Does vacuum saturation work for the higher dimensional vacuum condensates in
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 the QCD sum rules for the tetraquark (molecular) states, the higher
dimensional vacuum condensates play an important role in extracting the
tetraquark masses. We carry out the operator product expansion up to the
vacuum condensates of dimension-10 and observe that the vacuum condensates of
dimensions $6$, $8$ and $10$ have the same expressions but opposite signs for
the $C\gamma_{5}\otimes\gamma_{\mu}C$-type and $C\otimes\gamma_{\mu}C$ type
four-quark currents, which make their influences distinguishable, and they are
excellent channels to examine the vacuum saturation approximation. We
introduce a parameter $\kappa$ to parameterize the derivation from the vacuum
saturation or factorization approximation, and choose two sets parameters to
examine the influences on the predicted tetraquark masses, which can be
confronted to the experimental data in the future. In all the channels,
smaller value of the $\kappa$ leads to better convergent behavior in the
operator product expansion, which favors the vacuum saturation approximation.

PACS number: 12.39.Mk, 12.38.Lg

Key words: Tetraquark states, QCD sum rules

##  1 Introduction

In the QCD sum rules, we construct the quark (or quark-gluon or gluon)
currents to interpolate the hadrons with the same quantum numbers, then
introduce the two-point correlation functions and accomplish the operator
product expansion in the deep Euclidean space, the perturbative contributions
are embodied in the Wilson’s coefficients, while the nonperturbative
contributions are absorbed in the vacuum condensates. We have to resort to
non-zero vacuum expectation values for the normal-ordered quark-gluon
operators to describe the hadron properties in a satisfactory way.

Except for the quark condensates $\langle\bar{q}q\rangle$ with $q=u$, $d$,
which indicate spontaneous breaking of the Chiral symmetry through the Gell-
Mann-Oakes-Renner relation
$f^{2}_{\pi}m^{2}_{\pi}=-2(m_{u}+m_{d})\langle\bar{q}q\rangle$ [1], where the
$f_{\pi}$ is the decay constant of the pion, other vacuum condensates, such as
the $\langle\bar{q}g_{s}\sigma Gq\rangle$, $\langle\bar{q}q\bar{q}q\rangle$,
$\cdots$ are just parameters introduced by hand to describe the
nonperturbative QCD vacuum, we can parameterize the non-perturbative
properties in one way or the other.

There are a large number of four-quark condensates which can have a nonzero
vacuum expectation, such as the $\langle\bar{q}q\bar{q}q\rangle$,
$\langle\bar{q}i\gamma_{5}q\bar{q}i\gamma_{5}q\rangle$,
$\langle\bar{q}\gamma_{\mu}q\bar{q}\gamma^{\mu}q\rangle$,
$\langle\bar{q}\gamma_{\mu}t^{a}q\bar{q}\gamma^{\mu}t^{a}q\rangle$,
$\langle\bar{q}\gamma_{\mu}\gamma_{5}t^{a}q\bar{q}\gamma^{\mu}\gamma_{5}t^{a}q\rangle$,
etc, where $t^{a}=\frac{\lambda^{a}}{2}$, the $\lambda^{a}$ is the Gell-Mann
matrix. Up to now, none of those four-quark vacuum condensates are well
constrained or understood in any meaningful way. The commonly adopted method
is the so-called vacuum saturation (or factorization) approximation, we insert
a complete set of intermediate states between two $\bar{q}$ and $q$ quarks and
assume the vacuum contribution dominates the sum of all the intermediate
states, and obtain the squared Chiral condensate $\langle
0|:\bar{q}q:|0\rangle^{2}$,

$\displaystyle\langle
0|:\bar{q}_{\alpha}^{i}q_{\beta}^{j}\bar{q}_{\lambda}^{m}q_{\tau}^{n}:|0\rangle$
$\displaystyle=$ $\displaystyle\frac{1}{16N_{c}^{2}}\langle
0|:\bar{q}q:|0\rangle^{2}\left(\delta_{ij}\delta_{mn}\delta_{\alpha\beta}\delta_{\lambda\tau}-\delta_{in}\delta_{jm}\delta_{\alpha\tau}\delta_{\beta\lambda}\right)\,,$
(1)

where the $i$, $j$, $m$ and $n$ are color indexes, the $\alpha$, $\beta$,
$\lambda$ and $\tau$ are Dirac spinor indexes. In the original works, Shifman,
Vainshtein and Zakharov took the vacuum saturation hypothesis according to two
reasons [2, 3], one reason is the rather large value of the quark (or Chiral)
condensate $\langle\bar{q}q\rangle$, the other is the duality between the
quark and physical states, which implies that counting both the quark and
physical states may well become a double counting because they reproduce each
other [2, 3]. In additional, according to the arguments of Shifman, Vainshtein
and Zakharov and Ioffe [2, 3, 4], the accuracy of factorization hypothesis is
of order $\frac{1}{N_{c}^{2}}$, the vacuum saturation (or factorization) works
well in the large $N_{c}$ limit $\frac{1}{N_{c}^{2}}\sim 0$ [5], in reality,
$N_{c}=3$, $\frac{1}{N_{c}^{2}}\sim 10\%$.

We can introduce a parameter $\kappa$ to take into account the violation of
the vacuum saturation (or factorization) approximation [6],

$\displaystyle\langle 0|:\bar{q}q:|0\rangle^{2}$ $\displaystyle\to$
$\displaystyle\kappa\,\langle 0|:\bar{q}q:|0\rangle^{2}\,,$ (2)

the value $\kappa=1$ stands for the vacuum saturation approximation, while the
value $\kappa>1$ parameterizes its violation. A number of values have been
obtained, which depend on the studied channels. The existing estimations range
from $\kappa=1$ [7] to $2\sim 3$ [8] and even up to 6 [9].

In the QCD sum rules for the traditional mesons, we usually carry out the
operator product expansion up to the vacuum condensates of dimension $6$, the
four-quark condensate $\langle\bar{q}q\rangle^{2}$ is always companied with
the strong fine-structure constant $\alpha_{s}(\mu)$, the net effect is
greatly depressed, and plays a minor important role, the deviation from the
value $\kappa=1$ cannot make much difference, although the value $\kappa>1$
can lead to better QCD sum rules in some cases [9]. While in the QCD sum rules
for the hidden-charm (or hidden-bottom) tetraquark (molecular) states, we
usually carry out the operator product expansion up to the vacuum condensates
of dimension $10$, the vacuum condensate $\langle\bar{q}q\rangle^{2}$ plays a
very important role, the derivations from $\kappa=1$ make much diffidence on
the predictions [10].

In the present work, we choose the hidden-charm axialvector and vector four-
quark currents as an example to examine the validity of the vacuum saturation
(or factorization) approximation in details.

The article is arranged in the form: we obtain the QCD sum rules for the
axialvector and vector tetraquark states in section 2; in section 3, we
present the numerical results and discussions; section 4 is reserved for our
conclusion.

##  2 QCD sum rules for the hidden-charm tetraquark states

Let us write down the correlation functions $\Pi_{\mu\nu}(p)$ in the QCD sum
rules firstly,

$\displaystyle\Pi_{\mu\nu}(p)$ $\displaystyle=$ $\displaystyle i\int
d^{4}xe^{ip\cdot x}\langle
0|T\left\\{J_{\mu}(x)J_{\nu}^{\dagger}(0)\right\\}|0\rangle\,,$ (3)

where $J_{\mu}(x)=J_{\mu}^{A}(x)$ and $J_{\mu}^{V}(x)$,

$\displaystyle J_{\mu}^{A}(x)$ $\displaystyle=$
$\displaystyle\varepsilon^{ijk}\varepsilon^{imn}u^{Tj}(x)C\gamma_{5}c^{k}(x)\bar{d}^{m}(x)\gamma_{\mu}C\bar{c}^{Tn}(x)\,,$
$\displaystyle J_{\mu}^{V}(x)$ $\displaystyle=$
$\displaystyle\varepsilon^{ijk}\varepsilon^{imn}u^{Tj}(x)Cc^{k}(x)\bar{d}^{m}(x)\gamma_{\mu}C\bar{c}^{Tn}(x)\,,$
(4)

where the $i$, $j$, $k$, $m$ and $n$ are color indexes, the charge conjugation
matrix $C=i\gamma^{2}\gamma^{0}$. The simple four-quark currents
$J_{\mu}^{A}(x)$ and $J_{\mu}^{V}(x)$, which lead to simple analytical
expressions of the QCD sum rules, have both negative and positive charge-
conjugation components. Now we write down the two components explicitly,
$\sqrt{2}J_{\mu}^{A}(x)=J^{SA}_{-,\mu}(x)+J^{SA}_{+,\mu}(x)$,
$\sqrt{2}J_{\mu}^{V}(x)=J^{PA}_{-,\mu}(x)+J^{PA}_{+,\mu}(x)$, where

$\displaystyle J^{SA}_{-,\mu}(x)$ $\displaystyle=$
$\displaystyle\frac{\varepsilon^{ijk}\varepsilon^{imn}}{\sqrt{2}}\Big{[}u^{Tj}(x)C\gamma_{5}c^{k}(x)\bar{d}^{m}(x)\gamma_{\mu}C\bar{c}^{Tn}(x)-u^{Tj}(x)C\gamma_{\mu}c^{k}(x)\bar{d}^{m}(x)\gamma_{5}C\bar{c}^{Tn}(x)\Big{]}\,,$
$\displaystyle J^{SA}_{+,\mu}(x)$ $\displaystyle=$
$\displaystyle\frac{\varepsilon^{ijk}\varepsilon^{imn}}{\sqrt{2}}\Big{[}u^{Tj}(x)C\gamma_{5}c^{k}(x)\bar{d}^{m}(x)\gamma_{\mu}C\bar{c}^{Tn}(x)+u^{Tj}(x)C\gamma_{\mu}c^{k}(x)\bar{d}^{m}(x)\gamma_{5}C\bar{c}^{Tn}(x)\Big{]}\,,$
$\displaystyle J^{PA}_{-,\mu}(x)$ $\displaystyle=$
$\displaystyle\frac{\varepsilon^{ijk}\varepsilon^{imn}}{\sqrt{2}}\Big{[}u^{Tj}(x)Cc^{k}(x)\bar{d}^{m}(x)\gamma_{\mu}C\bar{c}^{Tn}(x)-u^{Tj}(x)C\gamma_{\mu}c^{k}(x)\bar{d}^{m}(x)C\bar{c}^{Tn}(x)\Big{]}\,,$
$\displaystyle J^{PA}_{+,\mu}(x)$ $\displaystyle=$
$\displaystyle\frac{\varepsilon^{ijk}\varepsilon^{imn}}{\sqrt{2}}\Big{[}u^{Tj}(x)Cc^{k}(x)\bar{d}^{m}(x)\gamma_{\mu}C\bar{c}^{Tn}(x)+u^{Tj}(x)C\gamma_{\mu}c^{k}(x)\bar{d}^{m}(x)C\bar{c}^{Tn}(x)\Big{]}\,,$
(6)

and the superscripts $S$, $A$ and $P$ represent the scalar, axialvector and
pseudoscalar (anti)diquark operators, respectively. The axialvector currents
$J^{SA}_{+,\mu}(x)$ and $J^{SA}_{-,\mu}(x)$ couple potentially to the hidden-
charm tetraquark states with the $J^{PC}=1^{++}$ and $1^{+-}$ respectively
which have almost degenerated masses but slightly different pole residues [11,
12], while the vector currents $J^{PA}_{+,\mu}(x)$ and $J^{PA}_{-,\mu}(x)$
couple potentially to the hidden-charm tetraquark states with the
$J^{PC}=1^{-+}$ and $1^{--}$ respectively which also have almost degenerated
masses but slightly different pole residues [13, 14]. In the present work, we
will not distinguish the charge-conjugation, and choose the simple four-quark
currents $J_{\mu}^{A}(x)$ and $J_{\mu}^{V}(x)$, which couple potentially to
the axialvector and vector hidden-charm tetraquark states with the
$J^{P}=1^{+}$ and $1^{-}$, respectively, as we are only interested in the
tetraquark masses.

The axialvector current $J^{A}_{\mu}(x)$ is of the
$C\gamma_{5}\otimes\gamma_{\mu}C$ type, while the vector current
$J^{V}_{\mu}(x)$ is of the $C\otimes\gamma_{\mu}C$ type, the relative P-wave
is tacitly or implicitly embodied in the pseudoscalar diquark. On the other
hand, we can construct the $C\gamma_{5}\otimes\gamma_{5}\gamma_{\mu}C$ type
vector current, where the relative P-wave is tacitly or implicitly embodied in
the vector diquark [14], or introduce a relative P-wave between the diquark
and antidiquark explicitly, and construct the

$\displaystyle
C\gamma_{5}\otimes\stackrel{{\scriptstyle\leftrightarrow}}{{\partial}}_{\mu}\otimes\gamma_{5}C\,,$
$\displaystyle
C\gamma_{\alpha}\otimes\stackrel{{\scriptstyle\leftrightarrow}}{{\partial}}_{\mu}\otimes\gamma^{\alpha}C\,,$
$\displaystyle
C\gamma_{\mu}\otimes\stackrel{{\scriptstyle\leftrightarrow}}{{\partial}}_{\alpha}\otimes\gamma^{\alpha}C+C\gamma^{\alpha}\otimes\stackrel{{\scriptstyle\leftrightarrow}}{{\partial}}_{\alpha}\otimes\gamma_{\mu}C\,,$
$\displaystyle
C\gamma_{5}\otimes\stackrel{{\scriptstyle\leftrightarrow}}{{\partial}}_{\mu}\otimes\gamma_{\nu}C+C\gamma_{\nu}\otimes\stackrel{{\scriptstyle\leftrightarrow}}{{\partial}}_{\mu}\otimes\gamma_{5}C-C\gamma_{5}\otimes\stackrel{{\scriptstyle\leftrightarrow}}{{\partial}}_{\nu}\otimes\gamma_{\mu}C-C\gamma_{\mu}\otimes\stackrel{{\scriptstyle\leftrightarrow}}{{\partial}}_{\nu}\otimes\gamma_{5}C\,,$
(7)

type vector currents, where
$\stackrel{{\scriptstyle\leftrightarrow}}{{\partial}}_{\mu}=\overrightarrow{\partial}_{\mu}-\overleftarrow{\partial}_{\mu}$
[15, 16]. As the analytical expressions of the QCD spectral densities for
those currents are lengthy and do not have simple relations with that for the
$C\gamma_{5}\otimes\gamma_{\mu}C$ type current. We prefer the currents
$J^{A}_{\mu}(x)$ and $J^{V}_{\mu}(x)$, because they lead to the QCD spectral
densities which have simple relations with each other.

At the hadron side, we isolate the contributions of the ground state hidden-
charm tetraquark states $Z_{c}$,

$\displaystyle\Pi_{\mu\nu}(p)$ $\displaystyle=$
$\displaystyle\frac{\lambda_{Z}^{2}}{M_{Z}^{2}-p^{2}}\left(-g_{\mu\nu}+\frac{p_{\mu}p_{\nu}}{p^{2}}\right)+\cdots\,\,,$
(8) $\displaystyle=$
$\displaystyle\Pi(p^{2})\left(-g_{\mu\nu}+\frac{p_{\mu}p_{\nu}}{p^{2}}\right)+\cdots\,\,,$

where the $M_{Z}$ and $\lambda_{Z}$ are masses and pole residues of the
hidden-charm tetraquark states $Z_{c}$, respectively, the $\lambda_{Z}$ are
defined by $\langle
0|J_{\mu}(0)|Z_{c}(p)\rangle=\lambda_{Z}\,\varepsilon_{\mu}$, the
$\varepsilon_{\mu}$ are the polarization vectors of the axialvector and vector
tetraquark states. We resort to the component $\Pi(p^{2})$ to investigate the
axialvector and vector tetraquark states.

We accomplish the operator product expansion up to the vacuum condensates of
dimension-10 consistently, which are the vacuum expectations of the quark-
gluon operators of the order $\mathcal{O}(\alpha_{s}^{k})$ with $k\leq 1$. In
calculations, we assume vacuum saturation for the higher dimensional vacuum
condensates and factorize the higher dimensional vacuum condensates into the
product of the lower ones. Then we write the correlation functions
$\Pi(p^{2})$ in the form,

$\displaystyle\Pi(p^{2})$ $\displaystyle=$
$\displaystyle\int_{4m_{c}^{2}}^{\infty}ds\frac{\rho_{QCD}(s)}{s-p^{2}}\,,$
(9)

through dispersion relation, where the $\rho_{QCD}$ are the spectral densities
at the quark-gluon level.

We implement the quark-hadron duality below the continuum thresholds $s_{0}$
and accomplish Borel transform with respect to the variable $P^{2}=-p^{2}$ to
obtain the QCD sum rules,

$\displaystyle\lambda^{2}_{Z}\,\exp\left(-\frac{M^{2}_{Z}}{T^{2}}\right)=\int_{4m_{c}^{2}}^{s_{0}}ds\,\rho_{QCD}(s)\,\exp\left(-\frac{s}{T^{2}}\right)\,,$
(10)

where the $T^{2}$ are the Borel parameters, $\rho_{QCD}(s)=\rho_{A}(s)$ and
$\rho_{V}(s)$, $\rho_{A}(s)=\sum\limits_{i}\rho^{A}_{i}(s)$,
$\rho_{V}(s)=\sum\limits_{i}\rho^{V}_{i}(s)$ with $i=0$, $3$, $4$, $5$, $6$,
$7$, $8$ and $10$,

$\displaystyle\rho^{A}_{0}(s)$ $\displaystyle=$
$\displaystyle\frac{1}{1024\pi^{6}}\int
dydz\,{y}{z}(1-y-z)^{2}\left(s-\overline{m}_{c}^{2}\right)^{3}(5s-\bar{m}_{c}^{2})\,,$
(11)

$\displaystyle\rho^{A}_{3}(s)$ $\displaystyle=$
$\displaystyle-\frac{{m}_{c}\langle\bar{q}q\rangle}{32\pi^{4}}\int
dydz\,{y}(1-y-z)\left(s-\overline{m}_{c}^{2}\right)(7s-3\overline{m}_{c}^{2})\,,$
(12)

$\displaystyle\rho^{A}_{4}(s)$ $\displaystyle=$
$\displaystyle-\frac{{m}_{c}^{2}}{384\pi^{4}}\langle\frac{\alpha_{s}{G}{G}}{\pi}\rangle\int
dydz\,\frac{z(1-y-z)^{2}}{y^{2}}(2s-\overline{m}_{c}^{2})$ (13)
$\displaystyle+\frac{1}{384\pi^{4}}\langle\frac{\alpha_{s}{G}{G}}{\pi}\rangle\int
dydz\,z(1-y-z)s(s-\overline{m}_{c}^{2})\,,$

$\displaystyle\rho^{A}_{5}(s)$ $\displaystyle=$
$\displaystyle\frac{{m}_{c}\langle\bar{q}g_{s}\sigma Gq\rangle}{64\pi^{4}}\int
dydz\,y(5s-3\overline{m}_{c}^{2})$ (14)
$\displaystyle-\frac{{m}_{c}\langle\bar{q}g_{s}\sigma
Gq\rangle}{64\pi^{4}}\int
dydz\,\frac{z(1-y-z)}{y}(2s-\overline{m}_{c}^{2})\,,$

$\displaystyle\rho^{A}_{6}(s)$ $\displaystyle=$
$\displaystyle\frac{{m}_{c}^{2}\langle\bar{q}q\rangle^{2}}{12\pi^{2}}\int
dy\,,$ (15)

$\displaystyle\rho^{A}_{7}(s)$ $\displaystyle=$
$\displaystyle\frac{{m}_{c}^{3}\langle\bar{q}q\rangle}{288\pi^{2}}\langle\frac{\alpha_{s}{G}{G}}{\pi}\rangle\int
dydz\,\frac{(1-y-z)(y+z)}{y^{3}}\left(1+\frac{2s}{T^{2}}\right)\delta\left(s-\overline{m}_{c}^{2}\right)$
(16)
$\displaystyle-\frac{{m}_{c}\langle\bar{q}q\rangle}{96\pi^{2}}\langle\frac{\alpha_{s}{G}{G}}{\pi}\rangle\int
dydz\,\frac{z(1-y-z)}{y^{2}}\left[3+2s\,\delta\left(s-\overline{m}_{c}^{2}\right)\right]$
$\displaystyle-\frac{{m}_{c}\langle\bar{q}q\rangle}{576\pi^{2}}\langle\frac{\alpha_{s}{G}{G}}{\pi}\rangle\int
dydz\,\left[3+2s\,\delta\left(s-\overline{m}_{c}^{2}\right)\right]$
$\displaystyle-\frac{{m}_{c}\langle\bar{q}q\rangle}{576\pi^{2}}\langle\frac{\alpha_{s}{G}{G}}{\pi}\rangle\int
dy\,y\left[3+2s\,\delta\left(s-\widetilde{m}_{c}^{2}\right)\right]\,,$

$\displaystyle\rho^{A}_{8}(s)$ $\displaystyle=$
$\displaystyle-\frac{{m}_{c}^{2}\langle\bar{q}q\rangle\langle\bar{q}g_{s}\sigma
Gq\rangle}{24\pi^{2}}\int
dy\,\left(1+\frac{s}{T^{2}}\right)\delta\left(s-\widetilde{m}_{c}^{2}\right)$
(17)
$\displaystyle+\frac{{m}_{c}^{2}\langle\bar{q}q\rangle\langle\bar{q}g_{s}\sigma
Gq\rangle}{48\pi^{2}}\rangle\int
dy\,\frac{1}{y}\delta\left(s-\widetilde{m}_{c}^{2}\right)\,,$

$\displaystyle\rho^{A}_{10}(s)$ $\displaystyle=$
$\displaystyle\frac{{m}_{c}^{2}\langle\bar{q}g_{s}\sigma
Gq\rangle^{2}}{192\pi^{2}T^{6}}\int
dy\,s^{2}\,\delta\left(s-\widetilde{m}_{c}^{2}\right)$ (18)
$\displaystyle-\frac{{m}_{c}^{2}\langle\bar{q}g_{s}\sigma
Gq\rangle^{2}}{192\pi^{2}T^{4}}\int
dy\,\frac{s}{y}\delta\left(s-\widetilde{m}_{c}^{2}\right)$
$\displaystyle-\frac{{m}_{c}^{4}\langle\bar{q}q\rangle^{2}}{108T^{4}}\langle\frac{\alpha_{s}{G}{G}}{\pi}\rangle\int
dy\,\frac{1}{y^{3}}\delta\left(s-\widetilde{m}_{c}^{2}\right)$
$\displaystyle+\frac{{m}_{c}^{2}\langle\bar{q}q\rangle^{2}}{36T^{2}}\langle\frac{\alpha_{s}{G}{G}}{\pi}\rangle\int
dy\,\frac{1}{y^{2}}\delta\left(s-\widetilde{m}_{c}^{2}\right)$
$\displaystyle+\frac{{m}_{c}^{2}\langle\bar{q}q\rangle^{2}}{216T^{6}}\langle\frac{\alpha_{s}{G}{G}}{\pi}\rangle\int
dy\,s^{2}\,\delta\left(s-\widetilde{m}_{c}^{2}\right)\,,$

$\displaystyle\rho^{V}_{0}(s)$ $\displaystyle=$
$\displaystyle\rho^{A}_{0}(s)\,,$ (19)

$\displaystyle\rho^{V}_{3}(s)$ $\displaystyle=$
$\displaystyle-\frac{{m}_{c}\langle\bar{q}q\rangle}{32\pi^{4}}\int
dydz\,{y}(1-y-z)\left(s-\overline{m}_{c}^{2}\right)^{2}\,,$ (20)

$\displaystyle\rho^{V}_{4}(s)$ $\displaystyle=$
$\displaystyle\rho^{A}_{4}(s)\,,$ (21)

$\displaystyle\rho^{V}_{5}(s)$ $\displaystyle=$
$\displaystyle\frac{{m}_{c}\langle\bar{q}g_{s}\sigma Gq\rangle}{64\pi^{4}}\int
dydz\,y(s-\overline{m}_{c}^{2})$ (22)
$\displaystyle+\frac{{m}_{c}\langle\bar{q}g_{s}\sigma
Gq\rangle}{64\pi^{4}}\int
dydz\,\frac{z(1-y-z)}{y}(2s-\overline{m}_{c}^{2})\,,$

$\displaystyle\rho^{V}_{6}(s)$ $\displaystyle=$
$\displaystyle-\rho^{A}_{6}(s)\,,$ (23)

$\displaystyle\rho^{V}_{7}(s)$ $\displaystyle=$
$\displaystyle\frac{{m}_{c}^{3}\langle\bar{q}q\rangle}{288\pi^{2}}\langle\frac{\alpha_{s}{G}{G}}{\pi}\rangle\int
dydz\,\frac{(1-y-z)(y+z)}{y^{3}}\delta\left(s-\overline{m}_{c}^{2}\right)$
(24)
$\displaystyle-\frac{{m}_{c}\langle\bar{q}q\rangle}{96\pi^{2}}\langle\frac{\alpha_{s}{G}{G}}{\pi}\rangle\int
dydz\,\frac{z(1-y-z)}{y^{2}}$
$\displaystyle-\frac{{m}_{c}\langle\bar{q}q\rangle}{576\pi^{2}}\langle\frac{\alpha_{s}{G}{G}}{\pi}\rangle\int
dydz\,\left[9+4s\delta\left(s-\overline{m}_{c}^{2}\right)\right]$
$\displaystyle-\frac{{m}_{c}\langle\bar{q}q\rangle}{576\pi^{2}}\langle\frac{\alpha_{s}{G}{G}}{\pi}\rangle\int
dy\,y\,,$

$\displaystyle\rho^{V}_{8}(s)$ $\displaystyle=$
$\displaystyle-\rho^{A}_{8}(s)\,,$ (25)

$\displaystyle\rho^{V}_{10}(s)$ $\displaystyle=$
$\displaystyle-\rho^{A}_{10}(s)\,,$ (26)

here $y_{f}=\frac{1+\sqrt{1-4m_{c}^{2}/s}}{2}$,
$y_{i}=\frac{1-\sqrt{1-4m_{c}^{2}/s}}{2}$, $z_{i}=\frac{ym_{c}^{2}}{ys-
m_{c}^{2}}$, $\overline{m}_{c}^{2}=\frac{(y+z)m_{c}^{2}}{yz}$,
$\widetilde{m}_{c}^{2}=\frac{m_{c}^{2}}{y(1-y)}$, $\int
dydz=\int_{y_{i}}^{y_{f}}dy\int_{z_{i}}^{1-y}dz$ and $\int
dy=\int_{y_{i}}^{y_{f}}dy$. When the $\delta$ functions
$\delta\left(s-\overline{m}_{c}^{2}\right)$ and
$\delta\left(s-\widetilde{m}_{c}^{2}\right)$ appear,
$\int_{y_{i}}^{y_{f}}dy\to\int_{0}^{1}dy$,
$\int_{z_{i}}^{1-y}dz\to\int_{0}^{1-y}dz$.

From the analytical expressions of the spectral densities $\rho_{10}^{A/V}(s)$
involving the vacuum condensates of dimension $10$, the highest dimensional
vacuum condensates, we can see explicitly that the $\langle\bar{q}g_{s}\sigma
Gq\rangle^{2}$ and
$\langle\bar{q}q\rangle^{2}\langle\frac{\alpha_{s}GG}{\pi}\rangle$ are
companied with the inverted Borel parameters $\frac{1}{T^{2}}$,
$\frac{1}{T^{4}}$ or $\frac{1}{T^{6}}$, their contributions are greatly
amplified at the small values of the $T^{2}$, and they play a great important
role in determining the lowest values of the Borel parameters. In the spectral
densities $\rho_{8}^{A/V}(s)$, some terms involving the vacuum condensate
$\langle\bar{q}q\rangle\langle\bar{q}g_{s}\sigma Gq\rangle$ are companied with
the inverted Borel parameter $\frac{1}{T^{2}}$ besides the small numerical
denominators, and their contributions are also amplified at the small values
of the $T^{2}$ and also play a great important role in determining the lowest
values of the Borel parameters.

In the spectral density $\rho_{7}^{A}(s)$, there exists a term companied with
the inverted Borel parameter $\frac{1}{T^{2}}$, while in the spectral density
$\rho_{7}^{V}(s)$, there exists no term companied with the inverted Borel
parameter $\frac{1}{T^{2}}$ at all. Even in the case of small Borel
parameters, the contributions of the vacuum condensate
$\langle\bar{q}q\rangle\langle\frac{\alpha_{s}GG}{\pi}\rangle$ are very small
in all the QCD sum rules for the hidden-charm tetraquark (molecular) states
due to the small value of the
$\langle\bar{q}q\rangle\langle\frac{\alpha_{s}GG}{\pi}\rangle$ and the
associated large numerical denominators [10, 11, 12, 13, 14], the vacuum
condensate $\langle\bar{q}q\rangle\langle\frac{\alpha_{s}GG}{\pi}\rangle$
plays a tiny role both in and out the Borel windows. While in the spectral
densities $\rho_{6}^{A/V}(s)$, the contributions of the vacuum condensate
$\langle\bar{q}q\rangle^{2}$ are greatly enhanced due to the associated small
numerical denominators. For the conventional mesons, the vacuum condensate
$\langle\bar{q}q\rangle^{2}$ is always companied with the strong fine-
structure constant $\alpha_{s}(\mu)$ and its contribution is greatly
suppressed, and the conventional mesons are not the ideal channels to explore
the non-factorizable effects [9].

All in all, the higher dimensional vacuum condensates
$\langle\bar{q}q\rangle^{2}$, $\langle\bar{q}q\rangle\langle\bar{q}g_{s}\sigma
Gq\rangle$, $\langle\bar{q}g_{s}\sigma Gq\rangle^{2}$ and
$\langle\bar{q}q\rangle^{2}\langle\frac{\alpha_{s}GG}{\pi}\rangle$ play an
important role in determining the Borel windows, their positive or negative
contributions, say $\rho^{V}_{6}(s)=-\rho^{A}_{6}(s)$,
$\rho^{V}_{8}(s)=-\rho^{A}_{8}(s)$ and $\rho^{V}_{10}(s)=-\rho^{A}_{10}(s)$,
have great influences on the convergent behaviors of the operator product
expansion and the pole contributions. We can take account of the non-
factorizable effects by introducing a parameter $\kappa$,

$\displaystyle\langle\bar{q}q\rangle^{2}$ $\displaystyle\to$
$\displaystyle\kappa\,\langle\bar{q}q\rangle^{2}\,,$
$\displaystyle\langle\bar{q}q\rangle\langle\bar{q}g_{s}\sigma Gq\rangle$
$\displaystyle\to$
$\displaystyle\kappa\,\langle\bar{q}q\rangle\langle\bar{q}g_{s}\sigma
Gq\rangle\,,$ $\displaystyle\langle\bar{q}g_{s}\sigma Gq\rangle^{2}$
$\displaystyle\to$ $\displaystyle\kappa\,\langle\bar{q}g_{s}\sigma
Gq\rangle^{2}\,,$
$\displaystyle\langle\bar{q}q\rangle^{2}\langle\frac{\alpha_{s}GG}{\pi}\rangle$
$\displaystyle\to$
$\displaystyle\kappa\,\langle\bar{q}q\rangle^{2}\langle\frac{\alpha_{s}GG}{\pi}\rangle\,.$
(27)

We should bear in mind that the parameter $\kappa$ associated with the vacuum
condensates of dimensions $6$, $8$ and $10$ is not necessary to have the same
value. The four-quark vacuum condensate $\langle\bar{q}q\rangle^{2}$ serves as
a milestone for the hidden-charm (or hidden-bottom) tetraquark (molecular)
states [10, 11, 12, 13, 14], the dominant influence of the higher dimensional
vacuum condensates comes from the vacuum condensate
$\langle\bar{q}q\rangle^{2}$ near the Borel windows, the Borel windows are
sensitive to the coefficient $\kappa$ associated with the
$\langle\bar{q}q\rangle^{2}$.

In the present work, we take account of the non-factorizable effects of the
vacuum condensates $\langle\bar{q}q\rangle\langle\bar{q}g_{s}\sigma
Gq\rangle$, $\langle\bar{q}g_{s}\sigma Gq\rangle^{2}$ and
$\langle\bar{q}q\rangle^{2}\langle\frac{\alpha_{s}GG}{\pi}\rangle$ for a more
robust estimation, as they are also acquired via the vacuum saturation after
all, and we will simplify the analysis and set all the parameters $\kappa$ to
have the same values.

We differentiate Eq.(10) with respect to $\tau=\frac{1}{T^{2}}$, and eliminate
the pole residues $\lambda_{Z}$ to obtain the QCD sum rules for the tetraquark
masses,

$\displaystyle
M^{2}_{Z}=\frac{-\frac{d}{d\tau}\int_{4m_{c}^{2}}^{s_{0}}ds\,\rho_{QCD}(s)\,e^{-\tau
s}}{\int_{4m_{c}^{2}}^{s_{0}}ds\,\rho_{QCD}(s)\,e^{-\tau s}}\,.$ (28)

##  3 Numerical results and discussions

We adopt two sets of input parameters at the QCD side to explore the hidden-
charm tetraquark states.

Set I

We choose the standard values of the vacuum condensates
$\langle\bar{q}q\rangle=-(0.24\pm 0.01\,\rm{GeV})^{3}$,
$\langle\bar{q}g_{s}\sigma Gq\rangle=m_{0}^{2}\langle\bar{q}q\rangle$,
$m_{0}^{2}=(0.8\pm 0.1)\,\rm{GeV}^{2}$,
$\langle\frac{\alpha_{s}GG}{\pi}\rangle=0.012\pm 0.004\,\rm{GeV}^{4}$ at the
energy scale $\mu=1\,\rm{GeV}$ [2, 3, 17, 18, 19, 20], and take the
$\overline{MS}$ mass of the charm quark from the Particle Data Group,
$m_{c}(m_{c})=(1.275\pm 0.025)\,\rm{GeV}$ [21]. In addition, we take account
of the energy-scale dependence of all the input parameters [22],

$\displaystyle\langle\bar{q}q\rangle(\mu)$ $\displaystyle=$
$\displaystyle\langle\bar{q}q\rangle({\rm 1GeV})\left[\frac{\alpha_{s}({\rm
1GeV})}{\alpha_{s}(\mu)}\right]^{\frac{12}{33-2n_{f}}}\,,$
$\displaystyle\langle\bar{q}g_{s}\sigma Gq\rangle(\mu)$ $\displaystyle=$
$\displaystyle\langle\bar{q}g_{s}\sigma Gq\rangle({\rm
1GeV})\left[\frac{\alpha_{s}({\rm
1GeV})}{\alpha_{s}(\mu)}\right]^{\frac{2}{33-2n_{f}}}\,,$ $\displaystyle
m_{c}(\mu)$ $\displaystyle=$ $\displaystyle
m_{c}(m_{c})\left[\frac{\alpha_{s}(\mu)}{\alpha_{s}(m_{c})}\right]^{\frac{12}{33-2n_{f}}}\,,$
$\displaystyle\alpha_{s}(\mu)$ $\displaystyle=$
$\displaystyle\frac{1}{b_{0}t}\left[1-\frac{b_{1}}{b_{0}^{2}}\frac{\log
t}{t}+\frac{b_{1}^{2}(\log^{2}{t}-\log{t}-1)+b_{0}b_{2}}{b_{0}^{4}t^{2}}\right]\,,$
(29)

where $t=\log\frac{\mu^{2}}{\Lambda^{2}}$, $b_{0}=\frac{33-2n_{f}}{12\pi}$,
$b_{1}=\frac{153-19n_{f}}{24\pi^{2}}$,
$b_{2}=\frac{2857-\frac{5033}{9}n_{f}+\frac{325}{27}n_{f}^{2}}{128\pi^{3}}$,
$\Lambda=213\,\rm{MeV}$, $296\,\rm{MeV}$ and $339\,\rm{MeV}$ for the quark
flavor numbers $n_{f}=5$, $4$ and $3$, respectively [21]. In the present work,
we explore the hidden-charm tetraquark states and choose $n_{f}=4$, then
evolve all the input parameters to some typical energy scales $\mu$ to extract
the tetraquark masses. The value
$\langle\frac{\alpha_{s}GG}{\pi}\rangle=0.012\pm 0.004\,\rm{GeV}^{4}$ is still
frequently used in the current QCD sum rules analysis, as no significant
progress in its determination has ever been made and proven to be more
reliable.

Set II

We choose the updated parameters obtained by S. Narison, including the mass of
the charm quark and the gluon condensate, $m_{c}(m_{c})=(1.266\pm
0.006)\,\rm{GeV}$ and $\langle\frac{\alpha_{s}GG}{\pi}\rangle=0.021\pm
0.001\,\rm{GeV}^{4}$ [23], while the energy scales of the vacuum condensates
are set to be $\mu=1\,\rm{GeV}$.

For the parameter Set I, we take the energy scale formula
$\mu=\sqrt{M^{2}_{X/Y/Z}-(2{\mathbb{M}}_{c})^{2}}$ with the effective
$c$-quark mass, ${\mathbb{M}}_{c}=1.82\,\rm{GeV}$, to obtain the suitable
energy scales of the QCD spectral densities [13, 24]. We can rewrite the
energy scale formula in the form,

$\displaystyle M^{2}_{X/Y/Z}$ $\displaystyle=$ $\displaystyle\mu^{2}+{\rm
Constants}\,,$ (30)

where the Constants have the value $4{\mathbb{M}}_{c}^{2}$ and are fitted by
the QCD sum rules, the predicted tetraquark masses and the pertinent energy
scales of the QCD spectral densities have a Regge-trajectory-like relation
[10].

We usually consult the experimental data on the mass gaps between the ground
states and first radial excited states to adjust the continuum threshold
parameters $s_{0}$ to exclude the possible contaminations from the continuum
states. Although the tetraquark states have not been experimentally
established yet, there are several excellent candidates, such as the
$Z_{c}(3900)$, $Z_{c}(4020)$, $Z_{c}(4430)$, etc.

According to the possible quantum numbers, decay modes and mass gaps, in the
scenarios of tetraquark states, we can tentatively assign the $X(3915)$ and
$X(4500)$ as the 1S and 2S hidden-charm tetraquark states with the
$J^{PC}=0^{++}$ [25, 26], assign the $Z_{c}(3900)$ and $Z_{c}(4430)$ as the 1S
and 2S hidden-charm tetraquark states with the $J^{PC}=1^{+-}$, respectively
[27, 28, 29], assign the $Z_{c}(4020)$ and $Z_{c}(4600)$ as the 1S and 2S
hidden-charm tetraquark states with the $J^{PC}=1^{+-}$, respectively [30,
31], and assign the $X(4140)$ and $X(4685)$ the 1S and 2S hidden-charm
tetraquark states with the $J^{PC}=1^{++}$, respectively [32, 33]. The energy
gaps between the ground states (1S) and first radial excited states (2S) are
about $0.57\sim 0.59\,\rm{GeV}$. In the present work, we can take the
continuum threshold parameters as $\sqrt{s_{0}}=M_{Z}+0.4\sim 0.6\,\rm{GeV}$,
where the $M_{Z}$ are the ground state masses of the hidden-charm tetraquark
states.

Direct calculations based on the QCD sum rules indicate that the
$C\gamma_{5}\otimes\gamma_{\mu}C\pm C\gamma_{\mu}\otimes\gamma_{5}C$ type
hidden-charm tetraquark states $c\bar{c}u\bar{d}$ with the $J^{PC}=1^{++}$ and
$1^{+-}$ respectively have almost degenerated masses [11, 12], while the
$C\otimes\gamma_{\mu}C\pm C\gamma_{\mu}\otimes C$ type hidden-charm tetraquark
states $c\bar{c}u\bar{d}$ with the $J^{PC}=1^{-+}$ and $1^{--}$ respectively
have almost degenerated masses or slightly different masses [13, 14]. In the
present work, we do not distinguish the charge conjugation, and take it for
granted that the $C\gamma_{5}\otimes\gamma_{\mu}C$ type tetraquark state has a
mass about $3.9\,\rm{GeV}$, just like the $Z_{c}(3900)$, and tentatively
choose the continuum threshold parameter as $\sqrt{s_{0}}=4.4\,\rm{GeV}$ as a
guide to optimize the parameters.

In 2019, the LHCb collaboration explored the $m(J/\psi\pi^{-})$ versus
$m(K^{+}\pi^{-})$ plane in the decays $B^{0}\to J/\psi K^{+}\pi^{-}$, and
observed two possible structures near $m(J/\psi\pi^{-})=4200\,\rm{MeV}$ and
$4600\,\rm{MeV}$, respectively [34]. The structure near
$m(J/\psi\pi^{-})=4600\,\rm{MeV}$ is in very good agreement with our predicted
mass of the $C\otimes\gamma_{\mu}C-C\gamma_{\mu}\otimes C$ type tetraquark
state with the $J^{PC}=1^{--}$ [14]. In Ref.[35], we assign the $Z_{c}(4600)$
to be the 1S vector tetraquark state tentatively and study its two-body strong
decays with the QCD sum rules based on solid quark-hadron duality, and obtain
the total decay width $144.8^{+50.6}_{-33.9}\,{\rm{MeV}}$, which is reasonable
for the tetraquark state.

In the scenario of tetraquark states, there are two possible assignments for
the $Z_{c}(4600)$, one is the first radial excited state of the $Z_{c}(4020)$
with the $J^{PC}=1^{+-}$ [30, 31], the other is the 1S vector tetraquark state
with the $J^{PC}=1^{--}$ [35], more experimental data are still needed to
obtain a more reliable assignment. At the present time, we take it for granted
that the $C\otimes\gamma_{\mu}C$ type tetraquark state has a mass about
$4.6\,\rm{GeV}$, and choose the continuum threshold parameter as
$\sqrt{s_{0}}=5.1\,\rm{GeV}$ as a guide to optimize the parameters.

Figure 1:  The absolute contributions of the vacuum condensates, where the A
and V denote the axialvector and vector tetraquark states, respectively, the I
and II denote the parameters Set I and Set II, respectively.

Figure 2:  The absolute contributions of the vacuum condensates for
$\kappa=1$, where the A and V denote the axialvector and vector tetraquark
states, respectively, the I and II denote the parameters Set I and Set II,
respectively.

In Fig.1, we plot the absolute values of the contributions of the vacuum
condensates in the operator product expansion with the parameters
($\sqrt{s_{0}}=4.4\,\rm{GeV}$, $T^{2}=2.9\,\rm{GeV}^{2}$) and
($5.1\,\rm{GeV}$, $3.9\,\rm{GeV}^{2}$) for the axialvector and vector hidden-
charm tetraquark states, respectively. In both cases, the contributions of the
vacuum condensates of dimension $6$ serve as a milestone. Although the
operator product expansion converges for all the values $\kappa=1$, $2$,
$\cdots$, $5$, the convergent behavior becomes worse with increase of the
values of the $\kappa$. A good convergent behavior of the operator product
expansion requires a small value of the $\kappa$. In Fig.2, we plot the
absolute values of the contributions of the vacuum condensates in the operator
product expansion with the value $\kappa=1$ as an example to examine the
influences of the parameters Set I and Set II. From the figure, we can see
that the parameter Set I leads to a slightly better convergent behavior
compared to the parameter Set II.

In the QCD sum rules for the hidden-charm (or hidden-bottom) tetraquark
(molecular) states, we adopt the spectral densities
$\rho_{QCD}(s)\Theta(s-s_{0})$ to represent the contributions of the higher
resonances and continuum states, where $\Theta(s)=1$ for $s>0$, else
$\Theta(s)=0$ [10, 11, 12, 13, 14]. Such approximations cannot ensure the
largest contributions come from the perturbative terms, sometimes the
contributions of the vacuum condensates of dimensions of $3$, $5$ or $6$ are
even larger than the contributions of the perturbative terms, for example, in
the present case for the axialvector hidden-charm tetraquark state, $D(3)\gg
D(0)$ and $|D(5)|\approx D(0)$. In all the QCD sum rules, the vacuum
condensates $\langle\bar{q}q\rangle^{2}$ with $q=u$, $d$, $s$ serve a
milestone in judging the convergent behaviors of the operator product
expansion [10, 11, 12, 13, 14], we require the hierarchies
$|D(6)|>|D(8)|>|D(10)|$ and $|D(10)|\leq 1\%$ or $\ll 1\%$.

Figure 3:  The pole contributions with variations of the Borel parameters,
where the A and V denote the axialvector and vector tetraquark states,
respectively, the I and II denote the parameters Set I and Set II,
respectively.

Figure 4:  The pole contributions with variations of the Borel parameters for
$\kappa=1$, where the A and V denote the axialvector and vector tetraquark
states, respectively, the I and II denote the parameters Set I and Set II,
respectively.

In Fig.3, we plot the pole contributions with the parameters
($\sqrt{s_{0}}=4.4\,\rm{GeV}$, $T^{2}=2.9\,\rm{GeV}^{2}$) and
($5.1\,\rm{GeV}$, $3.9\,\rm{GeV}^{2}$) for the axialvector and vector hidden-
charm tetraquark states, respectively. From the figure, we can see that in the
case of the axialvector tetraquark state, the pole contributions increase
monotonically with increase of the values of the $\kappa$, while in the case
of the vector tetraquark state, the pole contributions decrease monotonically
with increase of the values of the $\kappa$, the pole contributions play an
important role in examining the values of the $\kappa$. However, the pole
contributions alone cannot optimize the parameter $\kappa$. In Fig.4, we plot
the pole contributions with the value $\kappa=1$ as an example to examine the
influences of the parameters Set I and Set II. From the figure, we can see
that the parameter Set I leads to a slightly (much) larger pole contribution
compared to the parameter Set II for the axialvector (vector) tetraquark
state, the pole contributions in the vector tetraquark channels also play an
important role in examining the parameters Set I and Set II.

Figure 5:  The tetraquark masses with variations of the Borel parameters,
where the A and V denote the axialvector and vector tetraquark states,
respectively, the I and II denote the parameters Set I and Set II,
respectively.

In Fig.5, we plot the predicted masses via the Borel parameters with the
continuum threshold parameters $\sqrt{s_{0}}=4.4\,\rm{GeV}$ and
$5.1\,\rm{GeV}$ for the axialvector and vector hidden-charm tetraquark states,
respectively. From the figure, we can see that in the case of the axialvector
tetraquark state, the predicted masses decrease monotonically and slowly with
increase of the values of the $\kappa$, while in the case of the vector
tetraquark state, the predicted masses increase monotonically and quickly with
increase of the values of the $\kappa$, furthermore, in the case of the vector
tetraquark state, we cannot obtain flat Borel platforms with increase of the
values of the $\kappa$. We can obtain the conclusion tentatively that the
vector channel is much better to determine the value of the $\kappa$ compared
to the axialvector channel. Moreover, from the Fig.5, we can see that the
parameter Set II leads to a slightly (much) larger predicted masses compared
to the parameter Set I for the axialvector (vector) tetraquark state, the
predicted masses in the vector channel also play an important role in
examining the parameters Set I and Set II.

According to above analysis, we search for the best Borel parameters $T^{2}$
and continuum threshold parameters $s_{0}$ via trial and error to satisfy the
two fundamental criteria of the QCD sum rules for the typical values
$\kappa=1$, $2$, $3$, $4$ and $5$. The resulting Borel parameters, continuum
threshold parameters, energy scales, pole contributions are shown plainly in
Table 1. In calculations, we choose the same pole contributions $(40-60)\%$ in
all the channels and the central values are larger than (or equal) $50\%$, the
pole dominance criterion is well satisfied. On the other hand, the
contributions of the higher dimensional vacuum condensates satisfy the
relation $|D(6)|\gg|D(8)|\gg|D(10)|$ for all the values of the $\kappa$, the
operator product expansion converges very well. As the two fundamental
criteria of the QCD sum rules are all satisfied, we expect to make reliable
predictions.

Now we take account of all uncertainties of the input parameters, and obtain
the values of the masses and pole residues of the axialvector and vector
hidden-charm tetraquark states, which are shown plainly in Fig.6 and Table 1.
In Fig.6, we plot the predicted masses with variations of the Borel parameters
at much larger ranges than the Borel windows. From the figure, we can see
explicitly that there appear very flat platforms in the Borel windows, the
uncertainties come from the Borel parameters are rather small, the predictions
are reasonable and reliable.

In Fig.7, we plot the central values of the tetraquark masses with variations
of the parameter $\kappa$. From Table 1 and Fig.7, we can see that for the
axialvector tetraquark state, the predicted mass decreases monotonically and
slowly with increase of the parameter $\kappa$, smaller tetraquark mass favors
larger value of the $\kappa$. Compared to the parameter Set I, the predicted
mass decreases slightly quicker for the parameter Set II. The axialvector
channel is not the best channel to distinguish the parameters Set I and Set
II, only by precisely measuring the mass of the axialvector tetraquark state,
we can obtain powerful constraint on the input parameters, for example, if the
ground state has a mass about $3.9\,\rm{GeV}$, the preferred parameters are
(Set I, $\kappa=1$) and (Set II, $\kappa=2$).

For the vector tetraquark state, the predicted mass increases monotonically
and quickly with increase of the parameter $\kappa$, larger tetraquark mass
favors larger value of the $\kappa$, furthermore, the parameter Set II leads
to much larger predicted mass compared to the parameter Set I, the vector
channel is the best channel to distinguish the input parameters. For example,
if the ground state vector tetraquark state has a mass about $4.6\,\rm{GeV}$,
which happens to coincide with the mass of the $Y(4660)$ or $Z_{c}(4600)$ (in
the isospin limit), the preferred parameters are (Set I, $\kappa=1$). On the
other hand, if it has mass about $5.4\,\rm{GeV}$, the preferred parameters are
(Set II, $\kappa=5$). We can confronted the predictions in Table 1 to the
experimental data in the future to select the best parameters.

$J^{P}$ | $T^{2}(\rm{GeV}^{2})$ | $\sqrt{s_{0}}(\rm{GeV})$ | $\mu(\rm{GeV})$ | pole |  | $\kappa$ | $M_{Z}(\rm{GeV})$ | $\lambda_{Z}(10^{-2}\rm{GeV}^{5})$
---|---|---|---|---|---|---|---|---
$1^{+}$ | $2.7-3.1$ | $4.4\pm 0.1$ | $1.4$ | $(40-63)\%$ | Set I | $1$ | $3.91\pm 0.08$ | $2.11\pm 0.34$
$1^{+}$ | $2.8-3.2$ | $4.4\pm 0.1$ | $1.3$ | $(39-61)\%$ | Set I | $2$ | $3.88\pm 0.08$ | $2.01\pm 0.32$
$1^{+}$ | $2.9-3.3$ | $4.4\pm 0.1$ | $1.3$ | $(40-61)\%$ | Set I | $3$ | $3.85\pm 0.09$ | $2.05\pm 0.31$
$1^{+}$ | $2.9-3.3$ | $4.4\pm 0.1$ | $1.2$ | $(40-61)\%$ | Set I | $4$ | $3.85\pm 0.09$ | $1.97\pm 0.28$
$1^{+}$ | $3.0-3.4$ | $4.4\pm 0.1$ | $1.2$ | $(40-60)\%$ | Set I | $5$ | $3.83\pm 0.09$ | $2.02\pm 0.28$
$1^{+}$ | $2.8-3.2$ | $4.5\pm 0.1$ |  | $(39-61)\%$ | Set II | $1$ | $4.02\pm 0.08$ | $2.24\pm 0.34$
$1^{+}$ | $2.7-3.1$ | $4.4\pm 0.1$ |  | $(40-62)\%$ | Set II | $2$ | $3.90\pm 0.08$ | $1.95\pm 0.29$
$1^{+}$ | $2.8-3.2$ | $4.4\pm 0.1$ |  | $(40-62)\%$ | Set II | $3$ | $3.87\pm 0.08$ | $1.98\pm 0.28$
$1^{+}$ | $2.9-3.3$ | $4.4\pm 0.1$ |  | $(40-61)\%$ | Set II | $4$ | $3.84\pm 0.08$ | $2.01\pm 0.28$
$1^{+}$ | $3.0-3.4$ | $4.4\pm 0.1$ |  | $(40-60)\%$ | Set II | $5$ | $3.81\pm 0.09$ | $2.05\pm 0.28$
$1^{-}$ | $3.6-4.2$ | $5.1\pm 0.1$ | $2.8$ | $(40-62)\%$ | Set I | $1$ | $4.62\pm 0.08$ | $6.69\pm 0.81$
$1^{-}$ | $4.0-4.6$ | $5.3\pm 0.1$ | $3.1$ | $(41-61)\%$ | Set I | $2$ | $4.81\pm 0.09$ | $9.14\pm 1.10$
$1^{-}$ | $4.3-4.9$ | $5.4\pm 0.1$ | $3.3$ | $(40-59)\%$ | Set I | $3$ | $4.92\pm 0.09$ | $10.8\pm 1.30$
$1^{-}$ | $4.5-5.1$ | $5.5\pm 0.1$ | $3.4$ | $(41-59)\%$ | Set I | $4$ | $5.02\pm 0.10$ | $12.4\pm 1.50$
$1^{-}$ | $4.6-5.4$ | $5.6\pm 0.1$ | $3.6$ | $(40-61)\%$ | Set I | $5$ | $5.11\pm 0.10$ | $14.1\pm 1.60$
$1^{-}$ | $3.7-4.3$ | $5.4\pm 0.1$ |  | $(40-62)\%$ | Set II | $1$ | $4.91\pm 0.07$ | $6.65\pm 0.83$
$1^{-}$ | $4.1-4.7$ | $5.6\pm 0.1$ |  | $(41-61)\%$ | Set II | $2$ | $5.10\pm 0.07$ | $8.86\pm 0.99$
$1^{-}$ | $4.3-4.9$ | $5.7\pm 0.1$ |  | $(41-61)\%$ | Set II | $3$ | $5.22\pm 0.08$ | $10.3\pm 1.20$
$1^{-}$ | $4.5-5.2$ | $5.8\pm 0.1$ |  | $(40-60)\%$ | Set II | $4$ | $5.32\pm 0.09$ | $11.8\pm 1.40$
$1^{-}$ | $4.6-5.4$ | $5.9\pm 0.1$ |  | $(41-62)\%$ | Set II | $5$ | $5.41\pm 0.09$ | $13.4\pm 1.50$
Table 1:  The Borel parameters, continuum threshold parameters, energy scales,
pole contributions, parameter $\kappa$, masses and pole residues for the
tetraquark states.

Figure 6:  The tetraquark masses with variations of the Borel parameters,
where the A and V denote the axialvector and vector tetraquark states,
respectively, the I and II denote the parameters Set I and Set II,
respectively.  Figure 7:  The tetraquark masses with variations of the value
of the $\kappa$, where the A and V denote the axialvector and vector
tetraquark states, respectively, the I and II denote the parameters Set I and
Set II, respectively.

##  4 Conclusion

In the QCD sum rules for the tetraquark (molecular) states, the higher
dimensional vacuum condensates play an important role in extracting the
tetraquark masses. In the present work, we construct the
$C\gamma_{5}\otimes\gamma_{\mu}C$ type and $C\otimes\gamma_{\mu}C$ type four-
quark currents to investigate the tetraquark states via the QCD sum rules so
as to examine the vacuum saturation or factorization approximation for the
higher dimensional vacuum condensates. We carry out the operator product
expansion up to the vacuum condensates of dimension-10 consistently, and
introduce a parameter $\kappa$ to parameterize the derivation from the vacuum
saturation or factorization approximation, and choose two sets parameters to
examine the tetraquark masses. In calculations, we observe that smaller value
of the $\kappa$ leads to better convergent behavior in the operator product
expansion.

For the axialvector tetraquark state, the predicted masses decrease
monotonically and slowly with increase of the parameter $\kappa$, larger value
of the $\kappa$ leads to smaller tetraquark mass, both the parameters Set I
and Set II can lead to a tetraquark mass about $3.9\,\rm{GeV}$, the
axialvector channel is not the best channel to distinguish the parameters Set
I and Set II. For the vector tetraquark state, the predicted masses increase
monotonically and quickly with increase of the parameter $\kappa$, larger
value of the $\kappa$ leads to larger tetraquark mass, moreover, the parameter
Set II leads to much larger predicted tetraquark mass compared to the
parameter Set I, the vector channel is the best channel to distinguish the
value of the $\kappa$ and the parameters Set I and Set II.

If the $Z_{c}(3900)$ can be assigned to be the lowest axialvector hidden-charm
tetraquark state, the preferred parameters are (Set I, $\kappa=1\sim 2$) or
(Set II, $\kappa=2\sim 3$). On the other hand, if the lowest axialvector
hidden-charm tetraquark state has a mass about $3.8\,\rm{GeV}$, the preferred
parameters are (Set I or Set II, $\kappa=5$). At the vector sector, if the
$Y(4660)$ can be assigned to be the vector hidden-charm tetraquark state with
an implicit P-wave in the (anti)diquark, the preferred parameters are (Set I,
$\kappa=1$); else, if such a vector hidden-charm tetraquark configuration has
a mass about $4.8\,\rm{GeV}$, the favored parameters are (Set I, $\kappa=2$).
We can confronted the predictions in Table 1 to the experimental data in the
future to select the best parameters, or put powerful constraints on the input
parameters.

## Acknowledgements

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

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