Doubly charmed 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 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 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 , 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 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] √2 ud [1,3] √2 us 1 T cc = T cc(3¯). (2) [2,3] √2 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 {i,j} 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. {1,2} √2 ud {1,3} √2 us 1 The explicit form of each Hamiltonian is written as T cc = T cc(6),T cc = T cc (6), {2,3} √2 ds {1,1} uu cc cc cc cc T{2,2} = Tdd(6),T{3,3} = 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) √2 − √2 − LL δδ 2 1 1 T cc(3¯) = (ds sd)¯cc,¯ T cc(6) = (ud + du)¯cc,¯ ds √2 − ud √2 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 √2 ds √2 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 = | 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¯ | i ↑ ↑ ↑ ↑ | δ i √3 ↑ ↑ ↓ ↓ ↓ ↓ ↑ ↑ 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 | 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. The corresponding splitting mass matrix for the J P = 1   0δ, 1δ¯0 , 1J = ( )q( )q0 ( )q( )q0 ( )c¯( )c¯, 0+ tetraquarks is | i √2 ↑ ↓ − ↓ ↑ ↑ ↑ 1    3  1δ, 0 ¯0 , 1J = ( )q( )q0 ( )c¯( )c¯ ( )c¯( )c¯ + ((κqq0 )¯ + (κc¯c¯)¯) 0 δ ∆M(0 ) = − 2 3 3 , (8) | i √2 ↑ ↑ ↑ ↓ − ↓ ↑ 0 h1 1   1δ, 1δ¯0 , 1J = ( )q( )q0 ( )c¯( )c¯ + ( )c¯( )c¯ where | i 2 ↑ ↑ ↑ ↓ ↓ ↑   ( )q( )q0 + ( )q( )q0 ( )c¯( )c¯ .(13) 1 − ↑ ↓ ↓ ↑ ↑ ↑ h1 = ((κqq0 )3¯ + (κc¯c¯)3¯ 4κqc¯ 4κq0c¯). 2 − − The tetraquarks with the quark content [qq0][¯cc¯] do not (9) have any definite charge parity if q = u and q0 = u. Thus the above three 1+ states can mix6 with each other.6 The mass matrix is given as The mass splitting matrix ∆M for J P = 1+ can be P P M(J ) = mδ + mδ0 + ∆M(J ). (10) obtained with the base vectors defined in Eq. (13).

 1  2 ((κc¯c¯)3¯ 3(κqq0 )3¯) 0 √2(κq0c¯ κqc¯) − 1 − ∆M =  0 2 ((κqq0 )3¯ 3(κc¯c¯)3¯) 0  , − 1 √2(κq0c¯ κqc¯) 0 ((κqq0 )¯ + (κc¯c¯)¯ 2κq0c¯ 2κqc¯) − 2 3 3 − − (14)

In flavor SU(3) symmetry, all tetraquark states with- hadron. In the paper, we adopt the heavy quark mass as out orbital angular momenta should have the identical mc = 1.670GeV [24, 37]. mass. Since the strange quark is different from the up Different diquark masses will obviously affect the and down quarks, the reasonable tetraquark masses could tetraquark’s mass. In Refs. [54–56], the diquark masses be obtained. In the calculation of the doubly charmed effects are studied. The scalar and axial-vector diquark tetraquark spectra, we will distinguish the strange quark masses are assumed to be equal in order to limit the from the up and down quarks. For the light diquark, mqq number of parameters. The diquark mass is chosen to is determined to be 0.395GeV by the f (500) particle [24], 0 be mfg = ξ(m + m ) where fg qq, qs, ss is the fla- m is determined to be 0.590GeV by the a (980) parti- δ f g sq 0 vor content of the diquark. Naturally,∈ { the diquark} mass cle [24], and the strange diquark mass m = 0.785GeV is ss parameter ξ is assumed to be ξ [0, 1] [54]. In order to estimated by the relation mss msq = msq mqq [37, 53], ∈ − − consider the effects from the diquark mass, we denote the where q stands for u or d. The heavy antidiquark mass fg fg running diquark mass m and we have m = mδ + ∆δ. mc¯c¯ is estimated by the relation mc¯c¯ 2mc, since the δ δ ' Thus we have ∆δ = ξ(mf + mg) mδ. heavy quark is highly static in the rest frame of the − 4

 For the spin-spin couplings, the strange quark is also 4.11GeV,J P = 0+, cc  P + treated differently from the up and down quarks [24, 53, m(Tus(6)) = 4.15GeV,J = 1 , (19) P + 57]. The couplings are chosen as: (κqq)3¯ = 103MeV,  4.22GeV,J = 2 , (κsq)3¯ = 64MeV, (κc¯c¯)3¯ = 39MeV, (κqc¯)0 = 70MeV and  P +  4.30GeV,J = 0 , (κsc¯)0 = 72MeV where q also stands for u or d. The m(T cc(6)) = 4.34GeV,J P = 1+, (20) relation κ = 1 (κ ) is satisfied for the quark-antiquark ss ij 4 ij 0  4.41GeV,J P = 2+. state, which is extracted from one gluon exchange model. The spin-orbit coupling Aδ is estimated as 30MeV, and the orbit-orbit coupling Bδδ¯0 is estimated as 278 MeV [24, Other tetraquarks’ masses can be obtained by cc cc cc cc 53]. m(Tuu(6)) = m(Tdd(6)) = m(Tud(6)) and m(Tds (6)) = m(T cc(6)). The wave function of a tetraquark consists of four us parts: space-coordinate, color, flavor, and spin sub- From the above calculation, a doubly charmed cc ¯ P + spaces, tetraquark Tud(3) with spin-parity J = 1 and mass M = 3.60GeV is predicted in a diquark-antidiquark model, which is around 140MeV below the DD¯ thresh- 0 Ψ(q, q , c,¯ c¯) = ψ(x1, x2, x3, x4) χc(c1, c2, c3, c4) ¯ ∗ ∗ ¯ ⊗ old and 270MeV below the DD (D D) threshold. Other χf (f1, f2, f3, f4) χs(s1, s2, s3, s4) , doubly charmed tetraquark states are predicted above ⊗ ⊗ ¯ (15) the DD threshold, which will have large decay widths cc ¯ compared with the tetraquark Tud(3) with spin-parity J P = 1+ and mass 3.60GeV. where ψ(xi), χc(ci), χf (fi), and χs(si) denote the space, color, flavor, and spin wave functions, respectively. The For the orbitally excited tetraquark states with L = 1, 0 here we just consider the tetraquarks with spin-parity sub-labels 1, 2, 3, 4 denote q, q ,c ¯,c ¯, respectively. The P − diquark is attractive only in the triplet representation in J = 1 . The masses of the tetraquarks with the spin- P − ¯ color space, thus the color wave function is antisymmet- parity J = 1 in 3 representation are ric. The antidiquark [¯cc¯] is symmetric in flavor space, cc ¯ P − thus the spin wave function of the antidiquark [¯cc¯] has to m(Tud(3)) = 3.82GeV,J = 1 , (21) cc ¯ cc ¯ P − be symmetric with Sδ¯0 = 1 when we do not consider the m(Tus(3)) = m(Tds (3)) = 4.07GeV,J = 1 . (22) orbital excitation in the inner diquark system. P First we focus on the tetraquarks with L = 0, and then The masses of the tetraquarks with the spin-parity J = − the space wave function is symmetric. If the spin wave 1 in 6 representation are function of the diquark system [qq0] is antisymmetric, i.e.  P − S = 0, the flavor function should be also antisymmetric. 4.21GeV,J = 1 , δ cc  P − In this case, the charmed tetraquarks can be decomposed m(Tud(6)) = 4.19GeV,J = 1 , (23) P − into the 3¯ representation, with the spin-parity J P = 1+.  4.14GeV,J = 1 , Inputting the parameter masses and using the fixed di-  P −  4.39GeV,J = 1 , quark mass with ∆δ = 0, their masses are determined to cc P − m(Tus(6)) = 4.36GeV,J = 1 , (24) be  4.31GeV,J P = 1−,  P −  4.58GeV,J = 1 , cc ¯ P + cc P − m(Tud(3)) = 3.60GeV,J = 1 , (16) m(Tss (6)) = 4.56GeV,J = 1 , (25) cc ¯ cc ¯ P +  4.51GeV,J P = 1−. m(Tus(3)) = m(Tds (3)) = 3.85GeV,J = 1 . (17)

If using the running diquark mass with ∆δ = 0, the cc 6 cc IV. DOUBLY CHARMED TETRAQUARKS tetraquark mass m(Tqq0 ) will become to be m(Tqq0 )+∆δ. In Eqs. (16-25), we have assumed ∆δ = 0. DECAYS TO TWO CHARMED MESONS If the spin wave function of the diquark system [qq0] is symmetric, i.e. Sδ = 1, the flavor function should be also When the doubly charmed tetraquarks lie above the symmetric. In this case, the charmed tetraquarks can DD¯ threshold, they can decay into two charmed mesons. be decomposed into the 6 representation, with the spin- For the tetraquarks with positive parity, the two-body parity J P = 0+, 1+, 2+. Their masses are determined to decay amplitudes can be written as: be cc + (T [0 ] D¯D¯) = F ¯ ¯ fT cc , (26) M → DD  3.94GeV,J P = 0+, cc cc  P + where fT is the decay constant of the tetraquark, FD¯ D¯ m(Tud(6)) = 3.97GeV,J = 1 , (18) denotes the effective coupling to diquark-antidiquark P +  4.04GeV,J = 2 , pair. 5

For the tetraquarks with J P = 1−−, the two body are decay amplitudes are written as: Γ(T cc[1−−] D¯D¯) 1 2 0 →2 3 = 4 , F (f cc ) p 6πm cc 0 D¯ D¯ T | | T cc −− 0 FD¯ D¯ fT cc cc −− ∗ (T [1 ] D¯D¯) =ε cc (P ¯ P ¯ ) , Γ(T [1 ] D¯D¯ ) 1 T D D 31/2m M → · − Tcc¯ 2 0 →2 3 = 4 , F (f cc ) p 12πm cc (27) D¯ D¯ ∗ T | | T cc −− ∗ ∗ 4 104 2 2 48 4 cc −− ¯ ¯ ∗ µ ν ρ 0σ Γ(T [1 ] D¯ D¯ ) m cc m cc p + p (T [1 ] DD ) =ε ε ∗ P P  T 9 T 9 Tcc¯ D¯ D¯ D¯ µνρσ = . M → 2 0 → 2 3 −4 2 | | 2 2| | 0 FD¯ ∗D¯ ∗ (fT cc ) p 2πmT cc (mT cc 4 p ) fT cc | | − | | F ¯ ¯ ∗ , (28) DD 1/2 2 (31) 3 (mTcc¯) The decay widths of 1−− tetraquarks to double charmed mesons will be suppressed compared to that of (T cc[1−−] D¯ ∗(P )D¯ ∗(P 0)) the tetraquarks with positive parity to double charmed M → mesons. The suppression factor for the D¯D¯ final states µ ρ 0ν FD¯ ∗D¯ ∗ fT cc µρ ν =ε ε ε ∗ (g ( P P ¯ ∗ ) Tcc¯ D¯ ∗ D¯ 1/2 T D can be estimated as 3 mT cc − − ρ cc −− ¯ ¯ 0 2 2 µν ρ µν 0 ρν 0 µ Γ(T [1 ] DD) 4(fT cc ) p + g PT + g P ¯ ∗ + g (P D¯ ∗ PD¯ ∗ ) ), (29) , (32) D cc + → ¯ ¯ 2 | 2| − Γ(T [0 ] DD) ' 3(fT cc ) m cc → T The decay width of T cc D¯(D¯ ∗)D¯(D¯ ∗) can be writ- The decay constant of the doubly charmed tetraquark → 0 ten as: fT cc are not easily to extract clearly through the hadronic decays. These nonperturbative information can be ex- p tract through the purely leptonic decays as T cc cc ¯ ¯ ∗ ¯ ¯ ∗ 2 − cc → Γ(T D(D ) + D(D )) = | |2 , (30) 2` + 2ν ¯` for the doubly charged tetraquarks, T → 8πmT cc |M| − + → 2` + ` +ν ¯` for the singly charged tetraquarks, and T cc 2`− + 2`+ for the neutral tetraquarks. where →

p 2 2 2 2 (m cc (m1 m2) )(m cc (m1 + m2) ) p = T − − T − , V. DOUBLY CHARMED TETRAQUARKS | | 2mT cc DECAYS TO A CHARMED BARYON AND A LIGHT BARYON which is the momentum modulus of final charmed meson in the tetraquark rest frame. m1 and m2 are the final charmed meson’s masses, respectively. In this section, we will study the doubly charmed tetraquarks weak decays into baryon and anti-baryon, where one of baryons is charmed. Due to larger phase space, these decays may provide more useful information q to discover the doubly charmed tetraquarks at experi- ment. c¯ The charm quark decays into light quarks are clas- sified into three types: c sdu¯ , c udd/¯ ss¯ , and c¯ c dsu¯ , namely Cabibbo-favored,→ singly→ Cabibbo- suppressed,→ and doubly Cabibbo-suppressed topologies, respectively. The Cabibbo-favored channels may provide a windows to discover the doubly charmed tetraquarks. q At the hadron level, the effective Hamiltonian in charm FIG. 1: Feynman Diagrams for doubly charmed tetraquarks decays behave as the certain representations under the decays to two charmed mesons. flavor SU(3) symmetry as 3¯ 3 3¯ = 3¯ 3¯ 6 15. So the Hamiltonian can be decomposed⊗ ⊗ in terms⊕ ⊕ of⊕ a vector Hi(3), a traceless tensor antisymmetric in upper indices, The decay ratio for the tetraquark with positive parity [ij] is Hk (6), and a traceless tensor symmetric in upper in- {ij} dices, Hk (15). Γ(T cc[0+] D¯D¯) 1 ¯ = . For the Cabibbo-favored c sdu decay, we have the 2 →2 2 → F ¯ ¯ (fT cc ) p 8πm cc nonzero components [58] DD | | T 31 13 H2 (6) = H2 (6) = 1, P −− 31 − 13 The similar ratios for the tetraquarks with J = 1 H2 (15) = H2 (15) = 1, (33) 6 where the representation Hi(3¯) will vanish in the SU(3) TABLE I: Decay amplitudes of T cc(3¯) → T¯c + T decays. flavor symmetry. 3¯ 8 Singly charmed baryons with two light quarks can form Decay channel amplitude cc ¯ ¯− 1 an anti-triplet or sextet, which can be classified into two Tud(3) → Ξc n 2 (a15 − a6 − 2b6) cc ¯ ¯− − 1 matrixes Tds (3) → Ξc Σ − 2 (a15 + a6 + 2b6) cc ¯− 0 1 0 0  + +  Tus(3¯) → Ξc Σ √ (a15 − 2a6 − a6 − 2b6) 0 Λc Ξc 2 2 1 cc − 0 1 0 0 c + 0 T (3¯) → Ξ¯ Λ √ (3a + 2a6 + a + 2b6) T3¯ =  Λc 0 Ξc  , (34) us c 2 6 15 6 √ − + 0 2 Ξ Ξ 0 T cc(3¯) → Ξ¯0Σ− 1 (a + a0 − a − a0 ) − c − c us c 2 15 15 6 6 cc ¯ ¯ − 1 0 0 Tus(3) → Λc n − 2 (a15 + a15 + a6 + a6)  Σ++ √1 Σ+ √1 Ξ0+  c 2 c 2 c c  √1 Σ+ Σ0 √1 Ξ00  T6 = 2 c c 2 c . (35) cc ¯c   TABLE II: Decay amplitudes of T (3¯) → T6 + T8 decays. √1 Ξ0+ √1 Ξ00 Ω0 2 c 2 c c Decay channel amplitude 0 T cc (3¯) → Σ¯ −n 1 (a − a ) The light baryons made of three light quarks with spin- ud c 2 15 6 + cc ¯ ¯ 0 − 1 P 1 Tud(3) → Ωc Σ − √ (a15 − a6) parity J = 2 can form an octet, which has the expres- 2 T cc(3¯) → Σ¯ −−n − √1 (a + a ) sion ds c 2 15 6 0 T cc(3¯) → Σ¯ −Σ− 1 (a + a )  √1 Σ0 + √1 Λ0 Σ+ p  ds c 2 15 6 2 6 T cc(3¯) → Σ¯ −n 1 (a + a0 + a + a0 ) Σ− √1 Σ0 + √1 Λ0 n us c 2 15 15 6 6 T8 =  2 6 (36). cc ¯ −− 1 0 0  −  Tus(3¯) → Σc p+ √ (a15 + a6) − 0 q 2 0 2 Ξ Ξ Λ 0 3 T cc(3¯) → Σ¯ −Λ0 − √1 (2a + a0 + 3a0 ) − us c 2 6 15 15 6 0 T cc(3¯) → Σ¯ −Σ0 √1 (2a + a0 − a0 ) us c 2 2 15 15 6 T cc(3¯) → Ω¯ 0Ξ− √1 (a0 − a0 ) q us c 2 15 6 0 q cc ¯ ¯ 0 − 1 0 0 Tus(3) → Σc Σ 2 (a15 + a15 − a6 − a6)

q q¯ c¯ u¯ charmed baryon and a light baryon are given in Tabs. I,

c¯ II, III and IV. For convenience, we give the ratio of the FIG. 2: Feynman Diagrams for doubly charmed tetraquarks decay widths of the doubly charmed tetraquarks to a decays to baryon and anti-baryon. The two circles “⊕ ⊕” charmed anti-baryon and a light baryon in the flavor denote the four fermion weak effective vertex. SU(3) symmetry in Tabs. V and VI.

The decay amplitudes of Cabibbo-favored channels Whatever the doubly charmed tetraquarks decay to cc c c cc two charmed mesons or decay to a charmed baryon and A(T T¯ + T8) = T¯ + T8 eff T can be writ- → ∗ T cc h c |H | i a light baryon, BESIII, BelleII, and LHCb are excel- ten as VcsV A (T T¯ + T8). ud → lent experiment platforms to search for it. BESIII has cc ¯ ¯c + − For the decays T (3) T3¯ + T8, the effective Hamil- accumulated large e e collision data samples at an tonian can be written as→ T cc ¯ ¯c A (T (3) T3¯ + T8) → cc ¯c cc ¯ c ikn ¯ m [jl] TABLE III: Decay amplitudes of T (6) → T¯ + T8 decays. = a6T[ij](3)T3¯ [k,l] (T8)n Hm (6) 3 0 cc ¯ c imn ¯ k [jl] Decay channel amplitude +a6T (3)T¯  (T8)nHm (6) [ij] 3 [k,l] cc − 1 T (6) → Ξ¯ n − (a15 − a6 + 2b6) cc ¯ c imn ¯ j [kl] ud c 2 +b6T[ij](3)T3¯ [k,l] (T8)nHm (6) cc ¯− − 1 Tds (6) → Ξc Σ − 2 (a15 + a6 − 2b6) cc c ikn m {jl} cc − 0 1 0 0 ¯ ¯ T (6) → Ξ¯ Σ √ (−2a15 − a + a + 2b6) +a15T[ij](3)T3¯ [k,l] (T8)n Hm (15) us c 2 2 15 6 T cc(6) → Ξ¯−Λ0 √1 (2a + a0 + 3a0 + 6b ) 0 cc ¯ c imn ¯ k {jl} us c 2 6 15 15 6 6 +a15T[ij](3)T¯  (T8)nHm (15). (37) 3 [k,l] cc ¯0 − 1 0 0 Tus(6) → Ξc Σ − 2 (a15 + a15 − a6 − a6) T cc(6) → Λ¯ −n − 1 (a + a0 + a + a0 ) Similarly, the effective Hamiltonian have the same for- us c 2 15 15 6 6 cc ¯0 1 0 0 cc ¯c Tuu(6) → Ξc n √ (a15 + a15 − a6 − a6) mulae for the decays T (6) T3¯ + T8 by the replace- 2 cc cc → cc ¯− 1 0 0 ment of T (3¯) T (6). The term proportional Tuu(6) → Ξc p √ (a15 − a6 − 2b6) [ij] → {ij} 2 cc ¯ ¯c T cc(6) → Ξ¯−Ξ− √1 (a0 + a0 + 2b ) to b6 will vanish for both the T (3) T6 + T8 and ss c 2 15 6 6 T cc(6) T¯c + T decays. The Cabibbo-allowed→ am- cc ¯ − − 1 0 0 6 8 Tss (6) → Λc Σ √ (a15 + a15 + a6 + b6) plitudes→ for doubly charmed tetraquarks decays to a 2 7

cc ¯c TABLE IV: Decay amplitudes of T (6) → T6 + T8 decays. TABLE VI: SU(3) relations for decay widths for the 6 dou- bly charmed tetraquark to a charmed anti-baryon and a light Decay channel amplitude baryon. R denotes the ratio of two decay widths. cc 0− 1 T (6) → Σ¯ n − (a15 − a6) ud c 2 Γ(channel )/Γ(channel ) R cc ¯ 0 − 1 1 2 Tud(6) → Ωc Σ √ (a15 − a6) cc ¯ − − 2 Γ(Tss (6)→Λc Σ ) 2 cc ¯ −− 1 Γ(T cc(6)→Λ¯ −n) Tds (6) → Σc n − √ (a15 + a6) us c 2 Γ(T cc(6)→Σ¯ −n) 0 us c cc ¯ − − 1 cc ¯ − 1 Tds (6) → Σc Σ 2 (a15 + a6) Γ(Tus(6)→Λc n) cc ¯ −− 0 cc ¯ − 1 0 0 Γ(Tss (6)→Σc Σ ) Tus(6) → Σc n 2 (a15 + a15 + a6 + a6) cc ¯ − 2 Γ(T (6)→Λc n) cc −− 1 0 0 us T (6) → Σ¯ p+ √ (a + a ) Γ(T cc(6)→Σ¯ −Σ−) us c 2 15 6 ss c 2 0 cc ¯ − cc ¯ − 0 1 0 0 Γ(Tus(6)→Λc n) Tus(6) → Σc Λ − √ (3a15 + 2a6 + a6) cc ¯0 2 6 Γ(Tuu(6)→Ξc n) 0 cc ¯0 − 2 cc ¯ − 0 1 0 0 Γ(Tus(6)→Ξc Σ ) T (6) → Σ Σ − √ (a − 2a6 − a ) 0 us c 2 2 15 6 cc ¯ 0 − Γ(Tus(6)→Σc Σ ) cc ¯ 0 − 1 0 0 cc ¯0 − 1 Tus(6) → Ωc Ξ − √ (a15 − a6) Γ(Tus(6)→Ξc Σ ) 2 0 0 cc ¯ 0 cc ¯ 0 − 1 0 0 Γ(Tuu(6)→Σc n) Tus(6) → Σc Σ − (a15 + a15 − a6 − a6) cc ¯0 − 2 2 Γ(Tus(6)→Ξc Σ ) 0 cc 0 1 0 0 Γ(T cc (6)→Ω¯ 0Σ−) T (6) → Σ¯ n √ (a15 + a − a6 − a ) ud c uu c 2 15 6 0 2 Γ(T cc (6)→Σ¯ −n) cc 0− + 1 0 0 ud c T (6) → Σ¯ p √ (a − a ) cc ¯ 0 − uu c 2 15 6 Γ(Tus(6)→Ωc Ξ ) 0 2 cc 0 0 1 0 0 Γ(T cc (6)→Σ¯ −n) T (6) → Ω¯ Λ √ (−a − 2a + a + 2a ) ud c uu c 6 15 15 6 6 0 Γ(T cc (6)→Σ¯ −p+) cc ¯ 0 0 1 uu c 2 Tuu(6) → Ωc Σ √ (a15 − a6) cc ¯ 0− 2 Γ(Tud(6)→Σc n) cc −− 0 1 0 0 cc ¯ 0 0 T (6) → Σ¯ Λ √ (a − a + a − a ) Γ(Tuu(6)→Ωc Σ ) ss c 6 15 15 6 6 0 2 Γ(T cc (6)→Σ¯ −n) cc −− 0 1 0 0 ud c T (6) → Σ¯ Σ − √ (a15 + a + a6 + a ) cc ¯ −− ss c 2 15 6 Γ(Tds (6)→Σc n) 0 0 2 cc ¯ − − 1 0 0 Γ(T cc(6)→Σ¯ −Σ−) Tss (6) → Σc Ξ − √ (a15 + a6) ds c 2 cc ¯ −− Γ(Tus(6)→Σc p+) cc ¯ − − 1 0 0 0 2 Tss (6) → Σc Σ − √ (a15 + a15 + a6 + a6) cc ¯ − − 2 Γ(Tds (6)→Σc Σ ) cc ¯ 0− − Γ(Tss (6)→Σc Ξ ) 0 2 cc ¯ − − Γ(Tds (6)→Σc Σ )

TABLE V: SU(3) relations for decay widths for the 3¯ dou- bly charmed tetraquark to a charmed anti-baryon and a light baryon. R denotes the ratio of two decay widths.

Γ(channel1)/Γ(channel2) R cc ¯ ¯ 0 − VI. CONCLUSION Γ(Tud(3)→Ωc Σ ) 2 cc ¯ ¯ 0− Γ(Tud(3)→Σc n) cc ¯ ¯ −− Γ(Tds (3)→Σc n) 0 2 cc ¯ ¯ − − Γ(Tds (3)→Σc Σ ) We have considered the possibility for the doubly cc ¯ ¯ −− Γ(Tus(3)→Σc p+) charmed tetraquark states with the quark configuration 0 2 Γ(T cc(3¯)→Σ¯ −Σ−) 0 ds c [qq ][¯cc¯]. These states are straightforward consequences cc ¯ ¯ − Γ(Tus(3)→Σc n) cc ¯ ¯ − 1 of the constituent diquark-antidiquark model. The dou- Γ(Tus(3)→Λc n) cc ¯ ¯0 − Γ(Tus(3)→Ξc Σ ) bly charmed tetraquarks form the antitriplet and sextet cc ¯ ¯ 00 − 1 Γ(Tus(3)→Σc Σ ) configuration in the flavor SU(3) symmetry. The mass spectrum and their spin-parities of tetraquark states have been investigated. We found that a doubly charmed cc ¯ P + tetraquark Tud(3) with spin-parity J = 1 is around 140MeV below the DD¯ threshold and 270MeV below the DD¯ ∗(D∗D¯) threshold. For T cc[0+] and T cc[1−−] energy range from 3.8 GeV to 4.6 GeV and will con- tetraquark states, the decay modes of the two-body tinue taking data at open-charm energy region [59]. It charmed mesons have also been presented. Furthermore, would be an interesting research at those energy points the doubly charmed tetraquarks decays to a charmed with the world’s top integrated luminosity just as pa- baryon and a light baryon have been studied in the SU(3) pers [60–63] et al. The two charmed mesons channels flavor symmetry. The search for such kind of states can T cc D¯(D¯ ∗)+D¯(D¯ ∗) can be studied through the whole be carried out at BESIII, BelleII, LHCb and other ex- energy→ region from 3.8 GeV to 4.6 GeV. A charmed periments with their large data samples of high luminos- baryon and a light baryon decay channels showed in ity. These Cabibbo-favored channels shall provide a wid- Tabs. I, II, III and IV can also be studied carefully at ows to discover the possible doubly charmed tetraquark the same energy region. states. 8

Acknowledgments the National Natural Science Foundation of China under Grant No. U1732105, and by the Research Foundation for Advanced Talents of Nanjing Normal University under This work was supported in part by Natural Science Grant No. 2014102XGQ0085. Foundation of Jiangsu under Grant No. BK20171471, by

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