# Symmetry analysis involving meson mixing for charmonium decay

X.H.Mo1,2
1 Institute of High Energy Physics, CAS, Beijing 100049, China
2 University of Chinese Academy of Sciences, Beijing 100049, China
<EMAIL_ADDRESS>
###### Abstract

In the light of $SU(3)$ flavor symmetry, the effective interaction Hamiltonian
in tensor form is obtained by virtue of group representation theory. The
strong and electromagnetic breaking effects are treated as a spurion octet so
that the flavor singlet principle can be utilized as the criterion to
determine the form of effective Hamiltonian for all charmonium two body
decays. Moreover, a synthetical nonet is introduced to include both octet and
singlet representations for meson description, and resorting to the mixing
angle the pure octet and singlet states are combined into the observable
pseudoscalar and vector particles, so that the empirically effective
Hamiltonian can be obtained in a concise way. As an application, by virtue of
this scenario the relative phase between the strong and electromagnetic
amplitudes is studied for vector-pseudoscalar meson final state. In data
analysis of samples taken in $e^{+}e^{-}$ collider, the details of
experimental effects, such as energy spread and initial state radiative
correction are taken into consideration in order to make full use of
experimental information and acquire the accurate and delicate results.

###### pacs:

12.38.Qk, 12.39.Hg, 13.25.Gv, 13.40.Gp, 14.20.-c,14.40.-n

##  I Introduction

Since the upgraded Beijing Electron-Positron Collider (BEPCII) and
spectrometer detector (BESIII) started data taking in 2008 bes ; yellow , the
colossal charm and charmonium data samples in the world were collected,
especially the data at $J/\psi$ and $\psi^{\prime}$ resonance peaks, which
provide an unprecedented opportunity to acquire useful information for
understanding the interaction dynamics by analyzing various decay final
states.

Although the Standard Model (SM) has been accepted as a universally
appreciated theory basis in the high energy community, it is still hard to
calculate the wanted experimental observable from the first principle of the
SM for a great many of processes, especially when the strong interaction is
involved. Quantum chromodynamics (QCD) as a widely appreciated theory of
strong interaction, has been proved to be very successful at high energy when
the calculation can be executed perturbatively. Nevertheless, its validity at
the nonperturbative regime needs more experimental guidance. As an exploratory
step, it is necessary to develop a reliable and extensively applicable
phenomenological model (PM).

The advantage of PM lies in that a well-defined PM contains few experimentally
determined parameters which have clear physical meaning; moreover, with only
few parameters determined from experiment, PM could produce concrete results
which can be directly confirmed or falsified by experiment and may guide
further experimental searches. Such a model has a good relation with
elementary principle of the theory, and if applicable, can be used for further
theoretical refinement. This point is noteworthy for the time being, since the
general QCD can hardly provide solutions for special problems; conversely, we
have to establish certain effective empirical model to advance our
understanding for generic QCD principle.

As a matter of fact, many models are constructed Kowalski:1976mc ;
Clavelli:1983 ; Haber ; Seiden88 ; nMorisita91 ; rBaldini98 ; zmy2015 ;
Baldini19 ; moxh2022 ; moxh2024 , the parametrization of various decay modes
are obtained, such as the pseudoscalar and pseudoscalar mesons (PP), vector
and pseudoscalar mesons (VP), octet baryon-pair, and so on. Especially in Ref.
moxh2024 , by virtue of $SU(3)$ flavor symmetry, the effective interaction
Hamiltonian in tensor form is obtained according to group representation
theory. In the light of flavor singlet principle, systematical parametrization
is realized for all charmonium two-body decay, including both baryonic and
mesonic final states. The parametrizations of $\psi^{\prime}$ or $J/\psi$
decaying to octet baryon pair, decuplet baryon pair, decuplet-octet baryon
final state, vector-pseudoscalar meson final state, and pseudoscalar-
pseudoscalar meson final state, are presented. However, in the previous study
moxh2024  mesons are merely treated as pure octet states while the actual
particles are the mixing of both octet and singlet states. Therefore, this
paper concentrates on the meson mixing issue. With nonet concept and mixing
angle, both octet and singlet states are mixed into mass eigenvalue states so
that the pragmatically effective Hamiltonian can be constructed in a concise
way. Such a treatment finalizes the parametrization scheme proposed in the
previous symmetry analysis of charmonium decay.

In the next section, the parametrization scheme will be expounded first, a
synthetical nonet is introduced to combine both octet and singlet
representations for meson description, then the effective Hamiltonian is
obtained consecutively. The section that follows discusses in detail the
experimental character of $e^{+}e^{-}$ collider, then the successive section
focuses on concrete data analysis for VP final states. A special section is
used to discuss another fit method and some open questions on understanding
fit results. After that is a summary section. Relegated to the appendix are
two kinds of materials which are mainly concerned on calculation details.

##  II Analysis framework

In $e^{+}e^{-}$ collider experiment, the initial state is obviously
flavorless, then the final state must be flavor singlet. Moreover, only the
Okubo-Zweig-Iizuka (OZI) rule suppressed processes are considered and the
final states merely involve light quarks, that is $u,d,s$ quarks. Therefore,
$SU(3)$ group is employed for symmetry analysis. The key rule herein is the so
called “flavor singlet principle” that determines what kinds of terms are
permitted in the effective interaction Hamiltonian. Resorting to the
perturbation language, the Hamiltonian is written as

${\cal H}_{eff}=H_{0}+\Delta H~{},$ (1)

where $H_{0}$ is the symmetry conserved term and $\Delta H$ the symmetry
breaking term, which is generally small compare to $H_{0}$. Since we focus on
two-body decay, merely two multiplets, say ${\mathbf{n}}$ and ${\mathbf{m}}$,
need to be considered. In the light of group representation theory, the
product of two multiplets can be decomposed into a series of irreducible
representations, that is

${\mathbf{n}}\otimes{\mathbf{m}}={\mathbf{l}_{1}}\oplus{\mathbf{l}_{2}}\oplus\cdots\oplus{\mathbf{l}_{k}}~{}.$
(2)

The singlet principle requires that among the
${\mathbf{l}_{j}}(j=1,\cdots,k)$, only the singlet term, i.e.,
${\mathbf{l}_{j}}={\mathbf{1}}$ for certain $j$, is allowed in the
Hamiltonian. Since this term is obviously $SU(3)$ invariant, it is called the
symmetry conserved term, i.e., $H_{0}$.

Now turn to the $SU(3)$-breaking term. Following the recipe of the proceeding
study Muraskin1963 ; Gupta1964a ; Gupta1964b , the $SU(3)$-breaking effect is
treated as a “spurion” octet. With this notion, in order to pin down the
breaking term in the Hamiltonian, the products of this spurion octet with the
irreducible representations ${\mathbf{l}_{j}}(j=1,\cdots,k)$ will be
scrutinized, only the singlet term in the decomposition is valid in the
Hamiltonian. Concretely,

${\mathbf{l}_{j}}\otimes{\mathbf{8}}={\mathbf{q}_{1}}\oplus{\mathbf{q}_{2}}\oplus\cdots\oplus{\mathbf{q}_{k}}~{},$
(3)

then if and only if ${\mathbf{q}_{i}=1}$, the corresponding term is allowed in
the Hamiltonian. Since such a kind of term violates $SU(3)$ invariance, it is
called the symmetry breaking term. In a word, with the singlet principle, the
effective interaction Hamiltonian can be determined unequivocally.

Generally speaking, the Hamiltonian term should be written as $\psi
M_{1}M_{2}$, where $\psi$ indicates the charmonium state and $M_{1}$ and
$M_{2}$ are two multiplet components. Since $\psi$ is the same for the whole
final state, and the relative strength of multiplet final state really
matters, $\psi$ dependence is left implicit henceforth.

The tensor denotation is adopted in order to express all kinds of multiplets
consistently. A tensor of rank $(u,v)$ reads

$T^{i_{1}i_{2}\cdots i_{u}}_{j_{1}j_{2}\cdots j_{v}}~{}~{},$

or denoted as $T(u,v)$. The irreducible tensor of $SU(3)$ representation is
traceless and totally symmetric in indices of the same type. The number of
independent components in a multiplet is called its dimension, which is
calculated by the formula

$d\\{T(u,v)\\}=\frac{1}{2}(u+1)(v+1)(u+v+2)~{}.$ (4)

The baryon and meson of a multiplet can be expressed by the corresponding
components of the irreducible tensor, the characteristics of which can be used
to calculate the eigenvalues of particle charge ($Q$), hypercharge ($Y$), and
the third component of isospin ($I_{3}$), that is

$\begin{array}[]{lcr}Q&=u_{1}-v_{1}-\frac{1}{3}(u-v)~{}~{},\\\
Y&=-u_{3}+v_{3}+\frac{1}{3}(u-v)~{}~{},\\\
I_{3}&=\frac{1}{2}(u_{1}-u_{2})-\frac{1}{2}(v_{1}-v_{2})~{}~{},\\\
\end{array}$ (5)

where $u_{i}$ denotes the number of upper indices with value $i$, and $v_{j}$
denotes the number of lower indices with value $j$. For $SU(3)$ group,
$i,j=1,2,3$ and $u=u_{1}+u_{2}+u_{3}$ and $v=v_{1}+v_{2}+v_{3}$. The values
$(Q,Y,I_{3})$ of tensor component indicate the corresponding physical
particle, as examples, some commonly used octets of baryon and meson are
displayed as follows quangpham ; lichtenberg ; aHosaka  :

${\mathbf{B}}=\left(\begin{array}[]{ccc}\Sigma^{0}/\sqrt{2}+\Lambda/\sqrt{6}&\Sigma^{+}&p\\\
\Sigma^{-}&-\Sigma^{0}/\sqrt{2}+\Lambda/\sqrt{6}&n\\\
\Xi^{-}&\Xi^{0}&-2\Lambda/\sqrt{6}\end{array}\right)~{}~{},$ (6)
$\overline{\mathbf{B}}=\left(\begin{array}[]{ccc}\overline{\Sigma}^{0}/\sqrt{2}+\overline{\Lambda/}\sqrt{6}&\overline{\Sigma}^{+}&\overline{\Xi}^{+}\\\
\overline{\Sigma}^{-}&-\overline{\Sigma}^{0}/\sqrt{2}+\overline{\Lambda/}\sqrt{6}&\overline{\Xi}^{0}\\\
\overline{p}&\overline{n}&-2\overline{\Lambda}/\sqrt{6}\end{array}\right)~{}~{},$
(7)
${\mathbf{V}}=\left(\begin{array}[]{ccc}\rho^{0}/\sqrt{2}+\omega/\sqrt{6}&\rho^{+}&K^{*+}\\\
\rho^{-}&-\rho^{0}/\sqrt{2}+\omega/\sqrt{6}&K^{*0}\\\
K^{*-}&\overline{K}^{*0}&-2\omega/\sqrt{6}\end{array}\right)~{}~{},$ (8)

and

${\mathbf{P}}=\left(\begin{array}[]{ccc}\pi^{0}/\sqrt{2}+\eta/\sqrt{6}&\pi^{+}&K^{+}\\\
\pi^{-}&-\pi^{0}/\sqrt{2}+\eta/\sqrt{6}&K^{0}\\\
K^{-}&\overline{K}^{0}&-2\eta/\sqrt{6}\end{array}\right)~{}~{}.$ (9)

The corresponding tensor notations are respectively $B^{i}_{j}$,
$\overline{B}^{i}_{j}$, $V^{i}_{j}$, and $P^{i}_{j}$, where the superscript
denotes the row index of matrix and the subscript the column index.

The $SU(3)$-breaking effect is treated as a “spurion” octet, its additive
quantum numbers, such as $Q,Y,I_{3}$, are set to be 0 while the multiplicative
quantum numbers are set to be 1. Specially speaking, there are two kinds of
spurion octet, one is denoted as ${\mathbf{S}}_{m}$ that indicates the strong
breaking effect and is $I$-spin conserved breaking, the other is denoted as
${\mathbf{S}}_{e}$ that indicates the electromagnetic breaking effect and is
$U$-spin conserved breaking. The tensor expressions for these two special
octets read

$({\mathbf{S}}_{m})^{i}_{j}=g_{m}\delta^{i}_{3}\delta^{3}_{j}~{},$ (10)

and

$({\mathbf{S}}_{e})^{i}_{j}=g_{e}\delta^{i}_{1}\delta^{1}_{j}~{}.$ (11)

It is noted that if $u$ and $v$ are switched in Eq. (5), the values of
$Q,Y,I_{3}$ just change their signs, which means a particle turns into its
antiparticle. If the charge conjugate operator is denoted as $\hat{C}$, then

$\hat{C}~{}T(u,v)\to T(v,u)~{}.$

This can be confirmed for ${\mathbf{V}}$ and ${\mathbf{P}}$. But for baryon,
since the particle and its antiparticle are in different multiplet, so the
charge conjugate not only changes the signs of values $(Q,Y,I_{3})$ but also
keeps the mass of particle unchanged. According to this requirement, it yields

$\hat{C}~{}T(u,v)\to\overline{T}(v,u)~{},$

which can be confirmed for ${\mathbf{B}}$ and $\overline{\mathbf{B}}$.

Since the effective Hamiltonian must be $\hat{C}$-parity conserved, which
means it should be unchanged under $\hat{C}$-parity transformation. This will
constrain the form of the effective Hamiltonian. Let’s take the baryon octet
as an example. According to group theory, the product of two octets can be
reduced as follows

${\mathbf{8}}\otimes{\mathbf{8}}={\mathbf{1}}\oplus{\mathbf{8}}\oplus{\mathbf{8}}\oplus{\mathbf{1}0}\oplus{\mathbf{1}0^{*}}\oplus{\mathbf{2}7}~{}.$
(12)

The singlet (denoted as $[\overline{B}B]_{0}$) building from two octets reads

$[\overline{B}B]_{0}=\overline{B}^{i}_{j}B^{j}_{i}~{}~{},$ (13)

Here Einstein summation convention is adopted, that is the repeated suffix,
once as a subscript and once as a superscript, implies the summation. Under
$\hat{C}$-parity transformation,
$\hat{C}~{}[\overline{B}B]_{0}\to[B\overline{B}]_{0}$. In order to keep
$\hat{C}$-parity, the effective Hamiltonian should take the form

$H_{0}=g_{0}\cdot([\overline{B}B]_{0}+[B\overline{B}]_{0})~{}~{}.$ (14)

Next, by virtue of Eq. (12) there are two types of octet: an antisymmetric, or
$f$-type, and a symmetric, or $d$-type, which are respectively constructed as

$([\overline{B}B]_{f})^{i}_{j}=\overline{B}^{i}_{k}B^{k}_{j}-\overline{B}^{k}_{j}B^{i}_{k}~{}~{},$
(15)

and

$([\overline{B}B]_{d})^{i}_{j}=\overline{B}^{i}_{k}B^{k}_{j}+\overline{B}^{k}_{j}B^{i}_{k}-\frac{2}{3}\delta^{i}_{j}\cdot\overline{B}^{i}_{j}B^{j}_{i}~{}~{}.$
(16)

Under $\hat{C}$-parity transformation,
$\hat{C}~{}([\overline{B}B]_{d})^{i}_{j}\to-([B\overline{B}]_{d})^{j}_{i}$ and
$\hat{C}~{}([\overline{B}B]_{f})^{i}_{j}\to([B\overline{B}]_{f})^{j}_{i}$.
Here the eigenvalues $\xi_{d,f}$ with values $\xi_{d}=+1$ and $\xi_{f}=-1$
(obviously $\xi_{0}=+1$) can be introduced to describe such effects. So in the
effective Hamiltonian, two terms involved $[\overline{B}B]_{f}$ and
$[\overline{B}B]_{d}$ should have the forms

$[\overline{B}B]_{d,f}+\xi_{d,f}[B\overline{B}]_{d,f}~{}~{}.$

Nevertheless, the ${C}$-parity does not change the cross section of certain
process, the terms with and without ${C}$-parity transformation furnish the
same measurement results, as far as the relative strength is concerned, it
proves expedient to keep only one kind of term. Therefore, the effective
interaction Hamiltonian for octet-octet final state reads

$\left.\begin{array}[]{rl}{\cal H}_{eff}=&g_{0}\cdot[\overline{B}B]_{0}\\\
&+g_{m}\cdot([\overline{B}B]_{f})^{3}_{3}+g_{m}^{\prime}\cdot([\overline{B}B]_{d})^{3}_{3}\\\
&+g_{e}\cdot([\overline{B}B]_{f})^{1}_{1}+g_{e}^{\prime}\cdot([\overline{B}B]_{d})^{1}_{1}~{}~{}~{}.\end{array}\right.~{}~{}$
(17)

The other results for decuplet-decuplet and decuplet-octet final states can be
obtained similarly, the detailed information can be referred to Refs. moxh2022
; moxh2024 , together with the corresponding parametrization forms.

The effective Hamiltonian for meson final state can be obtained in the similar
way, say, for ${\mathbf{V}}{\mathbf{P}}$, it reads

$\left.\begin{array}[]{rl}{\cal H}_{eff}^{VP}=&g_{0}\cdot[VP]_{0}\\\
&+g_{m}\cdot([VP]_{f})^{3}_{3}+g_{m}^{\prime}\cdot([VP]_{d})^{3}_{3}\\\
&+g_{e}\cdot([VP]_{f})^{1}_{1}+g_{e}^{\prime}\cdot([VP]_{d})^{1}_{1}~{}~{}~{}.\end{array}\right.~{}~{}$
(18)

However, unlike baryon, the particle and antiparticle are contained in the
same meson multiplet, so
$\hat{C}~{}{\mathbf{V}}/{\mathbf{P}}\to{\mathbf{V}}/{\mathbf{P}}$, which like
the neutral particle such as $\pi^{0}$ with $\hat{C}~{}\pi^{0}\to\pi^{0}$. It
is known that $\pi^{0}$ has the inherent ${C}$-parity, that is
$\hat{C}~{}\pi^{0}=\eta_{\pi^{0}}\pi^{0}$, with $\eta_{\pi^{0}}=+1$. Herein,
the generalized inherent ${C}$-parity for multiplet is introduced, and its
value is set to be equal to that of the neutral particle in the multiplet.
That is to say,

$\left.\begin{array}[]{rl}\hat{C}~{}{\mathbf{P}}=&\eta_{P}{\mathbf{P}},~{}\mbox{with}~{}\eta_{P}=\eta_{\pi^{0}}=+1~{}~{};\\\
\hat{C}~{}{\mathbf{V}}=&\eta_{V}{\mathbf{V}},~{}\mbox{with}~{}\eta_{V}=\eta_{\rho^{0}}=-1~{}~{}.\end{array}\right.~{}~{}$
(19)

With these convention, it has

$\left.\begin{array}[]{rl}\hat{C}~{}[VP]_{f}=&\eta_{V}\eta_{P}\xi_{f}[VP]_{f}~{}~{},\\\
\hat{C}~{}[VP]_{d}=&\eta_{V}\eta_{P}\xi_{d}[VP]_{d}~{}~{}.\end{array}\right.~{}~{}$
(20)

The invariant of the effective Hamiltonian under ${C}$-parity transformation
requires that $\eta_{V}\eta_{P}\xi_{f}$ or $\eta_{V}\eta_{P}\xi_{d}$ must be
equal to $\eta_{J/\psi,\psi^{\prime}}$. Since
$\eta_{J/\psi,\psi^{\prime}}=-1$, only
$\eta_{J/\psi,\psi^{\prime}}=\eta_{V}\eta_{P}\xi_{d}$ is allowed, which means
only the term $[VP]_{d}$ can exist in the effective Hamiltonian of
${\mathbf{V}}{\mathbf{P}}$ final state, that is

${\cal
H}_{eff}^{VP}=g_{0}\cdot[VP]_{0}+g_{m}\cdot([VP]_{d})^{3}_{3}+g_{e}\cdot([VP]_{d})^{1}_{1}~{}.$
(21)

By virtue of the same criterion, the effective Hamiltonian of
${\mathbf{P}}{\mathbf{P}}$ final state reads

${\cal
H}_{eff}^{PP}=g_{m}\cdot([PP]_{f})^{3}_{3}+g_{e}\cdot([PP]_{f})^{1}_{1}~{}.$
(22)

With the components given in Eqs. (8) and (9), the corresponding
parametrization can be obtained and summarized in Tables 1 and 2,
respectively. For the other octet final states, the parametrization scheme can
be referred to the appendix.

Table 1:  Amplitude parametrization form for decays of the $\psi^{\prime}$ or $J/\psi$ into $VP$ final states. General expressions in terms of singlet $A$ (by definition $A=g_{0}$), as well as charge-breaking ($D=g_{e}/3$) and mass-breaking terms ($D^{\prime}=g_{m}/3$). The table can also be used for more similar decays by appropriate change in labeling, refer to appendix for more details.  Final state | Amplitude parametrization form
---|---
$\rho^{\pm}\pi^{\mp}$, $\rho^{0}\pi^{0}$ | $A+D-2D^{\prime}$
$K^{*\pm}K^{\mp}$ | $A+D+D^{\prime}$
$K^{*0}\overline{K}^{0}$, $\overline{K}^{*0}K^{0}$ | $A-2D+D^{\prime}$
$\omega\eta$ | $A-D+2D^{\prime}$
$\omega\pi^{0}$ | $~{}~{}\sqrt{3}D~{}~{}~{}$
$\rho^{0}\eta$ | $~{}~{}\sqrt{3}D~{}~{}~{}$
Table 2:  Amplitude parametrization form for decays of the $\psi^{\prime}$ or $J/\psi$ into $PP$ final states. General expressions in terms of the charge-breaking one ($D=2g_{e}$) and the mass-breaking one ($D^{\prime}=-2g_{m}$). The table can also be used for more similar decays by appropriate change in labeling, refer to appendix for more details.  Final state | Amplitude parametrization form
---|---
$\pi^{+}\pi^{-}$ | $D$
$K^{+}K^{-}$ | $D+D^{\prime}$
$K^{0}\overline{K}^{0}$ | $~{}~{}~{}~{}~{}~{}D^{\prime}$

In the light of Eq. (21), the first term $g_{0}\cdot[VP]_{0}$ represents the
$SU(3)$-conserved effect while in Eq. (22) only the $SU(3)$-breaking terms
exist. Therefore, for VP-like final states, the decay branching fractions are
generally greater than those of PP-like final states.

In above analysis, it is prominent that in Eqs. (8) and (9), the mesons
$\omega$ and $\eta$ are treated as pure octet components, but the real or
observable $\omega$ and $\eta$ are actually the mixing of pure octet and
singlet components with a certain mixing angle. According to PDG pdg2020 ,

$\left.\begin{array}[]{rcl}\phi&=&\omega^{8}~{}\cos\theta_{V}-\omega^{1}~{}\sin\theta_{V}~{},\\\
\omega&=&\omega^{8}~{}\sin\theta_{V}+\omega^{1}~{}\cos\theta_{V}~{},\end{array}\right.$
(23)

or its reverse

$\left.\begin{array}[]{rcl}\omega^{8}&=&\cos\theta_{V}~{}\phi+\sin\theta_{V}~{}\omega~{},\\\
\omega^{1}&=&-\sin\theta_{V}~{}\phi+\cos\theta_{V}~{}\omega~{},\end{array}\right.$
(24)

and

$\left.\begin{array}[]{rcl}\eta&=&\eta^{8}~{}\cos\theta_{P}-\eta^{1}~{}\sin\theta_{P}~{},\\\
\eta^{\prime}&=&\eta^{8}~{}\sin\theta_{P}+\eta^{1}~{}\cos\theta_{P}~{},\end{array}\right.$
(25)

or its reverse

$\left.\begin{array}[]{rcl}\eta^{8}&=&\cos\theta_{P}~{}\eta+\sin\theta_{P}~{}\eta^{\prime}~{},\\\
\eta^{1}&=&-\sin\theta_{P}~{}\eta+\cos\theta_{P}~{}\eta^{\prime}~{}.\end{array}\right.$
(26)

In above equations, the superscript $8$ indicates the pure octet component and
the superscript $1$ the pure singlet component.

In order to obtain the actual effective Hamiltonian, both octet and singlet
components are to be included. The concise and effective way is to introduce a
nonet that merges a singlet with an octet Haber . Under such circumstances,
the matrixes in Eqs. (8) and (9) are recast as

${\mathbf{V}_{N}}=\left(\begin{array}[]{ccc}V^{1}_{1}&\rho^{+}&K^{*+}\\\
\rho^{-}&V^{2}_{2}&K^{*0}\\\
K^{*-}&\overline{K}^{*0}&V^{3}_{3}\end{array}\right)~{}~{},$ (27)

with

$\left.\begin{array}[]{rcl}V^{1}_{1}&=&\rho^{0}/\sqrt{2}+\omega^{8}/\sqrt{6}+\omega^{1}/\sqrt{3}~{},\\\
V^{2}_{2}&=&-\rho^{0}/\sqrt{2}+\omega^{8}/\sqrt{6}+\omega^{1}/\sqrt{3}~{},\\\
V^{3}_{3}&=&-2\omega^{8}/\sqrt{6}+\omega^{1}/\sqrt{3}~{};\end{array}\right.$
(28)

and

${\mathbf{P}_{N}}=\left(\begin{array}[]{ccc}P^{1}_{1}&\pi^{+}&K^{+}\\\
\pi^{-}&P^{2}_{2}&K^{0}\\\
K^{-}&\overline{K}^{0}&P^{3}_{3}\end{array}\right)~{}~{},$ (29)

with

$\left.\begin{array}[]{rcl}P^{1}_{1}&=&\pi^{0}/\sqrt{2}+\eta^{8}/\sqrt{6}+\eta^{1}/\sqrt{3}~{},\\\
P^{2}_{2}&=&-\pi^{0}/\sqrt{2}+\eta^{8}/\sqrt{6}+\eta^{1}/\sqrt{3}~{},\\\
P^{3}_{3}&=&-2\eta^{8}/\sqrt{6}+\eta^{1}/\sqrt{3}~{}.\end{array}\right.$ (30)

With these new nonets ${\mathbf{V}_{N}}$ and ${\mathbf{P}_{N}}$, the effective
Hamiltonian in Eq. (21) can be used formally to acquire the corresponding
parametrization for ${\mathbf{V}}{\mathbf{P}}$ final states, with both octet
components, say $\omega^{8}$ and $\eta^{8}$, and singlet components, say
$\omega^{1}$ and $\eta^{1}$. Then with Eqs. (24) and (26), the pure octet and
singlet components $\omega^{8}$, $\omega^{1}$, $\eta^{8}$, and $\eta^{1}$ will
be replaced by the actual particles $\omega$, $\phi$, $\eta$, and
$\eta^{\prime}$. The final parametrization results are summarized in Table 3.

Table 3:  Amplitude parametrization form for decays of the $\psi^{\prime}$ or $J/\psi$ into $V~{}P$ final states. General expressions in terms of singlet $A$ (by definition $A=g_{0}$), as well as charge-breaking ($D=g_{e}/\sqrt{3}$) and mass-breaking terms ($D^{\prime}=2g_{m}/\sqrt{3}$). The shorthand symbols are defined as $s_{\alpha}=\sin\theta_{\alpha}$, $c_{\alpha}=\cos\theta_{\alpha}$, $s^{\pm}_{\alpha\beta}=\sin(\theta_{\alpha}\pm\theta_{\beta})$,$c^{\pm}_{\alpha\beta}=\cos(\theta_{\alpha}\pm\theta_{\beta})$,$s_{\gamma\alpha\beta}=\sin(\theta_{\gamma}+\theta_{\alpha}+\theta_{\beta})$, $c_{\gamma\alpha\beta}=\cos(\theta_{\gamma}+\theta_{\alpha}+\theta_{\beta})$, $s_{\gamma}=\sin\theta_{\gamma}\equiv\sqrt{1/3}$, and $c_{\gamma}=\cos\theta_{\gamma}\equiv\sqrt{2/3}$. It should be noted that the definitions of $D$ and $D^{\prime}$ herein are different from those in Table 1. States | $A$ | $D$ | $D^{\prime}$
---|---|---|---
$\rho^{0}\pi^{0}$ | 1 | $1/\sqrt{3}$ | $-1/\sqrt{3}$
$\rho^{+}\pi^{-}$ | 1 | $1/\sqrt{3}$ | $-1/\sqrt{3}$
$\rho^{+}\pi^{-}$ | 1 | $1/\sqrt{3}$ | $-1/\sqrt{3}$
$K^{*+}K^{-}$ | 1 | $1/\sqrt{3}$ | $1/2\sqrt{3}$
$K^{*-}K^{+}$ | 1 | $1/\sqrt{3}$ | $1/2\sqrt{3}$
$K^{*0}\overline{K}^{0}$ | 1 | $-2/\sqrt{3}$ | $1/2\sqrt{3}$
$\overline{K}^{*0}K^{0}$ | 1 | $-2/\sqrt{3}$ | $1/2\sqrt{3}$
$\phi\eta$ | $c^{-}_{VP}$ | $-s_{\gamma VP}-s_{\gamma}s_{V}s_{P}$ | $s_{\gamma VP}+s_{\gamma}s_{V}s_{P}$
$\phi\eta^{\prime}$ | $-s^{-}_{VP}$ | $c_{\gamma VP}+s_{\gamma}s_{V}c_{P}$ | $-c_{\gamma VP}-s_{\gamma}s_{V}c_{P}$
$\omega\eta$ | $s^{-}_{VP}$ | $c_{\gamma VP}+s_{\gamma}c_{V}s_{P}$ | $-c_{\gamma VP}-s_{\gamma}c_{V}c_{P}$
$\omega\eta^{\prime}$ | $c^{-}_{VP}$ | $s_{\gamma VP}-s_{\gamma}c_{V}s_{P}$ | $-s_{\gamma VP}+s_{\gamma}c_{V}c_{P}$
$\rho^{0}\eta$ | 0 | $\sqrt{3}\cdot s^{-}_{\gamma P}$ | 0
$\rho^{0}\eta^{\prime}$ | 0 | $\sqrt{3}\cdot c^{-}_{\gamma P}$ | 0
$\phi\pi^{0}$ | 0 | $\sqrt{3}\cdot s^{-}_{\gamma V}$ | 0
$\omega\pi^{0}$ | 0 | $\sqrt{3}\cdot c^{-}_{\gamma V}$ | 0

Besides the nonet approach, the singlet component can be treated separately,
and the final results are essentially the same. The details are degenerated
into appendix.

##  III Experimental section

Electron-positron collider experiment has its special character. When
analyzing the data taken in $e^{+}e^{-}$ collider, the important experimental
effects such as the initial state radiative (ISR) correction and the effect
due to energy spread of accelerator must be dealt with carefully.

###  III.1 Born section

For $e^{+}e^{-}$ colliding experiments, there is the inevitable continuum
amplitude wangp03hepnp

$e^{+}e^{-}\rightarrow\gamma^{*}\rightarrow hadrons$

which may produce the same final state as the resonance does. In
$e^{+}e^{-}\rightarrow{\mathbf{V}}{\mathbf{P}}$ at $J/\psi$ or $\psi^{\prime}$
resonance, the Born order cross section for final state $f$ is rudaz ; wymcgam
; Wang:2005sk ; wymphase ; wymogpiapp

$\sigma_{Born}=\frac{4\pi\alpha^{2}}{s^{3/2}}|A_{f}(s)|^{2}{\cal
P}_{f}(s)~{}~{},$ (31)

where ${\cal P}_{f}(s)=q_{f}^{3}/3$, with $q_{f}$ being the momentum of either
the ${\mathbf{V}}$ or the ${\mathbf{P}}$ particle, viz.

$q_{f}=\frac{[(s-(m_{1}+m_{2})^{2})(s-(m_{1}-m_{2})^{2})]^{1/2}}{2\sqrt{s}}~{},$
(32)

where $\sqrt{s}$ is the center of mass energy, $m_{1}$ and $m_{2}$ are the
masses of two ${\mathbf{V}}{\mathbf{P}}$ final state mesons. The total
amplitude reads

$A_{f}(s)=a_{3g}(s)+a_{\gamma}(s)+a_{c}(s)~{},$ (33)

which consists of three kinds of amplitudes corresponding to (a) the strong
interaction [$a_{3g}(s)$] presumably through three-gluon annihilation, (b) the
electromagnetic interaction [$a_{\gamma}(s)$] through the annihilation of
$c\overline{c}$ pair into a virtual photon, and (c) the electromagnetic
interaction [$a_{c}(s)$] due to one-photon continuum process. For
${\mathbf{V}}{\mathbf{P}}$ final state, the amplitudes have the forms :

$a_{c}(s)=Y_{f}\cdot{\cal F}(s)~{},$ (34) $a_{\gamma}(s)=Y_{f}\cdot
B(s)\cdot{\cal F}(s)~{},$ (35) $a_{3g}(s)=X_{f}\cdot B(s)\cdot{\cal F}(s)~{},$
(36)

with the definition

$B(s)\equiv\frac{3\sqrt{s}\Gamma_{ee}/\alpha}{s-M^{2}+iM\Gamma_{t}}~{}~{},$
(37)

where $\alpha$ is the QED fine structure constant; $M$ and $\Gamma_{t}$ are
the mass and the total width of the $\psi^{\prime}$ or $J/\psi$; $\Gamma_{ee}$
is the partial width to $e^{+}e^{-}$. ${\cal F}(s)$ is the form factor and
take the form $1/s$ for ${\mathbf{V}}{\mathbf{P}}$ final state. $X_{f}$ and
$Y_{f}$ are the functions of the amplitude parameters of final state $f$, that
is $A,D$, $D^{\prime}$, $s_{P}$, and $s_{V}$, à la Table 3, viz.

$Y=Y(D,s_{P},s_{V})~{},$ (38) $X=X(A,D^{\prime},s_{P},s_{V})e^{i\phi}~{}.$
(39)

By virtue of Eq. (25) and Eq. (23), $s_{P}=\sin\theta_{P}$ and
$s_{V}=\sin\theta_{V}$, where $\theta_{P}$ and $\theta_{V}$ are respectively
the mixing angle of pseudoscalar meson between $\eta$ and $\eta^{\prime}$, and
that of vector meson between $\phi$ and $\omega$. The concrete form of $X$ or
$Y$ depends on the decay mode, as examples, for $\rho\pi$ final state,
$X_{\rho\pi}=A-D^{\prime}/\sqrt{3}$ and $Y_{\rho\pi}=D/\sqrt{3}$ while for
$\omega\pi^{0}$ final state, $X_{\omega\pi^{0}}=0$ and
$Y_{\omega\pi^{0}}=D\cdot\sqrt{3}\cdot c^{-}_{\gamma V}$, according to the
parametrization form in Table 3. In principle, the parameters $A,D$, and
$D^{\prime}$ could be complex arguments, each with a magnitude together with a
phase. Conventionally, it is assumed that there is not relative phases among
the strong-originated amplitudes $A$, $D^{\prime}$, and the sole phase
[denoted by $\phi$ in Eq. (39) ] is between the strong and electromagnetic
interactions, that is between $X$ and $Y$, as indicated in Eqs. (39) and (38),
where $A$, $D$, and $D^{\prime}$ are actually treated as real numbers.

Table 4: Breakdown of experiment conditions correspond to different detectors and accelerators. The energy spread is the effective one, according to which the calculated maximum cross section satisfies the relation $N_{tot}=\sigma_{max}\cdot{\cal L}$. The number with star ($\ast$) is the equivalent luminosity calculated by the relation ${\cal L}=N_{tot}/\sigma_{max}$.  |  | Center of Mass Energy | Data Taking | Maximum | Total | Integral |
---|---|---|---|---|---|---|---
Detector | Accelerator | Spread | Position111 The data taking position is the energy which yield the maximum inclusive hadronic cross section. | section | event | luminosity | References
|  | (MeV) | (GeV) | (nb) | ($\times 10^{6}$) | (pb-1) |
BES II | BEPC | 1.23 | 3.68623 | 712.9 | $14.0\pm 0.6$ | 19.72 | tnpsp2004
|  | 1.23 | 3.65 | $\cdots$ | $\cdots$ | 6.42 | lum2004
BESIII | BECPII | 1.343 | 3.68624 | 662.16 | $107.0\pm 0.8$ | 161.63 | tnpsp2018
|  | 1.343 | 3.65 | $\cdots$ | $\cdots$ | 43.88 | tnpsp2018
|  | 1.318 | 3.68624 | 672.74 | $341.1\pm 2.1$ | 506.92 | tnpsp2018
|  | 1.324 | 3.68624 | 670.17 | $448.1\pm 2.9$ | 668.55 | tnpsp2018
|  | 1.324 | 3.65 | $\cdots$ | $\cdots$ | 48.8 | besretp2017
CLEO-c | CESR | 1.68 | 3.68627 | 557.23 | 3.08 | 5.63 | cleovp2005
|  | 1.68 | 3.67 | $\cdots$ | $\cdots$ | 20.46 |
BESIII | BECPII | 1.131 | 3.097014 | 2808.63 | $223.7\pm 1.4$ | 79.63 | tnjps2012
|  | 1.131 | 3.08 | $\cdots$ | $\cdots$ | 0.282 | bes3pi2012
|  | 0.898 | 3.096990 | 3447.87 | $1086.9\pm 6.0$ | 315.02 | tnjps2017
|  | 0.937 | 3.096993 | 3320.35 | $1310.6\pm 7.0$ | 394.65 | tnjps2017
|  | 0.937 | 3.08 | $\cdots$ | $\cdots$ | 153.8 | besretp2017
BES II | BEPC | 0.85 | 3.09700 | 3631.8 | $57.7\pm 2.72$ | 15.89$\ast$ | tnjps2003
|  | 0.85 | 3.07 | $\cdots$ | $\cdots$ | 2.3473 | besjopaet06
DM II | DCI | 1.98 | 3.097114 | 1702.0 | $8.6\pm 1.3$ | 5.053$\ast$ | JJousset90jvp

###  III.2 Observed section

In $e^{+}e^{-}$ collision, the Born order cross section is modified by the
initial state radiation in the way rad.1

$\sigma_{r.c.}(s)=\int\limits_{0}^{x_{m}}dxF(x,s)\frac{\sigma_{Born}(s(1-x))}{|1-\Pi(s(1-x))|^{2}},$
(40)

where $x_{m}=1-s^{\prime}/s$. $F(x,s)$ is the radiative function been
calculated to an accuracy of 0.1% rad.1 ; rad.2 ; rad.3 , and $\Pi(s)$ is the
vacuum polarization factor. In the upper limit of the integration,
$\sqrt{s^{\prime}}$ is the experimentally required minimum invariant mass of
the final particles. If $x_{m}=1$, it corresponds to no requirement for
invariant mass; if $x_{m}=0.2$, it corresponds to invariant mass cut of 3.3
GeV for $\psi^{\prime}$ resonance. The concrete value of $x_{m}$ should be
determined by the cut of invariant mass, which is adopted in actual event
selection.

By convention, $\Gamma_{ee}$ has the QED vacuum polarization in its definition
Tsai ; Luth . Here it is natural to extend this convention to the partial
widths of other pure electromagnetic decays, that is

$\Gamma_{f}=\frac{\tilde{\Gamma}_{ee}q^{3}_{f}}{M}\left|{\cal
F}(M^{2})\right|^{2}~{},$ (41)

where

$\tilde{\Gamma}_{ee}\equiv\frac{\Gamma_{ee}}{|1-\Pi(M^{2})|^{2}}~{}$

with the vacuum polarization effect included.

The $e^{+}e^{-}$ collider has a finite energy resolution which is much wider
than the intrinsic width of narrow resonances such as $\psi^{\prime}$ and
$J/\psi$ besscan95 ; besscan02 . Such an energy resolution is usually a
Gaussian distribution awChao :

$G(W,W^{\prime})=\frac{1}{\sqrt{2\pi}\Delta}e^{-\frac{(W-W^{\prime})^{2}}{2{\Delta}^{2}}},$

where $W=\sqrt{s}$ and $\Delta$, a function of energy, is the standard
deviation of Gaussian distribution. The experimentally observed cross section
is the radiative corrected cross section folded with the energy resolution
function

$\sigma_{obs}(W)=\int\limits_{0}^{\infty}dW^{\prime}\sigma_{r.c.}(W^{\prime})G(W^{\prime},W)~{}.$
(42)

In fact, as pointed out in Ref. wymcgam , the radiative correction and the
energy spread of the collider are two important factors, both of which reduce
the height of the resonance and shift the position of the maximum cross
section. Although the ISR are the same for all $e^{+}e^{-}$ experiments, the
energy spread is quite different for different accelerators, even different
for the same accelerator at different running periods. As an example, for the
CLEO data used in this paper, the energy spread varies due to different
accelerator lattices YELLOWBOOK : one (for CLEO III detector) with a single
wiggler magnet and a center-of-mass energy spread $\Delta$=1.5 MeV, the other
(for CLEOc detector) with the first half of its full complement (12) of
wiggler magnets and $\Delta$=2.3 MeV. The two $\Delta$’s lead to two maximum
total cross sections 602 nb and 416 nb, respectively. All these subtle effects
must be taken into account in data analysis. In the following analysis all
data are assumed to be taken at the energy point which yields the maximum
inclusive hadron cross sections in stead of the nominal resonance mass wymcgam
; wymhepnp . Besides the factors considered above, the resonance parameters
can also affect the evaluation results. Since the present central values of
resonance parameters can be obviously distinct from those of some time before,
the calculated maximum inclusive hadron cross sections will consequently
different. In order to ensure the relation $N_{tot}=\sigma_{max}\cdot{\cal
L}$, some adjustments are mandatory. The principle is as follows: if the
luminosity is available, the energy spread will be tuned to give consistent
maximum cross section; otherwise, the effective luminosity is evaluated by the
relation ${\cal L}=N_{tot}/\sigma_{max}$ according to the corresponding
accelerator parameters. All relevant experimental details are summarized in
Table 4, which are crucial for the following data analysis. At last, the
resonance parameters adopted in this paper for $J/\psi$ and $\psi^{\prime}$
are respectively pdg2020

$\begin{array}[]{rcl}M_{R}&=&3096.900\pm 0.006~{}~{}\mbox{MeV },\\\
\Gamma_{t}&=&92.9\pm 2.8~{}~{}\mbox{keV },\\\ \Gamma_{ee}&=&5.53\pm
0.10~{}~{}\mbox{keV };\end{array}$ (43)

and

$\begin{array}[]{rcl}M_{R}&=&3686.10\pm 0.06~{}~{}\mbox{MeV },\\\
\Gamma_{t}&=&294\pm 8~{}~{}\mbox{keV },\\\ \Gamma_{ee}&=&2.33\pm
0.04~{}~{}\mbox{keV }.\end{array}$ (44)

Table 5: Experimental data of $\psi^{\prime}$ decaying to $VP$ final states. For branching ratios, the first uncertainties are statistical, and the second systematic. The peak position is assumed at $\sqrt{s}=3.686$ GeV, while the continuum position is assumed at $\sqrt{s}=3.65$ GeV for BES and $\sqrt{s}=3.67$ GeV for CLEO. The efficiency indicates the selection efficiency for resonance events. The number in parenthesis is not quoted in the original literature but is evaluated according to the information given therein. Mode | $N^{obs}$ | $N^{obs}$ | Efficiency | Branching Ratio | Detector
---|---|---|---|---|---
| (peak) | (continuum) | (%) | ($\times 10^{-5}$) |
$\pi^{+}\pi^{-}\pi^{0}$ | $7771\pm 88$ | $220.6\pm 14.8$ | $30.5$ | $2.14\pm 0.03^{+0.08}_{-0.07}$ | BESIII bes3pi2012
| $216.7$ | $85$ | $33.5$ | $18.8^{+1.6}_{-1.5}\pm 2.8$ | CLEO cleovp2005
| $260\pm 19$ | $10.0\pm 4.2$ | $9.02$ | $18.1\pm 1.8\pm 1.9$ | BESII besrp2005
$\rho\pi$ | $34.4$ | $47$ | $28.8$ | $2.4^{+0.8}_{-0.7}\pm 0.2$ | CLEO cleovp2005
| $64.12\pm 6.44$ | $\cdots$ | $9.02$ | $5.1\pm 0.7\pm 1.1$ | BESII besrp2005
$K^{*+}K^{-}+c.c.$ | $7.7$ | $4$ | $16.7$ | $1.3^{+1.0}_{-0.7}\pm 0.3$ | CLEO cleovp2005
| $9.6\pm 4.2$ | $\cdots$ | $7.3$ | $2.9^{+1.3}_{-1.7}\pm 0.4$ | BESII beskstk2005
| $224\pm 21$ | $(54.7\pm 7.6)$ | $20.25$ | $3.18\pm 0.30^{+0.26}_{-0.31}$ | BESIII besvps2012
$K^{*0}\overline{K}^{0}+c.c$ | $34.5$ | $36$ | $8.7$ | $9.2^{+2.7}_{-2.2}\pm 0.9$ | CLEO cleovp2005
| $65.6\pm 9.0$ | $2.5^{+2.6}_{-1.8}$ | $9.7$ | $12.3^{+2.4}_{-2.6}\pm 1.7$ | BESII beskstk2005
$\phi\eta$ | $6.6$ | $3$ | $9.4$ | $2.0^{+1.5}_{-1.1}\pm 0.4$ | CLEO cleovp2005
| $16.7\pm 5.6$ | $\cdots$ | $18.9$ | $3.3\pm 1.1\pm 0.5$ | BESII besfw2004
| $216\pm 16$ | $(7.0\pm 2.5)$ | $33.53$ | $3.14\pm 0.23\pm 0.23$ | BESIII besvps2012
$\phi\eta^{\prime}$ | $8.4\pm 3.7$ | $\cdots$ | $8.4$ | $3.1\pm 1.4\pm 0.7$ | BESII besfw2004
| $201\pm 15$ |  $221\pm 15$222 The number of event is obtained at $\sqrt{s}=3.773$ GeV. | $26.8$ | $1.51\pm 0.16\pm 0.12$ | BESIII besfet2019
$\omega\eta$ | $<0$ | $3$ | $10.2$ | $<1.1$ | CLEO cleovp2005
| $<9.7$ | $\cdots$ | $6.3$ | $<3.1$ | BESII besfw2004
$\omega\eta^{\prime}$ | $4.2^{+3.2}_{-2.7}$ | $\cdots$ | $2.3$ | $3.2^{+2.4}_{-2.0}\pm 0.7$ | BESII besfw2004
$\rho^{0}\eta$ | $28.1$ | $38$ | $19.3$ | $3.0^{+1.1}_{-0.9}\pm 0.2$ | CLEO cleovp2005
| $29.2^{+7.5}_{-6.8}$ | $2.3^{+2.1}_{-1.4}$ | $(12.06)$ | $1.87^{+0.68}_{-0.62}\pm 0.18$ | BESII besret2004
$\rho^{0}\eta^{\prime}$ | $5.4^{+3.3}_{-2.2}$ | $<4.4$ | $(4.92)$ | $1.87^{+1.64}_{-1.11}\pm 0.33$ | BESII besret2004
| ($211\pm 16$)333 The solution of destructive interference between $\rho$ and nonresonant components. | $5.06\pm 2.01$ | $18.7$ | $1.02\pm 0.11\pm 0.24$ | BESIII besretp2017
| ($148\pm 18$)444 The solution of constructive interference between $\rho$ and nonresonant components. | $5.06\pm 2.01$ | $18.7$ | $0.569\pm 0.128\pm 0.236$ | BESIII besretp2017
$\phi\pi^{0}$ | $<0$ | $3$ | $15.8$ | $<0.7$ | CLEO cleovp2005
| $<4.4$ | $\cdots$ | $16.1$ | $<0.4$ | BESII besfw2004
| $<6$ | $\cdots$ | $35.63$ | $<0.04$ | BESIII besvps2012
$\omega\pi^{0}$ | $29.1$ | $55$ | $19.1$ | $2.5^{+1.2}_{-1.0}\pm 0.2$ | CLEO cleovp2005
| $31.2^{+7.7}_{-6.9}$ | $7.3^{+3.3}_{-2.7}$ | $(5.45)$ | $1.78^{+0.67}_{-0.62}\pm 0.28$ | BESII besret2004

##  IV Data analysis

The Standard Model mainly consists of two parts. One is Salam-Weinberg model
that depicts the electroweak interaction, which can usually accommodate
accurate enough evaluation for certain process. Another part is QCD, the
validity of which at nonperturbative regime needs more experimental guidance.
The production and decay of charmonium states benefit such a study.

As one of important and interesting steps, it is a good start point to study
the relative phase between the strong and electromagnetic (EM) interaction
amplitudes, which provides us a new viewpoint to explore the quarkonium decay
dynamics and profound our understanding on QCD. Studies have been carried out
for many $J/\psi$ and $\psi^{\prime}$ two-body mesonic decay modes with
various spin-parities : $1^{-}0^{-}$ wymphase ; JJousset90jvp ; mk3jvp85 ;
mk3jvp88 , $0^{-}0^{-}$ a00 ; LopezCastro:1994xw ; wymppdk ; bes2klks04 ; a11
, and $1^{-}1^{-}$ a11 . These analyses reveal that there exists a relative
orthogonal phase between the EM and strong decay amplitudes. There is also a
conjecture to claim that such an orthogonal phase is universal for all
quarkonia decays Wang:2003zx ; gerard .

The systematical parametrization scheme of charmonium decay modes facilitates
the study of the relative phase. In Ref. moxh2024 , the phase is measured for
various charmonium decay modes, including $\psi^{\prime}$ and/or $J/\psi$
decay to octet baryon pair, decuplet baryon pair, decuplet-octet baryon final
state, and pseudoscalar-pseudoscalar meson final state. In this section, the
study is devoted to vector-pseudoscalar (${\mathbf{V}}{\mathbf{P}}$) meson
final state according to the parametrization of Table 3.

Since our analysis involves the experimental details as indicated by the
description in preceding section, some measurements are not suitable in the
following study due to the lack of necessary information of detectors and/or
accelerators. In addition, at different energy point, the status parameters of
accelerators are also distinctive, so the studies of phase angle for
$\psi^{\prime}$ and $J/\psi$ decay are performed separately for the sake of
clarity.

Table 6: Data of $\psi^{\prime}\to VP$ decays from CLEO and BES experiments. The error is merely the statistical and the efficiency is the effective one as depicted in the text. Mode | Energy | $N^{obs}$ | Efficiency | Detector
---|---|---|---|---
| (GeV) |  | (%) |
$\rho\pi$ | $3.686$ | $54.24\pm 11.53$ | $33.5$ | CLEO cleovp2005
| $3.686$ | $64.12\pm 6.44$ | $9.02$ | BESII besrp2005
$K^{*+}K^{-}+c.c.$ | $3.686$ | $7.7\pm 5.9$ | $16.7$ | CLEO cleovp2005
| $3.67$ | $4\pm 3.1$ | $16.7$ |
| $3.686$ | $9.6\pm 4.2$ | $2.34$ | BESII beskstk2005
| $3.686$ | $224\pm 21$ | $6.65$ | BESIII besvps2012
$K^{*0}\overline{K}^{0}+c.c$ | $3.686$ | $34.5\pm 10.1$ | $8.7$ | CLEO cleovp2005
| $3.67$ | $36\pm 10.6$ | $8.7$ |
| $3.686$ | $65.6\pm 9.0$ | $3.11$ | BESII beskstk2005
| $3.65$ | $2.5\pm 2.6$ | $3.11$ |
$\phi\eta$ | $3.686$ | $6.6\pm 5.0$ | $9.4$ | CLEO cleovp2005
| $3.67$ | $3\pm 2.3$ | $9.4$ |
| $3.686$ | $16.7\pm 5.6$ | $3.67$ | BESII besfw2004
| $3.686$ | $216\pm 16$ | $6.50$ | BESIII besvps2012
$\phi\eta^{\prime}$ | $3.686$ | $8.4\pm 3.7$ | $1.92$ | BESII besfw2004
| $3.686$ | $201\pm 15$ | $2.21$ | BESIII besfet2019
$\omega\eta$ | $3.686$ | $4.2\pm 3.2$ | $0.95$ | BESII besfw2004
$\rho^{0}\eta$ | $3.686$ | $28.1\pm 10.3$ | $19.3$ | CLEO cleovp2005
| $3.67$ | $38\pm 13.9$ | $19.3$ |
| $3.686$ | $29.2\pm 7.5$ | $4.75$ | BESII besret2004
| $3.65$ | $2.3\pm 2.1$ | $4.29$ |
$\rho^{0}\eta^{\prime}$ | $3.686$ | $5.4\pm 3.3$ | $0.86$ | BESII besret2004
| $3.686$ | ($211\pm 16$)555 The destructive solution. | $3.116$ | BESIII besretp2017
| $3.686$ | ($148\pm 18$)666 The constructive solution. | $3.116$ | BESIII besretp2017
| $3.65$ | $5.06\pm 2.01$ | $3.116$ | BESIII besretp2017
$\omega\pi^{0}$ | $3.686$ | $29.1\pm 14.0$ | $19.1$ | CLEO cleovp2005
| $3.67$ | $55\pm 26.4$ | $19.1$ |
| $3.686$ | $31.2\pm 7.7$ | $4.87$ | BESII besret2004
| $3.65$ | $7.3\pm 3.3$ | $4.55$ |

###  IV.1 $\psi^{\prime}\to VP$ decay

There are lots of measurements concerned with $\psi^{\prime}$ decaying to
${\mathbf{V}}{\mathbf{P}}$ final states. However, the results of Ref.
Franklin83  are obtained forty years ago, and moreover only the upper limits
of $K^{*\pm}K^{\mp}$ and $\rho\pi$ final states are given based on 1 million
$\psi^{\prime}$ events. The results from Ref. wBartel76  are merely the upper
limits for $\rho\pi$ and $\gamma\eta^{\prime}$ final states based on 0.2
million $\psi^{\prime}$ events. Therefore, these kinds of measurements are not
adopted in our study. The results of experimental measurements obtained in
this century are collected in Table 5, which are mainly due to CLEO and BES
collaborations.

First, let’s focus on $\rho\pi$ final state. By virtue of Table 5, the
branching fraction of $\psi^{\prime}\to\pi^{+}\pi^{-}\pi^{0}$ agrees very well
for both CLEO and BES experiments, while that of $\psi^{\prime}\to\rho\pi$ are
rather different. As pointed out in Ref. besrp2005 , the partial wave analysis
indicates that in all $\pi^{+}\pi^{-}\pi^{0}$ events, only $28\%$ are due to
$\rho\pi$ final state. It seems that the result from BES takes more
information into account. If we adopt the same proportion of $\rho\pi$ for
CLEO data, the branching fraction $\psi^{\prime}\to\rho\pi$ is around
$5.3\times 10^{-5}$, which is fairly consist with BES result. Therefore, the
modified data from CLEO are adopted in the following analysis, the details are
presented in Table 6.

Second, in Ref. besretp2017  based on 448 million $\psi^{\prime}$ events,
$\pi^{+}\pi^{-}\eta^{\prime}$ final state is studied resorting to partial wave
analysis technique. The interference between $\rho$ and nonresonant components
is observed. The constructive and destructive interferences lead to two
possible solutions of branching fraction, that is $(5.69\pm 1.28\pm
2.36)\times 10^{-6}$ and $(1.02\pm 0.11\pm 0.24)\times 10^{-5}$, respectively.
In the following fit, two results will be dealt with separately.

Third, for many ${\mathbf{V}}{\mathbf{P}}$ decay modes, there are the
intermediate states. Take $\phi\eta$ mode as an example, $\phi\to K^{+}K^{-}$
and $\eta\to\gamma\gamma$, in calculation of branching ratio, the intermediate
decay branching ratios must be taken into consideration. Such kinds of effects
could also be included in Monte Carlo simulation. Therefore, it must be
careful to figure out how such kinds of effects are taken into account.
Moreover, there are some efficiency correction factors due to detector or
Monte Carlo simulation, which should also be considered. In a word, the
efficiency in the following analysis is the one that includes all kinds of
necessary effects, and is termed as the effective efficiency.

Fourth, as far as the aforementioned principle is concerned, the energy spread
will be tuned to give the maximum cross section that can satisfy the relation
$N_{tot}=\sigma_{max}\cdot{\cal L}$. CLEO data cleovp2005  are composed of two
sets, one with luminosity 2.74 pb-1 and the other 2.89 pb-1, which are taken
with energy spreads 1.5 MeV and 2.3 MeV, respectively. In the following
analysis, the data is treated as one set with total luminosity 5.63 pb-1
corresponding to the effective energy spread 1.68 MeV as displayed in Table 4.
It it worthy of noticing that unlike branching fraction evaluation, the
contribution due to QED continuum should not be subtracted from the observed
number of events, since the QED contribution is included in the calculation of
observed cross section. At last, since the error of number of events are
needed in analysis, for CLEO data, the maximum relative statistical error of
branching fraction is used to evaluate the corresponding error of the number
of events.

Chi-square method is adopted to fit the experimental data. The estimator is
constructed as

$\chi^{2}=\sum\limits_{i}\frac{[N_{i}-n_{i}(\vec{\eta})]^{2}}{(\delta
N_{i})^{2}}~{},$ (45)

where $N$ with the corresponding error ($\delta N$) denotes the experimentally
measured number of events while $n$ the theoretically calculated number of
events :

$n={\cal L}\cdot\sigma_{obs}\cdot\epsilon~{},$ (46)

where ${\cal L}$ is the integrated luminosity, $\epsilon$ is the effective
efficiency, and $\sigma_{obs}$ the observed cross section calculated according
to formula (42), which contains the parameters to be fit, such as $A$, $D$,
$D^{\prime}$, $s_{P}$, $s_{V}$, and the phase angle $\phi$. All these
parameters are denoted by the parameter vector $\vec{\eta}$ in Eq. (45). The
concrete form is determined by the parametrization form in Table 3. All
observed numbers of events together with the corresponding efficiencies
displayed in Table 6 are employed as input information. The data can be
grouped into four sets: two from BESIII, one with total luminosity 668.55
pb-1, the other with luminosity 161.63 pb-1; one from CLEO, with total
luminosity 5.63 pb-1 and one from BESII with luminosity 19.72 pb-1. There
might be some systematic difference among those datasets, so normalization
factors are introduced to take into account these systermatic effects.
However, only three relative (relative to the greatest data set of BESIII)
factors of luminosity are introduced with the belief that the relative
relations of measurements of each experiment group is more reliable than the
corresponding absolute values. The fit values of three factors $f_{cleo}$,
$f_{bes2}$, and $f_{bes3a}$ indicate that there indeed exists certain obvious
differences, the inconsistencies of these experiments from the highest
precision one range from 10% to 70%.

The fitting yields a $\chi^{2}$ of 18.91 with the number of degrees of freedom
being 19 for the destructive inference case :

$\begin{array}[]{rcl}\phi&=&-131.55^{\circ}\pm 13.05^{\circ},\\\
A&=&~{}0.577\pm 0.053~{},\\\ D&=&~{}0.334\pm 0.026~{},\\\
D^{\prime}&=&-0.025\pm 0.078~{},\\\ s_{P}&=&-0.277\pm 0.055~{},\\\
\theta_{P}&=&-16.10^{\circ}~{},\\\ s_{V}&=&~{}0.279\pm 0.195~{},\\\
\theta_{V}&=&~{}16.19^{\circ}~{},\\\ f_{cleo}&=&~{}0.937\pm 0.167~{},\\\
f_{bes2}&=&~{}1.277\pm 0.208~{},\\\ f_{bes3a}&=&~{}1.361\pm
0.252~{};\end{array}$ (47)

and a $\chi^{2}$ of 15.82 with the number of degrees of freedom being 19 for
the constructive inference case :

$\begin{array}[]{rcl}\phi&=&-144.31^{\circ}\pm 20.93^{\circ},\\\
A&=&~{}0.545\pm 0.047~{},\\\ D&=&~{}0.300\pm 0.027~{},\\\
D^{\prime}&=&-0.030\pm 0.068~{},\\\ s_{P}&=&-0.307\pm 0.058~{},\\\
\theta_{P}&=&-17.88^{\circ}~{},\\\ s_{V}&=&~{}0.386\pm 0.221~{},\\\
\theta_{V}&=&~{}22.73^{\circ}~{},\\\ f_{cleo}&=&~{}1.141\pm 0.220~{},\\\
f_{bes2}&=&~{}1.516\pm 0.264~{},\\\ f_{bes3a}&=&~{}1.703\pm
0.364~{};\end{array}$ (48)

where $\theta_{\alpha}=\arcsin s_{\alpha}~{}(\alpha=P,V)$.

The scan for parameter $\phi$ displays merely one minimum, which is a rather
uncommon case for multiple solution theory. When fitting cross sections with
several resonances or interfering background and resonances, one usually
obtains multiple solutions of parameters with equal fitting quality. Such a
phenomenon was firstly noticed experimentally moxh2010prd ; yuancz2010ijmpa ,
then some studies are performed from a mathematical point of view
yuancz2011cpc ; zhuk2011ijmpa ; hanx2018cpc ; baiyu2019prd . Especially in
Ref. baiyu2019prd , the source of multiple solutions for a combination of
several resonances or interfering background and resonances is found by
analyzing the mathematical structure of the Breit-Wigner function. It is
proved that there are exactly $2^{n-1}$ fitting solutions with equal quality
for $n$ amplitudes, and the multiplicity of the interfering background
function and resonance amplitudes depends on zeros of the amplitudes in the
complex plane. Our study involves the interference between strong and
electromagnetic amplitudes, corresponding to $n=2$ case, therefore it is
expected that there are two solutions for the phase angle $\phi$. For the
present fit result, we are not sure if this is the special feature of
$\psi^{\prime}\to{\mathbf{V}}{\mathbf{P}}$ decay, or the current data are not
accurate enough to differentiate two solutions which are close to each other.

Besides the fit for all data in Table 6, we also perform the fit for part of
it. In the light of parametrization of Table 3, it is obvious that
$\rho^{0}\eta$, $\rho^{0}\eta^{\prime}$, $\phi\pi^{0}$, and $\omega\pi^{0}$
are only concerned with $D$, $s_{P}$, and $s_{V}$, therefore the fit of the
data relevant to these final states can be used to determine the three
parameters. Based on the information of Table 6 related to $\rho^{0}\eta$,
$\rho^{0}\eta^{\prime}$, and $\omega\pi^{0}$ final states, the fitting yields
a $\chi^{2}$ of 7.08 with the number of degrees of freedom being 8 for the
destructive inference case :

$\begin{array}[]{rcl}D&=&~{}0.369\pm 0.023~{},\\\ s_{P}&=&-0.335\pm
0.044~{},\\\ \theta_{P}&=&-19.57^{\circ}~{},\\\ s_{V}&=&~{}0.853\pm
0.136~{},\\\ \theta_{V}&=&~{}58.54^{\circ}~{};\end{array}$ (49)

and a $\chi^{2}$ of 4.68 with the number of degrees of freedom being 8 for the
constructive inference case :

$\begin{array}[]{rcl}D&=&~{}0.351\pm 0.025~{},\\\ s_{P}&=&-0.405\pm
0.044~{},\\\ \theta_{P}&=&-23.90^{\circ}~{},\\\ s_{V}&=&~{}0.342\pm
0.400~{},\\\ \theta_{V}&=&~{}20.02^{\circ}~{}.\end{array}$ (50)

It can be seen that the fit value of $\theta_{P}$ is marginally consist with
the overall fit, which means that the value of $\theta_{P}$ is dominantly
determined by $\rho^{0}\eta$ and $\rho^{0}\eta^{\prime}$ final states. On the
contrary, since only the data of $\omega\pi^{0}$ final state is available, the
fluctuation of fit value of $\theta_{V}$ is fairly prominent. Anyway, if the
precise measurements of $\rho^{0}\eta$, $\rho^{0}\eta^{\prime}$,
$\phi\pi^{0}$, and $\omega\pi^{0}$ final states can be obtained, the mixing
angle of $\theta_{P}$ and $\theta_{V}$ is expected to be determined
definitely.

Table 7: Experimental data of $J/\psi$ decaying to $VP$ final states. For branching ratios, the first uncertainties are statistical, and the second systematic. The peak position is assumed at $\sqrt{s}=3.097$ GeV, while the continuum position is assumed at $\sqrt{s}=3.07$ or $\sqrt{s}=3.08$ GeV for BES measurement. The effective efficiency is presented and the one with star ($\ast$) is evaluated by virtue of the observed number of events $N^{obs}$, the total number of resonance events, and the corresponding of branching fraction. The symbol “$n.g.b.$” indicates that the continuum background is negligible. Mode | $N^{obs}$ | $N^{obs}$ | Efficiency | Branching Ratio | Detector
---|---|---|---|---|---
| (peak) | (continuum) | (%) | ($\times 10^{-3}$) |
$\pi^{+}\pi^{-}\pi^{0}$ | $1849852\pm 1360$ | $31.0\pm 5.6$ | $38.13$ | $21.37\pm 0.04^{+0.58}_{-0.56}$ | BESIII bes3pi2012
| $219691.0\pm 503.0$ |  | $17.83$ | $21.84\pm 0.05\pm 2.01$ | BESII besj3pi04
| $166$ |  | $2.68$ | $15.0\pm 2.0$ | MARKII Franklin83
$\rho\pi$ | $149.7$ |  | $2.68$ | $13.0\pm 3.0$ | MARKII Franklin83
| $543.0\pm 105.6$ |  | $6.08$ | $10\pm 2$ | DESY-H. wBartelg1976
$K^{*+}K^{-}+c.c.$ | $2285.0\pm 43.1$ |  | $5.814^{\ast}$ | $4.57\pm 0.17\pm 0.70$ | DM2 JJousset90jvp
| $24$ |  | $2.13$ | $2.6\pm 0.8$ | MARKII Franklin83
$K^{*0}\overline{K}^{0}+c.c$ | $1192.0\pm 39.1$ |  | $3.500^{\ast}$ | $3.96\pm 0.15\pm 0.60$ | DM2 JJousset90jvp
$\phi\eta$ | $346.0\pm 21.2$ |  | $6.286^{\ast}$ | $0.64\pm 0.04\pm 0.11$ | DM2 JJousset90jvp
| $2418.0\pm 65.2$ |  | $4.667^{\ast}$ | $0.898\pm 0.024\pm 0.089$ | BESII besjfpaet05
$\phi\eta^{\prime}$ | $167.0\pm 13.5$ |  | $4.736^{\ast}$ | $0.41\pm 0.03\pm 0.08$ | DM2 JJousset90jvp
| $728.0\pm 40.5$ |  | $2.311^{\ast}$ | $0.546\pm 0.031\pm 0.056$ | BESII besjfpaet05
| $31321\pm 201$ |  | $4.690$ | $0.510\pm 0.003\pm 0.032$ | BESIII bes16fetp
$\omega\eta$ | $378.0\pm 26.9$ |  | $3.074^{\ast}$ | $1.43\pm 0.10\pm 0.21$ | DM2 JJousset90jvp
| $4927.0\pm 91.0$ | $n.g.b.$ | $3.631^{\ast}$ | $2.352\pm 0.273$ | BESII besjopaet06
$\omega\eta^{\prime}$ | $6.0\pm 2.5$ |  | $0.388^{\ast}$ | $0.18^{+0.10}_{-0.08}\pm 0.03$ | DM2 JJousset90jvp
| $218.0\pm 32.8$ | $n.g.b.$ | $1.672^{\ast}$ | $0.226\pm 0.043$ | BESII besjopaet06
| $137\pm 20$ | $n.g.b.$ | $0.050$ | $0.208\pm 0.030\pm 0.014$ | BESIII besretp2017
$\rho^{0}\eta$ | $299.0\pm 34.0$ |  | $17.921^{\ast}$ | $0.194\pm 0.017\pm 0.029$ | DM2 JJousset90jvp
$\rho^{0}\eta^{\prime}$ | $19.2\pm 7.5$ |  | $2.690^{\ast}$ | $0.083\pm 0.030\pm 0.012$ | DM2 JJousset90jvp
| $3621\pm 83$ | $57.13\pm 11.03$ | $3.381$ | $0.0790\pm 0.0019\pm 0.0049$ | BESIII besretp2017
$\phi\pi^{0}$ | $24$ |  | $8.08$ | $<6.4\times 10^{-3}~{}(90\%C.L.)$ | BESII besjfpaet05
| ($838.5\pm 45.8$)777 The solution of constructive interference between $J/\psi\to\phi\pi^{0}$ and $J/\psi\to K^{+}K^{-}\pi^{0}$ decays. | $n.g.b.$ | $21.79$ | $(2.94\pm 0.16\pm 0.16)\times 10^{-3}$ | BESIII bes15fpi
| ($35.3\pm 9.3$)888 The solution of destructive interference between $J/\psi\to\phi\pi^{0}$ and $J/\psi\to K^{+}K^{-}\pi^{0}$ decays. | $n.g.b.$ | $21.79$ | $(1.24\pm 0.33\pm 0.30)\times 10^{-4}$ | BESIII bes15fpi
$\omega\pi^{0}$ | $222.0\pm 19.0$ |  | $7.171^{\ast}$ | $0.360\pm 0.028\pm 0.054$ | DM2 JJousset90jvp
| $2090.0\pm 67.3$ | $6.88\pm 2.85$ | $6.57$ | $0.538\pm 0.012\pm 0.065$ | BESII besjopaet06

###  IV.2 $J/\psi\to VP$ decay

Table 8: Comparison of various fitting results for the destructive interference solution data and positive angle case. The data with star (${\star}$) exclude the measurement of $\psi^{\prime}\to\omega\eta$ decay from BESII besjopaet06 . $n_{d}$ indicates the degree of freedom. $\vec{\eta}$ | All | All | DM2 | DM2 & BESII | DM2 & BESIII
---|---|---|---|---|---
& $\chi^{2}$ | data | data⋆ | data | data⋆ | data
$\chi^{2}$ | $4302.444$ | $1962.792$ | $91.254$ | $345.825$ | $239.325$
$\chi^{2}/n_{d}$ | $330.6$ | $163.6$ | $45.63$ | $49.40$ | $29.92$
$\phi$ | $(71.42\pm 1.82)^{\circ}$ | $(64.47\pm 1.91)^{\circ}$ | $(70.92\pm 3.43)^{\circ}$ | $(61.60\pm 3.46)^{\circ}$ | $(69.52\pm 3.18)^{\circ}$
$A$ | $2.697\pm 0.011$ | $2.631\pm 0.011$ | $2.013\pm 0.209$ | $2.460\pm 0.033$ | $2.914\pm 0.018$
$D$ | $0.538\pm 0.006$ | $0.562\pm 0.006$ | $0.395\pm 0.044$ | $0.497\pm 0.010$ | $0.690\pm 0.017$
$D^{\prime}$ | $-0.489\pm 0.017$ | $-0.638\pm 0.018$ | $-0.223\pm 0.049$ | $-0.679\pm 0.032$ | $0.171\pm 0.069$
$s_{P}$ | $-0.341\pm 0.004$ | $-0.472\pm 0.005$ | $-0.530\pm 0.046$ | $-0.755\pm 0.010$ | $-0.208\pm 0.028$
$\theta_{P}$ | $-19.96^{\circ}$ | $-28.17^{\circ}$ | $-32.76^{\circ}$ | $-48.98^{\circ}$ | $-11.98^{\circ}$
$s_{V}$ | $0.557\pm 0.002$ | $0.557\pm 0.001$ | $0.308\pm 0.045$ | $0.221\pm 0.010$ | $0.657\pm 0.025$
$\theta_{V}$ | $33.82^{\circ}$ | $33.85^{\circ}$ | $17.94^{\circ}$ | $12.76^{\circ}$ | $41.10^{\circ}$
$f_{dm2}$ | $0.590\pm 0.010$ | $0.603\pm 0.011$ | $1.020\pm 0.212$ | $0.698\pm 0.022$ | $0.422\pm 0.014$
$f_{bes2}$ | $1.177\pm 0.010$ | $1.166\pm 0.010$ |  | $1.295\pm 0.034$ |
$f_{bes3a}$ | $0.579\pm 0.009$ | $0.686\pm 0.015$ |  |  | $0.281\pm 0.012$
$f_{bes3b}$ | $1.233\pm 0.010$ | $1.864\pm 0.01-$ |  |  | $1.326\pm 0.011$

For $J/\psi$ decaying to ${\mathbf{V}}{\mathbf{P}}$ final state, there are
lots of experimental results, some of which are summarized in Table 7.
However, a few of measurements are obtained forty years ago with low statistic
samples. In Refs. bJeanMarie76  and gAlexander1978 , the branching fraction of
$\rho\pi$ is measured to be $(1.3\pm 0.3)\times 10^{-2}$ and $(1.6\pm
0.4)\times 10^{-2}$ based on 50 and 84 thousand $J/\psi$ events, respectively.
In Refs. Franklin83  and wBartelg1976 , the branching fraction of $\rho\pi$ is
measured to be $(1.3\pm 0.3)\times 10^{-2}$ and $(1.0\pm 0.2)\times 10^{-2}$
based on 0.4 and 0.87 million $J/\psi$ events, respectively. In Ref.
Brandelik:1978 , the study is mainly focused on $J/\psi$ radative decay into
$\pi\pi\gamma$ and $KK\gamma$. With 1.71 million $J/\psi$ events, the
branching fraction of $\rho\pi$ is merely obtained as a consistency check.
These kinds of measurements involving $\rho\pi$ final state are excluded from
the following analysis. The measurements of $J/\psi\to K^{*+}K^{-}+c.c.$ from
Ref. Franklin83 , $J/\psi\to\phi\pi^{0}$ from Ref. besjfpaet05 ,
$J/\psi\to\rho^{\pm}\pi^{\mp},~{}K^{*\pm}K^{\mp}$ from Ref. wBraunschweig1976
, and $J/\psi\to K^{*+}K^{-}+c.c.,~{}K^{*0}\overline{K}^{0}+c.c$ from Ref.
fVannucci77  are not adopted either due to low statistic.

Both Ref. mk3jvp85  and Ref. mk3jvp88  made a systematical measurement for
$J/\psi$ decaying into ${\mathbf{V}}{\mathbf{P}}$ final state. The latter
data, consisting of $5.8\times 10^{6}$ produced $J/\psi$’s, represents a
twofold increase over the former data ($2.7\times 10^{6}$), therefore the
better accuracy is realized in the latter analysis. Although all
${\mathbf{V}}{\mathbf{P}}$ channels, viz., $\rho\pi$, $\rho^{0}\eta$,
$\rho^{0}\eta^{\prime}$, $\omega\pi^{0}$, $\omega\eta$, $\omega\eta^{\prime}$,
$\phi\pi^{0}$, $\phi\eta$, $\phi\eta^{\prime}$, $K^{*+}K^{-}+c.c.$, and
$K^{*0}\overline{K}^{0}+c.c$, are measured, but the number of events and the
corresponding efficiencies are not provided, which leads to impossibility to
include these results in this analysis.

In addition, in Ref. bes3pi96 , cross sections of $\rho\pi$ final state are
measured at 29 different energy points covering a 40 MeV interval spanning the
$J/\psi$ resonance. Based on this data sample, which corresponds to a total
integrated luminosity of 238 nb-1, the branching fraction is determined to be
$(1.21\pm 0.20)$%. Such information is too distinctive to be combined with
those in Table 7. In Ref. bes09ksk  resorting to partial wave analysis
technique measured is the branching fraction of $J/\psi\to K^{*+}K^{-}+c.c.$,
whose feature is too different to be merged with other information. In both
Ref. aAubert3pi04  and Ref. aAubert07oget  the initial state radiation
technique is used to obtain the branching fractions of
$J/\psi\to\pi^{+}\pi^{-}\pi^{0}$ and $J/\psi\to\omega\eta$. Such kinds of
results can not be utilized in the present scheme of analysis.

Based on a sample of 1.31 billion $J/\psi$ events bes15fpi , the $K^{+}K^{-}$
mass spectrum is scrutinized and observed is a clear structure due to the
interference between $J/\psi\to\phi\pi^{0}$ and $J/\psi\to K^{+}K^{-}\pi^{0}$
decays. Such a interference yields two possible solutions of branching
fraction, that is $(2.94\pm 0.16\pm 0.16)\times 10^{-6}$ and $(1.24\pm 0.33\pm
0.30)\times 10^{-7}$, which correspond to the constructive and destructive
interferences, respectively. In the following fit, two results will be dealt
with separately.

The minimization estimator for $J/\psi$ is similar to that of $\psi^{\prime}$
as defined in Eq. (45) and the fit yields

$\begin{array}[]{rcll}\phi&=&-66.52^{\circ}\pm 1.82^{\circ}~{},&\mbox{ or
}~{}+71.42^{\circ}\pm 1.82^{\circ}~{},\\\ A&=&~{}2.622\pm 0.011~{},&\mbox{ or
}~{}2.697\pm 0.011~{},\\\ D&=&~{}0.523\pm 0.006~{},&\mbox{ or }~{}0.538\pm
0.006~{},\\\ D^{\prime}&=&-0.475\pm 0.017~{},&\mbox{ or }-0.489\pm
0.017~{},\\\ s_{P}&=&-0.341\pm 0.004~{},&\mbox{ or }-0.341\pm 0.004~{},\\\
\theta_{P}&=&-20.17^{\circ}~{},&\mbox{ or }-19.96^{\circ}~{},\\\
s_{V}&=&~{}0.557\pm 0.002~{},&\mbox{ or }~{}0.557\pm 0.002~{},\\\
\theta_{V}&=&~{}33.86^{\circ}~{},&\mbox{ or }~{}33.82^{\circ}~{},\\\
f_{dm2}&=&~{}0.624\pm 0.011~{},&\mbox{ or }~{}0.590\pm 0.010~{},\\\
f_{bes2}&=&~{}1.246\pm 0.010~{},&\mbox{ or }~{}1.177\pm 0.010~{},\\\
f_{bes3a}&=&~{}0.613\pm 0.010~{},&\mbox{ or }~{}0.579\pm 0.009~{},\\\
f_{bes3b}&=&~{}1.305\pm 0.010~{},&\mbox{ or }~{}1.233\pm 0.010~{};\end{array}$
(51)

with $\chi^{2}=4302$ for the destructive interference solution, and

$\begin{array}[]{rcll}\phi&=&-66.92^{\circ}\pm 1.77^{\circ}~{},&\mbox{ or
}~{}+71.82^{\circ}\pm 1.77^{\circ}~{},\\\ A&=&~{}2.633\pm 0.011~{},&\mbox{ or
}~{}2.713\pm 0.011~{},\\\ D&=&~{}0.536\pm 0.006~{},&\mbox{ or }~{}0.552\pm
0.006~{},\\\ D^{\prime}&=&-0.501\pm 0.017~{},&\mbox{ or }-0.516\pm
0.018~{},\\\ s_{P}&=&-0.397\pm 0.004~{},&\mbox{ or }-0.397\pm 0.004~{},\\\
\theta_{P}&=&-23.38^{\circ}~{},&\mbox{ or }-23.38^{\circ}~{},\\\
s_{V}&=&~{}0.492\pm 0.002~{},&\mbox{ or }~{}0.492\pm 0.002~{},\\\
\theta_{V}&=&~{}29.47^{\circ}~{},&\mbox{ or }~{}29.47^{\circ}~{},\\\
f_{dm2}&=&~{}0.620\pm 0.011~{},&\mbox{ or }~{}0.584\pm 0.010~{},\\\
f_{bes2}&=&~{}1.224\pm 0.010~{},&\mbox{ or }~{}1.152\pm 0.010~{},\\\
f_{bes3a}&=&~{}0.664\pm 0.011~{},&\mbox{ or }~{}0.625\pm 0.010~{},\\\
f_{bes3b}&=&~{}1.282\pm 0.010~{},&\mbox{ or }~{}1.207\pm 0.010~{};\end{array}$
(52)

with $\chi^{2}=4200$ for the constructive interference solution, where
$\theta_{\alpha}=\arcsin s_{\alpha}~{}(\alpha=P,V)$.

From a pure viewpoint of hypothesis test IGHuges2010 ; AGFrodeson19790 , the
ratio of the chi square value to the number of degrees of freedom should
approximate one for a good fit, but the values of $\chi^{2}$ for fitting
results in Eqs. (51) and (52) is horrendously large. Since the accuracy of
$J/\psi$ data is generally higher than that of $\psi^{\prime}$, the
discrepancies between the different experiments become much more prominent.
The data summarized in Table 6 can be grouped into four sets: two from BESIII,
one with total luminosity 394.65 pb-1, the other with luminosity 79.63 pb-1;
one from BESII with luminosity 15.89 pb-1 and one from MD2, with total
luminosity 5.053 pb-1. Four normalization factors of luminosity are introduced
to alleviate the possible inconsistence among the data from different
experiment groups. Nevertheless, such a kind of treatment is obviously not
sufficient enough so that certain great deviations still exist which lead to
huge chisquare value.

In order to figure out the effect due to the discrepancy of different
experiments on the value of $\chi^{2}$, the fits for different datasets are
performed respectively, viz., DM2 data, DM2 and BESII data, DM2 and BESIII
data, and all dataset, the results are tabulated in Table 8. The value of
$\chi^{2}$ is minimum for a sole experiment dataset (DM2 data), and the values
increase when more experiment datasets are fit together (DM2 and BESII
datasets or DM2 and BESIII datasets). It can be seen when all data are fit
together, the value of $\chi^{2}/n_{d}$ enhances almost one order of
magnitude. Moreover, after some filtration trial it is found that the
inconsistence of the measurement of $\psi^{\prime}\to\omega\eta$ decay from
BESII besjopaet06  from the other measurements is rather obviously. The fit is
performed for all data except for this one, the results are presented in the
second column of Table 8. The comparison of chisquares of the first two
columns indicates that the sole measurement of $\omega\eta$ channel from BESII
contributes more than half of chi square value, which seriously deteriorates
the $\chi^{2}$ probability of the fit. If we compare the two branching
fractions from DM2 JJousset90jvp  and BESII besjopaet06 , that is $(1.43\pm
0.10\pm 0.21)\times 10^{-3}$ and $(2.352\pm 0.273)\times 10^{-3}$, the former
is only about the half of the latter. The great discrepancy renders huge chi
square, which can only be settled down by further more accurate experimental
measurement.

Of course, there is a factor that has been neglected for data analysis. In the
fitting solely considered are the statistic uncertainties, if the systematic
uncertainties are included as well, it is expected that the chisquare could be
decreased to one half or one third of the present value. As a matter of fact,
we also performed a fit with increased error of $\omega\eta$ channel. If
$4927.0\pm 571.0$ instead of $4927.0\pm 91.0$ is used, $\chi^{2}=2055$ instead
of $\chi^{2}=4302$. Anyway, even so the value of $\chi^{2}$ is still too large
to be satisfied from a point of statistical view.

It also exists the possibility that the present parametrization form is not
exquisite enough to describe all data perfectly, but only more precise and
consistent experimental data can furnish quantitative evidence for or against
the present phenomenology model and pin down the problem we come across here.
The absence of a set of ideally experimental data is keenly felt.

Last but not least, we also perform the fit for data related to
$\rho^{0}\eta$, $\rho^{0}\eta^{\prime}$, $\omega\pi^{0}$, and $\phi\pi^{0}$
final states, the fitting yields a $\chi^{2}$ of 3.91 with the number of
degrees of freedom being 3 for the destructive interference case :

$\begin{array}[]{rcl}D&=&~{}0.404\pm 0.037~{},\\\ s_{P}&=&-0.288\pm
0.068~{},\\\ \theta_{P}&=&-16.73^{\circ}~{},\\\ s_{V}&=&~{}0.595\pm
0.003~{},\\\ \theta_{V}&=&~{}36.50^{\circ}~{},\\\ f_{dm2}&=&~{}1.096\pm
0.248~{},\\\ f_{bes2}&=&~{}1.773\pm 0.329~{};\end{array}$ (53)

and a $\chi^{2}$ of 3.97 with the number of degrees of freedom being 3 for the
constructive interference case :

$\begin{array}[]{rcl}D&=&~{}0.403\pm 0.036~{},\\\ s_{P}&=&-0.285\pm
0.068~{},\\\ \theta_{P}&=&-16.57^{\circ}~{},\\\ s_{V}&=&~{}0.660\pm
0.010~{},\\\ \theta_{V}&=&~{}41.30^{\circ}~{},\\\ f_{dm2}&=&~{}1.112\pm
0.252~{},\\\ f_{bes2}&=&~{}1.805\pm 0.337~{}.\end{array}$ (54)

For the destructive interference case the fit values of $\theta_{P}$ and
$\theta_{V}$ are marginally consist with those of overall fit, which means
that these four channels are necessary, sufficient, and efficient for
measurement of the mixing angles of pseudoscalar and vector mesons.

Table 9: Branching ratios of $\psi^{\prime},J/\psi\to VP$ extracted from PDG2022 pdg2022 . $Q_{h}$ is defined in Eq. (57) and calculated by the ratio of $B(\psi^{\prime}\to VP)$ to $B(J/\psi\to VP)$. $(K^{*+}K^{-})_{c.c.}$ and $(K^{*0}\overline{K}^{0})_{c.c}$ indicate $K^{*+}K^{-}+c.c.$ and $K^{*0}\overline{K}^{0}+c.c.$ respectively.  Mode | $B(\psi^{\prime}\to VP)$ | $B(J/\psi\to VP)$ | $Q_{h}$
---|---|---|---
| ($\times 10^{-5}$) | ($\times 10^{-3}$) | (%)
$\rho\pi$ | $3.2\pm 1.2$ | $16.9\pm 1.5$ | $0.2\pm 0.1$
$(K^{*+}K^{-})_{c.c.}$ | $2.9\pm 0.4$ | $6.0\pm 1.0$ | $1.1\pm 0.2$
$(K^{*0}\overline{K}^{0})_{c.c}$ | $10.9\pm 2.0$ | $4.2\pm 0.4$ | $2.6\pm 0.5$
$\phi\eta$ | $3.10\pm 0.31$ | $0.74\pm 0.08$ | $4.2\pm 0.6$
$\phi\eta^{\prime}$ | $1.54\pm 0.20$ | $0.46\pm 0.05$ | $3.3\pm 0.6$
$\omega\eta$ | $<1.1$ | $1.74\pm 0.20$ | $<0.632$
$\omega\eta^{\prime}$ | $3.2\pm 2.5$ | $0.189\pm 0.018$ | $16.9\pm 13.3$
$\rho^{0}\eta$ | $2.2\pm 0.6$ | $0.193\pm 0.023$ | $11.4\pm 3.4$
$\rho^{0}\eta^{\prime}$ | $1.87\pm 1.67$ | $0.081\pm 0.008$ | $23.1\pm 20.7$
$\omega\pi^{0}$ | $<0.004$ | $0.45\pm 0.05$ | $<0.0089$
$\phi\pi^{0}$ | $2.1\pm 0.6$ |  $(2.94\pm 0.23)\cdot 10^{-3}$999 The constructive solution. | $(7.1\pm 2.1)\cdot 10^{2}$
|  |  $(1.24\pm 0.45)\cdot 10^{-4}$101010 The destructive solution. | $(1.7\pm 0.8)\cdot 10^{4}$
Table 10: Fit results of branching ratio method for $\psi^{\prime},J/\psi\to VP$ decay. $n_{d.o.f}$ indicates degree of freedom. $\vec{\eta}$ | $\psi^{\prime}\to VP$ | $J/\psi\to VP$
---|---|---
& $\chi^{2}$ |  | constructive solution | destructive solution
$\chi^{2}$ | 1.995 | 184.075 | 184.075 | 169.651
$n_{d.o.f}$ | 3 | 5 | 5 | 5
$\phi$ | $(-133.56\pm 13.54)^{\circ}$ | $(61.65\pm 31.87)^{\circ}$ | $(-61.65\pm 20.89)^{\circ}$ | $(-152.18\pm 44.56)^{\circ}$
$A$ | $0.887\pm 0.092$ | $0.796\pm 0.027$ | $0.796\pm 0.031$ | $0.714\pm 0.040$
$D$ | $0.618\pm 0.079$ | $-0.192\pm 0.008$ | $-0.192\pm 0.008$ | $0.202\pm 0.007$
$D^{\prime}$ | $0.247\pm 0.166$ | $-0.277\pm 0.088$ | $-0.277\pm 0.060$ | $-0.233\pm 0.071$
$s_{P}$ | $-0.227\pm 0.074$ | $-0.585\pm 0.018$ | $-0.585\pm 0.022$ | $-0.439\pm 0.051$
$\theta_{P}$ | $-13.10^{\circ}$ | $-35.84^{\circ}$ | $-35.84^{\circ}$ | $-26.04^{\circ}$
$s_{V}$ | $0.029\pm 0.141$ | $0.493\pm 0.002$ | $0.493\pm 0.005$ | $0.594\pm 0.003$
$\theta_{V}$ | $1.64^{\circ}$ | $29.52^{\circ}$ | $29.52^{\circ}$ | $36.44^{\circ}$

##  V Discussion

Besides the cross section approach (CSA) used to analyze the data, there is
another way to deal with the information involving
$\psi^{\prime},J/\psi\to{\mathbf{V}}{\mathbf{P}}$ decay, which is call
branching ratio method (BRM). The idea is fairly simple, the so called
“reduced branching ratio” is related to the square of amplitude directly Haber
; a11 , that is

$\tilde{B}(\psi\to f)=|X_{f}+Y_{f}|^{2}~{},$ (55)

where $X_{f}$ and $Y_{f}$ are defined in Eqs. (39) and (38) for a certain
${\mathbf{V}}{\mathbf{P}}$ final state $f$, and the reduced branching ratio is
defined as follows

$\tilde{B}(\psi\to f)=\frac{{B}(\psi\to f)}{q_{f}^{3}}~{},$ (56)

where ${B}(\psi\to f)$ is the branching ratio of
$\psi(\psi=J/\psi,\psi^{\prime})$ decays to the final state $f$, and $q_{f}$
defined in Eq. (32), is the momentum of either particle in the center of mass
system for two-body decay. According to the formula (55) together with the
information in Table 9, the fits are performed and the fitting results are
displayed in Table 10.

Comparing results of two kinds of fits, for $\psi^{\prime}$ case, the values
of $\chi^{2}$ are both reasonable, merely one solution is found. Some results
are similar such as $A,D,s_{P}$ while others are different such as
$D^{\prime},s_{V}$. For $J/\psi$ case, the similarity is much worse,
especially for the destruction solution, only one minimum instead of two is
found for the phase angle. As a matter of fact, since the branching ratios
herein are averaged results that combine the various measurements due to many
experiments, such an admixture blurs the discrepancy between distinctive
experimental analysis. Therefore, the following discussion is based on the
results due to CSA instead of BRM.

Table 11: The fit results of phase angle for $\psi^{\prime}$ and $J/\psi$ two-body decays, some of which are from Ref. moxh2024 . The subscripts $d$ and $c$ indicate the results for destructive and constructive cases respectively.  Decay mode | phase angle $\phi$ (in degree)
---|---
$\psi^{\prime}\to B_{8}\overline{B}_{8}$ | $-94.59\pm 1.31$ | $+85.42\pm 2.25$
$J/\psi\to B_{8}\overline{B}_{8}$ | $-84.81\pm 0.70$ | $+95.19\pm 0.70$
$\psi^{\prime}\to B_{10}\overline{B}_{10}$ | $-75.51\pm 4.87$ | $+104.49\pm 4.91$
$J/\psi\to B_{10}\overline{B}_{10}$ | $-96.28\pm 17.23$ | $+83.27\pm 11.38$
$J/\psi\to B_{10}\overline{B}_{8}$ | $-89.97\pm 37.17$ | $+101.20\pm 71.87$
$\psi^{\prime}\to PP$ | $-58.19\pm 5.47$ | $+92.82\pm 5.62$
$J/\psi\to PP$ | $-87.25\pm 8.60$ | $+92.14\pm 8.61$
$\psi^{\prime}\to VP$ | $(-131.55\pm 13.05)_{d}$ |
| $(-144.31\pm 20.93)_{c}$ |
$J/\psi\to VP$ | $(-66.52\pm 1.82)_{d}$ | $(+71.42\pm 1.82)_{d}$
| $(-66.92\pm 1.77)_{c}$ | $(+71.42\pm 1.77)_{c}$

Table 11 summarizes various fitting results of phase angle for $\psi^{\prime}$
and $J/\psi$ two-body decays. The most exceptional solution is due to
$\psi^{\prime}\to{\mathbf{V}}{\mathbf{P}}$ decay. Such a situation can not
help reminding of the famous “$\rho\pi$ puzzle” in charmonium decays.

Theoretically, the OZI (Okubo-Zweig-Iizuka) ozi  suppressed decays of $J/\psi$
and $\psi^{\prime}$ to hadrons are via three gluons or a photon, in either
case, the perturbative QCD (pQCD) provides a relation appelquist

$\displaystyle Q_{h}$ $\displaystyle=$ $\displaystyle\frac{{\cal
B}_{\psi^{\prime}\rightarrow h}}{{\cal B}_{J/\psi\rightarrow h}}=\frac{{\cal
B}_{\psi^{\prime}\rightarrow e^{+}e^{-}}}{{\cal B}_{J/\psi\rightarrow
e^{+}e^{-}}}\approx(13.28\pm 0.29)\%~{},$ (57)

where ${\cal B}_{\psi^{\prime}\rightarrow e^{+}e^{-}}$ and ${\cal
B}_{J/\psi\rightarrow e^{+}e^{-}}$ are taken from PDG2022 pdg2022 . This
relation is expected to be held to a reasonable good degree for both inclusive
and exclusive decays. The so-called “$\rho\pi$ puzzle” is that the prediction
by Eq. (57) is severely violated in the $\rho\pi$ and several other decay
channels. The first evidence for this effect was found by Mark-II
Collaboration in 1983 Franklin83 . From then on many theoretical explanations
have been put forth to decipher this puzzle, refer to the treatise moxh07rprv
for a detailed review.

From the theoretical point of view, since the $Q$-value is smaller than the
expectation for $\rho\pi$, it may be caused either by enhanced $J/\psi$ or
suppressed $\psi^{\prime}$ decay rate. Another possibility is by both. Then
the relevant theoretical speculations can be classified into three categories
:

  1. 1.

$J/\psi$-enhancement hypothesis, which attributes the small $Q$-value to the
enhanced branching fraction of $J/\psi$ decays brodsky81 -brokar .

  2. 2.

$\psi^{\prime}$-suppress hypothesis, which attributes the small $Q$-value to
the suppressed branching fraction of $\psi^{\prime}$ decays chen -rosnersd .

  3. 3.

Other hypotheses, which are not included in the above two categories lixq
-hoyer .

With more and more experimental data are obtained, many theoretical
explanations have been ruled out and some still need to be tested. However,
“$\rho\pi$ puzzle” seems still a puzzle since no propose can explain all
existing experimental data satisfactorily and naturally. If we scrutinize the
$Q_{h}$ values in Table 9, it can seen that many values are suppressed
relative to the expected value 13.28% while some are enhanced, a fortiori for
$\phi\pi^{0}$ channel, the $Q_{h}$ value is several orders of magnitude
greater than the expectation. Furthermore the deviation seems rather
arbitrary, no regularity can be found. At the same time, if we investigate
Table 11, it can be seen that the phase angle for
$\psi^{\prime}\to{\mathbf{V}}{\mathbf{P}}$ decay is rather abnormal from the
other decays. So more we think about various information, more problems spring
up. It seems to trigger a Pandora’s Box of questions :

  1. 1.

$\psi^{\prime}$ and $J/\psi$ are both $S$-state of chamonium, the decay
pattern is similar for some channels but rather different for the others, the
prominent example is about $\rho\pi$ final state, some studies of which have
been performed in Ref. besrp2005 . What is the reason for such grotesquery
situations?

  2. 2.

Is the abnormal feature only for ${\mathbf{V}}{\mathbf{P}}$ mode, or for all
$VP$-like modes, such as $SV$, $SA^{-}$, $PA^{-}$, $VT$, $VA^{+}$, $TA^{-}$,
and $A^{+}A^{-}$ ? ($S,P,V,A$, and $T$ denote scalar, pseudoscalar, vector,
axial vector, and tensor, respectively; refer to Table 12 and Table 13 for
more details. )

  3. 3.

Does the exceptional phase angle have a connection with the abnormal $Q_{h}$
value? Is there a profound rationale for such a connection?

  4. 4.

From the viewpoint of phase angle and by virtue of fit result, the baryonic
mode seems more normal than the mesonic mode. What is the reason for such a
difference?

  5. 5.

From the viewpoint of phase angle and by virtue of fit result, $J/\psi$ decay
seems more normal than $\psi^{\prime}$. What is the reason for such a
difference?

  6. 6.

Does such a bizarre situation also exist for bottomonium decay?

  7. 7.

There is a suggestion that $\psi^{\prime}$ is not a pure state but a mixing of
$\psi(2^{3}S_{1})$ state and $\psi(1^{3}D_{1})$. Such a opinion is adopted to
explain the decrease $Q_{h}$ value for $\rho\pi$ channel rosnersd  and the
increase $Q_{h}$ value for $K^{0}_{S}K^{0}_{L}$ channel psippkskl . Anyway,
could this suggestion explain decay behaviors for all kinds of modes?

  8. 8.

Glueball was once put forth to explain “$\rho\pi$ puzzle” freund -anselm , but
the experimental research is unfavorable to the detailed analyses and
deductions due to this kind of scenario. However, the eccentric decay pattern
of ${\mathbf{V}}{\mathbf{P}}$ mode shown by their various $Q_{h}$ values
indeed implies the peculiar feature of this mode. The doubt is aggravated by
the fitted mixing angles of $\theta_{P}$ and $\theta_{V}$, none of which is
consist with theoretical expectation. According to mass matrix analysis,
$\theta_{P}=-11.3^{\circ}$ or $-24.5^{\circ}$ and $\theta_{V}=39.2^{\circ}$ or
$36.5^{\circ}$ corresponding to the quadratic or linear mass assumptions
respectively. The deviation from expectation maybe indicate the admixture of
pseudoscalar meson with some glueball-like component. This is an issue need to
be studied further.

It is true that a lot of measurements are available now, but even more precise
and systematical measurements are needed to clarify the dynamics of charmonium
decay.

##  VI Summary

The flavor-singlet principle, with assuming the flavor symmetry breaking
effects (both strong and electromagnetic breaking effects) as a special
$SU(3)$ octet, furnishes a criterion to figure out the effective interaction
Hamiltonian in tensor form for all kinds of two-body final states decaying
from a charmonium resonance. The generalized inherent ${C}$-parity for
multiplet is introduced, which plays a crucial role for determining the form
of effective Hamiltonian, especially for mesonic final states. Resorting to
the nonet notion, both octet and singlet representations for meson description
are synthesized together to acquire the effective Hamiltonian in a concise
way, then solving the meson mixing problem in charmonium decay. As an
application, by virtue of this scenario the relative phase between the strong
and electromagnetic amplitudes is measured for vector-pseudoscalar meson final
state, more information involving the interaction coupling coefficients is
obtained, all of which deepen our understanding of the dynamics of charmonium
decay. Furthermore, the fit results indicate that the measurements of four
final states, that is $\rho^{0}\eta$, $\rho^{0}\eta^{\prime}$,
$\omega\pi^{0}$, and $\phi\pi^{0}$, are necessary, sufficient, and efficient
for the study of mixing angles of pseudoscalar and vector multiplets.

In analysis of data samples taken in $e^{+}e^{-}$ collider, the details of
experimental effects, such as energy spread and initial state radiative
correction are taken into consideration in order to make full advantage of
experimental information and acquire the comprehensive results. However, the
discrepancy between different experimental measurement leads to large chi
square, which is fairly unfavorable from the statistic viewpoint. The data
analysis of this paper makes it urgent that further more precisely and
systematically experimental measurements should be performed based on BESIII
colossal data sample of charmonium decay, in order to figure out the unclear
issues we come across here.

By virtue of present analysis, the uniform parametrization scheme provides a
general description for various kinds of charmonium two-body decays and lays
an extensive foundation for more profound dynamics exploration in the future.

##  VII Appendix

This appendix is devoted to two issues. The first one is about the
${C}$-parity transformation of meson octet.

For two meson octets, denoting respectively by $O_{1}$ and $O_{2}$, defined
are the following terms which may be allowed or forbidden in the effective
Hamiltonian:

$[O_{1}O_{2}]_{0}=(O_{1})^{i}_{j}(O_{2})^{j}_{i}~{}~{},$ (58)
$([O_{1}O_{2}]_{f})^{i}_{j}=(O_{1})^{i}_{k}(O_{2})^{k}_{j}-(O_{1})^{k}_{j}(O_{2})^{i}_{k}~{}~{},$
(59)

and

$([O_{1}O_{2}]_{d})^{i}_{j}=(O_{1})^{i}_{k}(O_{2})^{k}_{j}+(O_{1})^{k}_{j}(O_{2})^{i}_{k}-\frac{2}{3}\delta^{i}_{j}\cdot(O_{1})^{i}_{j}(O_{2})^{j}_{i}~{}~{}.$
(60)

The aforementioned generalized inherent ${C}$-parity for meson octet is
introduced as a criterion for Hamiltonian terms, and its value (denoted as
$\eta_{O}$ ) is set to be equal to that of the neutral particle of the
corresponding octet. In Table 12 listed are some observed light mesons that
are classified into distinctive octets. The value of generalized ${C}$-parity
equals to the ${C}$-parity presented in the table.

Table 12:  Some meson octet particles. For $I=0$ meson, the mixing between singlet and octet always exists. The $f^{\prime}$ and $f$ are mixing states as defined in Eq. (61) or in Eqs. (63) and (64). In addition, $K_{1A}$ and $K_{1B}$ are nearly equal ($45^{\circ}$) mixtures of the $K_{1}(1270)$ and $K_{1}(1400)$.  Octet | $J^{PC}$ | $I=1$ | $I=1/2$ | $I=0$ | $I=0$
---|---|---|---|---|---
|  | $u\bar{d},\bar{u}d,$ | $u\bar{s},d\bar{s};$ | $f^{\prime}$ | $f$
|  | $(d\bar{d}-u\bar{u})/\sqrt{2}$ | $\bar{d}s,-\bar{u}s$ |  |
$P$ | $0^{-+}$ | $\pi$ | $K$ | $\eta$ | $\eta^{\prime}(958)$
$V$ | $1^{--}$ | $\rho(770)$ | $K^{*}(892)$ | $\phi(1020)$ | $\omega(782)$
$A^{-}$ | $1^{+-}$ | $b_{1}(1235)$ | $K_{1B}$ | $h_{1}(1380)$ | $h_{1}(1170)$
$S$ | $0^{++}$ | $a_{0}(1450)$ | $K^{*}_{0}(1430)$ | $f_{0}(1710)$ | $f_{0}(1370)$
$A^{+}$ | $1^{++}$ | $a_{1}(1260)$ | $K_{1A}$ | $f_{1}(1420)$ | $f_{1}(1285)$
$T$ | $2^{++}$ | $a_{2}(1320)$ | $K^{*}_{2}(1430)$ | $f^{\prime}_{2}(1525)$ | $f_{2}(1270)$
Table 13:  The determination of interaction terms in the effective Hamiltonian. The symbol $[O_{1}O_{2}]_{x}$ is shorthand of $[O_{1}O_{2}]_{0},[O_{1}O_{2}]_{d}$, and $[O_{1}O_{2}]_{f}$. Herein there are essentially two types of Hamiltonian forms, that is “$yyn$” which means both $[O_{1}O_{2}]_{0}$ and $[O_{1}O_{2}]_{d}$ terms are allowed in the effective Hamiltonian, and “$nny$” which means only $[O_{1}O_{2}]_{f}$ term is allowed. The symbol “$y$” indicates the allowed term, while “$n$” indicates forbidden. $O_{1}$ and $O_{2}$ denote octets $S,P,V,T,A^{+}$, and $A^{-}$, which are shown in Table 12. The superscript of symbol indicates the generalized inherent ${C}$-parity of corresponding octet. $[O_{1}O_{2}]_{x}$ | $S^{+}$ | $P^{+}$ | $V^{-}$ | $T^{+}$ | $A^{+}$ | $A^{-}$
---|---|---|---|---|---|---
$S^{+}$ | $nny$ | $nny$ | $yyn$ | $nny$ | $nny$ | $yyn$
$P^{+}$ |  | $nny$ | $yyn$ | $nny$ | $nny$ | $yyn$
$V^{-}$ |  |  | $nny$ | $yyn$ | $yyn$ | $nny$
$T^{+}$ |  |  |  | $nny$ | $nny$ | $yyn$
$A^{+}$ |  |  |  |  | $nny$ | $yyn$
$A^{-}$ |  |  |  |  |  | $nny$
Table 14: Amplitude parametrization form for decays of the $\psi^{\prime}$ or $J/\psi$ into PT final states. The table can also be used for other similar decays by appropriate change in labeling.  Final state | Parametrization form
---|---
$\pi^{+}a^{-}_{2}$/$\pi^{-}a^{+}_{2}$ |  | $\pm g_{e}$
$K^{+}K^{\star-}_{2}$/$K^{-}K^{\star+}_{2}$ | $\mp g_{m}$ | $\pm g_{e}$
$K^{0}\overline{K}^{\star 0}_{2}$/$\overline{K}^{0}{K}^{\star 0}_{2}$ | $\mp g_{m}$ |
Table 15: Amplitude parametrization form for decays of the $\psi^{\prime}$ or $J/\psi$ into $V~{}P$ final states. The results are based on the effective Hamiltonian in Eq. (71). With the assumptions in the last three columns, the results are the same as those due to the nonet approach. States | $g_{0}^{88}$ | $g_{m}^{88}$ | $g_{e}^{88}$ | $g_{0}^{11}$ | $g_{m}^{11}$ | $g_{e}^{11}$ | $g_{0}^{18}$ | $g_{m}^{18}$ | $g_{e}^{18}$ | $g_{0}^{81}$ | $g_{m}^{81}$ | $g_{e}^{81}$ | $g_{0}^{88}=g_{0}^{11}$ | $g_{m}^{88}=g_{m}^{18}=g_{m}^{81}$ | $g_{e}^{88}=g_{e}^{18}=g_{e}^{81}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$\rho^{0}\pi^{0}$ | 1 | $-2/3$ | $1/3$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | $-2/3$ | $1/3$
$\rho^{+}\pi^{-}$ | 1 | $-2/3$ | $1/3$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | $-2/3$ | $1/3$
$\rho^{+}\pi^{-}$ | 1 | $-2/3$ | $1/3$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | $-2/3$ | $1/3$
$K^{*+}K^{-}$ | 1 | $1/3$ | $1/3$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | $1/3$ | $1/3$
$K^{*-}K^{+}$ | 1 | $1/3$ | $1/3$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | $1/3$ | $1/3$
$K^{*0}\overline{K}^{0}$ | 1 | $1/3$ | $-2/3$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | $1/3$ | $-2/3$
$\overline{K}^{*0}K^{0}$ | 1 | $1/3$ | $-2/3$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | $1/3$ | $-2/3$
$\omega^{8}\eta^{8}$ | 1 | $2/3$ | $-1/3$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | $2/3$ | $-1/3$
$\omega^{8}\pi^{0}$ | 0 | 0 | $1/\sqrt{3}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | $1/\sqrt{3}$
$\rho^{0}\eta^{8}$ | 0 | 0 | $1/\sqrt{3}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | $1/\sqrt{3}$
$\omega^{1}\eta^{1}$ | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0
$\omega^{1}\eta^{8}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | $-2\sqrt{2}/{3}$ | $\sqrt{2}/3$ | 0 | 0 | 0 | 0 | $-2\sqrt{2}/{3}$ | $\sqrt{2}/3$
$\omega^{1}\pi^{0}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | $\sqrt{2/3}$ | 0 | 0 | 0 | 0 | 0 | $\sqrt{2/3}$
$\omega^{8}\eta^{1}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | $-2\sqrt{2}/{3}$ | $\sqrt{2}/3$ | 0 | $-2\sqrt{2}/{3}$ | $\sqrt{2}/3$
$\rho^{0}\eta^{1}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | $\sqrt{2/3}$ | 0 | 0 | $\sqrt{2/3}$
Table 16: Amplitude parametrization form for decays of the $\psi^{\prime}$ or $J/\psi$ into $V~{}P$ final states. The mixing between octet and singlet are taken into account according to Eqs. (26) and (24). The shorthand symbols are defined as $s_{\alpha}=\sin\theta_{\alpha}$ and $c_{\alpha}=\cos\theta_{\alpha}$ $(\alpha=V,P)$.  Decay mode | Coupling constant
---|---
$\psi\to X$ | $g_{0}^{88}$ | $g_{0}^{11}$ | $g_{m}$ | $g_{e}$
$\rho^{0}\pi^{0}$ | 1 | 0 | $-2/3$ | $1/3$
$\rho^{+}\pi^{-}$ | 1 | 0 | $-2/3$ | $1/3$
$\rho^{+}\pi^{-}$ | 1 | 0 | $-2/3$ | $1/3$
$K^{*+}K^{-}$ | 1 | 0 | $1/3$ | $1/3$
$K^{*-}K^{+}$ | 1 | 0 | $1/3$ | $1/3$
$K^{*0}\overline{K}^{0}$ | 1 | 0 | $1/3$ | $-2/3$
$\overline{K}^{*0}K^{0}$ | 1 | 0 | $1/3$ | $-2/3$
$\phi\eta$ | $c_{V}c_{P}$ | $s_{V}s_{P}$ | $\frac{2}{3}c_{V}c_{P}+\frac{2\sqrt{2}}{3}(c_{V}s_{P}+s_{V}c_{P})$ | $-\frac{1}{3}c_{V}c_{P}-\frac{\sqrt{2}}{3}(c_{V}s_{P}+s_{V}c_{P})$
$\phi\eta^{\prime}$ | $c_{V}s_{P}$ | $-s_{V}c_{P}$ | $\frac{2}{3}c_{V}s_{P}-\frac{2\sqrt{2}}{3}(c_{V}c_{P}-s_{V}s_{P})$ | $-\frac{1}{3}c_{V}s_{P}+\frac{\sqrt{2}}{3}(c_{V}c_{P}-s_{V}s_{P})$
$\omega\eta$ | $s_{V}c_{P}$ | $-c_{V}s_{P}$ | $\frac{2}{3}s_{V}c_{P}-\frac{2\sqrt{2}}{3}(c_{V}c_{P}-s_{V}s_{P})$ | $-\frac{1}{3}s_{V}c_{P}+\frac{\sqrt{2}}{3}(c_{V}c_{P}-s_{V}s_{P})$
$\omega\eta^{\prime}$ | $s_{V}s_{P}$ | $c_{V}c_{P}$ | $\frac{2}{3}s_{V}s_{P}-\frac{2\sqrt{2}}{3}(c_{V}s_{P}+s_{V}c_{P})$ | $-\frac{1}{3}s_{V}s_{P}+\frac{\sqrt{2}}{3}(c_{V}s_{P}+s_{V}c_{P})$
$\rho^{0}\eta$ | 0 | 0 | 0 | $\sqrt{\frac{1}{3}}c_{P}-\sqrt{\frac{2}{3}}s_{P}$
$\rho^{0}\eta^{\prime}$ | 0 | 0 | 0 | $\sqrt{\frac{1}{3}}s_{P}+\sqrt{\frac{2}{3}}c_{P}$
$\phi\pi^{0}$ | 0 | 0 | 0 | $\sqrt{\frac{1}{3}}c_{V}-\sqrt{\frac{2}{3}}s_{V}$
$\omega\pi^{0}$ | 0 | 0 | 0 | $\sqrt{\frac{1}{3}}s_{V}+\sqrt{\frac{2}{3}}c_{V}$

A remark is in order here. The physical isoscalars are mixtures of the $SU(3)$
wave function $\psi_{8}$ and $\psi_{1}$

$\left.\begin{array}[]{rcl}f^{\prime}&=&\psi_{8}~{}\cos\theta-\psi_{1}~{}\sin\theta~{},\\\
f&=&\psi_{8}~{}\sin\theta+\psi_{1}~{}\cos\theta~{},\end{array}\right.$ (61)

where $\theta$ is the nonet mixing angle and

$\left.\begin{array}[]{rcl}\psi_{8}&=&(u\bar{u}+d\bar{d}-2s\bar{s})/\sqrt{6}~{},\\\
\psi_{1}&=&(u\bar{u}+d\bar{d}+s\bar{s})/\sqrt{3}~{}.\end{array}\right.$ (62)

These mixing relations are often rewritten to exhibit the $u\bar{u}+d\bar{d}$
and $s\bar{s}$ components which decouple for the “ideal” mixing angle, such
that $\tan\theta_{i}=1/\sqrt{2}$ (or $\theta_{i}=35.3^{\circ}$). Defining
$\alpha=\theta+54.7^{\circ}$, one obtains the physical isoscalar in the flavor
basis

$f^{\prime}=\frac{1}{\sqrt{2}}(u\bar{u}+d\bar{d})\cos\alpha-s\bar{s}~{}\sin\alpha~{},$
(63)

and

$f=\frac{1}{\sqrt{2}}(u\bar{u}+d\bar{d})\sin\alpha+s\bar{s}~{}\cos\alpha~{},$
(64)

which is the orthogonal partner of $f^{\prime}$ (replace $\alpha$ by
$\alpha-90^{\circ}$). Thus for ideal mixing ($\alpha_{i}=90^{\circ}$),
$f^{\prime}$ becomes pure $s\bar{s}$ and the $f$ pure $u\bar{u}+d\bar{d}$.

Let’s return to the ${C}$-parity transformation issue. Under the
transformation of the generalized inherent ${C}$-parity,
$\hat{C}[O_{1}O_{2}]_{x}\to\xi_{x}[O_{1}O_{2}]_{x}$, where $x=0,d,f$, that is
$\xi_{0}=+1,\xi_{d}=+1,\xi_{f}=-1$. In addition, under ${C}$-parity
transformation, $\hat{C}O_{i}\to\eta_{O_{i}}O_{i},(i=1,2)$, synthetically,

$\hat{C}~{}[O_{1}O_{2}]_{x}=\eta_{O_{1}}\eta_{O_{2}}\xi_{x}[O_{1}O_{2}]_{x}~{},\\\
$ (65)

At the same time for the initial state of $\psi$

$\hat{C}~{}\psi=\eta_{\psi}\psi~{}~{}.\\\ $ (66)

Therefore, the term $[O_{1}O_{2}]_{x}$ is allowed in the effective Hamiltonian
as long as $\eta_{\psi}=-1=\eta_{O_{1}}\eta_{O_{2}}\xi_{x}$. Otherwise, the
term is forbidden. With this criterion, it is easy to figure out what kind of
terms can be presented in the effective Hamiltonian for various kinds of final
states, the results are summarized in Table 13.

By virtue of Table 13, two types of Hamiltonian forms exist. One type contains
both $[O_{1}O_{2}]_{0}$ and $[O_{1}O_{2}]_{d}$ terms, while the other contains
only $[O_{1}O_{2}]_{f}$ term, that is

${\cal
H}_{eff}^{O_{1}O_{2}}=g_{0}\cdot[O_{1}O_{2}]_{0}+g_{m}\cdot([O_{1}O_{2}]_{d})^{3}_{3}+g_{e}\cdot([O_{1}O_{2}]_{d})^{1}_{1}~{},$
(67)

or

${\cal
H}_{eff}^{O_{1}O_{2}}=g_{m}\cdot([O_{1}O_{2}]_{f})^{3}_{3}+g_{e}\cdot([O_{1}O_{2}]_{f})^{1}_{1}~{}.$
(68)

Comparing with Eqs. (21) and (22), Eq. (67) can be called the $VP$-type
Hamiltonian while Eq. (68) the $PP$-type Hamiltonian. For the most general
case of $PP$-type Hamiltonian, the mesons of final state may belong to
distinctive octets, take PT mode as an example, by virtue of Eq. (68), the
parametrization form is obtained and displayed in Table 14.

The second issue of this appendix is about another approach to derive the
effective Hamiltonian.

Besides the nonet approach, the singlet component can be treated separately.
Corresponding to the matrixes in Eqs. (8) and (9), the singlet matrixes are
introduced as follows

${\mathbf{S}_{V}}=\left(\begin{array}[]{ccc}\omega^{1}/\sqrt{3}&&\\\
&\omega^{1}/\sqrt{3}&\\\ &&\omega^{1}/\sqrt{3}\end{array}\right)~{}~{},$ (69)

and

${\mathbf{S}_{P}}=\left(\begin{array}[]{ccc}\eta^{1}/\sqrt{3}&&\\\
&\eta^{1}/\sqrt{3}&\\\ &&\eta^{1}/\sqrt{3}\end{array}\right)~{}~{}.$ (70)

Therefore, besides the octet-octet effective Hamiltonian, it also has the
singlet-singlet effective Hamiltonian and the octet-singlet effective
Hamiltonian, as listed in the following equations,

$\left.\begin{array}[]{rcl}{\cal
H}_{eff}^{88}&=&g_{0}^{88}[VP]_{0}+g^{88}_{m}([VP]_{d})^{3}_{3}+g^{88}_{e}([VP]_{d})^{1}_{1}~{},\\\
{\cal
H}_{eff}^{11}&=&g_{0}^{11}[S_{V}S_{P}]_{0}+g^{11}_{m}([S_{V}S_{P}]_{d})^{3}_{3}+g^{11}_{e}([S_{V}S_{P}]_{d})^{1}_{1}~{},\\\
{\cal
H}_{eff}^{18}&=&g_{0}^{18}[S_{V}P]_{0}+g^{18}_{m}([S_{V}P]_{d})^{3}_{3}+g^{18}_{e}([S_{V}P]_{d})^{1}_{1}~{},\\\
{\cal
H}_{eff}^{81}&=&g_{0}^{81}[VS_{P}]_{0}+g^{81}_{m}([VS_{P}]_{d})^{3}_{3}+g^{81}_{e}([VS_{P}]_{d})^{1}_{1}~{}.\end{array}\right.$
(71)

The calculation results are displayed in Table 15. If the mixing between octet
and singlet are taken into account according to Eqs. (26) and (24), the
results are changed into Table 16. If replacing $s_{V}$ and $c_{V}$ with
$\sqrt{1/3}$ and $\sqrt{2/3}$ respectively, the Table II in Ref. Haber  will
be recovered. It is should be noted that for $g_{m}$, there is an overall
normalized factor $-2/\sqrt{3}$ between our calculation and that of Ref. Haber
. Furthermore, if we introduce a new angle and define
$\sin\theta_{\gamma}\equiv\sqrt{1/3}$ and
$\cos\theta_{\gamma}\equiv\sqrt{2/3}$, the more compact form of Table 16 can
be obtained as shown in Table 3.

## Acknowledgment

This work is supported in part by National Key Research and Development
Program of China under Contracts No. 2023YFA1606003, No. 2023YFA1606000, No.
2020YFA0406302 and No. 2020YFA0406403.

## References

  * (1) M. Ablikim et al., (BESIII Collaboration), Nucl. Instr. Meth. A 614: 345 (2010).
  * (2) Kuang-Ta Chao and Yi-Fang Wang, Int. J. Mod. Phys. A 24, iii (2009)
  * (3) H. Kowalski and T. F. Walsh, Phys. Rev. D 14, 852 (1976).
  * (4) L. J. Clavelli and G. W. Intemann, Phys. Rev. D 28, 2767 (1983).
  * (5) H. E. Haber and J. Perrier, Phys. Rev. D 32, 2961 (1985).
  * (6) A. Seiden, H. F.-W. Sadrozinski, and H. E. Haber, Phys. Rev. D 38, 824 (1988).
  * (7) N. Morisita, I. Kitamura and T. Teshima, Phys. Rev. D 44, 175 (1991).
  * (8) R. Baldini et al., Phys. Lett. B 444, 111 (1998).
  * (9) K. Zhu, X. H. Mo, and C. Z. Yuan, Int. J. Mod. Phys. A 30 (2015) 1550148
  * (10) R. B. Ferroli et al., Phys. Lett. B 799, 135041 (2019).
  * (11) X. H. Mo and J. Y. Zhang, Phys. Lett. B 826, 136927 (2022).
  * (12) X. H. Mo, P. Wang, and J. Y. Zhang, Phys. Rev. D 107, 094009 (2023).
  * (13) M. Muraskin and S. L. Glashow, Phys. Rev. 132, 482 (1963).
  * (14) V. Gupta and V. Singh, Phys. Rev. 135, B1442 (1964).
  * (15) V. Gupta and V. Singh, Phys. Rev. 136, B782 (1964).
  * (16) Quang Ho-Kim and Xuan-Yem Pham, Elementary Particles and Their Interactions (Spinger-Verlag, Berlin, 1998)
  * (17) D. B. Lichtenberg, Unitary Symmetry and Elementary Particles (Second Edition, Academic press, New York, 1978)
  * (18) A. Hosaka and H. Toki, Quarks, Baryons and Chiral Symmetry” (World Scientific Publishing Co.Pte.Ltd., Singapore, 2001).
  * (19) P. A. Zyla et al. (PDG Collaboration), Porg. Theor. Exp. Phys. 2020, 083C01 (2020).
  * (20) P. Wang, C. Z. Yuan, and X. H. Mo, High Energy Phys. Nucl. Phys. 27, 465 (2003).
  * (21) S. Rudaz, Phys. Rev. D 14, 298 (1976).
  * (22) P. Wang, C. Z. Yuan, X. H. Mo, and D. H. Zhang, Phys. Lett. B 593, 89 (2004).
  * (23) P. Wang, X. H. Mo, and C. Z. Yuan, Int. J. Mod. Phys. A 21, 5163 (2006).
  * (24) P. Wang, C. Z. Yuan, and X. H. Mo, Phys. Rev. D 69, 057502 (2004).
  * (25) P. Wang, X. H. Mo, and C. Z. Yuan, Phys. Lett. B 557, 192 (2003).
  * (26) E. A. Kuraev and V. S. Fadin, Yad. Fiz. 41 (1985) 733 [Sov. J. Nucl. Phys. 41 (1985) 466].
  * (27) G. Altarelli and G. Martinelli, CERN 86-02, 47 (1986); O. Nicrosini and L. Trentadue, Phys. Lett. B 196, 551 (1987).
  * (28) F. A. Berends, G. Burgers and W. L. Neerven, Nucl. Phys. B297, 429 (1988); ibid. 304, 921 (1988).
  * (29) Y. S. Tsai, SLAC-PUB-3129 (1983).
  * (30) P. Alexander et al., Nucl. Phys. B 320, 45 (1989).
  * (31) J. Z. Bai et al. (BES Collaboration), Phys. Lett. B 355, 374 (1995).
  * (32) J. Z. Bai et al. (BES Collaboration), Phys. Lett. B 550, 24 (2002).
  * (33) A.W. Chao, Lectures on Accelerator Physics (World Scientific Publishing Co.Pte.Ltd., Singapore, 2020).
  * (34) CLEO-c/CESR-c Taskforces & CLEO-c Collaboration, Cornell University LEPP Report No. CLNS 01/1742 (2001) (unpublished).
  * (35) P. Wang, C. Z. Yuan, and X. H. Mo, HEP & NP 27, 465 (2003).
  * (36) X. H. Mo et al., High Energy Phys. Nucl. Phys. 28, 455 (2004).
  * (37) S. P. Chi, X. H. Mo, and Y. S. Zhu, High Energy Phys. Nucl. Phys. 28, 1135 (2004).
  * (38) M. Ablikim et al. (BESIII Collaboration), Chin. Phys. C 42, 023001 (2018).
  * (39) M. Ablikim et al. (BESIII Collaboration), Phys. Rev. D 96, 112012 (2017)
  * (40) N. E. Adam et al. (CLEO Collaboration), Phys. Rev. Lett. 94, 012005 (2005).
  * (41) M. Ablikim et al. (BESIII Collaboration), CPC (HEP & NP), 36, 915 (2012).
  * (42) M. Ablikim et al. (BESIII Collaboration), Phys. Lett. B 710, 594 (2012).
  * (43) M. Ablikim et al. (BESIII Collaboration), Chin. Phys. C 41, 013001 (2017).
  * (44) S. S. Fang et al., High Energy Phys. Nucl. Phys. 27, 277 (2003).
  * (45) M. Ablikim et al. (BESII Collaboration), Phys. Rev. D 73, 052007 (2006)
  * (46) J. Jousset et al. (DM2 Collaboration), Phys. Rev. D 41, 1389 (1990).
  * (47) D. Coffman et al., (Mark III Collaboration), Phys. Rev. D32, 2883 (1985).
  * (48) D. Coffman et al., (Mark III Collaboration), Phys. Rev. D38, 2695 (1988).
  * (49) M. Suzuki, Phys. Rev. D60, 051501 (1999).
  * (50) G. López Castro, J. L. Lucio M., and J. Pestieau, AIP Conference Proceedings 342, 441 (1995); arXiv:hep-ph/9902300.
  * (51) C. Z. Yuan, P. Wang, and X. H. Mo, Phys. Lett. B 567, 73 (2003).
  * (52) J. Z. Bai et al. (BES Collaboration), Phys. Rev. D69, 012003 (2004).
  * (53) L. Köpke and N. Wermes, Phys. Rep. 174, 67 (1989).
  * (54) P. Wang, C. Z. Yuan, and X. H. Mo, Phys. Lett. B574, 41 (2003).
  * (55) J. M. Gérard and J. Weyers, Phys. Lett. B 462, 324 (1999).
  * (56) M. E. B. Franklin et al., Phys. Rev. Lett. 51 (1983) 963.
  * (57) W. Bartel et al., Phys. Lett. 64B, 483 (1976).
  * (58) M. Ablikim et al. (BESII Collaboration), Phys. Lett. B 619, 247 (2005).
  * (59) M. Ablikim et al. (BESII Collaboration), Phys. Lett. B 614, 37 (2005).
  * (60) M. Ablikim et al. (BESII Collaboration), Phys. Rev. D 70, 112003 (2004)
  * (61) M. Ablikim et al. (BESIII Collaboration), Phys. Rev. D 87, 072011 (2012)
  * (62) M. Ablikim et al. (BESIII Collaboration), Phys. Rev. D 100, 092003 (2019)
  * (63) M. Ablikim et al. (BESII Collaboration), Phys. Rev. D 70, 112007 (2004)
  * (64) X. H. Mo, C. Z. Yuan, and P. Wang, Phys. Rev. D 82, 077501 (2010).
  * (65) C. Z. Yuan, X. H. Mo, and P. Wang, Int. J. Mod. Phys. A, 25: 5963 (2010)
  * (66) C. Z. Yuan, X. H. Mo, and P. Wang, CPC (HEP&NP), 35, 543 (2011).
  * (67) K. Zhu, X. H. Mo, C. Z. Yuan, and P. Wang, Int. J. Mod. Phys. A 26, 4511 (2011).
  * (68) X. Han and C. P. Shen, Chin. Phys. C 42, 043001 (2018).
  * (69) Y. Bai and D. Y. Chen, Phys. Rev. D 99, 072007 (2019).
  * (70) M. Ablikim et al. (BES Collaboration), Phys. Rev. D 70, 012005 (2004)
  * (71) W. Bartelg et al. (DESY-Heidelberg Collaboration), Phys. Lett. 64 B, 483 (1976).
  * (72) M. Ablikim et al. (BESII Collaboration), Phys. Rev. D 71, 032003 (2005)
  * (73) M. Ablikim et al. (BESIII Collaboration), Phys. Rev. D 93, 072008 (2016)
  * (74) M. Ablikim et al. (BESIII Collaboration), Phys. Rev. D 91, 112001 (2015)
  * (75) B. Jean-Marie et al. (MARKI Collaboration), Phys. Rev. Lett. 36 (1976) 291.
  * (76) G. Alexander et al. (PLUTO Collaboration), Phys. Lett. 72 B, 493 (1978).
  * (77) R. Brandelik et al. (DASP Collaboration), Phys. Lett. 74 B, 292 (1978).
  * (78) W. Braunschweig et al. (DASP Collaboration), Phys. Lett. 63 B, 487 (1976).
  * (79) F. Vannucci et al. (MARKI Collaboration), Phys. Rev. D 15, 1814 (1977)
  * (80) J. Z. Bai et al. (BES Collaboration), Phys. Rev. D 54, 1221 (1996).
  * (81) M. Ablikim et al. (BESIII Collaboration), Phys. Rev. D 100, 032004 (2019)
  * (82) B. Aubert et al. (BaBar Collaboration), Phys. Rev. D70 (2004) 072004.
  * (83) B. Aubert et al. (BaBar Collaboration), Phys. Rev. D73 (2006) 052003.
  * (84) Ifan G. Hughes and Thomas P. A. Hase, Measurements and their Uncertainties (Oxford University Press 2010)
  * (85) A. G. Frodeson, O. Skjeggestad, and H. Tøfte, “Pobability and statitics in particle physics” (Universitetsforlaget, Bergen-Oslo-Tromsø, 1979)
  * (86) R. L. Workman et al. (Particle Data Group), Prog. Theor. Exp. Phys. 2022, 083C01 (2022)
  * (87) S. Okubo, Phys. Lett. 5, 165 (1963); G. Zweig, CERN-preprint CERN-TH-401, 402, 412 (1964); Iizuka, Prog. Theor. Phys. Suppl. 37-38, 21 (1996).
  * (88) T. Appelquist and H. D. Politzer, Phys. Rev. Lett. 34, 43 (1975); A. De Rújula and S. L. Glashow, Phys. Rev. Lett. 34, 46 (1975).
  * (89) X. H. Mo, C. Z. Yuan, and P. Wang, High Energy Phys. Nucl. Phys. 31, 686 (2007).
  * (90) S. J. Brodsky and G. P. Lepage, Phys. Rev. D 24, 2848 (1981).
  * (91) P. G. O. Freund and Y. Nambu, Phys. Rev. Lett. 34, 1645 (1975). Iizuka, Prog. Theor. Phys. Suppl. 37-38, 21 (1996).
  * (92) W. S. Hou and A. Soni, Phys. Rev. Lett. 50, 569 (1983).
  * (93) W. S. Hou and A. Soni, Phys. Rev. D 29, 101 (1984).
  * (94) S. J. Brodsky, G. P. Lepage, and S. F. Tuan, Phys. Rev. Lett. 59, 621 (1987).
  * (95) C. T. Chan and W. S. Hou, Nucl. Phys. A 675, 367c (2000).
  * (96) W. S. Hou, Phys. Rev. D 55, 6952 (1997).
  * (97) M. Anselmino, M. Genovese, and E. Predazzi, Phys. Rev. D 44, 1597 (1991); M. Anselmino, M. Genovese, and D. E. Kharzeev, ibid. 50, 595 (1994).
  * (98) M. Chaichian and N. A. Törnqvist, Phys. Lett. B 323, 75 (1989).
  * (99) S. J. Brodsky and M. Karliner, Phys. Rev. Lett. 78, 4682 (1997).
  * (100) Y. Q. Chen and E. Braaten, Phys. Rev. Lett. 80, 5060 (1998).
  * (101) G. Karl and W. Roberts, Phys. Lett. B 144, 263 (1984).
  * (102) G. Karl and S-F. Tuan, Phys. Rev. D 34, 1692 (1986).
  * (103) S. S. Pinsky, Phys. Lett. B 236, 479 (1990).
  * (104) P. Artoisenet, J. M. Gérard, and J. Weyers, Phys. Lett. B 628, 211 (2005).
  * (105) V. L. Chernyak and A. R. Zhitnitsky, Phys. Rep. 112, 173 (1984).
  * (106) J. L. Rosner, Phys. Rev. D 64, 094002 (2001).
  * (107) X. Q. Li, D. V. Bugg, and B. S. Zou, Phys. Rev. D 55, 1421 (1997).
  * (108) M. Suzuki,Phys. Rev. D 57, 5171 (1998).
  * (109) M. Suzuki, Phys. Rev. D 60, 051501 (1999).
  * (110) M. Suzuki, Phys. Rev. D 63, 054021 (2001).
  * (111) J. P. Ma, Phys. Rev. D 65, 097506 (2002).
  * (112) L. J. Clavelli and G. W. Intemann, Phys. Rev. D 28, 2767 (1983).
  * (113) T. Feldmann and P. Kroll, Phys. Rev. D 62, 074006 (2000).
  * (114) P. Hoyer and S. Peigné, Phys. Rev. D 61, 031501 (2000).
  * (115) P. Wang, X. H. Mo, and C. Z. Yuan, Phys. Rev. D 70, 077505 (2004).