Weak Decays of Doubly-Heavy Tetraquarks ${b\bar c}{q\bar q}$

We study the weak decays of exotic tetraquark states ${b\bar c}{q\bar q}$ with two heavy quarks. Under the SU(3) symmetry for light quarks, these tetraquarks can be classified into an octet plus a singlet: $3\bigotimes\bar 3=1\bigoplus8$. We will concentrate on the octet tetraquarks with $J^{P}=0^{+}$, and study their weak decays, both semileptonic and nonleptonic. Hadron-level effective Hamiltonian is constructed according to the irreducible representations of the SU(3) group. Expanding the Hamiltonian, we obtain the decay amplitudes parameterized in terms of a few irreducible quantities. Based on these amplitudes, relations for decay widths are derived, which can be tested in future. We also give a list of golden channels that can be used to look for these states at various colliders.


I. INTRODUCTION
Since the first discovery of X(3872) by Belle in 2003 [1], a large number of charmonium-like and bottomonium-like hadrons have been discovered in the past decade [2]. Many of these discovered states defy a standard quarkonium interpretation and likely have a pair of hidden flavored quarks, with the quark content QQqq ′ (for a recent review, see Refs. [3,4]). Here Q represents a heavy bottom/charm quark and q(q ′ ) denotes a light u, d, s quark. Extensive theoretical studies have been carried out to explore their structures, productions and decays . In 2016, the D0 collaboration has reported an evidence for the open-bottom tetraquark X(5568) [31], though it has not been confirmed by the other experimental groups [32][33][34][35]. Therefore the existence of open-flavored tetraquarks is an interesting question in hadronic physics, in particular the hadron spectroscopy.
Four-quark states with two different heavy quarks and two light quarks are of great interest since they can provide a unique platform to study strong interactions under two color static sources. In the diquark-antidiquark model [36], the four-quarks system [bq][cq] with orbital angular momentum L=0 can have J P = 0 + [37]. Since the 0 + tetraquarks are lowest lying, their weak decays can provide unique insights to unravel their internal structure. In this paper, we adopt the SU(3) flavor symmetry to handle these weak decays. The SU(3) approach has been successfully applied into the B meson and heavy baryon decays [38][39][40][41][42][43][44][45][46][47][48][49][50][51][52][53][54] and a global picture consistent data has been established. In the SU(3) symmetry, the tetraquarks with two light quarks can form an octet and a singlet. In this work, we will concentrate on the octet, abbreviated as X bc8 .
In the following, we will first construct the hadron-level effective Hamiltonian according to the irreducible representations of the SU(3) group. Expanding these Hamiltonian, we obtain the decay amplitudes parameterized in terms of a few SU(3) irreducible quantities. Based on the expanded amplitudes, relations for decay widths are derived, which can be examined in future. We also give a list of golden channels that can be used to look for these states at various colliders.
The rest of this paper is organized as follows. In Sec. II, we give the multiplets expressions under the SU(3) flavor symmetry. From Sec. III to Sec. IV, we mainly study the semi-leptonic and non-leptonic weak decays of the X bc8 states. In Sec. V, we will give a collection of the golden channels that can be used to discovery the doubly heavy tetraquarks in future experiments. we summarize in the last section.

II. PARTICLE MULTIPLETS
Based on the light flavor SU (3) symmetry, open-flavor tetraquark with the quark constituents bcqq can form an octet and a singlet, of which the octet can be represented as For simplicity, we will not consider the flavor singlet in this work. The decomposition can be reached by 3 3 = 1 8.
In the meson sector, light pseudoscalar mesons or vector mesons can also form an octet plus a singlet, generally, the octet is written as as the same quark content, vector meson octet will give the similar structure. Besides, we need the representation of bottom mesons which form an SU(3) anti-triplet given as: , and the representation of anti-triplet charmed mesons given as: In this section, we will discuss the possible semi-leptonic weak decay modes of the octet tetraquark T bc8 . Considering the decay modes at quark level, T bc8 will hold both b-quark andc-quark decays. For the b-quark, semi-leptonic weak decays are governed by For thec-quark, semi-leptonic decays are induced bȳ In the following, we will study the decays above in order.
In b quark decay, the general electro-weak Hamiltonian of the b → c/uℓ − ν ℓ transition can be expressed as with q ′ = u, c, in which the electro-weak vertex is suggested to be a V − A structure. As a contrast, the vertex forms a triplet representation H ′ 3 within SU(3) flavor symmetry, specifically (H ′ 3 ) 1 = 1 and (H ′ 3 ) 2,3 = 0. At the hadron level, the transition can be included into the process that X bc8 decays to a charmed meson and ℓν ℓ . Following the SU(3) analysis, the Hamiltonian of hadronic level is constructed as here, the coefficient a 1 represents the non-perturbative parameter. For completeness, we give the corresponding Feynman diagram at quark level shown in Fig. 1.(a). It is convenient to achieve the decay amplitudes given in Tab. I by expanding the Hamiltonian constructed above, in which all amplitudes are represented as a 1 . Therefor, we can directly obtain the relations between different decay channels given as follows.
It is should be note that the phase space difference will provide corrections to these relations. For the SU(3) singlet b → c transition, the process at the hadron level is that X bc8 decays into a light meson octet and ℓν ℓ . Consequently, the Hamiltonian at the hadron level is constructed as One then obtain the amplitudes of different decay channels listed in Tab. I, from which we derive that all the channels in the transition give the equal decay widths.

2.c →d/sℓ −ν decay into B meson and ℓ −ν
Inc quark decay, the electro-weak effective Hamiltonian are given as Feynman diagrams for the b-quark non-leptonic decays of tetraquark X bc8 . Panels (a-k) correspond to the decays into a pair of mesons. In panels (k), the final meson produced by gluons is the flavor singly state which we will not consider here. The diagrams in panels(c,d,g,h,i) are usually power suppressed as a pair of quark and anti-quark in the initial state can annihilate.
The decay amplitudes deduced from the Hamiltonian above are listed in Tab. II. For completeness, we give the corresponding Feynman diagrams given in Fig. 1.(b). One then obtain the relations between different channels as follows.
More technically, the relations between different channels may be modified with a view to the SU(3) symmetry breaking effects in the charmed or anti-charmed quark decays.

IV. NON-LEPTONIC T bc8 DECAYS
In b quark non-leptonic decays, the transitions can be classified into four different kinds in the light of CKM matrix: here q 1,2,3 represent the light quark. Inc quark non-leptonic decays, the pronounced classifications are given as: which are Cabibbo allowed, singly Cabibbo suppressed, and doubly Cabibbo suppressed respectively. In the following, we will study the X bc8 non-leptonic decays in these orders. The operator of the b → ccd/s transition can form an triplet under the SU(3) light quark symmetry, according to that we can write down the effective Hamiltonian of X bc8 producing two mesons as follows.
Consistently, the corresponding Feynman diagrams are given in Fig. 2(a-d). In particular, the diagrams in Fig. 2(a,b) represent X bc8 decays into D and J/ψ mesons, and the diagrams in Fig. 2(c,d) denote processes with D and light mesons final states. Expanding the two Hamiltonian above, one obtains the decay amplitudes which are listed in Tab. III. In addition, the relations between the different decay widths are given as follows.
In topological level, the relevant Feynman diagrams are shown in Fig. 2(a-i). Specifically, the Feynman diagrams of X bc8 decays into a light meson plus J/ψ are given in Fig. 2(b,e), and the Fig. 2(a,f,g) correspond with the processes of producing D plusD, the processes that X bc8 produces two light mesons are represented by Feynman diagrams in Fig. 2(c,d,h,i). One derives the decay amplitudes given in Tab. IV respectively. Accordingly, we obtain the relations between different decay widths for J/ψ and a light meson as follows.
The relations for producing the charmed meson and anti-charmed meson become The relations for producing two light mesons become

(III). b → ucd/s transition: two-body decays into mesons
The operator (ūb)(qc) can form an anti-symmetric3 and a symmetric 6 representations. In the transition b → ucs, the nonzero components of the anti-symmetric tensor H ′′ 3 and the symmetric tensor H 6 are given respectively as (H ′′ 3 ) 13 = −(H ′′ 3 ) 31 = V * cs , (H6) 13 = (H6) 31 = V * cs . In the transition b → ucd, the nonzero components can be obtained by interchanging the subscripts 2 ↔ 3, and replacing V cs by V cd . Therefor, the effective Hamiltonian at the hadron level for X bc8 producing two mesons is constructed as Also the Feynman diagrams corresponding with the Hamiltonian above are given in Fig. 2(a-d). One then deduce the decay amplitudes for different channels shown in Tab. V, which leads to the relations for decay widths as For 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. (15) with the exchange 2 ↔ 3. Accordingly, the hadron-level effective Hamiltonian for X bc8 decays into mesons is constructed as In the two-body decays of the transition, the decay amplitudes are given in Tab. VI for the transition b → d and Tab. VII for the transition b → s. We obtain no direct relation of these decay widths.