# Mixing of X and Y states from QCD Sum Rules analysis

Ze-Sheng Chen1, Zhuo-Ran Huang2, Hong-Ying Jin3, T.G. Steele4, and Zhu-Feng
Zhang5 Institute of Modern Physics, Department of Physics, Zhejiang
University, Hangzhou, 310027,China1,3
Institute of High Energy Physics, Chinese Academy of Sciences, Beijing,
100049, China2
Department of Physics and Engineering Physics, University of Saskatchewan,
Saskatoon, Saskatchewan, S7N 5E2, Canada4
Department of Physics, Ningbo University, Ningbo, 315211, China5

###### Abstract

We study $\bar{Q}Q\bar{q}q$ and $\bar{Q}qQ\bar{q}$ states as mixed states in
QCD sum rules. By calculating the two-point correlation functions of pure
states of their corresponding currents, we review the mass and coupling
constant predictions of $J^{PC}=1^{++}$, $1^{--}$, $1^{-+}$ states. By
calculating the two-point mixed correlation functions of $\bar{Q}Q\bar{q}q$
and $\bar{Q}qQ\bar{q}$ currents, and we estimate the mass and coupling
constants of the corresponding “physical state” that couples to both
$\bar{Q}Q\bar{q}q$ and $\bar{Q}qQ\bar{q}$ currents. Our results suggest that
$1^{++}$ states are more likely mixing from $\bar{Q}Q\bar{q}q$ and
$\bar{Q}qQ\bar{q}$ components, while for $1^{--}$ and $1^{-+}$ states, there
is less mixing between $\bar{Q}Q\bar{q}q$ and $\bar{Q}qQ\bar{q}$. Our results
suggest the $Y$ series of states have more complicated components.

††preprint: APS/123-QED

##  I Introduction

In 2003, Belle observed a new state, known as the $X(3872)$, which definitely
contained a charm-anticharm pair and cannot be explained by ordinary quark-
antiquark model [1]. Since then, more new hadrons containing heavy quarks have
been found and studied in numerous experiments [2]. These hadrons are known as
XYZ states, containing a heavy quark-antiquark pair and at least a light
quark-antiquark pair, are naturally exotics [3]. Many structures have been
proposed to describe XYZ states including molecular, tetraquark and hybrid
components [4, 5, 6]. Like the studies of other mesons with exotic quantum
numbers, the convincing explanations of observed XYZ states remains an open
question in phenomenological particle physics. Recently, a study of $X(3872)$
by LHCb argued that the compact component should be required in X states[7],
and this result is more likely to support the tetraquark model to XYZ states,
but not exclude the molecular model to all exotic states. In this paper we
focus on states in two simple $\bar{Q}qQ\bar{q}$ and $\bar{Q}Q\bar{q}q$
combinations to study XYZ states ($\bar{Q}$ represents a heavy quark $c$ or
$b$ quark while $q$ represents a light quark $u,d$ or $s$ quark). These two
forms have been extensively studied in previous papers [8, 9]. However,
$\bar{Q}qQ\bar{q}$ and $\bar{Q}Q\bar{q}q$ states are difficult to distinguish
straightforwardly from decay modes of XYZ states because XYZ states are
usually observed to have both $\bar{Q}Q+\bar{q}q$ like decay modes and
$\bar{Q}q+Q\bar{q}$ like decay modes. Many scenarios were studied to
distinguish $\bar{Q}qQ\bar{q}$ and $\bar{Q}Q\bar{q}q$ states qualitatively
[10, 11, 12, 13, 14]. Furthermore, the physical states are usually mixtures of
different structures, which makes the problem more complicated. In a previous
study we have developed a method to estimate mixing strength of different
currents from a QCD sum rule (QCDSR) approach [15, 16, 17, 18], and we use the
same technique to study the mixing of $\bar{Q}qQ\bar{q}$ and
$\bar{Q}Q\bar{q}q$ XY states.

In this paper, we study three kinds of vector states with different quantum
numbers $J^{PC}=1^{++}$, $1^{--}$, $1^{-+}$. These states have long been
considered to be $\bar{Q}qQ\bar{q}$ or $\bar{Q}Q\bar{q}q$ molecular states in
different studies [19, 20, 21, 22, 23, 24, 25]. However, since many of them
have abundant decay modes, mixing scenarios should be taken into account.
Besides, it is generally believed that there is a large background of two free
mesons spectrum in the two points correlation function of four-quark currents.
To avoid such large uncertainty, we especially estimate the mixing strength of
$\bar{Q}qQ\bar{q}$ and $\bar{Q}Q\bar{q}q$ currents, to investigate the
corresponding physical states. The calculations will show us whether the
physical states prefer to be $\bar{Q}qQ\bar{q}$ or $\bar{Q}Q\bar{q}q$
molecular states, or whether they are strongly mixed states.

As mentioned above, for the 1++ channel, $X(3872)$ has been extensively
studied for a wide variety of structures [26, 27]. In molecular state schemes,
$X(3872)$ has been usually considered as $D^{*}D$ molecular state [12, 13,
14]. However, although the pure $D^{*}D$ molecular state was predicted to have
mass close to $X(3872)$, it had a too large decay width to agree with the
experimental results [28, 29]. On the other hand, $J/\psi\rho$ or
$J/\psi\omega$ state also have similar mass since their sum of masses of two
constituent parts are close to $X(3872)$. Hence the mixing of these two
molecular states is naturally possible. Besides, in recent observation to
$X(3872)$ by LHCb[7], the compact component is required. Hence we will
consider another state $\bar{c}c$ in the mixing, which have been studied in
[28].

For the 1– channel, many 1– states are found in the range 4200-4700 MeV,
permitting abundant possible pure or mixed molecular states. Some 1– states
have very similar mass like $Y(4220)/Y(4260)$ and $Y(4360)/Y(4390)$ [30, 31].
Hence it is interesting and meaningful to investigate the possible mixing of
molecular state, which has not been previously studied.

For 1-+ sector, no confirmed heavy hadrons with 1-+ quantum numbers have been
observed. Some potential candidates include $X(3940),X(4160),X(4350)$ [2]. The
constructions of 1-+ molecular states in $\bar{Q}qQ\bar{q}$ and
$\bar{Q}Q\bar{q}q$ scenarios are possible. As outlined below, we calculate the
mass spectrum of these states and estimate their mixing strength both in $u,d$
and $s$ quark case to help guide searches for these states in 1-+ sector.

Our methodology is introduced in Section II. Then we discuss $1^{++}$ states,
$1^{--}$ states, and $1^{-+}$ states in Sections III, IV, V respectively. We
discuss the importance of non-perturbative terms in calculations of evaluating
mixing strength in Section VI. Finally we give our summary and conclusions in
the last section.

##  II QCDSR approach and mixing strength

In QCDSR, we normally construct a mixing current combining two state
interpretations. The two-point correlation function of the mixing currents can
be written as

$\displaystyle\Pi\left(q^{2}\right)$ $\displaystyle=i\int
d^{4}x\,e^{iqx}\left\langle
0\left|T\left\\{(j_{a}\left(x\right)+cj_{b}(x))(j_{a}^{+}(0)+cj_{b}^{+}(0))\right\\}\right|0\right\rangle$
(1) $\displaystyle=\Pi_{a}(q^{2})+2c\Pi_{ab}(q^{2})+c^{2}\Pi_{b}(q^{2}),$

where $j_{a}$ and $j_{b}$ have the same quantum numbers, $c$ is a real
parameter related to the mixing strength (not the mixing strength itself,
since $c$ may not been normalized), and

$\displaystyle\Pi_{a}\left(q^{2}\right)$ $\displaystyle=i\int
d^{4}x\,e^{iqx}\left\langle
0\left|T(j_{a}(x)j_{a}^{+}(0))\right|0\right\rangle,$ (2)
$\displaystyle\Pi_{b}\left(q^{2}\right)$ $\displaystyle=i\int
d^{4}x\,e^{iqx}\left\langle
0\left|T(j_{b}(x)j_{b}^{+}(0))\right|0\right\rangle,$
$\displaystyle\Pi_{ab}\left(q^{2}\right)$ $\displaystyle=\frac{i}{2}\int
d^{4}x\,e^{iqx}\left\langle
0\left|T(j_{a}(x)j_{b}^{+}(0)+j_{b}(x)j_{a}^{+}(0))\right|0\right\rangle.$

Here we consider the mixed correlator $\Pi_{ab}$ because it provides a signal
of which states couple to both currents . One can insert a complete set of
particle eigenstates between $j_{a}$ and $j_{b}$, and the state with
relatively strong coupling to both these currents will be selected out through
the QCDSR. By estimating the mass, coupling constants and taking into account
experimental results, one can get insight into the constituent composition of
the corresponding states. This method worked well in our previous study on
vector and scalar meson states [15], and has been successfully applied in
other systems [16, 17, 18].

$\Pi_{ab}$ usually can be decomposed into different Lorentz structure

$\displaystyle\Pi_{ab}(q^{2})$
$\displaystyle=\sum_{n}\Pi_{n}(q^{2})A_{n}\textrm{,}$ (3)

where $n=1,2,3...$, $\Pi_{n}(q^{2})$ is the mixing state correlation function
with specific quantum numbers, and $A_{n}$ is corresponding Lorentz structure.
The forms of $A_{n}$ are related to $j_{a}$ and $j_{b}$, and we will define
them in sections below. For simplicity, we assume a specific $\Pi_{H}(q^{2})$
represents a mixed-state correlation function, is one of the possible
$\Pi_{n}(q^{2})$, and that $\Pi_{H}(q^{2})$ obeys dispersion relation [32]

$\displaystyle\Pi_{H}(q^{2})$
$\displaystyle=\int_{s_{\textrm{min}}}^{\infty}ds\frac{\rho_{H}(s)}{s-q^{2}-i\epsilon}+\ldots\textrm{,}$
(4)

where the spectral density $\rho_{H}(s)=\frac{1}{\pi}\textrm{Im}\Pi_{H}(s)$,
$s_{\textrm{min}}$ represents the physical threshold of the corresponding
current and dots on the right hand side represent polynomial subtraction terms
to render $\Pi_{H}(q^{2})$ finite. The spectral density $\rho_{H}(s)$ can be
calculated using the operator product expansion (OPE). In this paper, we
calculate the spectral density $\rho_{H}(s)$ up to dimension-six operators,

$\displaystyle\rho_{H}^{\textrm{(OPE)}}(s)$
$\displaystyle=\rho_{H}^{\textrm{(pert)}}(s)+\rho_{H}^{\left\langle\bar{q}q\right\rangle}(s)+\rho_{H}^{\left\langle
G^{2}\right\rangle}(s)$ (5)
$\displaystyle+\rho_{H}^{\left\langle\bar{q}Gq\right\rangle}(s)+\rho_{H}^{\left\langle\bar{q}q\right\rangle^{2}}(s)+\ldots~{}\textrm{,}$

then

$\Pi_{H}^{\textrm{(OPE)}}(q^{2})=\int_{s_{\textrm{min}}}^{\infty}ds\frac{\rho^{\textrm{(OPE)}}_{H}(s)}{s-q^{2}-i\epsilon}+\ldots~{}.$
(6)

On the phenomenological side, by using the narrow resonance spectral density
model,

$\Pi^{\textrm{(phen)}}_{H}(q^{2})=\frac{\lambda_{a}\lambda_{b}^{*}+\lambda_{b}\lambda_{a}^{*}}{2}\frac{1}{M_{H}^{2}-q^{2}}+\int_{s_{0}}^{\infty}ds\frac{\rho^{\textrm{(cont)}}_{H}(s)}{s-q^{2}-i\epsilon},$
(7)

where $\lambda_{a}$ and $\lambda_{b}$ are the respective couplings of the
ground state to the corresponding currents, $M_{H}$ represents mass of mixed
state which has relatively strong coupling to the corresponding currents,
$\rho_{H}^{\textrm{(cont)}}$ represents continuum contributions to spectral
density, and $s_{0}$ is the continuum threshold. By using the QCDSR continuum
spectral density assumptions

$\displaystyle\rho_{H}^{(\textrm{cont})}(s)=\rho_{H}^{\textrm{(OPE)}}(s)\Theta(s-s_{0}),$
(8)

and equating the OPE side and the phenomenological side of the correlation
function, $\Pi^{\textrm{(phen)}}_{H}(q^{2})=\Pi_{H}^{\textrm{(OPE)}}(q^{2})$,
we obtain the QCDSR master equation

$\int_{s_{\textrm{min}}}^{s_{0}}ds\frac{\rho^{\textrm{(OPE)}}_{H}(s)}{s-q^{2}-i\epsilon}+\ldots=\frac{\lambda_{a}\lambda_{b}^{*}+\lambda_{b}\lambda_{a}^{*}}{2}\frac{1}{M_{H}^{2}-q^{2}}.$
(9)

After applying the Borel transformation operator $\hat{B}$ to both side of the
master equation, the subtraction terms are eliminated and the master equation
can be written as [32, 33]

$\int_{s_{\textrm{min}}}^{s_{0}}ds\,\rho^{\textrm{(OPE)}}_{H}\left(s\right)e^{-s\tau}=\frac{\lambda_{a}\lambda_{b}^{*}+\lambda_{a}^{*}\lambda_{b}}{2}e^{-M^{2}_{H}\tau}\textrm{,}$
(10)

where the Borel parameter $\tau=1/M^{2}$, and $M$ is the Borel mass. The
master equation (10) is the foundation of our analysis. By taking the $\tau$
logarithmic derivative of Eq. (10), we obtain

$M_{H}^{2}=\frac{\int_{s_{\textrm{min}}}^{\infty}ds\
s\rho_{H}^{\textrm{(OPE)}}\left(s\right)e^{-s\tau}}{\int_{s_{\textrm{min}}}^{\infty}ds\
\rho_{H}^{\textrm{(OPE)}}\left(s\right)e^{-s\tau}}.$ (11)

One can set $a=b$ in Eqs. (7), (9), (10), and get the original pure state
QCDSR.

Because of the OPE truncation and the simplified assumption for the
phenomenological spectral density, Eq. (10) is not valid for all values of
$\tau$, thus the determination of the sum rule window, in which the validity
of (10) can be established, is very important. In the literature, different
methods are used in the determination of the $\tau$ sum rule window [34, 35].
In this paper, we follow previous similar studies to restrict resonance and
high dimensions condensate contributions (HDC), i.e., the resonance part obeys
the relation

$\frac{\int_{s_{\textrm{min}}}^{s_{0}}ds\
\rho_{H}^{\textrm{(OPE)}}\left(s\right)e^{-s\tau}}{\int_{s_{\textrm{min}}}^{\infty}ds\
\rho_{H}^{\textrm{(OPE)}}\left(s\right)e^{-s\tau}}>40\%,$ (12)

while HDC (usually $\left\langle\bar{q}q\right\rangle^{2}$ in molecular
systems) obeys the relation

$\frac{|\int_{s_{\textrm{min}}}^{\infty}ds\
\rho_{H}^{\left\langle\bar{q}q\right\rangle^{2}}\left(s\right)e^{-s\tau}|}{|\int_{s_{\textrm{min}}}^{\infty}ds\
\rho_{H}^{\textrm{(OPE)}}\left(s\right)e^{-s\tau}|}<10\%.$ (13)

Furthermore, the value of $s_{0}$ is also very important in QCDSR methods. It
is often assumed that the threshold satisfies $\sqrt{s_{0}}=M_{H}+\Delta_{s}$,
with $\Delta_{s}\approx$ 0.5 GeV especially in molecular state QCDSR
calculations [36, 37].

The approximation $\sqrt{s_{0}}=M_{H}+\Delta_{s}$ can be understood in QCDSR
because the parameter $s_{0}$ separates the ground state and other excited
states’ contributions to spectral density. Hence one can set $s_{0}$ less than
the first excitation threshold in case of involving excited state
contributions in spectral density, and $\Delta_{s}$ represents the approximate
mass difference between the ground and first excited states. We assume that
the first excited state is approximately equal to an excited constituent meson
and another ground state constituent meson, then we can establish $s_{0}$ by
comparing the mass difference between ground constituent meson and the first
excited constituent meson of the corresponding state (like charmonium and $D$
meson family in our case). We have listed some experimental data for the
charmonium and $D$ meson families in Table 1 and Table 2. One can easily find
that the mass difference between the ground state and first excited state are
all around $0.5^{+0.1}_{-0.1}$ GeV and the fluctuations are all acceptable in
QCDSR approach.

Table 1: Charmed meson (c$=\pm 1$) states, where $q$ represents $u,d$ quark. The $\bullet$ symbol indicates particles that have confirmed quantum numbers. PDG name | Possible structure | Ground state | Possible 1st excited state |  $\Delta_{s}/$MeV
---|---|---|---|---
$D$ | $\bar{c}\gamma_{5}q$ | $\bullet D(1865)$ | $D(2550)$ |  $\sim$685
$D_{1}$ | $\bar{c}\gamma_{\mu}\gamma_{5}q$ | $\bullet D_{1}(2420)$ | - | -
$D_{0}^{*}$ | $\bar{c}q$ | $\bullet D_{0}^{*}(2300)$ | $D_{J}^{*}(2600)$ |  $\sim$300
$D^{*}$ | $\bar{c}\gamma_{\mu}q$ | $\bullet D^{*}(2007)$ | $D^{*}(2640)$ |  $\sim$633
Table 2: Charmonium (possibly non-$\bar{c}c$ states). The $\bullet$ symbol indicates particles that have confirmed quantum numbers. Possible structure | Ground state/MeV | Possible 1st excited state/MeV |  $\Delta_{s}/$MeV
---|---|---|---
$\bar{c}\gamma_{5}c$ |  $\bullet\eta_{c}(1S)$/2984 |  $\bullet\eta_{c}(2S)$/3637 |  $\sim$653
$\bar{c}\gamma_{\mu}\gamma_{5}c$ |  $\bullet\chi_{c1}(1P)$/3510 | $\bullet\chi_{c1}(3872)$ |  $\sim$362
$\bar{c}c$ |  $\bullet\chi_{c0}(1P)$/3415 | $\chi_{c0}(3860)$ |  $\sim$445
$\bar{c}\gamma_{\mu}c$ |  $\bullet J/\psi(1S)$/3097 |  $\bullet\psi(2S)$/3686 |  $\sim$589

In order to estimate the mixing strength of the physical state strongly
coupled to both the two different currents, we define

$\displaystyle N\equiv$
$\displaystyle\frac{|\lambda_{a}\lambda_{b}^{*}+\lambda_{a}^{*}\lambda_{b}|}{2|\lambda^{{}^{\prime}}_{a}\lambda^{{}^{\prime}}_{b}|}\textrm{,}$
(14)

where $\lambda^{{}^{\prime}}_{a}$ and $\lambda^{{}^{\prime}}_{b}$ are coupling
constants of the relevant current with a pure state (i.e., the coupling that
emerges in the diagonal correlation functions $\Pi_{a}$, $\Pi_{b}$). Eq. (14)
is analogous to the mixing parameter defined in Ref. [38]. By using
appropriate factors of mass in the definitions of $\lambda^{{}^{\prime}}_{a}$
and $\lambda^{{}^{\prime}}_{b}$, we can therefore compare the magnitude of
coupling constants and estimate the mixing strength self-consistently. The
mixing strength depends on the definition and normalization of mixed state.
For example, in Ref. [39] the definition of the mixed state is

$\left|\textrm{M}\right\rangle=\cos\theta\left|\textrm{A}\right\rangle+\sin\theta\left|\textrm{B}\right\rangle,$
(15)

where $\left|\textrm{M}\right\rangle$ is a mixed state composed of pure states
$\left|\textrm{A}\right\rangle$ and $\left|\textrm{B}\right\rangle$ and
$\theta$ is a mixing angle. In this definition and normalization of the mixed
state, we see that $N\approx\cos\theta\sin\theta$, and
$N\in\left(0,\frac{1}{2}\right)$. Because of the different possible
normalizations and mixed state definitions, we use Eq. (14) as a robust
parameter to quantify mixing effects. Furthermore, because the behavior of $N$
is not linear, we define $\widetilde{N}$ under scenario of Eq.(15),

$\widetilde{N}=\sin^{2}\left(\frac{\arcsin(2N)}{2}\right).$ (16)

The quantity $\widetilde{N}$ gives the approximate proportion of pure part of
the mixed state. By comparing the mixed state mass with two relevant pure
states, suggests that the mixed state is dominated by the part whose pure
state mass prediction is closest to the mixed state mass. Otherwise, different
decay widths can also help us to distinguish the dominant part of the mixed
state.

We use the following numerical values of vacuum condensates consistent with
other QCDSR analyses of XYZ states: $\left\langle\bar{q}q\right\rangle=$
$(-0.23\pm 0.03)^{3}$GeV3, $\left\langle\bar{q}g_{s}\sigma Gq\right\rangle=$
$m_{0}^{2}\left\langle\bar{q}q\right\rangle$, $m_{0}^{2}=$0.8 GeV2,
$\left\langle\alpha_{s}G^{2}\right\rangle=$0.07$\pm 0.02$ GeV4,
$\left\langle\bar{s}s\right\rangle=$ $(0.8\pm
0.2)\left\langle\bar{q}q\right\rangle$ [40, 41]. In addition, the quark masses
$m_{c}=$1.27 GeV, $m_{q}$=$\frac{1}{2}(m_{u}+m_{d})$=0.004 GeV, $m_{s}$= 0.096
GeV, at the energy scale $\mu=$ 2 GeV [2], are used.

##  III Mixed state in 1++ channel

We start from the following three forms of currents:

$\displaystyle j^{X_{A}}_{\mu\nu}(x)=$
$\displaystyle\frac{i}{\sqrt{2}}[\bar{c}(x)\gamma_{\mu}c(x)\bar{q}(x)\gamma_{\nu}q(x)$
(17) $\displaystyle-$
$\displaystyle\bar{c}(x)\gamma_{\nu}c(x)\bar{q}(x)\gamma_{\mu}q(x)],$
$\displaystyle j^{X_{B_{1}}}_{\mu}(x)=$
$\displaystyle\frac{i}{\sqrt{2}}[\bar{c}(x)\gamma_{\mu}q(x)\bar{q}(x)\gamma_{5}c(x)$
$\displaystyle-$
$\displaystyle\bar{q}(x)\gamma_{\mu}c(x)\bar{c}(x)\gamma_{5}q(x)],$
$\displaystyle j^{X_{B_{2}}}_{\mu}(x)=$
$\displaystyle\frac{i}{\sqrt{2}}[\bar{c}(x)\gamma_{\mu}\gamma_{5}q(x)\bar{q}(x)c(x)$
$\displaystyle+$
$\displaystyle\bar{q}(x)\gamma_{\mu}\gamma_{5}c(x)\bar{c}(x)q(x)],$
$\displaystyle j^{X_{C}}_{\mu}(x)=$
$\displaystyle\frac{1}{6\sqrt{2}}\left\langle\bar{q}q\right\rangle\bar{c}(x)\gamma_{\mu}\gamma_{5}c(x),$

where $X$ denotes the 1++ state, the subscript $A$ of $X$ denotes the
$\bar{Q}Q\bar{q}q$ scenario, $B$ denotes the $\bar{Q}qQ\bar{q}$ scenario, $C$
denotes the $\bar{Q}Q$ scenario, and the corresponding mesonic structures of
these currents are listed in Table 3. We note that the former two currents can
be decomposed into two constituent meson currents, and the mass prediction of
each corresponding pure state are usually close to the sum of masses of these
two constituent mesons. The current $j^{X_{A}}_{\mu\nu}$ can be decomposed
into $J/\psi(1S)(3097)$ and $\rho(770)$ currents, $j^{X_{B_{1}}}_{\mu}(x)$ can
be decomposed into $D^{*}(2007)$ and $D(1865)$ currents. The sum of masses of
two constituent mesons are both close to $X(3872)$. Hence we choose these two
currents to study $X(3872)$. The current $j^{X_{B_{2}}}_{\mu}$ has the same
quantum numbers and it cannot be excluded in $1^{++}$ mixing state structures.
Besides, $j^{X_{C}}_{\mu}(x)$ is normalized according to Ref. [28]. Since the
mixing between $j^{X_{C}}_{\mu}(x)$ and $\bar{Q}Q\bar{q}q$ is suppressed (
$\bar{q}q$ in $\bar{Q}Q\bar{q}q$ becomes a bubble and vanishes), we only
consider the mixing between $j^{X_{C}}_{\mu}(x)$ and $j^{X_{B_{1}}}_{\mu}(x)$
or $j^{X_{B_{2}}}_{\mu}(x)$.

To study the pure $\bar{Q}Q\bar{q}q$, $\bar{Q}qQ\bar{q}$ and $\bar{Q}Q$
states, the respective two-point correlation functions can be decomposed into
different Lorentz structures

$\displaystyle\Pi_{\mu\nu\rho\sigma}^{X_{A}}(q^{2})=$
$\displaystyle\Pi_{a}^{X_{A}}(q^{2})\frac{1}{q^{2}}(q^{2}g_{\mu\rho}g_{\nu\sigma}-q^{2}g_{\mu\sigma}g_{\nu\rho}$
(18) $\displaystyle-$ $\displaystyle
q_{\mu}q_{\rho}g_{\nu\sigma}+q_{\mu}q_{\sigma}g_{\nu\rho}-q_{\nu}q_{\sigma}g_{\mu\rho}+q_{\nu}q_{\rho}g_{\mu\sigma})$
$\displaystyle+$
$\displaystyle\Pi_{b}^{X_{A}}(q^{2})\frac{1}{q^{2}}(-q_{\mu}q_{\rho}g_{\nu\sigma}+q_{\mu}q_{\sigma}g_{\nu\rho}$
$\displaystyle-$ $\displaystyle
q_{\nu}q_{\sigma}g_{\mu\rho}+q_{\nu}q_{\rho}g_{\mu\sigma}),$
$\displaystyle\Pi_{\mu\nu}^{X_{B_{k}}/C}(q^{2})=$
$\displaystyle\Pi_{(1)}^{X_{B_{k}}/C}(q^{2})\left(-g_{\mu\nu}+\frac{q_{\mu}q_{\nu}}{q^{2}}\right)$
$\displaystyle+$
$\displaystyle\Pi_{(0)}^{X_{B_{k}}/C}(q^{2})\left(\frac{q_{\mu}q_{\nu}}{q^{2}}\right),$

where $k=1,2$, $\Pi_{a}^{X_{A}}$ and $\Pi_{b}^{X_{A}}$ describes pure
molecular state contribution with respective quantum numbers $1^{++}$ and
$1^{-+}$, $\Pi_{(1)}^{X_{B_{k}}/C}$ and $\Pi_{(0)}^{X_{B_{k}}/C}$ respectively
describes $1^{++}$ and $0^{-+}$ state contributions. In the mixing scenario,
we start from the off-diagonal mixed correlator described in the previous
section, i.e.,

$\displaystyle\Pi^{M_{X_{1}}}_{\mu\nu\sigma}(q^{2})=$
$\displaystyle\frac{i}{2}\int d^{4}x\,e^{iq\cdot x}\langle
0|T(j^{X_{A}}_{\mu\nu}(x)j^{X_{B_{1}}+}_{\sigma}(0)$ (19) $\displaystyle+$
$\displaystyle j^{X_{B_{1}}}_{\sigma}(x)j^{X_{A}+}_{\mu\nu}(0))|0\rangle,$
$\displaystyle\Pi^{M_{X_{2}}}_{\mu\nu\sigma}(q^{2})=$
$\displaystyle\frac{i}{2}\int d^{4}x\,e^{iq\cdot x}\langle
0|T(j^{X_{A}}_{\mu\nu}(x)j^{X_{B_{2}}+}_{\sigma}(0)$ $\displaystyle+$
$\displaystyle j^{X_{B_{2}}}_{\sigma}(x)j^{X_{A}+}_{\mu\nu}(0))|0\rangle,$

where $M_{X_{k}}$, $k=$1,2, are mixed states assumed to result from the
corresponding currents. These mixed correlators have the Lorentz structure

$\displaystyle\Pi_{\mu\nu\sigma}^{M_{X_{k}}}(q^{2})=\Pi^{M_{X_{k}}}(q^{2})(q_{\alpha}\epsilon_{\alpha\mu\nu\sigma}).$
(20)

Besides, when we consider the two-quark states $j^{X_{C}}_{\mu}(x)$, the mixed
correlator and its Lorentz structure are

$\displaystyle\Pi^{M_{C}}_{\mu\nu}(q^{2})=$ $\displaystyle\frac{i}{2}\int
d^{4}x\,e^{iq\cdot x}\langle 0|T(j^{X_{C}}_{\mu}(x)j^{X_{B_{1}}+}_{\nu}(0)$
(21) $\displaystyle+$ $\displaystyle
j^{X_{B_{1}}}_{\nu}(x)j^{X_{C}+}_{\mu}(0))|0\rangle,$
$\displaystyle\Pi_{\mu\nu}^{M_{C}}(q^{2})=$
$\displaystyle\Pi_{(1)}^{M_{C}}(q^{2})\left(-g_{\mu\nu}+\frac{q_{\mu}q_{\nu}}{q^{2}}\right)$
$\displaystyle+$
$\displaystyle\Pi_{(0)}^{M_{C}}(q^{2})\left(\frac{q_{\mu}q_{\nu}}{q^{2}}\right),$

where $\Pi_{(1)}^{C}$ and $\Pi_{(0)}^{C}$ respectively describes $1^{++}$ and
$0^{-+}$ state contributions. Here we just consider the state mixed with
$j^{X_{C}}_{\mu}$ and $j^{X_{B_{1}}}$, which is a candidate to $X(3872)$.

We follow the $40\%-10\%$ sum-rule window and $\Delta_{s}$ methods mentioned
in Section II. After establishing $\tau$ window by the $40\%-10\%$ method at a
specific $s_{0}$, for each state we use Eq. (11) to plot the $\tau$ behavior
of $M_{H}$ for the chosen $s_{0}$ (see Appendix A for details). The mass
prediction $M_{H}$ and $s_{0}$ are then compared with the constraint
$\sqrt{s_{0}}=M_{H}+\Delta_{s}$, then $s_{0}$ is adjusted and the analysis is
repeated until we find the best $(M_{H},s_{0})$ solutions satisfying the
relation $\sqrt{s_{0}}=M_{H}+\Delta_{s}$. The coupling constants are naturally
obtained through the predicted $M_{H}$ and $s_{0}$ according to Eq. (10). The
mass and coupling constant prediction and associated QCDSR parameters are
collected in Table 3. All the parameters are the average values in the
corresponding $\tau$ window.

The uncertainties mainly are from the input parameters . For instance
$\left\langle\alpha_{s}G^{2}\right\rangle=$ $0.07\pm
0.02$GeV4,$\left\langle\bar{s}s\right\rangle=$ $(0.8\pm
0.2)\left\langle\bar{q}q\right\rangle$ and
$\left\langle\bar{q}q\right\rangle=$ $(-0.23\pm 0.03)^{3}$GeV3. The quark
masses and other parameters we have included in calculations have less than 5%
uncertainties due to substantial numerical fittings by other researchers.
There is also an uncertainty about the value of the threshold $s_{0}$.
Analogue to the studies in Refs.[20, 42], the fluctuation of threshold is set
to be $\pm 0.1$GeV($\sqrt{s}_{0}$).

In the pure state calculations, a $\tau$ window of $X_{A}$(3798) state cannot
be determined under $40\%-10\%$ and $\Delta_{s}$ method and we rearrange the
limits of resonance and HDC to $35\%-15\%$ (one may naturally expect that the
pure-state analysis requires such adjustments because of mixing ). The states
$X_{A}$(3798), $X_{B_{1}}$(3857) both have mass predictions close to
$X(3872)$. However, the large mass prediction of the $X_{B_{2}}$(5310) state
is much beyond the $D_{1}$+$D_{0}^{*}$ threshold,and does not match the
observed $1^{++}$ states.

Table 3: Summary of results for $1^{++}$ states. $\lambda=\frac{\lambda_{a}\lambda_{b}^{*}+\lambda_{a}^{*}\lambda_{b}}{2}$ when mixed cases involved, the same below. State | Current structure | Mass/GeV |  $\lambda$/$10^{-4}$ GeV10 |  $\sqrt{s_{0}}$/GeV |  $\tau$ window/GeV-2
---|---|---|---|---|---
$X_{A}$ | $J/\psi\rho$ | $3.798^{+0.09}_{-0.09}$ | $1.49^{+0.51}_{-0.47}$ | 4.4 | 0.30 – 0.31
$X_{B_{1}}$ | $D^{*}\bar{D}$ | $3.857^{+0.06}_{-0.06}$ | $2.24^{+0.65}_{-0.53}$ | 4.4 | 0.31 – 0.39
$X_{B_{2}}$ | $D_{1}\bar{D}_{0}^{*}$ | $5.310^{+0.04}_{-0.04}$ | $69.0^{+15.0}_{-14.0}$ | 5.8 | 0.20 – 0.29
$X_{C}$ | $\chi_{c1}$ | $3.511^{+0.02}_{-0.03}$ | $0.0229^{+0.0017}_{-0.0018}$ | 4.5 | 0.29 – 0.31
$M_{X_{1}}$ |  $J/\psi\rho$ \- $D^{*}\bar{D}$ | $3.987^{+0.06}_{-0.06}$ |  $0.168^{+0.049}_{-0.042}$ GeV-1 | 4.4 | 0.30 – 0.32
$M_{X_{2}}$ |  $J/\psi\rho$ – $D_{1}\bar{D}_{0}^{*}$ | $4.945^{+0.08}_{-0.06}$ |  $0.760^{+0.29}_{-0.20}$ GeV-1 | 5.45 | 0.22 – 0.24
$M_{C}$ |  $\chi_{c1}$ \- $D^{*}\bar{D}$ | $3.818^{+0.03}_{-0.02}$ | $0.0282^{+0.0027}_{-0.0024}$ | 4.5 | 0.28 – 0.30

The mixing strength can then be estimated by computing the value of $N$ via
Eq. (14). Note that the coupling constants of two mixed state correlators have
the form

$\displaystyle\Pi^{M_{X_{k}}}_{\mu\nu\sigma}(q^{2})$
$\displaystyle=\frac{i}{2}\int d^{4}x\,e^{iq\cdot x}\langle
0|T(j^{X_{A}}_{\mu\nu}(x)j^{X_{B_{k}}+}_{\sigma}(0)$ (22)
$\displaystyle+j^{X_{B_{k}}}_{\sigma}(x)j^{X_{A}+}_{\mu\nu}(0))|0\rangle$
$\displaystyle\sim(\frac{\lambda_{X_{A}}}{M_{H}}\epsilon_{\mu\nu\alpha\beta}\varepsilon^{X_{A}}_{\alpha}q_{\beta})(\lambda^{*}_{X_{B_{k}}}\varepsilon^{*X_{B_{k}}}_{\sigma})$
$\displaystyle+(\frac{\lambda_{X_{A}}^{*}}{M_{H}}\epsilon_{\mu\nu\alpha\beta}\varepsilon^{*X_{A}}_{\alpha}q_{\beta})(\lambda_{X_{B_{k}}}\varepsilon^{X_{B_{k}}}_{\sigma})+...$
$\displaystyle=\frac{\lambda_{X_{A}}\lambda^{*}_{X_{B_{k}}}+\lambda^{*}_{X_{A}}\lambda_{X_{B_{k}}}}{M_{H}}q_{\alpha}\epsilon_{\mu\nu\sigma\alpha}+...,$

where $k=1,2$, $\varepsilon_{\alpha/\sigma}^{X_{A}/X_{B_{k}}}$ is polarization
vector, $M_{H}$ represents the ground state mass of $X_{A}$, dots represent
excited contributions to the spectral density and polynomial subtraction
terms. In the definition of the mixing strength Eq. (14), we have omitted the
Lorentz structures of corresponding currents. The dimension of the decay
constant depends on the Lorentz structure we extract in the diagonal
correlator . If the two currents have different Lorentz structure, we need to
compensate the mass dimension of the decay constants , which are obtained from
other works, to make the mixing strength Eq. (14) dimensionless. The normal
method is to make the Lorentz structures massless by multipling a factor
$M^{n}_{H}$ with a suitable $n$. Hence we define $\lambda_{X_{A}}/M_{H}$ as
the new coupling constant of $X_{A}$ state . The mixing strength can be
written as

$\displaystyle N_{M_{X_{1}}}=\frac{0.168\textrm{GeV}^{9}\times
M_{H}(3.798\textrm{GeV})}{\sqrt{1.49}\textrm{GeV}^{5}\times\sqrt{2.24}\textrm{GeV}^{5}}=0.349,$
(23)
$\displaystyle\widetilde{N}_{M_{X_{1}}}=\sin^{2}\left(\frac{\arcsin(0.349\times
2)}{2}\right)=14\%,$ $\displaystyle
N_{M_{X_{2}}}=\frac{0.760\textrm{GeV}^{9}\times
M_{H}(3.798\textrm{GeV})}{\sqrt{1.49}\textrm{GeV}^{5}\times\sqrt{69.0}\textrm{GeV}^{5}}=0.285,$
$\displaystyle\widetilde{N}_{M_{X_{2}}}=\sin^{2}\left(\frac{\arcsin(0.285\times
2)}{2}\right)=9.0\%,$ $\displaystyle
N_{M_{C}}=\frac{0.0282\textrm{GeV}^{10}}{\sqrt{0.0229}\textrm{GeV}^{5}\times\sqrt{2.24}\textrm{GeV}^{5}}=0.125,$
$\displaystyle\widetilde{N}_{M_{C}}=\sin^{2}\left(\frac{\arcsin(0.125\times
2)}{2}\right)=1.6\%,$

The state $M_{X_{1}}$(3987) is a mixture of $X_{A}$(3798) and
$X_{B_{1}}$(3857) which have similar mass predictions close to $X(3872)$, and
unsurprisingly has the same mass prediction. Due to $X(3872)$ observed decays
to $\pi^{+}\pi^{-}J/\psi(1S)$, $\omega J/\psi(1S)$ and $\bar{D}^{*0}D^{0}$,
$M_{X_{1}}$(3987) is a good candidate to describe the $X(3872)$ [2]. We can
estimate proportions of each constituent and decay width of corresponding
decay modes by using the parameter $N_{M_{X_{1}}}$. Experimental results of
$X(3872)$ decay width $\Gamma_{1}$ of $\bar{Q}q+Q\bar{q}$ like decay mode is
$>30\%$, while decay width $\Gamma_{2}$ of $\bar{Q}Q+\bar{q}q$ like decay mode
is $>5\%$.By comparison, the parameter $N_{M_{X_{1}}}$ shows that the
proportions of the $\bar{Q}qQ\bar{q}$ and $\bar{Q}Q\bar{q}q$ parts of
$M_{X_{1}}$(3987) are respectively 86$\%$ and 14$\%$. Considering similar
Lorentz-invariant phase-space of these two kinds of decay modes, we can
roughly equate $\Gamma_{1}/\Gamma_{2}$ to the ratio of these two parts,
$86\%/14\%\sim 6$, which is consistent with experimental results. It should be
noted that our method can not determine definitely which constitute dominates
the mixing state. We tend to the one whose pure mass is closer to the mixing
state.

When we consider the two-quark state $j^{X_{C}}_{\mu}$, the corresponding
mixing angle is $\text{arcsin(0.25)}/2=7^{\circ}$, which is consist with the
result in Ref. [28], and the dominant part of $M_{C}$ is $j^{X_{C}}_{\mu}$.
However, we found that this result strongly depends on the normalization of
$j^{X_{C}}_{\mu}$. Hence a proper normalized current is essential in
calculations.

For the state $M_{X_{2}}$(4945), it’s mass prediction is larger than all
observed $1^{++}$ states. However, our calculation suggests that it is
relatively strongly mixed . The dominant part of $M_{X_{2}}$(4945) is more
likely to be $X_{B_{2}}$(5310) by comparing mass predictions.

In Ref. [19], authors have calculated state $X_{B_{1}}$ with similar method,
and got results: the mass $m_{X_{B_{1}}}=3.89^{+0.09}_{-0.09}$ GeV, the decay
constant $\lambda_{X_{B_{1}}}=2.96^{+1.09}_{-0.79}\times 10^{-4}$ GeV10 with
$\sqrt{s_{0}}=4.41$ GeV, which is consistent with our results.

##  IV Mixed state in $1^{--}$ channel

We start from two forms of currents as follows:

$\displaystyle j^{Y_{A_{1}}/Y_{A_{s1}}}_{\mu}(x)=$
$\displaystyle\bar{c}(x)c(x)\bar{q}(x)\gamma_{\mu}q(x),$ (24) $\displaystyle
j^{Y_{A_{2}}/Y_{A_{s2}}}_{\mu}(x)=$
$\displaystyle\bar{c}(x)\gamma_{\mu}c(x)\bar{q}(x)q(x),$ $\displaystyle
j^{Y_{B_{1}}/Y_{B_{s1}}}_{\mu}(x)=$
$\displaystyle\frac{i}{\sqrt{2}}[\bar{c}(x)\gamma_{\mu}q(x)\bar{q}(x)c(x)$
$\displaystyle+$ $\displaystyle\bar{q}(x)\gamma_{\mu}c(x)\bar{c}(x)q(x)],$
$\displaystyle j^{Y_{B_{2}}/Y_{B_{s2}}}_{\mu}(x)=$
$\displaystyle\frac{i}{\sqrt{2}}[\bar{c}(x)\gamma_{\mu}\gamma_{5}q(x)\bar{q}(x)\gamma_{5}c(x)$
$\displaystyle-$
$\displaystyle\bar{q}(x)\gamma_{\mu}\gamma_{5}c(x)\bar{c}(x)\gamma_{5}q(x)],$

where $Y$ denotes the $1^{--}$ state, the subscript $A$ of $Y$ represents
$\bar{Q}Q\bar{q}q$ scenario while $B$ represents the $\bar{Q}qQ\bar{q}$
scenario, the additional subscript $s$ represents the $s$ quark case, and one
can straightforwardly replace $q$ with $s$ quark when $j^{Y_{A_{s1}}}_{\mu}$,
$j^{Y_{A_{s2}}}_{\mu}$, $j^{Y_{B_{s1}}}_{\mu}$ and $j^{Y_{B_{s2}}}_{\mu}$ are
involved. The $Y(4230)$ was observed to decay to $\chi_{c0}\omega$, while
$Y(4660)$ was observed to have both $\psi(2S)\pi^{+}\pi^{-}$ and
$D_{s}^{+}D_{s1}(2536)^{-}$ decay modes [2]. Hence we especially focus on
currents $j^{Y_{A_{1}}}_{\mu}$ and $j^{Y_{A_{s2}}}_{\mu}$ which are consistent
with the respective decay modes, to describe $Y(4230)$ and $Y(4660)$
respectively, and discuss the corresponding mixed states in both $u,d$ quark
and $s$ quark for simplicity.

The two-point correlator functions of pure states have Lorentz structures

$\displaystyle\Pi_{\mu\nu}^{Y_{A_{k}}/Y_{A_{sk}}}(q^{2})=$
$\displaystyle\Pi_{(1)}^{Y_{A_{k}}/Y_{A_{sk}}}(q^{2})\left(-g_{\mu\nu}+\frac{q_{\mu}q_{\nu}}{q^{2}}\right)$
(25) $\displaystyle+$
$\displaystyle\Pi_{(0)}^{Y_{A_{k}}/Y_{A_{sk}}}(q^{2})\left(\frac{q_{\mu}q_{\nu}}{q^{2}}\right),$
$\displaystyle\Pi_{\mu\nu}^{Y_{B_{k}}/Y_{B_{sk}}}(q^{2})=$
$\displaystyle\Pi_{(1)}^{Y_{B_{k}}/Y_{B_{sk}}}(q^{2})\left(-g_{\mu\nu}+\frac{q_{\mu}q_{\nu}}{q^{2}}\right)$
$\displaystyle+$
$\displaystyle\Pi_{(0)}^{Y_{B_{k}}/Y_{B_{sk}}}(q^{2})\left(\frac{q_{\mu}q_{\nu}}{q^{2}}\right),$

where $k=$1,2, $\Pi_{(1)}^{Y_{A_{k}}/Y_{A_{sk}}}$ and
$\Pi_{(1)}^{Y_{B_{k}}/Y_{B_{sk}}}$ describe pure state contributions with
quantum numbers $1^{--}$, $\Pi_{(0)}^{Y_{A_{k}}/Y_{A_{sk}}}$ and
$\Pi_{(0)}^{Y_{B_{k}}/Y_{B_{sk}}}$ describe pure state contribution with
quantum numbers $0^{+-}$.

To study the mixed state, the off-diagonal mixing two-point correlation
functions described in Section II are

$\displaystyle\Pi^{M_{Y_{1}}}_{\mu\nu}(q^{2})=$ $\displaystyle\frac{i}{2}\int
d^{4}x\,e^{iq\cdot x}\langle
0|T(j^{Y_{A_{1}}}_{\mu}(x)j^{Y_{B_{1}}+}_{\nu}(0)$ (26) $\displaystyle+$
$\displaystyle j^{Y_{B_{1}}}_{\nu}(x)j^{Y_{A_{1}}+}_{\mu}(0))|0\rangle,$
$\displaystyle\Pi^{M_{Y_{2}}}_{\mu\nu}(q^{2})=$ $\displaystyle\frac{i}{2}\int
d^{4}x\,e^{iq\cdot x}\langle
0|T(j^{Y_{A_{1}}}_{\mu}(x)j^{Y_{B_{2}}+}_{\nu}(0)$ $\displaystyle+$
$\displaystyle j^{Y_{B_{2}}}_{\nu}(x)j^{Y_{A_{1}}+}_{\mu}(0))|0\rangle,$
$\displaystyle\Pi^{M_{Y_{s1}}}_{\mu\nu}(q^{2})=$ $\displaystyle\frac{i}{2}\int
d^{4}x\,e^{iq\cdot x}\langle
0|T(j^{Y_{A_{s2}}}_{\mu}(x)j^{Y_{B_{s1}}+}_{\nu}(0)$ $\displaystyle+$
$\displaystyle j^{Y_{B_{s1}}}_{\nu}(x)j^{Y_{A_{s2}}+}_{\mu}(0))|0\rangle,$
$\displaystyle\Pi^{M_{Y_{s2}}}_{\mu\nu}(q^{2})=$ $\displaystyle\frac{i}{2}\int
d^{4}x\,e^{iq\cdot x}\langle
0|T(j^{Y_{A_{s2}}}_{\mu}(x)j^{Y_{B_{s2}}+}_{\nu}(0)$ $\displaystyle+$
$\displaystyle j^{Y_{B_{s2}}}_{\nu}(x)j^{Y_{A_{s2}}+}_{\mu}(0))|0\rangle,$

where $M_{Y_{k}}$ and $M_{Y_{sk}}$, $k=1,2$, both represent mixed states
coupled to their respective currents. These mixed correlators have same
Lorentz structures as pure state cases,

$\displaystyle\Pi_{\mu\nu}^{M_{Y_{k}}/M_{Y_{sk}}}(q^{2})=$
$\displaystyle\Pi_{(1)}^{M_{Y_{k}}/M_{Y_{sk}}}(q^{2})\left(-g_{\mu\nu}+\frac{q_{\mu}q_{\nu}}{q^{2}}\right)$
(27) $\displaystyle+$
$\displaystyle\Pi_{(0)}^{M_{Y_{k}}/M_{Y_{sk}}}(q^{2})\left(\frac{q_{\mu}q_{\nu}}{q^{2}}\right),$

where $k=$1,2, $\Pi_{(1)}^{M_{Y_{k}}/M_{Y_{sk}}}$ and
$\Pi_{(0)}^{M_{Y_{k}}/M_{Y_{sk}}}$ respectively describe mixed state with
quantum number $1^{--}$, $0^{+-}$.

We follow same method mentioned in $1^{++}$ channel. The mesonic structures,
mass and coupling constant predictions, and the related QCDSR parameters are
collected in Table 4.

Table 4: Summary of results for $1^{--}$ states. State | Current structure | Mass/GeV |  $\lambda$/$10^{-4}$GeV10 |  $\sqrt{s_{0}}$/GeV |  $\tau$ window/GeV-2
---|---|---|---|---|---
$Y_{A_{1}}$ | $\chi_{c0}\omega$ | $4.207^{+0.08}_{-0.09}$ | $1.64^{+0.63}_{-0.49}$ | 4.8 | 0.27 – 0.28
$Y_{A_{s2}}$ | $J/\psi f(980)$ | $4.621^{+0.05}_{-0.06}$ | $21.3^{+5.2}_{-4.7}$ | 5.1 | 0.25 – 0.32
$Y_{B_{1}}$ | $D_{0}^{*}\bar{D}^{*}$ | $4.922^{+0.04}_{-0.04}$ | $34.5^{+6.7}_{-6.0}$ | 5.4 | 0.21 – 0.34
$Y_{B_{2}}$ | $D_{1}\bar{D}$ | $4.385^{+0.06}_{-0.06}$ | $7.36^{+1.95}_{-1.67}$ | 4.9 | 0.27 – 0.35
$Y_{B_{s1}}$ | $D_{s0}^{*}\bar{D}_{s}^{*}$ | $4.952^{+0.03}_{-0.04}$ | $37.9^{+6.5}_{-6.3}$ | 5.45 | 0.21 – 0.36
$Y_{B_{s2}}$ | $D_{s1}\bar{D}_{s}$ | $4.494^{+0.08}_{-0.05}$ | $9.60^{+3.7}_{-2.1}$ | 5.0 | 0.26 – 0.39
$M_{Y_{1}}$ |  $\chi_{c0}\omega$ \- $D_{0}^{*}\bar{D}^{*}$ | $4.770^{+0.07}_{-0.06}$ | $1.16^{+0.37}_{-0.28}$ | 5.3 | 0.24 – 0.25
$M_{Y_{2}}$ |  $\chi_{c0}\omega$ \- $D_{1}\bar{D}$ | $4.266^{+0.08}_{-0.08}$ | $0.373^{+0.117}_{-0.093}$ | 4.95 | 0.26 – 0.27
$M_{Y_{s1}}$ |  $J/\psi f(980)$ \- $D_{s0}^{*}\bar{D}_{s}^{*}$ | $4.610^{+0.05}_{-0.06}$ | $2.56^{+0.67}_{-0.56}$ | 5.1 | 0.24 – 0.33
$M_{Y_{s2}}$ |  $J/\psi f(980)$ \- $D_{s1}\bar{D}_{s}$ | $4.450^{+0.05}_{-0.06}$ | $1.64^{+0.43}_{-0.22}$ | 4.95 | 0.26 – 0.33

In $Y$ family of states, $Y(4160)$, $Y(4260)$, $Y(4415)$, $Y(4660)$ are
reported have decay modes including a $s$ quark in final states, and
$Y(4230)$, $Y(4360)$, $Y(4390)$ have not been observed to have decay modes
including a $s$ quark in final states. Furthermore, $Y(4260)$ only decays to
$K$ meson while $Y(4415)$ and $Y(4660)$ only decays to $D_{s}$ meson when a
$s$ quark is directly involved in final states. The $Y(4160)$ has both decay
modes including $K$ and $D_{s}$ mesons in final states. On the other hand, all
$Y$ states have both $\bar{Q}Q+\bar{q}q$ like decay modes and
$\bar{Q}q+Q\bar{q}$ like decay modes except $Y(4390)$. The decay mode
$Y(4390)$ to $\pi^{+}\pi^{-}h_{c}$ were observed, but other decay modes of
$Y(4390)$ have not been seen yet. Because the $K$ meson may decay to $\pi$
meson and disappear in final states, we cannot exclude a $s$ quark in
$Y(4230)$, $Y(4360)$, $Y(4390)$ [2]. Hence we suggest that $Y(4230)$ has
candidates $Y_{A_{1}}$(4207), $M_{Y_{2}}$(4266), $Y(4360)$ or $Y(4390)$ has
candidate $Y_{B_{2}}$(4385), $Y(4415)$ has candidates $Y_{B_{s2}}$(4494) and
$M_{Y_{s2}}$(4450), $Y(4660)$ has candidates $Y_{A_{s2}}$(4621) and
$M_{Y_{s1}}$(4610). Although the remaining states are not compatible with
known $1^{--}$ states, they still possibly mix with other states, and their
contributions can be estimated.

For the $1^{--}$ states, the mixing strengths are given by the data in Table
4,

$\displaystyle
N_{M_{Y_{1}}}=\frac{1.16\textrm{GeV}^{10}}{\sqrt{1.64}\textrm{GeV}^{5}\times\sqrt{34.5}\textrm{GeV}^{5}}=0.15,$
(28) $\displaystyle
N_{M_{Y_{2}}}=\frac{0.373\textrm{GeV}^{10}}{\sqrt{1.64}\textrm{GeV}^{5}\times\sqrt{7.36}\textrm{GeV}^{5}}=0.11,$
$\displaystyle
N_{M_{Y_{s1}}}=\frac{2.56\textrm{GeV}^{10}}{\sqrt{21.3}\textrm{GeV}^{5}\times\sqrt{37.9}\textrm{GeV}^{5}}=0.09,$
$\displaystyle
N_{M_{Y_{s2}}}=\frac{1.64\textrm{GeV}^{10}}{\sqrt{21.3}\textrm{GeV}^{5}\times\sqrt{9.60}\textrm{GeV}^{5}}=0.11.$

All mixed states have much weaker mixing strength compared with $1^{++}$ mixed
states. We suggest that $1^{--}$ states are preferred to be pure state and
weakly mixed with other states. This becomes more clear when we convert $N$ to
$\widetilde{N}$,

$\displaystyle\widetilde{N}_{M_{Y_{1}}}=\sin^{2}\left(\frac{\arcsin(0.15\times
2)}{2}\right)=2.3\%,$ (29)
$\displaystyle\widetilde{N}_{M_{Y_{2}}}=\sin^{2}\left(\frac{\arcsin(0.11\times
2)}{2}\right)=1.2\%,$
$\displaystyle\widetilde{N}_{M_{Y_{s1}}}=\sin^{2}\left(\frac{\arcsin(0.09\times
2)}{2}\right)=0.82\%,$
$\displaystyle\widetilde{N}_{M_{Y_{s2}}}=\sin^{2}\left(\frac{\arcsin(0.11\times
2)}{2}\right)=1.2\%,$

where the values of $\widetilde{N}$ suggest that the assumed mixed states with
quantum numbers $1^{--}$ are actually very pure state. As mentioned above,
$M_{Y_{1}}$(4770) which contains no $s$ quark, is close to $Y(4660)$, cannot
be compatible with known $1^{--}$ states. $M_{Y_{2}}$(4266) which is a
possible candidate for $Y(4230)$, is a mixture of $Y_{A_{1}}$(4207) and
$Y_{B_{2}}$(4385). By comparing the two mass predictions, $M_{Y_{2}}$(4266) is
closer to $Y_{A_{1}}$(4207) rather than $Y_{B_{2}}$(4385), and it is possibly
dominated by $\bar{Q}Q\bar{q}q$ component. For the same reasons,
$M_{Y_{s1}}(4610)$ is possibly dominated by a $\bar{Q}Q\bar{q}q$ component
while $M_{Y_{s2}}$(4450) is possibly dominated by $\bar{Q}qQ\bar{q}$. Hence we
suggest that $Y(4230)$, $Y(4660)$ prefer a $\bar{Q}Q\bar{q}q$ state, and
$Y(4415)$ prefers a $\bar{Q}qQ\bar{q}$ state.

In Ref. [30], authors have calculated states $Y_{B_{1}}$ and $Y_{B_{2}}$ with
similar method and obtained the results: mass
$m_{Y_{B_{1}}}=4.78^{+0.07}_{-0.07}$ with $\sqrt{s_{0}}=5.3$ GeV-2, and mass
$m_{Y_{B_{2}}}=4.36^{+0.08}_{-0.08}$ with $\sqrt{s_{0}}=4.9$ GeV-2, which are
consistent with our results. The small difference of mass of $Y_{B_{1}}$ is
caused by different values of $\sqrt{s_{0}}$. Besides, in Ref. [30], authors
have discussed different results of similar states of $Y_{B_{1}}$ and
$Y_{B_{2}}$ in previous papers. For instance, authors in Ref. [42] did not
distinguish the charge conjugations and obtained mass of a $Y_{B_{1}}$ like
state $m_{D^{*}_{0}\bar{D^{*}}}=4.26$ GeV. Our results are more supportive to
the results in Ref. [30].

In Ref. [8], authors have calculated states $Y_{A_{s2}}$ with similar method
and obtained the result: mass $m_{Y_{A_{s2}}}=4.67^{+0.09}_{-0.09}$ with
$\sqrt{s_{0}}=5.1$ GeV-2, which is consistent with our result.

##  V  Mixed state in $1^{-+}$ channel

We start from two forms of currents as follows:

$\displaystyle j^{P_{A}}_{\mu\nu}(x)$ $\displaystyle=j^{X_{A}}_{\mu\nu}(x),$
(30) $\displaystyle j^{P_{A_{s}}}_{\mu\nu}(x)$
$\displaystyle=\frac{i}{\sqrt{2}}[\bar{c}(x)\gamma_{\mu}c(x)\bar{s}(x)\gamma_{\nu}s(x)$
$\displaystyle-\bar{c}(x)\gamma_{\nu}c(x)\bar{s}(x)\gamma_{\mu}s(x)],$
$\displaystyle j^{P_{B_{1}}/P_{B_{s1}}}_{\mu}(x)$
$\displaystyle=\frac{i}{\sqrt{2}}[\bar{c}(x)\gamma_{\mu}q(x)\bar{q}(x)c(x)$
$\displaystyle-\bar{q}(x)\gamma_{\mu}c(x)\bar{c}(x)q(x)],$ $\displaystyle
j^{P_{B_{2}}/P_{B_{s2}}}_{\mu}(x)$
$\displaystyle=\frac{i}{\sqrt{2}}[\bar{c}(x)\gamma_{\mu}\gamma_{5}q(x)\bar{q}(x)\gamma_{5}c(x)$
$\displaystyle+\bar{q}(x)\gamma_{\mu}\gamma_{5}c(x)\bar{c}(x)\gamma_{5}q(x)],$

where $P$ denotes the $1^{-+}$ state, the subscript $A$ of $P$ represents the
$\bar{Q}Q\bar{q}q$ scenario while $B$ represents the $\bar{Q}qQ\bar{q}$
scenario, the additional subscript $s$ represents the $s$ quark case, and one
can straightforwardly replace $q$ with $s$ when $j^{P_{B_{s1}}}_{\mu}$ and
$j^{P_{B_{s2}}}_{\mu}$ are involved. The structures of these currents are
similar to the $1^{++}$ and $1^{--}$ cases, and it is interesting to compare
the mass predictions of these currents to $1^{++}$ and $1^{--}$ states.

To study the pure $\bar{Q}Q\bar{q}q$ and $\bar{Q}qQ\bar{q}$ states, the two-
point correlation functions respectively have Lorentz structure

$\displaystyle\Pi_{\mu\nu\rho\sigma}^{P_{A}/P_{A_{s}}}(q^{2})=$
$\displaystyle\Pi_{a}^{P_{A}/P_{A_{s}}}\frac{1}{q^{2}}(q^{2}g_{\mu\rho}g_{\nu\sigma}-q^{2}g_{\mu\sigma}g_{\nu\rho}$
(31) $\displaystyle-$ $\displaystyle
q_{\mu}q_{\rho}g_{\nu\sigma}+q_{\mu}q_{\sigma}g_{\nu\rho}-q_{\nu}q_{\sigma}g_{\mu\rho}+q_{\nu}q_{\rho}g_{\mu\sigma})$
$\displaystyle+$
$\displaystyle\Pi_{b}^{P_{A}/P_{A_{s}}}\frac{1}{q^{2}}(-q_{\mu}q_{\rho}g_{\nu\sigma}+q_{\mu}q_{\sigma}g_{\nu\rho}$
$\displaystyle-$ $\displaystyle
q_{\nu}q_{\sigma}g_{\mu\rho}+q_{\nu}q_{\rho}g_{\mu\sigma}),$
$\displaystyle\Pi_{\mu\nu}^{P_{B_{k}}/P_{B_{sk}}}(q^{2})=$
$\displaystyle\Pi_{(1)}^{P_{B_{k}}/P_{B_{sk}}}(q^{2})\left(-g_{\mu\nu}+\frac{q_{\mu}q_{\nu}}{q^{2}}\right)$
$\displaystyle+$
$\displaystyle\Pi_{(0)}^{P_{B_{k}}/P_{B_{sk}}}(q^{2})\left(\frac{q_{\mu}q_{\nu}}{q^{2}}\right),$

where $k=$1,2, $\Pi_{a}^{P_{A}/P_{A_{s}}}$ and $\Pi_{b}^{P_{A}/P_{A_{s}}}$
describes pure state contributions with quantum numbers $1^{++}$, $1^{-+}$
respectively, and $\Pi_{(1)}^{P_{B_{k}}/P_{B_{sk}}}$ and
$\Pi_{(0)}^{P_{B_{k}}/P_{B_{sk}}}$ describes $1^{-+}$, $0^{++}$ respectively.
In mixing scenarios, we start from the off-diagonal mixed correlator described
in the previous sections, i.e.,

$\displaystyle\Pi^{M_{P_{1}}/M_{P_{s1}}}_{\mu\nu\sigma}(q^{2})$
$\displaystyle=\frac{i}{2}\int d^{4}x\,e^{iq\cdot x}\langle
0|T(j^{P_{A}/P_{A_{s}}}_{\mu\nu}(x)j^{P_{B_{1}}/P_{B_{s1}}+}_{\sigma}(0)$ (32)
$\displaystyle+j^{P_{B_{1}}/P_{B_{s1}}}_{\sigma}(x)j^{P_{A}/P_{A_{s}}+}_{\mu\nu}(0))|0\rangle,$
$\displaystyle\Pi^{M_{P_{2}}/M_{P_{s2}}}_{\mu\nu\sigma}(q^{2})$
$\displaystyle=\frac{i}{2}\int d^{4}x\,e^{iq\cdot x}\langle
0|T(j^{P_{A}/P_{A_{s}}}_{\mu\nu}(x)j^{P_{B_{2}}/P_{B_{s2}}+}_{\sigma}(0)$
$\displaystyle+j^{P_{B_{2}}/P_{B_{s2}}}_{\sigma}(x)j^{P_{A}/P_{A_{s}}+}_{\mu\nu}(0))|0\rangle,$

where $M_{P_{k}}/M_{P_{sk}}$, $k=$1,2, are mixed states assumed to result from
the corresponding currents. The correlators have Lorentz structure

$\displaystyle\Pi_{\mu\nu\sigma}^{M_{P_{k}}/M_{P_{sk}}}(q^{2})=\Pi^{M_{P_{k}}/M_{P_{sk}}}(q^{2})(-g_{\mu\sigma}q_{\nu}+g_{\nu\sigma}q_{\mu}).$
(33)

We follow same method used in previous sections. The mesonic structures, mass
and coupling constant predictions, and related QCDSR parameters are collected
in Table 5.

Table 5: Summary of results for $1^{-+}$ molecular states.  State | Current structure | Mass/GeV |  $\lambda$/$10^{-4}$GeV10 |  $\sqrt{s_{0}}$/GeV |  $\tau$ window/GeV-2
---|---|---|---|---|---
$P_{A}$ | $J/\psi\rho$ | $4.658^{+0.05}_{-0.06}$ | $13.1^{+3.2}_{-3.1}$ | 5.15 | 0.24 – 0.29
$P_{A_{s}}$ | $J/\psi f(980)$ | $4.694^{+0.05}_{-0.05}$ | $14.0^{+3.5}_{-2.9}$ | 5.2 | 0.24 – 0.32
$P_{B_{1}}$ | $D_{0}^{*}\bar{D}^{*}$ | $4.927^{+0.05}_{-0.04}$ | $34.1^{+7.7}_{-5.7}$ | 5.4 | 0.21 – 0.30
$P_{B_{2}}$ | $D_{1}\bar{D}$ | $4.528^{+0.06}_{-0.05}$ | $9.7^{+2.9}_{-2.1}$ | 5.05 | 0.26 – 0.31
$P_{B_{s1}}$ | $D_{s0}^{*}\bar{D}_{s}^{*}$ | $4.999^{+0.04}_{-0.03}$ | $42.8^{+8.2}_{-6.8}$ | 5.5 | 0.21 – 0.33
$P_{B_{s2}}$ | $D_{s1}\bar{D}_{s}$ | $4.642^{+0.05}_{-0.05}$ | $13.0^{+3.4}_{-2.8}$ | 5.15 | 0.25 – 0.34
$M_{P_{1}}$ |  $J/\psi\rho$ \- $D_{0}^{*}\bar{D}^{*}$ | $4.505^{+0.06}_{-0.04}$ |  $0.401^{+0.096}_{-0.073}$ GeV-1 | 5.05 | 0.21 – 0.30
$M_{P_{2}}$ |  $J/\psi\rho$ \- $D_{1}\bar{D}$ | $4.494^{+0.06}_{-0.06}$ |  $0.240^{+0.060}_{-0.052}$ GeV-1 | 5.05 | 0.23 – 0.27
$M_{P_{s1}}$ |  $J/\psi f(980)$ \- $D_{s0}^{*}\bar{D}_{s}^{*}$ | $4.544^{+0.05}_{-0.05}$ |  $0.405^{+0.085}_{-0.077}$ GeV-1 | 5.1 | 0.21 – 0.33
$M_{P_{s2}}$ |  $J/\psi f(980)$ \- $D_{s1}\bar{D}_{s}$ | $4.536^{+0.06}_{-0.05}$ |  $0.269^{+0.067}_{-0.055}$ GeV-1 | 5.1 | 0.22 – 0.31

All pure states have mass predictions over 4.5 GeV, and cannot be compatible
with those known states which are probably $1^{-+}$ candidates [2].

The mixing strength can then be estimated by computing the value of $N$. Note
that the coupling constants of the mixed state correlator has the form

$\displaystyle\Pi^{M_{P_{k}}}_{\mu\nu\sigma}(q^{2})$
$\displaystyle=\frac{i}{2}\int d^{4}x\,e^{iq\cdot x}\langle
0|T(j^{P_{A}}_{\mu\nu}(x)j^{P_{B_{k}}+}_{\sigma}(0)$ (34)
$\displaystyle+j^{P_{B_{k}}}_{\sigma}(x)j^{P_{A}+}_{\mu\nu}(0))|0\rangle$
$\displaystyle\sim\frac{\lambda_{P_{A}}}{M_{H}}(\varepsilon^{P_{A}}_{\mu}q_{\nu}-\varepsilon^{P_{A}}_{\nu}q_{\mu})(\lambda^{*}_{P_{B_{k}}}\varepsilon^{*P_{B_{k}}}_{\sigma})$
$\displaystyle+\frac{\lambda^{*}_{P_{A}}}{M_{H}}(\varepsilon^{*P_{A}}_{\mu}q_{\nu}-\varepsilon^{*P_{A}}_{\nu}q_{\mu})(\lambda_{P_{B_{k}}}\varepsilon^{P_{B_{k}}}_{\sigma})+...$
$\displaystyle=\frac{\lambda_{P_{A}}\lambda^{*}_{P_{B_{k}}}+\lambda^{*}_{P_{A}}\lambda_{P_{B_{k}}}}{M_{H}}(-g_{\mu\sigma}q_{\nu}+g_{\nu\sigma}q_{\mu})+...,$

where $k=1,2$, $\varepsilon^{P_{A}}$ and $\varepsilon^{P_{B_{k}}}$ are
polarization vectors, and $M_{H}$ represents the ground state mass of $P_{A}$.
Analogues to $1^{++}$ channel, the values of $N$ obtained from Table 5 can be
written as

$\displaystyle N_{M_{P_{1}}}=\frac{0.401\textrm{GeV}^{9}\times
M_{H}(4.658\textrm{GeV})}{\sqrt{13.1}\textrm{GeV}^{5}\times\sqrt{34.1}\textrm{GeV}^{5}}=0.09\textrm{,}$
(35) $\displaystyle N_{M_{P_{2}}}=\frac{0.240\textrm{GeV}^{9}\times
M_{H}(4.658\textrm{GeV})}{\sqrt{13.1}\textrm{GeV}^{5}\times\sqrt{9.7}\textrm{GeV}^{5}}=0.10\textrm{,}$
$\displaystyle N_{M_{P_{s1}}}=\frac{0.405\textrm{GeV}^{9}\times
M_{H}(4.694\textrm{GeV})}{\sqrt{14.0}\textrm{GeV}^{5}\times\sqrt{42.8}\textrm{GeV}^{5}}=0.08\textrm{,}$
$\displaystyle N_{M_{P_{s2}}}=\frac{0.269\textrm{GeV}^{9}\times
M_{H}(4.694\textrm{GeV})}{\sqrt{14.0}\textrm{GeV}^{5}\times\sqrt{13.0}\textrm{GeV}^{5}}=0.09\textrm{,}$

where $M_{H}$ represents the corresponding $\bar{Q}Q\bar{q}q$ ground state
mass. Like the $1^{--}$ channel, all the mixed states which have quantum
numbers $1^{-+}$ are weakly mixed with corresponding currents, which becomes
even more clear when we convert $N$ to $\widetilde{N}$,

$\displaystyle\widetilde{N}_{M_{P_{1}}}=\sin^{2}\left(\frac{\arcsin(0.09\times
2)}{2}\right)=0.82\%,$ (36)
$\displaystyle\widetilde{N}_{M_{P_{2}}}=\sin^{2}\left(\frac{\arcsin(0.10\times
2)}{2}\right)=1.0\%,$
$\displaystyle\widetilde{N}_{M_{P_{s1}}}=\sin^{2}\left(\frac{\arcsin(0.08\times
2)}{2}\right)=0.64\%,$
$\displaystyle\widetilde{N}_{M_{P_{s2}}}=\sin^{2}\left(\frac{\arcsin(0.09\times
2)}{2}\right)=0.82\%.$

For the same reasons mentioned in the $1^{--}$ channel, $M_{P_{1}}$(4505) and
$M_{P_{s1}}$(4494) are dominated by $\bar{Q}Q\bar{q}q$ components,
$M_{P_{2}}$(4544) and $M_{P_{s2}}$(4536) are more likely dominated by
$\bar{Q}qQ\bar{q}$ components.

In Ref. [30], authors have calculated states $P_{B_{1}}$ and $P_{B_{2}}$ with
similar method and obtained the results: mass
$m_{P_{B_{1}}}=4.73^{+0.07}_{-0.07}$ with $\sqrt{s_{0}}=5.2$ GeV-2, and mass
$m_{P_{B_{2}}}=4.60^{+0.08}_{-0.08}$ with $\sqrt{s_{0}}=5.1$ GeV-2, which are
consistent with our results. The small difference of mass of $P_{B_{1}}$ is
caused by different values of $\sqrt{s_{0}}$. Besides, authors in Ref. [43]
obtained mass of state $P_{B_{1}}$, $m_{P_{B_{1}}}=4.19$ GeV. Our results are
more supportive to the results in Ref. [30].

##  VI Non-perturbative effects of mixing strength

In fact, we can convert $\bar{Q}Q\bar{q}q$ and $\bar{Q}qQ\bar{q}$ states to
each other through Fierz transformation. Generally,

$\displaystyle(\bar{Q}\Gamma_{1}Q)(\bar{q}\Gamma_{2}q)$
$\displaystyle=\sum_{ijk}C_{i}(\bar{Q}\Gamma_{j}q)(\bar{q}\Gamma_{k}Q)$ (37)
$\displaystyle+\sum_{lmn}C_{l}(\bar{Q}\Gamma_{m}\lambda^{a}q)(\bar{q}\Gamma_{n}\lambda^{a}Q),$
$\displaystyle(\bar{Q}\Gamma_{1}q)(\bar{q}\Gamma_{2}Q)$
$\displaystyle=\sum_{ijk}C_{i}(\bar{Q}\Gamma_{j}Q)(\bar{q}\Gamma_{k}q)$
$\displaystyle+\sum_{lmn}C_{l}(\bar{Q}\Gamma_{m}\lambda^{a}Q)(\bar{q}\Gamma_{n}\lambda^{a}q),$

where $\Gamma_{i}$ are gamma matrices, $C_{i}$ are parameters corresponding to
the related currents, and $\lambda^{a}$ are Gell-Mann matrices. That is,
$\bar{Q}Q\bar{q}q$ currents can be decomposed into a series of
$\bar{Q}qQ\bar{q}$ currents and a series of $\bar{Q}qQ\bar{q}$ color-octet
currents, and vice versa. In this paper, we have computed two-point
correlation functions of $\bar{Q}Q\bar{q}q$ and $\bar{Q}qQ\bar{q}$ currents.
One can convert one current to a series of other kinds of currents, and make
calculations analogous to a series of calculations of pure currents. For
instance,

$\displaystyle j^{Y_{A_{1}}}_{\mu}$
$\displaystyle=(\bar{c}c)(\bar{q}\gamma_{\mu}q)$ (38)
$\displaystyle=\frac{-i}{2\sqrt{2}}\frac{i}{\sqrt{2}}[(\bar{c}\gamma_{\mu}q)(\bar{q}c)+(\bar{q}\gamma_{\mu}c)(\bar{c}q)]$
$\displaystyle+\frac{-i}{2\sqrt{2}}\frac{i}{\sqrt{2}}[(\bar{c}\gamma_{\mu}\gamma_{5}q)(\bar{q}\gamma_{5}c)-(\bar{q}\gamma_{\mu}\gamma_{5}c)(\bar{c}\gamma_{5}q)]$
$\displaystyle+...$
$\displaystyle=\frac{-i}{2\sqrt{2}}j_{\mu}^{Y_{B_{1}}}+\frac{-i}{2\sqrt{2}}j_{\mu}^{Y_{B_{2}}}+...,$

where $j^{Y_{A_{1}}}_{\mu}$, $j_{\mu}^{Y_{B_{1}/B_{2}}}$ are defined in
Section IV. When we compute two-point correlation functions of
$j_{\mu}^{Y_{A_{1}}}$ and $j_{\mu}^{Y_{B_{1}/B_{2}}}$, it seems that the
result may highlight states $Y_{B_{1}}$/$Y_{B_{2}}$, and the parameters of the
current decomposition are likely to be related directly to mixing strength.
However, our calculations show different results. Although the contributions
in perturbative terms from different currents (e.g., $Y_{B_{1}}$ and
$Y_{B_{2}}$) will be suppressed, QCDSR calculations are sensitive to the
changes of borel window and threshold $s_{0}$, which depend on contributions
of non-perturbative terms. Moreover, the mixing strength is related to both
decay constants and parameters of the corresponding currents from the Fierz
transformation, and the decay constants are also sensitive to the Borel
window, which again depend on non-perturbative terms. To clarify this we have
computed another two $\bar{Q}Q\bar{q}q$ and $\bar{Q}qQ\bar{q}$ currents and
their mixed state,

$\displaystyle j^{Z_{A}}_{\mu}(x)=$
$\displaystyle\bar{c}(x)\gamma_{\mu}c(x)\bar{q}(x)\gamma_{5}q(x)$ (39)
$\displaystyle j^{Z_{B}}_{\mu\nu}(x)=$
$\displaystyle\frac{i}{\sqrt{2}}[\bar{c}(x)\gamma_{\mu}q(x)\bar{q}(x)\gamma_{\nu}c(x)$
$\displaystyle-$
$\displaystyle\bar{q}(x)\gamma_{\mu}c(x)\bar{c}(x)\gamma_{\nu}q(x)],$

where $Z$ denotes the $1^{+-}$ state, and the subscript $A$ of $Z$ represents
the $\bar{Q}Q\bar{q}q$ scenario while $B$ represents the $\bar{Q}qQ\bar{q}$
scenario. The mixed state is described by

$\displaystyle\Pi^{M_{Z}}_{\mu\nu\sigma}(q^{2})$
$\displaystyle=\frac{i}{2}\int d^{4}x\,e^{iq\cdot x}\langle
0|T(j^{Z_{A}}_{\sigma}(x)j^{Z_{B}+}_{\mu\nu}(0)$ (40)
$\displaystyle+j^{Z_{B}}_{\mu\nu}(x)j^{P_{A}+}_{\sigma}(0))|0\rangle,$

where $M_{Z}$ is assumed mixing from corresponding currents. The hadronic
structures along with results of mass, coupling constant and mixing strength
predictions are collected in Table 6.

Table 6: Summary of results for $1^{+-}$ states. State | Current structure | Mass/GeV |  $\lambda$/$10^{-4}$GeV10 |  $\sqrt{s_{0}}$/GeV |  $\tau$ window/GeV-2
---|---|---|---|---|---
$Z_{A}$ | $J/\psi\eta$ | $3.578^{+0.08}_{-0.08}$ | $1.04^{+0.36}_{-0.27}$ | 4.2 | 0.33 – 0.34
$Z_{B}$ | $D^{*}\bar{D}^{*}$ | $4.018^{+0.06}_{-0.06}$ | $2.90^{+0.88}_{-0.67}$ | 4.55 | 0.29 – 0.36
$M_{Z}$ |  $J/\psi\eta$ \- $D^{*}\bar{D}^{*}$ | $3.563^{+0.07}_{-0.06}$ |  $0.054^{+0.017}_{-0.012}$ GeV-1 | 4.1 | 0.31 – 0.35

Compared to $1^{++}$ currents $j_{\mu}^{X_{A/B_{1}}}$ and their mixed two-
point correlator $\Pi^{M_{X_{1}}}_{\mu\nu\sigma}(q^{2})$, which are given in
Eq. (17) and Eq. (19), $j_{\mu}^{Z_{A/B}}$ and
$\Pi^{M_{Z}}_{\mu\nu\sigma}(q^{2})$ have similar structures. Due to our
previous calculations in Section II, $M_{X_{1}}$ is relatively strongly mixed
with different components, and $M_{Z}$ is supposed to have similar properties.
However, the resulting mixing strength of $M_{Z}$ is

$\displaystyle N_{M_{Z}}=\frac{0.054\textrm{GeV}^{9}\times
M_{H}(4.018\textrm{GeV})}{\sqrt{1.04}\textrm{GeV}^{5}\times\sqrt{2.90}\textrm{GeV}^{5}}=0.125\textrm{,}$
(41) $\displaystyle\widetilde{N}_{M_{Z}}=\sin^{2}(\frac{\arcsin(0.125\times
2)}{2})=1.6\%.$

Compared to $M_{X_{1}}$($N_{M_{X_{1}}}$=0.349,
$\widetilde{N}_{M_{X_{1}}}$=14%), the mass predictions of two parts of mixed
state $M_{Z}$ differ, and although the contributions of perturbative terms in
two-point correlator functions are similar, the mixing strength of two states
are quite different. Hence we suggest that mixing strength is much sensitive
to the Borel window, threshold $s_{0}$, mass prediction, and decay constant,
which are all influenced by non-perturbative terms in QCDSR calculations.

##  VII Summary

In this paper we used QCD sum-rules to calculate the mass spectrum of
$\bar{Q}Q\bar{q}q$ and $\bar{Q}qQ\bar{q}$ states. Such states strongly couple
to $\bar{Q}Q\bar{q}q$ or $\bar{Q}qQ\bar{q}$ currents. So state’s components of
$\bar{Q}Q\bar{q}q$ and $\bar{Q}qQ\bar{q}$ can be mixed with each other. Such
mixing can be studied via the mixed correlators of $\bar{Q}Q\bar{q}q$ and
$\bar{Q}qQ\bar{q}$ currents. Our studies focus on mixing strength which may
determine whether the mixing picture accommodates candidates which have more
than single dominant decay modes.

Table 7: Summary of mixed state results.  Mixed state | Mass/GeV | $N$ | $\widetilde{N}$ | Dominant part | Possible Candidate
---|---|---|---|---|---
$M_{X_{1}}$ | $3.987^{+0.06}_{-0.06}$ | 0.349 | 14% | $\bar{Q}qQ\bar{q}$ | $X(3872)$
$M_{X_{2}}$ | $4.945^{+0.08}_{-0.06}$ | 0.285 | 9.0% | $\bar{Q}qQ\bar{q}$ | -
$M_{C}$ | $3.818^{+0.03}_{-0.02}$ | 0.125 | 1.6% | $\bar{Q}Q$ | $X(3872)$
$M_{Y_{1}}$ | $4.770^{+0.07}_{-0.06}$ | 0.15 | 2.3% | $\bar{Q}qQ\bar{q}$ | -
$M_{Y_{2}}$ | $4.266^{+0.08}_{-0.08}$ | 0.11 | 1.2% | $\bar{Q}Q\bar{q}q$ | $Y(4230)$
$M_{Y_{s1}}$ | $4.610^{+0.05}_{-0.06}$ | 0.06 |  $<$1% | $\bar{Q}Q\bar{q}q$ | $Y(4660)$
$M_{Y_{s2}}$ | $4.450^{+0.05}_{-0.06}$ | 0.11 | 1.2% | $\bar{Q}qQ\bar{q}$ | $Y(4415)$
$M_{P_{1}}$ | $4.505^{+0.06}_{-0.04}$ | 0.09 |  $<$1% | $\bar{Q}Q\bar{q}q$ | -
$M_{P_{2}}$ | $4.494^{+0.06}_{-0.06}$ | 0.10 | 1.0% | $\bar{Q}qQ\bar{q}$ | -
$M_{P_{s1}}$ | $4.544^{+0.05}_{-0.05}$ | 0.08 |  $<$1% | $\bar{Q}Q\bar{q}q$ | -
$M_{P_{s2}}$ | $4.536^{+0.06}_{-0.05}$ | 0.09 |  $<$1% | $\bar{Q}qQ\bar{q}$ | -

We list all the mixed states results in Table 7. The uncertainties of masses
are less than 5%, and the uncertainties of coupling constants are about 25%,
which are induced by uncertainties of input parameters and threshold $s_{0}$.
The relations $\sqrt{s_{0}}=M_{H}+\Delta_{s}$ and 40%-10% are required to
determine the window of $\tau$. These two conditions are not always satisfied
well. In some cases, the windows of $\tau$ is very narrow. If higher dimension
condensates are considered, we may reconsider the constraint of 40%-10% and
the situation may change.

For the $1^{++}$ channel, we find that two states $M_{X_{1}}$(3987) and
$M_{X_{2}}$(4945) are relatively strongly mixed with $\bar{Q}Q\bar{q}q$ and
$\bar{Q}qQ\bar{q}$ components. Furthermore, we estimate the ratio of decay
width of two kinds of decay modes of $M_{X_{1}}$(3987), which is roughly
consistent with experimental results for the $X(3872)$. When we consider the
mixing state combined with $\bar{Q}Q$ and $\bar{Q}qQ\bar{q}$, we revisit the
result in Ref. [28] with the new technique in Ref. [15]. The result argues
that $\bar{Q}Q$ is the dominant part of $X(3872)$, which can explain the
latest observation to $X(3872)$ of LHCb[7]. Our calculations just support
these two components can relatively strongly mix with each other in quantum
numbers $1^{++}$.

In other quantum number channels, states are found to be weakly mixed.
However, the calculations of these states is still meaningful to help us
establish the physical structure of corresponding state. For instance, pure
$\bar{Q}Q\bar{q}q$ $Y_{A_{1}}$(4207) and $Y_{A_{s2}}$(4610) configurations are
good candidates for $Y(4230)$ and $Y(4660)$ respectively. However, by checking
assumed mixed states mixing with $\bar{Q}Q\bar{q}q$ and $\bar{Q}qQ\bar{q}$
molecular states, we find these candidates have small components of
$\bar{Q}qQ\bar{q}$ which is inconsistent with the fact that $Y(4230)$ and
$Y(4660)$ have more abundant decay modes. For the same reasons, we can
establish the dominant part of $Y(4415)$. Our result suggests that $Y(4415)$
is dominated by $\bar{Q}qQ\bar{q}$, and agrees with the absence of $K$ meson
in observed decay final states. But $Y(4415)$ still has a small component of
$\bar{Q}Q\bar{q}q$. These states may therefore have more a complicated
construction, for instance, $\bar{q}q$ could be a color-octet state. Other
models, such as the tetraquark model, maybe valuable. By using the Fierz
transformations, tetraquark currents can be decomposed into various molecular
currents and color-octet currents, to show more mixed effects of different
possible states [44]. Since the mixing effects are normally small, the studies
via tetraquark currents cannot distinguish details of the mixing between the
different currents and only give the average of those currents. So the
tetraquark model is not a self-verifying because it cannot show which parts
(via the Fierz transformations) interact with each other strongly and others
do not. It may also meet challenges for quantitative descriptions of XYZ
states.

The calculations based on pure molecular currents have been criticized because
there is a large background of two free mesons spectrum. If the states indeed
have an absolutely dominate decay mode[45], there is no problem(actually, the
mass of molecule state is close to that of two free mesons). Otherwise, the
mixing pattern must be taken into account. The mixing of typical molecular
currents $\bar{Q}Q\bar{q}q$ and $\bar{Q}qQ\bar{q}$ are suppressed
(perturbatively) by small coefficients of Fierz transformations, so the
background of two free mesons spectrum also is suppressed. Non-perturbative
corrections play more important roles in the mixing correlator, which can
distinguish the real four-quark resonance from two free mesons. It should be
the essential feature of the mixing pattern. Our calculations show that the
mixing pattern is consistent with some of XYZ states, but fail in many others.
Since the mixing correlator is normalized by two diagonal correlators which
may be affected by a large background of two free meson spectrum, the real
mixture may be larger than our estimate. How to remove the background of two
free meson spectrum is still a big problem.

###### Acknowledgements.

This work is supported by NSFC (under grants 11175153 and 11205093) and by the
Natural Sciences and Engineering Research Council of Canada (NSERC).

##  Appendix A QCDSR analysis results

Here we show the $\tau$ dependence of $M_{H}^{2}$ defined in Eq. (11) for all
mixed states.

(a) $M_{H}^{2}$ dependence on $\tau$ for $M_{X_{1}}$. The solid line
represents $\sqrt{s_{0}}=4.4$GeV, and the dashed line and the dotted lines
respectively represent $\sqrt{s_{0}}=4.4\pm 0.1$GeV .

(b) $M_{H}^{2}$ dependence on $\tau$ for $M_{X_{2}}$. The solid line
represents $\sqrt{s_{0}}=5.45$GeV, and the dashed line and the dotted lines
respectively represent $\sqrt{s_{0}}=5.45\pm 0.1$GeV.

Figure 1: $M_{H}^{2}$ behaviors on $\tau$ for $1^{++}$ mixed states.

(a) $M_{H}^{2}$ dependence on $\tau$ for $M_{Y_{1}}$. The solid line
represents $\sqrt{s_{0}}=5.3$GeV, and the dashed line and the dotted line
respectively represent $\sqrt{s_{0}}=5.3\pm 0.1$GeV.

(b) $M_{H}^{2}$ dependence on $\tau$ for $M_{Y_{2}}$. The solid line
represents $\sqrt{s_{0}}=4.95$GeV, and the dashed line and the dotted line
respectively represent $\sqrt{s_{0}}=4.95\pm 0.1$GeV.

(c) $M_{H}^{2}$ dependence on $\tau$ for $M_{Y_{s1}}$. The solid line
represents $\sqrt{s_{0}}=5.1$GeV, and the dashed line and the dotted line
respectively represent $\sqrt{s_{0}}=5.1\pm 0.1$GeV.

(d) $M_{H}^{2}$ dependence on $\tau$ for $M_{Y_{s2}}$. The solid line
represents $\sqrt{s_{0}}=4.95$GeV, and the dashed line and the dotted line
respectively represent $\sqrt{s_{0}}=4.95\pm 0.1$GeV.

Figure 2: $M_{H}^{2}$ behaviors on$\tau$ for $1^{--}$ mixed states.

(a) $M_{H}^{2}$ dependence on $\tau$ for $M_{P_{1}}$. The solid line
represents $\sqrt{s_{0}}=5.05$GeV, and the dashed line and the dotted line
respectively represent $\sqrt{s_{0}}=5.05\pm 0.1$GeV.

(b) $M_{H}^{2}$ dependence on $\tau$ for $M_{P_{2}}$. The solid line
represents $\sqrt{s_{0}}=5.05$GeV, and the dashed line and the dotted line
respectively represent $\sqrt{s_{0}}=5.05\pm 0.1$GeV.

(c) $M_{H}^{2}$ dependence on $\tau$ for $M_{P_{s1}}$. The solid line
represents $\sqrt{s_{0}}=5.1$GeV, and the dashed line and the dotted line
respectively represent $\sqrt{s_{0}}=5.1\pm 0.1$GeV.

(d) $M_{H}^{2}$ dependence on $\tau$ for $M_{P_{s2}}$. The solid line
represents $\sqrt{s_{0}}=5.1$GeV, and the dashed line and the dotted line
respectively represent $\sqrt{s_{0}}=5.1\pm 0.1$GeV.

Figure 3: $M_{H}^{2}$ behaviors on $\tau$ for $1^{-+}$ mixed states.

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