Decipher the width of the $X(3872)$ via the QCD sum rules

Zhi-Gang Wang 111E-mail<EMAIL_ADDRESS>
Department of Physics, North China Electric Power University, Baoding 071003,
P. R. China

###### Abstract

In this work, we take the $X(3872)$ as the hidden-charm tetraquark state with
both isospin $I=0$ and $1$ components, then investigate the strong decays
$X(3872)\to J/\psi\pi^{+}\pi^{-}$, $J/\psi\omega$, $\chi_{c1}\pi^{0}$,
$D^{*0}\bar{D}^{0}$ and $D^{0}\bar{D}^{0}\pi^{0}$ with the QCD sum rules. We
take account of all the Feynman diagrams, and acquire four QCD sum rules based
on rigorous quark-hadron duality. We obtain the total decay width about
$1\,\rm{MeV}$, which is in excellent agreement with the experiment data
$\Gamma_{X}=1.19\pm 0.21\,\rm{MeV}$ from the Particle Data Group, it is the
first time to reproduce the tiny width of the $X(3872)$ via the QCD sum rules,
which supports assigning the $X(3872)$ as the hidden-charm tetraquark state
with the $J^{PC}=1^{++}$.

PACS number: 12.39.Mk, 12.38.Lg

Key words: Tetraquark state, QCD sum rules

##  1 Introduction

In 2003, the Belle collaboration observed a narrow charmonium-like state, the
$X(3872)$, in the $\pi^{+}\pi^{-}J/\psi$ mass spectrum in the exclusive
process $B^{\pm}\to K^{\pm}\pi^{+}\pi^{-}J/\psi$, which has a mass of
$3872.0\pm 0.6\pm 0.5\,\rm{MeV}$ and a width less than $2.3\,\rm{MeV}$, the
$X(3872)$ lies very near the $D^{*}\bar{D}$ threshold [1], which stimulated
the interpretation in terms of the $D^{*}\bar{D}$ molecular state [2, 3, 4, 5,
6, 7, 8, 9]. At the same time, other interpretations were suggested, such as
the tetraquark state [10, 11, 12, 13, 14, 15], hybrid state [16], charmonium-
molecule mixing state [17, 18], charmonium state [19], etc. In fact, the
observations of the $X(3872)$ in the $pp$ and $p\bar{p}$ collisions by the
CDF, ATLAS, LHCb and CMS collaborations disfavor the pure molecule assignment
[20, 21, 22, 23, 24].

In 2015, the LHCb collaboration studied the angular correlations in the
$B^{+}\to X(3872)K^{+}$ decays with
$X(3872)\to\rho^{0}J/\psi\to\pi^{+}\pi^{-}\mu^{+}\mu^{-}$ to measure orbital
angular momentum contributions and to determine the $J^{PC}$ of the $X(3872)$
to be $1^{++}$ [25]. The $X(3872)$ state is probably the best known (and most
enigmatic) representative of the $X$, $Y$ and $Z$ states. One important
discriminant between different models is the width of the $X(3872)$. In 2020,
the LHCb collaboration updated the mass and width of the $X(3872)$, and
obtained the Breit-Wigner width $\Gamma=0.96^{+0.19}_{-0.18}\pm
0.21\,\rm{MeV}$ [26], or $1.39\pm 0.24\pm 0.10\,\rm{MeV}$ [27], which
indicates non-zero mass of the $X(3872)$ and leads to the average width
$\Gamma_{X}=1.19\pm 0.21\,\rm{MeV}$ listed in ”The Review of Particle Physics”
[28]. As known, we cannot assign a hadron with the mass alone, and we should
study the decays to obtain more robust interpretation. Up to today, only the
decays $X(3872)\to J/\psi\pi^{+}\pi^{-}$, $J/\psi\omega$, $J/\psi\gamma$,
$\psi^{\prime}\gamma$, $\chi_{c1}\pi^{0}$, $D^{*0}\bar{D}^{0}$ and
$D^{0}\bar{D}^{0}\pi^{0}$ are established [28].

In the present work, we will focus on the scenario of tetraquark states. In
Ref.[15], we take the pseudoscalar, scalar, axialvector, vector, tensor
(anti)diquarks as the basic constituents, and construct the scalar,
axialvector and tensor tetraquark currents to study the mass spectrum of the
ground state hidden-charm tetraquark states with the QCD sum rules in a
comprehensive way, and observe that the $X(3872)$ can be assigned to be the
hidden-charm tetraquark state with the quantum numbers $J^{PC}=1^{++}$.
According to the recent combined data analysis, the decays $X(3872)\to
J/\psi\rho\to J/\psi\pi^{+}\pi^{-}$ and $X(3872)\to J/\psi\omega\to
J/\psi\pi^{+}\pi^{-}\pi^{0}$ have almost the same branching fractions [29],
the isospin breaking effects in the decays are large enough and beyond the
naive $\rho-\omega$ mixing. In this work, we introduce the isospin breaking
effects explicitly and study the decays $X(3872)\to J/\psi\pi^{+}\pi^{-}$,
$J/\psi\omega$, $\chi_{c1}\pi^{0}$, $D^{*0}\bar{D}^{0}$ and
$D^{0}\bar{D}^{0}\pi^{0}$ with the QCD sum rules based on rigorous quark-
hadron duality, and try to decipher the width of the $X(3872)$.

The article is arranged as follows: we obtain the QCD sum rules for the
hadronic coupling constants in section 2; in section 3, we present numerical
results and discussions; section 4 is reserved for our conclusion.

##  2 QCD sum rules for the hadronic coupling constants

We write down the three-point correlation functions
$\Pi_{\mu\nu\alpha}^{1/2}(p,q)$ and $\Pi_{\mu\alpha}^{1/2}(p,q)$ in the QCD
sum rules,

$\displaystyle\Pi^{1}_{\mu\nu\alpha}(p,q)$ $\displaystyle=$ $\displaystyle
i^{2}\int d^{4}xd^{4}y\,e^{ip\cdot x}e^{iq\cdot y}\,\langle
0|T\left\\{J_{\mu}^{J/\psi}(x)J_{\nu}^{\rho}(y)J^{X}_{\alpha}{}^{\dagger}(0)\right\\}|0\rangle\,,$
$\displaystyle\Pi^{2}_{\mu\nu\alpha}(p,q)$ $\displaystyle=$ $\displaystyle
i^{2}\int d^{4}xd^{4}y\,e^{ip\cdot x}e^{iq\cdot y}\,\langle
0|T\left\\{J_{\mu}^{J/\psi}(x)J_{\nu}^{\omega}(y)J^{X}_{\alpha}{}^{\dagger}(0)\right\\}|0\rangle\,,$
$\displaystyle\Pi^{1}_{\mu\alpha}(p,q)$ $\displaystyle=$ $\displaystyle
i^{2}\int d^{4}xd^{4}y\,e^{ip\cdot x}e^{iq\cdot y}\,\langle
0|T\left\\{J_{\mu}^{\chi}(x)J_{5}^{\pi}(y)J^{X}_{\alpha}{}^{\dagger}(0)\right\\}|0\rangle\,,$
$\displaystyle\Pi^{2}_{\mu\alpha}(p,q)$ $\displaystyle=$ $\displaystyle
i^{2}\int d^{4}xd^{4}y\,e^{ip\cdot x}e^{iq\cdot y}\,\langle
0|T\left\\{J_{\mu}^{D^{*}}(x)J_{5}^{D}(y)J^{X}_{\alpha}{}^{\dagger}(0)\right\\}|0\rangle\,,$
(1)

where the currents

$\displaystyle J_{\mu}^{J/\psi}(x)$ $\displaystyle=$
$\displaystyle\bar{c}(x)\gamma_{\mu}c(x)\,,$ $\displaystyle J_{\mu}^{\chi}(x)$
$\displaystyle=$ $\displaystyle\bar{c}(x)\gamma_{\mu}\gamma_{5}c(x)\,,$
$\displaystyle J_{\mu}^{D^{*}}(x)$ $\displaystyle=$
$\displaystyle\bar{u}(x)\gamma_{\mu}c(x)\,,$ (2) $\displaystyle
J_{\nu}^{\rho}(y)$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left[\bar{u}(y)\gamma_{\nu}u(y)-\bar{d}(y)\gamma_{\nu}d(y)\right]\,,$
$\displaystyle J_{\nu}^{\omega}(y)$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left[\bar{u}(y)\gamma_{\nu}u(y)+\bar{d}(y)\gamma_{\nu}d(y)\right]\,,$
$\displaystyle J_{5}^{\pi}(y)$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left[\bar{u}(y)i\gamma_{5}u(y)-\bar{d}(y)i\gamma_{5}d(y)\right]\,,$
$\displaystyle J_{5}^{D}(y)$ $\displaystyle=$
$\displaystyle\bar{c}(y)i\gamma_{5}u(y)\,,$ (3)

$\displaystyle J^{X}_{\alpha}(0)$ $\displaystyle=$ $\displaystyle\cos\theta
J^{u\bar{u}}_{\alpha}(0)+\sin\theta J^{d\bar{d}}_{\alpha}(0)\,,$ (4)

$\displaystyle J^{u\bar{u}}_{\alpha}(0)$ $\displaystyle=$
$\displaystyle\frac{\varepsilon^{ijk}\varepsilon^{imn}}{\sqrt{2}}\Big{[}u^{T}_{j}(0)C\gamma_{5}c_{k}(0)\bar{u}_{m}(0)\gamma_{\alpha}C\bar{c}^{T}_{n}(0)-u^{T}_{j}(0)C\gamma_{\alpha}c_{k}(0)\bar{u}_{m}(0)\gamma_{5}C\bar{c}^{T}_{n}(0)\Big{]}\,,$
$\displaystyle J^{d\bar{d}}_{\mu}(0)$ $\displaystyle=$
$\displaystyle\frac{\varepsilon^{ijk}\varepsilon^{imn}}{\sqrt{2}}\Big{[}d^{T}_{j}(0)C\gamma_{5}c_{k}(0)\bar{d}_{m}(0)\gamma_{\alpha}C\bar{c}^{T}_{n}(0)-d^{T}_{j}(0)C\gamma_{\alpha}c_{k}(0)\bar{d}_{m}(0)\gamma_{5}C\bar{c}^{T}_{n}(0)\Big{]}\,,$
(5)

interpolate the mesons $J/\psi$, $\chi_{c1}$, $D^{*}$, $\rho$, $\omega$,
$\pi$, $D$ and $X(3872)$, respectively. As the decays $X(3872)\to
J/\psi\rho\to J/\psi\pi^{+}\pi^{-}$ and $X(3872)\to J/\psi\omega\to
J/\psi\pi^{+}\pi^{-}\pi^{0}$ have almost the same branching fractions [29],
which is beyond the naive expectation of the $\rho-\omega$ mixing, we have to
introduce mixing effects in the $X(3872)$ and abandon the obsession that the
$X(3872)$ has definite isospin $I=0$. We determine the mixing angle $\theta$
by the experimental data via trial and error.

At the hadron side, we insert a complete set of intermediate hadronic states
with the same quantum numbers as the currents into the three-point correlation
functions, and isolate the ground state contributions explicitly,

$\displaystyle\Pi^{1}_{\mu\nu\alpha}(p,q)$ $\displaystyle=$
$\displaystyle\frac{\lambda_{X}\,f_{J/\psi}m_{J/\psi}\,f_{\rho}m_{\rho}\,G^{\prime}_{XJ/\psi\rho}}{(m_{X}^{2}-p^{\prime
2})(m_{J/\psi}^{2}-p^{2})(m_{\rho}^{2}-q^{2})}i\varepsilon_{\mu\nu\alpha\sigma}q^{\sigma}p\cdot
q+\cdots\,,$ (6) $\displaystyle=$ $\displaystyle\Pi_{\rho}(p^{\prime
2},p^{2},q^{2})\,i\varepsilon_{\mu\nu\alpha\sigma}q^{\sigma}p\cdot
q+\cdots\,,$ $\displaystyle\Pi^{2}_{\mu\nu\alpha}(p,q)$ $\displaystyle=$
$\displaystyle\frac{\lambda_{X}\,f_{J/\psi}m_{J/\psi}\,f_{\omega}m_{\omega}\,G^{\prime}_{XJ/\psi\omega}}{(m_{X}^{2}-p^{\prime
2})(m_{J/\psi}^{2}-p^{2})(m_{\omega}^{2}-q^{2})}i\varepsilon_{\mu\nu\alpha\sigma}q^{\sigma}p\cdot
q+\cdots\,,$ (7) $\displaystyle=$ $\displaystyle\Pi_{\omega}(p^{\prime
2},p^{2},q^{2})\,i\varepsilon_{\mu\nu\alpha\sigma}q^{\sigma}p\cdot
q+\cdots\,,$

$\displaystyle\Pi^{1}_{\mu\alpha}(p,q)$ $\displaystyle=$
$\displaystyle\frac{\lambda_{X}\,f_{\chi_{c1}}m_{\chi_{c1}}\mu_{\pi}\,G^{\prime}_{X\chi\pi}}{(m_{X}^{2}-p^{\prime
2})(m_{\chi_{c1}}^{2}-p^{2})(m_{\pi}^{2}-q^{2})}\varepsilon_{\alpha\mu\nu\sigma}p^{\nu}q^{\sigma}+\cdots\,,$
(8) $\displaystyle=$ $\displaystyle\Pi_{\pi}(p^{\prime
2},p^{2},q^{2})\,\varepsilon_{\alpha\mu\nu\sigma}p^{\nu}q^{\sigma}+\cdots\,,$
$\displaystyle\Pi^{2}_{\mu\alpha}(p,q)$ $\displaystyle=$
$\displaystyle\frac{\lambda_{X}\,f_{D^{*}}m_{D^{*}}\mu_{D}\,G^{\prime}_{XD^{*}D}}{(m_{X}^{2}-p^{\prime
2})(m_{D^{*}}^{2}-p^{2})(m_{D}^{2}-q^{2})}ip\cdot qg_{\mu\alpha}+\cdots\,,$
(9) $\displaystyle=$ $\displaystyle\Pi_{D}(p^{\prime 2},p^{2},q^{2})\,ip\cdot
qg_{\mu\alpha}+\cdots\,,$

where $p^{\prime}=p+q$, $\mu_{\pi}=\frac{f_{\pi}m_{\pi}^{2}}{m_{u}+m_{d}}$,
$\mu_{D}=\frac{f_{D}m_{D}^{2}}{m_{c}}$, the hadronic coupling constants
$G^{\prime}_{XJ/\psi\rho}$, $G^{\prime}_{XJ/\psi\omega}$,
$G^{\prime}_{X\chi\pi}$ and $G^{\prime}_{XD^{*}D}$ are defined by

$\displaystyle\langle J/\psi(p)\rho(q)|X(p^{\prime})\rangle$ $\displaystyle=$
$\displaystyle-\varepsilon^{\sigma\alpha\mu\nu}p^{\prime}_{\sigma}\,\zeta_{\alpha}\,\xi^{*}_{\mu}\,\varsigma^{*}_{\nu}\,p\cdot
q\,G^{\prime}_{XJ/\psi\rho}\,,$ $\displaystyle\langle
J/\psi(p)\omega(q)|X(p^{\prime})\rangle$ $\displaystyle=$
$\displaystyle-\varepsilon^{\sigma\alpha\mu\nu}p^{\prime}_{\sigma}\,\zeta_{\alpha}\,\xi^{*}_{\mu}\,\varsigma^{*}_{\nu}\,p\cdot
q\,G^{\prime}_{XJ/\psi\omega}\,,$
$\displaystyle\langle\chi_{c1}(p)\pi(q)|X(p^{\prime})\rangle$ $\displaystyle=$
$\displaystyle
i\varepsilon^{\sigma\alpha\mu\nu}p^{\prime}_{\sigma}\,\zeta_{\alpha}\,p_{\mu}\,\xi^{*}_{\nu}\,G^{\prime}_{X\chi\pi}\,,$
$\displaystyle\langle D^{*}(p)D(q)|X(p^{\prime})\rangle$ $\displaystyle=$
$\displaystyle-\zeta\cdot\xi^{*}\,p\cdot q\,G^{\prime}_{XD^{*}D}\,,$ (10)

the decay constants $\lambda_{X}$, $f_{J/\psi}$, $f_{\chi_{c1}}$, $f_{\rho}$,
$f_{\omega}$, $f_{D^{*}}$, $f_{D}$ and $f_{\pi}$, are defined by,

$\displaystyle\langle 0|J_{\mu}^{X}(0)|X(p^{\prime})\rangle$ $\displaystyle=$
$\displaystyle\lambda_{X}\zeta_{\mu}\,\,,$ (11) $\displaystyle\langle
0|J_{\mu}^{J/\psi}(0)|J/\psi(p)\rangle$ $\displaystyle=$ $\displaystyle
f_{J/\psi}m_{J/\psi}\xi_{\mu}\,\,,$ $\displaystyle\langle
0|J_{\mu}^{\chi}(0)|\chi_{c1}(p)\rangle$ $\displaystyle=$ $\displaystyle
f_{\chi_{c1}}m_{\chi_{c1}}\xi_{\mu}\,\,,$ $\displaystyle\langle
0|J_{\mu}^{D^{*}}(0)|D^{*}(p)\rangle$ $\displaystyle=$ $\displaystyle
f_{D^{*}}m_{D^{*}}\xi_{\mu}\,\,,$ (12) $\displaystyle\langle
0|J_{\mu}^{\rho}(0)|\rho(q)\rangle$ $\displaystyle=$ $\displaystyle
f_{\rho}m_{\rho}\varsigma_{\mu}\,\,,$ $\displaystyle\langle
0|J_{\mu}^{\omega}(0)|\omega(q)\rangle$ $\displaystyle=$ $\displaystyle
f_{\omega}m_{\omega}\varsigma_{\mu}\,\,,$ (13) $\displaystyle\langle
0|J_{5}^{D}(0)|D(q)\rangle$ $\displaystyle=$
$\displaystyle\frac{f_{D}m_{D}^{2}}{m_{c}}\,\,,$ $\displaystyle\langle
0|J_{5}^{\pi}(0)|\pi(q)\rangle$ $\displaystyle=$
$\displaystyle\frac{f_{\pi}m_{\pi}^{2}}{m_{u}+m_{d}}\,\,,$ (14)

the $\zeta_{\mu}$, $\xi_{\mu}$ and $\varsigma_{\mu}$ are polarization vectors
of the axialvector or vector mesons.

We choose the components $\Pi_{\rho}(p^{\prime 2},p^{2},q^{2})$,
$\Pi_{\omega}(p^{\prime 2},p^{2},q^{2})$, $\Pi_{\pi}(p^{\prime
2},p^{2},q^{2})$ and $\Pi_{D}(p^{\prime 2},p^{2},q^{2})$ to study the hadronic
coupling constants $G^{\prime}_{XJ/\psi\rho}$, $G^{\prime}_{XJ/\psi\omega}$,
$G^{\prime}_{X\chi\pi}$ and $G^{\prime}_{XD^{*}D}$, respectively. In Ref.[13],
the tensor structures
$p_{\mu}\varepsilon_{\nu\alpha\sigma\tau}p^{\sigma}q^{\tau}$ and
$\varepsilon_{\mu\nu\alpha\sigma}q^{\sigma}$, which differ from the structure
$\varepsilon_{\mu\nu\alpha\sigma}q^{\sigma}p\cdot q$ in Eqs.(6)-(7) greatly,
are chosen to study the hadronic coupling constants $G_{XJ/\psi\rho}$ and
$G_{XJ/\psi\omega}$. It is not odd that different predictions may be obtained.
Then, we acquire the hadronic spectral densities $\rho_{H}(s^{\prime},s,u)$
through triple dispersion relation,

$\displaystyle\Pi_{H}(p^{\prime 2},p^{2},q^{2})$ $\displaystyle=$
$\displaystyle\int_{\Delta_{s}^{\prime
2}}^{\infty}ds^{\prime}\int_{\Delta_{s}^{2}}^{\infty}ds\int_{\Delta_{u}^{2}}^{\infty}du\frac{\rho_{H}(s^{\prime},s,u)}{(s^{\prime}-p^{\prime
2})(s-p^{2})(u-q^{2})}\,,$ (15)

where the $\Delta_{s}^{\prime 2}$, $\Delta_{s}^{2}$ and $\Delta_{u}^{2}$ are
thresholds, we add the subscript $H$ to stand for the components
$\Pi_{\rho}(p^{\prime 2},p^{2},q^{2})$, $\Pi_{\omega}(p^{\prime
2},p^{2},q^{2})$, $\Pi_{\pi}(p^{\prime 2},p^{2},q^{2})$ and $\Pi_{D}(p^{\prime
2},p^{2},q^{2})$ at the hadron side.

As the operator product expansion is concerned, we calculate the vacuum
condensates up to dimension 5, and obtain the QCD spectral densities through
double dispersion relation,

$\displaystyle\Pi_{QCD}(p^{\prime 2},p^{2},q^{2})$ $\displaystyle=$
$\displaystyle\int_{\Delta_{s}^{2}}^{\infty}ds\int_{\Delta_{u}^{2}}^{\infty}du\frac{\rho_{QCD}(p^{\prime
2},s,u)}{(s-p^{2})(u-q^{2})}\,,$ (16)

as

$\displaystyle{\rm lim}_{\epsilon\to 0}\frac{{\rm
Im}\,\Pi_{QCD}(s^{\prime}+i\epsilon,p^{2},q^{2})}{\pi}$ $\displaystyle=$
$\displaystyle 0\,.$ (17)

In calculations, we neglect the gluon condensates due to their tiny
contributions [30, 31]. We accomplish the integral over $ds^{\prime}$ firstly
at the hadron side, then match the hadron side with the QCD side bellow the
continuum thresholds $s_{0}$ and $u_{0}$ to obtain rigorous quark-hadron
duality [30, 31],

$\displaystyle\int_{\Delta_{s}^{2}}^{s_{0}}ds\int_{\Delta_{u}^{2}}^{u_{0}}du\left[\int_{\Delta_{s}^{\prime
2}}^{\infty}ds^{\prime}\frac{\rho_{H}(s^{\prime},s,u)}{(s^{\prime}-p^{\prime
2})(s-p^{2})(u-q^{2})}\right]$ $\displaystyle=$
$\displaystyle\int_{\Delta_{s}^{2}}^{s_{0}}ds\int_{\Delta_{u}^{2}}^{u_{0}}du\frac{\rho_{QCD}(s,u)}{(s-p^{2})(u-q^{2})}\,.$
(18)

In the following, we write down the hadron representation explicitly,

$\displaystyle\Pi_{\rho}(p^{\prime 2},p^{2},q^{2})$ $\displaystyle=$
$\displaystyle\frac{\lambda_{X}\,f_{J/\psi}m_{J/\psi}\,f_{\rho}m_{\rho}\,G^{\prime}_{XJ/\psi\rho}}{(m_{X}^{2}-p^{\prime
2})(m_{J/\psi}^{2}-p^{2})(m_{\rho}^{2}-q^{2})}+\frac{C_{\rho}^{\prime}}{(m_{J/\psi}^{2}-p^{2})(m_{\rho}^{2}-q^{2})}\,,$
$\displaystyle\Pi_{\omega}(p^{\prime 2},p^{2},q^{2})$ $\displaystyle=$
$\displaystyle\frac{\lambda_{X}\,f_{J/\psi}m_{J/\psi}\,f_{\omega}m_{\omega}\,G^{\prime}_{XJ/\psi\omega}}{(m_{X}^{2}-p^{\prime
2})(m_{J/\psi}^{2}-p^{2})(m_{\omega}^{2}-q^{2})}+\frac{C_{\omega}^{\prime}}{(m_{J/\psi}^{2}-p^{2})(m_{\omega}^{2}-q^{2})}\,,$
$\displaystyle\Pi_{\pi}(p^{\prime 2},p^{2},q^{2})$ $\displaystyle=$
$\displaystyle\frac{\lambda_{X}\,f_{\chi_{c1}}m_{\chi_{c1}}\,\mu_{\pi}\,G^{\prime}_{X\chi\pi}}{(m_{X}^{2}-p^{\prime
2})(m_{\chi_{c1}}^{2}-p^{2})(m_{\pi}^{2}-q^{2})}+\frac{C_{\pi}^{\prime}}{(m_{\chi_{c1}}^{2}-p^{2})(m_{\pi}^{2}-q^{2})}\,,$
$\displaystyle\Pi_{D}(p^{\prime 2},p^{2},q^{2})$ $\displaystyle=$
$\displaystyle\frac{\lambda_{X}\,f_{D^{*}}m_{D^{*}}\,\mu_{D}\,G^{\prime}_{XD^{*}D}}{(m_{X}^{2}-p^{\prime
2})(m_{D^{*}}^{2}-p^{2})(m_{D}^{2}-q^{2})}+\frac{C_{D}^{\prime}}{(m_{D^{*}}^{2}-p^{2})(m_{D}^{2}-q^{2})}\,,$
(19)

where we introduce the parameters $C^{\prime}_{\rho/\omega/\pi/D}$ to stand
for all the contributions concerning the higher resonances in the $s^{\prime}$
channel,

$\displaystyle C^{\prime}_{\rho}$ $\displaystyle=$
$\displaystyle\int_{s^{\prime}_{0}}^{\infty}ds^{\prime}\frac{\tilde{\rho}_{\rho}(s^{\prime},m_{J/\psi}^{2},m_{\rho}^{2})}{s^{\prime}-p^{\prime
2}}\,,$ $\displaystyle C^{\prime}_{\omega}$ $\displaystyle=$
$\displaystyle\int_{s^{\prime}_{0}}^{\infty}ds^{\prime}\frac{\tilde{\rho}_{\omega}(s^{\prime},m_{J/\psi}^{2},m_{\omega}^{2})}{s^{\prime}-p^{\prime
2}}\,,$ $\displaystyle C^{\prime}_{\pi}$ $\displaystyle=$
$\displaystyle\int_{s^{\prime}_{0}}^{\infty}ds^{\prime}\frac{\tilde{\rho}_{\pi}(s^{\prime},m_{\chi_{c1}}^{2},m_{\pi}^{2})}{s^{\prime}-p^{\prime
2}}\,,$ $\displaystyle C^{\prime}_{D}$ $\displaystyle=$
$\displaystyle\int_{s^{\prime}_{0}}^{\infty}ds^{\prime}\frac{\tilde{\rho}_{D}(s^{\prime},m_{D^{*}}^{2},m_{D}^{2})}{s^{\prime}-p^{\prime
2}}\,,$ (20)

where the densities
$\rho_{H}(s^{\prime},s,u)=\tilde{\rho}_{\rho}(s^{\prime},s,u)\delta(s-m_{J/\psi}^{2})\delta(u-m_{\rho}^{2})$,
$\tilde{\rho}_{\omega}(s^{\prime},s,u)\delta(s-m_{J/\psi}^{2})\delta(u-m_{\omega}^{2})$,
$\tilde{\rho}_{\pi}(s^{\prime},s,u)\delta(s-m_{\chi_{c1}}^{2})\delta(u-m_{\pi}^{2})$
and
$\tilde{\rho}_{D}(s^{\prime},s,u)\delta(s-m_{D^{*}}^{2})\delta(u-m_{D}^{2})$,
respectively. The densities
$\tilde{\rho}_{\rho}(s^{\prime},m_{J/\psi}^{2},m_{\rho}^{2})$,
$\tilde{\rho}_{\omega}(s^{\prime},m_{J/\psi}^{2},m_{\omega}^{2})$,
$\tilde{\rho}_{\pi}(s^{\prime},m_{\chi_{c1}}^{2},m_{\pi}^{2})$ and
$\tilde{\rho}_{D}(s^{\prime},m_{D^{*}}^{2},m_{D}^{2})$ are complex and we have
no knowledge about the higher resonant states, as the spectrum is vague. We
take the unknown functions $C^{\prime}_{\rho/\omega/\pi/D}$ as free parameters
and adjust the suitable values to obtain flat Borel platforms for the hadronic
coupling constants $G^{\prime}_{XJ/\psi\rho}$, $G^{\prime}_{XJ/\psi\omega}$,
$G^{\prime}_{X\chi\pi}$ and $G^{\prime}_{XD^{*}D}$, respectively [30, 31, 32,
33, 34, 35, 36, 37].

We set $p^{\prime 2}=p^{2}$ in the correlation functions $\Pi_{H}(p^{\prime
2},p^{2},q^{2})$, and perform double Borel transform in regard to
$P^{2}=-p^{2}$ and $Q^{2}=-q^{2}$, respectively, then we set
$T_{1}^{2}=T_{2}^{2}=T^{2}$ to obtain four QCD sum rules,

$\displaystyle\frac{\lambda_{XJ/\psi\rho}G^{\prime}_{XJ/\psi\rho}}{m_{X}^{2}-m_{J/\psi}^{2}}\left[\exp\left(-\frac{m_{J/\psi}^{2}}{T^{2}}\right)-\exp\left(-\frac{m_{X}^{2}}{T^{2}}\right)\right]\exp\left(-\frac{m_{\rho}^{2}}{T^{2}}\right)+C^{\prime}_{\rho}\exp\left(-\frac{m_{J/\psi}^{2}+m_{\rho}^{2}}{T^{2}}\right)$
$\displaystyle=\frac{\cos\theta-\sin\theta}{\sqrt{2}}\frac{m_{c}}{16\sqrt{2}\pi^{4}}\int_{4m_{c}^{2}}^{s_{J/\psi}^{0}}ds\int_{0}^{s^{0}_{\rho}}du\,\sqrt{1-\frac{4m_{c}^{2}}{s}}\exp\left(-\frac{s+u}{T^{2}}\right)\,,$
(21)

$\displaystyle\frac{\lambda_{XJ/\psi\omega}G^{\prime}_{XJ/\psi\omega}}{m_{X}^{2}-m_{J/\psi}^{2}}\left[\exp\left(-\frac{m_{J/\psi}^{2}}{T^{2}}\right)-\exp\left(-\frac{m_{X}^{2}}{T^{2}}\right)\right]\exp\left(-\frac{m_{\omega}^{2}}{T^{2}}\right)+C^{\prime}_{\omega}\exp\left(-\frac{m_{J/\psi}^{2}+m_{\omega}^{2}}{T^{2}}\right)$
$\displaystyle=\frac{\cos\theta+\sin\theta}{\sqrt{2}}\frac{m_{c}}{16\sqrt{2}\pi^{4}}\int_{4m_{c}^{2}}^{s_{J/\psi}^{0}}ds\int_{0}^{s^{0}_{\omega}}du\,\sqrt{1-\frac{4m_{c}^{2}}{s}}\exp\left(-\frac{s+u}{T^{2}}\right)\,,$
(22)

$\displaystyle\frac{\lambda_{X\chi\pi}G^{\prime}_{X\chi\pi}}{m_{X}^{2}-m_{\chi_{c1}}^{2}}\left[\exp\left(-\frac{m_{\chi_{c1}}^{2}}{T^{2}}\right)-\exp\left(-\frac{m_{X}^{2}}{T^{2}}\right)\right]\exp\left(-\frac{m_{\rho}^{2}}{T^{2}}\right)+C^{\prime}_{\pi}\exp\left(-\frac{m_{\chi_{c1}}^{2}+m_{\pi}^{2}}{T^{2}}\right)$
$\displaystyle=\frac{\cos\theta-\sin\theta}{\sqrt{2}}\frac{m_{c}\langle\bar{q}g_{s}\sigma
Gq\rangle}{16\sqrt{2}\pi^{4}}\int_{4m_{c}^{2}}^{s_{\chi_{c1}}^{0}}ds\,\frac{s-3m_{c}^{2}}{s\sqrt{s(s-4m_{c}^{2})}}\exp\left(-\frac{s}{T^{2}}\right)\,,$
(23)

$\displaystyle\frac{\lambda_{XD^{*}D}G^{\prime}_{XD^{*}D}}{m_{X}^{2}-m_{D^{*}}^{2}}\left[\exp\left(-\frac{m_{D^{*}}^{2}}{T^{2}}\right)-\exp\left(-\frac{m_{X}^{2}}{T^{2}}\right)\right]\exp\left(-\frac{m_{D}^{2}}{T^{2}}\right)+C^{\prime}_{D}\exp\left(-\frac{m_{D^{*}}^{2}+m_{D}^{2}}{T^{2}}\right)$
$\displaystyle=\cos\theta\frac{3m_{c}^{2}}{64\sqrt{2}\pi^{4}}\int_{m_{c}^{2}}^{s_{D^{*}}^{0}}ds\int_{m_{c}^{2}}^{s_{D}^{0}}du\,\left(1-\frac{m_{c}^{2}}{s}\right)^{2}\left(1-\frac{m_{c}^{2}}{u}\right)^{2}\exp\left(-\frac{s+u}{T^{2}}\right)$
$\displaystyle-\cos\theta\frac{m_{c}\langle\bar{q}q\rangle}{8\sqrt{2}\pi^{2}}\int_{m_{c}^{2}}^{s_{D^{*}}^{0}}ds\,\left(1-\frac{m_{c}^{2}}{s}\right)^{2}\exp\left(-\frac{s+m_{c}^{2}}{T^{2}}\right)$
$\displaystyle-\cos\theta\frac{m_{c}\langle\bar{q}q\rangle}{8\sqrt{2}\pi^{2}}\int_{m_{c}^{2}}^{s_{D}^{0}}du\,\left(1-\frac{m_{c}^{2}}{u}\right)^{2}\exp\left(-\frac{u+m_{c}^{2}}{T^{2}}\right)$
$\displaystyle+\cos\theta\frac{m_{c}\langle\bar{q}g_{s}\sigma
Gq\rangle}{32\sqrt{2}\pi^{2}T^{2}}\left(1+\frac{m_{c}^{2}}{T^{2}}\right)\int_{m_{c}^{2}}^{s_{D^{*}}^{0}}ds\,\left(1-\frac{m_{c}^{2}}{s}\right)^{2}\exp\left(-\frac{s+m_{c}^{2}}{T^{2}}\right)$
$\displaystyle+\cos\theta\frac{m_{c}\langle\bar{q}g_{s}\sigma
Gq\rangle}{96\sqrt{2}\pi^{2}T^{2}}\left(1+\frac{3m_{c}^{2}}{T^{2}}\right)\int_{m_{c}^{2}}^{s_{D}^{0}}du\,\left(1-\frac{m_{c}^{2}}{u}\right)^{2}\exp\left(-\frac{u+m_{c}^{2}}{T^{2}}\right)$
$\displaystyle-\cos\theta\frac{m_{c}\langle\bar{q}g_{s}\sigma
Gq\rangle}{32\sqrt{2}\pi^{2}}\int_{m_{c}^{2}}^{s_{D^{*}}^{0}}ds\,\frac{1}{s}\left(1-\frac{m_{c}^{2}}{s}\right)\exp\left(-\frac{s+m_{c}^{2}}{T^{2}}\right)$
$\displaystyle-\cos\theta\frac{m_{c}^{3}\langle\bar{q}g_{s}\sigma
Gq\rangle}{96\sqrt{2}\pi^{2}}\int_{m_{c}^{2}}^{s_{D^{*}}^{0}}ds\,\frac{1}{s^{2}}\exp\left(-\frac{s+m_{c}^{2}}{T^{2}}\right)$
$\displaystyle-\cos\theta\frac{m_{c}\langle\bar{q}g_{s}\sigma
Gq\rangle}{32\sqrt{2}\pi^{2}}\int_{m_{c}^{2}}^{s_{D}^{0}}du\,\frac{1}{u}\left(1-\frac{m_{c}^{2}}{u}\right)^{2}\exp\left(-\frac{u+m_{c}^{2}}{T^{2}}\right)$
$\displaystyle+\cos\theta\frac{m_{c}^{3}\langle\bar{q}g_{s}\sigma
Gq\rangle}{96\sqrt{2}\pi^{2}}\int_{m_{c}^{2}}^{s_{D}^{0}}du\,\frac{1}{u^{2}}\exp\left(-\frac{u+m_{c}^{2}}{T^{2}}\right)\,,$
(24)

where
$\lambda_{XJ/\psi\rho}=\lambda_{X}\,f_{J/\psi}m_{J/\psi}\,f_{\rho}m_{\rho}$,
$\lambda_{XJ/\psi\omega}=\lambda_{X}\,f_{J/\psi}m_{J/\psi}\,f_{\omega}m_{\omega}$,
$\lambda_{X\chi\pi}=\lambda_{X}f_{\chi_{c1}}m_{\chi_{c1}}\mu_{\pi}$, and
$\lambda_{XD^{*}D}=\lambda_{X}f_{D^{*}}m_{D^{*}}\mu_{D}$. In Eq.(2), there
appears divergence due to the endpoint $s=4m_{c}^{2}$, we can avoid the
endpoint divergence with the simple replacement
$\frac{1}{\sqrt{s-4m_{c}^{2}}}\to\frac{1}{\sqrt{s-4m_{c}^{2}+4m_{s}^{2}}}$ by
adding a small squared $s$-quark mass $4m_{s}^{2}$ [31, 32, 33, 34, 35, 36].

In calculations, we factorize out the mixing angle $\theta$ in Eqs.(2)-(2) so
as to facilitate determining the mixing effects, and redefine the hadronic
coupling constants $G$ and free parameters $C$,

$\displaystyle G^{\prime}_{XJ/\psi\rho}$ $\displaystyle=$ $\displaystyle
G_{XJ/\psi\rho}\frac{\cos\theta-\sin\theta}{\sqrt{2}}\,,$ $\displaystyle
G^{\prime}_{XJ/\psi\omega}$ $\displaystyle=$ $\displaystyle
G_{XJ/\psi\omega}\frac{\cos\theta+\sin\theta}{\sqrt{2}}\,,$ $\displaystyle
G^{\prime}_{X\chi\pi}$ $\displaystyle=$ $\displaystyle
G_{X\chi\pi}\frac{\cos\theta-\sin\theta}{\sqrt{2}}\,,$ $\displaystyle
G^{\prime}_{XD^{*}D}$ $\displaystyle=$ $\displaystyle
G_{XD^{*}D}\cos\theta\,,$ (25) $\displaystyle C^{\prime}_{\rho}$
$\displaystyle=$ $\displaystyle
C_{\rho}\frac{\cos\theta-\sin\theta}{\sqrt{2}}\,,$ $\displaystyle
C^{\prime}_{\omega}$ $\displaystyle=$ $\displaystyle
C_{\omega}\frac{\cos\theta+\sin\theta}{\sqrt{2}}\,,$ $\displaystyle
C^{\prime}_{\pi}$ $\displaystyle=$ $\displaystyle
C_{\pi}\frac{\cos\theta-\sin\theta}{\sqrt{2}}\,,$ $\displaystyle
C^{\prime}_{D}$ $\displaystyle=$ $\displaystyle C_{D}\cos\theta\,,$ (26)

then it is easy to study the dependence on the mixing angle $\theta$.

##  3 Numerical results and discussions

We take the conventional vacuum condensates, $\langle\bar{q}q\rangle=-(0.24\pm
0.01\,\rm{GeV})^{3}$, $\langle\bar{q}g_{s}\sigma
Gq\rangle=m_{0}^{2}\langle\bar{q}q\rangle$, $m_{0}^{2}=(0.8\pm
0.1)\,\rm{GeV}^{2}$ at the energy scale $\mu=1\,\rm{GeV}$ [38, 39, 40], and
take the $\overline{MS}$ mass $m_{c}(m_{c})=(1.275\pm 0.025)\,\rm{GeV}$ from
the Particle Data Group [28]. We set $m_{u}=m_{d}=0$ and take account of the
energy-scale dependence from re-normalization group equation,

$\displaystyle\langle\bar{q}q\rangle(\mu)$ $\displaystyle=$
$\displaystyle\langle\bar{q}q\rangle({\rm 1GeV})\left[\frac{\alpha_{s}({\rm
1GeV})}{\alpha_{s}(\mu)}\right]^{\frac{12}{33-2n_{f}}}\,,$
$\displaystyle\langle\bar{q}g_{s}\sigma Gq\rangle(\mu)$ $\displaystyle=$
$\displaystyle\langle\bar{q}g_{s}\sigma Gq\rangle({\rm
1GeV})\left[\frac{\alpha_{s}({\rm
1GeV})}{\alpha_{s}(\mu)}\right]^{\frac{2}{33-2n_{f}}}\,,$ $\displaystyle
m_{c}(\mu)$ $\displaystyle=$ $\displaystyle
m_{c}(m_{c})\left[\frac{\alpha_{s}(\mu)}{\alpha_{s}(m_{c})}\right]^{\frac{12}{33-2n_{f}}}\,,$
$\displaystyle\alpha_{s}(\mu)$ $\displaystyle=$
$\displaystyle\frac{1}{b_{0}t}\left[1-\frac{b_{1}}{b_{0}^{2}}\frac{\log
t}{t}+\frac{b_{1}^{2}(\log^{2}{t}-\log{t}-1)+b_{0}b_{2}}{b_{0}^{4}t^{2}}\right]\,,$
(27)

where $t=\log\frac{\mu^{2}}{\Lambda_{QCD}^{2}}$,
$b_{0}=\frac{33-2n_{f}}{12\pi}$, $b_{1}=\frac{153-19n_{f}}{24\pi^{2}}$,
$b_{2}=\frac{2857-\frac{5033}{9}n_{f}+\frac{325}{27}n_{f}^{2}}{128\pi^{3}}$,
$\Lambda_{QCD}=210\,\rm{MeV}$, $292\,\rm{MeV}$ and $332\,\rm{MeV}$ for the
flavors $n_{f}=5$, $4$ and $3$, respectively [28, 41], and we choose
$n_{f}=4$, and evolve all the input parameters to the energy scale
$\mu=1\,\rm{GeV}$.

At the hadron side, we take $m_{\pi^{\pm}}=0.13957\,\rm{GeV}$,
$m_{\pi^{0}}=0.13498\,\rm{GeV}$, $m_{J/\psi}=3.0969\,\rm{GeV}$,
$m_{\chi_{c1}}=3.51067\,\rm{GeV}$, $m_{\rho}=0.77526\,\rm{GeV}$,
$m_{\omega}=0.78266\,\rm{GeV}$, $f_{\pi}=0.130\,\rm{GeV}$ from the Particle
Data Group [28], $m_{D^{*}}=2.01\,\rm{GeV}$, $m_{D}=1.87\,\rm{GeV}$,
$f_{D^{*}}=263\,\rm{MeV}$, $f_{D}=208\,\rm{MeV}$,
$s^{0}_{D^{*}}=6.4\,\rm{GeV}^{2}$, $s^{0}_{D}=6.2\,\rm{GeV}^{2}$ [42],
$f_{J/\psi}=0.418\,\rm{GeV}$ [43], $f_{\chi_{c1}}=0.338\,\rm{GeV}$ [44],
$f_{\rho}=0.215\,\rm{GeV}$, $f_{\omega}=f_{\rho}$,
$\sqrt{s^{0}_{\rho}}=1.2\,\rm{GeV}$, $s^{0}_{\omega}=s^{0}_{\rho}$ [45],
$m_{X}=3.91\,\rm{GeV}$, $\lambda_{X}=2.10\times 10^{-2}\,\rm{GeV}^{5}$ [15]
from the QCD sum rules, and
$f_{\pi}m^{2}_{\pi}/(m_{u}+m_{d})=-2\langle\bar{q}q\rangle/f_{\pi}$ from the
Gell-Mann-Oakes-Renner relation.

In calculations, we fit the free parameters as
$C_{\rho}=0.000250(T^{2}-1.5\,\rm{GeV}^{2})\,\rm{GeV}^{3}$,
$C_{\omega}=0.000245(T^{2}-1.5\,\rm{GeV}^{2})\,\rm{GeV}^{3}$, $C_{\pi}=0$ and
$C_{D}=0.0000725(T^{2}-2.1\,\rm{GeV}^{2})\,\rm{GeV}^{4}$ to obtain the Borel
windows $T^{2}_{\rho}=(2.3-3.3)\,\rm{GeV}^{2}$,
$T^{2}_{\omega}=(2.3-3.3)\,\rm{GeV}^{2}$,
$T^{2}_{\pi}=(3.6-4.6)\,\rm{GeV}^{2}$ and $T^{2}_{D}=(4.0-5.0)\,\rm{GeV}^{2}$,
where the subscripts $\rho$, $\omega$, $\pi$ and $D$ denote the corresponding
channels. We obtain uniform enough flat platforms
$T^{2}_{max}-T^{2}_{min}=1\,\rm{GeV}^{2}$, where the max and min denote the
maximum and minimum, respectively. In Fig.1, we plot the hadronic coupling
constants $G_{XJ/\psi\rho}$, $G_{XJ/\psi\omega}$, $G_{X\chi\pi}$ and
$G_{XD^{*}D}$ with variations of the Borel parameters at large intervals. In
the Borel windows, there appear very flat platforms indeed, it is reliable to
extract the hadron coupling constants.

Now, we estimate the uncertainties in the following ways. For example, the
uncertainties of an input parameter $\xi$, $\xi=\bar{\xi}+\delta\xi$, result
in the uncertainties
$\lambda_{X}f_{J/\psi}f_{\rho}G_{XJ/\psi\rho}=\bar{\lambda}_{X}\bar{f}_{J/\psi}\bar{f}_{\rho}\bar{G}_{XJ/\psi\rho}+\delta\,\lambda_{X}f_{J/\psi}f_{\rho}G_{XJ/\psi\rho}$,
$C_{\rho}=\bar{C}_{\rho}+\delta C_{\rho}$,

$\displaystyle\delta\,\lambda_{X}f_{J/\psi}f_{\rho}G_{XJ/\psi\rho}$
$\displaystyle=$
$\displaystyle\bar{\lambda}_{X}\bar{f}_{J/\psi}\bar{f}_{\rho}\bar{G}_{XJ/\psi\rho}\left(\frac{\delta
f_{J/\psi}}{\bar{f}_{J/\psi}}+\frac{\delta
f_{\rho}}{\bar{f}_{\rho}}+\frac{\delta\lambda_{X}}{\bar{\lambda}_{X}}+\frac{\delta
G_{XJ/\psi\rho}}{\bar{G}_{XJ/\psi\rho}}\right)\,,$ (28)

we can set $\delta C_{\rho}=0$ and $\frac{\delta
f_{J/\psi}}{\bar{f}_{J/\psi}}=\frac{\delta
f_{\rho}}{\bar{f}_{\rho}}=\frac{\delta\lambda_{X}}{\bar{\lambda}_{X}}=\frac{\delta
G_{XJ/\psi\rho}}{\bar{G}_{XJ/\psi\rho}}$ approximately.

Finally, we obtain the values of the hadronic coupling constants,

$\displaystyle G_{XJ/\psi\rho}$ $\displaystyle=$ $\displaystyle
1.88^{+0.11}_{-0.10}\,\rm{GeV}^{-2}\,,$ $\displaystyle G_{XJ/\psi\omega}$
$\displaystyle=$ $\displaystyle 1.88^{+0.11}_{-0.10}\,\rm{GeV}^{-2}\,,$
$\displaystyle|G_{X\chi\pi}|$ $\displaystyle=$ $\displaystyle 0.11\,,$
$\displaystyle G_{XD^{*}D}$ $\displaystyle=$ $\displaystyle
1.875^{+0.064}_{-0.064}\,\rm{GeV}^{-2}\,,$ (29)

where we only present the central value of the $G_{X\chi\pi}$ due to the tiny
partial decay width of the $X(3872)\to\chi_{c1}\pi^{0}$.

Figure 1: The central values of the hadronic coupling constants with
variations of the Borel parameters $T^{2}$, where the (I), (II), (III) and
(IV) denote the $G_{XJ/\psi\rho}$, $G_{XJ/\psi\omega}$, $G_{X\chi\pi}$ and
$G_{XD^{*}D}$, respectively.

Now we take the hadron masses $m_{X}=3.87165\,\rm{GeV}$,
$m_{D^{*0}}=2.00685\,\rm{GeV}$, $m_{D^{0}}=1.86484\,\rm{GeV}$ and
$m_{J/\psi}=3.09690\,\rm{GeV}$ from the Particle Data Group to calculate the
partial decay widths [28]. As the $X(3872)$ lies near the thresholds of the
final states $J/\psi\rho$, $J/\psi\omega$ and $D^{*}\bar{D}$, we should take
account of the finite width effects of the $\rho$, $\omega$ and $D^{*}$
mesons, due to the decay cascades,

$\displaystyle X(3872)$ $\displaystyle\to$ $\displaystyle J/\psi\rho^{0}\to
J/\psi\pi^{+}\pi^{-}\,,$ $\displaystyle X(3872)$ $\displaystyle\to$
$\displaystyle J/\psi\omega\to J/\psi\pi^{+}\pi^{-}\pi^{0}\,,$ $\displaystyle
X(3872)$ $\displaystyle\to$ $\displaystyle D^{*0}\bar{D}^{0}\to
D^{0}\bar{D}^{0}\pi^{0}\,.$ (30)

Then we obtain the partial decay widths via trial and error, as there is an
additional parameter $\theta$, $\theta=0.12\pi=21.6^{\circ}$. The partial
widths are listed in the following,

$\displaystyle\Gamma(X\to J/\psi\pi\pi)$ $\displaystyle=$
$\displaystyle\frac{1}{24\pi^{2}m_{X}^{2}}\int_{\Delta_{2\pi}^{2}}^{(m_{X}-m_{J/\psi})^{2}}ds\,|T_{\rho}|^{2}\frac{m_{\rho}\Gamma_{\rho}\,p(m_{X},m_{J/\psi},\sqrt{s})}{(s-m_{\rho}^{2})^{2}+m_{\rho}^{2}\Gamma_{\rho}^{2}}\
,$ (31) $\displaystyle=$ $\displaystyle 0.132^{+0.016}_{-0.013}\,\rm{MeV}\,,$

$\displaystyle\Gamma(X\to J/\psi\pi\pi\pi)$ $\displaystyle=$
$\displaystyle\frac{1}{24\pi^{2}m_{X}^{2}}\int_{\Delta_{3\pi}^{2}}^{(m_{X}-m_{J/\psi})^{2}}ds\,|T_{\omega}|^{2}\frac{m_{\rho}\Gamma_{\rho}\,p(m_{X},m_{J/\psi},\sqrt{s})\,{\rm
Br}}{(s-m_{\omega}^{2})^{2}+m_{\omega}^{2}\Gamma_{\omega}^{2}}\ ,$ (32)
$\displaystyle=$ $\displaystyle 0.129^{+0.016}_{-0.013}\,\rm{MeV}\,,$

$\displaystyle\Gamma(X\to\chi_{c1}\pi)$ $\displaystyle=$
$\displaystyle\frac{|T_{\pi}|^{2}\,p(m_{X},m_{\chi_{c1}},m_{\pi})}{24\pi
m_{X}^{2}}\ ,$ (33) $\displaystyle=$ $\displaystyle 0.0016\,\rm{MeV}\,,$

$\displaystyle\Gamma(X\to D^{0}\bar{D}^{0}\pi^{0})$ $\displaystyle=$
$\displaystyle\frac{1}{24\pi^{2}m_{X}^{2}}\int_{(m_{D}+m_{\pi})^{2}}^{(m_{X}-m_{D})^{2}}ds\,|T_{D}|^{2}\frac{m_{D^{*}}\Gamma_{D^{*}}\,p(m_{X},m_{D},\sqrt{s})}{(s-m_{D^{*}}^{2})^{2}+m_{D^{*}}^{2}\Gamma_{D^{*}}^{2}}\
,$ (34) $\displaystyle=$ $\displaystyle
2.262^{+0.157}_{-0.152}\,\rm{MeV}\,\,\,{\rm
for}\,\,\,\Gamma_{D^{*0}}=2.0\,\rm{MeV}\,,$ $\displaystyle=$ $\displaystyle
1.795^{+0.124}_{-0.121}\,\rm{MeV}\,\,\,{\rm
for}\,\,\,\Gamma_{D^{*0}}=1.0\,\rm{MeV}\,,$ $\displaystyle=$ $\displaystyle
1.326^{+0.092}_{-0.089}\,\rm{MeV}\,\,\,{\rm
for}\,\,\,\Gamma_{D^{*0}}=0.5\,\rm{MeV}\,,$ $\displaystyle=$ $\displaystyle
0.485^{+0.034}_{-0.032}\,\rm{MeV}\,\,\,{\rm
for}\,\,\,\Gamma_{D^{*0}}=0.1\,\rm{MeV}\,,$

where

$\displaystyle|T_{\rho}|^{2}$ $\displaystyle=$ $\displaystyle
G_{XJ/\psi\rho}^{\prime
2}\frac{\left(m_{X}^{2}-m_{J/\psi}^{2}-s\right)^{2}}{4}\left[\frac{\left(m_{X}^{2}-s\right)^{2}}{2m_{J/\psi}^{2}}+\frac{\left(m_{X}^{2}-m_{J/\psi}^{2}\right)^{2}}{2s}+4m_{X}^{2}-\frac{m_{J/\psi}^{2}+s}{2}\right]\,,$

$\displaystyle|T_{\omega}|^{2}$ $\displaystyle=$ $\displaystyle
G_{XJ/\psi\omega}^{\prime
2}\frac{\left(m_{X}^{2}-m_{J/\psi}^{2}-s\right)^{2}}{4}\left[\frac{\left(m_{X}^{2}-s\right)^{2}}{2m_{J/\psi}^{2}}+\frac{\left(m_{X}^{2}-m_{J/\psi}^{2}\right)^{2}}{2s}+4m_{X}^{2}-\frac{m_{J/\psi}^{2}+s}{2}\right]\,,$

$\displaystyle|T_{\pi}|^{2}$ $\displaystyle=$ $\displaystyle
G_{X\chi\pi}^{\prime
2}\frac{\lambda(m_{X}^{2},m_{\chi_{c1}}^{2},m_{\pi}^{2})}{2}\,,$

$\displaystyle|T_{D}|^{2}$ $\displaystyle=$ $\displaystyle G_{XD^{*}D}^{\prime
2}\frac{\left(m_{X}^{2}-m_{D}^{2}-s\right)^{2}}{4}\left[3+\frac{\lambda(m_{X}^{2},s,m_{D}^{2})}{4m_{X}^{2}s}\right]\,,$

$\displaystyle G^{\prime}_{XJ/\psi\rho}$ $\displaystyle\to$
$\displaystyle\widetilde{G}^{\prime}_{XJ/\psi\rho}=G^{\prime}_{XJ/\psi\rho}\exp\left(-\frac{m_{\rho}^{2}-s}{s-\Delta_{2\pi}^{2}}\frac{m_{\rho}^{4}}{\Delta_{2\pi}^{4}}\right)\,,$
$\displaystyle G^{\prime}_{XJ/\psi\omega}$ $\displaystyle\to$
$\displaystyle\widetilde{G}^{\prime}_{XJ/\psi\omega}=G^{\prime}_{XJ/\psi\omega}\exp\left(-\frac{m_{\omega}^{2}-s}{s-\Delta_{3\pi}^{2}}\frac{m_{\omega}^{6}}{\Delta_{3\pi}^{6}}\right)\,,$
(35)

$\Delta_{2\pi}=m_{\pi^{+}}+m_{\pi^{-}}$,
$\Delta_{3\pi}=m_{\pi^{+}}+m_{\pi^{-}}+m_{\pi^{0}}$, ${\rm
Br(\omega\to\pi\pi\pi)}=0.892$ [28],
$\lambda(a,b,c)=a^{2}+b^{2}+c^{2}-2ab-2ac-2bc$,
$p(A,B,C)=\frac{\sqrt{\left[A^{2}-(B+C)^{2}\right]\left[A^{2}-(B-C)^{2}\right]}}{2A}$.
The hadronic coupling constants from the QCD sum rules in Eqs.(2)-(2) are
physical quantities under zero width approximation. The physical widths from
the Particle Data Group are $\Gamma_{\rho}=147.4\,\rm{MeV}$,
$\Gamma_{\omega}=8.68\,\rm{MeV}$, $\Gamma_{\chi_{c1}}=0.88\,\rm{MeV}$,
$\Gamma_{D^{*0}}<2.1\,\rm{MeV}$, respectively, we introduce exponential form-
factors
$\exp\left(-\frac{m_{\rho}^{2}-s}{s-\Delta_{2\pi}^{2}}\frac{m_{\rho}^{4}}{\Delta_{2\pi}^{4}}\right)$
and
$\exp\left(-\frac{m_{\omega}^{2}-s}{s-\Delta_{3\pi}^{2}}\frac{m_{\omega}^{6}}{\Delta_{3\pi}^{6}}\right)$
to parameterize the ”off-shell” effects due to the $J/\psi\rho$ and
$J/\psi\omega$ thresholds, as the $X(3872)$ lies near the $J/\psi\rho$ and
$J/\psi\omega$ thresholds. At the mass-shell $s=m_{\rho}^{2}$ and
$m_{\omega}^{2}$, they reduce to 1 to match with the zero width approximation
in the QCD sum rules. At the thresholds, $s=\Delta_{2\pi}^{2}$ and
$\Delta_{3\pi}^{2}$, the available phase-spaces are very small, the decays
$\rho\to\pi\pi$ and $\omega\to\pi\pi\pi$ only take place through the lower
tails, which can be taken as some intermediate sates with the same quantum
numbers as the $\rho$ and $\omega$ except for the masses, and are greatly
suppressed. On the other hand, the ”off-shell” effects on the hadronic
coupling constants are considerable, we should introduce some form-factors to
parameterize them.

The width $\Gamma_{D^{*\pm}}=0.0834\pm 0.0018\,\rm{MeV}$ from the Particle
Data Group [28], if we take the approximation
$\Gamma_{D^{*0}}\approx\Gamma_{D^{*\pm}}\approx 0.1\,\rm{MeV}$, then
$\Gamma(X\to
D^{*0}\bar{D}^{0})$+$\Gamma(X\to\bar{D}^{*0}D^{0})=0.970^{+0.068}_{-0.064}\,\rm{MeV}$
from Eq.(34), which is in excellent agreement with the branching fraction
$(52.4^{+25.3}_{-14.3})\%$ from the combined data analysis [29] and the total
width $\Gamma_{X}=1.19\pm 0.21\,\rm{MeV}$ from the Particle Data Group [28].
On the other hand, if we take the masses of the $X(3872)$, $D^{\pm}$, $D^{0}$,
$\pi^{\pm}$ and $\pi^{0}$ from the Particle Data Group [28], the decays
$X(3872)\to D^{*+}\bar{D}^{-}\to\bar{D}^{+}\bar{D}^{-}\pi^{0}$ and $X(3872)\to
D^{*+}\bar{D}^{-}\to\bar{D}^{0}\bar{D}^{-}\pi^{+}$ cannot take place due to
the negative phase-space.

The partial widths $\Gamma(X\to
J/\psi\pi\pi)=0.132^{+0.016}_{-0.013}\,\rm{MeV}$ and $\Gamma(X\to
J/\psi\pi\pi\pi)=0.129^{+0.016}_{-0.013}\,\rm{MeV}$ from Eq.(31) and Eq.(32)
are in excellent agreement with the branching fractions ${\rm Br}(X\to
J/\psi\pi\pi)=(4.1^{+1.9}_{-1.1})\%$ and ${\rm Br}(X\to
J/\psi\omega)=(4.4^{+2.3}_{-1.3})\%$ from the combined data analysis [29], so
the mixing angle $\theta=21.6^{\circ}$, which is compatible with the values
$\theta=20.0^{\circ}$ [10] and $\theta=23.5^{\circ}$ [13]. The significant
difference is that we take account of all the Feynman diagrams and take
rigorous quark-hadron duality, while in Ref.[13], only the connected diagrams
are taken into account to obtain small partial decay widths.

The partial decay width $\Gamma(X\to\chi_{c1}\pi)=0.0016\,\rm{MeV}$ from
Eq.(33) is much smaller than the branching fraction $(3.6^{+2.2}_{-1.6})\%$
from the combined data analysis [29] or the branching fraction $(3.4\pm
1.6)\%$ from the Particle Data Group [28], more precise measurement is still
needed.

All in all, in this work, we reproduce the small width of the $X(3872)$ via
the QCD sum rules for the first time.

##  4 Conclusion

In this work, we take the $X(3872)$ as the hidden-charm tetraquark state with
both isospin $I=0$ and $1$ components, then investigate the hadronic coupling
constants $G^{\prime}_{XJ/\psi\rho}$, $G^{\prime}_{XJ/\psi\omega}$,
$G^{\prime}_{X\chi\pi}$ and $G^{\prime}_{XD^{*}D}$ with the QCD sum rules in
details. We select the optimal tensor structures and take account of all the
Feynman diagrams, then acquire four QCD sum rules based on the rigorous quark-
hadron duality. After careful calculations, we obtain the hadronic coupling
constants, then determine the mixing angle via trial and error, and obtain the
partial decay widths for the $X(3872)\to J/\psi\pi^{+}\pi^{-}$,
$J/\psi\omega$, $\chi_{c1}\pi^{0}$, $D^{*0}\bar{D}^{0}$ and
$D^{0}\bar{D}^{0}\pi^{0}$. The total width is about $1\,\rm{MeV}$, which is in
excellent agreement with the experiment data $\Gamma_{X}=1.19\pm
0.21\,\rm{MeV}$ from the Particle Data Group, it is the first time to
reproduce the small width of the $X(3872)$ via the QCD sum rules. The present
calculations support assigning the $X(3872)$ as the mixed hidden-charm
tetraquark state with the quantum numbers $J^{PC}=1^{++}$.

## Acknowledgements

This work is supported by National Natural Science Foundation, Grant Number
12175068.

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