††thanks: Retired Professor

# Dalitz plot studies of $D^{0}\to K^{0}_{S}K^{+}K^{-}$ decays in a
factorization approach

J.-P. Dedonder1∗ R. Kamiński2 L. Leśniak2 B. Loiseau1

###### Abstract

The BABAR Collaboration data of the $D^{0}\to K^{0}_{S}K^{+}K^{-}$ process are
analyzed within a quasi-two-body factorization framework. Starting from the
weak effective Hamiltonian, one has to evaluate matrix elements of $D^{0}$
transitions to two kaons for the tree amplitudes and the transitions between
one kaon and two kaons for the annihilation ones ($W$-exchange). In earlier
studies, assuming these transitions to proceed through the dominant
intermediate resonances, we approximated them as being proportional to the
kaon form factors. Here, to obtain a good fit, one has to multiply the scalar-
kaon form factors, derived from unitary relativistic coupled-channel models or
in a dispersion relation approach, by phenomenological energy-dependent
functions. The final state kaon-kaon interactions in the $S$-, $P$\- and $D$\-
waves are taken into account. All $S$-wave channels are treated in a unitary
way. In other respects, it is shown in a model-independent manner that the
$K^{+}K^{-}$ and $\bar{K}^{0}K^{+}$ $S$-wave effective mass squared
distributions, corrected for phase space, are significantly different. At
variance with the BABAR analysis, it means that the $f_{0}(980)$ resonance
must be included in the phenomenological analysis of the $D^{0}\to
K^{0}_{S}K^{+}K^{-}$ data. The best fit described in the main text has 19 free
parameters and indicates i) the dominance of annihilation amplitudes, ii) a
large dominance of the $f_{0}(980)$ meson in the near threshold $K^{+}K^{-}$
invariant mass distribution, and iii) a sizable branching fraction to the
$[\rho(770)^{+}+\rho(1450)^{+}+\rho(1700)^{+}]K_{S}^{0}$ final states. A first
appendix provides an update of the determination of the isoscalar-scalar
meson-meson amplitudes based on an enlarged set of data embodying new precise
low energy $\pi\pi$ data. A second appendix proposes two alternative fits
using the scalar-kaon form factors calculated from the Muskhelishvili-Omnès
dispersion relation approach. These fits have $\chi^{2}$ quite close to that
of the best fit but they show important contributions from both the $f_{0}$
and $a_{0}^{0}$ mesons and a weaker role of the $\rho^{+}$ mesons.

###### pacs:

13.25.Hw, 13.75.Lb

##  I Introduction

Measurements of the $D^{0}$-$\overline{D}^{0}$ mixing parameters, through
Dalitz-plot time dependent amplitude analyses of the weak process $D^{0}\to
K^{0}_{S}K^{+}K^{-}$, have been performed by the Belle Zupanc2009  and BABAR
collaborations B10 . Such studies could help in the understanding of the
origin of mixing and may indicate the presence of new physics contribution. As
predicted by the standard model in the charm sector, the violation of the CP
symmetry should be small for these $D^{0}$ decays. In Refs. Zupanc2009 ; B10
the description of the $D^{0}\to K^{0}_{S}K^{+}K^{-}$ decay amplitude has been
performed using the isobar model developed in B5 , extended in Aubert2008  and
B10 ; supmaPRL105 . The isobar model has also been applied in the experimental
analysis based on the data taken from the BESIII experiment Weidenkaff ;
Ablikim2006.02800 .

The Cabibbo-Kobayashi-Maskawa (CKM) angle $\gamma$ (or $\phi_{3}$) has been
evaluated from the analyses of the $B^{\pm}\to D^{0}K^{\pm}$, with $D^{0}\to
K^{0}_{S}\pi^{+}\pi^{-}$ and $D^{0}\to K^{0}_{S}K^{+}K^{-}$decays
SanchezPRL105121801 ; Poluektov2010 ; J.P.Lees_PRD87_052015_BABAR ; LHCbNP2014
. This angle can be also measured using some knowledge on the strong-phase
difference between $D^{0}$ and $\bar{D}^{0}\to K^{0}_{S}K^{+}K^{-}$ decay
amplitudes obtained by the CLEO Collaboration J.Libby_PRD82_CLEO . This method
has been used by the Belle H.Aihara_Belle2012 , LHCb R.Aaij_JHEP2018  and
BESIII BES2007  collaborations.

A good knowledge of the final state meson interactions in the $D^{0}\to
K^{0}_{S}K^{+}K^{-}$ decays is important to reduce the uncertainties in the
determination of the $D^{0}$-$\overline{D}^{0}$ mixing parameters and of the
CKM angle $\gamma$. The structures seen in the Dalitz plot spectra point out
to the complexity of these final state strong interactions. Their studies can
provide a better understanding of the strange meson interactions and of the
$D^{0}$ decay mechanism into $K^{0}_{S}K^{+}K^{-}$.

The experimental analyses like that of Ref. B10  rely mainly upon the use of
the isobar model. For a given reaction, this model has basically two fitted
parameters for each part of the decay amplitude. In this approach one can take
into account many existing resonances coupled to the interacting pairs of
mesons. In Refs. B10 ; supmaPRL105 , the authors introduce explicitly eight
resonances $a_{0}(980)^{0}$ , $a_{0}(980)^{+}$ , $a_{0}(980)^{-}$ ,
$\phi(1020)$ , $f_{2}(1270)$ , $f_{0}(1370)$ , $a_{0}(1450)^{0}$ ,
$a_{0}(1450)^{+}$ . Their analysis rely on 17 free parameters. However, the
decay amplitudes are not unitary and unitarity is not preserved in the three-
body decay channels; it is also violated in the two-body subchannels.
Furthermore, it is difficult to differentiate the $S$-wave amplitudes from the
nonresonant background terms. Their interferences are noteworthy and then some
two-body branching fractions, extracted from the data, could be unreliable.
One of the difficulty in the experimental analyses based on the isobar model
is the choice of the resonances needed to reach a good agreement with the
Dalitz plot data. In Ref. Aubert2008  the BABAR collaboration authors have
added the scalar $a_{0}(1450)$ to their model developed in 2005 B5 . In the
recent BESIII analysis Ablikim2006.02800  the Dalitz plot is described with
six resonances: $a_{0}(980)^{0}$ , $a_{0}(980)^{+}$ , $\phi(1020)$ ,
$a_{2}(1320)^{+}$ , $a_{2}(1320)^{-}$ , $a_{0}(1450)^{-}$.

Extending our previous work on the $D^{0}\to K^{0}_{S}\pi^{+}\pi^{-}$ decays
JPD_PRD89 , we analyze, in the quasi-two-body factorization framework, the
$D^{0}\to K^{0}_{S}K^{+}K^{-}$ data provided by the BABAR collaboration B10 .
As in our earlier studies, we assume that two of the three final-state mesons
form a single state which originates from a quark-antiquark, $q\bar{q}$, pair
and then apply the factorization procedure to these quasi-two-body final
states. Starting from the weak effective Hamiltonian, we derive tree and
annihilation ($W$-meson exchange) amplitudes both being either Cabibbo favored
(CF) with $c\to s\bar{d}u$ transition or doubly Cabibbo suppressed (DCS) with
$c\to du\bar{s}$ transition.

In the factorization approach, the CF and DCS amplitudes are expressed as
superpositions of appropriate effective coefficients and two products of two
transition matrix elements. The kaon form factors do not appear explicitly
except the isovector ones that enter in only one term of the CF tree
amplitude111 See the $a_{1}(m_{c})$ term of Eq. (6).. In all other terms of
our amplitudes, one has to evaluate either, for the tree ones, the matrix
elements of $D^{0}$ transitions to two-kaon states or, for the annihilation
ones, the transitions between one kaon and two kaon-states. Similarly to
previous studies Boito , assuming these transitions to proceed through the
dominant intermediate resonances, we have approximated them as being
proportional to the isoscalar or isovector kaon form factors.

Taking advantage of the coupling between the $\pi\pi$ and the $K\overline{K}$
channels and extending the derivation of the unitary isoscalar-scalar pion
form factor DedonderPol  to that of the kaon one, two of the present authors
(L. L. and R. K.) together with two collaborators, have recently studied, in
the quasi-two-body QCD factorization approach, the $B^{\pm}\to
K^{+}K^{-}K^{\pm}$ decays PLB699_102 ; KKK . These $S$-wave form factors are
derived using a unitary relativistic three coupled-channel model including
$\pi\pi$, $K\bar{K}$ and effective $(2\pi)(2\pi)$ interactions together with
chiral symmetry constraints. They include the contributions of the
$f_{0}(980)$ and $f_{0}(1400)$ resonances and require the knowledge of the
isoscalar-scalar meson-meson amplitudes from the two kaon threshold to
energies above the $D^{0}$ mass. The parameters of these amplitudes derived in
the late nineties by three of us (R. K., L. L. and B. L.) KLL ; EPJ  have been
updated using new precise low energy $\pi\pi$ data together with an enlarged
set of data as is shown in Appendix A.

The calculation requires also the knowledge of a contribution proportional to
the isovector-scalar form factor; it is represented by a function calculated
from a unitary model with relativistic two-coupled channel $\pi\eta$ and
$K\bar{K}$ equations based on the two-channel model of the $a_{0}(980)$ and
$a_{0}(1450)$ resonances built in AFLL ; AFLL2 .

The vector form factors have been calculated using vector dominance, quark
model assumptions and isospin symmetry in Ref. Bruch2005 . They receive
contributions from the vector mesons: $\rho(770)$, $\rho(1450)$, $\rho(1700)$,
$\omega(782)$, $\omega(1420)$, $\omega(1850)$, $\phi(1020)$ and $\phi(1680)$.
The isoscalar-tensor amplitude, saturated by the $f_{2}(1270)$, is described
by a relativistic Breit-Wigner term. The isovector-tensor resonance
$a_{2}(1320)$ has a mass close to that of the $f_{2}(1270)$. If the
contribution of these tensor mesons are described by relativistic Breit-Wigner
components, it is difficult to disentangle them because of the degeneracy in
their masses, widths and partial decay widths into $K\overline{K}$ PDG2020 .
Consequently, as in Ref. B10 , we consider only the $f_{2}(1270)$ to represent
the $D$ wave. It is an “effective” $f_{2}(1270)$ which takes into account both
tensor mesons.

In the present approach, a best fit is achieved in which the data are
reproduced with amplitudes that are optimized notably by adjusting the
isoscalar- and isovector-scalar form factors. It is interesting then to see
what could be the outcome of a model based, for instance, on the scalar form
factors calculated from the Muskhelishvili-Omnès dispersion relation approach
MO ; Moussallam_2000 ; Moussallam_2019 ; Bachir2015 . As in our best fit model
($\chi^{2}/ndf=1.25$), we have to introduce energy dependent phenomenological
functions multiplying the scalar form factors to obtain two fits with almost
as good $\chi^{2}/ndf$. Branching fractions of these two alternative models
are compared to those of our best fit model.

The paper is organized as follows. A full derivation of the $D^{0}\to
K^{0}_{S}K^{+}K^{-}$ decay amplitude, in the framework of the quasi-two-body
factorization approach is given in Sec. II. Based only on the experimental
data of the BABAR Collaboration B10  and without any model assumptions, we
show in Sec. III that the $f_{0}(980)$ contribution, at variance with the
BABAR analysis, has to be included in the decay amplitude. Section IV presents
the result of our best fit to the Dalitz plot data sample of Ref. B10 . Some
discussion and concluding remarks can be found in Sec. V. Appendix A presents
the update of the description of the $\pi\pi$, $\bar{K}K$ and effective
$(2\pi)(2\pi)$ S-wave isospin zero scattering amplitudes. Two alternative
models for the $D^{0}\to K^{0}_{S}K^{+}K^{-}$ decay amplitude, using kaon
scalar-form factors derived from the dispersion relation approach, are
presented in Appendix B.

##  II The $D^{0}\to K^{0}_{S}K^{+}K^{-}$ decay amplitudes in a factorization
framework

The decay amplitudes for the $D^{0}\to K^{0}_{S}K^{+}K^{-}$ process can be
evaluated as matrix elements of the effective weak Hamiltonian Buchalla1996

$H_{eff}=\frac{G_{F}}{\sqrt{2}}V_{CKM}\Big{[}C_{1}(\mu)O_{1}(\mu)+C_{2}(\mu)O_{2}(\mu)\Big{]}+h.c.,$
(1)

where the coefficients $V_{CKM}$ are given in terms of Cabibbo-Kobayashi-
Maskawa quark-mixing matrix elements and $G_{F}$ denotes the Fermi coupling
constant. The $C_{i}(\mu)$ are the Wilson coefficients of the four-quark
operators $O_{i}(\mu)$ at a renormalization scale $\mu$, chosen to be equal to
the $c$-quark mass $m_{c}$. The left-handed current-current operators
$O_{1,2}(\mu)$ arise from $W$-boson exchange.

The transition matrix elements that occur in the present work require two
specific values of the $V_{CKM}$ coupling matrix elements JPD_PRD89 :

$\Lambda_{1}\equiv V^{*}_{cs}V_{ud}=1-\lambda^{2}\hskip 28.45274pt{\rm
and}\hskip 28.45274pt\Lambda_{2}\equiv V^{*}_{cd}V_{us}=-\lambda^{2},$ (2)

where $\lambda$ is the sine of the Cabibbo angle and where $\Lambda_{1}$ is
associated to Cabibbo favored (CF) transitions while $\Lambda_{2}$ is
associated to doubly Cabibbo suppressed (DCS) amplitudes. The amplitudes are
functions of the Mandelstam invariants

$s_{\pm}=m_{\pm}^{2}=(p_{0}+p_{\pm})^{2},\hskip
56.9055pts_{0}=m_{0}^{2}=(p_{+}+p_{-})^{2},$ (3)

where $p_{0}$, $p_{+}$ and $p_{-}$ are the four-momenta of the $K_{S}^{0}$,
$K^{+}$ and $K^{-}$ mesons, respectively. Energy-momentum conservation implies

$p_{D^{0}}=p_{0}+p_{+}+p_{-}\hskip 28.45274pt{\rm and}\hskip
28.45274pts_{0}+s_{+}+s_{-}=m_{D^{0}}^{2}+m_{K^{0}}^{2}+2m_{K}^{2},$ (4)

where $p_{D^{0}}$ is the $D^{0}$ four-momentum and $m_{D^{0}}=1864.83$ MeV,
$m_{K^{0}}=497.611$ MeV and $m_{K}=493.677$ MeV denote the $D^{0}$, $K^{0}$
and charged kaon masses (Ref. PDG2020 ).

###  II.1 Tree and annihilation CF and DCS amplitudes

The full amplitude is the superposition of two tree CF and DCS amplitudes,
$T^{CF}(s_{0},s_{-},s_{+})$ and $T^{DCS}(s_{0},s_{-},s_{+})$ and of two
annihilation (i.e., exchange of W meson between the $c$ and $\overline{u}$
quarks of the $D^{0}$) CF and DCS amplitudes, $A^{CF}(s_{0},s_{-},s_{+})$ and
$A^{DCS}(s_{0},s_{-},s_{+})$. Thus, one writes the full amplitude as

$\displaystyle\mathcal{M}(s_{0},s_{-},s_{+})$ $\displaystyle=$
$\displaystyle\left\langle K^{0}_{S}(p_{0})\
K^{+}(p_{+}){K^{-}}(p_{-})|H_{eff}|D^{0}(p_{D^{0}})\right\rangle$ (5)
$\displaystyle=$ $\displaystyle
T^{CF}(s_{0},s_{-},s_{+})+T^{DCS}(s_{0},s_{-},s_{+})+A^{CF}(s_{0},s_{-},s_{+})+A^{DCS}(s_{0},s_{-},s_{+}).$

Although the three variables $s_{0},s_{-},s_{+}$ appear as arguments of the
amplitudes, all amplitudes depend only on two of them because of the relation
(4).

Figure 1:  Tree diagrams for Cabibbo favored amplitudes proportional to
$\Lambda_{1}\ a_{1}$ with $K^{-}\left[\overline{K}^{0}\ K^{+}\right]$ final
states

Assuming that the factorization approach Beneke2003 ; BurasNPB434_606 ;
Buchalla1996 ; AliPRD58  holds, the diagram illustrated in Fig. 1 is
proportional to $\Lambda_{1}a_{1}(m_{c})$ with quasi-two-body
$K^{-}[\overline{K}^{0}K^{+}]_{L}^{I=1}$ final states and the diagram in Fig.
2, proportional to $\Lambda_{1}a_{2}(m_{c})$ with quasi-two-body
$\overline{K}^{0}[K^{+}K^{-}]_{L}^{I}$ with angular momentum $L=S,P,D$ and
isospin $I=0,1$ states, yield the tree CF amplitude
$T^{CF}(s_{0},s_{-},s_{+})$ which reads, with $|0\rangle$ denoting the vacuum
state,

$\displaystyle T^{CF}(s_{0},s_{-},s_{+})\\!$ $\displaystyle\simeq$
$\displaystyle\frac{G_{F}}{2}\Lambda_{1}\\!\sum_{L={S,P,D}}\left\\{a_{1}(m_{c})\langle\left[\overline{K}^{0}(p_{0})\
K^{+}(p_{+})\right]_{L}^{I=1}|(\overline{u}\ d)_{V-A}|0\rangle\langle
K^{-}(p_{-})|(\overline{s}\ c)_{V-A}|D^{0}(p_{D^{0}})\right.\rangle$ (6)
$\displaystyle+$ $\displaystyle\left.a_{2}(m_{c})\sum_{I={0,1}}\
\langle\overline{K}^{0}(p_{0})|(\overline{s}\
d)_{V-A}|0\rangle\langle[K^{+}(p_{+}){K^{-}}(p_{-})]_{L,u}^{I}|(\overline{u}\
c)_{V-A}|D^{0}(p_{D^{0}})\rangle\right\\}$ $\displaystyle=$
$\displaystyle\sum_{L={S,P,D}}\\!\\!\left[T^{CF}_{[\overline{K}^{0}K^{+}]_{L}^{1}K^{-}}(s_{0},s_{-},s_{+})+T^{CF}_{\overline{K}^{0}[K^{+}K^{-}]_{L,u}^{0}}(s_{0},s_{-},s_{+})+T^{CF}_{\overline{K}^{0}[K^{+}K^{-}]_{L,u}^{1}}(s_{0},s_{-},s_{+})\right]$
$\displaystyle=$ $\displaystyle
T^{CF}_{[\overline{K}^{0}K^{+}]^{1}K^{-}}(s_{0},s_{-},s_{+})+\sum_{I=0,1}T^{CF}_{\overline{K}^{0}[K^{+}K^{-}]_{u}^{I}}(s_{0},s_{-},s_{+}).$

The short hand notation of the last line of Eq. (6) implies the L summation222
e.g.,
$T^{CF}_{[\overline{K}^{0}K^{+}]^{1}K^{-}}(s_{0},s_{-},s_{+})=\sum_{L={S,P,D}}T^{CF}_{[\overline{K}^{0}K^{+}]_{L}^{1}K^{-}}(s_{0},s_{-},s_{+}),etc.$
. It will be used wherever necessary. In the case of the creation of a
$K^{+}K^{-}$ pair we indicate by a subscript $q$ the $q\overline{q}$ pair from
which it originates (here a $u\overline{u}$ one, as seen in Fig. 2). We shall
therefore use the notation $[K^{+}K^{-}]_{L,q}^{I}$ and/or
$[K^{+}K^{-}]_{q}^{I}$ whenever necessary. We have also introduced the short-
hand notation

$(\overline{q}\ q)_{V-A}=\overline{q}\gamma\ (1-\gamma_{5})\ q$ (7)

which will be used throughout the text ($\gamma$ and $\gamma_{5}$ are Dirac’s
matrices). In deriving Eq. (6) small $CP$ violation effects in $K^{0}_{S}$
decays are neglected and we use

$|K^{0}_{S}\rangle\approx\frac{1}{\sqrt{2}}\
\left(|K^{0}\rangle+|\overline{K}^{0}\rangle\right).$ (8)

At leading order in the strong coupling constant $\alpha_{S}$, the effective
QCD factorization coefficients $a_{1}(m_{c})$ and $a_{2}(m_{c})$ are expressed
as

$a_{1}(m_{c})=C_{1}(m_{c})+\frac{C_{2}(m_{c})}{N_{C}},\hskip
28.45274pta_{2}(m_{c})=C_{2}(m_{c})+\frac{C_{1}(m_{c})}{N_{C}},$ (9)

where $N_{C}=3$ is the number of colors. Higher order vertex and hard
scattering corrections are not discussed in the present work and we introduce
effective values for these coefficients. From now on, the simplified notation
$a_{1}\equiv a_{1}(m_{c})$ and $a_{2}\equiv a_{2}(m_{c})$ will be used. As in
Ref. JPD_PRD89 , we take $a_{1}=1.1$ and $a_{2}=-0.5$.

Figure 2:  Tree diagram for Cabibbo favored amplitudes proportional to
$\Lambda_{1}a_{2}$ with $\overline{K}^{0}\
\left[K^{+}K^{-}\right]_{u}^{I=0,1}$ final states

Similarly, the DCS tree amplitudes illustrated in Figs. 3 and 4, yield the
amplitude $T^{DCS}(s_{0},s_{-},s_{+})$

$\displaystyle T^{DCS}(s_{0},s_{-},s_{+})\\!$ $\displaystyle\simeq$
$\displaystyle\\!\frac{G_{F}}{2}\
\Lambda_{2}\sum_{L={S,P,D}}\\!\Big{\\{}a_{1}\langle
K^{+}(p_{+})|(\overline{u}\ s)_{V-A}|0\rangle\ \langle[K^{0}(p_{0})\
K^{-}(p_{-})]_{L}^{1}|(\overline{d}\ c)_{V-A}|D^{0}(p_{D^{0}})\rangle$ (10)
$\displaystyle+$ $\displaystyle a_{2}\sum_{I=0,1}\langle
K^{0}(p_{0})|(\overline{d}\ s)_{V-A}|0\rangle\ \langle[K^{+}(p_{+})\
K^{-}(p_{-})]_{L,u}^{I}|(\overline{u}\
c)_{V-A}|D^{0}(p_{D^{0}})\rangle\Big{\\}}$ $\displaystyle=$
$\displaystyle\\!\sum_{L={S,P,D}}\\!\left[T^{DCS}_{K^{+}[{K}^{0}K^{-}]_{L}^{1}}(s_{0},s_{-},s_{+})+T^{DCS}_{{K}^{0}[K^{+}K^{-}]_{L,u}^{0}}(s_{0},s_{-},s_{+})+T^{DCS}_{{K}^{0}[K^{+}K^{-}]_{L,u}^{1}}(s_{0},s_{-},s_{+})\right]$
$\displaystyle=$
$\displaystyle\\!T^{DCS}_{K^{+}[{K}^{0}K^{-}]^{1}}(s_{0},s_{-},s_{+})\\!+\sum_{I=0,1}T^{DCS}_{{K}^{0}[K^{+}K^{-}]_{u}^{I}}(s_{0},s_{-},s_{+}).$

Figure 3: Tree diagram for doubly Cabibbo suppressed amplitudes for
$K^{+}\left[{K}^{0}K^{-}\right]^{I=1}$ final states Figure 4: Tree diagram for
doubly Cabibbo suppressed amplitudes for ${K}^{0}\left[K^{+}\
K^{-}\right]_{u}^{I=0,1}$ final states

A similar derivation goes for the CF annihilation amplitudes illustrated by
the diagram of Fig. 5, $A^{CF}(s_{0},s_{-},s_{+})$, so that one has333In the
amplitudes (11) and (12) we neglect the terms with the quasi two-boby
$K^{+}(p_{+})[\overline{K}^{0}(p_{0})K^{-}(p_{-})]_{L}^{1}$ and
$K^{-}(p_{-})[K^{+}(p_{+})K^{0}(p_{0})]_{L}^{1}$ final states. One expects a
small contribution of these terms because there exist no strangeness -2 (
$[\overline{K}^{0}K^{-}]_{L}^{1}]$ state) and +2 ($[K^{+}K^{0}]_{L}^{1}]$
state) resonances.

$\displaystyle A^{CF}(s_{0},s_{-},s_{+})$ $\displaystyle\simeq$
$\displaystyle\frac{G_{F}}{2}\ \Lambda_{1}\
a_{2}\sum_{L={S,P,D}}\Big{\\{}\sum_{I={0,1}}\langle\overline{K}^{0}(p_{0})[K^{-}(p_{-})K^{+}(p_{+})]_{L,s}^{I}|(\overline{s}\
d)_{V-A}|0\rangle$ (11) $\displaystyle+$ $\displaystyle\langle
K^{-}(p_{+})[\overline{K}^{0}(p_{0})K^{+}(p_{-})]_{L}^{1}|(\overline{s}\
d)_{V-A}|0\rangle\Big{\\}}\ \langle 0|(\overline{c}\
u)_{V-A}|D^{0}(p_{D^{0}})\rangle$ $\displaystyle=$
$\displaystyle\sum_{L={S,P,D}}\Big{[}A^{CF}_{K^{-}[\overline{K}^{0}K^{+}]_{L}^{1}}(s_{0},s_{-},s_{+})+A^{CF}_{\overline{K}^{0}[K^{+}K^{-}]_{L,s}^{0}}(s_{0},s_{-},s_{+})\Big{]}$
$\displaystyle=$ $\displaystyle
A^{CF}_{K^{-}[\overline{K}^{0}K^{+}]^{1}}(s_{0},s_{-},s_{+})+A^{CF}_{\overline{K}^{0}[K^{+}K^{-}]_{s}^{0}}(s_{0},s_{-},s_{+}).$

Figure 5: Diagram for CF annihilation ($W$-exchange) amplitudes with
$\overline{K}^{0}\ \left[K^{+}\ K^{-}\right]_{s}^{I=0}$ or
$K^{-}[K^{+}\overline{K}^{0}]^{I=1}$ final states.  Figure 6: Diagram for DCS
annihilation ($W$-exchange) amplitudes with ${K}^{0}\ \left[K^{+}\
K^{-}\right]_{s}^{I=0}$ or $K^{+}\left[K^{0}K^{-}\right]^{I=1}$ final states.

The corresponding DCS annihilation amplitudes, $A^{DCS}(s_{0},s_{-},s_{+})$,
(see Fig. 6), are easily obtained from the CF amplitudes in Eq. (11)

$\displaystyle A^{DCS}(s_{0},s_{-},s_{+})$ $\displaystyle\approx$
$\displaystyle\frac{G_{F}}{2}\ \Lambda_{2}\ a_{2}\
\sum_{L={S,P,D}}\sum_{I=0,1}\Big{[}\langle[K^{-}(p_{-})K^{+}(p_{+})]_{L,s}^{I}{K}^{0}(p_{0})|(\overline{d}\
s)_{V-A}|0\rangle$ (12) $\displaystyle+$ $\displaystyle\langle
K^{+}(p_{+})[K^{0}(p_{0})K^{-}(p_{-})]_{L}^{1}|(\overline{d}\
s)_{V-A}|0\rangle\Big{]}\langle 0|(\overline{c}\
u)_{V-A}|D^{0}(p_{D^{0}})\rangle$ $\displaystyle=$
$\displaystyle\sum_{L={S,P,D}}\Big{[}A^{DCS}_{K^{+}[K^{0}K^{-}]_{L}^{1}}(s_{0},s_{-},s_{+})+A^{DCS}_{K^{0}[K^{+}K^{-}]_{L,s}^{0}}(s_{0},s_{-},s_{+})\Big{]}$
$\displaystyle=$ $\displaystyle
A^{DCS}_{K^{+}[K^{0}K^{-}]^{1}}(s_{0},s_{-},s_{+})+A^{DCS}_{K^{0}[K^{+}K^{-}]_{s}^{0}}(s_{0},s_{-},s_{+}).$

Let us now review in detail all the amplitudes that will have to be evaluated.
We will follow closely the construction detailed in Ref. JPD_PRD89 .

###  II.2 Explicit tree amplitudes

In the following, starting from the expressions given in Eqs. (6) for the CF
amplitudes and in (10) for the DCS ones, we will express the different three-
body matrix elements entering in the amplitudes in terms of vertex functions
noted $G_{R_{S,P,D}[\overline{K}^{0}K^{+}]^{1}}(s)$ in the case of a
$[\overline{K}^{0}K^{+}]^{1}$ final state,
$G_{R_{S,P,D}[{K}^{0}K^{-}]^{1}}(s)$ in the case of a $[{K}^{0}K^{-}]^{1}$
final state, and $G_{R_{S,P,D}[{K}^{+}K^{-}]_{q}^{0,1}}(s)$ in the case of a
$[{K}^{+}K^{-}]_{q}^{0,1}$ final state. The vertex functions describe the
decays into $K\overline{K}$ of the possibly present intermediate resonances
$R_{S,P,D}$ which contribute to the process.

Further on, we will need to introduce transition form factors for which, as in
Ref. JPD_PRD89 , we will assume isospin and charge conjugation symmetry so
that the following equations arise:

$\displaystyle F_{0}^{D^{0}a_{0}^{-}}(s)$ $\displaystyle=$
$\displaystyle\sqrt{2}\ F_{0}^{D^{0}a_{0}^{0}}(s),$ $\displaystyle
F_{0}^{[{\overline{K}^{0}K^{+}}]^{1}}(s)$ $\displaystyle=$ $\displaystyle
F_{0}^{[{{K}^{0}K^{-}}]^{1}}(s),$ $\displaystyle
F_{1}^{\overline{K}^{0}K^{+}}(s)$ $\displaystyle=$ $\displaystyle-
F_{1}^{{K}^{0}K^{-}}(s),$ $\displaystyle
F_{0}^{K^{-}a_{0}^{+}}(m_{D^{0}}^{2})$ $\displaystyle=$
$\displaystyle\sqrt{2}\ F_{0}^{K^{0}a_{0}^{0}}(m_{D^{0}}^{2}),$ $\displaystyle
A_{0}^{D^{0}\rho^{-}}(s)$ $\displaystyle=$ $\displaystyle\sqrt{2}\
A_{0}^{D^{0}\rho^{0}}(s),$ $\displaystyle
A_{0}^{\overline{K}^{0}\phi}(m_{D^{0}}^{2})$ $\displaystyle=$ $\displaystyle
A_{0}^{K^{0}\phi}(m_{D^{0}}^{2}).$ (13)

In the above equations $F_{0}$ and $F_{1}$ denote scalar and vector transition
form factors of two pseudoscalar mesons while $A_{0}$’s are transition form
factors of pseudoscalar and vector mesons.

####  II.2.1 Scalar amplitudes

Following a derivation similar to that developped in Ref. JPD_PRD89 , the
isoscalar-scalar CF amplitude associated to the $\overline{K}^{0}\
[K^{+}K^{-}]_{S,u}^{I}$ final states can be described by (see Fig. 2)

$\displaystyle T^{CF}_{\overline{K}^{0}\
[K^{+}K^{-}]_{S,u}^{0}}(s_{0},s_{-},s_{+})$ $\displaystyle=$
$\displaystyle-\frac{G_{F}}{2}\ \Lambda_{1}\ a_{2}\
\frac{f_{K^{0}}}{\sqrt{2}}\ (m_{D^{0}}^{2}-s_{0})\
\sum_{R_{S}}F_{0}^{D^{0}R_{S}[K^{+}K^{-}]_{u}^{0}}(m_{K^{0}}^{2})$ (14)
$\displaystyle\times$ $\displaystyle\ G_{R_{S}[K^{+}K^{-}]_{u}^{0}}(s_{0})\
\langle R_{S}[K^{+}K^{-}]_{u}^{0}|u\overline{u}\rangle,$

where $f_{K^{0}}$ is the $K^{0}$ decay constant and the sum over $R_{S}$ runs
over the possibly contributing resonances in the isoscalar-scalar channel. It
can be seen here that we have approximated the three-body matrix element
$\langle[K^{+}(p_{+}){K^{-}}(p_{-})]_{S,u}^{0}|(\overline{u}\
c)_{V-A}|D^{0}(p_{D^{0}})\rangle$ entering Eq. (6) by the above sum over
$R_{S}$. It thus includes the contributions of the $f_{0}(500)$, $f_{0}(980)$
and of the $f_{0}(1370)$ and $f_{0}(1500)$ resonances. The $D^{0}$ to $R_{S}$
transition form factor entering Eq. (14) could have a different value for each
resonance $R_{S}$; here we can assume that its variation from one resonance to
the other is small and we can choose for its value that of the transition to
$f_{0}(980)$. Unless otherwise specified, by $f_{0}$ in
$F_{0}^{D^{0}f_{0}}(m_{K^{0}}^{2})$ we mean $f_{0}(980)$. We may parametrize
the sum over $R_{S}$ by introducing the isoscalar-scalar form factor,
$\Gamma_{2}^{n*}(s_{0})$, where $n$ denotes a nonstrange quark pair and which
can be built following the method discussed in Ref. DedonderPol . We then
apply the following approximation

$\sum_{R_{S}}F_{0}^{D^{0}R_{S}[K^{+}K^{-}]_{u}^{0}}(m_{K^{0}}^{2})\
G_{R_{S}[K^{+}K^{-}]_{u}^{0}}(s_{0})\ \langle
R_{S}[K^{+}K^{-}]_{u}^{0}|u\overline{u}\rangle=\chi^{n}\
\frac{\Gamma_{2}^{n*}(s_{0})}{\sqrt{2}}\ F_{0}^{D^{0}f_{0}}(m_{K^{0}}^{2}),$
(15)

where $\chi^{n}$ is a constant complex factor. Hence

$T^{CF}_{\overline{K}^{0}\
[K^{+}K^{-}]_{S,u}^{0}}(s_{0},s_{-},s_{+})=-\frac{G_{F}}{2}\ \Lambda_{1}\
a_{2}\ \ f_{K^{0}}\ (m_{D^{0}}^{2}-s_{0})\ F_{0}^{D^{0}f_{0}}(m_{K^{0}}^{2})\
\frac{\chi^{n}}{2}\Gamma_{2}^{n*}(s_{0}).$ (16)

The real transition form factor, $F_{0}^{D^{0}f_{0}}(m_{K^{0}}^{2})$, can be
obtained from Ref. El-Bennich_PRD79 . This amplitude has to be associated with
the corresponding isoscalar-scalar $K^{0}[K^{+}K^{-}]_{S,u}^{0}$ DCS amplitude
(see Fig. 4) approximated by

$T_{K^{0}[K^{+}K^{-}]_{S,u}^{0}}^{DCS}(s_{0},s_{-},s_{+})=\frac{\Lambda_{2}}{\Lambda_{1}}~{}T^{CF}_{\overline{K}^{0}\
[K^{+}K^{-}]_{S,u}^{0}}(s_{0},s_{-},s_{+})$ (17)

Recombining the two amplitudes (16) and (17), we have

$\displaystyle T_{1}$ $\displaystyle=$ $\displaystyle
T_{K^{0}[K^{+}K^{-}]_{S,u}^{0}}^{CF}(s_{0},s_{-},s_{+})+T_{K^{0}[K^{+}K^{-}]_{S,u}^{0}}^{DCS}(s_{0},s_{-},s_{+})$
(18) $\displaystyle=$ $\displaystyle-\frac{G_{F}}{2}\
(\Lambda_{1}+\Lambda_{2})\ a_{2}\ f_{K^{0}}\ (m_{D^{0}}^{2}-s_{0})\
F_{0}^{D^{0}f_{0}}(m_{K^{0}}^{2})\ \frac{\chi^{n}}{2}\
\Gamma_{2}^{n*}(s_{0}).$

We now turn to the three isovector-scalar tree amplitudes. The isovector-
scalar $K^{+}[{K}^{0}K^{-}]_{S}$ DCS amplitude, associated to the
$a_{0}(980)^{-}$ and $a_{0}(1450)^{-}$ resonances can be written as (see Fig.
3)

$\displaystyle T^{DCS}_{K^{+}[{K}^{0}K^{-}]_{S}^{1}}(s_{0},s_{-},s_{+})$
$\displaystyle=$ $\displaystyle-\frac{G_{F}}{2}\ \Lambda_{2}\ a_{1}\
f_{K^{+}}\ (m_{D^{0}}^{2}-s_{-})\
\sum_{R_{S}}F_{0}^{D^{0}R_{S}[K^{0}K^{-}]^{1}}(m_{K}^{2})$ (19) $\displaystyle
G_{R_{S}[K^{0}K^{-}]^{1}}(s_{-})\ \langle
R_{S}[K^{0}K^{-}]^{1}|d\overline{u}\rangle,$

the $a_{0}^{-}$ resonances being built from $\overline{u}d$ pairs and $\langle
R_{S}[K^{0}K^{-}]^{1}|d\overline{u}\rangle=1$. In Eq. (19) $f_{K^{+}}$ denotes
the charged kaon decay constant. Parametrizing, as above, the sum over $R_{S}$
as

$\sum_{R_{S}}F_{0}^{D^{0}R_{S}[K^{0}K^{-}]_{u}^{1}}(m_{K}^{2})\
G_{R_{S}[K^{0}K^{-}]^{1}}(s_{-})\ \langle
R_{S}[K^{0}K^{-}]^{1}|d\overline{u}\rangle=G_{1}(s_{-})\
F_{0}^{D^{0}a_{0}^{-}}(m_{K}^{2}),\\\ $ (20)

we get

$T_{2}=T^{DCS}_{K^{+}[{K}^{0}K^{-}]_{S}^{1}}(s_{0},s_{-},s_{+})=-\frac{G_{F}}{2}\
\Lambda_{2}\ a_{1}\ f_{K^{+}}\ (m_{D^{0}}^{2}-s_{-})\
F_{0}^{D^{0}a_{0}^{-}}(m_{K}^{2})\ G_{1}(s_{-}).$ (21)

The function $G_{1}(s)$ can be built using the isospin 1 coupled
$K\overline{K}$ and $\pi\eta$ channel description of the $a_{0}(980)$ and
$a_{0}(1450)$ performed in Ref. AFLL . As previously for the isoscalar-scalar
case, we have assumed here that the variation of the $D^{0}\to R_{S}$
transition form factor from one resonance to the other is small and we choose
it to be that of the lowest resonance, i.e., $R_{S}\equiv a_{0}(980)^{-}$,
denoted simply $a_{0}^{-}$.

In the case ot the isovector-scalar $[\overline{K}^{0}K^{+}]_{S}^{1}\ K^{-}$
CF amplitude ($a_{1}$ term of Eq. (6), related to the $a_{0}(980)^{+}$ and
$a_{0}(1450)^{+}$ resonances, one has444Because of the presence of the very
small mass squared difference between the charged and neutral kaons the
amplitude (22) will give no constraint on the kaon isovector-scalar form
factor.

$\displaystyle
T^{CF}_{K^{-}[\overline{K}^{0}K^{+}]_{S}^{1}}(s_{0},s_{-},s_{+})$
$\displaystyle=$ $\displaystyle-\frac{G_{F}}{2}\ \Lambda_{1}\ a_{1}\
(m_{D^{0}}^{2}-m_{K}^{2})\ \frac{m_{K}^{2}-m_{K^{0}}^{2}}{s_{+}}$ (22)
$\displaystyle\times$ $\displaystyle F_{0}^{D^{0}K^{-}}(s_{+})\
F_{0}^{[\overline{K}^{0}K^{+}]^{1}}(s_{+}),$

where $F_{0}^{[\overline{K}^{0}K^{+}]^{1}}(s_{+})$ is the kaon isovector-
scalar form factor and denoted also as
$F_{0}^{[\overline{K}^{0}K^{+}]^{1}}(s)$ in the second relation of the Eqs.
(II.2). For the transition form factor $F_{0}^{D^{0}K^{-}}(s_{+})$, following
Ref. Melichow , we use the parameterization:

$F_{0}^{D^{0}K^{-}}(s_{+})=\frac{0.78}{1-0.38~{}s_{+}/M_{V}^{2}+0.46~{}s_{+}^{2}/M_{V}^{4}},$
(23)

where $M_{V}=2.11$ GeV. We have then

$T_{3}=T^{CF}_{[\overline{K}^{0}K^{+}]_{S}^{1}K^{-}}(s_{0},s_{-},s_{+})=-\frac{G_{F}}{2}\
\Lambda_{1}\ a_{1}\ (m_{D^{0}}^{2}-m_{K}^{2})\
\frac{m_{K}^{2}-m_{K^{0}}^{2}}{s_{+}}\
F_{0}^{[\overline{K}^{0}K^{+}]^{1}}(s_{+})\ F_{0}^{D^{0}K^{-}}(s_{+})$ (24)

We proceed similarly for the isovector-scalar CF $\overline{K}^{0}\
[K^{+}K^{-}]_{S,u}^{1}$ and DCS ${K}^{0}\ [K^{+}K^{-}]_{S,u}^{1}$ amplitudes
(see Figs. 2 and 4). It is given by

$\displaystyle T^{CF}_{\overline{K}^{0}\
[K^{+}K^{-}]_{S}^{1}}(s_{0},s_{-},s_{+})$ $\displaystyle=$
$\displaystyle-\frac{G_{F}}{2}\ \Lambda_{1}\ a_{2}\ f_{K^{0}}\
(m_{D^{0}}^{2}-s_{0})\sum_{R_{S}}F_{0}^{D^{0}R_{S}[K^{+}K^{-}]_{u}^{1}}(m_{K^{0}}^{2})$
(25) $\displaystyle\times$ $\displaystyle
G_{R_{S}[K^{+}K^{-}]_{u}^{1}}(s_{0})\ \langle
R_{S}[K^{+}K^{-}]_{u}^{1}|u\overline{u}\rangle,$

where the sum over $R_{S}$ runs over the contributing resonances in that
channel, i.e., $a_{0}(980)^{0}$ and $a_{0}(1450)^{0}$ for which $\langle
R_{S}[K^{+}K^{-}]_{u}^{1}|u\overline{u}\rangle=\frac{1}{\sqrt{2}}$. Then, we
get

$\sum_{R_{S}}F_{0}^{D^{0}R_{S}[K^{+}K^{-}]_{u}^{1}}(m_{K^{0}}^{2})\
G_{R_{S}[K^{+}K^{-}]_{u}^{I=1}}(s_{0})\ \langle
R_{S}[K^{+}K^{-}]_{u}^{1}|u\overline{u}\rangle=\frac{1}{2}G_{1}(s_{0})\
F_{0}^{D^{0}a_{0}^{0}}(m_{K^{0}}^{2}),$ (26)

where we assume (isospin invariance) that, to describe the $u\bar{u}$
transition to the isovector-scalar $K^{+}K^{-}$ state, it is the same function
$G_{1}(s)$ as that introduced in Eqs. (20) for the $d\bar{u}$ transitions to
the isovector-scalar $K^{0}K^{-}$ state. In Eq. (26) $a_{0}^{0}$ means
$a_{0}(980)^{0}$. We may now rewrite

$T^{CF}_{\overline{K}^{0}\
[K^{+}K^{-}]_{S}^{1}}(s_{0},s_{-},s_{+})=-\frac{G_{F}}{2}\ \Lambda_{1}\ a_{2}\
f_{K^{0}}\ (m_{D^{0}}^{2}-s_{0})\ \frac{1}{2}G_{1}(s_{0})\
F_{0}^{D^{0}a_{0}^{0}}(m_{K^{0}}^{2}).$ (27)

In a similar way, the related isovector-scalar DCS amplitude reads

$T^{DCS}_{K^{0}[K^{+}K^{-}]_{S}^{1}}(s_{0},s_{-},s_{+})=\frac{\Lambda_{2}}{\Lambda_{1}}~{}T^{CF}_{\overline{K}^{0}\
[K^{+}K^{-}]_{S}^{1}}(s_{0},s_{-},s_{+}).$ (28)

Recombining Eqs. (27) and (28) we get

$\displaystyle T_{4}$ $\displaystyle=$ $\displaystyle
T^{CF}_{\overline{K}^{0}[K^{+}K^{-}]_{S}^{1}}(s_{0},s_{-},s_{+})+T^{DCS}_{K^{0}[K^{+}K^{-}]_{S}^{1}}(s_{0},s_{-},s_{+})$
(29) $\displaystyle=$ $\displaystyle-\frac{G_{F}}{2}\
(\Lambda_{1}+\Lambda_{2})\ a_{2}\ f_{K^{0}}\ (m_{D^{0}}^{2}-s_{0})\
\frac{1}{2}G_{1}(s_{0})\ F_{0}^{D^{0}a_{0}^{0}}(m_{K^{0}}^{2}).$

####  II.2.2 Vector amplitudes

Let us now study the vector tree amplitudes associated to the
$\overline{K}^{0}\ [K^{+}K^{-}]_{P,u}^{I}$ channel. The isoscalar-vector CF
amplitude can be built from the $\omega$ and $\phi$ resonances (see Fig. 2).
It reads

$\displaystyle T^{CF}_{\overline{K}^{0}\
[K^{+}K^{-}]_{P}^{0}}(s_{0},s_{-},s_{+})$ $\displaystyle=$
$\displaystyle\frac{G_{F}}{2}\ \Lambda_{1}\ a_{2}\ f_{K^{0}}\ (s_{-}-s_{+})\
\sum_{R_{P}}A_{0}^{D^{0}R_{P}[K^{+}K^{-}]_{u}^{0}}(m_{K^{0}}^{2})$ (30)
$\displaystyle\times$ $\displaystyle m_{R_{P}[K^{+}K^{-}]}\
G_{R_{P}[K^{+}K^{-}]_{u}^{0}}(s_{0})\ \langle
R_{P}[K^{+}K^{-}]_{u}^{0}|u\overline{u}\rangle,$

where $m_{R_{P}[K^{+}K^{-}]}$ denotes the mass of the contributing resonances.
Now we introduce the following parametrization in terms of the kaon vector
form factor $F_{1}^{[K^{+}K^{-}]_{u}^{0}}(s_{0})$ and, for the same reasons as
introduced in the scalar case [Eqs. (14) and (21)]

$\displaystyle\sum_{R_{P}}m_{R_{P}[K^{+}K^{-}]}\
A_{0}^{D^{0}R_{P}[K^{+}K^{-}]_{u}^{0}}(m_{K^{0}}^{2})$ $\displaystyle
G_{R_{P}[K^{+}K^{-}]_{u}^{0}}(s_{0})\ \langle
R_{P}[K^{+}K^{-}]_{u}^{0}|u\overline{u}\rangle$ (31) $\displaystyle=$
$\displaystyle\frac{1}{f_{\omega}}\ A_{0}^{D^{0}\omega}(m_{K^{0}}^{2})\
F_{1}^{[K^{+}K^{-}]_{u}^{0}}(s_{0}).$

This approximation relies on the fact that the mixing angle of the vector
meson nonet is very close to the ideal mixing angle,
$\theta_{V}=35.3^{\circ}$, so that the $\phi$ resonance amplitude gives an
almost nul contribution. Note that $f_{\omega}$ denotes the decay constant for
the $\omega(782)$ meson. We have then

$T^{CF}_{\overline{K}^{0}\
[K^{+}K^{-}]_{P}^{0}}(s_{0},s_{-},s_{+})=\frac{G_{F}}{2}\ \Lambda_{1}\
{a_{2}}\ (s_{-}-s_{+})\frac{f_{K^{0}}}{f_{\omega}}\
A_{0}^{D^{0}\omega}(m_{K^{0}}^{2})\ F_{1}^{[K^{+}K^{-}]_{u}^{0}}(s_{0}).$ (32)

The associated isoscalar-vector $K^{0}[K^{+}K^{-}]_{P,u}^{0}$ DCS amplitude is
given by

$T_{K^{0}[K^{+}K^{-}]_{P}^{0}}^{DCS}(s_{0},s_{-},s_{+})=\frac{\Lambda_{2}}{\Lambda_{1}}~{}T^{CF}_{\overline{K}^{0}\
[K^{+}K^{-}]_{P}^{0}}(s_{0},s_{-},s_{+}).$ (33)

Thus, from Eqs. (32) and (33), the isoscalar-vector amplitude reads

$\displaystyle T_{5}$ $\displaystyle=$ $\displaystyle
T^{CF}_{\overline{K}^{0}\
[K^{+}K^{-}]_{P}^{0}}(s_{0},s_{-},s_{+})+T_{K^{0}[K^{+}K^{-}]_{P}^{0}}^{DCS}(s_{0},s_{-},s_{+})$
(34) $\displaystyle=$ $\displaystyle\frac{G_{F}}{2}\
(\Lambda_{1}+\Lambda_{2})\ a_{2}\ (s_{-}-s_{+})\frac{f_{K^{0}}}{f_{\omega}}\
A_{0}^{D^{0}\omega}(m_{K^{0}}^{2})\ F_{1}^{[K^{+}K^{-}]_{u}^{0}}(s_{0}).$

The isovector-vector $T^{CF}_{\overline{K}^{0}\ [K^{+}K^{-}]_{P}^{1}}$
amplitude related to the $\rho^{0}$ resonances is given by a similar
expression to Eq. (30) (see Fig. 2)

$\displaystyle T^{CF}_{\overline{K}^{0}\
[K^{+}K^{-}]_{P}^{1}}(s_{0},s_{-},s_{+})$ $\displaystyle=$
$\displaystyle\frac{G_{F}}{2}\ \Lambda_{1}\ a_{2}\ f_{K^{0}}\ (s_{-}-s_{+})\
\sum_{R_{P}}A_{0}^{D^{0}R_{P}[K^{+}K^{-}]_{u}^{1}}(m_{K^{0}}^{2})$ (35)
$\displaystyle\times$ $\displaystyle m_{R_{P}[K^{+}K^{-}]}\
G_{R_{P}[K^{+}K^{-}]_{u}^{1}}(s_{0})\ \langle
R_{P}[K^{+}K^{-}]^{1}|u\overline{u}\rangle,$

where $\langle R_{P}[K^{+}K^{-}]^{1}|u\overline{u}\rangle=1/{\sqrt{2}}$.
Again, parametrizing the sum over the vertex functions by

$\displaystyle\sum_{R_{P}}A_{0}^{D^{0}R_{P}[K^{+}K^{-}]_{u}^{1}}(m_{K^{0}}^{2})m_{R_{P}[K^{+}K^{-}]}\
f_{R_{P}[K^{+}K^{-}]^{1}}$ $\displaystyle
G_{R_{P}[K^{+}K^{-}]_{u}^{1}}(s_{0})\ \langle
R_{P}[K^{+}K^{-}]^{1}|u\overline{u}\rangle$ (36) $\displaystyle=$
$\displaystyle\frac{1}{f_{\rho}}\ A_{0}^{D^{0}\rho^{0}}(m_{K^{0}}^{2})\
F_{1}^{[K^{+}K^{-}]_{u}^{1}}(s_{0}),$

where $f_{\rho}$ is the charged $\rho(770)$ decay constant,

$T^{CF}_{\overline{K}^{0}\
[K^{+}K^{-}]_{P}^{1}}(s_{0},s_{-},s_{+})=\frac{G_{F}}{2}\ \Lambda_{1}\
{a_{2}}\ (s_{-}-s_{+})\ \frac{f_{K^{0}}}{f_{\rho}}\
A_{0}^{D^{0}\rho^{0}}(m_{K^{0}}^{2})\ F_{1}^{[K^{+}K^{-}]_{u}^{1}}(s_{0}).$
(37)

Then comes the contribution of the isovector-vector
$K^{0}[K^{+}K^{-}]_{P}^{1}$ DCS amplitude (see Fig. 4). It goes as

$T_{K^{0}[K^{+}K^{-}]_{P}^{1}}^{DCS}(s_{0},s_{-},s_{+})=\frac{\Lambda_{2}}{\Lambda_{1}}~{}T^{CF}_{\overline{K}^{0}\
[K^{+}K^{-}]_{P}^{1}}(s_{0},s_{-},s_{+}),$ (38)

so that the total isovector-vector amplitude is

$\displaystyle T_{6}$ $\displaystyle=$ $\displaystyle
T^{CF}_{\overline{K}^{0}\
[K^{+}K^{-}]_{P}^{1}}(s_{0},s_{-},s_{+})+T_{K^{0}[K^{+}K^{-}]_{P}^{1}}^{DCS}(s_{0},s_{-},s_{+})$
(39) $\displaystyle=$ $\displaystyle\frac{G_{F}}{2}\
(\Lambda_{1}+\Lambda_{2})\ a_{2}\
(s_{-}-s_{+})A_{0}^{D^{0}\rho^{0}}(m_{K^{0}}^{2})\frac{f_{K^{0}}}{f_{\rho}}\
F_{1}^{[K^{+}K^{-}]_{u}^{1}}(s_{0}).$

The isovector-vector $[\overline{K}^{0}K^{+}]_{P}^{1}\ K^{-}$ CF amplitude has
the expression (see Fig. 1)

$\displaystyle
T^{CF}_{[\overline{K}^{0}K^{+}]_{P}^{1}K^{-}}(s_{0},s_{-},s_{+})$
$\displaystyle=$ $\displaystyle-\frac{G_{F}}{2}\ \Lambda_{1}\ a_{1}\
\left[s_{0}-s_{-}+(m_{D^{0}}^{2}-m_{K}^{2})\
\frac{m_{K^{0}}^{2}-m_{K}^{2}}{s_{+}}\right]$ (40) $\displaystyle\times$
$\displaystyle\sum_{R_{P}}F_{1}^{D^{0}K^{-}}(s_{+})\
G_{R_{P}[\overline{K}^{0}K^{+}]^{1}}(s_{+})\ \langle
R_{P}[\overline{K}^{0}K^{+}]^{1}|u\overline{d}\rangle$

where $\langle R_{P}[\overline{K}^{0}K^{+}]^{1}|u\overline{d}\rangle=1$ since
it is associated to the $\rho(770)^{+}$, $\rho(1450)^{+}$ and $\rho(1700)^{+}$
resonances. The sum over the vertex functions
$G_{R_{P}[\overline{K}^{0}K^{+}]^{I=1}}(s_{+})$ is expressed in terms of the
isovector-vector form factor $F_{1}^{\overline{K}^{0}K^{+}}(s_{+})$

$\sum_{R_{P}}F_{1}^{D^{0}K^{-}}(s_{+})\
G_{R_{P}[\overline{K}^{0}K^{+}]^{1}}(s_{+})\ \langle
R_{P}[\overline{K}^{0}K^{+}]^{1}|u\overline{d}\rangle=F_{1}^{D^{0}K^{-}}(s_{+})\
F_{1}^{\overline{K}^{0}K^{+}}(s_{+}),$ (41) $\displaystyle T_{7}$
$\displaystyle=$ $\displaystyle
T^{CF}_{[\overline{K}^{0}K^{+}]_{P}^{1}K^{-}}(s_{0},s_{-},s_{+})=-\frac{G_{F}}{2}\
\Lambda_{1}\ a_{1}\ $ (42) $\displaystyle\times$
$\displaystyle\left[s_{0}-s_{-}+(m_{D^{0}}^{2}-m_{K}^{2})\
\frac{m_{K^{0}}^{2}-m_{K}^{2}}{s_{+}}\right]\ F_{1}^{D^{0}K^{-}}(s_{+})\
F_{1}^{[\overline{K}^{0}K^{+}]^{1}}(s_{+}).$

As in Ref. Melichow  we parametrize $F_{1}^{D^{0}K^{-}}(s_{+})$ as follows

$F_{1}^{D^{0}K^{-}}(s_{+})=\frac{0.78}{(1-s_{+}/M_{V}^{2})(1-0.24~{}s_{+}/M_{V}^{2})}$
(43)

with, as before in Eq. (23), $M_{V}=2.11$ GeV.

The isovector-vector $K^{+}[{K}^{0}K^{-}]_{P}$ DCS amplitude is given by (see
Fig. 3)

$\displaystyle T_{K^{+}[{K}^{0}K^{-}]_{P}^{1}}^{DCS}(s_{0},s_{-},s_{+})$
$\displaystyle=$ $\displaystyle\frac{G_{F}}{2}\ \Lambda_{2}\ a_{1}\
\left[s_{0}-s_{+}+(m_{D^{0}}^{2}-m_{K}^{2})\
\frac{m_{K^{0}}^{2}-m_{K}^{2}}{s_{-}}\right]\ f_{K^{+}}\ $ (44)
$\displaystyle\times$
$\displaystyle\sum_{R_{P}}A_{0}^{D^{0}R_{P}[K^{0}K^{-}]^{1}}(m_{K}^{2})\
m_{R_{P}}\ G_{R_{P}[K^{0}K^{-}]^{1}}(s_{-})\ \langle
R_{P}[K^{0}K^{-}]|d\overline{u}\rangle.$

It is linked to the $\rho(770)^{-}$, $\rho(1450)^{-}$ and $\rho(1700)^{-}$
resonances and can be reexpressed as

$\displaystyle T_{8}$ $\displaystyle=$ $\displaystyle
T_{K^{+}[{K}^{0}K^{-}]_{P}^{1}}^{DCS}(s_{0},s_{-},s_{+})=\frac{G_{F}}{2}\
\Lambda_{2}\ a_{1}\ \frac{f_{K^{+}}}{f_{\rho}}$ (45) $\displaystyle\times$
$\displaystyle\left[s_{0}-s_{+}+(m_{D^{0}}^{2}-m_{K}^{2})\frac{m_{K^{0}}^{2}-m_{K}^{2}}{s_{-}}\right]\
A_{0}^{D^{0}\rho^{-}}(m_{K}^{2})\ F_{1}^{[K^{0}K^{-}]^{1}}(s_{-}).$

where we have introduced the isovector-vector form factor
$F_{1}^{[K^{0}K^{-}]^{1}}(s_{-})$ related to the sum over the vertex functions
$G_{R_{P}[K^{0}K^{-}]^{1}}(s_{-})$ by

$\sum_{R_{P}}A_{0}^{D^{0}R_{P}[K^{0}K^{-}]^{1}}(m_{K}^{2})\ m_{R_{P}}\
G_{R_{P}[K^{0}K^{-}]^{1}}(s_{-})\ \langle
R_{P}[K^{0}K^{-}]_{u}|d\overline{u}\rangle=\frac{1}{f_{\rho}}\
F_{1}^{K^{0}K^{-}}(s_{-})\ A_{0}^{D^{0}\rho^{-}}(m_{K}^{2})$ (46)

with $\langle R_{P}[K^{0}K^{-}]_{u}|d\overline{u}\rangle=1$, and $\rho$ refers
to $\rho(770)$.

####  II.2.3 Tensor amplitudes

For the isoscalar-tensor amplitude $\overline{K}^{0}\ [K^{+}K^{-}]_{D,u}^{0}$
amplitude, one can write (see Fig. 2)

$\displaystyle T^{CF}_{\overline{K}^{0}\
[K^{+}K^{-}]_{D}^{0}}(s_{0},s_{-},s_{+})$ $\displaystyle=$
$\displaystyle-\frac{G_{F}}{2}\ \Lambda_{1}\ a_{2}\ f_{K^{0}}\
\sum_{R_{D}}F^{D^{0}R_{D}[K^{+}K^{-}]_{u}^{0}}(m_{K^{0}}^{2})$ (47)
$\displaystyle\times$ $\displaystyle
G_{R_{D}[K^{+}K^{-}]_{u}^{0}}(s_{0},s_{-},s_{+})\langle
R_{D}[K^{+}K^{-}]_{u}^{0}|u\overline{u}\rangle$

but it will be dominated by the $f_{2}(1270)$ resonance with mass $m_{f_{2}}$;
it will be described by a Breit-Wigner representation. Linking the vertex
function to the form factor and using

$\langle R_{D}[K^{+}K^{-}]_{u}^{0}|u\overline{u}\rangle\approx\langle
f_{2}(1270)|u\overline{u}\rangle=\frac{1}{\sqrt{2}},$ (48)

we have

$\displaystyle T^{CF}_{\overline{K}^{0}\
[K^{+}K^{-}]_{D}^{0}}(s_{0},s_{-},s_{+})$ $\displaystyle=$
$\displaystyle-\frac{G_{F}}{2}\ \Lambda_{1}\ a_{2}\ f_{K^{0}}\
\frac{1}{\sqrt{2}}$ (49) $\displaystyle\times$ $\displaystyle
F^{D^{0}f_{2}}(m_{K^{0}}^{2})\ g_{f_{2}K^{+}K^{-}}\ \frac{D({\bf p_{2}},{\bf
p_{0}})}{m_{f_{2}}^{2}-s_{0}-im_{f_{2}}\ \Gamma_{f_{2}}(s_{0})},$

where $g_{f_{2}K^{+}K^{-}}$ is the coupling constant of the $f_{2}\to
K^{+}K^{-}$ transition and the function ${D({\bf p_{2}},{\bf p_{0}})}$ is
defined by

${D({\bf p_{2}},{\bf p_{0}})}=\frac{1}{3}\ {(|{\bf p_{2}}|\ |{\bf
p_{0}}|)^{2}-({\bf p_{2}}\cdot{\bf p_{0}})^{2}}.$ (50)

The three-momenta ${\bf p_{2}},{\bf p_{0}}$ are defined in the $[K^{+}K^{-}]$
center-of-mass (c.m.) system. One has

${{\bf p_{2}}={\bf p_{+}}=-\bf{p_{-}}},\hskip 56.9055pt\hskip 56.9055pt{|{\bf
p_{2}}|}={\frac{1}{2}}\ \sqrt{s_{0}-4m_{K}^{2}}$ (51)

and

$|{\bf p_{0}}|=\frac{\sqrt{[m_{D^{0}}^{2}-(\sqrt{s_{0}}+m_{K^{0}})^{2}]\
[m_{D^{0}}^{2}-(\sqrt{s_{0}}-m_{K^{0}})^{2}]}}{2\ \sqrt{s_{0}}}.$ (52)

The scalar product ${\bf p_{2}}\cdot{\bf p_{0}}$ which enters the function
${D({\bf p_{2}},{\bf p_{0}})}$ is given by the relation

$4\ {\bf p_{2}}\cdot{\bf p_{0}}=s_{-}-s_{+}.$ (53)

One has

$g_{f_{2}K^{+}K^{-}}=m_{f_{2}}\ \sqrt{\frac{60\pi\
\Gamma_{f_{2}K^{+}K^{-}}}{q_{f_{2}}^{5}}},$ (54)

where $\Gamma_{f_{2}K^{+}K^{-}}$ is the $f_{2}(1270)$ decay constant into
$K^{+}K^{-}$ and the momentum $q_{f_{2}}=\frac{1}{2}\
\sqrt{m_{f_{2}}^{2}-4m_{K}^{2}}$. In Eq. (49), because of the large width of
the $f_{2}$ meson, an energy dependent total width $\Gamma_{f_{2}}(s_{0})$ has
been introduced (see Eqs. (121) and (122) in Ref. JPD_PRD89 ) such that

$\Gamma_{f_{2}}(s_{0})=\left(\frac{q}{q_{f_{2}}}\right)^{5}\
\frac{m_{f_{2}}}{\sqrt{s_{0}}}\
\frac{(q_{f_{2}}r)^{4}+3(q_{f_{2}}r)^{2}+9}{(qr)^{4}+3(qr)^{2}+9}\
\Gamma_{f_{2}},$ (55)

where $r=4.0$ GeV-1 and $q\equiv{|{\bf p_{2}}|}$. In Eq. (49) the $D^{0}\to
f_{2}$ effective transition form factor $F^{D^{0}f_{2}}(m_{K^{0}}^{2})$ will
be treated as a free complex parameter.

To this amplitude one has to add the isoscalar-tensor
$K^{0}[K^{+}K^{-}]_{D}^{0}$ DCS amplitude given by (see Fig. 4)

$T_{K^{0}[K^{+}K^{-}]_{D}^{0}}^{DCS}(s_{0},s_{-},s_{+})=\frac{\Lambda_{2}}{\Lambda_{1}}~{}T^{CF}_{\overline{K}^{0}\
[K^{+}K^{-}]_{D}^{0}}(s_{0},s_{-},s_{+}).$ (56)

The total isoscalar-tensor amplitude then reads

$\displaystyle T_{9}$ $\displaystyle=$ $\displaystyle
T^{CF}_{\overline{K}^{0}\
[K^{+}K^{-}]_{D}^{0}}(s_{0},s_{-},s_{+})+T_{K^{0}[K^{+}K^{-}]_{D}^{0}}^{DCS}(s_{0},s_{-},s_{+})$
(57) $\displaystyle=$ $\displaystyle-\frac{G_{F}}{2}\
(\Lambda_{1}+\Lambda_{2})\ a_{2}\ f_{K^{0}}\ \frac{1}{\sqrt{2}}\
F^{D^{0}f_{2}}(m_{K^{0}}^{2})\ g_{f_{2}K^{+}K^{-}}\ \frac{D({\bf p_{2}},{\bf
p_{0}})}{m_{f_{2}}^{2}-s_{0}-im_{f_{2}}\ \Gamma_{f_{2}}(s_{0})}.$

###  II.3 Annihilation amplitudes

####  II.3.1 Scalar amplitudes

The isoscalar-scalar CF annihilation amplitude corresponding to
$\overline{K}^{0}[K^{+}K^{-}]_{S,s}^{0}$ final states (Fig. 5) is given by

$\displaystyle
A_{\overline{K}^{0}[K^{+}K^{-}]_{S}^{0}}^{CF}(s_{0},s_{-},s_{+})$
$\displaystyle=$ $\displaystyle-\frac{G_{F}}{2}\ \Lambda_{1}\ a_{2}\
f_{D^{0}}\ (m_{K^{0}}^{2}-s_{0})\
\sum_{R_{S}}F_{0}^{\overline{K}^{0}R_{S}[K^{+}K^{-}]_{s}^{0}}(m_{D^{0}}^{2})\
$ (58) $\displaystyle\times$ $\displaystyle
G_{R_{S}[K^{+}K^{-}]_{s}^{0}}(s_{0})\ \langle
R_{S}[K^{+}K^{-}]|\overline{s}s\rangle$

while the isoscalar-scalar DCS amplitude corresponding to
${K}^{0}[K^{+}K^{-}]_{S,s}^{0}$ final states (Fig. 6) is

$A_{K^{0}[K^{+}K^{-}]_{S}^{0}}^{DCS}(s_{0},s_{-},s_{+})=\frac{\Lambda_{2}}{\Lambda_{1}}~{}A_{\overline{K}^{0}[K^{+}K^{-}]_{S}^{0}}^{CF}(s_{0},s_{-},s_{+}),$
(59)

where we have used the equality

$F_{0}^{{K}^{0}R_{S}[K^{+}K^{-}]_{s}^{0}}(m_{D^{0}}^{2})=F_{0}^{\overline{K}^{0}R_{S}[K^{+}K^{-}]_{s}^{0}}(m_{D^{0}}^{2}).$
(60)

Thus the total isoscalar-scalar annihilation amplitude reads

$\displaystyle A_{1}$ $\displaystyle=$ $\displaystyle
A_{\overline{K}^{0}[K^{+}K^{-}]_{S}^{0}}^{CF}(s_{0},s_{-},s_{+})+A_{K^{0}[K^{+}K^{-}]_{S}^{0}}^{DCS}(s_{0},s_{-},s_{+})$
(61) $\displaystyle=$ $\displaystyle-\frac{G_{F}}{2}\
(\Lambda_{1}+\Lambda_{2})\ a_{2}\ f_{D^{0}}\ (m_{K^{0}}^{2}-s_{0})$
$\displaystyle\times$
$\displaystyle\sum_{R_{S}}F_{0}^{{K}^{0}R_{S}[K^{+}K^{-}]_{s}^{0}}(m_{D^{0}}^{2})\
G_{R_{S}[K^{+}K^{-}]}(s_{0})\ \langle R_{S}[K^{+}K^{-}]|\overline{s}s\rangle.$

Following the steps in Sec.II.2 it leads to

$A_{1}=-\frac{G_{F}}{2}\ (\Lambda_{1}+\Lambda_{2})\ a_{2}\ f_{D^{0}}\
(m_{K^{0}}^{2}-s_{0})\ F_{0}^{K^{0}f_{0}}(m_{D^{0}}^{2})\
\frac{1}{\sqrt{2}}\chi^{s}\ \Gamma_{2}^{s*}(s_{0}),$ (62)

where $\chi^{s}$ is a complex constant and $\Gamma_{2}^{s*}(s_{0})$ is the
kaon strange scalar form factor.

The isovector-scalar annihilation DCS amplitude, associated to the
$K^{+}[K^{0}K^{-}]_{S}^{1}$ final states containing the $a_{0}(980)^{-}$ and
$a_{0}(1450)^{-}$, is given by

$\displaystyle A_{2}=A_{K^{+}[K^{0}K^{-}]_{S}^{1}}^{DCS}(s_{0},s_{-},s_{+})$
$\displaystyle=$ $\displaystyle-\frac{G_{F}}{2}\ \Lambda_{2}\ a_{2}\
f_{D^{0}}\
(m_{K}^{2}-s_{-})\sum_{R_{S}}F_{0}^{K^{+}R_{S}[{K}^{0}K^{-}]^{1}}(m_{D^{0}}^{2})\
$ (63) $\displaystyle\times$ $\displaystyle
G_{R_{S}[{K}^{0}K^{-}]^{1}}(s_{-})\ \langle
R_{S}[{K}^{0}K^{-}]^{1}|d\overline{u}\rangle$

and hence with,

$\displaystyle\sum_{R_{S}}F_{0}^{K^{+}R_{S}[{K}^{0}K^{-}]^{1}}(m_{D^{0}}^{2})\
G_{R_{S}[{K}^{0}K^{-}]^{1}}(s_{-})\ \langle
R_{S}[{K}^{0}K^{-}]^{1}|d\overline{u}\rangle=F_{0}^{K^{+}a_{0}^{-}}(m_{D^{0}}^{2})\
G_{1}(s_{-}),$ (64)

reads

$A_{2}=-\frac{G_{F}}{2}\ \Lambda_{2}\ a_{2}\ f_{D^{0}}\ (m_{K}^{2}-s_{-})\
F_{0}^{K^{+}a_{0}^{-}}(m_{D^{0}}^{2})\ G_{1}(s_{-}).$ (65)

The corresponding isovector-scalar annihilation CF amplitude associated to the
$K^{-}[\overline{K}^{0}K^{+}]_{S}^{1}$ reads

$\displaystyle
A_{3}=A_{K^{-}[\overline{K}^{0}K^{+}]_{S}^{1}}^{CF}(s_{0},s_{-},s_{+})$
$\displaystyle=$ $\displaystyle-\frac{G_{F}}{2}\ \Lambda_{1}\ a_{2}\
f_{D^{0}}\
(m_{K}^{2}-s_{+})\sum_{R_{S}}F_{0}^{K^{-}R_{S}[\overline{K}^{0}K^{+}]^{1}}(m_{D^{0}}^{2})\
$ (66) $\displaystyle\times$ $\displaystyle
G_{R_{S}[\overline{K}^{0}K^{+}]^{1}}(s_{+})\ \langle
R_{S}[\overline{K}^{0}K^{+}]^{1}|\overline{d}u\rangle$

and contains the $a_{0}(980)^{+}$ and $a_{0}(1450)^{+}$. With the
approximation

$\displaystyle\sum_{R_{S}}F_{0}^{K^{-}R_{S}[\overline{K}^{0}K^{+}]^{1}}(m_{D^{0}}^{2})\
G_{R_{S}[\overline{K}^{0}K^{+}]^{1}}(s_{+})\ \langle
R_{S}[\overline{K}^{0}K^{+}]^{1}|\overline{d}u\rangle=F_{0}^{K^{-}a_{0}^{+}}(m_{D^{0}}^{2})\
G_{1}(s_{+})$ (67)

we reach

$A_{3}=-\frac{G_{F}}{2}\ \Lambda_{1}\ a_{2}\ f_{D^{0}}\ (m_{K}^{2}-s_{+})\
F_{0}^{K^{-}a_{0}^{+}}(m_{D^{0}}^{2})\ G_{1}(s_{+}).$ (68)

The $[K^{+}K^{-}]_{S}^{1}$ final states which would contain the
$a_{0}(980)^{0}$ and $a_{0}(1450)^{0}$ mesons cannot be formed from a
$s\overline{s}$ pair and thus the corresponding ${K}^{0}[K^{+}K^{-}]_{S}^{1}$
isovector-scalar DCS amplitude is zero.

####  II.3.2 Vector amplitudes

We now turn to the vector-annihilation amplitudes. The isoscalar-vector CF
amplitude corresponding to $\overline{K}^{0}[K^{+}K^{-}]_{P,s}^{0}$ final
states read (see Fig. 5)

$\displaystyle
A_{\overline{K}^{0}[K^{+}K^{-}]_{P}^{0}}^{CF}(s_{0},s_{-},s_{+})$
$\displaystyle=$ $\displaystyle\frac{G_{F}}{2}\ \Lambda_{1}\ a_{2}\ f_{D^{0}}\
(s_{-}-s_{+})\sum_{R_{P}}A_{0}^{\overline{K}^{0}R_{P}[K^{+}K^{-}]_{s}^{0}}(m_{D^{0}}^{2})\
m_{R_{P}}$ (69) $\displaystyle\times$ $\displaystyle
G_{R_{P}[K^{+}K^{-}]_{s}^{0}}(s_{0})\ \langle
R_{P}[K^{+}K^{-}]|\overline{s}s\rangle,$

and is associated to the $\phi$ mesons. It may be reexpressed as

$A_{\overline{K}^{0}[K^{+}K^{-}]_{P}^{0}}^{CF}(s_{0},s_{-},s_{+})=\frac{G_{F}}{2}\
\Lambda_{1}\ a_{2}\ \frac{f_{D^{0}}}{f_{\phi}}\ (s_{-}-s_{+})\
A_{0}^{\overline{K}^{0}\phi}(m_{D^{0}}^{2})\
F_{1}^{[K^{+}K^{-}]_{s}^{0}}(s_{0}).$ (70)

One has to add the associated DCS amplitude corresponding to
${K}^{0}[K^{+}K^{-}]_{P,s}^{0}$ final states (see Fig. 6); since
$A_{0}^{\overline{K}^{0}\phi}(m_{D^{0}}^{2})=A_{0}^{K^{0}\phi}(m_{D^{0}}^{2})$
we have

$\displaystyle A_{4}=\frac{G_{F}}{2}\ (\Lambda_{1}+\Lambda_{2})\ a_{2}\
\frac{f_{D^{0}}}{f_{\phi}}\ (s_{-}-s_{+})\ A_{0}^{{K}^{0}\phi}(m_{D^{0}}^{2})\
F_{1}^{[K^{+}K^{-}]_{s}^{0}}(s_{0}).$ (71)

The isovector amplitude corresponding to
${K}^{-}[\overline{K}^{0}K^{+}]_{P}^{1}$ final states, which contains the
$\rho(770)^{+}$, $\rho(1450)^{+}$ and $\rho(1700)^{+}$ mesons,

$\displaystyle A_{5}$ $\displaystyle=$ $\displaystyle-\frac{G_{F}}{2}\
\Lambda_{1}\ a_{2}\ \frac{f_{D^{0}}}{f_{\rho}}\
\left[s_{0}-s_{-}+\frac{(m_{D^{0}}^{2}-m_{K}^{2})(m_{K^{0}}^{2}-m_{K}^{2})}{s_{+}}\right]$
(72) $\displaystyle\times$
$\displaystyle\sum_{R_{P}}A_{0}^{K^{-}R_{P}[\overline{K}^{0}K^{+}]^{1}}(m_{D^{0}}^{2})\
m_{R_{P}}\ G_{R_{P}[\overline{K}^{0}K^{+}]^{1}}(s_{+})\ \langle
R_{P}[\overline{K}^{0}K^{+}]|\overline{d}u\rangle$

may be written as

$A_{5}=-\frac{G_{F}}{2}\ \Lambda_{1}\ a_{2}\ \frac{f_{D^{0}}}{f_{\rho}}\
\left[s_{0}-s_{-}+\frac{(m_{D^{0}}^{2}-m_{K}^{2})(m_{K^{0}}^{2}-m_{K}^{2})}{s_{+}}\right]\
A_{0}^{K^{-}\rho^{+}}(m_{D^{0}}^{2})\
F_{1}^{[\overline{K}^{0}K^{+}]^{1}}(s_{+}).$ (73)

Similarly, the isovector-DCS amplitude corresponding to
${K}^{+}[K^{0}K^{-}]_{P}^{1}$ final states reads

$A_{6}=-\frac{G_{F}}{2}\ \Lambda_{2}\ a_{2}\ \frac{f_{D^{0}}}{f_{\rho}}\
\left[s_{0}-s_{+}+\frac{(m_{D^{0}}^{2}-m_{K}^{2})(m_{K^{0}}^{2}-m_{K}^{2})}{s_{-}}\right]\
A_{0}^{K^{+}\rho^{-}}(m_{D^{0}}^{2})\ F_{1}^{[K^{0}K^{-}]^{1}}(s_{-}).$ (74)

It contains the $\rho(770)^{-}$, $\rho(1450)^{-}$ and $\rho(1700)^{-}$ mesons.

####  II.3.3 Tensor amplitudes

Finally we present the tensor amplitudes. The two isoscalar CF and DCS
amplitudes associated to the $\overline{K}^{0}[K^{+}K^{-}]_{D,s}^{0}$ and
$K^{0}[K^{+}K^{-}]_{D,s}^{0}$ final states read respectively

$\displaystyle
A_{\overline{K}^{0}[K^{+}K^{-}]_{D}^{0}}^{CF}(s_{0},s_{-},s_{+})$
$\displaystyle=$ $\displaystyle\frac{G_{F}}{2}\ \Lambda_{1}\ a_{2}\ f_{D^{0}}\
{D({\bf p_{2}},{\bf
p_{0}})}\sum_{R_{D}}F^{\overline{K}^{0}R_{D}[K^{+}K^{-}]_{s}^{0}}(m_{D^{0}}^{2})$
(75) $\displaystyle\times$ $\displaystyle
G_{R_{D}[K^{+}K^{-}]_{s}^{0}}(s_{0})\ \langle
R_{D}[K^{+}K^{-}]|\overline{s}s\rangle$

and

$\displaystyle
A_{K^{0}[K^{+}K^{-}]_{D}^{0}}^{DCS}(s_{0},s_{-},s_{+})=\frac{\Lambda_{2}}{\Lambda_{1}}~{}A_{\overline{K}^{0}[K^{+}K^{-}]_{D}^{0}}^{CF}(s_{0},s_{-},s_{+})$
(76)

They contain the $f_{2}(1270)$ meson. In the last equation we have used the
relation

$F^{\overline{K}^{0}R_{D}[K^{+}K^{-}]_{s}^{0}}(m_{D^{0}}^{2})=F^{{K}^{0}R_{D}[K^{+}K^{-}]_{s}^{0}}(m_{D^{0}}^{2}).$
(77)

Hence the total isoscalar-tensor amplitude reads

$\displaystyle A_{7}$ $\displaystyle=$ $\displaystyle
A_{\overline{K}^{0}[K^{+}K^{-}]_{D}^{0}}^{CF}(s_{0},s_{-},s_{+})+A_{K^{0}[K^{+}K^{-}]_{D}^{0}}^{DCS}(s_{0},s_{-},s_{+})$
(78) $\displaystyle=$ $\displaystyle\frac{G_{F}}{2}\
(\Lambda_{1}+\Lambda_{2})\ a_{2}\
f_{D^{0}}\frac{1}{\sqrt{2}}F^{K^{0}f_{2}}(m_{D^{0}}^{2})~{}g_{f_{2}K^{+}K^{-}}\
\frac{D({\bf p_{2}},{\bf p_{0}})}{m_{f_{2}}^{2}-s_{0}-im_{f_{2}}\
\Gamma_{f_{2}}(s_{0})}.$

###  II.4 Combination of amplitudes

The full scalar amplitude ${\cal M}_{1}(s_{0})$ is built up by the isoscalar
and isovector amplitudes associated to the channel $[K^{+}K^{-}]_{S}K^{0}_{S}$
with the $f_{0}$ and $a_{0}^{0}$ resonances [Eqs. (18), (29) and (62)]

${\cal M}_{1}(s_{0})=T_{1}+A_{1}+T_{4}={\cal M}_{1}^{n,I=0}(s_{0})+{\cal
M}_{1}^{s,I=0}(s_{0})+{\cal M}_{1}^{I=1}(s_{0}),$ (79)

In Eq. (79) the ${\cal M}_{1}^{n,I=0}(s_{0})$ and ${\cal
M}_{1}^{s,I=0}(s_{0})$ amplitudes are associated with the isoscalars $f_{0}$,

${\cal M}_{1}^{n,I=0}(s_{0})=-\frac{G_{F}}{2}\ (\Lambda_{1}+\Lambda_{2})\
a_{2}\ f_{K^{0}}\ (m_{D^{0}}^{2}-s_{0})\ F_{0}^{D^{0}f_{0}}(m_{K^{0}}^{2})\
\frac{\chi^{n}}{2}\Gamma_{2}^{n*}(s_{0}),$ (80) ${\cal
M}_{1}^{s,I=0}(s_{0})=-\frac{G_{F}}{2}\ (\Lambda_{1}+\Lambda_{2})\ a_{2}\
f_{D^{0}}\ (m_{K^{0}}^{2}-s_{0})\
F_{0}^{K^{0}f_{0}}(m_{D^{0}}^{2})\frac{\chi^{s}}{\sqrt{2}}\Gamma_{2}^{s*}(s_{0}),$
(81)

while the ${\cal M}_{1}^{I=1}(s_{0})$ amplitude is associated with the
isovectors $a_{0}^{0}$

${\cal M}_{1}^{I=1}(s_{0})=-\frac{G_{F}}{2}\ (\Lambda_{1}+\Lambda_{2})\ a_{2}\
f_{K^{0}}\ (m_{D^{0}}^{2}-s_{0})\ F_{0}^{D^{0}a_{0}^{0}}(m_{K^{0}}^{2})\
\frac{1}{2}G_{1}(s_{0}).$ (82)

The isoscalar amplitudes corresponding to the $\omega$ mesons [Eqs. (34) and
(71)] can be recombined with the isovector amplitudes [Eq. (39)] related to
the $\rho^{0}$ mesons

$\displaystyle{\cal M}_{2}$ $\displaystyle=$ $\displaystyle T_{5}+T_{6}+A_{4}$
(83) $\displaystyle=$ $\displaystyle\frac{G_{F}}{2}\
(\Lambda_{1}+\Lambda_{2})\ a_{2}\
(s_{-}-s_{+})\left[\frac{f_{K^{0}}}{f_{\rho}}\
A_{0}^{D^{0}\rho}(m_{K^{0}}^{2})\ F_{u}^{[K^{+}K^{-}]}(s_{0})\right.$
$\displaystyle\left.\hskip 142.26378pt+\frac{f_{D^{0}}}{f_{\phi}}\
A_{0}^{K^{0}\phi}(m_{D^{0}}^{2})\ F_{1}^{[K^{+}K^{-}]^{0}_{s}}(s_{0})\right].$

Here we have used $A_{0}^{D^{0}\omega}(m_{K^{0}}^{2})/f_{\omega}\approx
A_{0}^{D^{0}\rho}(m_{K^{0}}^{2})/f_{\rho}$ and defined

$F_{u}^{K^{+}K^{-}}(s_{0})=F_{1}^{[K^{+}K^{-}]^{0}_{u}}+F_{1}^{[K^{+}K^{-}]^{1}_{u}}.$
(84)

The form factors $F_{1}^{[K^{+}K^{-}]^{0}_{s}}(s_{0})\equiv
F_{s}^{K^{+}K^{-}}(s_{+})$ (in Eq. 83) and $F_{u}^{K^{+}K^{-}}(s_{0})$ have
been written in the forms given by Eqs. (23) and (25) of Ref. PLB699_102 ,
respectively. The first form factor takes contributions from the $\phi(1020)$
and $\phi(1680)$ resonances while the second one from eight vector meson
resonances $\rho(770)$, $\rho(1450)$, $\rho(1700)$ for the isovector part and
$\omega(782)$, $\omega(1420)$, $\omega(1680)$ for the isoscalar part, as
determined in Ref. Bruch2005  for the constrained fit to kaon form factors.

Since the isovector-scalar form factor
$F_{0}^{[\overline{K}^{0}K^{+}]^{1}}(s_{+})$ is related to the function
$G_{1}(s_{+})$ by the relation

$F_{0}^{[\overline{K}^{0}K^{+}]^{1}}(s_{+})=\frac{G_{1}(s_{+})}{G_{1}(0)},$
(85)

the isovector amplitude associated to the $a_{0}^{+}$ resonances in the
channel $[K^{0}_{S}K^{+}]_{S}K^{-}$ [Eqs. (24) and (68)] can be expressed as

$\displaystyle{\cal M}_{3}(s_{+})=T_{3}+A_{3}=\frac{G_{F}}{2}\ \Lambda_{1}$
$\displaystyle\left[a_{1}\ (m_{D^{0}}^{2}-m_{K}^{2})\
\frac{m_{K}^{2}-m_{K^{0}}^{2}}{s_{+}}\ F_{0}^{D^{0}K^{-}}(s_{+})\
\frac{1}{G_{1}(0)}\right.$ (86) $\displaystyle\left.+a_{2}\ f_{D^{0}}\
(m_{K}^{2}-s_{+})F_{0}^{K^{-}a_{0}^{+}}(m_{D^{0}}^{2})\right]G_{1}(s_{+}).$

The amplitude associated to the $\rho^{+}$ resonances [Eqs. (42) and (72)]
reads

$\displaystyle{\cal M}_{4}$ $\displaystyle=$ $\displaystyle
T_{7}+A_{5}=-\frac{G_{F}}{2}\ \Lambda_{1}\
\left[s_{0}-s_{-}+(m_{D^{0}}^{2}-m_{K}^{2})\
\frac{(m_{K^{0}}^{2}-m_{K}^{2})}{s_{+}}\right]$ (87) $\displaystyle\hskip
85.35826pt\left\\{a_{1}\ F_{1}^{D^{0}K^{-}}(s_{+})+a_{2}\
\frac{f_{D^{0}}}{f_{\rho}}A_{0}^{K^{-}\rho^{+}}(m_{D^{0}}^{2})\right\\}\
F_{1}^{[\overline{K}^{0}K^{+}]^{1}}(s_{+}).$

The form factor
$F_{1}^{[\overline{K}^{0}K^{+}]^{1}}(s_{+})=2~{}F_{1}^{[K^{+}K^{-}]^{1}_{u}}(s_{+})$
gets contributions from the three $\rho$ resonances as explained below Eq.
(84).

The isovector amplitude associated to the $a_{0}^{-}$ resonances in the
channel $[K^{0}_{S}K^{-}]_{S}K^{+}$ [Eqs. (21) and (65)] is

$\displaystyle{\cal M}_{5}(s_{-})$ $\displaystyle=$ $\displaystyle
T_{2}+A_{2}=-\frac{G_{F}}{2}\ \Lambda_{2}\left[a_{1}\ f_{K^{+}}\
(m_{D^{0}}^{2}-s_{-})\ F_{0}^{D^{0}a_{0}^{-}}(m_{K}^{2})\right.$ (88)
$\displaystyle\hskip 113.81102pt\left.+a_{2}\ f_{D^{0}}\ (m_{K}^{2}-s_{-})\
F_{0}^{K^{+}a_{0}^{-}}(m_{D^{0}}^{2})\right]G_{1}(s_{-}).$

The isovector amplitude associated to the $\rho^{-}$ resonances is given by
Eqs. (45) and (74)

$\displaystyle{\cal M}_{6}=T_{8}+A_{6}$ $\displaystyle=$
$\displaystyle\frac{G_{F}}{2}\ \Lambda_{2}\
\left[s_{0}-s_{+}+(m_{D^{0}}^{2}-m_{K}^{2})\
\frac{(m_{K^{0}}^{2}-m_{K}^{2}}{s_{-}})\right]$ (89) $\displaystyle\times$
$\displaystyle\ \left\\{\
a_{1}\frac{f_{K^{+}}}{f_{\rho}}A_{0}^{D^{0}\rho^{-}}(m_{K}^{2})-a_{2}\frac{f_{D^{0}}}{f_{\rho}}\
A_{0}^{K^{+}\rho^{-}}(m_{D^{0}}^{2})\right\\}F_{1}^{[\overline{K}^{0}K^{+}]^{1}}(s_{-}),$

where we have applied the relation
$F_{1}^{[K^{0}K^{-}]^{1}}(s_{-})=F_{1}^{[\overline{K}^{0}K^{+}]^{1}}(s_{-})$.

Finally, the isoscalar-tensor amplitudes related to the $f_{2}$ [Eqs. (57) and
(78)] can be recombined to give

$\displaystyle{\cal M}_{7}$ $\displaystyle=$ $\displaystyle T_{9}+A_{7}$ (90)
$\displaystyle=$ $\displaystyle-\frac{G_{F}}{2}\ (\Lambda_{1}+\Lambda_{2})\
a_{2}\frac{1}{\sqrt{2}}f_{K^{0}}P_{D}~{}g_{f_{2}K^{+}K^{-}}\ \frac{D({\bf
p_{2}},{\bf p_{0}})}{m_{f_{2}}^{2}-s_{0}-im_{f_{2}}\ \Gamma_{f_{2}}(s_{0})},$

where

$P_{D}=F^{D^{0}f_{2}}(m_{K^{0}}^{2})-\frac{f_{D^{0}}}{f_{K^{0}}}F^{K^{0}f_{2}}(m_{D^{0}}^{2})$
(91)

can be treated as a complex constant parameter fitted to data.

##  III Near threshold comparison of the $S$-wave $K^{+}K^{-}$ and
$K^{0}_{S}K^{+}$ effective mass projections

Our study can provide information on the $S$-wave content of the
$\overline{K}K$ effective mass densities. In the region of low effective
masses, near the $\overline{K}K$ thresholds, one expects dominant
contributions of the $S$\- and $P$-wave amplitudes which simplifies the
partial wave analysis of the experimental Dalitz plot distribution. This
analysis has been performed by the BABAR Collaboration for the following three
decay reactions: $D^{0}\to\overline{K}^{0}K^{+}K^{-}$B5 , $D^{0}\to
K^{-}K^{+}\pi^{0}$ B7  and $D_{s}^{+}\to K^{+}K^{-}\pi^{+}$ B11 . In the
$K^{+}K^{-}$ $S$-wave effective mass distributions both scalar resonances
$f_{0}(980)$ and $a_{0}(980)$ contribute while in the $\overline{K}^{0}K^{+}$
case only the $a_{0}(980)^{+}$ resonance is present.

In the analyses of Refs. B10  and B5  the $f_{0}(980)$ contribution has not
been introduced. A possible argument in favor of this choice has been
formulated in Ref. B5 , namely the authors have expected that the presence of
the $f_{0}(980)$ resonance would lead to an excess in the $K^{+}K^{-}$ mass
spectrum with respect to $\overline{K}^{0}K^{+}$. However, based on the
limited statistics of 12540 events they have observed that both
$\overline{K}K$ spectra are approximately equal. Below, using a much larger
sample of about 80000 signal events B10 , we show that the $K^{+}K^{-}$ and
$\overline{K}K$ spectra in $D^{0}$ decays into $K^{0}_{S}K^{+}K^{-}$ are
significantly different for low $\overline{K}K$ effective masses. Thus, the
contribution of the $f_{0}(980)$ resonance is required to obtain a good
description of the data of Ref. B10 .

To proceed further, the definitions of the $\overline{K}K$ effective $S$-wave
mass distributions corrected for phase space are needed. If we denote by
$N(s_{0},s_{+})$ the number of events of the
$D^{0}~{}\to~{}\overline{K}^{0}K^{+}K^{-}$ reaction, the corresponding Dalitz-
plot density distribution is given by $d^{2}N(s_{0},s_{+})/ds_{0}ds_{+}$. The
$K^{0}_{S}K^{+}$ effective mass squared distribution corrected for phase space
can be then defined as

$\frac{dn_{K^{0}_{S}K^{+}}(s_{+})}{ds_{+}}=\frac{1}{s_{0max}-s_{0min}}\int_{s_{0min}}^{s_{0max}}\frac{d^{2}N(s_{0},s_{+})}{ds_{0}ds_{+}}ds_{0},$
(92)

Figure 7:  Comparison of $K^{+}K^{-}$ and $\overline{K}^{0}K^{+}$ $S$-wave
effective mass squared distributions corrected for phase space as functions of
the variable $m_{KK}$ in bins of 2 MeV [see Eqs. (92) and (96)]. The variable
$m_{KK}$ is equal to $m_{K^{0}_{S}K^{+}}$ for the $\overline{K}^{0}K^{+}$ data
points (open red circles) or equal to $m_{0}$ for the $K^{+}K^{-}$ data points
(filled black circles). In the left panel a) the $\overline{K}^{0}K^{+}$
distribution is normalized to the number of events of the $K^{+}K^{-}$
distribution when integrated over $m_{KK}$ from the $K^{+}K^{-}$ threshold up
to 1.05 GeV. In the right panel b) the $\overline{K}^{0}K^{+}$ distribution
has not been renormalized. The curves correspond to the theoretical model
calculations (see Sec. IV).

where $s_{0max}$ and $s_{0min}$ are the maximum and minimum $s_{0}$ values at
fixed $s_{+}$. If we limit ourselves to the low $s_{+}$ values (for example,
up to about 1.05 GeV2) then to a good accuracy the above distribution
corresponds to the $S$-wave part of the total decay amplitude related to the
isovector-scalar$a_{0}(980)^{+}$ resonance. The reason is that the dominant
$P$-wave contribution, related to the $\phi(1020)$ resonance, is only present
in the $K^{+}K^{-}$ decay channel. Moreover, the low-mass $K^{0}_{S}K^{+}$ and
$K^{+}K^{-}$ distributions are very well separated on the Dalitz plot B10 .

For the low effective $K^{+}K^{-}$ masses one has to subtract the $P$-wave
contribution. Following Ref. B5 , this can be done by calculating the
spherical harmonic moments

$\sqrt{4\pi}\langle
Y^{0}_{0}(s_{0})\rangle=\int_{s_{+min}}^{s_{+max}}\frac{d^{2}N}{ds_{0}ds_{+}}ds_{+}$
(93)

and

$\sqrt{4\pi}\langle
Y^{2}_{0}(s_{0})\rangle=\sqrt{5}\int_{s_{+min}}^{s_{+max}}P_{2}(cos\theta)\frac{d^{2}N}{ds_{0}ds_{+}}ds_{+},$
(94)

with

$P_{2}(cos\theta)=\frac{1}{2}(3~{}cos^{2}\theta-1),$ (95)

and where $\theta$ is the helicity angle of the $K_{S}^{0}$ meson defined with
respect to the $K^{+}$ direction in the $K^{+}K^{-}$ center-of-mass frame,
$s_{+max}$ and $s_{+min}$ being the maximum and minimum $s_{+}$ values at
fixed $s_{0}$. The $S$-wave $K^{+}K^{-}$ effective mass squared distribution
corrected for phase space is then defined as

$\frac{dn_{K^{+}K^{-}}(s_{0})}{ds_{0}}=\frac{\sqrt{4\pi}}{s_{+max}-s_{+min}}[\langle
Y^{0}_{0}(s_{0})\rangle-\frac{\sqrt{5}}{2}\langle Y^{2}_{0}(s_{0})\rangle].$
(96)

For completeness we give below the kinematical relation for the cosine of the
helicity angle

$cos\ \theta=\frac{s_{-}-s_{+}}{s_{+max}-s_{+min}},\hskip 28.45274pt{\rm
where}\hskip 28.45274pts_{+max}-s_{+min}=4\lvert{\bf p}_{+}\rvert\lvert{\bf
p}_{0}\rvert,$ (97)

with $|{\bf p}_{+}|$ and $|{\bf p}_{0}|$ defined by Eqs. (51) and (52),
respectively.

We have performed a simplified partial wave analysis of the BABAR data
published in Ref. B10 . As described above, only the $S$\- and $P$-waves have
been included and the effective $K^{0}_{S}K^{+}$ and $K^{+}K^{-}$ masses were
smaller than 1.05 GeV2. The number of signal events of the
$D^{0}\to\bar{K}^{0}K^{+}K^{-}$ decays was 79900$\pm$300\. Based on the Dalitz
plot density distributions corrected for reconstruction efficiency and
background, the $S$-wave $K^{0}_{S}K^{+}$ and $K^{+}K^{-}$ effective mass
distribution corrected for phase space are calculated using Eqs. (92) and
(96).

The comparison of the calculated $S$-wave $K^{+}K^{-}$ and
$\overline{K}^{0}K^{+}$ distributions is shown in Fig. 7. In the left panel a)
a clear surplus of the $\bar{K}^{0}K^{+}$ distribution over the $K^{+}K^{-}$
one is seen below 1.02 GeV. Above $m_{KK}=1.02$ GeV the open circles
corresponding to $\overline{K}^{0}K^{+}$ spectrum are in majority located
below the closed circles ($K^{+}K^{-}$ events), so we observe a crossing of
the two distributions. This effect is statistically significant. It was not so
clear in 2005 when the first set of the BABAR data was published. But even
then, in Fig. 8 of Ref. B5 , one can see the same cross-over tendency as in
Fig. 7 although the statistics was lower by a factor larger than 6. In the
right panel b), one sees that unrenormalized $\overline{K}^{0}K^{+}$
distribution is lower than the $K^{+}K^{-}$ distribution by a factor of about
4. The lines show the corresponding theoretical distributions based on the
best fit to the Dalitz plot density distributions described in the next
section555These distributions are also relatively well described by the two
alternative models given in the Appendix B except for the two first data point
of the $m_{KK}$ distribution..

In conclusion, the shape of the $K^{+}K^{-}$ and $\overline{K}^{0}K^{+}$
$S$-wave effective mass squared distributions, corrected for phase space, is
significantly different, so in the phenomenological analysis of the
$D^{0}\to\overline{K}^{0}K^{+}K^{-}$ data one cannot neglect the $f_{0}(980)$
contribution in the decay amplitude.

##  IV Results and discussion

The differential branching fraction or the Dalitz plot density distribution is
defined as

$\frac{d^{2}{\rm Br}}{ds_{+}ds_{0}}=\frac{|{\cal
M}|^{2}}{32(2\pi)^{3}m_{D^{0}}^{3}\Gamma_{D^{0}}},$ (98)

where ${\cal M}=\sum_{i=1}^{7}{\cal M}_{i}$ is the decay amplitude for the
process studied and $\Gamma_{D^{0}}$ is the $D^{0}$ width. The decay
amplitudes ${\cal M}_{i}$ have been derived in Sec. II. In Table 1 one can
find some constant parameters which appear in these amplitudes.

Table 1: Values of coupling constants (in GeV) and the fixed form factors. Parameter | Value | Reference
---|---|---
$f_{K^{0}}$=$f_{K^{+}}$ | 0.1561 | JPD_PRD89
$f_{\rho}$ | 0.209 | Beneke2003
$f_{\phi}$ | 0.22 | Ball2007
$f_{D^{0}}$ | 0.2067 | JPD_PRD89
$F_{0}^{D^{0}f_{0}}(m_{K^{0}}^{2})$=$F_{0}^{D^{0}a_{0}^{0}}(m_{K^{0}}^{2})$ | 0.18 | El-Bennich_PRD79
$A_{0}^{D^{0}\rho^{0}}(m_{K^{0}}^{2})$ | $0.7$ | JPD_PRD89

To make a comparison of experimental data with model predictions the Dalitz
diagram has been divided into five regions as shown in Fig. 8. The dimensions
in different regions have been adjusted to the density of experimental events.
This has been done in two steps. In the first step the units $u_{0}=8.94\times
10^{-4}$ GeV2 and $u_{+}=8.97\times 10^{-4}$ GeV2 corresponding to the one
thousand of the full kinematic range of the variables $s_{0}$ and $s_{+}$ have
been chosen. A small difference between $u_{0}$ and $u_{+}$ comes form the
difference between the $K^{0}_{S}$ and $K^{+}$ masses. The cells in the ranges
I, III and IV have the dimensions $11u_{0}\times 11u_{+}$ while in the range
II the cells are larger having the dimensions $41u_{0}\times 41u_{+}$. Because
of the high density of experimental events around the position of the
$\phi(1020)$ resonance, the cells in the narrow range V have the dimensions
$1u_{0}\times 35u_{+}$. In the second step we have checked whether the
experimental number of events in a given cell was higher than ten. When this
was not the case the adjacent cells have been combined together to group a
sufficient number of events in an enlarged cell. Altogether the total number
of cells was equal to $N_{cells}=1196$ (including 164 enlarged cells). The
cell numbers in the regions I, II, III, IV and V were equal to 135, 282, 242,
187 and 350, respectively.

The fit of the model parameters to the experimental data has been performed
using the $\chi^{2}_{tot}$ function defined as a sum of two components:

$\chi^{2}_{tot}=\sum_{i=1}^{N_{cells}}\chi^{2}_{i}+\chi^{2}_{{\rm Br}}.$ (99)

The value of $\chi^{2}_{i}$ for each cell $i$ has been defined as in Ref. chi2
:

$\chi^{2}_{i}=2\Big{[}N^{exp}_{i}-N^{th}_{i}+N^{th}_{i}ln(\frac{N^{th}_{i}}{N^{exp}_{i}})\Big{]},$
(100)

where $N^{exp}_{i}$ is a number of experimental signal events in the cell $i$
corrected for the reconstruction efficiency and $N^{th}_{i}$ is the
theoretical number of events in the same cell666The efficiency and the signal
distributions on the Dalitz diagram have been provided to us by Fernando
Martinez-Vidal B10 . The samples of the $D^{0}$ and $\bar{D}^{0}$ decays into
$K^{0}_{S}K^{+}K^{-}$ have been combined.. Including the above corrections one
gets the total number of experimental events equal to $N^{exp}=80379$. The
total number of theoretical events is then taken equal to $N^{exp}$.

The second component of the $\chi^{2}$ function is given by

$\chi^{2}_{{\rm Br}}=w~{}\Big{(}\frac{{\rm Br}^{exp}-{\rm Br}^{th}}{\Delta{\rm
Br}^{exp}}\Big{)}^{2}.$ (101)

In our fit the experimental branching ratio for the decay $D^{0}\to
K_{S}^{0}K^{+}K^{-}$ has been taken equal to ${\rm Br}^{exp}=4.45\times
10^{-3}$ and its error $\Delta{\rm Br}^{exp}=0.34\times 10^{-3}$. These values
agree well with recent values of the Particle Data Group PDG2020 . The
theoretical branching fraction ${\rm Br}^{th}$ is obtained from Eq. (98) after
integrations of $\frac{d^{2}{\rm Br}}{ds_{+}ds_{0}}$ over the variables
$s_{+}$ and $s_{0}$. The weight factor $w$ in our fit has been set to 100 in
order to obtain a good agreement of the theoretical branching fraction with
its experimental value.

Figure 8: Partition of the Dalitz contour into five regions. Different sizes
of the ($s_{+},s_{0}$) cells are shown.

We have performed many fits with our model using different parameter sets. The
best fit $\chi^{2}~{}=~{}1474.4$ has been obtained with the nineteen free
parameters which are displayed in Table 2. Since the number of degrees of
freedom is $ndf=1196-19=1177$, the $\chi^{2}$ per degree of freedom is
$\chi^{2}/ndf=1.25$.

Table 2: Parameters of our model amplitudes and their errors. Phases are given in radians. Parameter | modulus | phase
---|---|---
$\chi_{n}$ | 22.5 GeV-1 fixed | $2.22^{+0.82}_{-0.98}$
$F_{0}^{K^{0}f_{0}}(m_{D^{0}}^{2})$ | $2.22^{+0.26}_{-0.17}$ | $2.21\pm 0.10$
$r_{2}$ |  $(5634^{+509}_{-560})$ GeV3/2 |
$r_{1}/r_{2}$ | 0.88 $\pm 0.01$ |
$s^{\prime}$ |  $(1.558^{+0.016}_{-0.014})$ GeV2 |
$p_{1}$ | (-1.84 $\pm 0.01)$ GeV-2 |
$p_{2}$ | (1.09 $\pm 0.01)$ GeV-4 |
$p_{3}$ | -(0.212 $\pm 0.004)$ GeV-6 |
$F_{0}^{K^{-}a_{0}^{+}}(m_{D^{0}}^{2})$ | $0.25^{+0.02}_{-0.03}$ | $5.33^{+0.12}_{-0.08}$
$A_{0}^{K^{0}\phi}(m_{D^{0}}^{2})$ | 0.985 $\pm 0.007$ | $3.67^{+0.12}_{-0.09}$
$M_{\phi}$ | (1019.58 $\pm 0.02$) MeV |
$\Gamma_{\phi}$ | (4.72 $\pm 0.04$) MeV |
$A_{0}^{K^{-}\rho^{+}}(m_{D^{0}}^{2})$ | $9.38^{+0.63}_{-0.58}$ | $5.01^{+0.06}_{-0.05}$
$P_{D}$ | $5.52^{+1.25}_{-1.24}$ | $3.97^{+0.23}_{-0.25}$

The value of the constant $|\chi^{n}|$ has been estimated using a relation
derived similarly as Eq. (18) in Ref. AF  in which the coupling constants of
the $f_{0}(980)$ resonance to the $K^{+}K^{-}$ pair are taken into account
instead of the $f_{0}(980)$ coupling to the $\pi\pi$ system. However, in the
present case one has to include two close $f_{0}(980)$ poles sitting on the
sheets (-+-) and (-++) (for their complex energy positions, $E_{R_{1}}$ and
$E_{R_{2}}$, see Table 9 in Appendix A). One can generalize Eq.(18) from Ref.
AF , valid for the pole position of a single resonance, to the case of two
close resonances:

$|\chi^{n}|\approx\frac{1}{|\Gamma_{2}^{n}(s_{0})|}\Big{|}\frac{g_{R_{1}K^{+}K^{-}}}{E_{R_{1}}^{2}-s_{0}}+\frac{g_{R_{2}K^{+}K^{-}}}{E_{R_{2}}^{2}-s_{0}}\Big{|},$
(102)

where $g_{R_{1}K^{+}K^{-}}$ and $g_{R_{2}K^{+}K^{-}}$ are the coupling
constants of the two $f_{0}$ resonances to $K^{+}K^{-}$. If one takes the
effective $K^{+}K^{-}$ mass in the range between 960 MeV and 990 MeV then
using Eq. (102) the averaged value of $|\chi^{n}|$ calculated in this range is
22.5 GeV-1.

The magnitude of the $\chi^{s}$ parameter is taken equal to that of
$|\chi^{n}|$ and its phase is set to zero. The reason for this choice is the
presence of the undetermined complex value of the form factor
$F_{0}^{K^{0}f_{0}}(m^{2}_{D^{0}})$ which is multiplied by $\chi^{s}$ in the
${\cal M}_{1}$ amplitude. The $F_{0}^{K^{0}f_{0}}(m^{2}_{D^{0}})$ value
results from the fit to data.

The form factors $\Gamma_{2}^{n}(s_{0})$ and $\Gamma_{2}^{s}(s_{0})$ have been
calculated in a three-channel model of meson-meson interactions ($\pi\pi$,
$K\bar{K}$ and an effective $2\pi\ 2\pi$), introduced in Ref. DedonderPol .
These form factors depend not only on the values of the meson-meson parameters
listed in Table 8 in Appendix A but also on two other parameters $\kappa$ and
$c$ defined by Eqs. (28) and (39) in Ref. DedonderPol , respectively. Their
values $\kappa=2807.3$ MeV and $c=0.109$ GeV-4 have been fitted to the
$B^{\pm}\to K^{\pm}K^{+}K^{-}$ decay data analyzed in Ref. KKK .

In Fig. 9 we show the effective energy dependence $E$ of moduli and phases of
the $K\overline{K}$ isoscalar-scalar nonstrange $\Gamma_{2}^{n}(E)$ and
strange $\Gamma_{2}^{s}(E)$ form factors. The energy $E$ is equal to the
square root of $s$.

Figure 9:  a) moduli of the isoscalar-scalar kaon nonstrange
$\Gamma_{2}^{n}(E)$ (solid line) and strange $\Gamma_{2}^{s}(E)$ (dashed line)
form factors; b) the corresponding phases.

In the above model the kaon threshold energy was set equal to the sum of the
charged and neutral kaon masses. However, the ${\cal M}_{1}$ amplitude
corresponds to the isoscalar $K^{+}K^{-}$ $S$-wave state with a threshold
lower by about 3.9 MeV in comparison with the $K^{+}K^{0}_{S}$ threshold
energy. In order to take this effect into account in an approximate way, we
introduce the variable

$\bar{s}_{0}=s_{0}\ \frac{s_{av}}{s_{th}}$

with $s_{th}=4m_{K}^{2}$ and the correction factor is
$s_{av}/s_{th}=1.008172$. The kaon form factors have to be evaluated at this
argument, i.e., $\Gamma_{2}^{n}(\bar{s}_{0})$ and
$\Gamma_{2}^{s}(\bar{s}_{0})$. To improve the quality of the data fit the form
factors $\Gamma_{2}^{n}(\bar{s}_{0})$ and $\Gamma_{2}^{s}(\bar{s}_{0})$ have
been multiplied by the function

$P(s_{0})=\frac{1-\frac{s_{0}-s_{th}}{s^{\prime}-s_{th}}}{1+bs_{0}^{3}},$
(103)

where $s^{\prime}$ is a new parameter which is fitted to the data (see Table
2). It corresponds to a zero of $P(s_{0})$. The third order polynomial in the
denominator, with the constant $b$ fixed to 0.0154 GeV-6, is introduced in
order to control asymptotically the high energy behavior of the ${\cal M}_{1}$
amplitude. This denominator replaces the denominator $(1+cE^{4})$ with
$E=\sqrt{s_{0}}$ in Eq. (39) of DedonderPol . A plot of the function (103)
used in our fit is shown as the continuous black line denoted RBF in Fig. 16
(a) where it is compared to the corresponding functions used in the two
alternative fits MOP1(P2) described in Appendix B. This function reduces the
moduli of the amplitudes which depend on the isoscalar-scalar form factors.

The masses and widths of the isovector-scalar resonances $a_{0}(980)$ and
$a_{0}(1450)$ are presented in Table LABEL:Table_a0. They have been fixed
during the minimization of the $\chi^{2}$ function. The parameters of the
$a_{0}(1450)$ on sheet $(-~{}-)$ were taken from Ref. PDG2020 . However, we
have studied the influence of the position of the $a_{0}(980)$ pole on sheet
$(-~{}+)$ in the complex energy domain on the $\chi^{2}$ minimum curve. In
this way the $a_{0}(980)$ mass and width on sheet $(-~{}+)$ have been
determined together with an estimation of their errors. The masses and widths
of other two associated $a_{0}$ poles are also given in Table LABEL:Table_a0.

Table 3: Parameters of resonances $a_{0}(980)$ and $a_{0}(1450)$. | mass (MeV) | width (MeV) | Riemann sheet
---|---|---|---
$a_{0}(980)$ | $979^{+3}_{-2}$ | $25^{+8}_{-6}$ | $-~{}~{}+$
$a_{0}(980)$ | $959$ | $34$ | $-~{}~{}-$
$a_{0}(1450)$ | $1474$ | $132$ | $-~{}~{}-$
$a_{0}(1450)$ | $1470$ | $91$ | $-~{}~{}+$
Table 4: Potential parameters of the $K\overline{K}$ $S$-wave isospin one interaction. $\beta_{1}$ | 21.662 GeV
---|---
$\beta_{2}$ | 21.831 GeV
$\lambda_{1}$ | $-2.9850\times 10^{-2}$
$\lambda_{2}$ | $-6.7977\times 10^{-2}$
$\lambda_{12}^{2}$ | $2.2142\times 10^{-7}$

The coupled channel model of the $a_{0}(980)$ and $a_{0}(1450)$ resonances
described in Ref. AFLL  has been implemented. There, the separable $\pi\eta$
and $K\overline{K}$ interactions have been used in the calculation of the
$S$-wave isospin one scattering amplitudes. Altogether the model has five
parameters: two range parameters $\beta_{1}$ and $\beta_{2}$, two channel
coupling constants $\lambda_{1}$ and $\lambda_{2}$, and the interchannel
coupling constant $\lambda_{12}$ (here the channel $\pi\eta$ is labeled by $1$
and the channel $K\overline{K}$ by $2$). The potential parameters are given in
Table LABEL:pot_param. There exist direct numerical relations between the four
parameters describing the positions of the $a_{0}(980)$ and $a_{0}(1450)$
resonances in the complex energy plane (Table LABEL:Table_a0) and the four
potential parameters $\beta_{1}$, $\lambda_{1}$, $\lambda_{2}$ and
$\lambda_{12}$ at fixed value of the fifth parameter $\beta_{2}$. These
relations are given in Ref. LL .

The function $G_{1}(s)$ is introduced to describe a transition from the
$u\bar{u}$ pair to the $K\overline{K}$ spin zero isospin one state. Two
isovector-scalar resonances $a_{0}(980)$ and $a_{0}(1450)$ can be formed
during that transition. Both resonances are also coupled to the $\pi\eta$
state. Therefore it is natural to consider three cases for the transition from
the $u\bar{u}$ pair to the $K\overline{K}$ state. In the first case the
$K\overline{K}$ pair is directly formed from the $u\bar{u}$ pair. In the
second case the $K\overline{K}$ pair undergoes the elastic rescattering in the
final state. In the third case the intermediate $\pi\eta$ pair is formed and
then the inelastic transition to the $K\overline{K}$ state takes place. The
interaction between the meson-meson pairs is treated in the framework of the
separable potential model fully described in Ref. LL  and used to study the
properties of the $a_{0}$ resonances (Refs. AFLL , AFLL2 ).

Below we briefly derive the dependence of the $G_{1}(s)$ function on the
meson-meson transition amplitudes. Labelling by 1 the $\pi\eta$ channel and by
2 the $K\overline{K}$ channel, one can express $G_{1}(s)$ as a superposition
of three terms:

$G_{1}(s)=R_{2}(s)+I_{22}(s)+I_{12}(s),$ (104)

where

$\displaystyle R_{2}(s)$ $\displaystyle=$ $\displaystyle r_{2}\ W(s)\
g_{2}(k_{2}),$ (105) $\displaystyle I_{22}(s)$ $\displaystyle=$ $\displaystyle
r_{2}\ W(s)\ \frac{T_{22}(s)}{g_{2}(k_{2})}\ C_{2}(s),$ (106) $\displaystyle
I_{12}(s)$ $\displaystyle=$ $\displaystyle r_{1}\ W(s)\
\frac{T_{12}(s)}{g_{1}(k_{1})}\ C_{1}(s).$ (107)

Here $r_{1}$ and $r_{2}$ are the coupling constants corresponding to the
$u\bar{u}$ transitions to the $\pi\eta$ and $K\overline{K}$ states,
respectively. The function $W(s)$ is the third-degree polynomial

$W(s)=1+p_{1}s+p_{2}s^{2}+p_{3}s^{3}.$ (108)

where $p_{1}$, $p_{2}$ and $p_{3}$ are the real parameters included in the
list of the model free parameters (see Table 2). We keep the same
$p_{j},j=1,2,3$, parameters for both channels. The fitted polynomial
$W(s_{0})$ is plotted as the continuous black line in Fig. 16(b) where it is
compared to the polynomial $P_{F}(s_{0})$ defined by Eq. (113) and used in the
alternative MOP1(P2) fits discussed in Appendix B. The introduction of these
polynomials improves the quality of the $\chi^{2}$ fit, in particular in the
region II where the density of events is small. The functions
$g_{i}(k_{i}),i=1,2$, are the vertex functions

$g_{i}(k_{i})=\sqrt{\frac{2\pi}{m_{i}}}\frac{1}{k_{i}^{2}+\beta_{i}^{2}},$
(109)

where $m_{i}$ are the channel reduced masses, $k_{i}$ are the channel momenta
and $\beta_{i}$ are the range parameters. In the $\pi\eta$ channel
$m_{1}=m_{\pi}m_{\eta}/(m_{\pi}+m_{\eta})$, in the $K\overline{K}$ channel
$m_{2}=m_{K}/2$. We take the neutral $\pi$ mass $m_{\pi}=134.977$ MeV and
$m_{\eta}=547.862$ MeV. The function $T_{22}(s)$ in Eq. (106) is the elastic
$K\overline{K}$scattering amplitude and $T_{12}(s)$ in Eq. (107) denotes the
transition amplitude from the $\pi\eta$ channel to the $K\overline{K}$
channel. In Eq. (106) one finds the integral

$C_{2}(s)=\int\frac{d^{3}p}{(2\pi)^{3}}\frac{g_{2}^{2}(p)}{E+i\epsilon-2E_{K}(p)}$
(110)

and in Eq. (107) we have

$C_{1}(s)=\int\frac{d^{3}p}{(2\pi)^{3}}\frac{g_{1}^{2}(p)}{E+i\epsilon-
E_{\pi}(p)-E_{\eta}(p)},$ (111)

where the energies are defined as $E=\sqrt{s}$,
$E_{K}(p)=\sqrt{p^{2}+m_{K}^{2}}$, $E_{\pi}(p)=\sqrt{p^{2}+m_{\pi}^{2}}$ and
$E_{\eta}(p)=\sqrt{p^{2}+m_{\eta}^{2}}$. The modulus and the phase of the
resulting $G_{1}(s)$ function are plotted in Figs. 10 (a) and 10 (b),
respectively.

Figure 10: a) modulus of the $G_{1}(s)$ function normalized to 1 at the
$K^{+}K^{-}$ threshold; b) phase of the $G_{1}(s)$ function ($E={\sqrt{s}}$).
At threshold $G_{1}(s_{th})$ = 304.69 GeV-1.

The importance of the annihilation diagrams in the description of the
experimental data should be here underlined. The annihilation terms,
proportional to $f_{D^{0}}$, are present in all the decay amplitudes ${\cal
M}_{i}$ and their magnitudes strongly dominate over other amplitudes which
contribute to the total decay amplitude. The annihilation amplitudes depend on
the appropriate form factors calculated for the momentum transfer squared
$m_{D^{0}}^{2}$. These fitted form factors are given in the second, ninth,
tenth and thirteenth rows of Table 2.

The mass and the width of the $\phi(1020)$ resonance seen in Table 2 are in
agreement with the corresponding BABAR values of ($1019.55\pm 0.02$) MeV and
($4.60\pm 0.04$) MeV, respectively B10 . The obtained width is higher, by
about 0.5 MeV, than the averaged value of ($4.249\pm 0.013$) MeV given by the
Particle Data Group in Ref. PDG2020 . This can be explained by a finite
experimental energy resolution.

The branching fraction distributions $\frac{d^{2}{\rm Br}_{i}}{ds_{+}ds_{0}}$
corresponding to the amplitudes ${\cal M}_{i}$, $i=1,...,7$ are obtained if in
Eq. (98) the amplitude ${\cal M}$ is replaced by ${\cal M}_{i}$. One can also
define the off-diagonal elements $\frac{d^{2}{\rm Br}_{ij}}{ds_{+}ds_{0}}$,
$i\neq j$,

$\frac{d^{2}{\rm Br}_{ij}}{ds_{+}ds_{0}}=\frac{Re[{\cal M}_{i}^{*}{\cal
M}_{j}]}{32(2\pi)^{3}m_{D^{0}}^{3}\Gamma_{D^{0}}}.$ (112)

If we integrate over $s_{+}$ and $s_{0}$ the differential branching fractions
$\frac{d^{2}{\rm Br}}{ds_{+}ds_{0}}$, $\frac{d^{2}{\rm Br}_{i}}{ds_{+}ds_{0}}$
and $\frac{d^{2}{\rm Br}_{ij}}{ds_{+}ds_{0}}$ then we get the corresponding
branching fractions ${\rm Br}$, ${\rm Br}_{i}$, $i=1$ to 7 or the off-diagonal
elements ${\rm Br}_{ij}$, where $i\neq j$. The matrix ${\rm Br}_{ij}$ is
symmetric: ${\rm Br}_{ij}={\rm Br}_{ji}$.

Table 5: Branching fractions (Br) for different quasi-two-body channels in the best fit to the BABAR data B10 . Amplitude | channel | Bri ($\%$)
---|---|---
${\cal{M}}_{1}$ | $[K^{+}\,K^{-}]_{S}\,K^{0}_{S}$ | $60.9^{+24.4}_{-10.6}$
${\cal{M}}_{2}$ | $[K^{+}\,K^{-}]_{P}\,K^{0}_{S}$ | $45.5\pm{0.7}$
${\cal{M}}_{3}$ | $[K^{0}_{S}\,K^{+}]_{S}\,K^{-}$ | $20.7^{+9.4}_{-6.0}$
${\cal{M}}_{4}$ | $[K^{0}_{S}\,K^{+}]_{P}\,K^{-}$ | $21.5^{+3.1}_{-2.8}$
${\cal{M}}_{5}$ | $[K^{0}_{S}\,K^{-}]_{S}\,K^{+}$ |  $0.76^{+0.18}_{-0.15}$
${\cal{M}}_{6}$ | $[K^{0}_{S}\,K^{-}]_{P}\,K^{+}$ |  $0.08\pm 0.01$
${\cal{M}}_{7}$ | $[K^{+}\,K^{-}]_{D}\,K^{0}_{S}$ |  $0.05\pm 0.02$
$\sum_{i=1,7}$ Bri |  | $149.5^{+26.9}_{-12.3}$

In Table 5 we give uncertainties of the branching fractions. They have been
obtained by choosing 10 000 different combinations of the 19 model parameters.
The parameters values have been generated from the Gaussian distributions
taking into account the parameter uncertainties written in Table 2 and some
correlations between the parameters in the amplitudes ${\cal{M}}_{1}$ and
${\cal{M}}_{3}$. Then the branching fraction uncertainties have been obtained
from the distributions of the 10 000 values of each branching fraction and of
their sum.

Let us notice the particularly large uncertainties of the branching fraction
${\rm Br}_{1}=60.9^{+24.4}_{-10.6}\%$. This is due to the fact that the
amplitude ${\cal{M}}_{1}$ consists of three components and contains 9 free
parameters.

As seen in Table 5 the largest contribution (near 61 $\%$) to the summed
branching fraction ${\rm Br}~{}=~{}\sum_{i=1}^{7}{\rm Br}_{i}$ comes from the
first amplitude ${\cal M}_{1}$. It corresponds to the quasi-two body channel
consisting of $K^{0}_{S}$ and the $K^{+}K^{-}$ pair in the $S$-wave. The
second contribution (near 46 $\%$) to the integrated branching fraction ${\rm
Br}$ is due to the amplitude ${\cal M}_{2}$. In this case the $K^{+}K^{-}$
pair is in the $P$-wave and its major part is related to the $\phi(1020)$
resonance. This resonance largely dominates in the region V of the Dalitz
diagram.

There are two almost equal contributions of about 21$\%$ from the channels
$[K^{0}_{S}\,K^{+}]_{S}\,K^{-}$ and $[K^{0}_{S}\,K^{+}]_{P}\,K^{-}$
(amplitudes ${\cal M}_{3}$ and ${\cal M}_{4}$, respectively). The ${\cal
M}_{3}$ amplitude can be related to a presence of the two isovector-scalar
resonances $a_{0}(980)$ and $a_{0}(1450)$. As seen in Table LABEL:Table_a0 the
mass of the resonance $a_{0}(980)$ equal to $979^{+3}_{-2}$ on sheet $-+$ is
lower than the $K^{0}_{S}K^{+}$ threshold mass of about 991.3 MeV. However,
due its finite width of $25^{+8}_{-6}$ MeV, this resonance, together with the
second $a_{0}(980)$ resonance on sheet $--$ at ($959-i~{}34$) MeV, can
strongly influence the near threshold $s_{+}$ range of the Dalitz plot density
distribution. On the other hand, the mass of the $a_{0}(1450)$ resonance lies
above the upper range of the $K^{0}_{S}K^{+}$ effective mass close to 1371
MeV. However, the $a_{0}(1450)$ resonances are wide and they can also affect
the distribution of the $D^{0}\to K^{+}K^{-}K^{0}_{S}$ events on the Dalitz
plot.

The contribution of the quasi-two body channel $[K^{0}_{S}\,K^{+}]_{P}\,K^{-}$
is related to nonzero couplings of the $P$-wave resonances $\rho(770)^{+}$,
$\rho(1450)^{+}$ and $\rho(1700)^{+}$ to $K^{0}_{S}K^{+}$. Although the
$\rho(770)^{+}$ mass lies below the $K^{0}_{S}\,K^{+}$ threshold its width is
sufficiently large to influence the differential density distribution of the
Dalitz plot for $s_{+}$ values above the threshold. The $\rho(1450)^{+}$ width
is even larger than that of $\rho(770)^{+}$, so the whole $s_{+}$ range on the
Dalitz plot is sensitive to the strength of its coupling to
$K^{0}_{S}\,K^{+}$. The above three $\rho$ resonances, being wide, cannot
create a clear structure or a well distinguished band on the Dalitz plot. This
could be a reason why they have not been included in the isobar model analyses
B10  and B5 .

These results can be compared to the results of the experimental analysis
which finds a summed branching fraction of 163.4 %, mainly with 71.1 % coming
from the $a_{0}(980)^{0}$ and $a_{0}(1450)^{0}$, 44.1 % from the $\phi(1020)$
resonance and 45.1 % from the $a_{0}(980)^{+}$ and $a_{0}(1450)^{+}$.

Table 6: Matrix of the branching fractions components Brij (Eq. (112)) for the best fit to the BABAR data B10 . All numbers are in per cent.  | ${\cal{M}}_{1}$ | ${\cal{M}}_{2}$ | ${\cal{M}}_{3}$ | ${\cal{M}}_{4}$ | ${\cal{M}}_{5}$ | ${\cal{M}}_{6}$ | ${\cal{M}}_{7}$
---|---|---|---|---|---|---|---
${\cal{M}}_{1}$ | 60.92 | 0.00 | 2.99 | -20.76 | -2.49 | -0.69 | 0.00
${\cal{M}}_{2}$ |  | 45.52 | -3.37 | -1.29 | -0.66 | -0.06 | 0.00
${\cal{M}}_{3}$ |  |  | 20.73 | 0.00 | -0.21 | 0.52 | 0.13
${\cal{M}}_{4}$ |  |  |  | 21.47 | 0.61 | 0.58 | -0.06
${\cal{M}}_{5}$ |  |  |  |  | 0.76 | 0.00 | 0.01
${\cal{M}}_{6}$ |  |  |  |  |  | 0.08 | 0.00
${\cal{M}}_{7}$ |  |  |  |  |  |  | 0.05

In Table 6 the diagonal branching fraction terms already shown in Table 5 are
given together with the off-diagonal terms ${\rm Br}_{ij}$. The sum of the
off-diagonal terms equals to -$49.53~{}\%$. One should remark here that some
off-diagonal terms are exactly equal to zero. This is due to the orthogonality
of certain wave functions. For example, the interference term ${\rm Br}_{12}$
vanishes since the wave functions of the $S$\- and $P$-states of the
$K^{+}K^{-}$ system are orthogonal. Due to the matrix symmetry the elements of
the branching fractions below the diagonal are not written.

The amplitude ${\cal M}_{1}$ is a sum of three terms [see Eqs. (79)-(82)]. The
first isoscalar term is proportional to the conjugated kaon nonstrange scalar
form factor $\Gamma_{2}^{*n}(s_{0})$ and the second one to the conjugated kaon
strange scalar form factor $\Gamma_{2}^{*s}(s_{0})$. The third term is
proportional to the function $G_{1}(s_{0})$ describing the transition from the
$u\bar{u}$ pair of quarks into the $K^{+}K^{-}$ pair of mesons in the isospin
one and spin zero state.

Table 7: Matrix of the branching fraction components Br of the ${\cal{M}}_{1}$ amplitude calculated for the best fit to the BABAR data B10 . All numbers are in per cent. | ${\cal{M}}^{n,I=0}_{1}$ | ${\cal{M}}^{s,I=0}_{1}$ | ${\cal{M}}^{I=1}_{1}$
---|---|---|---
${\cal{M}}^{n,I=0}_{1}$ |  1.19 | -7.32 | -0.36
${\cal{M}}^{s,I=0}_{1}$ |  | 59.82 | 5.40
${\cal{M}}^{I=1}_{1}$ |  |  |  4.48

In Table LABEL:TabBr123 the diagonal and the off-diagonal components of the
branching fraction related to the ${\cal M}_{1}$ amplitude are given. They are
defined in a similar way as the ${\rm Br}_{ij}$ components in Eq. (112). From
this Table we see that the major contribution close to 60% is related to the
strange scalar isospin zero component of the annihilation ($W$-exchange)
amplitude ${\cal M}_{1}^{s,I=0}$. Here the isoscalar-scalar resonances like
$f_{0}(980)$ are formed from the strange-antistrange pair of quarks. Following
the result of the fit shown in the above Table the formation of the isoscalar-
scalar resonances from the $u\bar{u}$ quarks is suppressed (the diagonal
${\cal M}_{1}^{n,I=0}$ branching fraction is equal only to 1.19 %). Also the
branching fraction equal to 4.48%, corresponding to the isovector-scalar
amplitude ${\cal M}_{1}^{I=1}$, is much smaller than that related to the
${\cal M}_{1}^{s,I=0}$ amplitude. The sum of all the off-diagonal components
equals to -4.57 %. A comparison of the results for this best fit model with
those for the alternative MOP1(P2) ones can be found in the Appendix B.

Figure 11:  Dalitz plot projections for $D^{0}~{}\to K^{0}_{S}K^{+}K^{-}$. The
points indicate the BABAR data B10  together with their statistical errors.
Histograms represent the best fit theoretical distributions.

Dalitz plot projections or one-dimensional effective mass squared
distributions of events are calculated by a proper integration of the two-
dimensional density distributions. They are shown in Fig. 11. The errors of
the experimental signal weighted event number distributions are the
statistical ones. The histograms correspond to the theoretical distributions
normalized to the same total number of events.

The distribution in Fig. 11(a) is strongly dominated by the maximum
corresponding to the $\phi(1020)$ resonance decaying to the $K^{+}K^{-}$ pair.
This decay is in the $P$-wave and leads to a characteristic two-maximum shape
of the Dalitz plot distribution as a function of $s_{+}$ \- the square of the
$K^{+}K^{-}$ effective mass. Since the branching fraction for the channel
$[K^{+}\,K^{-}]_{P}\,K^{0}_{S}$ is large (45.5 %) the two Dalitz projections
in Figs. 11(b) and(c) have a two-maximum character. There are also two other
important contributions related to the amplitudes ${\cal M}_{1}$ and ${\cal
M}_{3}$. However, they do not produce any pronounced structures on the Dalitz
plot since both are due to the $S$-wave in the $K^{+}K^{-}$ or in the
$K^{0}_{S}\,K^{+}$ configuration.

Finally let us discuss the low effective mass parts of the $K^{+}K^{-}$ and
$K^{0}_{S}\,K^{+}$ distributions ($m_{KK}<1.06$ GeV). Since the differential
branching fraction [Eq. (98)] is proportional to the Dalitz plot density
distribution of events $\frac{d^{2}N}{ds_{+}ds_{0}}$, one can calculate the
theoretical one-dimensional distributions of the event numbers using Eqs.
(92)-(94) of Sec. III. They are displayed in Fig. 11 as solid histograms. One
can see that the BABAR data agree well with the corresponding lines. This
agreement enforces the statement about the significant difference between the
$K^{+}K^{-}$ and $K^{0}_{S}\,K^{+}$ effective mass distributions which is due
to the dominant $f_{0}(980)$ resonance contribution to the $K\bar{K}$ final
state interaction amplitude.

##  V CONCLUSIONS

A theoretical model of the $D^{0}\to K_{S}^{0}K^{+}K^{-}$ decay amplitude has
been constructed within a quasi-two-body QCD factorization approach
introducing scalar kaon form factors to describe the $S$-wave kaon-kaon final
state interaction. In doing so, the contribution of isoscalar-scalar $f_{0}$
resonance family, viz. $f_{0}(980),f_{0}(1370)$ and that of the isovector-
scalar $a_{0}$ one, viz.
$a_{0}^{0}(980),a_{0}^{\pm}(980),a_{0}^{0}(1450),a_{0}^{\pm}(1450)$ are taken
into account. The isospin zero and one kaon-kaon $S$-wave interactions have
been treated in a unitary way using either coupled channel relativistic
equations, or a dispersion relation framework. The $P$\- and $D$\- waves of
the final state kaon-kaon interactions have also been taken into account.

Independently of any model assumptions, we have shown that the $K^{+}K^{-}$
and $\bar{K}^{0}K^{+}$ $S$-wave effective mass squared distributions,
corrected for phase space, are significantly different. This means that, in
the analyses of the $D^{0}\to\bar{K}^{0}K^{+}K^{-}$ data, one cannot neglect
the contribution of the $f_{0}(980)$ resonance and retain only the
$a_{0}(980)$ contribution.

In Appendix A, we have updated the meson-meson $S$-wave isospin zero
scattering amplitudes. These include the three coupled, $\pi\pi$, $\bar{K}K$
and an effective $2\pi\ 2\pi$ channels. Using the above amplitudes the new
kaon nonstrange and strange form factors $\Gamma_{2}^{n}(s_{0})$ and
$\Gamma_{2}^{s}(s_{0})$ have been calculated following Ref. KKK  and
introduced in the data analysis. As seen Fig. 14, these form factors are quite
similar to those derived using the Muskhelishvili-Omnès dispersion relation
approach Moussallam_2000 ; Moussallam_2019 .

In the factorization framework, for the $D^{0}\to K^{0}_{S}K^{+}K^{-}$ process
one has to evaluate the matrix elements of the $D^{0}$ transitions to two-
kaons or the transitions between one kaon and two kaons. The knowledge of
these transitions requires that of the three-body strong interaction between
the $D^{0},K^{0}_{S}$ and $K^{\pm}$ mesons and that between the
$K^{0}_{S},K^{+}$ and $K^{-}$ mesons. Here, to describe these transitions with
the two final kaons in $S$-wave state, we had to go beyond the simple
multiplication of the scalar kaon form factors by a complex constant. And to
obtain good fits we have multiplied the isoscalar-scalar kaon form factor by a
one free parameter energy-dependent function and introduced into the
isovector-scalar function an energy-dependent phenomenological polynomial
involving three free parameters.

The undetermined free parameters of our seven $D^{0}\to K^{0}_{S}K^{+}K^{-}$
amplitudes are then related to the strength of the isoscalar-scalar kaon form
factor, to the function proportional to the isovector-scalar kaon form factor
and to the unknown meson to meson transition form factors. They are obtained
through a $\chi^{2}$ minimization to the BABAR Dalitz plot distribution B10 .
It should be pointed out that the low density of events in the central region
of this Dalitz plot distributions (see Fig. 8) is difficult to reproduce.
Using unitary relativistic equations to built the isoscalar-scalar form factor
and a function proportional to the isovector-scalar one, we obtain a best fit
(denoted RBF), with a $\chi^{2}/ndf$ of 1.25 with 19 free parameters to be
compared to that of 1.28 for Ref. B10  which uses 17 free parameters.

In Appendix B, we have studied two alternative fits with scalar-kaon form
factors derived in the Muskhelishvili-Omnès dispersion relation framework. All
other amplitudes are parametrized as in the best fit model. If the scalar form
factors are multiplied by energy dependent phenomenological functions, we
obtain two good fits, one, denoted MOP1 with a $\chi^{2}/ndf$ of 1.32 and 16
free parameters and another one, MOP2, with a $\chi^{2}/ndf$ of 1.31 and 16
free parameters.

Our fits indicate the dominance of the annihilation amplitudes and for the
best fit a large dominance of the $[K^{+}K^{-}]_{S}$ isospin 0 $S$-wave
contribution and a sizable branching fraction to the
$[K^{0}_{S}K^{+}]_{P}K^{-}$ final state with the $[K^{0}_{S}K^{+}]$ pair
coupled to $\rho(770)^{+}$, $\rho(1450)^{+}$ and $\rho(1700)^{+}$. The
alternative fits show important contributions from both the $f_{0}$ and
$a_{0}^{0}$ mesons and a weaker $\rho^{+}$ mesons role. For all our models,
the one-dimensional distributions agree well with that of the BABAR data.

One can estimate the strength of the contributions of the different amplitudes
by looking at their branching ratio compared to the sum of their branching
ratios. As can be seen in Table LABEL:TabBr45Req for the best fit model this
sum777The numbers in brackets are the corresponding values of the MOP1 and
MOP2 fits, respectively is 149.5 [126.3, 164.1] % (163.4% in Ref. B10 ), which
points to sizable interference contributions. The kaon-kaon S-wave
interactions, related to the $f_{0}$ and $a_{0}^{0}$ resonances, gives a large
branching of $\sim$61 [45, 63] % with a large value (for BF, MOP1 and MOP2,
see Table LABEL:TabBr45Reqa) of $\sim$60 [23, 46] %, for the amplitude
proportional to the strange isoscalar-scalar form factor ($f_{0}$
contributions) and smaller branching $\sim$5 [16,16] % for the amplitude
proportional to the isovector-scalar form factor ($a_{0}^{0}$ contributions).
Corresponding figures in the isobar BABAR analysis B10  are $\sim$71 %,
dominated by the $a_{0}(980)^{0}$ and $a_{0}(1450)^{0}$ with no $f_{0}(980)$
and a $f_{0}(1370)$ $\sim$ 2% .

The branching fraction of the isospin 0 $P$-wave $\sim$46 [45, 45] %,
dominated by the $\phi(1020)$ resonance, is similar to that found, $\sim$44 %,
in Ref. B10 . The branching of the isovector amplitude associated to the
$a_{0}^{+}$ resonances is $\sim$21 [26, 40] % to be compared to $\sim$ 45 % in
Ref. B10 . The branching fraction of the amplitude related to the
$[\rho(770)^{+}+\rho(1450)^{+}+\rho(1700)^{+}]K_{S}^{0}$ final state, not
introduced in Ref. B10 , has a value of $\sim$22 [8, 13] %. One could say
that, this contribution with no bumps in the Dalitz plot distribution, is in
Ref.B10  taken into account by a part of that of the $a_{0}^{+}$.

The charmless hadronic $B^{0}\to K^{+}K^{-}K_{S}^{0}$ studied by the Belle
Be2010  and BABAR BA2012  Collaborations has the same meson final states as
the $D^{0}$ decay we have been studied here. A quasi-two-body QCD
factorization analysis of this $B^{0}$ decay process should allow, to
constrain, not only the weak interaction observables but also the scalar kaon
form factors, the transitions between one kaon and two kaons and to learn
about the $B^{0}$ transition to two kaons.

## Acknowledgements

We are grateful to François Le Diberder who, at the early stage of this work,
has helped us in getting access to the BABAR data. We are deeply indebted to
Fernando Martinez-Vidal from the BABAR Collaboration who provided us with
experimental information for this study. We thank him for many fruitful
exchanges. We are also indebted to Bachir Moussallam for very profitable
correspondence and for the communication of the results of his calculation of
scalar form factors. We also acknowledge helpful discussions with Piotr
Żenczykowski.

This work has been partially supported by a grant from the French-Polish
exchange program COPIN/CNRS-IN2P3, collaboration 08-127.

##  Appendix A Updated $\pi\pi$, $\bar{K}K$ and effective $2\pi\ 2\pi$
$S$-wave amplitudes

Here we present updated results for the meson-meson $S$-wave isospin zero
scattering amplitudes. They include the following three coupled channels:
$\pi\pi$, channel 1, $\bar{K}K$, channel 2 and effective $2\pi\ 2\pi$, channel
3. Our previous fits to the meson-meson scattering data were obtained in the
late nineties KLL ; EPJ . Since, new precise low energy $\pi\pi$ data have
appeared NA48 . Moreover, as noticed by Bachir Moussallam Moussallam_Nov2018 ,
we used an assumption valid only below the opening of the third channel,
namely the phase of the $\pi\pi~{}\to~{}\bar{K}K$ transition amplitude was set
equal to the sum of the elastic $\pi\pi$ and $\bar{K}K$ phaseshifts. The
derivation of the kaon isoscalar-scalar form factors $\Gamma_{2}^{n}(s_{0})$
and $\Gamma_{2}^{s}(s_{0})$, used in the present analysis for $s_{0min}=0.98$
GeV${}^{2}~{}\leq~{}s_{0}~{}\leq~{}s_{0max}=1.87$ GeV2, requires the knowledge
of the meson-meson amplitudes at energies above $s_{0max}$.

Thus, dropping the above mentioned assumption, we have performed a new
analysis based on an enlarged set of data. Using the same three coupled-
channel separable potential model as developed in Refs. KLL  and EPJ , we fit
the following data:

a) for the effective $\pi\pi$ mass $E$ between 286 and 390 MeV, the 10 values
of the elastic $\pi\pi$ phase shifts from the NA48 data NA48 ,

b) for $610$ MeV$\leq~{}E~{}\leq 1580$ MeV, the 50 values of the $\pi\pi$
phase shifts $\delta_{\pi\pi}$ and for $1010$ MeV$\leq~{}E~{}\leq 1580$ MeV
the 30 values of the $\pi\pi$ inelasticities $\eta_{\pi\pi}$, both quantities
obtained in the experimental analysis of Ref. KLRyb ,

c) for $995$ MeV$\leq~{}E~{}\leq 1580$ MeV, the 23 values of the moduli of the
transition $\pi\pi\to\overline{K}K$ amplitude $T_{12}$ extracted from Fig. 27
of Ref. Cohen ,

d) for the $\delta_{\pi\pi\to K\overline{K}}$ phases of the $T_{12}$
amplitude, the 21 values extracted in the analysis of Ref. Cohen  for $1016$
MeV$\leq~{}E~{}\leq 1530$ MeV,

e) plus the 6 data points for these phases between 1538 and 1741 MeV
determined in Ref. Etkin .

The total number of fitted data is then equal to 140. As in Ref. KLL , the
fitting method is based on the $\chi^{2}$ function being a sum of five
components related to the five data sets enumerated above. The resulting
$\chi^{2}$ is equal to 135.04 which, for 14 free model parameters, gives the
value $\chi^{2}/ndf=1.07$ when divided by $ndf=140-14=126$ degrees of freedom.

Table 8: Model parameters fitted to data. Parameter | value
---|---
$\Lambda_{11,1}$ | -0.14434 $\cdot 10^{-3}$
$\Lambda_{11,2}$ | -0.21102
$\Lambda_{22}$ | -0.62730
$\Lambda_{33}$ | -0.81318$\cdot 10^{-3}$
$\Lambda_{12,1}$ | 0.25184$\cdot 10^{-4}$
$\Lambda_{12,2}$ | 0.033294
$\Lambda_{13,1}$ | 0.25063$\cdot 10^{-4}$
$\Lambda_{13,2}$ | -0.34913
$\Lambda_{23}$ | -5.4206
$\beta_{1,1}$ | 3.0366$\cdot 10^{3}$ GeV
$\beta_{1,2}$ | 1.1019 GeV
$\beta_{2}$ | 0.98412 GeV
$\beta_{3}$ | 0.047940 GeV
$m_{3}$ | 0.75200 GeV

Figure 12:  Comparison to the data of our fit (solid line): a) the $\pi\pi$ elastic phase shifts versus the $\pi\pi$ center-of-mass energy $E$, b) the $\pi\pi$ inelasticities and for energies $E$ higher than the $\overline{K}K$ mass threshold. The data below $E$ = 400 MeV are taken from Ref. NA48  and those above 600 MeV from Ref. KLRyb  for the “down-flat” solution.  Table 9: Positions of $S$ matrix poles ($E$ in MeV). |  | Sign of |
---|---|---|---
Re $E$ | Im $E$ | Im $k_{1}$ Im $k_{2}$ Im $k_{3}$ |  n∘
227 | 0 |  $-$ $-$ $-$ | I
230 | 0 |  $-$ $-$ $+$ | II
230 | 0 |  $+$ $-$ $-$ | III
232 | 0 |  $+$ $-$ $+$ | IV
485 | -233 |  $-$ $+$ $-$ | V
485 | -233 |  $-$ $+$ $+$ | VI
506 | -262 |  $-$ $-$ $-$ | VII
507 | -265 |  $-$ $-$ $+$ | VIII
967 | -10 |  $-$ $+$ $-$ | IX
982 | -8 |  $-$ $+$ $+$ | X
1442 | -100 |  $-$ $+$ $-$ | XI
1444 | -93 |  $-$ $+$ $+$ | XII
1448 | -97 |  $-$ $-$ $+$ | XIII
1465 | -98 |  $-$ $-$ $-$ | XIV
1553 | -211 |  $+$ $-$ $-$ | XV
1559 | -213 |  $-$ $-$ $-$ | XVI
1581 | -138 |  $-$ $-$ $+$ | XVII
1584 | -134 |  $+$ $-$ $+$ | XVIII
Table 10: Coupling constants $g_{i}$ in GeV for a few representative $S$-matrix poles ($E$ in MeV). Re $E$ | Im $E$ | $|g_{1}|^{2}/4\pi$ | $|g_{2}|^{2}/4\pi$ | $|g_{3}|^{2}/4\pi|$ |  n∘
---|---|---|---|---|---
506 | -262 | 0.86 | 0.03 | 0.00 | VII
967 | -10 | 0.08 | 1.68 | 0.05 | IX
982 | -8 | 0.07 | 1.27 | 0.04 | X
1448 | -97 | 1.02 | 0.08 | 0.16 | XIII
1559 | -213 | 0.07 | 2.24 | 0.77 | XVI

Figure 13:  Comparison to the data Cohen  of our fit (solid line): a) Modulus
of the $\pi\pi\to\bar{K}K$ transition amplitude $T_{12}(E)$ normalized as in
Ref. Cohen , b) $T_{12}(E)$ phase.

The quality of our fit for the $\pi\pi$ phase shifts and inelasticities is
illustrated in Fig. 12. As seen in Fig. 13 a good fit is achieved for the
moduli and the phases of the $T_{12}$ amplitude. All experimental data sets
are well reproduced by our phenomenological model. The resulting separable
interaction parameters are listed in Table 8, their notation being identical
to that of Ref. KLL .

Positions of the $S$-matrix poles in the complex energy $E$ plane are given in
Table 9. The signs of the imaginary parts of the channel complex momenta
$k_{i}$, i=1,2,3 are indicated in order to mark the corresponding pole
position on different Riemann sheets. The total width $\Gamma$ of a given pole
equals to twice $\lvert$Im$E\rvert$.

As in the case of solution A (see Table 3 of Ref. EPJ ) one finds 18 poles.
The first four (I to IV), lying on the real axis below the $\pi\pi$ threshold,
are related to the $S$-matrix poles in the absence of interchannel couplings.
The next four (V to VIII), located on different sheets, correspond to the wide
resonance $f_{0}(500)$. There are two close poles (IX and X) related to the
narrow resonance $f_{0}(980)$ and four poles (XI to XIV) attributed to the
wider resonance $f_{0}(1400)$. The four poles (XV to XVIII) located between
1553 and 1584 MeV are responsible for the structure in the phase of the
transition $\pi\pi\to K\overline{K}$ amplitude as can be seen in the right
panel of Fig. 13 and in Fig. 6 of Ref. Etkin ; there is a maximum near 1500
MeV, close to the opening of the third channel, followed by a dip at about
1600 MeV. These latter poles could be related to the $f_{0}(1370)$ and
$f_{0}(1500)$ resonances.

In Table 10 we present values of the moduli of the channel coupling constants
calculated for five typical $S$ matrix poles (for their definitions see Eq.
(34) of Ref. EPJ ). The $f_{0}(500)$ poles like that with n∘ VII are mainly
coupled to the $\pi\pi$ channel ($i=1$). Also the four poles close to Re $E$ =
1450 MeV have a strong coupling only to the $\pi\pi$ channel. The $f_{0}(980)$
poles n∘s IX and X are preferentially coupled to the $\bar{K}K$ channel
($i=2$) like the four other poles n∘s XV to XVIII. This last group of poles
has also a substantial coupling to the $(2\pi)(2\pi)$ channel ($i=3$) in
addition to the strong coupling to the $\bar{K}K$ (i=3) one. All these poles
lie above the opening of the third channel taking place at $2m_{3}=1504$ MeV.

##  Appendix B Fits using kaon scalar form factors derived from dispersion
relation approach

Figure 14:  Moduli, a) for 0 GeV $\leqslant E\leqslant 2$ GeV and b) for 0.9
GeV $\leqslant E\leqslant~{}1.4$ GeV, of the isoscalar-scalar kaon form
factors calculated using the Muskhelishvili-Omnès dispersion relation approach
Moussallam_2019 . They are compared to those (continuous black lines) derived
in the best fit in Sec. IV from a unitary relativistic three coupled-channel
model. Both approaches use the updated $T_{\pi\pi}$ matrix derived in Appendix
A. The physical $E$ region, 0.987 GeV $\lesssim E\lesssim$1.367 GeV, is
delimited by the two vertical dashed lines.

In this appendix we complete our study by describing the results of two fits
of the BABAR-Collaboration Dalitz-plot distribution B10  taking, in the
amplitudes with final kaon-kaon states in $S$ wave and isospin 0, the scalar
$K\overline{K}$ form factors derived from the Muskhelishvili-Omnès (MO)
approach MO . The same parametrizations as those described in Sec. II are used
for all other amplitudes. In the MO dispersion-relation framework the
isoscalar-scalar $\Gamma_{2}^{n,s}(s)$ form factors have been calculated by B.
Moussallam Moussallam_2000 ; Moussallam_2019  from the MO equation using the
updated $\pi\pi$-$T$ matrix of the $\pi\pi$, $K\overline{K}$ and effective
$(2\pi)(2\pi)$ coupled-channel model of Ref. EPJ  (see Appendix A). In Fig. 14
the moduli of these MO form factors are compared to those derived in Sec. IV
from a relativistic coupled-channel model. In Sec. II one has introduced for
the form factors $\Gamma_{2}^{n,s}(s)$ complex phenomenological coefficients
of proportionality $\chi^{n,s}$ and in Sec. IV, to achieve good fits, notably
to reproduce the low density of events in the central region of the Dalitz
distribution (see Fig. 8, region II), we have been led to multiply them by the
energy-dependent phenomenological functions $P_{i}(s_{0})$ defined below in
Eqs. (114) and (115).

Figure 15:  Moduli of the isovector-scalar kaon form factors calculated from
MO equations in Ref. Bachir2015  for different $\delta_{12}$ parameters which
correspond to the sum of the $\eta\pi\to\eta\pi$ and $K\bar{K}\to K\bar{K}$
phase shifts at $\sqrt{s}=m_{a_{0}(1450)}$. The two vertical dashed lines
delimit the physical $s_{0}$ region, 0.975 GeV2 $\lesssim s\lesssim$1.87 GeV2.

The isovector-scalar $F_{0}^{[\overline{K}^{0}K^{+}]^{1}}(s)$ form factor has
been calculated in Ref. Bachir2015  from coupled MO equations for $\pi\eta$
and $K\overline{K}$ channels. Its modulus, for the parameters $\delta_{12}$=
100∘, 150∘ and 200∘ which are equal to the sum of the $\eta\pi\to\eta\pi$ and
$K\bar{K}\to K\bar{K}$ phase shifts at $\sqrt{s}=m_{a_{0}(1450)}$ , is plotted
in Fig. 15. The isovector amplitudes associated to the isospin-1 $a_{0}^{0}$
and $a_{0}^{+}$ resonances can be expressed in terms of this form factor by
using, in the Eqs. (82) and (86), the relation (85) with $G_{1}(0)=\chi^{1}$.
The strength $\chi^{1}$ is real and to obtain good fits, it was necessary to
multiply it by the phenomenological polynomial

$P_{F}(s_{+})\equiv
1+c_{1}(s_{+}-s^{\prime\prime})+c_{2}(s_{+}-s^{\prime\prime})^{2}+c_{3}(s_{+}-s^{\prime\prime})^{3},$
(113)

where the free parameters $c_{i},\ i=1,2,3$ and $s^{\prime\prime}$ are real.

Figure 16:  Comparison of: a) the functions $P_{i}(s_{0})$ multiplying, for
the three models MOP1, MOP2 and RBF, the isoscalar-scalar form factors
$\Gamma_{2}^{n(s)}(s_{0})$; b) the polynomials $P_{F}(s_{0})$ multiplying, for
the two models MOP1(P2), the isovector-scalar form factor
$F_{0}^{[\overline{K}^{0}K^{+}]^{1}}(s_{0})$ and $W(s_{0})$ introduced in Eq.
(108) for the $G_{1}(s_{0})$ function of the best fit RBF. Vertical dashed
lines as in Fig. 15.

An estimation of the phenomenological strength parameters $\chi^{n,s}$ using
Eq. (102) with $|\Gamma_{2}^{n,s}(m^{2}_{f_{0}})|\simeq~{}19$ (see Fig. 14)
leads to $|\chi^{n,s}|\simeq 26\ \rm{GeV}^{-1}$. For the kaon isovector-scalar
form factor with
$\chi^{1}|F_{0}^{[\overline{K}^{0}K^{+}]^{1}}(m^{2}_{a_{0}})|={g_{a_{0}K^{+}K^{-}}}/[{m_{a_{0}}\,\Gamma_{tot}(a_{0})}]=89.66~{}\rm{GeV}^{-1}$,
using (see Ref. AFLL ), $\Gamma_{tot}(a_{0})=71\pm 14$ MeV,
$|g_{a_{0}K^{+}K^{-}}|^{2}/(4\pi)=0.275$ GeV2, $m_{a_{0}}=980$ MeV and for
$\delta_{12}=150^{\circ}$
$|F_{0}^{[\overline{K}^{0}K^{+}]^{1}}(m^{2}_{a_{0}})|\simeq 10.89$ (see Fig.
15), one obtains $\chi^{1}\simeq 8.2\rm\ {GeV}^{-1}$. As expected, from our
study in Sec. III of the near threshold comparison between the $K^{+}K^{-}$
$S$-wave effective mass projection with that of the $\overline{K}^{0}K^{+}$, a
good fit without the contribution associated with the isospin 0 $f_{0}$
resonances ($\chi^{n,s}\equiv 0$) cannot be obtained.

Table 11: Comparison between the MOP1, MOP2 parameters from the two fits using kaon scalar-form factors derived from dispersion relations (see the text) and the corresponding RBF ones, from the best fit with kaon scalar-form factors calculated from unitary relativistic equation (see Table 2). The number of free parameters is denoted by $N_{p}$. Parameters without uncertainties are kept fixed during the minimization procedure. Phases $\varphi$ are in radians and detailed definitions of all parameters are given in the text. Fit | MOP1 | MOP2 | RBF
---|---|---|---
$\chi^{2}(N_{p},\chi^{2}/ndf)$ | 1559.7(16, 1.32) | 1546.9(16, 1.31) | 1474.4(19, 1.25)
$|\chi^{n}|$(GeV${}^{-1})$ | 26. | 35. | 22.5
$\varphi_{\chi^{n}}$ | 1.63 ${}^{+0.20}_{-0.18}$ | 4.19 ${\pm 0.10}$ | 2.22${}^{+0.82}_{-0.98}$
$\chi^{s}$(GeV${}^{-1})$ | 26. | 26. | 22.5
$|F_{0}^{K^{0}f_{0}}(m_{D^{0}}^{2})|$ | 0.42${}^{+0.03}_{-0.04}$ | 0.35${\pm 0.03}$ | 2.22${}^{+0.26}_{-0.17}$
$\varphi_{F_{0}^{K^{0}f_{0}}}$ | 2.40${}^{+0.05}_{-0.04}$ | 1.94 ${\pm 0.05}$ | 2.21${\pm 0.10}$
$s^{\prime}$(GeV-2) | 0. | -3. | 1.56${}^{+0.02}_{-0.01}$
$s^{\prime\prime}$(GeV2) | 1.5 | 1.5 |
$\chi^{1}$(GeV${}^{-1})$ | 8.2 | 15 |
$c_{1}$(GeV2) | -15.38 ${}^{+0.41}_{-0.45}$ | -7.53 ${\pm 0.14}$ |
$c_{2}$(GeV-2) | 8.16${}^{+0.66}_{-0.65}$ | 2.88${}^{+0.35}_{-0.34}$ |
$c_{3}$(GeV-4) | 37.20 ${}^{+1.99}_{-1.94}$ | 18.24${}^{+0.91}_{-0.89}$ |
$|F_{0}^{K^{-}a^{+}}(m_{D^{0}}^{2})|$ | 0.23${\pm 0.01}$ | 0.28${\pm 0.01}$ | 0.25${}^{+0.02}_{-0.03}$
$\varphi_{F_{0}^{K^{-}a^{+}}}$ | 5.82${\pm 0.05}$ | 5.61${}^{+0.05}_{-0.04}$ | 5.33${}^{+0.12}_{-0.08}$
$|A_{0}^{K^{0}\phi}(m_{D^{0}}^{2})|$ | 0.99$\pm{0.01}$ | 0.99${\pm 0.01}$ | 0.99${\pm 0.01}$
$\varphi_{A_{0}^{K^{0}\phi}}$ | -0.89${\pm 0.02}$ | -0.97${\pm 0.02}$ | 3.67${}^{+0.12}_{-0.09}$
$M_{\phi}$(MeV) | 1019.55$\pm 0.02$ | 1019.56$\pm 0.02$ | 1019.58$\pm 0.02$
$\Gamma_{\phi}$(MeV) | 4.69$\pm$0.04 | 4.70$\pm$0.04 | 4.72$\pm$0.04
$|A_{0}^{K^{-}\rho^{+}}(m_{D^{0}}^{2})|$ | 5.78${}^{+0.22}_{-0.25}$ | 7.94${}^{+0.34}_{-0.37}$ | $9.38^{+0.63}_{-0.58}$
$\varphi_{A_{0}^{K^{-}\rho^{+}}}$ | 1.18 $\pm{0.03}$ | 1.06${\pm 0.02}$ | 5.01${}^{+0.06}_{-0.05}$
$|P_{D}|$ | 15.71${}^{+0.78}_{-0.80}$ | 14.16${}^{+0.69}_{-0.70}$ | $5.52^{+1.25}_{-1.24}$
$\varphi_{P_{D}}$ | 1.13$\pm{0.09}$ | 0.97${}^{+0.10}_{-0.09}$ | 3.97${}^{+0.23}_{-0.25}$

We also find that improved $\chi^{2}$ are obtained with the $\delta_{12}$
parameter of the isovector-scalar $F_{0}^{[\overline{K}^{0}K^{+}]^{1}}$ form
factor equal to 150∘ (see Fig. 15). With the $N_{p}$=16 free parameters
displayed in Table LABEL:Tabparam, we obtain a fit, denoted as MOP1, with a
total $\chi^{2}$ of 1559.7 which corresponds to a $\chi^{2}/ndf$=1.32, not as
good as that found in the best fit model of Sec. IV. In this fit, the
phenomenological function multiplying the $\Gamma_{2}^{n(s)}(s_{0})$ is chosen
to be

$P_{1}(s_{0})\equiv P(s_{0}),$ (114)

with the zero $s^{\prime}$ of the function $P(s_{0})$ [Eq. (103)] at 0 GeV2.
Fixing $|\chi^{n}|$ to 35 (GeV${}^{-1})$, $\chi^{1}$ to 15 (GeV${}^{-1})$ a
slightly better fit, denoted MOP2, with $N_{p}=16$ and a total $\chi^{2}$ of
1546.9 ($\chi^{2}/ndf$=1.31) is obtained with

$P_{2}(s_{0})=1-s^{\prime}(s_{0}-s_{th}).$ (115)

Table LABEL:Tabparam gives then a comparison of all parameter values with
their uncertainties (when these parameters are fitted) for the MOP1, MOP2
models together with the corresponding parameters of the best fit RBF
presented in Sec. IV. The variations, in the $s_{0}$ physical region, of the
different functions $P_{i}(s_{0})$ for the MOP1 ($i=1,\ s^{\prime}=0$ GeV2) ,
MOP2 ($i=2$, $s^{\prime}=~{}{-3}$ GeV2) and RBF ($i=1,\ s^{\prime}=1.56$
GeV-2) fits are displayed in Fig. 16(a). As already indicated in Sec. IV [see
second sentence below Eq. (108)] Fig. 16(b) compares the fitted polynomial
$W(s_{0})$ to the phenomenological polynomials $P_{F}(s_{0})$ [see Eq. (113)]
multiplying the isovector-scalar form factor
$F_{0}^{[\overline{K}^{0}K^{+}]^{1}}(s_{0})$ for the two solutions MOP1(P2).

Table 12: As in Table LABEL:Tabparam but for the branching fractions in percent of the amplitudes ${\cal{M}}_{i}$, $i=1$ to $7$. For each branching fraction we indicate the dominant resonances (see text) of the quasi-two-body channels of the different amplitudes (see Table 5). For the MO fits, the given uncertainties, $\Delta{\rm Br}_{i},i=1,7$ and of their sum are calculated from Eqs. (116) to (118) using the average positive and negative uncertainties of the free parameters displayed in Table LABEL:Tabparam. See Sec. IV for the calculation of the $\Delta{\rm Br}_{i}$ of the RBF model. The uncertainties of the sum of the ${\rm Br}_{i}$ are obtained through the formula given in Eq. (118).  Fit | MOP1 | MOP2 | RBF
---|---|---|---
$\chi^{2}(N_{p},\chi^{2}/ndf)$ | 1559.7(16, 1.32) | 1546.9(16, 1.31) | 1474.4(19, 1.25)
${\rm Br}_{1}$[$f_{0}$’$s$, $a_{0}^{0}$’$s$] | 44.9$\pm$8.3 | 63.0$\pm$ 15.8 | $60.9^{+24.4}_{-10.6}$
${\rm Br}_{2}$[$\phi$] | 44.9$\pm$0.5 | 44.8$\pm$ 0.5 | 45.5$\pm$ 0.7
${\rm Br}_{3}$[$a_{0}^{+}$’$s$] | 25.9$\pm$ 0.8 | 40.1$\pm$1.9 | $20.7^{+9.4}_{-6.0}$
${\rm Br}_{4}$[$\rho^{+}$’$s$] | 7.7$\pm$ 0.6 | 13.1 $\pm$1.2 | $21.5^{+3.1}_{-2.8}$
${\rm Br}_{5}$[$a_{0}^{-}$’$s$] | 2.6$\pm$ 0.1 | 2.7$\pm$ 0.1 | $0.76^{+0.18}_{-0.15}$
${\rm Br}_{6}$[$\rho^{-}$’$s$] | 0.03$\pm$0.002 | 0.06$\pm$ 0.005 | 0.08$\pm$ 0.01
${\rm Br}_{7}$[$f_{2}(1270)$] | 0.37$\pm$ 0.04 | 0.30$\pm$ 0.03 | 0.05$\pm$0.02
$\sum_{i=1,7}{\rm Br}_{i}$ | 126.3$\pm$7.6 | 164.1$\pm$ 13.7 | 149.5${}^{+26.9}_{-12.3}$

The different branching fractions Br${}_{i},i=1$ to 7, of these two fits are
compared to those of the best fit model in Table LABEL:TabBr45Req. Following
Eq. (98) their given uncertainties $\Delta{\rm Br}_{i}$ are calculated as

$\Delta{\rm
Br}_{i}=\frac{1}{32(2\pi)^{3}m_{D^{0}}^{3}\Gamma_{D^{0}}}\left[\sum_{k,l=1,N_{p}}\frac{\partial
f_{i}}{\partial r_{k}}\ \frac{\partial f_{i}}{\partial r_{l}}\ \rho_{kl}\
\Delta r_{k}\ \Delta r_{l}\right]^{1/2}$ (116)

where,

$\frac{\partial f_{i}}{\partial r_{k}}=\int\int ds_{+}ds_{0}\
2\left(R_{i}\frac{\partial R_{i}}{\partial r_{k}}+I_{i}\frac{\partial
I_{i}}{\partial r_{k}}\right)\ \rm{with}\ f_{i}=\int\int ds_{+}ds_{0}\ |{\cal
M}_{i}(s_{0},s_{+})|^{2},$ (117)

and ${\cal M}_{i}=R_{i}+i\ I_{i}$. In Eqs. (116) and (117) $r_{k(l)}$ are the
free parameters entering the amplitude ${\cal M}_{i}$. In Eq. (116), $\Delta
r_{k(l)}$ are the $r_{k(l)}$ average uncertainties and $\rho_{kl}$ are the
$kl$ correlation coefficients of the MINUIT program. These quantities are
given in the minimization output with $\rho_{kl}=\rho_{lk}$ and $\rho_{kl}=1$
if $k=l$. The uncertainty $\Delta$Br of the sum of the Bri is calculated
as888This, with $i=j$, reduces to $\Delta{\rm
Br}=\left[\sum_{i=1,7}(\Delta{\rm Br}_{i})^{2}\right]^{1/2}$.

$\Delta{\rm
Br}=\frac{1}{32(2\pi)^{3}m_{D^{0}}^{3}\Gamma_{D^{0}}}\left[\sum_{i,j=1,7}\left(\sum_{k,l=1,N_{p}}\frac{\partial
f_{i}}{\partial r_{k}}\ \frac{\partial f_{j}}{\partial r_{l}}\ \rho_{kl}\
\Delta r_{k}\ \Delta r_{l}\right)\right]^{1/2}.$ (118)

The branching fractions of the three components of the ${\cal{M}}_{1}$
amplitude, viz., ${\cal{M}}^{n,I=0}_{1}$, ${\cal{M}}^{s,I=0}_{1}$,
${\cal{M}}^{I=1}_{1}$ [see Eqs. (79)-(82)], given in Table LABEL:TabBr45Reqa
show that the isoscalar $f_{0}$ and isovector $a_{0}^{0}$ resonance
contributions can be quite different. However, taking into account the large
uncertainties in the Br1 values of the MOP1, MOP2 and RBF fits (see Table
LABEL:TabBr45Req) the total scalar-resonance contribution in the
${\cal{M}}_{1}$ amplitude is similar. The corresponding results can be
qualitatively interpreted from, the expressions of the amplitudes given in
Sec. II.4, the values of the different parameters given in Tables 2,
LABEL:Tabparam and those of the kaon scalar-form factors in use. For the three
terms of the ${\cal M}_{1}$ amplitude [see Eqs. (79)-(82)] one can define the
renormalized amplitudes,

${\overline{\cal{M}}}^{n,I=0}_{1}(s_{0})=\frac{{{\cal{M}}}^{n,I=0}_{1}(s_{0})}{F_{W}},\
\ \
{\overline{\cal{M}}}^{s,I=0}_{1}(s_{0})=\frac{{{\cal{M}}}^{s,I=0}_{1}(s_{0})}{F_{W}}\
\ \ {\rm and}\ \ \
{\overline{\cal{M}}}^{I=1}_{1}(s_{0})=\frac{{{\cal{M}}}^{I=1}_{1}(s_{0})}{F_{W}},$
(119)

where $F_{W}=-G_{F}\ (\Lambda_{1}+\Lambda_{2})\ a_{2}/2$. In the case of the
amplitude ${\cal{M}}_{3}(s_{+})$ [see Eq. (86)], with $F^{\prime}_{W}=G_{F}\
\Lambda_{1}\ a_{2}/2$, one define the renormalized amplitude

${\widetilde{\cal{M}}}_{3}(s_{+})=\frac{{\cal{M}}_{3}(s_{+})}{F^{\prime}_{W}}.$
(120)

The coupling $\Lambda_{2}$ (=$-0.05)$ is small and $\Lambda_{1}+\Lambda_{2}$
(=0.90) is close to $\Lambda_{1}$ (=0.95) [see Eq. (2)], consequently
$F^{\prime}_{W}$ is close to $F_{W}$. One can then compare the amplitudes
(119) and (120) because the contribution of the $a_{1}$ term in
${\cal{M}}_{3}(s_{+})$ is negligible since the factor
$m_{K}^{2}-m_{K^{0}}^{2}=-0.0039$ GeV2 is very small. The moduli of these
amplitudes are plotted, for the best fit as the black continuous lines denoted
by RBF, the red dashed curve for the MOP1 model and the blue dotted one for
the MOP2 one, in Figs. 17(a), (b), (c) and (d).

The comparison, shown in Fig. 17, of the resulting $s_{0}$ behavior of the
moduli $|{\overline{\cal{M}}}^{n,I=0}_{1}(s_{0})|$,
$|{\overline{\cal{M}}}^{s,I=0}_{1}(s_{0})|$ and
$|{\overline{\cal{M}}}^{I=1}_{1}(s_{0})|$ allows furthermore to understand
qualitatively the different branching fractions displayed in Table
LABEL:TabBr45Reqa. The branching fractions can also be partly compared to the
fit fractions999 Br[$f_{0}(1370)$]=1.7%,
Br[$a_{0}(980)^{0}$]+Br[$a_{0}(1450)^{0}$]=71.1%, Br[$\phi(1020)$]=44.1%,
Br[$a_{0}(980)^{+}$]+Br[$a_{0}(1450)^{+}$]=45.1%, Br[$a_{0}(980)^{-}$]=0.7%
and Br[$f_{2}(1270)$]=0.7% of the BABAR isobar-model experimental analysis
supmaPRL105 .

The dominance of the branching fraction associated to the isoscalar-scalar
amplitude ${\cal M}_{1}^{s,I=0}$ for the best fit and to a less extent for the
MOP1 one, can be understood as their moduli
$|{\overline{\cal{M}}}^{s,I=0}_{1}(s_{0})|$ [Fig. 17(b)] are larger than the
moduli $|{\overline{\cal{M}}}^{n,I=0}_{1}(s_{0})|$ [Fig. 17(a)] and
$|{\overline{\cal{M}}}^{I=1}_{1}(s_{0})|$ [Fig. 17(c)]. Comparison of Fig.
17(a) and Fig. 17(c) can also explain qualitatively, for the best fit, the
difference between the ${\cal M}_{1}^{n,I=0}$ (1.19%) and ${\cal M}_{1}^{I=1}$
(4.48%) branching fractions.

Table 13: As in Table LABEL:Tabparam but for the branching fractions in percent of the amplitudes ${\cal{M}}^{n,I=0}_{1}$, ${\cal{M}}^{s,I=0}_{1}$ and ${\cal{M}}^{I=1}_{1}$ [see Eqs. (79-82)]. Lines 8 to 10 give the contribution of the interferences between these amplitudes.  Fit | MOP1 | MOP2 | RBF
---|---|---|---
$\chi^{2}(N_{p},\chi^{2}/ndf)$ | 1559.7(16, 1.32) | 1546.9(16, 1.31) | 1474.4(19, 1.25)
${\rm Br}_{1}$[$f_{0},a_{0}^{0}$] | 44.88 | 63.01 | 60.93
${\rm Br}[{\cal{M}}^{n,I=0}_{1}$: $f_{0}$ in $\Gamma^{n}(s)$] | 4.75 | 20.66 | 1.19
${\rm Br}[{\cal{M}}^{s,I=0}_{1}$: $f_{0}$ in $F_{0}^{K^{0}f_{0}}(m_{D^{0}}^{2})$ $\Gamma^{s}(s)$] | 22.69 | 45.91 | 59.82
${\rm Br}[{\cal{M}}^{I=1}_{1}$: $a_{0}^{0}$ in $\chi^{1}P_{F}(s)$ $F_{0}^{[\overline{K}^{0}K^{+}]^{1}}(s)\ {\rm or}\ G_{1}(s)$] | 16.47 | 16.47 | 4.48
${\rm Br}({\cal{M}}^{n,I=0}_{1})+{\rm Br}({\cal{M}}^{s,I=0}_{1})+{\rm Br}({\cal{M}}^{I=1}_{1})$ | 43.90 | 83.04 | 65.49
2 ${\rm Br}_{1112}$ | -12.95 | -37.98 | -14.6
2 ${\rm Br}_{1113}$ | 1.86 | 16.44 | -0.72
2 ${\rm Br}_{1213}$ | 12.03 | 0.43 | 10.8
$\sum_{i,j=1,3}^{i\neq j}{\rm Br}_{1i1j}$ | 0.94 | -20.02 | -4.57

Figure 17:  The black continuous line, denoted by RBF represents in a)
$|{\overline{\cal{M}}}^{n,I=0}_{1}(s_{0})|$, in b)
$|{\overline{\cal{M}}}^{s,I=0}_{1}(s_{0})|$, and in c)
$|{\overline{\cal{M}}}^{I=1}_{1}(s_{0})|$ [see Eqs. (119)] and in d)
$|{\widetilde{\cal{M}}}_{3}(s_{+})|$ [see Eq. (120)]. The red dashed curve,
the blue dotted one represent the corresponding moduli for the alternative
MOP1, MOP2 fits, respectively. As in Fig. 15 for the two vertical dashed lines
(the $s_{+}$ limits are very close to the $s_{0}$ ones)

.

For the $S$-wave $K\overline{K}$ contributions there are important differences
between the models MOP1, MOP2 and RBF. The kaon isoscalar-scalar form factors
are similar (see Figs. 9 and 14) but multiplication by the function
$P_{i}(s_{0})$ [see Eqs. (103), (114 and (115)] implies different
modifications. For the MO fits, the kaon isovector-scalar form factor
$F_{0}^{[\overline{K}^{0}K^{+}]^{1}}(s_{0})$ is multiplied by different
phenomenological polynomials $P_{F}(s_{0})$ [Eq. (113)] compared, in the Fig.
16(b), to the fitted polynomial $W(s_{0})$ [Eq. (108)] entering the
$G_{1}(s_{0})$ function of the best fit.

In the MOP2 fit (${\rm Br}({\cal{M}}^{n,I=0}_{1})$= 20.66 %) the $f_{0}$
contribution in $\Gamma_{2}^{n}(s)$ is enhanced by a larger $|\chi^{n}|$ (35
GeV-1) and by the function $P_{1}(s_{0})$ while it is more suppressed in the
best fit (1.19 %) than in the MOP1 one (4.75 %).

A striking difference arises for the modulus of the phenomenological
transition form factor $F_{0}^{K^{0}f_{0}}(m_{D^{0}}^{2})$: it is quite large,
2.22, for the best fit solution RBF as compared to its magnitude for the two
other fits where it is smaller than 0.43. This leads to large value of the
branching fraction (59.82 %) for the ${\cal{M}}^{s,I=0}_{1}(s_{0})$ amplitude
arising from $D^{0}$ annihilation via $W$ exchange in the RBF best fit
solution (see Table LABEL:TabBr45Reqa). The $|\chi^{s}|$ being the same (26
GeV-1), the difference between the MOP1 (22.69 %) and MOP2 (45.91 %) branching
fractions arises from smaller $P_{1}(s_{0})$ enhancement (for $s_{0}\gtrsim
1.3$ GeV2) than that of $P_{2}(s_{0})$, as can be seen in Figs. 16(a) and
17(b).

The values of the ${\rm Br}({\cal{M}}^{I=1}_{1})$ in the fith line of Table
LABEL:TabBr45Reqa indicate the isospin-1 $a_{0}^{0}$ resonances content in
$\chi_{1}P_{F}(s)F_{0}^{[\overline{K}^{0}K^{+}]^{1}}$ or $G_{1}(s)$. This
branching fraction small (4.48 %) in the RBF fit (some suppression because of
$W(s_{0})$) is the same (16.47 %) in the MOP1 and MOP2, the larger
$P_{F}(s_{0})$ [Fig. 16(b)] is conpensated by a smaller $\chi^{1}$ (8.2 versus
15).

The branching fraction Br3, which indicates the isovector $a_{0}^{+}$
resonances contribution, has values of 25.9, 40.1 and 20.7 % for the MOP1,
MOP2 and RBF fits, respectively (see Table LABEL:TabBr45Req). The modulus of
the transition form factor $F_{0}^{K^{-}a_{0}^{+}}(m_{D^{0}}^{2})$, entering
in Eq. (86) is equal to 0.23, 0.28 and 0.25 for the MOP1, MOP2 and best fits,
respectively. The branching fractions depend on the $\chi^{1}$ values and
$P_{F}(s_{+})$ behavior for the MO fits and on the role of $G_{1}(s_{+})$ in
the RBF solution and their values are in qualitative agreement with the
corresponding $|{\widetilde{\cal{M}}}_{3}(s_{+})|$ curves shown in Fig. 17(d).

The role of the $[\rho(770)^{+}+\rho(1450)^{+}+\rho(1700)^{+}]$ resonances is
different in our models: the Br4 of the MOP1, MOP2 and RBF fits are equal to
7.7, 13.1 and 21.5 %, respectively (see Table LABEL:TabBr45Req). The large
contribution in the RBF solution is partly due to the magnitude, 9.38, of the
modulus of the transition form factor $|A_{0}^{K^{-}\rho^{+}}(m_{D^{0}}^{2})|$
to be compared to 5.78 and 7.94 for the MOP1 and MOP2 fits, respectively. The
Br4 ratio between that of the RBF and those of the MOP1 and MOP2 fits is close
to the square of the corresponding $|A_{0}^{K^{-}\rho^{+}}(m_{D^{0}}^{2})|$
ratios. It can be seen that, to improve the $\chi^{2}$ of the fits, it seems
necessary to increase the $\rho^{+}$ resonances contributions.

The small isospin-1 $a_{0}^{-}$ and $\rho^{-}$ resonances contents in ${\rm
Br_{5(6)}}$ come from the fact that the ${\cal M}_{5(6)}$ amplitudes [see Eqs.
(88) and (89)] are proportional to the $V_{CKM}$ coupling $\Lambda_{2}$ with