revtex4-1Repair the float

#  The strong coupling $g_{XJ/\psi\phi}$ of $X(4700)\to J/\psi\phi$ in the
light-cone sum rules

Yiling Xie Dazhuang He Institute of Theoretical Physics, School of Physics,
Dalian University of Technology,
No.2 Linggong Road, Dalian, Liaoning, 116024, People’s Republic of China  Xuan
Luo School of Physics and Optoelectronics Engineering, Anhui University,
Hefei, Anhui 230601, People’s Republic of China  Hao Sun<EMAIL_ADDRESS>Institute of Theoretical Physics, School of Physics, Dalian University of
Technology,
No.2 Linggong Road, Dalian, Liaoning, 116024, People’s Republic of China

###### Abstract

We assign the scalar tetraquark and the D-wave tetraquark state for $X(4700)$
and calculate the width of the decay $X(4700)$ $\to J/\psi\phi$ within the
framework of light-cone sum rules. The strong coupling $g_{XJ/\psi\phi}$ is
obtained by considering the technique of soft-meson approximation. We also
investigate the mass and the decay constant of $X(4700)$ in the framework of
SVZ sum rules. Our prediction for the mass is in agreement with the
experimental measurement. For the decay width of $X(4700)$ $\to J/\psi\phi$,
if we assigns $X$(4700) as a scalar $cs\bar{c}\bar{s}$ tetraquark state, we
obtain $\Gamma=(109^{+35}_{-24})$ MeV, which indicates that $X(4700)$ $\to
J/\psi\phi$ is a predominant decay channel. On the contrary, if $X$(4700) is
assigned as a D-wave tetraquark state, we obtain $\Gamma=(17.1^{+6.2}_{-4.0})$
MeV, so that $X$(4700) $\to$ $J/\psi\phi$ becomes a much smaller decay
channel.

###### Contents

  1. I Introduction
  2. II Calculation Framework
    1. II.1  The strong coupling $g_{XJ/\psi\phi}$ in the LCSR
    2. II.2 OPE side calculation
    3. II.3  The mass and the decay constant of $X(4700)$
  3. III Numerical calculation
    1. III.1 Input parameters
    2. III.2 The mass, decay constant
    3. III.3 The coupling constant and the decay width
  4. IV Summary
  5. V Appendix
    1. V.1 The relations between the light-cone distribution amplitudes (LCDAs) and the matrix elements
    2. V.2 Spectral densities

##  I Introduction

In 2016, the LHCb Collaboration analyzed the $B^{+}$ $\to J/\psi\phi K^{+}$
decay with $3\ \text{fb}^{-1}$ data of $pp$ collision at $\sqrt{s}=7$ and $8$
TeV Aaij  _et al._ (2017a, b), confirmed there are four resonances in the
$J/\psi\phi$ mass spectrum, i.e., $X(4140)$, $X(4274)$, $X(4500)$ and
$X(4700)$. The spin parity number of $X(4140)$ and $X(4274)$ states are
determined to be $J^{PC}=1^{++}$ with $5.7\sigma$ and $5.8\sigma$
significance, respectively. The $J^{PC}$ of $X(4500)$ and $X(4700)$ states are
$0^{++}$ with $5.2\sigma$ and $4.9\sigma$ significance, respectively. Their
masses and widths are Aaij  _et al._ (2017a)

$\displaystyle X(4140):$ $\displaystyle\text{M}=4146.5\pm 4.5^{+4.5}_{-2.8}$
$\displaystyle\text{MeV},$ (1) $\displaystyle\Gamma=83\pm 21^{+21}_{-14}$
$\displaystyle\text{MeV},$ $\displaystyle X(4274):$
$\displaystyle\text{M}=4273.3\pm 8.3^{+17.2}_{-3.6}$
$\displaystyle\text{MeV},$ $\displaystyle\Gamma=56\pm 11^{+8}_{-11}$
$\displaystyle\text{MeV},$ $\displaystyle X(4500):$
$\displaystyle\text{M}=4506\pm 11^{+12}_{-15}$ $\displaystyle\text{MeV},$
$\displaystyle\Gamma=92\pm 21^{+21}_{-20}$ $\displaystyle\text{MeV},$
$\displaystyle X(4700):$ $\displaystyle\text{M}=4704\pm 10^{+14}_{-24}$
$\displaystyle\text{MeV},$ $\displaystyle\Gamma=120\pm 31^{+42}_{-33}$
$\displaystyle\text{MeV}.$

Since these resonance-like peaks appear in the $J/\psi\phi$ invariant mass
spectrum, and $J/\psi\phi$ contains a $c\bar{c}$ pair and a $s\bar{s}$ pair,
which implies that these states may be charmonium. The predicted mass of
$\chi_{c0}(6P)$ is about $4669$ MeV in the screened potential (SP) model Gui
_et al._ (2018), which is very close to that of $X(4700)$. But its predicted
total width is only about 16 MeV, too narrow to be comparable with $120\pm
31^{+42}_{-33}$ MeV, the observed width of $X(4700)$. Such discrepancy makes
it difficult to understand the structure of $X(4700)$ in this way. In the
constituent quark model, the authors in Refs.Fernández  _et al._ (2016);
Ortega  _et al._ (2016) calculated the quark-antiquark spectrum for
$J^{PC}=0^{++}$ channels, and showed that $X(4700)$ could appear as
conventional charmonium states with quantum numbers $5^{3}P_{0}$ since it lies
in the predicted mass and width ranges for $5^{3}P_{0}$. According to Non-
Relativistic QCD (NRQCD) results, $X(4700)$ is also compatible with a
charmonium $\chi_{c0}(4P)$Oncala and Soto (2017). However, the higher
charmonium states like $\chi_{cJ}(nP)$ would have numerous decay modes to open
charm mesons, unfortunately there is no relevant mass spectrum appears in the
$B\to$ $D^{(*)}_{(s)}\bar{D}^{*}_{(s)}K$ decays. In addition, the coupling of
the higher charmonium states to $J/\psi\phi$ and to $J/\psi\omega$ should be
very similar, but the BaBar collaboration measured the $J/\psi\omega$ mass
spectrum in $B^{+}$ $\to J/\psi\omega K^{+}$ decay and did not find any
structures resembling the $J/\psi\phi$ mass peaks Aaij  _et al._ (2017a, b),
which contradicts a charmonium interpretation for the $X(4700)$.

Though $X(4700)$ can’t be a pure charmonium, it could be explained as a hybrid
state, a charmonium state contains excited gluon fields. According to NRQCD,
the mass of $0^{++}$ hybrid $1p_{0}(H_{3})$ state is $4566$ MeV which is close
to the spectrum of $X(4700)$. However, the charmonium fraction of $0^{++}$
hybrid is very small. So it is difficult to understand $X(4700)$ observed in
the $J/\psi\phi$ channel since $J/\psi$ mainly contains $c\bar{c}$ pair.

Besides hybrid, charmonium state can also be mixed with a light hadron to form
a bound states, called hadrocharmonium. Considering that $X(4700)$ is observed
in the $J/\psi\phi$ spectrum, it could be a bound state of $J/\psi-\phi$ or
$\Psi(2S)-\phi$. However, the $J/\psi-\phi$ potential was found too weak, in
lattice QCD, to form a bound states Ozaki and Sasaki (2013). Furthermore, the
$\Psi(2S)-\phi$ bound state has already been assigned to $X(4274)$ Panteleeva
_et al._ (2019) so that rules out the possibility that $X(4700)$ be a s-wave
$\Psi(2S)-\phi$ bound state. Anyway, $X(4700)$ could be other hadrocharmonium
states in specific regions of the parameter space. Such explanation may
require different binding mechanisms Panteleeva  _et al._ (2019).

To extrapolate the intrinsic structure of $X(4700)$, a myriad of theoretical
studies has turned to the $cs\bar{c}\bar{s}$ tetraquark state.

It was suggested in Ref.Maiani  _et al._ (2016); Lü and Dong (2016) that
$X(4700)$ can be assigned as the 2S excited $cs\bar{c}\bar{s}$ tetraquark
state. $X$(4700) could also be explained as the radial excitation tetraquark
state, for instance, of the hidden charm tetraquarks with quark content
$\frac{1}{\sqrt{6}}(u\bar{u}+d\bar{d}-2s\bar{s})c\bar{c}$ in a diquark-
antidiquark model Zhu (2016), or the S-wave radial excited compact tetraquark
states within the framework of the color flux-tube model with a multibody
confinement potential Deng  _et al._ (2018). In addition, $X$(4700) could be
assigned as an first radial excitation of $X$(4350) tetraquark state through
the color-magnetic interaction model Wu  _et al._ (2016), or a 2S radial
excited compact tetraquark state with $J^{PC}=0^{++}$ in the chiral quark
model Yang and Ping (2019).

Quite apart from the above excited tetraquark states explanation for
$X(4700)$, there are several others. For example, it has been considered as a
$0^{++}$ axial-vector diquark-antidiquark bound states in the diquark model
Anwar  _et al._ (2018), or a compact tetraquark state with $IJ^{P}=00^{+}$ in
the framework of the quark delocalization color screening model Liu  _et al._
(2021). In particular, it can also be treated as a D-wave $cs\bar{c}\bar{s}$
tetraquark states Chen  _et al._ (2017) or a ground tetraquark state Wang
(2017) by using the sum rule approach develop by Shifman, Vainshtein and
Zakharov (SVZ sum rules) Reinders  _et al._ (1985).

Although the above theories have claimed the resonance peak in the
$J/\psi\phi$ mass spectrum corresponds to the genuine resonance, there are
other opinionsNakamura (2021). In Ref.Liu (2017); Ge  _et al._ (2021), the
authors have investigated the open-charmed mesons, $J/\psi K^{*+}$,
$\psi^{\prime}K^{+}$, and $\psi^{\prime}\phi$ re-scattering effects or
threshold cusps in the process $B^{+}\to J/\psi\phi K^{+}$, in which $X$(4700)
can be simulated by the $\psi^{\prime}\phi$ re-scattering via the
$\psi^{\prime}K_{1}$ loops. Anyway, within the available experimental data,
none of these theoretical interpretations can be completely accepted or
excluded to the nature of $X$(4700). At this point, the structure of $X$(4700)
is not yet fully settled.

As we mentioned, with the method of SVZ sum rules, $X$(4700) has been
investigated in Ref.Chen  _et al._ (2017) and Ref.Wang (2017) in which its
mass is predicted. In this paper, we follow the same assumption that $X$(4700)
is a D-wave tetraquark state and a groud tetraquark state simultaneously. We
evaluate the mass of $X$(4700) in SVZ sum rules. The results are compared with
the prediction in Ref.Chen  _et al._ (2017); Wang (2017), and with the values
in PDG Zyla  _et al._ (2020) to ensure the credibility of our calculation. We
then extend to evaluate the decay constant of $X$(4700), which is needed in
the numerical calculation of the strong coupling $g_{XJ/\psi\phi}$. While
calculating $g_{XJ/\psi\phi}$, the interpolating currents are still taken from
Ref.Chen  _et al._ (2017); Wang (2017), and the method of light-cone sum rules
(LCSR) is used and the full technical details are presented, in particular,
the technique of soft-meson approximation is considered. Based on these
calculations, we finally obtain the decay width of $X\text{(4700)}\to$
$J/\psi\phi$. The results are compared with the the total decay widths of
$X$(4700) in experimental measurements. Our present study can be regarded as a
supplement to other previous works.

Our paper is organized as follows: In Section.II the strong coupling
$g_{XJ/\psi\phi}$ will be derived with the approach of light-cone sum rules.
And we also calculate the mass and decay constant of the $X$(4700) state
within two-point SVZ sum rule approach. The numerical results and discussions
are shown in Section.III. We reach our summary in Section.IV.

##  II Calculation Framework

###  II.1  The strong coupling $g_{XJ/\psi\phi}$ in the LCSR

Before starting to predict the width of $X(\,4700\,)$ $\to J/\psi\phi$, we
need to calculate the strong coupling $g_{XJ/\psi\phi}$, in the framework of
QCD LCSR. Let’s strat by defining the two-point correlation function:

$\displaystyle\Pi_{\text{I},\mu}^{\text{LC}}(\,p+q,q\,)$ (2)
$\displaystyle=i\int\mbox{d}^{4}xe^{ipx}\braket{\phi(\,q\,)}{\,T\\{\,J_{\mu}^{J/\psi}(\,x\,)J_{\text{I}}^{X\dagger}(\,0\,)\,\\}\,}{0},$

where $p$, $q$ are the four-momentum for $J/\psi$ and $\phi$ respectively and
$\text{I}=(1),(2)$. Therefore $X(4700)$ has four-momentum $p^{\prime}=p+q$
according to the momentum conservation. $J_{\mu}^{J/\psi}$ is the
interpolating current of $J/\psi$, and $J_{(1)}^{X}$, $J_{(2)}^{X}$ are that
of $X(4700)$ in two different structures. They are given by Albuquerque and
Nielsen (2009); Bečirević  _et al._ (2014)

$\displaystyle J_{\mu}^{J/\psi}(x)=$
$\displaystyle\bar{c}_{i}(x)\gamma_{\mu}c_{i}(x),$ (3) $\displaystyle
J_{(1)}^{X}(x)=$
$\displaystyle\varepsilon_{ijk}\varepsilon_{imn}[s^{T}_{j}(x)C\gamma_{\mu}\gamma_{5}c_{k}(x)][\bar{s}_{m}(x)\gamma_{5}\gamma^{\mu}C\bar{c}_{n}^{T}(x)],$
$\displaystyle J_{(2)}^{X}(x)=$ $\displaystyle c_{a}^{T}(x)C\gamma_{\mu
1}[D_{\mu 3}D_{\mu 4}s_{b}(x)]$ $\displaystyle(\bar{c}_{a}(x)\gamma_{\mu
2}Cs_{b}^{T}(x)+\bar{c}_{b}(x)\gamma_{\mu 2}C\bar{s}_{a}^{T}(x))$
$\displaystyle\times$ $\displaystyle(g^{\mu 1\mu 3}g^{\mu 2\mu 4}+g^{\mu 1\mu
4}g^{\mu 2\mu 3}-g^{\mu 1\mu 2}g^{\mu 3\mu 4}/2)$

where $a,b,i,j,k,m,n$ are the color indexes and $C$ is the charge conjugation
matrix.

Next we need to establish a relation between the correlation function
$\Pi_{\text{I},\mu}^{\text{LC}}(p^{\prime},q)$ and the strong coupling
$g_{XJ/\psi\phi}$. For a general dispersion relation, we have

$\displaystyle\Pi_{\text{I},\mu}^{\text{LC}}(p^{\prime},q)=\frac{1}{\pi^{2}}\int\int\frac{ds_{1}ds_{2}\text{Im}\Pi_{\text{I},\mu}^{\text{LC}}(s_{1},s_{2})}{(s_{1}-p^{2})(s_{2}-p^{\prime
2})}+\cdots$ (4)

where the subtraction terms and the single dispersion integrals are not shown,
all of them will vanish after applying the double Borel transformation to
Eq.(4). Inserting in Eq.(2) with two complete sets of hadronic states and
using Eq.(4), we obtain the phenomenological expression of the correlation
function noted by $\Pi^{\text{LC,phen}}_{\text{I},\mu}(p^{\prime},q)$

$\displaystyle\Pi^{\text{LC,phen}}_{\text{I},\mu}(p^{\prime},q)=$ (5)
$\displaystyle\frac{\braket{0}{J_{\mu}^{J/\psi}}{J/\psi(p)}\braket{\phi(q)J/\psi(p)}{X(p^{\prime})}\braket{X(p^{\prime})}{J_{\text{I}}^{{X}\dagger}}{0}}{(p^{\prime
2}-m_{\text{I},X}^{2})(p^{2}-m_{J/\psi}^{2})}$
$\displaystyle+\int_{s_{1}^{0\prime}}^{\infty}\int_{s_{2}^{0\prime}}^{\infty}\frac{ds_{1}ds_{2}\
\rho_{\text{I},\mu}^{\text{LC,phen}}(s_{1},s_{2})}{(s_{1}-p^{2})(s_{2}-p^{\prime
2})}+\int_{s_{1}^{0\prime}}^{\infty}\frac{ds_{1}\
\rho_{\text{I},1,\mu}^{\text{LC,phen}}(s_{1})}{(s_{1}-p^{2})}$
$\displaystyle+\int_{s_{2}^{0\prime}}^{\infty}\frac{ds_{2}\
\rho_{\text{I},2,\mu}^{\text{LC,phen}}(s_{2})}{(s_{2}-p^{\prime 2})}.$

Here contributions of the higher resonances and the continuum states are
denoted by $\rho_{\text{I},\mu}^{\text{LC,phen}}(s_{1},s_{2})$,
$s_{1}^{0\prime}$ and $s_{2}^{0\prime}$ denote the lowest thresholds of
continuum states. $\rho_{\text{I},1,\mu}^{\text{phen}}(s_{1})$ and
$\rho_{\text{I},2,\mu}^{\text{phen}}(s_{2})$ are the additional contributions
to make the double dispersion integral finiteBelyaev  _et al._ (1995). The
strong coupling $g_{XJ/\psi\phi}$ is defined as an invariant constant
parameterizing the hadronic matrix element

$\displaystyle\braket{\phi(q)J/\psi(p)}{X(p^{\prime})}=$ $\displaystyle
g_{XJ/\psi\phi}$ (6) $\displaystyle\times$ $\displaystyle[-(q\cdot
p)(\varepsilon^{*}\cdot\varepsilon^{\prime})+(q\cdot\varepsilon^{*})(p\cdot\varepsilon^{\prime})],$

and the decay constants here are defined as:

$\displaystyle\braket{X(p^{\prime})}{J_{I}^{X\dagger}}{0}$
$\displaystyle=m_{I,X}\,f_{I,X},$ (7)
$\displaystyle\braket{0}{J_{\mu}^{J/\psi}}{J/\psi(p)}$
$\displaystyle=m_{J/\psi}f_{J/\psi}\,\varepsilon_{\mu},$

where $\varepsilon$, $\varepsilon^{\prime}$ are the polarization vectors of
the $J/\psi$ and $\phi$ respectively, $m_{J/\psi(X)}$ and $f_{J/\psi(X)}$ are
the mass and decay constants of $J/\psi$($X(4700)$).

By inserting Eq.(6) and (7) back to Eq.(5), and performing the polarization
sum, we can derive

$\displaystyle\Pi^{\text{LC,phen}}_{\text{I},\mu}(p^{\prime},q)$
$\displaystyle=\frac{m_{J/\psi}f_{J/\psi}\ m_{\text{I,X}}f_{\text{I,X}}\
g_{\text{XJ}/\psi\phi}}{(p^{\prime
2}-m_{\text{I,X}}^{2})(p^{2}-m_{J/\psi}^{2})}$ (8)
$\displaystyle\times[(p\cdot q)\
\epsilon_{\mu}^{\prime}-(p\cdot\epsilon^{\prime})\ q_{\mu}]+\cdots$
$\displaystyle=\Pi^{\text{LC,phen}}_{\text{I}}(p^{\prime},q)[(p\cdot
q)\epsilon_{\mu}^{\prime}-p\cdot\epsilon^{\prime}q_{\mu}],$

where we choose the structure proportional to $\epsilon_{\mu}^{\prime}$ to
work with. The relevant form can be written as

$\displaystyle\Pi^{\text{LC,phen}}_{\text{I}}(p^{\prime},q)$
$\displaystyle=\frac{m_{J/\psi}m_{\text{I},X}f_{J/\psi}f_{\text{I},X}g_{XJ/\psi\phi}}{(p^{\prime
2}-m_{\text{I},X}^{2})(p^{2}-m_{J/\psi}^{2})}$ (9)
$\displaystyle+\int_{s_{1}^{0\prime}}^{\infty}\int_{s_{2}^{0\prime}}^{\infty}\frac{\mbox{d}s_{1}\mbox{d}s_{2}\rho^{\text{LC,phen}}_{\text{I}}(s_{1},s_{2})}{(s_{1}-p^{2})(s_{2}-p^{\prime
2})}+\cdots.$

Applying the Borel transformations on variables $p^{2}$ and $p^{\prime
2}=(p+q)^{2}$ to the correlation function yields

$\displaystyle\mathcal{B}_{p^{2}}(M_{1}^{2})\mathcal{B}_{p^{\prime
2}}(M_{2}^{2})\Pi^{\text{LC,phen}}_{\text{I}}(p^{\prime},q)=$ (10)
$\displaystyle
m_{J/\psi}m_{\text{I},X}f_{J/\psi}f_{\text{I},X}g_{XJ/\psi\phi}\exp[-\frac{m_{J/\psi}^{2}}{M_{1}^{2}}-\frac{m_{\text{I},X}^{2}}{M_{2}^{2}}]$
$\displaystyle+\int_{s_{1}^{0\prime}}^{\infty}\int_{s_{2}^{0\prime}}^{\infty}\mbox{d}s_{1}\mbox{d}s_{2}\exp[-\frac{s_{1}}{M_{1}^{2}}-\frac{s_{2}}{M_{2}^{2}}]\rho^{\text{LC,phen}}_{\text{I}}(s_{1},s_{2}).$

To proceed, we represent the OPE result for the correlation function in the
form of the double dispersion integral

$\displaystyle\Pi_{\text{I}}^{\text{LC,OPE}}(p^{\prime},q)=\int_{s_{1}^{\prime}}^{\infty}\int_{s_{2}^{\prime}}^{\infty}\frac{ds_{1}ds_{2}\
\rho_{\text{I}}^{\text{LC,OPE}}(s_{1},s_{2})}{(s_{1}-p^{2})(s_{2}-p^{\prime
2})}+\cdots$ (11)

with the double spectral density

$\displaystyle\rho_{\text{I}}^{\text{LC,OPE}}(s_{1},s_{2})=\frac{\text{Im}\Pi_{\text{I}}^{\text{LC,OPE}}(s_{1},s_{2})}{\pi^{2}}.$
(12)

By choosing the structure proportional to $\epsilon_{\mu}^{\prime}$ and
performing the Borel transformations on variables $p^{2}$ and $p^{\prime
2}=(p+q)^{2}$, we find out

$\displaystyle\Pi^{\text{LC,OPE}}_{\text{I}}$
$\displaystyle(M_{1}^{2},M_{2}^{2})=$ (13)
$\displaystyle\int_{s_{1}^{\prime}}^{\infty}\int_{s_{2}^{\prime}}^{\infty}\mbox{d}s_{1}\mbox{d}s_{2}\exp[-\frac{s_{1}}{M_{1}^{2}}-\frac{s_{2}}{M_{2}^{2}}]\rho^{\text{LC,OPE}}_{\text{I}}(s_{1},s_{2}).$

Then we employ the quark-hadron duality: assume the integral of the hadronic
spectral density $\rho^{\text{h}}_{\text{I}}(s_{1},s_{2})$ in the region
$\\{s_{1}\geq s_{1}^{0\prime},s_{2}\geq s_{2}^{0\prime}\\}$ is equal to the
integral of the OPE spectral density
$\rho^{\text{LC,OPE}}_{\text{I}}(s_{1},s_{2})$ in a certain region of
$\\{s_{1}\geq s_{1}^{0},s_{2}\geq s_{2}^{0}\\}$

$\displaystyle\int_{s_{1}^{0\prime}}^{\infty}\int_{s_{2}^{0\prime}}^{\infty}\mbox{d}s_{1}\mbox{d}s_{2}\exp[-\frac{s_{1}}{M_{1}^{2}}-\frac{s_{2}}{M_{2}^{2}}]\rho^{\text{LC,phen}}_{\text{I}}(s_{1},s_{2})=$
(14)
$\displaystyle\int_{s_{1}^{0}}^{\infty}\int_{s_{2}^{0}}^{\infty}\mbox{d}s_{1}\mbox{d}s_{2}\exp[-\frac{s_{1}}{M_{1}^{2}}-\frac{s_{2}}{M_{2}^{2}}]\rho^{\text{LC,OPE}}_{\text{I}}(s_{1},s_{2}).$

Equating the double dispersion (10) and (13), and substituting (14) to (10),
the result of LCSR for the strong coupling reads:

$\displaystyle
g_{XJ/\psi\phi}=\frac{1}{m_{J/\psi}m_{\text{I},X}f_{J/\psi}f_{\text{I},X}}\exp[\,\frac{m_{J/\psi}^{2}}{M_{1}^{2}}+\frac{m_{\text{I},X}^{2}}{M_{2}^{2}}\,]$
(15)
$\displaystyle\times\int^{s_{1}^{0}}_{s_{1}^{\prime}}\int^{s_{2}^{0}}_{s_{2}^{\prime}}\mbox{d}s_{1}\mbox{d}s_{2}\exp[\,-\frac{s_{1}}{M_{1}^{2}}-\frac{s_{2}}{M_{2}^{2}}\,]\rho^{\text{LC,OPE}}_{\text{I}}(\,s_{1},s_{2}\,).$

Nevertheless, our situation differs from the standard one. From Eq.(2), we see
that the interpolating currents of $X(\,4700\,)$ is located at the space-time
point $x=0$, and the interpolating current of $J/\psi$ is located at the point
$x$. Therefore by contracting the $\bar{c}$ and $c$ quark fields, there remain
two light quarks $s$ and $\bar{s}$ sandwiched between the $\phi$ state and the
vacuum states, i.e.,
$\braket{\phi(\,q\,)}{\,[\,\bar{s}(\,0\,)s(\,0\,)]\,}{0}$. We encounter the
situation that the correlation function depends not on
$\braket{\phi(\,q\,)}{\,[\,\bar{s}(\,x\,)s(\,0\,)\,]\,}{0}$ but
$\braket{\phi(\,q\,)}{\,[\,\bar{s}(\,0\,)s(\,0\,)\,]\,}{0}$, so that the
$\phi$ distribution amplitude disappears and reduces to normalization factor.
Such situation is possible to appear in the kinematical limit $q\rightarrow 0$
which means $(\,p+q\,)=p$, and the correlation function depends only on one
variable $p^{2}$. Here, following Ref.Belyaev  _et al._ (1995), we adopt the
approach of soft-meson approximation, taking the limit $q\rightarrow 0$ and
dealing with the double pole terms.

The approximation of $q\rightarrow 0$ simplifies the hadronic side of the sum
rules, but leads to a more complicated expression on its hadronic
representation. The ground state depends only on the variable $p^{2}$:

$\displaystyle\Pi^{\text{LC,phen}}_{\text{I}}(\,p\,)=\frac{m_{J/\psi}m_{\text{I},X}f_{J/\psi}f_{\text{I},X}}{(\,p^{2}-m^{2}_{\text{I,X}}\,)^{2}}g_{XJ/\psi\phi}+\cdots,$
(16)

where $m^{2}_{\text{I}}=\frac{m_{J/\psi}^{2}+m_{\text{I},X}^{2}}{2}$ and the
Borel transformation on the variable $p^{2}$ applied to this correlation
function yieldsBelyaev  _et al._ (1995)

$\displaystyle\Pi^{\text{LC,phen}}_{\text{I}}(\,p^{\prime},q\,)$ (17)
$\displaystyle=\frac{1}{M^{2}}(\,m_{J/\psi}m_{\text{I},X}f_{J/\psi}f_{\text{I},X}g_{XJ/\psi\phi}+AM^{2}\,)e^{\frac{-m^{2}_{\text{I}}}{M^{2}}}+C,$

where the constant A incorporates all unsuppressed contributions, and C
contains all exponentially suppressed contributions. To remove unsuppressed
contributions, we perform the operator that the script inIoffe and Smilga
(1984)

$\displaystyle(\,1-M^{2}\frac{d}{dM^{2}}\,)M^{2}e^{m_{\text{I}}^{2}/M^{2}}$
(18)

on both sides of the sum rules expression: the phenomenological side and the
OPE side. So Eq.(15) becomes

$\displaystyle
g_{XJ/\psi\phi}=\frac{1}{m_{J/\psi}m_{\text{I},X}f_{J/\psi}f_{\text{I},X}}(\,1-M^{2}\frac{d}{dM^{2}}\,)M^{2}$
(19)
$\displaystyle\times\int^{\hat{s}^{\prime}}_{\hat{s}_{0}}\mbox{d}\hat{s}\exp[\,\frac{m_{J/\psi}^{2}}{2M^{2}}+\frac{m_{\text{I},X}^{2}}{2M^{2}}-\frac{\hat{s}}{M^{2}}\,]\rho^{\text{OPE}}_{\text{I}}(\,\hat{s}\,).$

Because of soft-meson approximation, the continuum state depends not on two
variables $s_{1}$ and $s_{2}$ but one that relabel as $\hat{s}$.

###  II.2 OPE side calculation

We already know that $g_{XJ//psi/phi}$ is related to the OPE part of the
correlation function, so we are going to calculate it. According to the Wick
Theorem, we can obtain

$\displaystyle\braket{\phi(\,q\,)}{\,\text{T}\,\\{\,J_{\mu}^{J/\psi}(\,x\,)J_{(1)}^{\text{X}\dagger}(\,0\,)\,\\}\,}{0}$
(20)
$\displaystyle=\braket{\phi(q)}{\,[\,\bar{s}^{a}_{\alpha}(\,0\,)s^{d}_{\beta}(\,0\,)\,]\,}{0}$
$\displaystyle\times[\,\gamma_{5}\tilde{S}^{ib}_{c}(\,x\,)\gamma_{\mu}\tilde{S}^{ie}_{c}(\,-x\,)\gamma_{\nu}\gamma_{5}$
$\displaystyle-\gamma_{\nu}\gamma_{5}\tilde{S}^{ib}_{c}(\,x\,)\gamma_{\mu}\tilde{S}^{ie}_{c}(\,-x\,)\gamma_{5}\,]_{\alpha\beta}.$

and

$\displaystyle\braket{\phi(\,q\,)}{\,\text{T}\\{\,J_{\mu}^{J/\psi}(\,x\,)J_{(2)}^{\text{X}\dagger}(\,0\,)\,\\}\,}{0}$
(21)
$\displaystyle=\braket{\phi(\,q\,)}{\,[\,(\,\bar{s}^{b}\overleftarrow{D}_{\mu
4}\overleftarrow{D}_{\mu 3}\,)_{\alpha}(\,0\,)s^{b}_{\beta}(\,0\,)\,]\,}{0}$
$\displaystyle\times[\,\gamma_{\mu
1}\tilde{S}^{ac}_{c}(\,x\,)\gamma_{\nu}\tilde{S}^{ac}_{c}(\,-x\,)\gamma_{\mu
2}\,]_{\alpha\beta}$ $\displaystyle+\braket{\phi(\,q\,)}{\
[\,(\,\bar{s}^{b}\overleftarrow{D}_{\mu 4}\overleftarrow{D}_{\mu
3}\,)_{\alpha}(\,0\,)s^{a}_{\beta}(\,0\,)\,]\,}{0}$
$\displaystyle\times[\,\gamma_{\mu
1}\tilde{S}^{ac}_{c}(\,x\,)\gamma_{\nu}\tilde{S}^{bc}_{c}(\,-x\,)\gamma_{\mu
2}\,]_{\alpha\beta}\ .$

Therefore, the correlation functions become

$\displaystyle\Pi_{(1),\mu}^{\text{OPE}}(\,p^{\prime},q\,)$
$\displaystyle=i\int
d^{4}xe^{ipx}\braket{\phi(q)}{\,\text{T}\\{\,J_{\mu}^{J/\psi}(\,x\,)J^{\text{X}\dagger}_{(1)}(\,0\,)\,\\}\,}{0}$
(22) $\displaystyle=i\int
d^{4}xe^{ipx}\varepsilon_{ijk}\varepsilon_{imn}\braket{\phi(\,q\,)}{\,[\
\bar{s}^{j}_{\alpha}(\,0\,)s^{m}_{\beta}(\,0\,)\,]\,}{0}$
$\displaystyle\quad\times[\,\gamma_{\mu}\gamma_{5}\tilde{S}^{kl}_{c}(\,x\,)\gamma_{\nu}\tilde{S}^{nl}_{c}(\,-x\,)\gamma_{\nu}\gamma_{5}\,]_{\alpha\beta},$

and

$\displaystyle\Pi_{(2),\mu}^{\text{OPE}}(\,p^{\prime},q\,)=i\int
d^{4}xe^{ipx}\braket{\phi(\,q\,)}{\,\text{T}\\{\,J_{\mu}^{J/\psi}(\,x\,)J^{\text{X}\dagger}_{(2)}(\,0\,)\,\\}\,}{0}$
(23) $\displaystyle=i\int d^{4}xe^{ipx}\\{\braket{\phi(\,q\,)}{\
[\,(\bar{s}^{b}(\,0\,)\overleftarrow{D}_{\mu 4}\overleftarrow{D}_{\mu
3})_{\alpha}s^{b}_{\beta}(\,0\,)\,]\,}{0}$ $\displaystyle\times[\,\gamma_{\mu
1}\tilde{S}^{ac}_{c}(x)\gamma_{\nu}\tilde{S}^{ac}_{c}(-x)\gamma_{\mu 2}\
]_{\alpha\beta}$
$\displaystyle+\braket{\phi(\,q\,)}{\,[\,(\,\bar{s}^{b}(\,0\,)\overleftarrow{D}_{\mu
4}\overleftarrow{D}_{\mu 3}\,)_{\alpha}s^{a}_{\beta}(\,0\,)\,]\,}{0}$
$\displaystyle\times[\,\gamma_{\mu
1}\tilde{S}^{ac}_{c}(\,x\,)\gamma_{\nu}\tilde{S}^{bc}_{c}(\,-x\,)\gamma_{\mu
2}\,]_{\alpha\beta}\\},$

where we introduce the notation

$\displaystyle\tilde{S}_{q}(\,x\,)=CS^{T}_{q}(\,x\,)C,$ (24)

and $S_{q}(x)$ is the propagator of quark $q$. For the heavy quark propagator
on the light-cone we employ its expression in terms of Reinders  _et al._
(1985)

$\displaystyle S^{ab}_{c}(x)$ $\displaystyle=$ (25) $\displaystyle
i\int\frac{d^{4}k}{(2\pi)^{4}}e^{-ikx}[\frac{\delta_{ab}(\not{k}+m_{c})}{k^{2}-m^{2}_{c}}$
$\displaystyle-\frac{g_{s}G_{ab}^{\alpha\beta}}{4}\frac{\sigma_{\alpha\beta}(\not{k}+m_{c})+(\not{k}+m_{c})\sigma_{\alpha\beta}}{(k^{2}-m_{c}^{2})^{2}}]$
$\displaystyle+\frac{g_{s}D_{\alpha}G^{n}_{\beta\lambda}t^{n}_{ij}(f^{\lambda\beta\alpha}+f^{\lambda\alpha\beta})}{3(k^{2}-m_{c}^{2})^{4}}$
$\displaystyle-\frac{g_{s}^{2}(t^{a}t^{b})_{ij}G^{a}_{\alpha\beta}G^{b}_{\mu\nu}(f^{\alpha\beta\mu\nu}+f^{\alpha\mu\beta\nu}+f^{\alpha\mu\nu\beta})}{4(k^{2}-m_{c}^{2})^{5}},$

with

$\displaystyle
f^{\lambda\alpha\beta}=(\not{k}+m_{c})\gamma_{\lambda}(\not{k}+m_{c})\gamma_{\alpha}(\not{k}+m_{c})\gamma_{\beta}(\not{k}+m_{c}),$
(26) $\displaystyle
f^{\alpha\beta\mu\nu}=(\not{k}+m_{c})\gamma_{\alpha}(\not{k}+m_{c})\gamma_{\beta}(\not{k}+m_{c})$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\gamma_{\mu}(\not{k}+m_{c})\gamma_{\nu}(\not{k}+m_{c}),$

here we use the shorthand notation

$\displaystyle G_{ab}^{\mu\nu}\equiv G_{i}^{\mu\nu}t^{i}_{ab},\ \ \
i=1,2,\cdots,8.$ (27)

It is convenient to perform the summation over the color indices by performing
the replacement

$\displaystyle\bar{s}^{d}_{\alpha}(0)s^{d^{\prime}}_{\beta}(0)=\frac{1}{3}\delta_{dd^{\prime}}\bar{s}_{\alpha}(0)s_{\beta}(0),$
(28)

and using the expansion

$\displaystyle\bar{s}_{\alpha}(0)s_{\beta}(0)\equiv\frac{1}{4}\Gamma^{a}_{\alpha\beta}\bar{s}(0)\Gamma^{a}s(0),$
(29)

where the sum runs over the Dirac structures $a$

$\displaystyle\Gamma^{a}=1,\gamma_{5},\gamma_{\mu},i\gamma_{5}\gamma_{\mu},\frac{\sigma_{\mu\nu}}{\sqrt{2}}.$
(30)

Substituting the summation Eq.(28) and the expansion Eq.(29) into Eq.(22) and
Eq.(23), we obtain

$\displaystyle\Pi_{(1),\mu}(p^{\prime},q)$ $\displaystyle=i\int
d^{4}xe^{ipx}\braket{\phi(q)}{T\\{J_{\mu}^{J/\psi}(x)J^{X\dagger}_{(1)}(0)\\}}{0}$
(31) $\displaystyle=i\int
d^{4}xe^{ipx}\varepsilon_{ijk}\varepsilon_{imn}\braket{\phi(q)}{[\bar{s}^{j}(0)\Gamma^{a}s^{m}(0)]}{0}$
$\displaystyle\times\text{Tr}[\gamma_{\mu}\gamma_{5}\tilde{S}^{kl}_{c}(x)\gamma_{\nu}\tilde{S}^{nl}_{c}(-x)\gamma_{\nu}\gamma_{5}\Gamma^{a}],$

and

$\displaystyle\Pi_{(2),\mu}(p^{\prime},q)$ $\displaystyle=i\int
d^{4}xe^{ipx}\braket{\phi(q)}{T\\{J_{\mu}^{J/\psi}(x)J^{X\dagger}_{(2)}(0)\\}}{0}$
(32) $\displaystyle=i\int
d^{4}xe^{ipx}\\{\braket{\phi(q)}{[(\bar{s}^{b}(0)\overleftarrow{D}_{\mu
4}\overleftarrow{D}_{\mu 3})\Gamma^{e}s^{b}(0)]}{0}$
$\displaystyle\times\text{Tr}[\gamma_{\mu
1}\tilde{S}^{ac}_{c}(x)\gamma_{\nu}\tilde{S}^{ac}_{c}(-x)\gamma_{\mu 2}]$
$\displaystyle+\braket{\phi(q)}{[(\bar{s}^{b}(0)\overleftarrow{D}_{\mu
4}\overleftarrow{D}_{\mu 3})\Gamma^{e}s^{a}(0)]}{0}$
$\displaystyle\times\text{Tr}[\gamma_{\mu
1}\tilde{S}^{ac}_{c}(x)\gamma_{\nu}\tilde{S}^{bc}_{c}(-x)\gamma_{\mu 2}]\\},$

Figure 1:  The leading order and one-gluon exchange diagrams contribute to
$\Pi_{(1),\mu}(p^{\prime},q)$, which are the main contribution. Figure 2: The
leading order diagram contribute to $\Pi_{\mu(2)}(p^{\prime},q)$. The one-
gluon exchange vanish in $\Pi_{(2),\mu}(p^{\prime},q)$.

The Feynman diagram of $\Pi_{(1),\mu}(p^{\prime},q)$ is shown in FIG.1 and the
Feynman diagrams of $\Pi_{(2),\mu}(p^{\prime},q)$ is shown in FIG.2. FIG.1
corresponds to the leading order and next-leading order contribution of
$\Pi_{(1),\mu}(p^{\prime},q)$. FIG.2 corresponds to the leading order
contribution of $\Pi_{(2),\mu}(p^{\prime},q)$, which is the dominant one
compare to, for example, the one-gluon exchange contribution.

Now we can substitute the propagator in Eq.(31) by the perturbative term of
Eq.(25). Using the Particle Distribution Amplitudes (DAs) of $\phi$ in
Appendix V.1 and contract the color index by the SU(N) algebra

$\displaystyle\varepsilon_{abc}\varepsilon_{dec}\delta_{ad}\delta_{bi}\delta_{ei}=-C_{A}(1-C_{A})=6,$
(33)

we will encounter four-dimensional integrals in the momentum spaces. After
above operation, some terms in Eq.(31) will proportional to, for example

$\displaystyle\int\frac{d^{4}k_{1}}{(2\pi)^{4}}\int\frac{d^{4}k_{2}}{(2\pi)^{4}}\frac{e^{-i(k_{1}-k_{2})x}k_{1}\cdot
k_{2}}{(k_{1}^{2}-m^{2}_{c})(k_{2}^{2}-m^{2}_{c})}[(p\cdot
q)\epsilon_{\mu}^{\prime}-p\cdot\epsilon^{\prime}q_{\mu}].$ (34)

The main steps to calculate some four-dimensional integrals like (34) can be
done by dimension regulation. Choose the structure proportional to
$\epsilon_{\mu}^{\prime}$, we can derive the corresponding spectral density

$\displaystyle\rho^{\text{OPE}}_{(1)}(\hat{s})=$
$\displaystyle-\frac{f_{\phi}^{\|}m_{\phi}\sqrt{\hat{s}\left(\hat{s}-4m^{2}\right)}\left(2m^{2}+\hat{s}\right)}{12\pi^{2}\hat{s}}$
(35)
$\displaystyle-\frac{f_{\phi}^{\|}m_{c}^{2}m_{\phi}\braket{\frac{\alpha_{s}}{\pi}GG}\left(6m^{2}-\hat{s}\right)\sqrt{\hat{s}\left(\hat{s}-4m^{2}\right)}}{144\pi^{2}\hat{s}^{2}\left(\hat{s}-4m^{2}\right)^{2}}$
$\displaystyle\frac{-\zeta_{4\phi}^{\bot}f_{\phi}^{\bot}m_{c}m_{\phi}^{2}(2m_{c}^{2}-\hat{s})\sqrt{\hat{s}(\hat{s}-4m_{c}^{2})}}{2\pi^{2}\hat{s}^{2}(4m_{c}^{2}-\hat{s})}$
$\displaystyle+\frac{\tilde{\zeta}_{4\phi}^{\bot}f_{\phi}^{\bot}m_{c}m_{\phi}^{2}\sqrt{\hat{s}(\hat{s}-4m_{c}^{2})}}{2\pi^{2}\hat{s}(4m_{c}^{2}-\hat{s})},$

and

$\displaystyle\rho^{\text{OPE}}_{(2)}(\hat{s})=\int_{0}^{1}duu^{2}\phi_{2}^{\bot}(u)\frac{\sqrt{2}m_{c}m_{\phi}^{2}f_{\phi}^{\bot}\sqrt{\hat{s}(\hat{s}-4m_{c}^{2})}}{2\pi^{2}\hat{s}},$
(36)

where $m_{\phi}$ and $f_{\phi}^{\bot}$ are the mass and decay constant of
$\phi$ respectively. $\phi_{2}^{\bot}(u)$ is LCDA. $\zeta_{4}^{\bot}$ and
$\tilde{\zeta}_{4}^{\bot}$ are three-particle distribution parameters defined
in Appendix V.1, $m_{c}$ is charm-quark mass. The strong coupling is then
evaluated by Eq.(19). In addition, we can easily obtain the decay width of
$X(4700)$ $\to J/\psi\phi$ by applying the usual feynman diagram method Dias
_et al._ (2013)

$\displaystyle\Gamma_{\text{I}}(X(4700)\to
J/\psi\phi)=\frac{(g_{XJ/\psi\phi})^{2}}{24\pi m_{\text{I},X}^{2}}$ (37)
$\displaystyle\times\lambda(m_{\text{I},X},m_{J/\psi},m_{\phi})\left(3+\frac{\lambda(m_{\text{I},X},m_{J/\psi},m_{\phi})}{m^{2}_{J/\psi}}\right),$

where

$\displaystyle\lambda(a,b,c)=\frac{\sqrt{a^{4}+b^{4}+c^{4}-2*(a^{2}b^{2}+b^{2}c^{2}+c^{2}a^{2})}}{2a}.$
(38)

###  II.3  The mass and the decay constant of $X(4700)$

To calculate the mass and the decay constant, we start from the two-point
correlation function:

$\displaystyle\Pi_{\text{I}}^{\text{SVZ}}(p)=i\int\mbox{d}^{4}xe^{ipx}\braket{0}{T\\{J_{\text{I}}^{X}(x)J_{\text{I}}^{X\dagger}(0)\\}}{0}.$
(39)

We first calculate the correlation function in terms of the phenomenological
expression by inserting in Eq.(39) with a complete set of hadronic states:

$\displaystyle\Pi^{\text{SVZ,phen}}_{\text{I}}(p)$
$\displaystyle=\frac{\braket{0}{J_{\text{I}}^{X}}{X(p)}\braket{X(p)}{J_{\text{I}}^{X\dagger}}{0}}{m_{\text{I},X}^{2}-p^{2}}$
(40)
$\displaystyle+\int_{s^{\prime}}^{\infty}d\hat{s}\frac{\rho_{I}^{\text{SVZ,phen}}(\hat{s})}{\hat{s}-p^{2}},$

where $\rho_{\text{I}}^{\text{SVZ,phen}}(\hat{s})$ stands for the
contributions of the higher resonances and the continuum states. The
subtraction terms are not shown because they will vanish after the Borel
transformation. After performing polarization sum we can derive

$\displaystyle\Pi_{\text{I}}^{\text{SVZ,phen}}(p)=\frac{m_{\text{I},X}^{2}f_{\text{I},X}^{2}}{m_{\text{I},X}^{2}-p^{2}}+\int_{s^{\prime}}^{\infty}d\hat{s}\frac{\rho_{\text{I}}^{\text{SVZ,phen}}(\hat{s})}{\hat{s}-p^{2}}.$
(41)

As we can see, there is a pole appearing on the right-hand side of Eq.(41).
The way of removing the pole is to perform the Borel transformation on
Eq.(41), which yields

$\displaystyle\Pi_{\text{I}}^{\text{SVZ,phen}}(M^{2})=$ (42) $\displaystyle
m_{\text{I},X}^{2}f_{\text{I},X}^{2}e^{-m_{\text{I},X}^{2}/M^{2}}+\int_{s^{\prime}}^{\infty}d\hat{s}\rho_{\text{I}}^{\text{SVZ,phen}}(\hat{s})e^{-\hat{s}/M^{2}}.$

Now we turn to consider the correlation function in the OPE side, after
contracting the heavy and light quark in term of Wick Theorm, we obtain

$\displaystyle\Pi_{(1)}^{\text{SVZ,OPE}}(p)=$ $\displaystyle
i\int\mbox{d}^{4}xe^{ipx}\epsilon_{ijk}\epsilon_{imn}\epsilon_{i^{\prime}j^{\prime}k^{\prime}}\epsilon_{i^{\prime}m^{\prime}n^{\prime}}$
(43)
$\displaystyle\text{Tr}[\tilde{S}^{jj^{\prime}}_{s}(x)\gamma_{\mu}\gamma_{5}S_{c}^{kk^{\prime}}(x)\gamma_{5}\gamma_{\alpha}]$
$\displaystyle\text{Tr}[S^{mm^{\prime}}_{s}(-x)\gamma_{5}\gamma_{\mu}\tilde{S}_{c}^{nn^{\prime}}(-x)\gamma_{\alpha}\gamma_{5}],$

and

$\displaystyle\Pi_{(2)}^{\text{SVZ,OPE}}(p)=i\int\mbox{d}^{4}xe^{ipx}$ (44)
$\displaystyle\text{Tr}[\gamma_{\mu 1}(\overrightarrow{D}_{\mu
3}\overrightarrow{D}_{\mu 4}S^{bb^{\prime}}_{s}(x-y)\overleftarrow{D}_{\nu
3}\overleftarrow{D}_{\nu 4})\gamma_{\nu
1}\gamma_{5}\tilde{S}_{c}^{aa^{\prime}}(-x)]|_{y=0}$
$\displaystyle(\text{Tr}[\gamma_{\mu
2}\tilde{S}^{bb^{\prime}}_{s}(-x)\gamma_{\nu 2}S_{c}^{aa^{\prime}}(-x)]$
$\displaystyle+\text{Tr}[\gamma_{\mu
2}\tilde{S}^{ba^{\prime}}_{s}(-x)\gamma_{\nu 2}S_{c}^{ab^{\prime}}(-x)]$
$\displaystyle-\text{Tr}[\gamma_{\mu
2}\tilde{S}^{ab^{\prime}}_{s}(-x)\gamma_{\nu 2}S_{c}^{ba^{\prime}}(-x)]$
$\displaystyle+\text{Tr}[\gamma_{\mu
2}\tilde{S}^{aa^{\prime}}_{s}(-x)\gamma_{\nu 2}S_{c}^{bb^{\prime}}(-x)]$
$\displaystyle\times$ $\displaystyle(g^{\mu 1\mu 3}g^{\mu 2\mu 4}+g^{\mu 1\mu
4}g^{\mu 2\mu 3}-g^{\mu 1\mu 2}g^{\mu 3\mu 4}/2)$ $\displaystyle\times$
$\displaystyle(g^{\nu 1\nu 3}g^{\nu 2\nu 4}+g^{\nu 1\nu 4}g^{\nu 2\nu
3}-g^{\nu 1\nu 2}g^{\nu 3\nu 4}/2).$

where

$\displaystyle\tilde{S}_{q}^{ab}(x)=CS_{q}^{ab}(x)C.$ (45)

For propagators of the $u$, $d$ and $s$ quarks in coordinate-space, An
expression for propagators is as followsHuang  _et al._ (2011); Agaev  _et
al._ (2016a):

$\displaystyle S_{q,ab}(x)$
$\displaystyle=\frac{i\delta_{ab}\not{x}}{2\pi^{2}x^{4}}-\frac{\delta_{ab}m_{q}}{4\pi^{2}x^{2}}-\frac{{\langle\bar{q}q\rangle}}{12}$
(46)
$\displaystyle-\frac{i}{32\pi^{2}}\frac{\lambda^{n}}{2}g_{s}G_{\mu\nu}^{n}\frac{1}{x^{2}}(\sigma^{\mu\nu}\not{x}+\not{x}\sigma^{\mu\nu})$
$\displaystyle+\frac{i\delta_{ab}\not{x}m_{q}{\langle\bar{q}q\rangle}}{48}-\frac{\delta_{ab}{\langle\bar{q}g_{s}\sigma
Gq\rangle}x^{2}}{192}$
$\displaystyle+\frac{i\delta_{ab}x^{2}\not{x}m_{q}\langle\bar{q}g_{s}\sigma
Gq\rangle}{1152}$
$\displaystyle-\frac{i\delta_{ab}x^{2}\not{x}g_{s}^{2}\langle\bar{q}q\rangle^{2}}{7776}-\frac{\delta_{ab}x^{4}\langle\bar{q}q\rangle\langle
g_{s}^{2}GG\rangle}{27648}.$

Notice the situation here is different from LCSR, where we do not encounter
the light quark propagator expressed in the coordinate-space. We have to face
divergences in the double integrals like

$\displaystyle\int\frac{d^{4}x}{x^{2n}}\int\int\frac{d^{4}k_{1}d^{4}k_{2}e^{ipx-
ik_{1}x-ik_{2}x}}{(k_{1}^{2}-m_{c}^{2})(k_{2}^{2}-m_{c}^{2})}.$ (47)

As shown in Ref.Agaev  _et al._ (2016b), by using Fourier transformation

$\displaystyle\begin{aligned}
&\frac{1}{\left(x^{2}\right)^{n}}=\int\frac{d^{D}p}{(2\pi)^{D}}e^{-ip\cdot
x}i(-1)^{n+1}2^{D-2n}\pi^{D/2}\\\
&\times\frac{\Gamma(D/2-n)}{\Gamma(n)}\left(-\frac{1}{p^{2}}\right)^{D/2-n}\end{aligned}$
(48)

and perform dimension regulationMatheus  _et al._ (2007) and extract the
imaginary part of results, we can obtain the results without any divergences.

The correlation function $\Pi^{\text{SVZ,OPE}}_{\text{I}}(p)$ can be
represented as the dispersion integral

$\displaystyle\Pi^{\text{SVZ,OPE}}_{\text{I}}(p)=\int_{4m_{c}^{2}}^{\infty}\mbox{d}\hat{s}\frac{\rho^{\text{SVZ,OPE}}_{\text{I}}(\hat{s})}{\hat{s}-p^{2}},$
(49)

where $\rho^{\text{SVZ,OPE}}_{\text{I}}(\hat{s})$ is the corresponding
spectral density.

By performing the Borel transformation on $\Pi^{\text{SVZ,OPE}}_{\text{I}}(p)$
and adopting the quark-hadron duality, one can obtain

$\displaystyle\Pi^{SVZ,OPE}_{\text{I}}(M^{2},\infty)\equiv\int_{4m_{c}^{2}}^{\infty}\mbox{d}\hat{s}\tilde{\rho}^{\text{SVZ,OPE}}_{\text{I}}(\hat{s})e^{-\hat{s}/M^{2}}$
(50)
$\displaystyle=m_{\text{I},X}^{2}f_{\text{I},X}^{2}e^{-m_{\text{I},X}^{2}/M^{2}}+\int_{s_{0}}^{\infty}\rho^{\text{SVZ,phen}}_{\text{I}}(\hat{s})e^{-\hat{s}/M^{2}}.$

Then subtract the continuum contribution yields:

$\displaystyle
m_{\text{I},X}^{2}f_{\text{I},X}^{2}e^{-m_{\text{I},X}^{2}/M^{2}}$
$\displaystyle=\int_{4m_{c}^{2}}^{s_{0}}\mbox{d}\hat{s}\rho^{\text{SVZ,OPE}}_{\text{I}}(\hat{s})e^{-\hat{s}/M^{2}}$
(51)

The mass of the $X(4700)$ state can be evaluated from the sum rule

$\displaystyle
m_{\text{I},X}^{2}=\frac{\int_{4m_{c}^{2}}^{s_{0}}\mbox{d}\hat{s}\hat{s}\rho^{\text{SVZ,OPE}}_{\text{I}}(\hat{s})e^{-\hat{s}/M^{2}}}{\int_{4m_{c}^{2}}^{s_{0}}\mbox{d}\hat{s}\rho^{\text{SVZ,OPE}}_{\text{I}}(\hat{s})e^{-\hat{s}/M^{2}}}.$
(52)

where the spectral densities are provided in APPENDIX V.2.

##  III Numerical calculation

###  III.1 Input parameters

In this section, we analyze the numerical results for the coupling constant
and the decay width of $X(4700)\to J/\psi\phi$, and present the mass and decay
constant of $X(4700)$ as well. We adopt the following parameters for the
numerical calculation. The current charm-quark mass, $m_{c}=(1.275\pm 0.025)$
GeV, the $J/\psi$-meson mass $m_{J/\psi}=(3096.900\pm 0.006)$ MeV and
$\phi$(980) mass $m_{\phi}=(990\pm 20)$ MeV from the Particla Data Group (PDG)
Zyla  _et al._ (2020). The $J/\psi$ and $\phi$(980) decay constants are taken
as $f_{J/\psi}=0.405$ GeV Dias  _et al._ (2013), $f_{\phi}=0.18\pm 0.015$ GeV
Colangelo  _et al._ (2010). The current-quark-mass for the s-quark is
$m_{s}=93^{+11}_{-5}$ MeV from PDG. The decay constants of $\phi$ is taken as
$f_{\phi}^{\|}=0.215$ GeV. The Gegenbauer moments, $a_{1}^{\|}=a_{1}^{\bot}=0$
and $a_{2}^{\|}=0.18$, $a_{2}^{\bot}=0.14$ Ball  _et al._ (2007). The
parameters $\zeta_{4}^{\bot}$ and $\widetilde{\zeta}_{4}^{\bot}$ are taken as
$\zeta_{4}^{\bot}=-0.01$ and $\widetilde{\zeta}_{4}^{\bot}=-0.03$ Ball  _et
al._ (2007). In addition, we also need to know the values of the non-
perturbative vacuum condensates. The related parameters are Hu  _et al._
(2021)

$\displaystyle\langle{\bar{q}q}\rangle=-(0.24\pm 0.01)^{3}\ \text{GeV}^{3},$
(53) $\displaystyle\braket{\bar{s}s}=(0.8\pm 0.1)\times\braket{\bar{q}q},$
$\displaystyle\braket{\frac{\alpha_{s}}{\pi}GG}=(0.012)\ \text{GeV}^{4},$
$\displaystyle\braket{g_{s}\bar{s}\sigma
Gs}=m_{0}^{2}\times\braket{\bar{s}s},$ $\displaystyle m_{0}^{2}=0.8\ GeV^{2},$
$\displaystyle m_{c}=1.275\pm 0.025\ \text{MeV}.$

The sum rule predictions for the mass, the decay constant and the coupling
constant depend on two parameters: Borel mass $M^{2}$ and continuum threshold
$s_{0}$. The value of $s_{0}$ is being correlated with the onset of excited
states of $X(4700)$. But according to the experimental data, there is no
resonance activity related to the first excited states of $X(4700)$. We should
turn to annother way. From Table 1 which entail the masses calculations of
charmonia and bottomonia, we can find the mass discrepancy between the ground
state and first excited state are all around $0.5$ GeV. Besides, the
experimental data in Table 2 which extract from PDG prove most of the
calculations in Table 1. Furthermore, one can refer to previous QCD sum rules
caculations that assign $X(4140)$ and $X(4685)$ as the 1S and 2S tetraquark
states respectively and that assign $Z_{c}(3900)$ and $Z_{c}(4430)$ as the 1S
and 2S tetraquark states respectively and so on (see Table 3). The mass
difference between the 1S and 2S tetraquark states are about $0.4\sim 0.6$
GeV. So we accept the mass discrepancy and employ

$\displaystyle(4.70+0.40)^{2}\ \text{GeV}^{2}\leq s_{0}\leq(4.70+0.60)^{2}\
\text{GeV}^{2}.$ (54)

or rewrite it as

$\displaystyle(5.20-0.10)^{2}\ \text{GeV}^{2}\leq s_{0}\leq(5.20+0.10)^{2}\
\text{GeV}^{2}.$ (55)

Table 1: Quark model masses calculated for the first three levels of charmonia and bottomonia Olpak  _et al._ (2017). Masses | $c\bar{c}$ | $b\bar{b}$
---|---|---
M(GeV) $\backslash$ n | n=1 | n=2 | n=3 | n=1 | n=2 | n=3
$M_{{}^{3}P_{0}}\left(\chi_{q0}\right)$ | 3.37 | 3.88 | 4.30 | 9.81 | 10.2 | 10.7
$M_{{}^{3}P_{1}}\left(\chi_{q1}\right)$ | 3.54 | 3.97 | 4.33 | 9.89 | 10.3 | 10.6
$M_{{}^{1}P_{1}}\left(h_{q}\right)$ | 3.53 | 3.96 | 4.37 | 9.88 | 10.3 | 10.6
$M_{{}^{3}P_{2}}\left(\chi_{q2}\right)$ | 3.54 | 3.98 | 4.34 | 9.89 | 10.3 | 10.6
Table 2: Masses of experimentally observed states in Particle Data Group listingsBeringer  _et al._ (2012). Masses | $c\bar{c}$ | $b\bar{b}$
---|---|---
$M(MeV)\backslash n$ | $n=1$ | $n=2$ | $n=3$ | $n=1$ | $n=2$ | $n=3$
$M_{{}^{3}P_{0}}\left(\chi_{q0}\right)$ | $3414.75$ | $-$ | $-$ | $9859.44$ | $10232.5$ | $-$
$M_{{}^{3}P_{1}}\left(\chi_{q1}\right)$ | $3510.66$ | $-$ | $-$ | $9892.78$ | $10255.46$ | $10512.1$
$M_{{}^{1}P_{1}}\left(h_{q}\right)$ | $3525.38$ | $-$ | $-$ | $9899.3$ | $10259.8$ | $-$
$M_{{}^{3}P_{2}}\left(\chi_{q2}\right)$ | $3556.20$ | $3922.5$ | $-$ | $9912.21$ | $10268.65$ | $-$
Table 3: The mass difference between the 1S and 2S hidden-charm tetraquark states with the possible assignments Wang and Xin (2021). $J^{PC}$ | $1\mathrm{~{}S}$ | 2 $\mathrm{~{}S}$ | Mass difference | References
---|---|---|---|---
$1^{++}$ | $X(4140)$ | $X(4685)$ | 566 $\mathrm{MeV}$ | Wang (2021); Wang and Di (2019)
$1^{+-}$ | $Z_{c}(3900)$ | $Z_{c}(4430)$ | $591\mathrm{MeV}$ | Maiani  _et al._ (2014); Nielsen and Navarra (2014); Wang (2015)
$0^{++}$ | $X(3915)$ | $X(4500)$ | $588\mathrm{MeV}$ | Lebed and Polosa (2016); Wang (2017)
$1^{+-}$ | $Z_{c}(4020)$ | $Z_{c}(4600)$ | $576\mathrm{MeV}$ | Chen and Chen (2019); Wang (2020)

After fixing $s_{0}$, we use two extra criteria to constrain the Borel mass
$M^{2}$:

  1. 1.

To obtain the minimal value of $M^{2}$, we require that the contribution of
the condensates like $\braket{\bar{q}g_{s}Gq}$ and higher dimension
condensates in the OPE is smaller than $5\%$ of the total contribution:

$\displaystyle\mathrm{CVG}\equiv\left|\frac{\tilde{\Pi}_{I}^{\braket{\bar{q}g_{s}Gq}+\cdots}\left(M^{2},\infty\right)}{\tilde{\Pi}_{I}^{OPE}\left(M^{2},\infty\right)}\right|\leq
5\%,$ (56)

where dots denote higher dimension contributions.

  2. 2.

Ensure that the one-pole in Eq.(41) is valid, we require that the pole
contribution (PC) should be larger than $30\%$ to determine the upper limit on
$M^{2}$

$\displaystyle\text{PC}=\frac{\tilde{\Pi}_{I}^{OPE}(M^{2},s_{0})}{\tilde{\Pi}_{I}^{OPE}(M^{2},\infty)}\geq
30\%.$ (57)

Figure 3:  Convergence (CVG) and pole contribution (PC) for $J_{(1)}^{X}$.
Figure 4:  Convergence (CVG) and pole contribution (PC) for $J_{(2)}^{X}$.

As depicted in FIG.3 and FIG.4, CVG and PC decrease as $M^{2}$ grows higher.
The green dot betokens that CVG turns into $30\%$, where the maximal Borel
mass $M^{2}$ can be obtained. And the red dot present PC converges with $5\%$
horizontal for $J_{(1)}^{X}$ and $J_{(2)}^{X}$, from which point, we can
select the minimum Borel mass. Subsequently, we require the working region of
the Borel parameter for mass and decay constant of $J_{(1)}^{X}$ to be in the
region of

$\displaystyle 3.60\ \text{GeV}^{2}\leq M^{2}\leq 4.13\ \text{GeV}^{2},$ (58)

and $J_{(2)}^{X}$ to be in that of

$\displaystyle 3.63\ \text{GeV}^{2}\leq M^{2}\leq 4.17\ \text{GeV}^{2}.$ (59)

###  III.2 The mass, decay constant

In Fig.5 and Fig.6 we present the results of the mass $m_{X}^{I}$ and the
decay constant $f_{X}^{I}$ as functions of the parameters $M^{2}$ at fixed
values of $s_{0}$$\in$$\\{(5.20-0.10)^{2}$, $(5.20+0.0)^{2}$,
$(5.20+0.1)^{2}\\}$. As we seen in the first figure in Fig.5 and Fig.6, the
yellow curves correspond to the measurements of the Belle Collaboration
Ablikim  _et al._ (2017) as shown in PDG. The blue, red and black curves show
clear dependence of our prediction on $s_{0}$ and $M^{2}$. By choosing
appropriate parameters, our predictions are consistent with the measurements.
At a fixed point of $M^{2}=3.9$ $\rm GeV^{2}$, the masses of $J_{(1)}^{X}$ and
$J_{(2)}^{X}$ are

$\displaystyle\quad m_{X}^{(1)}=4.70^{+0.06}_{-0.05}\ \text{GeV},$ (60)

and

$\displaystyle\quad m_{X}^{(1)}=4.67^{+0.04}_{-0.07}\ \text{GeV},$ (61)

respectively.

Figure 5:  The mass [first] and the decay constant [second] of a scalar
tetraquark state $X(4700)$ as a function of the Borel parameter $M^{2}$ at
different fixed values of $s_{0}$.

Figure 6:  The mass [first] and the decay constant [second] of a D-wave
tetraquark state $X(4700)$ as a function of the Borel parameter $M^{2}$ at
different fixed values of $s_{0}$.

The uncertainty comes from the various condensates, and the strange and charm
quark masses. Base on the Belle Collaboration measurements Ablikim  _et al._
(2017), $X(4700)$ has mass of $4704\pm 10^{+14}_{-24}$ MeV. The predicted
results of the two assignments are consistent with the experimental results.
Therefore we can conclude that it might be a D-wave $cs\bar{c}\bar{s}$
tetraquark or a scalar tetraquark state. We then extend the same technique to
evaluate the decay constant of $X(4700)$, the results of $J_{(1)}^{X}$ and
$J_{(2)}^{X}$ at the same bench mark point reads

$\displaystyle\lambda_{X}^{(1)}\equiv
m_{X}^{(1)}f_{X}^{(1)}=0.19^{+0.013}_{-0.014}\ \text{GeV}^{5},$ (62)

and

$\displaystyle\lambda_{X}^{(2)}\equiv
m_{X}^{(2)}f_{X}^{(2)}=0.164^{+0.010}_{-0.011}\ \text{GeV}^{7}.$ (63)

The mass and decay constant given above will be used as input parameters to
find the decay width of $X$(4700)$\to J/\psi\phi$.

###  III.3 The coupling constant and the decay width

Figure 7:  The strong coupling $g_{XJ/\psi\phi}$ of a scalar tetraquark state
$X(4700)$ as a function of the Borel parameter $M^{2}$ at different fixed
values of $s_{0}$.

We start to evaluate $g_{XJ/\psi\phi}$ and give the decay width of
$X$(4700)$\to J/\psi\phi$ in order to confirm which hypotheses are more
suitable. For $M^{2}$ and $s_{0}$, we use the same values as in the analysis
of the mass. The result is shown in Fig.7 and Fig.8. Fig.7 provides the result
for the tetraquark state. The parameters $M^{2}$ and $s_{0}$ are varied inside
of the regions:$(3.60-4.13)^{2}\ \text{GeV}^{2}$ and $(5.10-5.30)^{2}\
\text{GeV}^{2}$. Our prediction for $g_{XJ/\psi\phi}$, take the average value,
is

$\displaystyle g_{XJ/\psi\phi}=6.7^{+1.0}_{-0.8}\ \text{GeV}.$ (64)

The width of this decay can be obtained by Eq.(19):

$\displaystyle\Gamma(X(4700)\rightarrow J/\psi\phi)=(109^{+35}_{-24})\
\text{MeV}.$ (65)

The total width of $X$(4700) from PDGZyla  _et al._ (2020) shows that

$\displaystyle\Gamma(4700)=120\pm 31^{+42}_{-33}\text{MeV},$ (66)

which is close to our prediction within the error. The result indicate that if
we assign $X$(4700) as a scalar $cs\bar{c}\bar{s}$ tetraquark state $X$(4700)
$\to$ $J/\psi\phi$ will the predominant process.

Figure 8:  The strong coupling $g_{XJ/\psi\phi}$ of a D-wave tetraquark state
$X(4700)$ as a function of the Borel parameter $M^{2}$ at different fixed
values of $s_{0}$.

For D-wave tetraquark state, the results are shown in Fig.8. The prediction
for $g_{XJ/\psi\phi}$ is

$\displaystyle g_{XJ/\psi\phi}=2.65^{+0.44}_{-0.33}\ \text{GeV}.$ (67)

The width of its decay can be obtained by Eq.(19) to be

$\displaystyle\Gamma(X(4700)\rightarrow J/\psi\phi)=(17.1^{+6.2}_{-4.0})\
\text{MeV}.$ (68)

which is much smaller than the width of $X$(4700) in PDG. Our results
illustrates that, in this case, if one assign $X$(4700) as a D-wave tetraquark
state, $X$(4700) $\to$ $J/\psi\phi$ will be the minor decay channel. Therefore
the result imply that there exist multiple decays that not yet found in
experiments.

##  IV Summary

In this work we assign $X$(4700) as a D-wave tetraquark state and a scalar
tetraquark state to study the mass and the decay constant of $X(4700)$, and
also it’s decay $X(4700)\to J/\psi\phi$. The mass of $X(4700)$ is evaluated
through two-point sum rules, and both the results of the two assignments are
in agreement with the mass of $X(4700)$ in PDG. We also calculate the decay
constant of $X(4700)$ in the SVZ sum rules. We then perform the calculation of
the coupling constant $g_{XJ/\psi\phi}$ and the decay width by using the
approach of the light-cone sum rules. We find that being a scalar tetraquark
state will make $X(4700)$ most likely decay into $J/\psi\phi$ since in this
case $X(4700)\to J/\psi\phi$ is the predominant decay process, while being a
D-wave tetraquark state, the decay of $X(4700)\to J/\psi\phi$ is much smaller
and become a non-significant decay channel.

###### Acknowledgements.

Hao Sun is supported by the National Natural Science Foundation of China
(Grant No.12075043, No.12147205).

##  V Appendix

###  V.1 The relations between the light-cone distribution amplitudes (LCDAs)
and the matrix elements

The matrix elements of the $\phi$ can be expanded in terms of the
corresponding distribution amplitudes. Here we provide the expressions for the
$\left\langle\phi(P)|\bar{q}(0)\Gamma^{a}q(0)|0\right\rangle$ type matrix
elements Ball and Braun (1996):

$\displaystyle\left\langle\phi(P,\lambda)\left|\bar{q}(0)\gamma_{\mu}q(0)\right|0\right\rangle=f_{\phi}^{\|}m_{\phi}\epsilon_{\mu}^{*(\lambda)}$
(69)
$\displaystyle\left\langle\phi(P,\lambda)\left|\bar{q}(0)\sigma_{\mu\nu}q(0)\right|0\right\rangle=if_{\phi}^{\perp}\left(e_{\mu}^{*(\lambda)}P_{\nu}-e_{\nu}^{*(\lambda)}P_{\mu}\right).\
\ \ \ \ \ \ \ $ (70)

We also provide the expressions for
$\langle\phi(P)|\bar{q}(0)gG_{\alpha\beta}\Gamma^{a}q(0)|0\rangle$ type matrix
elements Ball  _et al._ (2007); Ball and Jones (2007):

$\displaystyle\left\langle\phi(P,\lambda)\left|\bar{q}(0)g\widetilde{G}_{\alpha\beta}\gamma_{\mu}\gamma_{5}q(0)\right|0\right\rangle$
(71)
$\displaystyle=f_{\phi}^{\|}m_{\phi}\zeta_{3\phi}^{\|}[e_{\alpha}^{(\lambda)}\left(P_{\beta}P_{\mu}-\frac{1}{3}m_{\phi}^{2}g_{\beta\mu}\right)-(\alpha\leftrightarrow\beta)]$
$\displaystyle+\frac{1}{3}f_{\phi}^{\|}m_{\phi}^{3}\zeta_{4\phi}^{\|}\left(e_{\alpha}^{(\lambda)}g_{\beta\mu}-e_{\beta}^{(\lambda)}g_{\alpha\mu}\right),$
$\displaystyle\left\langle\phi(P,\lambda)\left|\bar{q}(0)gG_{\alpha\beta}i\gamma_{\mu}q(0)\right|0\right\rangle$
(72)
$\displaystyle=-f_{\phi}^{\|}m_{\phi}\kappa_{3\phi}^{\|}[e_{\alpha}^{(\lambda)}\left(P_{\beta}P_{\mu}-\frac{1}{3}m_{\phi}^{2}g_{\beta\mu}\right)-(\alpha\leftrightarrow\beta)]$
$\displaystyle-\frac{1}{3}f_{\phi}^{\|}m_{\phi}^{3}\kappa_{4\phi}^{\|}\left(e_{\alpha}^{(\lambda)}g_{\beta\mu}-e_{\beta}^{(\lambda)}g_{\alpha\mu}\right),$
$\displaystyle\left\langle\phi(P,\lambda)\left|\bar{q}(0)gG_{\alpha\beta}q(0)\right|\right\rangle$
(73)
$\displaystyle=-if_{\phi}^{\perp}m_{\phi}^{2}\zeta_{4\phi}^{\perp}\left(\epsilon_{\alpha}^{(\lambda)}P_{\beta}-\epsilon_{\beta}^{(\lambda)}P_{\alpha}\right),$
$\displaystyle\left\langle\phi(P,\lambda)\left|\bar{q}(0)g\widetilde{G}_{\alpha\beta}\gamma_{5}q(0)\right|0\right\rangle$
(74)
$\displaystyle=f_{\phi}^{\perp}m_{\phi}^{2}\widetilde{\zeta}_{4\phi}^{\perp}\left(\epsilon_{\alpha}^{(\lambda)}P_{\beta}-\epsilon_{\beta}^{(\lambda)}P_{\alpha}\right),$
$\displaystyle\left\langle\phi(P,\lambda)\left|\bar{q}(0)gG_{\alpha\mu}\sigma_{\beta\mu}q(0)\right|\right\rangle$
(75)
$\displaystyle=f_{\phi}^{\perp}m_{\phi}^{2}[\frac{1}{2}\kappa_{3\phi}^{\perp}\left(\epsilon_{\alpha}^{(\lambda)}P_{\beta}+\epsilon_{\beta}^{(\lambda)}P_{\alpha}\right)$
$\displaystyle+\kappa_{4\phi}^{\perp}\left(\epsilon_{\alpha}^{(\lambda)}P_{\beta}-\epsilon_{\beta}^{(\lambda)}P_{\alpha}\right)],$

where the dual gluon field strength tensor is defined as
$\widetilde{G}_{\mu\nu}=\frac{1}{2}\epsilon_{\mu\nu\rho\sigma}G^{\rho\sigma}$.
The generic notations of $\zeta$ are G-conserving and $\kappa$ are G-breaking
parameters. $\zeta_{3\phi}$, $\widetilde{\zeta}_{3\phi}$, $\kappa_{3\phi}$ are
twist-3 and $\zeta_{4\phi}$, $\widetilde{\zeta}_{4\phi}$, $\kappa_{4\phi}$ are
twist-4 parameters given in Ball  _et al._ (2007).

The covariant derivative is defined as

$\displaystyle\overleftarrow{D}_{\mu}=\partial_{\mu}+igT^{a}A_{\mu}^{a},$ (76)

and it is also valid in the Fock-Schwinger gauge

$\displaystyle x^{\mu}A_{\mu}^{a}(x)=0,$ (77)

after performing some deduction, we can derive the following relations Gubler
(2013):

$\displaystyle A_{\mu}^{a}(x)=$
$\displaystyle\frac{1}{2}x^{v}G_{v\mu}^{a}(0)+\frac{1}{3}x^{v}x^{\alpha}\left[D_{\alpha}G_{v\mu}(0)\right]^{a}$
(78)
$\displaystyle+\frac{1}{8}x^{v}x^{\alpha}x^{\beta}\left[D_{\alpha}D_{\beta}G_{v\mu}(0)\right]^{a}+\cdots.$

Therefore, insert the gluonic fields back to the covariant derivative and
calculate
$\braket{\phi(P)}{\bar{q}(0)\overleftarrow{D}_{\mu}\overleftarrow{D}_{\nu}\Gamma^{a}q(0)}{0}$,
we obtain the desired results:

$\displaystyle\braket{\phi(P)}{\bar{q}(0)\overleftarrow{D}_{\mu}\overleftarrow{D}_{\nu}\Gamma^{a}q(0)}{0}$
(79)
$\displaystyle=\braket{\phi(P)}{\partial_{\mu}\partial_{\nu}[\bar{q}(x)]}{{}_{x=0}\Gamma^{a}q(0)|0}$
$\displaystyle+\braket{\phi(P)}{\bar{q}(0)\frac{igT^{b}}{2}G^{b}_{\mu\nu}\Gamma^{a}q(0)}{0}$
$\displaystyle=\partial_{\mu}\partial_{\nu}[\braket{\phi(P)}{\bar{q}(x)\Gamma^{a}q(0)}{0}]|_{x=0}$
$\displaystyle+\frac{i}{2}\braket{\phi(P)}{\bar{q}(0)gG_{\mu\nu}\Gamma^{a}q(0)}{0},$

where $\braket{\phi(P)}{\bar{q}(0)gG_{\mu\nu}\Gamma^{a}q(0)}{0}$ type matrix
elements are given before. In addition,
$\braket{\phi(P)}{\bar{q}(x)\Gamma^{a}q(0)}{0}$ type expressions are provided
below Ball and Braun (1996):

$\displaystyle\left\langle\phi(P,\lambda)\left|\bar{q}(x)\sigma_{\rho\nu}q(0)\right|0\right\rangle$
(80)
$\displaystyle=-i\left(e_{\rho}^{(\lambda)}P_{\nu}-e_{\nu}^{(\lambda)}P_{\rho}\right)\times
f_{\phi}^{\perp}\int_{0}^{1}due^{iupx}\phi_{2}^{\perp}(u,\mu),$
$\displaystyle\left\langle\phi(P,\lambda)\left|\bar{q}(x)\gamma_{\theta}\gamma_{5}q(0)\right|0\right\rangle$
(81)
$\displaystyle=-\frac{1}{4}\epsilon_{\theta\nu\rho\sigma}e^{(\lambda)\nu}P^{\rho}x^{\sigma}f_{\phi}m_{\phi}\times\int_{0}^{1}due^{iupx}g_{\perp}^{(a)}(u,\mu),$
$\displaystyle\left\langle\phi(P,\lambda)\left|\bar{q}(x)\gamma_{\rho}q(0)\right|0\right\rangle$
(82)
$\displaystyle=P_{\rho}\left(e^{(\lambda)}x\right)f_{\phi}m_{\phi}\times\int_{0}^{1}due^{iupx}\Phi_{\|}(u,\mu)$
$\displaystyle+e_{\rho}^{(\lambda)}f_{\phi}m_{\phi}\int_{0}^{1}due^{iupx}g_{\perp}^{(v)}(u,\mu),$

where

$\displaystyle g_{\perp}^{(v),\text{ twist }2}(u,\mu)$ (83)
$\displaystyle=\frac{1}{2}\left[\int_{0}^{u}dy\frac{\phi^{\|}_{2}(y,\mu)}{\bar{y}}+\int_{u}^{1}dy\frac{\phi^{\|}_{2}(y,\mu)}{y}\right],$
$\displaystyle g_{\perp}^{(a),\text{ twist }2}(u,\mu)$ (84)
$\displaystyle=2\left[\bar{u}\int_{0}^{u}dy\frac{\phi^{\|}_{2}(y,\mu)}{\bar{y}}+u\int_{u}^{1}dy\frac{\phi^{\|}_{2}(y,\mu)}{y}\right],$

and

$\displaystyle\Phi_{\|}(u,\mu)$ (85)
$\displaystyle=\frac{1}{2}\left[\bar{u}\int_{0}^{u}dy\frac{\phi^{\|}_{2}(y,\mu)}{\bar{y}}-u\int_{u}^{1}dy\frac{\phi^{\|}_{2}(y,\mu)}{y}\right].$

Here $\bar{u}=1-u$, $\bar{y}=1-y$. In term of Gegenbauer polynomials,
$\phi_{2}^{\|,\perp}$ is given as Ball  _et al._ (2007)

$\phi_{2}^{\|,\perp}(u,\mu)=6u\bar{u}\left\\{1+\sum_{n=1}^{\infty}a_{n}^{\|,\perp}(\mu)C_{n}^{3/2}(2u-1)\right\\},$
(86)

the Gegenbauer polynomials $C_{n}^{3/2}(x)$ and coefficients $a_{n}$ at the
renormalisation scale $\mu$ are given in details in Ball and Braun (1996).

###  V.2 Spectral densities

In this section we provide the spectral densities for $J_{(1)}$ and $J_{(2)}$.
In the following expressions, $\mathrm{H}(x)$ is defined as:

$\displaystyle\mathrm{H}(x)=\begin{cases}0&x\geq 0\\\ 1&x<0.\end{cases}$ (87)

The spectral density for $J_{(1)}$ can be divide into:

$\displaystyle\tilde{\rho}_{(1)}^{OPE}(\hat{s})$ $\displaystyle=$
$\displaystyle\tilde{\rho}_{(1)}^{pert}(\hat{s})+\tilde{\rho}_{(1)}^{\braket{\bar{q}q}}(\hat{s})+\tilde{\rho}_{(1)}^{\braket{\frac{\alpha_{s}}{\pi}GG}}(\hat{s})$
(88) $\displaystyle+$
$\displaystyle\tilde{\rho}_{(1)}^{\braket{\bar{q}g_{s}\sigma
Gq}}(\hat{s})+\tilde{\rho}_{(1)}^{\braket{\bar{q}q}^{2}}(\hat{s})+\tilde{\rho}_{(1)}^{\braket{\frac{\alpha_{s}}{\pi}GG}\braket{\bar{q}q}}(\hat{s})\
\ ~{}~{}~{}~{}$

with

$\displaystyle\tilde{\rho}_{(1)}^{pert}(\hat{s})=\int_{0}^{1}dx\int_{0}^{1}dy\frac{y\left(m_{c}^{2}+\hat{s}x(x(-y)+x+y-1)\right)^{2}}{768\pi^{6}(x-1)^{3}x^{3}(y-1)^{3}}$
(89) $\displaystyle\times$
$\displaystyle\left(m^{4}y\left(3y^{3}+7y^{2}-3y+3\right)-12m_{c}^{3}m_{s}y\left(y^{2}-1\right)\right.$
$\displaystyle-$ $\displaystyle 2m_{c}^{2}(x-1)x(y-1)(36m_{s}^{2}(y-1)$
$\displaystyle+$ $\displaystyle\hat{s}y(9y^{3}+20y^{2}+9y+6))$
$\displaystyle+$ $\displaystyle 30m_{c}m_{s}\hat{s}(x-1)x(y-1)^{2}y(y+1)$
$\displaystyle+$
$\displaystyle\left.3\hat{s}^{2}(x-1)^{2}x^{2}(y-1)^{2}y(7y^{3}+15y^{2}+9y+3)\right)$
$\displaystyle\times$
$\displaystyle\mathrm{H}\left(y\left(x\hat{s}(x(-y)+x+y-1)+m_{c}^{2}\right)\right),$
$\displaystyle\tilde{\rho}_{(1)}^{\braket{\bar{q}q}}(\hat{s})=-\int_{0}^{1}dx\int_{0}^{1}dy\frac{\braket{\bar{q}q}}{16\pi^{4}(x-1)^{2}x^{2}(y-1)}$
(90) $\displaystyle\times$ $\displaystyle\left(2m^{5}y(y+1)\right.$
$\displaystyle-$ $\displaystyle
2m^{4}\text{mq}(x-1)x\left(3y^{3}+5y^{2}-3y+5\right)$ $\displaystyle+$
$\displaystyle
2m_{c}^{3}(x-1)x\left(y^{2}-1\right)\left(m_{s}^{2}-3\hat{s}y\right)$
$\displaystyle+$ $\displaystyle
2m_{c}^{2}m_{s}\hat{s}(x-1)^{2}x^{2}\left(12y^{4}+9y^{3}-15y^{2}+y-7\right)$
$\displaystyle+$ $\displaystyle
m\hat{s}(x-1)^{2}x^{2}(y-1)^{2}(y+1)\left(4\hat{s}y-3m_{q}^{2}\right)$
$\displaystyle-$
$\displaystyle\left.4m_{s}\hat{s}^{2}(x-1)^{3}x^{3}(y-1)^{2}\left(5y^{3}+9y^{2}+5y+1\right)\right)$
$\displaystyle\times$
$\displaystyle\mathrm{H}\left(y\left(x\hat{s}(x(-y)+x+y-1)+m_{c}^{2}\right)\right),$
$\displaystyle\tilde{\rho}_{(1)}^{\braket{\frac{\alpha_{s}}{\pi}GG}}(\hat{s})=\int_{0}^{1}dx\int_{0}^{1}dy\frac{\braket{\frac{\alpha_{s}}{\pi}GG}}{512\pi^{4}(x-1)x(y-1)}$
$\displaystyle\left(\times m_{c}^{4}\left(3y^{3}+7y^{2}+y+1\right)\right.$
$\displaystyle-$ $\displaystyle
m^{2}\hat{s}(x-1)x\left(12y^{4}+15y^{3}-17y^{2}-7y-3\right)$ $\displaystyle+$
$\displaystyle\left.2\hat{s}^{2}(x-1)^{2}x^{2}(y-1)^{2}\left(5y^{3}+11y^{2}+5y+1\right)\right)$
$\displaystyle\times$
$\displaystyle\mathrm{H}\left(y\left(x\hat{s}(x(-y)+x+y-1)+m_{c}^{2}\right)\right),$
$\displaystyle\tilde{\rho}_{(1)}^{\braket{\bar{q}g_{s}\sigma
Gq}}(\hat{s})=\int_{0}^{1}dx\int_{0}^{1}dy\frac{\braket{\bar{q}g_{s}\sigma
Gq}}{192\pi^{4}(x-1)x}$ (91) $\displaystyle\times$
$\displaystyle\left(2m_{c}^{3}(2x-7)(y+1)\right.$ $\displaystyle+$
$\displaystyle 8m_{c}^{2}m_{s}(x-1)x\left(3y^{2}+4y+3\right)$ $\displaystyle-$
$\displaystyle 3m_{c}\hat{s}\left(2x^{2}-9x+7\right)x\left(y^{2}-1\right)$
$\displaystyle-$
$\displaystyle\left.24m_{s}\hat{s}(x-1)^{2}x^{2}\left(2y^{3}+y^{2}-y-2\right)\right)$
$\displaystyle\times$
$\displaystyle\mathrm{H}\left(y\left(x\hat{s}(x(-y)+x+y-1)+m_{c}^{2}\right)\right)$
$\displaystyle+$
$\displaystyle\int_{0}^{1}dx\left(\frac{\braket{\bar{q}g_{s}\sigma
Gq}\left(64m_{c}^{2}m_{s}-8m_{s}\hat{s}x(x+3)\right)}{384\pi^{4}}\right.$
$\displaystyle+$ $\displaystyle\left.\frac{\braket{\bar{q}g_{s}\sigma
Gq}\left(m\left(-2m_{0}x+m_{0}+8m_{s}^{2}\right)\right)}{384\pi^{4}}\right)$
$\displaystyle\times$ $\displaystyle\mathrm{H}(m^{2}-x(x+1)s),$
$\displaystyle\tilde{\rho}_{(1)}^{\braket{\bar{q}q}^{2}}(\hat{s})=\int_{0}^{1}dx\int_{0}^{1}dy$
(92)
$\displaystyle\mathrm{H}\left(y\left(x\hat{s}(x(-y)+x+y-1)+m_{c}^{2}\right)\right)$
$\displaystyle\times$
$\displaystyle(-\frac{g_{s}^{2}\braket{\bar{q}q}^{2}\left(m_{c}^{2}\left(3y^{2}+4y+3\right)\right)}{162\pi^{4}}$
$\displaystyle-$
$\displaystyle\frac{g_{s}^{2}\braket{\bar{q}q}^{2}\left(-3\hat{s}(x-1)x\left(2y^{3}+y^{2}-y-2\right)\right)}{162\pi^{4}})$
$\displaystyle+$
$\displaystyle\int_{0}^{1}dx\frac{\braket{\bar{q}q}^{2}}{324\pi^{4}}\mathrm{H}(m^{2}-x(x+1)s)$
$\displaystyle\times$
$\displaystyle\left(g_{s}^{2}\left(-2m_{c}^{2}-m_{c}m_{s}+\hat{s}(x+3)x\right)\right.$
$\displaystyle+$
$\displaystyle\left.27\pi^{2}\left(4m_{c}^{2}+2m_{c}m_{s}-m_{s}^{2}(x-6)x\right)\right),$
$\displaystyle\tilde{\rho}_{(1)}^{\braket{\frac{\alpha_{s}}{\pi}GG}\braket{\bar{q}q}}(\hat{s})=\int_{0}^{1}dx\frac{im_{c}\braket{\frac{\alpha_{s}}{\pi}GG}\braket{\bar{q}q}}{144\pi^{3}}$
(93) $\displaystyle\mathrm{H}$ $\displaystyle(m^{2}-x(x+1)s).$

The spectral density for $J_{(2)}$ can be divide into:

$\displaystyle\tilde{\rho}^{OPE}_{(2)}(\hat{s})$ $\displaystyle=$
$\displaystyle\tilde{\rho}_{(2)}^{\braket{\frac{\alpha_{s}}{\pi}GG}}(\hat{s})+\tilde{\rho}_{(2)}^{\braket{\bar{q}g_{s}\sigma
Gq}}(\hat{s})$ (94) $\displaystyle+$
$\displaystyle\tilde{\rho}_{(2)}^{\braket{\frac{\alpha_{s}}{\pi}GG}\braket{\bar{q}q}}(\hat{s})+\tilde{\rho}_{(2)}^{\braket{\bar{q}g_{s}\sigma
Gq}\braket{\bar{q}q}}(\hat{s})$ $\displaystyle+$
$\displaystyle\tilde{\rho}_{2}^{\braket{\bar{q}g_{s}\sigma
Gq}\braket{\frac{\alpha_{s}}{\pi}GG}}(\hat{s})+\tilde{\rho}_{(2)}^{\braket{\bar{q}q}^{2}\braket{\frac{\alpha_{s}}{\pi}GG}}(\hat{s})$

with

$\displaystyle\tilde{\rho}_{(2)}^{\braket{\frac{\alpha_{s}}{\pi}GG}}(\hat{s})=\int_{0}^{1}\int_{0}^{1}dxdy\frac{i\braket{\frac{\alpha_{s}}{\pi}GG}y^{2}}{9216\pi^{4}(x-1)^{4}x^{3}(y-1)^{3}}$
(95) $\displaystyle\times$
$\displaystyle\left(m^{8}\left(2x\left(18y^{4}+54y^{3}+49y^{2}+33y-3\right)\right.\right.$
$\displaystyle+$ $\displaystyle\left.36y^{4}+117y^{3}+109y^{2}+34y+6\right)$
$\displaystyle-$ $\displaystyle
12m_{c}^{6}\hat{s}x(y-1)\left(x^{2}\left(30y^{4}+84y^{3}+79y^{2}+38y-3\right)\right.$
$\displaystyle+$ $\displaystyle 2x\left(3y^{3}+2y^{2}-7y+2\right)$
$\displaystyle-$ $\displaystyle\left.30y^{4}-90y^{3}-83y^{2}-24y-1\right)$
$\displaystyle+$ $\displaystyle 12m_{c}^{4}\hat{s}^{2}(x-1)^{2}x^{2}(y-1)^{2}$
$\displaystyle\times$
$\displaystyle\left(x\left(90y^{4}+240y^{3}+224y^{2}+90y-6\right)\right.$
$\displaystyle+$
$\displaystyle\left.\left(90y^{3}+255y^{2}+229y+64\right)y\right)$
$\displaystyle-$ $\displaystyle 4m_{c}^{2}\hat{s}^{3}(x-1)^{3}x^{3}(y-1)^{3}$
$\displaystyle\times$
$\displaystyle\left(x\left(315y^{4}+810y^{3}+740y^{2}+264y-15\right)\right.$
$\displaystyle+$ $\displaystyle\left.315y^{4}+855y^{3}+745y^{2}+202y-3\right)$
$\displaystyle+$ $\displaystyle 3\hat{s}^{4}(x-1)^{4}x^{4}(y-1)^{4}$
$\displaystyle\times$
$\displaystyle\left(2x\left(84y^{4}+210y^{3}+187y^{2}+61y-3\right)\right.$
$\displaystyle+$
$\displaystyle\left.\left.+168y^{4}+441y^{3}+373y^{2}+98y-2\right)\right)$
$\displaystyle\times$
$\displaystyle\mathrm{H}\left(y\left(x\hat{s}(x(-y)+x+y-1)+m_{c}^{2}\right)\right),$
$\displaystyle\tilde{\rho}_{(2)}^{\braket{\bar{q}g_{s}\sigma
Gq}}(\hat{s})=\int_{0}^{1}\int_{0}^{1}dxdy\frac{5m_{s}\braket{\bar{q}g_{s}\sigma
Gq}}{576\pi^{4}(x-1)^{2}x^{2}(y-1)^{2}}$ (96) $\displaystyle\times$
$\displaystyle 2\left(m_{c}^{6}\left(6y^{3}+7y^{2}+8y-1\right)y\right.$
$\displaystyle-$ $\displaystyle
m_{c}^{4}(x-1)x(y-1)\left(3sy\left(24y^{3}+33y^{2}+26y-3\right)\right.$
$\displaystyle-$
$\displaystyle\left.m_{0}\left(12y^{3}+15y^{2}+14y-1\right)\right)$
$\displaystyle+$ $\displaystyle
m_{c}^{2}\hat{s}(x-1)^{2}x^{2}(y-1)^{2}\left(12\hat{s}y\left(10y^{3}+15y^{2}+10y-1\right)\right.$
$\displaystyle-$
$\displaystyle\left.m_{0}\left(48y^{3}+69y^{2}+50y-3\right)\right)$
$\displaystyle+$
$\displaystyle\hat{s}^{2}(x-1)^{3}x^{3}(y-1)^{3}\left(m_{0}\left(40y^{3}+62y^{2}+40y-2\right)\right.$
$\displaystyle-$
$\displaystyle\left.\left.\hat{s}y\left(60y^{3}+95y^{2}+58y-5\right)\right)\right)$
$\displaystyle\times$
$\displaystyle\mathrm{H}\left(y\left(x\hat{s}(x(-y)+x+y-1)+m_{c}^{2}\right)\right),$
$\displaystyle\tilde{\rho}_{(2)}^{\braket{\frac{\alpha_{s}}{\pi}GG}\braket{\bar{q}q}}(\hat{s})$
(97) $\displaystyle=$
$\displaystyle\int_{0}^{1}\int_{0}^{1}dxdy\mathrm{H}\left(y\left(x\hat{s}(x(-y)+x+y-1)+m_{c}^{2}\right)\right)$
$\displaystyle\times$
$\displaystyle\frac{-m_{c}\braket{\frac{\alpha_{s}}{\pi}GG}\braket{\bar{q}q}\left(m_{c}^{2}+\hat{s}x(x(-y)+x+y-1)\right)}{16\pi^{2}(x-1)^{2}x(y-1)}$
$\displaystyle\times$
$\displaystyle\left(m_{c}^{2}(y+1)y+2m_{c}m_{s}(x-1)(y-1)\right.$
$\displaystyle-$
$\displaystyle\left.2\hat{s}(x-1)xy\left(y^{2}-1\right)\right),$
$\displaystyle\tilde{\rho}_{(2)}^{\braket{\bar{q}g_{s}\sigma
Gq}\braket{\bar{q}q}}(\hat{s})$ (98) $\displaystyle=$
$\displaystyle\int_{0}^{1}\int_{0}^{1}dxdy\mathrm{H}\left(y\left(x\hat{s}(x(-y)+x+y-1)+m_{c}^{2}\right)\right)$
$\displaystyle\times$
$\displaystyle\frac{-5g_{s}^{2}\braket{\bar{q}g_{s}\sigma
Gq}\braket{\bar{q}q}}{3888\pi^{4}(x-1)x(y-1)}$ $\displaystyle\times$
$\displaystyle\left(m_{c}^{4}\left(12y^{3}+15y^{2}+14y-1\right)\right.$
$\displaystyle-$ $\displaystyle
m_{c}^{2}\hat{s}(x-1)x\left(48y^{4}+21y^{3}-19y^{2}-53y+3\right)$
$\displaystyle+$
$\displaystyle\left.2\hat{s}^{2}(x-1)^{2}x^{2}(y-1)^{2}\left(20y^{3}+31y^{2}+20y-1\right)\right),\
\ \ \ \ ~{}~{}~{}$ $\displaystyle\tilde{\rho}_{2}^{\braket{\bar{q}g_{s}\sigma
Gq}\braket{\frac{\alpha_{s}}{\pi}GG}}(\hat{s})$ (99) $\displaystyle=$
$\displaystyle\int_{0}^{1}\int_{0}^{1}dxdy\mathrm{H}\left(y\left(x\hat{s}(x(-y)+x+y-1)+m_{c}^{2}\right)\right)$
$\displaystyle\times$
$\displaystyle\left(-\frac{5\braket{\frac{\alpha_{s}}{\pi}GG}m_{c}\braket{\bar{q}g_{s}\sigma
Gq}(y+1)\left(2m^{2}\right)}{768\pi^{2}(x-1)}\right.$ $\displaystyle-$
$\displaystyle\frac{5\braket{\frac{\alpha_{s}}{\pi}GG}m_{c}\braket{\bar{q}g_{s}\sigma
Gq}}{768\pi^{2}(x-1)}$ $\displaystyle\times$
$\displaystyle\left.(y+1)\left(3\hat{s}x(x(-y)+x+y-1)\right)\right),$
$\displaystyle\tilde{\rho}_{(2)}^{\braket{\bar{q}q}^{2}\braket{\frac{\alpha_{s}}{\pi}GG}}(\hat{s})$
(100) $\displaystyle=$
$\displaystyle\int_{0}^{1}dx\mathrm{H}\left(m_{c}^{2}-x(x+1)\hat{s}\right)$
$\displaystyle\times\frac{1}{6}\braket{\frac{\alpha_{s}}{\pi}GG}m_{c}^{2}\braket{\bar{q}q}^{2}-\frac{1}{12}\braket{\frac{\alpha_{s}}{\pi}GG}m_{c}m_{s}\braket{\bar{q}q}^{2}x.~{}~{}~{}~{}~{}~{}~{}~{}$

## References

  * Aaij  _et al._ (2017a) R. Aaij _et al._ (LHCb), Phys. Rev. Lett. 118, 022003 (2017a), arXiv:1606.07895 [hep-ex] .
  * Aaij  _et al._ (2017b) R. Aaij _et al._ (LHCb), Phys. Rev. D 95, 012002 (2017b), arXiv:1606.07898 [hep-ex] .
  * Gui  _et al._ (2018) L.-C. Gui, L.-S. Lu, Q.-F. Lü, X.-H. Zhong, and Q. Zhao, Phys. Rev. D 98, 016010 (2018), arXiv:1801.08791 [hep-ph] .
  * Fernández  _et al._ (2016) F. Fernández, D. R. Entem, P. G. Ortega, and J. Segovia, PoS CHARM2016, 054 (2016), arXiv:1611.08534 [hep-ph] .
  * Ortega  _et al._ (2016) P. G. Ortega, J. Segovia, D. R. Entem, and F. Fernández, Phys. Rev. D 94, 114018 (2016), arXiv:1608.01325 [hep-ph] .
  * Oncala and Soto (2017) R. Oncala and J. Soto, Phys. Rev. D 96, 014004 (2017), arXiv:1702.03900 [hep-ph] .
  * Ozaki and Sasaki (2013) S. Ozaki and S. Sasaki, Phys. Rev. D 87, 014506 (2013), arXiv:1211.5512 [hep-lat] .
  * Panteleeva  _et al._ (2019) J. Y. Panteleeva, I. A. Perevalova, M. V. Polyakov, and P. Schweitzer, Phys. Rev. C 99, 045206 (2019), arXiv:1802.09029 [hep-ph] .
  * Maiani  _et al._ (2016) L. Maiani, A. D. Polosa, and V. Riquer, Phys. Rev. D 94, 054026 (2016), arXiv:1607.02405 [hep-ph] .
  * Lü and Dong (2016) Q.-F. Lü and Y.-B. Dong, Phys. Rev. D 94, 074007 (2016), arXiv:1607.05570 [hep-ph] .
  * Zhu (2016) R. Zhu, Phys. Rev. D 94, 054009 (2016), arXiv:1607.02799 [hep-ph] .
  * Deng  _et al._ (2018) C. Deng, J. Ping, H. Huang, and F. Wang, Phys. Rev. D 98, 014026 (2018), arXiv:1801.00164 [hep-ph] .
  * Wu  _et al._ (2016) J. Wu, Y.-R. Liu, K. Chen, X. Liu, and S.-L. Zhu, Phys. Rev. D 94, 094031 (2016), arXiv:1608.07900 [hep-ph] .
  * Yang and Ping (2019) Y. Yang and J. Ping, Phys. Rev. D 99, 094032 (2019), arXiv:1903.08505 [hep-ph] .
  * Anwar  _et al._ (2018) M. N. Anwar, J. Ferretti, and E. Santopinto, Phys. Rev. D 98, 094015 (2018), arXiv:1805.06276 [hep-ph] .
  * Liu  _et al._ (2021) X. Liu, H. Huang, J. Ping, D. Chen, and X. Zhu, Eur. Phys. J. C 81, 950 (2021), arXiv:2103.12425 [hep-ph] .
  * Chen  _et al._ (2017) H.-X. Chen, E.-L. Cui, W. Chen, X. Liu, and S.-L. Zhu, Eur. Phys. J. C 77, 160 (2017), arXiv:1606.03179 [hep-ph] .
  * Wang (2017) Z.-G. Wang, Eur. Phys. J. C 77, 78 (2017), arXiv:1606.05872 [hep-ph] .
  * Reinders  _et al._ (1985) L. J. Reinders, H. Rubinstein, and S. Yazaki, Phys. Rept. 127, 1 (1985).
  * Nakamura (2021) S. X. Nakamura, (2021), arXiv:2111.05115 [hep-ph] .
  * Liu (2017) X.-H. Liu, Phys. Lett. B 766, 117 (2017), arXiv:1607.01385 [hep-ph] .
  * Ge  _et al._ (2021) Y.-H. Ge, X.-H. Liu, and H.-W. Ke,  (2021), 10.1140/epjc/s10052-021-09590-y, arXiv:2103.05282 [hep-ph] .
  * Zyla  _et al._ (2020) P. A. Zyla _et al._ (Particle Data Group), PTEP 2020, 083C01 (2020).
  * Albuquerque and Nielsen (2009) R. M. Albuquerque and M. Nielsen, Nucl. Phys. A 815, 53 (2009), [Erratum: Nucl.Phys.A 857, 48–49 (2011)], arXiv:0804.4817 [hep-ph] .
  * Bečirević  _et al._ (2014) D. Bečirević, G. Duplančić, B. Klajn, B. Melić, and F. Sanfilippo, Nucl. Phys. B 883, 306 (2014), arXiv:1312.2858 [hep-ph] .
  * Belyaev  _et al._ (1995) V. M. Belyaev, V. M. Braun, A. Khodjamirian, and R. Ruckl, Phys. Rev. D 51, 6177 (1995), arXiv:hep-ph/9410280 .
  * Ioffe and Smilga (1984) B. L. Ioffe and A. V. Smilga, Nucl. Phys. B 232, 109 (1984).
  * Dias  _et al._ (2013) J. M. Dias, F. S. Navarra, M. Nielsen, and C. M. Zanetti, Phys. Rev. D 88, 016004 (2013), arXiv:1304.6433 [hep-ph] .
  * Huang  _et al._ (2011) P.-Z. Huang, H.-X. Chen, and S.-L. Zhu, Phys. Rev. D 83, 014021 (2011), arXiv:1010.2293 [hep-ph] .
  * Agaev  _et al._ (2016a) S. S. Agaev, K. Azizi, and H. Sundu, Phys. Rev. D 93, 074024 (2016a), arXiv:1602.08642 [hep-ph] .
  * Agaev  _et al._ (2016b) S. S. Agaev, K. Azizi, and H. Sundu, Phys. Rev. D 93, 074002 (2016b), arXiv:1601.03847 [hep-ph] .
  * Matheus  _et al._ (2007) R. D. Matheus, S. Narison, M. Nielsen, and J. M. Richard, Phys. Rev. D 75, 014005 (2007), arXiv:hep-ph/0608297 .
  * Colangelo  _et al._ (2010) P. Colangelo, F. De Fazio, and W. Wang, Phys. Rev. D 81, 074001 (2010), arXiv:1002.2880 [hep-ph] .
  * Ball  _et al._ (2007) P. Ball, V. M. Braun, and A. Lenz, JHEP 08, 090 (2007), arXiv:0707.1201 [hep-ph] .
  * Hu  _et al._ (2021) D.-D. Hu, H.-B. Fu, T. Zhong, Z.-H. Wu, and X.-G. Wu, (2021), arXiv:2107.02758 [hep-ph] .
  * Olpak  _et al._ (2017) M. A. Olpak, A. Ozpineci, and V. Tanriverdi, Phys. Rev. D 96, 014026 (2017), arXiv:1608.04539 [hep-ph] .
  * Beringer  _et al._ (2012) J. Beringer _et al._ (Particle Data Group), Phys. Rev. D 86, 010001 (2012).
  * Wang and Xin (2021) Z.-G. Wang and Q. Xin, (2021), arXiv:2112.04776 [hep-ph] .
  * Wang (2021) Z.-G. Wang, Adv. High Energy Phys. 2021, 4426163 (2021), arXiv:2103.04236 [hep-ph] .
  * Wang and Di (2019) Z.-G. Wang and Z.-Y. Di, Eur. Phys. J. C 79, 72 (2019), arXiv:1811.12821 [hep-ph] .
  * Maiani  _et al._ (2014) L. Maiani, F. Piccinini, A. D. Polosa, and V. Riquer, Phys. Rev. D 89, 114010 (2014), arXiv:1405.1551 [hep-ph] .
  * Nielsen and Navarra (2014) M. Nielsen and F. S. Navarra, Mod. Phys. Lett. A 29, 1430005 (2014), arXiv:1401.2913 [hep-ph] .
  * Wang (2015) Z.-G. Wang, Commun. Theor. Phys. 63, 325 (2015), arXiv:1405.3581 [hep-ph] .
  * Lebed and Polosa (2016) R. F. Lebed and A. D. Polosa, Phys. Rev. D 93, 094024 (2016), arXiv:1602.08421 [hep-ph] .
  * Chen and Chen (2019) H.-X. Chen and W. Chen, Phys. Rev. D 99, 074022 (2019), arXiv:1901.06946 [hep-ph] .
  * Wang (2020) Z.-G. Wang, Chin. Phys. C 44, 063105 (2020), arXiv:1901.10741 [hep-ph] .
  * Ablikim  _et al._ (2017) M. Ablikim _et al._ (BESIII), Phys. Rev. Lett 118, 092001 (2017), arXiv:1611.01317 [hep-ex] .
  * Ball and Braun (1996) P. Ball and V. M. Braun, Phys. Rev. D 54, 2182 (1996), arXiv:hep-ph/9602323 .
  * Ball and Jones (2007) P. Ball and G. W. Jones, JHEP 03, 069 (2007), arXiv:hep-ph/0702100 .
  * Gubler (2013) P. Gubler,  _A Bayesian Analysis of QCD Sum Rules_ , Ph.D. thesis, Tokyo Inst. Tech., Tokyo (2013).