#  Hadronic loop effects on the radiative decays of the first radial
excitations of $\eta$ and $\eta^{\prime}$

Yin Cheng1,2111Email<EMAIL_ADDRESS>Qiang Zhao1,2 222E-mail:
<EMAIL_ADDRESS>1) Institute of High Energy Physics, Chinese Academy of
Sciences, Beijing 100049, P.R. China  2) University of Chinese Academy of
Sciences, Beijing 100049, P.R. China

###### Abstract

Based on the one state assumption of $\eta(1405)$ and $\eta(1475)$, thus,
$\eta(1295)$ and $\eta(1405/1475)$ are organized as the first radial
excitations of $\eta$ and $\eta^{\prime}$, respectively, we investigate the
productions and radiative decays of these two states in
$J/\psi\to\gamma\eta_{X}\to\gamma\gamma V$, where $\eta_{X}$ stands for
$\eta(1295)$ and $\eta(1405/1475)$ and $V$ for vector mesons $\rho^{0},\
\omega,\ \phi$. As we have learned from previous studies that the hadronic
decays of these two states receive important contributions from the
intermediate $\bar{K}K^{*}+c.c.$ meson loops due to the triangle singularity
mechanism, we show that some measurable effects can also arise from the
$\bar{K}K^{*}+c.c.$ meson loops in their radiative decays. Our calculation
shows that the impact of the $\bar{K}K^{*}+c.c.$ meson loops on the
$\eta(1405/1475)$ radiative decays is relatively smaller than on $\eta(1295)$
since the latter has a much larger coupling to $\bar{K}K^{*}+c.c.$ However,
the production of $\eta(1295)$ in the $J/\psi$ radiative decays will be
strongly suppressed. As a consequence of the $\bar{K}K^{*}+c.c.$ meson loop
contributions, we find that the mixing angle extracted in the radiative decays
of $\eta(1295)$ and $\eta(1405/1475)$ will be different from each other, and
both are different from the one determined in other processes.

##  I Introduction

The study of flavor singlet and octet mixing in the lightest pseudoscalar
nonet, i.e. between $\eta$ and $\eta^{\prime}$, has attracted a lot of
attention in the history. As it has been well-established that the $U(1)_{A}$
anomaly is the driving mechanism for many interesting phenomena for these two
states, it also raises interesting questions on its role for higher radial
excitation states in the isoscalar pseudoscalar spectrum, in particular, the
first radial excitation states. The present experimental data for the
$J^{P(C)}=0^{-(+)}$ spectrum are still for from satisfactory. For the first
radial excitation there are enough states to fill a nonet between $1.25\sim
1.50$ GeV, which includes $\pi(1300)$, $K(1460)$, $\eta(1295)$ and
$\eta(1405)/\eta(1475)$ Zyla:2020zbs . But for higher excitations, the
experimental evidences are far from well-established.

Even for the first radial excitations one can see that the question whether
there are two states, $\eta(1405)$ and $\eta(1475)$, present in the same mass
region would have strong impact on our understanding of low-energy QCD
phenomena. Historically, the first evidence for $\eta(1405)/\eta(1475)$ was
from $p\bar{p}$ annihilations at rest into ($K\bar{K}\pi$)$\pi^{+}\pi^{-}$
Baillon:1967zz , where a pseudoscalar of $J^{PC}=0^{-+}$ was seen in the
invariant mass spectrum of $K\bar{K}\pi$. As an SU(3) partner of the lighter
pseudoscalar $\eta(1295)$, it shows that its production strength in $p\bar{p}$
annihilations is much larger than $\eta(1295)$. This was regarded as an
evidence for its unusual flavor contents in the literature (see e.g. Ref.
Klempt:2007cp  for a review). Later, MARK III Bai:1990hs  and DM-2
Augustin:1990ki  reported possible two-state structures around 1.44 GeV mass
region with increased statistics. The Obelix collaboration at LEAR
Nichitiu:2002cj  seemed to confirm the MARK III result and introduced two
pseudoscalars, i.e. $\eta(1405)$ and $\eta(1475)$, in the description of the
invariant mass spectrum of $\eta\pi\pi$ 333One notices that the masses and
widths extracted from these three analyses Bai:1990hs ; Augustin:1990ki ;
Nichitiu:2002cj  are very different.. The splitting of one state $\eta(1440)$
into two states, $\eta(1405)$ and $\eta(1475)$, suggested an outnumbering of
the SU(3) nonet and could be an indication of exotic hadrons beyond the
conventional quark model. In line of this possibility there was theoretical
expectation from the flux tube model that the ground state pseudoscalar
glueball should have a mass around 1.4 GeV Faddeev:2003aw . It made one of
these two close states, $\eta(1405)$ and $\eta(1475)$, a possible candidate
for the pseudoscalar glueball, and initiated a lot of efforts on understanding
their structures Donoghue:1980hw ; Close:1980rv ; Barnes:1981kp ; Close:1987er
; Amsler:2004ps ; Masoni:2006rz . Moreover, its mixing with the ground state
$\eta$ and $\eta^{\prime}$ become an interesting topic in phenomenology
although the gluon contents inside $\eta$ and $\eta^{\prime}$ cannot be
dramatically large Cheng:2008ss ; Tsai:2011dp ; Qin:2017qes .

While the signal for the heavier one turned out to be more clear in the
$K\bar{K}\pi$ channel, and the lighter one seemed to favor the $\eta\pi\pi$
channel, $\eta(1405)$ has been assigned as the pseudoscalar glueball candidate
while $\eta(1475)$ was assigned as the SU(3) partner of $\eta(1295)$ (see e.g.
the mini-reviews on non-$q\bar{q}$ mesons in early editions of Particle Data
Group since 1990). However, the glueball assignment with a low mass around 1.4
GeV is not supported by the lattice QCD (LQCD) simulations which came to the
playground later. Both quenched Chen:2005mg ; Bali:1993fb ; Morningstar:1999rf
; Chowdhury:2014mra  and unquenched calculations Richards:2010ck ; Sun:2017ipk
suggest that the ground state pseudoscalar glueball should have a mass around
$2.4-2.6$ GeV. In experiment, more and more high-precision data from $J/\psi$
and $\psi(3686)$ decays at BESIII were published during the past decade. There
is no indication that two pseudoscalar states $\eta(1405)$ and $\eta(1475)$
are needed in the description of any exclusive channel such as $\eta\pi\pi$
Ablikim:2011pu ; Ablikim:2010au ; Ablikim:2019wei , $K\bar{K}\pi$
Ablikim:2013lxa , and the isospin-violating $3\pi$ channel BESIII:2012aa .
However, it seems to be true that the pseudoscalars observed in different
channels have slightly shifted masses. For instance, the mass extracted in the
$K\bar{K}\pi$ channel is $1452.7\pm 3.8$ MeV Ablikim:2013lxa , while those in
$\eta\pi\pi$ and $3\pi$ are about 1405 MeV Ablikim:2011pu ; Ablikim:2010au ;
Ablikim:2019wei ; BESIII:2012aa .

A breakthrough of the puzzling situation was the proposal by Ref. Wu:2011yx
in the interpretation of the abnormally large isospin-breaking effects
observed by BESIII in $J/\psi\to\gamma\eta(1405/1475)\to\gamma+3\pi$
BESIII:2012aa . The interference from the intermediate $K^{*}\bar{K}+c.c.$
rescattering via a triangle loop can contribute to the isospin breaking at
leading order due to the satisfaction of the triangle singularity (TS)
condition Landau:1959fi ; Cutkosky:1960sp ; bonnevay:1961aa ; Peierls:1961zz .
The TS mechanism can naturally explain the mass shift and decay patterns with
only one state around 1.4 GeV Wu:2011yx ; Wu:2012pg ; Aceti:2012dj . Further
detailed studies including the width effects were also investigated in the
literature Achasov:2015uua ; Du:2019idk . In Ref. Achasov:2015uua  it was
claimed that the TS contribution would be suppressed by the width effects of
the intermediate $K^{*}$. Therefore, the TS mechanism may not be sufficient
for accounting for the large isospin-breaking effects observed by BESIII
BESIII:2012aa . A comprehensive analysis in Ref. Du:2019idk  later showed that
one important transition process via the TS mechanism was overlooked by the
previous analyses. The TS mechanism can also enhance the direct production of
$a_{0}(980)$ in the isospin-conserving channel, and then enhance the isospin-
violating channel via the $a_{0}(980)$-$f_{0}(980)$ mixing. The analysis of
Ref. Du:2019idk  thus firms up the role played by the TS mechanism in the
understanding of the $\eta(1405)$ and $\eta(1475)$ puzzle.

It should be mentioned that in phenomenological studies of the pseudoscalar
glueball mixing with the $q\bar{q}$ states, i.e. $\eta$-$\eta^{\prime}$-$G$ or
$\eta$-$\eta^{\prime}$-$G$-$\eta_{c}$ mixings, the physical mass of the
pseudoscalar glueball was assigned by $\eta(1405)$. By doing so, the gluon
contents introduced into $\eta$ and $\eta^{\prime}$ seem to agree with the
experimental observables. However, as shown by a detailed analysis of Ref.
Qin:2017qes  following the axial vector anomaly dynamics Cheng:2008ss ;
Tsai:2011dp , the gluon contents inside $\eta$ and $\eta^{\prime}$ are not
sensitive to the physical mass of the pseudoscalar glueball. Furthermore, with
the LQCD pure gauge glueball mass as an input, the physical mass cannot get to
be lighter than 1.8 GeV Qin:2017qes . Similar conclusion was found by Refs.
Mathieu:2009sg ; Gabadadze:1997zc  in the framework of the axial vector
anomaly.

Motivated by these progresses on disentangling the $\eta(1405)$ and
$\eta(1475)$ puzzle, we will investigate the scenario of treating $\eta(1295)$
and $\eta(1405)$ (Hereafter, we use $\eta(1405)$ to denote all signals related
to either $\eta(1405)$ or $\eta(1475)$ in the previous two-state scenario) as
the first radial excitation states of $\eta$ and $\eta^{\prime}$. The
radiative decay of $\eta(1405)\rightarrow\gamma\rho$ and
$\eta(1405)\rightarrow\gamma\phi$ have been measured by BES-II Bai:2004qj  and
BESIII Collaboration Ablikim:2018hxj , respectively, in the $J/\psi$ radiative
decays. We will systematically study $J/\psi\to\gamma\eta_{X}\to\gamma\gamma
V$ with $\eta_{X}=\eta(1295),\ \eta(1405)$, and $V=\phi,\ \rho^{0},\ \omega$,
and examine the role played by the intermediate $K^{*}\bar{K}$ meson loops.
This should provide further experimental evidences for the one-state solution
for $\eta(1405)$ and $\eta(1475)$, and allow a natural categorization of
$\eta(1295)$ and $\eta(1405)$ as the first radial excitation states of $\eta$
and $\eta^{\prime}$.

To proceed, we first introduce the mixing between the SU(3) flavor singlet and
octet, and then present the formalism for
$J/\psi\to\gamma\eta_{X}\to\gamma\gamma V$ in the framework of the vector
meson dominance (VMD) model in Sec. II. We stress that this will allow a self-
consistent calculation of both tree-level transitions and loop corrections of
the radiative decays of $\eta_{X}\to\gamma\gamma V$. In Sec. III we will
present our numerical results for measurable branching fractions, and discuss
their phenomenological consequences. A brief summary will be given in Sec. IV.
In the Appendix the loop functions for each loop transition amplitude are
provided.

##  II Formalism

###  II.1 Flavor singlet and octet mixing, and parametrization for the
production mechanism

As the first radial excitation of $\eta$ and $\eta^{\prime}$, $\eta(1295)$ and
$\eta(1405)$ can be expressed on the quark-flavor basis similar to
$\eta$-$\eta^{\prime}$,

$\begin{split}\eta(1295)&=\cos\alpha_{P}n\bar{n}-\sin\alpha_{P}s\bar{s},\\\
\eta(1405)&=\sin\alpha_{P}n\bar{n}+\cos\alpha_{P}s\bar{s},\end{split}$ (1)

where $n\bar{n}\equiv(u\bar{u}+d\bar{d})/\sqrt{2}$, and
$\alpha_{P}\equiv\arctan\sqrt{2}+\theta_{p}$ with $\theta_{p}$ the flavor
singlet and octet mixing angle. Whether the mixing angle is the same as that
for the $\eta$-$\eta^{\prime}$ mixing is still an open question. In Ref.
Isgur:1975ib  Isgur proposed that although the $\eta$-$\eta^{\prime}$ mixing
angle deviated from the ideal mixing significantly, the higher states should
restore the ideal mixing angle given that the mass difference between the
flavor $n\bar{n}$ and $s\bar{s}$ could be neglected. In this study we can
either leave the mixing angle to be determined by experimental data, or test
the results by adopting the same mixing angle as that for
$\eta$-$\eta^{\prime}$.

Taking the advantage of the antisymmetric tensor structure of the $VVP$
coupling, we can parametrize the coupling strength for
$J/\psi\to\gamma(q\bar{q})_{0^{-+}}$, where $(q\bar{q})_{0^{-+}}$ stands for a
light quark-antiquark pair produced in the $J/\psi$ radiative decays, as

$\displaystyle g_{0}$ $\displaystyle\equiv$
$\displaystyle\langle(q\bar{q})_{0^{-+}}|\hat{H}_{\gamma}|J/\psi\rangle,$ (2)

where $\hat{H}_{\gamma}$ represents the corresponding potential for the
production of $(q\bar{q})_{0^{-+}}$. Based on the SU(3) flavor symmetry the
coupling strengths $\tilde{g}_{\eta_{X}}$ can be written as

$\displaystyle\tilde{g}_{\eta(1295)}$ $\displaystyle=$ $\displaystyle
g_{0}(\sqrt{2}\cos\alpha_{P}-R\sin\alpha_{P})\ ,$
$\displaystyle\tilde{g}_{\eta(1405)}$ $\displaystyle=$ $\displaystyle
g_{0}(\sqrt{2}\sin\alpha_{P}+R\cos\alpha_{P})\ ,$ (3)

for $\eta(1295)$ and $\eta(1405)$, respectively. In the above equation $R$ is
an SU(3) flavor symmetry breaking parameter. It distinguishes the production
of an $s\bar{s}$ from $u\bar{u}$ and $d\bar{d}$ and generally takes $R\simeq
m_{u/d}/m_{s}$.

It is clear here that the production of $\eta(1295)$ would be highly
suppressed in comparison with the production of $\eta(1405)$ due to the
destructive interference between the $n\bar{n}$ and $s\bar{s}$ component for
$\eta(1295)$ and constructive interference for $\eta(1405)$ in the radial
excitation scenario if the same mixing angle as that for the
$\eta$-$\eta^{\prime}$ mixing is adopted. This is consistent with the current
experimental observation that signals for $\eta(1405)$ are much stronger than
for $\eta(1295)$ Zyla:2020zbs .

Under the assumption, the ratio of $\Gamma(J/\psi\to\gamma\eta(1405))$ to
$\Gamma(J/\psi\to\gamma\eta(1295))$ can be expressed as,

$\frac{\Gamma(J/\psi\to\gamma\eta(1405))}{\Gamma(J/\psi\to\gamma\eta(1295))}=\bigg{(}\frac{|\mathbf{p}_{\eta(1405)}|}{|\mathbf{p}_{\eta(1295)}|}\bigg{)}^{3}\bigg{(}\frac{\sqrt{2}\sin\alpha_{P}+R\cos\alpha_{P}}{\sqrt{2}\cos\alpha_{P}-R\sin\alpha_{P}}\bigg{)}^{2}\
,$ (4)

where the partial momentum has been included in these two $P$-wave processes.
In principle, experimental data for exclusive decay branching ratios will
determine $g_{0}$ and the relation in Eq. (4) will provide a test of the
radial excitation picture as $\alpha_{P}$ and $R$ will share the same values
as for $\eta$ and $\eta^{\prime}$ Zhao:2006gw ; Li:2007ky . However, to
extract the $J/\psi$ exclusive decay branching ratios one has to subtract the
decay information of the intermediate pseudoscalar mesons in
$J/\psi\to\gamma\eta_{X}\to\gamma\gamma V$. As shown in Table 1, so far the
most precise data from BESIII are still combined branching ratios. It means
that a better understanding of the $\eta_{X}$ exclusive decay into $\gamma V$
is required.

Table 1: Branching ratios of the combined decays of $J/\psi\to\gamma\eta(1405/1475)$ and $\eta(1405/1475)$ decays into final states. For $J/\psi\to\gamma\eta(1405)\to\gamma\gamma\phi$, two solutions are provided by the BESIII analysis Ablikim:2018hxj .  Channel | Branching ratio
---|---
$BR(J/\psi\rightarrow\gamma\eta(1405/1475)\rightarrow\gamma K\bar{K}\pi)$ |  $(2.8\pm 0.6)\times 10^{-3}$ Tanabashi:2018oca
$BR(J/\psi\rightarrow\gamma\eta(1405/1475)\rightarrow\gamma\gamma\rho)$ |  $(7.8\pm 2.0)\times 10^{-5}$ Tanabashi:2018oca
$BR(J/\psi\rightarrow\gamma\eta(1405/1475)\rightarrow\gamma\gamma\phi)\quad(I)$ |  $(7.03\pm 0.92\pm 0.91)\times 10^{-6}$ Ablikim:2018hxj
$BR(J/\psi\rightarrow\gamma\eta(1405/1475)\rightarrow\gamma\gamma\phi)\quad(II)$ |  $(10.36\pm 1.51\pm 1.54)^{-6}$ Ablikim:2018hxj

Figure 1:  Schematic diagrams for the process $\eta_{X}\to\gamma V$ at the
tree and loop level in the VMD model. (a) stands for the tree-level
transitions; (b) stands for the hadronic triangle loop transitions; (c) and
(d) denote the contact diagrams with a photon induced by the minimum
substitution at the two strong interaction vertices, respectively.

The effective couplings for $\eta_{X}\to\gamma V$ are defined as:

$\mathcal{L}_{\eta_{X}\gamma V}=g_{\eta_{X}\gamma
V}\epsilon_{\alpha\beta\delta\lambda}\partial^{\alpha}V^{\delta}\partial^{\beta}V^{\lambda}_{\gamma}P,$
(5)

where $g_{\eta_{X}\gamma V}$ contains the contributions from all possible
mechanisms in $\eta_{X}\to\gamma V$. At the hadronic level the effective
coupling can be decomposed into the tree diagram contributions via the VMD
model, and meson loop transitions as higher order corrections. The transition
mechanism is illustrated by Fig. 1. Notice that the coupling vertices are well
defined in the SU(3) flavor symmetry limit. The detailed calculations of the
tree and loop amplitudes will be given in the next Subsection. With the
calculated $g_{\eta_{X}\to\gamma V}$ one can express the partial decay width
for $J/\psi\to\gamma\eta_{X}\to\gamma\gamma V$ as

$\displaystyle\Gamma_{J/\psi\to\gamma\eta_{X}\to\gamma\gamma V}$
$\displaystyle\propto$
$\displaystyle\int\dfrac{d^{3}\mathbf{p_{1}}d^{3}\mathbf{p_{2}}d^{3}\mathbf{p_{3}}}{(2\pi)^{9}2E_{1}2E_{2}2E_{3}}(2\pi)^{4}\delta^{4}(P_{J/\psi}-p_{1}-p_{2}-p_{3})$
(6) $\displaystyle\times$
$\displaystyle\left\\{\tilde{g}_{\eta_{X}}g_{\eta_{X}\gamma
V}\left[G_{\eta_{X}}(s=(p_{2}+p_{3})^{2})+G_{\eta_{X}}(s=(p_{1}+p_{3})^{2})\right]\right\\}^{2}2s\times
2m^{2}_{J/\psi}|\mathbf{p_{1}}|^{2}|\mathbf{p_{2}}|^{2}\ ,$

where $P_{J/\psi}$ is the four-vector momentum of the initial $J/\psi$;
$G_{\eta_{X}}(s)$ denotes the propagator of $\eta(1405)$ or $\eta(1295)$,
respectively, i.e.

$\begin{split}G_{\eta(1405)}(s)&=\frac{i}{s-m_{\eta(1405)}^{2}+im_{\eta(1405)}\Gamma_{\eta(1405)}},\\\
G_{\eta(1295)}(s)&=\frac{i}{s-m_{\eta(1295)}^{2}+im_{\eta(1295)}\Gamma_{\eta(1295)}}.\end{split}$
(7)

Note that the parametrization of Eq. (6) has neglected the energy or momentum
dependence with the couplings $\tilde{g}_{\eta_{X}}$ and $g_{\eta_{X}\gamma
V}$. For the kinematics near the pole masses of $\eta_{X}$, this approximation
is reasonable. The decay coupling $g_{\eta_{X}\gamma V}$ contains
contributions from both tree-level and loop amplitudes, i.e.
$g_{\eta_{X}\gamma V}=g^{T}_{\eta_{X}\gamma V}+g^{L}_{\eta_{X}\gamma V}$ with
the superscripts $T$ and $L$ indicating the tree and loop processes,
respectively. These two quantities, $g^{T}_{\eta_{X}\gamma V}$ and
$g^{L}_{\eta_{X}\gamma V}$, will then be calculated in our model.

###  II.2 Tree-level amplitude in the VMD model

In this work, we adopt the VMD model Bauer:1977iq  to describe the
electromagnetic vertices. With the Lagrangian of Eq. (5) and the VMD model,
the amplitude of the tree diagram shown in Fig. 1 (a) can be written as

$\displaystyle i\mathcal{M}_{T}=ig^{T}_{\eta_{X}\gamma
V}\epsilon_{\alpha\beta\delta\lambda}p^{\alpha}_{\gamma}p^{\beta}_{V}\epsilon^{\delta}_{\gamma}\epsilon^{\lambda}_{V},$
(8)

where the tree-level effective coupling $g^{T}_{\eta_{X}\gamma V}$ can be
expressed as:

$\displaystyle g^{T}_{\eta_{X}\gamma V}$ $\displaystyle\equiv$ $\displaystyle-
ig_{\eta_{X}VV}\frac{em^{2}_{V}}{f_{V}}G_{V},$ (9)

where $G_{V}$ is the propagator of the intermediate vector meson $V$:

$G_{V}\equiv\frac{-i}{p^{2}_{\gamma}-m^{2}_{V}+im_{V}\Gamma_{V}},$ (11)

with $V$ denotes the vector mesons $\rho^{0},\ \omega$ or $\phi$ which is the
same as the final state vector meson in $\eta_{X}\to\gamma V$ as required by
the isospin symmetry. The strong coupling constants $g_{\eta_{X}VV}$ can be
extracted by other independent processes. Then, the couplings for other
vectors within the same SU(3) multiplet can be related to each other by the
SU(3) symmetry. We will come back to this in the next Subsection with detailed
extraction of $g_{\eta_{X}VV}$.

The vector meson decay constant $e/f_{V}$ can be determined by $V\to
e^{+}e^{-}$ using the experimental data, which can be expressed as,

$\frac{e}{f_{V}}=\left(\frac{3\Gamma_{V\to e^{+}e^{-}}}{2\alpha_{e}|{\bf
p}_{e}|}\right)^{1/2}\ ,$ (12)

where ${\bf p}_{e}$ is the three-vector momentum of the electron in the
vector-meson rest frame. The values for different vector meson decays are
extracted by adopting the experimental data of the vector meson decays into
$e^{+}e^{-}$ Zyla:2020zbs , and they are listed in Table 2.

Table 2:  Vector meson decay constants determined by $V\to e^{+}e^{-}$. The data are taken from the PDG Zyla:2020zbs . Channel | Total width of V | BR($V\to e^{+}e^{-}$) | $e/f_{V}\ (\times 10^{-2})$
---|---|---|---
$\phi\to e^{+}e^{-}$ |  $4.25$ MeV | $(2.97\pm 0.04)\times 10^{-4}$ | $-2.26$
$\rho^{0}\to e^{+}e^{-}$ |  $147.8$ MeV | $(4.72\pm 0.05)\times 10^{-5}$ | $6.05$
$\omega\to e^{+}e^{-}$ |  $8.49$MeV | $(7.36\pm 0.15)\times 10^{-5}$ | $1.78$

###  II.3 Loop amplitudes in the VMD model

####  II.3.1 Lagrangians and coupling constants

The triangle loop amplitudes illustrated by Fig. 1 (b) can also be calculated
in the VMD model. The loop amplitudes can reduce to an effective coupling
which contributes to the $\eta_{X}\gamma V$ coupling in the end. Within the
triangle loops the vertices for the photon couplings to the kaon and/or
$K^{*}$ pairs can be described by the VMD model which have the following
structures:

$\displaystyle g_{K^{*+}K^{-}\gamma}$ $\displaystyle=$
$\displaystyle\frac{i}{\sqrt{2}}\bigg{(}g_{\omega
K^{*+}K^{-}}\frac{em^{2}_{\omega}}{f_{\omega}}G_{\omega}+g_{\rho
K^{*+}K^{-}}\frac{em^{2}_{\rho}}{f_{\rho}}G_{\rho}\bigg{)}+ig_{\phi
K^{*+}K^{-}}\frac{em^{2}_{\phi}}{f_{\phi}}RG_{\phi},$ (13) $\displaystyle
g_{K^{*0}\bar{K}^{0}\gamma}$ $\displaystyle=$
$\displaystyle\frac{i}{\sqrt{2}}\bigg{(}g_{\omega
K^{*0}\bar{K}^{0}}\frac{em^{2}_{\omega}}{f_{\omega}}G_{\omega}+g_{\rho
K^{*0}\bar{K}^{0}}\frac{em^{2}_{\rho}}{f_{\rho}}G_{\rho}\bigg{)}+ig_{\phi
K^{*0}\bar{K}^{0}}\frac{em^{2}_{\phi}}{f_{\phi}}RG_{\phi},$ (14)

for the $K^{*}K\gamma$ vertex,

$\displaystyle g_{K^{*+}K^{*-}\gamma}$ $\displaystyle=$
$\displaystyle\frac{i}{\sqrt{2}}\bigg{(}g_{\omega
K^{*+}K^{*+}}\frac{em^{2}_{\omega}}{f_{\omega}}G_{\omega}+g_{\rho
K^{*+}K^{*+}}\frac{em_{\rho}^{2}}{f_{\rho}}G_{\rho}\bigg{)}+ig_{\phi
K^{*+}K^{*-}}\frac{em^{2}_{\phi}}{f_{\phi}}RG_{\phi},$ (15) $\displaystyle
g_{K^{*0}\bar{K}^{*0}\gamma}$ $\displaystyle=$
$\displaystyle\frac{i}{\sqrt{2}}\bigg{(}g_{\omega
K^{*0}K^{*0}}\frac{em^{2}_{\omega}}{f_{\omega}}G_{\omega}+g_{\rho
K^{*0}K^{*0}}\frac{em^{2}_{\rho}}{f_{\rho}}G_{\rho}\bigg{)}+ig_{\phi
K^{*0}\bar{K}^{*0}}\frac{em^{2}_{\phi}}{f_{\phi}}RG_{\phi},$ (16)

for the $K^{*}\bar{K}^{*}\gamma$ vertex, and

$\displaystyle g_{K^{+}K^{-}\gamma}$ $\displaystyle=$
$\displaystyle\frac{i}{\sqrt{2}}\bigg{(}g_{\omega
K^{+}K^{-}}\frac{em^{2}_{\omega}}{f_{\omega}}G_{\omega}+g_{\rho
K^{+}K^{-}}\frac{em^{2}_{\rho}}{f_{\rho}}G_{\rho}\bigg{)}+ig_{\phi
K^{+}K^{-}}\frac{em^{2}_{\phi}}{f_{\phi}}RG_{\phi},$ (17) $\displaystyle
g_{K^{0}\bar{K}^{0}\gamma}$ $\displaystyle=$
$\displaystyle\frac{i}{\sqrt{2}}\bigg{(}g_{\omega
K^{0}\bar{K}^{0}}\frac{em^{2}_{\omega}}{f_{\omega}}G_{\omega}+g_{\rho
K^{0}\bar{K}^{0}}\frac{em^{2}_{\rho}}{f_{\rho}}G_{\rho}\bigg{)}+ig_{\phi
K^{0}\bar{K}^{0}}\frac{em^{2}_{\phi}}{f_{\phi}}RG_{\phi},$ (18)

for the $K\bar{K}\gamma$ vertex. In the above equations the ground state
vector meson decay constants $e/f_{V}$ ($V=\phi,\ \rho^{0},\ \omega$) have
been given by Eq. (12) and their values listed in Table 2.

For the hadronic vertices one can see that they can be arranged by the SU(3)
symmetry. Thus, their relative strengths and phases are fixed. There are three
types of hadronic coupling vertices in the loop amplitudes, i.e. $VPP$, $VVP$
and $VVV$, for which effective Lagrangians are adopted. The corresponding
effective Lagrangians are as follows:

$\displaystyle\mathcal{L}_{VPP}$ $\displaystyle=$ $\displaystyle
ig_{VPP}Tr[(P\partial_{\mu}P-\partial_{\mu}PP)V^{\mu}],$ (19)
$\displaystyle\mathcal{L}_{VVP}$ $\displaystyle=$ $\displaystyle
g_{VVP}\epsilon_{\alpha\beta\mu\nu}Tr[\partial^{\alpha}V^{\mu}\partial^{\beta}V^{\nu}P],$
(20) $\displaystyle\mathcal{L}_{VVV}$ $\displaystyle=$ $\displaystyle
ig_{VVV}\bigl{<}(\partial_{\mu}V^{f}_{\nu}-\partial_{\nu}V^{f}_{\mu})V^{\mu}V^{\nu}\bigr{>},$
(21)

where $V$ and $P$ stand for the vector and pseudoscalar fields for the flavor
SU(3) multiplets, respectively, and have the following forms:

$P=\left(\begin{array}[]{ccc}\frac{\sin\alpha_{P}\eta^{\prime}+\cos\alpha_{P}\eta+\pi^{0}}{\sqrt{2}}&\pi^{+}&K^{+}\\\
\pi^{-}&\frac{\sin\alpha_{P}\eta^{\prime}+\cos\alpha_{P}\eta-\pi^{0}}{\sqrt{2}}&K^{0}\\\
K^{-}&\bar{K^{0}}&\cos\alpha_{P}\eta^{\prime}-\sin\alpha_{P}\eta\\\
\end{array}\right),$ (22)

and

$V=\left(\begin{array}[]{ccc}\frac{\omega+\rho^{0}}{\sqrt{2}}&\rho^{+}&{K^{*}}^{+}\\\
\rho^{-}&\frac{\omega-\rho^{0}}{\sqrt{2}}&K^{*0}\\\
K^{*-}&\bar{K}^{*0}&\phi\\\ \end{array}\right),$ (23)

where the ideal mixing between $\omega$ ($=(u\bar{u}+d\bar{d})/\sqrt{2}$) and
$\phi$ ($=s\bar{s}$) is implied.

Note that we adopt the same form as Eq. (22) for the first radial excitation
states of the pseudoscalar mesons.

Considering the $\eta_{X}$ coupling to $K^{*}\bar{K}$, the effective
Lagrangians have the following expressions:

$\displaystyle\mathcal{L}_{\eta(1405)K^{*+}K^{-}}$ $\displaystyle=$
$\displaystyle
g_{\eta(1405)K^{*+}K^{-}}(K^{-}\partial_{\mu}\eta(1405)-\eta(1405)\partial_{\mu}K^{-})(K^{*+})^{\mu}$
(24) $\displaystyle\equiv$ $\displaystyle
g_{XVP}(\frac{\sin\alpha_{P}}{\sqrt{2}}R-\cos\alpha_{P})(K^{-}\partial_{\mu}\eta(1405)-\eta(1405)\partial_{\mu}K^{-})(K^{*+})^{\mu}\
,$

and

$\displaystyle\mathcal{L}_{\eta(1295)K^{*+}K^{-}}$ $\displaystyle=$
$\displaystyle
g_{\eta(1295)K^{*+}K^{-}}(K^{-}\partial_{\mu}\eta(1295)-\eta(1295)\partial_{\mu}K^{-})(K^{*+})^{\mu}$
(25) $\displaystyle\equiv$ $\displaystyle
g_{XVP}(\frac{\cos\alpha_{P}}{\sqrt{2}}R+\sin\alpha_{P})(K^{-}\partial_{\mu}\eta(1295)-\eta(1295)\partial_{\mu}K^{-})(K^{*+})^{\mu},$

where $g_{XVP}$ is the overall coupling between a radial excitation
pseudoscalar $q\bar{q}$ and $K^{*+}K^{-}$ and will be determined later; $R$ is
the SU(3) flavor symmetry breaking factor defined earlier.

Couplings between $\eta_{X}$ and $K^{*}\bar{K}^{*}$ also contribute to the
loop amplitudes. The corresponding Lagrangians are

$\displaystyle\mathcal{L}_{\eta(1405)K^{*+}K^{*-}}$ $\displaystyle=$
$\displaystyle
g_{XVV}\epsilon_{\alpha\beta\mu\nu}\bigg{[}\frac{\sin\alpha_{P}}{\sqrt{2}}R\
\partial^{\alpha}(K^{*+})^{\mu}\partial^{\beta}(K^{*-})^{\nu}\eta(1405)+\cos\alpha_{P}\partial^{\alpha}(K^{*-})^{\mu}\partial^{\beta}(K^{*+})^{\nu}\eta(1405)\bigg{]}$
(26) $\displaystyle=$ $\displaystyle
g_{XVV}\bigg{(}\frac{\sin\alpha_{P}}{\sqrt{2}}R+\cos\alpha_{P}\bigg{)}\epsilon_{\alpha\beta\mu\nu}\partial^{\alpha}(K^{*+})^{\mu}\partial^{\beta}(K^{*-})^{\nu}\eta(1405),$

for the $\eta(1405)K^{*+}K^{*-}$ coupling, and

$\displaystyle\mathcal{L}_{\eta(1295)K^{*+}K^{*-}}$ $\displaystyle=$
$\displaystyle
g_{XVV}\bigg{(}\frac{\cos\alpha_{P}}{\sqrt{2}}R-\sin\alpha_{P}\bigg{)}\epsilon_{\alpha\beta\mu\nu}\partial^{\alpha}(K^{*+})^{\mu}\partial^{\beta}(K^{*-})^{\nu}\eta(1295),$
(27)

for the $\eta(1295)K^{*+}K^{*-}$ coupling, respectively. The coupling
$g_{XVV}$ is the overall coupling strength of the first radial excitation
state of a $(q\bar{q})_{0^{-+}}$ to a vector meson pair. This quantity will be
determined by the combined analysis of the data for
$J/\psi\to\gamma\eta(1405/1475)\to\gamma K\bar{K}\pi$ and
$J/\psi\to\gamma\eta(1405/1475)\to\gamma\gamma\rho$ Zyla:2020zbs .

The other hadronic vertices, which involve the interactions between the ground
state vector and pseudoscalar mesons, can be obtained by expanding Eqs.
(19)-(21).

We adopt the following strategy to determine the coupling constants:

  * •

In our calculation we take the same sign for the ground state coupling
$g_{VPP}$ and $g_{XVP}$. They are defined as positive and real numbers, and
then the signs for the other couplings will be fixed.

  * •

The coupling $g_{VPP}$ between the ground state vector and pseudoscalar mesons
can be determined by $\phi\to K^{+}K^{-}$. Then, the other $VPP$ couplings can
be related to $g_{\phi K^{+}K^{-}}$ by the SU(3) flavor symmetry. We note that
one can also extract $g_{VPP}$ via $\rho\to\pi\pi$, and some SU(3) flavor
symmetry breaking effects can be found. By adopting the coupling extracted
from $\phi\to K\bar{K}$, we actually absorb some leading SU(3) flavor symmetry
breaking effects into this quantity since all the vertices in the loop
processes involve couplings with the strange mesons. The corresponding
couplings $g_{\phi K^{+}K^{-}}$ and $g_{VPP}$ are listed in Tab. 3.

  * •

The coupling $g_{VVP}$ in the loop between the ground state vector and
pseudoscalar mesons is determined by fitting the experimental data for
$V\to\gamma P$ and $\eta^{\prime}\to\gamma V$ in the VMD model. For these
transitions between the ground state vector and pseudoscalar mesons, we assume
that the intermediate ground state vector mesons saturate the transition
amplitudes. The corresponding channels and fitting results are listed in Tab.
4, and the best fitting gives $g_{VVP}=8.38\pm 0.1$ GeV-1. We then adopt
$g_{VVP}=8.38$ GeV-1 to extract other $VVP$ couplings in the loop amplitudes
which are listed in Tab. 5. The sign is determined to be consistent with the
$g_{VPP}$ following the ${}^{3}P_{0}$ model.

  * •

The strong couplings $g_{VVV}$, $g_{VVP}$ and $g_{VPP}$ can be related to each
other by matching the relation of their corresponding transition amplitudes in
ELA to the same relation which can be extracted explicitly in the quark pair
creation (QPC) model (i.e. ${}^{3}P_{0}$ model). The value of $g_{VVV}$ for
the relevant channels are also listed in Tab. 5. We mention that the two
couplings $g_{VPP}$ and $g_{VVP}$ holds a reasonable relation as that
extracted in the quark model, i.e. $g_{VPP}\simeq m_{V}g_{VVP}/2$.

  * •

The physical coupling $g_{\eta(1405)K^{*}\bar{K}}$ can be extracted from
experimental data by a combined analysis of the $\eta(1405)$ decays into $\to
K\bar{K}\pi$, $\eta\pi\pi$ and $3\pi$ Wu:2012pg ; Du:2019idk . By adopting the
branching ratio $B.R.(\eta(1405)\to K^{*}\bar{K}+c.c.)\simeq 50\%$ and the
total width $\Gamma_{\eta(1405)}=90$ MeV, we can extract $g_{XVP}$ with
$\alpha_{P}=42^{\circ}$ Du:2019idk . Then, both $g_{\eta(1405)K^{*+}K^{-}}$
and $g_{\eta(1295)K^{*+}K^{-}}$ can be determined. Their values are listed in
Tab. 6. The overall coupling $g_{XVP}$ can also be determined. The extracted
values are listed in Tab. 6.

  * •

One can determine $g_{XVV}$ by treating the tree-level transition as the
dominant amplitude in $\eta(1405)\to\gamma\gamma\rho$ 444In principle, one can
relate $g_{XVP}$ to $g_{XVV}$ in the ${}^{3}P_{0}$ model, which leads to
$g_{XVV}\simeq 2g_{XVP}/m_{X}$. However, we find that this analytic relation
is broken by about 30%. This is understandable since either $K^{*}$ and
$\bar{K}^{*}$ in the final state will be off-shell and the form factor
corrections would become significant.. We will show later this is reasonable
for $\eta(1405)$ since the loop contributions are much smaller than the tree-
level amplitude. Then, by taking the ratio of
$BR(J/\psi\to\gamma\eta(1405)\to\gamma\gamma\rho)/BR(J/\psi\to\gamma\eta(1405)\to\gamma
K^{*}\bar{K}+c.c.)$ and with the constraint from the experimental data (see
Tab. 1 and Refs. Tanabashi:2018oca ; Ablikim:2018hxj ), we can extract
$g_{XVV}=8.7^{+2.2}_{-1.8}$ as listed in Tab. 7.

Table 3: Strong couplings adopted for the $VPP$ vertices in the $K^{*}\bar{K}(K^{(*)})$ loops.  $VPP$ coupling const. | Values
---|---
$g_{\phi K^{+}K^{-}}$ | $4.47$
$g_{\rho K^{+}K^{-}}$ | $-\frac{g_{\phi K^{+}K^{-}}}{\sqrt{2}}$
$g_{\omega K^{+}K^{-}}$ | $-\frac{g_{\phi K^{+}K^{-}}}{\sqrt{2}}$
$g_{VPP}$ | $4.47$
Table 4: The fitted radiative transitions between the ground state vector and pseudoscalar mesons in comparison with the experimental data in the VMD model. The best fitting gives $g_{VVP}=8.38\pm 0.1$ GeV-1. Channels | Experiments (MeV) | Fitted values (MeV)
---|---|---
$\eta^{\prime}\to\gamma\rho$ | $(6.7\pm 0.7)\times 10^{-2}$ | $(8.6\pm 0.2)\times 10^{-2}$
$\rho\to\gamma\eta$ | $(4.5\pm 0.3)\times 10^{-2}$ | $(5\pm 0.1)\times 10^{-2}$
$\rho\to\gamma\pi$ | $(7\pm 1)\times 10^{-2}$ | $(6.1\pm 0.1)\times 10^{-2}$
$\eta^{\prime}\to\gamma\omega$ | $(5.8\pm 0.7)\times 10^{-3}$ | $(6.4\pm 0.2)\times 10^{-3}$
$\omega\to\gamma\eta$ | $(3.8\pm 0.4)\times 10^{-3}$ | $(5.2\pm 0.1)\times 10^{-3}$
$\omega\to\gamma\pi$ | $0.71\pm 0.03$ | $0.72\pm 0.02$
$\phi\to\gamma\eta^{\prime}$ | $(2.6\pm 0.1)\times 10^{-4}$ | $(2.9\pm 0.1)\times 10^{-4}$
$\phi\to\gamma\eta$ | $(5.5\pm 0.1)\times 10^{-2}$ | $(5.2\pm 0.1)\times 10^{-2}$
$K^{*\pm}\to\gamma K^{\pm}$ | $(5\pm 0.5)\times 10^{-2}$ | $(2.4\pm 0.05)\times 10^{-2}$
$K^{*0}(\bar{K}^{*0})\to\gamma K^{0}(\bar{K}^{0})$ | $0.12\pm 0.01$ | $0.081\pm 0.002$
Table 5: Strong couplings adopted for the vertices in the $K^{*}\bar{K}(K^{(*)})$ loops.  VVP | Values (GeV-1) | VVV | Values
---|---|---|---
$g_{\phi K^{*+}K^{-}}$ | $8.38$ | $g_{\phi K^{*+}K^{*-}}$ | $4.47$
$g_{\rho K^{*+}K^{-}}$ | $\frac{g_{\phi K^{*+}K^{-}}}{\sqrt{2}}$ | $g_{\rho K^{*+}K^{*-}}$ | $-\frac{g_{\phi K^{*+}K^{*-}}}{\sqrt{2}}$
$g_{\omega K^{*+}K^{-}}$ | $\frac{g_{\phi K^{*+}K^{-}}}{\sqrt{2}}$ | $g_{\omega K^{*+}K^{*-}}$ | $-\frac{g_{\phi K^{*+}K^{*-}}}{\sqrt{2}}$
$g_{VVP}$ | $8.38$ | $g_{VVV}$ | $4.47$
Table 6: Hadronic couplings for $\eta_{X}\to K^{*+}K^{-}$ based on $g_{\eta(1405)K^{*+}K^{-}}=-3.64$ determined by combined analyses of the decays of $\eta(1405)\to K^{*}\bar{K}+c.c.$ and $\eta\pi\pi$ Wu:2012pg ; Du:2019idk . $\alpha_{P}=42^{\circ}$ is adopted for the mixing between $\eta(1295)$ and $\eta(1405)$.  $\eta_{X}VP$ | Values
---|---
$g_{\eta(1405)K^{*+}K^{-}}$ |  $-3.64$ Wu:2012pg ; Du:2019idk
$g_{\eta(1295)K^{*+}K^{-}}$ | $10.9$
$g_{XVP}$ | $9.98$
Table 7:  Strong couplings for $\eta_{X}\to VV$ in the VMD model. The overall coupling $g_{XVV}=8.7^{+2.2}_{-1.8}$ is extracted by the combined analysis of $J/\psi\to\gamma\eta(1405)\to\gamma\gamma\rho$ and $J/\psi\to\gamma\eta(1405)\to\gamma K^{*}\bar{K}+c.c.$ We adopt the central value of $g_{XVV}=8.7$ in the numerical calculation.  Coupling const. | Expression | Value (GeV-1)
---|---|---
$g_{\eta(1295)K^{*+}K^{*-}}$ | $g_{XVV}(\cos\alpha_{P}/\sqrt{2}-\sin\alpha_{P})$ | $-2.2$
$g_{\eta(1405)K^{*+}K^{*-}}$ | $g_{XVV}(\sin\alpha_{P}/\sqrt{2}+\cos\alpha_{P})$ | $9.7$
$g_{\eta(1295)\rho\rho}$ | $\sqrt{2}g_{XVV}\cos\alpha_{P}$ | $9.1$
$g_{\eta(1405)\rho\rho}$ | $\sqrt{2}g_{XVV}\sin\alpha_{P}$ | $8.2$
$g_{\eta(1295)\omega\omega}$ | $\sqrt{2}g_{XVV}\cos\alpha_{P}$ | $9.1$
$g_{\eta(1405)\omega\omega}$ | $\sqrt{2}g_{XVV}\sin\alpha_{P}$ | $8.2$
$g_{\eta(1295)\phi\phi}$ | $-2g_{XVV}R\sin\alpha_{P}$ | $-9.3$
$g_{\eta(1405)\phi\phi}$ | $2g_{XVV}R\cos\alpha_{P}$ | $10.3$
$g_{XVV}$ | - | $8.7^{+2.2}_{-1.8}$

Combining the couplings collected in Tables 2 and 5, we can extract the
effective couplings defined in Eqs. (13)-(18) as a test of the VMD model.
Since the widths of the intermediate vector mesons are considered, the
couplings extracted in the VMD are complex numbers. We also provide the
magnitudes of the complex couplings in round brackets in order to compare with
the quantities extracted from experiment. One can see that the VMD model has
accounted for the experimental data reasonably well under the assumption of
the ground state vector meson saturation.

Table 8: The effective couplings for the $K^{*}K^{(*)}\gamma$ transitions. The values extracted from the VMD model are listed in the second column and their corresponding modules are presented in the round brackets. In the last column the signs of the $g_{K^{*}K\gamma}$ couplings are determined in the quark model. The couplings $g_{K^{*}K^{*}\gamma}$ and $g_{KK\gamma}$ are treated as pure QED couplings. Thus, their coupling strengths will be given by the charge of the hadron. Note that $e\simeq 0.3$.  Electromagnetic couplings | Values in VMD (Magnitude) | Experimental values
---|---|---
$g_{K^{*+}K^{+}\gamma}$ (GeV-1) |  $-0.17-i0.05$ $(0.18)$ | $-0.25$
$g_{K^{*0}K^{0}\gamma}$ (GeV-1) |  $0.33+0.05i$ $(0.33)$ | $0.38$
$g_{K^{*+}K^{*+}\gamma}$ |  $0.25+0.026i$ $(0.25)$ | $e$
$g_{K^{*0}K^{*0}\gamma}$ |  $-0.01-0.024i$ $(0.03)$ | $0$
$g_{K^{+}K^{+}\gamma}$ |  $0.25+0.026i$ $(0.25)$ | $e$
$g_{K^{0}K^{0}\gamma}$ |  $-0.01-0.024i$ $(0.03)$ | $0$

The relations for couplings among the flavor SU(3) multiplets will be
explicitly presented in the construction of each loop amplitude.

####  II.3.2 Loop amplitudes

With the above effective Lagrangians we can write down the loop transition
amplitudes for Fig. 1 (b)-(d). For the triangle loops (Fig. 1 (b)), we use the
notation $[M1,M3,(M2)]$ to denote the intermediate interaction between
particle $M1$ and $M3$ by exchanging $M2$, e.g. $[K^{*},\bar{K},(K)]$
represents the loop where the intermediate $K^{*}$ and $\bar{K}$ scattering
into the final states by exchanging $K$. The masses and four-vector momenta of
these internal particles are denoted by ($m_{1}$, $m_{3}$, $m_{2}$) and
($p_{1}$, $p_{3}$, $p_{2}$), respectively. The four-vector momenta of the
initial-state meson $X$, final-state photon and vector meson are labelled as
$p_{X}$, $p_{\gamma}$ and $p_{V}$, respectively. The polarizations of the
final-state photon and vector meson are $\epsilon_{\gamma}$ and
$\epsilon_{V}$, respectively. Similarly, we adopt notation $[M1,M2]$ for the
internal mesons in Fig. 1 (c)-(d).

In order to cut off the ultra-violet (UV) divergence in the loop integrals, we
include a commonly-adopted form factor to regularize the integrand:

$\mathcal{F}(p_{i}^{2})=\prod_{i}\bigg{(}\frac{\Lambda_{i}^{2}-m_{i}^{2}}{\Lambda_{i}^{2}-p_{i}^{2}}\bigg{)}\
,$ (28)

where $\Lambda_{i}\equiv m_{i}+\alpha\Lambda_{QCD}$ with $\Lambda_{QCD}=220$
MeV and $\alpha=1\sim 2$ as the cut-off parameter.

As follows, we will write down the detailed amplitude for each loop transition
with explicit phase conventions.

  * •

$[K^{*},\bar{K},(K)]$

The amplitude for the $[K^{*},\bar{K},(K)]$ loop is

$i\mathcal{M}=\int\frac{d^{4}p_{1}}{(2\pi)^{4}}V_{1\sigma}D^{\sigma\mu}(K^{*})V_{2\mu}V_{3}D(K)D(\bar{K})\mathcal{F}(p_{i}^{2})$
(29)

where the vertex functions have been expressed by a compact form and have the
following expressions,

$\displaystyle V_{1\sigma}$ $\displaystyle=$ $\displaystyle
ig_{\eta_{X}K^{*}\bar{K}}(p_{X}+p_{3})_{\sigma},$ $\displaystyle V_{2\mu}$
$\displaystyle=$ $\displaystyle
ig_{K^{*}K\gamma}\epsilon_{\alpha\beta\delta\mu}p_{\gamma}^{\alpha}p_{1}^{\beta}\epsilon_{\gamma}^{\delta},$
$\displaystyle V_{3}$ $\displaystyle=$ $\displaystyle
iRg_{VK\bar{K}}(p_{2}-p_{3})_{\lambda}\epsilon_{V}^{\lambda}\ ,$ (30)

with the SU(3) flavor symmetry breaking factor $R$ included if the interacting
vector meson $V=\rho$ and $\omega$. In Eq. (29) functions
$D^{\sigma\mu}(K^{*})$ and $D(K)$ are the propagators for $K^{*}$ and $K$
respectively, with four-vector momentum $p$, i.e.

$\displaystyle D^{\sigma\mu}(K^{*})$ $\displaystyle=$
$\displaystyle\frac{-i(g^{\sigma\mu}-\frac{p^{\sigma}p^{\mu}}{p^{2}})}{p^{2}-m_{K^{*}}^{2}+i\epsilon}$
$\displaystyle D(K)$ $\displaystyle=$
$\displaystyle\frac{i}{p^{2}-m_{K}^{2}+i\epsilon}.$ (31)

In the calculations, the propagators will carry the corresponding four-vector
momenta required by momentum conservation.

There are four isospin channels for this type of triangle diagram, which
include $[K^{*+},K^{+},(K^{-})]$, $[K^{*-},K^{-},(K^{+})]$,
$[K^{*0},K^{0},(\bar{K}^{0})]$, and $[\bar{K}^{*0},\bar{K}^{0}(K^{0})]$. The
vertex coupling constants are connected by the SU(3) flavor symmetry:

$\begin{split}g_{\eta_{X}K^{*+}K^{-}}=-g_{\eta_{X}K^{*-}K^{+}}&=g_{\eta_{X}K^{*0}\bar{K}^{0}}=-g_{\eta_{X}\bar{K}^{*0}K^{0}},\\\
g_{\phi K^{+}K^{-}}=-g_{\phi K^{-}K^{+}}&=g_{\phi
K^{0}\bar{K}^{0}}=-g_{\phi\bar{K}^{0}K^{0}},\\\ g_{\omega
K^{+}K^{-}}=-g_{\omega K^{-}K^{+}}&=g_{\omega
K^{0}\bar{K}^{0}}=-g_{\omega\bar{K}^{0}K^{0}},\\\ g_{\rho K^{+}K^{-}}=-g_{\rho
K^{-}K^{+}}&=-g_{\rho K^{0}\bar{K}^{0}}=g_{\rho\bar{K}^{0}K^{0}}\
.\end{split}$ (32)

The coupling constants for the sum of all the isospin channels of the
$[K^{*},\bar{K},(K)]$ loop can be written as the following forms for the
$\gamma\phi$, $\gamma\omega$, and $\gamma\rho$ channels, respectively,

$\displaystyle 2g_{\eta_{X}K^{*+}K^{-}}g_{\phi
K^{+}K^{-}}(g_{K^{*+}K^{-}\gamma}+g_{K^{*0}K^{0}\gamma}),$ (33) $\displaystyle
2g_{\eta_{X}K^{*+}K^{-}}g_{\omega
K^{+}K^{-}}(g_{K^{*+}K^{-}\gamma}+g_{K^{*0}K^{0}\gamma}),$ (34) $\displaystyle
2g_{\eta_{X}K^{*+}K^{-}}g_{\rho
K^{+}K^{-}}(g_{K^{*+}K^{-}\gamma}-g_{K^{*0}K^{0}\gamma}).$ (35)

As listed in Tab. 8, the couplings $g_{K^{*+}K^{-}\gamma}$ and
$g_{K^{*0}K^{0}\gamma}$ have a sign phase, which implies the constructive
phase between the charged and neutral meson loops for $\eta_{X}\to\gamma\rho$,
while the $\gamma\phi$ and $\gamma\omega$ channels involve a destructive
phase.

The loop integral will be given in the Appendix and the contributions of each
type of the loop transitions will be collected and compared with each other
among different processes.

  * •

$[K^{*},\bar{K},(K^{*})]$

Similarly, by denoting the masses and four-vector momenta of the intermediate
mesons ($K^{*}$, $K^{*}$, $\bar{K}$) as ($m_{1}$, $m_{2}$, $m_{3}$) and
($p_{1}$, $p_{2}$, $p_{3}$), respectively, the loop transition amplitude can
be written as

$i\mathcal{M}=\int\frac{d^{4}p_{1}}{(2\pi^{4})}V_{1\sigma}D^{\sigma\mu}(K^{*})V_{2\mu\rho}D^{\rho\nu}(K^{*})V_{3\nu}D(\bar{K})\mathcal{F}({p^{2}_{i}})\
,$ (36)

where the vertex function $V_{1\sigma}$ has the same form as that in Eq.
(II.3.2), and the other two functions are:

$\displaystyle V_{2\mu\rho}$ $\displaystyle=$ $\displaystyle-
ig_{K^{*}K^{*}\gamma}\bigg{[}\epsilon^{\delta}_{\gamma}g_{\delta\mu}p_{1\rho}+\epsilon^{\delta}_{\gamma}g_{\delta\rho}p_{2\mu}-g_{\mu\rho}(p_{1}+p_{2})_{\delta}\epsilon^{\delta}_{\gamma}\bigg{]},$
(37) $\displaystyle V_{3\nu}$ $\displaystyle=$ $\displaystyle
iRg_{VK^{*+}K^{-}}\epsilon_{\alpha\beta\nu\lambda}p_{2}^{\alpha}p_{V}^{\beta}\epsilon^{\lambda}_{V}.$
(38)

Similar to the previous loop amplitude, there are also four isospin channels
which involve the charged and neutral intermediate mesons. They can be
combined together with the coupling constants for the $\gamma\phi$,
$\gamma\omega$ and $\gamma\rho^{0}$ channels, respectively, i.e.

$\displaystyle 2g_{\eta_{X}K^{*+}K^{-}}g_{\phi
K^{*+}K^{-}}(g_{K^{*+}K^{*-}\gamma}+g_{K^{*0}\bar{K}^{*0}\gamma}),$ (39)
$\displaystyle 2g_{\eta_{X}K^{*+}K^{-}}g_{\omega
K^{*+}K^{-}}(g_{K^{*+}K^{*-}\gamma}+g_{K^{*0}\bar{K}^{*0}\gamma}),$ (40)
$\displaystyle 2g_{\eta_{X}K^{*+}K^{-}}g_{\rho
K^{*+}K^{-}}(g_{K^{*+}K^{*-}\gamma}-g_{K^{*0}\bar{K}^{*0}\gamma}).$ (41)

As listed in Tab. 8, the charge neutral coupling
$g_{K^{*0}\bar{K}^{*0}\gamma}$ is much smaller than the charged one.

  * •

$[K^{*},\bar{K}^{*},(K)]$

With the same notation convention for the masses and four-vector momenta of
the intermediate mesons, the loop amplitude can be expressed as

$i\mathcal{M}=\int\frac{d^{4}p_{1}}{(2\pi)^{2}}V_{1\mu\nu}D^{\mu\mu^{\prime}}(K*)V_{2\mu^{\prime}}D(K)V_{3\nu^{\prime}}D^{\nu^{\prime}\nu}(\bar{K}^{*})\mathcal{F}(p^{2}_{i})\
,$ (42)

where the vertex function $V_{2\mu^{\prime}}$ has been given in Eq. (II.3.2)
and the other two vertex functions are

$\displaystyle V_{1\mu\nu}$ $\displaystyle=$ $\displaystyle-
ig_{\eta_{X}K^{*}\bar{K}^{*}}\epsilon_{\alpha\beta\mu\nu}p^{\alpha}_{1}p^{\beta}_{3}$
(43) $\displaystyle V_{3\nu^{\prime}}$ $\displaystyle=$ $\displaystyle
iRg_{VK^{*}\bar{K}}\epsilon_{\alpha_{2}\beta_{2}\nu^{\prime}\lambda}p^{\alpha_{2}}_{3}p^{\beta_{2}}_{V}\epsilon^{\lambda}_{V}\
.$ (44)

Combining together the four isospin channels, we have the coupling constants
for the $\gamma\phi$, $\gamma\omega$, and $\gamma\rho^{0}$ decays,
respectively, as follows:

$\displaystyle 2g_{\eta_{X}K^{*+}K^{*-}}g_{\phi
K^{*+}K^{-}}(g_{K^{*+}K^{-}\gamma}+g_{K^{*0}K^{0}\gamma}),$ (45)
$\displaystyle 2g_{\eta_{X}K^{*+}K^{*-}}g_{\omega
K^{*+}K^{-}}(g_{K^{*+}K^{-}\gamma}+g_{K^{*0}K^{0}\gamma}),$ (46)
$\displaystyle 2g_{\eta_{X}K^{*+}K^{*-}}g_{\rho
K^{*+}K^{-}}(g_{K^{*+}K^{-}\gamma}-g_{K^{*0}K^{0}\gamma}).$ (47)

  * •

$[K^{*},\bar{K}^{*},(K^{*})]$

The transition amplitude can be written as

$i\mathcal{M}=\int\frac{d^{4}p_{1}}{(2\pi)^{4}}V_{1\mu\nu}D^{\mu\mu^{\prime}}(K^{*})V_{2\mu^{\prime}\rho}D^{\rho\sigma}(K^{*})V_{3\nu^{\prime}\sigma}D^{\nu^{\prime}\nu}(\bar{K}^{*})\mathcal{F}(p^{2}_{i}),$
(48)

where the vertex functions have been given earlier for the corresponding
couplings. Similar to the loop amplitudes discussed, we combine the isospin
channels together to give the couplings for the $\gamma\phi$, $\gamma\omega$,
and $\gamma\rho^{0}$ decays, respectively, as follows:

$\displaystyle 2g_{\eta_{X}K^{*+}K^{*-}}g_{\phi
K^{*+}K^{*-}}(g_{K^{*+}K^{*-}\gamma}+g_{K^{*0}\bar{K}^{*0}\gamma}),$ (49)
$\displaystyle 2g_{\eta_{X}K^{*+}K^{*-}}g_{\omega
K^{*+}K^{*-}}(g_{K^{*+}K^{*-}\gamma}+g_{K^{*0}\bar{K}^{*0}\gamma}),$ (50)
$\displaystyle 2g_{\eta_{X}K^{*+}K^{*-}}g_{\rho
K^{*+}K^{*-}}(g_{K^{*+}K^{*-}\gamma}-g_{K^{*0}\bar{K}^{*0}\gamma}).$ (51)

  * •

$[K,\bar{K}^{*},(K)]$

The transition amplitude can be written as

$i\mathcal{M}=\int\frac{d^{4}p_{1}}{(2\pi)^{4}}V_{1\mu}D(K)V_{2}D(K)V_{3\nu}D^{\mu\nu}(\bar{K}^{*})\mathcal{F}(p^{2}_{i})\
,$ (52)

where the vertex functions $V_{1\mu}$ and $V_{3\nu}$ have been given earlier,
and $V_{2}$ has the following form:

$\displaystyle V_{2}$ $\displaystyle=$ $\displaystyle
ig_{KK\gamma}(p_{1}+p_{2})_{\delta}\epsilon^{\delta}_{\gamma}\ .$ (53)

Again, combining the four isospin channels of this type of the loop
transitions, we obtain the coupling constants for the $\gamma\phi$,
$\gamma\omega$, and $\gamma\rho^{0}$ decays, respectively, as follows:

$\displaystyle 2g_{\eta_{X}K^{+}K^{*-}}g_{\phi
K^{+}K^{*-}}(g_{K^{+}K^{+}\gamma}+g_{K^{0}K^{0}\gamma}),$ (54) $\displaystyle
2g_{\eta_{X}K^{+}K^{*-}}g_{\omega
K^{+}K^{*-}}(g_{K^{+}K^{+}\gamma}+g_{K^{0}K^{0}\gamma}),$ (55) $\displaystyle
2g_{\eta_{X}K^{+}K^{*-}}g_{\rho
K^{+}K^{*-}}(g_{K^{+}K^{+}\gamma}-g_{K^{0}K^{0}\gamma}),$ (56)

where the charge neutral amplitudes vanish literally due to the suppressed
coupling $g_{K^{0}K^{0}\gamma}$.

  * •

$[K,\bar{K}^{*},(K^{*})]$

The transition amplitude can be written as

$i\mathcal{M}=\int\frac{d^{4}p_{1}}{(2\pi)^{4}}V_{1\mu}D(K)V_{2\nu}D^{\nu\nu^{\prime}}(K^{*})V_{3\mu^{\prime}\nu^{\prime}}D^{\mu\mu^{\prime}}(\bar{K}^{*})\mathcal{F}(p^{2}_{i})\
,$ (57)

where all the vertex functions have been given earlier. The combined couplings
for these four isospin channels can be expressed as the following forms for
the $\gamma\phi$, $\gamma\omega$, and $\gamma\rho^{0}$ decays, respectively,

$\displaystyle 2g_{\eta_{X}K^{+}K^{*-}}g_{\phi
K^{*+}K^{*-}}(g_{K^{*+}K^{+}\gamma}+g_{K^{*0}K^{0}\gamma}),$ (58)
$\displaystyle 2g_{\eta_{X}K^{+}K^{*-}}g_{\omega
K^{*+}K^{*-}}(g_{K^{*+}K^{+}\gamma}+g_{K^{*0}K^{0}\gamma}),$ (59)
$\displaystyle 2g_{\eta_{X}K^{+}K^{*-}}g_{\rho
K^{*+}K^{*-}}(g_{K^{*+}K^{+}\gamma}-g_{K^{*0}K^{0}\gamma}).$ (60)

  * •

Contact loop diagrams

Besides the triangle loop transitions, the photon can be produced by the
minimum substitution of the derivative at the hadronic interaction vertices,
i.e.

$\partial_{\mu}\rightarrow\partial_{\mu}+ie\hat{Q}A_{\mu}.$ (61)

Such contributions are illustrated by Fig. 1 (c) and (d) and are referred to
as the contact loop diagrams in this paper.

For Fig. 1 (c) or (d), the intermediate mesons $[M1,M2]$ both could be
$[K^{*},\bar{K}]$ or $[K^{*},\bar{K}^{*}]$. It can be easily proved that
contributions from Fig. 1 (c) with either $[K^{*},\bar{K}]$ or
$[K^{*},\bar{K}^{*}]$ should vanish Li:2007xr . For Fig. 1 (d) with
$[M1,M2]=[K^{*},\bar{K}^{*}]$, the antisymmetric tensor coupling for the
$\eta_{X}K^{*}\bar{K}^{*}$ vertex dictates that the induced photon can only
contribute via its longitudinal component. Thus, for the real photon decay,
this transition will be forbidden. As a result, only the transition of Fig. 1
(d) with the intermediate $K^{*}\bar{K}$ mesons will have non-vanishing
contributions and the corresponding amplitude can be expressed as follows:

$i\mathcal{M}=\int\frac{d^{4}p_{1}}{(2\pi)^{4}}V_{1\mu}D^{\mu\alpha}(K^{*+})V_{2\alpha}D(K^{-})\mathcal{F}(p^{2}_{i}),$
(62)

where the vertex function $V_{1\mu}$ has been defined in Eq. (II.3.2) and
$V_{2\alpha}$ has the following form:

$\displaystyle V_{2\alpha}$ $\displaystyle=$ $\displaystyle-
ie(\hat{Q}K^{*+})Rg_{VK^{*+}K^{-}}\epsilon_{\delta\beta\alpha\lambda}\epsilon^{\delta}_{\gamma}p_{V}^{\beta}\epsilon^{\lambda}_{V}.$
(63)

It is obvious that the intermediate charge neutral loops cannot contribute
since the amplitude is proportional to the charge of the pseudoscalar meson.
Moreover, taking into account the SU(3) relation among the couplings, one
finds that the two amplitudes with the intermediate $K^{*-}K^{+}$ and
$K^{*+}K^{-}$ loops are constructive for all the decay channels into
$\gamma\phi$, $\gamma\omega$ and $\gamma\rho^{0}$.

The detailed expressions of the loop amplitudes defined in this Subsection are
provided in the Appendix. Meanwhile, taking the advantage of the antisymmetric
tensor structure for the $VVP$ ($V\gamma P$) coupling, we can define effective
couplings for each transition amplitude and the total amplitude can be written
as a sum of all these amplitudes, i.e.

$\displaystyle i\mathcal{M}_{Total}$ $\displaystyle=$ $\displaystyle
i(g^{T}_{\eta_{X}\gamma V}+g^{L}_{\eta_{X}\gamma
V})\epsilon_{\alpha\beta\delta\lambda}p^{\alpha}_{\gamma}p^{\beta}_{V}\epsilon^{\delta}_{\gamma}\epsilon_{V}^{\lambda}\
,$ (64)

where $g^{T}_{\eta_{X}\gamma V}$ and $g^{L}_{\eta_{X}\gamma V}$ are the
effective couplings extracted from the tree and loop transitions,
respectively. In order to understand better the behavior of the loop
transitions, we introduce a switch $\delta_{C}$ to rewrite
$g^{L}_{\eta_{X}\gamma V}$ as

$g^{L}_{\eta_{X}\gamma V}\equiv g^{L}_{triangle}+\delta_{C}g^{L}_{contact}\ .$
(65)

With $\delta_{C}=1$ or 0, one is able to switch on/off the contributions from
the contact loop diagram in the calculation.

##  III Numerical results and discussions

The present experimental measurements of the exclusive decays of $\eta(1295)$
and $\eta(1405)$ are still far from satisfactory. In particular, the mass
degeneracy of $f_{1}(1285)$ and $f_{1}(1420)$ with $\eta(1295)$ and
$\eta(1405)$ has brought in a lot of challenges to the data analysis. At this
moment, the data from $J/\psi$ and $\psi(3686)$ decays provide the joint
branching ratios for the production and decay of these two states.

###  III.1 Partial decay widths for $\eta(1405)$ and $\eta(1295)\to\gamma V$

With the amplitudes and parameters provided in the previous Section, we can
directly calculate the radiative decays of these two states. In Tab. 9 the
exclusive contributions from the tree-level amplitudes are listed and compared
with the results with the loop contributions included. Two values for the cut-
off parameter $\alpha$ are adopted for the loop amplitudes to show the
sensitivities of the partial decay widths to the loop contributions. One can
see that the loop transitions have relatively small impact on the
$\eta_{X}\to\gamma\rho^{0}$ decay channels, but play a significantly role in
$\eta_{X}\to\gamma\omega$ and $\gamma\phi$. Such an interference effect can be
further investigated by the branching ratio fractions between two exclusive
decay channels in the next Subsection.

Table 9:  Partial decay widths of $\eta_{X}\to\gamma V$ (in unit of MeV) calculated by the exclusive tree-level amplitudes, and two values $\alpha=1$ and $2$ for the cut-off parameter are adopted as a comparison.  | Channels | Widths (tree amp.) | Widths (all) with $\alpha=1$ | Widths (all) with $\alpha=2$
---|---|---|---|---
$\eta(1295)$ | $\gamma\rho$ | $1.72$ | $1.76$ | $1.45$
$\gamma\omega$ | $0.15$ | $0.23$ | $0.38$
$\gamma\phi$ | $0.05$ | $0.09$ | $0.16$
$\eta(1405)$ | $\gamma\rho$ | $2.25$ | $2.26$ | $2.67$
$\gamma\omega$ | $0.19$ | $0.16$ | $0.2$
$\gamma\phi$ | $0.16$ | $0.24$ | $0.35$

We can also examine the exclusive contributions from each loop diagram in
order to clarify their roles in the interference with the tree-level
amplitude. The results are listed in Tab. 10 for two cut-off values, i.e.
$\alpha=1$ and $\alpha=2$. It shows that the triangle diagrams (Fig. 1 (b))
have much smaller contributions than the contact diagrams (Figs. 1 (c) and
(d)), and the decay channel $\eta(1295)\to\gamma\omega$ has the largest
contributions from the loop diagrams. This will lead to significant changes to
the the branching ratio fraction between the two channels of
$\eta(1295)\to\gamma\phi$ and $\gamma\omega$. In addition, we also see some
sensitivities of the loop contributions to the cut-off parameter $\alpha$. It
is slightly surprising but understandable that the loop transition corrections
are significant for $\eta(1295)$. Note that it has large couplings to
$K^{*}\bar{K}+c.c.$ in the mixing scheme which will strongly enhance the loop
amplitude. Moreover, the tree-level amplitude is suppressed by the
intermediate $\omega\to e^{+}e^{-}$ coupling in the VMD model.

Table 10:  Decay widths of each type of the hadronic loop diagrams with the cut-off parameter $\alpha=1$ and $2$. Diagrams | Decay channels (KeV) | $\alpha=1$ | $\alpha=2$
---|---|---|---
$\eta(1295)$ | $\eta(1405)$ | $\eta(1295)$ | $\eta(1405)$
$[K^{*},\bar{K},(K)]$ | $\gamma\rho$ | $0.37$ | $0.21$ | $4.09$ | $1.64$
$\gamma\omega$ | $0.03$ | $0.018$ | $0.35$ | $0.14$
$\gamma\phi$ | $0.26$ | $0.4$ | $1.23$ | $1.1$
$[K^{*},\bar{K},(K^{*})]$ | $\gamma\rho$ | $0.06$ | $0.03$ | $1.1$ | $0.38$
$\gamma\omega$ | $0.05$ | $0.024$ | $0.84$ | $0.3$
$\gamma\phi$ | $0.04$ | $0.03$ | $0.8$ | $0.4$
$[K^{*},\bar{K}^{*},(K)]$ | $\gamma\rho$ | $9.7\times 10^{-5}$ | $5.6\times 10^{-3}$ | $2.4\times 10^{-3}$ | $0.13$
$\gamma\omega$ | $8.6\times 10^{-6}$ | $5.0\times 10^{-4}$ | $2.1\times 10^{-4}$ | $0.012$
$\gamma\phi$ | $1.5\times 10^{-5}$ | $1.3\times 10^{-3}$ | $3.0\times 10^{-4}$ | $0.021$
$[K^{*},\bar{K}^{*},(K^{*})]$ | $\gamma\rho$ | $0.03$ | $1.29$ | $0.26$ | $10.5$
$\gamma\omega$ | $0.024$ | $1.03$ | $0.2$ | $8.4$
$\gamma\phi$ | $0.02$ | $1.37$ | $0.17$ | $10.4$
$[K,\bar{K}^{*},(K)]$ | $\gamma\rho$ | $0.19$ | $0.1$ | $2.6$ | $0.99$
$\gamma\omega$ | $0.15$ | $0.08$ | $2.1$ | $0.8$
$\gamma\phi$ | $0.2$ | $0.18$ | $2.4$ | $1.46$
$[K,\bar{K}^{*},(K^{*})]$ | $\gamma\rho$ | $0.23$ | $0.07$ | $5.4$ | $1.33$
$\gamma\omega$ | $0.02$ | $6.2\times 10^{-3}$ | $0.49$ | $0.12$
$\gamma\phi$ | $0.03$ | $0.012$ | $0.72$ | $0.25$
$[K^{*},\bar{K}]$ | $\gamma\rho$ | $25.1$ | $6.6$ | $188.2$ | $41.9$
$\gamma\omega$ | $23.8$ | $6.3$ | $178.7$ | $40.3$
$\gamma\phi$ | $15.9$ | $6.4$ | $119.1$ | $40.5$
All loops | $\gamma\rho$ | $24.9$ | $5.5$ | $193.8$ | $59.5$
$\gamma\omega$ | $25.7$ | $5.0$ | $179.4$ | $37.3$
$\gamma\phi$ | $15.68$ | $11.0$ | $105.6$ | $58.2$

###  III.2 Relative production rate between $\eta(1405)$ and $\eta(1295)$

In order to extract information about their internal structures, we define
several branching ratio fractions which can be directly compared with the
experimental data. For the production and decay of $\eta(1295)$ and
$\eta(1405)$ in the same channel, we define

$\displaystyle\mathcal{R}_{\rho}$ $\displaystyle\equiv$
$\displaystyle\frac{\Gamma_{J/\psi\to\gamma\eta(1405)\to\gamma\gamma\rho}}{\Gamma_{J/\psi\to\gamma\eta(1295)\to\gamma\gamma\rho}},$
$\displaystyle\mathcal{R}_{\omega}$ $\displaystyle\equiv$
$\displaystyle\frac{\Gamma_{J/\psi\to\gamma\eta(1405)\to\gamma\gamma\omega}}{\Gamma_{J/\psi\to\gamma\eta(1295)\to\gamma\gamma\omega}},$
$\displaystyle\mathcal{R}_{\phi}$ $\displaystyle\equiv$
$\displaystyle\frac{\Gamma_{J/\psi\to\gamma\eta(1405)\to\gamma\gamma\phi}}{\Gamma_{J/\psi\to\gamma\eta(1295)\to\gamma\gamma\phi}}.$
(66)

With the amplitudes given in the previous Section, we can first examine the
branching ratio fractions contributed by the tree diagram in Fig. 1. As an
example, the fraction $\mathcal{R}_{\rho}$ has the following expression from
the tree-level transitions:

$\displaystyle\mathcal{R}_{\rho}$ $\displaystyle=$
$\displaystyle\bigg{(}\frac{|\mathbf{p}_{\eta(1405)}|}{|\mathbf{p}_{\eta(1295)}|}\bigg{)}^{3}\bigg{(}\frac{|\mathbf{p}_{\rho}|}{|\mathbf{p}_{\rho}^{\prime}|}\bigg{)}^{3}\frac{m_{L}\Gamma_{L}}{m_{H}\Gamma_{H}}\bigg{(}\frac{\sqrt{2}\tan\alpha_{P}+R}{\sqrt{2}-R\tan\alpha_{P}}\bigg{)}^{2}\tan^{2}\alpha_{P}\
,$ (67)

where $\mathbf{p}_{\eta(1405)}$ and $\mathbf{p}_{\eta(1295)}$ denote the
three-vector momenta of $\eta(1405)$ and $\eta(1295)$ in the $J/\psi$ rest
frame, respectively; $\mathbf{p}_{\rho}$ and $\mathbf{p}_{\rho}^{\prime}$ are
the final state vector meson momenta in the rest frame of $\eta(1405)$ and
$\eta(1295)$, respectively. This ratio is likely to be larger than unity given
that $\alpha_{P}\simeq 42^{\circ}$ in the scenario of the first radial
excitations. Similarly, the ratios $\mathcal{R}_{\omega}$ and
$\mathcal{R}_{\phi}$ from the tree diagrams can be extracted.

In Tab. 11 these three branching ratio fractions from the tree diagrams are
listed. It shows that the combined branching ratio for $\eta(1405)$ is about
one order of magnitude larger than that for $\eta(1295)$. This is consistent
with the experimental observations that the signal for $\eta(1295)$ is
significantly suppressed in the $J/\psi$ radiative decays.

Table 11:  Numerical results of $\mathcal{R}_{\rho}$, $\mathcal{R}_{\omega}$ and $\mathcal{R}_{\phi}$ with SU(3) breaking factor $R=0.8$ and mixing angle $\alpha_{P}=42^{\circ}$. $\alpha=1$ | $\mathcal{R}_{\rho}$ | $\mathcal{R}_{\omega}$ | $\mathcal{R}_{\phi}$
---|---|---|---
Tree level | $10.8$ | $10.9$ | $20.8$
T+L, $\delta_{C}=0$ | $11.7$ | $10.6$ | $24.0$
T+L, $\delta_{C}=1$ | $9.2$ | $5.5$ | $16.3$

The inclusion of the loop processes introduce sizeable corrections to the
branching ratio fractions as we have learned earlier. In Tab. 11 the
calculation results of $\mathcal{R}_{\rho}$, $\mathcal{R}_{\omega}$ and
$\mathcal{R}_{\phi}$ with the loop contributions are also listed. The results
with $\delta_{C}=1$ and $\delta_{C}=0$ distinguish the situations whether or
not to include the contributions from Fig. 1 (d) in the loop amplitude,
respectively. Note again that Fig. 1 (c) always vanishes. It shows that the
inclusion of the contact loop diagram has led to significant corrections to
both $\mathcal{R}_{\omega}$ and $\mathcal{R}_{\phi}$. In contrast,
$\mathcal{R}_{\rho}$ appears to be a relatively stable quantity. This is due
to the relatively large contributions from the tree diagrams in the
$\gamma\rho$ channel for both $\eta(1405)$ and $\eta(1295)$. In particular,
the tree diagram contributions are dominant in $\eta(1405)\to\gamma\rho^{0}$.

###  III.3 Branching ratio fractions between different radiative decay
channels

For $\eta(1295)$ or $\eta(1405)$ radiative decays into different channels we
define the following branching ratio fractions:

$\displaystyle\mathcal{R}^{\rho/\phi}_{\eta(1405)}$ $\displaystyle\equiv$
$\displaystyle\frac{\Gamma_{\eta(1405)\to\gamma\rho}}{\Gamma_{\eta(1405)\to\gamma\phi}},$
$\displaystyle\mathcal{R}^{\phi/\omega}_{\eta(1405)}$ $\displaystyle\equiv$
$\displaystyle\frac{\Gamma_{\eta(1405)\to\gamma\phi}}{\Gamma_{\eta(1405)\to\gamma\omega}},$
$\displaystyle\mathcal{R}^{\rho/\phi}_{\eta(1295)}$ $\displaystyle\equiv$
$\displaystyle\frac{\Gamma_{\eta(1295)\to\gamma\rho}}{\Gamma_{\eta(1295)\to\gamma\phi}},$
$\displaystyle\mathcal{R}^{\omega/\phi}_{\eta(1295)}$ $\displaystyle\equiv$
$\displaystyle\frac{\Gamma_{\eta(1295)\to\gamma\omega}}{\Gamma_{\eta(1295)\to\gamma\phi}}\
.$ (68)

Note that the fractions in Eq. (III.3) is defined in such a way that the
values at leading order will be larger than unity if $\eta(1295)$ and
$\eta(1405)$ are the first radial excitations of $\eta$ and $\eta^{\prime}$,
respectively.

Supposing that only the tree-level amplitudes contribute in $\eta_{X}\to\gamma
V$, the branching ratio fractions defined in Eq. (III.3) would have simple
forms in terms of the mixing angle. As an example, the ratio between the
$\gamma\rho^{0}$ and $\gamma\phi$ decay channel can be written as:

$\displaystyle\mathcal{R}^{\rho/\phi}_{\eta(1405)}$ $\displaystyle\equiv$
$\displaystyle\frac{\Gamma_{\eta(1405)\to\gamma\rho}}{\Gamma_{\eta(1405)\to\gamma\phi}}=\left(\frac{|\mathbf{p}_{\rho}|}{|\mathbf{p}_{\phi}|}\right)^{3}\left[\frac{(em_{\rho}^{2}/f_{\rho})G_{\rho}}{(em_{\phi}^{2}/f_{\phi})G_{\phi}}\right]^{2}\frac{\tan^{2}\alpha_{P}}{2R^{2}}\
,$ (69)

where $\mathbf{p}_{\rho}$ and $\mathbf{p}_{\phi}$ are the three-vector momenta
of the final state $\rho^{0}$ and $\phi$ in the initial $\eta(1405)$ rest
frame. By including the loop amplitudes, the branching ratio fractions will
deviate from the above expectation.

As mentioned in the Introduction that the BESIII Collaboration recently
measured the radiative decay of $\eta(1405)\to\gamma\phi$ in
$J/\psi\to\gamma\gamma\phi$ Ablikim:2018hxj , we can thus calculate
$\mathcal{R}^{\rho/\phi}_{\eta(1405)}$ and compare it with the data. The
results are presented in Fig. 2 in terms of two parameters $\alpha_{P}$ and
$m_{\eta(1405)}$. These two variables are closely related to the
interpretation of these two pseudoscalars. Therefore, the dependence of
$\mathcal{R}^{\rho/\phi}_{\eta(1405)}$ on these two variables can illustrate
whether it is a reasonable picture to treat these two states as the first
radial excitation states.

#### $\eta(1405)\to\gamma V$

In Fig. 2 the two overlapping bands denote the ranges of the experimental
ratios extracted from the two solutions for
$J/\psi\to\gamma\eta(1405/1475)\to\gamma\gamma\rho^{0}$ at BESIII
Ablikim:2018hxj  as listed in Tab. 1. The central values, $7.53\pm 2.49$ and
$11.10\pm 3.50$, are denoted by the short-dashed and dotted lines,
respectively. Within the commonly adopted values for $\alpha_{P}$, i.e.
$\alpha_{P}\simeq 38^{\circ}\sim 44^{\circ}$ (Fig. 2 (a)), and within the mass
region of $m_{\eta(1405)}=1.405\sim 1.475$ GeV (Fig. 2 (b)) the ratios (solid
lines) are consistent with the data within the errors. We also plot the ratios
with only the tree amplitude (middle-dashed lines) and tree plus triangle loop
amplitudes (long-dashed lines) as a comparison. Again, the large discrepancies
between the solid and long-dashed lines indicate that the loop diagrams
involving the EM contact interactions (Fig. 1 (d)) provide the main
contributions from the loop transitions.

Figure 2:  (color online) (a) Dependence of
$\mathcal{R}^{\rho/\phi}_{\eta(1405)}$ on the mixing angle $\alpha_{P}$ with
$m_{\eta(1405)}$ fixed at $1.405$ GeV. (b) Dependence of
$\mathcal{R}^{\rho/\phi}_{\eta(1405)}$ on the mass of $\eta(1405)$ with the
mixing angle $\alpha_{P}$ set at $42^{\circ}$. In both figures the cut-off
parameter $\alpha=1$ is adopted; “T+L” denotes the full calculations including
the tree and loop contributions. $\delta_{C}$ is the coefficient of the
contact diagrams and the values of $\delta_{C}=1$ and 0 will modulate the
contributions from the contact loop diagrams. “ExpCent” denotes the central
value of the two solutions from the BESIII analysis of
$\mathcal{R}^{\rho/\phi}_{\eta(1405)}$, i.e. $11.10\pm 3.50$ and $7.53\pm
2.49$, respectively Ablikim:2018hxj . The light and dark grey bands indicate
the error bars of the two solutions.

Figure 3:  (a) The dependence of $\mathcal{R}^{\phi/\omega}_{\eta(1405)}$ on
the mixing angle $\alpha_{P}$ with $\alpha=1$ and $m_{\eta(1405)}=1.405$ GeV.
(b) The dependence of $\mathcal{R}^{\phi/\omega}_{\eta(1405)}$ on the mass of
$\eta(1405)$ with the mixing angle $\alpha_{P}=42^{\circ}$. In both figures
the cut-off parameter $\alpha=1$ is adopted. The legends of the curves are the
same as those in Fig. 2.

In Fig. 3 we present the predictions of $R^{\phi/\omega}_{\eta(1405)}$ for a
range of $\alpha_{P}$ (Fig. 3 (a)) and $m_{\eta(1405)}$ (Fig. 3 (b)),
respectively, similar to Fig. 2. The ratios turn out to be stable and these
two decays, i.e. $\eta(1405)\to\gamma\phi$ and $\gamma\omega$ are comparable
with each other. Again, we see that the full amplitude calculations (solid
lines) are significantly different from the results with only the tree diagram
contributions (short-dashed lines) or the tree plus the triangle loop
amplitudes (long-dashed lines). This also indicates the dominant role played
by the loop diagrams involving the EM contact interactions.

Figure 4: The dependence of $\mathcal{R}^{\rho/\phi}_{\eta(1405)}$ (Fig. (a))
and $\mathcal{R}^{\phi/\omega}_{\eta(1405)}$ (Fig. (b)) on the cut-off
parameter $\alpha$, respectively, with $\alpha_{P}=42^{\circ}$ and
$m_{\eta(1405)}=1.405$ GeV fixed in the calculations. The light and dark grey
bands in (a) indicate the two solutions from the BESIII analysis, $7.53\pm
2.49$ and $11.10\pm 3.50$, respectively Ablikim:2018hxj .

Another aspect to be examined is the cutoff dependence of the branching ratio
fractions. In Fig. 4 both ratios $R^{\rho/\phi}_{\eta(1405)}$ and
$R^{\phi/\omega}_{\eta(1405)}$ in terms of a range of the cut-off parameter
$\alpha$ is presented in (a) and (b), respectively. It shows that
$R^{\rho/\phi}_{\eta(1405)}$ is a stable quantity with the increasing
$\alpha$, although it drops slightly within the range of $\alpha=1\sim 2$. In
contrast, the ratio $R^{\phi/\omega}_{\eta(1405)}$ increase gradually in terms
of $\alpha$. This indicates the increasing contributions of the loop
amplitudes in $\eta(1405)\to\gamma\phi$ with the increasing $\alpha$.
Combining Fig. 4 (a) and (b) together, one can see that the relation
$\Gamma_{\eta(1405)\to\gamma\rho^{0}}>\Gamma_{\eta(1405)\to\gamma\phi}>\Gamma_{\eta(1405)\to\gamma\omega}$
which is consistent with the expectation of the first radial excitation
assignment Wu:2011yx ; Wu:2012pg .

#### $\eta(1295)\to\gamma V$

In Fig. 5 (a) the predicted ratio $R^{\rho/\phi}_{\eta(1295)}$ in terms of the
mixing angle $\alpha_{P}$ is plotted. The full calculation is denoted by the
solid line while as a comparison, we also include the results for the
exclusive tree-level transition (short-dashed line) and the tree-level
amplitude plus the triangle transitions (long-dashed line). The difference
between the long-dashed and solid lines indicates the significant
contributions from the loop diagrams involving the EM contact interactions
(see Fig. 1 (d)).

In Fig. 5 (b) the dependence of $R^{\rho/\phi}_{\eta(1295)}$ on the cut-off
parameter $\alpha$ is shown where the solid and dashed line denote the full
calculation and calculation without the the EM contact loop transitions,
respectively. Within the range of $\alpha=1\sim 2$, the ratios keep to be
stable although the ratios exhibit a decreasing tendency with the increasing
$\alpha$.

Figure 5:  (a) The dependence of $\mathcal{R}^{\rho/\phi}_{\eta(1295)}$ on the
mixing angle $\alpha_{P}$ with $\alpha=1$. The middle dashed line is given by
the tree-level transitions. (b) The dependence of
$\mathcal{R}^{\rho/\phi}_{\eta(1295)}$ on the cut-off parameter $\alpha$ with
$\alpha_{P}=42^{\circ}$. In both calculations the solid and long-dashed lines
denote the full calculations and calculations without the contact loop
contributions, respectively.

In Fig. 6 the ratio $R^{\omega/\phi}_{\eta(1295)}$ is calculated and presented
in the similar way as Fig. 5. We can see that the dependence of the ratios on
the mixing angle $\alpha_{P}$ (Fig. 6 (a)) is similar to
$R^{\rho/\phi}_{\eta(1295)}$. We also investigate the cut-off dependence of
the loop transition contributions and the results are presented in Fig. 6 (b).
It shows that the ratio without the EM contact loop contributions exhibits a
smooth increase with the increasing $\alpha$. However, when the tree and loop
amplitudes are properly included, one can see that the ratio keeps stable and
insensitive to $\alpha$.

Figure 6:  (a) The dependence of $R^{\omega/\phi}_{\eta(1295)}$ on the mixing
angle $\alpha_{P}$ with $\alpha=1$. (b) The dependence of
$R^{\omega/\phi}_{\eta(1295)}$ on parameter $\alpha$ with the mixing angle
$\alpha_{P}=42^{\circ}$. The curve legends are the same as Fig. 5.

###  III.4 Loop influence on the mixing angle

The above studies have shown the impact of the loop transitions on the
experimental observables defined in the productions and decays of $\eta(1295)$
and $\eta(1405)$ in $J/\psi\to\gamma\gamma V$. As a consequence, it implies
that the measured mixing angle between $\eta(1295)$ and $\eta(1405)$ may
possess different values in different processes if the detailed transition
mechanisms have not been properly included.

To illustrate this, we adopt Eq. (69) as an example. Without the loop
transitions Eq. (69) will define a mixing angle which can be extracted with
the experimental value of $\mathcal{R}^{\rho/\phi}_{\eta(1405)}$ as the input.
But if the loop transitions are included, the ratio should be expressed as

$\displaystyle\mathcal{R}^{\rho/\phi}_{\eta(1405)}$ $\displaystyle\equiv$
$\displaystyle\frac{\Gamma_{\eta(1405)\to\gamma\rho}}{\Gamma_{\eta(1405)\to\gamma\phi}}=\left(\frac{|\mathbf{p}_{\rho}|}{|\mathbf{p}_{\phi}|}\right)^{3}\left[\frac{g^{T}_{\eta(1405)\gamma\rho}(\alpha_{P})+g^{L}_{\eta(1405)\gamma\rho}(\alpha_{P})}{g^{T}_{\eta(1405)\gamma\phi}(\alpha_{P})+g^{L}_{\eta(1405)\gamma\phi}(\alpha_{P})}\right]^{2}\frac{\tan^{2}\tilde{\alpha}_{P}}{2R^{2}}\
.$ (70)

By matching it to the tree-level relation defined in Eq. (69), one has

$\displaystyle\left[\frac{(em_{\rho}^{2}/f_{\rho})G_{\rho}}{(em_{\phi}^{2}/f_{\phi})G_{\phi}}\right]^{2}\frac{\tan^{2}\tilde{\alpha}_{P}}{2R^{2}}$
$\displaystyle\equiv$
$\displaystyle\left[\frac{g^{T}_{\eta(1405)\gamma\rho}(\alpha_{P})+g^{L}_{\eta(1405)\gamma\rho}(\alpha_{P})}{g^{T}_{\eta(1405)\gamma\phi}(\alpha_{P})+g^{L}_{\eta(1405)\gamma\phi}(\alpha_{P})}\right]^{2}\
,$ (71)

and the “empirical” mixing angle $\tilde{\alpha}_{P}$ can be extracted. In
Table 12 we list the extracted values of $\tilde{\alpha}_{P}$ from the ratios
$R^{\rho/\phi}_{\eta(1405)}$ and $R^{\rho/\phi}_{\eta(1295)}$. With three
values for $\alpha_{P}$ in the calculations as an illustration, we see that
the extracted values for the mixing angle $\tilde{\alpha}_{P}$ are different
in the measurements of these two ratios. To some extent, the deviations of
$\tilde{\alpha}_{P}$ from the commonly adopted values could be acceptable in a
single channel. However, when put two channels together, such deviations
should be regarded as significant. Further experimental measurements of these
ratios would be able to clarify the role played by the loop transitions.

Table 12:  Extracting a new mixing angle $\tilde{\alpha}_{P}$ for the given mixing angle $\alpha_{P}$ with cut off parameter $\alpha=1$. $\alpha_{P}$ | $\tilde{\alpha}_{P}$
---|---
$R^{\rho/\phi}_{\eta(1295)}$ | $R^{\rho/\phi}_{\eta(1405)}$
$42^{\circ}$ | $49.2^{\circ}$ | $36.2^{\circ}$
$40^{\circ}$ | $46.7^{\circ}$ | $34.5^{\circ}$
$38^{\circ}$ | $44.1^{\circ}$ | $32.7^{\circ}$

##  IV Summary

In this work, based on the one state assumption for $\eta(1405)$ and
$\eta(1475)$, we systematically investigate the radiative decays of
$\eta(1295)$ and $\eta(1405)$ by treating them as the first radial excitations
of $\eta$ and $\eta^{\prime}$. In the framework of the VMD model, we include
the intermediate $\bar{K}K^{*}+c.c.$ meson loops as the leading correction to
the tree-level transition amplitudes for the $\eta(1295)$ and
$\eta(1405)\to\gamma V$. With only one parameter, i.e. the cut-off parameter
$\alpha$, we are able to to understand the production and decay behavior of
both states in the $J/\psi$ radiative decays. In particular, the radiative
decays of $\eta(1405)\to\gamma V$ can be described in agreement with the
BESIII measurement. It is interesting to note that the loop transitions can
produce significant effects in some decay channels. For instance, the loop
contributions in $\eta(1295)\to\gamma\omega$ are found to be compatible with
the tree-level contributions. This should not be surprising since the coupling
of $g_{\eta(1295)K^{*}K}$ in the loop amplitude is sizeable and the $\omega$
meson decay constant in the tree-level amplitude is much smaller than that of
the $\rho^{0}$ meson (see Tab. 2). We also note that, due to the interference
from the loop transition amplitudes, the production of $\eta(1295)$ in
$J/\psi\to\gamma\gamma\rho^{0}$ will be significantly enhanced. Experimental
study of $\eta(1295)$ in this channel is strongly recommended.

In the scenario of assigning $\eta(1295)$ and $\eta(1405)$ as the first radial
excitation states of $\eta$ and $\eta^{\prime}$, we show that the branching
ratio fractions between these two states in the same decay channel, or between
two exclusive decay channels for the same state, exhibit interesting patterns
when the intermediate $\bar{K}K^{*}+c.c.$ meson loops are properly included.
As a consequence of the loop corrections, if one tries to extract the mixing
angle from the radiative decay data without taking into account the loop
corrections, it may end up with values different from each other and different
from the commonly adopted one for the $\eta$ and $\eta^{\prime}$ mixing. We
also find that the contributions from the meson loops are relatively small in
$\eta(1405)\to\gamma\rho^{0}$. Thus, this channel will be dominated by the
tree-level transition and is ideal for extracting the mixing angle.

In brief, we find that the radiative decays of $J/\psi\to\gamma\gamma V$ can
serve as a probe for understanding the nature of $\eta(1295)$ and
$\eta(1405)$. With the large data sample of $J/\psi$ at BESIII, we can
disentangle the role played by the meson loop transitions and gain more
insights into the pseudoscalar meson spectrum.

###### Acknowledgements.

This work is supported, in part, by the National Natural Science Foundation of
China (Grant Nos. 11425525 and 11521505), DFG and NSFC funds to the Sino-
German CRC 110 “Symmetries and the Emergence of Structure in QCD” (NSFC Grant
No. 12070131001, DFG Project-ID 196253076), National Key Basic Research
Program of China under Contract No. 2020YFA0406300, and Strategic Priority
Research Program of Chinese Academy of Sciences (Grant No. XDB34030302).

## Appendix: Amplitudes of the hadronic loop diagrams

In this Section, we present the loop amplitudes for the convenience of
tracking the calculation details. For simplicity, we do not distinguish the
coupling constants at the hadronic vertices, but just denote them as $g_{i}$
with $i=1,2,3$.

  * •

$[K^{*},\bar{K},(K)]$

$\displaystyle i\mathcal{M}$ $\displaystyle=$ $\displaystyle
g_{1}g_{2}g_{3}\int\frac{d^{4}p_{1}}{(2\pi)^{4}}(p_{X}+p_{3})_{\sigma}\frac{(g^{\sigma\mu}-\frac{p_{1}^{\sigma}p_{1}^{\mu}}{p_{1}^{2}})}{p_{1}^{2}-m_{K^{*}}^{2}+i\epsilon}\epsilon_{\alpha\beta\delta\mu}p_{\gamma}^{\alpha}p_{1}^{\beta}\epsilon_{\gamma}^{\delta}\frac{(p_{2}-p_{3})_{\lambda}\epsilon_{V}^{\lambda}}{(p_{2}^{2}-m_{K}^{2}+i\epsilon)(p_{3}^{2}-m_{K}^{2}+i\epsilon)}\mathcal{F}(q_{i}^{2})$
(72) $\displaystyle=$ $\displaystyle
g_{1}g_{2}g_{3}\int\frac{d^{4}p_{1}}{(2\pi)^{4}}\frac{\epsilon_{\alpha\beta\delta\mu}p^{\alpha}_{\gamma}p_{1}^{\beta}\epsilon^{\delta}_{\gamma}(2p_{\gamma}+2p_{V}-p_{1})^{\mu}(2p_{1}-2p_{\gamma}-p_{V})_{\lambda}\epsilon^{\lambda}_{V}}{(p_{1}^{2}-m_{K^{*}}^{2}+i\epsilon)(p_{2}^{2}-m_{K}^{2}+i\epsilon)(p_{3}^{2}-m_{K}^{2}+i\epsilon)}\mathcal{F}(q_{i}^{2})$
$\displaystyle=$ $\displaystyle
2g_{1}g_{2}g_{3}\epsilon_{\alpha\beta\delta\mu}p_{\gamma}^{\alpha}p_{V}^{\mu}\epsilon_{\gamma}^{\delta}\epsilon_{V}^{\lambda}\int\frac{d^{4}p_{1}}{(2\pi)^{4}}\frac{p_{1}^{\beta}[2(p_{1})_{\lambda}-(2p_{\gamma}+p_{V})_{\lambda}]}{(p_{1}^{2}-m_{K^{*}}^{2}+i\epsilon)(p_{3}^{2}-m_{K}^{2}+i\epsilon)(p_{2}^{2}-m_{K}^{2}+i\epsilon)}\mathcal{F}(q_{i}^{2})$
$\displaystyle=$ $\displaystyle
2g_{1}g_{2}g_{3}\epsilon_{\alpha\beta\delta\mu}p_{\gamma}^{\alpha}p_{V}^{\mu}\epsilon_{\gamma}^{\delta}\epsilon_{V}^{\lambda}\int\frac{d^{4}p_{1}}{(2\pi)^{4}}\frac{2p_{1}^{\beta}p_{1\lambda}}{(p_{1}^{2}-m_{K^{*}}^{2}+i\epsilon)(p_{3}^{2}-m_{K}^{2}+i\epsilon)(p_{2}^{2}-m_{K}^{2}+i\epsilon)}\mathcal{F}(q_{i}^{2}).$

Note that the due to the property of the antisymmetric tensor, only the
$\delta^{\beta}_{\lambda}$ term could survive in the tensor integral.

  * •

$[K^{*},\bar{K},(K^{*})]$

$\displaystyle i\mathcal{M}$ $\displaystyle=$
$\displaystyle\int\frac{d^{4}p_{1}}{(2\pi)^{4}}\frac{g_{1}g_{2}g_{3}(p_{X}+p_{3})_{\sigma}(g^{\sigma\mu}-\frac{p^{\sigma}_{1}p^{\mu}_{1}}{p^{2}_{1}})(g^{\rho\nu}-\frac{p^{\rho}_{2}p^{\nu}_{2}}{p^{2}_{2}})\epsilon_{\alpha\beta\nu\lambda}p^{\alpha}_{2}p^{\beta}_{V}\epsilon^{\lambda}_{V}}{(p^{2}_{1}-m^{2}_{K^{*}})(p^{2}_{2}-m^{2}_{K^{*}})(p_{3}^{2}-m^{2}_{K})}\times[\epsilon^{\delta}_{\gamma}g_{\delta\mu}p_{1\rho}+\epsilon^{\delta}_{\gamma}g_{\delta\rho}p_{2\mu}-g_{\mu\rho}(p_{1}+p_{2})_{\delta}\epsilon^{\delta}_{\gamma}]\mathcal{F}(p_{i}^{2})$
(73) $\displaystyle=$ $\displaystyle
g_{1}g_{2}g_{3}\int\frac{d^{4}p_{1}}{(2\pi)^{4}}\frac{-2}{p^{2}_{1}(p^{2}_{1}-m^{2}_{K^{*}})(p^{2}_{2}-m^{2}_{K^{*}})(p_{3}^{2}-m^{2}_{K})}$
$\displaystyle\times\left\\{\epsilon_{\alpha\beta\nu\lambda}p_{\gamma}^{\alpha}p_{V}^{\beta}\epsilon_{\gamma}^{\nu}\epsilon_{V}^{\lambda}[(p_{1}\cdot
p_{\gamma})^{2}-p_{1}^{2}(p_{V}\cdot p_{\gamma})+(p_{1}\cdot
p_{\gamma})(p_{1}\cdot p_{V})]\right.$
$\displaystyle\left.-\epsilon_{\alpha\beta\nu\lambda}p_{1}^{\alpha}p_{V}^{\beta}\epsilon_{\gamma}^{\nu}\epsilon_{V}^{\lambda}[(p_{1}\cdot
p_{\gamma})^{2}-p_{1}^{2}(p_{V}\cdot p_{\gamma})+(p_{1}\cdot
p_{\gamma})(p_{1}\cdot p_{V})]\right.$
$\displaystyle-\left.\epsilon_{\alpha\beta\nu\lambda}p_{\gamma}^{\alpha}p_{V}^{\beta}p_{1}^{\nu}\epsilon_{V}^{\lambda}[(p_{1}\cdot\epsilon_{\gamma}-p_{\gamma}\cdot\epsilon_{\gamma})(2p_{1}^{2}-p_{1}\cdot
p_{\gamma}-p_{1}\cdot
p_{V})-p_{1}^{2}(p_{V}\cdot\epsilon_{\gamma})]\right\\}\mathcal{F}(p_{i}^{2}).$

In the above integral, after contracting the Lorentz indices the amplitude can
be simplified to a more compact form. One notices that the first term in the
big bracket is a scalar integral, while the second and third term will pick up
the linear terms containing $p_{\gamma}^{\alpha}$ and
$\epsilon_{\gamma}^{\nu}$, respectively, due to the property of the
antisymmetric tensor. The same analysis is also applied to the following loop
amplitudes.

  * •

$[K^{*},\bar{K}^{*},(K)]$

$\displaystyle i\mathcal{M}$ $\displaystyle=$ $\displaystyle
g_{1}g_{2}g_{3}\int\frac{d^{4}p_{1}}{(2\pi)^{4}}\frac{\epsilon_{\alpha\beta\mu\nu}p^{\alpha}_{1}p^{\beta}_{3}(g^{\mu\mu^{\prime}}-\frac{p^{\mu}_{1}p^{\mu^{\prime}}_{1}}{p^{2}_{1}})\epsilon_{\alpha_{1}\beta_{1}\mu^{\prime}\delta}p^{\alpha_{1}}_{1}p^{\beta_{1}}_{\gamma}\epsilon^{\delta}_{\gamma}\epsilon_{\alpha_{2}\beta_{2}\nu^{\prime}\lambda}p^{\alpha_{2}}_{3}p^{\beta_{2}}_{V}\epsilon^{\lambda}_{V}(g^{\nu\nu^{\prime}}-\frac{p^{\nu}_{3}p^{\nu^{\prime}}_{3}}{p^{2}_{3}})}{(p^{2}_{1}-m^{2}_{K^{*}})(p^{2}_{2}-m^{2}_{K})(p^{2}_{3}-m^{2}_{K^{*}})}\mathcal{F}(q^{2}_{i})$
(74) $\displaystyle=$ $\displaystyle
g_{1}g_{2}g_{3}\int\frac{d^{4}p_{1}}{(2\pi)^{4}}\frac{\epsilon_{\alpha\beta\mu\nu}p^{\alpha}_{1}p^{\beta}_{3}\times\epsilon_{\alpha_{1}\beta_{1}\mu\delta}p^{\alpha_{1}}_{1}p^{\beta_{1}}_{\gamma}\epsilon^{\delta}_{\gamma}\times\epsilon_{\alpha_{2}\beta_{2}\nu\lambda}p^{\alpha_{2}}_{3}p^{\beta_{2}}_{V}\epsilon^{\lambda}_{V}}{(p^{2}_{1}-m^{2}_{K^{*}})(p^{2}_{2}-m^{2}_{K})(p^{2}_{3}-m^{2}_{K^{*}})}\mathcal{F}(q^{2}_{i})$
$\displaystyle=$ $\displaystyle
g_{1}g_{2}g_{3}\int\frac{d^{4}p_{1}}{(2\pi)^{4}}\frac{\mathcal{F}(q^{2}_{i})}{(p^{2}_{1}-m^{2}_{K^{*}})(p^{2}_{2}-m^{2}_{K})(p^{2}_{3}-m^{2}_{K^{*}})}\times\left\\{\epsilon_{\alpha\beta\delta\lambda}p_{\gamma}^{\alpha}p_{V}^{\beta}\epsilon_{\gamma}^{\delta}\epsilon_{V}^{\lambda}[(p_{1}\cdot
p_{\gamma})^{2}+(p_{1}\cdot p_{\gamma})(p_{1}\cdot
p_{\gamma})-p_{1}^{2}(p_{V}\cdot p_{\gamma})]\right.$
$\displaystyle+\epsilon_{\alpha\beta\delta\lambda}p_{1}^{\alpha}p_{V}^{\beta}\epsilon_{\gamma}^{\delta}\epsilon_{V}^{\lambda}[-(p_{1}\cdot
p_{\gamma})^{2}]+\epsilon_{\alpha\beta\delta\lambda}p_{\gamma}^{\alpha}p_{1}^{\beta}\epsilon_{\gamma}^{\delta}\epsilon_{V}^{\lambda}[(p_{1}\cdot
p_{V})^{2}-p_{1}^{2}p_{V}^{2}]$
$\displaystyle-\epsilon_{\alpha\beta\delta\lambda}p_{\gamma}^{\alpha}p_{V}^{\beta}p_{1}^{\delta}\epsilon_{V}^{\lambda}[(p_{1}\cdot
p_{\gamma})(p_{1}\cdot\epsilon_{\gamma})-(p_{V}\cdot
p_{\gamma})(p_{1}\cdot\epsilon_{\gamma})+(p_{1}\cdot
p_{\gamma})(p_{V}\cdot\epsilon_{\gamma})]$
$\displaystyle\left.+\epsilon_{\alpha\beta\delta\lambda}p_{\gamma}^{\alpha}p_{V}^{\beta}\epsilon_{\gamma}^{\delta}p_{1}^{\lambda}[(p_{1}\cdot
p_{V})(p_{1}\cdot\epsilon_{V})]\right\\}$

  * •

$[K^{*},\bar{K}^{*},(K^{*})]$

$\displaystyle i\mathcal{M}$ $\displaystyle=$ $\displaystyle
g_{1}g_{2}g_{3}\int\frac{d^{4}p_{1}}{(2\pi)^{4}}\frac{\epsilon_{\alpha\beta\mu\nu}p^{\alpha}_{1}p^{\beta}_{3}(g^{\mu{\mu^{\prime}}}-\frac{p^{\mu}_{1}p^{\mu^{\prime}}_{1}}{p^{2}_{1}})(g^{\rho\sigma}-\frac{p^{\rho}_{2}p^{\sigma}_{2}}{p^{2}_{2}})(g^{\nu{\nu^{\prime}}}-\frac{p^{\nu}_{3}p^{\nu^{\prime}}_{3}}{p^{2}_{3}})}{(p^{2}_{1}-m^{2}_{K^{*}})(p^{2}_{2}-m^{2}_{K^{*}})(p^{2}_{3}-m^{2}_{K^{*}})}$
(75)
$\displaystyle\times\bigg{[}\epsilon^{\delta}_{\gamma}g_{\delta\mu^{\prime}}p_{1\rho}+\epsilon^{\delta}_{\gamma}g_{\delta\rho}p_{2\mu^{\prime}}-g_{\mu^{\prime}\rho}(p_{1}+p_{2})_{\delta}\epsilon^{\delta}_{\gamma}\bigg{]}\times\bigg{[}\epsilon^{\lambda}_{V}g_{\lambda\nu^{\prime}}p_{3\sigma}-\epsilon^{\lambda}_{V}g_{\lambda\sigma}p_{2\nu^{\prime}}+g_{\nu^{\prime}\sigma}(p_{2}-p_{3})_{\lambda}\epsilon^{\lambda}_{V}\bigg{]}\mathcal{F}(q^{2}_{i}),$
$\displaystyle=$ $\displaystyle
g_{1}g_{2}g_{3}\int\frac{d^{4}p_{1}}{(2\pi)^{4}}\frac{\epsilon_{\alpha\beta\mu\nu}p^{\alpha}_{1}p^{\beta}_{3}(g^{\rho\sigma}-\frac{p^{\rho}_{2}p^{\sigma}_{2}}{p^{2}_{2}})}{(p^{2}_{1}-m^{2}_{K^{*}})(p^{2}_{2}-m^{2}_{K^{*}})(p^{2}_{3}-m^{2}_{K^{*}})}$
$\displaystyle\times\bigg{[}\epsilon^{\delta}_{\gamma}g_{\delta\mu}p_{1\rho}+\epsilon^{\delta}_{\gamma}g_{\delta\rho}p_{2\mu}-g_{\mu\rho}(p_{1}+p_{2})_{\delta}\epsilon^{\delta}_{\gamma}\bigg{]}\times\bigg{[}\epsilon^{\lambda}_{V}g_{\lambda\nu}p_{3\sigma}-\epsilon^{\lambda}_{V}g_{\lambda\sigma}p_{2\nu}+g_{\nu\sigma}(p_{2}-p_{3})_{\lambda}\epsilon^{\lambda}_{V}\bigg{]}\mathcal{F}(q^{2}_{i}),$
$\displaystyle=$ $\displaystyle
g_{1}g_{2}g_{3}\int\frac{d^{4}p_{1}}{(2\pi)^{4}}\frac{\mathcal{F}(q^{2}_{i})}{p_{2}^{2}(p^{2}_{1}-m^{2}_{K^{*}})(p^{2}_{2}-m^{2}_{K^{*}})(p^{2}_{3}-m^{2}_{K^{*}})}$
$\displaystyle\times\left\\{-\epsilon_{\alpha\beta\delta\lambda}p_{1}^{\alpha}p_{V}^{\beta}\epsilon_{\gamma}^{\delta}\epsilon_{V}^{\lambda}[p_{2}^{2}(-p_{1}\cdot
p_{\gamma}-p_{1}\cdot p_{V}+p_{1}^{2})+(p_{1}\cdot
p_{\gamma}-p_{1}^{2})(-2(p_{1}\cdot p_{\gamma})+p_{V}\cdot
p_{\gamma}-p_{1}\cdot p_{V}+p_{1}^{2})]\right.$
$\displaystyle+\epsilon_{\alpha\beta\delta\lambda}p_{\gamma}^{\alpha}p_{1}^{\beta}\epsilon_{\gamma}^{\delta}\epsilon_{V}^{\lambda}[p_{2}^{2}(-p_{1}\cdot
p_{\gamma}-p_{1}\cdot p_{V}+p_{1}^{2})+(p_{1}\cdot
p_{\gamma}-p_{1}^{2})(-2(p_{1}\cdot p_{\gamma})+p_{V}\cdot
p_{\gamma}-p_{1}\cdot p_{V}+p_{1}^{2})]$
$\displaystyle-\epsilon_{\alpha\beta\delta\lambda}p_{\gamma}^{\alpha}p_{V}^{\beta}p_{1}^{\delta}\epsilon_{V}^{\lambda}[2p_{2}^{2}(p_{1}\cdot\epsilon_{\gamma})+p_{2}^{2}(p_{1}\cdot\epsilon_{\gamma}-p_{V}\cdot\epsilon_{\gamma})+(p_{1}\cdot\epsilon_{\gamma})(-2(p_{1}\cdot
p_{\gamma})+p_{V}\cdot p_{\gamma}-p_{1}\cdot p_{V}+p_{1}^{2})]$
$\displaystyle+\epsilon_{\alpha\beta\delta\lambda}p_{\gamma}^{\alpha}p_{V}^{\beta}\epsilon_{\gamma}^{\delta}p_{1}^{\lambda}[2p_{2}^{2}(p_{\gamma}\cdot\epsilon_{V})-3p_{2}^{2}(p_{1}\cdot\epsilon_{V})-2(p_{1}^{2}-p_{1}\cdot
p_{\gamma})(p_{1}\cdot\epsilon_{V})$
$\displaystyle\left.+(p_{1}^{2}-p_{1}\cdot
p_{\gamma})(p_{1}\cdot\epsilon_{V}-p_{\gamma}\cdot\epsilon_{V})+2(p_{1}^{2}-p_{1}\cdot
p_{\gamma})(p_{\gamma}\cdot\epsilon_{V})]\right\\}$

  * •

$[K,\bar{K}^{*},(K)]$

$\displaystyle i\mathcal{M}$ $\displaystyle=$
$\displaystyle\int\frac{d^{4}p_{1}}{(2\pi)^{4}}g_{1}g_{2}g_{3}\frac{(p_{X}+p_{1})_{\mu}(p_{1}+p_{2})_{\delta}\epsilon^{\delta}_{\gamma}\epsilon_{\alpha\beta\nu\lambda}p^{\alpha}_{3}p^{\beta}_{V}\epsilon^{\lambda}_{V}(g^{\mu\nu}-\frac{p^{\mu}_{3}p^{\nu}_{3}}{p^{2}_{3}})}{(p^{2}_{1}-m^{2}_{K})(p^{2}_{2}-m^{2}_{K})(p^{2}_{3}-m^{2}_{K^{*}})}\mathcal{F}(q^{2}_{i})$
(76) $\displaystyle=$ $\displaystyle
g_{1}g_{2}g_{3}\int\frac{d^{4}p_{1}}{(2\pi)^{4}}\frac{\epsilon_{\alpha\beta\mu\lambda}p^{\alpha}_{3}p^{\beta}_{V}\epsilon^{\lambda}_{V}(2p_{1}+p_{3})^{\mu}(2p_{1}-p_{\gamma})_{\delta}\epsilon^{\delta}_{\gamma}}{(p^{2}_{1}-m^{2}_{K})(p^{2}_{2}-m^{2}_{K})(p^{2}_{3}-m^{2}_{K^{*}})}\mathcal{F}(q^{2}_{i})$
$\displaystyle=$ $\displaystyle
g_{1}g_{2}g_{3}\int\frac{d^{4}p_{1}}{(2\pi)^{4}}\frac{\epsilon_{\alpha\beta\mu\lambda}p^{\alpha}_{3}p^{\beta}_{V}\epsilon^{\lambda}_{V}(2p_{1})^{\mu}(2p_{1})_{\delta}\epsilon^{\delta}_{\gamma}}{(p^{2}_{1}-m^{2}_{K})(p^{2}_{2}-m^{2}_{K})(p^{2}_{3}-m^{2}_{K^{*}})}\mathcal{F}(q^{2}_{i})$
$\displaystyle=$ $\displaystyle
g_{1}g_{2}g_{3}\int\frac{d^{4}p_{1}}{(2\pi)^{4}}\frac{\epsilon_{\alpha\beta\mu\lambda}(p_{\gamma}+p_{V}-p_{1})^{\alpha}p^{\beta}_{V}\epsilon^{\lambda}_{V}(2p_{1})^{\mu}(2p_{1})_{\delta}\epsilon^{\delta}_{\gamma}}{(p^{2}_{1}-m^{2}_{K})(p^{2}_{2}-m^{2}_{K})(p^{2}_{3}-m^{2}_{K^{*}})}\mathcal{F}(q^{2}_{i})$
$\displaystyle=$ $\displaystyle
4g_{1}g_{2}g_{3}\epsilon_{\alpha\beta\mu\lambda}p^{\beta}_{V}\epsilon^{\lambda}_{V}\epsilon^{\delta}_{\gamma}\int\frac{d^{4}p_{1}}{(2\pi)^{4}}\frac{p_{\gamma}^{\alpha}p_{1}^{\mu}p_{1\delta}-p_{1}^{\alpha}p_{1}^{\mu}p_{1}^{\delta}}{(p^{2}_{1}-m^{2}_{K})(p^{2}_{2}-m^{2}_{K})(p^{2}_{3}-m^{2}_{K^{*}})}\mathcal{F}(q^{2}_{i})$
$\displaystyle=$ $\displaystyle
4g_{1}g_{2}g_{3}\epsilon_{\alpha\beta\mu\lambda}p_{\gamma}^{\alpha}p^{\beta}_{V}\epsilon^{\lambda}_{V}\epsilon^{\delta}_{\gamma}\int\frac{d^{4}p_{1}}{(2\pi)^{4}}\frac{p_{1}^{\mu}p_{1\delta}}{(p^{2}_{1}-m^{2}_{K})(p^{2}_{2}-m^{2}_{K})(p^{2}_{3}-m^{2}_{K^{*}})}\mathcal{F}(q^{2}_{i})\
.$

  * •

$[K,\bar{K}^{*},(K^{*})]$

$\displaystyle i\mathcal{M}$ $\displaystyle=$ $\displaystyle
g_{1}g_{2}g_{3}\int\frac{d^{4}p_{1}}{(2\pi)^{4}}\frac{(p_{X}+p_{1})_{\mu}\epsilon_{\alpha\beta\nu\delta}p^{\alpha}_{2}p^{\beta}_{\gamma}\epsilon^{\delta}_{\gamma}(g^{\nu\nu^{\prime}}-\frac{p^{\nu}_{2}p^{\nu^{\prime}}_{2}}{p^{2}_{2}})(g^{\mu^{\prime}\mu}-\frac{p^{\mu^{\prime}}_{3}p^{\mu}_{3}}{p^{2}_{3}})}{(p^{2}_{1}-m^{2}_{K})(p^{2}_{2}-m^{2}_{K^{*}})(p^{2}_{3}-m^{2}_{K^{*}})}$
(77)
$\displaystyle\times[\epsilon^{\lambda}_{V}g_{\lambda\mu^{\prime}}p_{3\nu^{\prime}}-\epsilon^{\lambda}_{V}g_{\lambda\nu^{\prime}}p_{2\mu^{\prime}}+g_{\mu^{\prime}\nu^{\prime}}(p_{2}-p_{3})_{\lambda}\epsilon^{\lambda}_{V}]\mathcal{F}(q^{2}_{i}),$
$\displaystyle=$ $\displaystyle
g_{1}g_{2}g_{3}\int\frac{d^{4}p_{1}}{(2\pi)^{4}}\frac{(p_{X}+p_{1})_{\mu}\epsilon_{\alpha\beta\nu\delta}p^{\alpha}_{2}p^{\beta}_{\gamma}\epsilon^{\delta}_{\gamma}(g^{\mu^{\prime}\mu}-\frac{p^{\mu^{\prime}}_{3}p^{\mu}_{3}}{p^{2}_{3}})}{(p^{2}_{1}-m^{2}_{K})(p^{2}_{2}-m^{2}_{K^{*}})(p^{2}_{3}-m^{2}_{K^{*}})}\times[\epsilon^{\lambda}_{V}g_{\lambda\mu^{\prime}}p_{3\nu}-\epsilon^{\lambda}_{V}g_{\lambda\nu}p_{2\mu^{\prime}}+g_{\mu^{\prime}\nu}(p_{2}-p_{3})_{\lambda}\epsilon^{\lambda}_{V}]\mathcal{F}(q^{2}_{i}),$
$\displaystyle=$ $\displaystyle
g_{1}g_{2}g_{3}\int\frac{d^{4}p_{1}}{(2\pi)^{4}}\frac{\mathcal{F}(q^{2}_{i})}{p_{3}^{2}(p^{2}_{1}-m^{2}_{K})(p^{2}_{2}-m^{2}_{K^{*}})(p^{2}_{3}-m^{2}_{K^{*}})}$
$\displaystyle\times\left\\{-\epsilon_{\alpha\beta\delta\lambda}p_{\gamma}^{\alpha}p_{1}^{\beta}\epsilon_{\gamma}^{\delta}\epsilon_{V}^{\lambda}[2(p_{\gamma}\cdot
p_{V})^{2}+2p_{1}^{2}\left(p_{1}\cdot
p_{\gamma}\right)-p^{2}_{3}\left(p_{\gamma}\cdot
p_{V}\right)+p^{2}_{3}\left(p_{1}\cdot
p_{V}\right)+p^{2}_{3}p_{1}^{2}+p_{1}^{2}\left(p_{\gamma}\cdot
p_{V}\right)-2p_{V}^{2}\left(p_{1}\cdot p_{\gamma}\right)\right.$
$\displaystyle-2\left(p_{1}\cdot p_{V}\right)\left(p_{\gamma}\cdot
p_{V}\right)-4\left(p_{1}\cdot p_{\gamma}\right)\left(p_{\gamma}\cdot
p_{V}\right)+p_{V}^{2}\left(p_{\gamma}\cdot
p_{V}\right)+p_{1}^{2}p_{V}^{2}+p_{1}^{2}\left(p_{1}\cdot
p_{V}\right)-p_{V}^{2}\left(p_{1}\cdot p_{V}\right)-p_{1}^{4}]$
$\displaystyle-\epsilon_{\alpha\beta\delta\lambda}p_{\gamma}^{\alpha}p_{V}^{\beta}\epsilon_{\gamma}^{\delta}p_{1}^{\lambda}[-p^{2}_{3}\left(p_{\gamma}\cdot\epsilon_{V}\right)+3p^{2}_{3}\left(p_{1}\cdot\epsilon_{V}\right)-p_{1}^{2}\left(p_{\gamma}\cdot\epsilon_{V}\right)-2\left(p_{\gamma}\cdot
p_{V}\right)\left(p_{1}\cdot\epsilon_{V}\right)+p_{V}^{2}\left(p_{\gamma}\cdot\epsilon_{V}\right)$
$\displaystyle\left.+2\left(p_{\gamma}\cdot
p_{V}\right)\left(p_{\gamma}\cdot\epsilon_{V}\right)+p_{1}^{2}\left(p_{1}\cdot\epsilon_{V}\right)-p_{V}^{2}\left(p_{1}\cdot\epsilon_{V}\right)]\right\\}.$

In the above amplitudes the product of the propagators and the form factor can
be expanded as

$\displaystyle\frac{\mathcal{F}(q_{i}^{2})}{D_{1}D_{2}D_{3}}$ $\displaystyle=$
$\displaystyle\frac{1}{(p_{1}^{2}-m_{1}^{2})(p_{2}^{2}-m_{2}^{2})(p_{3}^{3}-m_{3}^{3})}\bigg{(}\frac{m_{1}^{2}-\Lambda_{1}^{2}}{p_{1}^{2}-\Lambda_{1}^{2}}\bigg{)}\bigg{(}\frac{m_{2}^{2}-\Lambda_{2}^{2}}{p_{2}^{2}-\Lambda_{2}^{2}}\bigg{)}\bigg{(}\frac{m_{3}^{2}-\Lambda_{3}^{2}}{p_{3}^{2}-\Lambda_{3}^{2}}\bigg{)}$
(78) $\displaystyle=$
$\displaystyle\bigg{(}\frac{1}{p_{1}^{2}-m_{1}^{2}}-\frac{1}{p_{1}^{2}-\Lambda_{1}^{2}}\bigg{)}\bigg{(}\frac{1}{p_{2}^{2}-m_{2}^{2}}-\frac{1}{p_{2}^{2}-\Lambda_{2}^{2}}\bigg{)}\bigg{(}\frac{1}{p_{3}^{2}-m_{3}^{2}}-\frac{1}{p_{3}^{2}-\Lambda_{3}^{2}}\bigg{)}$
$\displaystyle=$
$\displaystyle\frac{1}{(p_{1}^{2}-m_{1}^{2})(p_{2}^{2}-m_{2}^{2})(p_{3}^{2}-m_{3}^{2})}-\frac{1}{(p_{1}^{2}-\Lambda_{1}^{2})(p_{2}^{2}-m_{2}^{2})(p_{3}^{3}-m_{3}^{3})}-\frac{1}{(p_{1}^{2}-m_{1}^{2})(p_{2}^{2}-\Lambda_{2}^{2})(p_{3}^{2}-m_{3}^{2})}$
$\displaystyle-$
$\displaystyle\frac{1}{(p_{1}^{2}-m_{1}^{2})(p_{2}^{2}-m_{2}^{2})(p_{3}^{2}-\Lambda_{3}^{2})}+\frac{1}{(p_{1}^{2}-\Lambda_{1}^{2})(p_{2}^{2}-\Lambda_{2}^{2})(p_{3}^{2}-m_{3}^{2})}+\frac{1}{(p_{1}^{2}-\Lambda_{1}^{2})(p_{2}^{2}-m_{2}^{2})(p_{3}^{2}-\Lambda_{3}^{2})}$
$\displaystyle+$
$\displaystyle\frac{1}{(p_{1}^{2}-m_{1}^{2})(p_{2}^{2}-\Lambda_{2}^{2})(p_{3}^{2}-\Lambda_{3}^{2})}-\frac{1}{(p_{1}^{2}-\Lambda_{1}^{2})(p_{2}^{2}-\Lambda_{2}^{2})(p_{3}^{2}-\Lambda_{3}^{2})}\
.$

  * •

$[K^{*},\bar{K}]$

$\displaystyle i\mathcal{M}$ $\displaystyle=$ $\displaystyle
eg_{\eta_{X}K^{*+}K^{-}}g_{VK^{*+}K^{-}}\int\frac{d^{4}p_{1}}{(2\pi)^{4}}\frac{(p_{X}+p_{2})_{\mu}(g^{\mu\alpha}-\frac{p^{\mu}_{1}p^{\alpha}_{1}}{p^{2}_{1}})\epsilon_{\delta\beta\alpha\lambda}\epsilon^{\delta}_{\gamma}p^{\beta}_{V}\epsilon^{\lambda}_{V}}{(p^{2}_{1}-m^{2}_{K^{*}})(p^{2}_{2}-m^{2}_{K})}\mathcal{F}(q^{2}_{i}).$
(79) $\displaystyle=$ $\displaystyle
eg_{1}g_{2}\int\frac{d^{4}p_{1}}{(2\pi)^{4}}\frac{\epsilon_{\delta\beta\alpha\lambda}\epsilon^{\delta}_{\gamma}p^{\beta}_{V}\epsilon^{\lambda}_{V}(p_{X}+p_{2})_{\mu}(g^{\mu\alpha}-\frac{p^{\mu}_{1}p^{\alpha}_{1}}{p^{2}_{1}})}{(p^{2}_{1}-m^{2}_{K^{*}})(p^{2}_{2}-m^{2}_{K})}\mathcal{F}(q^{2}_{i}).$
$\displaystyle=$ $\displaystyle
eg_{1}g_{2}\int\frac{d^{4}p_{1}}{(2\pi)^{4}}\left\\{\epsilon_{\alpha\beta\delta\lambda}p_{1}^{\alpha}p_{V}^{\beta}\epsilon_{\gamma}^{\delta}\epsilon_{V}^{\lambda}\frac{2(p_{1}\cdot
p_{\gamma}+p_{1}\cdot
p_{V})}{p_{1}^{2}(p^{2}_{1}-m^{2}_{K^{*}})(p^{2}_{2}-m^{2}_{K})}-\frac{2\epsilon_{\alpha\beta\delta\lambda}p_{\gamma}^{\alpha}p_{V}^{\beta}\epsilon_{\gamma}^{\delta}\epsilon_{V}^{\lambda}}{(p^{2}_{1}-m^{2}_{K^{*}})(p^{2}_{2}-m^{2}_{K})}\right\\}\mathcal{F}(q^{2}_{i}).$

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