# Two-photon production of $f_{0}$ and $a_{0}$ resonances as hadronic
molecules composed of two vector mesons

Li-Ke Yang<EMAIL_ADDRESS>CAS Key Laboratory of Theoretical Physics,
Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190,
China  School of Physics, University of Chinese Academy of Sciences (UCAS),
Beijing 100049, China  Zheng-Li Wang<EMAIL_ADDRESS>School of
Physics, University of Chinese Academy of Sciences (UCAS), Beijing 100049,
China  Jia-Jun Wu<EMAIL_ADDRESS>School of Physics, University of Chinese
Academy of Sciences (UCAS), Beijing 100049, China  Bing-Song Zou
<EMAIL_ADDRESS>Department of Physics, Tsinghua University, Beijing 100084,
China  CAS Key Laboratory of Theoretical Physics, Institute of Theoretical
Physics, Chinese Academy of Sciences, Beijing 100190, China  School of
Physics, University of Chinese Academy of Sciences (UCAS), Beijing 100049,
China

###### Abstract

Ascribed as $\rho\rho$ and $K^{*}\bar{K}^{*}$ molecular states, respectively,
iso-scalar $f_{0}(1500)$ and $f_{0}(1710)$ states are expected to have iso-
vector partners, potentially identified as $a_{0}(1450)$ and $a_{0}(1710)$.
The predicted dominant decay modes for these two $a_{0}$ resonances are
$a_{0}(1450)\to\omega\pi\pi$ and $a_{0}(1710)\to\omega\pi\pi,\,\phi\pi\pi$. We
estimate cross sections for two-photon production of these four resonances
within hadronic molecular picture, and demonstrate that SuperKEKB’s luminosity
is sufficient for their observation and more precise parameter measurements.

##  I Introduction

Understanding the hadron spectrum is one of the most important challenges in
hadron physics. There are various models interpreting the observed states, and
selecting the right one is a key issue. These models can be differentiated by
examining their predictions of the complete spectrum, with consideration of
fundamental symmetries. For example, in the physical depiction of $K\bar{K}$
molecules, both iso-scalar and iso-vector states are symmetrically generated,
corresponding to the existing states $f_{0}(980)$ and $a_{0}(980)$ [1, 2].

The $f_{0}(1370)$ was previously considered to be an iso-scalar $\rho\rho$
molecular state, dynamically generated by the $\rho\rho$ interaction [3, 4].
However, the inclusion of coupled-channels of pseudoscalar mesons suggests
that the pole and partial decay widths are more closely aligned with those of
the $f_{0}(1500)$ listed in the Review of Particle Physics (RPP) [5].

Similarly, $f_{0}(1710)$ is interpreted as the iso-scalar $K^{*}\bar{K}^{*}$
molecular state, dynamically generated by the vector meson-vector meson
interaction [3, 6]. Including the coupled-channels of pseudoscalar mesons does
not significantly shift the pole, and the mass, width, and partial widths are
consistent with those of $f_{0}(1710)$ reported in the RPP [16].

However, the measurement as well as exploration of the iso-vector partners of
$f_{0}(1500)$ and $f_{0}(1710)$ as hadronic molecules are less explicit. Given
that $\rho\rho$ cannot form an $a_{0}$ state within the isospin formalism due
to Bose-Einstein statistics, which requires $l+S+I=\text{even}$, $\rho\omega$
emerges as the most promising candidate for forming the iso-vector partner of
$f_{0}(1500)$. This partner is postulated to be the $a_{0}(1450)$ [8].
Meanwhile, the iso-vector partner of $f_{0}(1710)$ is still missing,
potentially identifiable as $a_{0}(1710)$ [8].

Therefore, to validate the vector meson-vector meson molecule model of
$f_{0}(1500)$ and $f_{0}(1710)$ , it is essential to confirm the nature of
both $a_{0}(1450)$ and $a_{0}(1710)$. However, up to now, different
laboratories have not reached a consensus on the measured masses of
$a_{0}(1450)$ and $a_{0}(1710)$. The situation is even worse when it comes to
the measurement of the width. For $a_{0}(1450)$, measurements vary across
different experiments, while for $a_{0}(1710)$, only two results have been
reported so far.

The latest measurement of $a_{0}(1450)$ is from Belle in the study of the
reaction $\gamma\gamma\to\eta\pi^{0}$ [12]. The partial-wave analysis of the
differential cross section indicates the presence of a resonance that could be
possibly attributed to $a_{0}(1450)$. However, its nominal fit mass is
$1316.8\,\text{MeV}$, with no corresponding peak for $a_{0}(1450)$ in the
total cross section spectrum. In fact, if $a_{0}(1450)$ is a hadronic
molecule, its dominant decay channel is predicted to be $\omega\pi\pi$ [8], so
investigating $\gamma\gamma\to\omega\pi\pi$ is crucial to finding
$a_{0}(1450)$.

The BaBar Collaboration observed $a_{0}(1700)\to\eta\pi$ in their
$\gamma\gamma\to\eta_{c}\to\eta\pi^{+}\pi^{-}$ analysis [9]. The BESIII
Collaboration also found evidence supporting the existence of an iso-vector
partner of $f_{0}(1710)$ [10], reinforcing the existence of $a_{0}(1710)$ with
a mass close to that of $f_{0}(1710)$. However, a newly observed $a_{0}$-like
state with a mass of $1.817\,\text{GeV}$ is also suspected to be $a_{0}(1710)$
[11].

Therefore, it is important to identify suitable reactions for the search of
$a_{0}(1450)$ and $a_{0}(1710)$ to obtain more accurate measurements. The two-
photon production reaction could be an ideal place to detect $a_{0}(1450)$,
$a_{0}(1710)$, as well as $f_{0}(1500)$ and $f_{0}(1710)$. In this process,
one can exclude additional hadrons in the final state, except for the decay
products of the target resonance, such as observing $a_{0}(1450)$ in
$\gamma\gamma\to\omega\pi^{+}\pi^{-},\eta\pi^{0}$. Moreover, with the full
upgrade of KEKB to SuperKEKB, it is expected to provide 40 times more two-
photon production events compared to KEKB [13].

In this paper, we calculate the cross sections for the two-photon production
of $a_{0}(1450)$, $a_{0}(1710)$, $f_{0}(1500)$, and $f_{0}(1710)$, followed by
their subsequent decays into main decay channels, exemplified by
$\gamma\gamma\to a_{0}(1450)\to\omega\pi\pi$. We also present the expected
$N_{p}$s for these reactions at SuperKEKB. The paper is organized as follows:
In Sec. II, we mainly present the formalism for the Lagrangians and the
double-photon luminosity function. In Sec. III, we show the numerical results
and discussions. Finally we conclude with a summary in Sec. IV.

##  II Formalism

###  II.1 The Interaction Vertices

The effective Lagrangians involved are as follows [14]:

$\displaystyle\mathcal{L}_{VVV}$
$\displaystyle=ig\langle(\partial_{\mu}V_{\nu}-\partial_{\nu}V_{\mu})V^{\mu}V^{\nu}\rangle,$
(1) $\displaystyle\mathcal{L}_{V\gamma}$
$\displaystyle=-M_{V}^{2}\frac{e}{g}A_{\mu}\langle V^{\mu}Q\rangle,$ (2)
$\displaystyle\mathcal{L}_{VPP}$ $\displaystyle=-ig\langle
V^{\mu}[P,\partial_{\mu}P]\rangle,$ (3)
$\displaystyle\mathcal{L}_{V_{1}V_{2}a_{0}}$
$\displaystyle=g_{a_{0}V_{1}V_{2}}\phi_{a_{0}}V_{1}^{\mu}V_{2\mu},$ (4)
$\displaystyle\mathcal{L}_{V_{1}V_{2}f_{0}}$
$\displaystyle=g_{f_{0}V_{1}V_{2}}\phi_{f_{0}}V_{1}^{\mu}V_{2\mu}.$ (5)

$g=M_{V}/2f$ is the hidden gauge coupling constant, where $f$ is the pion
decay constant and $M_{V}$ is the mass of the vector meson $V$;
$\langle...\rangle$ denotes the trace in SU(3) space; $Q=diag(2,-1,-1)/3$, and
$e=-|e|$ is the electron charge; $\phi_{S}$ is the field of the scalar
resonance $S$; $V_{i}^{\mu}$, with $i=1,2$, are the fields of the vector
mesons $V_{i}$. Additionally, $V_{\mu}$ is the vector field matrix defined as
follows:

$\left(\begin{array}[]{ccc}\frac{1}{\sqrt{2}}\rho^{0}+\frac{1}{\sqrt{2}}\omega&\rho^{+}&K^{*+}\\\
\rho^{-}&-\frac{1}{\sqrt{2}}\rho^{0}+\frac{1}{\sqrt{2}}\omega&K^{*0}\\\
K^{*-}&\bar{K}^{*0}&\phi\end{array}\right)_{\mu},$ (6)

and $P$ is the pseudoscalar field matrix defined as follows:

$\left(\begin{array}[]{ccc}\frac{1}{\sqrt{2}}\pi^{0}+\frac{1}{\sqrt{6}}\eta&\pi^{+}&K^{+}\\\
\pi^{-}&-\frac{1}{\sqrt{2}}\pi^{0}+\frac{1}{\sqrt{6}}\eta&K^{0}\\\
K^{-}&\bar{K}^{0}&-\frac{2}{\sqrt{6}}\eta\end{array}\right).$ (7)

According to the Lagrangians, we can define the following tensors
corresponding to different vertex structures:

$\displaystyle(V_{SV_{1}V_{2}})^{\mu\nu}$
$\displaystyle=g_{SV_{1}V_{2}}g^{\mu\nu},$ (8) $\displaystyle V_{SP_{1}P_{2}}$
$\displaystyle=g_{SP_{1}P_{2}},$ (9)
$\displaystyle(V_{V_{1}V_{2}V_{3}})^{\alpha\mu\nu}$
$\displaystyle=g_{V_{1}V_{2}V_{3}}\left\\{(k_{V_{1}}^{\nu}-k_{V_{2}}^{\nu})g^{\alpha\mu}+\right.$
$\displaystyle\left.(k_{V_{2}}^{\alpha}-k_{V_{3}}^{\alpha})g^{\mu\nu}+(k_{V_{3}}^{\mu}-k_{V_{1}}^{\mu})g^{\nu\alpha}\right\\},$
(10) $\displaystyle(V_{\gamma V})^{\mu\nu}$ $\displaystyle=g_{\gamma
V}g^{\mu\nu},$ (11) $\displaystyle(V_{VP_{1}P_{2}})^{\mu}$
$\displaystyle=g_{VP_{1}P_{2}}(k^{\mu}_{P_{1}}-k^{\mu}_{P_{2}}).$ (12)

Note that we assume all particles in $(V_{V_{1}V_{2}V_{3}})^{\alpha\mu\nu}$ to
be incident particles approaching the vertex. In practice, note that for
$(V_{\rho\pi\pi})^{\mu}$ we have:

$\displaystyle(V_{\rho\pi\pi})^{\mu}$
$\displaystyle=(V_{\rho^{0}\pi^{+}\pi^{-}})^{\mu}+(V_{\rho^{+}\pi^{0}\pi^{+}})^{\mu}+(V_{\rho^{-}\pi^{0}\pi^{-}})^{\mu}.$
(13)

Based on these vertices, we can construct the corresponding Feynman diagrams,
as shown in Fig. 3(a)$\sim$2(b). P refers to pseudoscalar meson and V refers
to vector meson. All the detailed involved channels can be found in Sec. III.

(a)

(b)

Figure 1: Feynman diagrams for two-photon production of $a_{0}(1450)$

(a)

(b)

Figure 2: Feynman diagrams for two-photon production of $a_{0}(1710)$

(a)

(b)

Figure 3: Feynman diagrams for two-photon production of $f_{0}(1500)$

(a)

(b)

Figure 4: Feynman diagrams for two-photon production of $f_{0}(1710)$

###  II.2 Propogators

The propogators for spin 0 resonance and vector meson are defined as follows:

$\displaystyle G_{R}(k)=\frac{1}{k^{2}-m_{R}^{2}+im_{R}\Gamma_{R}},$ (14)
$\displaystyle(G_{V}(k))_{\mu\nu}=\frac{-\tilde{g}_{\mu\nu}}{k^{2}-m_{V}^{2}+im_{V}\Gamma_{R}},$
(15)

where k is the 4-momentum, and $m,\,\Gamma$ are the mass and decay width of
the particle. Particularly, in the case of a vector meson in $\gamma V$
vertex, where the 4-momentum satisfies $k^{2}=0$, the propagator is as
follows:

$\displaystyle(G^{\prime}_{V}(k))_{\mu\nu}=\frac{g_{\mu\nu}}{m_{V}^{2}}.$ (16)

Definition of $\tilde{g}_{\mu\nu}$ is as follows [15]:

$\displaystyle\sum\limits_{m=1}^{3}\epsilon_{\mu}(m)\epsilon^{*}_{\nu}(m)$
$\displaystyle=-g_{\mu\nu}+\frac{p_{\mu}p_{\nu}}{p^{2}}$
$\displaystyle\equiv-\tilde{g}_{\mu\nu}(p),$ (17)

where $\epsilon_{\mu}(m)$ is the polarization vector for a massive vector
meson with its 4-momentum to be $p_{\mu}$ and its thrid component of spin in
its rest frame to be $m$.

Next, consider the photon polarization vector $A_{\mu}$ with 4-momentum
$q_{\mu}$. We take into account both the Lorentz orthogonality condition
$A_{\mu}q^{\mu}=0$ and an additional gauge invariance condition. We assume the
Coulomb gauge $A_{\mu}Q^{\mu}=0$ in the c.m. frame, where $Q^{\mu}$ is the
summation of the momentum of two incoming photons. Let
$K_{\mu}=Q_{\mu}-q_{\mu}$, the 4-momentum of the other incoming photon. The
summation of polarization vectors is then given as follows [15]:

$\displaystyle\sum_{m}A_{\mu}(m)A^{*}_{\nu}(m)$
$\displaystyle=-g_{\mu\nu}+\frac{q_{\mu}K_{\nu}+K_{\mu}q_{\nu}}{q\cdot K}$
$\displaystyle\equiv-g_{\mu\nu}^{\bot\bot}.$ (18)

###  II.3 Form Factors

For each triangle loop in the Feynman diagram as shown in Fig. 4 and Fig. 5,
we introduce two types of form factors. The first type is a form factor with a
smooth cutoff that serves to regularize the loop integration:

$f_{\Lambda}(k)=\exp\left[{-\frac{|\vec{k}|^{2}}{(\Lambda_{1})^{2}}}\right],$
(19)

where $\Lambda_{1}=1.0\,\text{GeV}$[16]. The second type is a dipole form
factor for the off-shell exchanged meson $\bar{K}^{*}$:

$F_{M}(k)=\frac{(\Lambda_{2})^{4}}{(\Lambda_{2})^{4}+(k^{2}-m^{2})^{2}},$ (20)

where $\Lambda_{2}=1.0\,\text{GeV}$.

For the off-shell mesons $\rho$, $\omega$, and $\phi$ in the $\gamma V$
vertex, we also introduce a dipole form factor with cutoff parameter
$\Lambda_{3}=0.63\text{--}0.83\,\text{GeV}$. The determination of
$\Lambda_{3}$ will be discussed in Sec. II.6.

###  II.4 The Scattering Amplitudes

We can define a function $G(k_{1},k_{2},k_{5})$, which combines the
propogators in the loop with the aforementioned form factors, to calculate the
triangle loop $K^{*}(k_{3})-\bar{K}^{*}(k_{4})-\bar{K}^{*}(k_{5})$, where
$\bar{K}^{*}(k_{5})$ is the exchanged meson, and $k_{3}=k_{1}+k_{5}$,
$k_{4}=k_{2}-k_{5}$:

$\displaystyle
G(k_{1},k_{2},k_{5})=\frac{1}{(k_{1}+k_{5})^{2}-m_{K^{*}}^{2}+i\epsilon}$
$\displaystyle\times\frac{1}{(k_{2}-k_{5})^{2}-m_{\bar{K}^{*}}^{2}+i\epsilon}\frac{1}{k_{5}^{2}-m_{\bar{K}^{*}}^{2}+i\epsilon}$
$\displaystyle\times\frac{(\Lambda_{2})^{4}}{(\Lambda_{2})^{4}+(k_{5}^{2}-m_{\bar{K}^{*}}^{2})^{2}}\exp\left[{-\frac{|\vec{k}_{5}|^{2}}{(\Lambda_{1})^{2}}}\right].$
(21)

Next, we proceed to write down the scattering amplitude. In this section, we
present one scattering amplitude as an example, while the remaining scattering
amplitudes can be found in Appendix B.

For the Feynman diagram as shown in Fig. 4(a), the corresponding scattering
amplitude of $\gamma(k_{1})\gamma(k_{2})\to
f_{0}(1710)\to\pi(p_{1})\pi(p_{2})$ is as follows:

$\displaystyle\mathcal{M}_{2a}$
$\displaystyle=A_{1\mu}(m_{1},k_{1})A_{2\rho}(m_{2},k_{2})$
$\displaystyle\times(V_{\gamma(k_{1})\rho})^{\mu\nu}(V_{\gamma(k_{2})\rho})^{\rho\sigma}(G^{\prime}_{\rho}(k_{1}))_{\nu\alpha}(G^{\prime}_{\rho}(k_{2}))_{\sigma\kappa}$
$\displaystyle\times(V_{f_{0}K^{*}\bar{K}^{*}})^{\delta\eta}G_{f_{0}}(k_{1}+k_{2})V_{f_{0}P_{1}P_{2}}$
$\displaystyle\times\int\frac{id^{4}k_{5}}{(2\pi)^{4}}(-\tilde{g}_{\beta\delta}(k_{3}))(-\tilde{g}_{\lambda\eta}(k_{4}))(-\tilde{g}_{\gamma\theta}(k_{5}))$
$\displaystyle\times G(k_{1},k_{2},k_{5})\left\\{(V_{\rho
K^{*+}K^{*-}})^{\alpha\beta\gamma}(V_{\rho
K^{*-}K^{*+}})^{\kappa\lambda\theta}\right.$ $\displaystyle+\left.(V_{\rho
K^{*-}K^{*+}})^{\alpha\beta\gamma}(V_{\rho
K^{*+}K^{*-}})^{\kappa\lambda\theta}\right\\}$ $\displaystyle+\left.(V_{\rho
K^{*0}\bar{K}^{*0}})^{\alpha\beta\gamma}(V_{\rho\bar{K}^{*0}K^{*0}})^{\kappa\lambda\theta}\right\\}$
$\displaystyle+\left.(V_{\rho\bar{K}^{*0}K^{*0}})^{\alpha\beta\gamma}(V_{\rho
K^{*0}\bar{K}^{*0}})^{\kappa\lambda\theta}\right\\},$ (22)

where $A_{1\mu},A_{2\rho}$ are the polarization vectors of the two incoming
photons.

###  II.5 Coupling Constants

The coupling constants $g_{\gamma V}$ involved in the tensor $(V_{\gamma
V})^{\mu\nu}$ are given as follows:

$\displaystyle g_{\gamma\rho}$
$\displaystyle=-\frac{1}{\sqrt{2}}M_{\rho}^{2}\frac{e}{g},$ (23)
$\displaystyle g_{\gamma\omega}$
$\displaystyle=-\frac{1}{3\sqrt{2}}M_{\omega}^{2}\frac{e}{g},$ (24)
$\displaystyle g_{\gamma\phi}$
$\displaystyle=\frac{1}{3}M_{\phi}^{2}\frac{e}{g}.$ (25)

The coupling constants $g_{V_{1}V_{2}V_{3}}$ involved in the tensor
$(V_{V_{1}V_{2}V_{3}})^{\mu\nu\alpha}$ are given as follows:

$\displaystyle
g_{\rho^{0}K^{*+}K^{*-}}=g_{\rho^{0}K^{*-}K^{*+}}=\frac{1}{\sqrt{2}}ig,$ (26)
$\displaystyle
g_{\rho^{0}K^{*0}\bar{K}^{*0}}=g_{\rho^{0}K^{*0}\bar{K}^{*0}}=-\frac{1}{\sqrt{2}}ig,$
(27) $\displaystyle g_{\omega K^{*+}K^{*-}}=g_{\omega
K^{*-}K^{*+}}=\frac{1}{\sqrt{2}}ig,$ (28) $\displaystyle g_{\omega
K^{*0}\bar{K}^{*0}}=g_{\omega\bar{K}^{*0}K^{*0}}=\frac{1}{\sqrt{2}}ig,$ (29)
$\displaystyle g_{\phi K^{*+}K^{*-}}=g_{\phi K^{*-}K^{*+}}=ig,$ (30)
$\displaystyle g_{\phi K^{*0}\bar{K}^{*0}}=g_{\phi\bar{K}^{*0}K^{*0}}=ig.$
(31)

The coupling constant $g_{\rho\pi\pi}$ involved in the tensor
$(V_{\rho\pi\pi})^{\mu}$ is given as follows:

$g_{\rho\pi\pi}=-\sqrt{2}ig.$ (32)

We now come to discuss the coupling constants involved in the tensors
$(V_{SV_{1}V_{2}})^{\mu\nu}$ and $V_{SP_{1}P_{2}}$.

For a pure molecular state composed of two constituent particles, its coupling
constant with the constituent particles is given by the following formula[17]:

$\frac{g_{RV_{1}V_{2}}^{2}}{4\pi}=\frac{(m_{1}+m_{2})^{\frac{5}{2}}}{(m_{1}m_{2})^{\frac{1}{2}}}\sqrt{32\epsilon},$
(33)

where $m_{1},\,m_{2}$ are the masses of the two component particles
$V_{1},\,V_{2}$, and $\epsilon$ is the binding energy. With this formula we
can obtain $g_{a_{0}(1450)\rho\omega}$, $g_{a_{0}(1710)K^{*}\bar{K}^{*}}$,
$g_{a_{0}(1710)\rho\phi}$, $g_{f_{0}(1500)\rho\rho}$ and
$g_{f_{0}(1710)K^{*}\bar{K}^{*}}$. Note that $g_{f_{0}(1500)\rho\rho}$ gives
the coupling constant of $f_{0}(1500)$ to both $\rho^{0}\rho^{0}$ and
$\rho^{+}\rho^{-}$. Specifically, we have
$g_{f_{0}(1500)\rho^{0}\rho^{0}}=\frac{1}{\sqrt{2}}g_{f_{0}(1500)\rho^{+}\rho^{-}}=\frac{1}{\sqrt{3}}g_{f_{0}(1500)\rho\rho}$.

Generally, we can determine the coupling constants through data fitting,
provided that the corresponding partial decay widths are accessible. For two-
body decays, this determination can be performed using the following formula
[18]:

$\Gamma=\frac{1}{8\pi}|\mathcal{M}|^{2}\frac{|\vec{p}_{1}|}{M^{2}}.$ (34)

For our calculations, the partial widths of $f_{0}(1500)$ are based on the RPP
recommended values or average values [18], whereas the partial widths of
$f_{0}(1710)$ are derived using the unitary coupled channel approach, as
predicted in [16]. Similarly, the partial widths for $a_{0}(1450)$ and
$a_{0}(1710)$, given by the same approach, are sourced from [16] and [8]
respectively.

Additionally, it is worth mentioning that the mass and total width of the
resonances $f_{0}(1500)$, $f_{0}(1710)$, $a_{0}(1450)$, and $a_{0}(1710)$,
used for cross section calculations in Sec. III, are presented in Table 3. The
data for $f_{0}(1500)$ and $f_{0}(1710)$ come from the RPP [18], while the
data for $a_{0}(1450)$ and $a_{0}(1710)$ are sourced from [8].

Table 1: Partial widths and coupling constants relevant to $f_{0}(1500)$ and $f_{0}(1710)$ Channels | Partial Width [MeV] | Coupling Constant [GeV]
---|---|---
| $f_{0}(1500)$ | $f_{0}(1710)$ | $f_{0}(1500)$ | $f_{0}(1710)$
$\pi\pi$ | $37.3$ | $12.7$ | $1.70$ | $1.06$
$K\bar{K}$ | $9.2$ | $44.0$ | $0.96$ | $2.16$
$\eta\eta$ | $\backslash$ | $12.9$ | $\backslash$ | $1.21$
Table 2: Partial widths and coupling constants relevant to $a_{0}(1450)$ and $a_{0}(1710)$ Channels | Partial Width [MeV] | Coupling Constant [GeV]
---|---|---
| $a_{0}(1450)$ | $a_{0}(1710)$ | $a_{0}(1450)$ | $a_{0}(1710)$
$\rho\omega$ | $\backslash$ | $61.0$ | $\backslash$ | $3.50$
$\eta\pi$ | $14.7$ | $66.9$ | $1.14$ | $2.55$
$K\bar{K}$ | $13.2$ | $74.4$ | $1.15$ | $2.81$
Table 3: Mass and total width of $f_{0}$ and $a_{0}$ resonances Resonances | mass[GeV] | total width[GeV]
---|---|---
$f_{0}(1500)$ | $1.522$ | $0.108$
$f_{0}(1710)$ | $1.733$ | $0.150$
$a_{0}(1450)$ | $1.500$ | $0.128$
$a_{0}(1710)$ | $1.720$ | $0.220$

###  II.6 Cutoff parameter $\Lambda_{3}$

Belle has investigated $\gamma\gamma\to VV$ and presented the cross section
spectrum for $\gamma\gamma\to\phi\omega$ [19]. The phase space is expected to
drop significantly near the threshold, leading to a decrease in the cross
section. However, the contribution from the $0^{+}$ state does not seem to
decrease near $2.0\,\text{GeV}$, likely due to the contribution from
$f_{0}(1710)$. Therefore, by fitting the partial wave data of $0^{+}$
contribution from Belle for $\gamma\gamma\to\phi\omega$ [19], we can determine
the value of $\Lambda_{3}$, which represents the cutoff parameter applied in
the dipole form factor for the vector mesons $\rho,\,\omega,\,\phi$ in the
$\gamma V$ vertex. The corresponding Feynman diagram is shown in Fig. 5.

Figure 5: Feynman diagram for $\gamma\gamma\to f_{0}(1710)\to\phi\omega$

Note that the two vector mesons $(V(k_{1}),V(k_{2}))$ represented at the
$\gamma V$ vertex in Fig. 5 include the following pairs:
$(\rho^{0}(k_{1}),\rho^{0}(k_{2}))$, $(\phi(k_{1}),\phi(k_{2}))$,
$(\omega(k_{1}),\omega(k_{2}))$, $(\phi(k_{1}),\omega(k_{2}))$,
$(\omega(k_{1}),\phi(k_{2}))$. The corresponding scattering amplitude is
presented in Appendix B. The total width of $f_{0}(1710)$ is taken to be
$0.150\,\text{GeV}$ [18]. As for the mass, we adopt a range from
$1.71\,\text{GeV}$ [16] to $1.78\,\text{GeV}$, which is the mass threshold of
$K^{*}\bar{K}^{*}$.

We have set the cutoff parameters $\Lambda_{1}=1.0\,\text{GeV}$ and
$\Lambda_{2}=1.0\,\text{GeV}$ in $G(k_{1},k_{2},k_{5})$ and
$G(p_{1},p_{2},p_{5})$. Through data fitting, we obtained
$\Lambda_{3}=0.63\,\text{GeV}$ when the mass of $f_{0}(1710)$ is
$1.71\,\text{GeV}$, and $\Lambda_{3}=0.83\,\text{GeV}$ when the mass is
$1.78\,\text{GeV}$. Note that here $\Lambda_{3}$ is especially sensitive to
mass variations. This sensitivity arises because the coupling constant
$g_{f_{0}K^{*}\bar{K}^{*}}$, calculated with Eq. (33), decreases significantly
as the binding energy $\epsilon$ approaches 0.

###  II.7 Double-Photon Luminosity Function

If the cross section $\sigma_{\gamma\gamma}$ for a two-photon production
reaction is known, and the integrated luminosity $\int Ldt$ for $e^{+}e^{-}$
scattering is provided, then we can obtain the event yield for the two-photon
production as follows:

$Y_{\gamma\gamma}(W)=\sigma_{\gamma\gamma}(W)L_{\gamma\gamma}(W)\int Ldt\Delta
W,$ (35)

where $W$ is the c.m. energy of the two incident photons.
$L_{\gamma\gamma}(W)$ is the double-photon luminosity function. $\Delta W$ is
the bin width. $Y_{\gamma\gamma}(W)$ represents the event yield for two-photon
production in a $W$ range between $W$ and $W+\Delta W$.

Let $E_{0}$ be the initial energy of the electron in the $e^{+}e^{-}$ c.m.
frame. In the same frame, assume two photons with energies $E_{1}$ and
$E_{2}$, and four-momentum squares $-Q^{2}$ and $-P^{2}$, are produced in the
$e^{+}e^{-}$ scattering. For $Q^{2}\to 0$ and $P^{2}\to 0$, the luminosity
function $L_{\gamma\gamma}$ is given as follows [20]:

$L_{\gamma\gamma}(W)=\int
f_{\gamma/e}(y,W,Q^{2}_{\text{max}})f_{\gamma/e}(z,W,Q^{2}_{\text{max}})\frac{W}{2E_{0}^{2}}\frac{1}{2r}dr,$
(36)

where $y\approx E_{1}/E_{0}$, $z=E_{2}/E_{0}$, and $r=y/z$.

The definition of $f_{\gamma/e}$ is given as follows[12]:

$\displaystyle f_{\gamma/e}(z,W,Q^{2}_{\text{max}})=\frac{\alpha}{\pi
z}\left\\{(1+(1-z)^{2})\frac{1}{2}\right.$
$\displaystyle\times\left.\int_{ln\left(m_{e}^{2}z^{2}/(1-z)\right)}^{lnQ^{2}_{\text{max}}}F(e^{\nu},W)d\nu-1+z\right\\},$
(37)

where $\alpha\approx\frac{1}{137}$ is the fine-structure constant. $\nu=\ln
Q^{2}$ and $F(Q^{2},W)=1/(1+Q^{2}/W^{2})^{2}$[12].
$Q^{2}_{\text{max}}=1\,\text{GeV}^{2}$ gives a maximum virtuality of the
incident photons. Details on the derivation of the expression for
$L_{\gamma\gamma}$ can be found in Appendix A.

##  III Numerical results and discussion

Figure 6 shows the distribution of cross sections as a function of the c.m.
energy $W$ of the incoming $\gamma\gamma$ system for the two-photon production
of $a_{0}(1450)$, $a_{0}(1710)$, $f_{0}(1500)$, and $f_{0}(1710)$.

Figure 6: Cross sections for two-photon production of $a_{0}(1450)$(Log
scale), $a_{0}(1710)$, $f_{0}(1500)$, and $f_{0}(1710)$ vs. c.m. energy $W$.
In each figure, the lower group of lines corresponds to
$\Lambda_{3}=0.63\,\text{ GeV}$, and the upper group corresponds to
$\Lambda_{3}=0.83\,\text{ GeV}$.

In this work, we use a fixed $e^{+}e^{-}$ c.m. energy of $10.58\,\text{GeV}$
for the double-photon luminosity function calculation. The total integrated
luminosity is set to be $8000\,\text{fb}^{-1}$, about $40$ times that of
Belle’s $223\,\text{fb}^{-1}$ in their $\gamma\gamma\to\eta\pi^{0}$ analysis
[12].

Table 4: Peak information related to $a_{0}(1450)$ | $\omega\pi\pi$ | $\eta\pi$ | $K\bar{K}$
---|---|---|---
$E_{p}$ (Peak energy) [GeV] | $1.54$ | $1.50$ | $1.50$
$\sigma_{p}$ (Peak cross section) [nb] | $0.39\text{--}4.77$ | $0.04\text{--}0.52$ | $0.04\text{--}0.46$
$N_{p}$ (Peak event yield) [$10^{5}/(10\,\text{MeV})$] | $1.53\text{--}18.8$ | $0.17\text{--}2.15$ | $0.16\text{--}1.92$

Table 5: Peak information related to $a_{0}(1710)$ | $\phi\pi\pi$ | $\omega\pi\pi$ | $\eta\pi$ | $K\bar{K}$
---|---|---|---|---
$E_{p}$ (Peak energy) [GeV] | $1.84$ | $1.74$ | $1.71$ | $1.71$
$\sigma_{p}$ (Peak cross section) [nb] | $0.06\text{--}1.10$ | $0.06\text{--}1.07$ | $0.02\text{--}0.37$ | $0.02\text{--}0.41$
$N_{p}$ (Peak event yield) [$10^{5}/(10\,\text{MeV})$] | $0.18\text{--}3.08$ | $0.20\text{--}3.35$ | $0.07\text{--}1.20$ | $0.08\text{--}1.34$

Table 6: Peak information related to $f_{0}(1500)$ | $\pi\pi$ | $K\bar{K}$ | $4\pi$
---|---|---|---
$E_{p}$ (Peak energy) [GeV] | $1.52$ | $1.52$ | $1.52$
$\sigma_{p}$ (Peak cross section) [nb] |  $0.13$–$1.59$ |  $0.03$–$0.39$ |  $0.29$–$3.43$
$N_{p}$ (Peak event yield) [$10^{5}/(10\,\text{MeV})$] |  $0.55$–$6.44$ |  $0.13$–$1.58$ |  $1.18$–$13.9$

Table 7: Peak information related to $f_{0}(1710)$ | $\pi\pi$ | $K\bar{K}$ | $\eta\eta$
---|---|---|---
$E_{p}$ (Peak energy) [GeV] | $1.74$ | $1.74$ | $1.74$
$\sigma_{p}$ (Peak cross section) [nb] |  $0.02$–$0.05$ |  $0.06$–$0.17$ |  $0.02$–$0.05$
$N_{p}$ (Peak event yield) [$10^{5}/(10\,\text{MeV})$] |  $0.05$–$0.16$ |  $0.19$–$0.54$ |  $0.06$–$0.16$

Tables 7 through 7 present the peak information for $E_{p}$ (peak energy in
GeV), $\sigma_{p}$ (peak cross section in nb), and $N_{p}$ (peak event yield
in $10^{5}/(10\,\text{MeV})$) across reactions, with the values varying based
on $\Lambda_{3}$. To avoid any confusion, it should be noted that the
previously determined range for $\Lambda_{3}$, based on two sets of
$f_{0}(1710)$ masses at $1.710\,\text{GeV}$ and $1.780\,\text{GeV}$, serves as
a foundation for our calculations. However, the specific mass of $f_{0}(1710)$
employed in our cross section calculations is the average value in RPP, which
is situated between these two sets, rather than at the extremes. Furthermore,
given that the partial widths of $f_{0}(1710)$ used in our calculations are
derived from the unitary coupled channel approach, we have also adopted a set
of mass and width values for $f_{0}(1710)$ from [16]. Consequently, the
results are approximately 2.4 times those presented in Table 7, yet they
remain within the same order of magnitude.

In Belle’s analysis of $\gamma\gamma\to\eta\pi^{0}$, the efficiency $\eta$ for
$W\in[1.0,2.0]\,\text{GeV}$ generally lies between $0.01$ and $0.02$ [12]. The
event yield of $\gamma\gamma\to a_{0}(980)/a_{0}(Y)\to\eta\pi^{0}$ displays a
notable peak at $W=0.9823\,\text{GeV}$ at the order of
$1000/(10\,\text{MeV})$, denoted as $N_{0}$. Here $a_{0}(Y)$ represents an
unidentified scalar resonance.

For comparison, the calculated $N_{p}$ for the reaction $\gamma\gamma\to
a_{0}(1450)\to\omega\pi\pi$ at SuperKEKB ranges between
$153\text{--}1880N_{0}$. By setting $\eta=0.01$, the range of $N_{p}$ narrows
to $1.5\text{--}18.8N_{0}$, still at or above the order of magnitude of
$N_{0}$. Hence, Belle-II is expected to clearly observe the resonance peak of
$a_{0}(1450)$ through the reaction $\gamma\gamma\to\omega\pi\pi$ at SuperKEKB.
Similarly, the calculated $N_{p}$ for the reaction $\gamma\gamma\to
a_{0}(1450)\to\eta\pi$ at SuperKEKB ranges between $17\text{--}215N_{0}$.
Setting $\eta=0.01$, $N_{p}$ narrows to $0.2\text{--}2.2N_{0}$, which
approaches or reaches the order of magnitude of $N_{0}$. This suggests that
observing the $a_{0}(1450)$ resonance peak through the reaction
$\gamma\gamma\to\eta\pi^{0}$ at SuperKEKB is likely. Similarly, the expected
$N_{p}$ of $16\text{--}192N_{0}$ for the reaction $\gamma\gamma\to K\bar{K}$
indicates a similar potential for observation.

From analogous analyses, the observation prospects for other resonances at
SuperKEKB can be inferred. Table 7 indicates that $a_{0}(1710)$ is likely
observable via the reactions
$\gamma\gamma\to\phi\pi\pi,\,\omega\pi\pi,\,\eta\pi,\,K\bar{K}$, with expected
$N_{p}$ ranging between $18\text{--}308N_{0}$, $20\text{--}335N_{0}$, and
$7\text{--}120N_{0}$, $8\text{--}134N_{0}$, respectively. Table 7 indicates
that $f_{0}(1500)$ is observable via the reactions
$\gamma\gamma\to\pi\pi,\,4\pi$, with expected $N_{p}$ ranging between
$55\text{--}644N_{0}$, $118\text{--}1390N_{0}$, and suggests a possible
observation in the $K\bar{K}$ channel, estimated at $13\text{--}158N_{0}$.
Tables 7 indicates a modest likelihood of observing $f_{0}(1710)$ in the
reactions $\gamma\gamma\to\pi\pi,\,K\bar{K},\,\eta\eta$, with expected $N_{p}$
ranging between $5\text{--}16N_{0}$, $19\text{--}54N_{0}$,
$6\text{--}16N_{0}$.

Transitioning from Belle II’s potential observations at SuperKEKB, we now
consider the scenario at KEKB’s original integrated luminosity. By setting the
integrated luminosity to $223\,\text{fb}^{-1}$, the expected $N_{p}$ for
$\gamma\gamma\to a_{0}(1450)\to\eta\pi$ falls within
$0.009\text{--}0.120N_{0}$ at $\eta=0.02$. For context, Belle has documented
resonance peaks for $a_{0}(980)$, $a_{0}(Y)$, and $a_{2}(1320)$ with event
yields surpassing $N_{0}$, and the number of all candidate events is
approximately $10N_{0}$. Thus, Belle’s failure to observe an $a_{0}(1450)$
resonance peak in the reaction $\gamma\gamma\to\eta\pi^{0}$ aligns with our
projection.

We proceed to compare our calculations with Belle’s measurements on
$f_{0}(1500)$ and $f_{0}(1710)$. In a detailed analysis of the reaction
$\gamma\gamma\to\pi^{0}\pi^{0}$, Belle found no distinct resonance peak for
the $f_{0}(1500)$ around $1.5\,\text{GeV}$. Nevertheless, the nominal fit
suggests a possible contribution from $f_{0}(1370)$, $f_{0}(1500)$, or a mixed
contribution from both, denoted as $f_{0}(Y)$. This fit yields a mass of
$1469.7\pm 4.7\,\text{MeV}/c^{2}$, a total decay width of
$89.7^{+8.1}_{-6.6}\,\text{MeV}$, and
$\Gamma_{\gamma\gamma}\mathcal{B}(f_{0}(Y)\to\pi^{0}\pi^{0})=11.2^{+5.0}_{-4.0}\,\text{eV}$.

Now we could estimate the cross section for the reaction $\gamma\gamma\to
f_{0}(Y)\to\pi^{0}\pi^{0}$ with these parameters. Firstly, note that Belle
adopts the relativistic Breit-Wigner resonance amplitude, denoted as
$A_{R}(W)$, to fit the peak of a spin-J resonance $R$ with mass $m_{R}$, as
described in the following equation [21]:

$A_{R}^{J}(W)=\sqrt{\frac{8\pi(2J+1)m_{R}}{W}}\frac{\sqrt{\Gamma_{\gamma\gamma}(W)\Gamma_{\pi^{0}\pi^{0}}(W)}}{m_{R}^{2}-W^{2}-im_{R}\Gamma_{R}}.$
(38)

Then we could obtain $\sigma_{p}=0.71\,\text{nb}$ at $W=1.47\,\text{GeV}$ with
the following formula:

$\sigma(W)=\int d\Omega|A_{f_{0}(Y)}(W)|^{2}|Y^{0}_{0}|^{2},$ (39)

where $N_{0}^{0}=\sqrt{\frac{1}{4\pi}}$, and $A_{f_{0}(Y)}(W)$ is given by Eq.
(38).

From our calculations, the $\sigma_{p}$ for the reaction $\gamma\gamma\to
f_{0}(1500)\to\pi\pi$ is $0.13\text{--}1.59\,\text{nb}$ at
$W=1.52\,\text{GeV}$. Therefore, the $\sigma_{p}$ for the reaction
$\gamma\gamma\to f_{0}(1500)\to\pi^{0}\pi^{0}$ would be
$0.04\text{--}0.40\,\text{nb}$, which is possibly comparable to the fit result
$0.71\,\text{nb}$.

Belle has performed an analysis on the reaction $\gamma\gamma\to K\bar{K}$ to
check the existence of the $f_{0}(1710)$ resonance within two fixed resonances
$a_{2}(1320)$, $f_{2}(1270)$ and a floating resonance $f_{2}^{\prime}(1525)$
[22]. Six acceptable fits were identified for the $f_{0}(1710)$ resonance in
this analysis. For the first four solutions, Belle performed a fit for the
c.m. energy region $1.2\,\text{GeV}\leq W\leq 2.0\,\text{GeV}$ by floating the
mass, width, $\Gamma_{\gamma\gamma}\mathcal{B}(K\bar{K})$ and the relative
phase of both the $f_{2}^{\prime}(1525)$ and $f_{J}(1710)$ ($J=0$ or $J=2$).
This approach is referred to as Fit-All, with the four solutions denoted as
fit-1, fit-2, fit-3, and fit-4. Belle provided another two fits with different
approach. For these two solutions, Belle first obtained the
$f_{2}^{\prime}(1525)$ parameters by fitting in the range
$1.15\,\text{GeV}\leq W\leq 1.65\,\text{GeV}$ and ignoring the contribution of
the $f_{J}(1710)$. Then Belle included the contribution of the $f_{0}(1710)$
and fitted the region $1.2\,\text{GeV}\leq W\leq 2.0\,\text{GeV}$ by fixing
the $f_{2}^{\prime}(1525)$ parameters. This approach is referred to as Fit-
Part, with the two solutions denoted as fit-H and fit-L. We estimate the
$\sigma_{p}$ for the reaction $\gamma\gamma\to f_{0}(1710)\to K\bar{K}$ based
on these parameters, as detailed in Table 8.

Table 8: Peak cross sections for the reaction $\gamma\gamma\to f_{0}(1710)\to K\bar{K}$ in different fits. Mass is given in MeV/$c^{2}$, total width in MeV, and $\Gamma_{\gamma\gamma}\mathcal{B}(K\bar{K})_{f_{0}}$ in eV. | Fit-All | Fit-Part
---|---|---
Sol. | fit-1 | fit-2 | fit-3 | fit-4 | fit-H | fit-L
Mass($f_{0}$) | 1781 | 1780 | 1783 | 1761 | $1750^{+5+29}_{-6-18}$ | $1749^{+5+31}_{-6-42}$
$\Gamma_{\text{tot}}(f_{0})$ | 99 | 110 | 96 | 119 | $138^{+12+96}_{-11-50}$ | $145^{+11+31}_{-10-54}$
$\Gamma_{\gamma\gamma}\mathcal{B}(K\bar{K})_{f_{0}}$ | 216 | 6.3 | 189 | 10.3 | $12^{+3+227}_{-2-8}$ | $21^{+6+38}_{-4-26}$
$\sigma_{p}$ [nb] | 6.7 | 0.2 | 6.1 | 0.3 | $0.3^{+0.07+5.3}_{-0.05-0.2}$ | $0.5^{+0.14+0.8}_{-0.09-0.6}$

From our calculations, the $\sigma_{p}$ for the reaction $\gamma\gamma\to
f_{0}(1710)\to K\bar{K}$ is $0.17\,\text{nb}$ when
$\Lambda_{3}=0.83\,\text{GeV}$, which falls within the same order of magnitude
as the estimates from fit-2, fit-4, fit-H, and fit-L.

The two-photon decay width of $f_{0}(1710)$ could offer another valuable
perspective. Belle [22] and CELLO [23] both measured
$\Gamma_{K\bar{K}}\Gamma_{\gamma\gamma}/\Gamma_{\text{total}}$ in the reaction
$\gamma\gamma\to K_{S}^{0}K_{S}^{0}$. Utilizing
$\Gamma_{K\bar{K}}/\Gamma_{\text{total}}=0.38^{+0.09}_{-0.19}$ [24], we could
obtain the corresponding $\Gamma_{\gamma\gamma}$, as listed in Table. 9. On
the other hand, $f_{0}(1710)$ has been considered to be dynamically generated
by the vector meson-vector meson interaction within the hidden gauge
formalism, and calculations of the two-photon decay width of $f_{0}(1710)$ as
well as its iso-vector partner are presented in [25]. The results are listed
alongside ours in Table. 9 and Table. 10.

Table 9: $\Gamma_{\gamma\gamma}$ for $f_{0}(1710)$ from experimental and theoretical studies Source | $\Gamma_{\gamma\gamma}\mathcal{B}(K\bar{K})\,[\text{eV}]$ | $\Gamma_{\gamma\gamma}\,[\text{eV}]$
---|---|---
Exp | BELLE | $12^{+3+227}_{-2-8}$ | $32^{+8+597}_{-5-21}$
CELLO | $<110$ | $<289$
Th | Ref [25] | $\backslash$ | $50$
Present work | $\backslash$ | $2\text{--}8$
Table 10: $\Gamma_{\gamma\gamma}$ for $a_{0}(1710)$ from theoretical studies Source | $\Gamma_{\gamma\gamma}\,[\text{keV}]$
---|---
Th | Ref [25] | $1.61$
Present work | $0.03\text{--}0.57$

The error in Belle’s measurement might be too large to be convincing, yet both
our calculations and those presented in [25] align with the CELLO data,
despite our results being one to two orders of magnitude lower. This
difference may in part be due to the form factor for the $\rho^{0}$, $\omega$,
and $\phi$ mesons’ coupling to photons introduced in our work, as shown in Eq.
(40) for the estimation of the total cross section. Such a form factor could
notably reduce the two-photon decay width by a factor $r$, ranging between
$0.006\text{--}0.08$.

$r=\left(\frac{(\Lambda_{3})^{4}}{(\Lambda_{3})^{4}+m_{V}^{4}}\right)^{4},$
(40)

where we have assumed a general mass of vector mesons,
$m_{V}=0.8\,\text{GeV}$.

The situation is similar for the $f_{0}(1500)$. Ref [26] initially considered
the $f_{0}(1370)$ to be dynamically generated by the $\rho\rho$ interaction,
providing its two-photon decay width as $1.62\,\text{keV}$, without
considering the form factor of the $\rho^{0}$, $\omega$, and $\phi$ mesons
that couple to the photon. This identification of $f_{0}(1370)$ shifts to
$f_{0}(1500)$ after including the coupled-channel effects of pseudoscalar
mesons [5]. In comparison, our results for the two-photon decay width of
$f_{0}(1500)$ are in the range of $0.01\text{--}0.12\,\text{keV}$, which are
also one to two orders of magnitude lower.

It should be noted that omitting the form factor would lead to even larger
predictions for the cross section of the two-photon production process,
thereby increasing the potential for observing the aforementioned $a_{0}$ and
$f_{0}$ resonances.

##  IV Summary

We consider $a_{0}(1450)$ and $a_{0}(1710)$ as hadronic molecules and
calculate the cross sections for the two-photon production reactions
$\gamma\gamma\to a_{0}(1450)\to\omega\pi\pi,\,\eta\pi^{0},\,K\bar{K}$,
$\gamma\gamma\to
a_{0}(1710)\to\phi\pi\pi,\,\omega\pi\pi,\,\eta\pi^{0},\,K\bar{K}$,
$\gamma\gamma\to f_{0}(1500)\to\pi\pi,\,K\bar{K},\,4\pi$, and $\gamma\gamma\to
f_{0}(1710)\to\pi\pi,\,K\bar{K},\,\eta\eta$, along with the expected event
yields at SuperKEKB.

Based on our calculations, the peak cross sections for the reactions
$\gamma\gamma\to f_{0}(1500)\to\pi\pi$ and $\gamma\gamma\to f_{0}(1710)\to
K\bar{K}$ might be comparable to the experimental values reported by Belle in
their $\gamma\gamma\to\pi^{0}\pi^{0}$ and $\gamma\gamma\to K_{S}^{0}K_{S}^{0}$
analysis [21, 22]. Moreover, our calculations indicate that the event yield
for $\gamma\gamma\to a_{0}(1450)\to\eta\pi^{0}$ is not sufficient to form an
observable peak in KEKB, which is consistent with the observation by Belle
[12].

Based on our predictions, $a_{0}(1450)$ can be observed in the reaction
$\gamma\gamma\to\omega\pi\pi$ and may be possibly observed in
$\gamma\gamma\to\eta\pi^{0}$. $a_{0}(1710)$ could be possibly observed in
$\gamma\gamma\to\phi\pi\pi,\,\omega\pi\pi,\,\eta\pi^{0},\,K\bar{K}$ at
SuperKEKB. Furthermore, observation of $a_{0}(1710)$ in reaction
$\gamma\gamma\to\phi\pi\pi$ could serve as confirmation of its hidden strange
quark content.

The reaction $\gamma\gamma\to\omega\pi\pi$ is expected to be the most suitable
for observing $a_{0}(1450)$, while $\gamma\gamma\to\phi\pi\pi$ is expected to
be most suitable for observing $a_{0}(1710)$. If Belle-II successfully
utilizes SuperKEKB to observe these reactions and measure the masses and decay
widths of $a_{0}(1450)$ and $a_{0}(1710)$ more accurately, it will help us
verify their hadronic molecule interpretations as $\rho\omega$ and
$K^{*}\bar{K}^{*}$ molecular states, respectively.

The $f_{0}(1500)$ and $f_{0}(1710)$ were previously considered as glueball
candidates [27, 28, 29]. As iso-scalar glueballs, they would have no iso-
vector partners. Therefore, confirming $a_{0}(1450)$ and $a_{0}(1710)$ as the
iso-vector partners of $f_{0}(1500)$ and $f_{0}(1710)$ is crucial in
determining their nature.

If the above hypotheses are successfully verified, it could be confirmed that
the resonances $f_{0}(980)$, $f_{0}(1500)$, and $f_{0}(1710)$ in the $I=0$
sector are $K\bar{K}$, $\rho\rho$, and $K^{*}\bar{K}^{*}$ iso-scalar molecular
states, respectively. Similarly, the resonances $a_{0}(980)$, $a_{0}(1450)$,
and $a_{0}(1710)$ in the $I=1$ sector are $K\bar{K}$, $\rho\omega$, and
$K^{*}\bar{K}^{*}$ iso-vector molecular states, respectively. These results
are summarized in Table 11.

Table 11: Hadronic molecule table of $f_{0}$ and $a_{0}$ resonances
$\begin{array}[]{|c|c|c|c|}\hline\cr\text{}&\smash{\raisebox{-1.5pt}{$K$}}\smash{\raisebox{-1.5pt}{$\bar{K}$}}&\rho\rho/\rho\omega&\smash{\raisebox{-1.5pt}{$K^{*}$}}\smash{\raisebox{-1.5pt}{$\bar{K}^{*}$}}\\\
\hline\cr I=0&f_{0}(980)&f_{0}(1500)&f_{0}(1710)\\\ \hline\cr
I=1&a_{0}(980)&a_{0}(1450)&a_{0}(1710)\\\ \hline\cr\end{array}$

## Acknowledgments

We thank useful discussions and valuable comments from Feng-Kun Guo and Hao-
jie Jing. This work is supported by the NSFC and the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) through the funds
provided to the Sino-German Collaborative Research Center TRR110 “Symmetries
and the Emergence of Structure in QCD” (NSFC Grant No. 12070131001, DFG
Project-ID 196253076 - TRR 110), by the NSFC Grant No.11835015, No.12047503,
No. 12175239, No. 12221005 and by the Chinese Academy of Sciences (CAS) under
Grant No.XDB34030000, and by the Chinese Academy of Sciences under Grant No.
YSBR-101, and by the Xiaomi Foundation / Xiaomi Young Talents Program.

## References

  * [1] J. A. Oller and E. Oset, Phys. Rev. D 60, 074023 (1999) doi:10.1103/PhysRevD.60.074023 [arXiv:hep-ph/9809337 [hep-ph]].
  * [2] G. Janssen, B. C. Pearce, K. Holinde and J. Speth, Phys. Rev. D 52, 2690-2700 (1995) doi:10.1103/PhysRevD.52.2690 [arXiv:nucl-th/9411021 [nucl-th]].
  * [3] L. S. Geng and E. Oset, Phys. Rev. D 79, 074009 (2009) doi:10.1103/PhysRevD.79.074009 [arXiv:0812.1199 [hep-ph]].
  * [4] R. Molina, D. Nicmorus and E. Oset, Phys. Rev. D 78, 114018 (2008) doi:10.1103/PhysRevD.78.114018 [arXiv:0809.2233 [hep-ph]].
  * [5] Z. L. Wang and B. S. Zou, Phys. Rev. D 99, no.9, 096014 (2019) doi:10.1103/PhysRevD.99.096014 [arXiv:1901.10169 [hep-ph]].
  * [6] M. L. Du, D. Gülmez, F. K. Guo, U. G. Meißner and Q. Wang, Eur. Phys. J. C 78, no.12, 988 (2018) doi:10.1140/epjc/s10052-018-6475-8 [arXiv:1808.09664 [hep-ph]].
  * [7] Z. L. Wang and B. S. Zou, Phys. Rev. D 104, no.11, 114001 (2021) doi:10.1103/PhysRevD.104.114001 [arXiv:2107.14470 [hep-ph]].
  * [8] Z. L. Wang and B. S. Zou, Eur. Phys. J. C 82, no.6, 509 (2022) doi:10.1140/epjc/s10052-022-10460-4 [arXiv:2203.02899 [hep-ph]].
  * [9] J. P. Lees et al. [BaBar], Phys. Rev. D 104, no.7, 072002 (2021) doi:10.1103/PhysRevD.104.072002 [arXiv:2106.05157 [hep-ex]].
  * [10] M. Ablikim et al. [BESIII], Phys. Rev. D 105, no.5, L051103 (2022) doi:10.1103/PhysRevD.105.L051103 [arXiv:2110.07650 [hep-ex]].
  * [11] M. Ablikim et al. [BESIII], Phys. Rev. Lett. 129, no.18, 18 (2022) doi:10.1103/PhysRevLett.129.182001 [arXiv:2204.09614 [hep-ex]].
  * [12] S. Uehara et al. [Belle], Phys. Rev. D 80, 032001 (2009) doi:10.1103/PhysRevD.80.032001 [arXiv:0906.1464 [hep-ex]].
  * [13] Y. Funakoshi [SuperKEKB], JACoW eeFACT2022, 171-177 (2023) doi:10.18429/JACoW-eeFACT2022-WEZAS0101
  * [14] H. Nagahiro, L. Roca, A. Hosaka and E. Oset, Phys. Rev. D 79, 014015 (2009) doi:10.1103/PhysRevD.79.014015 [arXiv:0809.0943 [hep-ph]].
  * [15] B. S. Zou and D. V. Bugg, Eur. Phys. J. A 16, 537-547 (2003) doi:10.1140/epja/i2002-10135-4 [arXiv:hep-ph/0211457 [hep-ph]].
  * [16] Z. L. Wang and B. S. Zou, Phys. Rev. D 104, no.11, 114001 (2021) doi:10.1103/PhysRevD.104.114001 [arXiv:2107.14470 [hep-ph]].
  * [17] F. K. Guo, C. Hanhart and U. G. Meissner, Phys. Lett. B 665, 26-29 (2008) doi:10.1016/j.physletb.2008.05.057 [arXiv:0803.1392 [hep-ph]].
  * [18] R. L. Workman et al. [Particle Data Group], PTEP 2022, 083C01 (2022) doi:10.1093/ptep/ptac097
  * [19] Z. Q. Liu et al. [Belle], Phys. Rev. Lett. 108, 232001 (2012) doi:10.1103/PhysRevLett.108.232001 [arXiv:1202.5632 [hep-ex]].
  * [20] C. Berger and W. Wagner, Phys. Rept. 146, 1-134 (1987) doi:10.1016/0370-1573(87)90012-3
  * [21] S. Uehara et al. [Belle], Phys. Rev. D 78, 052004 (2008) doi:10.1103/PhysRevD.78.052004 [arXiv:0805.3387 [hep-ex]].
  * [22] S. Uehara et al. [Belle], PTEP 2013, no.12, 123C01 (2013) doi:10.1093/ptep/ptt097 [arXiv:1307.7457 [hep-ex]].
  * [23] H. J. Behrend et al. [CELLO], Z. Phys. C 43, 91 (1989) doi:10.1007/BF02430613
  * [24] R. S. Longacre, A. Etkin, K. J. Foley, W. A. Love, T. W. Morris, A. C. Saulys, E. D. Platner, S. J. Lindenbaum, J. R. Bensinger and L. E. Kirsch, et al. Phys. Lett. B 177, 223-227 (1986) doi:10.1016/0370-2693(86)91061-0
  * [25] T. Branz, L. S. Geng and E. Oset, Phys. Rev. D 81, 054037 (2010) doi:10.1103/PhysRevD.81.054037 [arXiv:0911.0206 [hep-ph]].
  * [26] H. Nagahiro, J. Yamagata-Sekihara, E. Oset, S. Hirenzaki and R. Molina, Phys. Rev. D 79, 114023 (2009) doi:10.1103/PhysRevD.79.114023 [arXiv:0809.3717 [hep-ph]].
  * [27] F. Giacosa, T. Gutsche, V. E. Lyubovitskij and A. Faessler, Phys. Rev. D 72, 094006 (2005) doi:10.1103/PhysRevD.72.094006 [arXiv:hep-ph/0509247 [hep-ph]].
  * [28] M. Chanowitz, Phys. Rev. Lett. 95, 172001 (2005) doi:10.1103/PhysRevLett.95.172001 [arXiv:hep-ph/0506125 [hep-ph]].
  * [29] K. T. Chao, X. G. He and J. P. Ma, Phys. Rev. Lett. 98, 149103 (2007) doi:10.1103/PhysRevLett.98.149103 [arXiv:0704.1061 [hep-ph]].
  * [30] S. Uehara, [arXiv:1310.0157 [hep-ph]].

##  Appendix A Supplement to Double-Photon Luminosity Function

In this appendix we detail the derivation of the double-photon luminosity
function.

$L_{\gamma\gamma}(W)$ is the probability distribution function of a two-photon
production with a total energy of $W$ from a pair of beam particles, which
refer to $e^{+}e^{-}$ in the context. Its physical significance can be seen
from the following equation [30]:

$\sigma_{e^{+}e^{-}}=\int\sigma_{\gamma\gamma}(W)L_{\gamma\gamma}(W)dW.$ (41)

Here, $\sigma_{e^{+}e^{-}}$ denotes the cross section for the reaction
$e^{+}e^{-}\to e^{+}e^{-}X$, where $X$ can represent any final-state
particles.

To obtain the expression for $L_{\gamma\gamma}$, we first need to derive the
expression for $\sigma_{e^{+}e^{-}}$. Under the condition that the two
incident photons are nearly on-shell ($Q^{2}\to 0$, $P^{2}\to 0$), Eq. (2.29)
in [20] provides a simplified equation:

$\sigma_{e^{+}e^{-}}=\int
f_{\gamma/e}(y,W,Q^{2}_{\text{max}})f_{\gamma/e}(z,W,Q^{2}_{\text{max}})\sigma_{\gamma\gamma}(W)dydz.$
(42)

Define $s=yz=W^{2}/4E_{0}^{2}$ and recall that $r=y/z$. To simplify
calculations, we can perform the integration variable transformation as
follows: $dy\,dz\to J_{rs}\,dr\,ds$ and subsequently $ds\to J_{sW}\,dW$. This
introduces two Jacobian factors:

$J_{rs}=\left|\frac{\partial(y,z)}{\partial(r,s)}\right|=\frac{1}{2r},$ (43)

and

$J_{sW}=\left|\frac{\partial s}{\partial W}\right|=\frac{W}{2E_{0}^{2}}.$ (44)

Thus, we have:

$L_{\gamma\gamma}(W)=\int
f_{\gamma/e}(y,W,Q^{2}_{\text{max}})f_{\gamma/e}(z,W,Q^{2}_{\text{max}})\frac{W}{2E_{0}^{2}}\frac{1}{2r}dr,$
(45)

as have shown in Sec. II.

The original formula of Eq. (37) is from Eq. (2.19) in [20], and
$\ln\frac{E(1-z)}{mz}\theta_{2,max}$ in Eq. (2.19) has been replaced with the
integral [12] :

$\frac{1}{2}\int_{\ln\left(m_{e}^{2}z^{2}/(1-z)\right)}^{\ln
Q^{2}_{\text{max}}}F(e^{\nu},W)d\nu.$ (46)

$F(Q^{2},W)$ is introduced to describe the factorized relation between the
cross sections for virtual photons and the real photons:

$\sigma_{\gamma^{*}\gamma^{*}}(W)=F(Q_{1}^{2},W)F(Q_{2}^{2},W)\sigma_{\gamma\gamma}(W),$
(47)

where $\sigma_{\gamma^{*}\gamma^{*}}$ is the cross section for virtual
photons, and $\sigma_{\gamma\gamma}$ is the cross section for real photons.

##  Appendix B Supplement to Scattering Amplitudes

In this appendix we present all the scattering amplitudes for each Feynman
diagram given in Sec. II. For the reaction $\gamma(k_{1})\gamma(k_{2})\to
f_{0}(1500)\to P(p_{1})P(p_{2})$ as shown in Fig. 3(a),

$\displaystyle\mathcal{M}_{1a}$
$\displaystyle=A_{1\mu}(m_{1},k_{1})A_{2\rho}(m_{2},k_{2})$
$\displaystyle\times(V_{\gamma(k_{1})\rho})^{\mu\nu}(V_{\gamma(k_{2})\rho})^{\rho\sigma}(G^{\prime}_{\rho}(k_{1}))_{\nu\alpha}(G^{\prime}_{\rho}(k_{2}))_{\sigma\beta}$
$\displaystyle\times(V_{f_{0}\rho\rho})^{\alpha\beta}G_{f_{0}}(k_{1}+k_{2})V_{f_{0}PP}.$
(48)

For the reaction $\gamma(k_{1})\gamma(k_{2})\to
f_{0}(1500)\to\pi(p_{1})\pi(p_{2})\pi(p_{3})\pi(p_{4})$ as shown in Fig. 3(b),

$\displaystyle\mathcal{M}_{1b}$
$\displaystyle=A_{1\mu}(m_{1},k_{1})A_{2\rho}(m_{2},k_{2})$
$\displaystyle\times(V_{\gamma(k_{1})\rho})^{\mu\nu}(V_{\gamma(k_{2})\rho})^{\rho\sigma}(G^{\prime}_{\rho}(k_{1}))_{\nu\alpha}(G^{\prime}_{\rho}(k_{2}))_{\sigma\beta}$
$\displaystyle\times(V_{f_{0}\rho\rho})^{\alpha\beta}G_{f_{0}}(k_{1}+k_{2})(V_{f_{0}\rho\rho})^{\kappa\lambda}$
(49)
$\displaystyle\times(G_{\rho}(p_{5}))_{\kappa\delta}(G_{\rho}(p_{6}))_{\lambda\theta}(V_{\rho\pi\pi})^{\delta}(V_{\rho\pi\pi})^{\theta}.$
(50)

For the reactions $\gamma(k_{1})\gamma(k_{2})\to f_{0}(1710)\to
P(p_{1})P(p_{2})$ as shown in Fig. 4(a) and Fig. 4(b), the scattering
amplitudes $\mathcal{M}_{2a}$ and $\mathcal{M}_{2b}$ are nearly identical,
with the exception that the $\gamma V$ vertex mesons
$(\rho(k_{1}),\rho(k_{2}))$ in $\mathcal{M}_{2a}$ are replaced by
$(\phi(k_{1}),\phi(k_{2}))$, $(\phi(k_{1}),\omega(k_{2}))$,
$(\omega(k_{1}),\phi(k_{2}))$, and $(\omega(k_{1}),\omega(k_{2}))$ in
$\mathcal{M}_{2b}$.

For the reaction $\gamma(k_{1})\gamma(k_{2})\to
a_{0}(1450)\to\eta(p_{1})\pi^{0}(p_{2})$ as shown in Fig. 1(a),

$\displaystyle\mathcal{M}_{3a}$
$\displaystyle=A_{1\mu}(m_{1},k_{1})A_{2\rho}(m_{2},k_{2})$
$\displaystyle\times(V_{\gamma(k_{1})\rho})^{\mu\nu}(V_{\gamma(k_{2})\rho})^{\rho\sigma}(G^{\prime}_{V}(k_{1}))_{\nu\alpha}(G^{\prime}_{V}(k_{2}))_{\sigma\beta}$
$\displaystyle\times(V_{a_{0}VV})^{\alpha\beta}G_{a_{0}}(k_{1}+k_{2})V_{a_{0}\eta\pi^{0}},$
(51)

where $(V(k_{1}),\,V(k_{2}))$ includes $(\rho(k_{1}),\,\omega(k_{2}))$,
$(\omega(k_{1}),\,\rho(k_{2}))$.

For the reaction $\gamma(k_{1})\gamma(k_{2})\to
a_{0}(1450)\to\pi^{+}(p_{1})\pi^{-}(p_{2})\omega(p_{3})$ as shown in Fig.
1(b),

$\displaystyle\mathcal{M}_{3b}$
$\displaystyle=A_{1\mu}(m_{1},k_{1})A_{2\rho}(m_{2},k_{2})$
$\displaystyle\times(V_{\gamma(k_{1})\rho})^{\mu\nu}(V_{\gamma(k_{2})\rho})^{\rho\sigma}(G^{\prime}_{V}(k_{1}))_{\nu\alpha}(G^{\prime}_{V}(k_{2}))_{\sigma\beta}$
$\displaystyle\times(V_{a_{0}VV})^{\alpha\beta}G_{a_{0}}(k_{1}+k_{2})(V_{a_{0}\rho\omega})^{\kappa\lambda}$
$\displaystyle\times\epsilon_{\kappa}(m_{3},p_{3})(G_{\rho^{0}}(p_{4}))_{\lambda\delta}(V_{\rho^{0}\pi^{+}\pi^{-}})^{\delta},$
(52)

where $\epsilon_{\kappa}(m_{3},p_{3})$ is the polarization vector of the meson
$\omega$.

For the reaction $\gamma(k_{1})\gamma(k_{2})\to a_{0}(1710)\to
P(p_{1})P(p_{2})$ as shown in Fig. 2(a),

$\displaystyle\mathcal{M}_{4a}$
$\displaystyle=A_{1\mu}(m_{1},k_{1})A_{2\rho}(m_{2},k_{2})$
$\displaystyle\times(V_{\gamma(k_{1})\rho})^{\mu\nu}(V_{\gamma(k_{2})\rho})^{\rho\sigma}(G^{\prime}_{V}(k_{1}))_{\nu\alpha}(G^{\prime}_{V}(k_{2}))_{\sigma\beta}$
$\displaystyle\times(V_{a_{0}VV})^{\alpha\beta}G_{a_{0}}(k_{1}+k_{2})V_{a_{0}PP},$
(53)

where $(V(k_{1}),\,V(k_{2}))$ includes $(\rho(k_{1}),\,\rho(k_{2}))$,
$(\rho(k_{1}),\,\phi(k_{2}))$, $(\phi(k_{1}),\,\rho(k_{2}))$,
$(\rho(k_{1}),\,\omega(k_{2}))$, $(\omega(k_{1}),\,\rho(k_{2}))$.

For the reaction $\gamma(k_{1})\gamma(k_{2})\to
a_{0}(1710)\to\pi^{+}(p_{1})\pi^{-}(p_{2})\omega(p_{3})$ as shown in Fig.
2(b),

$\displaystyle\mathcal{M}_{4b}$
$\displaystyle=A_{1\mu}(m_{1},k_{1})A_{2\rho}(m_{2},k_{2})$
$\displaystyle\times(V_{\gamma(k_{1})\rho})^{\mu\nu}(V_{\gamma(k_{2})\rho})^{\rho\sigma}(G^{\prime}_{V}(k_{1}))_{\nu\alpha}(G^{\prime}_{V}(k_{2}))_{\sigma\beta}$
$\displaystyle\times(V_{a_{0}VV})^{\alpha\beta}G_{a_{0}}(k_{1}+k_{2})(V_{a_{0}\rho\omega})^{\kappa\lambda}$
$\displaystyle\times\epsilon_{\kappa}(m_{3},p_{3})(G_{\rho^{0}}(p_{4}))_{\lambda\delta}(V_{\rho^{0}\pi^{+}\pi^{-}})^{\delta},$
(54)

where $\epsilon_{\kappa}(m_{3},p_{3})$ is the polarization vector of the
mesons $\phi$ or $\omega$.

For the reaction $\gamma(k_{1})\gamma(k_{2})\to
f_{0}(1710)\to\phi(p_{1})\omega(p_{2})$ as shown in Fig. 5,

$\displaystyle\mathcal{M}_{5}$
$\displaystyle=A_{1\mu}(m_{1},k_{1})A_{2\rho}(m_{2},k_{2})$
$\displaystyle\times(V_{\gamma(k_{1})\rho})^{\mu\nu}(V_{\gamma(k_{2})\rho})^{\rho\sigma}(G^{\prime}_{\rho}(k_{1}))_{\nu\alpha}(G^{\prime}_{\rho}(k_{2}))_{\sigma\kappa}$
$\displaystyle\times(V_{f_{0}K^{*}\bar{K}^{*}})^{\delta\eta}G_{f_{0}}(k_{1}+k_{2})(V_{f_{0}K^{*}\bar{K}^{*}})^{\delta^{\prime}\eta^{\prime}}$
$\displaystyle\times\epsilon_{\alpha^{\prime}}(m_{1}^{\prime},p_{1})\epsilon_{\kappa^{\prime}}(m_{2}^{\prime},p_{2})$
$\displaystyle\times\int\frac{id^{4}k_{5}}{(2\pi)^{4}}(-\tilde{g}_{\beta\delta}(k_{3}))(-\tilde{g}_{\lambda\eta}(k_{4}))(-\tilde{g}_{\gamma\theta}(k_{5}))$
$\displaystyle\times G(k_{1},k_{2},k_{5})\left\\{(V_{\rho
K^{*+}K^{*-}})^{\alpha\beta\gamma}(V_{\rho
K^{*-}K^{*+}})^{\kappa\lambda\theta}\right.$ $\displaystyle+\left.(V_{\rho
K^{*-}K^{*+}})^{\alpha\beta\gamma}(V_{\rho
K^{*+}K^{*-}})^{\kappa\lambda\theta}\right\\}$ $\displaystyle+\left.(V_{\rho
K^{*0}\bar{K}^{*0}})^{\alpha\beta\gamma}(V_{\rho\bar{K}^{*0}K^{*0}})^{\kappa\lambda\theta}\right\\}$
$\displaystyle+\left.(V_{\rho\bar{K}^{*0}K^{*0}})^{\alpha\beta\gamma}(V_{\rho
K^{*0}\bar{K}^{*0}})^{\kappa\lambda\theta}\right\\}$ (55)
$\displaystyle\times\int\frac{id^{4}p_{5}}{(2\pi)^{4}}(-\tilde{g}_{\beta^{\prime}\delta^{\prime}}(p_{3}))(-\tilde{g}_{\lambda^{\prime}\eta^{\prime}}(p_{4}))(-\tilde{g}_{\gamma^{\prime}\theta^{\prime}}(p_{5}))$
$\displaystyle\times G(p_{1},p_{2},p_{5})\left\\{(V_{\phi
K^{*+}K^{*-}})^{\alpha^{\prime}\beta^{\prime}\gamma^{\prime}}(V_{\omega
K^{*-}K^{*+}})^{\kappa^{\prime}\lambda^{\prime}\theta^{\prime}}\right.$
$\displaystyle+\left.(V_{\phi
K^{*-}K^{*+}})^{\alpha^{\prime}\beta^{\prime}\gamma^{\prime}}(V_{\omega
K^{*+}K^{*-}})^{\kappa^{\prime}\lambda^{\prime}\theta^{\prime}}\right\\}$
$\displaystyle+\left.(V_{\phi
K^{*0}\bar{K}^{*0}})^{\alpha^{\prime}\beta^{\prime}\gamma^{\prime}}(V_{\omega\bar{K}^{*0}K^{*0}})^{\kappa^{\prime}\lambda^{\prime}\theta^{\prime}}\right\\}$
$\displaystyle+\left.(V_{\phi\bar{K}^{*0}K^{*0}})^{\alpha^{\prime}\beta^{\prime}\gamma^{\prime}}(V_{\omega
K^{*0}\bar{K}^{*0}})^{\kappa^{\prime}\lambda^{\prime}\theta^{\prime}}\right\\}$
(56)

where $\epsilon_{\alpha^{\prime}}(m_{1}^{\prime},p_{1})$ and
$\epsilon_{\kappa^{\prime}}(m_{2}^{\prime},p_{2})$ are the polarization
vectors of the mesons $\phi$ and $\omega$.