# Study of $D_{s}^{+}\rightarrow K^{+}K^{-}\pi^{+}$ decay

Zhong-Yu Wang  School of Physics and Electronics, Hunan Key Laboratory of
Nanophotonics and Devices, Central South University, Changsha 410083, China
Jing-Yu Yi  School of Physics and Electronics, Hunan Key Laboratory of
Nanophotonics and Devices, Central South University, Changsha 410083, China
Zhi-Feng Sun  School of Physical Science and Technology, Lanzhou University,
Lanzhou 730000, China  Research Center for Hadron and CSR Physics, Lanzhou
University and Institute of Modern Physics of CAS, Lanzhou 730000, China
Lanzhou Center for Theoretical Physics, Key Laboratory of Theoretical Physics
of Gansu Province, and Frontiers Science Center for Rare Isotopes, Lanzhou
University, Lanzhou 730000, China  C. W. Xiao<EMAIL_ADDRESS>School of
Physics and Electronics, Hunan Key Laboratory of Nanophotonics and Devices,
Central South University, Changsha 410083, China

###### Abstract

We investigate the $D_{s}^{+}\rightarrow K^{+}K^{-}\pi^{+}$ decay
theoretically with the final state interactions, which is based on the chiral
unitary approach and takes into account the external and internal $W$-emission
mechanisms at the quark level. Only considering three resonances
contributions, the $f_{0}(980)$ in $S$-wave, the $\bar{K}^{*}(892)^{0}$ and
$\phi(1020)$ in $P$-wave, one can make a good description of the recent
experimental data from BESIII Collaboration, where the contribution from
$S$-wave is found to be small. Besides, we also make a calculation of the
corresponding branching fractions, which are consistent with the results of
BESIII Collaboration and Particle Data Group.

##  I Introduction

The weak decays of heavy mesons have drawn much attention in experiments for
understanding the hadronic decay properties and the resonance productions of
the decay processes. In particular, some non-leptonic three-body weak decays
of $D_{(s)}$ mesons were measured, where many resonances were produced in the
final state interactions. The Dalitz plot analysis for the decay of
$D_{s}^{+}\to\pi^{+}\pi^{-}\pi^{+}$ was performed in Ref. Aubert:2008ao ,
where the resonance contributions from $f_{0}(980)$, $f_{0}(1370)$,
$f_{0}(1500)$, etc, were considered, and it was found that the contributions
from $S$-wave resonances in $\pi^{+}\pi^{-}$ channel were dominant 111These
findings was confirmed by the recent measurements of BESIII Collaboration
BESIII:2021jnf .. The doubly Cabibbo-suppressed decay $D_{s}^{+}\to
K^{+}K^{+}\pi^{-}$ was firstly reported in Ref. Ko:2009tc , where the
branching ratios for corresponding channels were measured. The resonance
contributions from $S$\- and $P$-waves, such as $f_{0}(980)$,
$\bar{K}^{*}(892)^{0}$, $\phi(1020)$, etc, were analyzed in Ref.
delAmoSanchez:2010yp  for the decay $D_{s}^{+}\to K^{+}K^{-}\pi^{+}$. Note
that its decay fractions for different decay modes were consistent with the
former measurements of the E687 Frabetti:1995sg  and CLEO Mitchell:2009aa
Collaborations. In Ref. Aaij:2018hik , the branching fractions were measured
precisely for the doubly Cabibbo-suppressed decays $D^{+}\to K^{-}K^{+}K^{+}$,
$D^{+}\to\pi^{-}\pi^{+}K^{+}$ and $D_{s}^{+}\to K^{+}K^{+}\pi^{-}$ compared to
the normal decays $D^{+}\to K^{-}\pi^{+}\pi^{+}$ and $D_{s}^{+}\to
K^{-}K^{+}\pi^{+}$. Ref. Ablikim:2019pit  reported the first observation of
the $W$-annihilation dominant decays $D_{s}^{+}\to a_{0}(980)^{+}\pi^{0}$ and
$D_{s}^{+}\to a_{0}(980)^{0}\pi^{+}$ in the analysis of the reaction
$D_{s}^{+}\to\pi^{+}\pi^{0}\eta$, where the effect of
$a_{0}(980)^{0}-f_{0}(980)$ mixing was found to be negligible. For the doubly
Cabibbo-suppressed decay $D^{+}\to K^{-}K^{+}K^{+}$, Ref. Aaij:2019lwx
investigated the resonant contributions in detail and found that the dominant
contributions came from $S$-wave component of the $K^{-}K^{+}$ system, which
observed a large contribution of the $a_{0}(980)$ with the destructive
interference from the $f_{0}(980)$. In Ref. Ablikim:2019ibo , the singly
Cabibbo-suppressed decays $D^{+}\to\eta\eta\pi^{+}$ and
$D^{+(0)}\to\eta\pi^{+}\pi^{0(-)}$ were observed and the corresponding
absolute branching fractions were measured.

On the other hand, the non-leptonic three-body weak decays of $D_{(s)}$ catch
much theoretical attention Chau:1982da ; Chau:1987tk . The Cabibbo-favored
charmed-meson decay $D^{+}\to K^{-}\pi^{+}\pi^{+}$ was investigated in Ref.
Niecknig:2015ija  using dispersion theory based on the Khuri-Treiman
formalism, where the final state interactions between all three decay products
were explained well by the constarints from analyticity and unitarity. In this
framework, more higher partial waves were taken into account, and it was found
that the contribution from $K\pi$ components in $D$-wave was small. With the
same framework, a further study on the decays $D^{+}\to K^{-}\pi^{+}\pi^{+}$
and $D^{+}\to\bar{K}^{0}\pi^{0}\pi^{+}$ was done in Ref. Niecknig:2017ylb ,
which made a consistent description on the Daltz plot data from the CLEO,
FOCUS and BESIII collaborations. Applying chiral effective Lagrangians to
extract the interaction information of $K\bar{K}$, Ref. Aoude:2018zty  studied
the decay $D^{+}\to K^{-}K^{+}K^{+}$ with the isobar model and the coupled
channel $K$-matrix approach, where the resonance contributions from
$f_{0}(980)$ and $a_{0}(980)$ could be distinguished with different isospins
of the two-body rescattering amplitudes. With the short-distance $W$-boson
annihilation mechanism and the final state interaction of triangle
rescattering process, Ref. Yu:2021euw  discussed the $a_{0}(980)$ resonance
contribution in the $D_{s}^{+}\to a_{0}(980)\rho(\omega)$ decays and made
predictions for their branching fractions. Using the chiral unitary approach
(ChUA) Oller:1997ti ; Oset:1997it ; Oller:1997ng ; Kaiser:1998fi ;
Oller:2000fj ; Oset:2008qh , which characterizes by the Bethe-Salpeter (BS)
equation in coupled channels, Ref. Xie:2014tma  studied the resonance
productions of $f_{0}(500)$, $f_{0}(980)$ and $a_{0}(980)$ in the final state
interaction of weak decay $D^{0}\to\bar{K}^{0}\pi^{+}\pi^{-}$ and
$D^{0}\to\bar{K}^{0}\pi^{0}\eta$, respectively. For the first observed
$W$-annihilation decays $D_{s}^{+}\to a_{0}(980)^{+}\pi^{0}$ and $D_{s}^{+}\to
a_{0}(980)^{0}\pi^{+}$ Ablikim:2019pit , Ref. Molina:2019udw  proposed an
internal $W$-emission mechanism to explain the process
$D_{s}^{+}\to\pi^{+}\pi^{0}\eta$, where the resonances $a_{0}(980)^{+}$ and
$a_{0}(980)^{0}$ were dynamically generated utilizing the coupled channel
interaction of $K\bar{K}$ and $\pi\eta$ in the final state interactions. These
$W$-annihilation decays Ablikim:2019pit  were also explained in Ref.
Hsiao:2019ait  by the triangle rescattering mechanism of
$\rho\pi\eta^{(\prime)}$, where the $\pi\eta$ invariant mass spectra were well
described and the corresponding branching ratios were in good agreement with
the experimental measurement. Furthermore, with the internal and external
$W$-emission mechanisms instead of the $W$-annihilation, and considering the
contributions from the $\rho^{+}$ and the intermediate $\rho^{+}\eta$ and
$K^{*}\bar{K}/K\bar{K}^{*}$ triangle diagrams, Ref. Ling:2021qzl  obtained a
good description of the experimental data for the decay
$D_{s}^{+}\to\pi^{+}\pi^{0}\eta$ Ablikim:2019pit , where the $a_{0}(980)$ was
produced in the final state interaction of $\pi\eta$. Analogously, applying
the internal and external $W$-emission mechanisms, the single Cabibbo-
suppressed $D^{+}\rightarrow\pi^{+}\pi^{0}\eta$ decay Ablikim:2019ibo  was
investigated in Ref. Duan:2020vye  with the contribution of $a_{0}(980)$ in
the two-body $\pi\eta$ final state interaction. The double Cabibbo-suppressed
$D^{+}\rightarrow K^{-}K^{+}K^{+}$ decay Aaij:2019lwx  was studied in Ref.
Roca:2020lyi , which used N/D method to extend the applicability range up to
1.4 GeV and got the invariant mass distributions in good agreement with
experimental data. Exploiting the external $W$-emission mechanism, Ref.
Toledo:2020zxj  studied the Cabibbo favoured decay $D^{0}\rightarrow
K^{-}\pi^{+}\eta$ reported in Ref. Belle:2020fbd  and described the
experimental data well only with the contributions from $a_{0}(980)$ and
$\kappa$ (or called $K^{*}_{0}(700)$ state). Moreover, how the two-body
invariant mass distributions could be affected by the different weights of the
internal and external $W$-emission mechanisms in the study of the reactions
$D^{+}\to\pi^{+}\eta\eta$ and $D^{+}\to\pi^{+}\pi^{0}\eta$ Ikeno:2021kzf ,
where the future experimental data were expected to be helpful to pin down the
reaction mechanisms and the contribution of the $a_{0}(980)$ resonance. And
more discussions about non-leptonic three body decays of $D^{0}$ or
$D_{(s)}^{+}$ based on the ChUA can be found in Ref. Oset:2016lyh .

Recently, the BESIII Collaboration reported the precise measurement of the
branching fraction for the $D_{s}^{+}\to K^{+}K^{-}\pi^{+}$ decay and obtained
$\mathcal{B}(D_{s}^{+}\to K^{+}K^{-}\pi^{+})=(5.47\pm 0.08\pm 0.13)\%$
Ablikim:2020xlq , which was consistent with the former measurements
delAmoSanchez:2010yp ; Frabetti:1995sg ; Mitchell:2009aa . In Ref.
Ablikim:2020xlq , performing amplitude analysis with high-statistics sample,
the $K^{+}K^{-}$ invariant mass distribution was studied with a model-
independent method. However, from the extracted $S$-wave lineshape of the
$K^{+}K^{-}$ mass spectrum, the contributions from $a_{0}(980)$ and/or
$f_{0}(980)$ were not identified in the low $K^{+}K^{-}$ mass region. This is
the motivation of the present work, where we try to investigate that which
contribution of the $f_{0}(980)$ and $a_{0}(980)$ resonances is dominant or
unique for the decay $D_{s}^{+}\to K^{+}K^{-}\pi^{+}$. In our former work
Ahmed:2020qkv , the resonance contributions from the $f_{0}(980)$ and
$f_{0}(500)$ were investigated in final state interactions of the decays
$B^{0}_{(s)}\to\phi\pi^{+}\pi^{-}$. Thus, in the present work, we also focus
on the resonance contributions in the final state interactions. Based on the
ChUA and using the external $W$-emission mechanism, the Cabibbo favored weak
decays $D_{s}^{+}\rightarrow\pi^{+}\pi^{+}\pi^{-}$ and $D_{s}^{+}\rightarrow
K^{+}K^{-}\pi^{+}$ were analyzed in Ref. Dias:2016gou , where the experimental
data for the invariant mass distributions of $K^{+}K^{-}$ delAmoSanchez:2010yp
and $\pi^{+}\pi^{-}$ BaBar:2008nlp  was well described and the $f_{0}(980)$
resonance contribution was found in both the $K^{+}K^{-}$ and $\pi^{+}\pi^{-}$
invariant mass distributions. In a recent work of Wang:2021naf , the invariant
mass distributions of $K^{+}K^{-}$ from both BABAR delAmoSanchez:2010yp  and
BESIII Ablikim:2020xlq  Collaborations’ measurements were studied by both the
external and internal $W$-emission mechanisms, and the dominant contribution
from the $f_{0}(980)$ was found in the $K^{+}K^{-}$ spectrum. Different from
their works Dias:2016gou ; Wang:2021naf , which only focused on the invariant
mass distributions of $K^{+}K^{-}$, a full analysis of both $K\bar{K}$ and
$K\pi$ mass spectra is done in our work including the $S$\- and $P$-waves’
contributions.

In the present work, we will introduce the formalism of final state
interaction and consider the resonance productions for the decay of
$D_{s}^{+}\rightarrow K^{+}K^{-}\pi^{+}$ in Section II. Then, the calculation
results and discussions are presented in Section III. Finally, we make a short
conclusion in Section IV.

##  II Formalism

(a) The external $W$-emission mechanism.

(b) The internal $W$-emission mechanism.

Figure 1: Diagrammatic representation of the $D_{s}^{+}\rightarrow
K^{+}K^{-}\pi^{+}$ decay.

In the present work, we investigate the decay of $D_{s}^{+}\rightarrow
K^{+}K^{-}\pi^{+}$. As discussed in Refs. Molina:2019udw ; Roca:2020lyi , the
contribution for the topology diagrams of weak decay was classified by
ordering their importance as follows: external $W$-emission, internal
$W$-emission, $W$-exchange and annihilation, horizontal $W$-loop and vertical
$W$-loop, see more details in Refs. Chau:1982da ; Chau:1987tk ;
Morrison:1989xq . As implied in the experimental results Ablikim:2020xlq , the
resonance contributions mainly involved the scalar one $S(980)$ (representing
the $f_{0}(980)$ and $a_{0}(980)$ states) in $S$-wave, and the vector ones
$\bar{K}^{*}(892)^{0}$ and $\phi(1020)$ in $P$-wave, whereas the contributions
from the others were minor. Note that the external $W$-emission in Fig. 1(a)
can not produce the state $a_{0}(980)$ ($I=1$ ), since the $s\bar{s}$ pair has
isospin $I=0$ as shown in the results of Ref. Dias:2016gou , and of course not
for the $\bar{K}^{*}(892)^{0}$ with $s\bar{d}$, which was ignored in Ref.
Wang:2021naf . Thus, to consider the contribution of the
$\bar{K}^{*}(892)^{0}$, we need to take into account the internal $W$-emission
mechanism, as shown in Fig. 1(b). First, let’s look at the contribution of
$S$-wave, as shown in Fig. 1(a) for the external $W$-emission, where the
$\bar{s}$ quark remains a spectator and the $c$ quark decays into an $s$ quark
through the emission of a $W^{+}$ boson. Then the emission $W^{+}$ boson
creates the $u\bar{d}$ quarks, and then eventually forms a $\pi^{+}$ meson. In
order to produce the final states $K^{+}$ and $K^{-}$ alongside with the
$\pi^{+}$ meson, the $s\bar{s}$ quark pair needs to hadronize to create a pair
of pseudoscalar mesons, which can be fulfilled by the hadronization procedure
through another pair of quarks generated from the vacuum,
$q\bar{q}(\bar{u}u+\bar{d}d+\bar{s}s)$, as depicted in Fig. 2. Unlike Fig.
1(a), for the internal $W$-emission mechanism shown in Fig. 1(b), there are
two possible hadronization processes. One is that the $u\bar{s}$ quarks form a
$K^{+}$ meson and the ones $s\bar{d}$ undergo hadronization process. The other
one is that the $s\bar{d}$ quarks form a $\bar{K}^{0}$ meson and the ones
$u\bar{s}$ undergo the hadronization. Thus, the $D_{s}^{+}$ weak decay
mechanisms in Fig. 1 can be formulated as follows

$\displaystyle D^{+}_{s}$ $\displaystyle\Rightarrow
V_{cs}V_{ud}(u\bar{d}\rightarrow\pi^{+})[s\bar{s}\rightarrow
s\bar{s}\cdot(\bar{u}u+\bar{d}d+\bar{s}s)]$ (1) $\displaystyle\Rightarrow
V_{cs}V_{ud}(u\bar{d}\rightarrow\pi^{+})[M_{33}\rightarrow(M\cdot M)_{33}],$
$\displaystyle D^{+}_{s}$ $\displaystyle\Rightarrow
V_{cs}V_{ud}\left((u\bar{s}\rightarrow K^{+})[s\bar{d}\rightarrow
s\bar{d}\cdot(\bar{u}u+\bar{d}d+\bar{s}s)]\right.$ (2)
$\displaystyle\left.+(s\bar{d}\rightarrow\bar{K}^{0})[u\bar{s}\rightarrow
u\bar{s}\cdot(\bar{u}u+\bar{d}d+\bar{s}s)]\right)$ $\displaystyle\Rightarrow
V_{cs}V_{ud}\left((u\bar{s}\rightarrow K^{+})[M_{32}\rightarrow(M\cdot
M)_{32}]\right.$
$\displaystyle\left.+(s\bar{d}\rightarrow\bar{K}^{0})[M_{13}\rightarrow(M\cdot
M)_{13}]\right),$

where $V_{q_{1}q_{2}}$ is the element of the Cabibbo-Kobayashi-Maskawa (CKM)
matrix from $q_{1}$ to $q_{2}$ quark, and the $q\bar{q}$ matrix element $M$ in
SU(3) is defined as

$M=\left(\begin{array}[]{lll}{u\bar{u}}&{u\bar{d}}&{u\bar{s}}\\\
{d\bar{u}}&{d\bar{d}}&{d\bar{s}}\\\
{s\bar{u}}&{s\bar{d}}&{s\bar{s}}\end{array}\right).$ (3)

Then we can write the elements of matrix $M$ in terms of the physical mesons,
which are given by

$\Phi=\left(\begin{array}[]{ccc}{\frac{1}{\sqrt{2}}\pi^{0}+\frac{1}{\sqrt{6}}\eta}&{\pi^{+}}&{K^{+}}\\\
{\pi^{-}}&{-\frac{1}{\sqrt{2}}\pi^{0}+\frac{1}{\sqrt{6}}\eta}&{K^{0}}\\\
{K^{-}}&{\bar{K}^{0}}&{-\frac{2}{\sqrt{6}}\eta}\end{array}\right),$ (4)

where we take $\eta\equiv\eta_{8}$. The hadronization processes at the quark
level in Eqs. (1) and (2) can be accomplished to the hadron level in terms of
two pseudoscalar mesons

$(M\cdot
M)_{33}=(\Phi\cdot\Phi)_{33}=K^{+}K^{-}+K^{0}\bar{K}^{0}+\frac{2}{3}\eta\eta,$
(5) $(M\cdot
M)_{32}=(\Phi\cdot\Phi)_{32}=K^{-}\pi^{+}-\frac{1}{\sqrt{2}}\bar{K}^{0}\pi^{0}-\frac{1}{\sqrt{6}}\bar{K}^{0}\eta,$
(6) $(M\cdot
M)_{13}=(\Phi\cdot\Phi)_{13}=\frac{1}{\sqrt{2}}K^{+}\pi^{0}-\frac{1}{\sqrt{6}}K^{+}\eta+K^{0}\pi^{+}.$
(7)

Figure 2: Schematic representation of the hadronization $q\bar{q}\rightarrow
q\bar{q}(\bar{u}u+\bar{d}d+\bar{s}s)$.

Then, we get all the final states with $\pi^{+}$, $K^{+}$ or $\bar{K}^{0}$
produced in the hadronization processes,

$\displaystyle H^{(a)}$
$\displaystyle=V_{P}V_{cs}V_{ud}\left(K^{+}K^{-}+K^{0}\bar{K}^{0}+\frac{2}{3}\eta\eta\right)\pi^{+}$
(8)
$\displaystyle=V_{P}V_{cs}V_{ud}\left(K^{+}K^{-}\pi^{+}+K^{0}\bar{K}^{0}\pi^{+}+\frac{2}{3}\eta\eta\pi^{+}\right),$
$\displaystyle H^{(b)}$ $\displaystyle=\beta\times
V_{P}V_{cs}V_{ud}\left[(K^{-}\pi^{+}-\frac{1}{\sqrt{2}}\bar{K}^{0}\pi^{0}-\frac{1}{\sqrt{6}}\bar{K}^{0}\eta)K^{+}\right.$
(9)
$\displaystyle\left.+(\frac{1}{\sqrt{2}}K^{+}\pi^{0}-\frac{1}{\sqrt{6}}K^{+}\eta+K^{0}\pi^{+})\bar{K}^{0}\right]$
$\displaystyle=\beta\times
V_{P}V_{cs}V_{ud}\left(K^{+}K^{-}\pi^{+}-\frac{2}{\sqrt{6}}K^{+}\bar{K}^{0}\eta+K^{0}\bar{K}^{0}\pi^{+}\right),$

where $H^{(a)}$ and $H^{(b)}$ are the contributions of Fig. 1(a) and Fig.
1(b), respectively. The $V_{P}$ is the production vertex factor of weak decay
process in Fig. 1(a), which contains all the dynamical factors in the weak
interaction and is an unknown parameter in our formalism, see the discussion
later. In our calculation, we take $V_{P}$ as a constant Wang:2020pem ;
Ahmed:2020qkv . For the internal $W$-emission in Fig. 1(b), the production
vertex factor should be different, and thus one more coefficient $\beta$ in
$H^{(b)}$ is introduced for the relative weight of the internal $W$-emission
mechanism with respect to the external one Dai:2018nmw ; Wang:2020pem . In
Fig. 1(a), the $u$ and $\bar{d}$ quarks from the external $W$-emission are
constrained to form the color singlet $\pi^{+}$. Whereas, in Fig. 1(b), the
$u$ and $\bar{d}$ quarks from internal $W$-emission have the fixed colors.
Thus, the absolute value of $\beta$ should be less than $1$. Then, we obtain
the total contributions

$\displaystyle H$ $\displaystyle=H^{(a)}+H^{(b)}$ (10)
$\displaystyle=V_{P}V_{cs}V_{ud}\left[(1+\beta)K^{+}K^{-}\pi^{+}+(1+\beta)K^{0}\bar{K}^{0}\pi^{+}+\frac{2}{3}\eta\eta\pi^{+}-\frac{2\beta}{\sqrt{6}}K^{+}\bar{K}^{0}\eta\right].$

(a) Tree-level production.

(b) Rescattering of $K^{+}K^{-}$, $K^{0}\bar{K}^{0}$ and $\eta\eta$.

(c) Rescattering of $K^{-}\pi^{+}$ and $\bar{K}^{0}\eta$.

Figure 3: Diagrammatic representation of the $D_{s}^{+}\rightarrow
K^{+}K^{-}\pi^{+}$ decay.

One can see in Eq. (10) that the final states $K^{+}K^{-}\pi^{+}$ are produced
directly in the hadronization processes. For the other ones, taking into
account the final state interactions, we can also generate the final states
$K^{+}K^{-}\pi^{+}$ by the rescattering procedure, as depicted in Fig. 3, see
more discussions in Ref. Miyahara:2015cja . Thus, the amplitudes for Fig. 3
with these final state productions at the tree level and rescattering
processes can be written as

$\displaystyle t(s_{12},s_{23})=\mathcal{D}$
$\displaystyle\left[(1+\beta)+(1+\beta)G_{K^{+}K^{-}}(s_{12})T_{K^{+}K^{-}\rightarrow
K^{+}K^{-}}(s_{12})\right.$ (11)
$\displaystyle\left.+(1+\beta)G_{K^{0}\bar{K}^{0}}(s_{12})T_{K^{0}\bar{K}^{0}\rightarrow
K^{+}K^{-}}(s_{12})\right.$
$\displaystyle\left.+\frac{2}{3}G_{\eta\eta}(s_{12})T_{\eta\eta\rightarrow
K^{+}K^{-}}(s_{12})\right.$
$\displaystyle\left.+(1+\beta)G_{K^{-}\pi^{+}}(s_{23})T_{K^{-}\pi^{+}\rightarrow
K^{-}\pi^{+}}(s_{23})\right.$
$\displaystyle\left.-\frac{2\beta}{\sqrt{6}}G_{\bar{K}^{0}\eta}(s_{23})T_{\bar{K}^{0}\eta\rightarrow
K^{-}\pi^{+}}(s_{23})\right],$

where the factors $V_{P}$, $V_{cs}$ and $V_{ud}$ have been absorbed in the
constant $\mathcal{D}$, which is also included the normalization factor when
we fit the invariant mass distributions later. And the energy of two-body
system is defined as $s_{ij}=(p_{i}+p_{j})^{2}$, with $p_{i}$ and $p_{j}$ the
four-momenta of the two particles, where the indices $i,\ j=1,\ 2,\ 3$ denote
the three final states of $K^{+}$, $K^{-}$, $\pi^{+}$, respectively. Besides,
$G_{ij}$ is the loop function of two mesons propagators, which will be
discussed later. It is worth to mention that, in Eq. (11), in principle there
is a factor of $2$ in the term related with the identical particles $\eta\eta$
because of the two possibilities in the operators of Eq. (10) to create them,
which has been cancelled with the factor of $1/2$ in their propagators within
our normalization scheme, see more discussions in Ref. Liang:2015qva . Note
that in Eq. (11), the channels $K^{+}K^{-}$ and $K^{0}\bar{K}^{0}$ in $S$-wave
can be decomposition with isospins $I=0$ and $I=1$, written

$T_{K^{+}K^{-}\to K^{+}K^{-}}=\frac{1}{2}\left(T^{I=0}_{K\bar{K}\to
K\bar{K}}+T^{I=1}_{K\bar{K}\to K\bar{K}}\right),$ (12) $T_{K^{0}\bar{K}^{0}\to
K^{+}K^{-}}=\frac{1}{2}\left(T^{I=0}_{K\bar{K}\to
K\bar{K}}-T^{I=1}_{K\bar{K}\to K\bar{K}}\right),$ (13)

where we have used the convention of the physical states
$\left|\pi^{+}\rangle=-\right|1,1\rangle$ and
$\left|K^{+}\rangle=-\right|1/2,1/2\rangle$ for the isospin basis Oller:1997ti
. But, from the second and third parts of Eq. (11) and Eqs. (12), (13), one
can see that the summation finally only with isospin $I=0$ contributes to the
$K^{+}K^{-}$ final state interaction for the $D_{s}^{+}\rightarrow
K^{+}K^{-}\pi^{+}$ decay, and thus there is no $I=1$ component, which is
consistent with the findings of Refs. delAmoSanchez:2010yp ; Wang:2021naf .
Therefore, there should be only the resonance $f_{0}(980)$ contribution in the
$K^{+}K^{-}$ invariant mass distribution, and without the one of $a_{0}(980)$.
Indeed, as found in Ref. Xiao:2019lrj , the $f_{0}(980)-a_{0}(980)$ mixing
would not appear in the scattering amplitudes of $T_{K^{+}K^{-}\to
K^{+}K^{-}}$ and $T_{K^{0}\bar{K}^{0}\to K^{+}K^{-}}$, see Eqs. (12) and (13),
which can also be seen in our results later. Moreover, the ingredients
$\pi^{+}K^{-}$ in $S$-wave can be in $I=1/2$ and $I=3/2$. But as found in Ref.
Toledo:2020zxj , the $I=1/2$ is dominant, and thus we omit the contribution of
$I=3/2$.

Furthermore, the two-body scattering amplitudes in Eq. (11), such as
$T_{K^{+}K^{-}\to K^{+}K^{-}}$, can be evaluated by the coupled channel Bethe-
Salpeter equation of the ChUA,

$\displaystyle T=[1-VG]^{-1}V,$ (14)

where the matrix $V$ is constructed by the scattering potentials for each
coupled channel. In the ChUA, the interaction potentials can be calculated
from the chiral Lagrangians. Since there is no $I=1$ contribution, there are
five channels coupled to $K^{+}K^{-}$, $\pi^{+}\pi^{-}$, $\pi^{0}\pi^{0}$,
$K^{+}K^{-}$, $K^{0}\bar{K}^{0}$, and $\eta\eta$, which are denoted as 1 to 5
accordingly and given by (after applying the $S$-wave projection)
Ahmed:2020qkv ,

$\displaystyle V_{11}=-\frac{1}{2f^{2}}s,\quad
V_{12}=-\frac{1}{\sqrt{2}f^{2}}\left(s-m_{\pi}^{2}\right),\quad
V_{13}=-\frac{1}{4f^{2}}s,$ (15) $\displaystyle
V_{14}=-\frac{1}{4f^{2}}s,\quad
V_{15}=-\frac{1}{3\sqrt{2}f^{2}}m_{\pi}^{2},\quad
V_{22}=-\frac{1}{2f^{2}}m_{\pi}^{2},$ $\displaystyle
V_{23}=-\frac{1}{4\sqrt{2}f^{2}}s,\quad
V_{24}=-\frac{1}{4\sqrt{2}f^{2}}s,\quad V_{25}=-\frac{1}{6f^{2}}m_{\pi}^{2},$
$\displaystyle V_{33}=-\frac{1}{2f^{2}}s,\quad V_{34}=-\frac{1}{4f^{2}}s,$
$\displaystyle
V_{35}=-\frac{1}{12\sqrt{2}f^{2}}\left(9s-6m_{\eta}^{2}-2m_{\pi}^{2}\right),\quad
V_{44}=-\frac{1}{2f^{2}}s,$ $\displaystyle
V_{45}=-\frac{1}{12\sqrt{2}f^{2}}\left(9s-6m_{\eta}^{2}-2m_{\pi}^{2}\right),$
$\displaystyle
V_{55}=-\frac{1}{18f^{2}}\left(16m_{K}^{2}-7m_{\pi}^{2}\right),$

where $f$ is the pion decay constant, taken as $f=0.093$ GeV Oller:1997ti ,
and $m_{P}$ the corresponding mass of pseudoscalar meson ($P$). For the
$I=1/2$ sector, there are three coupled channels, $K^{-}\pi^{+}$,
$\bar{K}^{0}\pi^{0}$, and $\bar{K}^{0}\eta$, which are specified as 1 to 3
channels, respectively, and given by Toledo:2020zxj  222Note that in Ref.
Toledo:2020zxj  the coupled channels are $K^{+}\pi^{-}$, ${K}^{0}\pi^{0}$, and
${K}^{0}\eta$, which are the same for our case due to the charge symmetry.,

$\displaystyle
V_{11}=\frac{-1}{6f^{2}}\left(\frac{3}{2}s-\frac{3}{2s}\left(m_{\pi}^{2}-m_{K}^{2}\right)^{2}\right),$
(16) $\displaystyle
V_{12}=\frac{1}{2\sqrt{2}f^{2}}\left(\frac{3}{2}s-m_{\pi}^{2}-m_{K}^{2}-\frac{\left(m_{\pi}^{2}-m_{K}^{2}\right)^{2}}{2s}\right),$
$\displaystyle
V_{22}=\frac{-1}{4f^{2}}\left(-\frac{s}{2}+m_{\pi}^{2}+m_{K}^{2}-\frac{\left(m_{\pi}^{2}-m_{K}^{2}\right)^{2}}{2s}\right),$
$\displaystyle
V_{13}=\frac{1}{2\sqrt{6}f^{2}}\left(\frac{3}{2}s-\frac{7}{6}m_{\pi}^{2}-\frac{1}{2}m_{\eta}^{2}-\frac{1}{3}m_{K}^{2}+\frac{3}{2s}\left(m_{\pi}^{2}-m_{K}^{2}\right)\left(m_{\eta}^{2}-m_{K}^{2}\right)\right),$
$\displaystyle
V_{23}=-\frac{1}{4\sqrt{3}f^{2}}\left(\frac{3}{2}s-\frac{7}{6}m_{\pi}^{2}-\frac{1}{2}m_{\eta}^{2}-\frac{1}{3}m_{K}^{2}+\frac{3}{2s}\left(m_{\pi}^{2}-m_{K}^{2}\right)\left(m_{\eta}^{2}-m_{K}^{2}\right)\right),$
$\displaystyle
V_{33}=-\frac{1}{4f^{2}}\left(-\frac{3}{2}s-\frac{2}{3}m_{\pi}^{2}+m_{\eta}^{2}+3m_{K}^{2}-\frac{3}{2s}\left(m_{\eta}^{2}-m_{K}^{2}\right)^{2}\right).$

On the other hand, the diagonal matrix $G$ is made up by the loop functions,
of which the element is two intermediate meson propagators, given by

$G_{kk}(s)=i\int\frac{d^{4}q}{(2\pi)^{4}}\frac{1}{q^{2}-m_{1}^{2}+i\varepsilon}\frac{1}{\left(p_{1}+p_{2}-q\right)^{2}-m_{2}^{2}+i\varepsilon}\text{
,}$ (17)

where $p_{1}$ and $p_{2}$ are the four-momenta of the two initial particles in
the $k$-th channel, $m_{1}$ and $m_{2}$ are the masses of two intermediate
particles, and $s=(p_{1}+p_{2})^{2}$. Since the integral of this equation is
logarithmically divergent, we take the explicit form of dimensional
regularization method Oller:2000fj ,

$\displaystyle G_{kk}(s)=$
$\displaystyle\frac{1}{16\pi^{2}}\left\\{a_{\mu}+\ln\frac{m_{1}^{2}}{\mu^{2}}+\frac{m_{2}^{2}-m_{1}^{2}+s}{2s}\ln\frac{m_{2}^{2}}{m_{1}^{2}}\right.$
(18)
$\displaystyle+\frac{q_{cm\,k}(s)}{\sqrt{s}}\left[\ln\left(s-\left(m_{2}^{2}-m_{1}^{2}\right)+2q_{cm\,k}(s)\sqrt{s}\right)\right.$
$\displaystyle+\ln\left(s+\left(m_{2}^{2}-m_{1}^{2}\right)+2q_{cm\,k}(s)\sqrt{s}\right)$
$\displaystyle-\ln\left(-s-\left(m_{2}^{2}-m_{1}^{2}\right)+2q_{cm\,k}(s)\sqrt{s}\right)$
$\displaystyle\left.\left.-\ln\left(-s+\left(m_{2}^{2}-m_{1}^{2}\right)+2q_{cm\,k}(s)\sqrt{s}\right)\right]\right\\},$

where $\mu$ is the regularization scale, chosen as 0.6 GeV from Refs.
Duan:2020vye ; Ahmed:2020qkv , and $a_{\mu}$ the subtraction constant.
Following the method of Ref. Oset:2001cn , we will determine the values of
$a_{\mu}$ for different channels, see the discussion later. Besides,
$q_{cm\,k}(s)$ is the three momentum of the particle in the center-of-mass
frame, given by

$q_{cm\,k}(s)=\frac{\lambda^{1/2}\left(s,m_{1}^{2},m_{2}^{2}\right)}{2\sqrt{s}},$
(19)

with the usual Källen triangle function
$\lambda(a,b,c)=a^{2}+b^{2}+c^{2}-2(ab+ac+bc)$.

Note that, in Fig. 1(a), the $s\bar{s}$ quarks are not only in $S$-wave to
form $K^{+}K^{-}$ through hadronization 333One should keep in mind that the
$f_{0}(980)$ resonance is generated in the final state interaction., but also
in $P$-wave to produce the vector meson $\phi(1020)$, which decays into
$K^{+}K^{-}$ finally. We consider a full relativistic amplitude for the decay
$D_{s}^{+}\rightarrow\phi(1020)\pi^{+}\rightarrow K^{+}K^{-}\pi^{+}$ as done
in Refs. Toledo:2020zxj ; Roca:2020lyi ,

$\mathcal{M}_{\phi}(s_{12},s_{23})={\mathcal{D}_{\phi}}e^{i\alpha_{\phi}}\frac{s_{23}-s_{13}}{s_{12}-m_{\phi}^{2}+im_{\phi}\Gamma_{\phi}},$
(20)

where $\mathcal{D}_{\phi}$ is a normalization constant and $\alpha_{\phi}$ a
phase, which will be fitted by the experimental data later. Besides,
$\Gamma_{\phi}$ is the total width of the $\phi(1020)$, taking as
$\Gamma_{\phi}=4.25$ MeV. One thing should be mentioned that the $s_{ij}$ are
not independent totally and fulfill the constraint condition,

$s_{12}+s_{23}+s_{13}=m_{D_{s}^{+}}^{2}+m_{K}^{2}+m_{K}^{2}+m_{\pi}^{2},$ (21)

which means that only two of $s_{ij}$ variables are independent.

Analogously, in Fig. 1(b), the $s\bar{d}$ quarks can form the
$\bar{K}^{*}(892)^{0}$ meson in $P$-wave, and then $\bar{K}^{*}(892)^{0}$
decays into $K^{-}\pi^{+}$. The full relativistic amplitude for the process
$D_{s}^{+}\rightarrow K^{+}\bar{K}^{*}(892)^{0}\rightarrow K^{+}K^{-}\pi^{+}$
is given by Toledo:2020zxj

$\displaystyle\mathcal{M}_{\bar{K}^{*}}(s_{12},s_{23})=\frac{\mathcal{D}_{\bar{K}^{*}}e^{i\alpha_{\bar{K}^{*}}}}{s_{23}-m_{\bar{K}^{*}}^{2}+im_{\bar{K}^{*}}\Gamma_{\bar{K}^{*}}}\left[(m_{K}^{2}-m_{\pi}^{2})\frac{m_{D_{s}^{+}}^{2}-m_{K}^{2}}{m_{\bar{K}^{*}}^{2}}+s_{13}-s_{12}\right],$
(22)

where $\mathcal{D}_{\bar{K}^{*}}$ is a constant, $\alpha_{\bar{K}^{*}}$ a
phase, $\Gamma_{\bar{K}^{*}}$ the total width of the $\bar{K}^{*}(892)^{0}$,
taking as $\Gamma_{\bar{K}^{*}}=50.80$ MeV. Then the total amplitude with the
contributions of $S$\- and $P$-waves is obtained as

$t^{\prime}(s_{12},s_{23})=t(s_{12},s_{23})+\mathcal{M}_{\phi}(s_{12},s_{23})+\mathcal{M}_{\bar{K}^{*}}(s_{12},s_{23}).$
(23)

Finally, we get the double differential width distribution of three-body decay
Zyla:2020pdg ,

$\frac{d^{2}\Gamma}{ds_{12}ds_{23}}=\frac{1}{(2\pi)^{3}}\frac{1}{32m_{D_{s}^{+}}^{3}}\left|t^{\prime}(s_{12},s_{23})\right|^{2}.$
(24)

Thus, the single invariant mass distributions $d\Gamma/ds_{12}$ and
$d\Gamma/ds_{23}$ can be obtained by integrating the other invariant mass
variable in Eq. (24). Furthermore, one can obtain $d\Gamma/ds_{13}$ through
Eq. (21). For integrating $s_{23}$, the limits of integration are given in
Particle Data Group (PDG) Zyla:2020pdg , written

$\left(s_{23}\right)_{\max}=\left(E_{2}^{*}+E_{3}^{*}\right)^{2}-\left(\sqrt{E_{2}^{*2}-m_{2}^{2}}-\sqrt{E_{3}^{*2}-m_{3}^{2}}\right)^{2},$
(25)
$\left(s_{23}\right)_{\min}=\left(E_{2}^{*}+E_{3}^{*}\right)^{2}-\left(\sqrt{E_{2}^{*2}-m_{2}^{2}}+\sqrt{E_{3}^{*2}-m_{3}^{2}}\right)^{2},$
(26)

where

$E_{2}^{*}=\frac{s_{12}-m_{1}^{2}+m_{2}^{2}}{2\sqrt{s_{12}}},$ (27)
$E_{3}^{*}=\frac{m_{D_{s}^{+}}^{2}-s_{12}-m_{3}^{2}}{2\sqrt{s_{12}}}.$ (28)

And for integrating $s_{12}$, the limits of integration are given by

$\left(s_{12}\right)_{\max}=\left(E_{2}^{*^{\prime}}+E_{1}^{*^{\prime}}\right)^{2}-\left(\sqrt{E_{2}^{*^{\prime\;2}}-m_{2}^{2}}-\sqrt{E_{1}^{*^{\prime\;2}}-m_{1}^{2}}\right)^{2},$
(29)
$\left(s_{12}\right)_{\min}=\left(E_{2}^{*^{\prime}}+E_{1}^{*^{\prime}}\right)^{2}-\left(\sqrt{E_{2}^{*^{\prime\;2}}-m_{2}^{2}}+\sqrt{E_{1}^{*^{\prime\;2}}-m_{1}^{2}}\right)^{2},$
(30)

where

$E_{2}^{*^{\prime}}=\frac{s_{23}-m_{3}^{2}+m_{2}^{2}}{2\sqrt{s_{23}}},$ (31)
$E_{1}^{*^{\prime}}=\frac{m_{D_{s}^{+}}^{2}-s_{23}-m_{1}^{2}}{2\sqrt{s_{23}}}.$
(32)

Note that in the ChUA, we can only make reliable predictions up to $1.1\sim
1.2$ GeV in the coupled channel interaction. The limits of the integral
variable of Eq. (24) for the invariant masses are higher than $1.2$ GeV. Of
course, we are more interested in the resonance region of the invariant masses
below $1.1$ GeV, where the region above $1.1$ GeV has little impact, see the
results of Ref. Wang:2021naf . Thus, as done in Refs. Debastiani:2016ayp ;
Toledo:2020zxj , we smoothly extrapolate $G\left(s\right)T\left(s\right)$
above the energy cut $\sqrt{s}\geq\sqrt{s_{cut}}=1.1$ GeV using

$G\left(s\right)T\left(s\right)=G\left(s_{cut}\right)T\left(s_{cut}\right){e}^{-\alpha\left(\sqrt{s}-\sqrt{s_{cut}}\right)},\text{
for }\sqrt{s}>\sqrt{s_{cut}},$ (33)

where $G$ is the loop function of two meson propagators in Eq. (18), and $T$
the amplitude obtained by Eq. (14). Besides, $\alpha$ is a smoothing
extrapolation parameter, of which the value will be discussed in the next
section.

##  III Results

In our calculations, we first determine the values of subtraction constants
$a_{\mu}$ in the loop functions. Following the method of Ref. Oset:2001cn ,
one can make the loop functions to have the same value at the threshold of
each coupled channel using the dimensional regularization and the cut off
formulae, and then determine the values of the subtraction constants $a_{\mu}$
for each coupled channel. For $I=0$ sector, taking the cutoff $q_{max}=0.6$
GeV Duan:2020vye ; Ahmed:2020qkv , we obtain

$\displaystyle a_{\pi^{+}\pi^{-}}=-1.30,\ a_{\pi^{0}\pi^{0}}=-1.29,\
a_{K^{+}K^{-}}=-1.63,\ a_{K^{0}\bar{K}^{0}}=-1.63,\ a_{\eta\eta}=-1.68.$ (34)

For $I=1/2$ sector, based on the cutoff obtained in Ref. Guo:2005wp , which is
also $q_{max}=0.6$ GeV, we have

$\displaystyle a_{\pi^{+}K^{-}}=-1.57,\ a_{\pi^{0}\bar{K}^{0}}=-1.57,\
a_{\eta\bar{K}^{0}}=-1.66.$ (35)

Note that, as discussed in Ref. Ahmed:2020qkv , in order to investigate the
properties of the resonances, we do not treat $a_{\mu}$ as free parameter in
our fits. Thus, the resonances $f_{0}(980)$ and $K_{0}^{*}(700)$ (also called
$\kappa$) are naturally generated in the coupled channel interactions of the
ChUA.

Table I: Values of the parameters from the fit.

Parameters |  $\mathcal{D}$ |  $\beta$ |  $\alpha$ |  $\mathcal{D}_{\phi}$ |  $\alpha_{\phi}$ |  $\mathcal{D}_{\bar{K}^{*}}$ |  $\alpha_{\bar{K}^{*}}$
---|---|---|---|---|---|---|---
Fit I |  $6635.79$ |  $0.16$ |  $19.62$ |  $1201.86$ |  $0.00$(fixed) |  $825.97$ |  $0.15$
Fit II |  $3151.32$ |  $-0.90$ |  $7.13$ |  $158.32$ |  $0.39$ |  $148.56$ |  $0.05$

(a) Results for $S$-wave with $\chi^{2}/dof.=37.51/(40-3)=1.01$.

(b) Results for $P$-wave with $\chi^{2}/dof.=206.61/(40-3)=5.58$.

Figure 4: $K^{+}K^{-}$ invariant mass distributions in $S$-wave and $P$-wave
of the $D_{s}^{+}\to K^{+}K^{-}\pi^{+}$ decay (with the reduced chi-square).
Data are taken from Ref. Ablikim:2020xlq .

Therefore, we have seven parameters in the fit of the experimental data,
$\mathcal{D}$, $\beta$ and $\alpha$ in $S$-wave, ${\mathcal{D}_{\phi}}$,
$\alpha_{\phi}$, ${\mathcal{D}_{\bar{K}^{*}}}$ and $\alpha_{\bar{K}^{*}}$ in
$P$-wave. But, some of these parameters are uncorrelated, since there is no
interference effect between $S$\- and $P$-waves. First, we fit the
$K^{+}K^{-}$ invariant mass distributions in $S$\- and $P$-waves separately
(the parameters for the other one are set as zero), which are shown in Fig. 4.
From Fig. 4(a), we can obtain the parameters of $S$-wave, and the ones of
$P$-wave from Fig. 4(b). The fitting parameters are shown in Fit I results of
Table I, where the phase of $\phi(1020)$, $\alpha_{\phi}$, is fixed as $0$ in
the $P$-wave fitting. Besides, we can see that the fitting value of $\beta$ is
$0.16$, which is less than $1$ as discussed before. Note that, the
uncertainties for the parameters of $S$-wave are quite large due to its small
contribution as found later. One can see from Fig. 4 that the fitting results
for the $K^{+}K^{-}$ invariant mass distributions are in good agreement with
the experimental data both in $S$\- and $P$-waves. Our results of Fig. 4(a)
for $S$-wave are consistent with those obtained in Ref. Wang:2021naf . Note
that we use the dimensional regularization method to solve singular
integration of the loop function, see Eq. (18), which is different with the
one of cutoff method used in Ref. Wang:2021naf . On the other hand, we use the
double differential width distribution of three-body decay in Eq. (24),
whereas, only the two-body $K^{+}K^{-}$ invariant mass distribution was
concerned in Ref. Wang:2021naf . Thus, our results include the contribution of
$K^{-}\pi^{+}$ channel, although it is very small, see the results later. As
shown in Fig. 4(b), indeed the main contribution is the $\phi$ in the $P$-wave
$K^{+}K^{-}$ invariant mass distribution.

Figure 5: Effects of the different extrapolation parameter $\alpha$ on the
$S$-wave $K^{+}K^{-}$ invariant mass distribution.

In our fitting, the value of $\alpha$ in Eq. (33) is $19.62$ GeV-1 (see Table
I), which means that $G\left(s\right)T\left(s\right)$ is reduced by seven
times at the position of $\sqrt{s_{cut}}+0.1$ GeV. To check the effects of the
smooth extrapolation of the amplitudes, we also take $\alpha=10$ GeV-1 and
$30$ GeV-1, which reduce $G\left(s\right)T\left(s\right)$ by a factor about 3
and 19, respectively, at $\sqrt{s_{cut}}+0.1$ GeV. All the other parameters
are taken the same values as the ones in Fit I of Table I. Our results are
shown in Fig. 5, where one can see that the results barely change in the
$S$-wave $K^{+}K^{-}$ invariant mass distribution with different extrapolation
factors (different value of $\alpha$). This indicates that the influence of
$\alpha$ on our fitting results is trivial, since it only affects the data
above the cut at 1.1 GeV.

(a) $s_{K^{+}K^{-}}$

(b) $s_{K^{+}K^{-}}$ near the $\phi(1020)$ peak.

(c) $s_{K^{-}\pi^{+}}$ ($\chi^{2}/dof.=192.89/(87-7)=2.41$)

(d) $s_{K^{+}\pi^{+}}$

Figure 6: Fitting for different invariant mass distributions of Dalitz plot
projections data Ablikim:2020xlq . The solid (red) line is the total
contributions of $S$\- and $P$-waves, the dash (blue) line the contribution of
$\phi(1020)$, the dash-dot (green) line the one of $\bar{K}^{*}(892)^{0}$, the
dot (magenta) line the one from $S$-wave (mainly $f_{0}(980)$), and dot
(black) points are the experimental data taken from Ref. Ablikim:2020xlq .

Next, we perform a fit to the Dalitz plot projections data Ablikim:2020xlq
for different invariant mass distributions, which is shown in Fig. 6(c). The
parameters from the fit are shown in Fit II of Table I. Because there is only
one experimental data for the peak near $1.02$ GeV in Fig. 6(a), and the bins
of the data in Fig. 6(b) are different from those of the other three figures,
we choose the data of $s_{K^{-}\pi^{+}}$ in Fig. 6(c) to do the fit 444Note
that we have done a fit with the combined data of $s_{K^{-}\pi^{+}}$ and
$s_{K^{+}\pi^{+}}$ (in Figs. 6(c) and 6(d)), and found that there is no
significant improvement to the fit only with the $s_{K^{-}\pi^{+}}$ data..
With the fitting parameters obtained, we can directly get the results of Figs.
6(a) and 6(d), except for Fig. 6(b). Since the sampling interval and the
number of events in Fig. 6(b) are different from the ones in Fig. 6(c), apart
from using the fitting parameters obtained, we add a global factor for the
overall strength of the curve to match the different event numbers in Fig.
6(b), of which the value is $0.054$. From Fig. 6, one can see that our fitting
results are in good agreement with the experimental data. It is remarkable
that we only fit the data of Fig. 6(c) and obtain good description of the
other data in Figs. 6(a), 6(b) and 6(d), which is analogous to the case of
$D^{+}\to K^{-}K^{+}K^{+}$ decay discussed in Ref. Roca:2020lyi . In Fig.
6(a), the contribution of $\phi(1020)$ is obvious, and the two small bumps in
the middle and high energy regions are contributed by $\bar{K}^{*}(892)^{0}$,
see dash-dot (green) line. Fig. 6(b) shows the detailed structure of the
$\phi(1020)$ state. For the $K^{-}\pi^{+}$ invariant mass distribution in Fig.
6(c), besides the contribution of $\bar{K}^{*}(892)^{0}$, the two peaks in the
middle and high energy regions are dominantly contributed by the $\phi(1020)$,
see dash (blue) line. Similarly, for the $K^{+}\pi^{+}$ invariant mass
distribution in Fig. 6(d), the lower peak near $0.5$ GeV is contributed by
$\bar{K}^{*}(892)^{0}$, and the other two are mainly contributed by the
$\phi(1020)$ too. Note that we only put two resonances’ contribution, the
$\bar{K}^{*}(892)^{0}$ and $\phi(1020)$ states added by hand, and obtain the
results of Figs. 6(c) and 6(d) in good agreement with experimental data rather
than considering more higher resonances’ contribution in the experimental
analysis Ablikim:2020xlq . This is a curious result of our work, which is
similar to the one obtained in Ref. Toledo:2020zxj  for the investigation of
$D^{0}\to K^{-}\pi^{+}\eta$ decay with only two resonances’ contribution
describing the data well. Furthermore, compared with the $P$-wave, the
contribution of $S$-wave to Dalitz plot projection data is smaller, which is
mainly concentrated in the $K^{-}\pi^{+}$ and $K^{+}\pi^{+}$ channels, see the
dot (magenta) line in Fig. 6. Thus, there is no clear signal for the
$K^{*}_{0}(700)$ resonance in $S$-wave, as found in the experiment
Ablikim:2020xlq . Besides, it is obvious that the contribution of
$\bar{K}^{*}(892)^{0}$ is very small compared with $\phi(1020)$ in the
$K^{+}K^{-}$ invariant mass distribution, see Figs. 4(b) and 6(b). Thus, the
fitting parameters $\mathcal{D}_{\bar{K}^{*}}$ and $\alpha_{\bar{K}^{*}}$ may
have large uncertainties in Fit I of Table I, which is analogous to the one of
$\mathcal{D}$ in Fit II of Table I. Using the $P$-wave fitting parameters of
Fit II, and the ratio of strength parameter $\mathcal{D}_{\phi}$ of
$\phi(1020)$ between Fit I and Fit II, $1201.86/158.32=7.59$, we can easily
get the corresponding $K^{-}\pi^{+}$ invariant mass distribution, which is
shown in Fig. 7 and can be used to evaluate the branching ratio below.

Figure 7: $K^{-}\pi^{+}$ invariant mass distribution in $P$-wave of the
$D_{s}^{+}\rightarrow K^{+}K^{-}\pi^{+}$ decay.

In addition, we make a calculation for the ratios of branching fractions in
different decay channels. By integrating the resonance contributions in
$D_{s}^{+}\rightarrow K^{+}K^{-}\pi^{+}$ decays (recall that there is no
contribution of the $a_{0}(980)$ state), we find

$\displaystyle\frac{\mathcal{B}[D_{s}^{+}\rightarrow
f_{0}(980)\pi^{+},f_{0}(980)\rightarrow
K^{+}K^{-}]}{\mathcal{B}[D_{s}^{+}\rightarrow\phi(1020)\pi^{+},\phi(1020)\rightarrow
K^{+}K^{-}]}=0.28^{+0.03}_{-0.08}\;,$ (36)

where the integral limit is $[2m_{K},1.1]$ GeV for both $D_{s}^{+}\rightarrow
f_{0}(980)\pi^{+}$ and $D_{s}^{+}\rightarrow\phi(1020)\pi^{+}$ decays, with
the uncertainty from the upper limits $1.1\pm 0.05$ GeV. Analogously, we get

$\displaystyle\frac{\mathcal{B}[D_{s}^{+}\rightarrow\bar{K}^{*}(892)^{0}K^{+},\bar{K}^{*}(892)^{0}\rightarrow
K^{-}\pi^{+}]}{\mathcal{B}[D_{s}^{+}\rightarrow\phi(1020)\pi^{+},\phi(1020)\rightarrow
K^{+}K^{-}]}=1.18^{+0.02}_{-0.05}\;,$ (37)

where the integral limit is $[(m_{K}+m_{\pi}),1.1]$ GeV for
$D_{s}^{+}\rightarrow\bar{K}^{*}(892)^{0}K^{+}$ decay, and for the
contributions of $\phi(1020)$ and $\bar{K}^{*}(892)^{0}$, we integrate the
$P$-wave contribution of dash (blue) line in Fig. 4(b) and dash-dot (green)
line in Fig. 7, respectively. Taking the experimental measurement of the
branching fraction from BESIII Collaboration Ablikim:2020xlq
$\mathcal{B}[D_{s}^{+}\rightarrow\phi(1020)\pi^{+},\phi(1020)\rightarrow
K^{+}K^{-}]=(2.21\pm 0.05\pm 0.07)\%$ as input, and combining Eqs. (36) and
(37), one can easily obtain the branching fractions,

$\displaystyle\mathcal{B}[D_{s}^{+}\rightarrow
f_{0}(980)\pi^{+},f_{0}(980)\rightarrow K^{+}K^{-}]=(0.61\pm
0.02^{+0.06}_{-0.17})\,\%,$ (38)
$\displaystyle\mathcal{B}[D_{s}^{+}\rightarrow\bar{K}^{*}(892)^{0}K^{+},\bar{K}^{*}(892)^{0}\rightarrow
K^{-}\pi^{+}]=(2.61\pm 0.10^{+0.05}_{-0.12})\,\%,$

where the first uncertainties are estimated from the experimental error of
$\mathcal{B}[D_{s}^{+}\rightarrow\phi(1020)\pi^{+},\ \\\ \phi(1020)\rightarrow
K^{+}K^{-}]$, and the second ones come from the integration limits of Eqs.
(36) and (37). Note that the corresponding branching ratios from BESIII
Collaboration were reported as Ablikim:2020xlq

$\displaystyle\mathcal{B}[D_{s}^{+}\rightarrow S(980)\pi^{+},S(980)\rightarrow
K^{+}K^{-}]=(1.05\pm 0.04\pm 0.06)\%,$ (39)
$\displaystyle\mathcal{B}[D_{s}^{+}\rightarrow\bar{K}^{*}(892)^{0}K^{+},\bar{K}^{*}(892)^{0}\rightarrow
K^{-}\pi^{+}]=(2.64\pm 0.06\pm 0.07)\%,$

where $S(980)$ represents the states of $f_{0}(980)$ and $a_{0}(980)$.
Moreover, the ones reported in PDG Zyla:2020pdg  are given by,

$\displaystyle\mathcal{B}[D_{s}^{+}\rightarrow
f_{0}(980)\pi^{+},f_{0}(980)\rightarrow K^{+}K^{-}]=(1.14\pm 0.31)\%,$ (40)
$\displaystyle\mathcal{B}[D_{s}^{+}\rightarrow\bar{K}^{*}(892)^{0}K^{+},\bar{K}^{*}(892)^{0}\rightarrow
K^{-}\pi^{+}]=(2.58\pm 0.08)\%.$

One can see that our branching fraction of
$\mathcal{B}[D_{s}^{+}\rightarrow\bar{K}^{*}(892)^{0}K^{+},\bar{K}^{*}(892)^{0}\rightarrow
K^{-}\pi^{+}]$ is consistent with the one obtained in the experimental results
of Ref. Ablikim:2020xlq  and PDG Zyla:2020pdg  within the uncertainties.
Whereas, the one of $\mathcal{B}[D_{s}^{+}\rightarrow
f_{0}(980)\pi^{+},f_{0}(980)\rightarrow K^{+}K^{-}]$ is about 40% smaller than
theirs.

##  IV Conclusions

Based on the chiral unitary method for two-body final state interactions, we
investigate the $D_{s}^{+}\rightarrow K^{+}K^{-}\pi^{+}$ decay with the final
state interaction approach. In the decay process of hadronization, we have
considered the mechanisms of external and internal $W$-emission. When we sum
all the contributions from the final states in $S$-wave, we find only
$f_{0}(980)$ with isospin $I=0$ contributed and without the one of
$a_{0}(980)$ as indicated in both the experiment delAmoSanchez:2010yp  and
theories Dias:2016gou ; Wang:2021naf . Note that the contribution of
$f_{0}(980)$ is produced in the interactions of $K^{+}K^{-}$ with its coupled
channels, which means that the $f_{0}(980)$ is a bound state of $K\bar{K}$ and
locates below the threshold of $K\bar{K}$. On the other hand, as found in our
fitting results, the contribution from $S$-wave is small. This is why the
contribution from $f_{0}(980)$ or/and $a_{0}(980)$ could not be distinguished
in recent experimental analysis Ablikim:2020xlq .

With only seven free parameters and three resonances’ contribution in both
$S$\- and $P$-waves, we obtain the results of the invariant mass distribution
in good agreement with the experimental data. It is remarkable that only
taking into account three resonances’ contribution, the $f_{0}(980)$,
$\bar{K}^{*}(892)^{0}$ and $\phi(1020)$ states, the experimental invariant
mass distributions of $K^{+}K^{-}$, $K^{+}\pi^{+}$ and $K^{-}\pi^{+}$ can be
described well without any higher resonances’ contribution as done in the
experimental analysis Ablikim:2020xlq . The other feature of our results is
that only fitting with the $K^{-}\pi^{+}$ invariant mass distribution, one can
get good description of other invariant mass distributions of the Dalitz plot
projection data. As shown in these fitting results, for the $K^{+}K^{-}$
invariant mass distribution, except of the clear $\phi(1020)$ peak, the two
small bumps in the middle and high energy regions are caused by the
$\bar{K}^{*}(892)^{0}$. For the $K^{+}\pi^{+}$ and $K^{-}\pi^{+}$ invariant
mass distributions, the lower peak is mainly contributed by the
$\bar{K}^{*}(892)^{0}$, while the other two peaks in the middle and high
energy regions are dominated by the $\phi(1020)$. Furthermore, we also
calculate the branching fractions of the dominant decay channels with the
scalar and vector resonances produced in the final states. We obtain the
result of the branching ratio of
$\mathcal{B}[D_{s}^{+}\rightarrow\bar{K}^{*}(892)^{0}K^{+},\bar{K}^{*}(892)^{0}\rightarrow
K^{-}\pi^{+}]=(2.61\pm 0.10^{+0.05}_{-0.12})\,\%$, which is consistent with
the experimental measurement from BESIII Collaboration and Particle Data Group
within the uncertainties, and the one of $\mathcal{B}[D_{s}^{+}\rightarrow
f_{0}(980)\pi^{+},f_{0}(980)\rightarrow K^{+}K^{-}]=(0.61\pm
0.02^{+0.06}_{-0.17})\,\%$ a bit smaller.

## Acknowledgments

We thank Prof. Eulogio Oset for careful reading the manuscript and valuable
comments, and acknowledge Profs. Bastian Kubis, Yu-Kuo Hsiao for helpful
comments. ZFS is partly supported by the National Natural Science Founadtion
of China (NSFC) under Grants No. 11965016, and the Fundamental Research Funds
for the Central Universities under Grants No. Lzujbky-2021-sp24.

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