Analysis of the $QQ\bar{Q}\bar{Q}$ tetraquark states with QCD sum rules

TL;DR

This paper uses QCD sum rules to predict the masses of specific fully heavy tetraquark states with different quantum numbers, providing theoretical benchmarks for future experimental verification.

Contribution

It presents novel mass predictions for $QQ\bar{Q}\bar{Q}$ tetraquark states with $J^{PC}=0^{++}$ and $2^{++}$ using QCD sum rules, and discusses their internal structure as superpositions of meson-meson pairs.

Findings

Predicted masses: 5.99 GeV and 6.09 GeV for $cc\bar{c}\bar{c}$ states.

Predicted masses: 18.84 GeV and 18.85 GeV for $bb\bar{b}\bar{b}$ states.

Tetraquark states can be viewed as superpositions of meson-meson pairs.

Abstract

In this article, we study the $J^{PC}=0^{++}$ and $2^{++}$ $QQ\bar{Q}\bar{Q}$ tetraquark states with the QCD sum rules, and obtain the predictions $M_{X(cc\bar{c}\bar{c},0^{++})} =5.99\pm0.08\,\rm{GeV}$, $M_{X(cc\bar{c}\bar{c},2^{++})} =6.09\pm0.08\,\rm{GeV}$, $M_{X(bb\bar{b}\bar{b},0^{++})} =18.84\pm0.09\,\rm{GeV}$ and $M_{X(bb\bar{b}\bar{b},2^{++})}=18.85\pm0.09\,\rm{GeV}$, which can be confronted to the experimental data in the future. Furthermore, we illustrate that the diquark-antidiquark type tetraquark state can be taken as a special superposition of a series of meson-meson pairs and embodies the net effects.

Tables

Table 1. Table 1: The Borel parameters, continuum threshold parameters, energy scales, pole contributions, masses and pole residues of the tetraquark states.

	$T^2\left({\mathrm{GeV}}^2\right)$	$s_0\left({\mathrm{GeV}}^2\right)$	$\mu \left(\mathrm{GeV}\right)$	pole	$M_X\left(\mathrm{GeV}\right)$	${\lambda }_X\left({\mathrm{GeV}}^5\right)$
$cc\overline{c}\overline{c}\left(0^{++}\right)$	$4.2-4.6$	$42\pm 1$	$2.0$	$\left(46-62\right)\%$	$5.99\pm 0.08$	$\left(3.72\pm 0.54\right)\times {10}^{-1}$
$cc\overline{c}\overline{c}\left(2^{++}\right)$	$4.6-5.2$	$44\pm 1$	$2.0$	$\left(46-63\right)\%$	$6.09\pm 0.08$	$\left(3.36\pm 0.45\right)\times {10}^{-1}$
$bb\overline{b}\overline{b}\left(0^{++}\right)$	$13.0-13.6$	$374\pm 3$	$3.1$	$\left(49-61\right)\%$	$18.84\pm 0.09$	$6.79\pm 1.27$
$bb\overline{b}\overline{b}\left(2^{++}\right)$	$13.0-13.6$	$375\pm 3$	$3.1$	$\left(51-63\right)\%$	$18.85\pm 0.09$	$5.55\pm 1.00$

Equations

$$(3_c\otimes 3_c)\otimes ({\overline{3}}_c\otimes {\overline{3}}_c)$$

$${\Pi }_{\mu \nu \alpha \beta }(p)$$

$$\Pi (p)$$

$$J_{\mu \nu }(x)$$

$$J(x)$$

$${\eta }^i(x)$$

$${\eta }_{\mu \nu }^j(x)$$

$${\Pi }_{\mu \nu \alpha \beta }(p)$$

$$\Pi (p)$$

$$⟨0|J_{\mu \nu }(0)|X(p)⟩$$

$$⟨0|J(0)|X(p)⟩$$

$$\sum _{\lambda }{\epsilon }_{\alpha \beta }^{\ast }(\lambda ,p){\epsilon }_{\mu \nu }(\lambda ,p)$$

$${\Pi }_{\mu \nu \alpha \beta }(p)$$

$$\Pi (p)$$

$$S_{ij}(x)$$

$${\lambda }_X^2\exp \left(-\frac{M_X^2}{T^2}\right)={\int }_{16m_Q^2}^{s_0}ds{\int }_{z_i}^{z_f}dz{\int }_{t_i}^{t_f}dt{\int }_{r_i}^{r_f}dr\rho (s,z,t,r)\exp \left(-\frac{s}{T^2}\right),$$

$$r_{f/i}$$

$$t_{f/i}$$

$$z_{f/i}$$

$$M_X^2$$

$${\overline{M}}_n(P_0^2)$$

$$r(n,P_0^2)$$

$$m_c(\mu )$$

$$m_b(\mu )$$

$${\alpha }_s(\mu )$$

$$\mathcal{L}$$

$$M_{X(cc\bar{c}\bar{c},0^{++})}$$

$$M_{X(cc\bar{c}\bar{c},2^{++})}$$

$$M_{X(bb\bar{b}\bar{b},0^{++})}$$

$$M_{X(bb\bar{b}\bar{b},2^{++})}$$

$$X(cc\bar{c}\bar{c},0^{++}/2^{++})$$

$$X(bb\bar{b}\bar{b},0^{++}/2^{++})$$

$$X(cc\bar{c}\bar{c},0^{++}/2^{++})$$

$$X(bb\bar{b}\bar{b},0^{++}/2^{++})$$

$$X(cc\bar{c}\bar{c},0^{++}/2^{++})$$

$$X(bb\bar{b}\bar{b},0^{++}/2^{++})$$

$$J_{\mu \nu }$$

$$J$$

Full text

PACS number: 12.39.Mk, 12.38.Lg

Key words: Tetraquark states, QCD sum rules

1 Introduction

The observations of the charmonium-like and bottomonium-like states have provided us with a good opportunity to study the exotic states and understand the strong interactions, especially those charged states $Z_{c}(3885)$, $Z_{c}(3900)$, $Z_{c}(4020)$, $Z_{c}(4025)$, $Z_{c}(4200)$, $Z(4430)$, $Z_{b}(10610)$, $Z_{b}(10650)$, they are excellent candidates for the multiquark states [1]. If they are tetraquark states, they consist two heavy quarks and two light quarks, we have to deal with both the heavy and light degrees of freedom of the dynamics. On the other hand, if there exist tetraquark configurations consist of four heavy quarks, the dynamics is much simple. There have been several works on the mass spectrum of the $QQ\bar{Q}\bar{Q}$ tetraquark states, such as the non-relativistic potential models [2, 3, 4, 5], the Bethe-Salpeter equation [6], the constituent diquark model with spin-spin interaction [7, 8], the constituent quark model with color-magnetic interaction [9], the moment QCD sum rules [10], etc. In this article, we study the tetraquark states consist of four heavy quarks with the Borel QCD sum rules.

The QCD sum rules is a powerful theoretical tool in studying the ground state tetraquark states and molecular states, and has given many successful descriptions of the masses and hadronic coupling constants [11]. In this article, we study the $J^{PC}=0^{++}$ and $2^{++}$ $QQ\bar{Q}\bar{Q}$ tetraquark states, which may be observed in the $e^{+}e^{-}$ and $pp$ collisions, for example, $e^{+}e^{-}\to J/\psi\bar{c}c$, $pp\to\bar{c}c\bar{c}c$. The ATLAS, CMS and LHCb collaborations have measured the cross section for double charmonium production [12], the CMS collaboration has observed the $\Upsilon$ pair production [13].

The quarks have color $SU(3)$ symmetry, we can construct the tetraquark states according to the routine ${\rm quark}\to{\rm diquark}\to{\rm tetraquark}$,

[TABLE]

where the ${\bf 1}_{c}$, ${\bf 3}_{c}$ ($\overline{{\bf 3}}_{c}$), ${\bf 6}_{c}$ and ${\bf 8}_{c}$ denote the color singlet, triplet (antitriplet), sextet and octet, respectively. The one-gluon exchange leads to attractive (repulsive) interaction in the color antitriplet (sextet) channel, which favors (disfavors) the formation of diquark states in the color antitriplet (sextet) [14]. The diquarks $\varepsilon^{ijk}q^{T}_{j}C\Gamma q^{\prime}_{k}$ in color antitriplet have five structures in Dirac spinor space, where the $i$, $j$ and $k$ are color indexes, $C\Gamma=C\gamma_{5}$, $C$, $C\gamma_{\mu}\gamma_{5}$, $C\gamma_{\mu}$ and $C\sigma_{\mu\nu}$ for the scalar, pseudoscalar, vector, axialvector and tensor diquarks, respectively. The stable diquark configurations are the scalar ($C\gamma_{5}$) and axialvector ($C\gamma_{\mu}$) diquark states from the QCD sum rules [15, 16]. The double-heavy diquark states $\varepsilon^{ijk}Q^{T}_{j}C\gamma_{5}Q_{k}$ cannot exist due to the Pauli principle. In this article, we take the double-heavy diquark states $\varepsilon^{ijk}Q^{T}_{j}C\gamma_{\mu}Q_{k}$ as basic constituents to construct the tetraquark states.

The article is arranged as follows: we derive the QCD sum rules for the masses and pole residues of the $QQ\bar{Q}\bar{Q}$ 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 $QQ\bar{Q}\bar{Q}$ tetraquark states

We write down the two-point correlation functions $\Pi_{\mu\nu\alpha\beta}(p)$ and $\Pi(p)$ in the QCD sum rules,

[TABLE]

where

[TABLE]

$Q=c,b$, the $i$, $j$, $k$, $m$, $n$ are color indexes, the $C$ is the charge conjunction matrix. We choose the currents $J(x)$ and $J_{\mu\nu}(x)$ to interpolate the $J^{PC}=0^{++}$ and $2^{++}$ diquark-antidiquark type $QQ\bar{Q}\bar{Q}$ tetraquark states, respectively. In Ref.[10], Chen et al choose the currents $\eta^{i}(x)$ and $\eta^{j}_{\mu\nu}(x)$ with $i=1,2,3,4,5$ and $j=1,2$ to interpolate the $0^{++}$ and $2^{++}$ $QQ\bar{Q}\bar{Q}$ tetraquark states, respectively,

[TABLE]

where $\Gamma^{1}=\gamma_{5}$, $\Gamma^{2}=\gamma_{\mu}\gamma_{5}$, $\Gamma^{3}=\sigma_{\mu\nu}$, $\Gamma^{4}=\gamma_{\mu}$, $\Gamma^{5}=1$, $\Gamma^{1}_{\mu}=\gamma_{\mu}$, $\Gamma^{2}_{\mu}=\gamma_{\mu}\gamma_{5}$, the $a$ and $b$ are color indexes. The $C\gamma_{5}$, $C$, $C\gamma_{\mu}\gamma_{5}$ are antisymmetric, while the $C\gamma_{\mu}$, $C\sigma_{\mu\nu}$ are symmetric. So the currents $\eta^{1/2/5}(x)$ and $\eta^{2}_{\mu\nu}(x)$ are in color ${\bf 6}_{c}\otimes{\bf\bar{6}}_{c}$ representation, while the currents $\eta^{3/4}(x)$ and $\eta^{1}_{\mu\nu}(x)$ have both color ${\bf 6}_{c}\otimes{\bf\bar{6}}_{c}$ and ${\bf\bar{3}}_{c}\otimes{\bf 3}_{c}$ components. The currents $J(x)$ and $J_{\mu\nu}(x)$ chosen in this article are in the color ${\bf\bar{3}}_{c}\otimes{\bf 3}_{c}$ representation, and significantly differ from the currents $\eta^{4}(x)$ and $\eta^{1}_{\mu\nu}(x)$ chosen in Ref.[10], respectively. The one-gluon exchange leads to attractive (repulsive) interaction in the color ${\bf\bar{3}}_{c}$ (${\bf 6}_{c}$) channel [14], the currents or quark structures chosen in the present work and in Ref.[10] couple potentially to the tetraquark states with different masses.

At the phenomenological side, we can insert a complete set of intermediate hadronic states with the same quantum numbers as the current operators $J_{\mu\nu}(x)$ and $J(x)$ into the correlation functions $\Pi_{\mu\nu\alpha\beta}(p)$ and $\Pi(p)$ to obtain the hadronic representation [17, 18]. After isolating the ground state contributions of the scalar and tensor $QQ\bar{Q}\bar{Q}$ tetraquark states (denoted by $X$), we get the following results,

[TABLE]

where $\widetilde{g}_{\mu\nu}=g_{\mu\nu}-\frac{p_{\mu}p_{\nu}}{p^{2}}$, the pole residues $\lambda_{X}$ are defined by

[TABLE]

the $\varepsilon_{\mu\nu}(\lambda,p)$ is the polarization vector of the tensor tetraquark states,

[TABLE]

Now we briefly outline the operator product expansion for the correlation functions $\Pi_{\mu\nu\alpha\beta}(p)$ and $\Pi(p)$ in perturbative QCD. We contract the heavy quark fields in the correlation functions $\Pi_{\mu\nu\alpha\beta}(p)$ and $\Pi(p)$ with Wick theorem, and obtain the results:

[TABLE]

where the $S_{ij}(x)$ is the full $Q$ quark propagator,

[TABLE]

and $t^{n}=\frac{\lambda^{n}}{2}$, the $\lambda^{n}$ is the Gell-Mann matrix [18]. Then we compute the integrals both in the coordinate and momentum spaces to obtain the correlation functions $\Pi_{\mu\nu\alpha\beta}(p)$ and $\Pi(p)$, therefore the QCD spectral densities through dispersion relation. The calculations are straightforward but tedious.

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

[TABLE]

where $\rho(s,z,t,r)=\rho_{S}(s,z,t,r)$ and $\rho_{T}(s,z,t,r)$ for the scalar and tensor tetraquark states, respectively, the explicit expressions are given in the Appendix,

[TABLE]

and $\hat{s}=\frac{s}{m_{Q}^{2}}$.

We derive Eq.(11) with respect to $\tau=\frac{1}{T^{2}}$, then eliminate the pole residues $\lambda_{X}$, and obtain the QCD sum rules for the masses of the scalar and tensor $QQ\bar{Q}\bar{Q}$ tetraquark states,

[TABLE]

In the moment QCD sum rules, the moments $\overline{M}_{n}(P_{0}^{2})$ at the phenomenological side are defined by

[TABLE]

where the $X^{\prime}$ denotes the first radial excited state of the $X$, the $P_{0}^{2}$ is a particular value for the parameter $P^{2}=-p^{2}$. We can extract the mass $M_{X}$ according to the ratio $r(n,P_{0}^{2})$ at large values of $n$,

[TABLE]

where the small values $\delta_{n}(P_{0}^{2})\approx\delta_{n+1}(P_{0}^{2})$. In Refs.[19, 20], we observe that $\frac{\lambda_{X^{\prime}}^{2}}{\lambda_{X}^{2}}=6.6$ (or $9.6$) for the central values of the pole residues for $X=Z_{c}(3900)$, $X^{\prime}=Z(4430)$ (or $X=X(3915)$, $X^{\prime}=X(4500)$) in the scenario of tetraquark states. So the $n$ has to be postponed to very large values [10]. In the present work, the contributions of the high resonances and continuum states are depressed by the weight function $\exp\left(-\frac{s}{T^{2}}\right)$. The differences between the predicted masses in the present work and in Ref.[10] originate from the different currents or quark structures.

3 Numerical results and discussions

We take the gluon condensate to be the standard value [17, 18, 21], and take the $\overline{MS}$ masses $m_{c}(m_{c})=(1.275\pm 0.025)\,\rm{GeV}$ and $m_{b}(m_{b})=(4.18\pm 0.03)\,\rm{GeV}$ from the Particle Data Group [22]. We take into account the energy-scale dependence of the $\overline{MS}$ masses from the renormalization group equation,

[TABLE]

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 flavors $n_{f}=5$, $4$ and $3$, respectively [22].

The values of the thresholds are $2M_{\eta_{c}}=5966.8\,\rm{MeV}$, $2M_{J/\psi}=6193.8\,\rm{MeV}$, $2M_{\eta_{b}}=18798.0\,\rm{MeV}$, $2M_{\Upsilon}=18920.6\,\rm{MeV}$ from the Particle Data Group [22]. The masses of the $0^{++}$ and $2^{++}$ $QQ\bar{Q}\bar{Q}$ tetraquark states from the phenomenological quark models lie above or below those thresholds [2, 3, 4, 5, 6, 7, 8, 9, 10]. In Ref.[23], we study the vector and axialvector $B_{c}$ mesons with the QCD sum rules and obtain the masses $M_{B_{c}^{*}}=6.337\pm 0.052\,\rm{GeV}$ and $M_{B_{c1}}=6.730\pm 0.061\,\rm{GeV}$ at the typical energy scale $\mu=2\,\rm{GeV}$. The $B_{c}$ mesons have two heavy quarks, and the mass $M_{B_{c}^{*}}=6.337\pm 0.052\,\rm{GeV}$ lies slightly above the threshold $2M_{J/\psi}=6193.8\,\rm{MeV}$, so we expect the ideal energy scale to extract masses of the $cc\bar{c}\bar{c}$ tetraquark states from the QCD sum rules is about $\mu=2\,\rm{GeV}$, it is indeed the case.

In Fig.1, we plot the masses of the $cc\bar{c}\bar{c}$ tetraquark states with variations of the energy scales and Borel parameters for the threshold parameters $s_{S}^{0}=42\,\rm{GeV}^{2}$ and $S_{T}^{0}=44\,\rm{GeV}^{2}$. From the figure, we can see that the predicted masses decrease monotonously and slowly with increase of the energy scales, the QCD sum rules are stable with variations of the Borel parameters at the energy scales $1.2\,{\rm{GeV}}<\mu<2.2\,{\rm GeV}$. At the energy scale $\mu=2.0\,\rm{GeV}$, the relation $S_{S/T}^{0}=\left(M_{S/T}+0.5\,\rm{GeV}\right)^{2}$ is satisfied, naively, we expect that the energy gap between the ground states and the first radial excited states ia about $0.5\,\rm{GeV}$. In the QCD sum rules, we usually take the continuum threshold parameters as $\sqrt{s_{0}}=M_{gr}+(0.4\sim 0.6)\,\rm{GeV}$ for the conventional mesons, where the $gr$ denotes the ground states. Experimentally, the energy gaps $M_{\psi^{\prime}}-M_{J/\psi}=589\,\rm{MeV}$ and $M_{\eta_{c}^{\prime}}-M_{\eta_{c}}=656\,\rm{MeV}$ from the Particle Data Group [22]. Now we revisit the mass gaps of the tetraquark states. In Ref.[19], we tentatively assign the $Z_{c}(3900)$ and $Z(4430)$ to be the ground state and the first radial excited state of the axial-vector tetraquark states with $J^{PC}=1^{+-}$, respectively, and reproduce the experimental values of the masses with the QCD sum rules. In Ref.[20], we tentatively assign the $X(3915)$ and $X(4500)$ to be the ground state and the first radial excited state of the scalar $cs\bar{c}\bar{s}$ tetraquark states with $J^{PC}=0^{++}$, respectively, and reproduce the experimental values of the masses with the QCD sum rules. The mass gaps are $M_{Z(4430)}-M_{Z_{c}(3900)}=576\,\rm{MeV}$ and $M_{X(4500)}-M_{X(3915)}=588\,\rm{MeV}$, which also satisfy the relation $\sqrt{s_{0}}=M_{gr}+(0.4\sim 0.6)\,\rm{GeV}$, if only the ground states are taken into account in the QCD sum rules. In this article, we take the relation $\sqrt{s_{0}}=M_{gr}+(0.4\sim 0.6)\,\rm{GeV}$ as a constraint, and search for the optimal values of the $s_{0}$ to reproduce the $M_{gr}$. At the regions $S_{S/T}^{0}\leq\left(M_{S/T}+0.6\,\rm{GeV}\right)^{2}$, the contributions of the excited states are expected not to be included in.

In Fig.2, we plot the masses of the $bb\bar{b}\bar{b}$ tetraquark states with variations of the energy scales and Borel parameters for the threshold parameters $s_{S}^{0}=374\,\rm{GeV}^{2}$ and $S_{T}^{0}=375\,\rm{GeV}^{2}$. From the figure, we can see that the predicted masses also decrease monotonously and slowly with increase of the energy scales, the QCD sum rules are stable with variations of the Borel parameters at the energy scales $2.5\,{\rm{GeV}}<\mu<3.3\,{\rm GeV}$. At the energy scale $\mu=3.1\,\rm{GeV}$, the relation $S_{S/T}^{0}=\left(M_{S/T}+0.5\,\rm{GeV}\right)^{2}$ is satisfied. For the conventional mesons, the energy gaps $M_{\Upsilon^{\prime}}-M_{\Upsilon}=563\,\rm{MeV}$ and $M_{\eta_{b}^{\prime}}-M_{\eta_{b}}=600\,\rm{MeV}$ from the Particle Data Group [22]. It is also reasonable to choose the relation $S_{S/T}^{0}\leq\left(M_{S/T}+0.6\,\rm{GeV}\right)^{2}$.

We search for the Borel parameters $T^{2}$ and continuum threshold parameters $s_{0}$ to satisfy the two criteria of the QCD sum rules: pole dominance at the phenomenological side and convergence of the operator product expansion at the QCD side. Furthermore, we take the relation $\sqrt{S_{S/T}^{0}}=M_{S/T}+(0.4\sim 0.6)\,\rm{GeV}$ as an additional constraint to obey. The resulting Borel parameters, continuum threshold parameters, energy scales, pole contributions are shown explicitly in Table 1. From the Table, we can see that the pole dominance at the phenomenological side is well satisfied. In the Borel windows, the dominant contributions come from the perturbative terms, the contributions of the gluon condensate are about $-10\%$, the operator product expansion is well convergent. Now the two criteria of the QCD sum rules are all satisfied, we expect to make reasonable predictions.

In Ref.[24], we tentatively assign the $D_{s3}^{*}(2860)$ to be a D-wave $c\bar{s}$ meson, and study the mass and decay constant of the $D_{s3}^{*}(2860)$ with the QCD sum rules by calculating the contributions of the vacuum condensates up to dimension-6 in the operator product expansion. In calculations, we observe that only the perturbative term, gluon condensate and three-gluon condensate have contributions. At the Borel window, the contributions are about $(107-109)\%$, $-(7-9)\%$ and $\ll 1\%$, respectively, see the first diagram in Fig.3 [24], the three-gluon condensate can be neglected safely. In the present case, the contributions of the gluon condensate are about $-10\%$, just like in the case of the QCD sum rules for the $D_{s3}^{*}(2860)$, so neglecting the three-gluon condensate cannot impair the predictive ability. As the dominant contributions come from the perturbative terms, perturbative $\mathcal{O}(\alpha_{s})$ corrections amount to multiplying the pertubative terms by a factor $\kappa$, which can be absorbed into the pole residues and cannot impair the predicted masses remarkably.

We take into account all uncertainties of the input parameters, and obtain the values of the ground state masses and pole residues, which are also shown explicitly in Table 1 and Figs.3-4. From Table 1, we can see that the additional constraint is also satisfied. In Figs.3-4, we plot the masses and pole residues with variations of the Borel parameters at much larger intervals than the Borel windows shown in Table 1. From Figs.3-4, we can see that the predicted masses and pole residues are rather stable with variations of the Borel parameters, the uncertainties originate from the Borel parameters in the Borel windows are very small.

From Table 1, we can see that the mass splitting between the $0^{++}$ and $2^{++}$ $bb\bar{b}\bar{b}$ tetraquark states is much smaller than that for the $cc\bar{c}\bar{c}$ tetraquark states. The heavy quark effective Lagrangian can be written as

[TABLE]

where $D^{\mu}_{\perp}=D^{\mu}-v^{\mu}v\cdot D$, the $D_{\mu}$ is the covariant derivative, and the $h_{v}$ is the effective heavy quark field. The heavy quark spin symmetry breaking terms appear at the order $\mathcal{O}(1/m_{Q})$ [25]. The $\overline{MS}$ masses are $m_{c}(m_{c})=(1.275\pm 0.025)\,\rm{GeV}$ and $m_{b}(m_{b})=(4.18\pm 0.03)\,\rm{GeV}$ from the Particle Data Group [22], the heavy quark spin symmetry breaking effects in the $c$-quark systems are much larger than that in the $b$-quark systems.

In 2002, the SELEX collaboration reported the first observation of a signal for the double-charm baryon state $\Xi_{cc}^{+}$ in the decay mode $\Xi_{cc}^{+}\rightarrow\Lambda_{c}^{+}K^{-}\pi^{+}$ [26], and confirmed later by the same collaboration in the decay mode $\Xi_{cc}^{+}\rightarrow pD^{+}K^{-}$ with the measured mass $M_{\Xi}=(3518.9\pm 0.9)\,\rm{MeV}$ [27]. In Ref.[28], we study the ${1\over 2}^{+}$ doubly heavy baryon states $\Omega_{QQ}$ and $\Xi_{QQ}$ by subtracting the contributions from the corresponding ${1\over 2}^{-}$ doubly heavy baryon states with the QCD sum rules, and obtain the value $M_{\Xi_{cc}}=3.57\pm 0.14\,\rm{GeV}$. If exciting additional quark-antiquark pair $q\bar{q}$ with $J^{PC}=0^{++}$ costs energy about $1\,\rm{GeV}$, then $M_{X(cc\bar{c}\bar{c},0^{++}/2^{++})}\approx 2M_{\Xi_{cc}}-1\,\rm{GeV}=6.14\pm 0.14\,\rm{GeV}$, which is consistent with the present prediction.

The predicted masses are

[TABLE]

The decays

[TABLE]

are kinematically allowed, but the available spaces are small. The decays

[TABLE]

are kinematically forbidden, but the decays

[TABLE]

can take place, we can search for the $X(cc\bar{c}\bar{c},0^{++}/2^{++})$ and $X(bb\bar{b}\bar{b},0^{++}/2^{++})$ in the mass spectrum of the $\mu^{+}\mu^{-}\mu^{+}\mu^{-}$ in the future.

In the following, we perform Fierz re-arrangement to the tensor current $J_{\mu\nu}$ and scalar current $J$ both in the color and Dirac-spinor spaces to obtain the results,

[TABLE]

[TABLE]

Now we can see that the diquark-antidiquark type current can be changed to a current as a special superposition of color singlet-singlet type currents, which couple potentially to the meson-meson pairs. The diquark-antidiquark type tetraquark state can be taken as a special superposition of a series of meson-meson pairs, and embodies the net effects. The decays to its components (meson-meson pairs) are Okubo-Zweig-Iizuka super-allowed, but the re-arrangements in the color-space are highly non-trivial.

We take the current $J$ as an example to illustrate that the scalar tetraquark state can embody the net effects of all the meson-meson pairs. At the phenomenological side, we can insert a complete set of intermediate hadronic states with the same quantum numbers as the current operator $J(x)$ into the correlation function $\Pi(p)$ to obtain the hadronic representation [17, 18]. After isolating the lowest meson-meson pairs, we get the following result,

[TABLE]

where the decay constants $f_{\eta_{c}}$, $f_{J/\psi}$, $f_{\chi_{c0}}$ and $f_{\chi_{c1}}$ are defined by

[TABLE]

the $\varepsilon_{\mu}$ are the polarization vectors of the $J/\psi$ and $\chi_{c1}$.

We can rewrite the correlation function $\Pi(p)$ into the following form through dispersion relation,

[TABLE]

In this article, we choose the value $s_{0}<4M_{\chi_{c0}}^{2},\,4M_{\chi_{c1}}^{2}$, the meson pairs $\chi_{c0}\chi_{c0}$ and $\chi_{c1}\chi_{c1}$ have no contributions, the QCD sum rules can be written as

[TABLE]

[TABLE]

where we introduce a coefficient $\kappa$, if $\kappa=1$, the QCD sum rules can be saturated by the meson pairs $\eta_{c}\eta_{c}$ and $J/\psi J/\psi$.

We choose the input parameters as $M_{\eta_{c}}=2.9834\,\rm{GeV}$, $M_{J/\psi}=3.0969\,\rm{GeV}$ [22], $f_{\eta_{c}}=0.387\,\rm{GeV}$, $f_{J/\psi}=0.418\,\rm{GeV}$ [29], $s_{0}=42\,\rm{GeV}^{2}$. In Fig.5, we plot the coefficient $\kappa$ comes from Eq.(28) with variation of the Borel parameter $T^{2}$, from the figure, we can see that the values of the $\kappa$ are rather stable with variation of the Borel parameter. Now we choose the special value $T^{2}=4.4\,\rm{GeV}^{2}$, and plot the coefficient $\kappa$ with variation of the energy scale $\mu$ in Fig.6. From the figure, we can see that the coefficient $\kappa$ decreases monotonously and quickly with increase of the energy scale $\mu$ at the region $\mu\leq 1.6\,\rm{GeV}$. At the vicinity of the energy scale $\mu=1.5\,\rm{GeV}$, $\kappa\approx 1$, however, the reliable QCD sum rules do not depend heavily on the energy scale $\mu$. So the QCD sum rules cannot be saturated by the meson pairs $\eta_{c}\eta_{c}$ and $J/\psi J/\psi$.

Now we saturate the QCD sum rules by the meson pairs $\eta_{c}\eta_{c}$, $J/\psi J/\psi$ plus a scalar tetraquark state $X(cc\bar{c}\bar{c},0^{++})$ at the phenomenological side,

[TABLE]

[TABLE]

In Fig.7, we plot the mass $M_{X(cc\bar{c}\bar{c},0^{++})}$ comes from Eqs.(29-30) with variation of the Borel parameter $T^{2}$, from the figure, we can see that the values of the $M_{X(cc\bar{c}\bar{c},0^{++})}$ are rather stable with variation of the Borel parameter at the energy scale $\mu\geq 1.8\,\rm{GeV}$. Now we choose the special value $T^{2}=4.4\,\rm{GeV}^{2}$, and plot the mass $M_{X(cc\bar{c}\bar{c},0^{++})}$ with variation of the energy scale $\mu$ in Fig.8. From the figure, we can see that the mass $M_{X(cc\bar{c}\bar{c},0^{++})}$ increases monotonously and quickly with increase of the energy scale $\mu$ at the region $\mu\leq 1.8\,\rm{GeV}$, and decreases monotonously and slowly at the region $\mu\geq 2.0\,\rm{GeV}$. In the range $\mu=(1.8-2.0)\,\rm{GeV}$, the predicted mass $M_{X(cc\bar{c}\bar{c},0^{++})}$ is rather stable, $M_{X(cc\bar{c}\bar{c},0^{++})}=5.76\,\rm{GeV}$, which is below the threshold $2M_{\eta_{c}}=5966.8\,\rm{MeV}$. On the other hand, the pole residue $\lambda_{X(cc\bar{c}\bar{c},0^{++})}$ increases monotonously and quickly with increase of the energy scale $\mu$ at the region $\mu\geq 1.6\,\rm{GeV}$, no stable QCD sum rules can be obtained, see Fig.9. So the QCD sum rules cannot be saturated by the meson pairs $\eta_{c}\eta_{c}$, $J/\psi J/\psi$ plus a scalar tetraquark state $X(cc\bar{c}\bar{c},0^{++})$.

In this article, the diquark-antidiquark type tetraquark state is taken as a special superposition of a series of meson-meson pairs, and embodies the net effects. The decays to its components (meson-meson pairs) are Okubo-Zweig-Iizuka super-allowed, but the re-arrangements in the color-space are highly non-trivial. In other words, the lowest states $X(QQ\bar{Q}\bar{Q},0^{++}/2^{++})$ can saturate the QCD sum rules satisfactorily.

4 Conclusion

In this article, we study the $J^{PC}=0^{++}$ and $2^{++}$ $QQ\bar{Q}\bar{Q}$ tetraquark states with the QCD sum rules by constructing the diquark-antidiquark type currents and calculating the contributions of the vacuum condensate up to dimension 4 in the operator product expansion. We obtain the predictions $M_{X(cc\bar{c}\bar{c},0^{++})}=5.99\pm 0.08\,\rm{GeV}$, $M_{X(cc\bar{c}\bar{c},2^{++})}=6.09\pm 0.08\,\rm{GeV}$, $M_{X(bb\bar{b}\bar{b},0^{++})}=18.84\pm 0.09\,\rm{GeV}$, $M_{X(bb\bar{b}\bar{b},2^{++})}=18.85\pm 0.09\,\rm{GeV}$, which can be confronted to the experimental data in the future. Furthermore, we illustrate that the diquark-antidiquark type tetraquark state can be taken as a special superposition of a series of meson-meson pairs and embodies the net effects. We can search for the $J^{PC}=0^{++}$ and $2^{++}$ $QQ\bar{Q}\bar{Q}$ tetraquark states in the mass spectrum of the $\mu^{+}\mu^{-}\mu^{+}\mu^{-}$.

Appendix

The explicit expression of the QCD spectral densities $\rho_{S/T}(s)$,

[TABLE]

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where

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Acknowledgements

This work is supported by National Natural Science Foundation, Grant Numbers 11375063, and Natural Science Foundation of Hebei province, Grant Number A2014502017.