Eur. Phys. J. C (2019) 79:645 https://doi.org/10.1140/epjc/s10052-019-7150-4

Regular Article - Theoretical Physics

Fully heavy bbc¯c¯: lifetimes and weak decays

Gang Li1,a, Xiao-Feng Wang1,YeXing2,b 1 School of Physics and Engineering, Qufu Normal University, Qufu 273165, China 2 INPAC, SKLPPC, MOE Key Laboratory for Physics, School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China

Received: 30 April 2019 / Accepted: 18 July 2019 / Published online: 7 August 2019 © The Author(s) 2019

Abstract We study the lifetime and weak decays of the valuable. Fully-heavy four- state with no light quark + {bb} full-heavy S-wave 0 tetraquark T{¯cc¯} . Using the operator degrees of freedom is of this type and might be an ideal product expansion rooted in heavy quark expansion, we find probe to study the interplay between perturbative QCD and a rather short lifetime, at the order (0.1 − 0.3) × 10−12s non-perturbative QCD. depending on the inputs. With the flavor SU(3) , we Generally speaking, more heavy correspond to a then construct the effective Hamiltonian at the level, larger mass. For instance, there have been some phenomeno- and derive relations between decay widths of different chan- logical studies to determine the mass and the spectrum prop- nels. According to the electro-weak effective operators, we erties of the fully-heavy tetraquark bcb¯ c¯, including the con- classify different decay modes, and make a collection of the stituent quark and model [38,39], simple quark {bb} → − 0 − golden channels, such as T{¯cc¯} B K Bc for the charm model [40], nonrelativistic effective field theory(NREFT) {bb} → − − [41], QCD sum rules [42,43], and quark potential model [44]. quark decay and T{¯cc¯} B D for the decay. Our results for the lifetime and golden channels are helpful to In Ref. [41], the authors utilize the NREFT to determine the . search for the fully-heavy tetraquark in future experiments. mass with the upper bound as 12 58 GeV, consistent with the mass calculated in the simple [40]. Despite of these studies, it is still not conclusive that whether the bcb¯ c¯ (or its charge conjugate cbc¯ b¯) is above or below the B B 1 Introduction c c threshold. It is likely that the bcb¯ c¯ lies below the threshold of the B B pair, which means that such a state is stable against In the past decades, quark model has achieved great suc- c c the strong . In this case, the dominant decay modes cesses in the hadron spectroscopy study. In addition to would be induced by . In a diquark-diquark the quark-anti-quark assignment for a and three- model [45], the S-wave fully-heavy tetraquark state bcb¯ c¯ can quark interpretation of a , it allows the existence of form 0+ and 2+. In this paper we will mainly focus on the non-standard exotic states [1Ð6]. Since the observation of lowest lying state 0+, which might be assigned as a weakly- X(3872) in 2003 [1], many exotic candidates have been coupled state. announced on the experimental side in the heavy quarko- In this paper, we will first use the operator product expan- nium sector in various processes [7]. Charged heavy quarko- ± ± ± sion (OPE) technique and calculate the lifetime of the S-wave niumlike states Zc(3900) , Zc(4020) , Zb(10610) , and + ± 0 bcb¯ c¯. The light flavor SU(3) symmetry is a useful tools to Z (10650) observed by BES-III and Belle collaborations b analyze weak decays of a heavy quark, and has been success- [2Ð4] have already experimentally established as being fully applied to the meson or baryon system [46Ð61]. Though exotic, since they contain at least two quarks and two anti- the SU(3) breaking effects in transition might be quarks with the hidden QQ¯ . Until now, extensive theoretical sizable, the results from the flavor symmetry can describe the studies have been carried out to explore their internal struc- experimental data in a global viewpoint. To be more explicit, tures, production and decay behaviors [8Ð37]. Most of the one can write down the Hamiltonian at the hadron level with established states tend to contain a pair of heavy quark, and hadron fields and transition operators. Some limited amount thus the discovery of exotic states of new categories will be of input parameters will be introduced to describe the non- perturbative transitions. With the SU(3) amplitudes, one can a e-mail: [email protected] obtain relations between decay widths of different processes, b e-mail: [email protected] 123 645 Page 2 of 12 Eur. Phys. J. C (2019) 79 :645

⎛ −− − − ⎞ which can be examined in experiment. Such an analysis is  √1  √1  ⎜ c 2 c 2 c ⎟ also helpful to identify the decay modes that will be mostly ⎜ 1 − 0 1 0 ⎟ (F ¯ ){ } = ⎜ √   √  ⎟ useful to discover the fully-heavy tetraquark state. Since the c6 ij ⎝ 2 c c 2 c ⎠ −  SU(3) analysis is based on the light quark flavor symmetry, √1  √1  0 0 2 c 2 c c thus the analytical results can work well in all states with ⎛ ⎞ ¯ ¯ 0 0 the same bcbc flavor constituents but with different quantum 0 b b [ ] ⎜ − ⎟ numbers, even for those molecular states Bc Bcs. Generally, ( ) ij = ⎝ −0  ⎠ , Fb3¯ b 0 b the molecular states may decay against the −0 −− b b 0 than the weak interaction. ⎛ ⎞ The rest of this paper is organized as follows. In Sect. 2, + √1 0 √1 0 ⎜ b 2 b 2 b ⎟ we give the particle multiplets under the SU(3) symmetry. { } − − (F ) ij = ⎜ √1 0  √1  ⎟ . (2) Section 3 is devoted to calculate the lifetime of the tetraquark b6 ⎝ 2 b b 2 b ⎠ − − state using the OPE. In Sect. 4, we discuss the weak decays √1 0 √1   2 b 2 b b of many-body final states, including mesonic two-body or three-body decays and baryonic two-body decays. In Sect. In the meson sector, singly heavy form an SU(3) 5, we present a collection of the golden channels. Finally, we triplet or anti-triplet, while the light mesons form an octet provide a short summary. plus a flavor singlet. These multiplets can be written as ⎛ ⎞ π0 η + + ⎛ ⎞ √ + √ π K B− ⎜ 2 6 ⎟ 0 ⎜ 0 ⎟ = ⎜ π − − √π + √η 0 ⎟ , T = ⎝ ⎠ , 2 Particle multiplets in SU(3) M8 ⎝ K ⎠ Bi B 2 6 0 − ¯ 0 √η K K −2 Bs ⎛ ⎞ ⎛ ⎞ 6 The tetraquark with the quark constituents bcb¯ c¯ does not 0 D0 D contain any light quark and thus is an SU(3) singlet. Recalling T = ⎝ + ⎠ , i = ⎝ − ⎠ . Di D D D (3) that diquark [QQ] or [qq] live in Acolor ⊗ Sflavor ⊗ Sspin + D D− spaces, with A and S representing the symmetry and anti- s s symmetry representation respectively, we find the allowed The weight diagrams of the multiplets are given in Figs. 1 ⊗ = ⊕ quantum numbers are 1 1 0 2. In this paper, we and 2. will mainly focus on the lowest lying state with J P = 0+, {bb} which is abbreviated as T{¯cc¯} . In the baryon sector, we give the SU(3) representations 3 Lifetime for with different charm quantum numbers (C)or bottom quantum numbers (B) as follows. The triply heavy {bb} In this section we will discuss the lifetime of T{¯cc¯} using the baryon with C =−3 denoted as Fccc can form an SU(3) sin- −− {bb} →  =− OPE [63,64]. The decay width of T{¯cc¯} X are as follows: glet ccc . Baryons with doubly heavy quarks(i.e. C 2, , = = B C 1, B 2) are supposed to be an anti-triplet(triplet)  3−→ { } 1 d p given as (T bb → X) = i (2π)4δ4 {¯c1¯c} 2m (2π)32E ⎛ ⎞ T X i i −− ⎛ + ⎞    (¯ ¯ ¯)  ( ) { } cc ccu bc bcu ( − ) | |H| bb |2, ⎜ ⎟ ⎜ ⎟ pT pi X T{¯cc¯} (4) ( T ) = ⎜ − (¯ ¯ ¯) ⎟ , i = ⎝ 0 ( ) ⎠ , Fcc i ⎝ cc ccd ⎠ Fbc bc bcd i λ −  (¯ ¯¯) 0 (bcs) μ cc ccs bc λ ⎛ ⎞ where mT , pT , and are the mass, four-momentum and spin 0 (bbu) {bb} bb of T{¯cc¯} , respectively. The electro-weak effective Hamilto- ⎜ ⎟ w i = ⎝ − ( ) ⎠ . nian He is given as Fbb bb bbd (1) ef f − ( ) ⎡ ⎤ bb bbs G   Hew = √F ⎣ q ( q + q ) − ⎦ Consistently, the singly heavy baryons with C =−1(B = ef f Vc C1 O1 C2 O2 Vp C j O j 2 = , 1) are expected to form a triplet(anti-triplet) and a anti- q u c j=3 sextet(sextet) as [62] (5) ⎛ − − ⎞ 0   here, Ci and Oi are Wilson coefficients and operators. V sare ⎜ c c ⎟ ⎜ − 0 ⎟ the combinations of CabibboÐKobayashiÐMaskawa(CKM) (F )[ ] = −  , c3 ij ⎝ c 0 c ⎠ elements. Using the optical theorem, the total decay width of { } −− −0 ( bb → ) c c 0 T{¯cc¯} X can be rewritten as 123 Eur. Phys. J. C (2019) 79 :645 Page 3 of 12 645

(a) (b) (c) (d)

Fig. 1 The weight diagrams for the anti-charmed meson triplet, charmed meson anti-triplet, bottom meson triplet and light meson octet

(a) (b) (c) (d) (e)

(f) (g)

Fig. 2 The weight diagrams for the doubly heavy baryon are given in aÐc, which anti-triplet Fcc to be (a), triplet Fbc to be (b), or triplet Fbb to be , , (c). The singly anti-charm baryon multiplets are Fc3 Fc6¯ shown in d, e, and the singly bottom baryon multiplets are given in f, g signed as Fb3¯ Fb6

   { } 1 { } { } ( bb → ) = bb |T | bb with G F being the Fermi constant and VCKM being the T{¯cc¯} X T{¯cc¯} T{¯cc¯} (6) 2mT CKM mixing matrix. The coefficients c , are the pertur- λ i Q bative short-distance coefficients. The contribution to decay T = 4 {H ( )H ( )} Im i d xT ef f x ef f 0 (7) width from the lowest dimension operator is given as

 G2 m5 {bb} F Q 2 In the heavy quark expansion (HQE), the transition operators (T → X) = |VCKM| c ,Q {¯cc¯} 192π 3 3 up to dimension 6 contribute: Q=b,c { } { } T bb |QQ¯ |T bb  {¯cc¯} {¯cc¯} ,  G2 m5 (9) T = F Q |V |2 2mT π 3 CKM = , 192 Q b c where the matrix element { } { } c , bb ¯ bb ¯ 5 Q ¯ μν T{¯¯} |QQ|T{¯¯}  × c3,Q(QQ) + (Qgsσμν G Q) cc cc m2 (10) Q  2mT c6,Q corresponds to the bottom and charmed number in the + ( ¯ )( ¯ ) , 2 3 Qq qQ (8) ¯ m Q tetraquark state. The matrix elements of the bb operator and 123 645 Page 4 of 12 Eur. Phys. J. C (2019) 79 :645 the cc¯ operator give the bottom-quark and charm-quark num- forms a triplet H3, particularly (H3)1 = 0,(H3)2 = {bb} ,( ) = ber in the T{¯cc¯} tetraquark respectively given as Vcd H3 3 Vcs. • { } { } The c quark non-leptonic decays are classified as T bb |bb¯ |T bb  {¯cc¯} {¯cc¯} = + O( / ), 2 1 mb ¯ →¯ ¯, ¯ →¯ ¯/ ¯, ¯ → ¯ ¯, 2mT c sdu c udd ss c dsu (17)  {bb}|¯ | {bb} T{¯cc¯} cc T{¯cc¯} = 2 + O(1/mc). (11) The three kinds of decays are Cabibbo allowed, singly 2m T Cabibbo suppressed, and doubly Cabibbo suppressed The short distance coefficients c3,Qs have been calculated respectively. Under the flavor SU(3) symmetry, the tran- = . ± . , = . ± . ¯ ¯ as c3,b 5 29 0 35 c3,c 6 29 0 72 at the leading sition c¯ →¯q1q2q¯3 can be decomposed as 3 ⊗ 3 ⊗ 3 = order(LO) and c3,b = 6.88 ± 0.74, c3,c = 11.61 ± 1.55 at 3¯ ⊕ 3¯ ⊕ 6 ⊕ 15. For the Cabibbo allowed transition the next-to-leading order(NLO) [63]. Therefore we expect c¯ →¯sdu¯, the nonzero tensor components are given as {bb} that the total decay width and lifetime of the T{¯cc¯} tetraquark as ( )2 =−( )2 = ,( )2 = ( )2 = .  H6 31 H6 13 1 H15 31 H15 13 1 − { } (2.44 ± 0.23) × 10 12 GeV, LO (T bb ) = , (12) (18) {¯cc¯} (3.97 ± 1.50) × 10−12 GeV, NLO  − ¯ { } (0.27 ± 0.02) × 10 12 s, LO For the singly Cabibbo suppressed transition c¯ →¯udd τ(T bb ) = , (13) {¯cc¯} (0.17 ± 0.02) × 10−12 s, NLO and c¯ →¯uss¯, the combination of tensor components are given as where we use the heavy quark masses mc = 1.4 GeV and {bb} mb = 4.8 GeV. The lifetime of T{¯¯} is much smaller than 3 3 2 2 cc (H6) =−(H6) = (H6) =−(H6) = sin(θC ), that of 31 13 12 21 c ( )3 = ( )3 =−( )2 =−( )2 = (θ ). H15 31 H15 13 H15 12 H15 21 sin C τ( +) = . × −12 , Bc 0 507 10 s (14) (19) and in particular, their ratio is about one third. while for the doubly Cabibbo suppressed transition c¯ → ds¯ u¯,wehave 4 Weak decays ( )3 =−( )3 =− 2 θ , H6 21 H6 12 sin C ( )3 = ( )3 =− 2 θ . In this section, we will discuss the possible weak decay H15 21 H15 12 sin C (20) modes of the tetraquark. Usually, the b and c quark in tetraquark state can decay weakly. For simplicity, we will • The b quark non-leptonic decays are classified as: classify the decays modes by the quantities of CKM matrix elements. b → ccd¯ /s, b → cud¯ /s, b → ucd¯ /s, b → q1q¯2q3, (21) • For the b/c quark decays into pair, semi-leptonic decay process, we consider the following groups. here q1,2,3 represent the light quark(d/s). → / −ν¯ , ¯ → ¯/¯−ν¯ . b c u  c d s  (15) The transition operator for the b → ccd¯ /s forms an ( )2 = ∗ ,( )3 = ∗ triplet, with H3 Vcd H3 Vcs. The oper- The general electro-weak Hamiltonian for the above ator of the transition b → cud¯ /s can form an octet semi-leptonic transition can be expressed as ( )2 = 8, whose nonzero composition followed as H8 1 V ∗ ,(H )3 = V ∗ b → ucs¯  ud 8 1 us . For the transition ,the G F  μ ¯  13  ¯ operator can form an anti-symmetric 3 with (H ) = Hef f = √ Vq bq¯ γ (1 − γ5)bγμ(1 − γ5)ν 3¯  2  −(H )31 = V ∗ plus a symmetric 6 tensors with μ ¯ 3¯ cs +Vcqc¯γ (1 − γ5)qγμ(1 − γ5)ν + h.c., ( )13 = ( )31 = ∗ H6 H6 Vcs. It is straightforward to obtain (16) the similar transition b → ucd¯ by exchanging the index 2 → 3 and the Vcs → Vcd in previous transition. with q = (u, c), q = (d, s), in which the operator of − b → u/c ν¯ transition forms an SU(3) flavor triplet The charmless transition b → q1q¯2q3 (qi = d, s) can  ( )1 = ( )2,3 = ⊗ ¯ ⊗ = ⊕ ⊕ ¯ ⊕ H3 or singlet, with H3 1 and H3 0. Fur- be decomposed as 3 3 3 3 3 6 15, where − thermore, it is easy to see that the c¯ →¯q ν¯ transition the triplet H3 behave as the penguin level operator. In 123 Eur. Phys. J. C (2019) 79 :645 Page 5 of 12 645

− − − ν¯ − ν¯ ¯ c¯ ν¯ c¯ ν¯ b u/c c¯ d/s¯

c¯ c¯ c¯ b b c¯ c¯ c¯ b b b b

(a) (b) (c) (d)

{bb} /¯ Fig. 3 Feynman diagrams for T{¯cc¯} semi-leptonic decays within b c quark decay. Panel a represents a meson final state, while the panel bÐd correspond with two mesons final states. In panel a, b, the quarks in initial state interact by the exchange of W boson

 = ( → ) {bb} the S 0 b d decays, the nonzero components of Table 1 Amplitudes for tetraquark T{¯cc¯} decays into two mesons and these irreducible tensors are given as three mesons for the transition b → c−ν¯ Channel Amplitude Channel Amplitude 2 12 21 (H3) = 1,(H ) =−(H ) { } { } 6 1 6 1 T bb → B l−ν¯ a V T bb → B J/ψl−ν¯ a V = ( )23 =−( )32 = , {¯cc¯} c 2 cb {¯cc¯} c 3 cb H H 1 { } 0 { } 0 6 3 6 3 T bb → D B−l−ν¯ a V T bb → D− B l−ν¯ a V ( )12 = ( )21 =− ( )22 {¯cc¯} 4 cb {¯cc¯} 4 cb 2 H15 1 2 H15 1 3 H15 2 { } 0 T bb → D− B l−ν¯ a V =− ( )23 =− ( )32 = . {¯cc¯} s s 4 cb 6 H15 3 6 H15 3 6 (22)

For the S = 1(b → s) decays, the nonzero entries in Feynman diagrams are shown in Fig. 3a, b. One then obtain the irreducible tensor H , H , H can be obtained from 3 6 15 the amplitudes of different decay channels listed in Table 1, Eq. (22) with the exchange 2 ↔ 3. from which we derive that the simple relations between ( {bb} → 0 − −ν)¯ = different decay widths as: T{¯cc¯} D B l In the following, we will study the possible decay modes { } 0 { } 0 {bb} ( bb → − −ν)¯ = ( bb → − −ν)¯ T{¯¯} D B l T{¯¯} Ds Bs l . of T{¯cc¯} in order. cc cc ¯ − {bb} 4.1.2 c¯ → d/s¯ ν transition 4.1 Semi-leptonic T{¯cc¯} decays

− Similarly, one can find the allowed process in hadronic level 4.1.1 b → c/u ν transition ¯ − for the c¯ → d/s¯ ν transition. For the channels with the B meson plus B meson in the final state, we con- → c At the hadron level, the b u transition can be real- {bb}( ) i ν¯ {bb} struct the Hamiltonian as c1T{¯cc¯} H3 i B Bc . Then the ized by the process that T{¯¯} decays to a anti-charmed cc M( {bb} → 0 − −ν)¯ = meson plus Bc meson and ν. Following the SU(3) anal- decay amplitudes are deduced as T{¯cc¯} B Bc l ysis, the Hamiltonian at the hadronic level is constructed as , M( {bb} → 0 − −ν)¯ = c1Vcd T{¯¯} Bs Bc l c1Vcs. For completeness, {bb}( )i ν,¯ cc a1T{¯cc¯} H3 Di Bc with the coefficient a1 representing we give the corresponding Feynman diagram given in Fig. 3d. the non-perturbative parameter. For completeness, we give the corresponding Feynman diagram at quark level shown in {bb} 4.2 T{¯cc¯} : non-leptonic multi-body decays Fig. 3c. It is convenient to obtain the decay amplitudes by expanding the Hamiltonian constructed above and the ampli- M( {bb} → 0 − −ν)¯ = tude T{¯cc¯} D Bc l a1Vub. 4.2.1 b → ccd¯ /s transition For the SU(3) singlet b → c transition, the final {bb} of the many-body semileptonic decays of T{¯cc¯} can be a Bc The operators in the transition can form an triplet under the /ψ meson, Bc plus J , charmed meson plus bottom meson SU(3) light quark symmetry, and accordingly, we can write respectively. Consequently, the Hamiltonian at the hadron {bb} down the effective Hamiltonian of T{¯cc¯} producing two or level is constructed as three final states as follows: H = {bb} ν¯ + {bb} /ψν¯ H = {bb}( )i , ef f a2T{¯cc¯} Bc a3T{¯cc¯} Bc J ef f a1T{¯cc¯} H3 Di Bc + {bb} i ν.¯ H = {bb}( )i /ψ + {bb}( )i j a4T{¯cc¯} Di B (23) ef f a2T{¯cc¯} H3 Di J Bc a3T{¯cc¯} H3 D j Mi Bc 123 645 Page 6 of 12 Eur. Phys. J. C (2019) 79 :645

(a) (b) (c) (d) (e) (f)

(g) (h) (i) (j) (k)

{bb} Fig. 4 The Feynman diagrams for T{¯cc¯} non-leptonic decays within b quark decay. Panels a, b represent the two-body mesonic decays, two-body baryonic decays are shown in panels j, k and the panels cÐi indicate the three-body mesonic decays

+ {bb}( )i j , ( {bb} → − 0 −) = ( {bb} → 0 − −) a4T{¯cc¯} H3 Di D j B T{¯cc¯} B D Ds T{¯cc¯} B D Ds { } bb i j 1 { } 0 − − Hef f = a5T{¯¯} (H3) (Fcc) (Fb3)[ij] = ( bb → ), cc T{¯cc¯} Bs Ds Ds {bb} 2 + ( )i ( ) j ( ) { } −− a6T{¯cc¯} H3 Fcc Fb6¯ {ij} ( bb →  0) T{¯cc¯} cc b {bb} i +a7T (H3) (Fbc)i Fccc. (24) {¯cc¯} = 1( {bb} → − −) = ( {bb} → − −), T{¯cc¯} cc b T{¯cc¯} cc b The corresponding Feynman diagrams are given in Fig. 4. 2 { } −− { } − − {bb} (T bb →  0) = (T bb →   ) In particular, the diagrams in Fig. 4a, b represent T{¯cc¯} two- {¯cc¯} cc b {¯cc¯} cc b body mesonic decays into anti-charmed and Bc mesons, and 1 { } − − = (T bb →   ), the diagrams in Fig. 4c, d denote the three-body final states 2 {¯cc¯} cc b −− with anti-charmed meson plus Bc meson and a light meson. ( {bb} →  0) T{¯cc¯} cc b In addition, the a3 term in Hamiltonian with the final states = ( {bb} → − −), ( {bb} → −−0) of two anti-charmed mesons plus B meson and the a4 term T{¯cc¯} cc b T{¯cc¯} cc b with the final states of a anti-charmed meson plus B meson {bb} − − c = (T →   ). and J/ψ are represented in several Feynman diagrams which {¯cc¯} cc b given in Fig. 4e, f; gÐi respectively. The two-body baryonic processes induced from a5, a6, a7 terms are shown in Fig. 4j, 4.2.2 b → cud¯ /s transition k. Expanding the Hamiltonian above, one obtains the decay amplitudes which are listed in Table 2, Table 3. Besides, The hadron-level effective Hamiltonian of two-body and the relations between the different decay widths are given as three-body decays can be constructed as follows. { } { } j H = b T bb (H )i M j B + b T bb (H )i B D , {bb} 0 − − {bb} − − ef f 1 {¯cc¯} 8 j i c 2 {¯cc¯} 8 j i (T{¯¯} → D B π ) = 6(T{¯¯} → D B η) cc c cc c H = {bb}( )i j /ψ + {bb}( )i j { } − − { } − − ef f b3T{¯¯} H8 j M J Bc b4T{¯¯} H8 j D Di Bc = (T bb → D B K 0) = 2(T bb → D B π 0), cc i cc {¯cc¯} s c {¯cc¯} c { } j + b T bb (H )i B D J/ψ { } 0 3 { } 5 {¯cc¯} 8 j i (T bb → D B− K −) = (T bb → D− B−η) {¯cc¯} c 2 {¯cc¯} s c + {bb}( )i j k + {bb}( )i j k b6T{¯cc¯} H8 j B Dk Mi b7T{¯cc¯} H8 k B D j Mi = ( {bb} → − − 0), T{¯cc¯} D Bc K + {bb}( )i j k b8T{¯cc¯} H8 k B Di M j {bb} − 0 − {bb} 0 − − ( → ) = ( → ) { } T{¯cc¯} B D D T{¯cc¯} Bs D Ds + bb ( )i k j , b9T{¯cc¯} H8 j Bc Mi Mk 1 { } 0 = ( bb → − −), {bb} i [ jk] T{¯¯} B D D H = b T (H ) (F ¯ ) (F )[ ] 2 cc ef f 10 {¯cc¯} 8 j c3 b3 ik 123 Eur. Phys. J. C (2019) 79 :645 Page 7 of 12 645

{bb} = ( {bb} → −−), Table 2 Tetraquark T{¯cc¯} decays into two and three mesons for the 2 T{¯cc¯} c b transition b → ccd¯ /s {bb} −− 0 {bb} − − (T{¯¯} →   ) = 2(T{¯¯} →   ) Channel Amplitude Channel Amplitude cc c b cc c b {bb} − − = (T{¯¯} →   ), {bb} → − − ∗ {bb} → − − ∗ cc c b T{¯¯} D Bc a1V T{¯¯} Ds Bc a1Vcs cc cd cc ( {bb} → −−) = ( {bb} → −−), {bb} → − − /ψ ∗ {bb} → − − /ψ ∗ T{¯cc¯} c b 2 T{¯cc¯} c b T{¯cc¯} D Bc J a2Vcd T{¯cc¯} Ds Bc J a2Vcs {bb} − − {bb} → 0 − − ∗ {bb} → 0 − − ∗ (T →   ) T{¯cc¯} D D B a4Vcd T{¯cc¯} D Ds B a4Vcs {¯cc¯} c b { } { } bb → − − 0 ∗ bb → − − 0 ∗ 1 {bb} − − T{¯cc¯} D D B 2a4Vcd T{¯cc¯} D Ds B a4Vcs = ( →   ), T{¯cc¯} c b {bb} → − − 0 ∗ {bb} → − − 0 ∗ 2 T{¯cc¯} D Ds Bs a4Vcd T{¯cc¯} Ds Ds Bs 2a4Vcs { } −− { } − − (T bb →  0) = 2(T bb →   ), {bb} → 0 −π − ∗ {bb} → 0 − − ∗ {¯cc¯} c b {¯cc¯} c b T{¯cc¯} D Bc a3Vcd T{¯cc¯} D Bc K a3Vcs {bb} −− 0 { } a V ∗ { } 0 (T →   ) T bb → D− B−π 0 − 3√ cd T bb → D− B− K a V ∗ {¯cc¯} c b {¯cc¯} c 2 {¯cc¯} c 3 cs {bb} − − { } a V ∗ { } = ( →   ). bb → − −η 3√ cd bb → − − 0 ∗ 2 T{¯¯} c T{¯¯} D Bc T{¯¯} Ds Bc K a3V cc b cc 6 cc cd {bb} → − −η − 2 ∗ T{¯cc¯} Ds Bc 3 a3Vcs 4.2.3 b → ucd¯ /s transition

{bb} Table 3 Tetraquark T{¯cc¯} decays into doubly charmed baryon plus {bb} singly bottom baryon for the transition of b → ccd¯ /s The effective Hamiltonian at the hadron level for T{¯cc¯} pro- ducing three mesons or two baryons are constructed as Channel Amplitude Channel Amplitude H = {bb}( ){ij} , −− −− ef f c1T{¯¯} H6 Di D j Bc {bb} 0 ∗ {bb} 0 ∗ cc T{¯¯} →   −a5V T{¯¯} →   −a5V cc cc b cd cc cc b cs H = {bb}( )[ij]( ) {bb} → − − − ∗ {bb} → − − ∗ ef f c2T{¯cc¯} H3¯ Fb3 [ij] Fccc T{¯cc¯} cc b a5Vcs T{¯cc¯} cc b a5Vcd ∗ {bb} { } {bb} −− a V {bb} −−  a V ∗ + ( ) ij ( ) . T →  0 6√ cd T →   0 √6 cs c3T{¯cc¯} H6 Fb6¯ {ij} Fccc (26) {¯cc¯} cc b 2 {¯cc¯} cc b 2 { } − − { } − − ∗ T bb →   a V ∗ T bb →   a√6 Vcs It should be noticed that the operator H¯ in mesonic process {¯cc¯} cc b 6 cd {¯cc¯} cc b 2 3 { } − − a V ∗ { } − − vanishs as the two antisymmetry superscripts contract with T bb →   6√ cd T bb →   a V ∗ {¯cc¯} cc b 2 {¯cc¯} cc b 6 cs the two symmetry anti-charmed fields. Though the Hamilto- −− −− {bb} → 0  ∗ {bb} → 0  ∗ T{¯cc¯} bc ccc a7Vcd T{¯cc¯} bc ccc a7Vcs nian for the mesonic process follows only c1 term, the corre- sponding Feynman diagrams can be allowed with different topologies given in Fig. 4gÐi. One then proceed to obtain {bb} 0 − − ∗ {bb} [ ] M( → ) = + ( )i ( ) jk ( ) the decay amplitudes T{¯cc¯} D D Bc 2c1V , b11T{¯cc¯} H8 j Fc3¯ Fb6¯ {ik} cd {bb} 0 − − ∗ { } M( → ) = + bb ( )i ( ){ jk}( ) T{¯cc¯} D Ds Bc 2c1Vcs for the mesonic pro- b12T{¯cc¯} H8 j Fc6 Fb3 [ik] { } −− { } cesses and M(T bb → 0 ) = 2c V ∗ , M(T bb → {bb} i { jk} {¯cc¯} b ccc 2 cd {¯cc¯} + b T (H ) (F ) (F ¯ ){ } −− −− √ 13 {¯cc¯} 8 j c6 b6 ik 0 ) = ∗ M( {bb} → 0 ) = ∗ ccc 2c2V , T{¯¯} ccc 2c3V , {bb} b cs c√c b cd + ( )i ( ) j ( ) . {bb}  −− ∗ b14T{¯cc¯} H8 j Fcc Fbc i (25) M( →  0 ) = T{¯cc¯} b ccc 2c3Vcs for the baryonic pro- cesses, from which we derive the equation as At the topological level, the relevant Feynman diagrams are { } 0 { } −− shown in Fig. 4. One derives the decay amplitudes given (T bb → D D− B−) (T bb → 0 ) {¯cc¯} c = {¯cc¯} b ccc in Tables 4 and 5, respectively. Accordingly, we obtain the { } { } −− ( bb → 0 − −) (T bb → 0 ) relations between different decay widths as follows: T{¯cc¯} D Ds Bc {¯cc¯} b ccc −− {bb} 0 (T{¯¯} →   ) |V |2 ( {bb} → −π 0 −) = ( {bb} → − −η) = cc b ccc = cd . T{¯cc¯} Bc K 3 T{¯cc¯} Bc K {bb}  −− 2 ( →  0 ) |Vcs| T{¯cc¯} b ccc 1 { } 0 = (T bb → B−π − K ), 2 {¯cc¯} c 4.2.4 b → q1q¯2q3 charmless transition ( {bb} → − 0 −) = 3( {bb} T{¯cc¯} Bc K K T{¯cc¯} 2 {bb} At the hadron level, the effective Hamiltonian for T → −π −η), ( {bb} → − −π 0) {¯cc¯} Bc T{¯cc¯} B Ds decaying into mesons or baryons is constructed as follows, { } 1 bb 0 − − {bb} j {bb} {ij} k = (T{¯¯} → B D π ), H = ( )i + ( ) , 2 cc s ef f d1T{¯cc¯} H3 B Di D j d2T{¯cc¯} H15 k B Di D j ( {bb} → −−0) = ( {bb} → −−) H = {bb}( )i j + {bb}( )[ij] k T{¯cc¯} c b T{¯cc¯} c b ef f d3T{¯cc¯} H3 Mi D j Bc d4T{¯cc¯} H6¯ k Mi D j Bc 123 645 Page 8 of 12 Eur. Phys. J. C (2019) 79 :645

{bb} Table 4 Tetraquark T{¯cc¯} Channel Amplitude Channel Amplitude decays into two mesons and three mesons for the transition {bb} → π − − ∗ {bb} → − − ∗ T{¯¯} Bc b1V T{¯¯} K Bc b1Vus b → cud¯ /s cc ud cc {bb} → − − ∗ {bb} → − − ∗ T{¯cc¯} B D b2Vud T{¯cc¯} B Ds b2Vus {bb} → π − /ψ − ∗ {bb} → − /ψ − ∗ T{¯cc¯} J Bc b3Vud T{¯cc¯} K J Bc b3Vus {bb} → 0 − − ∗ {bb} → 0 − − ∗ T{¯cc¯} D D Bc b4Vud T{¯cc¯} D Ds Bc b4Vus {bb} → − − /ψ ∗ {bb} → − − /ψ ∗ T{¯cc¯} B D J b5Vud T{¯cc¯} B Ds J b5Vus {bb} → − 0π − ( + ) ∗ {bb} → − 0 − ( + ) ∗ T{¯cc¯} B D b6 b7 Vud T{¯cc¯} B D K b6 b7 Vus { } (b −b )V ∗ { } 0 T bb → B− D−π 0 8 √6 ud T bb → B− D− K b V ∗ {¯cc¯} 2 {¯cc¯} 6 us { } (b +b )V ∗ { } ∗ T bb → B− D−η 6 √8 ud T bb → B− D−π 0 b8√Vus {¯cc¯} 6 {¯cc¯} s 2 { } { } ( − ) ∗ T bb → B− D− K 0 b V ∗ T bb → B− D−η b8 √2b6 Vus {¯cc¯} s 6 ud {¯cc¯} s 6 {bb} → 0 −π − ( + ) ∗ {bb} → 0 − − ∗ T{¯cc¯} B D b7 b8 Vud T{¯cc¯} B D K b7Vus {bb} → 0 −π − ∗ {bb} → 0 − − ∗ T{¯cc¯} B Ds b8Vus T{¯cc¯} Bs D K b8Vud {bb} → 0 −π − ∗ {bb} → 0 − − ( + ) ∗ T{¯cc¯} Bs Ds b7Vud T{¯cc¯} Bs Ds K b7 b8 Vus { } ∗ { } 0 bb → π 0 − − b9√Vus bb → π − − ∗ T{¯¯} K Bc T{¯¯} K Bc b9V cc  2 cc us {bb} → π −η − 2 ∗ {bb} → 0 − − ∗ T{¯cc¯} Bc 3 b9Vud T{¯cc¯} K K Bc b9Vud { } ∗ T bb → K −ηB− − b9√Vus {¯cc¯} c 6

(a) (b) (c) (d)

{bb} ¯ Fig. 5 The Feynman diagrams for T{¯cc¯} non-leptonic decays within c quark decay. The lowest allowed many-body charmed decays are three-body mesonic decays shown in panels aÐc and two-body baryonic decays given in panel d

{bb} Table 5 Tetraquark T{¯cc¯} Channel Amplitude Channel Amplitude decays into singly charmed baryon plus singly bottom {bb} → −− − ∗ {bb} → −− ∗ T{¯¯} c b10Vus T{¯¯} c b10V baryon for the transition cc b cc b ud { } − − { } − − ∗ → ¯ / T bb →   b V ∗ T bb →   b11√Vus b cud s {¯cc¯} c b 11 ud {¯cc¯} c b 2 { } − − b V ∗ { } − − T bb →   11√ ud T bb →   b V ∗ {¯cc¯} c b 2 {¯cc¯} c b 11 us −− −− {bb} →  0 − ∗ {bb} →  0 − ∗ T{¯cc¯} c b b12Vud T{¯cc¯} c b b12Vus { } − − ∗ { } − − b V ∗ T bb →   − b12√Vus T bb →   12√ ud {¯cc¯} c b 2 {¯cc¯} c b 2 { } −− b V ∗ { } −− ∗ T bb →  0 13√ ud T bb →  0 b13√Vus {¯cc¯} c b 2 {¯cc¯} c b 2 { } − − b V ∗ { } − − T bb →   13√ ud T bb →   1 b V ∗ {¯cc¯} c b 2 {¯cc¯} c b 2 13 us { } − − { } − − ∗ T bb →   1 b V ∗ T bb →   b13√Vus {¯cc¯} c b 2 13 ud {¯cc¯} c b 2 −− −− {bb} →  0 ∗ {bb} →  0 ∗ T{¯cc¯} cc bc b14Vud T{¯cc¯} cc bc b14Vus

123 Eur. Phys. J. C (2019) 79 :645 Page 9 of 12 645

{bb} Table 6 Tetraquark T{¯cc¯} Channel Amplitude Channel Amplitude decays into three mesons { } { } induced by the charmless bb 0 − − bb − 0 − −d3+√d4+5d5 T{¯¯} → D π B d3 − d4 + 3d5 T{¯¯} → D π B b → d transition cc c cc c 2 { } + ( + ) { } T bb → D−ηB− d3 3√d4 d5 T bb → D− K 0 B− d + d − d {¯cc¯} c 6 {¯cc¯} s c 3 4 5 {bb} → 0 − − ( + ) {bb} → − − 0 ( − ) T{¯cc¯} D D B d1 6d2 T{¯cc¯} D D B 2 d1 2d2 {bb} → − − 0 ( − ) T{¯cc¯} D Ds Bs d1 2d2

{bb} Table 7 Tetraquark T{¯cc¯} Channel Amplitude Channel Amplitude decays into three mesons induced by the charmless b → s {bb} → 0 − − − + {bb} → − 0 − + − T{¯cc¯} D K Bc d3 d4 3d5 T{¯cc¯} D K Bc d3 d4 d5 transition √  {bb} → −π 0 − ( + ) {bb} → −η − − 2 ( − ) T{¯cc¯} Ds Bc 2 d4 2d5 T{¯cc¯} Ds Bc 3 d3 3d5 {bb} → 0 − − ( + ) {bb} → − − 0 ( − ) T{¯cc¯} D Ds B d1 6d2 T{¯cc¯} D Ds B d1 2d2 {bb} → − − 0 ( − ) T{¯cc¯} Ds Ds Bs 2 d1 2d2

{bb} Table 8 Tetraquark T{¯cc¯} Channel Amplitude Channel Amplitude decays into doubly charmed −− − baryon plus singly bottom {bb} →  0 − {bb} →  − + T{¯cc¯} cc b 2d7 d6 T{¯cc¯} cc b d6 2d7 baryon induced by the { } −− + { } − − → T bb →  0 d8√6d9 T bb →   d − 2d charmless b d transition {¯cc¯} cc b 2 {¯cc¯} cc b 8 9 { } − − − T bb →   d8√2d9 {¯cc¯} cc b 2

+ {bb}( ){ij} k , ( {bb} → − −) d5T{¯cc¯} H15 k Mi D j Bc T{¯cc¯} cc b H = {bb}( )i ( ) j ( ) = ( {bb} → − −), ( {bb} → − −) ef f d6T{¯cc¯} H3 Fcc Fb3 [ij] 2 T{¯cc¯} cc b T{¯cc¯} cc b { } [ ] + bb ( ) ij ( )k( ) 1 {bb} − − d7T{¯cc¯} H6¯ k Fcc Fb3 [ij] = (T →   ). 2 {¯cc¯} cc b + {bb}( )i ( ) j ( ) d8T{¯cc¯} H3 Fcc Fb6¯ {ij} + {bb}( ){ij}( )k( ) . ¯ →¯ ¯ d9T{¯cc¯} H15 k Fcc Fb6¯ {ij} (27) 4.2.5 c q1q2q3 transition

In the three-body mesonic decays, the decay amplitudes are {bb} The effective Hamiltonian at the hadron-level for T{¯cc¯} pro- given in Table 6 for the transition b → d and Table 7 for the ducing two or three body final states can be constructed as transition b → s. In the two-body baryonic decays, the cor- follows, responding amplitudes are listed in Table 8 for the transition b → d and Table 9 for the transition b → s. We obtain the {bb} k i j Hef f = f1T{¯¯} (H ){ } B B Dk, relations of these decay widths given as cc 15 ij H = {bb}( )k i j ef f f2T{¯cc¯} H6 [ij] Mk B Bc ( {bb} → 0 − −) = ( {bb} T{¯cc¯} B D D 2 T{¯cc¯} + {bb}( )k i j , f3T{¯cc¯} H15 {ij} Mk B Bc 0 − − {bb} 0 − − → ), ( → ) { } Bs D Ds T{¯cc¯} B D Ds H = bb ( )k ( )[ij]( ) ef f f4T{¯cc¯} H6 [ij] Fc3¯ Fbb k 1 { } 0 = ( bb → − −), {bb} k {ij} T{¯¯} Bs Ds Ds + f T (H ) (F ) (F ) . (28) 2 cc 5 {¯cc¯} 15 {ij} c6 bb k

{bb} Table 9 Tetraquark T{¯cc¯} Channel Amplitude Channel Amplitude decays into doubly charmed −− − baryon plus singly bottom {bb} →  0 − {bb} →  − − − T{¯cc¯} cc b 2d7 d6 T{¯cc¯} cc b d6 2d7 baryon induced by the { } −− + { } − − − → T bb →  0 d8√6d9 T bb →   d8√2d9 charmless b s transition {¯cc¯} cc b 2 {¯cc¯} cc b 2 {bb} → − − − T{¯cc¯} cc b d8 2d9

123 645 Page 10 of 12 Eur. Phys. J. C (2019) 79 :645

{bb} Table 10 Tetraquark T{¯cc¯} Channel Amplitude Channel Amplitude decays into three mesons for the { } { } transition c¯ →¯q1q2q¯3.In bb − 0 − bb − 0 − T{¯¯} → B B D 2 f1 T{¯¯} → B K B f2 + f3 particular, the amplitudes are cc s cc c {bb} → 0π − − − shown as Cabibbo allowed, T{¯cc¯} Bs Bc f3 f2 singly Cabibbo suppressed, { } 0 { } 0 T bb → B− B D− −2 f sC T bb → B− B D− 2 f sC doubly suppressed respectively {¯cc¯} 1 {¯cc¯} s s 1 { } ( + ) { } T bb → B−π 0 B− f2 √f3 sC T bb → B−ηB− − 3 ( f + f ) sC {¯cc¯} c 2 {¯cc¯} c 2 2 3 {bb} → 0π − − ( − ) {bb} → 0 − − ( − ) T{¯cc¯} B Bc f2 f3 sC T{¯cc¯} Bs K Bc f3 f2 sC {bb} → − 0 − 2 {bb} → − 0 − ( + ) 2 T{¯cc¯} B B Ds 2 f1sC T{¯cc¯} B K Bc f2 f3 sC   {bb} → 0 − − ( − ) − 2 T{¯cc¯} B K Bc f2 f3 sC

{bb} PC Table 11 Tetraquark T{¯cc¯} decays into singly charmed baryon and dou- with the same quark constituent but with the different J ¯ →¯ ¯ 0 bly bottom baryon for the transition c q1q2q3. In particular, the quantum numbers. For instance, one can replace a K by amplitudes are shown as Cabibbo allowed, singly Cabibbo suppressed, ∗0 doubly suppressed respectively K . Following the criteria [65], we can obtain the golden decay Channel Amplitude Channel Amplitude √ channels in Table 12. {bb} → −− − {bb} → −− T{¯cc¯} c bb 2 f4 T{¯cc¯} c bb 2 f5 {bb} − − {bb} − − • Branching fractions: For c¯-quark decays, one should T{¯¯} →   2 f4sC T{¯¯} →   −2 f4sC cc c bb √ cc c bb √ {bb} − − {bb} − − choose the corresponding channels with the transition T{¯¯} →   − 2 f5sC T{¯¯} →   2 f5sC − cc c bb cc c bb ¯ →¯ ¯ ¯ →¯ ν¯ − − √ of c sdu or c s , while for b-quark decays, {bb} →  − − 2 {bb} →  − 2 − T{¯cc¯} c bb 2 f4sC T{¯cc¯} c bb 2 f5sC the process with the quark level transition b → c ν¯ or b → ccs¯ or b → cud¯ should be chosen. • Detection efficiency: At hadron colliders like LHC, Here, it should be noticed that the above effective Hamil- charged have higher rates to be detected than tonian can not lead to the two-body mesonic decays of neutral states. So we will remove the channels with {bb} π 0 η φ ρ±(→ π ±π 0 ∗±(→ T{¯¯} . Further more, the corresponding Feynman diagrams the final states , , , n, ), K cc ± ± are given in Fig. 5. Expanding the Hamiltonian above and we K π 0) and ω, but keep the modes with π , K 0(→ + − + − can obtain the decay amplitudes shown in Tables 10 and 11. π π ), ρ0(→ π π ). The relations between different decay widths are given as ( {bb} → − − 0) = ( {bb} → − − 0), ( {bb} T{¯cc¯} D B B T{¯cc¯} Ds B Bs T{¯cc¯} 6 Conclusions 1 { } → B−π 0 B−) = (T bb → B−ηB−), c 3 {¯cc¯} c Although many charmonium-like and bottomonium-like {bb} 0 − − {bb} 0 − − states have been found on experimental side, our current (T{¯¯} → B π B ) = (T{¯¯} → B K B ), cc c cc s c knowledge on hadron exotics is still far from mature. The ( {bb} → −− ) = ( {bb} → −− ), ( {bb} T{¯cc¯} c bb T{¯cc¯} c bb T{¯cc¯} understanding on the hadron spectroscopy can be deepen by → −− ) = ( {bb} → −− ). the study of exotic states of new categories. In this direc- c bb T{¯cc¯} c bb {bb} tion, the fully-heavy tetraquark T{¯cc¯} are of great interest. In this paper, we have discussed the lifetime and the weak {bb} decays. From our calculation, the lifetime of T{¯cc¯} is found 5 Golden decay channels about 0.1Ð0.3 ps. We have systematically discussed the pos- sible weak decay modes, such as two- or three-body mesonic decays and two-body baryonic decays. Finally, we have col- In this section, we will discuss the golden channels to recon- { } {bb} lected the golden channels of T bb with the largest branching struct the T{¯cc¯} . Our previous classifications are mainly based {¯cc¯} on the CKM elements. In principle, the amplitudes of b-quark fraction and experimental detector efficiency. Certainly, the − decay transitions such as b → c ν¯, b → ccs¯ and b → cud¯ framework and analysis can be applicable to the states with −2 ¯ ¯ will receive the largest contribution as Vcb ∼ 10 .Forthe the same quark structure bbcc. Our results for the lifetime − c¯-quark decay, the c¯ →¯sdu¯ and c¯ →¯s ν¯ transition has and golden channels are helpful to search for the fully-heavy ∗ ∼ tetraquark in future experiments. the largest decay widths as Vcs 1. In our analysis, the final meson can be replaced by its corresponding counterpart 123 Eur. Phys. J. C (2019) 79 :645 Page 11 of 12 645

Table 12 Cabibbo allowed { } T bb c¯-quark and b-quark {¯cc¯} {bb} → 0 − −ν¯ {bb} → − 0 − {bb} → − 0 − {bb} → 0π − − decays respectively T{¯cc¯} Bs Bc l T{¯cc¯} B Bs D T{¯cc¯} B K Bc T{¯cc¯} Bs Bc {bb} → −− {bb} → −− T{¯cc¯} c bb T{¯cc¯} c bb {bb} → −ν¯ {bb} → π − − {bb} → − − {bb} → − − T{¯cc¯} Bcl T{¯cc¯} Bc T{¯cc¯} B D T{¯cc¯} Ds Bc {bb} → /ψ −ν¯ {bb} → 0 − −ν¯ {bb} → − 0 −ν¯ {bb} → − 0 −ν¯ T{¯cc¯} Bc J l T{¯cc¯} D B l T{¯cc¯} D B l T{¯cc¯} Ds Bs l {bb} → − − /ψ {bb} → 0 − − {bb} → − − 0 {bb} → − − 0 T{¯cc¯} Ds Bc J T{¯cc¯} D Ds B T{¯cc¯} D Ds B T{¯cc¯} Ds Ds Bs {bb} → 0 − − {bb} → − − 0 {bb} → π − /ψ − {bb} → 0 − − T{¯cc¯} D Bc K T{¯cc¯} D Bc K T{¯cc¯} J Bc T{¯cc¯} D D Bc {bb} → − − /ψ {bb} → − 0π − {bb} → − − 0 {bb} → 0 −π − T{¯cc¯} B D J T{¯cc¯} B D T{¯cc¯} B Ds K T{¯cc¯} B D {bb} → 0 − − {bb} → 0 −π − {bb} → 0 − − {bb} → 0 − − T{¯cc¯} Bs D K T{¯cc¯} Bs Ds T{¯cc¯} K K Bc T{¯cc¯} D Ds Bc −− − −− − {bb} →  0 {bb} →  − {bb} →  0 {bb} →  − T{¯cc¯} cc b T{¯cc¯} cc b T{¯cc¯} cc b T{¯cc¯} cc b − −− − − {bb} →  − {bb} → 0  {bb} →  − {bb} →  − T{¯cc¯} cc b T{¯cc¯} bc ccc T{¯cc¯} c b T{¯cc¯} c b −− − − −− {bb} →  0 {bb} →  − {bb} →  − {bb} →  0 T{¯cc¯} c b T{¯cc¯} c b T{¯cc¯} c b T{¯cc¯} c b − − −− −− {bb} →  − {bb} →  − {bb} →  0 {bb} → 0 T{¯cc¯} c b T{¯cc¯} c b T{¯cc¯} cc bc T{¯cc¯} b ccc −− {bb} → 0 T{¯cc¯} b ccc

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