reference amplitude
with real and imaginary parts of its complex coefficient $c_{k}$ fixed to 1
and 0, respectively. Parameters such as magnitudes and phases for each
amplitude, the fit fractions and interference fit fractions are calculated
from these quantities. The statistical uncertainties are determined using
large samples of pseudoexperiments to ensure that correlations between
parameters are accounted for.

###  7.4 Dalitz plot fit results

The masses and widths of the ${D}^{*}_{2}(2460)^{0}$, ${D}^{*}_{1}(2680)^{0}$,
${D}^{*}_{3}(2760)^{0}$ and ${D}^{*}_{2}(3000)^{0}$ resonances are determined
from the fit and are given in Table 4.

Table 4: Masses and widths determined in the fit to data, with statistical uncertainties only. Contribution | Mass (MeV) | Width (MeV)
---|---|---
${D}^{*}_{2}(2460)^{0}$ | $2463.7\pm 0.4$ | $\phantom{1}47.0\pm 0.8$
${D}^{*}_{1}(2680)^{0}$ | $2681.1\pm 5.6$ | $186.7\pm 8.5$
${D}^{*}_{3}(2760)^{0}$ | $2775.5\pm 4.5$ | $\phantom{1}95.3\pm 9.6$
${D}^{*}_{2}(3000)^{0}$ | $\phantom{1}3214\pm 29$ | $\phantom{1}186\pm 38$

The floated complex coefficients at each knot position and the splines
describing the total ${{D}^{+}}{{\pi}^{-}}$ S-wave are shown in Fig. 8. The
phase motion at low $m({{D}^{+}}{{\pi}^{-}})$ is consistent with that expected
due to the presence of the ${D}^{*}_{0}(2400)^{0}$ state. There is, however,
an ambiguous solution with the opposite phase motion in this region, which
occurs since there are significant contributions only from S- and P-waves and
thus only $\cos(\delta_{0}-\delta_{1})$ can be determined as seen in Eq. (3).
Since the P-wave in this region is described by the ${D}^{*}_{v}(2007)^{0}$
amplitude, and hence has slowly varying phase, the entire
${{D}^{+}}{{\pi}^{-}}$ S-wave has a sign ambiguity. Similar ambiguities have
been observed previously [23]. Only results consistent with the expected phase
motion are reported.

Figure 8: Real and imaginary parts of the S-wave amplitude, shown in an Argand
diagram. The knots are shown with statistical uncertainties only, connected by
the cubic spline interpolation used in the fit. The leftmost point is that at
the lowest value of $m({{D}^{+}}{{\pi}^{-}})$, with mass increasing along the
connected points. Each point, labelled 1–13, corresponds to the position of a
knot in the spline, at values of
$m({{D}^{+}}{{\pi}^{-}})=\\{2.01,2.10,2.20,2.30,2.40,2.50,2.60,2.70,2.80,2.90,3.10,4.10,5.14\\}\,\mathrm{\,Ge\kern-0.90005ptV}$.
The points at $(0.5,0.0)$ and $(0.0,0.0)$ are fixed. The anticlockwise
rotation of the phase at low $m({{D}^{+}}{{\pi}^{-}})$ is as expected due to
the presence of the ${D}^{*}_{0}(2400)^{0}$ resonance.

Table 5 shows the values of the complex coefficients and fit fractions for
each amplitude. The interference fit fractions are given in Appendix A.

Table 5: Complex coefficients and fit fractions determined from the Dalitz
plot fit. Uncertainties are statistical only.

Contribution |  | Isobar model coefficients
---|---|---
Fit fraction (%) | Real part | Imaginary part | Magnitude | Phase (rad)
${D}^{*}_{2}(2460)^{0}$ | $35.7\pm 0.6$ | $1.00$ | $0.00$ | $1.00$ | $0.00$
${D}^{*}_{1}(2680)^{0}$ | $\phantom{1}8.3\pm 0.6$ | $-0.38\pm 0.02$ | $\phantom{-}0.30\pm 0.02$ | $0.48\pm 0.02$ | $\phantom{-}2.47\pm 0.09$
${D}^{*}_{3}(2760)^{0}$ | $\phantom{1}1.0\pm 0.1$ | $\phantom{-}0.17\pm 0.01$ | $\phantom{-}0.00\pm 0.01$ | $0.17\pm 0.01$ | $\phantom{-}0.01\pm 0.20$
${D}^{*}_{2}(3000)^{0}$ | $\phantom{1}0.23\pm 0.07$ | $\phantom{-}0.05\pm 0.02$ | $-0.06\pm 0.02$ | $0.08\pm 0.01$ | $-0.84\pm 0.28$
${D}^{*}_{v}(2007)^{0}$ | $10.8\pm 0.7$ | $\phantom{-}0.51\pm 0.03$ | $-0.20\pm 0.05$ | $0.55\pm 0.02$ | $-0.38\pm 0.19$
${B}^{*0}_{v}$ | $\phantom{1}2.7\pm 1.0$ | $\phantom{-}0.27\pm 0.03$ | $\phantom{-}0.04\pm 0.04$ | $0.27\pm 0.05$ | $\phantom{-}0.14\pm 0.38$
Total S-wave | $57.0\pm 0.8$ | $\phantom{-}1.21\pm 0.02$ | $-0.35\pm 0.04$ | $1.26\pm 0.01$ | $-0.28\pm 0.05$
Total fit fraction | 115.7 |  |  |  |

Given the complexity of the DP fit, the minimisation procedure may find local
minima in the likelihood function. To try to ensure that the global minimum is
found, the fit is performed many times with randomised initial values for the
$c_{j}$ terms. No other minima are found with negative log-likelihood values
close to that of the global minimum so they are not considered further.

Figure 9: Differences between the SDP distribution of the data and fit model,
in terms of the normalised residual in each bin. No bin lies outside the
$z$-axis limits.

The consistency of the fit model and the data is evaluated in several ways.
Numerous one-dimensional projections comparing the data and fit model
(including several shown below and those from the moments study in Sec. 5)
show good agreement. Additionally, a two-dimensional $\chi^{2}$ value is
calculated by comparing the data and the fit model distributions across the
SDP in $484$ equally populated bins. Figure 9 shows the normalised residual in
each bin. The distribution of the $z$-axis values from Fig. 9 is consistent
with a unit Gaussian centered on zero. Further checks using unbinned fit
quality tests [68] show satisfactory agreement between the data and the fit
model.

One-dimensional projections of the baseline fit model and data onto
$m({{D}^{+}}{{\pi}^{-}})_{\rm min}$, $m({{D}^{+}}{{\pi}^{-}})_{\rm max}$ and
$m({{\pi}^{-}}{{\pi}^{-}})$ are shown in Fig. 10. The model is seen to give a
good description of the data sample, with the most evident discrepancy at low
values of $m({{D}^{+}}{{\pi}^{-}})_{\rm max}$, a region of the DP (that
corresponds to high values of $m({{\pi}^{-}}{{\pi}^{-}})$ and
$m({{D}^{+}}{{\pi}^{-}})_{\rm min}\approx 3.2\mathrm{\,Ge\kern-1.00006ptV}$)
in which many different amplitudes contribute. In Fig. 11, zoomed views of the
$m({{D}^{+}}{{\pi}^{-}})_{\rm min}$ invariant mass projection are provided for
regions at threshold and around the ${D}^{*}_{2}(2460)^{0}$,
${D}^{*}_{1}(2680)^{0}$–${D}^{*}_{3}(2760)^{0}$ and ${D}^{*}_{2}(3000)^{0}$
resonances. Projections of the cosine of the ${{D}^{+}}{{\pi}^{-}}$ helicity
angle in the same regions of $m({{D}^{+}}{{\pi}^{-}})_{\rm min}$ are also
shown in Fig. 11. Good agreement is seen in all these projections, suggesting
that the model gives an acceptable description of the data and the spin
assignments of the ${D}^{*}_{1}(2680)^{0}$, ${D}^{*}_{3}(2760)^{0}$ and
${D}^{*}_{2}(3000)^{0}$ states are correct.

Figure 10: Projections of the data and amplitude fit onto (top)
$m({{D}^{+}}{{\pi}^{-}})_{\rm min}$, (middle) $m({{D}^{+}}{{\pi}^{-}})_{\rm
max}$ and (bottom) $m({{\pi}^{-}}{{\pi}^{-}})$, with the same projections
shown (right) with a logarithmic $y$-axis scale. Components are described in
the legend.

Figure 11: Projections of the data and amplitude fit onto (left)
$m({{D}^{+}}{{\pi}^{-}})$ and (right) the cosine of the helicity angle for the
${{D}^{+}}{{\pi}^{-}}$ system in (top to bottom) the low mass threshold
region, the ${D}^{*}_{2}(2460)^{0}$ region, the
${D}^{*}_{1}(2680)^{0}$–${D}^{*}_{3}(2760)^{0}$ region and the
${D}^{*}_{2}(3000)^{0}$ region. Components are as shown in Fig. 10.

##  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, and possible fit bias. Model uncertainties arise due to the fixed
parameters in the amplitude model, the addition of amplitudes not included in
the baseline fit, the modelling of the amplitudes from virtual resonances, and
the effect of removing the least well modelled part of the phase space. 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.
The total uncertainty on each yield, including systematic effects due to the
modelling of the components in the $B$ candidate mass fit, is calculated, and
the yields varied accordingly in the DP fit. The deviations from the baseline
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 bin contents of the histograms used to model the
shapes within their statistical uncertainties. For ${{{B}^{-}}}\rightarrow
D^{(*)+}{{K}^{-}}{{\pi}^{-}}$ 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. Changes in the results
compared to the baseline DP fit result are again assigned as systematic
uncertainties.

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 $3\times 3$ cluster is varied by its
statistical uncertainty and the surrounding bins in the cluster 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. An ensemble of DP fits are performed, each with a unique
efficiency histogram, and the effects on the 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.

The only fixed parameter in the lineshapes of resonant amplitudes is the
Blatt–Weisskopf barrier radius, $r_{\rm BW}$. To account for potential
systematic effects, this is varied between 3 and 5
$\mathrm{\,Ge\kern-1.00006ptV}^{-1}$ [16], and the difference compared to the
baseline fit model is assigned as an uncertainty. The choice of knot positions
in the quasi-model-independent description of the ${{D}^{+}}{{\pi}^{-}}$
S-wave is another source of possible systematic uncertainty. This is evaluated
from the change in the fit results when more knots are added at low
$m({{D}^{+}}{{\pi}^{-}})$. As discussed in Sec. 6, it is not possible to add
more knots at high $m({{D}^{+}}{{\pi}^{-}})$ without destabilising the fit.

As discussed in Sec. 1, it is possible that there is more than one spin 1
resonance in the range
$2.6<m({{D}^{+}}{{\pi}^{-}})<2.8\mathrm{\,Ge\kern-1.00006ptV}$. The measured
parameters of the ${D}^{*}_{1}(2680)^{0}$ resonance are most consistent with
those given for the $D^{*}(2650)$ state in Table 1, therefore the effect of
including an additional $D^{*}(2760)$ contribution is considered as a source
of systematic uncertainty. Separate fits are performed with the parameters of
the $D^{*}(2760)$ state fixed to the values determined by BaBar [6] and LHCb
[7] and the larger of the deviations from the baseline results is taken as the
associated uncertainty. Additional fits are performed with the value of the
${D}^{*}_{v}(2007)^{0}$ width given in Table 3, which corresponds to the
current experimental upper limit [19], replaced by the measured central value
for the ${{D}^{*}}(2010)^{+}$ ($83.4\mathrm{\,ke\kern-1.00006ptV}$); the
associated systematic uncertainty is negligible. The dependence of the results
on the effective pole mass description of Eq. (16) that is used for the
virtual resonance contributions is found by using a fixed width in Eq. (14),
removing the dependence on $m^{\rm eff}_{0}$.

A discrepancy between the model and the data is seen in the low
$m({{D}^{+}}{{\pi}^{-}})_{\rm max}$ region, as discussed in Sec. 7.4. Since
this may not be accounted for by the other sources of systematic uncertainty,
the effect on the results is determined by performing fits where this region
of the DP is vetoed by removing separately candidates with either
$m({{D}^{+}}{{\pi}^{-}})_{\rm max}<3.3\mathrm{\,Ge\kern-1.00006ptV}$ or
$m({{\pi}^{-}}{{\pi}^{-}})>3.05\mathrm{\,Ge\kern-1.00006ptV}$. Systematic
uncertainties are assigned as the difference in the fitted parameters compared
to the baseline fit.

Contributions to the experimental and model systematic uncertainties for the
fit fractions, masses and widths are broken down in Tables 6 and 7. The
largest source of experimental systematic uncertainty for many parameters is
the knowledge of the efficiency variation across the Dalitz plot. The various
parameters are affected differently by the sources of model uncertainty, with
some being affected by the variation of fixed parameters in the model, others
(notably the parameters associated with the ${D}^{*}_{1}(2680)^{0}$ amplitude)
by the introduction of an additional $D_{1}^{*}(2760)^{0}$ resonance, and some
changing when the poorly-modelled region of phase space is vetoed. The effect
of the finite mass resolution, described in Sec. 7.3, on the measurements of
the masses and widths of resonances is found to be negligible.

Table 6: Breakdown of experimental systematic uncertainties on the fit fractions (%) and masses and widths $(\mathrm{Me\kern-0.90005ptV})$. | Nominal | S/B frac. | Eff. | Bkgd. | Fit bias | Total
---|---|---|---|---|---|---
${D}^{*}_{2}(2460)^{0}$ |  $35.7$$\,\pm\,$ | $0.6$ | 0.1 | 1.3 | 0.0 | 0.2 | 1.4
${D}^{*}_{1}(2680)^{0}$ |  $8.3$$\,\pm\,$ | $0.6$ | 0.0 | 0.7 | 0.1 | 0.1 | 0.7
${D}^{*}_{3}(2760)^{0}$ |  $1.0$$\,\pm\,$ | $0.1$ | 0.0 | 0.1 | 0.0 | 0.0 | 0.1
${D}^{*}_{2}(3000)^{0}$ |  $0.2$$\,\pm\,$ | $0.1$ | 0.0 | 0.1 | 0.0 | 0.0 | 0.1
${D}^{*}_{v}(2007)^{0}$ |  $10.8$$\,\pm\,$ | $0.7$ | 0.0 | 0.7 | 0.1 | 0.1 | 0.7
$B^{*}_{v}$ |  $2.7$$\,\pm\,$ | $1.0$ | 0.0 | 1.4 | 0.1 | 0.2 | 1.4
Total S-wave |  $57.0$$\,\pm\,$ | $0.8$ | 0.1 | 0.6 | 0.1 | 0.1 | 0.6
$m\left({D}^{*}_{2}(2460)^{0}\right)$ |  $2463.7$$\,\pm\,$ | $0.4$ | 0.0 | 0.3 | 0.1 | 0.1 | 0.3
$\Gamma\left({D}^{*}_{2}(2460)^{0}\right)$ |  $47.0$$\,\pm\,$ | $0.8$ | 0.1 | 0.9 | 0.1 | 0.0 | 0.9
$m\left({D}^{*}_{1}(2680)^{0}\right)$ |  $2681.1$$\,\pm\,$ | $5.6$ | 0.1 | 4.8 | 0.9 | 0.2 | 4.9
$\Gamma\left({D}^{*}_{1}(2680)^{0}\right)$ |  $186.7$$\,\pm\,$ | $8.5$ | 0.5 | 8.4 | 1.0 | 1.2 | 8.6
$m\left({D}^{*}_{3}(2760)^{0}\right)$ |  $2775.5$$\,\pm\,$ | $4.5$ | 0.4 | 4.4 | 0.6 | 0.4 | 4.5
$\Gamma\left({D}^{*}_{3}(2760)^{0}\right)$ |  $95.3$$\,\pm\,$ | $9.6$ | 0.9 | 5.9 | 1.5 | 4.9 | 7.9
$m\left({D}^{*}_{2}(3000)^{0}\right)$ |  $3214$$\,\pm\,$ | $29$ | 3 | 29 | 13 | 9 | 33
$\Gamma\left({D}^{*}_{2}(3000)^{0}\right)$ |  $186$$\,\pm\,$ | $38$ | 2 | 31 | 8 | 12 | 34
Table 7: Breakdown of model uncertainties on the fit fractions (%) and masses and widths $(\mathrm{Me\kern-0.90005ptV})$. | Nominal | Fixed | Add | Alternative | DP veto | Total
---|---|---|---|---|---|---
|  | params. | $D_{1}^{*}(2760)^{0}$ | models |  |
${D}^{*}_{2}(2460)^{0}$ |  $35.7$$\,\pm\,$ | $0.6$ | 0.9 | 0.0 | 0.0 | 0.1 | 0.9
${D}^{*}_{1}(2680)^{0}$ |  $8.3$$\,\pm\,$ | $0.6$ | 0.2 | 0.9 | 0.0 | 1.5 | 1.8
${D}^{*}_{3}(2760)^{0}$ |  $1.0$$\,\pm\,$ | $0.1$ | 0.0 | 0.0 | 0.0 | 0.2 | 0.2
${D}^{*}_{2}(3000)^{0}$ |  $0.2$$\,\pm\,$ | $0.1$ | 0.0 | 0.0 | 0.0 | 0.1 | 0.1
${D}^{*}_{v}(2007)^{0}$ |  $10.8$$\,\pm\,$ | $0.7$ | 2.3 | 0.1 | 0.0 | 0.2 | 2.3
$B^{*}_{v}$ |  $2.7$$\,\pm\,$ | $1.0$ | 1.2 | 0.2 | 0.0 | 1.0 | 1.6
Total S-wave |  $57.0$$\,\pm\,$ | $0.8$ | 0.8 | 0.4 | 0.0 | 0.1 | 0.9
$m\left({D}^{*}_{2}(2460)^{0}\right)$ |  $2463.7$$\,\pm\,$ | $0.4$ | 0.4 | 0.1 | 0.0 | 0.4 | 0.6
$\Gamma\left({D}^{*}_{2}(2460)^{0}\right)$ |  $47.0$$\,\pm\,$ | $0.8$ | 0.2 | 0.0 | 0.0 | 0.1 | 0.3
$m\left({D}^{*}_{1}(2680)^{0}\right)$ |  $2681.1$$\,\pm\,$ | $5.6$ | 4.7 | 11.8 | 0.1 | 3.0 | 13.1
$\Gamma\left({D}^{*}_{1}(2680)^{0}\right)$ |  $186.7$$\,\pm\,$ | $8.5$ | 3.2 | 4.5 | 0.3 | 6.0 | 8.2
$m\left({D}^{*}_{3}(2760)^{0}\right)$ |  $2775.5$$\,\pm\,$ | $4.5$ | 3.4 | 0.4 | 0.0 | 3.3 | 4.7
$\Gamma\left({D}^{*}_{3}(2760)^{0}\right)$ |  $95.3$$\,\pm\,$ | $9.6$ | 2.8 | 3.2 | 0.0 | 32.9 | 33.1
$m\left({D}^{*}_{2}(3000)^{0}\right)$ |  $3214$$\,\pm\,$ | $29$ | 25 | 1 | 1 | 26 | 36
$\Gamma\left({D}^{*}_{2}(3000)^{0}\right)$ |  $186$$\,\pm\,$ | $38$ | 7 | 19 | 0 | 60 | 63

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. All fits
give consistent results.

##  9 Results and summary

Results for the complex coefficients multiplying each amplitude are reported
in Table 8, and those that describe the ${{D}^{+}}{{\pi}^{-}}$ S-wave
amplitude are shown in Table 9. These complex numbers are reported in terms of
real and imaginary parts and also in terms of magnitude and phase as, due to
correlations, the propagation of uncertainties from one form to the other may
not be trivial. Results for the interference fit fractions are given in
Appendix A.

The fit fractions, summarised in Table 10, for resonant contributions are
converted into quasi-two-body product branching fractions by multiplying by
the ${{{B}^{-}}}\rightarrow{{D}^{+}}{{\pi}^{-}}{{\pi}^{-}}$ branching
fraction. This value is taken from the world average after a correction for
the relative branching fractions of ${{{B}^{+}}}{{{B}^{-}}}$ and
${{B}^{0}}{{\kern 1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}}$ pairs at the
${\mathchar 28935\relax{(4S)}}$ resonance, $\Gamma({\mathchar
28935\relax{(4S)}}\rightarrow{{{B}^{+}}}{{{B}^{-}}})/\Gamma({\mathchar
28935\relax{(4S)}}\rightarrow{{B}^{0}}{{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}})=1.055\pm 0.025$ [19], giving
${\cal
B}\left({{{B}^{-}}}\rightarrow{{D}^{+}}{{\pi}^{-}}{{\pi}^{-}}\right)=(1.014\pm
0.054)\times 10^{-3}$. The product branching fractions are shown in Table 11;
they cannot be converted into absolute branching fractions because the
branching fractions for the resonance decays to ${{D}^{+}}{{\pi}^{-}}$ are
unknown.

Table 8: Results for the complex amplitudes. The three quoted errors are statistical, experimental systematic and model uncertainties. Resonance | Isobar model coefficients
---|---
| Real part | Imaginary part
${D}^{*}_{2}(2460)^{0}$ | $1.00$ | $0.00$
${D}^{*}_{1}(2680)^{0}$ | $-0.38\pm 0.02\pm 0.05\pm 0.08$ | $\phantom{-}0.30\pm 0.02\pm 0.08\pm 0.03$
${D}^{*}_{3}(2760)^{0}$ | $\phantom{-}0.17\pm 0.01\pm 0.01\pm 0.02$ | $\phantom{-}0.00\pm 0.01\pm 0.05\pm 0.02$
${D}^{*}_{2}(3000)^{0}$ | $\phantom{-}0.05\pm 0.02\pm 0.02\pm 0.04$ | $-0.06\pm 0.02\pm 0.05\pm 0.03$
${D}^{*}_{v}(2007)^{0}$ | $\phantom{-}0.51\pm 0.03\pm 0.02\pm 0.05$ | $-0.20\pm 0.05\pm 0.11\pm 0.05$
$B^{*}_{v}$ | $\phantom{-}0.27\pm 0.03\pm 0.11\pm 0.10$ | $\phantom{-}0.04\pm 0.04\pm 0.12\pm 0.05$
Total S-wave | $\phantom{-}1.21\pm 0.02\pm 0.01\pm 0.02$ | $-0.35\pm 0.04\pm 0.07\pm 0.03$
|  |
| Magnitude | Phase
${D}^{*}_{2}(2460)^{0}$ | $1.00$ | $0.00$
${D}^{*}_{1}(2680)^{0}$ | $0.48\pm 0.02\pm 0.01\pm 0.06$ | $\phantom{-}2.47\pm 0.09\pm 0.18\pm 0.12$
${D}^{*}_{3}(2760)^{0}$ | $0.17\pm 0.01\pm 0.01\pm 0.02$ | $\phantom{-}0.01\pm 0.20\pm 0.11\pm 0.09$
${D}^{*}_{2}(3000)^{0}$ | $0.08\pm 0.01\pm 0.01\pm 0.01$ | $-0.84\pm 0.28\pm 0.52\pm 0.63$
${D}^{*}_{v}(2007)^{0}$ | $0.55\pm 0.02\pm 0.01\pm 0.06$ | $-0.38\pm 0.19\pm 0.15\pm 0.08$
$B^{*}_{v}$ | $0.27\pm 0.05\pm 0.13\pm 0.09$ | $\phantom{-}0.14\pm 0.38\pm 0.19\pm 0.25$
Total S-wave | $1.26\pm 0.01\pm 0.02\pm 0.02$ | $-0.28\pm 0.05\pm 0.05\pm 0.03$
Table 9: Results for the ${{D}^{+}}{{\pi}^{-}}$ S-wave amplitude at the spline knots. The three quoted errors are statistical, experimental systematic and model uncertainties. Knot mass |  ${{D}^{+}}{{\pi}^{-}}$ S-wave amplitude
---|---
$(\mathrm{Ge\kern-1.00006ptV})$ | Real part | Imaginary part
2.01 | $-0.11\pm 0.05\pm 0.07\pm 0.09$ | $-0.04\pm 0.03\pm 0.05\pm 0.11$
2.10 | $\phantom{-}0.00\pm 0.05\pm 0.11\pm 0.05$ | $-0.58\pm 0.02\pm 0.03\pm 0.03$
2.20 | $\phantom{-}0.39\pm 0.05\pm 0.08\pm 0.05$ | $-0.62\pm 0.04\pm 0.07\pm 0.04$
2.30 | $\phantom{-}0.62\pm 0.02\pm 0.03\pm 0.01$ | $-0.28\pm 0.05\pm 0.10\pm 0.03$
2.40 | 0.50 | 0.00
2.50 | $\phantom{-}0.23\pm 0.01\pm 0.01\pm 0.01$ | $-0.00\pm 0.02\pm 0.04\pm 0.01$
2.60 | $\phantom{-}0.21\pm 0.01\pm 0.01\pm 0.01$ | $-0.10\pm 0.02\pm 0.03\pm 0.06$
2.70 | $\phantom{-}0.14\pm 0.01\pm 0.01\pm 0.01$ | $-0.05\pm 0.01\pm 0.02\pm 0.02$
2.80 | $\phantom{-}0.14\pm 0.01\pm 0.01\pm 0.01$ | $-0.10\pm 0.01\pm 0.02\pm 0.04$
2.90 | $\phantom{-}0.13\pm 0.01\pm 0.02\pm 0.01$ | $-0.16\pm 0.01\pm 0.02\pm 0.02$
3.10 | $\phantom{-}0.05\pm 0.01\pm 0.02\pm 0.02$ | $-0.12\pm 0.01\pm 0.01\pm 0.01$
4.10 | $\phantom{-}0.04\pm 0.01\pm 0.01\pm 0.01$ | $\phantom{-}0.07\pm 0.01\pm 0.01\pm 0.01$
5.14 | 0.00 | 0.00
|  |
| Magnitude | Phase
2.01 | $\phantom{-}0.12\pm 0.05\pm 0.07\pm 0.06$ | $-2.82\pm 0.22\pm 0.28\pm 1.47$
2.10 | $\phantom{-}0.58\pm 0.02\pm 0.03\pm 0.03$ | $-1.56\pm 0.09\pm 0.17\pm 0.08$
2.20 | $\phantom{-}0.73\pm 0.01\pm 0.03\pm 0.02$ | $-1.00\pm 0.08\pm 0.15\pm 0.08$
2.30 | $\phantom{-}0.68\pm 0.01\pm 0.03\pm 0.01$ | $-0.42\pm 0.08\pm 0.14\pm 0.05$
2.40 | 0.50 | 0.00
2.50 | $\phantom{-}0.23\pm 0.01\pm 0.01\pm 0.01$ | $-0.00\pm 0.06\pm 0.07\pm 0.05$
2.60 | $\phantom{-}0.23\pm 0.01\pm 0.01\pm 0.03$ | $-0.42\pm 0.09\pm 0.13\pm 0.24$
2.70 | $\phantom{-}0.15\pm 0.01\pm 0.01\pm 0.01$ | $-0.31\pm 0.07\pm 0.11\pm 0.15$
2.80 | $\phantom{-}0.17\pm 0.01\pm 0.01\pm 0.01$ | $-0.63\pm 0.08\pm 0.10\pm 0.19$
2.90 | $\phantom{-}0.20\pm 0.01\pm 0.01\pm 0.01$ | $-0.87\pm 0.09\pm 0.12\pm 0.10$
3.10 | $\phantom{-}0.14\pm 0.00\pm 0.01\pm 0.01$ | $-1.16\pm 0.10\pm 0.13\pm 0.13$
4.10 | $\phantom{-}0.08\pm 0.00\pm 0.01\pm 0.01$ | $\phantom{-}1.02\pm 0.12\pm 0.20\pm 0.16$
5.14 | 0.00 | 0.00
Table 10: Results for the fit fractions. The three quoted errors are statistical, experimental systematic and model uncertainties. Resonance | Fit fraction (%)
---|---
${D}^{*}_{2}(2460)^{0}$ | $35.69\pm 0.62\pm 1.37\pm 0.89$
${D}^{*}_{1}(2680)^{0}$ | $\phantom{3}8.32\pm 0.62\pm 0.69\pm 1.79$
${D}^{*}_{3}(2760)^{0}$ | $\phantom{3}1.01\pm 0.13\pm 0.13\pm 0.25$
${D}^{*}_{2}(3000)^{0}$ | $\phantom{3}0.23\pm 0.07\pm 0.07\pm 0.08$
${D}^{*}_{v}(2007)^{0}$ | $10.79\pm 0.68\pm 0.74\pm 2.34$
$B^{*}_{v}$ | $\phantom{3}2.69\pm 1.01\pm 1.43\pm 1.61$
Total S-wave | $56.96\pm 0.78\pm 0.62\pm 0.87$
Table 11: Results for the product branching fractions ${\cal B}({{{B}^{-}}}\rightarrow R{{\pi}^{-}})\times{\cal B}(R\rightarrow{{D}^{+}}{{\pi}^{-}})$. The four quoted errors are statistical, experimental systematic, model and inclusive branching fraction uncertainties. Resonance | Branching fraction ($10^{-4}$)
---|---
${D}^{*}_{2}(2460)^{0}$ | $3.62\pm 0.06\pm 0.14\pm 0.09\pm 0.25$
${D}^{*}_{1}(2680)^{0}$ | $0.84\pm 0.06\pm 0.07\pm 0.18\pm 0.06$
${D}^{*}_{3}(2760)^{0}$ | $0.10\pm 0.01\pm 0.01\pm 0.02\pm 0.01$
${D}^{*}_{2}(3000)^{0}$ | $0.02\pm 0.01\pm 0.01\pm 0.01\pm 0.00$
${D}^{*}_{v}(2007)^{0}$ | $1.09\pm 0.07\pm 0.07\pm 0.24\pm 0.07$
$B^{*}_{v}$ | $0.27\pm 0.10\pm 0.14\pm 0.16\pm 0.02$
Total S-wave | $5.78\pm 0.08\pm 0.06\pm 0.09\pm 0.39$

The masses and widths of the ${D}^{*}_{2}(2460)^{0}$, ${D}^{*}_{1}(2680)^{0}$,
${D}^{*}_{3}(2760)^{0}$ and ${D}^{*}_{2}(3000)^{0}$ resonances are determined
to be

$\displaystyle m({D}^{*}_{2}(2460)^{0})$ $\displaystyle=$ $\displaystyle
2463.7\pm\phantom{1}0.4\pm\phantom{2}0.4\pm\phantom{2}0.6\mathrm{\,Me\kern-1.00006ptV}\,,$
$\displaystyle\Gamma({D}^{*}_{2}(2460)^{0})$ $\displaystyle=$
$\displaystyle\phantom{24}47.0\pm\phantom{1}0.8\pm\phantom{2}0.9\pm\phantom{2}0.3\mathrm{\,Me\kern-1.00006ptV}\,,$
$\displaystyle m({D}^{*}_{1}(2680)^{0})$ $\displaystyle=$ $\displaystyle
2681.1\pm\phantom{1}5.6\pm\phantom{2}4.9\pm
13.1\mathrm{\,Me\kern-1.00006ptV}\,,$
$\displaystyle\Gamma({D}^{*}_{1}(2680)^{0})$ $\displaystyle=$
$\displaystyle\phantom{2}186.7\pm\phantom{1}8.5\pm\phantom{2}8.6\pm\phantom{2}8.2\mathrm{\,Me\kern-1.00006ptV}\,,$
$\displaystyle m({D}^{*}_{3}(2760)^{0})$ $\displaystyle=$ $\displaystyle
2775.5\pm\phantom{1}4.5\pm\phantom{2}4.5\pm\phantom{2}4.7\mathrm{\,Me\kern-1.00006ptV}\,,$
$\displaystyle\Gamma({D}^{*}_{3}(2760)^{0})$ $\displaystyle=$
$\displaystyle\phantom{27}95.3\pm\phantom{1}9.6\pm\phantom{2}7.9\pm
33.1\mathrm{\,Me\kern-1.00006ptV}\,,$ $\displaystyle m({D}^{*}_{2}(3000)^{0})$
$\displaystyle=$
$\displaystyle\phantom{1.}3214\pm\phantom{1.}29\pm\phantom{1.}33\pm\phantom{1.}36\mathrm{\,Me\kern-1.00006ptV}\,,$
$\displaystyle\Gamma({D}^{*}_{2}(3000)^{0})$ $\displaystyle=$
$\displaystyle\phantom{13.}186\pm\phantom{1.}38\pm\phantom{1.}34\pm\phantom{1.}63\mathrm{\,Me\kern-1.00006ptV}\,,$

where the three quoted errors are statistical, experimental systematic and
model uncertainties. The results for the $D^{*}_{2}(2460)^{0}$ are consistent
with the PDG averages [19] given in Table 1. The ${D}^{*}_{1}(2680)^{0}$ state
has parameters close to those measured for the $D^{*}(2650)$ resonance
observed by LHCb in prompt production in $pp$ collisions [7]. As discussed in
Sec. 1, both 2S and 1D states with spin-parity $J^{P}=1^{-}$ are expected in
this region. Similarly, the ${D}^{*}_{3}(2760)^{0}$ state has parameters close
to those for the $D^{*}(2760)$ states reported in Refs. [7, 6] and for the
charged $D_{3}^{*}(2760)^{+}$ state [11]. It appears likely to be a member of
the 1D family. The ${D}^{*}_{2}(3000)^{0}$ state has parameters that are not
consistent with any previously observed resonance, although due to the large
uncertainties it cannot be ruled out that it has a common origin with the
$D^{*}(3000)$ state that was reported, without evaluation of systematic
uncertainties, in Ref. [7]. It could potentially be a member of the 2P or 1F
family.

Removal of any of the ${D}^{*}_{1}(2680)^{0}$, ${D}^{*}_{3}(2760)^{0}$ and
${D}^{*}_{2}(3000)^{0}$ states from the baseline fit model results in large
changes of the likelihood value. To investigate the effect of the systematic
uncertainties, a similar likelihood ratio test is performed in the alternative
models that give the largest uncertainties on the parameters of these
resonances. Accounting for the four degrees of freedom associated with each
resonance, the significances of the ${D}^{*}_{1}(2680)^{0}$ and
${D}^{*}_{3}(2760)^{0}$ states including systematic uncertainties are found to
be above $10\,\sigma$, while that for the ${D}^{*}_{2}(3000)^{0}$ state is
$6.6\,\sigma$. Assigning alternative spin hypotheses to these states results
in similarly large changes in likelihood.

In summary, an analysis of the amplitudes contributing to
${{{B}^{-}}}\rightarrow{{D}^{+}}{{\pi}^{-}}{{\pi}^{-}}$ decays has been
performed using a data sample corresponding to $3.0\mbox{\,fb}^{-1}$ of $pp$
collision data recorded by the LHCb experiment. The Dalitz plot fit model
containing resonant contributions from the ${D}^{*}_{2}(2460)^{0}$,
${D}^{*}_{1}(2680)^{0}$, ${D}^{*}_{3}(2760)^{0}$ and ${D}^{*}_{2}(3000)^{0}$
states, virtual ${D}^{*}_{v}(2007)^{0}$ and $B^{*0}_{v}$ resonances and a
quasi-model-independent description of the full ${{D}^{+}}{{\pi}^{-}}$ S-wave
was found to give a good description of the data. These results constitute the
first observations of the ${D}^{*}_{3}(2760)^{0}$ and ${D}^{*}_{2}(3000)^{0}$
resonances and may be useful to develop improved models of the dynamics in the
${{D}^{+}}{{\pi}^{-}}$ system.

## 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 and MPG (Germany); INFN (Italy); FOM
and NWO (The Netherlands); MNiSW and NCN (Poland); MEN/IFA (Romania); MinES
and FASO (Russia); MinECo (Spain); SNSF and SER (Switzerland); NASU (Ukraine);
STFC (United Kingdom); NSF (USA). We acknowledge the computing resources that
are provided by CERN, IN2P3 (France), KIT and DESY (Germany), INFN (Italy),
SURF (The Netherlands), PIC (Spain), GridPP (United Kingdom), RRCKI and Yandex
LLC (Russia), CSCS (Switzerland), IFIN-HH (Romania), CBPF (Brazil), PL-GRID
(Poland) and OSC (USA). We are indebted to the communities behind the multiple
open source software packages on which we depend. Individual groups or members
have received support from AvH Foundation (Germany), EPLANET, Marie
Skłodowska-Curie Actions and ERC (European Union), Conseil Général de Haute-
Savoie, Labex ENIGMASS and OCEVU, Région Auvergne (France), RFBR and Yandex
LLC (Russia), GVA, XuntaGal and GENCAT (Spain), Herchel Smith Fund, The Royal
Society, Royal Commission for the Exhibition of 1851 and the Leverhulme Trust
(United Kingdom).

##  Appendix A Results for interference fit fractions

The central values and statistical errors for the interference fit fractions
are shown in Table 12. The experimental systematic and model uncertainties are
given in Tables 13.

Table 12: Interference fit fractions (%) and statistical uncertainties. The
amplitudes are: ($A_{0}$) ${D}^{*}_{v}(2007)^{0}$, ($A_{1}$)
${{D}^{+}}{{\pi}^{-}}$ S-wave, ($A_{2}$) ${D}^{*}_{2}(2460)^{0}$, ($A_{3}$)
${D}^{*}_{1}(2680)^{0}$, ($A_{4}$) $B^{*0}_{v}$, ($A_{5}$)
${D}^{*}_{3}(2760)^{0}$, ($A_{6}$) ${D}^{*}_{2}(3000)^{0}$. The diagonal
elements are the same as the conventional fit fractions.

| $A_{0}$ | $A_{1}$ | $A_{2}$ | $A_{3}$ | $A_{4}$ | $A_{5}$ | $A_{6}$
---|---|---|---|---|---|---|---
$A_{0}$ | $\phantom{-}10.8\pm 0.7$ | $\phantom{-}3.1\pm 1.0$ | $-0.8\pm 0.0$ | $\phantom{-}0.7\pm 1.9$ | $-6.2\pm 1.3$ | $\phantom{-}0.1\pm 0.0$ | $-0.2\pm 0.0$
$A_{1}$ |  | $\phantom{-}57.0\pm 0.8$ | $-2.4\pm 0.2$ | $-5.5\pm 0.4$ | $-1.9\pm 1.4$ | $-0.0\pm 0.0$ | $-0.3\pm 0.1$
$A_{2}$ |  |  | $\phantom{-}35.7\pm 0.6$ | $-0.3\pm 0.1$ | $-0.7\pm 0.4$ | $-0.2\pm 0.0$ | $-0.5\pm 0.2$
$A_{3}$ |  |  |  | $\phantom{-}8.3\pm 0.6$ | $-0.9\pm 1.8$ | $\phantom{-}0.1\pm 0.0$ | $\phantom{-}0.1\pm 0.0$
$A_{4}$ |  |  |  |  | $\phantom{-}2.7\pm 1.0$ | $-0.0\pm 0.0$ | $\phantom{-}0.1\pm 0.0$
$A_{5}$ |  |  |  |  |  | $\phantom{-}1.0\pm 0.1$ | $\phantom{-}0.0\pm 0.0$
$A_{6}$ |  |  |  |  |  |  | $\phantom{-}0.2\pm 0.1$

Table 13: (Top) Experimental and (bottom) model systematic uncertainties on
the interference fit fractions (%). The amplitudes are: ($A_{0}$)
${D}^{*}_{v}(2007)^{0}$, ($A_{1}$) ${{D}^{+}}{{\pi}^{-}}$ S-wave, ($A_{2}$)
${D}^{*}_{2}(2460)^{0}$, ($A_{3}$) ${D}^{*}_{1}(2680)^{0}$, ($A_{4}$)
$B^{*0}_{v}$, ($A_{5}$) ${D}^{*}_{3}(2760)^{0}$, ($A_{6}$)
${D}^{*}_{2}(3000)^{0}$. The diagonal elements are the same as the
conventional fit fractions.

| $A_{0}$ | $A_{1}$ | $A_{2}$ | $A_{3}$ | $A_{4}$ | $A_{5}$ | $A_{6}$
---|---|---|---|---|---|---|---
$A_{0}$ | 0.74 | 0.42 | 0.04 | 1.46 | 1.42 | 0.01 | 0.06
$A_{1}$ |  | 0.62 | 0.21 | 0.34 | 0.58 | 0.03 | 0.13
$A_{2}$ |  |  | 1.37 | 0.13 | 0.14 | 0.01 | 0.24
$A_{3}$ |  |  |  | 0.69 | 2.11 | 0.00 | 0.06
$A_{4}$ |  |  |  |  | 1.43 | 0.15 | 0.05
$A_{5}$ |  |  |  |  |  | 0.13 | 0.01
$A_{6}$ |  |  |  |  |  |  | 0.07

| $A_{0}$ | $A_{1}$ | $A_{2}$ | $A_{3}$ | $A_{4}$ | $A_{5}$ | $A_{6}$
---|---|---|---|---|---|---|---
$A_{0}$ | 2.34 | 0.91 | 0.21 | 1.01 | 3.11 | 0.04 | 0.12
$A_{1}$ |  | 0.87 | 0.21 | 0.48 | 1.74 | 0.02 | 0.16
$A_{2}$ |  |  | 0.89 | 0.07 | 0.53 | 0.08 | 0.34
$A_{3}$ |  |  |  | 1.79 | 0.87 | 0.02 | 0.04
$A_{4}$ |  |  |  |  | 1.61 | 0.04 | 0.05
$A_{5}$ |  |  |  |  |  | 0.25 | 0.03
$A_{6}$ |  |  |  |  |  |  | 0.08

## References

  * [1] S. Godfrey and N. Isgur, _Mesons in a relativized quark model with chromodynamics_ , Phys. Rev. D32 (1985) 189
  * [2] N. Isgur and M. B. Wise, _Spectroscopy with heavy quark symmetry_ , Phys. Rev. Lett. 66 (1991) 1130
  * [3] P. Colangelo, F. De Fazio, F. Giannuzzi, and S. Nicotri, _New meson spectroscopy with open charm and beauty_ , Phys. Rev. D86 (2012) 054024, arXiv:1207.6940
  * [4] D. Mohler, S. Prelovsek, and R. M. Woloshyn, _$D\pi$ scattering and $D$ meson resonances from lattice QCD_, Phys. Rev. D87 (2013) 034501, arXiv:1208.4059
  * [5] G. Moir et al., _Coupled-Channel $D\pi$, $D\eta$ and $D_{s}\bar{K}$ Scattering from Lattice QCD_, arXiv:1607.07093, submitted to JHEP
  * [6] 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
  * [7] 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
  * [8] 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
  * [9] BaBar collaboration, B. Aubert et al., _Dalitz plot analysis of ${{{B}^{-}}}\rightarrow{{D}^{+}}{{\pi}^{-}}{{\pi}^{-}}$_, Phys. Rev. D79 (2009) 112004, arXiv:0901.1291
  * [10] Belle collaboration, A. Kuzmin et al., _Study of ${{\kern 1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}}\rightarrow D^{0}\pi^{+}\pi^{-}$ decays_, Phys. Rev. D76 (2007) 012006, arXiv:hep-ex/0611054
  * [11] LHCb collaboration, R. Aaij et al., _Dalitz plot analysis of $B^{0}\rightarrow\overline{D}^{0}\pi^{+}\pi^{-}$ decays_, Phys. Rev. D92 (2015) 032002, arXiv:1505.01710
  * [12] LHCb collaboration, R. Aaij et al., _First observation and amplitude analysis of the $B^{-}\rightarrow D^{+}K^{-}\pi^{-}$ decay_, Phys. Rev. D91 (2015) 092002, Erratum ibid. D93 (2016) 119901, arXiv:1503.02995
  * [13] LHCb collaboration, R. Aaij et al., _Amplitude analysis of $B^{0}\rightarrow\overline{D}^{0}K^{+}\pi^{-}$ decays_, Phys. Rev. D92 (2015) 012012, arXiv:1505.01505
  * [14] Belle collaboration, J. Brodzicka et al., _Observation of a new $D_{sJ}$ meson in ${{{B}^{+}}}\rightarrow{{\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{D}^{0}}{{K}^{+}}$ decays_, Phys. Rev. Lett. 100 (2008) 092001, arXiv:0707.3491
  * [15] LHCb collaboration, R. Aaij et al., _Observation of overlapping spin- $1$ and spin-$3$ $\overline{D}^{0}K^{-}$ resonances at mass $2.86$ GeV/$c^{2}$_, Phys. Rev. Lett. 113 (2014) 162001, arXiv:1407.7574
  * [16] LHCb collaboration, R. Aaij et al., _Dalitz plot analysis of $B^{0}_{s}\rightarrow\overline{D}^{0}K^{-}\pi^{+}$ decays_, Phys. Rev. D90 (2014) 072003, arXiv:1407.7712
  * [17] 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
  * [18] LHCb collaboration, R. Aaij et al., _Study of $D_{sJ}^{+}$ mesons decaying to $D^{*+}K^{0}_{S}$ and $D^{*0}K^{+}$ final states_, JHEP 02 (2015) 133, arXiv:1601.01495
  * [19] Particle Data Group, K. A. Olive et al., _Review of particle physics_ , Chin. Phys. C38 (2014) 090001
  * [20] B. Chen, X. Liu, and A. Zhang, _Combined study of $2S$ and $1D$ open-charm mesons with natural spin-parity_, Phys. Rev. D92 (2015) 034005, arXiv:1507.02339
  * [21] S. Godfrey and K. Moats, _Properties of excited charm and charm-strange mesons_ , Phys. Rev. D93 (2016) 034035, arXiv:1510.08305
  * [22] LASS collaboration, D. Aston et al., _A study of $K^{-}\pi^{+}$ scattering in the reaction $K^{-}p\rightarrow K^{-}\pi^{+}n$ at $11\,Ge\kern-1.00006ptV\\!/c$_, Nucl. Phys. B296 (1988) 493
  * [23] E791 collaboration, E. M. Aitala _et al._ and W. M. Dunwoodie, _Model independent measurement of S-wave ${{K}^{-}}{{\pi}^{+}}$ systems using ${{D}^{+}}\rightarrow K\pi\pi$ decays from Fermilab E791_, Phys. Rev. D73 (2006) 032004, Erratum ibid. D74 (2006) 059901, arXiv:hep-ex/0507099
  * [24] CLEO collaboration, G. Bonvicini et al., _Dalitz plot analysis of the ${{D}^{+}}\rightarrow{{K}^{-}}{{\pi}^{+}}{{\pi}^{+}}$ decay_, Phys. Rev. D78 (2008) 052001, arXiv:0802.4214
  * [25] FOCUS collaboration, J. M. Link et al., _The $K^{-}\pi^{+}$ S-wave from the $D^{+}\rightarrow K^{-}\pi^{+}\pi^{+}$ decay_, Phys. Lett. B681 (2009) 14, arXiv:0905.4846
  * [26] BaBar collaboration, P. del Amo Sanchez et al., _Analysis of the $D^{+}\rightarrow K^{-}\pi^{+}e^{+}\nu_{e}$ decay channel_, Phys. Rev. D83 (2011) 072001, arXiv:1012.1810
  * [27] BaBar collaboration, J. P. Lees et al., _Measurement of the I=1/2 $K\pi$ S-wave amplitude from Dalitz plot analyses of $\eta_{c}\rightarrow K\bar{K}\pi$ in two-photon interactions_, Phys. Rev. D93 (2016) 012005, arXiv:1511.02310
  * [28] BaBar collaboration, B. Aubert et al., _Amplitude analysis of the decay ${{D}^{0}}\rightarrow{{K}^{-}}{{K}^{+}}{{\pi}^{0}}$_, Phys. Rev. D76 (2007) 011102, arXiv:0704.3593
  * [29] BaBar collaboration, B. Aubert et al., _Dalitz plot analysis of ${{D}^{+}_{s}}\rightarrow\pi^{+}\pi^{-}\pi^{+}$_, Phys. Rev. D79 (2009) 032003, arXiv:0808.0971
  * [30] LHCb collaboration, R. Aaij et al., _A model-independent confirmation of the $Z(4430)^{-}$ state_, Phys. Rev. D92 (2015) 112009, arXiv:1510.01951
  * [31] LHCb collaboration, R. Aaij et al., _Observation of $J/\psi p$ resonances consistent with pentaquark states in $\Lambda_{b}^{0}\rightarrow J/\psi pK^{-}$ decays_, Phys. Rev. Lett. 115 (2015) 072001, arXiv:1507.03414
  * [32] E. E. Kolomeitsev and M. F. M. Lutz, _On heavy light meson resonances and chiral symmetry_ , Phys. Lett. B582 (2004) 39, arXiv:hep-ph/0307133
  * [33] J. Vijande, F. Fernandez, and A. Valcarce, _Open-charm meson spectroscopy_ , Phys. Rev. D73 (2006) 034002, Erratum ibid. D74 (2006) 059903, arXiv:hep-ph/0601143
  * [34] F.-K. Guo et al., _Dynamically generated $0^{+}$ heavy mesons in a heavy chiral unitary approach_, Phys. Lett. B641 (2006) 278, arXiv:hep-ph/0603072
  * [35] D. Gamermann, E. Oset, D. Strottman, and M. J. Vicente Vacas, _Dynamically generated open and hidden charm meson systems_ , Phys. Rev. D76 (2007) 074016, arXiv:hep-ph/0612179
  * [36] LHCb collaboration, A. A. Alves Jr. et al., _The LHCb detector at the LHC_, JINST 3 (2008) S08005
  * [37] LHCb collaboration, R. Aaij et al., _LHCb detector performance_ , Int. J. Mod. Phys. A30 (2015) 1530022, arXiv:1412.6352
  * [38] 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
  * [39] T. Sjöstrand, S. Mrenna, and P. Skands, _PYTHIA 6.4 physics and manual_ , JHEP 05 (2006) 026, arXiv:hep-ph/0603175
  * [40] T. Sjöstrand, S. Mrenna, and P. Skands, _A brief introduction to PYTHIA 8.1_ , Comput. Phys. Commun. 178 (2008) 852, arXiv:0710.3820
  * [41] I. Belyaev et al., _Handling of the generation of primary events in Gauss, the LHCb simulation framework_ , J. Phys. Conf. Ser. 331 (2011) 032047
  * [42] D. J. Lange, _The EvtGen particle decay simulation package_ , Nucl. Instrum. Meth. A462 (2001) 152
  * [43] 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
  * [44] Geant4 collaboration, J. Allison et al., _Geant4 developments and applications_ , IEEE Trans. Nucl. Sci. 53 (2006) 270
  * [45] Geant4 collaboration, S. Agostinelli et al., _Geant4: a simulation toolkit_ , Nucl. Instrum. Meth. A506 (2003) 250
  * [46] M. Clemencic et al., _The LHCb simulation application, Gauss: Design, evolution and experience_, J. Phys. Conf. Ser. 331 (2011) 032023
  * [47] M. Feindt and U. Kerzel, _The NeuroBayes neural network package_ , Nucl. Instrum. Meth. A 559 (2006) 190
  * [48] M. Pivk and F. R. Le Diberder, _sPlot: A statistical tool to unfold data distributions_ , Nucl. Instrum. Meth. A555 (2005) 356, arXiv:physics/0402083
  * [49] M. Adinolfi et al., _Performance of the LHCb RICH detector at the LHC_, Eur. Phys. J. C73 (2013) 2431, arXiv:1211.6759
  * [50] 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
  * [51] LHCb collaboration, R. Aaij et al., _Precision measurement of $D$ meson mass differences_, JHEP 06 (2013) 065, arXiv:1304.6865
  * [52] W. D. Hulsbergen, _Decay chain fitting with a Kalman filter_ , Nucl. Instrum. Meth. A552 (2005) 566, arXiv:physics/0503191
  * [53] 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
  * [54] LHCb collaboration, R. Aaij et al., _Search for the decay $B^{0}_{s}\rightarrow\overline{D}^{0}f_{0}(980)$_, JHEP 08 (2015) 005, arXiv:1505.01654
  * [55] R. H. Dalitz, _On the analysis of tau-meson data and the nature of the tau-meson_ , Phil. Mag. 44 (1953) 1068
  * [56] G. N. Fleming, _Recoupling effects in the isobar model. 1. General formalism for three-pion scattering_ , Phys. Rev. 135 (1964) B551
  * [57] 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
  * [58] D. Herndon, P. Soding, and R. J. Cashmore, _A generalised isobar model formalism_ , Phys. Rev. D11 (1975) 3165
  * [59] J. Blatt and V. E. Weisskopf, Theoretical nuclear physics, J. Wiley (New York), 1952.  page 362
  * [60] 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
  * [61] C. Zemach, _Three pion decays of unstable particles_ , Phys. Rev. 133 (1964) B1201
  * [62] C. Zemach, _Use of angular-momentum tensors_ , Phys. Rev. 140 (1965) B97
  * [63] Laura++ Dalitz plot fitting package, http://laura.hepforge.net
  * [64] N. N. Achasov and G. N. Shestakov, _$\pi\pi$ scattering S wave from the data on the reaction ${{\pi}^{-}}{p}\rightarrow{{\pi}^{0}}{{\pi}^{0}}{n}$_, Phys. Rev. D67 (2003) 114018, arXiv:hep-ph/0302220
  * [65] J. R. Pelaez and F. J. Yndurain, _The pion-pion scattering amplitude_ , Phys. Rev. D71 (2005) 074016, arXiv:hep-ph/0411334
  * [66] R. Kaminski, J. R. Pelaez, and F. J. Yndurain, _The pion-pion scattering amplitude. III. Improving the analysis with forward dispersion relations and Roy equations_ , Phys. Rev. D77 (2008) 054015, arXiv:0710.1150
  * [67] R. Garcia-Martin et al., _The pion-pion scattering amplitude. IV: Improved analysis with once subtracted Roy-like equations up to $1100\mathrm{\,Me\kern-1.00006ptV}$_, Phys. Rev. D83 (2011) 074004, arXiv:1102.2183
  * [68] M. Williams, _How good are your fits? Unbinned multivariate goodness-of-fit tests in high energy physics_ , JINST 5 (2010) P09004, arXiv:1006.3019

LHCb collaboration

R. Aaij40, B. Adeva39, M. Adinolfi48, Z. Ajaltouni5, S. Akar6, J. Albrecht10,
F. Alessio40, M. Alexander53, S. Ali43, G. Alkhazov31, P. Alvarez Cartelle55,
A.A. Alves Jr59, S. Amato2, S. Amerio23, Y. Amhis7, L. An41, L. Anderlini18,
G. Andreassi41, M. Andreotti17,g, J.E. Andrews60, R.B. Appleby56, O. Aquines
Gutierrez11, F. Archilli1, P. d’Argent12, J. Arnau Romeu6, A. Artamonov37, M.
Artuso61, E. Aslanides6, G. Auriemma26, M. Baalouch5, I. Babuschkin56, S.
Bachmann12, J.J. Back50, A. Badalov38, C. Baesso62, S. Baker55, W. Baldini17,
R.J. Barlow56, C. Barschel40, S. Barsuk7, W. Barter40, V. Batozskaya29, B.
Batsukh61, V. Battista41, A. Bay41, L. Beaucourt4, J. Beddow53, F. Bedeschi24,
I. Bediaga1, L.J. Bel43, V. Bellee41, N. Belloli21,i, K. Belous37, I.
Belyaev32, E. Ben-Haim8, G. Bencivenni19, S. Benson40, J. Benton48, A.
Berezhnoy33, R. Bernet42, A. Bertolin23, F. Betti15, M.-O. Bettler40, M. van
Beuzekom43, I. Bezshyiko42, S. Bifani47, P. Billoir8, T. Bird56, A.
Birnkraut10, A. Bitadze56, A. Bizzeti18,u, T. Blake50, F. Blanc41, J. Blouw11,
S. Blusk61, V. Bocci26, T. Boettcher58, A. Bondar36, N. Bondar31,40, W.
Bonivento16, A. Borgheresi21,i, S. Borghi56, M. Borisyak35, M. Borsato39, F.
Bossu7, M. Boubdir9, T.J.V. Bowcock54, E. Bowen42, C. Bozzi17,40, S. Braun12,
M. Britsch12, T. Britton61, J. Brodzicka56, E. Buchanan48, C. Burr56, A.
Bursche2, J. Buytaert40, S. Cadeddu16, R. Calabrese17,g, M. Calvi21,i, M.
Calvo Gomez38,m, A. Camboni38, P. Campana19, D. Campora Perez40, D.H. Campora
Perez40, L. Capriotti56, A. Carbone15,e, G. Carboni25,j, R. Cardinale20,h, A.
Cardini16, P. Carniti21,i, L. Carson52, K. Carvalho Akiba2, G. Casse54, L.
Cassina21,i, L. Castillo Garcia41, M. Cattaneo40, Ch. Cauet10, G. Cavallero20,
R. Cenci24,t, M. Charles8, Ph. Charpentier40, G. Chatzikonstantinidis47, M.
Chefdeville4, S. Chen56, S.-F. Cheung57, V. Chobanova39, M. Chrzaszcz42,27, X.
Cid Vidal39, G. Ciezarek43, P.E.L. Clarke52, M. Clemencic40, H.V. Cliff49, J.
Closier40, V. Coco59, J. Cogan6, E. Cogneras5, V. Cogoni16,40,f, L.
Cojocariu30, G. Collazuol23,o, P. Collins40, A. Comerma-Montells12, A.
Contu40, A. Cook48, S. Coquereau8, G. Corti40, M. Corvo17,g, C.M. Costa
Sobral50, B. Couturier40, G.A. Cowan52, D.C. Craik52, A. Crocombe50, M. Cruz
Torres62, S. Cunliffe55, R. Currie55, C. D’Ambrosio40, E. Dall’Occo43, J.
Dalseno48, P.N.Y. David43, A. Davis59, O. De Aguiar Francisco2, K. De Bruyn6,
S. De Capua56, M. De Cian12, J.M. De Miranda1, L. De Paula2, M. De Serio14,d,
P. De Simone19, C.-T. Dean53, D. Decamp4, M. Deckenhoff10, L. Del Buono8, M.
Demmer10, D. Derkach35, O. Deschamps5, F. Dettori40, B. Dey22, A. Di Canto40,
H. Dijkstra40, F. Dordei40, M. Dorigo41, A. Dosil Suárez39, A. Dovbnya45, K.
Dreimanis54, L. Dufour43, G. Dujany56, K. Dungs40, P. Durante40, R.
Dzhelyadin37, A. Dziurda40, A. Dzyuba31, N. Déléage4, S. Easo51, M. Ebert52,
U. Egede55, V. Egorychev32, S. Eidelman36, S. Eisenhardt52, U. Eitschberger10,
R. Ekelhof10, L. Eklund53, Ch. Elsasser42, S. Ely61, S. Esen12, H.M. Evans49,
T. Evans57, A. Falabella15, N. Farley47, S. Farry54, R. Fay54, D. Fazzini21,i,
D. Ferguson52, V. Fernandez Albor39, A. Fernandez Prieto39, F. Ferrari15,40,
F. Ferreira Rodrigues1, M. Ferro-Luzzi40, S. Filippov34, R.A. Fini14, M.
Fiore17,g, M. Fiorini17,g, M. Firlej28, C. Fitzpatrick41, T. Fiutowski28, F.
Fleuret7,b, K. Fohl40, M. Fontana16, F. Fontanelli20,h, D.C. Forshaw61, R.
Forty40, V. Franco Lima54, M. Frank40, C. Frei40, J. Fu22,q, E. Furfaro25,j,
C. Färber40, A. Gallas Torreira39, D. Galli15,e, S. Gallorini23, S.
Gambetta52, M. Gandelman2, P. Gandini57, Y. Gao3, L.M. Garcia Martin68, J.
García Pardiñas39, J. Garra Tico49, L. Garrido38, P.J. Garsed49, D. Gascon38,
C. Gaspar40, L. Gavardi10, G. Gazzoni5, D. Gerick12, E. Gersabeck12, M.
Gersabeck56, T. Gershon50, Ph. Ghez4, S. Gianì41, V. Gibson49, O.G. Girard41,
L. Giubega30, K. Gizdov52, V.V. Gligorov8, D. Golubkov32, A. Golutvin55,40, A.
Gomes1,a, I.V. Gorelov33, C. Gotti21,i, M. Grabalosa Gándara5, R. Graciani
Diaz38, L.A. Granado Cardoso40, E. Graugés38, E. Graverini42, G. Graziani18,
A. Grecu30, P. Griffith47, L. Grillo21, B.R. Gruberg Cazon57, O. Grünberg66,
E. Gushchin34, Yu. Guz37, T. Gys40, C. Göbel62, T. Hadavizadeh57, C.
Hadjivasiliou5, G. Haefeli41, C. Haen40, S.C. Haines49, S. Hall55, B.
Hamilton60, X. Han12, S. Hansmann-Menzemer12, N. Harnew57, S.T. Harnew48, J.
Harrison56, M. Hatch40, J. He63, T. Head41, A. Heister9, K. Hennessy54, P.
Henrard5, L. Henry8, J.A. Hernando Morata39, E. van Herwijnen40, M. Heß66, A.
Hicheur2, D. Hill57, C. Hombach56, W. Hulsbergen43, T. Humair55, M.
Hushchyn35, N. Hussain57, D. Hutchcroft54, M. Idzik28, P. Ilten58, R.
Jacobsson40, A. Jaeger12, J. Jalocha57, E. Jans43, A. Jawahery60, F. Jiang3,
M. John57, D. Johnson40, C.R. Jones49, C. Joram40, B. Jost40, N. Jurik61, S.
Kandybei45, W. Kanso6, M. Karacson40, J.M. Kariuki48, S. Karodia53, M.
Kecke12, M. Kelsey61, I.R. Kenyon47, M. Kenzie40, T. Ketel44, E. Khairullin35,
B. Khanji21,40,i, C. Khurewathanakul41, T. Kirn9, S. Klaver56, K.
Klimaszewski29, S. Koliiev46, M. Kolpin12, I. Komarov41, R.F. Koopman44, P.
Koppenburg43, A. Kozachuk33, M. Kozeiha5, L. Kravchuk34, K. Kreplin12, M.
Kreps50, P. Krokovny36, F. Kruse10, W. Krzemien29, W. Kucewicz27,l, M.
Kucharczyk27, V. Kudryavtsev36, A.K. Kuonen41, K. Kurek29, T.
Kvaratskheliya32,40, D. Lacarrere40, G. Lafferty56,40, A. Lai16, D. Lambert52,
G. Lanfranchi19, C. Langenbruch9, T. Latham50, C. Lazzeroni47, R. Le Gac6, J.
van Leerdam43, J.-P. Lees4, A. Leflat33,40, J. Lefrançois7, R. Lefèvre5, F.
Lemaitre40, E. Lemos Cid39, O. Leroy6, T. Lesiak27, B. Leverington12, Y. Li7,
T. Likhomanenko35,67, R. Lindner40, C. Linn40, F. Lionetto42, B. Liu16, X.
Liu3, D. Loh50, I. Longstaff53, J.H. Lopes2, D. Lucchesi23,o, M. Lucio
Martinez39, H. Luo52, A. Lupato23, E. Luppi17,g, O. Lupton57, A. Lusiani24, X.
Lyu63, F. Machefert7, F. Maciuc30, O. Maev31, K. Maguire56, S. Malde57, A.
Malinin67, T. Maltsev36, G. Manca7, G. Mancinelli6, P. Manning61, J.
Maratas5,v, J.F. Marchand4, U. Marconi15, C. Marin Benito38, P. Marino24,t, J.
Marks12, G. Martellotti26, M. Martin6, M. Martinelli41, D. Martinez Santos39,
F. Martinez Vidal68, D. Martins Tostes2, L.M. Massacrier7, A. Massafferri1, R.
Matev40, A. Mathad50, Z. Mathe40, C. Matteuzzi21, A. Mauri42, B. Maurin41, A.
Mazurov47, M. McCann55, J. McCarthy47, A. McNab56, R. McNulty13, B. Meadows59,
F. Meier10, M. Meissner12, D. Melnychuk29, M. Merk43, A. Merli22,q, E.
Michielin23, D.A. Milanes65, M.-N. Minard4, D.S. Mitzel12, A. Mogini8, J.
Molina Rodriguez62, I.A. Monroy65, S. Monteil5, M. Morandin23, P. Morawski28,
A. Mordà6, M.J. Morello24,t, J. Moron28, A.B. Morris52, R. Mountain61, F.
Muheim52, M. Mulder43, M. Mussini15, D. Müller56, J. Müller10, K. Müller42, V.
Müller10, P. Naik48, T. Nakada41, R. Nandakumar51, A. Nandi57, I. Nasteva2, M.
Needham52, N. Neri22, S. Neubert12, N. Neufeld40, M. Neuner12, A.D. Nguyen41,
C. Nguyen-Mau41,n, S. Nieswand9, R. Niet10, N. Nikitin33, T. Nikodem12, A.
Novoselov37, D.P. O’Hanlon50, A. Oblakowska-Mucha28, V. Obraztsov37, S.
Ogilvy19, R. Oldeman49, C.J.G. Onderwater69, J.M. Otalora Goicochea2, A.
Otto40, P. Owen42, A. Oyanguren68, P.R. Pais41, A. Palano14,d, F. Palombo22,q,
M. Palutan19, J. Panman40, A. Papanestis51, M. Pappagallo14,d, L.L.
Pappalardo17,g, W. Parker60, C. Parkes56, G. Passaleva18, A. Pastore14,d, G.D.
Patel54, M. Patel55, C. Patrignani15,e, A. Pearce56,51, A. Pellegrino43, G.
Penso26, M. Pepe Altarelli40, S. Perazzini40, P. Perret5, L. Pescatore47, K.
Petridis48, A. Petrolini20,h, A. Petrov67, M. Petruzzo22,q, E. Picatoste
Olloqui38, B. Pietrzyk4, M. Pikies27, D. Pinci26, A. Pistone20, A. Piucci12,
S. Playfer52, M. Plo Casasus39, T. Poikela40, F. Polci8, A. Poluektov50,36, I.
Polyakov61, E. Polycarpo2, G.J. Pomery48, A. Popov37, D. Popov11,40, B.
Popovici30, S. Poslavskii37, C. Potterat2, E. Price48, J.D. Price54, J.
Prisciandaro39, A. Pritchard54, C. Prouve48, V. Pugatch46, A. Puig Navarro41,
G. Punzi24,p, W. Qian57, R. Quagliani7,48, B. Rachwal27, J.H. Rademacker48, M.
Rama24, M. Ramos Pernas39, M.S. Rangel2, I. Raniuk45, G. Raven44, F. Redi55,
S. Reichert10, A.C. dos Reis1, C. Remon Alepuz68, V. Renaudin7, S.
Ricciardi51, S. Richards48, M. Rihl40, K. Rinnert54,40, V. Rives Molina38, P.
Robbe7,40, A.B. Rodrigues1, E. Rodrigues59, J.A. Rodriguez Lopez65, P.
Rodriguez Perez56, A. Rogozhnikov35, S. Roiser40, V. Romanovskiy37, A. Romero
Vidal39, J.W. Ronayne13, M. Rotondo19, M.S. Rudolph61, T. Ruf40, P. Ruiz
Valls68, J.J. Saborido Silva39, E. Sadykhov32, N. Sagidova31, B. Saitta16,f,
V. Salustino Guimaraes2, C. Sanchez Mayordomo68, B. Sanmartin Sedes39, R.
Santacesaria26, C. Santamarina Rios39, M. Santimaria19, E. Santovetti25,j, A.
Sarti19,k, C. Satriano26,s, A. Satta25, D.M. Saunders48, D. Savrina32,33, S.
Schael9, M. Schellenberg10, M. Schiller40, H. Schindler40, M. Schlupp10, M.
Schmelling11, T. Schmelzer10, B. Schmidt40, O. Schneider41, A. Schopper40, K.
Schubert10, M. Schubiger41, M.-H. Schune7, R. Schwemmer40, B. Sciascia19, A.
Sciubba26,k, A. Semennikov32, A. Sergi47, N. Serra42, J. Serrano6, L.
Sestini23, P. Seyfert21, M. Shapkin37, I. Shapoval17,45,g, Y. Shcheglov31, T.
Shears54, L. Shekhtman36, V. Shevchenko67, A. Shires10, B.G. Siddi17, R. Silva
Coutinho42, L. Silva de Oliveira2, G. Simi23,o, S. Simone14,d, M. Sirendi49,
N. Skidmore48, T. Skwarnicki61, E. Smith55, I.T. Smith52, J. Smith49, M.
Smith55, H. Snoek43, M.D. Sokoloff59, F.J.P. Soler53, D. Souza48, B. Souza De
Paula2, B. Spaan10, P. Spradlin53, S. Sridharan40, F. Stagni40, M. Stahl12, S.
Stahl40, P. Stefko41, S. Stefkova55, O. Steinkamp42, S. Stemmle12, O.
Stenyakin37, S. Stevenson57, S. Stoica30, S. Stone61, B. Storaci42, S.
Stracka24,t, M. Straticiuc30, U. Straumann42, L. Sun59, W. Sutcliffe55, K.
Swientek28, V. Syropoulos44, M. Szczekowski29, T. Szumlak28, S. T’Jampens4, A.
Tayduganov6, T. Tekampe10, G. Tellarini17,g, F. Teubert40, C. Thomas57, E.
Thomas40, J. van Tilburg43, M.J. Tilley55, V. Tisserand4, M. Tobin41, S.
Tolk49, L. Tomassetti17,g, D. Tonelli40, S. Topp-Joergensen57, F. Toriello61,
E. Tournefier4, S. Tourneur41, K. Trabelsi41, M. Traill53, M.T. Tran41, M.
Tresch42, A. Trisovic40, A. Tsaregorodtsev6, P. Tsopelas43, A. Tully49, N.
Tuning43, A. Ukleja29, A. Ustyuzhanin35,67, U. Uwer12, C. Vacca16,40,f, V.
Vagnoni15,40, S. Valat40, G. Valenti15, A. Vallier7, R. Vazquez Gomez19, P.
Vazquez Regueiro39, S. Vecchi17, M. van Veghel43, J.J. Velthuis48, M.
Veltri18,r, G. Veneziano41, A. Venkateswaran61, M. Vernet5, M. Vesterinen12,
B. Viaud7, D. Vieira1, M. Vieites Diaz39, X. Vilasis-Cardona38,m, V. Volkov33,
A. Vollhardt42, B. Voneki40, D. Voong48, A. Vorobyev31, V. Vorobyev36, C.
Voß66, J.A. de Vries43, C. Vázquez Sierra39, R. Waldi66, C. Wallace50, R.
Wallace13, J. Walsh24, J. Wang61, D.R. Ward49, H.M. Wark54, N.K. Watson47, D.
Websdale55, A. Weiden42, M. Whitehead40, J. Wicht50, G. Wilkinson57,40, M.
Wilkinson61, M. Williams40, M.P. Williams47, M. Williams58, T. Williams47,
F.F. Wilson51, J. Wimberley60, J. Wishahi10, W. Wislicki29, M. Witek27, G.
Wormser7, S.A. Wotton49, K. Wraight53, S. Wright49, K. Wyllie40, Y. Xie64, Z.
Xing61, Z. Xu41, Z. Yang3, H. Yin64, J. Yu64, X. Yuan36, O. Yushchenko37, M.
Zangoli15, K.A. Zarebski47, M. Zavertyaev11,c, L. Zhang3, Y. Zhang7, Y.
Zhang63, A. Zhelezov12, Y. Zheng63, A. Zhokhov32, X. Zhu3, V. Zhukov9, S.
Zucchelli15.

1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université Savoie Mont-Blanc, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9I. Physikalisches Institut, RWTH Aachen University, Aachen, Germany
10Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
11Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
12Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
13School of Physics, University College Dublin, Dublin, Ireland
14Sezione INFN di Bari, Bari, Italy
15Sezione INFN di Bologna, Bologna, Italy
16Sezione INFN di Cagliari, Cagliari, Italy
17Sezione INFN di Ferrara, Ferrara, Italy
18Sezione INFN di Firenze, Firenze, Italy
19Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
20Sezione INFN di Genova, Genova, Italy
21Sezione INFN di Milano Bicocca, Milano, Italy
22Sezione INFN di Milano, Milano, Italy
23Sezione INFN di Padova, Padova, Italy
24Sezione INFN di Pisa, Pisa, Italy
25Sezione INFN di Roma Tor Vergata, Roma, Italy
26Sezione INFN di Roma La Sapienza, Roma, Italy
27Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
28AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
29National Center for Nuclear Research (NCBJ), Warsaw, Poland
30Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
31Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
32Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
33Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
34Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
35Yandex School of Data Analysis, Moscow, Russia
36Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
37Institute for High Energy Physics (IHEP), Protvino, Russia
38ICCUB, Universitat de Barcelona, Barcelona, Spain
39Universidad de Santiago de Compostela, Santiago de Compostela, Spain
40European Organization for Nuclear Research (CERN), Geneva, Switzerland
41Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
42Physik-Institut, Universität Zürich, Zürich, Switzerland
43Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
44Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
45NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
46Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
47University of Birmingham, Birmingham, United Kingdom
48H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
49Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
50Department of Physics, University of Warwick, Coventry, United Kingdom
51STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
52School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
53School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
54Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
55Imperial College London, London, United Kingdom
56School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
57Department of Physics, University of Oxford, Oxford, United Kingdom
58Massachusetts Institute of Technology, Cambridge, MA, United States
59University of Cincinnati, Cincinnati, OH, United States
60University of Maryland, College Park, MD, United States
61Syracuse University, Syracuse, NY, United States
62Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
63University of Chinese Academy of Sciences, Beijing, China, associated to 3
64Institute of Particle Physics, Central China Normal University, Wuhan,
Hubei, China, associated to 3
65Departamento de Fisica , Universidad Nacional de Colombia, Bogota, Colombia,
associated to 8
66Institut für Physik, Universität Rostock, Rostock, Germany, associated to 12
67National Research Centre Kurchatov Institute, Moscow, Russia, associated to
32
68Instituto de Fisica Corpuscular (IFIC), Universitat de Valencia-CSIC,
Valencia, Spain, associated to 38
69Van Swinderen Institute, University of Groningen, Groningen, The
Netherlands, associated to 43

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