EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-PH-EP-2015-238 LHCb-PAPER-2015-026 September 22, 2015

Studies of the resonance structure in
${{\mathrm{D}}^{0}}\\!\rightarrow{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{{\mathrm{K}}^{\pm}}{{\uppi}^{\mp}}$ decays

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

Amplitude models are applied to studies of resonance structure in
${{\mathrm{D}}^{0}}\\!\rightarrow{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{{\mathrm{K}}^{-}}{{\uppi}^{+}}$ and
${{\mathrm{D}}^{0}}\\!\rightarrow{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{{\mathrm{K}}^{+}}{{\uppi}^{-}}$ decays using ${\mathrm{p}}{\mathrm{p}}$
collision data corresponding to an integrated luminosity of
$3.0\mbox{\,fb}^{-1}$ collected by the LHCb experiment. Relative magnitude and
phase information is determined, and coherence factors and related observables
are computed for both the whole phase space and a restricted region of
$100{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ around the
${\mathrm{K}}^{*}({892})^{{\pm}}$ resonance. Two formulations for the
${\mathrm{K}}{\uppi}$ $S$-wave are used, both of which give a good description
of the data. The ratio of branching fractions ${\cal
B}({{\mathrm{D}}^{0}}\\!\rightarrow{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{{\mathrm{K}}^{+}}{{\uppi}^{-}})/{\cal
B}({{\mathrm{D}}^{0}}\\!\rightarrow{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{{\mathrm{K}}^{-}}{{\uppi}^{+}})$ is measured to be $0.655\pm
0.004\mathrm{\,(stat)}\pm 0.006\mathrm{\,(syst)}$ over the full phase space
and $0.370\pm 0.003\mathrm{\,(stat)}\pm 0.012\mathrm{\,(syst)}$ in the
restricted region. A search for $C\\!P$ violation is performed using the
amplitude models and no significant effect is found. Predictions from
$\mathrm{SU}(3)$ flavor symmetry for ${\mathrm{K}}^{*}({892}){\mathrm{K}}$
amplitudes of different charges are compared with the amplitude model results.

Published in Phys. Rev. D 93, 052018 (2016)

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

##  1 Introduction

A large variety of physics can be accessed by studying the decays111The
inclusion of charge-conjugate processes is implied, except in the definition
of $C\\!P$ asymmetries.
${{\mathrm{D}}^{0}}\\!\rightarrow{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{{\mathrm{K}}^{-}}{{\uppi}^{+}}$ and
${{\mathrm{D}}^{0}}\\!\rightarrow{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{{\mathrm{K}}^{+}}{{\uppi}^{-}}$. Analysis of the relative amplitudes of
intermediate resonances contributing to these decays can help in understanding
the behavior of the strong interaction at low energies. These modes are also
of interest for improving knowledge of the Cabibbo-Kobayashi-Maskawa (CKM) [1,
2] matrix, and $C\\!P$-violation measurements and mixing studies in the
${\mathrm{D}}^{0}$–${\overline{\mathrm{D}}{}}{}^{0}$ system. Both modes are
singly Cabibbo-suppressed (SCS), with the
${{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}{{\mathrm{K}}^{-}}{{\uppi}^{+}}$
final state favored by approximately $\times 1.7$ with respect to its
${{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}{{\mathrm{K}}^{+}}{{\uppi}^{-}}$
counterpart [3]. The main classes of Feynman diagrams, and the sub-decays to
which they contribute, are shown in Fig. 1.

Flavor symmetries are an important phenomenological tool in the study of
hadronic decays, and the presence of both charged and neutral
${\mathrm{K}}^{*}$ resonances in each
${{\mathrm{D}}^{0}}\\!\rightarrow{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{{\mathrm{K}}^{\pm}}{{\uppi}^{\mp}}$ mode allows several tests of
$\mathrm{SU}(3)$ flavor symmetry to be carried out [4, 5]. The
${{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{{\mathrm{K}}^{\pm}}{{\uppi}^{\mp}}$ final states also provide
opportunities to study the incompletely understood ${\mathrm{K}}{\uppi}$
$S$-wave systems [6], and to probe several resonances in the
${{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}{{\mathrm{K}}^{\pm}}$ decay
channels that are poorly established.

An important goal of flavor physics is to make a precise determination of the
CKM unitarity-triangle angle
$\gamma\equiv\arg({-{V_{{\mathrm{u}}{\mathrm{d}}}}{V_{{\mathrm{u}}{\mathrm{b}}}^{\ast}}/{V_{{\mathrm{c}}{\mathrm{d}}}}{V_{{\mathrm{c}}{\mathrm{b}}}^{\ast}}})$.
Information on this parameter222Another notation, $\phi_{3}\equiv\gamma$,
exists in the literature. can be obtained by studying $C\\!P$-violating
observables in the decays
${{{\mathrm{B}}^{-}}}\\!\rightarrow\shortstack{{(\rule[2.15277pt]{12.50002pt}{0.51212pt})}
\\\ [-.7ex] $\mathrm{D}$}{}^{0}{{\mathrm{K}}^{-}}$, where the
${\mathrm{D}}^{0}$ and ${\overline{\mathrm{D}}{}}{}^{0}$ are reconstructed in
a set of common final states [7, 8], such as the modes
${{\mathrm{D}}^{0}}\\!\rightarrow{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{{\mathrm{K}}^{-}}{{\uppi}^{+}}$ and
${{\mathrm{D}}^{0}}\\!\rightarrow{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{{\mathrm{K}}^{+}}{{\uppi}^{-}}$ [9]. Optimum statistical power is achieved
by studying the dependence of the $C\\!P$ asymmetry on where in three-body
phase space the $\mathrm{D}$-meson decay occurs, provided that the decay
amplitude from the intermediate resonances is sufficiently well described.
Alternatively, an inclusive analysis may be pursued, as in Ref. [10], with a
‘coherence factor’ [11] parameterizing the net effect of these resonances. The
coherence factor of these decays has been measured by the CLEO collaboration
using quantum-correlated ${\mathrm{D}}^{0}$ decays at the open-charm threshold
[12], but it may also be calculated from knowledge of the contributing
resonances. In both cases, therefore, it is valuable to be able to model the
variation of the magnitude and phase of the ${\mathrm{D}}^{0}$-decay
amplitudes across phase space.

The search for $C\\!P$ violation in the charm system is motivated by the fact
that several theories of physics beyond the Standard Model (SM) predict
enhancements above the very small effects expected in the SM [13, 14, 15].
Singly Cabibbo-suppressed decays provide a promising laboratory in which to
perform this search for direct $C\\!P$ violation because of the significant
role that loop diagrams play in these processes [16]. Multi-body SCS decays,
such as
${{\mathrm{D}}^{0}}\\!\rightarrow{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{{\mathrm{K}}^{-}}{{\uppi}^{+}}$ and
${{\mathrm{D}}^{0}}\\!\rightarrow{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{{\mathrm{K}}^{+}}{{\uppi}^{-}}$, have in addition the attractive feature
that the interfering resonances may lead to $C\\!P$ violation in local regions
of phase space, again motivating a good understanding of the resonant
substructure. The same modes may also be exploited to perform a
${\mathrm{D}}^{0}$–${\overline{\mathrm{D}}{}}{}^{0}$ mixing measurement, or to
probe indirect $C\\!P$ violation, either through a time-dependent measurement
of the evolution of the phase space of the decays, or the inclusive
${{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}{{\mathrm{K}}^{-}}{{\uppi}^{+}}$
and ${{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{{\mathrm{K}}^{+}}{{\uppi}^{-}}$ final states [17].

In this paper time-integrated amplitude models of these decays are constructed
and used to test $\mathrm{SU}(3)$ flavor symmetry predictions, search for
local $C\\!P$ violation, and compute coherence factors and associated
parameters. In addition, a precise measurement is performed of the ratio of
branching fractions of the two decays. The data sample is obtained from
${\mathrm{p}}{\mathrm{p}}$ collisions corresponding to an integrated
luminosity of $3.0\mbox{\,fb}^{-1}$ collected by the LHCb detector [18, 19]
during 2011 and 2012 at center-of-mass energies
$\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$ and
$8\mathrm{\,Te\kern-1.00006ptV}$, respectively. The sample contains around one
hundred times more signal decays than were analyzed in a previous amplitude
study of the same modes performed by the CLEO collaboration [12].

The paper is organized as follows. In Sect. 2, the detector, data and
simulation samples are described, and in Sect. 3 the signal selection and
backgrounds are discussed. The analysis formalism, including the definition of
the coherence factor, is presented in Sect. 4. The method for choosing the
composition of the amplitude models, fit results and their systematic
uncertainties are described in Sect. 5. The ratio of branching fractions,
coherence factors, $\mathrm{SU}(3)$ flavor symmetry tests and $C\\!P$
violation search results are presented in Sect. 6. Finally, conclusions are
drawn in Sect. 7.

flavourless

(150,50) Hbot1,Rbotdummy,Hbot2,Rdummy,Htop1,Rtopdummy,Htop2 Qbot,Dz,Qtop
label=(a) ${\mathrm{D}}^{0}$,label.dist=-0.1wDz
fermion,label=$\mathrm{c}$,label.side=left,tension=2.0Qtop,Wtop
fermion,tension=0.5,label=${\mathrm{d}},,{\mathrm{s}}$,label.side=rightWtop,Hbot2
fermion,label=$\overline{\mathrm{u}}$,label.side=leftHbot1,Qbot
fermion,label=$\mathrm{u}$,label.side=rightWbot,Htop1
fermion,label=$\overline{\mathrm{s}}$,,$\overline{\mathrm{d}}$,label.side=rightHtop2,Wbot
photon,label=$\mathrm{W}^{+}$,tension=2,label.side=right,right=0.15Wbot,Wtop
phantomQbot,Wtop phantom,tension=4Hbot1,Rbotdummy,Hbot2
phantom,tension=4Htop1,Rtopdummy,Htop2 phantom,tension=-2Hbot2,Rdummy
phantom,tension=-2Htop1,Rdummy 0.12wRbotdummy,Rtopdummy (150,50) Qbot,Dz,Qtop
Hbot1,Rbotdummy,Hbot2,Rdummy,Htop1,Rtopdummy,Htop2 label=(b)
${\mathrm{D}}^{0}$,label.dist=-0.1wDz
fermion,label=$\mathrm{c}$,label.side=left,tension=1Qtop,Wtop
fermion,label=$\overline{\mathrm{u}}$,label.side=left,tension=1Wbot,Qbot
plain,tension=2Wtop,Gtop plain,tension=2dummy1,Wbot
fermion,label=$\mathrm{d}$,label.side=left,tension=0.5Gtop,Htop2
fermion,label=$\overline{\mathrm{d}}$,label.side=left,tension=0.5Hbot1,dummy1
fermion,tension=0.5,label=$\overline{\mathrm{u}}$,,$\overline{\mathrm{s}}$,label.side=right,label.dist=0.1hHtop1,Gend
fermion,tension=0.5,label=$\mathrm{u}$,,$\mathrm{s}$,label.side=right,label.dist=0.1hGend,Hbot2
gluon,tension=0,right=0.5Gtop,Gend
photon,label=$\mathrm{W}$,label.side=right,tension=0Wtop,Wbot
phantom,tension=0.5Qbot,Gend,Qtop phantom,tension=4Hbot1,Rbotdummy,Hbot2
phantom,tension=4Htop1,Rtopdummy,Htop2 phantom,tension=-2Hbot2,Rdummy
phantom,tension=-2Htop1,Rdummy 0.12wRbotdummy,Rtopdummy

(150,50) Qbot,Dz,Qtop Hbot1,Rbotdummy,Hbot2,Rdummy,Htop1,Rtopdummy,Htop2
label=(c) ${\mathrm{D}}^{0}$,label.dist=-0.1wDz
fermion,label=$\mathrm{c}$,label.side=left,tension=2Qtop,Wleft
fermion,label=$\overline{\mathrm{u}}$,tension=2,label.side=leftPbot,Qbot
photon,label=$\mathrm{W}^{+}$,left=0.5Wleft,Wright
fermion,label=${\mathrm{d}},,{\mathrm{s}}$,label.side=rightWleft,Ptop
fermionPtop,Wright gluon,tension=0.1,right=0.2Pbot,Ptop
fermion,label=$\mathrm{u}$,label.side=left,tension=1Wright,Htop2
fermion,label=$\overline{\mathrm{u}}$,label.side=leftHbot1,Gtop
fermion,tension=3Gtop,Pbot
fermion,label=$\overline{\mathrm{d}}$,,$\overline{\mathrm{s}}$,label.side=right,label.dist=0.08hHtop1,Gint
fermion,tension=3,label=$\mathrm{d}$,,$\mathrm{s}$,label.side=right,label.dist=0.08hGint,Hbot2
gluon,left=0.3,tension=0.5Gtop,Gint phantom,tension=4Hbot1,Rbotdummy,Hbot2
phantom,tension=4Htop1,Rtopdummy,Htop2 phantom,tension=-2Hbot2,Rdummy
phantom,tension=-2Htop1,Rdummy phantomQtop,Gint,Qbot 0.12wRbotdummy,Rtopdummy
(150,50) Qbot,Dz,Qtop Hbot1,Rbotdummy,Hbot2,Rdummy,Htop1,Rtopdummy,Htop2
label=(d) ${\mathrm{D}}^{0}$,label.dist=-0.1wDz
fermion,label=$\mathrm{c}$,tension=2,label.side=leftQtop,Gleft
fermion,label=$\mathrm{c}$,tension=1,label.side=leftGleft,Wtop
fermion,label.side=left,label=${\mathrm{d}},,{\mathrm{s}}$Wtop,Pright
fermion,label.side=left,label=${\overline{\mathrm{d}}},,{\overline{\mathrm{s}}}$,tension=2Pright,Wbot
fermion,label=$\overline{\mathrm{u}}$,tension=2Wbot,Qbot
photon,tension=-0.2,label=$\mathrm{W}^{-}$,label.side=rightWtop,Wbot
gluon,tension=3Pright,Gint gluon,left=0.5,tension=3Gleft,Gright
fermionGright,Htop2 fermionHtop1,Gint fermionHbot2,Gright fermionGint,Hbot1
phantom,tension=-1Gint,Gright phantom,tension=4Hbot1,Rbotdummy,Hbot2
phantom,tension=4Htop1,Rtopdummy,Htop2 phantom,tension=-2Hbot2,Rdummy
phantom,tension=-2Htop1,Rdummy 0.12wRbotdummy,Rtopdummy

Figure 1: SCS classes of diagrams contributing to the decays
${{\mathrm{D}}^{0}}\\!\rightarrow{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{{\mathrm{K}}^{\pm}}{{\uppi}^{\mp}}$. The color-favored (tree) diagrams (a)
contribute to the
${\mathrm{K}}^{*{\pm}}_{0,1,2}\\!\rightarrow{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{{\uppi}^{\pm}}$ and
$(\mathrm{a}_{0,2},\uprho)^{\pm}\\!\rightarrow{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{{\mathrm{K}}^{\pm}}$ channels, while the color-suppressed exchange
diagrams (b) contribute to the
$(\mathrm{a}_{0,2},\uprho)^{\pm}\\!\rightarrow{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{{\mathrm{K}}^{\pm}}$,
${\mathrm{K}}^{*{0}}_{0,1,2}\\!\rightarrow{{\mathrm{K}}^{+}}{{\uppi}^{-}}$ and
${\overline{\mathrm{K}}{}}^{*{0}}_{0,1,2}\\!\rightarrow{{\mathrm{K}}^{-}}{{\uppi}^{+}}$
channels. Second-order loop (penguin) diagrams (c) contribute to the
$(\mathrm{a}_{0,2},\uprho)^{\pm}\\!\rightarrow{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{{\mathrm{K}}^{\pm}}$ and
${\mathrm{K}}^{*{\pm}}_{0,1,2}\\!\rightarrow{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{{\uppi}^{\pm}}$ channels, and, finally, OZI-suppressed penguin
annihilation diagrams (d) contribute to all decay channels.

##  2 Detector and simulation

The LHCb detector is a single-arm forward spectrometer covering the
pseudorapidity range $2<\eta<5$, designed for the study of particles
containing $\mathrm{b}$ or $\mathrm{c}$ quarks. The detector includes a high-
precision tracking system consisting of a silicon-strip vertex detector
surrounding the ${\mathrm{p}}{\mathrm{p}}$ interaction region, a large-area
silicon-strip detector located upstream of a dipole magnet with a bending
power of about $4{\rm\,Tm}$, and three stations of silicon-strip detectors and
straw drift tubes placed downstream of the magnet. The tracking system
provides a measurement of momentum, $p$, of charged particles with a relative
uncertainty that varies from 0.5% at low momentum to 1.0% at
$200{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The minimum distance of a track to
a primary ${\mathrm{p}}{\mathrm{p}}$ interaction vertex (PV), the impact
parameter, is measured with a resolution of $(15+29/\mbox{$p_{\rm
T}$}){\,\upmu\rm m}$, where $p_{\rm T}$ is the component of the momentum
transverse to the beam, in ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. Different
types of charged hadrons are distinguished using information from two ring-
imaging Cherenkov (RICH) detectors. Photons, electrons and hadrons are
identified by a calorimeter system consisting of scintillating-pad and
preshower detectors, an electromagnetic calorimeter and a hadronic
calorimeter.

The trigger [20] consists of a hardware stage, based on information from the
calorimeter and muon systems, followed by a software stage, in which all
charged particles with $\mbox{$p_{\rm
T}$}>500\,(300){\mathrm{\,Me\kern-1.00006ptV\\!/}c}$ are reconstructed for
2011 (2012) data. At the hardware trigger stage, events are required to have a
muon with high $p_{\rm T}$ or a hadron, photon or electron with high
transverse energy in the calorimeters. For hadrons, the transverse energy
threshold is $3.5\mathrm{\,Ge\kern-1.00006ptV}$. Two software trigger
selections are combined for this analysis. The first reconstructs the decay
chain
${\mathrm{D}}^{*}({{2010}})^{{+}}\\!\rightarrow{{\mathrm{D}}^{0}}\uppi^{+}_{\text{slow}}$
with ${{\mathrm{D}}^{0}}\\!\rightarrow\mathrm{h}^{+}\mathrm{h}^{-}\mathrm{X}$,
where $\mathrm{h}^{\pm}$ represents a pion or a kaon and $\mathrm{X}$ refers
to any number of additional particles. The charged pion originating in the
${\mathrm{D}}^{*}({{2010}})^{{+}}$ decay is referred to as ‘slow’ due to the
small $Q$-value of the decay. The second selection fully reconstructs the
decay
${{\mathrm{D}}^{0}}\\!\rightarrow{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{{\mathrm{K}}^{\pm}}{{\uppi}^{\mp}}$, without flavor tagging. In both cases
at least one charged particle in the decay chain is required to have a
significant impact parameter with respect to any PV.

In the offline selection, trigger signals are associated with reconstructed
particles. Selection requirements can therefore be made on the trigger
selection itself and on whether the decision was due to the signal candidate,
other particles produced in the ${\mathrm{p}}{\mathrm{p}}$ collision, or both.
It is required that the hardware hadronic trigger decision is due to the
signal candidate, or that the hardware trigger decision is due solely to other
particles produced in the ${\mathrm{p}}{\mathrm{p}}$ collision.

Decays ${{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}\\!\rightarrow{{\uppi}^{+}}{{\uppi}^{-}}$ are reconstructed in two
different categories: the first involves
${\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}$ mesons that decay early enough
for the pions to be reconstructed in the vertex detector; the second contains
${\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}$ mesons that decay later such that
track segments of the pions cannot be formed in the vertex detector. These
categories are referred to as _long_ and _downstream_ , respectively. The long
category has better mass, momentum and vertex resolution than the downstream
category, and in 2011 was the only category available in the software trigger.

In the simulation, ${\mathrm{p}}{\mathrm{p}}$ collisions are generated using
Pythia [21, *Sjostrand:2007gs] with a specific LHCb configuration [23]. Decays
of hadronic particles are described by EvtGen [24], in which final-state
radiation is generated using Photos [25]. The interaction of the generated
particles with the detector, and its response, are implemented using the
Geant4 toolkit [26, *Agostinelli:2002hh] as described in Ref. [28].

##  3 Signal selection and backgrounds

The offline selection used in this analysis reconstructs the decay chain
${\mathrm{D}}^{*}({{2010}})^{{+}}\\!\rightarrow{{\mathrm{D}}^{0}}\uppi^{+}_{\text{slow}}$
with ${{\mathrm{D}}^{0}}\\!\rightarrow{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{{\mathrm{K}}^{\pm}}{{\uppi}^{\mp}}$, where the charged pion
$\uppi^{+}_{\text{slow}}$ from the ${\mathrm{D}}^{*}({{2010}})^{{+}}$ decay
tags the flavor of the neutral $\mathrm{D}$ meson. Candidates are required to
pass one of the two software trigger selections described in Sect. 2, as well
as several offline requirements. These use information from the RICH detectors
to ensure that the charged kaon is well-identified, which reduces the
background contribution from the decays
${{\mathrm{D}}^{0}}\\!\rightarrow{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{{\uppi}^{+}}{{\uppi}^{-}}{{\uppi}^{0}}$ and
${{\mathrm{D}}^{0}}\\!\rightarrow{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{{\uppi}^{-}}{\upmu^{+}}{{\upnu}_{\mu}}$. In addition the
${\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}$ decay vertex is required to be
well-separated from the ${\mathrm{D}}^{0}$ decay vertex in order to suppress
the
${{\mathrm{D}}^{0}}\\!\rightarrow{{\mathrm{K}}^{-}}{{\uppi}^{+}}{{\uppi}^{+}}{{\uppi}^{-}}$
background, where a ${{\uppi}^{+}}{{\uppi}^{-}}$ combination is close to the
${\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}$ mass. ${\mathrm{D}}^{0}$
candidates are required to have decay vertices well-separated from any PV, and
to be consistent with originating from a PV. This selection suppresses the
semileptonic and
${{\mathrm{D}}^{0}}\\!\rightarrow{{\mathrm{K}}^{-}}{{\uppi}^{+}}{{\uppi}^{+}}{{\uppi}^{-}}$
backgrounds to negligible levels, while a small contribution from
${{\mathrm{D}}^{0}}\\!\rightarrow{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{{\uppi}^{+}}{{\uppi}^{-}}{{\uppi}^{0}}$ remains in the ${\Delta m}\equiv
m({{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{\mathrm{K}}{\uppi}{\uppi}_{\text{slow}})-m({{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{\mathrm{K}}{\uppi})$ distribution. A kinematic fit [29] is applied to the
reconstructed ${\mathrm{D}}^{*}({{2010}})^{{+}}$ decay chain to enhance the
resolution in $m({{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{\mathrm{K}}{\uppi})$, $\Delta m$ and the two-body invariant masses
$m({{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}{\mathrm{K}})$,
$m({{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}{\uppi})$ and
$m({\mathrm{K}}{\uppi})$ that are used to probe the resonant structure of
these decays. This fit constrains the ${\mathrm{D}}^{*}({{2010}})^{{+}}$ decay
vertex to coincide with the closest PV with respect to the
${\mathrm{D}}^{*}({{2010}})^{{+}}$ candidate, fixes the
${\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}$ candidate mass to its nominal
value, and is required to be of good quality.

Signal yields and estimates of the various background contributions in the
signal window are determined using maximum likelihood fits to the
$m({{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}{\mathrm{K}}{\uppi})$ and
$\Delta m$ distributions. The signal window is defined as the region less than
$18{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$
($0.8{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$) from the peak value of
$m({{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}{\mathrm{K}}{\uppi})$ ($\Delta
m$), corresponding to approximately three standard deviations of each signal
distribution. The three categories of interest are: signal decays, mistagged
background where a correctly reconstructed ${\mathrm{D}}^{0}$ meson is
combined with a charged pion that incorrectly tags the ${\mathrm{D}}^{0}$
flavor, and a combinatorial background category, which also includes a small
peaking contribution in $\Delta m$ from the decay
${{\mathrm{D}}^{0}}\\!\rightarrow{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{{\uppi}^{+}}{{\uppi}^{-}}{{\uppi}^{0}}$. These fits use candidates in the
ranges $139<{\Delta m}<153{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and
$1.805<m({{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{\mathrm{K}}{\uppi})<1.925{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$. The
sidebands of the $m({{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{\mathrm{K}}{\uppi})$ distribution are defined as those parts of the fit
range where $m({{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{\mathrm{K}}{\uppi})$ is more than
$30{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ from the peak value. The $\Delta
m$ ($m({{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}{\mathrm{K}}{\uppi})$)
distribution in the signal region of
$m({{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}{\mathrm{K}}{\uppi})$ ($\Delta
m$) is fitted to determine the ${\mathrm{D}}^{*}({{2010}})^{{+}}$
(${\mathrm{D}}^{0}$) yield in the two-dimensional signal region [30]. The
${\mathrm{D}}^{*}({{2010}})^{{+}}$ (${\mathrm{D}}^{0}$) signal shape in the
$\Delta m$ ($m({{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{\mathrm{K}}{\uppi})$) distribution is modeled using a Johnson $S_{U}$ [31]
(Cruijff [32]) function. In the $m({{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{\mathrm{K}}{\uppi})$ distribution the combinatorial background is modeled
with an exponential function, while in $\Delta m$ a power law function is
used, $f({\Delta m};m_{\pi},p,P,b)=(\frac{{\Delta
m}-m_{\pi}}{m_{\pi}})^{p}-b^{p-P}(\frac{{\Delta m}-m_{\pi}}{m_{\pi}})^{P}$,
with the parameters $p$, $P$ and $b$ determined by a fit in the
$m({{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}{\mathrm{K}}{\uppi})$
sidebands. The small
${{\mathrm{D}}^{0}}\\!\rightarrow{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{{\uppi}^{+}}{{\uppi}^{-}}{{\uppi}^{0}}$ contribution in the $\Delta m$
distribution is described by a Gaussian function, and the component
corresponding to ${\mathrm{D}}^{0}$ mesons associated with a random slow pion
is the sum of an exponential function and a linear term. These fits are shown
in Fig. 2. The results of the fits are used to determine the yields of
interest in the two-dimensional signal region. These yields are given in Table
1 for both decay modes, together with the fractions of backgrounds.

A second kinematic fit that also constrains the ${\mathrm{D}}^{0}$ mass to its
known value is performed and used for all subsequent parts of this analysis.
This fit further improves the resolution in the two-body invariant mass
coordinates and forces all candidates to lie within the kinematically allowed
region of the Dalitz plot. The Dalitz plots [33] for data in the two-
dimensional signal region are shown in Fig. 3. Both decays are dominated by a
${\mathrm{K}}^{*}({892})^{{\pm}}$ structure. The
${\mathrm{K}}^{*}({892})^{{0}}$ is also visible as a destructively interfering
contribution in the
${{\mathrm{D}}^{0}}\\!\rightarrow{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{{\mathrm{K}}^{-}}{{\uppi}^{+}}$ mode and the
low-$m^{2}_{{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}{\uppi}}$ region of
the ${{\mathrm{D}}^{0}}\\!\rightarrow{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{{\mathrm{K}}^{+}}{{\uppi}^{-}}$ mode, while a clear excess is seen in the
high-$m^{2}_{{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}{\uppi}}$ region.
Finally, a veto is applied to candidates close to the kinematic boundaries;
this is detailed in Sect. 4.3.

Figure 2: Mass (left) and $\Delta m$ (right) distributions for the ${{\mathrm{D}}^{0}}\\!\rightarrow{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}{{\mathrm{K}}^{-}}{{\uppi}^{+}}$ (top) and ${{\mathrm{D}}^{0}}\\!\rightarrow{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}{{\mathrm{K}}^{+}}{{\uppi}^{-}}$ (bottom) samples with fit results superimposed. The long-dashed (blue) curve represents the ${\mathrm{D}}^{*}({{2010}})^{{+}}$ signal, the dash-dotted (green) curve represents the contribution of real ${\mathrm{D}}^{0}$ mesons combined with incorrect $\uppi^{+}_{\text{slow}}$ and the dotted (red) curve represents the combined combinatorial and ${{\mathrm{D}}^{0}}\\!\rightarrow{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}{{\uppi}^{+}}{{\uppi}^{-}}{{\uppi}^{0}}$ background contribution. The vertical solid lines show the signal region boundaries, and the vertical dotted lines show the sideband region boundaries. Table 1: Signal yields and estimated background rates in the two-dimensional signal region. The larger mistag rate in the ${{\mathrm{D}}^{0}}\\!\rightarrow{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}{{\mathrm{K}}^{+}}{{\uppi}^{-}}$ mode is due to the different branching fractions for the two modes. Only statistical uncertainties are quoted. |  | Mistag | Combinatorial
---|---|---|---
Mode | Signal yield | background [%] | background [%]
${{\mathrm{D}}^{0}}\\!\rightarrow{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}{{\mathrm{K}}^{-}}{{\uppi}^{+}}$ | 113 290$\,\pm\,$ | 130 | 0.89$\,\pm\,$ | 0.09 | 3.04$\,\pm\,$ | 0.14
${{\mathrm{D}}^{0}}\\!\rightarrow{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}{{\mathrm{K}}^{+}}{{\uppi}^{-}}$ | 76 380$\,\pm\,$ | 120 | 1.93$\,\pm\,$ | 0.16 | 2.18$\,\pm\,$ | 0.15

Figure 3: Dalitz plots of the
${{\mathrm{D}}^{0}}\\!\rightarrow{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{{\mathrm{K}}^{-}}{{\uppi}^{+}}$ (left) and
${{\mathrm{D}}^{0}}\\!\rightarrow{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{{\mathrm{K}}^{+}}{{\uppi}^{-}}$ (right) candidates in the two-dimensional
signal region.

##  4 Analysis formalism

The dynamics of a decay
${{\mathrm{D}}^{0}}\\!\rightarrow\mathrm{A}\mathrm{B}\mathrm{C}$, where
${\mathrm{D}}^{0}$, $\mathrm{A}$, $\mathrm{B}$ and $\mathrm{C}$ are all
pseudoscalar mesons, can be completely described by two variables, where the
conventional choice is to use a pair of squared invariant masses. This paper
will use $m^{2}_{{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}{\uppi}}\equiv
m^{2}({{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}{\uppi})$ and
$m^{2}_{{\mathrm{K}}{\uppi}}\equiv m^{2}({\mathrm{K}}{\uppi})$ as this choice
highlights the dominant resonant structure of the
${{\mathrm{D}}^{0}}\\!\rightarrow{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{{\mathrm{K}}^{\pm}}{{\uppi}^{\mp}}$ decay modes.

###  4.1 Isobar models for
${{\mathrm{D}}^{0}}\\!\rightarrow{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{{\mathrm{K}}^{\pm}}{{\uppi}^{\mp}}$

The signal isobar models decompose the decay chain into
${{\mathrm{D}}^{0}}\\!\rightarrow(\mathrm{R}\\!\rightarrow(\mathrm{A}\mathrm{B})_{J})\mathrm{C}$
contributions, where $\mathrm{R}$ is a resonance with spin $J$ equal to 0, 1
or 2. Resonances with spin greater than 2 should not contribute significantly
to the
${{\mathrm{D}}^{0}}\\!\rightarrow{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{{\mathrm{K}}^{\pm}}{{\uppi}^{\mp}}$ decays. The corresponding 4-momenta
are denoted $p_{{{\mathrm{D}}^{0}}}$, $p_{\mathrm{A}}$, $p_{\mathrm{B}}$ and
$p_{\mathrm{C}}$. The reconstructed invariant mass of the resonance is denoted
$m_{\mathrm{A}\mathrm{B}}$, and the nominal mass $m_{\mathrm{R}}$. The matrix
element for the
${{\mathrm{D}}^{0}}\\!\rightarrow{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{{\mathrm{K}}^{\pm}}{{\uppi}^{\mp}}$ decay is given by

$\mathcal{M}_{{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{{\mathrm{K}}^{\pm}}{{\uppi}^{\mp}}}(m^{2}_{{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{\uppi}},m^{2}_{{\mathrm{K}}{\uppi}})\\\
=\sum_{\mathrm{R}}a_{\mathrm{R}}e^{i\phi_{\mathrm{R}}}\mathcal{M}_{\mathrm{R}}(m^{2}_{\mathrm{A}\mathrm{B}},m^{2}_{\mathrm{A}\mathrm{C}}),$
(1)

where $a_{\mathrm{R}}e^{i\phi_{\mathrm{R}}}$ is the complex amplitude for
$\mathrm{R}$ and the contributions $\mathcal{M}_{\mathrm{R}}$ from each
intermediate state are given by

$\mathcal{M}_{\mathrm{R}}(m^{2}_{\mathrm{A}\mathrm{B}},m^{2}_{\mathrm{A}\mathrm{C}})=B_{J}^{{{\mathrm{D}}^{0}}}(p,|p_{0}|,d_{{{\mathrm{D}}^{0}}})\Omega_{J}(m^{2}_{\mathrm{A}\mathrm{B}},m^{2}_{\mathrm{A}\mathrm{C}})T_{\mathrm{R}}(m^{2}_{\mathrm{A}\mathrm{B}})B_{J}^{\mathrm{R}}(q,q_{0},d_{\mathrm{R}}),$
(2)

where $B_{J}^{{{\mathrm{D}}^{0}}}(p,|p_{0}|,d_{{{\mathrm{D}}^{0}}})$ and
$B_{J}^{\mathrm{R}}(q,q_{0},d_{\mathrm{R}})$ are the Blatt-Weisskopf
centrifugal barrier factors for the production and decay, respectively, of the
resonance $\mathrm{R}$ [34]. The parameter $p$ ($q$) is the momentum of
$\mathrm{C}$ ($\mathrm{A}$ or $\mathrm{B}$) in the $\mathrm{R}$ rest frame,
and $p_{0}$ ($q_{0}$) is the same quantity calculated using the nominal
resonance mass, $m_{\mathrm{R}}$. The meson radius parameters are set to
$d_{{{\mathrm{D}}^{0}}}=5.0{\,(\mathrm{Ge\kern-1.00006ptV\\!/}c)^{-1}}$ and
$d_{\mathrm{R}}=1.5{\,(\mathrm{Ge\kern-1.00006ptV\\!/}c)^{-1}}$ consistent
with the literature [12, 35]; the systematic uncertainty due to these choices
is discussed in Sect. 5.2. Finally,
$\Omega_{J}(m^{2}_{\mathrm{A}\mathrm{B}},m^{2}_{\mathrm{A}\mathrm{C}})$ is the
spin factor for a resonance with spin $J$ and $T_{\mathrm{R}}$ is the
dynamical function describing the resonance $\mathrm{R}$. The functional forms
for $B_{J}(q,q_{0},d)$ are given in Table 2 and those for
$\Omega_{J}(m^{2}_{\mathrm{A}\mathrm{B}},m^{2}_{\mathrm{A}\mathrm{C}})$ in
Table 3 for $J=0,1,2$. As the form for $\Omega_{1}$ is antisymmetric in the
indices $\mathrm{A}$ and $\mathrm{B}$, it is necessary to define the particle
ordering convention used in the analysis; this is done in Table 4.

Table 2: Blatt-Weisskopf centrifugal barrier penetration factors, $B_{J}(q,q_{0},d)$ [34]. $J$ | $B_{J}(q,q_{0},d)$
---|---
0 | $1$
1 | $\sqrt{\frac{1+(q_{0}d)^{2}}{1+(qd)^{2}}}$
2 | $\sqrt{\frac{9+3(q_{0}d)^{2}+(q_{0}d)^{4}}{9+3(qd)^{2}+(qd)^{4}}}$
Table 3: Angular distribution factors, $\Omega_{J}(p_{{{\mathrm{D}}^{0}}}+p_{\mathrm{C}},p_{\mathrm{B}}-p_{\mathrm{A}})$. These are expressed in terms of the tensors $T^{\mu\nu}=-g^{\mu\nu}+\frac{p^{\mu}_{\mathrm{A}\mathrm{B}}p^{\nu}_{\mathrm{A}\mathrm{B}}}{m_{\mathrm{R}}^{2}}$ and $T^{\mu\nu\alpha\beta}=\frac{1}{2}(T^{\mu\alpha}T^{\nu\beta}+T^{\mu\beta}T^{\nu\alpha})-\frac{1}{3}T^{\mu\nu}T^{\alpha\beta}$. $J$ | $\Omega_{J}(p_{{{\mathrm{D}}^{0}}}+p_{\mathrm{C}},p_{\mathrm{B}}-p_{\mathrm{A}})$
---|---
0 | $1$
1 | $(p_{{{\mathrm{D}}^{0}}}^{\mu}+p_{\mathrm{C}}^{\mu})T_{\mu\nu}(p_{\mathrm{B}}^{\nu}-p_{\mathrm{A}}^{\nu})/(\mathrm{Ge\kern-1.00006ptV\\!/}c)^{2}$
2 | $(p_{{{\mathrm{D}}^{0}}}^{\mu}+p_{\mathrm{C}}^{\mu})(p_{{{\mathrm{D}}^{0}}}^{\nu}+p_{\mathrm{C}}^{\nu})T_{\mu\nu\alpha\beta}(p_{\mathrm{A}}^{\alpha}-p_{\mathrm{B}}^{\alpha})(p_{\mathrm{A}}^{\beta}-p_{\mathrm{B}}^{\beta})/(\mathrm{Ge\kern-1.00006ptV\\!/}c)^{4}$
Table 4: Particle ordering conventions used in this analysis. Decay | $\mathrm{A}$ | $\mathrm{B}$ | $\mathrm{C}$
---|---|---|---
${{\mathrm{D}}^{0}}\\!\rightarrow{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}\shortstack{{(\rule[2.15277pt]{12.50002pt}{0.51212pt})} \\\ [-.7ex] $\mathrm{K}$}{}^{\kern-1.39998pt*0}$, | $\shortstack{{(\rule[2.15277pt]{12.50002pt}{0.51212pt})} \\\ [-.7ex] $\mathrm{K}$}{}^{\kern-1.39998pt*0}\\!\rightarrow{{\mathrm{K}}^{\pm}}{{\uppi}^{\mp}}$ | $\uppi$ | $\mathrm{K}$ | ${\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}$
${{\mathrm{D}}^{0}}\\!\rightarrow{{\mathrm{K}}^{\mp}}{{\mathrm{K}}^{*\pm}}$, | ${{\mathrm{K}}^{*\pm}}\\!\rightarrow{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}{{\uppi}^{\pm}}$ | ${\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}$ | $\uppi$ | $\mathrm{K}$
${{\mathrm{D}}^{0}}\\!\rightarrow{{\uppi}^{\mp}}({\uprho}^{\pm},\mathrm{a}^{\pm})$, | ${\uprho}^{\pm},\mathrm{a}^{\pm}\\!\rightarrow{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}{{\mathrm{K}}^{\pm}}$ | $\mathrm{K}$ | ${\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}$ | $\uppi$

The dynamical function $T_{\mathrm{R}}$ chosen depends on the resonance
$\mathrm{R}$ in question. A relativistic Breit-Wigner form is used unless
otherwise noted

$T_{\mathrm{R}}(m_{\mathrm{A}\mathrm{B}})=\frac{1}{(m_{\mathrm{R}}^{2}-m^{2}_{\mathrm{A}\mathrm{B}})-im_{\mathrm{R}}\Gamma_{\mathrm{R}}(m_{\mathrm{A}\mathrm{B}})},$
(3)

where the mass-dependent width is

$\Gamma_{\mathrm{R}}(m_{\mathrm{A}\mathrm{B}})=\Gamma_{\mathrm{R}}\left[B_{J}^{\mathrm{R}}(q,q_{0},d_{\mathrm{R}})\right]^{2}\frac{m_{\mathrm{R}}}{m_{\mathrm{A}\mathrm{B}}}\left(\frac{q}{q_{0}}\right)^{2J+1}.$
(4)

Several alternative forms are used for specialized cases. The Flatté [36] form
is a coupled-channel function used to describe the
$\mathrm{a}_{0}(980)^{{\pm}}$ resonance [37, 38, 39, 40, 12],

$T_{\mathrm{R}}=\frac{1}{(m_{\mathrm{a}_{0}(980)^{{\pm}}}^{2}-m_{{\mathrm{K}}{\overline{\mathrm{K}}{}}}^{2})-i[\rho_{{\mathrm{K}}{\overline{\mathrm{K}}{}}}g_{{\mathrm{K}}{\overline{\mathrm{K}}{}}}^{2}+\rho_{\upeta{\uppi}}g_{\upeta{\uppi}}^{2}]},$
(5)

where the phase space factor is given by

$\rho_{\mathrm{A}\mathrm{B}}=\frac{1}{m_{{\mathrm{K}}{\overline{\mathrm{K}}{}}}^{2}}\sqrt{(m_{{\mathrm{K}}{\overline{\mathrm{K}}{}}}^{2}-(m_{\mathrm{A}}+m_{\mathrm{B}})^{2})(m_{{\mathrm{K}}{\overline{\mathrm{K}}{}}}^{2}-(m_{\mathrm{A}}-m_{\mathrm{B}})^{2})},$
(6)

and the coupling constants $g_{{\mathrm{K}}{\overline{\mathrm{K}}{}}}$ and
$g_{\upeta{\uppi}}$ are taken from Ref. [37], fixed in the isobar model fits
and tabulated in Appendix A. The Gounaris-Sakurai [41] parameterization is
used to describe the ${\uprho}(1450)^{{\pm}}$ and ${\uprho}(1700)^{{\pm}}$
states [42, 43, 44, 39, 45],

$T_{\mathrm{R}}=\frac{1+d(m_{{\uprho}})\frac{\Gamma_{{\uprho}}}{m_{{\uprho}}}}{(m_{{\uprho}}^{2}-m_{{\mathrm{K}}{\overline{\mathrm{K}}{}}}^{2})+f(m_{{\mathrm{K}}{\overline{\mathrm{K}}{}}}^{2},m_{{\uprho}}^{2},\Gamma_{{\uprho}})-im_{{\uprho}}\Gamma_{{\uprho}}(m_{{\mathrm{K}}{\overline{\mathrm{K}}{}}})},$
(7)

where

$d(m_{{\uprho}})=\frac{3m_{{\mathrm{K}}}^{2}}{\pi
q_{0}^{2}}\log\left(\frac{m_{{\uprho}}+2q_{0}}{2m_{{\mathrm{K}}}}\right)+\frac{m_{{\uprho}}}{2\pi
q_{0}}-\frac{m_{{\mathrm{K}}}^{2}m_{{\uprho}}}{\pi q_{0}^{3}},$ (8)

and

$f(m_{{\mathrm{K}}{\overline{\mathrm{K}}{}}}^{2},m_{{\uprho}}^{2},\Gamma_{{\uprho}})=\Gamma_{{\uprho}}\frac{m_{{\uprho}}^{2}}{q_{0}^{3}}\left\\{q_{0}^{2}\left[h(m_{{\mathrm{K}}{\overline{\mathrm{K}}{}}}^{2})-h(m_{{\uprho}}^{2})\right]+q_{0}^{2}h^{\prime}(m_{{\uprho}}^{2})(m_{{\uprho}}^{2}-m_{{\mathrm{K}}{\overline{\mathrm{K}}{}}}^{2})\right\\}.$
(9)

The parameter $m_{{\mathrm{K}}}$ is taken as the mean of
$\rm\,m_{{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}}$ and
$m_{{{\mathrm{K}}^{\pm}}}$, and
$h^{\prime}(m_{{\uprho}}^{2})\equiv\frac{dh(m_{{\uprho}}^{2})}{dm_{{\uprho}}^{2}}$
is calculated from

$h(m^{2})=\frac{2q(m^{2})}{\pi
m}\log\left(\frac{m+2q(m^{2})}{2m_{{\mathrm{K}}}}\right),$ (10)

in the limit that
$m_{{\mathrm{K}}}=m_{{{\mathrm{K}}^{\pm}}}=m_{{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}}$. Parameters for the $\uprho$ resonances ${\uprho}(1450)^{{\pm}}$ and
${\uprho}(1700)^{{\pm}}$ are taken from Ref. [46] and tabulated in Appendix A.

This analysis uses two different parameterizations for the
${\mathrm{K}}{\uppi}$ $S$-wave contributions, dubbed GLASS and LASS, with
different motivations. These forms include both
${\mathrm{K}}^{*}_{{0}}({1430})$ resonance and nonresonant
${\mathrm{K}}{\uppi}$ $S$-wave contributions. The LASS parameterization takes
the form

$T_{\mathrm{R}}=f\left(\frac{m_{{\mathrm{K}}{\uppi}}}{m_{{\mathrm{K}}^{*}_{{0}}({1430})}}\right)\frac{m_{{\mathrm{K}}{\uppi}}}{q}\sin(\delta_{S}+\delta_{F})e^{i(\delta_{S}+\delta_{F})},$
(11)

where $f(x)=A\exp\left(b_{1}x+b_{2}x^{2}+b_{3}x^{3}\right)$ is an empirical
real production form factor, and the phases are defined by

$\tan\delta_{F}=\frac{2aq}{2+arq^{2}},\hskip
28.45274pt\tan\delta_{S}=\frac{m_{\mathrm{R}}\Gamma_{\mathrm{R}}(m_{{\mathrm{K}}{\uppi}})}{m_{{\mathrm{K}}^{*}_{{0}}({1430})}^{2}-m_{{\mathrm{K}}{\uppi}}^{2}}.$
(12)

The scattering length $a$, effective range $r$ and
${\mathrm{K}}^{*}_{{0}}({1430})$ mass and width are taken from measurements
[47] at the LASS experiment [48] and are tabulated in Appendix A. With the
choice $f(x)=1$ this form has been used in previous analyses e.g. Refs. [49,
50, 51], and if $\delta_{F}$ is additionally set to zero the relativistic
$S$-wave Breit-Wigner form is recovered. The Watson theorem [52] states that
the phase motion, as a function of ${\mathrm{K}}{\uppi}$ invariant mass, is
the same in elastic scattering and decay processes, in the absence of final-
state interactions (i.e. in the isobar model). Studies of
${\mathrm{K}}{\uppi}$ scattering data indicate that the $S$-wave remains
elastic up to the ${\mathrm{K}}\upeta^{\prime}$ threshold [47]. The magnitude
behavior is not constrained by the Watson theorem, which motivates the
inclusion of the form factor $f(x)$, but the LASS parameterization preserves
the phase behavior measured in ${\mathrm{K}}{\uppi}$ scattering. The real form
factor parameters are allowed to take different values for the neutral and
charged ${\mathrm{K}}^{*}_{{0}}({1430})$ resonances, as the production
processes are not the same, but the parameters taken from LASS measurements,
which specify the phase behavior, are shared between both
${\mathrm{K}}{\uppi}$ channels. A transformed set
$\mathbf{b^{\prime}}=\mathbf{U}^{-1}\mathbf{b}$ of the parameters
$\mathbf{b}=(b_{1},b_{2},b_{3})$ are also defined for use in the isobar model
fit, which is described in detail in Sect. 5. The constant matrix $\mathbf{U}$
is chosen to minimize fit correlations, and the form factor is normalized to
unity at the center of the accessible kinematic range, e.g.
$\frac{1}{2}(m_{{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}}+m_{{{\uppi}^{\pm}}}+m_{{{\mathrm{D}}^{0}}}-m_{{{\mathrm{K}}^{\pm}}})$ for
the charged ${\mathrm{K}}{\uppi}$ $S$-wave.

The GLASS (Generalized LASS) parameterization has been used by several recent
amplitude analyses, e.g. Refs. [39, 40, 12],

$T_{\mathrm{R}}=\left[F\sin(\delta_{F}+\phi_{F})e^{i(\delta_{F}+\phi_{F})}+\sin(\delta_{S})e^{i(\delta_{S}+\phi_{S})}e^{2i(\delta_{F}+\phi_{F})}\right]\frac{m_{{\mathrm{K}}{\uppi}}}{q},$
(13)

where $\delta_{F}$ and $\delta_{S}$ are defined as before, and $F$, $\phi_{F}$
and $\phi_{S}$ are free parameters in the fit. It should be noted that this
functional form can result in phase behavior significantly different to that
measured in LASS scattering data when its parameters are allowed to vary
freely. This is illustrated in Fig. 10 in Sect. 5.3.

###  4.2 Coherence factor and $C\\!P$-even fraction

The coherence factor $R_{\mathrm{f}}$ and mean strong-phase difference
$\delta_{\mathrm{f}}$ for the multi-body decays
$\mathrm{D}\\!\rightarrow\mathrm{f}$ and
$\mathrm{D}\\!\rightarrow\overline{\mathrm{f}}$ quantify the similarity of the
two decay structures [11]. In the limit $R_{\mathrm{f}}\rightarrow 1$ the
matrix elements for the two decays are identical. For
${{\mathrm{D}}^{0}}\\!\rightarrow{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{{\mathrm{K}}^{\pm}}{{\uppi}^{\mp}}$ the coherence factor and mean strong-
phase difference are defined by [12, 10]

$R_{{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{\mathrm{K}}{\uppi}}e^{-i\delta_{{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{\mathrm{K}}{\uppi}}}\equiv\frac{\int\mathcal{M}_{{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{{\mathrm{K}}^{+}}{{\uppi}^{-}}}(m^{2}_{{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{\uppi}},m^{2}_{{\mathrm{K}}{\uppi}})\mathcal{M}^{*}_{{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{{\mathrm{K}}^{-}}{{\uppi}^{+}}}(m^{2}_{{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{\uppi}},m^{2}_{{\mathrm{K}}{\uppi}})dm^{2}_{{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{\uppi}}dm^{2}_{{\mathrm{K}}{\uppi}}}{M_{{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{{\mathrm{K}}^{+}}{{\uppi}^{-}}}M_{{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{{\mathrm{K}}^{-}}{{\uppi}^{+}}}},$ (14)

where

$M^{2}_{{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{{\mathrm{K}}^{\pm}}{{\uppi}^{\mp}}}\equiv\int|\mathcal{M}_{{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{{\mathrm{K}}^{\pm}}{{\uppi}^{\mp}}}(m^{2}_{{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{\uppi}},m^{2}_{{\mathrm{K}}{\uppi}})|^{2}dm^{2}_{{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{\uppi}}dm^{2}_{{\mathrm{K}}{\uppi}},$ (15)

and the integrals are over the entire available phase space. The restricted
phase space coherence factor
$R_{{{\mathrm{K}}^{*}}{\mathrm{K}}}e^{-i\delta_{{{\mathrm{K}}^{*}}{\mathrm{K}}}}$
is defined analogously but with all integrals restricted to an area of phase
space close to the ${\mathrm{K}}^{*}({892})^{{\pm}}$ resonance. The restricted
area is defined by Ref. [12] as the region where the
${{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}{{\uppi}^{\pm}}$ invariant mass
is within $100{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the
${\mathrm{K}}^{*}({892})^{{\pm}}$ mass. The four observables
$R_{{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}{\mathrm{K}}{\uppi}}$,
$\delta_{{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}{\mathrm{K}}{\uppi}}$,
$R_{{{\mathrm{K}}^{*}}{\mathrm{K}}}$ and
$\delta_{{{\mathrm{K}}^{*}}{\mathrm{K}}}$ were measured using quantum-
correlated
${\uppsi(3770)}\\!\rightarrow{{\mathrm{D}}^{0}}{{\overline{\mathrm{D}}{}}{}^{0}}$
decays by the CLEO collaboration [12], and the coherence was found to be large
for both the full and the restricted regions. This analysis is not sensitive
to the overall phase difference between
${{\mathrm{D}}^{0}}\\!\rightarrow{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{{\mathrm{K}}^{+}}{{\uppi}^{-}}$ and
${{\overline{\mathrm{D}}{}}{}^{0}}\\!\rightarrow{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{{\mathrm{K}}^{+}}{{\uppi}^{-}}$. However, since it cancels in
$\delta_{{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{\mathrm{K}}{\uppi}}-\delta_{{{\mathrm{K}}^{*}}{\mathrm{K}}}$, this
combination, as well as $R_{{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{\mathrm{K}}{\uppi}}$ and $R_{{{\mathrm{K}}^{*}}{\mathrm{K}}}$, can be
calculated from isobar models and compared to the respective CLEO results.

An associated parameter that it is interesting to consider is the $C\\!P$-even
fraction [53],

$\begin{split}F_{+}&\equiv\frac{|\langle\mathrm{D}_{+}|{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{{\mathrm{K}}^{\pm}}{{\uppi}^{\mp}}\rangle|^{2}}{|\langle\mathrm{D}_{+}|{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{{\mathrm{K}}^{\pm}}{{\uppi}^{\mp}}\rangle|^{2}+|\langle\mathrm{D}_{-}|{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{{\mathrm{K}}^{\pm}}{{\uppi}^{\mp}}\rangle|^{2}}\\\
&=\frac{1}{2}\left[1+2R_{{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{\mathrm{K}}{\uppi}}\cos(\delta_{{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{\mathrm{K}}{\uppi}})\sqrt{{\cal
B}_{{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}{\mathrm{K}}{\uppi}}}(1+{\cal
B}_{{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{\mathrm{K}}{\uppi}})^{-1}\right],\end{split}$ (16)

where the $C\\!P$ eigenstates $|\mathrm{D}_{\pm}\rangle$ are given by
$\frac{1}{\sqrt{2}}\left[|{{\mathrm{D}}^{0}}\rangle\pm|{{\overline{\mathrm{D}}{}}{}^{0}}\rangle\right]$
and ${\cal B}_{{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{\mathrm{K}}{\uppi}}$ is the ratio of branching fractions of the two
${{\mathrm{D}}^{0}}\\!\rightarrow{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{{\mathrm{K}}^{\pm}}{{\uppi}^{\mp}}$ modes. As stated above, the relative
strong phase $\delta_{{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{\mathrm{K}}{\uppi}}$ is not predicted by the amplitude models and requires
external input.

###  4.3 Efficiency modeling

The trigger strategy described in Sect. 2, and to a lesser extent the offline
selection, includes requirements on variables such as the impact parameter and
$p_{\rm T}$ of the various charged particles correlated with the 2-body
invariant masses $m^{2}_{{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}{\uppi}}$
and $m^{2}_{{\mathrm{K}}{\uppi}}$. There is, therefore, a significant
variation in reconstruction efficiency as a function of
$m^{2}_{{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}{\uppi}}$ and
$m^{2}_{{\mathrm{K}}{\uppi}}$. This efficiency variation is modeled using
simulated events generated with a uniform distribution in these variables and
propagated through the full LHCb detector simulation, trigger emulation and
offline selection. Weights are applied to the simulated events to ensure that
various subsamples are present in the correct proportions. These weights
correct for known discrepancies between the simulation and real data in the
relative reconstruction efficiency for long and downstream tracks, and take
into account the ratios of $\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$ to
$\sqrt{s}=8\mathrm{\,Te\kern-1.00006ptV}$ and
${{\mathrm{D}}^{0}}\\!\rightarrow{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{{\mathrm{K}}^{-}}{{\uppi}^{+}}$ to
${{\mathrm{D}}^{0}}\\!\rightarrow{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{{\mathrm{K}}^{+}}{{\uppi}^{-}}$ simulated events to improve the
description of the data. The efficiencies of offline selection requirements
based on information from the RICH detectors are calculated using a data-
driven method based on calibration samples [54] of
${\mathrm{D}}^{*}({{2010}})^{{+}}\\!\rightarrow{{\mathrm{D}}^{0}}\uppi^{+}_{\text{slow}}$
decays, where
${{\mathrm{D}}^{0}}\\!\rightarrow{{\mathrm{K}}^{-}}{{\uppi}^{+}}$. These
efficiencies are included as additional weights. A non-parametric kernel
estimator [55] is used to produce a smooth function
${\varepsilon}(m^{2}_{{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{\uppi}},m^{2}_{{\mathrm{K}}{\uppi}})$ describing the efficiency variation
in the isobar model fits. The average model corresponding to the full dataset
recorded in 2011 and 2012, which is used unless otherwise noted, is shown in
Fig. 4. Candidates very near to the boundary of the allowed kinematic region
of the Dalitz plot are excluded, as the kinematics in this region lead to
variations in efficiency that are difficult to model. It is required that
$\max(|\cos(\theta_{{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{\uppi}})|,|\cos(\theta_{{\uppi}{\mathrm{K}}})|,|\cos(\theta_{{\mathrm{K}}{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}})|)<0.98$, where $\theta_{\mathrm{A}\mathrm{B}}$ is the angle between the
$\mathrm{A}$ and $\mathrm{B}$ momenta in the $\mathrm{A}\mathrm{C}$ rest
frame. This criterion removes $5\%$ of the candidates.

Figure 4: Efficiency function used in the isobar model fits, corresponding to
the average efficiency over the full dataset. The coordinates
$m^{2}_{{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}{\uppi}}$ and
$m^{2}_{{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}{\mathrm{K}}}$ are used to
highlight the approximate symmetry of the efficiency function. The $z$ units
are arbitrary.

The simulated events are also used to verify that the resolution in
$m^{2}_{{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}{\uppi}}$ and
$m^{2}_{{\mathrm{K}}{\uppi}}$ is around
$0.004{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$, corresponding to ${\cal
O}(2{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}})$ resolution in
$m({{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}{{\uppi}^{\pm}})$. Although
this is not explicitly accounted for in the isobar model fits, it has a small
effect which is measurable only on the parameters of the
${\mathrm{K}}^{*}({892})^{{\pm}}$ resonance and is accounted for in the
systematic uncertainties.

###  4.4 Fit components

There are three event categories described in Sect. 3 that must be treated
separately in the isobar model fits. The signal and mistagged components are
described by terms proportional to
${\varepsilon}(m^{2}_{{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{\uppi}},m^{2}_{{\mathrm{K}}{\uppi}})|\mathcal{M}_{{{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{{\mathrm{K}}^{\pm}}{{\uppi}^{\mp}}}}(m^{2}_{{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{\uppi}},m^{2}_{{\mathrm{K}}{\uppi}})|^{2}$, while the combinatorial
component is described by a smooth function,
$c_{{{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{{\mathrm{K}}^{\pm}}{{\uppi}^{\mp}}}}(m^{2}_{{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{\uppi}},m^{2}_{{\mathrm{K}}{\uppi}})$, obtained by applying to data in the
$m({{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}{\mathrm{K}}{\uppi})$ sidebands
the same non-parametric kernel estimator used to model the efficiency
variation. The same combinatorial background model is used for both
${\mathrm{D}}^{0}$ flavors, and the same efficiency function is used for both
modes and ${\mathrm{D}}^{0}$ flavors. The overall function used in the fit to
${{\mathrm{D}}^{0}}\\!\rightarrow{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{{\mathrm{K}}^{\pm}}{{\uppi}^{\mp}}$ decays is therefore

$\displaystyle P_{{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{{\mathrm{K}}^{\pm}}{{\uppi}^{\mp}}}(m^{2}_{{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{\uppi}},m^{2}_{{\mathrm{K}}{\uppi}})$
$\displaystyle=(1-f_{\text{m}}-f_{\text{c}}){\varepsilon}(m^{2}_{{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{\uppi}},m^{2}_{{\mathrm{K}}{\uppi}})|\mathcal{M}_{{{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{{\mathrm{K}}^{\pm}}{{\uppi}^{\mp}}}}(m^{2}_{{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{\uppi}},m^{2}_{{\mathrm{K}}{\uppi}})|^{2}$
$\displaystyle+f_{\text{m}}{\varepsilon}(m^{2}_{{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{\uppi}},m^{2}_{{\mathrm{K}}{\uppi}})|\overline{\mathcal{M}}_{{{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{{\mathrm{K}}^{\pm}}{{\uppi}^{\mp}}}}(m^{2}_{{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{\uppi}},m^{2}_{{\mathrm{K}}{\uppi}})|^{2}$ (17)
$\displaystyle+f_{\text{c}}c_{{{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{{\mathrm{K}}^{\pm}}{{\uppi}^{\mp}}}}(m^{2}_{{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{\uppi}},m^{2}_{{\mathrm{K}}{\uppi}}),$

where the mistagged contribution consists of
${{\overline{\mathrm{D}}{}}{}^{0}}\\!\rightarrow{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{{\mathrm{K}}^{\pm}}{{\uppi}^{\mp}}$ decays and $f_{\text{m}}$
($f_{\text{c}}$) denotes the mistagged (combinatorial) fraction tabulated in
Table 1.

All parameters except the complex amplitudes
$a_{\mathrm{R}}e^{i\phi_{\mathrm{R}}}$ are shared between the PDFs for both
modes and both ${\mathrm{D}}^{0}$ flavors. For the other parameters, Gaussian
constraints are included unless stated otherwise. The nominal values used in
the constraints are tabulated in Appendix A. No constraints are applied for
the ${\mathrm{K}}{\uppi}$ $S$-wave parameters $b_{1..3}$, $F$, $\phi_{S}$ and
$\phi_{F}$, as these have no suitable nominal values. The
${\mathrm{K}}{\uppi}$ $S$-wave parameters $a$ and $r$ are treated differently
in the GLASS and LASS models. In the LASS case these parameters are shared
between the neutral and charged ${\mathrm{K}}{\uppi}$ channels and a Gaussian
constraint to the LASS measurements [47] is included. In the GLASS case these
are allowed to vary freely and take different values for the two channels.

##  5 Isobar model fits

This section summarizes the procedure by which the amplitude models are
constructed, describes the various systematic uncertainties considered for the
models and finally discusses the models and the coherence information that can
be calculated from them.

Amplitude models are fitted using the isobar formalism and an unbinned
maximum-likelihood method, using the GooFit [56] package to exploit massively-
parallel Graphics Processing Unit (GPU) architectures. Where
$\chi^{2}/\mathrm{bin}$ values are quoted these are simply to indicate the fit
quality. Statistical uncertainties on derived quantities, such as the
resonance fit fractions, are calculated using a pseudoexperiment method based
on the fit covariance matrix.

###  5.1 Model composition

Initially, 15 resonances are considered for inclusion in each of the isobar
models: ${\mathrm{K}}^{*}({892,1410,1680})^{{0,\pm}}$,
${\mathrm{K}}^{*}_{{0,2}}({1430})^{{0,\pm}}$,
$\mathrm{a}_{0}(980,1450)^{{\pm}}$, $\mathrm{a}_{2}(1320)^{{\pm}}$ and
${\uprho}(1450,1700)^{{\pm}}$. Preliminary studies showed that models
containing the ${\mathrm{K}}^{*}({1680})$ resonances tend to include large
interference terms, which are cancelled by other large components. Such fine-
tuned interference effects are in general unphysical, and are therefore
disfavored in the model building [57, 38]. The ${\mathrm{K}}^{*}({1680})$
resonances are not considered further, and additionally the absolute value of
the sum of interference fractions [58] is required to be less than $30\%$ in
all models. In the absence of the ${\mathrm{K}}^{*}({1680})$ resonances, large
interference terms are typically generated by the ${\mathrm{K}}{\uppi}$
$S$-wave contributions. The requirement on the sum of interference fractions,
while arbitrary, allows an iterative procedure to be used to search for the
best amplitude models. This procedure explores a large number of possible
starting configurations and sets of resonances; it begins with the most
general models containing all 13 resonances and considers progressively
simpler configurations, trying a large number of initial fit configurations
for each set of resonances, until no further improvement in fit quality is
found among models simple enough to satisfy the interference fraction limit.
Higher values of this limit lead to a large number of candidate models with
similar fit quality.

A second procedure iteratively removes resonances from the models if they do
not significantly improve the fit quality. In this step a resonance must
improve the value of $-2\log\mathcal{L}$, where $\mathcal{L}$ is the
likelihood of the full dataset, by at least 16 units in order to be retained.
Up to this point, the ${\mathrm{K}}^{*}({892})$ mass and width parameters and
${\mathrm{K}}{\uppi}$ $S$-wave parameters have been allowed to vary in the
fit, but mass and width parameters for other resonances have been fixed. To
improve the quality of fit further, in a third step, $S$ and $P$-wave
resonance parameters are allowed to vary. The tensor resonance parameters are
known precisely [3], so remain fixed. At this stage, resonances that no longer
significantly improve the fit quality are removed, with the threshold
tightened so that each resonance must increase $-2\log\mathcal{L}$ by 25 units
in order to be retained.

Finally, parameters that are consistent with their nominal values to within
$1\sigma$ are fixed to the nominal value. The nominal values used are
tabulated in Appendix A. The entire procedure is performed in parallel using
the GLASS and LASS parameterizations of the ${\mathrm{K}}{\uppi}$ $S$-wave.
The data are found to prefer a solution where the GLASS parameterization of
the charged ${\mathrm{K}}{\uppi}$ $S$-wave has a poorly constrained degree of
freedom. The final change to the GLASS models is, therefore, to fix the
charged ${\mathrm{K}}{\uppi}$ $S$-wave $F$ parameter in order to stabilize the
uncertainty calculation for the two corresponding $a_{\mathrm{R}}$ parameters
by reducing the correlations among the free parameters.

###  5.2 Systematic uncertainties

Several sources of systematic uncertainty are considered. Those due to
experimental issues are described first, followed by uncertainties related to
the amplitude model formalism. Unless otherwise stated, the uncertainty
assigned to each parameter using an alternative fit is the absolute difference
in its value between the nominal and alternative fit.

As mentioned in Sect. 4.3, candidates extremely close to the edges of the
allowed kinematic region of the Dalitz plot are excluded. The requirement made
is that the largest of the three $|\cos(\theta_{\mathrm{A}\mathrm{B}})|$
values is less than 0.98. A systematic uncertainty due to this process is
estimated by changing the threshold to $0.96$, as this excludes a similar
additional area of the Dalitz plot as the original requirement.

The systematic uncertainty related to the efficiency model
${\varepsilon}(m^{2}_{{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{\uppi}},m^{2}_{{\mathrm{K}}{\uppi}})$ is evaluated in four ways. The first
probes the process by which a smooth curve is produced from simulated events;
this uncertainty is evaluated using an alternative fit that substitutes the
non-parametric estimator with a polynomial parameterization. The second
uncertainty is due to the limited sample size of simulated events. This is
evaluated by generating several alternative polynomial efficiency models
according to the covariance matrix of the polynomial model parameters; the
spread in parameter values from this ensemble is assigned as the uncertainty
due to the limited sample size. The third contribution is due to possible
imperfections in the description of the data by the simulation. This
uncertainty is assigned using an alternative simultaneous fit that separates
the sample into three categories according to the year in which the data were
collected and the type of ${\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}$
candidate used. As noted in Sect. 2, the sample recorded during 2011 does not
include downstream ${\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}$ candidates.
These sub-samples have different kinematic distributions and
${\varepsilon}(m^{2}_{{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{\uppi}},m^{2}_{{\mathrm{K}}{\uppi}})$ behavior, so this procedure tests
the ability of the simulation process to reproduce the variation seen in the
data. The final contribution is due to the re-weighting procedures used to
include the effect of offline selection requirements based on information from
the RICH detectors, and to correct for discrepancies between data and
simulation in the reconstruction efficiencies of long and downstream
${\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}$ candidates. This is evaluated
using alternative efficiency models where the relative proportion of the track
types is altered, and the weights describing the efficiency of selection
requirements using information from the RICH detectors are modified to account
for the limited calibration sample size. Additional robustness checks have
been performed to probe the description of the efficiency function by the
simulated events. In these checks the data are divided into two equally
populated bins of the ${\mathrm{D}}^{0}$ meson $p$, $p_{\rm T}$ or $\eta$ and
the amplitude models are re-fitted using each bin separately. The fit results
in each pair of bins are found to be compatible within the assigned
uncertainties, indicating that the simulated ${\mathrm{D}}^{0}$ kinematics
adequately match the data.

An uncertainty is assigned due to the description of the hardware trigger
efficiency in simulated events. Because the hardware trigger is not only
required to fire on the signal decay, it is important that the underlying
${\mathrm{p}}{\mathrm{p}}$ interaction is well described, and a systematic
uncertainty is assigned due to possible imperfections. This uncertainty is
obtained using an alternative efficiency model generated from simulated events
that have been weighted to adjust the fraction where the hardware trigger was
fired by the signal candidate.

The uncertainty due to the description of the combinatorial background is
evaluated by recomputing the $c_{{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{{\mathrm{K}}^{\pm}}{{\uppi}^{\mp}}}(m^{2}_{{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{\uppi}},m^{2}_{{\mathrm{K}}{\uppi}})$ function using
$m_{{{\mathrm{D}}^{0}}}$ sideband events to which an alternative kinematic fit
has been applied, without a constraint on the ${\mathrm{D}}^{0}$ mass. The
alternative model is expected to describe the edges of the phase space less
accurately, while providing an improved description of peaking features.

An alternative set of models is produced using a threshold of 9 units in the
value of $-2\log\mathcal{L}$ instead of the thresholds of 16 and 25 used for
the model building procedure. These models contain more resonances, as fewer
are removed during the model building process. A systematic uncertainty is
assigned using these alternative models for those parameters which are common
between the two sets of models.

Two parameters of the Flatté dynamical function, which is used to describe the
$\mathrm{a}_{0}(980)^{{\pm}}$ resonance, are fixed to nominal values in the
isobar model fits. Alternative fits are performed, where these parameters are
fixed to different values according to their quoted uncertainties, and the
largest changes to the fit parameters are assigned as systematic
uncertainties.

The effect of resolution in the
$m^{2}_{{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}{\uppi}}$ and
$m^{2}_{{\mathrm{K}}{\uppi}}$ coordinates is neglected in the isobar model
fits, and this is expected to have an effect on the measured
${\mathrm{K}}^{*}({892})^{{\pm}}$ decay width. An uncertainty is calculated
using a pseudoexperiment method, and is found to be small.

The uncertainty due to the yield determination process described in Sect. 3 is
measured by changing the fractions $f_{\text{m}}$ and $f_{\text{c}}$ in the
isobar model fit according to their statistical uncertainties, and taking the
largest changes with respect to the nominal result as the systematic
uncertainty.

There are two sources of systematic uncertainty due to the amplitude model
formalism considered. The first is that due to varying the meson radius
parameters $d_{{{\mathrm{D}}^{0}}}$ and $d_{\mathrm{R}}$, defined in Sect.
4.1. These are changed from
$d_{{{\mathrm{D}}^{0}}}=5.0{\,(\mathrm{Ge\kern-1.00006ptV\\!/}c)^{-1}}$ and
$d_{\mathrm{R}}=1.5{\,(\mathrm{Ge\kern-1.00006ptV\\!/}c)^{-1}}$ to
$2.5{\,(\mathrm{Ge\kern-1.00006ptV\\!/}c)^{-1}}$ and
$1.0{\,(\mathrm{Ge\kern-1.00006ptV\\!/}c)^{-1}}$, respectively. The second is
due to the dynamical function $T_{\mathrm{R}}$ used to describe the
${\uprho}(1450,1700)^{{\pm}}$ resonances. These resonances are described by
the Gounaris-Sakurai functional form in the nominal models, which is replaced
with a relativistic $P$-wave Breit-Wigner function to calculate a systematic
uncertainty due to this choice.

The uncertainties described above are added in quadrature to produce the total
systematic uncertainty quoted for the various results. For most quantities the
dominant systematic uncertainty is due to the meson radius parameters
$d_{{{\mathrm{D}}^{0}}}$ and $d_{\mathrm{R}}$. The largest sources of
experimental uncertainty relate to the description of the efficiency variation
across the Dalitz plot. The fit procedure and statistical uncertainty
calculation have been validated using pseudoexperiments and no bias was found.

Tables summarizing the various sources of systematic uncertainty and their
relative contributions are included in Appendix C.

###  5.3 Isobar model results

|
---|---
|

Figure 5: Distributions of $m^{2}_{{\mathrm{K}}{\uppi}}$ (upper left),
$m^{2}_{{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}{\uppi}}$ (upper right)
and $m^{2}_{{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}{\mathrm{K}}}$ (lower
left) in the
${{\mathrm{D}}^{0}}\\!\rightarrow{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{{\mathrm{K}}^{-}}{{\uppi}^{+}}$ mode with fit curves from the best GLASS
model. The solid (blue) curve shows the full PDF
$P_{{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{{\mathrm{K}}^{-}}{{\uppi}^{+}}}(m^{2}_{{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{\uppi}},m^{2}_{{\mathrm{K}}{\uppi}})$, while the other curves show the
components with the largest integrated fractions.

|
---|---
|

Figure 6: Distributions of $m^{2}_{{\mathrm{K}}{\uppi}}$ (upper left),
$m^{2}_{{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}{\uppi}}$ (upper right)
and $m^{2}_{{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}{\mathrm{K}}}$ (lower
left) in the
${{\mathrm{D}}^{0}}\\!\rightarrow{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{{\mathrm{K}}^{-}}{{\uppi}^{+}}$ mode with fit curves from the best LASS
model. The solid (blue) curve shows the full PDF
$P_{{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{{\mathrm{K}}^{-}}{{\uppi}^{+}}}(m^{2}_{{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{\uppi}},m^{2}_{{\mathrm{K}}{\uppi}})$, while the other curves show the
components with the largest integrated fractions.

|
---|---
|

Figure 7: Distributions of $m^{2}_{{\mathrm{K}}{\uppi}}$ (upper left),
$m^{2}_{{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}{\uppi}}$ (upper right)
and $m^{2}_{{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}{\mathrm{K}}}$ (lower
left) in the
${{\mathrm{D}}^{0}}\\!\rightarrow{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{{\mathrm{K}}^{+}}{{\uppi}^{-}}$ mode with fit curves from the best GLASS
model. The solid (blue) curve shows the full PDF
$P_{{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{{\mathrm{K}}^{+}}{{\uppi}^{-}}}(m^{2}_{{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{\uppi}},m^{2}_{{\mathrm{K}}{\uppi}})$, while the other curves show the
components with the largest integrated fractions.

|
---|---
|

Figure 8: Distributions of $m^{2}_{{\mathrm{K}}{\uppi}}$ (upper left),
$m^{2}_{{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}{\uppi}}$ (upper right)
and $m^{2}_{{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}{\mathrm{K}}}$ (lower
left) in the
${{\mathrm{D}}^{0}}\\!\rightarrow{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{{\mathrm{K}}^{+}}{{\uppi}^{-}}$ mode with fit curves from the best LASS
model. The solid (blue) curve shows the full PDF
$P_{{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{{\mathrm{K}}^{+}}{{\uppi}^{-}}}(m^{2}_{{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{\uppi}},m^{2}_{{\mathrm{K}}{\uppi}})$, while the other curves show the
components with the largest integrated fractions.

Figure 9: Decay rate and phase variation across the Dalitz plot. The top row
shows $|\mathcal{M}_{{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{{\mathrm{K}}^{\pm}}{{\uppi}^{\mp}}}(m^{2}_{{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{\uppi}},m^{2}_{{\mathrm{K}}{\uppi}})|^{2}$ in the best GLASS isobar
models, the center row shows the phase behavior of the same models and the
bottom row shows the same function subtracted from the phase behavior in the
best LASS isobar models. The left column shows the
${{\mathrm{D}}^{0}}\\!\rightarrow{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{{\mathrm{K}}^{-}}{{\uppi}^{+}}$ mode with
${{\mathrm{D}}^{0}}\\!\rightarrow{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{{\mathrm{K}}^{+}}{{\uppi}^{-}}$ on the right. The small inhomogeneities
that are visible in the bottom row relate to the GLASS and LASS models
preferring slightly different values of the ${\mathrm{K}}^{*}({892})^{{\pm}}$
mass and width.

Table 5: Isobar model fit results for the
${{\mathrm{D}}^{0}}\\!\rightarrow{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{{\mathrm{K}}^{-}}{{\uppi}^{+}}$ mode. The first uncertainties are
statistical and the second systematic.

| $a_{\mathrm{R}}$ |  $\phi_{\mathrm{R}}$ $(^{\circ})$ | Fit fraction [%]
---|---|---|---
Resonance | GLASS | LASS | GLASS | LASS | GLASS | LASS
${\mathrm{K}}^{*}({892})^{{+}}$ | 1.0 (fixed) | 1.0 (fixed) | 0.0 (fixed) | 0.0 (fixed) |  $57.0$$\,\pm\,$ | $0.8\pm 2.6$ |  $56.9$$\,\pm\,$ | $0.6\pm 1.1$
${\mathrm{K}}^{*}({1410})^{{+}}$ |  $4.3$$\,\pm\,$ | $0.3\pm 0.7$ |  $5.83$$\,\pm\,$ | $0.29\pm 0.29$ |  $-160$$\,\pm\,$ | $6\pm 24$ |  $-143$$\,\pm\,$ | $3\pm 6$ |  $5$$\,\pm\,$ | $1\pm 4$ |  $9.6$$\,\pm\,$ | $1.1\pm 2.9$
$({{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}{\uppi})^{+}_{S\text{-wave}}$ |  $0.62$$\,\pm\,$ | $0.05\pm 0.18$ |  $1.13$$\,\pm\,$ | $0.09\pm 0.21$ |  $-67$$\,\pm\,$ | $5\pm 15$ |  $-59$$\,\pm\,$ | $4\pm 13$ |  $12$$\,\pm\,$ | $2\pm 9$ |  $11.7$$\,\pm\,$ | $1.0\pm 2.3$
${\overline{\mathrm{K}}{}}^{*}({892})^{{0}}$ |  $0.213$$\,\pm\,$ | $0.007\pm 0.018$ |  $0.210$$\,\pm\,$ | $0.006\pm 0.010$ |  $-108$$\,\pm\,$ | $2\pm 4$ |  $-101.5$$\,\pm\,$ | $2.0\pm 2.8$ |  $2.5$$\,\pm\,$ | $0.2\pm 0.4$ |  $2.47$$\,\pm\,$ | $0.15\pm 0.23$
${\overline{\mathrm{K}}{}}^{*}({1410})^{{0}}$ |  $6.0$$\,\pm\,$ | $0.3\pm 0.5$ |  $3.9$$\,\pm\,$ | $0.2\pm 0.4$ |  $-179$$\,\pm\,$ | $4\pm 17$ |  $-174$$\,\pm\,$ | $4\pm 7$ |  $9$$\,\pm\,$ | $1\pm 4$ |  $3.8$$\,\pm\,$ | $0.5\pm 2.0$
${\overline{\mathrm{K}}{}}^{*}_{{2}}({1430})^{{0}}$ |  $3.2$$\,\pm\,$ | $0.3\pm 1.0$ | — |  $-172$$\,\pm\,$ | $5\pm 23$ | — |  $3.4$$\,\pm\,$ | $0.6\pm 2.7$ | —
$({\mathrm{K}}{\uppi})^{0}_{S\text{-wave}}$ |  $2.5$$\,\pm\,$ | $0.2\pm 1.3$ |  $1.28$$\,\pm\,$ | $0.12\pm 0.23$ |  $50$$\,\pm\,$ | $10\pm 80$ |  $75$$\,\pm\,$ | $3\pm 8$ |  $11$$\,\pm\,$ | $2\pm 10$ |  $18$$\,\pm\,$ | $2\pm 4$
$\mathrm{a}_{0}(980)^{{-}}$ | — |  $1.07$$\,\pm\,$ | $0.09\pm 0.14$ | — |  $82$$\,\pm\,$ | $5\pm 7$ | — |  $4.0$$\,\pm\,$ | $0.7\pm 1.1$
$\mathrm{a}_{2}(1320)^{{-}}$ |  $0.19$$\,\pm\,$ | $0.03\pm 0.09$ |  $0.17$$\,\pm\,$ | $0.03\pm 0.05$ |  $-129$$\,\pm\,$ | $8\pm 17$ |  $-128$$\,\pm\,$ | $10\pm 8$ |  $0.20$$\,\pm\,$ | $0.06\pm 0.21$ |  $0.15$$\,\pm\,$ | $0.06\pm 0.13$
$\mathrm{a}_{0}(1450)^{{-}}$ |  $0.52$$\,\pm\,$ | $0.04\pm 0.15$ |  $0.43$$\,\pm\,$ | $0.05\pm 0.10$ |  $-82$$\,\pm\,$ | $7\pm 31$ |  $-49$$\,\pm\,$ | $11\pm 19$ |  $1.2$$\,\pm\,$ | $0.2\pm 0.6$ |  $0.74$$\,\pm\,$ | $0.15\pm 0.34$
${\uprho}(1450)^{{-}}$ |  $1.6$$\,\pm\,$ | $0.2\pm 0.5$ |  $1.3$$\,\pm\,$ | $0.1\pm 0.4$ |  $-177$$\,\pm\,$ | $7\pm 32$ |  $-144$$\,\pm\,$ | $7\pm 9$ |  $1.3$$\,\pm\,$ | $0.3\pm 0.7$ |  $1.4$$\,\pm\,$ | $0.2\pm 0.7$
${\uprho}(1700)^{{-}}$ |  $0.38$$\,\pm\,$ | $0.08\pm 0.15$ | — |  $-70$$\,\pm\,$ | $10\pm 60$ | — |  $0.12$$\,\pm\,$ | $0.05\pm 0.14$ | —

Table 6: Isobar model fit results for the
${{\mathrm{D}}^{0}}\\!\rightarrow{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{{\mathrm{K}}^{+}}{{\uppi}^{-}}$ mode. The first uncertainties are
statistical and the second systematic.

| $a_{\mathrm{R}}$ |  $\phi_{\mathrm{R}}$ $(^{\circ})$ | Fit fraction [%]
---|---|---|---
Resonance | GLASS | LASS | GLASS | LASS | GLASS | LASS
${\mathrm{K}}^{*}({892})^{{-}}$ | 1.0 (fixed) | 1.0 (fixed) | 0.0 (fixed) | 0.0 (fixed) |  $29.5$$\,\pm\,$ | $0.6\pm 1.6$ |  $28.8$$\,\pm\,$ | $0.4\pm 1.3$
${\mathrm{K}}^{*}({1410})^{{-}}$ |  $4.7$$\,\pm\,$ | $0.5\pm 1.1$ |  $9.1$$\,\pm\,$ | $0.6\pm 1.5$ |  $-106$$\,\pm\,$ | $6\pm 25$ |  $-79$$\,\pm\,$ | $3\pm 7$ |  $3.1$$\,\pm\,$ | $0.6\pm 1.6$ |  $11.9$$\,\pm\,$ | $1.5\pm 2.2$
$({{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}{\uppi})^{-}_{S\text{-wave}}$ |  $0.58$$\,\pm\,$ | $0.05\pm 0.11$ |  $1.16$$\,\pm\,$ | $0.11\pm 0.32$ |  $-164$$\,\pm\,$ | $6\pm 31$ |  $-101$$\,\pm\,$ | $6\pm 21$ |  $5.4$$\,\pm\,$ | $0.9\pm 1.7$ |  $6.3$$\,\pm\,$ | $0.9\pm 2.1$
${\mathrm{K}}^{*}({892})^{{0}}$ |  $0.410$$\,\pm\,$ | $0.010\pm 0.021$ |  $0.427$$\,\pm\,$ | $0.010\pm 0.013$ |  $176$$\,\pm\,$ | $2\pm 9$ |  $-175.0$$\,\pm\,$ | $1.7\pm 1.4$ |  $4.82$$\,\pm\,$ | $0.23\pm 0.35$ |  $5.17$$\,\pm\,$ | $0.21\pm 0.32$
${\mathrm{K}}^{*}({1410})^{{0}}$ |  $6.2$$\,\pm\,$ | $0.5\pm 1.4$ |  $4.2$$\,\pm\,$ | $0.5\pm 0.9$ |  $175$$\,\pm\,$ | $4\pm 14$ |  $165$$\,\pm\,$ | $5\pm 10$ |  $5.2$$\,\pm\,$ | $0.7\pm 1.6$ |  $2.2$$\,\pm\,$ | $0.6\pm 2.1$
${\mathrm{K}}^{*}_{{2}}({1430})^{{0}}$ |  $6.3$$\,\pm\,$ | $0.5\pm 1.7$ | — |  $-139$$\,\pm\,$ | $5\pm 21$ | — |  $7$$\,\pm\,$ | $1\pm 4$ | —
$({\mathrm{K}}{\uppi})^{0}_{S\text{-wave}}$ |  $3.7$$\,\pm\,$ | $0.3\pm 1.8$ |  $1.7$$\,\pm\,$ | $0.2\pm 0.4$ |  $100$$\,\pm\,$ | $10\pm 70$ |  $144$$\,\pm\,$ | $3\pm 6$ |  $12$$\,\pm\,$ | $1\pm 8$ |  $17$$\,\pm\,$ | $2\pm 6$
$\mathrm{a}_{0}(980)^{{+}}$ |  $1.8$$\,\pm\,$ | $0.1\pm 0.6$ |  $3.8$$\,\pm\,$ | $0.2\pm 0.7$ |  $64$$\,\pm\,$ | $5\pm 24$ |  $126$$\,\pm\,$ | $3\pm 6$ |  $11$$\,\pm\,$ | $1\pm 6$ |  $26$$\,\pm\,$ | $2\pm 10$
$\mathrm{a}_{0}(1450)^{{+}}$ |  $0.44$$\,\pm\,$ | $0.05\pm 0.13$ |  $0.86$$\,\pm\,$ | $0.10\pm 0.12$ |  $-140$$\,\pm\,$ | $9\pm 35$ |  $-110$$\,\pm\,$ | $8\pm 7$ |  $0.45$$\,\pm\,$ | $0.09\pm 0.34$ |  $1.5$$\,\pm\,$ | $0.3\pm 0.4$
${\uprho}(1450)^{{+}}$ |  $2.3$$\,\pm\,$ | $0.4\pm 0.8$ | — |  $-60$$\,\pm\,$ | $6\pm 18$ | — |  $1.5$$\,\pm\,$ | $0.5\pm 0.9$ | —
${\uprho}(1700)^{{+}}$ |  $1.04$$\,\pm\,$ | $0.12\pm 0.32$ |  $1.25$$\,\pm\,$ | $0.15\pm 0.33$ |  $4$$\,\pm\,$ | $11\pm 20$ |  $39$$\,\pm\,$ | $9\pm 15$ |  $0.5$$\,\pm\,$ | $0.1\pm 0.5$ |  $0.53$$\,\pm\,$ | $0.11\pm 0.23$

Table 7: Modulus and phase of the relative amplitudes between resonances that appear in both the ${{\mathrm{D}}^{0}}\\!\rightarrow{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}{{\mathrm{K}}^{-}}{{\uppi}^{+}}$ and ${{\mathrm{D}}^{0}}\\!\rightarrow{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}{{\mathrm{K}}^{+}}{{\uppi}^{-}}$ modes. Relative phases are calculated using the value of $\delta_{{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}{\mathrm{K}}{\uppi}}$ measured in $\uppsi(3770)$ decays [12], and the uncertainty on this value is included in the statistical uncertainty. The first uncertainties are statistical and the second systematic. Relative |  |  |  |  |
---|---|---|---|---|---
amplitude |  | GLASS | LASS
$\frac{\mathcal{A}({\mathrm{K}}^{*}({892})^{{-}})}{\mathcal{A}({\mathrm{K}}^{*}({892})^{{+}})}$ | mod |  $0.582$$\,\pm\,$ | $0.007\pm 0.008$ |  $0.576$$\,\pm\,$ | $0.005\pm 0.010$
arg $(^{\circ})$ |  $-2$$\,\pm\,$ | $15\pm 2$ |  $-2$$\,\pm\,$ | $15\pm 1$
$\frac{\mathcal{A}({\mathrm{K}}^{*}({1410})^{{-}})}{\mathcal{A}({\mathrm{K}}^{*}({1410})^{{+}})}$ | mod |  $0.64$$\,\pm\,$ | $0.08\pm 0.22$ |  $0.90$$\,\pm\,$ | $0.08\pm 0.15$
arg $(^{\circ})$ |  $52$$\,\pm\,$ | $17\pm 20$ |  $62$$\,\pm\,$ | $16\pm 6$
$\frac{\mathcal{A}(({{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}{\uppi})^{-}_{S-\text{wave}})}{\mathcal{A}(({{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}{\uppi})^{+}_{S-\text{wave}})}$ | mod |  $0.54$$\,\pm\,$ | $0.06\pm 0.26$ |  $0.59$$\,\pm\,$ | $0.05\pm 0.08$
arg $(^{\circ})$ |  $-100$$\,\pm\,$ | $20\pm 40$ |  $-44$$\,\pm\,$ | $17\pm 10$
$\frac{\mathcal{A}({\mathrm{K}}^{*}({892})^{{0}})}{\mathcal{A}({\overline{\mathrm{K}}{}}^{*}({892})^{{0}})}$ | mod |  $1.12$$\,\pm\,$ | $0.05\pm 0.11$ |  $1.17$$\,\pm\,$ | $0.04\pm 0.05$
arg $(^{\circ})$ |  $-78$$\,\pm\,$ | $16\pm 10$ |  $-75$$\,\pm\,$ | $15\pm 2$
$\frac{\mathcal{A}({\mathrm{K}}^{*}({1410})^{{0}})}{\mathcal{A}({\overline{\mathrm{K}}{}}^{*}({1410})^{{0}})}$ | mod |  $0.60$$\,\pm\,$ | $0.05\pm 0.12$ |  $0.62$$\,\pm\,$ | $0.09\pm 0.12$
arg $(^{\circ})$ |  $-9$$\,\pm\,$ | $16\pm 14$ |  $-23$$\,\pm\,$ | $17\pm 11$
$\frac{\mathcal{A}({\mathrm{K}}^{*}_{{2}}({1430})^{{0}})}{\mathcal{A}({\overline{\mathrm{K}}{}}^{*}_{{2}}({1430})^{{0}})}$ | mod |  $1.1$$\,\pm\,$ | $0.1\pm 0.5$ | —
arg $(^{\circ})$ |  $31$$\,\pm\,$ | $17\pm 12$ | —
$\frac{\mathcal{A}(({{\mathrm{K}}^{+}}{{\uppi}^{-}})_{S-\text{wave}})}{\mathcal{A}(({{\mathrm{K}}^{-}}{{\uppi}^{+}})_{S-\text{wave}})}$ | mod |  $0.87$$\,\pm\,$ | $0.08\pm 0.14$ |  $0.78$$\,\pm\,$ | $0.06\pm 0.18$
arg $(^{\circ})$ |  $49$$\,\pm\,$ | $25\pm 16$ |  $68$$\,\pm\,$ | $16\pm 6$
$\frac{\mathcal{A}(\mathrm{a}_{0}(980)^{{+}})}{\mathcal{A}(\mathrm{a}_{0}(980)^{{-}})}$ | mod | — |  $2.1$$\,\pm\,$ | $0.2\pm 0.6$
arg $(^{\circ})$ | — |  $42$$\,\pm\,$ | $16\pm 5$
$\frac{\mathcal{A}(\mathrm{a}_{0}(1450)^{{+}})}{\mathcal{A}(\mathrm{a}_{0}(1450)^{{-}})}$ | mod |  $0.49$$\,\pm\,$ | $0.06\pm 0.28$ |  $1.14$$\,\pm\,$ | $0.16\pm 0.30$
arg $(^{\circ})$ |  $-60$$\,\pm\,$ | $19\pm 34$ |  $-63$$\,\pm\,$ | $20\pm 19$
$\frac{\mathcal{A}({\uprho}(1450)^{{+}})}{\mathcal{A}({\uprho}(1450)^{{-}})}$ | mod |  $0.86$$\,\pm\,$ | $0.16\pm 0.26$ | —
arg $(^{\circ})$ |  $110$$\,\pm\,$ | $20\pm 50$ | —
$\frac{\mathcal{A}({\uprho}(1700)^{{+}})}{\mathcal{A}({\uprho}(1700)^{{-}})}$ | mod |  $1.6$$\,\pm\,$ | $0.4\pm 0.4$ | —
arg $(^{\circ})$ |  $70$$\,\pm\,$ | $20\pm 70$ | —

The fit results for the best isobar models using the GLASS and LASS
parameterizations of the ${\mathrm{K}}{\uppi}$ $S$-wave are given in Tables 5
and 6. Distributions of $m^{2}_{{\mathrm{K}}{\uppi}}$,
$m^{2}_{{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}{\uppi}}$ and
$m^{2}_{{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}{\mathrm{K}}}$ are shown
alongside the best model of the
${{\mathrm{D}}^{0}}\\!\rightarrow{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{{\mathrm{K}}^{-}}{{\uppi}^{+}}$ mode using the GLASS parameterization in
Fig. 5. In Fig. 5 and elsewhere the nomenclature
$\mathrm{R}_{1}\times\mathrm{R}_{2}$ denotes interference terms. The
corresponding distributions showing the best model using the LASS
parameterization are shown in Fig. 6. Distributions for the
${{\mathrm{D}}^{0}}\\!\rightarrow{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{{\mathrm{K}}^{+}}{{\uppi}^{-}}$ mode are shown in Figs. 7 and 8. Figure 9
shows the GLASS isobar models in two dimensions, and demonstrates that the
GLASS and LASS choices of ${\mathrm{K}}{\uppi}$ $S$-wave parameterization both
lead to similar descriptions of the overall phase variation. Figures 5–8 show
distributions distorted by efficiency effects, while Fig. 9 shows the decay
rate without distortion. Lookup tables for the complex amplitude variation
across the Dalitz plot in all four isobar models are available in the
supplemental material.

The data are found to favor solutions that have a significant neutral
${\mathrm{K}}{\uppi}$ $S$-wave contribution, even though the exchange (Fig.
1b) and penguin annihilation (Fig. 1d) processes that contribute to the
neutral channel are expected to be suppressed. The expected suppression is
observed for the $P$-wave ${\mathrm{K}}^{*}({892})$ resonances, with the
neutral mode fit fractions substantially lower. The models using the LASS
parameterization additionally show this pattern for the
${\mathrm{K}}^{*}({1410})$ states. The sums of the fit fractions [58],
excluding interference terms, in the
${{\mathrm{D}}^{0}}\\!\rightarrow{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{{\mathrm{K}}^{-}}{{\uppi}^{+}}$ and
${{\mathrm{D}}^{0}}\\!\rightarrow{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{{\mathrm{K}}^{+}}{{\uppi}^{-}}$ models are, respectively, 103% (109%) and
81% (99%) using the GLASS (LASS) ${\mathrm{K}}{\uppi}$ $S$-wave
parameterization.

Using measurements of the mean strong-phase difference between the
${{\mathrm{D}}^{0}}\\!\rightarrow{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{{\mathrm{K}}^{\pm}}{{\uppi}^{\mp}}$ modes available from $\uppsi(3770)$
decays [12], the relative complex amplitudes between each resonance in one
${\mathrm{D}}^{0}$ decay mode and its conjugate contribution to the other
${\mathrm{D}}^{0}$ decay mode are computed. These values are summarized in
Table 7.

Additional information about the models is listed in Appendices B and C,
including the interference fractions and decomposition of the systematic
uncertainties. The best models also contain contributions from the
${\uprho}(1450)^{{\pm}}$ and ${\uprho}(1700)^{{\pm}}$ resonances in the
${{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}{{\mathrm{K}}^{\pm}}$ channels,
supporting evidence in Ref. [46] of the
${\mathrm{K}}{\overline{\mathrm{K}}{}}$ decay modes for these states.
Alternative models are fitted where one ${\uprho}^{\pm}$ contribution is
removed from the best models; in these the value of $-2\log\mathcal{L}$ is
found to degrade by at least 162 units. Detailed results are tabulated in
Appendix B.

The ${\mathrm{K}}{\uppi}$ $S$-wave systems are poorly understood [6], and
there is no clear theoretical guidance regarding the correct description of
these systems in an isobar model. As introduced in Sect. 4.1, the LASS
parameterization is motivated by the Watson theorem, but this assumes that
three-body interactions are negligible and is not, therefore, expected to be
precisely obeyed in nature.

Figure 10: Comparison of the phase behavior of the various
${\mathrm{K}}{\uppi}$ $S$-wave parameterizations used. The solid (red) curve
shows the LASS parameterization, while the dashed (blue) and dash-dotted
(green) curves show, respectively, the GLASS functional form fitted to the
charged and neutral $S$-wave channels. The final two curves show the GLASS
forms fitted to the charged ${\mathrm{K}}{\uppi}$ $S$-wave in
${{\mathrm{D}}^{0}}\\!\rightarrow{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{{\uppi}^{+}}{{\uppi}^{-}}$ decays in Ref. [39] (triangular markers,
purple) and Ref. [40] (dotted curve, black). The latter of these was used in
the analysis of
${{\mathrm{D}}^{0}}\\!\rightarrow{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{{\mathrm{K}}^{\pm}}{{\uppi}^{\mp}}$ decays by the CLEO collaboration [12].

The isobar models using the GLASS parameterization favor solutions with
qualitatively similar phase behavior to those using the LASS parameterization.
This is illustrated in Fig. 10, which also shows the GLASS forms obtained in
fits to
${{\mathrm{D}}^{0}}\\!\rightarrow{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{{\uppi}^{+}}{{\uppi}^{-}}$ decays by the BaBar collaboration [39, 40] and
previously used in fits to the
${{\mathrm{D}}^{0}}\\!\rightarrow{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{{\mathrm{K}}^{\pm}}{{\uppi}^{\mp}}$ decay modes [12]. This figure shows
that the GLASS functional form has substantial freedom to produce different
phase behavior to the LASS form, but that this is not strongly favored in the
${{\mathrm{D}}^{0}}\\!\rightarrow{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{{\mathrm{K}}^{\pm}}{{\uppi}^{\mp}}$ decays. The good quality of fit
obtained using the LASS parameterization indicates that large differences in
phase behavior with respect to ${\mathrm{K}}{\uppi}$ scattering data [47, 48]
are not required in order to describe the
${{\mathrm{D}}^{0}}\\!\rightarrow{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{{\mathrm{K}}^{\pm}}{{\uppi}^{\mp}}$ decays. A similar conclusion was drawn
in Ref. [59] for the decay
${{\mathrm{D}}^{+}}\\!\rightarrow{{\mathrm{K}}^{-}}{{\uppi}^{+}}{{\uppi}^{+}}$,
while Ref. [60] found behavior inconsistent with scattering data using the
same ${\mathrm{D}}^{+}$ decay mode but a slightly different technique. Ref.
[61] studied the ${\mathrm{K}}{\uppi}$ $S$-wave in
${\uptau^{-}}\\!\rightarrow{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{{\uppi}^{-}}{{\upnu}_{\tau}}$ decays and found that a parameterization
based on the LASS ${\mathrm{K}}{\uppi}$ scattering data, but without a real
production form factor, gave a poor description of the $\uptau^{-}$ decay
data.

The quality of fit for each model is quantified by calculating $\chi^{2}$
using a dynamic binning scheme. The values are summarized in Table 8, while
the binning scheme and two-dimensional quality of fit are shown in Appendix B.
This binning scheme is generated by iteratively sub-dividing the Dalitz plot
to produce new bins of approximately equal population until further sub-
division would result in a bin population of fewer than 15 candidates, or a
bin dimension smaller than $0.02{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$
in $m^{2}_{{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}{\uppi}}$ or
$m^{2}_{{\mathrm{K}}{\uppi}}$. This minimum size corresponds to five times the
average resolution in these variables.

Table 8: Values of $\chi^{2}/\mathrm{bin}$ indicating the fit quality obtained using both ${\mathrm{K}}{\uppi}$ $S$-wave parameterizations in the two decay modes. The binning scheme for the ${{\mathrm{D}}^{0}}\\!\rightarrow{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}{{\mathrm{K}}^{-}}{{\uppi}^{+}}$ (${{\mathrm{D}}^{0}}\\!\rightarrow{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}{{\mathrm{K}}^{+}}{{\uppi}^{-}}$) mode contains 2191 (2573) bins. | Isobar model
---|---
| GLASS | LASS
${{\mathrm{D}}^{0}}\\!\rightarrow{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}{{\mathrm{K}}^{-}}{{\uppi}^{+}}$ | 1.12 | 1.10
${{\mathrm{D}}^{0}}\\!\rightarrow{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}{{\mathrm{K}}^{+}}{{\uppi}^{-}}$ | 1.07 | 1.09

The overall fit quality is slightly better in the isobar models using the
GLASS ${\mathrm{K}}{\uppi}$ $S$-wave parameterization, but this is not a
significant effect and it should be noted that these models contain more
degrees of freedom, with 57 parameters fitted in the final GLASS model
compared to 50 when using the LASS parameterization.

##  6 Additional measurements

In this section, several additional results, including those derived from the
amplitude models, are presented.

###  6.1 Ratio of branching fractions measurement

The ratio of branching fractions

${\cal B}_{{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{\mathrm{K}}{\uppi}}\equiv\frac{{\cal
B}({{\mathrm{D}}^{0}}\\!\rightarrow{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{{\mathrm{K}}^{+}}{{\uppi}^{-}})}{{\cal
B}({{\mathrm{D}}^{0}}\\!\rightarrow{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{{\mathrm{K}}^{-}}{{\uppi}^{+}})},$ (18)

and the restricted region ratio ${\cal B}_{{{\mathrm{K}}^{*}}{\mathrm{K}}}$,
defined in the same region near the ${\mathrm{K}}^{*}({892})^{{\pm}}$
resonance as the coherence factor $R_{{{\mathrm{K}}^{*}}{\mathrm{K}}}$ (Sect.
4.2), are also measured. The efficiency correction due to the reconstruction
efficiency ${\varepsilon}(m^{2}_{{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{\uppi}},m^{2}_{{\mathrm{K}}{\uppi}})$ is evaluated using the best isobar
models, and the difference between the results obtained with the two
${\mathrm{K}}{\uppi}$ $S$-wave parameterizations is taken as a systematic
uncertainty in addition to those effects described in Sect. 5.2. This
efficiency correction modifies the ratio of yields quoted in Table 1 by
approximately 3%.

The two ratios are measured to be

$\displaystyle{\cal B}_{{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{\mathrm{K}}{\uppi}}$ $\displaystyle=0.655\pm 0.004\mathrm{\,(stat)}\pm
0.006\mathrm{\,(syst)},$ $\displaystyle{\cal
B}_{{{\mathrm{K}}^{*}}{\mathrm{K}}}$ $\displaystyle=0.370\pm
0.003\mathrm{\,(stat)}\pm 0.012\mathrm{\,(syst)}.$

These are the most precise measurements to date.

###  6.2 Coherence factor and $C\\!P$-even fraction results

Table 9: Coherence factor observables to which the isobar models are sensitive. The third column summarizes the CLEO results measured in quantum-correlated decays [12], where the uncertainty on $\delta_{{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}{\mathrm{K}}{\uppi}}-\delta_{{{\mathrm{K}}^{*}}{\mathrm{K}}}$ is calculated assuming maximal correlation between $\delta_{{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}{\mathrm{K}}{\uppi}}$ and $\delta_{{{\mathrm{K}}^{*}}{\mathrm{K}}}$. Variable | GLASS | LASS | CLEO
---|---|---|---
$R_{{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}{\mathrm{K}}{\uppi}}$ | 0.573$\,\pm\,$ | $0.007\pm 0.019$ | 0.571$\,\pm\,$ | $0.005\pm 0.019$ | 0.73$\pm$ | 0.08
$R_{{{\mathrm{K}}^{*}}{\mathrm{K}}}$ | 0.831$\,\pm\,$ | $0.004\pm 0.010$ | 0.835$\,\pm\,$ | $0.003\pm 0.011$ | 1.00$\pm$ | 0.16
$\delta_{{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}{\mathrm{K}}{\uppi}}-\delta_{{{\mathrm{K}}^{*}}{\mathrm{K}}}$ |  $(0.2$$\,\pm\,$ | $0.6\pm 1.1)^{\circ}$ |  $(-0.0$$\,\pm\,$ | $0.5\pm 0.7)^{\circ}$ |  $(-18$$\pm$ | $31)^{\circ}$

The amplitude models are used to calculate the coherence factors
$R_{{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}{\mathrm{K}}{\uppi}}$ and
$R_{{{\mathrm{K}}^{*}}{\mathrm{K}}}$, and the strong-phase difference
$\delta_{{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{\mathrm{K}}{\uppi}}-\delta_{{{\mathrm{K}}^{*}}{\mathrm{K}}}$, as described
in Sect. 4.2. The results are summarized in Table 9, alongside the
corresponding values measured in $\uppsi(3770)$ decays by the CLEO
collaboration. Lower, but compatible, coherence is calculated using the isobar
models than was measured at CLEO, with the discrepancy larger for the
coherence factor calculated over the full phase space. The results from the
GLASS and LASS isobar models are very similar, showing that the coherence
variables are not sensitive to the ${\mathrm{K}}{\uppi}$ $S$-wave
parameterization.

The coherence factor $R_{{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{\mathrm{K}}{\uppi}}$, and the ratio of branching fractions ${\cal
B}_{{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}{\mathrm{K}}{\uppi}}$, are
combined with the mean phase difference between the two final states measured
in $\uppsi(3770)$ decays [12] to calculate the $C\\!P$-even fraction $F_{+}$,
defined in Eq. 16, which is determined to be

$F_{+}=0.777\pm 0.003\mathrm{\,(stat)}\pm 0.009\mathrm{\,(syst)},$

using the GLASS amplitude models. A consistent result is obtained using the
alternative (LASS) amplitude models. This model-dependent value is compatible
with the direct measurement using only $\uppsi(3770)$ decay data [12, 53].

###  6.3 $\mathrm{SU}(3)$ flavor symmetry tests

$\mathrm{SU}(3)$ flavor symmetry can be used to relate decay amplitudes in
several $\mathrm{D}$ meson decays, such that a global fit to many such
amplitudes can provide predictions for the neutral and charged
${\mathrm{K}}^{*}({892})$ complex amplitudes in
${{\mathrm{D}}^{0}}\\!\rightarrow{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{{\mathrm{K}}^{\pm}}{{\uppi}^{\mp}}$ decays [4, 5]. Predictions are
available for the ${{\mathrm{K}}^{*+}}{{\mathrm{K}}^{-}}$,
${{\mathrm{K}}^{*-}}{{\mathrm{K}}^{+}}$,
${{\mathrm{K}}^{*0}}{{\overline{\mathrm{K}}{}}{}^{0}}$ and
${{\overline{\mathrm{K}}{}}{}^{*0}}{{\mathrm{K}}^{0}}$ complex amplitudes,
where ${\mathrm{K}}^{*}$ refers to the ${\mathrm{K}}^{*}({892})$ resonances.
There are therefore three relative amplitudes and two relative phases that can
be determined from the isobar models, with an additional relative phase
accessible if the isobar results are combined with the CLEO measurement of the
mean strong phase difference [12]. The results are summarized in Table 10.

The isobar model results are found to follow broadly the patterns predicted by
$\mathrm{SU}(3)$ flavor symmetry. The amplitude ratio between the
${\mathrm{K}}^{*}({892})^{{+}}$ and
${\overline{\mathrm{K}}{}}^{*}({892})^{{0}}$ resonances, which is derived from
the ${{\mathrm{D}}^{0}}\\!\rightarrow{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{{\mathrm{K}}^{-}}{{\uppi}^{+}}$ isobar model alone, shows good agreement.
The two other amplitude ratios additionally depend on the ratio ${\cal
B}_{{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}{\mathrm{K}}{\uppi}}$, and
these are more discrepant with the $\mathrm{SU}(3)$ predictions. The relative
phase between the charged and neutral ${\mathrm{K}}^{*}({892})$ resonances
shows better agreement with the flavor symmetry prediction in the
${{\mathrm{D}}^{0}}\\!\rightarrow{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{{\mathrm{K}}^{+}}{{\uppi}^{-}}$ mode, where both resonances have clear
peaks in the data. The GLASS and LASS isobar models are found to agree well,
suggesting the problems are not related to the ${\mathrm{K}}{\uppi}$ $S$-wave.

Table 10: $\mathrm{SU}(3)$ flavor symmetry predictions [5] and results. The
uncertainties on phase difference predictions are calculated from the quoted
magnitude and phase uncertainties. Note that some theoretical predictions
depend on the $\upeta$–$\upeta^{\prime}$ mixing angle
$\theta_{\upeta-\upeta^{\prime}}$ and are quoted for two different values. The
bottom entry in the table relies on the CLEO measurement [12] of the coherence
factor phase $\delta_{{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{\mathrm{K}}{\uppi}}$, and the uncertainty on this phase is included in the
statistical uncertainty, while the other entries are calculated directly from
the isobar models and relative branching ratio. Where two uncertainties are
quoted the first is statistical and the second systematic.

| Theory | Experiment
---|---|---
Ratio | $\theta_{\upeta-\upeta^{\prime}}=19.5^{\circ}$ | $\theta_{\upeta-\upeta^{\prime}}=11.7^{\circ}$ | GLASS | LASS
$\frac{|\mathcal{A}({\mathrm{K}}^{*}({892})^{{-}}{{\mathrm{K}}^{+}})|}{|\mathcal{A}({\mathrm{K}}^{*}({892})^{{+}}{{\mathrm{K}}^{-}})|}$ | 0.685$\,\pm\,$ | 0.032 | 0.685$\,\pm\,$ | 0.032 |  $0.582$$\,\pm\,$ | $0.007\pm 0.007$ |  $0.576$$\,\pm\,$ | $0.005\pm 0.010$
$\frac{|\mathcal{A}({\overline{\mathrm{K}}{}}^{*}({892})^{{0}}{{\mathrm{K}}^{0}})|}{|\mathcal{A}({\mathrm{K}}^{*}({892})^{{+}}{{\mathrm{K}}^{-}})|}$ | 0.138$\,\pm\,$ | 0.033 | 0.307$\,\pm\,$ | 0.035 |  $0.297$$\,\pm\,$ | $0.010\pm 0.024$ |  $0.295$$\,\pm\,$ | $0.009\pm 0.014$
$\frac{|\mathcal{A}({\mathrm{K}}^{*}({892})^{{0}}{{\overline{\mathrm{K}}{}}{}^{0}})|}{|\mathcal{A}({\mathrm{K}}^{*}({892})^{{+}}{{\mathrm{K}}^{-}})|}$ | 0.138$\,\pm\,$ | 0.033 | 0.307$\,\pm\,$ | 0.035 |  $0.333$$\,\pm\,$ | $0.008\pm 0.016$ |  $0.345$$\,\pm\,$ | $0.007\pm 0.010$
Argument | Theory (∘) | Experiment (∘)
$\frac{\mathcal{A}({\overline{\mathrm{K}}{}}^{*}({892})^{{0}}{{\mathrm{K}}^{0}})}{\mathcal{A}({\mathrm{K}}^{*}({892})^{{+}}{{\mathrm{K}}^{-}})}$ | 151$\,\pm\,$ | 14 | 112$\,\pm\,$ | 8 |  $72$$\,\pm\,$ | $2\pm 4$ |  $78.5$$\,\pm\,$ | $2.0\pm 2.8$
$\frac{\mathcal{A}({\mathrm{K}}^{*}({892})^{{0}}{{\overline{\mathrm{K}}{}}{}^{0}})}{\mathcal{A}({\mathrm{K}}^{*}({892})^{{-}}{{\mathrm{K}}^{+}})}$ |  $-9$$\,\pm\,$ | 13 |  $-37$$\,\pm\,$ | 6 |  $-4$$\,\pm\,$ | $2\pm 9$ |  $5.0$$\,\pm\,$ | $1.7\pm 1.4$
$\frac{\mathcal{A}({\mathrm{K}}^{*}({892})^{{0}}{{\overline{\mathrm{K}}{}}{}^{0}})}{\mathcal{A}({\overline{\mathrm{K}}{}}^{*}({892})^{{0}}{{\mathrm{K}}^{0}})}$ | 180 | 180 |  $-78$$\,\pm\,$ | $16\pm 10$ |  $-75$$\,\pm\,$ | $15\pm 2$

###  6.4 $C\\!P$ violation tests

Searches for time-integrated $C\\!P$-violating effects in the resonant
structure of these decays are performed using the best isobar models. The
resonance amplitude and phase parameters $a_{\mathrm{R}}$ and
$\phi_{\mathrm{R}}$ are substituted with $a_{\mathrm{R}}(1\pm\Delta
a_{\mathrm{R}})$ and $\phi_{\mathrm{R}}\pm\Delta\phi_{\mathrm{R}}$,
respectively, where the signs are set by the flavor tag. The convention
adopted is that a positive sign produces the ${\overline{\mathrm{D}}{}}{}^{0}$
complex amplitude. The full fit results are tabulated in Appendix D.

A subset of the $\Delta$ parameters is used to perform a $\chi^{2}$ test
against the no-$C\\!P$ violation hypothesis: only those parameters
corresponding to resonances that are present in the best isobar models using
both the GLASS and LASS ${\mathrm{K}}{\uppi}$ $S$-wave parameterizations are
included. The absolute difference
$|\Delta_{{\footnotesize\mbox{{GLASS}}}}-\Delta_{{\footnotesize\mbox{{LASS}}}}|$
is assigned as the systematic uncertainty due to dependence on the choice of
isobar model. This subset of parameters is shown in Table 11(b), where the
change in fit fraction between the ${\mathrm{D}}^{0}$ and
${\overline{\mathrm{D}}{}}{}^{0}$ solutions is included for illustrative
purposes. In the $\chi^{2}$ test the statistical and systematic uncertainties
are added in quadrature.

Using the best GLASS (LASS) isobar models the test result is
$\chi^{2}/\mathrm{ndf}=30.5/32=0.95$ ($32.3/32=1.01$), corresponding to a
$p$-value of 0.54 (0.45). Therefore, the data are compatible with the
hypothesis of $C\\!P$-conservation.

Table 11: $C\\!P$ violation fit results. Results are only shown for those
resonances that appear in both the GLASS and LASS models. The first
uncertainties are statistical and the second systematic; the only systematic
uncertainty is that due to the choice of isobar model.

(a) Results for the
${{\mathrm{D}}^{0}}\\!\rightarrow{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{{\mathrm{K}}^{-}}{{\uppi}^{+}}$ mode.

| $\Delta a_{\mathrm{R}}$ |  $\Delta\phi_{\mathrm{R}}$ $(^{\circ})$ |  $\Delta(\text{Fit fraction})$ [%]
---|---|---|---
Resonance | GLASS | LASS | GLASS | LASS | GLASS | LASS
${\mathrm{K}}^{*}({892})^{{+}}$ | 0.0 (fixed) | 0.0 (fixed) | 0.0 (fixed) | 0.0 (fixed) |  $0.6$$\,\pm\,$ | $1.0\pm 0.3$ |  $0.9$$\,\pm\,$ | $1.0\pm 0.3$
${\mathrm{K}}^{*}({1410})^{{+}}$ |  $0.07$$\,\pm\,$ | $0.06\pm 0.04$ |  $0.03$$\,\pm\,$ | $0.06\pm 0.04$ |  $3.9$$\,\pm\,$ | $3.5\pm 1.9$ |  $2.0$$\,\pm\,$ | $2.9\pm 1.9$ |  $1.4$$\,\pm\,$ | $0.8\pm 0.2$ |  $1.2$$\,\pm\,$ | $1.6\pm 0.2$
$({{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}{\uppi})^{+}_{S\text{-wave}}$ |  $0.02$$\,\pm\,$ | $0.08\pm 0.07$ |  $-0.05$$\,\pm\,$ | $0.08\pm 0.07$ |  $2.0$$\,\pm\,$ | $1.7\pm 0.0$ |  $2.0$$\,\pm\,$ | $1.7\pm 0.0$ |  $1$$\,\pm\,$ | $4\pm 3$ |  $-2.3$$\,\pm\,$ | $3.5\pm 3.3$
${\overline{\mathrm{K}}{}}^{*}({892})^{{0}}$ |  $-0.046$$\,\pm\,$ | $0.031\pm 0.005$ |  $-0.051$$\,\pm\,$ | $0.030\pm 0.005$ |  $1.2$$\,\pm\,$ | $1.6\pm 0.3$ |  $1.5$$\,\pm\,$ | $1.7\pm 0.3$ |  $-0.43$$\,\pm\,$ | $0.30\pm 0.03$ |  $-0.47$$\,\pm\,$ | $0.29\pm 0.03$
${\overline{\mathrm{K}}{}}^{*}({1410})^{{0}}$ |  $0.006$$\,\pm\,$ | $0.034\pm 0.017$ |  $0.02$$\,\pm\,$ | $0.04\pm 0.02$ |  $2$$\,\pm\,$ | $5\pm 5$ |  $-3$$\,\pm\,$ | $6\pm 5$ |  $0.3$$\,\pm\,$ | $1.0\pm 0.1$ |  $0.4$$\,\pm\,$ | $0.7\pm 0.1$
$({\mathrm{K}}{\uppi})^{0}_{S\text{-wave}}$ |  $0.05$$\,\pm\,$ | $0.04\pm 0.02$ |  $0.03$$\,\pm\,$ | $0.04\pm 0.02$ |  $0.4$$\,\pm\,$ | $1.6\pm 0.6$ |  $1.0$$\,\pm\,$ | $1.4\pm 0.6$ |  $2.2$$\,\pm\,$ | $1.3\pm 0.4$ |  $2.6$$\,\pm\,$ | $2.2\pm 0.4$
$\mathrm{a}_{2}(1320)^{{-}}$ |  $-0.25$$\,\pm\,$ | $0.14\pm 0.01$ |  $-0.24$$\,\pm\,$ | $0.13\pm 0.01$ |  $2$$\,\pm\,$ | $9\pm 3$ |  $-1$$\,\pm\,$ | $9\pm 3$ |  $-0.20$$\,\pm\,$ | $0.13\pm 0.05$ |  $-0.15$$\,\pm\,$ | $0.10\pm 0.05$
$\mathrm{a}_{0}(1450)^{{-}}$ |  $-0.01$$\,\pm\,$ | $0.14\pm 0.12$ |  $-0.13$$\,\pm\,$ | $0.14\pm 0.12$ |  $0$$\,\pm\,$ | $5\pm 4$ |  $-4$$\,\pm\,$ | $6\pm 4$ |  $-0.0$$\,\pm\,$ | $0.4\pm 0.4$ |  $-0.4$$\,\pm\,$ | $0.4\pm 0.4$
${\uprho}(1450)^{{-}}$ |  $0.06$$\,\pm\,$ | $0.13\pm 0.11$ |  $-0.05$$\,\pm\,$ | $0.12\pm 0.11$ |  $-13$$\,\pm\,$ | $10\pm 9$ |  $-5$$\,\pm\,$ | $9\pm 9$ |  $0.3$$\,\pm\,$ | $0.7\pm 0.6$ |  $-0.3$$\,\pm\,$ | $0.7\pm 0.6$

(b) Results for the
${{\mathrm{D}}^{0}}\\!\rightarrow{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{{\mathrm{K}}^{+}}{{\uppi}^{-}}$ mode.

| $\Delta a_{\mathrm{R}}$ |  $\Delta\phi_{\mathrm{R}}$ $(^{\circ})$ |  $\Delta(\text{Fit fraction})$ [%]
---|---|---|---
Resonance | GLASS | LASS | GLASS | LASS | GLASS | LASS
${\mathrm{K}}^{*}({892})^{{-}}$ | 0.0 (fixed) | 0.0 (fixed) | 0.0 (fixed) | 0.0 (fixed) |  $-1.1$$\,\pm\,$ | $0.7\pm 0.2$ |  $-0.9$$\,\pm\,$ | $0.7\pm 0.2$
${\mathrm{K}}^{*}({1410})^{{-}}$ |  $0.05$$\,\pm\,$ | $0.12\pm 0.08$ |  $-0.03$$\,\pm\,$ | $0.10\pm 0.08$ |  $-6$$\,\pm\,$ | $4\pm 3$ |  $-3.0$$\,\pm\,$ | $3.6\pm 2.8$ |  $0.6$$\,\pm\,$ | $2.7\pm 2.4$ |  $-2$$\,\pm\,$ | $4\pm 2$
$({{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}{\uppi})^{-}_{S\text{-wave}}$ |  $0.10$$\,\pm\,$ | $0.25\pm 0.24$ |  $-0.14$$\,\pm\,$ | $0.25\pm 0.24$ |  $-7.7$$\,\pm\,$ | $3.4\pm 0.0$ |  $-8$$\,\pm\,$ | $4\pm 0$ |  $2$$\,\pm\,$ | $6\pm 6$ |  $-4$$\,\pm\,$ | $6\pm 6$
${\mathrm{K}}^{*}({892})^{{0}}$ |  $-0.010$$\,\pm\,$ | $0.024\pm 0.001$ |  $-0.012$$\,\pm\,$ | $0.022\pm 0.001$ |  $-1.4$$\,\pm\,$ | $2.9\pm 2.2$ |  $0.8$$\,\pm\,$ | $2.8\pm 2.2$ |  $-0.4$$\,\pm\,$ | $0.4\pm 0.0$ |  $-0.4$$\,\pm\,$ | $0.4\pm 0.0$
${\mathrm{K}}^{*}({1410})^{{0}}$ |  $0.10$$\,\pm\,$ | $0.10\pm 0.09$ |  $0.19$$\,\pm\,$ | $0.13\pm 0.09$ |  $-1$$\,\pm\,$ | $9\pm 8$ |  $-9$$\,\pm\,$ | $9\pm 8$ |  $1.9$$\,\pm\,$ | $1.1\pm 0.2$ |  $1.6$$\,\pm\,$ | $0.8\pm 0.2$
$({\mathrm{K}}{\uppi})^{0}_{S\text{-wave}}$ |  $-0.07$$\,\pm\,$ | $0.06\pm 0.05$ |  $-0.12$$\,\pm\,$ | $0.06\pm 0.05$ |  $-2$$\,\pm\,$ | $4\pm 4$ |  $2$$\,\pm\,$ | $4\pm 4$ |  $-4$$\,\pm\,$ | $5\pm 5$ |  $-9$$\,\pm\,$ | $6\pm 5$
$\mathrm{a}_{0}(980)^{{+}}$ |  $0.06$$\,\pm\,$ | $0.04\pm 0.01$ |  $0.052$$\,\pm\,$ | $0.025\pm 0.008$ |  $-3$$\,\pm\,$ | $5\pm 2$ |  $-0.9$$\,\pm\,$ | $3.1\pm 2.2$ |  $2.2$$\,\pm\,$ | $2.8\pm 2.4$ |  $4.6$$\,\pm\,$ | $3.3\pm 2.4$
$\mathrm{a}_{0}(1450)^{{+}}$ |  $-0.11$$\,\pm\,$ | $0.10\pm 0.04$ |  $-0.07$$\,\pm\,$ | $0.07\pm 0.04$ |  $10$$\,\pm\,$ | $8\pm 5$ |  $5$$\,\pm\,$ | $6\pm 5$ |  $-0.21$$\,\pm\,$ | $0.30\pm 0.23$ |  $-0.4$$\,\pm\,$ | $0.4\pm 0.2$
${\uprho}(1700)^{{+}}$ |  $-0.03$$\,\pm\,$ | $0.13\pm 0.09$ |  $-0.12$$\,\pm\,$ | $0.13\pm 0.09$ |  $4$$\,\pm\,$ | $6\pm 2$ |  $2$$\,\pm\,$ | $5\pm 2$ |  $-0.07$$\,\pm\,$ | $0.25\pm 0.19$ |  $-0.27$$\,\pm\,$ | $0.27\pm 0.19$

##  7 Conclusions

The decay modes
${{\mathrm{D}}^{0}}\\!\rightarrow{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{{\mathrm{K}}^{\pm}}{{\uppi}^{\mp}}$ have been studied using unbinned,
time-integrated, fits to a high purity sample of 189 670 candidates, and two
amplitude models have been constructed for each decay mode. These models are
compared to data in a large number of bins in the relevant Dalitz plots and a
$\chi^{2}$ test indicates a good description of the data.

Models are presented using two different parameterizations of the
${\mathrm{K}}{\uppi}$ $S$-wave systems, which have been found to be an
important component of these decays. These systems are poorly understood, and
comparisons have been made to previous results and alternative
parameterizations, but the treatment of the ${\mathrm{K}}{\uppi}$ $S$-wave is
found to have little impact on the other results presented in this paper. The
large fractions attributed to the neutral ${\mathrm{K}}{\uppi}$ $S$-wave
channels could indicate larger than expected contributions from the penguin
annihilation diagrams shown in Fig. 1d.

The models are seen to favor small, but significant, contributions from the
${\uprho}(1450,1700)^{{\pm}}\\!\rightarrow{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{{\mathrm{K}}^{\pm}}$ resonances, modes which were seen by the OBELIX
experiment [46] but are not well established. All models contain clear
contributions from both the ${\mathrm{K}}^{*}({892})^{{\pm}}$ and
${\mathrm{K}}^{*}({892})^{{0}}$ resonances, with the
${\mathrm{K}}^{*}({892})^{{0}}$ contribution found to be suppressed as
expected from the diagrams shown in Fig. 1. This allows the full set of
amplitudes in these decays that are predicted by $\mathrm{SU}(3)$ flavor
symmetry to be tested, in contrast to the previous analysis by the CLEO
collaboration [12]. Partial agreement is found with these predictions.

The ratio of branching fractions between the two
${{\mathrm{D}}^{0}}\\!\rightarrow{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{{\mathrm{K}}^{\pm}}{{\uppi}^{\mp}}$ modes is also measured, both across
the full Dalitz plot area and in a restricted region near the
${\mathrm{K}}^{*}({892})^{{\pm}}$ resonance, with much improved precision
compared to previous results.

Values for the
${{\mathrm{D}}^{0}}\\!\rightarrow{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{\mathrm{K}}{\uppi}$ coherence factor are computed using the amplitude
models, again both for the whole Dalitz plot area and in the restricted
region, and are found to be in reasonable agreement with direct measurements
by CLEO [12] using quantum-correlated
${\uppsi(3770)}\\!\rightarrow{{\mathrm{D}}^{0}}{{\overline{\mathrm{D}}{}}{}^{0}}$
decays. The $C\\!P$-even fraction of the
${{\mathrm{D}}^{0}}\\!\rightarrow{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{\mathrm{K}}{\uppi}$ decays is also computed, using input from the quantum-
correlated decays, and is found to be in agreement with the direct measurement
[12, 53]. A search for time-integrated $C\\!P$ violation is carried out using
the amplitude models, but no evidence is found with either choice of
parameterization for the ${\mathrm{K}}{\uppi}$ $S$-wave.

The models presented here will be useful for future
${\mathrm{D}}^{0}$–${\overline{\mathrm{D}}{}}{}^{0}$ mixing, indirect $C\\!P$
violation and CKM angle $\gamma$ studies, where knowledge of the strong-phase
variation across the Dalitz plot can improve the attainable precision. These
improvements will be particularly valuable for studies of the large dataset
that is expected to be accumulated in Run 2 of the LHC.

## Acknowledgements

We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at the LHCb institutes. We acknowledge support from CERN
and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); NSFC
(China); CNRS/IN2P3 (France); BMBF, DFG and MPG (Germany); INFN (Italy); FOM
and NWO (The Netherlands); MNiSW and NCN (Poland); MEN/IFA (Romania); MinES
and FANO (Russia); MinECo (Spain); SNSF and SER (Switzerland); NASU (Ukraine);
STFC (United Kingdom); NSF (USA). We acknowledge the computing resources that
are provided by CERN, IN2P3 (France), KIT and DESY (Germany), INFN (Italy),
SURF (The Netherlands), PIC (Spain), GridPP (United Kingdom), RRCKI (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. We are also thankful for the computing
resources and the access to software R&D tools provided by Yandex LLC
(Russia). Individual groups or members have received support from AvH
Foundation (Germany), EPLANET, Marie Skłodowska-Curie Actions and ERC
(European Union), Conseil Général de Haute-Savoie, Labex ENIGMASS and OCEVU,
Région Auvergne (France), RFBR (Russia), XuntaGal and GENCAT (Spain), The
Royal Society and Royal Commission for the Exhibition of 1851 (United
Kingdom). We acknowledge the use of the Advanced Research Computing (ARC)
facilities (Oxford) in carrying out this work.

Appendices

##  Appendix A Additional isobar formalism information

This appendix contains Table 12, which summarizes the nominal values used for
the various resonance and form factor parameters.

Table 12: Nominal values for isobar model parameters that are fixed in the model fits, or used in constraint terms. These values are taken from Refs. [3, 37, 46, 47] as described in Sect. 4. Parameter | Value |
---|---|---
${\mathrm{K}}^{*}({892})^{{\pm}}$ | $m_{\mathrm{R}}$ | 891.66$\pm$ | 0.26 | ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$
$\Gamma_{\mathrm{R}}$ | 50.8$\pm$ | 0.9 | ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$
${\mathrm{K}}^{*}({1410})^{{\pm}}$ | $m_{\mathrm{R}}$ | 1.414$\pm$ | 0.015 | ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$
$\Gamma_{\mathrm{R}}$ | 0.232$\pm$ | 0.021 | ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$
$({{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}{\uppi})^{\pm}_{S\text{-wave}}$ | $m_{\mathrm{R}}$ | 1.435$\pm$ | 0.005 | ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$
$\Gamma_{\mathrm{R}}$ | 0.279$\pm$ | 0.006 | ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$
${\mathrm{K}}^{*}({892})^{{0}}$ | $m_{\mathrm{R}}$ | 895.94$\pm$ | 0.22 | ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$
$\Gamma_{\mathrm{R}}$ | 48.7$\pm$ | 0.8 | ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$
${\mathrm{K}}^{*}({1410})^{{0}}$ | $m_{\mathrm{R}}$ | 1.414$\pm$ | 0.015 | ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$
$\Gamma_{\mathrm{R}}$ | 0.232$\pm$ | 0.021 | ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$
${\mathrm{K}}^{*}_{{2}}({1430})^{{0}}$ | $m_{\mathrm{R}}$ | 1.4324$\pm$ | 0.0013 | ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$
$\Gamma_{\mathrm{R}}$ | 0.109$\pm$ | 0.005 | ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$
$({\mathrm{K}}{\uppi})^{0}_{S\text{-wave}}$ | $m_{\mathrm{R}}$ | 1.435$\pm$ | 0.005 | ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$
$\Gamma_{\mathrm{R}}$ | 0.279$\pm$ | 0.006 | ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$
${\mathrm{K}}{\uppi}$ $S$-wave | $r$ | 1.8$\pm$ | 0.4 | ${\,(\mathrm{Ge\kern-1.00006ptV\\!/}c)^{-1}}$
$a$ | 1.95$\pm$ | 0.09 | ${\,(\mathrm{Ge\kern-1.00006ptV\\!/}c)^{-1}}$
$\mathrm{a}_{0}(980)^{{\pm}}$ | $m_{\mathrm{R}}$ | 0.980$\pm$ | 0.020 | ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$
$g_{\upeta{\uppi}}$ | 324$\pm$ | 15 | $\mathrm{\,Me\kern-1.00006ptV}$
$\frac{g^{2}_{{\mathrm{K}}{\overline{\mathrm{K}}{}}}}{g^{2}_{\upeta{\uppi}}}$ | 1.03$\pm$ | 0.14 |
$\mathrm{a}_{2}(1320)^{{\pm}}$ | $m_{\mathrm{R}}$ | 1.3181$\pm$ | 0.0007 | ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$
$\Gamma_{\mathrm{R}}$ | 0.1098$\pm$ | 0.0024 | ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$
$\mathrm{a}_{0}(1450)^{{\pm}}$ | $m_{\mathrm{R}}$ | 1.474$\pm$ | 0.019 | ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$
$\Gamma_{\mathrm{R}}$ | 0.265$\pm$ | 0.013 | ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$
${\uprho}(1450)^{{\pm}}$ | $m_{\mathrm{R}}$ | 1.182$\pm$ | 0.030 | ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$
$\Gamma_{\mathrm{R}}$ | 0.389$\pm$ | 0.020 | ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$
${\uprho}(1700)^{{\pm}}$ | $m_{\mathrm{R}}$ | 1.594$\pm$ | 0.020 | ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$
$\Gamma_{\mathrm{R}}$ | 0.259$\pm$ | 0.020 | ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$

##  Appendix B Additional isobar model information

This appendix contains additional information about the various isobar model
parameters that are used and allowed to vary freely in the model fits, e.g.
resonance mass and width parameter values, and parameters of the GLASS and
LASS ${\mathrm{K}}{\uppi}$ $S$-wave functional forms.

Tables 13(b) and 14(b) summarize the most significant interference terms in
the ${{\mathrm{D}}^{0}}\\!\rightarrow{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{{\mathrm{K}}^{-}}{{\uppi}^{+}}$ and
${{\mathrm{D}}^{0}}\\!\rightarrow{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{{\mathrm{K}}^{+}}{{\uppi}^{-}}$ models, respectively. Table 15 defines the
matrices $\mathbf{U}$ used to define the LASS ${\mathrm{K}}{\uppi}$ $S$-wave
form factor. Tables 16 (GLASS) and 17 (LASS) summarize the various resonance
and form factor parameters. The nominal values that are used in Gaussian
constraint terms are given in Appendix A.

Figure 11 shows the smooth functions that describe the combinatorial
background in the isobar model fits. Figure 12 illustrates the two-dimensional
quality of fit achieved in the four isobar models and shows the binning scheme
used to derive $\chi^{2}/\mathrm{bin}$ values.

Table 13: Interference fractions for the
${{\mathrm{D}}^{0}}\\!\rightarrow{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{{\mathrm{K}}^{-}}{{\uppi}^{+}}$ mode. The first uncertainties are
statistical and the second systematic. Only the 25 largest terms are shown.

(a) Best GLASS isobar model. Resonances | Fit fraction [%]
---|---
$({\mathrm{K}}{\uppi})^{0}_{S\text{-wave}}$ $\times$ | ${\mathrm{K}}^{*}({892})^{{+}}$ |  $12$$\,\pm\,$ | $1\pm 4$
$({{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}{\uppi})^{+}_{S\text{-wave}}$ $\times$ | ${\overline{\mathrm{K}}{}}^{*}({1410})^{{0}}$ |  $10.4$$\,\pm\,$ | $0.7\pm 2.0$
${\overline{\mathrm{K}}{}}^{*}({1410})^{{0}}$ $\times$ | ${\mathrm{K}}^{*}({892})^{{+}}$ |  $-9.0$$\,\pm\,$ | $0.4\pm 1.0$
$({{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}{\uppi})^{+}_{S\text{-wave}}$ $\times$ | $({\mathrm{K}}{\uppi})^{0}_{S\text{-wave}}$ |  $-8$$\,\pm\,$ | $2\pm 11$
$({{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}{\uppi})^{+}_{S\text{-wave}}$ $\times$ | ${\overline{\mathrm{K}}{}}^{*}_{{2}}({1430})^{{0}}$ |  $-7.8$$\,\pm\,$ | $1.1\pm 1.6$
${\mathrm{K}}^{*}({1410})^{{+}}$ $\times$ | ${\overline{\mathrm{K}}{}}^{*}({1410})^{{0}}$ |  $-4.6$$\,\pm\,$ | $0.5\pm 3.4$
${\overline{\mathrm{K}}{}}^{*}_{{2}}({1430})^{{0}}$ $\times$ | ${\overline{\mathrm{K}}{}}^{*}({1410})^{{0}}$ |  $-4$$\,\pm\,$ | $1\pm 4$
$({{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}{\uppi})^{+}_{S\text{-wave}}$ $\times$ | ${\mathrm{K}}^{*}({892})^{{+}}$ |  $4.0$$\,\pm\,$ | $0.4\pm 2.4$
$({{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}{\uppi})^{+}_{S\text{-wave}}$ $\times$ | ${\mathrm{K}}^{*}({1410})^{{+}}$ |  $-4.0$$\,\pm\,$ | $0.5\pm 2.1$
${\mathrm{K}}^{*}({892})^{{+}}$ $\times$ | ${\uprho}(1450)^{{-}}$ |  $3.7$$\,\pm\,$ | $0.4\pm 1.0$
$({\mathrm{K}}{\uppi})^{0}_{S\text{-wave}}$ $\times$ | ${\overline{\mathrm{K}}{}}^{*}({1410})^{{0}}$ |  $-3.4$$\,\pm\,$ | $0.5\pm 3.5$
${\mathrm{K}}^{*}({892})^{{+}}$ $\times$ | $\mathrm{a}_{0}(1450)^{{-}}$ |  $3.2$$\,\pm\,$ | $0.3\pm 1.3$
${\overline{\mathrm{K}}{}}^{*}({1410})^{{0}}$ $\times$ | ${\overline{\mathrm{K}}{}}^{*}({892})^{{0}}$ |  $-2.7$$\,\pm\,$ | $0.2\pm 0.5$
${\overline{\mathrm{K}}{}}^{*}({1410})^{{0}}$ $\times$ | $\mathrm{a}_{0}(1450)^{{-}}$ |  $2.6$$\,\pm\,$ | $0.2\pm 0.8$
$({{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}{\uppi})^{+}_{S\text{-wave}}$ $\times$ | $\mathrm{a}_{0}(1450)^{{-}}$ |  $2.3$$\,\pm\,$ | $0.4\pm 1.6$
$({\mathrm{K}}{\uppi})^{0}_{S\text{-wave}}$ $\times$ | ${\uprho}(1450)^{{-}}$ |  $2.1$$\,\pm\,$ | $0.3\pm 1.2$
$({\mathrm{K}}{\uppi})^{0}_{S\text{-wave}}$ $\times$ | ${\mathrm{K}}^{*}({1410})^{{+}}$ |  $1.9$$\,\pm\,$ | $0.5\pm 2.9$
${\mathrm{K}}^{*}({1410})^{{+}}$ $\times$ | $\mathrm{a}_{0}(1450)^{{-}}$ |  $-1.8$$\,\pm\,$ | $0.2\pm 1.1$
${\mathrm{K}}^{*}({1410})^{{+}}$ $\times$ | ${\mathrm{K}}^{*}({892})^{{+}}$ |  $1.7$$\,\pm\,$ | $0.7\pm 3.5$
$({{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}{\uppi})^{+}_{S\text{-wave}}$ $\times$ | $\mathrm{a}_{2}(1320)^{{-}}$ |  $-1.6$$\,\pm\,$ | $0.3\pm 1.5$
${\mathrm{K}}^{*}({1410})^{{+}}$ $\times$ | ${\overline{\mathrm{K}}{}}^{*}({892})^{{0}}$ |  $1.36$$\,\pm\,$ | $0.11\pm 0.32$
${\overline{\mathrm{K}}{}}^{*}_{{2}}({1430})^{{0}}$ $\times$ | $\mathrm{a}_{0}(1450)^{{-}}$ |  $-1.3$$\,\pm\,$ | $0.2\pm 1.0$
${\overline{\mathrm{K}}{}}^{*}_{{2}}({1430})^{{0}}$ $\times$ | ${\mathrm{K}}^{*}({1410})^{{+}}$ |  $-1.3$$\,\pm\,$ | $0.2\pm 0.5$
${\mathrm{K}}^{*}({892})^{{+}}$ $\times$ | ${\overline{\mathrm{K}}{}}^{*}({892})^{{0}}$ |  $1.2$$\,\pm\,$ | $0.2\pm 0.4$
$({{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}{\uppi})^{+}_{S\text{-wave}}$ $\times$ | ${\uprho}(1450)^{{-}}$ |  $1.1$$\,\pm\,$ | $0.3\pm 0.9$

(b) Best LASS isobar model. Resonances | Fit fraction [%]
---|---
$({\mathrm{K}}{\uppi})^{0}_{S\text{-wave}}$ $\times$ | ${\mathrm{K}}^{*}({892})^{{+}}$ |  $11.7$$\,\pm\,$ | $0.5\pm 1.5$
$({\mathrm{K}}{\uppi})^{0}_{S\text{-wave}}$ $\times$ | ${\mathrm{K}}^{*}({1410})^{{+}}$ |  $-11.5$$\,\pm\,$ | $1.7\pm 3.4$
$({{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}{\uppi})^{+}_{S\text{-wave}}$ $\times$ | $({\mathrm{K}}{\uppi})^{0}_{S\text{-wave}}$ |  $-10.8$$\,\pm\,$ | $1.6\pm 3.4$
${\overline{\mathrm{K}}{}}^{*}({1410})^{{0}}$ $\times$ | ${\mathrm{K}}^{*}({892})^{{+}}$ |  $-5.7$$\,\pm\,$ | $0.4\pm 0.8$
$({{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}{\uppi})^{+}_{S\text{-wave}}$ $\times$ | ${\mathrm{K}}^{*}({892})^{{+}}$ |  $5.6$$\,\pm\,$ | $0.4\pm 1.2$
$({{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}{\uppi})^{+}_{S\text{-wave}}$ $\times$ | ${\overline{\mathrm{K}}{}}^{*}({1410})^{{0}}$ |  $5.2$$\,\pm\,$ | $0.5\pm 0.9$
$({\mathrm{K}}{\uppi})^{0}_{S\text{-wave}}$ $\times$ | $\mathrm{a}_{0}(980)^{{-}}$ |  $-5.1$$\,\pm\,$ | $0.8\pm 0.8$
${\mathrm{K}}^{*}({892})^{{+}}$ $\times$ | ${\uprho}(1450)^{{-}}$ |  $4.0$$\,\pm\,$ | $0.3\pm 1.2$
${\mathrm{K}}^{*}({1410})^{{+}}$ $\times$ | ${\overline{\mathrm{K}}{}}^{*}({1410})^{{0}}$ |  $-3.7$$\,\pm\,$ | $0.4\pm 2.0$
$({\mathrm{K}}{\uppi})^{0}_{S\text{-wave}}$ $\times$ | ${\uprho}(1450)^{{-}}$ |  $3.1$$\,\pm\,$ | $0.4\pm 0.9$
${\mathrm{K}}^{*}({892})^{{+}}$ $\times$ | $\mathrm{a}_{0}(1450)^{{-}}$ |  $2.7$$\,\pm\,$ | $0.3\pm 0.5$
$({{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}{\uppi})^{+}_{S\text{-wave}}$ $\times$ | ${\uprho}(1450)^{{-}}$ |  $2.5$$\,\pm\,$ | $0.2\pm 0.8$
$({{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}{\uppi})^{+}_{S\text{-wave}}$ $\times$ | ${\mathrm{K}}^{*}({1410})^{{+}}$ |  $-2.5$$\,\pm\,$ | $0.4\pm 0.9$
${\mathrm{K}}^{*}({1410})^{{+}}$ $\times$ | $\mathrm{a}_{0}(1450)^{{-}}$ |  $-2.1$$\,\pm\,$ | $0.2\pm 0.9$
${\mathrm{K}}^{*}({1410})^{{+}}$ $\times$ | ${\overline{\mathrm{K}}{}}^{*}({892})^{{0}}$ |  $1.91$$\,\pm\,$ | $0.12\pm 0.14$
${\mathrm{K}}^{*}({1410})^{{+}}$ $\times$ | $\mathrm{a}_{0}(980)^{{-}}$ |  $-1.76$$\,\pm\,$ | $0.27\pm 0.28$
$({{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}{\uppi})^{+}_{S\text{-wave}}$ $\times$ | ${\overline{\mathrm{K}}{}}^{*}({892})^{{0}}$ |  $-1.75$$\,\pm\,$ | $0.10\pm 0.19$
$({{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}{\uppi})^{+}_{S\text{-wave}}$ $\times$ | $\mathrm{a}_{0}(1450)^{{-}}$ |  $1.7$$\,\pm\,$ | $0.5\pm 1.5$
${\overline{\mathrm{K}}{}}^{*}({892})^{{0}}$ $\times$ | $\mathrm{a}_{0}(980)^{{-}}$ |  $-1.68$$\,\pm\,$ | $0.14\pm 0.29$
${\overline{\mathrm{K}}{}}^{*}({1410})^{{0}}$ $\times$ | ${\overline{\mathrm{K}}{}}^{*}({892})^{{0}}$ |  $-1.7$$\,\pm\,$ | $0.1\pm 0.5$
${\overline{\mathrm{K}}{}}^{*}({1410})^{{0}}$ $\times$ | $\mathrm{a}_{0}(980)^{{-}}$ |  $1.49$$\,\pm\,$ | $0.16\pm 0.21$
${\mathrm{K}}^{*}({1410})^{{+}}$ $\times$ | ${\mathrm{K}}^{*}({892})^{{+}}$ |  $-1.5$$\,\pm\,$ | $0.9\pm 1.2$
$({\mathrm{K}}{\uppi})^{0}_{S\text{-wave}}$ $\times$ | $\mathrm{a}_{0}(1450)^{{-}}$ |  $1.3$$\,\pm\,$ | $0.8\pm 1.8$
$({{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}{\uppi})^{+}_{S\text{-wave}}$ $\times$ | $\mathrm{a}_{2}(1320)^{{-}}$ |  $-1.2$$\,\pm\,$ | $0.3\pm 0.8$
$({\mathrm{K}}{\uppi})^{0}_{S\text{-wave}}$ $\times$ | ${\overline{\mathrm{K}}{}}^{*}({892})^{{0}}$ |  $1.21$$\,\pm\,$ | $0.07\pm 0.19$

Table 14: Interference fractions for the
${{\mathrm{D}}^{0}}\\!\rightarrow{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{{\mathrm{K}}^{+}}{{\uppi}^{-}}$ mode. The first uncertainties are
statistical and the second systematic. Only the 25 largest terms are shown.

(a) Best GLASS isobar model. Resonances | Fit fraction [%]
---|---
${\mathrm{K}}^{*}_{{2}}({1430})^{{0}}$ $\times$ | $\mathrm{a}_{0}(980)^{{+}}$ |  $10.3$$\,\pm\,$ | $0.7\pm 3.5$
$({{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}{\uppi})^{-}_{S\text{-wave}}$ $\times$ | $({\mathrm{K}}{\uppi})^{0}_{S\text{-wave}}$ |  $6$$\,\pm\,$ | $1\pm 5$
${\mathrm{K}}^{*}({892})^{{-}}$ $\times$ | $\mathrm{a}_{0}(980)^{{+}}$ |  $-5$$\,\pm\,$ | $1\pm 4$
${\mathrm{K}}^{*}({1410})^{{0}}$ $\times$ | ${\mathrm{K}}^{*}({892})^{{-}}$ |  $-5.0$$\,\pm\,$ | $0.3\pm 1.0$
$({\mathrm{K}}{\uppi})^{0}_{S\text{-wave}}$ $\times$ | ${\mathrm{K}}^{*}({892})^{{-}}$ |  $5$$\,\pm\,$ | $1\pm 4$
${\mathrm{K}}^{*}_{{2}}({1430})^{{0}}$ $\times$ | ${\mathrm{K}}^{*}({1410})^{{0}}$ |  $-4.1$$\,\pm\,$ | $0.7\pm 2.2$
${\mathrm{K}}^{*}({1410})^{{0}}$ $\times$ | $\mathrm{a}_{0}(980)^{{+}}$ |  $3.8$$\,\pm\,$ | $0.2\pm 1.0$
$({{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}{\uppi})^{-}_{S\text{-wave}}$ $\times$ | ${\mathrm{K}}^{*}_{{2}}({1430})^{{0}}$ |  $4$$\,\pm\,$ | $1\pm 7$
${\mathrm{K}}^{*}({892})^{{-}}$ $\times$ | ${\mathrm{K}}^{*}({892})^{{0}}$ |  $3.61$$\,\pm\,$ | $0.10\pm 0.32$
$({\mathrm{K}}{\uppi})^{0}_{S\text{-wave}}$ $\times$ | ${\uprho}(1450)^{{+}}$ |  $3.4$$\,\pm\,$ | $0.6\pm 1.4$
${\mathrm{K}}^{*}({1410})^{{-}}$ $\times$ | ${\mathrm{K}}^{*}({892})^{{-}}$ |  $-3.4$$\,\pm\,$ | $0.4\pm 0.6$
${\mathrm{K}}^{*}_{{2}}({1430})^{{0}}$ $\times$ | ${\mathrm{K}}^{*}({892})^{{-}}$ |  $3.2$$\,\pm\,$ | $0.4\pm 1.3$
$({\mathrm{K}}{\uppi})^{0}_{S\text{-wave}}$ $\times$ | ${\mathrm{K}}^{*}_{{2}}({1430})^{{0}}$ |  $-3.1$$\,\pm\,$ | $1.2\pm 1.7$
${\mathrm{K}}^{*}_{{2}}({1430})^{{0}}$ $\times$ | ${\uprho}(1450)^{{+}}$ |  $-2.6$$\,\pm\,$ | $0.5\pm 1.6$
${\mathrm{K}}^{*}({1410})^{{0}}$ $\times$ | ${\uprho}(1450)^{{+}}$ |  $2.3$$\,\pm\,$ | $0.4\pm 0.8$
$({{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}{\uppi})^{-}_{S\text{-wave}}$ $\times$ | ${\mathrm{K}}^{*}({892})^{{-}}$ |  $1.9$$\,\pm\,$ | $0.2\pm 1.3$
$({{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}{\uppi})^{-}_{S\text{-wave}}$ $\times$ | ${\mathrm{K}}^{*}({1410})^{{0}}$ |  $-1.9$$\,\pm\,$ | $0.6\pm 2.6$
${\mathrm{K}}^{*}({892})^{{0}}$ $\times$ | $\mathrm{a}_{0}(980)^{{+}}$ |  $-1.8$$\,\pm\,$ | $0.3\pm 1.8$
$\mathrm{a}_{0}(1450)^{{+}}$ $\times$ | $\mathrm{a}_{0}(980)^{{+}}$ |  $1.7$$\,\pm\,$ | $0.4\pm 0.8$
$({{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}{\uppi})^{-}_{S\text{-wave}}$ $\times$ | ${\mathrm{K}}^{*}({1410})^{{-}}$ |  $1.7$$\,\pm\,$ | $0.3\pm 1.1$
$({\mathrm{K}}{\uppi})^{0}_{S\text{-wave}}$ $\times$ | $\mathrm{a}_{0}(980)^{{+}}$ |  $-1$$\,\pm\,$ | $1\pm 4$
$({{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}{\uppi})^{-}_{S\text{-wave}}$ $\times$ | ${\uprho}(1450)^{{+}}$ |  $-1.4$$\,\pm\,$ | $0.2\pm 0.4$
${\mathrm{K}}^{*}({1410})^{{0}}$ $\times$ | ${\mathrm{K}}^{*}({892})^{{0}}$ |  $-1.3$$\,\pm\,$ | $0.2\pm 0.8$
$({{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}{\uppi})^{-}_{S\text{-wave}}$ $\times$ | $\mathrm{a}_{0}(1450)^{{+}}$ |  $1.33$$\,\pm\,$ | $0.17\pm 0.29$
${\mathrm{K}}^{*}_{{2}}({1430})^{{0}}$ $\times$ | ${\uprho}(1700)^{{+}}$ |  $-1.3$$\,\pm\,$ | $0.2\pm 1.2$

(b) Best LASS isobar model. Resonances | Fit fraction [%]
---|---
$({{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}{\uppi})^{-}_{S\text{-wave}}$ $\times$ | $\mathrm{a}_{0}(980)^{{+}}$ |  $11.5$$\,\pm\,$ | $0.8\pm 1.8$
$({\mathrm{K}}{\uppi})^{0}_{S\text{-wave}}$ $\times$ | ${\mathrm{K}}^{*}({1410})^{{-}}$ |  $-11$$\,\pm\,$ | $2\pm 5$
${\mathrm{K}}^{*}({892})^{{-}}$ $\times$ | $\mathrm{a}_{0}(980)^{{+}}$ |  $9.6$$\,\pm\,$ | $0.7\pm 2.2$
${\mathrm{K}}^{*}({1410})^{{-}}$ $\times$ | ${\mathrm{K}}^{*}({892})^{{-}}$ |  $-9.4$$\,\pm\,$ | $0.7\pm 1.0$
$\mathrm{a}_{0}(1450)^{{+}}$ $\times$ | $\mathrm{a}_{0}(980)^{{+}}$ |  $7.0$$\,\pm\,$ | $0.6\pm 1.1$
$({\mathrm{K}}{\uppi})^{0}_{S\text{-wave}}$ $\times$ | $\mathrm{a}_{0}(980)^{{+}}$ |  $-5.9$$\,\pm\,$ | $1.2\pm 2.6$
$({{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}{\uppi})^{-}_{S\text{-wave}}$ $\times$ | $({\mathrm{K}}{\uppi})^{0}_{S\text{-wave}}$ |  $-5$$\,\pm\,$ | $2\pm 5$
$({\mathrm{K}}{\uppi})^{0}_{S\text{-wave}}$ $\times$ | $\mathrm{a}_{0}(1450)^{{+}}$ |  $-5.0$$\,\pm\,$ | $0.6\pm 1.3$
$({{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}{\uppi})^{-}_{S\text{-wave}}$ $\times$ | ${\mathrm{K}}^{*}({1410})^{{-}}$ |  $4.4$$\,\pm\,$ | $0.5\pm 0.9$
${\mathrm{K}}^{*}({892})^{{-}}$ $\times$ | ${\mathrm{K}}^{*}({892})^{{0}}$ |  $3.73$$\,\pm\,$ | $0.08\pm 0.30$
${\mathrm{K}}^{*}({892})^{{0}}$ $\times$ | $\mathrm{a}_{0}(980)^{{+}}$ |  $3.7$$\,\pm\,$ | $0.3\pm 1.0$
${\mathrm{K}}^{*}({1410})^{{-}}$ $\times$ | $\mathrm{a}_{0}(980)^{{+}}$ |  $-3.6$$\,\pm\,$ | $0.4\pm 1.0$
${\mathrm{K}}^{*}({1410})^{{0}}$ $\times$ | ${\mathrm{K}}^{*}({892})^{{-}}$ |  $-3.5$$\,\pm\,$ | $0.5\pm 0.9$
$({\mathrm{K}}{\uppi})^{0}_{S\text{-wave}}$ $\times$ | ${\mathrm{K}}^{*}({1410})^{{0}}$ |  $-2.8$$\,\pm\,$ | $0.3\pm 1.1$
$({{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}{\uppi})^{-}_{S\text{-wave}}$ $\times$ | $\mathrm{a}_{0}(1450)^{{+}}$ |  $2.6$$\,\pm\,$ | $0.5\pm 1.6$
$({\mathrm{K}}{\uppi})^{0}_{S\text{-wave}}$ $\times$ | ${\mathrm{K}}^{*}({892})^{{0}}$ |  $-2.0$$\,\pm\,$ | $0.1\pm 0.4$
${\mathrm{K}}^{*}({1410})^{{-}}$ $\times$ | $\mathrm{a}_{0}(1450)^{{+}}$ |  $1.9$$\,\pm\,$ | $0.4\pm 0.6$
$({{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}{\uppi})^{-}_{S\text{-wave}}$ $\times$ | ${\mathrm{K}}^{*}({892})^{{0}}$ |  $-1.86$$\,\pm\,$ | $0.13\pm 0.18$
${\mathrm{K}}^{*}({892})^{{-}}$ $\times$ | $\mathrm{a}_{0}(1450)^{{+}}$ |  $1.6$$\,\pm\,$ | $0.3\pm 0.4$
$({{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}{\uppi})^{-}_{S\text{-wave}}$ $\times$ | ${\mathrm{K}}^{*}({892})^{{-}}$ |  $1.5$$\,\pm\,$ | $0.3\pm 1.3$
${\mathrm{K}}^{*}({1410})^{{-}}$ $\times$ | ${\uprho}(1700)^{{+}}$ |  $1.5$$\,\pm\,$ | $0.2\pm 0.5$
${\mathrm{K}}^{*}({892})^{{-}}$ $\times$ | ${\uprho}(1700)^{{+}}$ |  $-1.45$$\,\pm\,$ | $0.16\pm 0.34$
${\mathrm{K}}^{*}({1410})^{{-}}$ $\times$ | ${\mathrm{K}}^{*}({892})^{{0}}$ |  $-1.4$$\,\pm\,$ | $0.2\pm 0.5$
${\mathrm{K}}^{*}({1410})^{{-}}$ $\times$ | ${\mathrm{K}}^{*}({1410})^{{0}}$ |  $1.4$$\,\pm\,$ | $0.3\pm 0.6$
${\mathrm{K}}^{*}({1410})^{{0}}$ $\times$ | ${\mathrm{K}}^{*}({892})^{{0}}$ |  $-1.4$$\,\pm\,$ | $0.2\pm 0.7$

Table 15: Matrices $\mathbf{U}$ relating the fit coordinates $\mathbf{b^{\prime}}$ to the LASS form factor coordinates $\mathbf{b}=\mathbf{Ub^{\prime}}$ defined in Sect. 4. $({{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}{\uppi})^{\pm}_{S\text{-wave}}$ | $({\mathrm{K}}{\uppi})^{0}_{S\text{-wave}}$
---|---
$\begin{pmatrix}[r]-0.460&0.702&-0.543\\\ 0.776&0.197&-0.631\\\ -0.433&-0.711&-0.554\\\ \end{pmatrix}$ | $\begin{pmatrix}[r]-0.452&-0.676\phantom{0}&-0.582\\\ 0.776&0.0243&-0.631\\\ -0.440&0.737\phantom{0}&-0.513\\\ \end{pmatrix}$
|
---|---
Figure 11: Smooth functions, $c_{{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{{\mathrm{K}}^{\pm}}{{\uppi}^{\mp}}}(m^{2}_{{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{\mathrm{K}}},m^{2}_{{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{\uppi}})$, used to describe the combinatorial background component in the
${{\mathrm{D}}^{0}}\\!\rightarrow{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{{\mathrm{K}}^{-}}{{\uppi}^{+}}$ (left) and
${{\mathrm{D}}^{0}}\\!\rightarrow{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle
S}}{{\mathrm{K}}^{+}}{{\uppi}^{-}}$ (right) amplitude model fits.

Figure 12: Two-dimensional quality-of-fit distributions illustrating the dynamic binning scheme used to evaluate $\chi^{2}$. The variable shown is $\frac{d_{i}-p_{i}}{\sqrt{p_{i}}}$ where $d_{i}$ and $p_{i}$ are the number of events and the fitted value, respectively, in bin $i$. The ${{\mathrm{D}}^{0}}\\!\rightarrow{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}{{\mathrm{K}}^{-}}{{\uppi}^{+}}$ (${{\mathrm{D}}^{0}}\\!\rightarrow{{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}{{\mathrm{K}}^{+}}{{\uppi}^{-}}$) mode is shown in the left (right) column, and the GLASS (LASS) isobar models are shown in the top (bottom) row. Table 16: Additional fit parameters for GLASS models. This table does not include parameters that are fixed to their nominal values. The first uncertainties are statistical and the second systematic. Parameter | Value |
---|---|---
${\mathrm{K}}^{*}({892})^{{\pm}}$ | $m_{\mathrm{R}}$ |  $893.1$$\,\pm\,$ | $0.1\pm 0.9$ | ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$
$\Gamma_{\mathrm{R}}$ |  $46.9$$\,\pm\,$ | $0.3\pm 2.5$ | ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$
${\mathrm{K}}^{*}({1410})^{{\pm}}$ | $\Gamma_{\mathrm{R}}$ |  $210$$\,\pm\,$ | $20\pm 60$ | ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$
$({{\mathrm{K}}^{0}_{\rm\scriptscriptstyle S}}{\uppi})^{\pm}_{S\text{-wave}}$ | $F$ | 1.785 (fixed) |
$a$ |  $4.7$$\,\pm\,$ | $0.4\pm 1.0$ | ${\,(\mathrm{Ge\kern-1.00006ptV\\!/}c)^{-1}}$
$\phi_{F}$ |  $0.28$$\,\pm\,$ | $0.05\pm 0.19$ | $\rm\,rad$
$\phi_{S}$ |  $2.8$$\,\pm\,$ | $0.2\pm 0.5$ | $\rm\,rad$
$r$ |  $-5.3$$\,\pm\,$ | $0.4\pm 1.9$ | ${\,(\mathrm{Ge\kern-1.00006ptV\\!/}c)^{-1}}$
${\mathrm{K}}^{*}({1410})^{{0}}$ | $m_{\mathrm{R}}$ |  $1426$$\,\pm\,$ | $8\pm 24$ | ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$
$\Gamma_{\mathrm{R}}$ |  $270$$\,\pm\,$ | $20\pm 40$ | ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$
$({\mathrm{K}}{\uppi})^{0}_{S\text{-wave}}$ | $F$ |  $0.15$$\,\pm\,$ | $0.03\pm 0.14$ |
$a$ |  $4.2$$\,\pm\,$ | $0.3\pm 2.8$ | ${\,(\mathrm{Ge\kern-1.00006ptV\\!/}c)^{-1}}$
$\phi_{F}$ |  $-2.5$$\,\pm\,$ | $0.2\pm 1.0$ | $\rm\,rad$
$\phi_{S}$ |  $-1.1$$\,\pm\,$ | $0.6\pm 1.3$ | $\rm\,rad$
$r$ |  $-3.0$$\,\pm\,$ | $0.4\pm 1.7$ | ${\,(\mathrm{Ge\kern-1.00006ptV\\!/}c)^{-1}}$
$\mathrm{a}_{0}(1450)^{{\pm}}$ | $m_{\mathrm{R}}$ |  $1430$$\,\pm\,$ | $10\pm 40$ | ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$
${\uprho}(1450)^{{\pm}}$ | $\Gamma_{\mathrm{R}}$ |  $410$$\,\pm\,$ | $19\pm 35$ | ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$
${\uprho}(1700)^{{\pm}}$ | $m_{\mathrm{R}}$ |  $1530$$\,\pm\,$ | $10\pm 40$ | ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$