Doubly hidden-charm/bottom QQ Q ¯ Q ¯ tetraquark states
Wei Chen, Hua-Xing Chen, Xiang Liu, T. G. Steele, Shi-Lin Zhu
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Doubly hidden-charm/bottom QQ ¯ Q ¯ Q tetraquark states Wei Chen , a , b , Hua-Xing Chen , Xiang Liu , , T. G. Steele , and Shi-Lin Zhu , , School of Physics, Sun Yat-Sen University, Guang Zhou 510275, China School of Physics and Beijing Key Laboratory of Advanced Nuclear Materials and Physics, Beihang Uni-versity, Beijing 100191, China School of Physical Science and Technology, Lanzhou University, Lanzhou 730000, China ResearchCenterforHadronandCSRPhysics, LanzhouUniversityandInstituteofModernPhysicsofCAS,Lanzhou 730000, China Department of Physics and Engineering Physics, University of Saskatchewan, Saskatoon, Saskatchewan,S7N 5E2, Canada School of Physics and StateKey Laboratory of Nuclear Physics and Technology, Peking University, Beijing100871, China Collaborative Innovation Center of Quantum Matter, Beijing 100871, China Center of High Energy Physics, Peking University, Beijing 100871, China
Abstract.
We study the mass spectra for the cc ¯ c ¯ c and bb ¯ b ¯ b tetraquark states by develop-ing a moment sum rule method. Our results show that the bb ¯ b ¯ b tetraquarks lie below thethreshold of η b (1 S ) η b (1 S ). They are probably stable and very narrow. The masses forthe doubly hidden-charm states cc ¯ c ¯ c are higher than the spontaneous dissociation thresh-olds of two charmonium mesons. We suggest to search for such states in the J /ψ J /ψ and η c (1 S ) η c (1 S ) channels. The configurations of multiquark states were proposed by Gell-Mann [1] and Zweig [2] at the birthof quark model (QM). In the past fifty years, it has been an extremely intriguing research issue ofsearching for multiquark matter. The light tetraquark qq ¯ q ¯ q state has been used to investigate thescalar mesons below 1 GeV [3]. Since 2003, plenty of charmoniumlike states have been observedand the hidden-charm qc ¯ q ¯ c tetraquark fomalism is extensively discussed to explain the nature of thesenew XYZ states [4–11].The doubly hidden-charm / bottom tetraquark QQ ¯ Q ¯ Q is composed of four heavy quarks. Suchtetraquark states did not receive much attention in both experimental and theoretical aspects [12–21]. Recently, there are some discussions about the masses and decays of the QQ ¯ Q ¯ Q states [22–31].The masses of these QQ ¯ Q ¯ Q tetraquarks are far away from the mass regions of the conventional Q ¯ Q mesons and the XYZ states. It will be very easy to distinguish them from the XYZ and Q ¯ Q statesin the spectroscopy. On the other hand, the QQ ¯ Q ¯ Q states favor the compact tetraquark configurationthan the loosely bound hadron molecular configuration, since the light mesons can not be exchanged a Speaker b e-mail: [email protected] PJ Web of Conferences between the two charmonium / bottomonium states. In this paper, we develop a moment QCD sum rulemethod to calculate the mass spectra for the doubly hidden-charm / bottom cc ¯ c ¯ c and bb ¯ b ¯ b tetraquarkstates. In this section we briefly introduce the method of QCD sum rules [32–34]. Comparing to the tradi-tional SVZ QCD sum rules, we use another version of QCD sum rules, the moment QCD sum rules inour analyses for the doubly hidden-charm / bottom QQ ¯ Q ¯ Q tetraquark systems. The moment QCD sumrules have been very successfully used for studying the charmonium and bottomonium mass spectra[32, 33, 35–37] and determining the heavy quark masses and the strong coupling constant [38–40].We start by considering the following two-point correlation functions Π ( q ) = i Z d xe iq · x h | T [ J ( x ) J † (0)] | i , Π µν ( q ) = i Z d xe iq · x h | T [ J µ ( x ) J † ν (0)] | i , Π µν, ρσ ( q ) = i Z d xe iq · x h | T [ J µν ( x ) J † ρσ (0)] | i , (1)in which the interpolating currents J ( x ), J µ ( x ) and J µν ( x ) couple to the scalar, vector and tensor statesrespectively.To study the doubly hidden-charm / bottom tetraquarks, we construct the QQ ¯ Q ¯ Q interpolating cur-rents with four heavy quarks in the compact diquark-antidiquark configuration. We use all diquarkfields Q Ta CQ b , Q Ta C γ Q b , Q Ta C γ µ γ Q b , Q Ta C γ µ Q b , Q Ta C σ µν Q b and Q Ta C σ µν γ Q b and consider thePauli principle to determine the color and flavor structures for the tetraquark operators. FollowingRefs. [22, 41], we obtain the QQ ¯ Q ¯ Q tetraquark interpolating currents as the following. The interpo-lating currents with J PC = ++ are J = Q Ta C γ Q b ¯ Q a γ C ¯ Q Tb , J = Q Ta C γ µ γ Q b ¯ Q a γ µ γ C ¯ Q Tb , J = Q Ta C σ µν Q b ¯ Q a σ µν C ¯ Q Tb , J = Q Ta C γ µ Q b ¯ Q a γ µ C ¯ Q Tb , J = Q Ta CQ b ¯ Q a C ¯ Q Tb , (2)where J , J , J belong to the symmetric [ c ] QQ ⊗ [ ¯6 c ] ¯ Q ¯ Q color structure while J , J belong to theantisymmetric [ ¯3 c ] QQ ⊗ [ c ] ¯ Q ¯ Q color structure. The interpolating currents with J PC = − + and 0 −− are J ± = Q Ta CQ b ¯ Q a γ C ¯ Q Tb ± Q Ta C γ Q b ¯ Q a C ¯ Q Tb , J + = Q Ta C σ µν Q b ¯ Q a σ µν γ C ¯ Q Tb , (3)in which J + and J + couple to the states with J PC = − + , and J − couples to the states with J PC = −− .The currents J ± belong to the symmetric color structure while J + belongs to antisymmetric colorstructure. The interpolating currents with J PC = ++ and 1 + − are J ± µ = Q Ta C γ µ γ Q b ¯ Q a C ¯ Q Tb ± Q Ta CQ b ¯ Q a γ µ γ C ¯ Q Tb , J ± µ = Q Ta C σ µν γ Q b ¯ Q a γ ν C ¯ Q Tb ± Q Ta C γ ν Q b ¯ Q a σ µν γ C ¯ Q Tb , (4) CNFP 2016 in which J + µ and J + µ couple to the states with J PC = ++ , and J − µ and J − µ couple to the states with J PC = + − . The currents J ± µ belong to the symmetric color structure while J ± µ belongs to antisym-metric color structure. The interpolating currents with J PC = − + and 1 −− are J ± µ = Q Ta C γ µ γ Q b ¯ Q a γ C ¯ Q Tb ± Q Ta C γ Q b ¯ Q a γ µ γ C ¯ Q Tb , J ± µ = Q Ta C σ µν Q b ¯ Q a γ ν C ¯ Q Tb ± Q Ta C γ ν Q b ¯ Q a σ µν C ¯ Q Tb , (5)in which J + µ and J + µ couple to the states with J PC = − + , and J − µ and J − µ couple to the states with J PC = −− . The currents J ± µ belong to the symmetric color structure while J ± µ belongs to antisym-metric color structure. The interpolating currents with J PC = ++ are J µν = Q Ta C γ µ Q b ¯ Q a γ ν C ¯ Q Tb + Q Ta C γ ν Q b ¯ Q a γ µ C ¯ Q Tb , J µν = Q Ta C γ µ γ Q b ¯ Q a γ ν γ C ¯ Q Tb + Q Ta C γ ν γ Q b ¯ Q a γ µ γ C ¯ Q Tb , (6)where current J µν belongs to the antisymmetric color structure while J µν belongs to symmetric colorstructure.At the hadronic level, the correlation functions in Eq.(1) can be described by the dispersion relation Π ( q ) = ( q ) N π Z ∞ M H Im Π ( s ) s N ( s − q − i ǫ ) ds + N − X n = b n ( q ) n , (7)where M H is the hadron mass and b n are unknown subtraction constants. A narrow resonance approx-imation is usually used to describe the spectral function ρ ( s ) = π Im Π ( s ) = X n δ ( s − m n ) h | J | n ih n | J † | i + · · · = f X δ ( s − m X ) + · · · , (8)where “ · · · " represents the excited higher states and continuum contributions and f X is a couplingconstant between the interpolating current and hadron state h | J | X i = f X , h | J µ | X i = f X ǫ µ , h | J µν | X i = f X ǫ µν , (9)in which ǫ µ and ǫ µν are the polarization vector and tensor, respectively. To pick out the contribution ofthe lowest lying resonance in Eq. (8), we define moments in Euclidean region Q = − q > M n ( Q ) = n ! (cid:18) − ddQ (cid:19) n Π ( Q ) | Q = Q = Z ∞ m Q ρ ( s )( s + Q ) n + ds (10) = f X ( m X + Q ) n + (cid:2) + δ n ( Q ) (cid:3) , (11)in which δ n ( Q ) contains the contributions of higher states and continuum. It tends to zero as n goesto infinity. We consider the following ratio to eliminate f X in Eq. (11) r ( n , Q ) ≡ M n ( Q ) M n + ( Q ) = (cid:0) m X + Q (cid:1) + δ n ( Q )1 + δ n + ( Q ) . (12) PJ Web of Conferences
One expects δ n ( Q ) (cid:27) δ n + ( Q ) for su ffi ciently large n to suppress the contributions of higher statesand continuum [33]. Then hadron mass of the lowest lying resonance m X is then extracted as m X = q r ( n , Q ) − Q . (13)Using the operator production expansion (OPE) method, the two-point function can also be evalu-ated at the quark-gluonic level as a function of various QCD parameters. In the fully heavy tetraquarksystems, we only need to calculate the perturbative term and the gluon condensate contributions to thecorrelation functions. One can find the results of the moments M n ( Q ) in Ref. [22]. We perform the numerical analyses by using the following values of parameters [43–46] m c (MS) = . ± .
03 GeV , m b (MS) = . ± .
03 GeV , h g s GG i = (0 . ± .
14) GeV . (14)To provide reliable moment sum rule analyses, one needs to find suitable working regions of the twoparameters n and Q in the ratio r ( n , Q ). We define ξ = Q / m c for cc ¯ c ¯ c and Q / m b for bb ¯ b ¯ b systems for convenience. These two parameters will a ff ect the pole contribution and the OPE conver-gence. For small value of ξ , the high dimension condensates in OPE will give large contributions, andthus leading to bad OPE convergence [33, 37]. However, a larger value of ξ means slower conver-gence of δ n ( Q ) in Eq. (11). Such behavior can be compensated by n : the OPE convergence becomesgood for small n while δ n ( Q ) tends to zero for su ffi ciently large n . One needs to find suitable workingregions for ( n , ξ ) where the lowest lying resonance dominates the moments and the OPE convergeswell.
12 20 28 36 44 52 60 68 7618.118.518.919.319.720.1 18.45n m X @ G e V D Ξ= Ξ= Ξ= Ξ= Figure 1.
Hadron mass m X b for J ( bb ¯ b ¯ b ) with J PC = ++ , as a function of n for di ff erent value of ξ . As an example, we use the interpolating current J with J PC = ++ in Eq. (2) to perform numericalanalyses. Requiring the perturbative term to be larger than the gluon condensate term, we obtain upper CNFP 2016 J PC Currents m X c (GeV) m X b (GeV)0 ++ J . ± .
15 18 . ± . J . ± .
17 18 . ± . J . ± .
16 18 . ± . J . ± .
16 18 . ± . J . ± .
18 19 . ± . − + J + . ± .
18 18 . ± . J + . ± .
18 18 . ± . −− J − . ± .
18 18 . ± . ++ J + µ . ± .
19 18 . ± . J + µ . ± .
19 18 . ± . + − J − µ . ± .
18 18 . ± . J + µ . ± .
15 18 . ± . − + J + µ . ± .
18 18 . ± . J + µ . ± .
18 18 . ± . −− J − µ . ± .
18 18 . ± . J − µ . ± .
18 18 . ± . ++ J µν . ± .
15 18 . ± . J µν . ± .
19 18 . ± . Table 1.
Mass spectra for the cc ¯ c ¯ c and bb ¯ b ¯ b tetraquarks. limits n max , which increases with respect to the value of ξ . We show the hadron mass m X b as a functionof n for ξ = . , . , . , . m X b = (18 . ± .
15) GeV , (15)in which the error comes from the uncertainties of ξ , the heavy quark mass and the gluon condensatein Eq. (14). Using the interpolating currents in Eqs. (2)–(6), we perform numerical analyses for all cc ¯ c ¯ c and bb ¯ b ¯ b systems with various quantum numbers. We collect the numerical results in Table 1. Itis shown that the negative parity states ( J PC = − + , −− , − + , −− ) are slightly heavier than the positiveparity states ( J PC = ++ , ++ , + − , ++ ).It is interesting to compare the mass spectra with the corresponding two-meson mass thresholds.As shown in Fig. 2, the masses of bb ¯ b ¯ b tetraquarks are below the η b (1 S ) η b (1 S ) threshold while all cc ¯ c ¯ c tetraquarks lie above the η c (1 S ) η c (1 S ) threshold. The two bottomonium mesons decays for the bb ¯ b ¯ b tetraquarks are thus forbidden by the kinematics. For the doubly hidden-charm cc ¯ c ¯ c tetraquarks,they can decay via the spontaneous dissociation mechanism by considering the restrictions of thesymmetries. In Table 2, we collect the possible S -wave and P -wave dissociation decay channels forthe cc ¯ c ¯ c states.In principle, the bb ¯ b ¯ b tetraquark can also decay into B ( ∗ ) ¯ B ( ∗ ) via a heavy quark pair annihilationand a light quark pair creation processes. The suppression by the annihilation of a heavy quark pairwill be compensated by the large phase space factor. Such B ( ∗ ) ¯ B ( ∗ ) decay modes may dominate thetotal width of the doubly hidden-bottom bb ¯ b ¯ b tetraquark state. PJ Web of Conferences − + ++ −− ++ + − −− − + . . . . . . M a ss ( G e V ) ++ Figure 2.
Summary of the doubly hidden-charm / bottom tetraquark spectra labelled by J PC . The green and redsolid (dashed) lines indicate the η c (1 S ) η c (1 S ) ( η b (1 S ) η b (1 S )) and J /ψ J /ψ ( Υ (1 S ) Υ (1 S )) thresholds, respectively. J PC S-wave P-wave0 ++ η c (1 S ) η c (1 S ), J /ψ J /ψ η c (1 S ) χ c (1 P ), J /ψ h c (1 P )0 − + η c (1 S ) χ c (1 P ), J /ψ h c (1 P ) J /ψ J /ψ −− J /ψχ c (1 P ) J /ψη c (1 S )1 ++ − J /ψ h c (1 P ), η c (1 S ) χ c (1 P ), η c (1 S ) χ c (1 P )1 + − J /ψη c (1 S ) J /ψχ c (1 P ), J /ψχ c (1 P ), η c (1 S ) h c (1 P )1 − + J /ψ h c (1 P ), η c (1 S ) χ c (1 P ) − −− J /ψχ c (1 P ), J /ψχ c (1 P ), J /ψη c (1 S ) η c (1 S ) h c (1 P ) Table 2.
Possible decay modes of the cc ¯ c ¯ c states by spontaneous dissociation into two charmonium mesons. In this paper, we have calculated the mass spectra for the doubly hidden-charm / bottom cc ¯ c ¯ c and bb ¯ b ¯ b tetraquark states by using the moment QCD sum rule method. Our results show that the cc ¯ c ¯ c tetraquarks lie above the two charmonium spontaneous dissociation thresholds and thus can mainlydecay into two charmonium mesons. We suggest to search for these doubly hidden-charm cc ¯ c ¯ c states CNFP 2016 in the J /ψ J /ψ and η c (1 S ) η c (1 S ) channels. For the bb ¯ b ¯ b tetraquarks, their masses are lower thanthe η b (1 S ) η b (1 S ) threshold so that the two bottomonium mesons decays are kinematical forbidden.These bb ¯ b ¯ b tetraquark, if exist, may be very narrow and stable. In the near future, these doublyhidden-charm / bottom cc ¯ c ¯ c and bb ¯ b ¯ b tetraquark states can be searched for at facilities such as LHCb,CMS, RHIC and the forthcoming BelleII. Acknowledgments
This project is supported by the Chinese National Youth Thousand Talents Program; the Natural Sci-ences and Engineering Research Council of Canada (NSERC); the National Natural Science Founda-tion of China under Grants No. 11475015, 373 No. 11375024, No. 11222547, No. 11175073, 374No. 11575008, and No. 11621131001; the 973 program; 375 the Ministry of Education of China(SRFDP under Grant 376 No. 20120211110002 and the Fundamental Research 377 Funds for theCentral Universities); and the National 378 Program for Support of Top-Notch Youth Professionals.
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