# The effective radius for production of baryon-antibaryon pairs from $\psi$
decays

Shu-Ming Wu<EMAIL_ADDRESS>CAS Key Laboratory of Theoretical
Physics, Institute of Theoretical Physics, Chinese Academy of Sciences,
Beijing 100190, China  University of Chinese Academy of Sciences (UCAS),
Beijing 100049, China  Jia-Jun Wu<EMAIL_ADDRESS>University of Chinese
Academy of Sciences (UCAS), Beijing 100049, China  Bing-Song Zou
<EMAIL_ADDRESS>CAS Key Laboratory of Theoretical Physics, Institute of
Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China
University of Chinese Academy of Sciences (UCAS), Beijing 100049, China
Synergetic Innovation Center for Quantum Effects and Applications (SICQEA),
Hunan Normal University, Changsha 410081, China

###### Abstract

By using the covariant L-S Scheme for the partial wave analysis, we deduce the
ratios between the S-wave and D-wave contributions from the recent data of
$\psi(1^{-})\to B_{8}(1/2^{+})\bar{B}_{8}(1/2^{-})$ from the BESIII
collaboration. For the $J/\psi\to\Lambda\bar{\Lambda}$ and
$J/\psi(\psi(2S))\to\Sigma^{+}\bar{\Sigma}^{-}$, the ratios are fixed and the
average angular momenta are computed to estimate the effective radii of these
processes. The results show that the effective radii of these decays of
$J/\psi(\psi(2S))$ are very small, which are around 0.04 fm. Thus, it is a
nice place to search excited baryon resonances with lower spin in the decays
of $J/\psi(\psi(2S))$. Furthermore, for the other $\psi(1^{-})\to
B_{8}(1/2^{+})\bar{B}_{8}(1/2^{-})$ reactions, we propose some methods to get
such effective radius.

##  I Introduction

The study of the baryon resonance is one of the most important topic in the
hadron physics. The long standing puzzle of “ missing resonances ” is still
unsolved Klempt:2009pi ; Crede:2013kia . A lot of excited nucleon resonances
predicted in the quark model have not been observed in the experiments. One
possible reason is that these resonances may be too broad to observe, or the
proper reactions which have significant signal of the resonances have not been
explored experimentally yet. Especially, the information of the nucleon
resonances with mass heavier than 2 GeV is still very limited.

In the review of Particle Data Group Zyla:2020zbs , all of the nucleon
resonances with mass larger than 2150 MeV are listed as follows, $N(2190)$,
$N(2200)$, $N(2250)$, $N(2300)$, $N(2570)$, $N(2600)$ and $N(2700)$. Among
these nucleon resonances, the $N(2190)$, $N(2200)$, $N(2250)$, $N(2600)$, and
$N(2700)$ resonances are found from experiments based on $\pi N,\gamma N$
reactions Sokhoyan:2015fra ; Svarc:2014zja ; Anisovich:2011fc ; Arndt:2006bf ;
Cutkosky:1980rh ; Hohler:1979yr , while $N(2300)$ and $N(2570)$ are from the
decay final states of $\psi(2S)$ Ablikim:2012zk . In the analysis of $\pi N$,
$\gamma N$ reactions, Bonn-Jülich group Ronchen:2012eg  and ANL-Osaka group
Kamano:2013iva  used dynamical coupled-channels approach to extract the
$N^{*}$ resonances with various effective interactions. Bonn-Gatchina group
Sarantsev:2009zz  and GWU/SAID group Arndt:2006bf  used a K-matrix approach
with Breit-Wigner resonances to analyze related channels. And the KSU group
Hunt:2018wqz  used a generalized energy-dependent Breit-Wigner parametrization
of amplitudes to treat these channels. The spins of all these $N^{*}$
resonances with mass heavier than 2.15 GeV are found to be larger than 5/2.
However, from the $\psi$ decay, the two new $N^{*}$ resonances found by BES
collaboration from partial wave analysis (PWA) of
$\psi^{\prime}\to\bar{N}N^{*}+c.c.\to\bar{N}N\pi$ have spin less than or equal
to 5/2. It is interesting to find that the spins of the heavy $N^{*}$
resonances generated from $\pi N,\gamma N$ reaction are obviously larger than
these from the $\psi$ decay.

In fact, the effective radii for these $N^{*}$ production processes play an
important role for this kind of spin selection. With larger effective radius
($r_{\text{eff}}$), the involved orbit angular momentum ($L$) should also be
larger because of $L=r_{\text{eff}}\times p$, where $p$ is the relative
momentum of two hadrons in the final states or the initial states. Since the
size of $N$ is around $1$ fm, the effective radius of the $\pi N$ and $\gamma
N$ reaction will be around $1$ fm. For the $N^{*}$ with mass larger than 2
GeV, the momentum of $N$ is around $1$ GeV at the rest frame of the $N^{*}$.
Thus the angular momentum of $\pi N$ will be $L\sim r_{\text{eff}}\times p\sim
1{\text{fm}}\cdot{\text{GeV}}\sim 5$. For the $\pi N$ and $\gamma N$
reactions, the total angular momentum ($J$) of the initial state is from
$L-1/2$ to $L+1/2$ and $L-3/2$ to $L+3/2$, respectively. It leads to the spins
of the $N^{*}$ resonances generated from $\pi N$ and $\gamma N$ being roughly
from $7/2$ to $13/2$. Correspondingly, the spins of $N^{*}$ resonances
observed from $\pi N$ and $\gamma N$ reactions in Refs. Sokhoyan:2015fra ;
Svarc:2014zja ; Anisovich:2011fc ; Arndt:2006bf ; Cutkosky:1980rh ;
Hohler:1979yr  are all larger than $5/2$. However, in the $\psi$ decay, the
$\bar{N}N^{*}$ only can be generated from the annihilation of the charm and
anti-charm quarks, therefore, the effective radius is small, hence leads to a
small orbit angular momentum between $\bar{N}N^{*}$. Thus, qualitatively
speaking, the spin of $N^{*}$ generated from the $\psi$ decay cannot be very
large. Indeed only $N^{*}$ with low spins were observed by BES collaboration
in their PWA of various $N^{*}$ production processes from $J/\psi$ and
$\psi(2S)$ decays Ablikim:2012zk ; Bai:2001ua ; Ablikim:2004ug ;
Ablikim:2009iw ; Ablikim:2013vtm .

To give a quantitative estimation of the upper limit for the spin of $N^{*}$
produced from $J/\psi$ and $\psi(2S)$ decays, it is important to study the
effective interaction radius $(r_{{\text{eff}}})$ for these processes. With
$\bar{L}=r_{{\text{eff}}}\times p$, $r_{{\text{eff}}}$ can be determined by
calculating the average orbital angular momentum $(\bar{L})$ and the relative
momentum $(p)$. This article mainly studies the interaction radius of the
$\psi(1^{-})\to B_{8}\bar{B}_{8}$. In this work, we construct the amplitudes
within the covariant L-S scheme Zou:2002yy ; Dulat:2011rn ; Anisovich:2004zz ;
Chung:1993da , and then we do a PWA based on the BESIII data Ablikim:2018zay ;
Ablikim:2020lpe ; Ablikim:2017tys ; Ablikim:2012eu ; Ablikim:2018ttl ;
Ablikim:2016sjb ; Ablikim:2016iym . From the PWA, the ratio between the S-wave
and D-wave contributions can be computed, and the average orbital angular
momentum $(\bar{L})$ of $B\bar{B}$ can be deduced. At last, the effective
ranges of these reactions can be obtained.

This article is organized as follows. In Sec. II, we present the theoretical
framework of our calculation. Our results are shown in Sec. III as well as
some discussions. Then a brief summary in Sec. IV of this work is followed and
Appendix A, B is presented at last.

##  II Theoretical framework

The covariant L-S scheme is applied for studying the partial wave analysis of
$\psi(1^{-})\to B_{8}(1/2^{+})\bar{B}_{8}(1/2^{-})$ as follows Zou:2002yy ;
Dulat:2011rn ; Anisovich:2004zz ; Chung:1993da :

$\displaystyle A$
$\displaystyle=g_{S}\Psi^{(1)}_{\mu}\epsilon^{\mu}+g_{D}e^{i\delta}\Psi^{(1)}_{\mu}\epsilon_{\nu}\tilde{t}^{(2)\mu\nu},$
(1)

where $g_{S}$ and $g_{D}$ are real number and have the same sign, stand for
the S-wave and D-wave coupling constants, $\delta$ is the relative phase
between S-wave and D-wave, $\epsilon_{\nu}$ is the polarization vector of
$\psi$, $\Psi^{(1)}_{\mu}$ and $\tilde{t}^{(2)\mu\nu}$ are the covariant spin
wave function $(\boldsymbol{S}_{B\bar{B}}=1)$ and D-wave orbital angular
momentum covariant tensors, respectively. Here the explicit expressions of
them are as follows,

$\displaystyle\Psi^{(1)}_{\mu}$
$\displaystyle=\bar{u}_{B}(\gamma_{\mu}-\frac{r_{\mu}}{m_{\psi}+2m_{B}})v_{\bar{B}},$
(2) $\displaystyle\tilde{t}^{(2)\,\mu\nu}$
$\displaystyle=r^{\mu}r^{\nu}-\frac{r^{2}}{3}g^{\mu\nu}+\frac{r^{2}}{3m_{\psi}^{2}}p_{\psi}^{\mu}p_{\psi}^{\nu},$
(3)

where $r=p_{B}-p_{\bar{B}}$, $m_{i}$ and $p_{i}$ are the mass and four
momentum of particle $i$, and $u_{B}(v_{\bar{B}})$ is the spinor wave function
of baryon $B$($\bar{B}$). The detailed process is in Appendix A.

In the rest frame of the particle $\psi$, the differential decay rate of
$\psi\to B\bar{B}$ is determined by the amplitude $A$ as:

$\displaystyle\frac{d\Gamma}{d\Omega}=\frac{1}{32\pi^{2}}\overline{|A|^{2}}\frac{|\vec{p}_{B}|}{m_{\psi}^{2}}.$
(4)

Because of the positron-electron collision, the $\psi$ spin projection is
limited to be $\pm 1$ along the beam direction. With Eq. (4), the angular
distribution for $\psi(1^{-})\to B_{8}\bar{B}_{8}$ can be written as a
function of the angle $\theta$ between $B$ direction and the electron beam as
follows:

$\displaystyle\frac{d\Gamma}{d\cos{\theta}}=C(1+\alpha\cos^{2}{\theta}),$ (5)
$\displaystyle\alpha=\frac{-6(g_{D}/g_{S})^{2}|\vec{p}_{B}|^{4}\\!+\\!36(g_{D}/g_{S})|\vec{p}_{B}|^{2}\cos{\delta}}{9\\!+\\!10(g_{D}/g_{S})^{2}|\vec{p}_{B}|^{4}\\!-\\!12(g_{D}/g_{S})|\vec{p}_{B}|^{2}\cos{\delta}},$
(6)

where C is an overall normalization.

Furthermore, to determine the amplitude $A$ in Eq.(1), at least three inputs
are needed to control the parameters, $g_{S}$, $g_{D}$ and $\delta$. In the
experimental side, not only $\Gamma$ and $\alpha$ are measured, but also
another parameter $\Delta\Phi$ can be got by the cascade decay of $B$. The
$\Delta\Phi$ is defined as the relative phase between the electric
($G_{E}^{\psi}$) and magnetic ($G_{M}^{\psi}$) form factors as follows,

$\displaystyle\Delta\Phi\equiv{\text{Arg}}(\frac{G_{E}^{\psi}}{G_{M}^{\psi}}).$
(7)

By using $G_{E}^{\psi}$ and $G_{M}^{\psi}$, the amplitude of the decay
$\psi(1^{-})\to B_{8}(1/2^{+})\bar{B}_{8}(1/2^{-})$ can be re-defined as
Faldt:2016qee ; Faldt:2017kgy :

$\displaystyle\tilde{A}=-ie_{g}\bar{u}_{B}O_{\mu}v_{\bar{B}}\epsilon^{\mu},$
(8) $\displaystyle
O_{\mu}=G_{M}^{\psi}\gamma_{\mu}-\frac{2m_{B}}{r^{2}}(G^{\psi}_{M}-G^{\psi}_{E})r_{\mu},$
(9)

where $e_{g}$ is an overall couping constant.

By comparing two amplitudes defined in Eqs.(1, 8), the $\Delta\Phi$ can be
expressed as:

$\displaystyle
e^{i\Delta\Phi}=\sqrt{\frac{1+\alpha}{1-\alpha}}\left(\frac{2m_{B}}{m_{\psi}}-\frac{r^{2}}{m_{\psi}(m_{\psi}+2m_{B})}\right)\frac{3+2r^{2}\boldsymbol{y}}{3-r^{2}\boldsymbol{y}},$
(10)

where $\boldsymbol{y}\equiv e^{i\delta}g_{D}/g_{S}$,
$r^{2}=-4|\vec{p}_{B}|^{2}$. We have derived them in Appendix B in detail. Now
through Eqs.(6, 10), the $g_{D}/g_{S}$ and $\delta$ can be calculated from the
measured $\alpha$ and $\Delta\Phi$.

The percentage of S-wave contribution can be estimated as the ratio between
pure S-wave partial width and total width, $\Gamma_{S}/\Gamma_{\text{Total}}$,
which is determined by the value of $g_{D}/g_{S}$:

$\displaystyle\frac{\Gamma_{S}}{\Gamma_{Total}}$
$\displaystyle=\frac{9}{9+2(g_{D}/g_{S})^{2}(m_{\psi}^{2}-4m_{B}^{2})^{2}}.$
(11)

Then, the average angular momentum $\bar{l}$ can be approximated to

$\displaystyle\bar{l}$
$\displaystyle=l_{S}\frac{\Gamma_{S}}{\Gamma_{Total}}+l_{D}\frac{\Gamma_{D}}{\Gamma_{Total}}$
$\displaystyle=\frac{4(g_{D}/g_{S})^{2}(m_{\psi}^{2}-4m_{B}^{2})^{2}}{9+2(g_{D}/g_{S})^{2}(m_{\psi}^{2}-4m_{B}^{2})^{2}}.$
(12)

At last, the effective decay range $r_{{\text{eff}}}$ can be estimated from
$\bar{l}=r_{{\text{eff}}}\times p$.

##  III Results and discussions

For the decays of $J/\psi\to\Lambda\bar{\Lambda}$ and
$\Sigma^{+}\bar{\Sigma}^{-}$, $\psi(2S)\to\Sigma^{+}\bar{\Sigma}^{-}$, the
parameters $\alpha$ and $\Delta\Phi$ have been both well measured
Ablikim:2018ttl ; Ablikim:2020lpe  as shown in Table III. The ratio between
S-wave and D-wave coupling constants, $g_{D}/g_{S}$ (GeV-2), and the relative
phase, $\delta$, can be determined in our calculation as shown in Table III.
It shows that in the all three reactions the S-wave contribution is larger
than $85\%$, and the effective radius of the $\psi$ decay is around $0.04$ fm.

Table 1: The parameters of
$J/\psi\to\Lambda\bar{\Lambda}$,$\Sigma^{+}\bar{\Sigma}^{-}$,
$\psi(2S)\to\Sigma^{+}\bar{\Sigma}^{-}$ reactions are including the measured
$\alpha$ and $\Delta\Phi$(radian), the computed ratio between the coupling
constants of S-wave and D-wave ($g_{D}/g_{S}$ (GeV-2)), the relative phase
between S-wave and D-wave ($\delta$), the ratio between partial width of
S-wave and total width ($\Gamma_{S}/\Gamma_{\text{Total}}$), and the effective
radius ($r_{\text{eff}}$(fm)).

Mode | $\alpha$ | $\Delta\Phi$ | $g_{D}/g_{S}$ | $\delta$ | $\Gamma_{S}/\Gamma_{\text{Total}}$ | $r_{{\text{eff}}}$
---|---|---|---|---|---|---
$J/\psi\to\Lambda\bar{\Lambda}$ |  $0.461\pm 0.013$Ablikim:2018zay  |  $0.74\pm 0.019$Ablikim:2018zay  | $0.180\pm 0.005$ | $-0.804\pm 0.024$ | $85.7\pm 0.6\%$ | $0.0488\pm 0.0021$
$J/\psi\to\Sigma^{+}\bar{\Sigma}^{-}$ |  $-0.508\pm 0.010$Ablikim:2020lpe  |  $-0.270\pm 0.021$Ablikim:2020lpe  | $0.171\pm 0.006$ | $2.67\pm 0.04$ | $90.9\pm 0.6\%$ | $0.0362\pm 0.0024$
$\psi(2S)\to\Sigma^{+}\bar{\Sigma}^{-}$ |  $0.682\pm 0.041$Ablikim:2020lpe  |  $0.379\pm 0.084$Ablikim:2020lpe  | $0.097\pm 0.009$ | $-0.33\pm 0.10$ | $88.3\pm 2.0\%$ | $0.033\pm 0.006$

For the other decay channels, the $\Delta\phi$ have not been measured, but we
still can estimate the range of the $r_{{\text{eff}}}$ from the measured
$\alpha$. At first, by using Eq.(6) and the value of $\alpha$, the
relationship between $g_{D}/g_{S}$ (GeV-2) and $\delta$ will be fixed as shown
in the Fig.III and Fig.III. From Fig.III and Fig. III, we find the possible
values of $g_{D}/g_{S}$ (GeV-2) are limited in a range. It will lead to a
range of the $r_{{\text{eff}}}$ as shown in Table III, and the values are all
below the 0.4 fm which is obviously much smaller than the size of proton.
Especially, for the $\alpha$ of $\psi(2S)\to p\bar{p}$, there are
$\alpha=1.03\pm 0.09$, however the value of $\alpha$ should satisfy
$|\alpha|\leq 1$. Thus, it is failed to extract any information from the
current value of the $\alpha$ of $\psi(2S)\to p\bar{p}$.

Table 2: The $\alpha$ and the estimation of $r_{{\text{eff}}}$(fm) in the
reactions of $\psi\to B_{8}\bar{B}_{8}$ without $\Delta\Phi$ measurement.

Figure 1: The relationship between $g_{D}/g_{S}({\text{GeV}}^{-2})$ and
$\delta$ in $J/\psi\to B_{8}\bar{B}_{8}$ with the fixed value of $\alpha$ from
experiment.

Figure 2: The relationship between $g_{D}/g_{S}({\text{GeV}}^{-2})$ and
$\delta$ in $J/\psi\to B_{8}\bar{B}_{8}$ with the fixed value of $\alpha$ from
experiment.

From the above results, the effective radius of $\psi$ decay is very small. As
discussed in the introduction, the small effective radius will lead to the
small orbital angular momentum of the process of $\psi\to\bar{N}N^{*}$,
resulting in the $N^{*}$ produced by this process to be dominated by the low
spin $N^{*}$. Therefore, when dealing with $N^{*}$ partial wave analysis in
$\psi$ decay, the highest partial wave can be taken as an angular momentum
$l=5/2$. Furthermore, some decay channels have not enough experimental inputs,
only a range of $r_{\text{eff}}$ is estimated. It is still necessary to
confirm the effective reaction radii of them, which will accurately tell us
how large spin of the resonance can be found in the decay of $\psi$.

Obviously, more observables are needed to be measured if we want to fix the
effective ranges of all decay channels. For the $\Sigma\bar{\Sigma}$,
$\Xi\bar{\Xi}$ and $\Lambda\bar{\Lambda}$ decay channels, the $\Delta\Phi$ can
be got from the measurement of differential decay of cascade decay. For
example, from the data of the final states’ angular distribution of decay of
$J/\psi\to\Xi\bar{\Xi}\to\Lambda\pi\bar{\Lambda}\pi$, the $\Delta\Phi$ of this
reaction can be measured. However, for the decay of $J/\psi$ or $\psi(2S)\to
p\bar{p}$ or $n\bar{n}$, there are no cascade decay information of proton or
neutron, since proton is stable and the life time of neutron is about 15
minutes, respectively. As a result, the $\Delta\Phi$ of these two reactions
can not be obtained easily. It is worthy to mention that the polarization
information of $p\bar{p}$ or $n\bar{n}$ can provide completely control of the
$g_{D}/g_{S}$(Gev-2). We can define $\Gamma_{\parallel}$ to stand for the
decay width of the process where $p(n)$ and $\bar{p}(\bar{n})$ have the same
polarization:

$\displaystyle\Gamma_{\parallel}=\frac{|\vec{p}_{B}|}{32\pi^{2}m_{\psi}^{2}}\int\frac{1}{2}\sum_{s_{\Psi},s_{B}=s_{\bar{B}}}|A(s_{\Psi},s_{B},s_{\bar{B}})|^{2}d\Omega$
$\displaystyle=\Gamma_{S}+0.7\Gamma_{D}.$ (13)

In other word, once the $\Gamma_{{\text{Total}}}$ and $\Gamma_{\parallel}$ are
both fixed, the $\Gamma_{S}$ and $\Gamma_{D}$ can be computed. In Fig.3, it
describes the relation between the
$\Gamma_{\parallel}/\Gamma_{{\text{Total}}}$ and the
$g_{D}/g_{S}$(GeV${}^{-2})$.

Figure 3: The ratio between $\Gamma_{\parallel}$ and $\Gamma_{{\text{Total}}}$
vs the $g_{D}/g_{S}({\text{GeV}}^{-2})$ in $J/\psi\to p\bar{p}$.

In all of the reactions mentioned above, the phase ($\delta$) between S-wave
and D-wave plays an important role. Generally speaking, if we assume that the
$B_{8}\bar{B}_{8}$ is generated from the $c\bar{c}$ annihilation through three
gluons, to the leading order, the relative phase $\delta$ should be zero.
However, there are a lot of other mechanisms which will provide different
phase between the S-wave and D-wave, a free parameter for relative phase is
acceptable. In our Tables III and III, all of the values of the $\alpha$ for
different processes are listed and among these data we can find only the
values of $\alpha$ for $J/\psi\to\Sigma^{+}\bar{\Sigma}^{-}$ and
$\Sigma^{0}\bar{\Sigma}^{0}$ are negative while others are all positive. It
implies that there is a relative phases ($\delta$) difference of $\pi$ between
the $J/\psi\to\Sigma^{+}\bar{\Sigma}^{-}(\Sigma^{0}\bar{\Sigma}^{0})$
processes and others as shown in Table III.

From the other point of view, the electric and magnetic form factors
expressing the amplitude as shown in Eqs.(8-9) can also reflect the value of
$\alpha$ and branching ratio ($Br$) as follows,

$\displaystyle Br$ $\displaystyle=|G_{E}|^{2}+|G_{M}|^{2},$ (14)
$\displaystyle\alpha$
$\displaystyle=\frac{|G_{M}|^{2}-2|G_{E}|^{2}}{|G_{M}|^{2}+2|G_{E}|^{2}}.$
(15)

Then from the branching ratios and the values of $\alpha$ for each channel,
their $|G_{E}|$ and $|G_{M}|$ can be computed as shown in Table III. As
described in Ref. Ferroli:2018yad , the sign difference of $\alpha$ between
$\Sigma^{0}\bar{\Sigma}^{0}$ and $\Lambda\bar{\Lambda}$ can be explained by
using a SU(3) symmetry breaking model Ferroli:2019nex . In Ref.
Ferroli:2019nex , they provide all amplitudes listed in Table III, where
$G_{0}$ is related to the three gluons interaction within SU(3) symmetry,
$D_{e}$ and $F_{e}$ for the EM breaking effects, $D_{m}$ and $F_{m}$ describe
the mass difference breaking effects, $\phi$ is relative phase between strong
interaction and EM interaction, $R$ is the ratio between $gg\gamma$ and $ggg$
contributions. By following the method of Ref. Ferroli:2018yad , we can
decompose the electric and magnetic amplitudes as the different parts,
respectively. At last for electric and magnetic amplitudes, we can solve them
independently, the results are shown in Table III. From these solutions, we
can find that the $g_{0}$ of electric amplitudes is even smaller than that of
other symmetry breaking terms, while $g_{0}$ is dominant in the magnetic
amplitudes. Of course such parameters are not proper, especially for the
electric amplitudes. However, in Ref. Ferroli:2019nex , this model is able to
explain the branching ratios since most channels are dominated by the magnetic
amplitudes. In Ref. Ferroli:2018yad , they use this model explain the $\alpha$
for the $\Sigma^{0}\bar{\Sigma}^{0}$ and $\Lambda\bar{\Lambda}$, but once all
octets baryon channels are considered it is not so reasonable to explain the
values of $\alpha$ which need electric and magnetic amplitudes both well
described. Thus, beside the SU(3) breaking effect mentioned above, other
mechanisms are needed to solve this problem. For example, for the $\Sigma$ the
total spin and isospin of light quark pair are both 1, i.e., so called “bad
diquark” Dobbs:2017hyd ; Jaffe:2003sg , while for the $\Lambda$ the [ud] quark
pair has both spin and isospin to be 0, i.e., so called “good diquark”. It may
be a reason causing the angular distribution difference for the case of
$\Sigma\bar{\Sigma}$ from others. The hadronic loops may be another possible
mechanism. The study of these possible mechanisms are beyond the scope of this
paper and should be explored in the future. An important point is : although
the detailed production mechanisms may be different to cause the difference of
angular distributions, the deduced small effective radii for their production
from vector charmonium annihilation are nearly the same and lead to the S-wave
dominant final states for both cases.

Table 3: The values of $|G_{E}|^{2}$ and $|G_{M}|^{2}$.

Mode | $|G_{E}|^{2}(10^{-3})$ | $|G_{M}|^{2}(10^{-3})$
---|---|---
$p\bar{p}$ | $0.238\pm 0.018$ | $1.87\pm 0.04$
$n\bar{n}$ | $0.30\pm 0.17$ | $1.77\pm 0.23$
$\Sigma^{+}\bar{\Sigma}^{-}$ | $0.91\pm 0.19$ | $0.59\pm 0.13$
$\Sigma^{0}\bar{\Sigma}^{0}$ | $0.661\pm 0.025$ | $0.503\pm 0.023$
$\Lambda\bar{\Lambda}$ | $0.303\pm 0.010$ | $1.640\pm 0.032$
$\Xi^{0}\bar{\Xi}^{0}$ | $0.108\pm 0.028$ | $1.06\pm 0.05$
$\Xi^{-}\bar{\Xi}^{+}$ | $0.12\pm 0.04$ | $0.92\pm 0.08$

Table 4: Amplitudes parameterization.

Mode | Amplitudes
---|---
$p\overline{p}$ | $(G_{0}-D_{m}+F_{m})(1+R)e^{i\varphi}+D_{e}+F_{e}$
$n\overline{n}$ | $(G_{0}-D_{m}+F_{m})e^{i\varphi}-2\,D_{e}$
$\Sigma^{+}\overline{\Sigma}^{-}$ | $(G_{0}+2D_{m})(1+R)e^{i\varphi}+D_{e}+F_{e}$
$\Sigma^{0}\overline{\Sigma}^{0}$ | $(G_{0}+2D_{m})e^{i\varphi}+D_{e}$
$\Lambda\overline{\Lambda}$ | $(G_{0}-2D_{m})e^{i\varphi}-D_{e}$
$\Xi^{0}\overline{\Xi}^{0}$ | $(G_{0}-D_{m}-F_{m})e^{i\varphi}-2\,D_{e}$
$\Xi^{-}\overline{\Xi}^{+}$ | $(G_{0}-D_{m}-F_{m})(1+R)e^{i\varphi}+D_{e}-F_{e}$

Table 5: The values of amplitudes parameters.

Mode | $g_{0}(10^{-3})$ | $d_{e}(10^{-3})$ | $d_{m}(10^{-3})$ | $f_{e}(10^{-3})$ | $f_{m}(10^{-3})$ | $r$ | $\cos\phi$
---|---|---|---|---|---|---|---
the electric part | $0.132\pm 0.009$ | $0.097\pm 0.068$ | $0.368\pm 0.021$ | $-0.323\pm 0.106$ | $-0.408\pm 0.123$ | $-0.081\pm 0.104$ | $-0.612\pm 0.152$
the magnetic part | $0.957\pm 0.062$ | $0.262\pm 0.217$ | $-0.165\pm 0.069$ | $0.130\pm 0.149$ | $0.168\pm 0.041$ | $-0.021\pm 0.154$ | $0.124\pm 0.359$

##  IV Summary

In this article, we use the covariant L-S scheme for the PWA to construct the
coupling of $\psi(1^{-})\to B_{8}(1/2^{+})\bar{B}_{8}(1/2^{-})$ and fit
experiment’s measured parameters to calculate the effective radii of these
processes. For the processes $J/\psi\to\Lambda\bar{\Lambda}$,
$J/\psi\to\Sigma^{+}\bar{\Sigma}^{-}$ and
$\psi(2S)\to\Sigma^{+}\bar{\Sigma}^{-}$, the effective radii are deduced to be
around 0.04 fm. However, for other processes of $\psi\to
B_{8}(1/2^{+})\bar{B}_{8}(1/2^{-})$, only the range of the effective radii are
computed since there is only one measured parameter $\alpha$. To fix the radii
of these processes, we give some proposals. Especially, for the $\psi\to
p\bar{p}$ or $n\bar{n}$, the polarization observables could help to extract
the effective radii of them. From our calculation, the effective radius of
$\psi$ decay is much smaller than the normal size of nucleon, therefore, the
angular momenta between baryon and anti-baryon are very small. Thus, only
excited resonances with low spins can be produced and observed in the $\psi$
decay. It hence provides a unique window to look for low spin nucleon and
hyperon resonances with masses above 2 GeV in the $\psi$ decay.

## Acknowledgments

We thank useful discussions and valuable comments from Fengkun Guo, Xiaorui
Lyu, Ronggang Ping, Jifeng Hu and Jianbin Jiao. This work is supported by the
NSFC and the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)
through the funds provided to the Sino-German Collaborative Research Center
TRR110 “Symmetries and the Emergence of Structure in QCD” (NSFC Grant No.
12070131001, DFG Project-ID 196253076 - TRR 110), by the NSFC Grant
No.11835015, No.12047503, and by the Chinese Academy of Sciences (CAS) under
Grant No.XDB34030000, also by the Fundamental Research Funds for the Central
Universities (J.J.W), and National Key R$\&$D Program of China under Contract
No. 2020YFA0406400 (J.J.W).

##  Appendix A THE COVARIANT L-S SCAME

In this appendix we briefly introduce the covariant L-S scheme for the
effective $N^{*}NM$ couplings.

For a hadron deacy $a\to b+c$, the conservation relation of total angular
momentum lead to:

$\displaystyle\boldsymbol{S}_{a}=\boldsymbol{S}_{bc}+\boldsymbol{L}_{bc},$
(16)

where $\boldsymbol{L}_{bc}$ and $\boldsymbol{S}_{bc}$ is the relative orbital
angular momentum and the total spin between the particle b and c.

Then we separate first the orbital angular momentum wave function
$\tilde{t}^{{\mu}}$ and covariant spin wave functions $\psi(\Psi)$ or
$\phi(\Phi)$.

The orbital angular momentum $L_{bc}$ state can be represented by covariant
tensor wave functions $\tilde{t}_{\mu_{1}\cdots\mu_{L}}^{(L)}$Zou:2002yy ;
Dulat:2011rn ; Anisovich:2004zz ; Chung:1993da ,

$\displaystyle\tilde{g}_{\mu\nu}(p)\equiv
g_{\mu\nu}-\frac{p_{\mu}p_{\nu}}{p^{2}}=-\sum_{S=\pm
1,0}\epsilon_{\mu}(p,S)\epsilon_{\nu}^{\star}(p,S),$ (17)
$\displaystyle\tilde{t}^{(0)}=1,$ (18)
$\displaystyle\tilde{t}^{(1)}_{\mu}=\tilde{g}_{\mu\nu}(p_{a})r^{\nu}\equiv\tilde{r}_{\mu},$
(19) $\displaystyle\tilde{t}^{(2)}_{\mu\nu}=\tilde{r}_{\mu}\tilde{r}_{\nu}$
$\displaystyle-\frac{1}{3}(\tilde{r}\cdot\tilde{r})\tilde{g}_{\mu\nu},$ (20)
$\displaystyle...$

where $r=p_{b}-p_{c}$ is the relative four momentum of the two decay products
in the parent particle rest frame.

For the covariant spin wave functions, we use function $\psi(\Psi)$ or
$\phi(\Phi)$ to describe with four cases $M\to B^{*}\bar{B}$,
$M\to\bar{B}^{*}B$, $B^{*}\to BM$ and $\bar{B}^{*}\to\bar{B}M$.Zou:2002yy ;
Dulat:2011rn ; Chung:1993da

For the case of $M\to B^{*}(n+\frac{1}{2})\bar{B}(\frac{1}{2})$ and
$M\to\bar{B}^{*}(n+\frac{1}{2})B(\frac{1}{2})$, $S_{bc}$ can be n or (n+1):

$\displaystyle\psi^{(n)}_{\mu_{1}\cdots\mu_{n}}$
$\displaystyle=\bar{u}_{\mu_{1}\cdots\mu_{n}}(p_{b},S_{b})\gamma_{5}v(p_{c},S_{c}),$
(21) $\displaystyle\Psi^{(n+1)}_{\mu_{1}\cdots\mu_{n+1}}$
$\displaystyle=\bar{u}_{\mu_{1}\cdots\mu_{n}}(p_{b},S_{b})\bar{\gamma}_{\mu_{n+1}}v(p_{c},S_{c})$
$\displaystyle+(\mu_{1}\leftrightarrow\mu_{n+1})+\cdots+(\mu_{n}\leftrightarrow\mu_{n+1}),$
(22) $\displaystyle\psi^{C(n)}_{\mu_{1}...\mu_{n}}$
$\displaystyle=-\bar{u}(p_{c},S_{c})\gamma_{5}v_{\mu_{1}...\mu_{n}}(p_{b},S_{b}),$
(23) $\displaystyle\Psi^{C(n+1)}_{\mu_{1}...\mu_{n+1}}$
$\displaystyle=\bar{u}(p_{c},S_{c})\bar{\gamma}_{\mu_{n+1}}v_{\mu_{1}...\mu_{n}}(p_{b},S_{b})$
$\displaystyle+(\mu_{1}\leftrightarrow\mu_{n+1})+...+(\mu_{n}\leftrightarrow\mu_{n+1}),$
(24)

where $\bar{\gamma}_{\mu}\equiv\gamma_{\mu}-r_{\mu}/(m_{a}+m_{b}+m_{c})$.

And use the orbital angular momentum covariant tensors
$\tilde{t}_{\mu_{1}\cdots\mu_{L}}^{(L)}$, covariant spin wave functions
$\psi(\Psi)$, metric tensor $g_{\mu\nu}$, totally antisymmetric Levi-Civita
tensor $\epsilon_{\mu\nu\lambda\sigma}$ and momentum of the parent particle
$p_{\mu}$ we can get the effective couplings.

##  Appendix B CALCULATING THE TWO FORMS OF AMPLITUDE

In this appendix we will derive the relationship between the two parameters in
Eq. (10).

$\displaystyle A_{S}$
$\displaystyle=\Psi^{(1)}_{\mu}\epsilon^{\mu}=\bar{u}_{B}(\gamma_{\mu}-\frac{r_{\mu}}{m_{\psi}+2m_{B}})v_{\bar{B}}\epsilon^{\mu},$
(25) $\displaystyle A_{D}$
$\displaystyle=\Psi^{(1)}_{\mu}\epsilon_{\nu}\tilde{t}^{(2)\mu\nu}=\bar{u}_{B}(\gamma^{\nu}-\frac{r^{\nu}}{m_{\psi}+2m_{B}})$
$\displaystyle\cdot(r_{\mu}r_{\nu}-\frac{r^{2}}{3}g_{\mu\nu}+\frac{r^{2}}{3m_{\psi}^{2}}p^{\psi}_{\mu}p^{\psi}_{\nu})v_{\bar{B}}\epsilon^{\mu}$
(26) $\displaystyle=$
$\displaystyle\bar{u}_{B}[-\frac{r^{2}}{3}\gamma_{\mu}+(2m_{B}-\frac{2}{3}\frac{r^{2}}{m_{\psi}+2m_{B}})r_{\mu}]v_{\bar{B}}\epsilon^{\mu}.$
(27)

So,

$\displaystyle A$
$\displaystyle=g_{S}\bar{u}_{B}[(1-\frac{r^{2}}{3}\boldsymbol{y})\gamma_{\mu}-(\frac{1}{m_{\psi}+2m_{B}}$
$\displaystyle-2m_{B}\boldsymbol{y}+\frac{2}{3}\frac{r^{2}}{m_{\psi}+2m_{B}}\boldsymbol{y})r_{\mu}]v_{\bar{B}}\epsilon^{\mu},$
(28) $\displaystyle\tilde{A}$
$\displaystyle=-ie_{g}G_{M}^{\psi}\bar{u}_{B}[\gamma_{\mu}-\frac{2m_{B}}{r^{2}}(1-\boldsymbol{x})r_{\mu}]v_{\bar{B}}\epsilon^{\mu},$
(29)

where $\boldsymbol{y}\equiv e^{i\delta}g_{D}/g_{S}$, $\boldsymbol{x}\equiv
G_{E}^{\psi}/G_{M}^{\psi}$.

Compared with Eqs. (28,29),we can get:

$\displaystyle(1-\frac{r^{2}}{3}\boldsymbol{y})\frac{2m_{B}}{r^{2}}(1-\boldsymbol{x})$
$\displaystyle=$
$\displaystyle\frac{1}{m_{\psi}+2m_{B}}-2m_{B}\boldsymbol{y}+\frac{2}{3}\frac{r^{2}}{m_{\psi}+2m_{B}}\boldsymbol{y}.$
(30)

So, the parameters in two forms of amplitude must satisfy:

$\displaystyle\boldsymbol{y}=-\frac{3}{2r^{2}}\frac{2m_{B}(m_{\psi}+2m_{B})-r^{2}-2m_{B}(m_{\psi}+2m_{B})\boldsymbol{x}}{2m_{B}(m_{\psi}+2m_{B})-r^{2}+m_{B}(m_{\psi}+2m_{B})\boldsymbol{x}}.$

It equals Eq. (10).

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