# Testing the light scalar meson as a non-$q\bar{q}$ state
in semileptonic $D$ decays

Yu-Kuo Hsiao  Corresponding author<EMAIL_ADDRESS>School of Physics and
Information Engineering, Shanxi Normal University, Taiyuan 030031, China  Shu-
Qi Yang<EMAIL_ADDRESS>School of Physics and Information Engineering,
Shanxi Normal University, Taiyuan 030031, China  Wen-Juan Wei
<EMAIL_ADDRESS>School of Physics and Information Engineering, Shanxi Normal
University, Taiyuan 030031, China  Bai-Cian Ke<EMAIL_ADDRESS>School of
Physics and Microelectronics, Zhengzhou University, Zhengzhou, Henan 450001,
China

###### Abstract

To distinguish between the normal $q\bar{q}$ and exotic diquark-antidiqark
($q^{2}\bar{q}^{2}$) contents of the lowest-lying scalar meson ($S_{0}$), we
investigate the semileptonic $D\to S_{0}e^{+}\nu_{e},S_{0}\to M_{1}M_{2}$
decays, where $M_{1(2)}$ represents a pseudoscalar meson. With the form
factors extracted from the current data, we calculate ${\cal
B}(D_{s}^{+}\to\sigma_{0}e^{+}\nu_{e},\sigma_{0}\to\pi^{0}\pi^{0})=(12.9^{+6.3}_{-4.9})\times
10^{-4}$ and $(0.8^{+1.2}_{-0.7})\times 10^{-4}$ for the $q\bar{q}$ and
$q^{2}\bar{q}^{2}$ quark structures, respectively, and compare them to the
experimental upper limit: $6.4\times 10^{-4}$. It is clearly seen that $S_{0}$
prefers to be the $q^{2}\bar{q}^{2}$ bound state. Particularly, ${\cal
B}_{q\bar{q}}(D_{s}^{+}\to\sigma_{0}e^{+}\nu_{e},\sigma_{0}\to\pi^{+}\pi^{-})=(25.8^{+12.5}_{-\;\,9.8})\times
10^{-4}$ and ${\cal
B}_{q^{2}\bar{q}^{2}}(D_{s}^{+}\to\sigma_{0}e^{+}\nu_{e},\sigma_{0}\to\pi^{+}\pi^{-})=(1.5^{+2.4}_{-1.3})\times
10^{-4}$ are predicted to deviate far from each other, useful for a clear
experimental investigation.

##  I introduction

The lowest-lying scalar meson ($S_{0}$), such as $f_{0}\equiv f_{0}(980)$,
$a_{0}\equiv a_{0}(980)$, or $\sigma_{0}\equiv f_{0}(500)$, has an unclear
quark content. One might think of $S_{0}$ the normal p-wave $q\bar{q}$ meson
Anisovich:2000wb ; Bediaga:2003zh ; Aliev:2007uu ; Colangelo:2010bg ;
Shi:2015kha ; Soni:2020sgn ; Klempt:2021nuf , the compact diquark-antidiquark
($q^{2}\bar{q}^{2}$) tetraquark Jaffe:1976ig ; Jaffe:1976ih ; Close:2002zu ;
Pelaez:2003dy ; Maiani:2004uc ; Amsler:2004ps ; Jaffe:2004ph ; Achasov:2005hm
; tHooft:2008rus ; Fariborz:2009cq ; Weinberg:2013cfa ; Agaev:2018fvz , or the
molecular $M_{1}M_{2}$ bound state Weinstein:1990gu ; Branz:2007xp ;
Dai:2012kf ; Dai:2014lza ; Sekihara:2014qxa ; Yao:2020bxx ; Wang:2022vga ,
where $M_{1(2)}$ represents a pseudoscalar meson. The nature of the light
scalar meson thus remains a puzzle of the hadronization. Several decay
processes have been used to solve the puzzle Wang:2009azc ; Stone:2013eaa ;
Achasov:2021dvt ; Achasov:2020aun ; Cheng:2017fkw ; Achasov:2020qfx .
Particularly, the semileptonic $D\to S_{0}e^{+}\nu_{e}$ weak decays involve
the $S_{0}$ formation, and advantageously, there is no QCD effect caused by
the lepton pair besides the $D\to S_{0}$ transition. A clean test-bed for
exploring the scalar meson is hence provided.

BESIII collaboration has recently measured the semileptonic $D_{s}^{+}$
decays, whose branching fractions are reported as BESIII:2023wgr ;
BESIII:2021drk

$\displaystyle{\cal B}(D_{s}^{+}\to f_{0}e^{+}\nu_{e},f_{0}\to\pi^{+}\pi^{-})$
$\displaystyle=(1.72\pm 0.13\pm 0.10)\times 10^{-3}\,,$ (1)
$\displaystyle{\cal B}(D_{s}^{+}\to f_{0}e^{+}\nu_{e},f_{0}\to\pi^{0}\pi^{0})$
$\displaystyle=(7.9\pm 1.4\pm 0.3)\times 10^{-4}\;,$ $\displaystyle{\cal
B}(D_{s}^{+}\to\sigma_{0}e^{+}\nu_{e},\sigma_{0}\to\pi^{0}\pi^{0})$
$\displaystyle<6.4\times 10^{-4}\;(\text{90\% C.L.})\,.$

As a parameterization of the $D_{s}^{+}$ to $f_{0}$ transition Soni:2020sgn ,
the form factor $f^{+}(p^{2})$ with $p^{2}=(p_{D_{s}}-p_{f_{0}})^{2}$ plays a
key role in interpreting the branching fraction, which has been extracted as
$f^{+}(0)|V_{cs}|=0.504\pm 0.017\pm 0.035$ by BESIII BESIII:2023wgr  with
$f^{+}(0)=f^{+}(p^{2})$ at $p^{2}=0$ and $|V_{cs}|$ the $c\to s$ Cabibbo-
Kobayashi-Maskawa (CKM) matrix element. In addition, the fitted $f^{+}(0)$ has
been used to test the model calculations BESIII:2023wgr , which is believed to
help to investigate the constituents of the scalar meson.

In fact, $f^{+}(0)$ extracted from a single decay of $D_{s}^{+}\to
f_{0}e^{+}\nu,f_{0}\to\pi^{+}\pi^{-}$ can barely be applied to distinguish one
quark content for $S_{0}$ from the other. Only when $f^{+}(0)$ is related to
those of the other $D_{s}^{+}\to S_{0}e^{+}\nu_{e},S_{0}\to M_{1}M_{2}$
decays, it becomes possible to discern the $q\bar{q}$ or $q^{2}\bar{q}^{2}$
nature of the scalar meson. The reason is that the light scalar mesons
actually mix with each other, and the two possible quark structures have
different scenarios for the mixing. Taking the $f_{0}-\sigma_{0}$ mixing as
our example, the mixing angle for $q\bar{q}$ is commonly studied as
$|\theta|\simeq 20^{\circ}$ Klempt:2021nuf , whereas the one for
$q^{2}\bar{q}^{2}$ is less than $10^{\circ}$ Maiani:2004uc , which would cause
sizable distinction for the branching fractions.

In this report, we propose to re-fit $f^{+}(0)$ in the $D_{s}^{+}\to
f_{0}e^{+}\nu_{e},f_{0}\to\pi^{+}\pi^{-}$ decay, which needs to agree with the
experimental extraction. Then, we will relate $f^{+}(0)$ to those of the other
semileptonic $D_{s}^{+}$ decays, and calculate the branching fractions for the
different quark structures. Likewise, the scalar formation will be
investigated in the semileptonic $D^{+(0)}$ decays. Since the mixing scenarios
can cause sizable distinction for the branching fractions, by comparing our
calculations to the current data, it is anticipated that one of the quark
contents will be preferred.

##  II Formalism

According to the effective Hamiltonians of the quark-level $c\to
qe^{+}\nu_{e}$ decays Buras:1998raa , the amplitude of $D\to
S_{0}e^{+}\nu_{e}$ is written as

$\displaystyle{\cal M}(D\to S_{0}e^{+}\nu_{e})$ $\displaystyle=$
$\displaystyle\frac{G_{F}}{\sqrt{2}}V_{cq}\langle
S_{0}|\bar{q}\gamma_{\mu}(1-\gamma_{5})c|D\rangle\bar{u}_{\nu}\gamma_{\mu}(1-\gamma_{5})v_{e}\,,$
(2)

where $G_{F}$ is the Fermi constant, $V_{cq}$ the CKM matrix element, and $q$
denotes $s$ or $d$. Specifically, we explore $S_{0}$ as $a_{0}$, $f_{0}$, and
$\sigma_{0}$. In Eq. (2), one parameterizes the matrix elements of the $D$ to
$S_{0}$ transition as Becirevic:1999kt

$\displaystyle\langle S_{0}|(\bar{q}c)|D\rangle$ $\displaystyle=$
$\displaystyle
i(p_{D}+p_{S_{0}})_{\mu}f^{+}(p^{2})+i\frac{m^{2}_{D}-m^{2}_{S_{0}}}{p^{2}}p_{\mu}[f^{0}(p^{2})-f^{+}(p^{2})]\,,$
(3)

where $(\bar{q}c)=\bar{q}\gamma_{\mu}(1-\gamma_{5})c$,
$p_{\mu}=(p_{D}-p_{S_{0}})_{\mu}$, and $f^{+,0}(p^{2})$ are the form factors.
For the momentum dependence, $f^{+,0}(p^{2})$ are presented with the double
and single pole representations, respectively, given by BESIII:2023wgr ;
Becirevic:1999kt

$\displaystyle
f^{+}(p^{2})=\frac{f(0)}{(1-p^{2}/m^{2}_{\text{A}})^{2}}\,,\;f^{0}(p^{2})=\frac{f(0)}{1-p^{2}/m^{2}_{\text{B}}}\,,$
(4)

with $f(0)$ at $p^{2}=0$ and the pole masses $m_{\text{A}}$ and
$m_{\text{B}}$.

The scalar mesons appear as resonances in $D\to S_{0}e^{+}\nu_{e},S_{0}\to
M_{1}M_{2}$. We follow Ref. Flatte:1976xu  to use the Flatt$\acute{\text{e}}$
formula for the two-channel resonances of $f_{0}$ and $a_{0}$, whereas the
resonance of $\sigma_{0}$ is by the Breit-Wigner model pdg . The resonant
amplitudes of $S_{0}\to M_{1}M_{2}$ are thus presented as

$\displaystyle{\cal R}_{f_{0}}\equiv{\cal
R}(f_{0}\to\pi\pi)=\frac{C_{f_{0}\to\pi\pi}}{m_{f_{0}}^{2}-t-im_{f_{0}}(g_{f_{0}\pi\pi}\rho_{\pi\pi}+g_{f_{0}KK}\rho_{KK})}\,,$
$\displaystyle{\cal R}_{a_{0}}\equiv{\cal
R}(a_{0}\to\pi\eta)=\frac{C_{a_{0}\to\pi\eta}}{m_{a_{0}}^{2}-t-i(g_{a_{0}\pi\eta}^{2}\rho_{\pi\eta}+g_{a_{0}KK}^{2}\rho_{KK})}\,,$
$\displaystyle{\cal R}_{\sigma_{0}}\equiv{\cal
R}(\sigma_{0}\to\pi\pi)=\frac{C_{\sigma_{0}\to\pi\pi}}{m_{\sigma_{0}}^{2}-t-i\,m_{\sigma_{0}}\Gamma_{\sigma_{0}}}\,,$
(5)

with the denominators from LHCb:2015klp ; LHCb:2012ae , BaBar:2005vhe ;
BESIII:2020hfw , and LHCb:2013dkk ; pdg , respectively. Moreover, we define
$t\equiv(p_{M_{1}}+p_{M_{2}})^{2}$,
$\rho_{M_{1}M_{2}}\equiv[(1-m_{+}^{2}/t)(1-m_{-}^{2}/t)]^{1/2}$ with
$m_{\pm}=m_{M_{1}}\pm m_{M_{2}}$, and
$\Gamma_{\sigma_{0}}\equiv(\rho_{\pi\pi}/\bar{\rho}_{\pi\pi})\Gamma^{0}_{\sigma_{0}}$
with $\bar{\rho}_{\pi\pi}=\rho_{\pi\pi}(t)$ at $t=m_{\sigma_{0}}^{2}$, while
$\langle M_{1}M_{2}|S_{0}\rangle\equiv C_{S_{0}\to M_{1}M_{2}}$ explains the
$S_{0}\to M_{1}M_{2}$ strong decay. Subsequently, the amplitude of $D\to
S_{0}e^{+}\nu_{e},S_{0}\to M_{1}M_{2}$ decay is given by ${\cal M}(D\to
S_{0}e^{+}\nu,S_{0}\to M_{1}M_{2})={\cal R}_{S_{0}}{\cal M}(D\to
S_{0}e^{+}\nu_{e})$.

The two possible quark contents of $S_{0}$ need the investigation, written as

$\displaystyle|f_{0}\rangle=\cos\theta_{I}|s\bar{s}\rangle+\sin\theta_{I}|n\bar{n}\rangle\,,$
$\displaystyle|\sigma_{0}\rangle=-\sin\theta_{I}|s\bar{s}\rangle+\cos\theta_{I}|n\bar{n}\rangle\,,$
$\displaystyle|a_{0}^{0}\rangle=|\sqrt{1/2}(u\bar{u}-d\bar{d})\rangle\,,|a_{0}^{-}\rangle=|d\bar{u}\rangle\,,$
(6)

for the $q\bar{q}$ structures Stone:2013eaa ; Maiani:2004uc , and

$\displaystyle|f_{0}\rangle=\cos\theta_{II}|n\bar{n}s\bar{s}\rangle+\sin\theta_{II}|u\bar{u}d\bar{d}\rangle\,,$
$\displaystyle|\sigma_{0}\rangle=-\sin\theta_{II}|n\bar{n}s\bar{s}\rangle+\cos\theta_{II}|u\bar{u}d\bar{d}\rangle\,,$
$\displaystyle|a_{0}^{0}\rangle=|\sqrt{1/2}(u\bar{u}-d\bar{d})s\bar{s}\rangle\,,|a_{0}^{-}\rangle=|d\bar{u}s\bar{s}\rangle\,,$
(7)

for the $q^{2}\bar{q}^{2}$ structures Maiani:2004uc , where
$|n\bar{n}\rangle\equiv|\sqrt{1/2}(u\bar{u}+d\bar{d})\rangle$ and
$|n\bar{n}s\bar{s}\rangle\equiv|\sqrt{1/2}(u\bar{u}+d\bar{d})s\bar{s}\rangle$,
and $\theta_{I,II}$ are the $f_{0}-\sigma_{0}$ mixing angles. For the
$q\bar{q}$ structures, we derive that

$\displaystyle\langle
f_{0},\sigma_{0}|(\bar{d}c)|D^{+}\rangle=(\sin\theta_{I},\cos\theta_{I})\times\langle
f_{n}|(\bar{d}c)|D^{+}\rangle\,,$ $\displaystyle\langle
a_{0}^{0}|(\bar{d}c)|D^{+}\rangle=\sqrt{1/2}\langle
a_{0}^{-}|(\bar{d}c)|D^{0}\rangle=-\langle f_{n}|(\bar{d}c)|D^{+}\rangle\,,$
$\displaystyle\langle
f_{0},\sigma_{0},a_{0}^{0}|(\bar{s}c)|D^{+}_{s}\rangle=(\cos\theta_{I},-\sin\theta_{I},0)\times\langle
f_{s}|(\bar{s}c)|D^{+}_{s}\rangle\,,$ (8)

with $f_{n}\equiv|n\bar{n}\rangle$ and $f_{s}\equiv|s\bar{s}\rangle$. The
matrix elements of the $D\to S_{0}$ transition for the $q^{2}\bar{q}^{2}$
structures can be a little bit more complicated, given by

$\displaystyle\langle f_{0}|(\bar{d}c)|D^{+}\rangle=\cos\theta_{II}\langle
F_{ns}|(\bar{d}c)|D^{+}\rangle+\sin\theta_{II}\langle
F_{ud}|(\bar{d}c)|D^{+}\rangle\,,$
$\displaystyle\langle\sigma_{0}|(\bar{d}c)|D^{+}\rangle=-\sin\theta_{II}\langle
F_{ns}|(\bar{d}c)|D^{+}\rangle+\cos\theta_{II}\langle
F_{ud}|(\bar{d}c)|D^{+}\rangle\,,$ $\displaystyle\langle
a_{0}^{0}|(\bar{d}c)|D^{+}\rangle=\sqrt{1/2}\langle
a_{0}^{-}|(\bar{d}c)|D^{0}\rangle=-\langle F_{ns}|(\bar{d}c)|D^{+}\rangle\,,$
$\displaystyle\langle
f_{0},\sigma_{0},a_{0}^{0}|(\bar{s}c)|D^{+}_{s}\rangle=(\cos\theta_{II},-\sin\theta_{II},0)\times\langle
F_{ns}|(\bar{s}c)|D^{+}_{s}\rangle\,,$ (9)

with $F_{ns}\equiv|n\bar{n}s\bar{s}\rangle$ and
$F_{ud}\equiv|u\bar{u}d\bar{d}\rangle$. By taking $S_{0}$ as the tetraquark,
the $D^{+}\to d\bar{d}\to F_{ns(ud)}$ transition needs an additional quark
pair from $g\to s\bar{s}(u\bar{u})$, where $g$ denotes a gluon occurring from
the vacuum. In the $D_{s}^{+}\to s\bar{s}\to F_{ns}$ transition, $g\to
u\bar{u}$ or $g\to d\bar{d}$ is needed.

Figure 1: $\theta_{\bf M}$, $\theta_{\bf L}$ and $\phi$: the angular variables
of the four-body $D\to S_{0}e^{+}\nu_{e},S_{0}\to M_{1}M_{2}$ decay.

For the resonant four-body $D(p_{D})\to
S_{0}(p_{S_{0}})e^{+}(p_{e})\nu(p_{\nu}),S_{0}(p_{S_{0}})\to
M_{1}(p_{1})M_{2}(p_{2})$ decay, one has $s\equiv(p_{e}+p_{\nu})^{2}\equiv
m_{e\nu}^{2}$, $t$, and the angular variables $(\theta_{\bf M},\theta_{\bf
L},\phi)$ in the phase space Kl4 ; Wise ; Tsai:2021ota . We illustrate the
angular variables in Fig. 1, where $\theta_{\bf M(L)}$ is the angle between
$\vec{p}_{1}$ ($\vec{p}_{e}$) in the $M_{1}M_{2}$ ($e\nu$) rest frame and the
resonant $S_{0}$ moving direction (the line of flight of $e\nu$ system) in the
$D$ meson rest frame, while $\phi$ is the angle between the $M_{1}M_{2}$ and
$e\nu$ planes, formed by the momenta of the $M_{1}M_{2}$ and $e\nu$ systems,
respectively, in the $D$ meson rest frame. The partial decay width thus reads
Hsiao:2022tfj ; Hsiao:2022uzx

$\displaystyle d\Gamma=\frac{|\bar{\cal
M}|^{2}}{4(4\pi)^{6}m_{D}^{3}}X\alpha_{\bf M}\alpha_{\bf
L}\,ds\,dt\,d\text{cos}\,\theta_{\bf M}\,d\text{cos}\,\theta_{\bf
L}\,d\phi\,,$ (10)

where $X=[(m_{D}^{2}-s-t)^{2}/4-st]^{1/2}$, $\alpha_{\bf
M}=\lambda^{1/2}(t,m_{1}^{2},m_{2}^{2})/t$, and $\alpha_{\bf
L}=\lambda^{1/2}(s,m_{e}^{2},m_{\nu}^{2})/s$, with
$\lambda(a,b,c)=a^{2}+b^{2}+c^{2}-2ab-2bc-2ca$. The allowed regions of the
variables are $(m_{e}+m_{\nu})^{2}\leq s\leq(m_{D}-\sqrt{t})^{2}$,
$(m_{1}+m_{2})^{2}\leq t\leq(m_{D}-m_{e}-m_{\nu})^{2}$, $0\leq\theta_{\bf
M,L}\leq\pi$, and $0\leq\phi\leq 2\pi$.

##  III Numerical Results

In the numerical analysis, we use $V_{cs}=1-\lambda^{2}$ and $V_{cd}=-\lambda$
with $\lambda=0.22453\pm 0.00044$ in the Wolfenstein parameterization pdg ,
and the mixing angles in the $q\bar{q}$ and $q^{2}\bar{q}^{2}$ pictures are
given by Klempt:2021nuf ; Maiani:2004uc

$\displaystyle\theta_{I}=(158.7\pm
4.0)^{\circ}\,,\;\theta_{II}=(174.6^{+3.4}_{-3.2})^{\circ}\,.$ (11)

We follow Refs. Becirevic:1999kt ; pdg  to use
$(m_{\text{A}},m_{\text{B}})=(2.46,2.32)$ GeV and $(2.42,2.34)$ GeV for the
$D_{s}^{+}\to S_{0}$ and $D^{+(0)}\to S_{0}$ transitions, respectively. In Eq.
(II), the denominators involve the resonant masses (decay width) and the
coupling constants, which are fitted to agree with the partial distributions
of $D\to S_{0}e^{+}\nu,S_{0}\to M_{1}M_{2}$ in the $M_{1}M_{2}$ invariant mass
spectra. The fitted values of $f_{0}$, $a_{0}$, and $\sigma_{0}$ can be found
in Refs. LHCb:2015klp ; LHCb:2012ae , Abele:1998qd ; Yang:2022jog ;
Chai:2021pyp , and pdg ; LHCb:2015klp , respectively, given by

$\displaystyle(m_{f_{0}},g_{f_{0}\pi\pi})=(940,199\pm
30)~{}\text{MeV}\,,\;g_{f_{0}KK}=(3.0\pm 0.3)g_{f_{0}\pi\pi}\,,$
$\displaystyle(m_{a_{0}},g_{a_{0}\pi\eta})=(999,324\pm
15)~{}\text{MeV}\,,\;g^{2}_{a_{0}KK}=(1.03\pm 0.14)g^{2}_{a_{0}\pi\eta}\,,$
$\displaystyle(m_{\sigma_{0}},\Gamma_{\sigma_{0}}^{0})=(500,500)~{}\text{MeV}\,.$
(12)

One determines $C_{S_{0}\to M_{1}M_{2}}$ to be BESIII:2020hfw ; Cheng:2022ysn

$\displaystyle
C_{f_{0}\to\pi^{+}\pi^{-}}=\sqrt{2}C_{f_{0}\to\pi^{0}\pi^{0}}=(1.5\pm
0.1)~{}\text{GeV}\,,$ $\displaystyle C_{a_{0}^{-(0)}\to\pi^{-(0)}\eta}=(2.5\pm
0.2)~{}\text{GeV}\,,$ $\displaystyle
C_{\sigma_{0}\to\pi^{+}\pi^{-}}=\sqrt{2}C_{\sigma_{0}\to\pi^{0}\pi^{0}}=(3.9\pm
0.1)~{}\text{GeV}\,.$ (13)

Table 1: Branching fractions of the resonant $D\to S_{0}e^{+}\nu_{e},S_{0}\to M_{1}M_{2}$ decays in the $q\bar{q}$ and $q^{2}\bar{q}^{2}$ structures to be compared to the data. The error receives the uncertainties from the mixing angle, form factor, and strong coupling constants.  decay channel |  our work: $(q\bar{q},q^{2}\bar{q}^{2})$ | experimental data
---|---|---
$10^{3}{\cal B}(D_{s}^{+}\to f_{0}e^{+}\nu_{e},f_{0}\to\pi^{+}\pi^{-})$ | (input, input) |  $1.7\pm 0.2$ BESIII:2023wgr
$10^{4}{\cal B}(D_{s}^{+}\to\sigma_{0}e^{+}\nu_{e},\sigma_{0}\to\pi^{+}\pi^{-})$ | $(25.8^{+12.5}_{-\;\,9.8},1.5^{+2.4}_{-1.3})$ | —–
$10^{4}{\cal B}(D_{s}^{+}\to f_{0}e^{+}\nu_{e},f_{0}\to\pi^{0}\pi^{0})$ | $(8.3\pm 1.5,8.3\pm 1.5)$ |  $7.9\pm 1.4$ BESIII:2021drk
$10^{4}{\cal B}(D_{s}^{+}\to\sigma_{0}e^{+}\nu_{e},\sigma_{0}\to\pi^{0}\pi^{0})$ | $(12.9^{+6.3}_{-4.9},0.8^{+1.2}_{-0.7})$ |  $<6.4$ BESIII:2021drk
$10^{4}{\cal B}(D^{0}\to a_{0}^{-}e^{+}\nu_{e},a_{0}^{-}\to\pi^{-}\eta)$ | (input, input) |  $1.3\pm 0.3$ pdg
$10^{4}{\cal B}(D^{+}\to a_{0}^{0}e^{+}\nu_{e},a_{0}^{0}\to\pi^{0}\eta)$ | $(1.6\pm 0.5,1.6\pm 0.5)$ |  $2.0\pm 0.9$ pdg
$10^{5}{\cal B}(D^{+}\to f_{0}e^{+}\nu_{e},f_{0}\to\pi^{+}\pi^{-})$ | $(0.5\pm 0.2,3.8\pm 1.2)$ |  $<2.8$ BESIII:2018qmf
$10^{4}{\cal B}(D^{+}\to\sigma_{0}e^{+}\nu_{e},\sigma_{0}\to\pi^{+}\pi^{-})$ | $(3.7^{+1.1}_{-0.9},8.4^{+2.4}_{-2.1})$ |  $6.3\pm 0.5$ pdg

Using the measured ${\cal B}(D_{s}^{+}\to
f_{0}e^{+}\nu_{e},f_{0}\to\pi^{+}\pi^{-})$ in Table 1 and the parameters in
Eqs. (11, III), and (III), we extract $F^{D_{s}^{+}\to f_{0}}\equiv f(0)$ for
$D_{s}^{+}\to f_{0}$. Likewise, we determine $F^{D^{0}\to a_{0}^{-}}$ from
${\cal B}(D^{0}\to a_{0}^{-}e^{+}\nu_{e},a_{0}^{-}\to\pi^{-}\eta)$. The fitted
form factors are given by

$\displaystyle F^{D_{s}^{+}\to f_{0}}=0.55\pm 0.03\,,\;F^{D^{0}\to
a_{0}^{-}}=0.46\pm 0.06\,,$ (14)

where the errors come from the data. In terms of the mixing relations in Eqs.
(II, II), we relate $F^{D_{s}^{+}\to f_{0}}$ and $F^{D^{0}\to a_{0}^{-}}$ to
those of the other decays, and calculate the branching fractions in Table 1 in
comparison with the data.

##  IV Discussions and Conclusions

We consider the four-body $D\to S_{0}e^{+}\nu,S_{0}\to M_{1}M_{2}$ decay for
the branching fraction, instead of using ${\cal B}(D\to S_{0}e^{+}\nu,S_{0}\to
M_{1}M_{2})\simeq{\cal B}(D\to S_{0}e^{+}\nu){\cal B}(S_{0}\to M_{1}M_{2})$,
such that the $S_{0}\to M_{1}M_{2}$ resonant effect is taken into account. In
Eq. (14), $F^{D_{s}^{+}\to f_{0}}=0.55\pm 0.03$ being close to the value of
$0.52\pm 0.04$ by BESIII justifies our determination, with which we obtain
$F^{D_{s}^{+}\to\sigma_{0}}_{(q\bar{q},q^{2}\bar{q}^{2})}=-(\tan\theta_{I},\tan\theta_{II})\times
F^{D_{s}^{+}\to f_{0}}=(0.22\pm 0.01,0.052\pm 0.003)$ for the $q\bar{q}$ and
$q^{2}\bar{q}^{2}$ structures, respectively. We thus calculate the branching
fractions as

$\displaystyle{\cal B}_{(q\bar{q},q^{2}\bar{q}^{2})}^{D_{s}^{+}\to
f_{0}\to\pi^{0}\pi^{0}}=(8.3\pm 1.5,8.3\pm 1.5)\times 10^{-4}\,,$
$\displaystyle{\cal
B}_{(q\bar{q},q^{2}\bar{q}^{2})}^{D_{s}^{+}\to\sigma_{0}\to\pi^{0}\pi^{0}}=(12.9^{+6.3}_{-4.9},0.8^{+1.2}_{-0.7})\times
10^{-4}\,,$ (15)

with ${\cal B}_{(q\bar{q},q^{2}\bar{q}^{2})}^{D\to S_{0}\to
M_{1}M_{2}}\equiv{\cal B}_{(q\bar{q},q^{2}\bar{q}^{2})}(D\to
S_{0}e^{+}\nu_{e},S_{0}\to M_{1}M_{2})$, where both ${\cal
B}_{(q\bar{q},q^{2}\bar{q}^{2})}^{D_{s}^{+}\to f_{0}\to\pi^{0}\pi^{0}}$ are
consistent with the data. By comparing ${\cal
B}_{(q\bar{q},q^{2}\bar{q}^{2})}^{D_{s}^{+}\to\sigma_{0}\to\pi^{0}\pi^{0}}$
with the experimental upper bound in Table 1: ${\cal B}_{ex}<6.4\times
10^{-4}$, it is clearly seen that the $S_{0}$ meson favors to be the
tetraquark.

In the semileptonic $D^{+}$ decays, it is given that $F_{q\bar{q}}^{D^{+}\to
a_{0}^{0}}=F_{q^{2}\bar{q}^{2}}^{D^{+}\to a_{0}^{0}}$ $=-\sqrt{1/2}F^{D^{0}\to
a_{0}^{-}}$ with the value of $F^{D^{0}\to a_{0}^{-}}$ in Eq. (14). We hence
obtain ${\cal B}_{(q\bar{q},q^{2}\bar{q}^{2})}^{D^{+}\to
a_{0}^{0}\to\pi^{0}\eta}=(1.6\pm 0.5,1.6\pm 0.5)\times 10^{-4}$ interpreting
the data. We also obtain

$\displaystyle{\cal B}^{D^{+}\to
f_{0}\to\pi^{+}\pi^{-}}_{(q\bar{q},q^{2}\bar{q}^{2})}=(0.5\pm 0.2,3.8\pm
1.2)\times 10^{-5}\,,$ $\displaystyle{\cal
B}^{D^{+}\to\sigma_{0}\to\pi^{+}\pi^{-}}_{(q\bar{q},q^{2}\bar{q}^{2})}=(3.7^{+1.1}_{-0.9},8.4^{+2.4}_{-2.1})\times
10^{-4}\,,$ (16)

where we have used $(F_{q\bar{q}}^{D^{+}\to
f_{0}},F_{q\bar{q}}^{D^{+}\to\sigma_{0}})=(\sin\theta_{I},\cos\theta_{I})\times
F^{D^{+}\to a_{0}^{0}}$ and $F^{D^{+}\to
f_{0}}_{q^{2}\bar{q}^{2}}=-\cos\theta_{II}$ $\times F^{D^{+}\to a_{0}^{0}}$
with $\sin\theta_{II}\simeq 0$ being safely neglected, whereas
$F^{D^{+}\to\sigma_{0}}_{q^{2}\bar{q}^{2}}$ cannot be simply related to
$F^{D^{+}\to a_{0}^{0}}$, due to that $\sigma_{0}=\cos\theta_{II}F_{ud}$ and
$a_{0}^{0}=F_{ns}$ involve the different quark pairs from $g\to u\bar{u}$ and
$g\to s\bar{s}$, respectively, in the $D^{+}\to d\bar{d}\to S_{0}$
transitions. To estimate ${\cal
B}^{D^{+}\to\sigma_{0}\to\pi^{+}\pi^{-}}_{(q^{2}\bar{q}^{2})}$, the $SU(3)$
flavor $[SU(3)_{f}]$ symmetry has been applied, such that $g\to u\bar{u}$ is
not distinguished from $g\to s\bar{s}$. We hence assume
$F_{ud}=\sqrt{2}F_{ns}$, which leads to
$F^{D^{+}\to\sigma_{0}}=-\cos\theta_{II}\sqrt{2}F^{D^{+}\to a_{0}^{0}}$.

In Eq. (IV), ${\cal B}_{(q\bar{q})}^{D^{+}\to f_{0}\to\pi^{+}\pi^{-}}=(0.5\pm
0.2)\times 10^{-5}$ corresponds to $\sin^{2}\theta_{I}$ as small as 0.13,
reflecting the fact that $f_{0}$ is mainly a $s\bar{s}$ bound state in the
$q\bar{q}$ structure. It seems that ${\cal B}_{(q^{2}\bar{q}^{2})}^{D^{+}\to
f_{0}\to\pi^{+}\pi^{-}}=3.8\times 10^{-5}$ deviates from ${\cal
B}_{ex}(D^{+}\to f_{0}e^{+}\nu_{e},f_{0}\to\pi^{+}\pi^{-})<2.8\times 10^{-5}$,
which is, however, still allowed to be within the upper limit by considering
the uncertainty. For the $\sigma_{0}$ decay modes, ${\cal
B}^{D^{+}\to\sigma_{0}\to\pi^{+}\pi^{-}}_{(q\bar{q})}=(3.7^{+1.1}_{-0.9})\times
10^{-4}$ is in tension with ${\cal B}_{ex}=(6.3\pm 0.5)\times 10^{-4}$,
whereas ${\cal
B}^{D^{+}\to\sigma_{0}\to\pi^{+}\pi^{-}}_{(q^{2}\bar{q}^{2})}=(8.4^{+2.4}_{-2.1})\times
10^{-4}$ shows its consistency. Therefore, we conclude that it is too early to
say which of the two quark constituents can interpret the semileptonic $D^{+}$
decays, while the calculations have not presented decisive disagreements with
the data.

Since $\bar{B}^{0}_{s}\to J/\Psi(f_{0},\sigma_{0})$ and $\bar{B}^{0}\to
J/\Psi(f_{0},\sigma_{0})$, respectively, proceed through the
$\bar{B}^{0}_{s}\to s\bar{s}\to(f_{0},\sigma_{0})$ and $\bar{B}^{0}\to
d\bar{d}\to(f_{0},\sigma_{0})$ transitions, their relative decay rates have
been proposed to distinguish between the quark content of the scalars being
quark-antiquark or tetraquark Stone:2013eaa . Indeed, the following
measurement of $\bar{B}^{0}_{s}\to J/\Psi f_{0},f_{0}\to\pi^{+}\pi^{-}$ by
LHCb suggests that the mixing angle is less than $7.7^{\circ}$ LHCb:2014ooi ,
consistent with the tetraquark interpretation Maiani:2004uc .

We demonstrate that the semileptonic $D^{+}$ decays are not as good as the
$D_{s}^{+}$ ones to discern the $q\bar{q}$ or $q^{2}\bar{q}^{2}$ structures
for $S_{0}$. One possible reason is that the resonant peaks from
$D^{+}\to\rho^{0}\to e^{+}\nu,\rho^{0}\to\pi^{+}\pi^{-}$ and $D^{+}\to
S_{0}\to e^{+}\nu,S_{0}\to\pi^{+}\pi^{-}$ might overlap in the
$\pi^{+}\pi^{-}$ invariant mass spectrum, such that the interference could
cause a less accurate analysis. In contrast, the semileptonic $D_{s}^{+}$
decays of
$D_{s}^{+}\to\sigma_{0}\ell^{+}\nu_{\ell},\sigma_{0}\to\pi^{+(0)}\pi^{+(0)}$
with $\ell=(e,\mu)$ are able to avoid the resonant peak from $\rho^{0}$.
Particularly, ${\cal
B}_{(q\bar{q})}^{D_{s}^{+}\to\sigma_{0}\to\pi^{+}\pi^{-}}=(25.8^{+12.5}_{-\;\,9.8})\times
10^{-4}$ and ${\cal
B}_{(q^{2}\bar{q}^{2})}^{D_{s}^{+}\to\sigma_{0}\to\pi^{+}\pi^{-}}=(1.5^{+2.4}_{-1.3})\times
10^{-4}$ are predicted to be far from each other, benefiting an experimental
clarification.

In summary, we have studied the semileptonic $D^{+,0},D^{+}_{s}\to
S_{0}e^{+}\nu_{e},S_{0}\to M_{1}M_{2}$ decays, where the resonant effect from
the broad decay width of $S_{0}$ has been considered. To calculate the
branching fractions, we have used the current data to extract the form factors
of the $D^{+}_{s}\to f_{0}$ and $D^{0}\to a_{0}^{-}$ transitions, and have
related them to those of the other semileptonic $D^{+}_{(s)}$ decays. We have
hence obtained ${\cal
B}(D_{s}^{+}\to\sigma_{0}e^{+}\nu_{e},\sigma_{0}\to\pi^{0}\pi^{0})=(12.9^{+6.3}_{-4.9})\times
10^{-4}$ and $(0.8^{+1.2}_{-0.7})\times 10^{-4}$ for the $q\bar{q}$ and
$q^{2}\bar{q}^{2}$ structures, respectively, compared to the experimental
upper limit of $6.4\times 10^{-4}$. Clearly, it has been shown that $S_{0}$
prefers to be the tetraquark state. We have studied the
$D_{s}^{+}\to\sigma_{0}e^{+}\nu_{e},\sigma_{0}\to\pi^{+}\pi^{-}$ with ${\cal
B}_{q\bar{q}}=(25.8^{+12.5}_{-\;\,9.8})\times 10^{-4}$ and ${\cal
B}_{q^{2}\bar{q}^{2}}(1.5^{+2.4}_{-1.3})\times 10^{-4}$. With the two
predicted numbers deviating far from each other, it indicated that future
measurements can make a clarification for the nature of the scalar meson.

## ACKNOWLEDGMENTS

YKH was supported in parts by NSFC (Grants No. 11675030 and No. 12175128). BCK
was supported in parts by NSFC (Grants No. 11875054 and No. 12192263) and
Joint Large-Scale Scientific Facility Fund of the NSFC and CAS (Grant No.
U2032104).

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