Fully Heavy Tetraquark ${bb \bar c \bar c}$: Lifetimes and Weak Decays

We study the lifetime and weak decays of the full-heavy S-wave $0^+$ tetraquark $T^{\{bb\}}_{\{\bar c\bar c\}}$. Using the operator product expansion rooted in heavy quark expansion, we find a rather short lifetime, at the order $(0.1-0.3)\times 10^{-12}s$ depending on the inputs. With the flavor SU(3) symmetry, we then construct the effective Hamiltonian at the hadron level, and derive relations between decay widths of different channels. According to the electro-weak effective operators, we classify different decay modes, and make a collection of the golden channels, such as $T_{\{\bar c \bar c\}}^{\{bb\}}\to B^- K^0 B_c^-$ for the charm quark decay and $T_{\{\bar c \bar c\}}^{\{bb\}}\to B^-D^-$ for the bottom quark decay. Our results for the lifetime and golden channels are helpful to search for the fully-heavy tetraquark in future experiments.


I. INTRODUCTION
In the past decades, quark model has achieved great successes in the hadron spectroscopy study. In addition to the quark-anti-quark assignment for a meson and three-quark interpretation of a baryon, it allows the existence of nonstandard exotic states [1][2][3][4][5][6]. Since the observation of X(3872) in 2003 [1], many exotic candidates have been announced on the experimental side in the heavy quarkonium sector in various processes [7]. Charged heavy quarkoniumlike states Z c (3900) ± , Z c (4020) ± , Z b (10610) ± , and Z b (10650) ± observed by BES-III and Belle collaborations [2][3][4] have already experimentally established as being exotic, since they contain at least two quarks and two antiquarks with the hidden QQ. Until now, extensive theoretical studies have been carried out to explore their internal structures, production and decay behaviors . Most of the established states tend to contain a pair of heavy quark, and thus the discovery of exotic states of new categories will be valuable. Fully-heavy four-quark state with no light quark degrees of freedom is of this type and might be an ideal probe to study the interplay between perturbative QCD and non-perturbative QCD.
Generally speaking, more heavy quarks correspond to a larger mass. For instance, there have been some phenomenological studies to determine the mass and the spectrum properties of the fully-heavy tetraquark bcbc, including the constituent quark and diquark model [34,35], chiral quark model [36], nonrelativistic effective field theory(NREFT) [37], and QCD sum rules [38,39]. In Ref. [37], the authors utilize the NREFT to determine the mass with the upper bound as 12.58 GeV, consistent with the mass calculated in the chiral quark model [36]. Despite of these studies, it is still not conclusive that whether the bcbc (or its charge conjugate cbcb) is above or below the B c B c threshold. It is likely that the bcbc lies below the threshold of the B c B c pair, which means that such a state is stable against the strong interaction. In this case, the dominant decay modes would be induced by weak interaction. In a diquark-diquark model [40], the S-wave fully-heavy tetraquark state bcbc can form 0 + and 2 + . In this paper we will mainly focus on the lowest lying state 0 + , which might be assigned as a weakly-coupled state.
In this paper, we will first use the operator product expansion (OPE) technique and calculate the lifetime of the S-wave 0 + bcbc. The light flavor SU(3) symmetry is a useful tools to analyze weak decays of a heavy quark, and has been successfully applied to the meson or baryon system [41][42][43][44][45][46][47][48][49][50][51][52][53][54][55][56]. Though the SU(3) breaking effects in charm quark transition might be sizable, the results from the flavor symmetry can describe the experimental data in a global viewpoint. To be more explicit, one can write down the Hamiltonian at the hadron level with hadron fields and transition operators. Some limited amount of input parameters will be introduced to describe the non-perturbative transitions. With the SU(3) amplitudes, one can obtain relations between decay widths of different processes, which can be examined in experiment. Such an analysis is also helpful to identify the decay modes that will be mostly useful to discover the fully-heavy tetraquark state.
The rest of this paper is organized as follows. In Sec. II, we give the particle multiplets under the SU(3) symmetry. Section III is devoted to calculate the lifetime of the tetraquark state using the OPE. In Section IV, we discuss the weak decays of many-body final states, including mesonic two-body or three-body decays and baryonic two-body decays. In section V, we present a collection of the golden channels. Finally, we provide a short summary.

II. PARTICLE MULTIPLETS IN SU(3)
The tetraquark with the quark constituents bcbc does not contain any light quark and thus is an SU(3) singlet.
Recalling that diquark [QQ] or [qq] live in A color ⊗ S flavor ⊗ S spin spaces, with A and S representing the symmetry and anti-symmetry representation respectively, we find the allowed spin quantum numbers are 1 ⊗ 1 = 0 ⊕ 2. In this paper, we will mainly focus on the lowest lying state with J P = 0 + , which is abbreviated as T Consistently, the singly heavy baryons with C = −1(B = 1) are expected to form a triplet(anti-triplet) and a anti-sextet(sextet) as [57] (d) The weight diagrams for the doubly heavy baryon are given in (a,b,c), which anti-triplet F cc to be (a), triplet F bc to be (b), or triplet F bb to be (c). The singly anti-charm baryon multiplets are F c3, F c6 shown in (d,e), and the singly bottom baryon multiplets are given in (f,g) signed as F b3 , F b6 .

III. LIFETIME
In this section we will discuss the lifetime of T {bb} {cc} using the OPE [58,59]. The decay width of T {bb} {cc} → X are as follows: where m T , p µ T , and λ are the mass, four-momentum and spin of T {bb} {cc} , respectively. The electro-weak effective Hamiltonian H ew ef f is given as In the heavy quark expansion (HQE), the transition operators up to dimension 6 contribute: with G F being the Fermi constant and V CKM being the CKM mixing matrix. The coefficients c i,Q are the perturbative short-distance coefficients. The contribution to decay width from the lowest dimension operator is given as where the matrix element The short distance coefficients c 3,Q s have been calculated as c 3,b = 5.29 ± 0.35, c 3,c = 6.29 ± 0.72 at the leading order(LO) and c 3,b = 6.88 ± 0.74, c 3,c = 11.61 ± 1.55 at the next-to-leading order(NLO) [58]. Therefore we expect that the total decay width and lifetime of the T and in particular, their ratio is about one third.

IV. WEAK DECAYS
In this section, we will discuss the possible weak decay modes of the tetraquark. Usually, the b and c quark in tetraquark state can decay weakly. For simplicity, we will classify the decays modes by the quantities of CKM matrix elements.
• For the b/c quark decays into lepton pair, semi-leptonic decay process, we consider the following groups.
The general electro-weak Hamiltonian for the above semi-leptonic transition can be expressed as • The c quark non-leptonic decays are classified as The three kinds of decays are Cabibbo allowed, singly Cabibbo suppressed, and doubly Cabibbo suppressed respectively. Under the flavor SU(3) symmetry, the transitionc →q 1 q 2q3 can be decomposed as3 For the Cabibbo allowed transitionc →sdū, the nonzero tensor components are given as For the singly Cabibbo suppressed transitionc →ūdd andc →ūss, the combination of tensor components are given as while for the doubly Cabibbo suppressed transitionc →dsū, we have • The b quark non-leptonic decays are classified as: here q 1,2,3 represent the light quark(d/s).
The transition operator for the b → ccd/s forms an triplet, with ( The operator of the transition b → cūd/s can form an octet 8, whose nonzero composition followed as ( It is straightforward to obtain the similar transition b → ucd by exchanging the index 2 → 3 and the V cs → V cd in previous transition. where the triplet H 3 behave as the penguin level operator. In the ∆S = 0(b → d) decays, the nonzero components of these irreducible tensors are given as For the ∆S = 1(b → s) decays, the nonzero entries in the irreducible tensor H 3 , H 6 , H 15 can be obtained from Eq. (22) with the exchange 2 ↔ 3.
In the following, we will study the possible decay modes of T {bb} {cc} in order.
At the hadron level, the b → u transition can be realized by the process that T Feynman diagrams are shown in Fig. 3.(a,b). One then obtain the amplitudes of different decay channels listed in Tab. I, from which we derive that the simple relations between different decay widths as: Similarly, one can find the allowed process in hadronic level for thec →d/sℓ − ν ℓ transition. For the channels with the B meson plus B c meson in the final state, we construct the Hamiltonian as Then the decay For completeness, we give the corresponding Feynman diagram given in Fig. 3.(d).
The corresponding Feynman diagrams are given in Fig. 4. In particular, the diagrams in Fig. 4.(a,b) represent T  Fig. 4.(e,f) and Fig. 4.(g,h,i) respectively. The two-body baryonic processes induced from a 5 , a 6 , a 7 terms are shown in Fig. 4.(j,k). Expanding the Hamiltonian above, one obtains the decay amplitudes which are listed in Tab. II, Tab. III. Besides, the relations between the different decay widths are given as follows.

b → cūd/s transition
The hadron-level effective Hamiltonian of two-body and three-body decays can be constructed as At the topological level, the relevant Feynman diagrams are shown in Fig. 4. One derives the decay amplitudes given in Tab. IV, Tab. V respectively. Accordingly, we obtain the relations between different decay widths as follows: It should be noticed that the operator H3 in mesonic process vanishs as the two antisymmetry superscripts contract with the two symmetry anti-charmed fields. Though the Hamiltonian for the mesonic process follows only c 1 term, the corresponding Feynman diagrams can be allowed with different topologies given in Fig. 4.(g,h,i). One then proceed to obtain the decay amplitudes M(T cs for the baryonic processes, from which we derive the equation as

b → q1q2q3 Charmless transition
At the hadron level, the effective Hamiltonian for T {bb} {cc} decaying into mesons or baryons is constructed as follows, In the three-body mesonic decays, the decay amplitudes are given in Tab. VI for the transition b → d and Tab. VII for the transition b → s. In the two-body baryonic decays, the corresponding amplitudes are listed in Tab. VIII for the transition b → d and Tab. IX for the transition b → s. We obtain the relations of these decay widths given as

5.c →q1q2q3 transition
The effective Hamiltonian at the hadron-level for T {bb} {cc} producing two or three body final states can be constructed as follows, Here, it should be noticed that the above effective Hamiltonian can not lead to the two-body mesonic decays of T {bb} {cc} . Further more, the corresponding Feynman diagrams are given in Fig. 5. Expanding the Hamiltonian above and we can obtain the decay amplitudes shown in Tab. X, and Tab. XI. The relations between different decay widths are given as In this section, we will discuss the golden channels to reconstruct the T {bb} {cc} . Our previous classifications are mainly based on the CKM elements. In principle, the amplitudes of b-quark decay transitions such as b → cℓ −ν ℓ , b → ccs and b → cūd will receive the largest contribution as V cb ∼ 10 −2 . For thec-quark decay, thec →sdū andc →sℓ −ν ℓ transition has the largest decay widths as V * cs ∼ 1. In our analysis, the final meson can be replaced by its corresponding counterpart with the same quark constituent but with the different J P C quantum numbers. For instance, one can replace a K 0 by K * 0 .
Following the criteria [60], we can obtain the golden decay channels in Table XII.
• Branching fractions: Forc-quark decays, one should choose the corresponding channels with the transition of c →sdū orc →sℓ −ν ℓ , while for b-quark decays, the process with the quark level transition b → cℓ −ν ℓ or b → ccs or b → cūd should be chosen.

VI. CONCLUSIONS
Although many charmonium-like and bottomonium-like states have been found on experimental side, our current knowledge on hadron exotics is still far from mature. The understanding on the hadron spectroscopy can be deepen by the study of exotic states of new categories. In this direction, the fully-heavy tetraquark T {bb} {cc} are of great interest. In this paper, we have discussed the lifetime and the weak decays. From our calculation, the lifetime of T {bb} {cc} is found about 0.1 − 0.3 ps. We have systematically discussed the possible weak decay modes, such as two-or three-body mesonic decays and two-body baryonic decays. Finally, we have collected the golden channels of T {bb} {cc} with the largest branching fraction and experimental detector efficiency. Our results for the lifetime and golden channels are helpful to search for the fully-heavy tetraquark in future experiments.