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Doubly Charmed Tetraquarks in a Diquark-Antidiquark Model

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Doubly Charmed Tetraquarks in a Diquark-Antidiquark Model Doubly charmed tetraquarks in a diquark-antidiquark model Xiaojun Yan, Bin Zhong and Ruilin Zhu ∗ Department of Physics and Institute of Theoretical Physics, Nanjing Normal University, Nanjing, Jiangsu 210023, China We study the spectra of the doubly charmed tetraquark states in a diquark-antidiquark model. The doubly charmed tetraquark states form an antitriplet and a sextet configurations according to flavor SU(3) symmetry. For the tetraquark state [qq0][¯cc¯], we show the mass for both bound and excited states. The two-body decays of tetraquark states T cc[0+] and T cc[1−−] to charmed mesons have also been studied. In the end,the doubly charmed tetraquarks decays to a charmed baryon and a light baryon have been studied in the SU(3) flavor symmetry. Keywords: Exotic states, tetraquark, diquark I. INTRODUCTION may give an interpretation to such XYZ states [44]. A tetraquark state can also be represented with four quark configuration [qq0][¯cc¯](two charm antiquarksc ¯'s and two Since last decade, Belle, BABAR, BESIII, LHCb and light quarks with up, down and strange quarks u, d and other experimental data have indicated a considerable s), which is called as doubly charmed (C=2) tetraquark number of exotic hadronic resonances with charm or T cc. From a theoretical point of view, T cc may be a more beauty, the so-called XYZ and Pc particles including interesting hadronic state because the two heavy charm tetraquark states and pentaquark states [1{17]. All of quark are more likely to form a lower energy diquark these exotic states have many unexpected properties, and two light quarks will rotate round the charmed di- such as masses, decay widths and cross-sections, which quark. Besides, its flavor quantum numbers can give the are hard to be explained in the conventional quark model. possibility to clearly distinguish these tetraquarks from The nature of these states is one of the most interesting conventional quarkonia. subjects in hadron physics. The doubly bottomed tetraquark states with four The study of heavy flavor tetraquark states consist- quark configuration [bb][¯qq¯0](two bottom quarks and two ing of two quarks and two antiquarks has a long history. light antiquarks) have been studied in Refs. [45{49]. In Hidden charm and bottom tetraquark states have been Ref. [45], a doubly bottomed tetraquark with spin-parity investigated in the past decades [18{38]. The recent re- J P = 1+ and mass m = 10389MeV was predicted, which views of the exotic states can be found in Refs. [39{41]. is around 215 MeV below BB¯∗ threshold. In Ref. [46], a Currently, the exact inner physical picture of these exotic doubly bottomed tetraquark around 135 MeV below BB¯∗ states have not been reached an agreement in both the threshold was predicted. The Lattice calculations were theoretical and experimental communities. The possible performed in Refs. [47, 48], where the doubly bottomed explanations of the XYZ resonances can be classified tetraquark below BB¯∗ threshold were also predicted. In into hadronic molecules, compact tetraquarks, diquark- Ref. [49], the doubly bottomed and charmed tetraquarks antidiquark states, conventional quarkonium, triangle were discussed in the heavy quark symmetry. anomaly, and kinematics effects. In this paper, we discuss the possibility of the doubly In Ref. [40], the multiquark resonances are reviewed charmed tetraquark states, and attempt to investigate and a coherent description of the so called X and Z res- the mass spectrum and decay widths of doubly charmed onances is presented. The first suggestion to use light tetraquarks in a diquark-antidiquark model. The dou- diquarks to explain the exotic states is from Jaffe and bly charmed tetraquark states can form an antitriplet arXiv:1804.06761v1 [hep-ph] 18 Apr 2018 Wilczek [42]. Maiani et al. then introduced heavy- configuration and a sextet configuration according to light diquarks to explain the charmonium-like states in flavor SU(3) symmetry. We hope the doubly charmed Ref. [24]. The evidence that in a tetraquark system the tetraquark states T cc can be discovered at BES III, Belle two quarks arrange their color in a diquark before inter- II, LHCb and other experiments with their large data acting with the antiquarks has been found on the lat- samples of high luminosity. tice [43]. The paper is organized as the following. In Sec. II, Based on the diquark-antidiquark model, tetraquark we classify the doubly charmed tetraquark states into 0 states [cq ][¯cq¯] can form an octet representation and a anti-triplet and sextet representations in flavor SU(3) singlet representation in flavor SU(3) symmetry which symmetry. In Sec. III, the spectra of the doubly charmed tetraquark states are calculated from a diquark- antidiquark model. In Sec. V, we study the two-body ∗ charmed mesons decays of doubly charmed tetraquarks. Email:[email protected] We summarize and conclude in the end. 2 II. DOUBLY CHARMED TETRAQUARKS IN III. DOUBLY CHARMED TETRAQUARKS FLAVOR SU(3) SYMMETRY SPECTRA Considering the flavor SU(3) symmetry, the dou- Because the heavy charm quarks interact with each cc 0 bly charmed tetraquarks T [qq ][¯cc¯] with two light other at the momentum scale mcv with the heavy charm 0 ∼ quarks, q and q , can be conventionally classified into two quark relative velocity v, and this scale mcv is a large groups. The three light quarks, (u; d; s) form a triplet 3 quantity compared with the typical hadron scale, the representation and the charm quark c is a singlet [50{52]. two heavy charm quarks will have a small distance be- The doubly charmed tetraquarks T cc [qq0][¯cc¯] can have tween each other and are easily to form an attractive the following irreducible representations∼ diquark. Considering a diquark-antidiquark picture δδ¯0 with δ = [qq0] and δ¯0 = [¯cc¯], the effective Hamiltonian in- 3 3 = 3¯ 6 : (1) cludes three kinds of interactions: spin-spin interactions ⊗ ⊕ of quarks in the diquark, antidiquark and between them; spin-orbital interactions; orbit-orbital interactions. The ¯ cc We label 3 representation by T[i;j]. Here the flavor effective Hamiltonian then can be written as [24]: components are antisymmetric under the exchange of i cc cc and j, and thus T[i;j] is traceless as T[i;i] = 0. The com- ponents can be given explicitly as δ δ¯0 δδ¯0 H = mδ + mδ0 + HSS + HSS + HSS + HSL + HLL; 1 1 T cc = T cc(3¯);T cc = T cc(3¯); (5) [1;2] p2 ud [1;3] p2 us 1 T cc = T cc(3¯): (2) [2;3] p2 ds where mδ and mδ0 are the constituent masses of the di- quark [qq0] and the antidiquark [¯cc¯], respectively. Hδ We label 6 representation by T cc . Here the flavor SS fi;jg and Hδ¯0 denote the spin-spin interaction inside the di- components are symmetric under the exchange of i and SS δδ¯0 j. In this case, the components are given by quark and antidiquark, respectively. HSS denotes the spin-spin interaction of quarks between diquark and an- 1 1 tidiquark. HSL and HLL represent the spin-orbital and T cc = T cc(6);T cc = T cc(6); purely orbital interactions. f1;2g p2 ud f1;3g p2 us 1 The explicit form of each Hamiltonian is written as T cc = T cc(6);T cc = T cc (6); f2;3g p2 ds f1;1g uu cc cc cc cc Tf2;2g = Tdd(6);Tf3;3g = Tss (6); (3) δ H = 2(κqq0 )¯(Sq Sq0 ); SS 3 · To summarize, the flavor components of tetraquarks δ¯0 H = 2(κc¯c¯)¯(Sc¯ Sc¯); in flavor SU(3) symmetry can be explicitly obtained as SS 3 · δδ¯0 H = 4κq0c¯(Sq0 Sc¯) + 4κqc¯(Sq Sc¯); below SS · · HSL = 2Aδ(Sδ L) + 2Aδ¯0 (Sδ¯0 L); 1 1 · · cc ¯ cc ¯ L(L + 1) Tud(3) = (ud du)¯cc;¯ Tus(3) = (us su)¯cc;¯ H = B ¯0 : (6) p2 − p2 − LL δδ 2 1 1 T cc(3¯) = (ds sd)¯cc;¯ T cc(6) = (ud + du)¯cc;¯ ds p2 − ud p2 cc 1 cc 1 T (6) = (us + su)¯cc;¯ T (6) = (ds + sd)¯cc;¯ where S 0 and S are the spin operators of light and us p2 ds p2 q(q ) c(¯c) cc cc heavy quarks, respectively. Sδ and Sδ¯0 denote the spin Tuu(6) = uuc¯c;¯ Tdd(6) = ddc¯c;¯ operators of the diquark and antidiquark, respectively. cc Tss (6) = ssc¯c:¯ (4) L is the orbital angular momentum operator. The other parameters are all coefficients. Aδ(δ¯0) and Bδδ¯0 are the The orbitally excited tetraquark states also form anti- spin-orbit and orbit-orbit couplings, respectively. (κqq0 )3¯ and (κc¯c¯)3¯ are the spin-spin couplings for diquark in color triplet and sextet representations. Considering the first ¯ orbital excitation with L = 1, the orbitally excited antitriplet 3. κq0c¯ and κqc¯ are the spin-spin couplings for tetraquark states will have the spin-parity 1−. For the a quark-antiquark pair. cc neutral tetraquarks, Tuu can have the definite charge- The orbital angular momentum of ground states of parity, thus their J PC quantum numbers can be 1−− or tetraquark is zero. In this case, there are two possible 1−+. tetraquark configurations with the spin-parity J P = 0+ 3 in the spin space, The above mass matrix is naturally diagonalized, and one can easily obtain two different eigenvalues. 1 P 0δ; 0δ¯0 ; 0J = ( )q( )q0 ( )q( )q0 ( )c¯( )c¯ The corresponding tetraquark configuration for J = j i 2 " # − # " " # 2+ is 1 ( )q( )q0 ( )q( )q0 ( )c¯( )c¯; −2 " # − # " # " 1 ; 1 ¯0 ; 2 = ( ) ( ) 0 ( ) ( ) ; (11) 1 δ δ J q q c¯ c¯ 1δ; 1 ¯0 ; 0J = ( )q( )q0 ( )c¯( )c¯ + ( )q( )q0 ( )c¯( )c¯ j i " " " " j δ i p3 " " # # # # " " with the mass 1 ( )q( )q0 + ( )q( )q0 ( )c¯( )c¯ −2 " # # " " # + 1 1 M(2 ) = mδ + mδ0 + ((κqq0 )3¯ + (κc¯c¯)3¯) ( )q( )q0 + ( )q( )q0 ( )c¯( )c¯ : (7) 2 −2 " # # " # " 1 + (2κqc¯ + 2κq0c¯) : (12) where Sδ;S ¯0 ;SJ denotes the doubly charmed 2 j δ i tetraquark; the Sδ and Sδ¯0 represent the spin of 0 diquark [qq ] and antidiquark [¯cc¯], respectively; the The possible configurations for the tetraquark with SJ represents the total angular momentum of the J P = 1+ are tetraquark.
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