#  Are there the VP couplings in the $\psi(3770)$ non-charmed decays hidden
behind the current measurements?

D. Zhang, G. Rong, J.C. Chen
Institute of High Energy Physics, Beijing 100049, China
<EMAIL_ADDRESS>
###### Abstract

A global analysis of the full amplitudes for $e^{+}e^{-}\rightarrow$ VP
(Vector and Pseudoscalar) channels at $\sqrt{s}$ =3.773 GeV and 3.670 GeV,
which were measured by the CLEO-c Collaboration, shows that those measurements
are essentially nontrivial for searching for the $\psi(3770)$
non-$D\overline{D}$ decays. Unlike the nearly negative verdict on the
$\psi(3770)$ strong decays to the VP channels in the original analysis of the
CLEO-c data, there exist some unusual solutions that predict the remarkable
strength of $SU(3)$ symmetry VP decay of $\psi(3770)$ resonance, which give
some clue to understand the mechanism of $\psi(3770)$ non-$D\overline{D}$
decays and to reexplain the well-known $\rho-\pi$ puzzle in the J/$\psi$ and
$\psi(3686)$ decays.

###### pacs:

13.20.Gd, 13.66.Bc, 14.40.Gx

##  I Motivation

There is a long-standing puzzle in understanding the exist measurements for
$\psi(3770)$ and $D\overline{D}$ production cross sections at the peak of
$\psi(3770)$ production in $e^{+}e^{-}$ annihilation nondd ; R-Z-C . Potential
Models predict that $\psi(3770)$ decays into $D\overline{D}$ with branching
fraction of $\sim 100\%$. Recently careful investigation shows that the
branching fraction of $\psi(3770)$ non-$D\overline{D}$ decay would be up to
more than $10\%$ PhysRevLett_97_121801 ; PhysLettB_641_145 . It is very
interesting to know what are the exclusive $\rm{non-}D\overline{D}$ final
states of $\psi(3770)$ decays. Except the electromagnetic transitions and
hadronic transitions of $\psi(3770)$ to lower charmonium states, are there
indeed other significant exclusive non-charmed decay modes from $\psi(3770)$
decays?

In the charmonium decays, there is another long-standing puzzle in
understanding the $\rho\pi$ decays of J/$\psi$ and $\psi(3686)$. The partial
widths of $\rho\pi$ channel and other VP channels in the $\psi(3686)$ decays
are unexpectedly lower than those in J/$\psi$ decays. This is so called
“$\rho\pi$” puzzle. Are the J/$\psi$ decay rates enhanced by some unknown
mechanism or the $\psi(3686)$ decay rates are suppressed abnormally? To
investigate the possible source of this puzzle, it is also important to
measure the $\psi(3770)$ VP decay amplitude.

Recently, the BES Collaboration BES_KK  observed a large production cross
section for $e^{+}e^{-}\rightarrow K^{*}(892)^{0}\overline{K}^{0}$+c.c.

$\sigma(e^{+}e^{-}\rightarrow{K^{*0}\overline{K}^{0}}+c.c.)=(15.0\pm 4.6\pm
3.3)~{}~{}{\rm{pb}},$

at center-of-mass energy of $\sqrt{s}$=3.773 GeV and found that the
$K^{*\pm}(892)K^{\mp}$ production is suppressed. Taking into account the
possible interference between the strong decay amplitude and the continuum
production amplitude at $\sqrt{s}$=3.773 GeV, the BES Collaboration set an
upper limit on the strong decay partial width for $\psi(3770)\rightarrow
K^{*}(892)\overline{K}$+c.c. to be

$\Gamma(\psi(3770)\rightarrow
K^{*}(892)\overline{K}~{}+~{}\rm{c.c.})<29.0~{}~{}{\rm keV}$

at $90\%$ confidence level.

The CLEO-c Collaboration made more careful studies of twelve exclusive VP
decay channels for $\psi(3770)\rightarrow\rho\pi$,
$K^{*}(892)\overline{K}+c.c.$, $\omega\pi^{0}$, $\rho\eta$,
$\rho\eta^{\prime}$, $\omega\eta$, $\omega\eta^{\prime}$, $\phi\eta$,
$\phi\eta^{\prime}$ and $\phi\pi^{0}$ reported in Ref. CLEO_c_VP . The CLEO-c
Collaboration measured the cross sections for all of the channels at the
energies $\sqrt{s}$=3.773 GeV and $\sqrt{s}$=3.670 GeV. The CLEO-c results
show that the measured cross sections at $\sqrt{s}$=3.773 GeV are almost equal
to or even less than the ones measured at $\sqrt{s}$=3.670 GeV, which mean
that the net cross sections for the $\psi(3770)$ decays are consistent with
zero except only for the channel $\psi(3770)\rightarrow\phi\eta$. The negative
results about the $\psi(3770)$ strong VP decays led people ignoring the
important strong decay component existing in $\psi(3770)$ and only focusing
their attention on the form factors of those channels as well as the isospin
violation in electromagnetic interaction CLEO_c_VP .

In this paper, we develop a model to account for both the amplitudes of
electromagnetic (E-M) production and $\psi(3770)$ strong decay in the process
of $e^{+}e^{-}\rightarrow$ VP. By analyzing the cross sections for the
exclusive VP channels, which were measured by the CLEO-c Collaboration, we
extract out the branching fractions for $\psi(3770)$ decay to these VP final
states.

##  II The Model and the Formulae

In the $\psi(3686)$ decay sector, because of the smallness of the strong VP
decay coupling, the E-M decay component as well as the continuum (E-M)
component of the VP channel would be no longer the small amounts comparing
with those of strong decay. People have to deal with the two components
properly Wang_2003 . At the resonance peak, the production amplitude consists
of two parts, one is the decay amplitude of charmonium resonance and the other
is continuum E-M production amplitude. In the resonance decay part, there are
two components as well. They are the E-M decay amplitudes and the strong decay
amplitude. Totally, there are three components involved in the $e^{+}e^{-}$
annihilation process at the resonance peak, which are the strong decay
component, the E-M decay component and the continuum production component.

Unlike the VP decays of J/$\psi$ and $\psi(3686)$, the E-M decay amplitudes of
$\psi(3770)$ can be neglected due to the little tiny dileptonic decay
branching fraction. There are only the strong decay amplitude and the
continuum production amplitude in the $e^{+}e^{-}$ collision production at
$\sqrt{s}$=3.773 GeV. Typically, according to the conventional point of view,
the partial widths of the $\psi(3770)$ VP decay channels could be up to keV
order of magnitude, like their cousins in J/$\psi$ decays. However, due to the
large width of $\psi(3770)$ the decay amplitudes of those channels can not get
large amplification as the ones at the narrow resonance states. Associated
with the measurements of the form factors of channel $\omega\pi^{0}$ at the
energies of $\psi(3686)$ and $\psi(3770)$ resonance vicinities BES_ompi0 ;
CLEO_3686_VP ; CLEO_c_VP  the decay process with only a few keV partial width
of the rather wide resonance is really hard to be measured if one does not
consider the interference between the amplitudes of the strong decay and the
continuum production. As the measurements by CLEO-c CLEO_c_VP , both the
evident yield excess of channel $\phi\eta$ and the rather large yield deficit
of the $\rho\pi$ channel at the resonance peak hint that there must be rather
complex interference between the two kinds of amplitudes acting globally on
the VP channels. Some destructive interference just shows up at $\rho\pi$
channel in the “deficit” way. And more complex interferences cause the
$\phi\eta$ yields enhanced at resonance peak. In fact, in such complicate
interference case the decay contributions may easily be covered up by the
continuum contribution and the interference contributions. If one completely
neglects the buried decay contribution, the single E-M amplitude assumption
would not describe the measured cross sections well. In practice, it is
dangerous to measure the branching fractions for the $\psi(3770)$
non-$D\overline{D}$ decays by simply considering the net yields for the
channels observed at the peak of $\psi(3770)$ over that at the nearby off
resonance region. In this analysis we introduce the strong decay amplitudes in
the analysis formalism to see how the strong decay affects the VP production
at the $\psi(3770)$ resonance peak.

We describe the global decay of $\psi(3770)$ and the continuum production
process still based on the flavor $SU(3)$ invariant model, which was developed
thirty years ago haber_su3 , In this model the strange quark mass correction
in both of the strong coupling and the E-M coupling, the wave function nonet
symmetry breaking and the double Okubo-Zweig-Iizuka (DOZI) suppression effects
are all taken into account. As for the continuum production at the two energy
points $\sqrt{s}$=3.773 GeV and 3.670 GeV, except the coherent strong decay
amplitudes from the J/$\psi$ and $\psi(3686)$ tails, which can safely be
neglected from the calculations, there is only the continuum E-M amplitudes
itself. In addition, we guess that the incoherent component contributions
which are mainly from the initial state radiative (ISR) return to J/$\psi$ and
$\psi(3686)$ resonances have efficiently been rejected in the work reported in
Ref. CLEO_c_VP  and can be neglected in our analysis too.

Following the convention given in Ref. Seiden_1988 , we define that ${\bf g}$
represents the VP strong decay amplitude in the flavor $SU(3)$ symmetry limit;
${{\bf{g}}_{s}}$ represents the strong decay amplitude from $s$ quark,

$s_{g}=1-({\bf{g}}_{s}/|{\bf{g}}|)/2,$ (1)

characterizes the $SU(3)$ mass violation, which is as the same as the
parameter “$s$” given in the Tab. VIII of Ref. Seiden_1988 ; $\theta_{P}$
represents the $\eta-\eta^{\prime}$ mixing angle; the product $r\cdot{\bf{g}}$
represents the amplitude correction of the $SU(3)$ nonet symmetry violation
with the factor $(1-s_{\rm P})$ for a strange pseudoscalar production and with
the factor $(1-s_{\rm V})$ for a strange vector production. If $s_{\rm
V}=s_{\rm P}$=0, (exactly $s_{\rm V}+s_{\rm P}$=0), $r\cdot{\bf{g}}$ measures
the pure DOZI amplitude correction. Unlike the case in Ref. Seiden_1988 ,
because the E-M amplitude is no longer small comparing with the strong
amplitude, we have to consider both the isoscalar and isovector components of
the E-M amplitude. We define the E-M amplitude in the form of $SU(3)$ octet
matrix representation as

${\bf E}={\bf e}_{1}\cdot\rm{I}_{3}+{\bf e}_{0}\cdot\rm Y$

in which ${\bf e}_{0}$ and ${\bf e}_{1}$ are the isoscalar and isovector
components, respectively, and $\rm{I}_{3}$ and Y are, respectively, the
isospin third component and the hypercharge matrices in flavor $SU(3)$ octet
space. We define $\theta_{0}$ as the phase of ${\bf e}_{0}$ relative to
${\bf{g}}$, $\delta_{1}$ as the phase shift difference of ${\bf e}_{1}$ to
${\bf e}_{0}$ and a factor $(1/2-s_{e})$ as the correction for strange quark
coupling to E-M isoscalar part ${\bf{e}}_{0}$. We assume that the couplings
${\bf e}$’s and their phases do not change in the all VP channels, and their
moduli at the two different energy points $\sqrt{s}$=3.773 GeV and
$\sqrt{s}$=3.671 GeV only change with a $1/s^{3}$ dependence. If ${\bf
e}_{1}={\bf e}_{0}$, we return to common definition as Refs. haber_su3 ;
Seiden_1988  did.

Table 1: The Amplitudes for VP production in $e^{+}e^{-}$ annihilation at $\sqrt{s}=3.773$ GeV. The coupling ${\bf{g}}_{K}$ defined in channels $K^{*}\bar{K}+c.c.$ can be considered as a free parameter if one of ${\bf{g}}$ and ${\bf{g}}_{s}$ is fixed. Channel($ch$) | $M^{ch}_{res,3770}$ | $M^{ch}_{ctm,3770}$
---|---|---
$\rho^{0}\pi^{0},\rho^{\pm}\pi^{\mp}$ | ${\bf{g}}$ | ${\bf{e}}_{0}$
$\omega\eta$ | ${\bf{g}}X_{\eta}+\sqrt{2}r{\bf{g}}[\sqrt{2}X_{\eta}+(1-s_{\rm P})Y_{\eta}]$ | ${\bf{e}}_{0}X_{\eta}$
$\phi\eta$ | ${\bf{g}}_{s}Y_{\eta}+r{\bf{g}}(1-s_{\rm V})[\sqrt{2}X_{\eta}+(1-s_{\rm P})Y_{\eta}]$ | $-2{\bf{e}}_{0}(1-s_{e})Y_{\eta}$
$\omega\eta^{\prime}$ | ${\bf{g}}X_{\eta^{\prime}}+\sqrt{2}r{\bf{g}}[\sqrt{2}X_{\eta^{\prime}}+(1-s_{\rm P})Y_{\eta^{\prime}}]$ | ${\bf{e}}_{0}X_{\eta^{\prime}}$
$\phi\eta^{\prime}$ | ${\bf{g}}_{s}Y_{\eta^{\prime}}+r{\bf{g}}(1-s_{\rm V})[\sqrt{2}X_{\eta^{\prime}}+(1-s_{\rm{P}})Y_{\eta^{\prime}}]$ | $-2{\bf{e}}_{0}(1-s_{e})Y_{\eta^{\prime}}$
$\omega\pi^{0}$ | 0 | $3{\bf{e}}_{1}$
$\phi\pi^{0}$ | 0 | 0
$\rho^{0}\eta$ | 0 | $3{\bf{e}}_{1}X_{\eta}$
$\rho^{0}\eta^{\prime}$ | 0 | $3{\bf{e}}_{1}Y_{\eta}$
$K^{*0}\bar{K}^{0}+c.c.$ | ${\bf{g}}_{K}=({\bf{g}}+{\bf{g}}_{s})/2$ | $-{\bf{e}}_{0}(1/2-s_{e})-3/2{\bf e}_{1}$
$K^{*\pm}K^{\mp}$ | ${\bf{g}}_{K}=({\bf{g}}+{\bf{g}}_{s})/2$ | $-{\bf{e}}_{0}(1/2-s_{e})+3/2{\bf e}_{1}$

For the channels “$ch$”, ($ch$=$\rho^{0}\pi^{0},K^{*0}\overline{K}^{0}+c.c.$,
etc) at energy $\sqrt{s}$, $M^{ch}_{res,\sqrt{s}}$ denotes the resonance decay
amplitude and $M^{ch}_{ctm,\sqrt{s}}$ denotes the continuum production
amplitude. The total production amplitudes are then written as

$M^{ch}_{\sqrt{s}}=M^{ch}_{res,\sqrt{s}}+M^{ch}_{ctm,\sqrt{s}},$ (2)

in which

$M^{ch}_{res,3670}=0,$

and

$M^{ch}_{ctm,3670}=M^{ch}_{ctm,3770}\cdot f_{d}$

at the energy of $\sqrt{s}$=3.670 GeV, where $f_{d}=3.773^{3}/3.670^{3}$ is
the scaling factor for the $1/s^{3}$ energy dependence of the cross section.
The amplitudes $M^{ch}_{ctm,3770}$ and $M^{ch}_{res,3770}$ defined at
$\sqrt{s}$=3.773 GeV for all of the channels are listed in Tab. 1. In the
table, $X_{\eta}={\rm{cos}}(54.736^{o}+\theta_{P})$,
$Y_{\eta}={\rm{sin}}(54.736^{o}+\theta_{P})$,
$X_{\eta^{\prime}}=-{\rm{sin}}(54.736^{o}+\theta_{P})$ and
$Y_{\eta^{\prime}}={\rm{cos}}(54.736^{o}+\theta_{P})$, which are the same as
those given in Ref. Seiden_1988 . Those amplitudes completely control the
correlations among the VP channel productions. If any significant decay
amplitude $M^{ch}_{res,3773}$ given in Tab. 1 has been measured to be non-
zero, which means that the measured couplings ${\bf{g}}$, ${\bf{g}}_{s}$ etc.
are non-zero, this indicates that $\psi(3770)$ has a significant branching
fraction for decay to the non-charmed channel “$ch$”.

Table 2: The numbers of the events observed by CLEO-c, the subscripts $sw$ and $sb$ indicate the signal window and the side band window, respectively; the upper script 3.67 and 3.77 indicate the c.m. energies. Channel($ch$) | $N^{3.67}_{sw}$ | $N^{3.67}_{sb}$ | $N^{3.77}_{sw}$ | $N^{3.77}_{sb}$ | $\epsilon(\%)$
---|---|---|---|---|---
$\rho\pi$ | 43 | 5.4 | 314 | 44.8 | 26.3
$~{}\rho^{0}\pi^{0}$ | 21 | 3.4 | 130 | 33.0 | 32.5
$~{}\rho^{\pm}\pi^{\mp}$ | 22 | 2.0 | 184 | 11.8 | 23.1
$\omega\pi^{0}$ | 54 | 6.2 | 696 | 39.2 | 19.0
$\phi\pi^{0}$ | 1 | 1.6 | 2 | 40 | 16.5
$\rho^{0}\eta$ | 36 | 3.1 | 508 | 31.0 | 19.6
$\omega\eta$ | 4 | 0.0 | 15 | 6.0 | 9.9
$\phi\eta$ | 5 | 1.0 | 132 | 15.9 | 11.0
$\rho^{0}\eta^{\prime}$ | 1 | 0.0 | 27 | 0.9 | 2.9
$\omega\eta^{\prime}$ | 0 | 0.0 | 2 | 0.0 | 1.5
$\phi\eta^{\prime}$ | 0 | 0.0 | 9 | 2.0 | 1.2
$K^{*0}\bar{K}^{0}+c.c.$ | 38 | 0.4 | 501 | 18.1 | 8.8
$K^{*\pm}K^{\mp}$ | 4 | 1.0 | 36 | 32.4 | 16.0

We start the analysis from the observed numbers of the events for the VP
channels, their errors and their corresponding detection efficiencies at the
energies of $\sqrt{s}$=3.773 GeV and 3.670 GeV, which were published by CLEO-c
Collaboration CLEO_c_VP . The numbers and errors of the events are obtained in
both the signal windows and the side bands. Tab. 2 shows those numbers and
detection efficiencies which are given in Ref. CLEO_c_VP . Taking the numbers
of the events from Tab. 2, we obtain the numbers,

$N^{obs,ch}_{\sqrt{s}}=(N^{\sqrt{s}}_{sw}-N^{\sqrt{s}}_{sb})_{ch},$ (3)

of the observed events at $\sqrt{s}$ for the channel “$ch$”. Usually the
determinations of the detection efficiencies and the ISR corrections are all
energy dependent and relate to the production line shapes for those channels.
We guess that the determinations of the detection efficiencies given in Ref.
CLEO_c_VP  were done under the assumption of that the continuum cross section
line shape is in the $1/s^{3}$ energy dependence and the energy cut is at
$\sqrt{s^{\prime}}\geq J/\psi$ mass, where $\sqrt{s^{\prime}}$ is the center
of mass energy of the ISR return system. It should be stressed that, for the
ISR and FSR (Final State Radiative) corrections in the continuum processes,
our calculation gives $\eta_{ctm}$=1.19 at $\sqrt{s}$=3773 GeV and
$\eta_{ctm}$=1.11 at $\sqrt{s}$=3670 GeV, while Ref. CLEO_c_VP  gives
$\eta_{ctm}$=1/1.20=0.833 at both of the two energy points. As for the
resonance decay, the ISR correction is quite different from the one for the
continuum process. In our calculation, the ISR correction factor for the
resonance is $\eta_{res}$=0.824 including the FSR correction at $\sqrt{s}$
=3.773 GeV for all channels. In our ISR correction calculations, the $v^{3}$
phase space dependences have been taken into account, where
$v=\sqrt{[1-(m_{V}+m_{P})^{2}/s][(1-(m_{V}-m_{P})^{2}/s]}$ is the velocity of
the vector daughter in the CM decay system. However, in the calculation of
$\eta_{ctm}$’s, a mean production threshold of the channels has been set to
serve as the common threshold for all of those channels. For this reason, the
channel dependences of the corrections $\eta_{ctm}$’s are ignored in this
analysis. For the calculation of the contribution of the interference terms
among the amplitudes describing different processes, we have to know their own
detection efficiencies and ISR corrections. In the analysis we simply take the
geometric average of the related coefficients as the effective ones.

Using the amplitudes of VP channels in Eq.(2) and Tab. 1, the efficiencies for
those channels, the ISR and FSR correction as given above, the $v^{3}$ phase
space dependences and the luminosities $L_{\sqrt{s}}$ accumulated at the two
collision energy points of $\sqrt{s}$ =3.773 GeV and 3.670 GeV, we can
calculate the expected numbers $N^{exp,ch}_{3670}$ and $N^{exp,ch}_{3773}$ of
the event yields for those channels under the simplification assumption
concerning the interference terms mentioned above. For channel “$ch$”

$N^{exp,ch}_{3670}=L_{3670}\cdot
v^{3}_{3670}\cdot\epsilon^{ch}_{ctm}\cdot\eta_{ctm,3670}^{ch}\cdot|M^{ch}_{ctm,3670}|^{2}$

and

$\displaystyle N^{exp,ch}_{3773}=L_{3773}\cdot v^{3}_{3773}\times$
$\displaystyle\left|\sqrt{\epsilon^{ch}_{res}\cdot\eta_{res}^{ch}}\cdot
M^{ch}_{res,3773}+\sqrt{\epsilon^{ch}_{ctm}\cdot\eta_{ctm}^{ch}}\cdot
M^{ch}_{ctm,3773}\right|^{2}.$

Comparing the numbers $N^{obs,ch}_{\sqrt{s}}$ defined in Eq.(3) with the
expected one $N^{exp,ch}_{\sqrt{s}}$, we get the equation set

$N^{exp,ch}_{\sqrt{s}}=N^{obs,ch}_{\sqrt{s}}$ (4)

in which $ch$ = $\rho\pi$, $K^{*}(892)\overline{K}$+c.c., $\omega\pi^{0}$,
$\rho\eta$, $\rho\eta^{\prime}$,
$\omega\eta$,$\omega\eta^{\prime}$,$\phi\eta$, $\phi\eta^{\prime}$ and
$\phi\pi^{0}$, $\sqrt{s}$=3.670 and 3.773 GeV. Because of the zero observation
and zero expectation for the $\phi\pi^{0}$ channel, listed in Tab. 2 and Tab.
1, we can get rid of this channel in our analysis. So we only focus our
attention on the rest eleven channels.

In the maximum likelihood fit, leaving the parameters ${\bf g}$, ${\bf
g}_{s}$, ${\bf e_{0}}$, ${\bf e_{1}}$, $r$, $s_{e}$, $s_{\rm V}$, $s_{\rm P}$,
$\theta_{P}$ $\delta_{1}$ and ${\rm{cos}}\theta_{0}$ free, we can solve the
Eq.(4) by maximizing the probability function

$Prob=\prod_{ch,\sqrt{s}}^{2N_{ch}}P_{ch}(N^{obs,ch}_{\sqrt{s}},N^{exp,ch}_{\sqrt{s}})$
(5)

where $P_{ch}(N^{obs,ch}_{\sqrt{s}},N^{exp,ch}_{\sqrt{s}})$ is the probability
of finding $N^{obs,ch}_{\sqrt{s}}$ events with the assumed mean number
$N^{exp,ch}_{\sqrt{s}}$ at the energy $\sqrt{s}$, and get the solution of
Eq.(4) with most probable values of the parameters which control the VP
channel production at $\sqrt{s}$=3.773 and 3.670 GeV.

From the solutions of ${\bf{g}}$, $r$, ${\bf{e}}_{0}$ and/or ${\bf{e}}_{1}$
etc, one can get the Born cross sections for the VP channel production at
$\sqrt{s}$=3.773 or 3.670 GeV. For example, the Born cross section for decay
channel $\psi(3770)\rightarrow$ “$ch$” at $\sqrt{s}$ =3.773 GeV can be written
as

$\sigma^{ch}_{res}=v^{3}_{3773}\cdot|M^{ch}_{res,3773}|^{2}.$ (6)

The total Born cross section for $e^{+}e^{-}$ annihilation to the channel
“$ch$” at $\sqrt{s}$=3.773 GeV is given by

$\sigma^{ch}_{T}=v^{3}_{3773}\cdot|M^{ch}_{res,3773}+M^{ch}_{ctm,3773}|^{2}.$
(7)

While the Born cross section for the channel “$ch$” in continuum production at
$\sqrt{s}$ is then given by

$\sigma^{ch}_{ctm,\sqrt{s}}=v^{3}_{\sqrt{s}}\cdot|M^{ch}_{ctm,\sqrt{s}}|^{2}.$
(8)

##  III The results

According to the different coupling configurations of the amplitudes for the
VP channels, shown in Eq.(2) and Tab. 1, the twelve VP channels can be divided
into three sub-sets. The first one consists of the channels without E-M
isovector components, such as $\rho\pi$, $\omega\eta$, $\phi\eta$,
$\omega\eta^{\prime}$ and $\phi\eta^{\prime}$. The second one consists of the
E-M isovector component only, which is the pure E-M channels $\omega\pi^{0}$,
$\rho\eta$, and $\rho\eta^{\prime}$. The third subset includes only the
channels $K^{*}\overline{K}+c.c.$, for which the amplitudes involve all of the
two E-M coupling parts and the strong coupling component as well as their
interferences in the VP production. Since there is no common coupling
parameter in the first two sub-sets despite of the pseudoscalar mixing angle
$\theta_{P}$, one can simply try to solve Eq.(4) separately in the two sub-
sets at first.

###  III.1 The channels without isovector E-M amplitude

We start the analysis from the first set in which the channels are without the
isovector E-M amplitude contribution. From the numbers of events,
$N^{obs,ch}_{\sqrt{s}}$, observed at the two energy points of $\sqrt{s}$=3.671
GeV and 3.773 GeV for the six channels $\rho^{0}\pi^{0}$,
$\rho^{\pm}\pi^{\mp}$, $\omega\eta$, $\phi\eta$, $\omega\eta^{\prime}$ and
$\phi\eta^{\prime}$, we solve the Eq.(4). Leaving all of the related
parameters, such as $|{\bf g}|$, ${\bf g}_{s}$, $|{\bf e}_{0}|$,
$cos\theta_{0}$, $s_{e}$, $\theta_{P}$, $r$, $s_{V}$ and $s_{P}$ free, the
fitting yields
$|{\bf{g}}|=2.752\pm{0.291}$,
${\bf{g}}_{s}=1.279\pm{0.706}$,
$\rm{cos}\theta_{0}=-0.935\pm{0.141}$,
$|{\bf{e}}_{0}|=1.495\pm{0.126}$,
$s_{e}=0.264\pm{0.247}$,
$r=-0.236\pm{0.070}$,
$\theta_{\rm P}=(-17.83\pm{12.12})^{o}$,
$s_{V}=0.092\pm{0.941}$
and
$s_{P}=-0.091\pm{0.587}$.
Indeed, as hinted by the measurements from the CLEO-c CLEO_c_VP  and guessed
above, the solution of $\rm{cos}\theta_{0}=-0.935\pm{0.141}$ shows that
${\bf{g}}$ and ${\bf{e}}_{0}$ are almost opposite in the VP production at
$\psi(3770)$ resonance peak. Associated with earlier measurements in the
J/$\psi$ production and decays, in which the E-M decay amplitudes and the
strong decay amplitude are more likely with the phase difference of
$\phi_{e,res}-\phi_{g}\sim+90^{o}$ kopke ; achasov , and the phase difference
between the continuum amplitude and the one of resonance E-M decay at the
resonance peak is $\phi_{e,ctm}-\phi_{e,res}=+90^{o}$ too BES_95 ; Wang_03 ,
the measurement of the phase difference
$\theta_{0}=\phi_{e,ctm}-\phi_{g}=(160^{+20}_{-17})^{o}$ between
${\bf{e}}_{0}$ and ${\bf{g}}$ here is reasonable. The $90^{o}$ phase
difference between the E-M decay amplitudes and the strong decay amplitude
shown in J/$\psi$ decays kopke ; achasov  is mainly due to the exist of an
original $90^{o}$ phase difference from the short distance force range in the
on-shell three gluon annihilation of $1^{--}$ quarkonium states. This argument
should be kept in the $\psi(3770)$ decays. Out of the hard quark-gluon
interaction level the strong phase shift of long distance final state
interaction should essentially be small zheng . The resolutions of
$s_{V}=0.092\pm{0.941}$ and $s_{P}=-0.091\pm{0.587}$, which are consistent
with zero, seem to mean the small nonet symmetry breaking between the singlet
and octet wave functions.

Table 3: The fitted parameters. |  | Solution 1 | Solution 2 | Solution 3 | Solution 4
---|---|---|---|---|---
|  | the channels | the channels | all channels | all channels
|  | without isovector | with isovector | with | without
|  | E-M component | E-M component | counter terms | counter terms
$|{\bf g}|$ |  | $2.64^{+0.41}_{-0.54}$ | 2.64(fixed) | $2.67^{+0.38}_{-0.32}$ | $2.43^{+0.31}_{-0.29}$
${\bf g}_{s}$ |  | $1.23^{+0.67}_{-0.74}$ | $-1.79^{+0.42}_{-0.36}$ | $1.18^{+0.71}_{-0.78}$ | $-1.23^{+0.52}_{-0.49}$
$s_{g}$ |  | $0.27^{+0.14}_{-0.13}$ | $0.84^{+0.17}_{-0.20}$ | $0.28\pm{0.16}$ | $0.75\pm{0.42}$
${\bf g}_{s}^{\rm{add}}$ |  |  |  | $-1.51^{+0.43}_{-0.40}$ | 0.0(fixed)
$\rm{cos}\theta_{0}$ |  | $-0.91^{+0.12}_{-0.13}$ | -0.91(fixed) | $-0.91\pm{0.07}$ | $-0.86^{+0.06}_{-0.04}$
$|{\bf e}_{0}|$ |  | $1.49^{+0.12}_{-0.13}$ | 1.49(fixed) | $1.49^{+0.12}_{-0.13}$ | $1.44^{+0.13}_{-0.14}$
$s_{e}$ |  | $0.25^{+0.19}_{-0.22}$ | $-0.35\pm{0.11}$ | $0.20^{+0.20}_{-0.23}$ | $-0.29^{+0.13}_{-0.14}$
$a^{\mu}$ |  |  |  | $-0.55^{+2.38}_{-0.22}$ | 0.0(fixed)
$|{\bf e}_{1}|$ |  |  | $1.223^{+0.016}_{-0.018}$ | $1.226^{+0.017}_{-0.019}$ | $1.22\pm{0.02}$
$\delta_{1}$ |  |  | $(7.39^{+5.90}_{-7.69})^{o}$ | $(6.70^{+6.59}_{-8.19})^{o}$ | $(7.62^{+8.88}_{-6.02})^{o}$
$\theta_{P}$ |  | $(-18.7^{+5.8}_{-4.7})^{o}$ | $(-23.2\pm{2.2})^{o}$ | $(-22.5^{+1.9}_{-2.0})^{o}$ | $(-24.1^{+2.1}_{-2.3})^{o}$
$r$ |  | $-0.24\pm{0.06}$ |  | $-0.26\pm{0.06}$ | $-0.32^{+0.09}_{-0.08}$
$s_{V}$ |  | 0.0(fixed) |  | 0.0(fixed) | $-0.42^{+0.36}_{-0.59}$
$s_{P}$ |  | 0.0(fixed) |  | 0.0(fixed) | $0.45\pm{0.70}$
$\chi^{2}/n_{dof}$ |  | 4.63/5=0.93 | 3.98/5=0.80 | 9.26/11=0.84 | 13.2/11=1.20

So we can assume that the nonet symmetry maintains for the wave functions,
which means that the nonet symmetry violation is only in the DOZI coupling. If
we fix $s_{V}$= $s_{P}$ =0 in the fit, we get the parameters $|{\bf g}|$,
${\bf g}_{s}$, $|{\bf e}_{0}|$, $cos\theta_{0}$, $s_{e}$, $\theta_{P}$ and
$r$, which are listed in the second column (“Solution 1”) of Tab. 3, where
$\theta_{\rm P}=(-18.73^{+5.8}_{-4.7})^{o}$ indicates

$X_{\eta}=Y_{\eta^{\prime}}=0.809^{+0.045}_{-0.064}~{}\rm{and}~{}Y_{\eta}=-X_{\eta^{\prime}}=-0.588^{+0.068}_{-0.079}.$

The fit gives $\chi^{2}/n_{dof}=4.63/5=0.93$, which is also listed in Tab. 3.
With the ${\bf{g}}$ and ${\bf{g}}_{s}$, we obtain the $SU(3)$ mass correction

$s_{g}=0.268^{+0.140}_{-0.127}.$

Except for the three coupling strengths, $|{\bf e}_{0}|$, $|{\bf g}|$ and
$|{\bf g}_{s}|$, which have their own dynamics in the higher energy position,
the other relative correction parameters, $s_{g}$, $s_{e}$, $r$ and the mixing
angle $\theta_{\rm P}$ obtained from the fit are all reasonable comparing with
those obtained from J/$\psi$ decays measured by Mark-III and DM2 MARK3_DM2 .
However, from Eqs.(2) and (6), we find that the strong decay coupling $|{\bf
g}|$ gives quite large cross section, branching fraction and the partial width
for $\psi(3770)\rightarrow\rho\pi$ decay, which are

$\left.\begin{aligned}
&\sigma^{\rho\pi}_{res}=(18.2^{+6.11}_{-6.70})~{}~{}{\rm{pb}}\\\
&B^{\rho\pi}=(1.83^{+0.62}_{-0.67})\times 10^{-3}\\\
&\Gamma^{\rho\pi}=49.7^{+16.9}_{-18.3}~{}\rm{keV}.\end{aligned}\right\\}$ (9)

The partial width is almost in two order of magnitude higher than that of the
conventional typical partial width of the J/$\psi$ VP decays. The latter one
is at the order of 1 keV. If we assume that the fraction of the width of
$\psi(3770)\rightarrow\rho\pi$ to the width of $\psi(3770)\rightarrow
light~{}hadrons$ is roughly as the same as the one in the J/$\psi$ decays, the
huge partial width of $\psi(3770)\rightarrow\rho\pi$ would predict that about
$10\%$ of $\psi(3770)$ decays to non-$D\overline{D}$ final states. This
predicted branching fraction for $\psi(3770)\rightarrow{non-}D\overline{D}$ is
almost as the same as the one measured by BES-II Collaboration
PhysRevLett_97_121801 ; PhysLettB_641_145 . The large cross section
$\sigma^{\rho\pi}_{res}$ as shown in Eq.(9) is a factor of more than 3 of the
total $\rho\pi$ production cross section,

$\sigma^{\rho\pi}_{T}=5.35^{+1.45}_{-1.58}~{}~{}{\rm{pb}},$ (10)

obtained from Eq.(7). This cross section is consistent with the measurement
reported in Ref. CLEO_c_VP , (see Tab. 4). In the Tab. 4 we list all of the
production cross sections and the branching fractions for
$\psi(3770)\rightarrow$ VP predicted in this work, and also listed the
production cross sections of the VP channels given in Ref. CLEO_c_VP  as the
comparison. As for the $\rho\pi$ production cross section at $\sqrt{s}=3.670$
GeV in this measurement, we get

$\sigma^{\rho\pi}_{T,3670}=6.34^{+1.05}_{-1.03}~{}~{}{\rm{pb}},$ (11)

which is indeed higher than the one given in Eq.(10) at resonance peak.
Because of the different determinations of the ISR corrections in Ref.
CLEO_c_VP  and in our work, the decrease of the Born cross section of
$\rho\pi$ channel at $\psi(3770)$ resonance peak is not so large as that
obtained by CLEO-c Collaboration CLEO_c_VP  (see also Tab. 4). Owing to the
large cancellation between the two amplitudes ${\bf g}$ and ${\bf e}_{0}$, the
large cross section $\sigma^{\rho\pi}_{res}$ for
$\psi(3770)\rightarrow\rho\pi$ disappeared without the global amplitude
analysis.

###  III.2 The pure E-M channels $\omega\pi^{0}$, $\rho^{0}\eta$ and
$\rho^{0}\eta^{\prime}$ plus the channel $K^{*}\overline{K}$+c.c

We consider together the last two sub-sets of the channels in which the
amplitudes of E-M production contain isovector component. The isoscalar E-M
component and the strong decay coupling only serve in the channel
$K^{*}\overline{K}$+c.c., and the strong coupling here is with a combined form
${\bf{g}}_{K}=({\bf g}+{\bf g}_{s})/2$. Inserting the ten numbers of the
events observed at the two energy points of 3.670 GeV and 3.773 GeV for the
rest five channels into Eq.(4), leaving ${\bf e}_{1}$, $\delta_{1}$, $s_{e}$,
$\theta_{\rm P}$ and ${\bf g}_{K}$ free (or instead of ${\bf g}_{K}$, leaving
${\bf g}_{s}$ free but fixing ${\bf g}$ at some reasonable value such as
$|{\bf g}|$=2.635 obtained from last solution independently), and fixing
$|{\bf e}_{0}|$=1.494 and cos$\theta_{0}$=-0.907 obtained also from last
solution in assumption of that there is no more correction added to the
couplings ${\bf g}$ and $|{\bf e}_{0}|$ measured in last solution in the
channel $K^{*}\overline{K}$, we fit the numbers of events observed in the five
channels and obtain the solution of the free parameters. The results are
listed in the third column (“Solution 2”) of Tab. 3. The fit gives
$\chi^{2}/n_{dof}=3.98/5=0.80$.

From above two solutions, we see that the measured pseudoscalar mixing angles
$\theta_{\rm P}$ in the two independent measurements are consistent with each
other. The isovector E-M component, ${\bf{e}}_{1}$, is really split from the
isoscalar one, ${\bf{e}}_{0}$, with almost 2$\sigma$ deviation in magnitude
and with a non-zero phase shift difference. However, we note that the $s$
quark strong decay coupling ${\bf{g}}_{s}$ and its E-M coupling correction
$s_{e}$ in the solution 2 are both with the negative values, while they are
supposed to be positive in the conventional $SU$(3) invariant model with
simple static mass corrections. The minus $s_{e}$ is formally very likely to
be some anomalous “magnetic moments” term added to the $s$ quark E-M coupling.
As for the minus ${\bf g}_{s}$, which is obviously irrelative to the three
pure E-M channels, it seems that the $s$ quark strong coupling undergo almost
$180^{o}$ phase shift from the conventional $SU(3)$ strong interaction wave
function. The odd behavior of the minus ${\bf g}_{s}$ as well as the E-M
“anomalous magnetic moments” term for the $s$ quark indicate that there might
be some other dynamic sources or more complicated interaction correction
contributing to the production of $K^{*}\overline{K}$+c.c. in $e^{+}e^{-}$
annihilation.

Owing to the cancellation of ${\bf{g}}$ and the opposite ${\bf{g}}_{s}$ (small
${\bf g}_{K}$), the decay cross section, branching fraction and partial width

$\left.\begin{aligned}
&\sigma^{K^{*}\overline{K}~{}}_{res}=(0.558^{+0.702}_{-0.379})~{}~{}{\rm{pb}}\\\
&B^{K^{*}\overline{K}~{}}=(0.56^{+0.70}_{-0.38})\times 10^{-4}\\\
&\Gamma^{K^{*}\overline{K}~{}}=(1.52^{+1.92}_{-1.03})~{}~{}{\rm{keV}},\end{aligned}\right\\}$
(12)

for $\psi(3770)\rightarrow K^{*}\overline{K}$+c.c in this solution are quite
small comparing with that of $\psi(3770)\rightarrow\rho\pi$ measured in last
solution given in Eq.(9). These results are also listed in Tab. 4. The
remarkable increase of the E-M coupling $e_{0}$ by negative correction $s_{e}$
and the large interferences between the two E-M amplitudes and the strong
amplitudes result a serious asymmetry between the total production cross
sections of channels $K^{*0}\overline{K}^{0}$+c.c. and $K^{*\pm}K^{\mp}$. For
example, at $\sqrt{s}=3.773$ GeV the cross sections

$\left.\begin{aligned}
&\sigma^{K^{*0}\overline{K}^{0}~{}+c.c.}_{T}=19.30^{+3.55}_{-1.85}~{}~{}{\rm{pb}}\\\
&\sigma^{K^{*\pm}K^{\mp}~{}}_{T}<0.50~{}~{}({\rm{pb,~{}~{}at~{}90\%~{}confidence~{}level}}),\end{aligned}\right\\}$
(13)

are consistent with the observed values reported in Refs. CLEO_c_VP  and
CLEO_3686_VP . Using the parameters $|{\bf e}_{1}|$ and $\theta_{\rm P}$, and
from Eq.(8) we get the production cross sections of the three pure E-M
channels,

$\left.\begin{aligned}
&\sigma^{\omega\pi^{0}}_{ctm}=(11.74^{+0.31}_{-0.34})~{}~{}{\rm{pb}}\\\
&\sigma^{\rho\eta}_{ctm}=(8.00{\pm{0.22}})~{}~{}{\rm{pb}}\\\
&\sigma^{\rho\eta^{\prime}}_{ctm}=(2.57{\pm{0.28}})~{}~{}{\rm{pb}}.\end{aligned}\right\\}$
(14)

These are also listed in Tab. 4. The E-M production cross sections and the
total production cross sections obtained in this subsection are all
systematically lower than those measured by CLEO-c Collaboration CLEO_c_VP  by
about $30\%$, (see Tab. 4). Those differences are also due to the different
determinations of ISR corrections in the two works as mentioned above.

###  III.3 The global fit including all of the measured VP channels

We can introduce two additional effective counter terms,
${\bf{g}}^{\rm{add}}_{s}$ and ${\bf{a}}^{\mu}$, to compensate the odd behavior
appeared in the channel $K^{*}(892)\overline{K}$+c.c. for both the strong
coupling and E-M coupling of $s$ quark. We can simply assume that
${\bf{g}}^{\rm{add}}_{s}$ and ${\bf g}$, ${\bf{a}}^{\mu}$ and ${\bf e}_{0}$
are collinear, respectively. In this case, we define the amplitudes of
channels $K^{*}(892)\overline{K}$+c.c. as

$\displaystyle
M^{K^{*+}K^{-},K^{*-}K^{+}}_{res,3770}=({\bf{g}}+{\bf{g}}_{s})/2+{\bf{g}}^{\rm{add}}_{s},$
(15) $\displaystyle
M^{K^{*+}K^{-},K^{*-}K^{+}}_{ctm,3770}=-{\bf{e}}_{0}(1/2-s_{e})+3/2{\bf
e}_{1}+{\bf{a}}^{\mu},$ $\displaystyle
M^{K^{*0}\overline{K}^{0}+c.c.}_{res,3770}=({\bf{g}}+{\bf{g}}_{s})/2+{\bf
g}^{\rm{add}}_{s},$ $\displaystyle
M^{K^{*0}\overline{K}^{0}+c.c.}_{ctm,3770}=-{\bf{e}}_{0}(1/2-s_{e})-3/2{\bf
e}_{1}+{\bf{a}}^{\mu},$

as given in Tab. 1. With those amplitudes we globally solve the Eq.(4) with
all of the observed channels. Fixing $s_{V}$= $s_{P}$ =0 and leaving all of
the other parameters including ${\bf{g}}^{\rm{add}}_{s}$ and ${\bf{a}}^{\mu}$
free, the fit gives the most probable values of parameters $|{\bf{g}}|$,
${\bf{g}}_{s}$, ${\bf{g}}^{\rm{add}}_{s}$, $\rm{cos}\theta_{0}$,
$|{\bf{e}}_{0}|$, $s_{e}$, ${\bf{a}}^{\mu}$ $|{\bf{e}}_{1}|$, $\delta_{1}$,
$r$, $\theta_{\rm P}$. Those results are listed in the fourth column
(“Solution 3”) of Tab. 3 with $\chi^{2}/n_{dof}$=9.26/11=0.84. From the
parameters of solution 3 and Eqs.(2),(6),(7), (8) and (15), we can get the
production cross sections of the twelve VP channels, including the zero
measurement of channel $\phi\pi^{0}$. For example, for the channels $\rho\pi$
and $K^{*}(892)\overline{K}$+c.c., the decay cross sections from $\psi(3770)$
resonance, and their decay branching fractions and partial widths can be
calculated as

$\left.\begin{aligned}
&\sigma^{\rho\pi}_{res}=(18.68^{+4.12}_{-4.19})~{}~{}{\rm{pb}}\\\
&B^{\rho\pi}=(1.87^{+0.41}_{-0.42})\times 10^{-3}\\\
&\Gamma^{\rho\pi}=51.1^{+11.2}_{-11.5}~{}\rm{keV}\\\
&\sigma^{K^{*}\overline{K}~{}}_{res}=(0.56^{+1.28}_{-0.22})~{}~{}{\rm{pb}}\\\
&B^{K^{*}\overline{K}~{}}=(0.56^{+1.28}_{-0.22})\times 10^{-4}\\\
&\Gamma^{K^{*}\overline{K}~{}}=(1.52^{+3.50}_{-0.60})~{}~{}{\rm{keV}}.\end{aligned}\right\\}$
(16)

And the production cross sections of the three pure E-M channels can be
calculated as

$\left.\begin{aligned}
&\sigma^{\omega\pi^{0}}_{ctm}=(11.79^{+0.34}_{-0.36})~{}~{}{\rm{pb}}\\\
&\sigma^{\rho\eta}_{ctm}=(7.92\pm{0.41})~{}~{}{\rm{pb}}\\\
&\sigma^{\rho\eta^{\prime}}_{ctm}=(2.69\pm{{0.29}})~{}~{}{\rm{pb}}.\end{aligned}\right\\}$
(17)

The measured values in Eqs.(16),(17) are all consistent with those measured in
the last two subsections. As for the measurements of other channels, we have

$\left.\begin{aligned} &B^{\omega\eta}=(2.12^{+0.75}_{-0.48})\times 10^{-4}\\\
&B^{\omega\eta^{\prime}}<0.25\times 10^{-4},~{}({\rm~{}{at~{}90\%~{}c.l.}})\\\
&B^{\phi\eta}=(0.87^{+0.75}_{-0.58})\times 10^{-4}\\\
&B^{\phi\eta^{\prime}}<0.60\times
10^{-4},~{}({\rm~{}{at~{}90\%~{}c.l.}}),\end{aligned}\right\\}$ (18)

obtained in the assumption of that there exist the counter terms
${\bf{g}}^{\rm{add}}_{s}$ and ${\bf{a}}^{\mu}$ and $s_{V}$=$s_{P}$=0. If we
fix the strong couplings ${\bf g}$ and ${\bf g}_{s}$ to be the values which
force the partial widths of $\rho\pi$ and $K^{*}\overline{K}$ etc. to be at
the order of 1 keV which is at the same order of the J/$\psi$ VP decay
coupling dynamics, the fitted $\chi^{2}$ is 47.27 for 15 degree of freedom
which corresponds to 5.3 standard deviation worse than that of solution 3.

If there essentially were no the two counter terms specially for channel
$K^{*}\overline{K}$+c.c (${\bf{g}}^{\rm{add}}_{s}={\bf{a}}^{\mu}=0$), i.e.
there existed the negative coupling ${\bf{g}}_{s}$ and negative correction
$s_{e}$ universally allowed on all of other relative channels with the value
$s$ quark, the fit gives a somehow poor solution with large $\chi^{2}$=19.2
for 13 degree of freedom, which gives 2.7 standard deviation away from the
counter term assumption (solution 3). However, in this case, the effects of
the opposite ${\bf{g}}_{s}$ and negative $s_{e}$ in the related channels of
${\phi\eta}$ and ${\phi\eta}^{\prime}$ might be compensated by somehow larger
nonet symmetry breaking with non-zero $s_{V}$ and $s_{P}$. Ignoring the terms
${\bf{g}}^{\rm{add}}_{s}$ and ${\bf{a}}^{\mu}$ used in the last solution and
leaving the nonet symmetry breaking parameters $s_{V}$ and $s_{P}$ free, we
solve the Eq.(4) again to obtain a new solution. The results are listed in the
fifth column (“Solution 4”) of Tab. 3. The fit gives
$\chi^{2}/{\rm{n_{d.o.f.}}}$=13.2/11=1.2, which is a little bit higher than
those in other solutions. However, the solution still has the statistical
significance more than 5.6 standard deviation to the one obtained in the
assumption that only the E-M components (i.e. the ${\bf e}_{0}$, ${\bf e}_{1}$
and $\delta_{1}$, and the $\eta-\eta^{\prime}$ mixing angle) act on the VP
channel production. As for the solution 4, we obtain the unusual solution
again, which is like solution 2 with the negative ${\bf{g}}_{s}$ and negative
E-M correction $s_{e}$ for the $s$ quark couplings. The parameters of $r$,
$s_{\rm V}$ and $s_{\rm P}$ which associate with the nonet symmetry breaking
measurements are now with unexpected larger values. Nevertheless, as guessed
above, those unusual numbers may suggest more complicate dynamics for the $s$
quark production either in the channel $K^{*}\overline{K}$ or in all of the
relative channels, involving value $s$ quark production. This solution
predicts the decay cross sections, branching fractions and decay width of
channels $\rho\pi$ and $K^{*}(892)\overline{K}$ and the production cross
sections of three pure E-M production channels $\omega\pi^{0}$, $\rho\eta$ and
$\rho\eta^{\prime}$,

$\left.\begin{aligned}
&\sigma^{\rho\pi}_{res}=(15.50_{-3.52}^{+4.21})~{}~{}{\rm{pb}}\\\
&B^{\rho\pi}=(1.55^{+0.42}_{-0.35})\times 10^{-3}\\\
&\Gamma^{\rho\pi}=42.4^{+11.5}_{-9.6}~{}\rm{keV}\\\
&\sigma^{K^{*}\overline{K}~{}}_{res}=(1.14^{+1.12}_{-0.57})~{}~{}{\rm{pb}}\\\
&B^{K^{*}\overline{K}~{}}=(1.14^{+1.12}_{-0.57})\times 10^{-4}\\\
&\Gamma^{K^{*}\overline{K}~{}}=(3.11^{+3.06}_{-1.56})~{}~{}{\rm{keV}}\\\
&\sigma^{\omega\pi^{0}}_{ctm}=(11.67^{+0.35}_{-0.38})~{}~{}{\rm{pb}}\\\
&\sigma^{\rho\eta}_{ctm}=(8.10^{+0.60}_{-0.61})~{}~{}{\rm{pb}}\\\
&\sigma^{\rho\eta^{\prime}}_{ctm}=(2.43^{0.39}_{-0.37})~{}~{}{\rm{pb}},\end{aligned}\right\\}$
(19)

which are consistent with those obtained in the last solution with the counter
terms. However, for the channels $\omega\eta$, $\omega\eta^{\prime}$,
$\phi\eta$ and $\phi\eta^{\prime}$, we obtain the branching fractions of

$\left.\begin{aligned} &B^{\omega\eta}=(0.92^{+0.75}_{-0.92})\times 10^{-4}\\\
&B^{\omega\eta^{\prime}}<1.66\times 10^{-4},~{}({\rm~{}{at~{}90\%~{}c.l.}})\\\
&B^{\phi\eta}<1.57\times 10^{-4},~{}({\rm~{}{at~{}90\%~{}c.l.}})\\\
&B^{\phi\eta^{\prime}}=3.46^{+8.93}_{-1.75}\times
10^{-4},\end{aligned}\right\\}$ (20)

which are quite different comparing with those given in Eq.(18). As for the
branching fractions for $\psi(3770)\rightarrow\phi\eta$, the measured values
obtained by this analysis are also somehow inconsistent with that measured by
CLEO-c Collaboration CLEO_c_VP , (see Tab. 4). However both the results show
the large $\psi(3770)$ VP decay couplings. The summed total cross sections
over all VP channels of $\sigma^{ch}_{T}$ at both energy points of
$\sqrt{s}$=3.670 GeV and 3.773 GeV are listed in Tab. 4. The summed values of
solution 3 and solution 4 as well as the results of CLEO-c measurement show in
almost equals at the two energy points. The real resonance decays have been
“hidden”. This is mainly owing to the destructive nature of interference
between the couplings ${\bf g}$ and ${\bf e}_{0}$.

Table 4: Summary of the results. The errors for our measurements are statistical only. The branching fraction of channel $\phi\eta$ measured by CLEO-c Collaboration is directly from work CLEO_c_VP  and the upper limits of other channels measured by CLEO-c Collaboration are from the “Method II” of paper CLEO_c_VP  |  | Solution 1 | Solution 2 | Solution 3 | Solution 4 | CLEO-c
---|---|---|---|---|---|---
$\rho\pi$ |  $\sigma_{3670}$ (pb) | $6.86^{+1.14}_{-1.12}$ |  | $6.89^{+1.61}_{-1.60}$ | $6.43^{+1.53}_{-1.35}$ | $8.0^{+1.7}_{-1.4}\pm 0.9$
|  $\sigma_{T}$ (pb) | $5.35^{+1.45}_{-1.58}$ |  | $5.42^{+3.38}_{-2.00}$ | $5.08^{+3.57}_{-1.26}$ | $4.4\pm 0.3\pm 0.5$
|  $\sigma_{R}$ (pb) | $18.24^{+6.11}_{-6.70}$ |  | $18.68^{+4.56}_{-3.36}$ | $15.50^{+5.04}_{-2.96}$ | $<0.04$
|  Br $10^{-3}$ | $1.83^{+0.61}_{-0.67}$ |  | $1.87^{+0.46}_{-0.34}$ | $1.55^{+0.50}_{-0.30}$ | $<0.004$
$K^{*0}\bar{K}^{0}+c.c.$ |  $\sigma_{T}$ (pb) |  | $19.30^{+3.55}_{-1.85}$ | $19.34^{+11.29}_{-6.78}$ | $19.44\pm 3.61$ | $23.3\pm 1.1\pm 3.1$
|  $\sigma_{R}$ (pb) |  | $0.28^{+0.37}_{-0.19}$ | $0.28^{+0.64}_{-0.11}$ | $0.57^{+0.56}_{-0.29}$ | $<20.8$
|  Br $10^{-4}$ |  | $0.28^{+0.37}_{-0.19}$ | $0.28^{+0.64}_{-0.11}$ | $0.57^{+0.56}_{-0.29}$ | $<20.8$
$K^{*\pm}K^{\mp}$ |  $\sigma_{T}$ (pb) |  | $<0.502$ | $<2.59$ | $<0.77$ | $<0.6$
|  $\sigma_{R}$ (pb) |  | $0.28^{+0.37}_{-0.19}$ | $0.28^{+0.64}_{-0.11}$ | $0.57^{+0.56}_{-0.29}$ | $<0.1$
|  Br $10^{-4}$ |  | $0.28^{+0.37}_{-0.19}$ | $0.28^{+0.64}_{-0.11}$ | $0.57^{+0.56}_{-0.29}$ | $<0.1$
$\omega\pi^{0}$ |  $\sigma_{T}$ (pb) |  | $11.74^{+0.31}_{-0.34}$ | $11.79^{+0.33}_{-0.36}$ | $11.67^{+0.36}_{-0.38}$ | $14.6\pm 0.6\pm 1.5$
|  $\sigma_{R}$ (pb) |  | $0.0$ | $0.0$ | $0.0$ | $<0.06$
|  Br $10^{-4}$ |  | $0.0$ | $0.0$ | $0.0$ | $<0.06$
$\rho^{0}\eta$ |  $\sigma_{T}$ (pb) |  | $8.00\pm 0.22$ | $7.92\pm 0.41$ | $8.10\pm 0.44$ | $10.3\pm 0.5\pm 1.0$
|  $\sigma_{R}$ (pb) |  | $0.0$ | $0.0$ | $0.0$ | $<1.3$
|  Br $10^{-4}$ |  | $0.0$ | $0.0$ | $0.0$ | $<1.3$
$\rho^{0}\eta^{\prime}$ |  $\sigma_{T}$ (pb) |  | $2.57\pm 0.28$ | $2.69\pm 0.29$ | $2.43\pm 0.32$ | $3.8^{+0.9}_{-0.8}\pm 0.6$
|  $\sigma_{R}$ (pb) |  | $0.0$ | $0.0$ | $0.0$ | $<0.4$
|  Br $10^{-4}$ |  | $0.0$ | $0.0$ | $0.0$ | $<0.4$
$\omega\eta$ |  $\sigma_{T}$ (pb) | $0.45^{+1.07}_{-0.23}$ |  | $0.38^{+0.62}_{-0.20}$ | $0.32^{+0.42}_{-0.10}$ | $0.4\pm 0.2\pm 0.1$
|  $\sigma_{R}$ (pb) | $2.19^{+1.13}_{-0.74}$ |  | $2.12\pm 0.21$ | $0.92^{+0.92}_{-0.77}$ | $<0.1$
|  Br $10^{-4}$ | 2.19 |  | $2.12\pm 0.21$ | $0.92^{+0.92}_{-0.77}$ | $<0.1$
$\omega\eta^{\prime}$ |  $\sigma_{T}$ (pb) | $0.44^{+0.46}_{-0.26}$ |  | $0.59^{+0.59}_{-0.39}$ | $0.44^{+1.84}_{-0.41}$ | $0.6^{+0.8}_{-0.3}\pm 0.6$
|  $\sigma_{R}$ (pb) | $<0.33$ |  | $<0.27$ | $<0.47$ | $<1.9$
|  Br $10^{-4}$ | $<0.33$ |  | $<0.27$ | $<0.47$ | $<1.9$
$\phi\eta$ |  $\sigma_{T}$ (pb) | $3.99^{+2.75}_{-1.27}$ |  | $3.91^{+2.75}_{-1.27}$ | $3.70^{+2.78}_{-1.02}$ | $4.5\pm 0.5\pm 0.5$
|  $\sigma_{R}$ (pb) | $0.84^{+0.79}_{-0.58}$ |  | $0.87^{+0.85}_{-0.52}$ | $<1.45$ | $2.4\pm{0.6}$
|  Br $10^{-4}$ | $0.84^{+0.79}_{-0.58}$ |  | $0.87^{+0.85}_{-0.52}$ | $<1.45$ | $3.1\pm{0.7}$
$\phi\eta^{\prime}$ |  $\sigma_{T}$ (pb) | $1.95^{+4.49}_{-2.83}$ |  | $2.29^{+4.20}_{-1.76}$ | $1.69^{+1.37}_{-0.51}$ | $2.5^{+1.5}_{-1.1}\pm 0.4$
|  $\sigma_{R}$ (pb) | $<0.52$ |  | $<0.66$ | $3.46^{+4.90}_{-1.76}$ | $<3.8$
|  Br $10^{-4}$ | $<0.52$ |  | $<0.66$ | $3.46^{+4.90}_{-1.76}$ | $<3.8$
All “$ch$” summed |  $\sigma^{all}_{3670}$ (pb) |  |  | $58.3\pm 6.2$ | $62.3\pm 6.9$ | $64.2^{+15.8}_{-6.8}$
|  $\sigma^{all}_{T,3770}$ (pb) |  |  | $54.4^{+13.0}_{-7.4}$ | $52.9^{+6.3}_{-4.1}$ | $64.6^{+4.5}_{-4.3}$

##  IV Discussion and Summary

From above global analyses of the production cross sections for twelve
channels of $e^{+}e^{-}\rightarrow$ VP measured by CLEO-c Collaboration at the
$\psi(3770)$ resonance peak of $\sqrt{s}$ =3.773 GeV and at the energy of
$\sqrt{s}$=3.670 GeV in the continuum region, we obtained four different
solutions of the parameters to control the VP channel production. From Tab. 4
we see that the measured branching fractions or the production cross sections
except for the channels involving the nonet symmetry breaking are all
consistent with each other in the four solutions.

It is remarkable that the large $SU(3)$ symmetry strong decay strength $|{\bf
g}|$ leads to the huge branching fraction of a level of $10^{-3}$ for the
typical channel $\rho\pi$, which corresponds to the decay width of two order
of magnitude higher than that in J/$\psi$ decays. The large strong decay
coupling hidden behind the VP channel production in the $e^{+}e^{-}$
annihilation at the $\psi(3770)$ resonance peak might help people to
understand the sources of the $\psi(3770)$ non-$D\overline{D}$ decays and give
people some useful information to reexplain the long-standing $\rho\pi$ puzzle
in the $1^{--}$ charmonium state VP decays. Furthermore, the large strong
decay coupling with the opposite $s$-quark strong coupling and the minus E-M
correction $s_{e}$ or equivalently, the $s$-quark “anomalous magnetic moments”
${\bf{a}}^{\mu}$, required by the $K^{*}\overline{K}$ production as presented
in those solutions with different treatments and different assumptions in this
work are all unusual comparing with the conventional hard gluon annihilation
picture plus single $\psi(3770)$ resonance assumption. If those measurements
and analyses are all correct, one has to re-understand the strong interaction
dynamics which leads to the large Okubo-Zweig-Iizuka rule violation in the
vector meson $\psi(3770)$ decays and the strange behavior of the $s$ quark
couplings to the light hadron production in the energy region around
$\psi(3770)$ resonance. It seems that people has to seriously consider the
role of the long distance strong interaction corrections including the
$D(D_{s})$ meson exchange scheme to describe those anomalous phenomena and the
large Okubo-Zweig-Iizuka breaking in this energy region, like many authors did
Lipkin ; Q.Zhao .

The destructive interferences mechanism might manifest the possibility of the
significant existences of buried $\psi(3770)$ non-$D\overline{D}$ decays.
However, how do people understand the large net inclusive non- $D\overline{D}$
hadron branching fraction of $\psi(3770)$ decays measured by BES Collaboration
PhysRevLett_97_121801 ; PhysLettB_641_145  recently. Phenomenologically, for
example, if the behavior of the reversed $s$ quark strong coupling appeared in
VP channel is still maintained in those cases, it would lead to constructive
interference with the parallel continuum E-M amplitudes and cause abundant
strangeness meson production at the $\psi(3770)$ peak, resulting in the large
net cross section excess. The one of the exceptions is the channel
$K^{*}\overline{K}$+c.c., in which the E-M production is due to the magnetic
moments coupling. The minus E-M coupling correction for $s$ quark or the
counter term ${\bf{a}}^{\mu}$ as a special “anomalous magnetic moments”
enhances the E-M production at the continuum region and leads to the
observation of the equal cross sections of channel $K^{*}\overline{K}$+c.c. at
the two energy points $\sqrt{s}$=3.670 and 3.773 GeV in work CLEO_c_VP . This
argument can be cleared up by coming more precisely experimental measurements.
Of course, another probable outlet would be that there are more complicated
structures or contents in the $\psi(3770)$ resonance scope which are evident
in a measurement of cross sections for $e^{+}e^{-}\rightarrow$ hadrons by the
BES Collaboration BES_two ; BES_two_DD . This measurement of the cross section
indicates that there are somehow complicate “diresonance” structure instead of
the conventional single $\psi(3770)$ resonance assumption BES_two ; voloshin .
The exist of the extra substances might respond to the unusual behavior of
$\psi(3770)$ VP decays and the measured large non-$D\overline{D}$ branching
fraction of $\psi(3770)$ decays PhysRevLett_97_121801 ; PhysLettB_641_145 .

##  V Acknowledgments

This work is supported in part by the National Natural Science Foundation of
China under contract No. 10935007.

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