EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-EP-2019-157 LHCb-PAPER-2019-017 23 January 2020

Amplitude analysis of the
${{{B}^{+}}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{+}}{{\pi}^{-}}$ decay

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

The results of an amplitude analysis of the charmless three-body decay
${{{B}^{+}}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{+}}{{\pi}^{-}}$, in which
$C\\!P$-violation effects are taken into account, are reported. The analysis
is based on a data sample corresponding to an integrated luminosity of
$3\text{\,fb}^{-1}$ of ${p}{p}$ collisions recorded with the LHCb detector.
The most challenging aspect of the analysis is the description of the
behaviour of the $\pi^{+}\pi^{-}$ S-wave contribution, which is achieved by
using three complementary approaches based on the isobar model, the K-matrix
formalism, and a quasi-model-independent procedure. Additional resonant
contributions for all three methods are described using a common isobar model,
and include the $\rho(770)^{0}$, $\omega(782)$ and $\rho(1450)^{0}$ resonances
in the ${{\pi}^{+}}{{\pi}^{-}}$ P-wave, the $f_{2}(1270)$ resonance in the
${{\pi}^{+}}{{\pi}^{-}}$ D-wave, and the $\rho_{3}(1690)^{0}$ resonance in the
${{\pi}^{+}}{{\pi}^{-}}$ F-wave. Significant $C\\!P$-violation effects are
observed in both S- and D-waves, as well as in the interference between the S-
and P-waves. The results from all three approaches agree and provide new
insight into the dynamics and the origin of $C\\!P$-violation effects in
${{{B}^{+}}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{+}}{{\pi}^{-}}$ decays.

Published in Phys. Rev. D 101 (2020) 012006

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

##  1 Introduction

In the Standard Model (SM), $C\\!P$ violation originates from a single
irreducible complex phase in the Cabibbo–Kobayashi–Maskawa (CKM) matrix [1,
2]. Thus far, all measurements of $C\\!P$ violation in particle decays are
consistent with this explanation. Nevertheless, the degree of $C\\!P$
violation permitted in the SM is inconsistent with the macroscopic matter-
antimatter asymmetry observed in the Universe [3], motivating further studies
and searches for sources of $C\\!P$ violation beyond the SM.

For the manifestation of $C\\!P$ violation in decay, at least two interfering
amplitudes with different strong and weak phases are required. In the SM, weak
phases are associated with the complex elements of the CKM matrix and have
opposite sign between charge-conjugate processes, while strong phases are
associated with hadronic final-state effects and do not change sign under
$C\\!P$ conjugation. In decays of $b$ hadrons to charmless hadronic final
states, contributions from both tree and loop (so-called “penguin”) diagrams,
which can provide the relative weak phase that is necessary for $C\\!P$
violation to manifest, are possible with comparable magnitudes. Indeed,
significant $C\\!P$ asymmetries have been observed in both
${{B}^{0}}\rightarrow{{K}^{+}}{{\pi}^{-}}$ [4, 5, 6, 7] and
${{B}^{0}}\rightarrow{{\pi}^{+}}{{\pi}^{-}}$ decays [4, 8, 9]. In multibody
decays, variation across the phase space of strong phases, caused by hadronic
resonances, allows for further enhancement of $C\\!P$ violation effects and a
richer phenomenology compared to two-body decays. Large $C\\!P$ asymmetries
localised in regions of phase space of charmless three-body $B$-meson decays
have been observed in model-independent analyses [10, 11, 12, 13], but until
recently there has been no description of these effects with an accurate model
of the contributing resonances. An amplitude analysis of
${{{B}^{+}}}\\!\rightarrow{{K}^{+}}{{K}^{+}}{{\pi}^{-}}$ decays [14] has shown
that $\pi\pi\leftrightarrow{K}{\kern 1.79993pt\overline{\kern-1.79993ptK}}$
rescattering plays an important role in the observed $C\\!P$ violation, and it
is anticipated that similar effects will occur in other charmless three-body
$B$-meson decays.

This paper documents an analysis of the
${{{B}^{+}}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{+}}{{\pi}^{-}}$ decay amplitude
in the two-dimensional phase space known as the Dalitz plot [15, 16]. The
inclusion of charge-conjugate processes is implied, except where asymmetries
are discussed. Previous studies of this decay mode indicate that the amplitude
contains a sizable $\rho(770)^{0}$ component [17, 18, 19, 20]. The amplitude
analysis performed by the BaBar collaboration [20] additionally observed a
large S-wave contribution, however, measurements of $C\\!P$-violating
quantities were limited by statistical precision.

Phenomenological studies [21, 22, 23, 24, 25, 26] have focussed on
investigating the localised $C\\!P$ asymmetries seen in the model-independent
analysis of ${{{B}^{+}}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{+}}{{\pi}^{-}}$
decays [12], with some works indicating the potential importance of the
$\rho$–$\omega$ mixing effect between the $\rho(770)^{0}$ and $\omega(782)$
resonances [27, 28, 29], and the interference between the $\rho(770)^{0}$
resonance and the broad S-wave contribution [30, 31, 32, 33]. Furthermore, the
relative pattern of $C\\!P$ asymmetries between
${{{B}^{+}}}\\!\rightarrow{h}^{+}{h}^{+}{h}^{-}$ decays, where ${h}$ is a kaon
or pion, could be indicative of $C\\!P$ violation induced by
$\pi\pi\leftrightarrow{K}{\kern 1.79993pt\overline{\kern-1.79993ptK}}$
rescattering [34, 30, 32, 35].

The present analysis is performed on data corresponding to 3$\text{\,fb}^{-1}$
collected by the LHCb experiment, of which $1\text{\,fb}^{-1}$ was collected
in 2011 with a $p$ $p$ collision centre-of-mass energy of
$7\text{\,Te\kern-1.00006ptV}$ and $2\text{\,fb}^{-1}$ was collected in 2012
with a centre-of-mass energy of $8\text{\,Te\kern-1.00006ptV}$. A model of the
Dalitz plot distribution is constructed in terms of the intermediate resonant
and nonresonant structures. Due to its magnitude and potential importance to
the observed $C\\!P$ violation in this decay, particular attention is given to
the ${{\pi}^{+}}{{\pi}^{-}}$ S-wave contribution, which is known to consist of
numerous overlapping resonances and open decay channels [36]. Three different
state-of-the-art approaches to the modelling of the S-wave are used to ensure
that any inaccuracies in the description of this part of the amplitude do not
impact the interpretation of the physical quantities reported.

This paper is organised as follows: Section 2 gives a brief description of the
LHCb detector and the event reconstruction and simulation software; the signal
candidate selection procedure is described in Section 3; Section 4 describes
the procedure for estimating the signal and background yields that enter into
the amplitude fit; Section 5 outlines the formalism used for the construction
of the amplitude models, as well as a description of the mass lineshapes used
to parameterise the intermediate structures; Section 6 describes the
systematic uncertainties associated with the analysis procedure; Section 7
documents the physics parameters of interest obtained from the amplitude
models and presents projections of the fit models on the selected data; these
results are then discussed in Section 8; and a summary of the work as a whole
can be found in Section 9. A shorter description of the analysis, more
focussed on the first observations of different sources of $C\\!P$-violation
effects, can be found in a companion article [37].

##  2 Detector and simulation

The LHCb detector [38, 39] 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 [40], a large-area silicon-strip detector located upstream
of a dipole magnet with a bending power of about $4{\mathrm{\,Tm}}$, and three
stations of silicon-strip detectors and straw drift tubes [41] placed
downstream of the magnet. The tracking system provides a measurement of the
momentum $p$ of charged particles with relative uncertainty that varies from
0.5% at low momentum to 1.0% at 200$\text{\,Ge\kern-1.00006ptV\\!/}c$. The
minimum distance of a track to a primary vertex (PV), or impact parameter
(IP), is measured with a resolution of
$(15+29/p_{\mathrm{T}})\,\upmu\text{m}$, where $p_{\mathrm{T}}$ is the
component of the momentum transverse to the beam (in
$\text{\,Ge\kern-1.00006ptV\\!/}c$). Different types of charged hadrons are
distinguished using information from two ring-imaging Cherenkov detectors
[42]. Photons, electrons and hadrons are identified by a calorimeter system
consisting of scintillating-pad and preshower detectors, an electromagnetic
and a hadronic calorimeter. Muons are identified by a system composed of
alternating layers of iron and multiwire proportional chambers [43]. The
magnetic field deflects oppositely charged particles in opposite directions
and this can lead to detection asymmetries. Periodically reversing the
magnetic field polarity throughout the data-taking reduces this effect to a
negligible level. Approximately $60\%$ of 2011 data and $52\%$ of 2012 data
was collected in the “down” polarity configuration, and the rest in the “up”
configuration.

The online event selection is performed by a trigger [44] which consists of a
hardware stage followed by a software stage. The hardware stage is based on
information from the calorimeter and muon systems in which events are required
to contain a muon with high $p_{\mathrm{T}}$, or a hadron, photon or electron
with high transverse energy in the calorimeters. The software trigger requires
a two- or three-track secondary vertex with significant displacement from all
primary $pp$ interaction vertices. All charged particles with
$p_{\mathrm{T}}>500\,(300)\text{\,Me\kern-1.00006ptV\\!/}c$ are reconstructed,
for data collected in 2011 (2012), in events where at least one charged
particle has transverse momentum
$p_{\mathrm{T}}>1.7\,(1.6)\text{\,Ge\kern-1.00006ptV\\!/}c$ and is
inconsistent with originating from a PV. A multivariate algorithm [45] is used
for the identification of secondary vertices consistent with the decay of a
$b$ hadron.

Simulated data samples are used to investigate backgrounds from other
$b$-hadron decays and also to study the detection and reconstruction
efficiency of the signal. In the simulation, $pp$ collisions are generated
using Pythia [46, *Sjostrand:2007gs] with a specific LHCb configuration [48].
Decays of unstable particles are described by EvtGen [49], in which final-
state radiation is generated using Photos [50]. The interaction of the
generated particles with the detector and its response are implemented using
the Geant4 toolkit [51, *Agostinelli:2002hh] as described in Ref. [53].

##  3 Selection

The selection of signal candidates follows closely the procedure used in the
model-independent analysis of the same data sample [12]. Signal ${{B}^{+}}$
candidates are formed from three good-quality tracks that are consistent with
originating from the same secondary vertex (SV), which must also be at least
$3\text{\,mm}$ away from any PV. The reconstructed ${{B}^{+}}$ candidate is
associated with the PV that is most consistent with its flight direction. A
requirement is also imposed on the angle between the ${{B}^{+}}$ momentum and
the vector between the PV and SV, that must be less than approximately
$6\text{\,mrad}$.

To reject random associations of tracks (combinatorial background), a boosted
decision-tree classifier [54] is trained to discriminate between simulated
signal candidates and candidates in collision data residing in a region where
this background dominates,
$5.4<m({{\pi}^{+}}{{\pi}^{+}}{{\pi}^{-}})<5.8\text{\,Ge\kern-1.00006ptV\\!/}c^{2}$.
The variables that enter this classifier are ${{B}^{+}}$ and decay product
kinematic properties, quantities based on the quality of the reconstructed
tracks and decay vertices, as well as the ${{B}^{+}}$ displacement from the
PV. The requirement on the output of this classifier is optimised to maximise
the expected approximate signal significance, $N_{s}/\sqrt{N_{s}+N_{b}}$,
where $N_{s}$ is the expected signal yield within
$40\text{\,Me\kern-1.00006ptV\\!/}c^{2}$ of the known ${{B}^{+}}$ mass [55],
and $N_{b}$ is the corresponding combinatorial background level within the
same region.

To suppress backgrounds that arise when any number of kaons are misidentified
as pions, requirements are placed on the particle-identification information
associated with each final-state track. Furthermore, tracks associated with a
hit in the muon system are also removed, as are tracks that are outside the
fiducial region of the particle-identification system.

A veto in both combinations of the opposite-sign dipion mass, where
$1.740<m({{\pi}^{+}}{{\pi}^{-}})<1.894\text{\,Ge\kern-1.00006ptV\\!/}c^{2}$,
is applied to remove ${{{B}^{+}}}\\!\rightarrow({{\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}\rightarrow{{\pi}^{+}}{{\pi}^{-}}){{\pi}^{+}}$
decays, along with partially reconstructed decays involving intermediate
${\kern 1.79993pt\overline{\kern-1.79993ptD}}{}^{0}$ mesons and decays of
${\kern 1.79993pt\overline{\kern-1.79993ptD}}{}^{0}$ mesons where one or more
kaons are misidentified as pions. Approximately $2\%$ of events contain
multiple ${{B}^{+}}$ decay candidates following the aforementioned selection
procedure, of which one is chosen at random.

The Dalitz-plot variables are calculated following a kinematic mass
constraint, fixing the ${{B}^{+}}$ candidate mass to the known value to
improve resolution and to ensure that all decays remain within the Dalitz-plot
boundary. In consequence, the experimental resolution in the region with the
narrowest resonance considered in this analysis, the $\omega(782)$ state, is
better than $3\text{\,Me\kern-1.00006ptV\\!/}c^{2}$, with a corresponding full
width at half maximum of better than $7\text{\,Me\kern-1.00006ptV\\!/}c^{2}$.
This is smaller than the $\omega(782)$ width, and therefore effects related to
the finite resolution in the Dalitz plot are not considered further.

##  4 ${{B}^{+}}$ candidate invariant-mass fit

An extended, unbinned, maximum-likelihood fit is performed to the
$m({{\pi}^{+}}{{\pi}^{+}}{{\pi}^{-}})$ invariant-mass spectrum to extract
yields and charge asymmetries of the
${{{B}^{+}}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{+}}{{\pi}^{-}}$ signal and
various contributing backgrounds. The fit is performed to candidates in the
range
$5.080<m({{\pi}^{+}}{{\pi}^{+}}{{\pi}^{-}})<5.580\text{\,Ge\kern-1.00006ptV\\!/}c^{2}$,
and its results are used to obtain signal and background yields in the signal
region,
$5.249<m({{\pi}^{+}}{{\pi}^{+}}{{\pi}^{-}})<5.317\text{\,Ge\kern-1.00006ptV\\!/}c^{2}$,
in which the subsequent Dalitz-plot fit is performed. All shape parameters of
the probability density functions (PDFs) comprising the fit model are shared
between ${{B}^{+}}$ and ${{B}^{-}}$ candidates and only the yields are
permitted to vary between these categories, which are fitted simultaneously.
The data are also subdivided by data-taking year, and whether the hardware
trigger decision is due to hadronic calorimeter deposits associated with the
signal candidate, or due to other particles in the event, to permit correction
for possible differences in efficiency between subsamples (see Section 5.5).

The shape of the ${{{B}^{+}}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{+}}{{\pi}^{-}}$
signal decay is parameterised by the sum of a core Gaussian with two Crystal
Ball functions [56], with tails on opposite sides of the peak in order to
describe the asymmetric tails of the distribution due to detector resolution
and final-state radiation. The tail mean and width parameters of the Crystal
Ball functions are determined from simulation relative to the core mean and
width, which are left free in the fit to collision data to account for small
differences between simulation and data. All remaining parameters, apart from
the total yield, are obtained from a fit to simulated events.

Partially reconstructed backgrounds, which predominantly arise from four-body
$B$-meson decays where a charged hadron or neutral particle is not
reconstructed, are modelled with an ARGUS function [57] convolved with a
Gaussian resolution function. The smooth combinatorial background is modelled
with a falling exponential function. The only significant source of cross-feed
background, where one or more kaons are misidentified as pions, is the
${{{B}^{+}}}\\!\rightarrow{{K}^{+}}{{\pi}^{+}}{{\pi}^{-}}$ decay. To obtain an
accurate model for this background, simulated
${{{B}^{+}}}\\!\rightarrow{{K}^{+}}{{\pi}^{+}}{{\pi}^{-}}$ decays are weighted
according to the amplitude model obtained by the BaBar collaboration [58],
also accounting for the probability to be reconstructed as a
${{{B}^{+}}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{+}}{{\pi}^{-}}$ candidate. A
model for ${{{B}^{+}}}\\!\rightarrow{{K}^{+}}{{\pi}^{+}}{{\pi}^{-}}$ decays
based on a similar-sized data sample has also been obtained by the Belle
collaboration [59]; the details of the model used do not impact this analysis.
This shape is modelled in the invariant-mass fit as a sum of a Gaussian with
two Crystal Ball functions, where all parameters are determined from a fit to
the weighted simulation. Furthermore, the yield of this component is
constrained to the
${{{B}^{+}}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{+}}{{\pi}^{-}}$ signal yield
multiplied by the product of the relative branching fractions of these decays
and the inverse of the relative overall reconstruction and selection
efficiencies, which are described in Section 5.5.

The mass fit results are shown in Fig. 1, while Table 1 quotes the component
yields and phase-space-integrated raw detection asymmetries in the ${{B}^{+}}$
signal region, which are subsequently used in the Dalitz-plot fit. The quoted
uncertainties account for systematic effects evaluated with the procedures
outlined in Section 6.

(a)(b)

Figure 1:  Invariant-mass fit model for (a) ${{B}^{-}}$ and (b) ${{B}^{+}}$ candidates reconstructed in the ${\pi}^{\mp}$ ${\pi}^{+}$ ${\pi}^{-}$ final state for the combined 2011 and 2012 data taking samples. Points with error bars represent the data while the components comprising the model are listed in the plot legend.  Table 1: Component yields and phase-space-integrated raw detection asymmetries in the ${{B}^{+}}$ signal region, calculated from the results of the invariant-mass fit. The uncertainties include both statistical and systematic effects. Parameter | Value
---|---
Signal yield | $20\,600\pm 1\,600$
Combinatorial background yield | $4\,400\pm 1\,600$
${{{B}^{+}}}\\!\rightarrow{{K}^{+}}{{\pi}^{+}}{{\pi}^{-}}$ background yield | $143\pm\phantom{0\,0}11$
Combinatorial background asymmetry | $+0.005\pm 0.010$
${{{B}^{+}}}\\!\rightarrow{{K}^{+}}{{\pi}^{+}}{{\pi}^{-}}$ background asymmetry | $\phantom{+}0.000\pm 0.008$

##  5 Dalitz-plot model

The ${{{B}^{+}}}\\!\rightarrow\pi^{+}_{1}\pi^{+}_{2}\pi^{-}_{3}$ decay
amplitude can be expressed fully in terms of the invariant-mass-squared of two
pairs of the decay products $m^{2}_{13}$ and $m^{2}_{23}$. As no resonances
are expected to decay to ${{\pi}^{+}}{{\pi}^{+}}$, the squared invariant-
masses of the two combinations of oppositely charged pions are used as these
two pairs. Due to Bose symmetry, the amplitude is invariant under exchange of
the two like-sign pions, $A(m_{13}^{2},m_{23}^{2})\equiv
A(m_{23}^{2},m_{13}^{2})$, meaning that the assignments of $\pi^{+}_{1}$ and
$\pi^{+}_{2}$ are arbitrary.

Due to this symmetry, a natural “folding” occurs in the Dalitz plot about the
axis $m_{13}^{2}=m_{23}^{2}$. Since the majority of the resonant structure is
expected to be at low mass
$m({{\pi}^{+}}{{\pi}^{-}})<2\text{\,Ge\kern-1.00006ptV\\!/}c^{2}$, the data
and its projections are presented with the two axes being the squares of the
low-mass $m_{\rm low}$ and high-mass $m_{\rm high}$ combinations of the
opposite-sign pion pairs, for visualisation purposes. Plots of this kind are
therefore similar to those found in other analyses with Dalitz plots that are
expected to contain resonances along only one axis. In this case, structure
resulting predominantly from the mass lineshape appears in $m_{\rm low}$,
while $m_{\rm high}$ is influenced by the angular momentum eigenfunctions. The
Dalitz-plot distributions of the selected candidates can be seen in Fig. 2(a)
and (b).

(a)(b)(c)(d)

Figure 2:  Conventional Dalitz-plot distributions for (a) ${{B}^{+}}$ and (b)
${{B}^{-}}$, and square Dalitz-plot (defined in Section 5.1.1) distributions
for (c) ${{B}^{+}}$ and (d) ${{B}^{-}}$ candidate decays to
${{\pi}^{\pm}}{{\pi}^{+}}{{\pi}^{-}}$. Depleted regions are due to the ${\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}$ veto.

###  5.1 Amplitude analysis formalism

In general, the isobar model is used to define the total amplitude for
${{B}^{+}}$ decays as a coherent sum over $N$ components, each described by a
function $F_{j}$ that parameterises the intermediate resonant or nonresonant
processes:

$A^{+}(m_{13}^{2},m_{23}^{2})=\sum_{j}^{N}A^{+}_{j}(m_{13}^{2},m_{23}^{2})=\sum_{j}^{N}c^{+}_{j}F_{j}(m_{13}^{2},m_{23}^{2})\,,$
(1)

and similarly for ${{B}^{-}}$:

$A^{-}(m_{13}^{2},m_{23}^{2})=\sum_{j}^{N}A^{-}_{j}(m_{13}^{2},m_{23}^{2})=\sum_{j}^{N}c^{-}_{j}F_{j}(m_{13}^{2},m_{23}^{2})\,,$
(2)

where the complex coefficients $c_{j}$ represent the relative contribution of
component $j$. These are expressed in the “Cartesian” $C\\!P$-violating
convention,

$c_{j}^{\pm}=(x\pm\delta x)+i(y\pm\delta y)\,,$ (3)

for $c_{j}^{+}$ ($c_{j}^{-}$) coefficients corresponding to ${{B}^{+}}$
(${{B}^{-}}$) decays. The function $F$ contains only strong dynamics and, for
a resonant or nonresonant contribution to the $m_{13}$ spectrum, is
parameterised as

$F(m_{13}^{2},m_{23}^{2})\propto R(m_{13})\cdot T(\vec{p},\vec{q})\cdot
X(|\vec{p}\,|r_{\rm BW}^{P})\cdot X(|\vec{q}\,|r_{\rm BW}^{R})\,,$ (4)

where $R$ describes the mass lineshape, $T$ describes the angular dependence,
and $X$ are Blatt–Weisskopf barrier factors [60] depending on a radius
parameter $r_{\rm BW}$. Here and in the following, the momentum of one of the
$m_{13}$ decay products is denoted as $\vec{q}$ and the momentum of the third
pion ($\pi_{2}$) as $\vec{p}$, where both momenta are evaluated in the rest
frame of the dipion system.

Using the Zemach tensor formalism [61, 62], the angular probability
distribution terms $T(\vec{p},\vec{q})$ are given by

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

These are related to the Legendre polynomials $P_{L}(\cos\theta_{\rm hel})$,
where the helicity angle $\theta_{\rm hel}$ is the angle between $\vec{p}$ and
$\vec{q}$ and provide a good visual indicator of the spin of the intermediate
state.

The Blatt–Weisskopf barrier factors account for the finite size of the
decaying hadron and correct for the unphysical increase in the amplitude above
the angular momentum barrier introduced by the form of angular momentum
distributions given in Eq. (5.1). They are expressed in terms of
$z=|\vec{q}\,|r^{P}_{\rm BW}$ for the ${{B}^{+}}$ decay and
$z=|\vec{p}\,|r^{R}_{\rm BW}$ for the intermediate state,

$\displaystyle L=0\ :\ X(z)$ $\displaystyle=$ $\displaystyle 1\,,$
$\displaystyle L=1\ :\ X(z)$ $\displaystyle=$
$\displaystyle\sqrt{\frac{1+z_{0}^{2}}{1+z^{2}}}\,,$ $\displaystyle L=2\ :\
X(z)$ $\displaystyle=$
$\displaystyle\sqrt{\frac{z_{0}^{4}+3z_{0}^{2}+9}{z^{4}+3z^{2}+9}}\,,$
$\displaystyle L=3\ :\ X(z)$ $\displaystyle=$
$\displaystyle\sqrt{\frac{z_{0}^{6}+6z_{0}^{4}+45z_{0}^{2}+225}{z^{6}+6z^{4}+45z^{2}+225}}\,,$
(6)

where $L$ is the relative angular momentum between the ${{{B}^{+}}}$ meson and
the resonance, which is equal to the spin of the resonance since the
${{{B}^{+}}}$ meson is spinless. The variable $z_{0}$ is equal to the value of
$z$ when the mass of the propagator is equal to the mass of the resonance.
Unless otherwise stated, the barrier radius of the ${{B}^{+}}$ meson and the
intermediate resonance is taken to be $r_{\rm BW}^{P}=r_{\rm
BW}^{R}=4.0\text{\,Ge\kern-1.00006ptV}^{-1}$ ($\approx 0.8\text{\,fm}$), for
all resonances. (To simplify expressions, natural units with $c=\hbar=1$ are
used throughout Sections 5.1–5.3.)

####  5.1.1 Square Dalitz plot

Since resonances tend to populate the edges of the conventional Dalitz plot in
charmless $B$ decays, it is useful to define the so-called “square” Dalitz
plot [19], which provides improved resolution in these critical regions when
using uniform binning, for example when modelling efficiency effects.
Furthermore, the mapping to a square space aligns the bin boundaries to the
kinematic boundaries of the phase space. As such, all efficiencies and
backgrounds described in Sections 5.5 and 5.6 are determined as a function of
the square Dalitz-plot variables, however the data and fit projections in
Section 7 are displayed as a function of the squares of the invariant-mass
pairs.

The square Dalitz plot is defined in terms of $m^{\prime}$ and
$\theta^{\prime}$

$m^{\prime}\equiv\frac{1}{\pi}\arccos\left(2\,\frac{m({{\pi}^{+}}{{\pi}^{+}})-m({{\pi}^{+}}{{\pi}^{+}})^{\rm
min}}{m({{\pi}^{+}}{{\pi}^{+}})^{\rm max}-m({{\pi}^{+}}{{\pi}^{+}})^{\rm
min}}-1\right),\ \ \ \
\theta^{\prime}\equiv\frac{1}{\pi}\theta({{\pi}^{+}}{{\pi}^{+}}),$ (7)

where $m({{\pi}^{+}}{{\pi}^{+}})^{\rm max}=m_{B^{+}}-m_{{{\pi}^{-}}}$ and
$m({{\pi}^{+}}{{\pi}^{+}})^{\rm min}=2m_{{{\pi}^{+}}}$ represent the kinematic
limits permitted in the
${{{B}^{+}}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{+}}{{\pi}^{-}}$ decay and
$\theta({{\pi}^{+}}{{\pi}^{+}})$ is the angle between ${\pi}^{+}$ and
${\pi}^{-}$ in the ${{\pi}^{+}}{{\pi}^{+}}$ rest frame. The Bose symmetry of
the final state requires that distributions are symmetric with respect to
$\theta^{\prime}=0.5$. The square Dalitz-plot distributions of the selected
candidates can be seen in Fig. 2(c) and (d).

###  5.2 Mass lineshapes

Resonant contributions are mostly described by the relativistic Breit–Wigner
lineshape

$R(m)=\frac{1}{(m_{0}^{2}-m^{2})-im_{0}\Gamma(m)}\,,$ (8)

with a mass-dependent decay width

$\Gamma(m)=\Gamma_{0}\left(\frac{q}{q_{0}}\right)^{2L+1}\left(\frac{m_{0}}{m}\right)X^{2}(qr^{R}_{\rm
BW})\,,$ (9)

where $q_{0}$ is the value of $q=|\vec{q}\,|$ when the invariant mass, $m$, is
equal to the pole mass, $m_{0}$, of the resonance. Here, the nominal resonance
width is given by $\Gamma_{0}$.

For the broad $\rho(770)^{0}$ resonance, an analytic dispersive term is
included to ensure unitarity far from the pole mass, known as the
Gounaris–Sakurai model [63]. It takes the form

$R(m)=\frac{1+D\,\Gamma_{0}/m_{0}}{(m_{0}^{2}-m^{2})+f(m)-i\,m_{0}\Gamma(m)}\,,$
(10)

with an additional mass dependence

$f(m)=\Gamma_{0}\,\frac{m_{0}^{2}}{q_{0}^{3}}\,\left[\;q^{2}\left[h(m)-h(m_{0})\right]+\left(\,m_{0}^{2}-m^{2}\,\right)\,q^{2}_{0}\,\frac{{\rm
d}h}{{\rm d}m^{2}}\bigg{|}_{m_{0}}\;\right]\,,$ (11)

where

$h(m)=\frac{2}{\pi}\,\frac{q}{m}\,\log\left(\frac{m+2q}{2m_{\pi}}\right)\,,$
(12)

and

$\frac{{\rm d}h}{{\rm
d}m^{2}}\bigg{|}_{m_{0}}=h(m_{0})\left[(8q_{0}^{2})^{-1}-(2m_{0}^{2})^{-1}\right]\,+\,(2\pi
m_{0}^{2})^{-1}\,.$ (13)

The constant parameter $D$ is given by

$D=\frac{3}{\pi}\frac{m_{\pi}^{2}}{q_{0}^{2}}\,\log\left(\frac{m_{0}+2q_{0}}{2m_{\pi}}\right)+\frac{m_{0}}{2\pi\,q_{0}}-\frac{m_{\pi}^{2}m_{0}}{\pi\,q_{0}^{3}}\,.$
(14)

Isospin-violating $\omega(782)$ decays to two charged pions can occur via
$\omega(782)$ mixing with the $\rho(770)^{0}$ state. To account for such
effects, a model is constructed that directly parameterises the interference
between these two contributions following Ref. [64],

$R_{\rho\omega}(m)=R_{\rho}(m)\left[\frac{1+R_{\omega}(m)\,\Delta\,|\zeta|\exp(i\phi_{\zeta})}{1-\Delta^{2}\,R_{\rho}(m)\,R_{\omega}(m)}\right]\,,$
(15)

where $R_{\rho}(m)$ is the Gounaris–Sakurai $\rho(770)^{0}$ lineshape,
$R_{\omega}(m)$ is the relativistic Breit–Wigner $\omega(782)$ lineshape,
$|\zeta|$ and $\phi_{\zeta}$ are free parameters of the fit that denote the
respective magnitude and phase of the production amplitude of $\omega(782)$
with respect to that for the $\rho(770)^{0}$ state, and
$\Delta\equiv\delta\left(m_{\rho}+m_{\omega}\right)$, where $\delta$ governs
the electromagnetic mixing of $\rho(770)^{0}$ and $\omega(782)$ and $m_{\rho}$
and $m_{\omega}$ represent the known particle masses [55]. This is equivalent
to the parameterisation described in Ref. [65] if the small $\Delta^{2}$ term
in the denominator is ignored. The value for $\delta$ is fixed in the fit to
$\delta=0.00215\pm 0.00035\text{\,Ge\kern-1.00006ptV}$ [64].

###  5.3 S-wave models

The S-wave ($L=0$) component of the
${{{B}^{+}}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{+}}{{\pi}^{-}}$ amplitude is both
large in magnitude and contains many overlapping resonances and decay channel
thresholds, i.e., where increasing two-body invariant mass opens additional
decay channels that were previously inaccessible, thereby modulating the
observed intensity. This analysis includes three distinct treatments of the
S-wave component in
${{{B}^{+}}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{+}}{{\pi}^{-}}$ in an attempt to
better understand its behaviour. They also increase confidence that parameters
reported for the non-S-wave contributions are robust and provide additional
information for further study.

As such, three sets of results are presented here, corresponding to the cases
where the ${{\pi}^{+}}{{\pi}^{-}}$ S-wave is described by: (i) a coherent sum
of specific resonant contributions (isobar); (ii) a monolithic, unitarity-
preserving model informed by historical scattering data (K-matrix); and (iii)
a quasi-model-independent binned approach (QMI). All approaches contain
identical contributions to higher partial waves, where $L>0$.

####  5.3.1 Isobar model

The isobar model S-wave amplitude is represented by the coherent sum of
contributions from the $\sigma$, or $f_{0}(500)$, meson and a
$\pi\pi\leftrightarrow{K}{\kern 1.79993pt\overline{\kern-1.79993ptK}}$
rescattering amplitude within the mass range
$1.0<m({{\pi}^{+}}{{\pi}^{-}})<1.5\text{\,Ge\kern-1.00006ptV}$. The $\sigma$
meson is represented as a simple pole [66, 36], parameterised as

$A_{\sigma}(m)=\frac{1}{s_{\sigma}-m^{2}}\,,$ (16)

with $s_{\sigma}$ the square of the pole position
$\sqrt{s_{\sigma}}=m_{\sigma}-\,i\,\Gamma_{\sigma}$, extracted from the fit.

The concept of $\pi\pi\leftrightarrow{K}{\kern
1.79993pt\overline{\kern-1.79993ptK}}$ rescattering was originally developed
inside the context of two-body interactions. For three-body decays,
rescattering means that a pair of mesons produced in one channel will appear
in the final state of a coupled channel. Therefore, a model is used that
describes the source of the rescattering [35, 32],

$A_{\rm{source}}(m)=[1+(m/\Delta^{2}_{\pi\pi})]^{-1}[1+(m/\Delta^{2}_{KK})]^{-1}\,,$
(17)

where $\Delta^{2}_{\pi\pi}=\Delta^{2}_{KK}=1\text{\,Ge\kern-1.00006ptV}$. The
total rescattering amplitude in the three-body ${{B}^{+}}$ decay is then

$A_{\rm{scatt}}(m)=A_{\rm{source}}(m)\,f_{\rm{rescatt}}(m)\,.$ (18)

The amplitude $f_{\rm{rescatt}}(m)=\sqrt{1-\eta(m)^{2}}e^{2i\delta(m)}$ is
described in terms of the inelasticity, $\eta(m)$, and a phase shift,
$\delta(m)$. Functional forms of these are used that combine constraints from
unitarity and analyticity with dispersion relation techniques [67]. The
inelasticity is described by

$\eta(m)=1-\left(\epsilon_{1}\dfrac{k_{2}(m)}{m}+\epsilon_{2}\dfrac{k_{2}^{2}(m)}{m^{2}}\right)\,\dfrac{M^{\prime
2}-m^{2}}{m^{2}}\,,$ (19)

with

$k_{2}(m)=\frac{1}{2}\sqrt{m^{2}-4m^{2}_{K}}\,,$ (20)

where $m_{K}=0.494\text{\,Ge\kern-1.00006ptV}$ is the charged kaon mass,
$\epsilon_{1}=2.4$, $\epsilon_{2}=-5.5$, and
$M^{\prime}=1.5\text{\,Ge\kern-1.00006ptV}$. The phase shift is given by

$\cot\delta(m)=c_{0}\,\dfrac{(m^{2}-M^{2}_{s})(M^{2}_{f}-m^{2})}{M^{2}_{f}m}\,\dfrac{|k_{2}(m)|}{k_{2}^{2}(m)}\,,$
(21)

where $M_{f}=1.32\text{\,Ge\kern-1.00006ptV}$, $c_{0}=1.3$ and
$M_{s}=0.92\text{\,Ge\kern-1.00006ptV}$. Except for $m_{K}$, values of all
parameters are taken from Ref. [67].

####  5.3.2 K-matrix model

The coherent sum of resonant contributions modelled with Breit–Wigner
lineshapes can be used to describe the dynamics of three-body decays when the
quasi-two-body resonances are relatively narrow and isolated. However, when
there are broad, overlapping resonances (with the same isospin and spin-parity
quantum numbers) or structures that are near open decay channels, this model
does not satisfy $S$-matrix unitarity, thereby violating the conservation of
quantum mechanical probability current.

Assuming that the dynamics is dominated by two-body processes (i.e. that the
S-wave does not interact with other decay products in the final state), then
two-body unitarity is naturally conserved within the K-matrix approach [68].
This approach was originally developed for two-body scattering [69] and the
study of resonances in nuclear reactions [70, 71], but was extended to
describe resonance production and $n$-body decays in a more general way [72].
This section provides a brief introduction to the K-matrix approach as applied
in this analysis; for more detail see Ref. [73].

From unitarity conservation, the form-factor for two-body production is
related to the scattering amplitude for the same channel, when including all
coupled channels. In this way, the K-matrix model describes the amplitude,
$F_{u}$, of a channel $u$ in terms of the initial $\hat{P}$-vector preparation
of channel states $v$, that has the same form as $\hat{K}$, “scattering” into
the final state $u$ via the propagator term
$(\hat{I}-i\hat{K}\hat{\rho})^{-1}$,

$F_{u}=\sum_{v=1}^{n}[(\hat{I}-i\hat{K}\hat{\rho})^{-1}]_{uv}\,\hat{P}_{v}\,.$
(22)

The diagonal matrix $\hat{\rho}$ accounts for phase space, where the element
for the two-body channel $u$ is given by [55]

$\hat{\rho}_{uu}=\sqrt{\left(1-\frac{(m_{1}+m_{2})^{2}}{s}\right)\left(1-\frac{(m_{1}-m_{2})^{2}}{s}\right)}\,,$
(23)

where $s=m_{\pi^{+}\pi^{-}}^{2}$ and $m_{1}$ and $m_{2}$ are the rest masses
of the two decay products. This expression is analytically continued by
setting $\hat{\rho}_{uu}$ to be $i|\hat{\rho}_{uu}|$ when the channel is below
its mass threshold (provided it does not cross into another channel). For the
coupled multi-meson final states, the corresponding expression can be found in
Ref. [73].

The scattering matrix, $\hat{K}$, can be parameterised as a combination of the
sum of $N_{p}$ poles with real bare masses $m_{\alpha}$, together with
nonresonant “slowly-varying” parts (SVPs). These slowly-varying parts are so-
called as they have a $1/s$ dependence, and incorporate real coupling
constants, $f^{\rm{scatt}}_{uv}$ [74]. The scattering matrix is symmetric in
$u$ and $v$,

$\hat{K}_{uv}(s)=\left(\sum_{\alpha=1}^{N_{p}}{\frac{g^{\alpha}_{u}g^{\alpha}_{v}}{m^{2}_{\alpha}-s}}+f^{\rm{scatt}}_{uv}\frac{m^{2}_{0}+s^{\rm{scatt}}_{0}}{s+s^{\rm{scatt}}_{0}}\right)f_{\\!A0}(s)\,,$
(24)

where $g_{u}^{\alpha}$ and $g_{v}^{\alpha}$ denote the real coupling constants
of the pole $m_{\alpha}$ to the channels $u$ and $v$, respectively. The factor

$f_{\\!A0}(s)=\left(\frac{1\text{\,Ge\kern-1.00006ptV}^{2}-s_{\\!A0}}{s-s_{\\!A0}}\right)\left(s-\frac{1}{2}m_{\pi}^{2}\right)$
(25)

is the Adler zero term, which suppresses the false kinematic singularity due
to left-hand cuts when $s$ goes below the $\pi\pi$ production threshold [75].
The parameters $m_{0}^{2}$, $s^{\rm{scatt}}_{0}$ and $s_{\\!A0}$ are real
constants. Note that the masses $m_{\alpha}$ are those of the poles, or the
so-called _bare_ states of the system, which do not correspond to the masses
and widths of _resonances_ – mixtures of bare states.

Extension to three-body decays is achieved by fitting for the complex
coefficients $\beta_{\alpha}$ and $f^{\rm{prod}}_{v}$ of the production pole
and SVP terms in the production vector, $\hat{P}$,

$\hat{P}_{v}=\sum_{\alpha=1}^{N_{p}}\frac{\beta_{\alpha}g_{v}^{\alpha}}{m_{\alpha}^{2}-s}+\frac{m_{0}^{2}+s_{0}^{\rm{prod}}}{s+s_{0}^{\rm{prod}}}f^{\rm{prod}}_{v}\,,$
(26)

where these coefficients are different for ${{B}^{+}}$ and ${{B}^{-}}$ decays
to allow for $C\\!P$ violation. The parameter $s_{0}^{\rm{prod}}$ is dependent
on the production environment and is taken from Ref. [76].

Using the above expressions, the amplitude for each production pole $\alpha$
to the $\pi\pi$ channel (denoted by the subscript $u=1$) is given by

$A_{\alpha}(s)=\frac{\beta_{\alpha}}{m^{2}_{\alpha}-s}\sum_{v=1}^{n}\,[(\hat{I}-i\hat{K}\hat{\rho})^{-1}]_{1v}\,g^{\alpha}_{v}\,,$
(27)

where the sum is over the intermediate channels, $v$, while the SVP production
amplitudes are separated out for each individual channel as

$A_{{\rm{SVP}},v}(s)=\frac{m_{0}^{2}+s_{0}^{\rm{prod}}}{s+s_{0}^{\rm{prod}}}\,[(\hat{I}-i\hat{K}\hat{\rho})^{-1}]_{1v}\,f^{\rm{prod}}_{v}\,.$
(28)

All of these contributions are then summed to give the total S-wave amplitude

$F_{1}=\sum_{\alpha=1}^{N_{p}}A_{\alpha}+\sum_{v=1}^{n}A_{{\rm{SVP}},v}\,.$
(29)

The $\hat{K}$ matrix elements in Eq. (24) are completely defined using the
values quoted in Ref. [77] and given in Table 2 (for five channels $n=5$ and
five poles $N_{p}=5$), which are obtained from a global analysis of $\pi\pi$
scattering data [74].

Table 2: K-matrix parameters quoted in Ref. [77], which are obtained from a global analysis of $\pi\pi$ scattering data [74]. Only $f_{1v}$ parameters are listed here, since only the dipion final state is relevant to the analysis. Masses $m_{\alpha}$ and couplings $g_{u}^{\alpha}$ are given in  GeV, while units of $\\!\text{\,Ge\kern-0.90005ptV}^{2}$ for $s$-related quantities are implied; $s^{\rm{prod}}_{0}$ is taken from Ref. [76]. $\alpha$ | $m_{\alpha}$ | $g^{\alpha}_{1}[\pi\pi]$ | $g^{\alpha}_{2}[{K}{\kern 1.79993pt\overline{\kern-1.79993ptK}}]$ | $g^{\alpha}_{3}[4\pi]$ | $g^{\alpha}_{4}[\eta\eta]$ | $g^{\alpha}_{5}[\eta\eta^{\prime}]$
---|---|---|---|---|---|---
$1$ | $0.65100$ | $0.22889$ | $-0.55377$ |  $-$$0.00000$ | $-0.39899$ | $-0.34639$
$2$ | $1.20360$ | $0.94128$ |  $-$$0.55095$ |  $-$$0.00000$ |  $-$$0.39065$ |  $-$$0.31503$
$3$ | $1.55817$ | $0.36856$ |  $-$$0.23888$ |  $-$$0.55639$ |  $-$$0.18340$ |  $-$$0.18681$
$4$ | $1.21000$ | $0.33650$ |  $-$$0.40907$ |  $-$$0.85679$ |  $-$$0.19906$ | $-0.00984$
$5$ | $1.82206$ | $0.18171$ | $-0.17558$ | $-0.79658$ | $-0.00355$ |  $-$$0.22358$
| $s_{0}^{\rm{scatt}}$ | $f_{11}^{\rm{scatt}}$ | $f_{12}^{\rm{scatt}}$ | $f_{13}^{\rm{scatt}}$ | $f_{14}^{\rm{scatt}}$ | $f_{15}^{\rm{scatt}}$
| $3.92637$ | $0.23399$ |  $-$$0.15044$ | $-0.20545$ |  $-$$0.32825$ |  $-$$0.35412$
| $s_{0}^{\rm{prod}}$ | $m_{0}^{2}$ | $s_{\\!A0}$ |  |  |
| $3.0$ | $1.0$ | $-0.15$ |  |  |

####  5.3.3 Quasi-model-independent analysis

In the quasi-model-independent (QMI) analysis, the amplitude for the $\pi\pi$
S-wave is described by individual magnitudes and phases within each of $17$
bins in $m({{\pi}^{+}}{{\pi}^{-}})$. The QMI approach exploits the fact that
the S-wave amplitude is constant in $\cos\theta_{\rm hel}$ (Eq. (5.1)) to
disentangle this component from other contributions to the Dalitz plot,
assuming the higher-order waves to be well modelled by the isobar approach.

The bins are defined identically for ${{B}^{+}}$ and ${{B}^{-}}$: $13$ below
the charm veto and $4$ above. The binning scheme is chosen ad hoc as
optimisation requires a priori knowledge of the S-wave and total
${{{B}^{+}}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{+}}{{\pi}^{-}}$ amplitude model.
The bin boundaries are chosen adaptively, by requiring initially roughly equal
numbers of candidates in each region and using the isobar model subsequently
to tune the bins in order to reduce intrinsic bias in the method’s ability to
reproduce a known S-wave.

Similar QMI approaches have previously been performed in amplitude analyses of
various decays of $B$ and $D$ mesons to study the $K\pi$ [78, 79, 80, 81],
$\pi\pi$ [82] and $D\pi$ [83] S-waves. In contrast to these previous
approaches, no interpolation between bins is performed in this analysis — the
amplitudes are constant within each bin. This choice is made as interpolation
is based on a premise of smoothness, which is appropriate for goals such as
confirmation of a new resonance. However, interpolation is not optimal in the
description of the $\pi\pi$ S-wave due to the opening of various decay
channels that become kinematically allowed as the mass increases, and which
cause sharp changes in the amplitude on scales less than the bin width. A key
difference between this and the other two S-wave approaches is that the QMI
absorbs the average contribution from potential interactions with the other
decay products in the final state. These final state interactions may be
quantified by comparing with similar dipion S-wave distributions obtained by
rescattering experiments.

###  5.4 Measurement quantities

The primary outputs of the Dalitz-plot fit are the complex isobar coefficients
$c_{j}^{\pm}$ defined in Eq. (1). However, since these depend on the choice of
phase convention, amplitude formalism and normalisation, they are not
straightforwardly comparable between analyses and have limited physical
meaning. Instead, it is useful to compare fit fractions, defined as the
integral of the absolute value of the amplitude squared for each intermediate
component, $j$, divided by that of the coherent matrix-element squared for all
intermediate contributions,

${\cal F}^{\pm}_{j}=\frac{\int_{\rm
DP}|A^{\pm}_{j}(m_{13}^{2},m_{23}^{2})|^{2}~{}{\rm d}m_{13}^{2}~{}{\rm
d}m_{23}^{2}}{\int_{\rm DP}|A^{\pm}(m_{13}^{2},m_{23}^{2})|^{2}~{}{\rm
d}m_{13}^{2}~{}{\rm d}m_{23}^{2}}\,.$ (30)

These fit fractions will not sum to unity if there is net constructive or
destructive interference. The interference fit fractions are given by

${\cal I}^{\pm}_{i\,>\,j}=\frac{\int_{\rm DP}2{\rm
Re}[A_{i}^{\pm}(m_{13}^{2},m_{23}^{2})A_{j}^{\pm*}(m_{13}^{2},m_{23}^{2})]~{}{\rm
d}m_{13}^{2}~{}{\rm d}m_{23}^{2}}{\int_{\rm
DP}|A^{\pm}(m_{13}^{2},m_{23}^{2})|^{2}~{}{\rm d}m_{13}^{2}~{}{\rm
d}m_{23}^{2}}\,,$ (31)

where the sum of the fit fractions and interference fit fractions must be
unity. The fit fractions are defined for the ${{B}^{+}}$ and ${{B}^{-}}$ decay
amplitudes separately, however a measure of the $C\\!P$-averaged contribution
can be obtained from the $C\\!P$-conserving fit fraction

${\cal F}_{j}=\frac{\int_{\rm
DP}\left(|A^{+}_{j}(m_{13}^{2},m_{23}^{2})|^{2}+|A^{-}_{j}(m_{13}^{2},m_{23}^{2})|^{2}\right)~{}{\rm
d}m_{13}^{2}~{}{\rm d}m_{23}^{2}}{\int_{\rm
DP}\left(|A^{+}(m_{13}^{2},m_{23}^{2})|^{2}+|A^{-}(m_{13}^{2},m_{23}^{2})|^{2}\right)~{}{\rm
d}m_{13}^{2}~{}{\rm d}m_{23}^{2}}\,.$ (32)

Since the $\rho(770)^{0}$ and $\omega(782)$ components cannot be completely
decoupled in the mixing amplitude, their effective lineshapes are defined for
the purpose of calculating fit fractions. This is achieved by separating Eq.
(15) into the two respective terms with a common denominator.

Another important physical quantity is the quasi-two-body parameter of $C\\!P$
violation in decay associated with a particular intermediate contribution

$\mathcal{A}_{{C\\!P}}^{j}=\frac{|c^{-}_{j}|^{2}-|c^{+}_{j}|^{2}}{|c^{-}_{j}|^{2}+|c^{+}_{j}|^{2}}\,.$
(33)

The $C\\!P$ asymmetry associated with the S-wave cannot be determined using
Eq. (33) since this component involves several contributions. Instead, it is
determined as the asymmetry of the relevant ${{B}^{-}}$ and ${{B}^{+}}$ decay
rates,

$\mathcal{A}_{{C\\!P}}^{\rm S}=\frac{\int_{\rm DP}|A^{-}_{\rm
S}(m_{13}^{2},m_{23}^{2})|^{2}~{}{\rm d}m_{13}^{2}~{}{\rm
d}m_{23}^{2}-\int_{\rm DP}|A^{+}_{\rm S}(m_{13}^{2},m_{23}^{2})|^{2}~{}{\rm
d}m_{13}^{2}~{}{\rm d}m_{23}^{2}}{\int_{\rm DP}|A^{-}_{\rm
S}(m_{13}^{2},m_{23}^{2})|^{2}~{}{\rm d}m_{13}^{2}~{}{\rm
d}m_{23}^{2}+\int_{\rm DP}|A^{+}_{\rm S}(m_{13}^{2},m_{23}^{2})|^{2}~{}{\rm
d}m_{13}^{2}~{}{\rm d}m_{23}^{2}}\,,$ (34)

where $A^{\pm}_{\rm S}$ is the coherent sum of all contributions to the
S-wave.

###  5.5 Efficiency model

The efficiency of selecting a signal decay is parameterised in the two-
dimensional square Dalitz plot and determined separately for ${{B}^{+}}$ and
${{B}^{-}}$ decays. Non-uniformities in these distributions arise as a result
of the detector geometry, reconstruction and trigger algorithms, particle
identification selections and other background rejection requirements such as
that imposed on the boosted decision-tree classifier to discriminate against
combinatorial background. The efficiency map is primarily obtained using
simulated decays, however effects arising from the hardware trigger and
particle identification efficiency are determined using data calibration
samples.

The hardware-trigger efficiency correction is calculated using pions from
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$ decays, arising from promptly
produced ${{D}^{*+}}\\!\rightarrow{{D}^{0}}({{K}^{-}}{{\pi}^{+}}){{\pi}^{+}}$
decays, and affects two disjoint subsets of the selected candidates: those
where the trigger requirements were satisfied by hadronic calorimeter deposits
as a result of the signal decay and those where the requirements were
satisfied only by deposits from the rest of the event. In the first case, the
probability to satisfy the trigger requirements is calculated using
calibration data as a function of the transverse energy of each final-state
particle of a given species, the dipole-magnet polarity, and the hadronic
calorimeter region. In the second subset, a smaller correction is applied
following the same procedure in order to account for the requirement that
these tracks did not fire the hardware trigger. These corrections are combined
according to the relative abundance of each category in data.

Figure 3: Square Dalitz-plot distributions for the (left) ${{B}^{+}}$ and
(right) ${{B}^{-}}$ signal efficiency models, smoothed using a two-dimensional
cubic spline. Depleted regions are due to the ${\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}$ veto.

The particle identification efficiency is calculated from calibration data
also corresponding to the
${{D}^{*+}}\\!\rightarrow{{D}^{0}}({{K}^{-}}{{\pi}^{+}}){{\pi}^{+}}$ decay,
where pions and kaons can be identified without the use of the LHCb particle
identification system [84]. The particle identification efficiencies for the
background-subtracted pions and kaons are parameterised in terms of their
total and transverse momentum, and the number of tracks in the event. This
efficiency is assumed to factorise with respect to the final-state tracks and
therefore the efficiency for each track is multiplied to form the overall
efficiency.

The effect of an asymmetry between the production rates of ${{{B}^{-}}}$ and
${{{B}^{+}}}$ mesons is indistinguishable in this analysis from a global
detection efficiency asymmetry. Therefore, the ${{B}^{+}}$ production
asymmetry, as measured within the LHCb acceptance [85], is taken into account
by introducing a global asymmetry of approximately $-0.6\%$ into the
efficiency maps. This is obtained as an average of the measured production
asymmetries, weighted by the relative integrated luminosity obtained in 2011
and 2012. The overall efficiency, as a function of square Dalitz-plot
position, can be seen in Fig. 3 for ${{B}^{+}}$ and ${{B}^{-}}$ decays
separately. These histograms are smoothed by a two-dimensional cubic spline to
mitigate effects of discontinuity at the bin edges, with bins abutting
kinematic boundaries repeated to ensure good behaviour at the edge of the
phase space. The signal PDF for ${{B}^{+}}$ or ${{B}^{-}}$ decays is then
given by

${\cal P}^{\pm}_{\rm
sig}(m^{2}_{13},m^{2}_{23})=\frac{\epsilon^{\pm}(m^{\prime},\theta^{\prime})|A^{\pm}(m^{2}_{13},m^{2}_{23})|^{2}}{\int_{\rm
DP}\epsilon^{\pm}(m^{\prime},\theta^{\prime})|A^{\pm}(m^{2}_{13},m^{2}_{23})|^{2}~{}{\rm
d}m^{2}_{13}{\rm d}m^{2}_{23}},$ (35)

where $\epsilon^{\pm}$ represents the Dalitz-plot dependent efficiency for the
$B^{\pm}$ decay.

###  5.6 Background model

The dominant source of background in the signal region is combinatorial in
nature. In the Dalitz-plot fit, the distribution of this background is
modelled separately for ${{B}^{+}}$ and ${{B}^{-}}$ decays using square
Dalitz-plot histograms of upper sideband data, from the region
$5.35<m({{\pi}^{+}}{{\pi}^{+}}{{\pi}^{-}})<5.68\text{\,Ge\kern-1.00006ptV\\!/}c^{2}$,
with a uniform $16\times 16$ binning in $m^{\prime}$ and $\theta^{\prime}$. In
this region, a feature is observed that can be identified as arising from real
${{B}^{0}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{-}}$ decays combined with a random
track from the rest of the event. However, this background does not enter the
signal region due to kinematics. As such, this feature is modelled in the
$m({{\pi}^{+}}{{\pi}^{-}})$ spectrum using a Gaussian function located at the
known ${B}^{0}$ mass, and events are subtracted from the combinatorial
background histograms accordingly.

The corresponding combinatorial background distributions can be seen in Fig.
4. For use in the fit, these histograms are smoothed using a two-dimensional
cubic spline to mitigate effects of discontinuity at the bin edges. In the
Dalitz-plot fit, the charge asymmetry in the combinatorial background yield is
fixed to that obtained in the ${{B}^{+}}$ invariant-mass fit described in
Section 4.

Figure 4: Square Dalitz-plot distributions for the (left) ${{B}^{+}}$ and
(right) ${{B}^{-}}$ combinatorial background models, scaled to represent their
respective yields in the signal region.

A secondary source of background arises from misidentified
${{{B}^{+}}}\\!\rightarrow{{K}^{+}}{{\pi}^{+}}{{\pi}^{-}}$ decays. This
background is modelled using simulated
${{{B}^{+}}}\\!\rightarrow{{K}^{+}}{{\pi}^{+}}{{\pi}^{-}}$ events,
reconstructed under the
${{{B}^{+}}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{+}}{{\pi}^{-}}$ decay hypothesis.
To account for the phase-space distribution of this background, the events are
weighted according to the amplitude model obtained by the BaBar collaboration
[58]. Similarly to the combinatorial background, this contribution is
described in terms of a uniform $16\times 16$ binned square Dalitz-plot
histogram, smoothed with a two-dimensional cubic spline. The corresponding
distribution, without the smoothing, can be seen in Fig. 5.

Figure 5: Square Dalitz-plot distribution for the misidentified
${{{B}^{+}}}\\!\rightarrow{{K}^{+}}{{\pi}^{+}}{{\pi}^{-}}$ background model,
scaled to represent its yield in the signal region.

###  5.7 Fit procedure

Of the three approaches to the S-wave, the isobar and K-matrix fits are
performed using the `Laura++` Dalitz-plot fitter package v3.5 [73], which
interfaces to the `MINUIT` function minimisation algorithm [86, 87]. In
contrast, the QMI approach relies on the `Mint2` [88] amplitude-analysis
interface to `Minuit2` [87]. The fundamental difference between these
amplitude-analysis software packages is in the handling of the normalisation.
The former approximates the definite integral by employing a Gaussian
quadrature approach, while the latter invokes a Monte Carlo technique.
Additionally, due to the size of its parameter space, the QMI greatly benefits
from the use of GPU-accelerated solutions.

In all cases, the combined $\rho$–$\omega$ mixing component is set to be the
“reference” amplitude. In practice, this means that the average magnitude of
the ${{B}^{+}}$ and ${{B}^{-}}$ coefficients for this component is set to
unity (in terms of Eq. (3), $x=1$), while the $\delta x$ parameter is left
free to vary to allow for $C\\!P$ violation. Since there is no sensitivity to
the phase difference between the ${{B}^{+}}$ and ${{B}^{-}}$ amplitudes, the
imaginary part of the $\rho$–$\omega$ component is set to zero for both
${{B}^{+}}$ and ${{B}^{-}}$ ($y=\delta y=0$), which means that all other
contributions to the model are measured relative to this component.

The extended likelihood function that is optimised is of the form

$\mathcal{L}=e^{-N}\prod_{i=1}^{N_{\rm
cand}}\left[\sum_{k}N_{k}\mathcal{P}_{k}^{i}\right]$ (36)

where $N_{k}$ is the yield for the candidate category $k$ (given in Table 1),
$N$ is equal to $\sum_{k}N_{k}$, $N_{\rm cand}$ is the total number of
candidates, and $\mathcal{P}_{k}^{i}$ is the probability density function for
candidates in category $k$ in terms of the Dalitz-plot coordinates. The
optimal values of the fitted parameters are found by minimising twice the
negative log-likelihood, $-2\log\mathcal{L}$.

Since Dalitz-plot analyses involve multidimensional parameter spaces,
depending upon the initial parameter values the results may correspond to a
local, rather than global, minimum of the $-2\log\mathcal{L}$ function. To
attempt to find the global minimum, a large number of fits are performed where
the initial values of the complex isobar coefficients $c_{j}$ are randomised.
The fit result with the smallest $-2\log\mathcal{L}$ value out of this
ensemble is then taken to be the nominal result for each S-wave method, and
solutions near to this are also inspected (see Appendix B.1 for the K-matrix
model fit).

###  5.8 Model selection

The inclusion or exclusion of various resonant contributions to the amplitude
is studied using the isobar and K-matrix S-wave approaches. This is not
practical with the QMI approach as the large S-wave parameter space requires a
detailed search for the global minimum given each model hypothesis. Starting
with resonant contributions identified during previous analyses of the
${{{B}^{+}}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{+}}{{\pi}^{-}}$ decay [17, 18,
19, 20], additional resonances are examined iteratively, in the order that
maximises the change in log-likelihood between the current and proposed model
with respect to the data. This procedure is terminated when the log-likelihood
gain from including any contribution not yet in the model is less than 10.
Only resonances that have been observed by two or more experiments and have
been seen to decay to two charged pions are considered initially. Scalar and
vector nonresonant contributions, and possible virtual excited ${B}^{*}$
states [89], are then investigated as possible improvements to the model,
however none are found to have a significant contribution.

After this initial iterative procedure, a second step is performed that
involves ad hoc trials of alternative mass lineshapes for resonances already
included in the model, and investigation of additional, more speculative,
states. These form the basis of several important systematic uncertainties
listed in Section 6 and are further discussed in Section 8. Lastly, tests are
performed for “latent” resonant contributions up to spin $4$, where a
resonance is inserted as a relativistic Breit–Wigner shape with a width of
$0.025\text{\,Ge\kern-1.00006ptV}$, $0.050\text{\,Ge\kern-1.00006ptV}$,
$0.100\text{\,Ge\kern-1.00006ptV}$, or $0.150\text{\,Ge\kern-1.00006ptV}$, in
mass steps of $0.2\text{\,Ge\kern-1.00006ptV}$. No significant evidence of any
resonant structure not captured by the previously established model is
observed.

The goodness of fit is assessed by comparing the fit model with the data in
square Dalitz-plot bins and determining an associated $\chi^{2}$ value (see
Appendix A for the distribution for the isobar model fit, Appendix B for the
distribution for the K-matrix model fit, and Appendix C for the distribution
for the QMI model fit). The binning is chosen through an adaptive procedure
that requires an approximately constant number of candidates from the data
sample in each bin. For various values of the required number of candidates
per bin, the ratio of the $\chi^{2}$ to the number of bins is approximately
1.5 accounting for statistical uncertainties only. Given the impact of the
systematic uncertainties on the results, as shown in Section 7, the agreement
of the fit models with the data is reasonable. Smaller $\chi^{2}$ values are
obtained for the S-wave models with larger numbers of free parameters, such
that all three approaches have comparable goodness-of-fit overall. The
distribution in the square Dalitz plot of bins that contribute significantly
to the $\chi^{2}$ does not reveal any clear source of mismodelling.
Nevertheless, a discrepancy between all of the models and the data is observed
in the region around the $f_{2}(1270)$ resonance, which is investigated
further in Section 8.3.

Resonant contributions with spin greater than zero that were identified
through the model selection procedure are common to all three approaches and
are listed in Table 3. Furthermore, the mass and width of the dominant
$\rho(770)^{0}$ contribution are left free to vary, which results in a
significantly better fit quality.

Table 3:  Non-S-wave resonances and their default lineshapes as identified by the model selection procedure. These are common to all S-wave approaches.  Resonance | Spin | Mass lineshape
---|---|---
$\rho(770)^{0}$ | $1$ | Gounaris–Sakurai ($\rho$–$\omega$ mixing)
$\omega(782)$ | $1$ | Relativistic Breit–Wigner ($\rho$–$\omega$ mixing)
$f_{2}(1270)$ | $2$ | Relativistic Breit–Wigner
$\rho(1450)^{0}$ | $1$ | Relativistic Breit–Wigner
$\rho_{3}(1690)^{0}$ | $3$ | Relativistic Breit–Wigner

##  6 Systematic uncertainties

Sources of systematic uncertainty are separated into two categories: those
that arise from experimental effects and those from the inherent lack of
knowledge on the amplitude models. The experimental systematic uncertainties
comprise those from the uncertainty on the signal and background yields, the
phase-space-dependent efficiency description, the combinatorial and
${{{B}^{+}}}\\!\rightarrow{{K}^{+}}{{\pi}^{+}}{{\pi}^{-}}$ background models
in the Dalitz plot, and the intrinsic fit bias. Model systematic uncertainties
comprise those introduced by the uncertainty on the known resonance masses and
widths, the radius parameter of the ad hoc Blatt–Weisskopf barrier factors and
from potential additional resonant contributions to the amplitude.
Furthermore, this latter category also includes sources of uncertainty that
are specific to each S-wave approach. The effects in each category are
considered to be uncorrelated and are therefore combined in quadrature to
obtain the total systematic uncertainty.

The uncertainties on the signal yield and the background yields and
asymmetries, given in Table 1, comprise a statistical component as well as
systematic effects due to the invariant-mass fit procedure. The uncertainty
arising from assumptions regarding the signal parameterisation is found by
replacing the model with two Crystal Ball functions with a common mean and
width, but independent tail parameters. Similarly, the model for the
combinatorial background is replaced with a first-order polynomial. The
uncertainty on the cross-feed
${{{B}^{+}}}\\!\rightarrow{{K}^{+}}{{\pi}^{+}}{{\pi}^{-}}$ background shape in
the three-body invariant-mass fit is negligible, however the yield of this
component is varied by three times the nominal uncertainty on the expectation
from the simulation to account for possible inaccuracies in the constraint.
Additionally, effects associated with allowing different relative signal and
partially reconstructed background yields in the data subcategories separated
by source of hardware trigger decision are investigated by constraining them
to be common in both subcategories. The combined statistical and systematic
uncertainties on the signal and background yields and the background
asymmetries are then propagated to the Dalitz-plot fit, where those variations
causing the largest upward and downward deviations with respect to the nominal
yield values are taken to assign the systematic uncertainty relating to the
three-body invariant-mass fit.

To account for the statistical uncertainty on the efficiency description, an
ensemble of efficiency maps is created by sampling bin-by-bin from the
baseline description, according to uncorrelated Gaussian distributions with
means corresponding to the central value of the nominal efficiency in each
bin, and widths corresponding to the uncertainty. The standard deviation of
the distribution of resulting Dalitz-plot fit parameters obtained when using
this ensemble is then taken to be the associated systematic uncertainty. To
account for potential biases in the method used to correct the hardware
trigger efficiency, an alternative method using
${{B}^{0}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}({\mu^{+}\mu^{-}}){{K}^{+}}{{\pi}^{-}}$ decays, requiring a positive
trigger decision on the muons from the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$ decay, is used to apply corrections to the simulation [90, 91]. The
effect on the baseline results is assigned as an uncertainty.

Additionally, to account for potential variation of the efficiency within a
nominal square Dalitz-plot bin, the efficiency map is constructed using a
finer binning scheme, and the total deviation of the results is taken as the
systematic uncertainty. The effect arising from the uncertainty on the
measured ${{B}^{+}}$ production asymmetry is also considered, but is found to
be negligible.

The statistical uncertainty on the combinatorial background distribution is
propagated to the Dalitz-plot fit results in a procedure similar to that for
the efficiency map. Uncertainty associated with the Dalitz-plot model of the
${{{B}^{+}}}\\!\rightarrow{{K}^{+}}{{\pi}^{+}}{{\pi}^{-}}$ decay is also
assigned. This is calculated by fluctuating the parameters obtained in the
${{{B}^{+}}}\\!\rightarrow{{K}^{+}}{{\pi}^{+}}{{\pi}^{-}}$ fit according to
their uncertainties [58], taking into account the reported correlations on the
statistical uncertainties. The standard deviation in the variation of the
subsequent Dalitz-plot fit results is taken to be the systematic uncertainty
due to this effect.

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 generally
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.

Sources of model uncertainty independent of the S-wave approach are those
arising from the uncertainties on the masses and widths of resonances in the
baseline model, the Blatt–Weisskopf barrier factors and contributions from
additional resonances. The systematic uncertainty due to resonance masses and
widths are again assigned with an ensemble technique, where the parameter
values, excluding those that appear in the isobar S-wave model, are fluctuated
according to the uncertainties listed in the Particle Data Group tables [55].
Where appropriate, these are taken to be those from combinations only
considering decays to ${\pi}^{+}$ ${\pi}^{-}$. The uncertainty arising from
the lack of knowledge of the radius parameter of the Blatt–Weisskopf barrier
factors is estimated by modifying the value of this between $3$ and
$5\text{\,Ge\kern-1.00006ptV}^{-1}$, with the maximum deviation of the fit
parameters taken to be the systematic uncertainty.

To account for mismodelling in the $f_{2}(1270)$ region, discussed in Section
8, an additional systematic uncertainty is assigned as the maximal variation
in fit parameters when either an additional spin-$2$ component with mass and
width parameters determined by the fit is included into the model, or when the
$f_{2}(1270)$ resonance mass and width are permitted to vary in the fit.
Furthermore, a possible contribution from the $\rho(1700)^{0}$ resonance
cannot be excluded. Using perturbative QCD calculations, the branching
fraction of the ${{{B}^{+}}}\\!\rightarrow\rho(1700)^{0}{{\pi}^{+}}$ decay has
been calculated to be around $3\times 10^{-7}$ [92], which is plausibly within
the sensitivity of this analysis. Therefore a systematic uncertainty is
assigned as the deviation of the fit parameters with respect to the nominal
values, when the $\rho(1700)^{0}$ contribution is included.

For fits related to the isobar approach, the nominal rescattering
parametrisation relies on a source term with two components as given in Eq.
(17). The fits have little sensitivity to the values chosen for the
$\Delta^{2}_{\pi\pi}$ and $\Delta^{2}_{KK}$ parameters, so the robustness of
this parametrisation is investigated by using instead a source term with only
one component, $A_{\rm{source}}=[1+(m/\Delta^{2}_{KK})]^{-1}$, and the
difference in the results obtained assigned as a systematic uncertainty. The
parameters of the $\sigma$ contribution to the S-wave are also varied within
the uncertainties on the world-average mass and width, and the effect on the
results taken as a systematic uncertainty.

For fits using the K-matrix approach, both the fourth $\hat{P}$-vector pole
and the fourth slowly varying part result in a negligible change to the total
likelihood when removed, and therefore a systematic uncertainty is assigned
that corresponds to the maximum deviation of the parameters, with respect to
the nominal values, when these components are removed from the K-matrix model.
Furthermore, in the baseline fit the $s_{0}^{\rm prod}$ parameter appearing in
the slowly varying parts of Eq. (28) is fixed to a value of
$-3\text{\,Ge\kern-1.00006ptV}^{2}/c^{4}$. However as this comprises part of
the production component of the K-matrix, this is not fixed by scattering data
and can depend on the production environment. As such, this value is varied
between $-1$ and $-5\text{\,Ge\kern-1.00006ptV}^{2}/c^{4}$ based on the
likelihood profile and the maximum deviation from the nominal fit results
taken to be the systematic uncertainty due to this effect.

For the fits involving the QMI approach, an additional bias may arise from the
intrinsic ability of the approach to reproduce the underlying analytic S-wave.
Causes of such a bias can include the definition of the binning scheme, the
extent to which the S-wave interferes with other partial waves in a particular
bin, and the approximation of an analytic lineshape by a constant amplitude in
each bin. This systematic uncertainty is evaluated reusing the ensemble of
pseudoexperiments generated for estimating the K-matrix fit bias, fitting them
with the QMI model, and determining the difference between the obtained and
true bin-averaged values of the S-wave amplitude. The QMI intrinsic bias is by
far the dominant systematic uncertainty on the S-wave magnitude and phase
motion. Previous quasi-model-independent partial wave analyses have not
recognised such an effect as a possible source of bias; an important
conclusion of this study is that the associated systematic uncertainty must be
accounted for in analyses in which quantitative results from binned partial-
wave amplitude models are obtained.

The systematic uncertainties for the $C\\!P$-averaged fit fractions and quasi-
two-body $C\\!P$ asymmetries are summarised in Tables 4 and 5 for the isobar
approach, Tables 6 and 7 for the K-matrix approach, and Tables 8 and 9 for the
QMI approach. In general the largest sources of systematic uncertainty are due
to variations in the model, which tend to dominate the total uncertainties for
the $C\\!P$-averaged fit fractions while the $C\\!P$ asymmetries for well
established resonances are somewhat more robust against these effects. In
particular, the inclusion of an additional tensor or vector resonance, i.e.
the $f_{2}(1430)$ or $\rho(1700)^{0}$ states, can have a large effect on
parameters associated with other resonances, particularly when they are in the
same partial wave. With larger data samples it may be possible to clarify the
contributions from these amplitudes and thereby reduce these uncertainties.
Intrinsic fit bias is also an important source of uncertainty for several
measurements, in particular those using the QMI description of the S-wave.

Table 4: Systematic uncertainties on the $C\\!P$-averaged fit fractions, given
in units of $10^{-2}$, for the isobar method. Uncertainties are given both for
the total S-wave, and for the individual components due to the $\sigma$ pole
and the rescattering amplitude. For comparison, the statistical uncertainties
are also listed at the bottom.

Category | $\rho(770)^{0}$ | $\omega(782)$ | $f_{2}(1270)$ | $\rho(1450)^{0}$ | $\rho_{3}(1690)^{0}$ | S-wave | Rescattering | $\sigma$
---|---|---|---|---|---|---|---|---
$B$ mass fit | $0.23$ | $0.01$ | $0.68$ | $0.07$ | $0.03$ | $0.40$ | $0.16~{}~{}$ | $0.02$
Efficiency |  |  |  |  |  |  |  |
Simulation sample size | $0.10$ | $<0.01$ | $0.06$ | $0.05$ | $0.01$ | $0.08$ | $0.02~{}~{}$ | $0.09$
Binning | $0.07$ | $<0.01$ | $0.03$ | $0.08$ | $0.02$ | $0.09$ | $0.01~{}~{}$ | $0.08$
L0 Trigger | $0.02$ | $<0.01$ | $<0.01$ | $<0.01$ | $<0.01$ | $0.02$ | $<0.01~{}~{}$ | $0.02$
Combinatorial bkgd | $0.26$ | $<0.01$ | $0.14$ | $0.15$ | $0.03$ | $0.28$ | $0.04~{}~{}$ | $0.31$
$B^{+}\rightarrow K^{+}{{\pi}^{+}}{{\pi}^{-}}$ | $0.01$ | $<0.01$ | $<0.01$ | $<0.01$ | $<0.01$ | $0.01$ | $<0.01~{}~{}$ | $0.01$
Fit bias | $0.03$ | $<0.01$ | $0.01$ | $0.01$ | $0.01$ | $0.04$ | $<0.01~{}~{}$ | $0.04$
Total experimental |  $0.37$ |  $0.01$ |  $0.69$ |  $0.19$ |  $0.05$ |  $0.5$ |  $0.17$ |  $0.34$
Amplitude model |  |  |  |  |  |  |  |
Resonance properties | $0.63$ | $0.01$ | $0.17$ | $0.39$ | $0.05$ | $0.29$ | $0.01~{}~{}$ | $0.41$
Barrier factors | $0.82$ | $0.01$ | $0.18$ | $0.40$ | $0.01$ | $0.05$ | $0.04~{}~{}$ | $0.17$
Alternative lineshapes |  |  |  |  |  |  |  |
$f_{2}(1270)$ | $0.23$ | $<0.01$ | $0.68$ | $0.07$ | $0.03$ | $0.40$ | $0.16~{}~{}$ | $0.02$
$f_{2}(1430)$ | $0.40$ | $<0.01$ | $0.88$ | $0.25$ | $0.10$ | $0.90$ | $0.21~{}~{}$ | $0.66$
$\rho(1700)^{0}$ | $0.88$ | $0.02$ | $0.09$ | $1.28$ | $0.01$ | $0.01$ | $<0.01~{}~{}$ | $<0.01$
Isobar specifics |  |  |  |  |  |  |  |
$\sigma$ from PDG | $2.00$ | $0.03$ | $0.69$ | $1.18$ | $0.32$ | $3.40$ | $0.35~{}~{}$ | $4.90$
Rescattering | $0.01$ | $<0.01$ | $0.19$ | $0.03$ | $<0.01$ | $0.11$ | $0.07~{}~{}$ | $0.24$
Total model |  $2.46$ |  $0.04$ |  $1.35$ |  $1.85$ |  $0.34$ |  $3.55$ |  $0.45$ |  $4.97$
Statistical uncertainty |  $0.60$ |  $0.03$ |  $0.28$ |  $0.29$ |  $0.08$ |  $0.49$ |  $0.15$ |  $0.48$

Table 5: Systematic uncertainties on ${\cal A}_{{C\\!P}}$ values, given in
units of $10^{-2}$, for the isobar method. Uncertainties are given both for
the total S-wave, and for the individual components due to the $\sigma$ pole
and the rescattering amplitude. For comparison, the statistical uncertainties
are also listed at the bottom.

Category | $\rho(770)^{0}$ | $\omega(782)$ | $f_{2}(1270)$ | $\rho(1450)^{0}$ | $\rho_{3}(1690)^{0}$ | S-wave | Rescattering | $\sigma$
---|---|---|---|---|---|---|---|---
$B$ mass fit | $0.12$ | $0.10$ | $0.89$ | $0.40$ | $4.19$ | $0.58$ | $4.20\quad$ | $0.54$
Efficiency |  |  |  |  |  |  |  |
Simulation sample size | $0.34$ | $0.71$ | $0.61$ | $0.92$ | $1.24$ | $0.36$ | $1.00\quad$ | $0.35$
Binning | $0.27$ | $0.87$ | $0.23$ | $1.19$ | $0.52$ | $0.28$ | $1.43\quad$ | $0.22$
L0 Trigger | $0.02$ | $0.37$ | $0.17$ | $0.31$ | $0.28$ | $0.14$ | $0.32\quad$ | $0.19$
Combinatorial bkgd | $0.40$ | $0.50$ | $1.02$ | $3.06$ | $5.75$ | $0.75$ | $3.16\quad$ | $0.75$
$B^{+}\rightarrow K^{+}{{\pi}^{+}}{{\pi}^{-}}$ | $<0.01$ | $0.01$ | $0.02$ | $0.03$ | $0.05$ | $0.01$ | $0.04\quad$ | $0.01$
Fit bias | $0.05$ | $0.35$ | $0.25$ | $1.10$ | $2.95$ | $0.04$ | $0.96\quad$ | $0.09$
Total experimental |  $0.60$ |  $1.33$ |  $1.49$ |  $3.61$ |  $7.75$ |  $0.99$ |  $5.54$ |  $0.99$
Amplitude model |  |  |  |  |  |  |  |
Resonance properties | $0.20$ | $0.53$ | $0.55$ | $2.66$ | $5.58$ | $0.41$ | $1.58\quad$ | $0.29$
Barrier factors | $0.18$ | $0.95$ | $0.80$ | $3.84$ | $1.56$ | $1.27$ | $0.34\quad$ | $1.25$
Alternative lineshapes |  |  |  |  |  |  |  |
$f_{2}(1270)$ | $0.11$ | $0.10$ | $0.82$ | $0.30$ | $4.05$ | $0.49$ | $4.07\quad$ | $0.45$
$f_{2}(1430)$ | $0.02$ | $0.04$ | $2.84$ | $1.76$ | $12.05$ | $0.98$ | $6.39\quad$ | $1.05$
$\rho(1700)^{0}$ | $1.49$ | $0.81$ | $0.75$ | $27.78$ | $4.57$ | $0.73$ | $6.32\quad$ | $0.66$
Isobar specifics |  |  |  |  |  |  |  |
$\sigma$ from PDG | $0.01$ | $3.26$ | $2.97$ | $21.83$ | $19.04$ | $0.11$ | $12.9\quad$ | $0.53$
Rescattering | $0.02$ | $0.14$ | $0.81$ | $0.19$ | $1.97$ | $0.29$ | $1.24\quad$ | $0.17$
Total model |  $1.52$ |  $3.54$ |  $4.44$ |  $35.68$ |  $24.13$ |  $1.90$ |  $16.37$ |  $1.92$
Statistical uncertainty |  $1.07$ |  $6.51$ |  $6.10$ |  $3.25$ |  $11.36$ |  $1.79$ |  $8.59$ |  $1.73$

Table 6: Systematic uncertainties on the $C\\!P$-averaged fit fractions, given
in units of $10^{-2}$, for the K-matrix method. For comparison, the
statistical uncertainties are also listed at the bottom.

Category | $\rho(770)^{0}$ | $\omega(782)$ | $f_{2}(1270)$ | $\rho(1450)^{0}$ | $\rho_{3}(1690)^{0}$ | S-wave
---|---|---|---|---|---|---
$B$ mass fit | $1.31$ | $0.01$ | $0.51$ | $0.65$ | $0.04$ | $2.53$
Efficiency |  |  |  |  |  |
Simulation sample size | $0.13$ | $<0.01$ | $0.07$ | $0.09$ | $0.02$ | $0.09$
Binning | $0.46$ | $<0.01$ | $0.08$ | $0.32$ | $0.07$ | $0.33$
L0 trigger | $0.02$ | $<0.01$ | $0.01$ | $0.04$ | $<0.01$ | $0.03$
Combinatorial bkgd | $0.41$ | $0.01$ | $0.15$ | $0.35$ | $0.08$ | $0.24$
$B^{+}\rightarrow K^{+}{{\pi}^{+}}{{\pi}^{-}}$ | $0.01$ | $<0.01$ | $<0.01$ | $0.02$ | $<0.01$ | $0.01$
Fit bias | $0.06$ | $<0.01$ | $0.05$ | $0.09$ | $0.04$ | $0.06$
Total experimental |  $1.45$ | $0.01$ |  $0.55$ |  $0.81$ |  $0.13$ |  $2.56$
Amplitude model |  |  |  |  |  |
Resonance properties | $1.02$ | $0.01$ | $0.18$ | $1.41$ | $0.09$ | $0.32$
Barrier factors | $0.24$ | $<0.01$ | $0.34$ | $0.19$ | $0.06$ | $0.57$
Alternative lineshapes |  |  |  |  |  |
$f_{2}(1270)$ | $0.29$ | $0.01$ | $0.62$ | $0.60$ | $0.03$ | $0.05$
$f_{2}(1430)$ | $2.30$ | $<0.01$ | $2.24$ | $4.17$ | $0.36$ | $0.01$
$\rho(1700)^{0}$ | $1.66$ | $0.01$ | $0.08$ | $0.55$ | $0.02$ | $0.97$
K-matrix specifics |  |  |  |  |  |
$s^{0}_{\rm prod}$ | $0.63$ | $<0.01$ | $0.06$ | $0.21$ | $0.03$ | $0.48$
K-matrix components | $0.48$ | $0.01$ | $0.04$ | $0.36$ | $0.01$ | $0.57$
Total model |  $3.14$ | $0.02$ |  $2.36$ |  $4.50$ |  $0.38$ |  $1.39$
Statistical uncertainty |  $0.8$ | $0.04$ |  $0.4$ |  $0.7$ |  $0.1$ |  $0.6$

Table 7: Systematic uncertainties on ${\cal A}_{{C\\!P}}$ values, given in
units of $10^{-2}$, for the K-matrix method. For comparison, the statistical
uncertainties are also listed at the bottom.

Category | $\rho(770)^{0}$ | $\omega(782)$ | $f_{2}(1270)$ | $\rho(1450)^{0}$ | $\rho_{3}(1690)^{0}$ | S-wave
---|---|---|---|---|---|---
$B$ mass fit | $1.97$ | $0.12$ | $1.42$ | $9.74$ | $5.77$ | $1.03$
Efficiency |  |  |  |  |  |
Simulation sample size | $0.22$ | $0.88$ | $0.73$ | $0.97$ | $1.34$ | $0.42$
Binning | $1.53$ | $5.48$ | $0.15$ | $2.89$ | $1.72$ | $1.54$
L0 trigger | $0.15$ | $0.59$ | $0.19$ | $0.32$ | $0.30$ | $0.02$
Combinatorial bkgd | $0.61$ | $0.60$ | $1.31$ | $3.45$ | $5.82$ | $0.93$
$B^{+}\rightarrow K^{+}{{\pi}^{+}}{{\pi}^{-}}$ | $0.01$ | $0.03$ | $0.03$ | $0.04$ | $0.12$ | $0.03$
Fit bias | $0.02$ | $0.04$ | $0.24$ | $0.85$ | $0.40$ | $0.36$
Total experimental |  $2.60$ |  $5.60$ |  $2.09$ |  $10.81$ |  $8.49$ |  $2.13$
Amplitude model |  |  |  |  |  |
Resonance properties | $0.62$ | $0.91$ | $1.08$ | $4.35$ | $5.34$ | $1.27$
Barrier factors | $1.97$ | $3.54$ | $0.04$ | $12.53$ | $2.79$ | $3.50$
Alternative lineshapes |  |  |  |  |  |
$f_{2}(1270)$ | $0.58$ | $0.56$ | $0.48$ | $2.96$ | $4.41$ | $1.13$
$f_{2}(1430)$ | $3.04$ | $1.69$ | $8.78$ | $41.78$ | $33.96$ | $4.77$
$\rho(1700)^{0}$ | $3.38$ | $1.17$ | $0.39$ | $8.82$ | $8.80$ | $1.60$
K-matrix specifics |  |  |  |  |  |
$s^{0}_{\rm prod}$ | $2.08$ | $4.42$ | $0.20$ | $3.42$ | $0.98$ | $2.41$
K-matrix components | $2.11$ | $5.31$ | $0.01$ | $8.11$ | $0.21$ | $1.03$
Total model |  $5.84$ |  $8.10$ |  $8.87$ |  $45.67$ |  $35.88$ |  $6.88$
Statistical uncertainty |  $1.5$ |  $8.4$ |  $4.3$ |  $8.4$ |  $11.8$ |  $2.6$

Table 8: Systematic uncertainties on the $C\\!P$-averaged fit fractions, given
in units of $10^{-2}$, for the QMI method. For comparison, the statistical
uncertainties are also listed at the bottom.

Category | $\rho(770)^{0}$ | $\omega(782)$ | $f_{2}(1270)$ | $\rho(1450)^{0}$ | $\rho_{3}(1690)^{0}$ | S-wave
---|---|---|---|---|---|---
$B$ mass fit | $1.03$ | $0.03$ | $0.29$ | $0.42$ | $0.01$ | $1.34$
Efficiency |  |  |  |  |  |
Simulation sample size | $0.15$ | $0.01$ | $0.22$ | $0.12$ | $0.02$ | $0.25$
Binning | $0.03$ | $0.01$ | $0.03$ | $0.21$ | $0.04$ | $0.01$
L0 trigger | $0.04$ | $<0.01$ | $0.01$ | $0.01$ | $<0.01$ | $0.04$
Combinatorial bkgd | $0.60$ | $0.01$ | $0.19$ | $0.67$ | $0.06$ | $0.62$
$B^{+}\rightarrow K^{+}{{\pi}^{+}}{{\pi}^{-}}$ | $0.03$ | $<0.01$ | $0.01$ | $0.02$ | $<0.01$ | $0.03$
Fit bias | $1.06$ | $0.10$ | $0.46$ | $0.61$ | $0.14$ | $0.68$
Total experimental |  $1.72$ |  $0.12$ |  $0.65$ |  $1.26$ |  $0.16$ |  $1.77$
Amplitude model |  |  |  |  |  |
Resonance properties | $0.63$ | $0.04$ | $0.21$ | $0.73$ | $0.03$ | $0.18$
Barrier factors | $0.95$ | $0.05$ | $0.58$ | $0.80$ | $0.04$ | $0.78$
Alternative lineshapes |  |  |  |  |  |
$f_{2}(1270)$ | $0.10$ | $0.04$ | $0.30$ | $0.34$ | $0.04$ | $0.36$
$f_{2}(1430)$ | $0.28$ | $0.01$ | $3.83$ | $0.49$ | $0.04$ | $0.36$
$\rho(1700)^{0}$ | $0.24$ | $0.01$ | $0.07$ | $0.52$ | $<0.01$ | $0.45$
QMI specifics |  |  |  |  |  |
QMI bias | $0.89$ | $0.11$ | $0.32$ | $3.65$ | $0.47$ | $0.93$
Total model |  $1.35$ |  $0.12$ |  $3.90$ |  $3.82$ |  $0.48$ |  $1.39$
Statistical uncertainty |  $0.61$ |  $0.09$ |  $0.38$ |  $0.42$ |  $0.13$ |  $0.56$

Table 9: Systematic uncertainties on ${\cal A}_{{C\\!P}}$ values, given in
units of $10^{-2}$, for the QMI method. For comparison, the statistical
uncertainties are also listed at the bottom.

Category | $\rho(770)^{0}$ | $\omega(782)$ | $f_{2}(1270)$ | $\rho(1450)^{0}$ | $\rho_{3}(1690)^{0}$ | S-wave
---|---|---|---|---|---|---
$B$ mass fit | $0.40$ | $1.02$ | $0.23$ | $0.92$ | $0.31$ | $0.04$
Efficiency |  |  |  |  |  |
Simulation sample size | $0.54$ | $1.59$ | $2.29$ | $1.19$ | $0.67$ | $0.46$
Binning | $0.26$ | $1.46$ | $0.25$ | $1.31$ | $0.87$ | $0.24$
L0 trigger | $0.15$ | $0.75$ | $0.14$ | $0.07$ | $0.12$ | $0.04$
Combinatorial bkgd | $0.91$ | $3.05$ | $1.96$ | $10.99$ | $2.88$ | $2.72$
$B^{+}\rightarrow K^{+}{{\pi}^{+}}{{\pi}^{-}}$ | $0.01$ | $0.04$ | $0.11$ | $0.33$ | $0.30$ | $0.07$
Fit bias | $1.92$ | $13.45$ | $5.14$ | $8.24$ | $7.07$ | $2.86$
Total experimental |  $2.29$ |  $14.20$ |  $6.04$ |  $14.25$ |  $8.00$ |  $4.17$
Amplitude model |  |  |  |  |  |
Resonance properties | $0.47$ | $2.31$ | $0.88$ | $3.23$ | $2.06$ | $1.26$
Barrier factors | $0.17$ | $3.39$ | $1.99$ | $12.01$ | $3.03$ | $5.12$
Alternative lineshapes |  |  |  |  |  |
$f_{2}(1270)$ | $0.02$ | $0.68$ | $0.70$ | $0.98$ | $0.32$ | $0.67$
$f_{2}(1430)$ | $0.51$ | $0.72$ | $0.08$ | $2.96$ | $1.52$ | $0.67$
$\rho(1700)^{0}$ | $0.63$ | $2.37$ | $0.97$ | $4.09$ | $0.29$ | $1.39$
QMI specifics |  |  |  |  |  |
QMI bias | $1.35$ | $5.56$ | $4.70$ | $29.40$ | $37.89$ | $4.40$
Total model |  $1.58$ |  $7.00$ |  $5.24$ |  $32.17$ |  $38.05$ |  $6.96$
Statistical uncertainty |  $1.27$ |  $15.44$ |  $3.63$ |  $5.55$ |  $17.01$ |  $1.52$

##  7 Results

Numerical results for the fit fractions and quasi-two-body $C\\!P$ asymmetries
are given in this section; Appendices A–C also provide the correlation
matrices for the $C\\!P$-averaged fit fractions and $C\\!P$ asymmetries, while
Appendix D presents the the combined values for the three S-wave approaches.
The complex coefficients are fitted in terms of Cartesian parameters as shown
in Eq. (3), but it can also be convenient to interpret them in terms of
magnitudes and phases. A comparison of the phases of the non-S-wave
contributions between the three approaches can be found in Appendix E. The
fitted complex coefficients are recorded for completeness in Appendices A, B
and C for the isobar, K-matrix and QMI approaches, respectively. Throughout
the tables in this section, separate parameters for the $\rho(770)^{0}$ and
$\omega(782)$ resonances are presented, which are extracted from the combined
component described by Equation 15. In general, the statistical uncertainty is
lowest for the model with the fewest parameters (isobar), and highest for the
model with the largest number of parameters (QMI), as expected.

A comparison of the data and all three fit models, projected onto $m_{\rm
low}$ for a large dipion mass range, along with the asymmetries between
${{{B}^{-}}}$ and ${{{B}^{+}}}$ decays, can be seen in Fig. 6. In these and
subsequent figures, the difference between the data and model expectation,
divided by the uncertainty on this quantity (the “pull”) is also shown below
in the same binning scheme. Projections focussing on the low $m_{\rm low}$ and
the $\rho(770)^{0}$ regions are shown in Fig. 7, while the $f_{2}(1270)$ and
high $m_{\rm high}$ regions are displayed in Fig. 8. The fit result projected
on the helicity angle in each of the $\rho(770)^{0}$ and $f_{2}(1270)$
regions, shown in Fig. 9, is also given separately above and below the
$\rho(770)^{0}$ pole in Fig. 10. The projection on the helicity angle in the
vicinity of the $\rho_{3}(1690)^{0}$ resonance is shown in Fig. 11. Figure 12
shows the raw difference in the number of ${{{B}^{-}}}$ and ${{{B}^{+}}}$
candidates in the low $m_{\rm low}$ region for negative and positive helicity
angle cosines. Additional projections separating the contributions for various
components of the amplitude model are shown in Appendices F, G and H.

(a)(b)(c)(d)

Figure 6: Fit projections of each model (a) in the low $m_{\rm low}$ region
and (b) in the full range of $m_{\rm high}$, with the corresponding
asymmetries shown beneath in (c) and (d). The normalised residual or pull
distribution, defined as the difference between the bin value less the fit
value over the uncertainty on the number of events in that bin, is shown below
each fit projection.

(a)(b)(c)(d)

Figure 7: Fit projections of each model on $m_{\rm low}$ (a) in the region
below the $\rho(770)^{0}$ resonance and (b) in the $\rho(770)^{0}$ region,
with the corresponding asymmetries shown beneath in (c) and (d). The pull
distribution is shown below each fit projection.

(a)(b)(c)(d)

Figure 8: Fit projections of each model on $m_{\rm low}$ (a) in the region
around the $f_{2}(1270)$ resonance and (b) in the high $m_{\rm high}$ region,
with the corresponding asymmetries shown beneath in (c) and (d). The pull
distribution is shown below each fit projection.

(a)(b)(c)(d)

Figure 9: Fit projections of each model on $\cos\,\theta_{\rm hel}$ (a) in the
region around the $\rho(770)^{0}$ resonance and (b) in the $f_{2}(1270)$
region, with the corresponding asymmetries shown beneath in (c) and (d). The
pull distribution is shown below each fit projection.

(a)(b)(c)(d)

Figure 10: Fit projections of each model on $\cos\,\theta_{\rm hel}$ in the
regions (a) below and (b) above the $\rho(770)^{0}$ resonance pole, with the
corresponding asymmetries shown beneath in (c) and (d). The pull distribution
is shown below each fit projection.

(a)(b)

Figure 11: Fit projections of each model (a) on $\cos\,\theta_{\rm hel}$ in
the $\rho_{3}(1690)$ region, with (b) the corresponding asymmetry shown
beneath. The pull distribution is shown below each fit projection.

(a)(b)

Figure 12: Raw difference in the number of ${{{B}^{-}}}$ and ${{{B}^{+}}}$
candidates in the low $m_{\rm low}$ region, for (a) positive, and (b) negative
cosine of the helicity angle. The pull distribution is shown below each fit
projection.

###  7.1 Fit fractions

The $C\\!P$-averaged fit fractions are given in Table 11 for all three S-wave
approaches. The fit fractions and interference fit fractions, separated by
${B}^{\pm}$ charge for each S-wave approach, are given in Tables 12–17. In all
cases, statistical uncertainties are calculated using $68\%$ confidence
intervals obtained from the results of fits performed to data sets sampled
from the nominal fit models. Throughout this paper, if three uncertainties are
listed, they are separated into statistical, systematic and amplitude model
sources, whereas if only two are listed, the systematic and models sources
have been combined in quadrature for brevity. The total sums of fit fractions
for the ${{B}^{+}}$ and ${{B}^{-}}$ amplitudes can be found in Table 10.

Table 10: The $C\\!P$-separated sum of fit fractions in units of $10^{-2}$, for each approach, where the first uncertainty is statistical, the second the experimental systematic and the third is the model systematic.  S-wave approach | ${{B}^{+}}$ | ${{B}^{-}}$
---|---|---
Isobar | $93.7\pm 2.6\pm 1.5\pm 4.5$ | $100.7\pm 2.7\pm 1.7\pm 6.0$
K-Matrix | $99.2\pm 1.8\pm 4.1\pm 5.7$ | $108.3\pm 1.7\pm 3.3\pm 9.3$
QMI | $92.2\pm 1.2\pm 7.7\pm 3.2$ | $108.0\pm 1.7\pm 3.7\pm 6.3$
Table 11: The $C\\!P$-averaged fit fractions in units of $10^{-2}$, for each
approach, where the first uncertainty is statistical, the second the
experimental systematic and the third is the model systematic.

Component | Isobar | K-matrix | QMI
---|---|---|---
$\rho(770)^{0}$ | $55.5\phantom{0}\pm 0.6\phantom{0}\pm 0.4\phantom{0}\pm 2.5\phantom{0}$ | $56.5\phantom{0}\pm 0.7\phantom{0}\pm 1.5\phantom{0}\pm 3.1\phantom{0}$ | $54.8\phantom{0}\pm 1.0\phantom{0}\pm 1.9\phantom{0}\pm 1.0\phantom{0}$
$\omega(782)$ | $0.50\pm 0.03\pm 0.01\pm 0.04$ | $0.47\pm 0.04\pm 0.01\pm 0.03$ | $\phantom{0}0.57\pm 0.10\pm 0.12\pm 0.12$
$f_{2}(1270)$ | $9.0\phantom{0}\pm 0.3\phantom{0}\pm 0.7\phantom{0}\pm 1.4\phantom{0}$ | $9.3\phantom{0}\pm 0.4\phantom{0}\pm 0.6\phantom{0}\pm 2.4\phantom{0}$ | $9.6\phantom{0}\pm 0.4\phantom{0}\pm 0.7\phantom{0}\pm 3.9\phantom{0}$
$\rho(1450)^{0}$ | $5.2\phantom{0}\pm 0.3\phantom{0}\pm 0.2\phantom{0}\pm 1.9\phantom{0}$ | $10.5\phantom{0}\pm 0.7\phantom{0}\pm 0.8\phantom{0}\pm 4.5\phantom{0}$ | $\phantom{0}7.4\phantom{0}\pm 0.5\phantom{0}\pm 3.9\phantom{0}\pm 1.1\phantom{0}$
$\rho_{3}(1690)^{0}$ | $0.5\phantom{0}\pm 0.1\phantom{0}\pm 0.1\phantom{0}\pm 0.3\phantom{0}$ | $1.5\phantom{0}\pm 0.1\phantom{0}\pm 0.1\phantom{0}\pm 0.4\phantom{0}$ | $\phantom{0}1.0\phantom{0}\pm 0.1\phantom{0}\pm 0.5\phantom{0}\pm 0.1\phantom{0}$
S-wave | $25.4\phantom{0}\pm 0.5\phantom{0}\pm 0.5\phantom{0}\pm 3.6\phantom{0}$ | $25.7\phantom{0}\pm 0.6\phantom{0}\pm 2.6\phantom{0}\pm 1.4\phantom{0}$ | $26.8\phantom{0}\pm 0.7\phantom{0}\pm 2.0\phantom{0}\pm 1.0\phantom{0}$

Table 12: Fit (diagonal) and interference (off-diagonal) fractions for
${{B}^{+}}$ decay in units of $10^{-2}$, between amplitude components in the
isobar approach. The first uncertainty is statistical and the second the
quadratic sum of systematic and model sources.

|  $\rho(770)^{0}$–$\omega(782)$ | $f_{2}(1270)$ | $\rho(1450)^{0}$ | $\rho_{3}(1690)^{0}$ | rescattering | $\sigma$
---|---|---|---|---|---|---
$\rho(770)^{0}$–$\omega(782)$ | $57.9\pm 0.8\pm 1.6$ | $-1.8\pm 0.1\pm 0.3$ | $+8.3\pm 0.6\pm 4.1$ | $+0.8\pm 0.1\pm 0.1$ | $-0.7\pm 0.1\pm 0.1$ | $+1.3\pm 0.2\pm 0.4$
$f_{2}(1270)$ |  | $5.1\pm 0.4\pm 1.1$ | $-0.4\pm 0.1\pm 0.3$ | $-0.2\pm 0.0\pm 0.0$ | $+0.2\pm 0.0\pm 0.1$ | $-0.2\pm 0.1\pm 0.3$
$\rho(1450)^{0}$ |  |  | $6.2\pm 0.5\pm 1.1$ | $+0.1\pm 0.0\pm 0.1$ | $-0.1\pm 0.0\pm 0.1$ | $-2.7\pm 0.1\pm 0.4$
$\rho_{3}(1690)^{0}$ |  |  |  | $1.0\pm 0.2\pm 0.2$ | $0.0\pm 0.0\pm 0.0$ | $+0.3\pm 0.1\pm 0.1$
rescattering |  |  |  |  | $0.8\pm 0.1\pm 0.2$ | $0.0\pm 0.2\pm 0.6$
$\sigma$ |  |  |  |  |  | $22.2\pm 0.6\pm 0.9$

Table 13: Fit (diagonal) and interference (off-diagonal) fractions for
${{B}^{-}}$ decay in units of $10^{-2}$, between amplitude components in the
isobar approach. The first uncertainty is statistical and the second the
quadratic sum of systematic and model sources.

|  $\rho(770)^{0}$–$\omega(782)$ | $f_{2}(1270)$ | $\rho(1450)^{0}$ | $\rho_{3}(1690)^{0}$ | rescattering | $\sigma$
---|---|---|---|---|---|---
$\rho(770)^{0}$–$\omega(782)$ | $53.3\pm 0.8\pm 2.1$ | $-1.4\pm 0.1\pm 0.6$ | $+1.8\pm 0.6\pm 1.4$ | $+0.2\pm 0.1\pm 0.2$ | $-0.5\pm 0.1\pm 0.1$ | $+5.7\pm 0.4\pm 0.3$
$f_{2}(1270)$ |  | $12.6\pm 0.4\pm 1.6$ | $-0.9\pm 0.1\pm 0.3$ | $0.0\pm 0.0\pm 0.0$ | $+0.5\pm 0.0\pm 0.1$ | $-1.1\pm 0.1\pm 0.6$
$\rho(1450)^{0}$ |  |  | $4.3\pm 0.4\pm 2.9$ | $0.0\pm 0.0\pm 0.0$ | $-0.2\pm 0.1\pm 0.1$ | $-2.1\pm 0.2\pm 0.7$
$\rho_{3}(1690)^{0}$ |  |  |  | $0.1\pm 0.0\pm 0.1$ | $0.0\pm 0.0\pm 0.0$ | $+0.2\pm 0.1\pm 0.1$
rescattering |  |  |  |  | $2.0\pm 0.2\pm 0.6$ | $-2.9\pm 0.3\pm 0.5$
$\sigma$ |  |  |  |  |  | $27.9\pm 0.7\pm 1.5$

Table 14: Fit (diagonal) and interference (off-diagonal) fractions for
${{B}^{+}}$ decay in units of $10^{-2}$, between amplitude components in the
K-matrix approach. The first uncertainty is statistical and the second the
quadratic sum of systematic and model sources.

|  $\rho(770)^{0}$–$\omega(782)$ | $f_{2}(1270)$ | $\rho(1450)^{0}$ | $\rho_{3}(1690)^{0}$ | S-wave
---|---|---|---|---|---
$\rho(770)^{0}$–$\omega(782)$ | $59.5\pm 2.1\pm 9.8$ | $-1.7\pm 0.1\pm 0.5$ | $+2.8\pm 1.5\pm 7.7$ | $+1.0\pm 0.1\pm 0.3$ | $+0.6\pm 0.5\pm 2.3$
$f_{2}(1270)$ |  | $5.6\pm 0.5\pm 2.5$ | $-0.3\pm 0.1\pm 0.4$ | $-0.3\pm 0.0\pm 0.1$ | $0.0\pm 0.2\pm 0.5$
$\rho(1450)^{0}$ |  |  | $10.2\pm 0.8\pm 3.1$ | $+0.4\pm 0.1\pm 0.4$ | $-3.0\pm 0.3\pm 1.0$
$\rho_{3}(1690)^{0}$ |  |  |  | $2.1\pm 0.2\pm 0.3$ | $0.0\pm 0.2\pm 0.6$
S-wave |  |  |  |  | $23.1\pm 0.9\pm 3.7$

Table 15: Fit (diagonal) and interference (off-diagonal) fractions for
${{B}^{-}}$ decay in units of $10^{-2}$, between amplitude components in the
K-matrix approach. The first uncertainty is statistical and the second the
quadratic sum of systematic and model sources.

|  $\rho(770)^{0}$–$\omega(782)$ | $f_{2}(1270)$ | $\rho(1450)^{0}$ | $\rho_{3}(1690)^{0}$ | S-wave
---|---|---|---|---|---
$\rho(770)^{0}$–$\omega(782)$ | $56.1\pm 2.0\pm 14.0$ | $-0.2\pm 0.2\pm 0.9$ | $-8.6\pm 1.3\pm 10.9$ | $+0.7\pm 0.1\pm 0.3$ | $+4.4\pm 0.4\pm 1.4$
$f_{2}(1270)$ |  | $12.5\pm 0.6\pm 2.4$ | $-1.1\pm 0.1\pm\phantom{0}0.3$ | $-0.2\pm 0.0\pm 0.2$ | $-1.2\pm 0.2\pm 0.9$
$\rho(1450)^{0}$ |  |  | $10.8\pm 1.0\pm\phantom{0}7.4$ | $-0.1\pm 0.1\pm 0.2$ | $-3.0\pm 0.3\pm 1.4$
$\rho_{3}(1690)^{0}$ |  |  |  | $0.9\pm 0.2\pm 0.6$ | $+0.8\pm 0.1\pm 0.5$
S-wave |  |  |  |  | $28.1\pm 0.7\pm 3.1$

Table 16: Fit (diagonal) and interference (off-diagonal) fractions for
${{B}^{+}}$ decay in units of $10^{-2}$, between amplitude components in the
QMI approach. The first uncertainty is statistical and the second the
quadratic sum of systematic and model sources.

|  $\rho(770)^{0}$–$\omega(782)$ | $f_{2}(1270)$ | $\rho(1450)^{0}$ | $\rho_{3}(1690)^{0}$ | S-wave
---|---|---|---|---|---
$\rho(770)^{0}$–$\omega(782)$ | $52.4\pm 1.3\pm 4.4$ | $-1.7\pm 0.1\pm 0.7$ | $+5.6\pm 1.3\pm 6.1$ | $+0.8\pm 0.1\pm 0.3$ | $-1.1\pm 0.6\pm 3.3$
$f_{2}(1270)$ |  | $6.0\pm 0.5\pm 3.5$ | $-0.4\pm 0.0\pm 0.4$ | $-0.4\pm 0.0\pm 0.1$ | $+0.2\pm 0.1\pm 0.5$
$\rho(1450)^{0}$ |  |  | $8.5\pm 0.8\pm 3.4$ | $+0.1\pm 0.0\pm 0.4$ | $-2.3\pm 0.6\pm 2.0$
$\rho_{3}(1690)^{0}$ |  |  |  | $1.9\pm 0.3\pm 0.5$ | $-0.3\pm 0.1\pm 0.4$
S-wave |  |  |  |  | $22.8\pm 1.1\pm 3.4$

Table 17: Fit (diagonal) and interference (off-diagonal) fractions for
${{B}^{-}}$ decay in units of $10^{-2}$, between amplitude components in the
QMI approach. The first uncertainty is statistical and the second the
quadratic sum of systematic and model sources.

|  $\rho(770)^{0}$–$\omega(782)$ | $f_{2}(1270)$ | $\rho(1450)^{0}$ | $\rho_{3}(1690)^{0}$ | S-wave
---|---|---|---|---|---
$\rho(770)^{0}$–$\omega(782)$ | $57.2\pm 1.3\pm 4.8$ | $-0.6\pm 0.2\pm 0.9$ | $-3.8\pm 1.3\pm 5.1$ | $+0.1\pm 0.1\pm 0.1$ | $+6.0\pm 0.7\pm 2.3$
$f_{2}(1270)$ |  | $13.2\pm 0.6\pm 7.6$ | $-0.7\pm 0.1\pm 0.5$ | $-0.1\pm 0.0\pm 0.2$ | $+0.1\pm 0.2\pm 1.9$
$\rho(1450)^{0}$ |  |  | $6.2\pm 0.7\pm 5.5$ | $0.0\pm 0.0\pm 0.1$ | $-2.9\pm 0.4\pm 2.3$
$\rho_{3}(1690)^{0}$ |  |  |  | $0.1\pm 0.1\pm 0.6$ | $0.0\pm 0.1\pm 0.4$
S-wave |  |  |  |  | $30.8\pm 0.8\pm 2.7$

###  7.2 $C\\!P$ asymmetries

Quasi-two-body $C\\!P$ asymmetries associated to each component are shown in
Table 18. A detailed discussion of these results is given in Section 8,
however it should be stressed that $C\\!P$-violation effects can manifest in
Dalitz-plot distributions through interference effects that leave values of
the quasi-two-body $C\\!P$ asymmetries consistent with zero, and indeed this
occurs in ${{{B}^{+}}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{+}}{{\pi}^{-}}$ decays.
The phase-space integrated $C\\!P$ asymmetry is consistent, in all three
models, with the value previously determined through model-independent
analysis [12].

Table 18: Quasi-two-body $C\\!P$ asymmetries in units of $10^{-2}$, for each
approach. The first uncertainty is statistical, the second the experimental
systematic and the third is the model systematic.

Component | Isobar | K-matrix | QMI
---|---|---|---
$\rho(770)^{0}$ | $+0.7\pm\phantom{0}1.1\pm\phantom{0}0.6\pm\phantom{0}1.5$ | $+4.2\pm\phantom{0}1.5\pm\phantom{0}2.6\pm\phantom{0}5.8$ | $+4.4\pm\phantom{0}1.7\pm\phantom{0}2.3\pm\phantom{0}1.6$
$\omega(782)$ | $-4.8\pm\phantom{0}6.5\pm\phantom{0}1.3\pm\phantom{0}3.5$ | $-6.2\pm\phantom{0}8.4\pm\phantom{0}5.6\pm\phantom{0}8.1$ | $-7.9\pm 16.5\pm 14.2\pm\phantom{0}7.0$
$f_{2}(1270)$ | $+46.8\pm\phantom{0}6.1\pm\phantom{0}1.5\pm\phantom{0}4.4$ | $+42.8\pm\phantom{0}4.1\pm\phantom{0}2.1\pm\phantom{0}8.9$ | $+37.6\pm\phantom{0}4.4\pm\phantom{0}6.0\pm\phantom{0}5.2$
$\rho(1450)^{0}$ | $-12.9\pm\phantom{0}3.3\pm\phantom{0}3.6\pm 35.7$ | $+9.0\pm\phantom{0}6.0\pm 10.8\pm 45.7$ | $-15.5\pm\phantom{0}7.3\pm 14.3\pm 32.2$
$\rho_{3}(1690)^{0}$ | $-80.1\pm 11.4\pm 7.8\pm 24.1$ | $-35.7\pm 10.8\pm\phantom{0}8.5\pm 35.9$ | $-93.2\pm\phantom{0}6.8\pm\phantom{0}8.0\pm 38.1$
S-wave | $+14.4\pm\phantom{0}1.8\pm\phantom{0}1.0\pm\phantom{0}1.9$ | $+15.8\pm\phantom{0}2.6\pm\phantom{0}2.1\pm\phantom{0}6.9$ | $+15.0\pm\phantom{0}2.7\pm\phantom{0}4.2\pm\phantom{0}7.0$

###  7.3 S-wave projections

The squared amplitude and phase motion of the S-wave models as a function of
$m({{\pi}^{+}}{{\pi}^{-}})$ can be seen in Fig. 13(a) and (b) for the isobar
approach, Fig. 13(c) and (d) for the K-matrix approach and Fig. 13(e) and (f)
for the QMI approach. A comparison of all three models, for the
$C\\!P$-averaged S-wave projections, can be seen in Fig. 14. The QMI S-wave is
recorded in Table 19, while the statistical and systematic correlation
matrices obtained with this approach are given in the Supplemental Material.

(a)(b)

(c)(d)

(e)(f)

Figure 13:  The (top) isobar, (middle) K-matrix and (bottom) QMI S-wave
results where (a), (c) and (e) show the magnitude squared while (b), (d) and
(f) show the phase motion. Discontinuities in the phase motion are due to
presentation in the range $[-180^{\circ},180^{\circ}]$. Red curves indicate
${{B}^{+}}$ while blue curves represent ${{B}^{-}}$ decays, with the
statistical and total uncertainties bounded by the dark and light bands,
respectively (incorporating only the dominant systematic uncertainties). Note
that the overall scale of the squared magnitude contains no physical meaning,
but is simply a manifestation of the different scale factors and conventions
adopted by each of the three amplitude analysis approaches.

(a)(b)

Figure 14:  Comparison of results for the $C\\!P$-averaged S-wave obtained in
the three different approaches, where (a) shows the magnitude squared while
(b) shows the phase motion. Discontinuities in the phase motion are due to
presentation in the range $[-180^{\circ},180^{\circ}]$. The blue curve
indicates the isobar S-wave, the amber curve indicates the K-matrix S-wave,
and the green points with error bars represent the QMI S-wave. The band or
error bars in each case represent the total uncertainty, incorporating the
dominant systematic uncertainties. As the integral of the $|A|^{2}$ plot in
each approach is proportional to its respective S-wave fit fraction, the
overall scale of the K-matrix and QMI plots are set relative to the isobar
S-wave fit fraction in order to facilitate comparison between the three
approaches.  Table 19: QMI S-wave fit results where the first uncertainty is
statistical and the second the quadratic sum of systematic and model sources.

Region, $i$ ($\text{\,Ge\kern-1.00006ptV\\!/}c^{2}$) |  $|A^{+}_{i}|^{2}$ ($10^{-2}$) | $\delta^{+}_{i}\,(^{\circ})$ |  $|A^{-}_{i}|^{2}$ ($10^{-2}$) | $\delta^{-}_{i}\,(^{\circ})$
---|---|---|---|---
$0.28\leq m(\pi^{+}\pi^{-})<0.51$ | $5.87\pm 0.38\pm 0.93$ | $-170\pm 10\pm\phantom{0}24$ | $7.86\pm 0.43\pm 1.35$ | $\phantom{0}{-}96\pm\phantom{0}6\pm\phantom{0}10$
$0.51\leq m(\pi^{+}\pi^{-})<0.63$ | $5.06\pm 0.34\pm 0.92$ | $+172\pm\phantom{0}4\pm\phantom{0}47$ | $7.14\pm 0.44\pm 1.08$ | $\phantom{0}{-}63\pm\phantom{0}3\pm\phantom{00}6$
$0.63\leq m(\pi^{+}\pi^{-})<0.70$ | $4.34\pm 0.42\pm 0.90$ | $-168\pm\phantom{0}5\pm\phantom{0}24$ | $4.28\pm 0.48\pm 1.26$ | $\phantom{0}{-}59\pm\phantom{0}3\pm\phantom{00}6$
$0.70\leq m(\pi^{+}\pi^{-})<0.77$ | $4.37\pm 0.48\pm 0.90$ | $-131\pm\phantom{0}6\pm\phantom{0}12$ | $4.06\pm 0.54\pm 1.51$ | $\phantom{0}{-}50\pm\phantom{0}4\pm\phantom{00}9$
$0.77\leq m(\pi^{+}\pi^{-})<0.84$ | $4.15\pm 0.51\pm 0.98$ | $-118\pm\phantom{0}3\pm\phantom{00}7$ | $2.34\pm 0.23\pm 1.35$ | $\phantom{0}{-}61\pm 11\pm\phantom{0}29$
$0.84\leq m(\pi^{+}\pi^{-})<0.90$ | $3.46\pm 0.45\pm 0.88$ | $\phantom{0}{-}97\pm\phantom{0}3\pm\phantom{00}7$ | $2.76\pm 0.29\pm 0.95$ | $\phantom{0}{-}31\pm\phantom{0}9\pm\phantom{0}24$
$0.90\leq m(\pi^{+}\pi^{-})<0.99$ | $2.36\pm 0.31\pm 0.59$ | $\phantom{0}{-}77\pm\phantom{0}4\pm\phantom{0}11$ | $1.36\pm 0.18\pm 1.33$ | $\phantom{0}{-}36\pm\phantom{0}6\pm\phantom{0}71$
$0.99\leq m(\pi^{+}\pi^{-})<1.11$ | $1.04\pm 0.18\pm 0.41$ | $-105\pm\phantom{0}5\pm\phantom{0}11$ | $0.24\pm 0.07\pm 0.37$ | $\phantom{0}{-}61\pm 10\pm\phantom{0}26$
$1.11\leq m(\pi^{+}\pi^{-})<1.21$ | $1.00\pm 0.17\pm 0.43$ | $\phantom{0}{-}99\pm\phantom{0}6\pm\phantom{0}12$ | $0.54\pm 0.10\pm 0.47$ | $\phantom{0}{-}45\pm\phantom{0}8\pm\phantom{0}22$
$1.21\leq m(\pi^{+}\pi^{-})<1.30$ | $0.58\pm 0.14\pm 0.33$ | $\phantom{0}{-}91\pm\phantom{0}8\pm\phantom{0}15$ | $0.68\pm 0.11\pm 0.45$ | $\phantom{0}{-}15\pm 10\pm\phantom{0}29$
$1.30\leq m(\pi^{+}\pi^{-})<1.40$ | $0.33\pm 0.09\pm 0.17$ | $\phantom{0}{-}65\pm 10\pm\phantom{0}17$ | $0.75\pm 0.10\pm 0.38$ | $\phantom{00}{+}8\pm\phantom{0}9\pm\phantom{0}21$
$1.40\leq m(\pi^{+}\pi^{-})<1.56$ | $0.17\pm 0.05\pm 0.15$ | $\phantom{0}{-}20\pm 12\pm\phantom{0}48$ | $0.45\pm 0.07\pm 0.24$ | $\phantom{0}{+}28\pm\phantom{0}8\pm\phantom{0}32$
$1.56\leq m(\pi^{+}\pi^{-})<1.74$ | $0.01\pm 0.01\pm 0.07$ | $\phantom{0}{-}86\pm 44\pm\phantom{0}74$ | $0.41\pm 0.07\pm 0.19$ | $\phantom{00}{+}5\pm\phantom{0}9\pm\phantom{0}30$
$1.74\leq m(\pi^{+}\pi^{-})<2.00$ | $0.20\pm 0.06\pm 0.11$ | $-111\pm 13\pm\phantom{0}30$ | $0.43\pm 0.06\pm 0.14$ | $\phantom{0}{-}11\pm 13\pm\phantom{0}42$
$2.00\leq m(\pi^{+}\pi^{-})<2.50$ | $0.02\pm 0.02\pm 0.05$ | $-128\pm 21\pm 109$ | $0.19\pm 0.02\pm 0.06$ | $\phantom{0}{-}25\pm\phantom{0}8\pm 109$
$2.50\leq m(\pi^{+}\pi^{-})<3.50$ | $0.01\pm 0.00\pm 0.05$ | $-149\pm 19\pm\phantom{0}73$ | $0.08\pm 0.01\pm 0.05$ | $\phantom{0}{-}36\pm\phantom{0}8\pm 165$
$3.50\leq m(\pi^{+}\pi^{-})<5.14$ | $0.00\pm 0.01\pm 0.03$ | $+100\pm 72\pm 173$ | $0.02\pm 0.01\pm 0.07$ | $-144\pm 15\pm 121$

###  7.4 $\rho(770)^{0}$ mass and width

The $\rho(770)^{0}$ mass and width are allowed to vary freely in each fit as
mentioned in Section 5.8. The fitted results are consistent with the world-
average values, and can be seen in Table 20.

Table 20: The obtained $\rho(770)^{0}$ mass and width parameters, for each approach, where the uncertainty is statistical.  S-wave approach | Mass ($\text{\,Me\kern-1.00006ptV\\!/}c^{2}$) | Width ( MeV)
---|---|---
Isobar | $770.8\pm 1.3$ | $153.4\pm 3.2$
K-Matrix | $766.7\pm 1.4$ | $147.3\pm 3.1$
QMI | $766.3\pm 1.5$ | $148.2\pm 3.5$

###  7.5 Multiple fit solutions

A search for secondary solutions with negative log-likelihood values worse
than, but close to, that of the best fit is performed for each S-wave approach
by setting the initial values of the complex coefficients of the model to
random values and repeating the fit to data. In both the isobar and QMI
approaches, no secondary solutions are found within 25 units of $-2\log{\cal
L}$. For the K-matrix approach, however, secondary solutions are found in
which some of the pole or SVP amplitude coefficients are rotated in the Argand
plane with respect to the best fit result. Studies using data sampled from the
nominal model indicate that these could potentially be resolved with larger
data samples, and further improvements may also be possible by fitting for the
scattering parameters along with the amplitude coefficients. Isobar
coefficients and S-wave projections corresponding to the secondary minimum
closest to the most-negative minimum, with a change in log-likelihood of
$0.8$, are given in Appendix B.

##  8 Discussion

The results and figures presented in Section 7 show that the three models
exhibit good overall agreement with the data and with each other, both in
$C\\!P$-average projections and in the variation of the asymmetries across the
phase space. In this section the main features observed in the data and in the
models are discussed in more detail.

Many of the interference fit fractions are close to zero, as expected, since
interference effects between partial waves with even and odd values of the
relative angular momentum cancel when integrated over the helicity angle. The
largest interference fit fraction is between the combined $\rho$–$\omega$
component and the $\rho(1450)^{0}$ resonance; since each of these is spin-$1$,
the interference does not vanish when integrated over the Dalitz plot. No
significant interference fit-fraction asymmetries are observed, however this
does not preclude sizeable asymmetries in localised regions of the Dalitz
plot.

###  8.1 The $\rho(770)^{0}$–$\omega(782)$ region

The interference between the spin-1 $\rho(770)^{0}$ and $\omega(782)$
resonances is well described by the models, as shown in Fig. 7(b). No
significant asymmetry is observed in this region when integrating over
$\cos\theta_{\rm hel}$ as shown in Fig. 7(d), and also seen in the
$\rho(770)^{0}$ and $\omega(782)$ quasi-two-body $C\\!P$ asymmetry parameters
in Table 18. A number of theoretical calculations of these quantities are
available in the literature, with some authors [93, 94, 95, 96] predicting
values for
$\mathcal{A}_{{C\\!P}}\left(\mbox{${{{B}^{+}}}\\!\rightarrow\rho(770)^{0}{{\pi}^{+}}$}\right)$
that are consistent with the measured result, albeit sometimes with large
uncertainties. Other approaches [97, 98, 99, 100] give predictions for this
quantity which appear to now be ruled out. There is also no evident
$C\\!P$-violation effect associated with $\rho$–$\omega$ mixing, contrary to
some theoretical predictions [27, 28, 29].

A significant $C\\!P$ asymmetry in the $\rho(770)^{0}$ region can, however, be
seen in the $\cos\theta_{\rm hel}$ projections shown in Figs. 10(c) and (d)
when dipion masses below and above the known $\rho(770)^{0}$ mass are
inspected separately. The same effect can be seen in Fig. 12 where the data
are separated by the sign of the value of $\cos\theta_{\rm hel}$. This
feature, previously observed through a model-independent analysis [12], is
characteristic of $C\\!P$ violation originating from a sizeable interference
between the spin-$1$ $\rho(770)^{0}$ resonance and the broad spin-$0$
contribution present in this region. This effect cancels when integrating over
the helicity angle, since the interference term is proportional to
$\cos\theta_{\rm hel}$. In addition, the change in the asymmetry below and
above the $\rho(770)^{0}$ peak indicates that the effect is mediated by a
strong phase difference dominated by the evolution of the $\rho(770)^{0}$
Breit–Wigner amplitude phase. All three approaches to the modelling of the
${{\pi}^{+}}{{\pi}^{-}}$ S-wave describe this effect well.

###  8.2 The ${{\pi}^{+}}{{\pi}^{-}}$ S-wave

A notable feature in Fig. 7(c) is the small but approximately constant
asymmetry at $m({{\pi}^{+}}{{\pi}^{-}})$ values below the $\rho(770)^{0}$
mass. This region is dominated by the S-wave component with a small
contribution from the $\rho(770)^{0}$ low-mass tail; the $C\\!P$ asymmetry in
the S-wave in this region is also seen in all three approaches in Fig. 13(a),
(c) and (e). A $C\\!P$ asymmetry in the S-wave below the inelastic (${K}{\kern
1.79993pt\overline{\kern-1.79993ptK}}$) threshold cannot be explained via
$\pi\pi\leftrightarrow{K}{\kern 1.79993pt\overline{\kern-1.79993ptK}}$
rescattering, and therefore has a different origin to effects seen elsewhere
in the Dalitz plot.

The combined significance of $C\\!P$ violation in the S-wave and in the
interference between the S- and P-waves is evaluated from the change in log-
likelihood between the baseline fit and in fits where all relevant
$C\\!P$-violation parameters are fixed to be zero. Since the $\rho$–$\omega$
component serves as the reference amplitude in the baseline fit, this means
that all $\delta x$ and $\delta y$ parameters, defined in Eq. (3), associated
with the S-wave are fixed to zero. This is done in fits with each of the
approaches to the S-wave model, with the resulting change in log-likelihood
converted into a $p$-value, and subsequently into the number of Gaussian
standard deviations ($\sigma$), accounting for the number of fixed parameters.
The values obtained are around $30\sigma$ in all cases, despite the very
different number of degrees of freedom associated with the S-wave in the
different approaches. While this method can only be considered to give an
approximation to the significance, it is sufficient to establish the presence
of $C\\!P$ violation far beyond any reasonable doubt.

In order to separate the effects of $C\\!P$ violation in the S-wave and in the
interference between the S- and P-waves, additional fits are performed in
which the reference amplitude is changed to the S-wave. The $\delta x$ and
$\delta y$ parameters associated with the S-wave are then fixed to zero, while
those associated with the P-wave are allowed to vary in the fits. In this
case, $C\\!P$ violation in the interference between the S- and P-waves is
allowed, while none is possible in the S-wave itself, and hence the
significance of each effect individually can be assessed. The values obtained
are above $10\sigma$ in each of the S-wave modelling approaches, thus
establishing that both $C\\!P$-violation effects are present.

At low $m({{\pi}^{+}}{{\pi}^{-}})$ values, the S-wave magnitude and phase
motion of the three approaches broadly agree, particularly for the
$C\\!P$-averaged $|A|^{2}$, and all models capture similar behaviour around
$1\text{\,Ge\kern-1.00006ptV\\!/}c^{2}$. However, in the ${K}{\kern
1.79993pt\overline{\kern-1.79993ptK}}$ threshold region shown in Fig. 12, the
change in sign of the difference between the number of ${{B}^{+}}$ and
${{B}^{-}}$ candidates between positive and negative $\cos\theta_{\rm hel}$ is
captured only by the K-matrix model. It is worth noting that this is the only
model with an explicit $f_{0}(980)$ term: the isobar model includes only
$\pi\pi\leftrightarrow{K}{\kern 1.79993pt\overline{\kern-1.79993ptK}}$
rescattering above the ${K}{\kern 1.79993pt\overline{\kern-1.79993ptK}}$
threshold and the QMI binning is not sufficiently fine in this region to
resolve a narrow structure.

At $1.5\text{\,Ge\kern-1.00006ptV\\!/}c^{2}$, the K-matrix has a clear phase
motion, seen in Fig. 13, that is associated with the $f_{0}(1500)$
contribution. Consistent behaviour is seen in the QMI approach, although the
uncertainties preclude a definite corroboration of the presence of the
$f_{0}(1500)$ state. The isobar model does not include this component
explicitly, and therefore it is expected that the phase is broadly constant
here, continuing to the upper kinematic boundary. Above
$3\text{\,Ge\kern-1.00006ptV\\!/}c^{2}$, the magnitude of the S-wave component
in all three approaches is consistent with zero and therefore the phase values
are dominated by statistical and systematic uncertainties.

###  8.3 The $f_{2}(1270)$ region

Despite broad consistency between the three fit models, a clear discrepancy
with the data is apparent for all of them in the $f_{2}(1270)$ region shown in
Fig. 8(a). All fit model projections lie under (over) the data below (above)
the $f_{2}(1270)$ peak, which is set to the known value of $1275.5\pm
0.8\text{\,Me\kern-1.00006ptV\\!/}c^{2}$ [55]. Better agreement with the data
is obtained when the $f_{2}(1270)$ mass and width are allowed to vary in the
fits, as shown in Fig. 15(a). However, the obtained masses, equal to $1256\pm
4\text{\,Me\kern-1.00006ptV\\!/}c^{2}$, $1252\pm
4\text{\,Me\kern-1.00006ptV\\!/}c^{2}$ and $1260\pm
4\text{\,Me\kern-1.00006ptV\\!/}c^{2}$ for the isobar, K-matrix and QMI S-wave
approaches, respectively, where the uncertainties are statistical only, are at
least four standard deviations away from the world average. The values
obtained for the width are, however, consistent with the world average.
Moreover, if the $f_{2}(1270)$ mass and width are allowed to vary
independently in the ${{B}^{-}}$ and ${{B}^{+}}$ subsamples, inconsistent
values are obtained.

(a)(b)

Figure 15:  Data and fit model projections in the $f_{2}(1270)$ region with
(a) freely varied $f_{2}(1270)$ resonance parameters, and (b) with an
additional spin-$2$ component with mass and width parameters determined by the
fit.

Alternatively, the discrepancy between the data and the models can be reduced
by adding another spin-2 resonance in the $f_{2}(1270)$ region, as shown in
Fig. 15(b). The established states $f_{2}^{\prime}(1525)$, with
$m_{f_{2}^{\prime}(1525)}=1525\pm 5\text{\,Me\kern-1.00006ptV\\!/}c^{2}$ and
$\Gamma_{f_{2}^{\prime}(1525)}=73\,^{+6}_{-5}\text{\,Me\kern-1.00006ptV}$, and
$f_{2}(1565)$, with $m_{f_{2}(1565)}=1562\pm
13\text{\,Me\kern-1.00006ptV\\!/}c^{2}$ and $\Gamma_{f_{2}(1565)}=134\pm
8\text{\,Me\kern-1.00006ptV}$, are too high in mass and too narrow to be
likely to induce a significant effect in the $f_{2}(1270)$ region. Therefore,
an additional spin-2 resonance with mass and width parameters free to vary in
the fit is introduced, with initial values corresponding to those of the not-
well-established $f_{2}(1430)$ resonance. The fit results for the mass are
consistent between each S-wave approach, with $m_{f_{2}(X)}=1600\pm 60$,
$1541\pm 24$ and $1560\pm 14\text{\,Me\kern-1.00006ptV\\!/}c^{2}$ for the
isobar, K-matrix and QMI fits, respectively. However, the obtained values for
the width are inconsistent, varying between $\Gamma_{f_{2}(X)}=367\pm 97$,
$204\pm 78$ and $115\pm 37\text{\,Me\kern-1.00006ptV}$, where the
uncertainties are statistical only. Therefore the addition of a second spin-2
state does not appear in the baseline model, but rather is considered as a
source of systematic uncertainty on the model composition.

Consequently, the baseline model includes in the $\pi\pi$ D-wave only the
$f_{2}(1270)$ resonance, with its mass and width fixed to the known values.
Analysis of larger data samples will be required to obtain a more detailed
understanding of the $\pi\pi$ D-wave in
${{{B}^{+}}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{+}}{{\pi}^{-}}$ decays.

The effect of additional decay channels on the energy-dependent width of the
$f_{2}(1270)$ is also considered as another possibility for the baseline fit
model discrepancy. A global analysis is performed to express the known
branching fractions of $f_{2}(1270)$ decays to $\pi\pi$, $K$ $\kern
1.79993pt\overline{\kern-1.79993ptK}$, $\eta\eta$,
${{\pi}^{+}}{{\pi}^{-}}{{\pi}^{0}}{{\pi}^{0}}$,
${{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}$ and
${{\pi}^{0}}{{\pi}^{0}}{{\pi}^{0}}{{\pi}^{0}}$ in terms of their respective
couplings in $f_{2}(1270)$ decays to these final states. Subsequent fits
accounting for the energy-dependent width in a similar way as in the Flatté
lineshape [101] are found to have minimal impact on the model and therefore do
not contribute to the systematic uncertainties.

Despite the considerations outlined above, the $C\\!P$ violation associated
with the $f_{2}(1270)$ resonance is robust with respect to the experimental
and model systematic uncertainties documented in Section 6. This can be seen
by comparing the coefficients that describe the $f_{2}(1270)$ resonance with
those obtained during systematic variations as shown in Fig. 16. The fact that
the contours for ${{{B}^{+}}}$ and ${{{B}^{-}}}$ coefficients do not overlap
is a visual representation of the significantly non-zero values of the $\delta
x$ and $\delta y$ parameters of Eq. (3). The quasi-two-body $C\\!P$ asymmetry,
defined in Eq. (33), is related to the difference in the magnitudes of the
${{B}^{+}}$ and ${{B}^{-}}$ complex coefficients, i.e. the difference in
distance from $(0,0)$ to the centres of the corresponding ellipses in Fig. 16.
In addition to this, $C\\!P$ violation can also be observed in the difference
in the phases relative to the $\rho(770)^{0}$–$\omega(782)$ reference
component in the ${{B}^{+}}$ and ${{B}^{-}}$ amplitudes, which would manifest
in the Dalitz plot as a difference between ${{B}^{+}}$ and ${{B}^{-}}$ decays
in the interference pattern between $\rho(770)^{0}$ and $f_{2}(1270)$
resonances. This is indicated by the difference in the polar angle in the
Argand plane of Fig. 16. It is evident that systematic variations do not
significantly modify the distance between the solid and dashed ellipses in the
$(x,y)$ plane, meaning that the significant overall $C\\!P$ violation
associated with the $f_{2}(1270)$ resonance is robust.

(a) |  (b)
---|---
(c) |
Figure 16: Central values (points) and statistical $68\%$ Gaussian confidence
regions (ellipses) for the complex coefficients associated with the
$f_{2}(1270)$ resonance under various systematic assumptions for the
${{B}^{+}}$ (solid) and ${{B}^{-}}$ (dashed) decay amplitude models. The
nominal result and statistical uncertainty is given in black, while the
results of the dominant systematic variations to the nominal model (per
Section 6) are given by the coloured ellipses, as noted in the legend, for
each of the (a) isobar, (b) K-matrix and (c) QMI S-wave approaches. The model
specific systematic uncertainties are discussed in Sec. 6.

When interpreting Fig. 16, it should be noted that in the QMI approach the
amplitude components are not individually normalised, unlike the case for the
isobar and K-matrix approaches. Since some systematic variations can change
the overall scale of various lineshapes, their respective deviations as
depicted in this plot cannot be directly interpreted as entirely systematic in
origin, and as such naturally manifest as the largest deviations from the
nominal model.

The statistical significance of $C\\!P$ violation in ${{{B}^{+}}}\rightarrow
f_{2}(1270){{\pi}^{+}}$ is estimated by finding, for each S-wave method, the
difference of twice the log-likelihood between two fits: the nominal one and
another where the $C\\!P$-violating parameters of the $f_{2}(1270)$ are fixed
to zero. Since this quantity behaves like a $\chi^{2}$ distribution with two
degrees of freedom, it is converted into a $p$-value from which confidence
intervals are derived. The significance of $C\\!P$ violation is found to be
$20\sigma$, $15\sigma$ and $14\sigma$ for the isobar, K-matrix and QMI
approaches, respectively. This corresponds to the first observation of $C\\!P$
violation in ${{{B}^{+}}}\\!\rightarrow f_{2}(1270){{\pi}^{+}}$ decay, which
is the first observation of $C\\!P$ violation in any process with a final
state containing a tensor resonance. The measured central value of
$\mathcal{A}_{{C\\!P}}\left(\mbox{${{{B}^{+}}}\\!\rightarrow
f_{2}(1270){{\pi}^{+}}$}\right)$ is consistent with some theoretical
predictions [26, 102, 103] that, however, have large uncertainties.

###  8.4 The $\rho_{3}(1690)^{0}$ region

The contribution from a spin-$3$ $\rho_{3}(1690)^{0}$ component is evident in
Fig. 11, with a dip in intensity at $\cos\theta_{\rm hel}=0$, characteristic
of an odd-spin resonance, as well as multiple “lobes” associated with a spin
greater than $1$. The central value of the $C\\!P$ asymmetry of this component
is large and positive, however its systematic uncertainty is also large,
mostly driven by ambiguities in the amplitude model. These uncertainties
preclude any conclusive statement about $C\\!P$ violation in
${{{B}^{+}}}\\!\rightarrow\rho_{3}(1690)^{0}{{\pi}^{+}}$ decays; an analysis
with a larger data sample will be necessary to clarify this point.

##  9 Conclusions

An amplitude analysis of the
${{{B}^{+}}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{+}}{{\pi}^{-}}$ decay is
performed, using a data sample corresponding to $3\text{\,fb}^{-1}$ collected
by the LHCb experiment during Run 1. Three complementary approaches are used
to describe the large S-wave contribution to this decay. Overall good
agreement is found between all three models and the data, although some
discrepancies in the region around the $f_{2}(1270)$ region persist in the
baseline models. Significant $C\\!P$ violation associated with the
$f_{2}(1270)$ resonance is observed, the first observation of $C\\!P$
violation in any process containing a tensor resonance, which is robust with
respect to systematic uncertainties related to both experimental effects and
to the composition of the amplitude model.

The quasi-two-body $C\\!P$ asymmetry in the
${{{B}^{+}}}\\!\rightarrow\rho(770)^{0}{{\pi}^{+}}$ decay is found to be
compatible with zero. However, $C\\!P$-violation effects that are
characteristic of interference between the spin-$1$ $\rho(770)^{0}$ resonance
and the spin-$0$ S-wave contribution are observed, and are well described in
all three approaches to the S-wave. This is the first observation of $C\\!P$
violation mediated entirely by interference between hadronic resonances.

All three approaches to the description of the ${{\pi}^{+}}{{\pi}^{-}}$ S-wave
broadly agree on the variation of its magnitude and phase. One striking
feature is the presence of a significant $C\\!P$ asymmetry in the S-wave that
is not associated with the aforementioned interference effect, of which a
substantial component is at low $m({{\pi}^{+}}{{\pi}^{-}})$. Further
phenomenological and experimental investigations will be required to better
understand the underlying dynamics of these and other effects in
${{{B}^{+}}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{+}}{{\pi}^{-}}$ decays, and to
elucidate connections with $C\\!P$-violation effects observed in
${{{B}^{+}}}\\!\rightarrow{{K}^{+}}{{K}^{+}}{{\pi}^{-}}$ decays [14].

The results of this analysis provide valuable input to phenomenological work
on the underlying nature of the remarkably large $C\\!P$ violation observed in
the charmless three-body decays of the charged $B$ meson, and in $B$-meson
decays in general. Furthermore, the robust description of low mass S-wave
achieved with the approaches documented here gives insight into the effects of
low-energy QCD in $B$-meson decays.

## Acknowledgements

We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at the LHCb institutes. We acknowledge support from CERN
and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); MOST
and NSFC (China); CNRS/IN2P3 (France); BMBF, DFG and MPG (Germany); INFN
(Italy); NWO (Netherlands); MNiSW and NCN (Poland); MEN/IFA (Romania); MSHE
(Russia); MinECo (Spain); SNSF and SER (Switzerland); NASU (Ukraine); STFC
(United Kingdom); DOE NP and NSF (USA). We acknowledge the computing resources
that are provided by CERN, IN2P3 (France), KIT and DESY (Germany), INFN
(Italy), SURF (Netherlands), PIC (Spain), GridPP (United Kingdom), RRCKI and
Yandex LLC (Russia), CSCS (Switzerland), IFIN-HH (Romania), CBPF (Brazil), PL-
GRID (Poland) and OSC (USA). We are indebted to the communities behind the
multiple open-source software packages on which we depend. Individual groups
or members have received support from AvH Foundation (Germany); EPLANET, Marie
Skłodowska-Curie Actions and ERC (European Union); ANR, Labex P2IO and OCEVU,
and Région Auvergne-Rhône-Alpes (France); Key Research Program of Frontier
Sciences of CAS, CAS PIFI, and the Thousand Talents Program (China); RFBR, RSF
and Yandex LLC (Russia); GVA, XuntaGal and GENCAT (Spain); the Royal Society
and the Leverhulme Trust (United Kingdom).

Appendices

##  Appendix A Isobar model tables

The results for the Cartesian coefficients obtained from the fit with the
isobar description of the S-wave are reported in Table 21, with the fitted
pole parameters given in Table 22. The correlation matrices for the
$C\\!P$-averaged fit fractions and quasi-two-body decay $C\\!P$ asymmetries,
corresponding to those presented in Tables 11 and 18, can be found in Tables
from 21 to 26. As an indication of fit quality, signed $\chi^{2}$
distributions in the square Dalitz plot, separated by charge, are produced
with an adaptive binning procedure requiring at least 15 events per bin. These
are shown in Fig. 17.

Table 21:  Cartesian coefficients, $c_{j}$, for the components of the isobar
model fit.

Component | $x$ | $y$ | $\delta x$ | $\delta y$
---|---|---|---|---
$\rho(770)^{0}$ | $1.000~{}({\rm fixed})$ | $0.000~{}({\rm fixed})$ |  $-0.003$$\,\pm\,$ |  $0.000$$\,\pm\,$ | $0.010$ | $0.000~{}({\rm fixed})$
$\omega(782)$ |  $0.091$$\,\pm\,$ |  $0.004$$\,\pm\,$ | $0.000$ |  $-0.007$$\,\pm\,$ |  $0.007$$\,\pm\,$ | $0.010$ |  $0.000$$\,\pm\,$ |  $0.003$$\,\pm\,$ | $0.000$ |  $-0.022$$\,\pm\,$ |  $0.006$$\,\pm\,$ | $0.010$
$f_{2}(1270)$ |  $0.291$$\,\pm\,$ |  $0.011$$\,\pm\,$ | $0.058$ |  $0.204$$\,\pm\,$ |  $0.009$$\,\pm\,$ | $0.067$ |  $-0.002$$\,\pm\,$ |  $0.009$$\,\pm\,$ | $0.044$ |  $-0.179$$\,\pm\,$ |  $0.009$$\,\pm\,$ | $0.044$
$\rho(1450)^{0}$ |  $-0.223$$\,\pm\,$ |  $0.015$$\,\pm\,$ | $0.010$ |  $0.191$$\,\pm\,$ |  $0.010$$\,\pm\,$ | $0.001$ |  $0.031$$\,\pm\,$ |  $0.014$$\,\pm\,$ | $0.020$ |  $0.068$$\,\pm\,$ |  $0.009$$\,\pm\,$ | $0.020$
$\rho_{3}(1690)^{0}$ |  $0.073$$\,\pm\,$ |  $0.009$$\,\pm\,$ | $0.022$ |  $-0.045$$\,\pm\,$ |  $0.010$$\,\pm\,$ | $0.022$ |  $0.044$$\,\pm\,$ |  $0.009$$\,\pm\,$ | $0.014$ |  $-0.013$$\,\pm\,$ |  $0.010$$\,\pm\,$ | $0.014$
Rescattering |  $0.142$$\,\pm\,$ |  $0.009$$\,\pm\,$ | $0.025$ |  $-0.040$$\,\pm\,$ |  $0.009$$\,\pm\,$ | $0.042$ |  $-0.047$$\,\pm\,$ |  $0.008$$\,\pm\,$ | $0.031$ |  $-0.027$$\,\pm\,$ |  $0.008$$\,\pm\,$ | $0.031$
$\sigma$ |  $-0.485$$\,\pm\,$ |  $0.013$$\,\pm\,$ | $0.130$ |  $0.284$$\,\pm\,$ |  $0.017$$\,\pm\,$ | $0.091$ |  $0.231$$\,\pm\,$ |  $0.011$$\,\pm\,$ | $0.022$ |  $0.270$$\,\pm\,$ |  $0.015$$\,\pm\,$ | $0.022$

Table 22:  Fitted values of the pole parameters in the isobar model fit.  Parameter | Value
---|---
$m_{\sigma}$ | $0.563\pm 0.010$
$\Gamma_{\sigma}$ | $0.350\pm 0.013$
Table 23: Statistical correlation matrix for the isobar model $C\\!P$-averaged
fit fractions.

| $\rho(770)^{0}$ | $\omega(782)$ | $f_{2}(1270)$ | $\rho(1450)^{0}$ | $\rho_{3}(1690)^{0}$ | Rescattering | $\sigma$
---|---|---|---|---|---|---|---
$\rho(770)^{0}$ | $+1.00$ | $-0.12$ | $-0.31$ | $+0.07$ | $-0.02$ |  $-0.11$ | $-0.62$
$\omega(782)$ |  | $+1.00$ | $+0.04$ | $+0.05$ | $\phantom{+}0.00$ |  $-0.09$ | $-0.02$
$f_{2}(1270)$ |  |  | $+1.00$ | $-0.11$ | $+0.04$ |  $-0.12$ | $+0.11$
$\rho(1450)^{0}$ |  |  |  | $+1.00$ | $-0.07$ |  $-0.16$ | $-0.41$
$\rho_{3}(1690)^{0}$ |  |  |  |  | $+1.00$ |  $+0.10$ | $-0.05$
Rescattering |  |  |  |  |  |  $+1.00$ | $+0.15$
$\sigma$ |  |  |  |  |  |  | $+1.00$

Table 24: Systematic correlation matrix for the isobar model $C\\!P$-averaged
fit fractions.

| $\rho(770)^{0}$ | $\omega(782)$ | $f_{2}(1270)$ | $\rho(1450)^{0}$ | $\rho_{3}(1690)^{0}$ | Rescattering | $\sigma$
---|---|---|---|---|---|---|---
$\rho(770)^{0}$ | $+1.00$ | $+0.77$ | $+0.58$ | $-0.72$ | $+0.86$ |  $-0.53$ | $-0.75$
$\omega(782)$ |  | $+1.00$ | $+0.55$ | $-0.46$ | $+0.73$ |  $-0.25$ | $-0.49$
$f_{2}(1270)$ |  |  | $+1.00$ | $-0.34$ | $+0.71$ |  $-0.78$ | $-0.47$
$\rho(1450)^{0}$ |  |  |  | $+1.00$ | $-0.53$ |  $+0.57$ | $+0.68$
$\rho_{3}(1690)^{0}$ |  |  |  |  | $+1.00$ |  $-0.71$ | $-0.89$
Rescattering |  |  |  |  |  |  $+1.00$ | $+0.77$
$\sigma$ |  |  |  |  |  |  | $+1.00$

Table 25: Statistical correlation matrix for the isobar model quasi-two-body
decay $C\\!P$ asymmetries.

| $\rho(770)^{0}$ | $\omega(782)$ | $f_{2}(1270)$ | $\rho(1450)^{0}$ | $\rho_{3}(1690)^{0}$ | Rescattering | $\sigma$
---|---|---|---|---|---|---|---
$\rho(770)^{0}$ | $+1.00$ | $-0.07$ | $-0.25$ | $\phantom{+}0.00$ | $-0.02$ |  $-0.21$ | $-0.55$
$\omega(782)$ |  | $+1.00$ | $-0.02$ | $-0.07$ | $-0.02$ |  $+0.01$ | $-0.08$
$f_{2}(1270)$ |  |  | $+1.00$ | $-0.01$ | $+0.23$ |  $-0.03$ | $-0.02$
$\rho(1450)^{0}$ |  |  |  | $+1.00$ | $-0.03$ |  $-0.26$ | $-0.28$
$\rho_{3}(1690)^{0}$ |  |  |  |  | $+1.00$ |  $\pm 0.00$ | $-0.17$
Rescattering |  |  |  |  |  |  $+1.00$ | $+0.15$
$\sigma$ |  |  |  |  |  |  | $+1.00$

Table 26: Systematic correlation matrix for the isobar model quasi-two-body
decay $C\\!P$ asymmetries.

| $\rho(770)^{0}$ | $\omega(782)$ | $f_{2}(1270)$ | $\rho(1450)^{0}$ | $\rho_{3}(1690)^{0}$ | Rescattering | $\sigma$
---|---|---|---|---|---|---|---
$\rho(770)^{0}$ | $+1.00$ | $+0.58$ | $+0.50$ | $-0.46$ | $+0.39$ |  $+0.06$ | $+0.27$
$\omega(782)$ |  | $+1.00$ | $+0.28$ | $-0.20$ | $+0.71$ |  $+0.08$ | $+0.53$
$f_{2}(1270)$ |  |  | $+1.00$ | $+0.38$ | $-0.28$ |  $+0.82$ | $+0.57$
$\rho(1450)^{0}$ |  |  |  | $+1.00$ | $-0.46$ |  $+0.77$ | $+0.40$
$\rho_{3}(1690)^{0}$ |  |  |  |  | $+1.00$ |  $-0.49$ | $+0.02$
Rescattering |  |  |  |  |  |  $+1.00$ | $+0.60$
$\sigma$ |  |  |  |  |  |  | $+1.00$

Figure 17:  Signed $\chi^{2}$ distributions indicating the agreement between
the isobar model fit and the data for (a) ${{B}^{+}}$ and (b) ${{B}^{-}}$
decays.

##  Appendix B K-matrix model tables

The results for the Cartesian coefficients, and other fitted parameters,
obtained with the K-matrix S-wave approach can be found in Table 27.
Correlation matrices for these parameters are reported in Ref. [104]. Here, as
the reference amplitude is the $\rho$–$\omega$ mixing component, rather than a
component only representing the $\rho(770)^{0}$ resonance, the magnitude of
the positive isobar coefficient describing the $\rho(770)^{0}$ resonance is