Doubly hidden-charm/bottom $QQ\bar Q\bar Q$ tetraquark states

We study the mass spectra for the $cc\bar c\bar c$ and $bb\bar b\bar b$ tetraquark states by developing a moment sum rule method. Our results show that the $bb\bar b\bar b$ tetraquarks lie below the threshold of $\eta_b(1S)\eta_b(1S)$. They are probably stable and very narrow. The masses for the doubly hidden-charm states $cc\bar c\bar c$ are higher than the spontaneous dissociation thresholds of two charmonium mesons. We suggest to search for such states in the $J/\psi J/\psi$ and $\eta_c(1S)\eta_c(1S)$ channels.


Introduction
The configurations of multiquark states were proposed by Gell-Mann [1] and Zweig [2] at the birth of quark model (QM). In the past fifty years, it has been an extremely intriguing research issue of searching for multiquark matter. The light tetraquark qqqq state has been used to investigate the scalar mesons below 1 GeV [3]. Since 2003, plenty of charmoniumlike states have been observed and the hidden-charm qcqc tetraquark fomalism is extensively discussed to explain the nature of these new XYZ states [4][5][6][7][8][9][10][11].
The doubly hidden-charm/bottom tetraquark QQQQ is composed of four heavy quarks. Such tetraquark states did not receive much attention in both experimental and theoretical aspects [12][13][14][15][16][17][18][19][20][21]. Recently, there are some discussions about the masses and decays of the QQQQ states [22][23][24][25][26][27][28][29][30][31]. The masses of these QQQQ tetraquarks are far away from the mass regions of the conventional QQ mesons and the XYZ states. It will be very easy to distinguish them from the XYZ and QQ states in the spectroscopy. On the other hand, the QQQQ states favor the compact tetraquark configuration than the loosely bound hadron molecular configuration, since the light mesons can not be exchanged a Speaker b e-mail: chenwei29@mail.sysu.edu.cn between the two charmonium/bottomonium states. In this paper, we develop a moment QCD sum rule method to calculate the mass spectra for the doubly hidden-charm/bottom cccc and bbbb tetraquark states.

QCD sum rules
In this section we briefly introduce the method of QCD sum rules [32][33][34]. Comparing to the traditional SVZ QCD sum rules, we use another version of QCD sum rules, the moment QCD sum rules in our analyses for the doubly hidden-charm/bottom QQQQ tetraquark systems. The moment QCD sum rules have been very successfully used for studying the charmonium and bottomonium mass spectra [32,33,[35][36][37] and determining the heavy quark masses and the strong coupling constant [38][39][40].
We start by considering the following two-point correlation functions in which the interpolating currents J(x), J µ (x) and J µν (x) couple to the scalar, vector and tensor states respectively.
To study the doubly hidden-charm/bottom tetraquarks, we construct the QQQQ interpolating currents with four heavy quarks in the compact diquark-antidiquark configuration. We use all diquark fields Q T a CQ b , Q T a Cγ 5 Q b , Q T a Cγ µ γ 5 Q b , Q T a Cγ µ Q b , Q T a Cσ µν Q b and Q T a Cσ µν γ 5 Q b and consider the Pauli principle to determine the color and flavor structures for the tetraquark operators. Following Refs. [22,41], we obtain the QQQQ tetraquark interpolating currents as the following. The interpolating currents with J PC = 0 ++ are where in which J + 1 and J + 2 couple to the states with J PC = 0 −+ , and J − 1 couples to the states with J PC = 0 −− . The currents J ± 1 belong to the symmetric color structure while J + 2 belongs to antisymmetric color structure. The interpolating currents with J PC = 1 ++ and 1 +− are ICNFP 2016 in which J + 1µ and J + 2µ couple to the states with J PC = 1 ++ , and J − 1µ and J − 2µ couple to the states with J PC = 1 +− . The currents J ± 1µ belong to the symmetric color structure while J ± 2µ belongs to antisymmetric color structure. The interpolating currents with J PC = 1 −+ and 1 −− are in which J + 1µ and J + 2µ couple to the states with J PC = 1 −+ , and J − 1µ and J − 2µ couple to the states with J PC = 1 −− . The currents J ± 1µ belong to the symmetric color structure while J ± 2µ belongs to antisymmetric color structure. The interpolating currents with J PC = 2 ++ are where current J 1µν belongs to the antisymmetric color structure while J 2µν belongs to symmetric color structure.
At the hadronic level, the correlation functions in Eq.(1) can be described by the dispersion relation where M H is the hadron mass and b n are unknown subtraction constants. A narrow resonance approximation is usually used to describe the spectral function where "· · · " represents the excited higher states and continuum contributions and f X is a coupling constant between the interpolating current and hadron state 0|J|X = f X , in which ǫ µ and ǫ µν are the polarization vector and tensor, respectively. To pick out the contribution of the lowest lying resonance in Eq. (8), we define moments in Euclidean region Q 2 = −q 2 > 0 [33,42]: in which δ n (Q 2 0 ) contains the contributions of higher states and continuum. It tends to zero as n goes to infinity. We consider the following ratio to eliminate f X in Eq. (11) EPJ Web of Conferences One expects δ n (Q 2 0 ) δ n+1 (Q 2 0 ) for sufficiently large n to suppress the contributions of higher states and continuum [33]. Then hadron mass of the lowest lying resonance m X is then extracted as Using the operator production expansion (OPE) method, the two-point function can also be evaluated at the quark-gluonic level as a function of various QCD parameters. In the fully heavy tetraquark systems, we only need to calculate the perturbative term and the gluon condensate contributions to the correlation functions. One can find the results of the moments M n (Q 2 0 ) in Ref. [22].

Numerical results
We perform the numerical analyses by using the following values of parameters [43][44][45][46] To provide reliable moment sum rule analyses, one needs to find suitable working regions of the two parameters n and Q 2 0 in the ratio r(n, Q 2 0 ). We define ξ = Q 2 0 /16m 2 c for cccc and Q 2 0 /m 2 b for bbbb systems for convenience. These two parameters will affect the pole contribution and the OPE convergence. For small value of ξ, the high dimension condensates in OPE will give large contributions, and thus leading to bad OPE convergence [33,37]. However, a larger value of ξ means slower convergence of δ n (Q 2 0 ) in Eq. (11). Such behavior can be compensated by n: the OPE convergence becomes good for small n while δ n (Q 2 0 ) tends to zero for sufficiently large n. One needs to find suitable working regions for (n, ξ) where the lowest lying resonance dominates the moments and the OPE converges well.  As an example, we use the interpolating current J 1 with J PC = 0 ++ in Eq. (2) to perform numerical analyses. Requiring the perturbative term to be larger than the gluon condensate term, we obtain upper limits n max , which increases with respect to the value of ξ. We show the hadron mass m X b as a function of n for ξ = 0.2, 0.4, 0.6, 0.8 in Fig. 1. One notes that the mass curves have plateaus which provide stable mass prediction in which the error comes from the uncertainties of ξ, the heavy quark mass and the gluon condensate in Eq. (14). Using the interpolating currents in Eqs. (2)-(6), we perform numerical analyses for all cccc and bbbb systems with various quantum numbers. We collect the numerical results in Table 1. It is shown that the negative parity states (J PC = 0 −+ , 0 −− , 1 −+ , 1 −− ) are slightly heavier than the positive parity states (J PC = 0 ++ , 1 ++ , 1 +− , 2 ++ ). It is interesting to compare the mass spectra with the corresponding two-meson mass thresholds. As shown in Fig. 2, the masses of bbbb tetraquarks are below the η b (1S )η b (1S ) threshold while all cccc tetraquarks lie above the η c (1S )η c (1S ) threshold. The two bottomonium mesons decays for the bbbb tetraquarks are thus forbidden by the kinematics. For the doubly hidden-charm cccc tetraquarks, they can decay via the spontaneous dissociation mechanism by considering the restrictions of the symmetries. In Table 2, we collect the possible S -wave and P-wave dissociation decay channels for the cccc states.
In principle, the bbbb tetraquark can also decay into B ( * )B( * ) via a heavy quark pair annihilation and a light quark pair creation processes. The suppression by the annihilation of a heavy quark pair will be compensated by the large phase space factor. Such B ( * )B( * ) decay modes may dominate the total width of the doubly hidden-bottom bbbb tetraquark state.