# Pionic transitions from $Z_{c}(4020)$ to $D$ wave charmonia

Xiao-Yu Qi1 Qi Wu2 Dian-Yong Chen1,3111Corresponding author<EMAIL_ADDRESS>1 School of Physics, Southeast University, Nanjing 210094, People’s Republic
of China  2 School of Physics and Center of High Energy Physics, Peking
University, Beijing 100871, China  3 Lanzhou Center for Theoretical Physics,
Lanzhou University, Lanzhou 730000, China

###### Abstract

In the present work, we investigate the charmed meson loops contributions to
the pionic transitions from $Z_{c}(4020)$ to the $D$ wave triplets charmonia
by using an effective Lagrangian approach. Our estimations indicate that the
predicted branching fraction of $Z_{c}(4020)\to\psi(3770)\pi$ is sizable and
comparable with the one of $Z_{c}(4020)\to h_{c}\pi$, while the one of
$Z_{c}(4020)\to\psi_{2}(3823)\pi$ and $Z_{c}(4020)\to\psi_{3}(3842)\pi$ are
one and two order of magnitude smaller than the one of $Z_{c}(4020)\to
h_{c}\pi$, respectively. The present estimations for
$Z_{c}(4020)\to\psi(3770)\pi$ is consistent with the preliminary results of
BES III Collaboration and could be further test by the experimental analysis
of the process $e^{+}e^{-}\to\psi(3770)\pi^{+}\pi^{-}$ with more accurate data
in future.

##  I introduction

As the first confirmed charmonium-like state, $Z_{c}^{\pm}(3900)$, was first
observed by the BES III BESIII:2013ris  and Belle Belle:2013yex
Collaborations in the $\pi^{\pm}J/\psi$ invariant mass spectrum of process
$e^{+}e^{-}\to\pi^{+}\pi^{-}J/\psi$ at $\sqrt{s}=4.260$ GeV one decade ago.
Later on, the authors of Ref. Xiao:2013iha  further confirmed the existence of
$Z_{c}^{\pm}(3900)$ in the $\pi^{\pm}J/\psi$ invariant mass spectrum of the
same process and also reported the first evidence of $Z_{c}^{0}(3900)$ in the
$\pi^{0}J/\psi$ invariant mass spectrum of the process
$e^{+}e^{-}\to\pi^{0}\pi^{0}J/\psi$ by using the data taken with the CLEO-c
detector at $\sqrt{s}$=4.170 GeV. Later, the neutral partner,
$Z_{c}(3900)^{0}$ was observed in the process
$e^{+}e^{-}\to\pi^{0}\pi^{0}J/\psi$ by the BES III Collaboration
BESIII:2015cld ; BESIII:2020oph , and then the isospin triplets of
$Z_{c}(3900)$ are established. Using the data samples of the process
$e^{+}e^{-}\to\pi^{+}\pi^{-}\pi^{0}\eta_{c}$ at $\sqrt{s}=4.226$ GeV, the BES
III Collaboration observed the evidence for the decay
$Z_{c}(3900)^{\pm}\to\rho^{\pm}\eta_{c}$ BESIII:2019rek . Besides these hidden
charm processes, the charmonium-like state $Z_{c}(3900)$ has also been
observed in the open charm process. In the $D^{\ast}\bar{D}$ invariant mass
distributions of the process
$e^{+}e^{-}\to\pi^{\pm,0}(D\bar{D}^{*}+c.c)^{\mp,0}$, the BES III
Collaboration further confirmed the existence of $Z_{c}(3900)$ BESIII:2013qmu
; BESIII:2015pqw ; BESIII:2015ntl ; Belle:2015yoa ; BESIII:2015aym ;
BESIII:2017bua .

As a cousin of $Z_{c}(3900)$, the charged charmonium-like state $Z_{c}(4020)$
was first observed in the $h_{c}\pi^{\pm}$ invariant mass distributions of the
process $e^{+}e^{-}\to\pi^{+}\pi^{-}h_{c}$ in the year of 2013 BESIII:2013ouc
, and later, the neutral partner, $Z_{c}(4020)^{0}$ was observed in the
$h_{c}\pi^{0}$ invariant mass distributions of the process
$e^{+}e^{-}\to\pi^{0}\pi^{0}h_{c}$ BESIII:2014gnk . Similar to the case of
$Z_{c}(3900)$, the charmonium like state $Z_{c}(4020)$ has also been observed
in the open charm process. In the year of 2014, the BES III Collaboration
observed the charged $Z_{c}(4020)^{\pm}$ in the invariant mass distribution of
$(D^{*}\bar{D}^{*})^{\pm}$ of the process
$e^{+}e^{-}\to(D^{*}\bar{D}^{*})^{\pm}\pi^{\mp}$ at $\sqrt{s}=4.26$ GeV
BESIII:2013mhi , later the neutral one was also observed in the open charm
channel BESIII:2015tix .

The observations of $Z_{c}(3900)$ and $Z_{c}(4020)$ show that their
$I^{G}(J^{P})$ quantum numbers are $1^{+}(1^{+})$. The isospin triplets nature
indicate that there are at least four constituent quarks in both $Z_{c}(3900)$
and $Z_{c}(4020)$, thus, some tetraquark interpretations have been proposed
Chen:2010ze ; Voloshin:2013dpa ; Chen:2015fsa ; Chen:2016qju ; Liu:2019zoy ;
Wang:2013llv ; Agaev:2017tzv . Moreover, the observed mass of $Z_{c}(3900)$
and $Z_{c}(4020)$ are in the vicinities of the thresholds of $D^{\ast}\bar{D}$
and $D^{\ast}\bar{D}^{\ast}$, respectively, which indicate that $Z_{c}(3900)$
and $Z_{c}(4020)$ could be good candidates of molecular states composed of
$D^{\ast}\bar{D}+c.c$ and $D^{\ast}\bar{D}^{\ast}$ Liu:2009qhy ; Sun:2012zzd ;
Wang:2013qwa ; Cui:2013xla ; Wang:2013daa ; Chen:2013omd ; Wang:2014gwa ;
Aceti:2014uea ; Zhao:2014gqa ; Gutsche:2014zda ; Aceti:2014kja ; He:2015mja ;
Karliner:2015ina ; Chen:2015igx , respectively. However, the estimations in
Ref. He:2015mja  indicates the interaction between $D^{\ast}\bar{D}$ is not
strong enough to form a bound state and the observed $Z_{c}(3900)$ could be
interpreted as a $D^{\ast}\bar{D}$ resonance. Besides the resonance
interpretations, some kinematical mechanisms have also been proposed
Chen:2011xk ; Chen:2013coa ; Swanson:2014tra ; Szczepaniak:2015eza ;
Swanson:2015bsa ; Albaladejo:2015lob ; BESIII:2015wge .

To data, the nature of the $Z_{c}(3900)$ and $Z_{c}(4020)$ is still in debate.
From the experimental side, searching for the evidence of $Z_{c}(3900)$ and
$Z_{c}(4020)$ in more process are essential for decoding their nature. With
the accumulation of the experimental data, more channels have been precisely
measured in the electron-positron annihilation process by BES III and Belle
Collaborations ParticleDataGroup:2022pth . For example, in the year of 2014,
the Belle Collaboration reported their precise measurement of
$e^{+}e^{-}\to\pi^{+}\pi^{-}\psi(3686)$ via initial state radiation, and the
evidence for a charged charmonium-like structure at 4.05 GeV was observed in
the $\pi^{\pm}\psi(3686)$ intermediate state in the $Y(4360)$ decays
Belle:2014wyt . Later, the BES III Collaboration reported the measurement of
the same process, and in the $\pi^{\pm}\psi(3686)$ invariant mass distribution
a charged structure was observed with a mass $m=(4032.1\pm 2.4)$ MeV
BESIII:2017tqk , but the experimental data can not be well described with a
fixed width in different kinematic region. Moreover, the BES III Collaboration
measured the cross sections for $e^{+}e^{-}\to\pi^{+}\pi^{-}\psi(3770)$, and
the $\pi^{\pm}\psi(3770)$ invariant distributions were also reported
BESIII:2019tdo . The hints for peaks at $4.04$ and $4.13$ GeV was observed in
the $\sqrt{s}=4.42$ GeV data, but the statistical significance is rather low.
In addition, a large data sample of
$e^{+}e^{-}\to\pi^{+}\pi^{-}\psi_{2}(3828)/\pi^{0}\pi^{0}\psi_{2}(3842)$ has
also been collected by BES III detector in recent years BESIII:2022yga ;
BESIII:2022cyq , which may provide us a good opportunity of searching charged
charmonium-like states in $\pi^{\pm}\psi_{2}(3823)$.

Together with $\psi_{3}(3842)$ observed by LHCb Collaboration LHCb:2019lnr ,
the $D$-wave spin triplets charmonia have been well established. Considering
the $J^{P}$ quantum numbers conservations and the kinematical limitation, one
find that $Z_{c}(4020)$ can transit to $\psi(1^{3}D_{J}),\ (J=1,2,3)$ by
emitting a pion. Thus, in the present work, we estimate the branching
fractions of $Z_{c}(4020)^{+}\to\pi^{+}\psi(1^{3}D_{J}),\ (J=1,2,3)$ and
discuss the experimental potential of observing $Z_{c}(4020)^{+}$ in
$\pi^{+}\psi(1^{3}D_{J})$ invariant mass distributions.

This paper is organized as follows. After introduction, we present our
estimations of the branching fractions of the process
$Z_{c}(4020)\to\psi(1^{3}D_{J})\pi^{+}$ in Section II, where the final state
interactions plays the dominant role. The numerical results and related
discussions are presented in Section III and the last section is devoted to a
short summary.

##  II THE HIDDEN CHARM DECAYS of $Z_{C}(4020)$

|  |
---|---|---
$(a)$ | $(b)$ | $(c)$
|  |
$(d)$ | $(e)$ |
Figure 1: Diagrams contributing to $Z_{c}(4020)\to\psi(3770)\pi$ (diagrams (a)
and (b)), $Z_{c}(4020)\to\psi_{2}(3823)\pi$ (diagrams (c) and (d)) and
$Z_{c}(4020)\to\psi_{3}(3842)\pi$ (diagrams (e)) at the hadron level.

The experimental measurement indicates that $Z_{c}(4020)$ dominantly decays
into $D^{\ast}\bar{D}^{\ast}$BESIII:2013ouc ; BESIII:2013mhi , the
$D^{\ast}\bar{D}^{\ast}$ pair can transits into charmonium and a light meson
by exchanging a proper charmed meson, such kind of meson loop mechanism can
well reproduce the decay properties of $Z_{c}(3900)$ and $Z_{c}(4020)$
Xiao:2018kfx ; Chen:2015igx ; Chen:2017abq . Here, we further extend such
decay mechanism to investigate the processes
$Z_{c}(4020)\to\psi(1^{3}D_{J})\pi,\ (J=1,2,3)$. In Fig. 1, the meson loop
contributing to the relevant decay processes are presented.

###  II.1 Effective Lagrangians

In the present work, the meson loops are evaluated in the hadronic level, and
the effective Lagrangian approach is adopted to depict the interactions
between the mesons. As for the interactions between $Z_{c}(4020)$
(hereinafter, $Z_{c}^{\prime}$ refers to $Z_{c}(4020)$) and
$D^{\ast}\bar{D}^{\ast}$ meson pair, the corresponding effective Lagrangians
can be expressed as,

$\displaystyle\mathcal{L}=ig_{Z_{c}^{\prime}D^{*}D^{*}}\varepsilon_{\mu\nu\alpha\beta}\partial^{\mu}Z_{c}^{\prime\nu}D^{\ast\alpha}\bar{D}^{\ast\beta}.$
(1)

As for the effective interactions relevant to the $D$ wave charmonia and
charmed meson pair, they are constructed in the heavy quark limit. For the
heavy-light meson, the wave function should be independent on the flavor and
spin of the heavy quarks in the heavy quark limit. Thus, the heavy light meson
can be characterized by the light degree of freedom, which is defined as
$\vec{s}_{\ell}=\vec{l}+\vec{s}_{q}$ with $\vec{\ell}$ and $\vec{s}_{q}$ to be
orbital angular momentum and the spin of light quark, respectively. For each
value of $s_{\ell}$ corresponds to a degenerated doublets with total spin
$S=s_{\ell}\pm 1/2$ and in the infinite heavy quark mass limit, the doublet
degenerate in mass. For the $S$-wave charmed mesons doublets,
$\left(D,D^{*}\right)$ and $\left(\bar{D},\bar{D}^{*}\right)$ can be expressed
in the matrix form, which are Lipkin:1986av ; Casalbuoni:1996pg ;
Kaymakcalan:1983qq ; Oh:2000qr ,

$\displaystyle H_{1}$ $\displaystyle=$
$\displaystyle\frac{1+v\\!\\!\\!/}{2}\left[D^{\ast\mu}\gamma_{\mu}-D\gamma_{5}\right]$
$\displaystyle H_{2}$ $\displaystyle=$
$\displaystyle\left[\bar{D}^{\ast\mu}\gamma_{\mu}-\bar{D}\gamma_{5}\right]\frac{1+v\\!\\!\\!/}{2}$
(2)

respectively, and $H_{1,2}$ satisfy
$\bar{H}_{1,2}=\gamma^{0}H^{\dagger}_{1,2}\gamma^{0}$.

For the heavy quarkonium, the heavy flavor symmetry is violated, while the
degeneracy is still expected under the rotations of the two heavy quark spins,
which allows us to build up multiplets for each value of the orbital angular
momentum. For $D$ wave charmonia, the matrix form of the multiplets can be
expressed Lipkin:1986av ,

$\displaystyle\mathcal{J}^{\mu\lambda}$ $\displaystyle=$
$\displaystyle\frac{1+v\\!\\!\\!/}{2}\Bigg{[}\psi^{\mu\alpha\lambda}_{3}\gamma_{\alpha}+\frac{1}{\sqrt{6}}\left(\psi^{\mu\alpha\beta\rho}v_{\alpha}\gamma_{\beta}\psi^{\lambda}_{2\rho}+\psi^{\lambda\alpha\beta\rho}v_{\alpha}\gamma_{\beta}\psi^{\mu}_{2\rho}\right)$
(3)
$\displaystyle+\frac{\sqrt{15}}{10}\left[\Big{(}\gamma^{\mu}-v^{\mu}\Big{)}\psi^{\lambda}_{1}+\left(\gamma^{\lambda}-v^{\lambda}\right)\psi^{\mu}_{1}\right]$
$\displaystyle-\frac{1}{\sqrt{15}}\left(g^{\mu\lambda}-v^{\mu}v^{\lambda}\right)\gamma_{\alpha}\psi^{\alpha}_{1}+\eta^{\mu\lambda}_{c}\gamma_{5}\Bigg{]}\frac{1-v\\!\\!\\!/}{2}.$

With the above matrix expression of heavy light meson and the $D$-wave heavy
quarkonia, the general effective Lagrangian describing the coupling between
$D$-wave charmonia and charmed meson pair can be expressed as follows
Li:2013xia ,

$\displaystyle\mathcal{L}=ig_{2}\mathrm{Tr}\left[\mathcal{J}^{\mu\lambda}\bar{H}_{2}{\stackrel{{\scriptstyle\leftrightarrow}}{{\partial_{\mu}}}}\gamma_{\lambda}\bar{H}_{1}\right]+\mathrm{H.c.}.$
(4)

After further expanding the above Lagrangian, we can obtain the specific
effective interactions relevant to the current calculation, which are,

$\displaystyle\mathcal{L}$ $\displaystyle=$ $\displaystyle
g_{\psi_{1}DD}\psi^{\mu}_{1}\left(D\partial_{\mu}D^{\dagger}-D^{\dagger}\partial_{\mu}D\right)$
$\displaystyle+g_{\psi_{1}DD^{\ast}}\varepsilon^{\mu\nu\alpha\beta}\Big{[}D{\stackrel{{\scriptstyle\leftrightarrow}}{{\partial_{\mu}}}}D^{\ast\dagger}_{\beta}-D^{\ast}_{\beta}{\stackrel{{\scriptstyle\leftrightarrow}}{{\partial_{\mu}}}}D^{\dagger}\Big{]}\partial_{\nu}\psi_{1\alpha}$
$\displaystyle+g_{\psi_{1}D^{\ast}D^{\ast}}\Big{[}-4\left(\psi^{\mu}_{1}D^{\ast\nu\dagger}\partial_{\mu}D^{\ast}_{\nu}-\psi^{\mu}_{1}D^{\ast}_{\nu}\partial_{\mu}D^{\ast\nu\dagger}\right)+\psi^{\mu}_{1}D^{\ast\nu\dagger}\partial_{\nu}D^{\ast}_{\mu}$
$\displaystyle-\psi^{\mu}_{1}D^{\ast\nu}\partial_{\nu}D^{\ast\dagger}_{\mu}\Big{]}+ig_{\psi_{2}DD^{\ast}}\psi^{\mu\nu}_{2}\left(D{\stackrel{{\scriptstyle\leftrightarrow}}{{\partial_{\nu}}}}D^{\ast\dagger}_{\mu}-D^{\ast}_{\mu}{\stackrel{{\scriptstyle\leftrightarrow}}{{\partial_{\nu}}}}D^{\dagger}\right)$
$\displaystyle+ig_{\psi_{2}D^{\ast}D^{\ast}}\varepsilon_{\alpha\beta\mu\nu}\Big{[}D^{\ast\nu}{\stackrel{{\scriptstyle\leftrightarrow}}{{\partial^{\beta}}}}D^{\ast\dagger}_{\lambda}-D^{\ast\nu\dagger}{\stackrel{{\scriptstyle\leftrightarrow}}{{\partial^{\beta}}}}D^{\ast}_{\lambda}\Big{]}\partial^{\mu}\psi^{\alpha\lambda}_{2}$
$\displaystyle+g_{\psi_{3}D^{\ast}D^{\ast}}\psi^{\mu\nu\alpha}_{3}\Big{[}D^{\ast}_{\alpha}{\stackrel{{\scriptstyle\leftrightarrow}}{{\partial_{\mu}}}}D^{\ast\dagger}_{\nu}+D^{\ast}_{\nu}{\stackrel{{\scriptstyle\leftrightarrow}}{{\partial_{\mu}}}}D^{\ast\dagger}_{\alpha}\Big{]}.$

Considering the heavy quark limit and the chiral symmetry, the effective
interaction related to light pseudoscalar mesons and charm mesons are
constructed as follows Falk:1992cx ; Chen:2014sra ; Yan:1992gz ; Cheng:1992xi
; Wise:1992hn ,

$\displaystyle\mathcal{L}_{D^{(*)}D^{(*)}\mathcal{P}}$ $\displaystyle=$
$\displaystyle-
ig_{D^{\ast}D\mathcal{P}}\left(D^{i\dagger}\partial^{\mu}\mathcal{P}_{ij}D^{\ast
j}_{\mu}-D^{\ast i\dagger}_{\mu}\partial^{\mu}\mathcal{P}_{ij}D^{j}\right)$
(6)
$\displaystyle+\frac{1}{2}g_{D^{\ast}D^{\ast}\mathcal{P}}\varepsilon_{\mu\nu\alpha\beta}D^{\ast\mu\dagger}_{i}\partial^{\nu}\mathcal{P}_{ij}{\stackrel{{\scriptstyle\leftrightarrow}}{{\partial^{\alpha}}}}D^{\ast\beta}_{j},$

where $D=(D^{0},D^{+},D_{s}^{+})$ is the charmed meson triplets and the
concrete expression of the matrices form of pseudoscalar mesons is,

$\displaystyle\mathcal{P}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}\frac{\pi^{0}}{\sqrt{2}}+\alpha\eta+\beta\eta^{\prime}&\pi^{+}&K^{+}\\\
\pi^{-}&-\frac{\pi^{0}}{\sqrt{2}}+\alpha\eta+\beta\eta^{\prime}&K^{0}\\\
K^{-}&\bar{K}^{0}&\gamma\eta+\delta\eta^{\prime}\end{array}\right),\ \ $ (10)

where the parameter $\alpha$, $\beta$, $\gamma$ and $\delta$ can be related to
the mixing angle by,

$\displaystyle\alpha$ $\displaystyle=$
$\displaystyle\frac{\cos\theta-\sqrt{2}\sin\theta}{\sqrt{6}},\quad\beta=\frac{\sin\theta+\sqrt{2}\cos\theta}{\sqrt{6}},$
$\displaystyle\gamma$ $\displaystyle=$
$\displaystyle\frac{-2\cos\theta-\sqrt{2}\sin\theta}{\sqrt{6}},\quad\delta=\frac{-2\sin\theta+\sqrt{2}\cos\theta}{\sqrt{6}},\
\ \ \ $ (11)

with the mixing angle to be $\theta=-19.1^{\circ}$ MARK-III:1988crp ;
DM2:1988bfq .

###  II.2 Decay Amplitudes

With the above effective Lagrangians, one can obtain the amplitudes of
$Z_{c}(4020)\to\psi(1^{3}D_{J})\pi$ corresponding to the diagrams in Fig. 1,
which are,

$\displaystyle\mathcal{M}_{a}$ $\displaystyle=$ $\displaystyle
i^{3}\int\frac{d^{4}q}{(2\pi)^{4}}\left[ig_{Z_{c}^{\prime}}\varepsilon_{\mu\nu\alpha\beta}(-i)p^{\mu}\epsilon_{Z_{c}^{\nu}}^{\nu}\right]\left[ig_{D^{*}D\pi}ip_{3\theta}\right]$
$\displaystyle\left[-g_{\mathrm{\psi}_{1}D^{*}D}\varepsilon_{\rho\tau\sigma\xi}(-i)\left(p_{2}^{\rho}-q^{\rho}\right)ip_{4}^{\tau}\epsilon_{\mathrm{\psi}_{1}}^{\sigma}\right]\frac{-g^{\alpha\theta}+p_{1}^{\alpha}p_{1}^{\theta}/m_{1}^{2}}{p_{1}^{2}-m_{1}^{2}}$
$\displaystyle\times\frac{-g^{\beta\xi}+p_{2}^{\beta}p_{2}^{\xi}/m_{2}^{2}}{p_{2}^{2}-m_{2}^{2}}\frac{1}{q^{2}-m_{q}^{2}}\mathcal{F}^{2}\left(q^{2},m_{q}^{2}\right),$
$\displaystyle\mathcal{M}_{b}$ $\displaystyle=$ $\displaystyle
i^{3}\int\frac{d^{4}q}{(2\pi)^{4}}\left[g_{Z_{c}^{\prime}}\varepsilon_{\mu\nu\alpha\beta}(-i)p^{\mu}\epsilon_{Z_{c}^{\prime}}^{\nu}\right]$
$\displaystyle\times\left[\frac{1}{2}g_{D^{*}D^{*}\pi}\varepsilon_{\theta\phi\kappa\lambda}ip_{3}^{\phi}i\left(p_{1}^{\kappa}+q^{\kappa}\right)\right]$
$\displaystyle\times\left[g_{\psi_{1}D^{*}D^{*}}\epsilon_{\psi_{1}}^{\rho}\left(-4\left(q_{\rho}-p_{2\rho}\right)g_{\sigma\xi}+q_{\sigma}g_{\rho\xi}\right.\right.\left.\left.-p_{2\xi}g_{\sigma\rho}\right)\right]$
$\displaystyle\times\frac{-g^{\alpha\theta}+p_{1}^{\alpha}p_{1}^{\theta}/m_{1}^{2}}{p_{1}^{2}-m_{1}^{2}}\frac{-g^{\beta\sigma}+p_{2}^{\beta}p_{2}^{\sigma}/m_{2}^{2}}{p_{2}^{2}-m_{2}^{2}}$
$\displaystyle\times\frac{-g^{\lambda\xi}+q^{\lambda}q^{\xi}/m_{q}^{2}}{q^{2}-m_{q}^{2}}\mathcal{F}^{2}\left(q^{2},m_{q}^{2}\right),$
$\displaystyle\mathcal{M}_{c}$ $\displaystyle=$ $\displaystyle
i^{3}\int\frac{d^{4}q}{(2\pi)^{4}}\left[ig_{Z_{c}^{\prime}}\varepsilon_{\mu\nu\alpha\beta}(-i)p^{\mu}\epsilon_{Z_{c}^{\prime}}^{\nu}\right]\left[ig_{D^{*}D\pi}ip_{3\rho}\right]$
(12)
$\displaystyle\times\left[ig_{\psi_{2}D^{*}D}\epsilon_{\psi_{2}}^{\tau\xi}(-i)\left(p_{2\xi}-q_{\xi}\right)\right]\frac{-g^{\alpha\rho}+p_{1}^{\alpha}p_{1}^{\rho}/m_{1}^{2}}{p_{1}^{2}-m_{1}^{2}}$
$\displaystyle\times\frac{-g_{\tau}^{\beta}+p_{2}^{\beta}p_{2\tau}/m_{2}^{2}}{p_{2}^{2}-m_{2}^{2}}\frac{1}{q^{2}-m_{q}^{2}}\mathcal{F}^{2}\left(q^{2},m_{q}^{2}\right)$

$\displaystyle\mathcal{M}_{d}$ $\displaystyle=$ $\displaystyle
i^{3}\int\frac{d^{4}q}{(2\pi)^{4}}\left[ig_{Z_{c}^{\prime}}\varepsilon_{\mu\nu\alpha\beta}(-i)p^{\mu}\epsilon_{Z_{c}^{\prime}}^{\nu}\right]$
$\displaystyle\times\left[\frac{1}{2}g_{D^{*}D^{*}\pi}\varepsilon_{\rho\tau\sigma\xi}ip_{3}^{\tau}i\times\left(p_{1}^{\sigma}+q^{\sigma}\right)\right]$
$\displaystyle\times\left[ig_{\psi_{2}D^{*}D^{*}}\varepsilon_{\theta\phi\kappa\lambda}\left(g_{\eta}^{\lambda}g_{\delta\omega}\left(p_{2}^{\phi}-q^{\phi}\right)-g_{\omega}^{\lambda}g_{\delta\eta}\left(q^{\phi}\right.\right.\right.$
$\displaystyle\left.\left.\left.-p_{2}^{\phi}\right)\right)p_{4}^{\kappa}\epsilon_{\psi_{2}}^{\theta\delta}\right]\frac{-g^{\alpha\rho}+p_{1}^{\alpha}p_{1}^{\rho}/m_{1}^{2}}{p_{1}^{2}-m_{1}^{2}}\frac{-g^{\beta\omega}+p_{2}^{\beta}p_{2}^{\omega}/m_{2}^{2}}{p_{2}^{2}-m_{2}^{2}}$
$\displaystyle\times\frac{-g^{\xi\eta}+q^{\xi}q^{\eta}/m_{q}^{2}}{q^{2}-m_{q}^{2}}\mathcal{F}^{2}\left(q^{2},m_{q}^{2}\right),$
$\displaystyle\mathcal{M}_{e}$ $\displaystyle=$ $\displaystyle
i^{3}\int\frac{d^{4}q}{(2\pi)^{4}}\left[ig_{Z_{c}^{\prime}}\varepsilon_{\mu\nu\alpha\beta}(-i)p^{\mu}\epsilon_{Z_{c}^{\prime}}^{\nu}\right]\left[\frac{1}{2}g_{D^{*}D^{*}\pi}\varepsilon_{\rho\tau\sigma\xi}ip_{3}^{\tau}i\right.$
(13)
$\displaystyle\left.\times\left(p_{1}^{\sigma}+q^{\sigma}\right)\right]\left[g_{\psi_{3}D^{*}D^{*}}\epsilon_{\psi_{3}}^{\theta\phi\kappa}(-i)\left(p_{2\theta}-q_{\theta}\right)\right.$
$\displaystyle\left.\times\left(g_{\kappa\lambda}g_{\phi\eta}+g_{\phi\lambda}g_{k\eta}\right)\right]$
$\displaystyle\times\frac{-g^{\alpha\rho}+p_{1}^{\alpha}p_{1}^{\rho}/m_{1}^{2}}{p_{1}^{2}-m_{1}^{2}}\frac{-g^{\beta\eta}+p_{2}^{\beta}p_{2}^{\eta}/m_{2}^{2}}{p_{2}^{2}-m_{2}^{2}}$
$\displaystyle\times\frac{-g^{\beta\lambda}+q^{\xi}q^{\lambda}/m_{q}^{2}}{q^{2}-m_{q}^{2}}\mathcal{F}^{2}\left(q^{2},m_{q}^{2}\right).$

In the above amplitudes, a form factor ${F}(q^{2},m_{q}^{2})$ is introduced to
reflect the off-shell effect of the exchange mesons and to make the amplitude
convergent in the ultraviolet region. In the present work, a form factor in
the monopole form is employed, which is,

$\displaystyle\mathcal{F}\left(q^{2},m_{q}^{2}\right)=\left(\frac{m_{q}^{2}-\Lambda^{2}}{q^{2}-\Lambda^{2}}\right),$
(14)

where the parameter $\Lambda$ is reparameterized as
$\Lambda=m_{q}+\alpha\Lambda_{QCD}$ with $\Lambda_{QCD}=220$ MeV and $m_{q}$
to be the mass of the exchanged meson. The model parameter $\alpha$ should be
of order one, However, its concrete value cannot be derived by the first
principle methods Tornqvist:1993vu ; Tornqvist:1993ng ; Locher:1993cc ;
Li:1996yn . In practice, the exact value of $\alpha$ is usually determined by
comparing the experimental measurements with the theoretical estimates. In the
present work, the relevant processes have not been measured, thus, we take
$\alpha$ as a free model parameter and varies it from one to three to check
$\alpha$ dependences of the estimated branching fractions.

With the above amplitudes corresponding the meson loop diagrams in Fig. 1, we
can obtain the amplitudes for $Z_{c}^{\prime}\to\psi(1^{3}D_{J})\pi$, which
are,

$\displaystyle\mathcal{M}_{Z_{c}^{\prime}\to\psi(1^{3}D_{1})\pi}$
$\displaystyle=$ $\displaystyle 2(\mathcal{M}_{a}+\mathcal{M}_{b}),$
$\displaystyle\mathcal{M}_{Z_{c}^{\prime}\to\psi(1^{3}D_{2})\pi}$
$\displaystyle=$ $\displaystyle 2(\mathcal{M}_{c}+\mathcal{M}_{d}),$
$\displaystyle\mathcal{M}_{Z_{c}^{\prime}\to\psi(1^{3}D_{3})\pi}$
$\displaystyle=$ $\displaystyle 2\mathcal{M}_{e},$ (15)

where the factor 2 comes from the charge symmetry. Then the decay widths of
$Z_{c}^{\prime}\to\psi(1^{3}D_{J})\pi$ could be estimated by

$\displaystyle\Gamma(Z_{c}^{\prime}\to\psi_{J}\pi)=\frac{1}{3}\frac{1}{8\pi}\frac{|\vec{p}|}{m_{Z_{c}^{\prime}}^{2}}\overline{\left|\mathcal{M}_{Z_{c}^{\prime}\to\psi(1^{3}D_{J})\pi}\right|^{2}},$
(16)

where $\vec{p}$ denotes the momentum of daughter particles in the initial rest
frame.

##  III Numerical Results and discussion

###  III.1 Coupling constants

So far, $Z_{c}(4020)$ has only been observed in the $D^{\ast}\bar{D}^{\ast}$
and $h_{c}\pi$ channels. At $\sqrt{s}=4.26$ GeV, the Born cross sections for
$e^{+}e^{-}\to\pi^{\pm}(D^{\ast}\bar{D}^{\ast})^{\mp}$ are measured to be
$(137\pm 9\pm 15)$ pb and the fraction from the quasi-two body cascade decay
process
$e^{+}e^{-}\to\pi^{\pm}Z_{c}(4020)^{\mp}\to\pi^{\pm}(D^{\ast}\bar{D}^{\ast})^{\mp}$
was measured to be $0.65\pm 0.09\pm 0.06$ BESIII:2013mhi , and the cross
sections for the quasi-two body process
$e^{+}e^{-}\to\pi^{\pm}Z_{c}(4020)^{\mp}\to\pi^{+}\pi^{-}h_{c}$ was measured
to be $(7.4\pm 1.7\pm 2.1)$ pb BESIII:2013ouc . Then, the ratio of the widths
of $Z_{c}(4020)\to D^{\ast}\bar{D}^{\ast}$ and $Z_{c}(4020)\to h_{c}\pi$ is
estimated to be,

$\displaystyle\frac{\Gamma[Z_{c}(4020)\to
D^{\ast}\bar{D}^{\ast}]}{\Gamma[Z_{c}(4020)\to h_{c}\pi]}=12.0\pm 3.68\pm
3.48.$ (17)

With the center values of the estimated ratio and the width of $Z_{c}(4020)$,
one can find the center value of $\Gamma[Z_{c}(4020)\to
D^{\ast}\bar{D}^{\ast}]$ is about 12 MeV by assuming that $Z_{c}(4020)$
dominantly decays into $D^{\ast}\bar{D}^{\ast}$ and $h_{c}\pi$. By comparing
the width of $Z_{c}(4020)\to D^{\ast}\bar{D}^{\ast}$ estimated by the
effective Lagrangian in Eq. (1) and the one deduced by the experimental
measurements, one can obtain the coupling constant $g_{Z_{c}^{\prime}}$=5.63
Xiao:2018kfx .

In the heavy quark limit, the coupling constants of the $D-$wave charmonia and
the charmed meson pair satisfy,

$\displaystyle g_{\psi_{1}DD}$ $\displaystyle=$
$\displaystyle-2g_{2}\frac{\sqrt{15}}{3}\sqrt{m_{\psi_{1}}m_{D}m_{D}},$
$\displaystyle g_{\psi_{1}DD^{\ast}}$ $\displaystyle=$ $\displaystyle
g_{2}\frac{\sqrt{15}}{3}\sqrt{m_{D}m_{D^{\ast}}/m_{\psi_{1}}},$ $\displaystyle
g_{\psi_{1}D^{\ast}D^{\ast}}$ $\displaystyle=$ $\displaystyle-
g_{2}\frac{\sqrt{15}}{15}\sqrt{m_{\psi_{1}}m_{D^{\ast}}m_{D^{\ast}}},$
$\displaystyle g_{\psi_{2}DD^{\ast}}$ $\displaystyle=$ $\displaystyle
2g_{2}\sqrt{\frac{3}{2}}\sqrt{m_{\psi_{2}}m_{D}m_{D^{\ast}}},$ $\displaystyle
g_{\psi_{2}D^{\ast}D^{\ast}}$ $\displaystyle=$ $\displaystyle
2g_{2}\sqrt{\frac{1}{6}}\sqrt{m_{D^{\ast}}m_{D^{\ast}}/m_{\psi_{2}}},$
$\displaystyle g_{\psi_{3}D^{\ast}D^{\ast}}$ $\displaystyle=$ $\displaystyle
2g_{2}\sqrt{m_{\psi_{3}}m_{D^{\ast}}m_{D^{\ast}}},$

where $g_{2}$ is a gauge coupling and its value can be estimated by the
measured width of $\psi(3770)\to D\bar{D}$, which is $g_{2}=1.39$.

Considering the heavy quark limit and chiral symmetry, the coupling constants
related to the pion meson satisfy,

$\displaystyle g_{D^{\ast}D\pi}$ $\displaystyle=$
$\displaystyle\frac{2g}{f_{\pi}}\sqrt{m_{D^{\ast}}m_{D}},$ $\displaystyle
g_{D^{\ast}D^{\ast}\pi}$ $\displaystyle=$ $\displaystyle\frac{2g}{f_{\pi}}.$
(19)

with $f_{\pi}=132\mathrm{MeV}$ to be the decay constant of pion meson,
$g=0.5$, which was determined by using the partial width of
$D^{\ast}\rightarrow D\pi$ Colangelo:2003sa .

###  III.2 branching fractions of $Z_{c}(4020)\to\psi(1^{3}D_{J})\pi$

With the above preparations, All the relevant parameters have been fixed
expect for the model parameter $\alpha$, which is introduced by the form
factor in the amplitudes. In Ref. Cheng:2004ru , the branching fraction of
$Z_{c}(4020)\to h_{c}\pi$ was estimated with the meson loop mechanism and we
found the branching fraction of $Z_{c}(4020)$ could be well reproduce in the
range $1.0<\alpha<3.0$. Thus, in the present work, we take the same model
parameter range to estimate the partial widths of
$Z_{c}(4020)^{+}\to\psi(1^{3}D_{J})\pi^{+}$, with the center value of the
width of $Z_{c}(4020)$, we can further estimate the branching fractions of
$Z_{c}(4020)^{+}\to\psi(1^{3}D_{J})\pi^{+}$ depending on the model parameter
$\alpha$.

Figure 2: (Color online). The branching fractions of
$Z_{c}(4020)\to\psi(1^{3}D_{J})\pi$ depending on the model parameter $\alpha$
(J=1,2,3).

The branching fractions of $Z_{c}(4020)^{+}\to\psi(1^{3}D_{J})\pi^{+}$
depending on the model parameter $\alpha$ are presented in Fig 2. From the
figure one can find the branching fraction for
$Z_{c}(4020)\to\psi(1^{3}D_{J})\pi$ increase with the model paramerter
$\alpha$. In particular, the branching fractions of
$Z_{c}(4020)\to\psi(1^{3}D_{1})\pi$ are estimated to be,

$\displaystyle B[Z_{c}(4020)\to\psi(1^{3}D_{1})\pi]$ $\displaystyle=$
$\displaystyle(0.31-3.22)\times 10^{-2}$ $\displaystyle
B[Z_{c}(4020)\to\psi(1^{3}D_{2})\pi]$ $\displaystyle=$
$\displaystyle(3.26-8.07)\times 10^{-3}$ $\displaystyle
B[Z_{c}(4020)\to\psi(1^{3}D_{3})\pi]$ $\displaystyle=$
$\displaystyle(0.23-1.41)\times 10^{-3}.\ \ \ \ $ (20)

Figure 3: (Color online). The ratios of branching fractions of
$Z_{c}(4020)\to\psi(1^{3}D_{J})\pi$ depending on the model parameter $\alpha$.

Our estimations indicate that the model parameter dependences of these
branching fractions are similar, thus, one can expect the ratios of these
branching fractions should be weakly dependent on the model parameter. The
ratios are defined as,

$\displaystyle
R_{21}=\frac{B\left[Z_{c}(4020)\to\psi(1^{3}D_{2})\pi\right]}{B\left[Z_{c}(4020)\to\psi(1^{3}D_{1})\pi\right]},$
$\displaystyle
R_{31}=\frac{B\left[Z_{c}(4020)\to\psi(1^{3}D_{3})\pi\right]}{B\left[Z_{c}(4020)\to\psi(1^{3}D_{1})\pi\right]}.$
(21)

In Fig. 3, we present our estimations of the $\alpha$ dependence of the
branching fractions ratios defined above. From the figure, one can find that
the ratio $R_{21}$ decrease from 1.05 to 0.25 with $\alpha$ increasing from 1
to 3. As for $R_{31}$, it’s estimated to be $(4.4\sim 7.4)\times 10^{-2}$,
which indicates that the branching ratio of $Z_{c}(4020)\to\psi_{3}(3842)\pi$
is about two order smaller than the one of $Z_{c}(4020)\to\psi(3770)\pi$.

Moreover, from Eq. (17) one can conclude the branching fraction of
$Z_{c}(4020)\to h_{c}\pi$ is about $7.7\%$. Here we can further discuss the
experimental potential of observing $Z_{c}(4020)\to\psi(1^{3}D_{J})\pi$ by the
ratios of the branching fractions of $Z_{c}(4020)\to\psi(1^{3}D_{J})\pi$ and
$Z_{c}(4020)\to h_{c}\pi$, and the ratios are estimated to be,

$\displaystyle r_{1}$ $\displaystyle=$
$\displaystyle\frac{B\left[Z_{c}(4020)\to\psi(3770)\pi\right]}{B\left[Z_{c}(4020)\to
h_{c}\pi\right]}=(0.43-4.46)\times 10^{-1},$ $\displaystyle r_{2}$
$\displaystyle=$
$\displaystyle\frac{B\left[Z_{c}(4020)\to\psi_{2}(3823)\pi\right]}{B\left[Z_{c}(4020)\to
h_{c}\pi\right]}=(0.45-1.12)\times 10^{-1},$ $\displaystyle r_{3}$
$\displaystyle=$
$\displaystyle\frac{B\left[Z_{c}(4020)\to\psi_{3}(3842)\pi\right]}{B\left[Z_{c}(4020)\to
h_{c}\pi\right]}=(0.32-1.95)\times 10^{-2}.$

From the above estimation, one can find $r_{3}$ is of order of $10^{-2}$,
which indicate that the experimental observation of
$Z_{c}(4020)\to\psi_{3}(3842)\pi$ should be impossible at this stage. As for
$r_{2}$, we find that its maximum is about 0.1,, which indicate the branching
fraction of $Z_{c}(4020)\to\psi_{2}(3823)\pi$ is at least one order smaller
than the one of $Z_{c}(4020)\to h_{c}\pi$ and should also be hard to be
detected experimentally. As for $r_{1}$, we find its maximum is about 0.5.
Considering the large branching fraction of $Z_{c}(4020)\to h_{c}\pi$, the
branching fraction of $Z_{c}(4020)\to\psi(3770)\pi$ are also sizable, which
indicate that the experimental observations of $Z_{c}(4020)\to\psi(3770)\pi$
should be possible. Such results are also consistent with the experimental
hints for the peak at 4.04 GeV in the $\psi(3770)\pi$ invariant mass
distributions of the process $e^{+}e^{-}\to\psi(3770)\pi^{+}\pi^{-}$, and
further experimental analysis of this channel with more data should provide a
crucial test to the present estimations.

##  IV SUMMARY

About one decade ago, the BES III Collaboration reported two near thresholds
charmonium like states, $Z_{c}(3900)$ and $Z_{c}(4020)$. However, their
internal structure have not been decoded until now. Searching for more decay
and production process will help us to reveal the nature of these charmonium-
like states. With the accumulation of the experimental data, the BES III
Collaboration have observed the cross sections for
$e^{+}e^{-}\to\psi(3770)\pi^{+}\pi^{-}$ and
$e^{+}e^{-}\to\pi^{+}\pi^{-}\psi_{2}(3823)/\pi^{0}\pi^{0}\psi_{2}(3823)$. The
hints for the peaks at 4.04 and 4.13 GeV has been observed. If we consider the
peak around 4.04 GeV is the contributions from $Z_{c}(4020)$, then the peak at
4.13 GeV should be the reflection of $Z_{c}(4020)$.

Inspired the observation of BES III Collaboration, we investigate charmed
meson loop contributions to the decay process of
$Z_{c}(4020)\to\psi(1^{3}D_{J})\pi$ by using an effective Lagrangian approach.
Our estimations indicate that the branching fractions of
$Z_{c}(4020)\to\psi(3770)\pi$ can reach up to $3.22\%$, which is comparable
with the one of $Z_{c}(4020)\to h_{c}\pi$. Considering the observations of
$Z_{c}(4020)$ in the $h_{c}\pi$ channel, we conclude that the observation of
$Z_{c}(4020)$ in the $\psi(3770)\pi$ channel is possible, which is consistent
with the preliminary analysis of the BES III Collaboration. As for
$Z_{c}(4020)\to\psi_{2}(3823)\pi$, the maximum of the branching fraction is
$8.07\times 10^{-2}$, which is one order smaller than the one of
$Z_{c}(4020)\to h_{c}\pi$. Thus, it should be rather difficult to detect the
$Z_{c}(4020)$ in the $\psi_{2}(3823)\pi$ channel at this stage. In addition,
the branching fraction for $Z_{c}(4020)\to\psi_{3}(3842)\pi$ is at least two
order smaller than the one of $Z_{c}(4020)\to h_{c}\pi$, thus it is impossible
to observe $Z_{c}(4020)$ in the $\psi_{3}(3842)\pi$ decay mode.

## ACKNOWLEDGMENTS

This work is supported by the National Natural Science Foundation of China
under the Grant No. 12175037 and 11775050.

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