1 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 \(Q\bar{Q} q\bar{q}'\) (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 [5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34]. In 2016, the D0 collaboration has reported an evidence for the open-bottom hadron X(5568) [35], though it has not been confirmed by the other experimental groups [36,37,38,39]. Therefore the existence of open-flavored hadrons is an interesting question in hadronic physics, in particular the hadron spectroscopy.

Four-quark states, such as tetraquark states, molecular states, are candidates of the exotic states, by exotic we mean a more complex structure than the simple quark-antiquark scenario for a meson and three-quark structure for a baryon. The idea of unconventional quark structures dates back to Gellman original proposal of quark model [40]. In general ordinary mesons \(Q_i \bar{Q}_j\) could mix with the four-quark states \(Q_i \bar{Q}_j q \bar{q}\), by creating and annihilating light quark-antiquark pairs. Therefore, pure physical exotic state can be the mixture between meson and tetraquark (or molecular) or tetraquark and molecular, see some reviews [41,42,43,44]. Despite of many progresses, the structure of hadron exotics is still under debate, such as the X(3872), and more work is certainly needed to clarify this issue. In this paper, we choose the tetraquark model and focus on the four-quark states with two different heavy quarks and two light quarks, Since they can provide a unique platform to study strong interactions under two color static sources. In the diquark-antidiquark model [45], the four-quarks system \([b q] [\bar{c}\bar{q}]\) with orbital angular momentum L = 0 can have \(J^{P}=0^{+}\) [46]. 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 [47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62] 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 \(T_{b\bar{c}8}\).

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 the 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 Sect. 2, we give the multiplets expressions under the SU(3) flavor symmetry. From Sects. 3 and 4, we mainly study the semi-leptonic and non-leptonic weak decays of the \(X_{b\bar{c}8}\) states. In Sect. 5, we will give a collection of the golden channels that can be used to discover the doubly heavy tetraquarks in future experiments. we make a summary in the last section.

2 Particle multiplets

Based on the light flavor SU(3) symmetry, open-flavor tetraquark with the quark constituents \({b\bar{c}}{q\bar{q}}\) can form an octet and a singlet, \(3\bigotimes \bar{3}=1\bigoplus 8\)., of which the octet and singlet can be represented as

$$\begin{aligned} {T_{{\mathbf{b}\bar{\mathbf{c}}{} \mathbf{8}}}}= & {} \left( {{\begin{array}{ccc} \frac{T^{Bc}_{\pi ^0}}{\sqrt{2}}+\frac{T^{Bc}_{\eta _8}}{\sqrt{6}} &{} T^{Bc}_{\pi ^+} &{} T^{Bc}_{K^+} \\ T^{Bc}_{\pi ^-}&{} -\frac{T^{Bc}_{\pi ^0}}{\sqrt{2}}+\frac{T^{Bc}_{\eta _8}}{\sqrt{6}} &{} T^{Bc}_{K^0} \\ T^{Bc}_{K^-} &{} T^{Bc}_{\overline{K}^0} &{} -\frac{2}{\sqrt{6}}T^{Bc}_{\eta _8} \end{array}}} \right) ,\nonumber \\ {T_{{\mathbf{b}\bar{\mathbf{c}} \mathbf{1}}}}= & {} \left( {{\begin{array}{ccc} \frac{T^{Bc}_{\eta _1}}{\sqrt{3}}&{}0&{}0\\ 0&{}\frac{T^{Bc}_{\eta _1}}{\sqrt{3}}&{}0\\ 0&{}0&{}\frac{T^{Bc}_{\eta _1}}{\sqrt{3}} \end{array}}} \right) \,. \end{aligned}$$
(1)

In the light meson sector, pseudoscalar mesons or vector mesons can also form an octet plus a singlet, generally, the octet and singlet are written as

$$\begin{aligned} M=\begin{pmatrix} \frac{\pi ^0}{\sqrt{2}}+\frac{\eta _8}{\sqrt{6}}+\frac{\eta _1}{\sqrt{3}} &{}\pi ^+ &{} K^+\\ \pi ^-&{}-\frac{\pi ^0}{\sqrt{2}}+\frac{\eta _8}{\sqrt{6}}+\frac{\eta _1}{\sqrt{3}}&{}{K^0}\\ K^-&{}\bar{K}^0 &{}-2\frac{\eta _8}{\sqrt{6}}+\frac{\eta _1}{\sqrt{3}} \end{pmatrix}.\nonumber \\ \end{aligned}$$
(2)

In the flavor space, \(\eta _8=(\bar{u}u+ \bar{d}d-2\bar{s}s)/\sqrt{6}\), and \(\eta _1= (\bar{u}u+ \bar{d}d+ \bar{s}s)/\sqrt{3}\). With the same quark content, the vector meson octet and singlet will take a similar structure:

$$\begin{aligned} V=\begin{pmatrix} \frac{\rho ^0+\omega }{\sqrt{2}} &{}\rho ^+ &{} K^{*+}\\ \rho ^-&{}-\frac{\rho ^0+\omega }{\sqrt{2}}&{}{K^{*0}}\\ K^{*-}&{}\bar{K}^{*0} &{}\phi \end{pmatrix}, \end{aligned}$$
(3)

where \(\omega \) and \(\phi \) are the mixtures of \(V_8\) and \(V_1\):

$$\begin{aligned} \begin{pmatrix} \omega \\ \phi \end{pmatrix} = \begin{pmatrix} \cos \theta &{} \sin \theta \\ -\sin \theta &{} \cos \theta \end{pmatrix} \begin{pmatrix} V_8 \\ V_1 \end{pmatrix}. \end{aligned}$$
(4)

An ideal mixing exists for \(\omega \) and \(\phi \) and thus \(\theta \simeq 54.74^\circ \). Besides, we need the representation of bottom mesons which form an SU(3) anti-triplet given as: \(B_i=\left( \begin{array}{ccc} B^-,&\overline{B}^0,&\overline{B}^0_s \end{array} \right) ,\) and the representation of anti-triplet charmed mesons given as: \(D_i=\left( \begin{array}{ccc} D^0,&D^+,&D^+_s \end{array} \right) , \;\;\; \overline{D}^i=\left( \begin{array}{ccc}\overline{D}^0,&D^-,&D^-_s \end{array} \right) .\) Actually, in the SU(3) symmetry, one should add a minus sign to \(D^+\) and \(\overline{K}^0\), but this will not affect the total decay widths.

3 Semi-leptonic \(T_{b\bar{c}8}\)/\(T_{b\bar{c}1}\) decays

In this section, we will discuss the possible semi-leptonic weak decay modes of the tetraquark \(T_{b\bar{c}8}\)/\(T_{b\bar{c}1}\). At quark level, both b-quark and \(\bar{c}\)-quark in \(T_{b\bar{c}8}\)/\(T_{b\bar{c}1}\) can decay. For the b-quark, semi-leptonic weak decays are governed by

$$\begin{aligned} b\rightarrow c/u \ell ^- \bar{\nu }_{\ell }. \end{aligned}$$
(5)

For the \(\bar{c}\)-quark, semi-leptonic decays are induced by

$$\begin{aligned} \bar{c}\rightarrow \bar{d}/\bar{s} \ell ^- \bar{\nu }_{\ell }. \end{aligned}$$
(6)
Fig. 1
figure 1

Topological diagrams for semileptonic decays of tetraquark \(T_{b\bar{c}8}\)/\(T_{b\bar{c}1}\). a–c Correspond to the b quark decay and d, e denotes the \(\bar{c}\) quark decay. It is necessary to point out that any gluon exchange can introduce new Feynman diagrams, but these Feynman diagrams have the same SU(3) topology, and thus can be classified as the same topological diagram

Full size image

3.1 \(b\rightarrow c/u \ell ^- \overline{\nu }_{\ell }\): decays into a meson and \(\ell ^- \overline{\nu }_{\ell }\)

The general electro-weak Hamiltonian of the \(b\rightarrow c/u \ell ^- \overline{\nu }_{\ell }\) transition is given as

$$\begin{aligned} {\mathcal {H}}_{eff}= & {} \frac{G_F}{\sqrt{2}} \left[ V_{q'b} \bar{q}' \gamma ^\mu (1-\gamma _5)b \bar{\ell }\gamma _\mu (1-\gamma _5) \nu _{\ell }\right] +h.c.,\nonumber \\ \end{aligned}$$
(7)

with \(q'=u,c\), in which the electro-weak vertex denotes the \(V-A\) structure. For \(b\rightarrow u\ell ^- \overline{\nu }_{\ell }\), the electro-weak vertex forms a triplet \(H_{3}'\) 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 \(T_{b\bar{c}8}\)/\(T_{b\bar{c}1}\) decays to a charmed meson and \(\ell \overline{\nu }_{\ell }\). The lepton pair does not carry any SU(3) index, and thus is an SU(3) singlet. Following the SU(3) analysis, the Hamiltonian at the hadronic level is constructed as

$$\begin{aligned}&\mathcal {H}_{eff}&=a_1(T_{b\bar{c}8})^i_j (H_3')^j D_i \bar{\ell } \nu +a'_1 (T_{b\bar{c}1})^i_i (H_3')^j D_j \bar{\ell }\nu \;,\nonumber \\ \end{aligned}$$
(8)

here, the coefficients \(a_1\) and \(a'_{1}\) represent the non-perturbative parameters. For completeness, we give the corresponding topological diagram at quark level shown in Fig. 1a, b. The diagrams given in Fig. 1 are topological diagrams, which are collections of Feynman diagrams with the same SU(3) topology. It is convenient to achieve the decay amplitudes given in Table 1 by expanding the Hamiltonian constructed above, in which all amplitudes are represented as \(a_1\) and \(a'_1\). Therefore, we can directly obtain the relations between different decay channels given as follows:

$$\begin{aligned} \Gamma \left( T^{Bc}_{\pi ^-}\rightarrow D^- l^-\bar{\nu }\right)= & {} 2\Gamma \left( T^{Bc}_{\pi ^0}\rightarrow \overline{D}^0 l^-\bar{\nu }\right) \\= & {} \Gamma \left( T^{Bc}_{K^-}\rightarrow D^-_s l^-\bar{\nu }\right) \\= & {} 6\Gamma \left( T^{Bc}_{\eta _8}\rightarrow \overline{D}^0 l^-\bar{\nu }\right) . \end{aligned}$$

It should be noted that the phase space difference will provide corrections to these relations.

Table 1 Amplitudes for tetraquark \(T_{b\bar{c}8}\)/\(T_{b\bar{c}1}\) decays into anti-charmed meson, a light meson or \(J/\Psi \)
Full size table

For the SU(3) singlet \(b\rightarrow c\) transition, the process at the hadron level is that \(T_{b\bar{c}8}\)/\(T_{b\bar{c}1}\) decays into a light meson octet or \(J/\psi \) and \(\ell \overline{\nu }_{\ell } \). Consequently, the Hamiltonian at the hadron level is constructed as

$$\begin{aligned} \mathcal {H}_{eff}= & {} a_2 (T_{b\bar{c}8})^i_j M^j_i \bar{\ell } \nu +a'_2 (T_{b\bar{c}1})^i_i M^j_j \bar{\ell } \nu \nonumber \\&+\, a'_3 (T_{b\bar{c}1})^i_i J/\Psi \bar{\ell } \nu \;. \end{aligned}$$
(9)

One then obtains the amplitudes of different decay channels listed in Table 1, from which we derive that all the channels of the octet tetraquark in the transition give equal decay widths.

3.2 \(\bar{c}\rightarrow \bar{d}/\bar{s} \ell ^- \bar{\nu }\) decay into B meson and \(\ell ^- \bar{\nu }\)

The electro-weak effective Hamiltonian for \(\bar{c}\) quark decay is given as

$$\begin{aligned} {\mathcal {H}}_{eff}= & {} \frac{G_F}{\sqrt{2}} \left[ V_{cq} \bar{c} \gamma ^\mu (1-\gamma _5)q \bar{\ell } \gamma _\mu (1-\gamma _5) \nu _{\ell }\right] +h.c.,\nonumber \\ \end{aligned}$$
(10)

where \(q=d,s\), \(V_{cd}\) and \(V_{cs}\) are CKM matrix elements. Under the SU(3) symmetry, the \(\bar{c}\rightarrow \bar{q} \ell ^-\bar{\nu }\) transition can form an SU(3) triplet, denoted as \(H_{ 3}\) with \((H_{ 3})^1=0,~(H_{ 3})^2=V_{cd},~(H_{ 3})^3=V_{cs}\). Accordingly, we construct the Hamiltonian at the hadron level as follows:

$$\begin{aligned} \mathcal {H}_{eff}=c_1(T_{b\bar{c}8})^i_j (H_3)_i \overline{B}^j \bar{\ell } \nu +c'_1(T_{b\bar{c}1})^i_i (H_3)_j \overline{B}^j \bar{\ell } \nu \;.\nonumber \\ \end{aligned}$$
(11)

The decay amplitudes deduced from the Hamiltonian above are listed in Table 2. For completeness, we give the corresponding topological diagrams in Fig. 1d, e. One then obtains the relations between different channels as follows:

$$\begin{aligned} \Gamma \left( T^{Bc}_{\pi ^-}\rightarrow B^- l^-\bar{\nu }\right)= & {} 2\Gamma \left( T^{Bc}_{\pi ^0}\rightarrow \overline{B}^0 l^-\bar{\nu }\right) \\= & {} \Gamma \left( T^{Bc}_{K^0}\rightarrow \overline{B}^0_s l^-\bar{\nu }\right) \\= & {} 6\Gamma \left( T^{Bc}_{\eta _8}\rightarrow \overline{B}^0 l^-\bar{\nu }\right) ,\\ \Gamma \left( T^{Bc}_{K^-}\rightarrow B^- l^-\bar{\nu }\right)= & {} \Gamma \left( T^{Bc}_{\overline{K}^0}\rightarrow \overline{B}^0 l^-\bar{\nu }\right) \\= & {} \frac{3}{2}\Gamma \left( T^{Bc}_{\eta _8}\rightarrow \overline{B}^0_s l^-\bar{\nu }\right) . \end{aligned}$$

It is necessary to notice that the relations between different channels can be modified with a view to the SU(3) symmetry breaking effects in the charmed or anti-charmed quark decays.

Table 2 Amplitudes for tetraquark \(T_{b\bar{c}8}\)/\(T_{b\bar{c}1}\) decays into a B meson
Full size table

4 Non-leptonic \(T_{b\bar{c}8}\)/\(T_{b\bar{c}1}\) decays

Fig. 2
figure 2

Topological diagrams for the b-quark non-leptonic decays of tetraquark \(T_{b\bar{c}8}\)/\(T_{b\bar{c}1}\). a–p Correspond to the decays into a pair of mesons. In j, l, m, n, p, the final meson produced by gluons is the flavor singly state. In k, o, p, the initial state is the flavor singlet \(T_{b\bar{c}1}\). The diagrams in c, d, g, h, i are usually power suppressed as a pair of quark and anti-quark in the initial state can annihilate

Full size image

The b quark non-leptonic decays can be divided into four categories:

$$\begin{aligned} b\rightarrow c\bar{c} d/s, \; b\rightarrow c \bar{u} d/s, \; b\rightarrow u \bar{c} d/s, \; b\rightarrow q_1 \bar{q}_2 q_3,\nonumber \\ \end{aligned}$$
(12)

where \(q_{1,2,3}\) represent the light quark. In \(\bar{c}\) quark non-leptonic decays, the classifications are given as:

$$\begin{aligned} \bar{c}\rightarrow \bar{s} d \bar{u}, \; \bar{c}\rightarrow \bar{u} d \bar{d}/s \bar{s}, \; \bar{c}\rightarrow \bar{d} s \bar{u}, \; \end{aligned}$$
(13)

which are Cabibbo allowed, singly Cabibbo suppressed, and doubly Cabibbo suppressed, respectively. In the following, we will study the \(T_{b\bar{c}8}\)/\(T_{b\bar{c}1}\) non-leptonic decays in order.

4.1 \(b\rightarrow c\bar{c} d/s\) transition: two-body decays into mesons

The operators for the \(b\rightarrow c\bar{c} d/s\) transition can form a triplet under the SU(3) light quark symmetry, according to that we can write down the effective Hamiltonian of \(T_{b\bar{c}8}\)/\(T_{b\bar{c}1}\) producing two mesons as follows:

$$\begin{aligned} \mathcal {H}_{eff}= & {} a_1 (T_{b\bar{c}8})^i_j (H_3)^j D_i J/\psi +a'_1 (T_{b\bar{c}1})^i_i (H_3)^j D_j J/\psi ,\nonumber \\ \mathcal {H}_{eff}= & {} a_2 (T_{b\bar{c}8})^i_j (H_3)^j D_k M^k_i + a_3 (T_{b\bar{c}8})^i_j (H_3)^k D_k M^j_i \nonumber \\&+\,a_4 (T_{b\bar{c}8})^i_j (H_3)^k D_i M^j_k\nonumber \\&+\,a'_2 (T_{b\bar{c}1})^i_i (H_3)^j D_k M^k_j+a'_3 (T_{b\bar{c}1})^i_i (H_3)^k D_k M^j_j \nonumber \\&+\,a'_4 (T_{b\bar{c}8})^i_k (H_3)^k D_i M^j_j, \end{aligned}$$
(14)
Table 3 Tetraquark \(T_{b\bar{c}8}\)/\(T_{b\bar{c}1}\) decays into anti-charmed meson plus light meson or anti-charmed meson plus \(J/\psi \)
Full size table

with \((H_{ 3})^{2}=V_{cd}^*\) and \((H_{ 3})^{3}=V_{cs}^*\). Consequently, the corresponding topological diagrams are given in Fig. 2a–d for the octet tetraquark \(T_{b\bar{c}8}\), the diagrams in Fig. 2k, o, m, p represent the processes which correspond with the singlet tetraquark or light singlet meson. In particular, the diagrams in Fig. 2a, b represent \(T_{b\bar{c}8}\) decays into D and \(J/\psi \) mesons, and the diagrams in Fig. 2c, d denote processes with D and light mesons final states. Expanding the two Hamiltonian above, one obtains the decay amplitudes which are listed in Table 3. In addition, the relations between the different decay widths are given as follows:

$$\begin{aligned} \Gamma \left( T^{Bc}_{\pi ^0}\rightarrow K^- \overline{D}^0\right)= & {} \frac{1}{2}\Gamma \left( T^{Bc}_{\pi ^-}\rightarrow K^- D^-\right) \\= & {} \Gamma \left( T^{Bc}_{\pi ^0}\rightarrow \overline{K}^0 D^-\right) \\= & {} \frac{1}{2}\Gamma \left( T^{Bc}_{\pi ^+}\rightarrow \overline{K}^0 \overline{D}^0\right) ,\\ \Gamma \left( T^{Bc}_{\pi ^+}\rightarrow \eta _8\overline{D}^0\right)= & {} 2\Gamma \left( T^{Bc}_{\pi ^0}\rightarrow \eta _8D^-\right) \\= & {} 2\Gamma \left( T^{Bc}_{\eta _8}\rightarrow \pi ^0 D^-\right) \\= & {} \Gamma \left( T^{Bc}_{\eta _8}\rightarrow \pi ^- \overline{D}^0\right) ,\\ \Gamma \left( T^{Bc}_{K^0}\rightarrow \pi ^- \overline{D}^0\right)= & {} 2\Gamma \left( T^{Bc}_{K^0}\rightarrow \pi ^0 D^-\right) \\= & {} \Gamma \left( T^{Bc}_{K^+}\rightarrow \pi ^+ D^-\right) \\= & {} 2\Gamma \left( T^{Bc}_{K^+}\rightarrow \pi ^0 \overline{D}^0\right) ,\\ \Gamma \left( T^{Bc}_{\overline{K}^0}\rightarrow K^- \overline{D}^0\right)= & {} 2\Gamma \left( T^{Bc}_{\pi ^0}\rightarrow K^0 D^-_s\right) \\= & {} \Gamma \left( T^{Bc}_{\pi ^+}\rightarrow K^+ D^-_s\right) ,\\ \Gamma \left( T^{Bc}_{\pi ^0}\rightarrow \pi ^0 D^-_s\right)= & {} { }\Gamma \left( T^{Bc}_{\pi ^-}\rightarrow \pi ^- D^-_s\right) \\= & {} \Gamma \left( T^{Bc}_{\pi ^+}\rightarrow \pi ^+ D^-_s\right) ,\\ \Gamma \left( T^{Bc}_{K^+}\rightarrow K^0 \overline{D}^0\right)= & {} { }\Gamma \left( T^{Bc}_{K^-}\rightarrow \pi ^- D^-_s\right) \\= & {} 2\Gamma \left( T^{Bc}_{\overline{K}^0}\rightarrow \pi ^0 D^-_s\right) ,\\ \Gamma \left( T^{Bc}_{K^+}\rightarrow \eta _8\overline{D}^0\right)= & {} { }\Gamma \left( T^{Bc}_{K^0}\rightarrow \eta _8D^-\right) ,\\ \Gamma \left( T^{Bc}_{\eta _8}\rightarrow K^- \overline{D}^0\right)= & {} { }\Gamma \left( T^{Bc}_{\eta _8}\rightarrow \overline{K}^0 D^-\right) ,\\ \Gamma \left( T^{Bc}_{\pi ^-}\rightarrow \pi ^- D^-\right)= & {} { }\Gamma \left( T^{Bc}_{K^0}\rightarrow K^0 D^-\right) ,\\ \Gamma \left( T^{Bc}_{\pi ^+}\rightarrow \pi ^+ D^-\right)= & {} { }\Gamma \left( T^{Bc}_{\overline{K}^0}\rightarrow \overline{K}^0 D^-\right) ,\\ \Gamma \left( T^{Bc}_{\pi ^+}\rightarrow \pi ^0 \overline{D}^0\right)= & {} { }\Gamma \left( T^{Bc}_{\pi ^0}\rightarrow \pi ^- \overline{D}^0\right) ,\\ \Gamma \left( T^{Bc}_{\overline{K}^0}\rightarrow \overline{K}^0 D^-_s\right)= & {} { }\Gamma \left( T^{Bc}_{K^-}\rightarrow K^- D^-_s\right) ,\\ \Gamma \left( T^{Bc}_{K^+}\rightarrow K^+ D^-\right)= & {} { }\Gamma \left( T^{Bc}_{K^-}\rightarrow K^- D^-\right) ,\\ \Gamma \left( T^{Bc}_{K^+}\rightarrow K^+ D^-_s\right)= & {} { }\Gamma \left( T^{Bc}_{K^0}\rightarrow K^0 D^-_s\right) ,\\ \Gamma \left( T^{Bc}_{\pi ^+}\rightarrow \eta _1 \overline{D}^0\right)= & {} {2}\Gamma \left( T^{Bc}_{\pi ^0}\rightarrow \eta _1 D^-\right) \\= & {} \Gamma \left( T^{Bc}_{\overline{K}^0}\rightarrow \eta _1 D^-_s\right) \\= & {} {6}\Gamma \left( T^{Bc}_{\eta _8}\rightarrow \eta _1 D^-\right) ,\\ \Gamma \left( T^{Bc}_{K^+}\rightarrow \eta _1 \overline{D}^0\right)= & {} \Gamma \left( T^{Bc}_{K^0}\rightarrow \eta _1 D^-\right) \\= & {} {\frac{3}{2}}\Gamma \left( T^{Bc}_{\eta _8}\rightarrow \eta _1 \overline{D}^-_s\right) ,\\ \Gamma \left( T^{Bc}_{\eta _1}\rightarrow \pi ^- \overline{D}^0\right)= & {} {2}\Gamma \left( T^{Bc}_{\eta _1}\rightarrow \pi ^0 D^-\right) \\= & {} \Gamma \left( T^{Bc}_{\eta _1}\rightarrow K^0 D^-_s\right) \\= & {} {6}\Gamma \left( T^{Bc}_{\eta _1}\rightarrow \eta _8D^-\right) ,\\ \Gamma \left( T^{Bc}_{\eta _1}\rightarrow \overline{K}^0 D^-\right)= & {} \Gamma \left( T^{Bc}_{\eta _1}\rightarrow K^- \overline{D}^0\right) \\= & {} {\frac{3}{2}}\Gamma \left( T^{Bc}_{\eta _1}\rightarrow \eta _8D^-_s\right) ,\\ 2\Gamma \left( T^{Bc}_{\pi ^0}\rightarrow D^- J/\psi \right)= & {} \Gamma \left( T^{Bc}_{\pi ^+}\rightarrow \overline{D}^0 J/\psi \right) \\= & {} \Gamma \left( T^{Bc}_{\overline{K}^0}\rightarrow D^-_s J/\psi \right) \\= & {} 6\Gamma \left( T^{Bc}_{\eta _8}\rightarrow D^- J/\psi \right) ,\\ \Gamma \left( T^{Bc}_{K^0}\rightarrow D^- J/\psi \right)= & {} \Gamma \left( T^{Bc}_{K^+}\rightarrow \overline{D}^0 J/\psi \right) \\= & {} \frac{3}{2}\Gamma \left( T^{Bc}_{\eta _8}\rightarrow D^-_s J/\psi \right) . \end{aligned}$$

It is necessary to notice that relations for channels involving \(\eta (\eta ')\) should be further modified due to the mixing effects. Here and in the following, our calculation of decay amplitudes using the SU(3) invariant effective Hamiltonian can be applied to the vector meson, following the replacement \(\pi \rightarrow \rho \), \(K\rightarrow K^*\), \(\eta _8 \rightarrow V_8\), and \(\eta _1 \rightarrow V_1\), and taking into account the \(\omega -\phi \) mixing effects.

Table 4 Tetraquark \(T_{b\bar{c}8}\)/\(T_{b\bar{c}1}\) decays into \(J/\psi \) plus light meson or charmed meson plus anti-charmed meson or two light mesons
Full size table

4.2 \(b\rightarrow c \bar{u} d/s\) transition: two body decays into mesons

The operators of the \(b\rightarrow c \bar{u} d/s\) transition can form an octet according to the SU(3) symmetry, of which nonzero entry \((H_{\mathbf{8}})^2_1 =V_{ud}^*\) for the \(b\rightarrow c\bar{u}d\) transition, and \((H_{\mathbf{8}})^3_1 =V_{us}^*\) for the \(b\rightarrow c\bar{u}s\) transition. As usual, the hadron-level effective Hamiltonian can be constructed as

$$\begin{aligned} \mathcal {H}_{eff}= & {} a_6(T_{b\bar{c}8})^i_j (H_8)^k_i M^j_k J/\psi + a_7(T_{b\bar{c}8})^i_j (H_8)^j_k M^k_i J/\psi \nonumber \\&+\, a'_6(T_{b\bar{c}1})^i_i (H_8)^k_j M^j_k J/\psi + a'_7(T_{b\bar{c}8})^i_j (H_8)^j_i M^k_k J/\psi ,\nonumber \\ \mathcal {H}_{eff}= & {} a_8(T_{b\bar{c}8})^i_j (H_8)^j_i D_k \overline{D}^k +a_9 (T_{b\bar{c}8})^i_j (H_8)^k_i D_k \overline{D}^j \nonumber \\&+\, a_{10}(T_{b\bar{c}8})^i_j (H_8)^j_k D_i \overline{D}^k +a'_8(T_{b\bar{c}1})^i_i (H_8)^j_k D_j \overline{D}^k,\nonumber \\ \mathcal {H}_{eff}= & {} a_{11}(T_{b\bar{c}8})^i_j (H_8)^j_i M^l_k M^k_l + a_{12}(T_{b\bar{c}8})^i_j (H_8)^l_i M^j_k M^k_l \nonumber \\&+\, a_{13}(T_{b\bar{c}8})^i_j (H_8)^j_k M^l_i M^k_l + a_{14}(T_{b\bar{c}8})^i_j (H_8)^l_k M^j_i M^k_l \nonumber \\&+\, a_{15}(T_{b\bar{c}8})^i_j (H_8)^l_k M^k_i M^j_l+a'_{11}(T_{b\bar{c}1})^i_i (H_8)^l_j M^j_k M^k_l\nonumber \\&+\, a'_{12}(T_{b\bar{c}1})^i_i (H_8)^l_k M^j_j M^k_l+ a'_{13}(T_{b\bar{c}8})^i_j (H_8)^j_l M^l_i M^k_k\nonumber \\&+\, a'_{14}(T_{b\bar{c}8})^i_j (H_8)^l_i M^j_l M^k_k+a'_{15}(T_{b\bar{c}8})^i_j (H_8)^j_i M^l_l M^k_k.\nonumber \\ \end{aligned}$$
(15)

At quark level, the relevant topological diagrams are shown in Fig. 2a–i for the processes without singlet state, in Fig. 2j–p for the processes including the singlet state. Specifically, the topological diagrams of \(T_{b\bar{c}8}\) decays into a light meson plus \(J/\psi \) are given in Fig. 2b, e, and the Fig. 2a, f, g correspond with the processes of producing D plus \(\bar{D}\), the processes that \(T_{b\bar{c}8}\) produces two light mesons are represented by topological diagrams in Fig. 2c, d, h, i. One derives the decay amplitudes given in Table 4 respectively. Accordingly, we obtain the relations between different decay widths for \(J/\psi \) and a light meson as follows:

$$\begin{aligned}&\Gamma \left( T^{Bc}_{\pi ^+}\rightarrow \pi ^0 J/\psi \right) = { }\Gamma \left( T^{Bc}_{\pi ^0}\rightarrow \pi ^- J/\psi \right) ,\\&\Gamma \left( T^{Bc}_{\pi ^+}\rightarrow \overline{K}^0 J/\psi \right) = 2\Gamma \left( T^{Bc}_{\pi ^0}\rightarrow K^- J/\psi \right) ,\\&\Gamma \left( T^{Bc}_{K^+}\rightarrow \pi ^0 J/\psi \right) = \frac{1}{2}\Gamma \left( T^{Bc}_{K^0}\rightarrow \pi ^- J/\psi \right) . \end{aligned}$$

The relations for producing the charmed meson and anti-charmed meson become

$$\begin{aligned} \Gamma \left( T^{Bc}_{\pi ^0}\rightarrow D^0 D^-_s\right) = \frac{1}{2}\Gamma \left( T^{Bc}_{\pi ^+}\rightarrow D^+ D^-_s\right) . \end{aligned}$$

The relations for producing two light mesons become

$$\begin{aligned} \Gamma \left( T^{Bc}_{\pi ^-}\rightarrow \pi ^- \pi ^- \right)= & {} 2\Gamma \left( T^{Bc}_{K^-}\rightarrow K^- \pi ^- \right) \\= & {} 2\Gamma \left( T^{Bc}_{K^0}\rightarrow K^0 \pi ^- \right) \\= & {} 2\Gamma \left( T^{Bc}_{\pi ^0}\rightarrow \pi ^0 \pi ^- \right) ,\\ \Gamma \left( T^{Bc}_{\pi ^-}\rightarrow \pi ^- K^- \right)= & {} \frac{1}{2}\Gamma \left( T^{Bc}_{K^-}\rightarrow K^- K^- \right) \\= & {} \Gamma \left( T^{Bc}_{\overline{K}^0}\rightarrow \overline{K}^0 K^- \right) ,\\ \Gamma \left( T^{Bc}_{\pi ^0}\rightarrow \pi ^- \overline{K}^0 \right)= & {} 3\Gamma \left( T^{Bc}_{\pi ^+}\rightarrow \eta _8\overline{K}^0 \right) \\= & {} \Gamma \left( T^{Bc}_{\pi ^+}\rightarrow \pi ^0 \overline{K}^0 \right) ,\\ \Gamma \left( T^{Bc}_{K^+}\rightarrow \pi ^0 \pi ^0 \right)= & {} \frac{1}{2}\Gamma \left( T^{Bc}_{K^+}\rightarrow \pi ^+ \pi ^- \right) ,\\ \Gamma \left( T^{Bc}_{\pi ^+}\rightarrow \pi ^0 \eta _8\right)= & {} \Gamma \left( T^{Bc}_{\pi ^0}\rightarrow \pi ^- \eta _8\right) ,\\ \Gamma \left( T^{Bc}_{\pi ^+}\rightarrow \overline{K}^0 \eta _8\right)= & {} 2\Gamma \left( T^{Bc}_{\pi ^0}\rightarrow K^- \eta _8\right) ,\\ \Gamma \left( T^{Bc}_{\overline{K}^0}\rightarrow \pi ^0 K^- \right)= & {} 3\Gamma \left( T^{Bc}_{\overline{K}^0}\rightarrow \eta _8K^- \right) ,\\ \Gamma \left( T^{Bc}_{K^+}\rightarrow \pi ^0 K^0 \right)= & {} 3\Gamma \left( T^{Bc}_{K^+}\rightarrow \eta _8K^0 \right) ,\\ \Gamma \left( T^{Bc}_{K^+}\rightarrow \pi ^0 \eta _8\right)= & {} \frac{1}{2}\Gamma \left( T^{Bc}_{K^0}\rightarrow \pi ^- \eta _8\right) ,\\ \Gamma \left( T^{Bc}_{\eta _1}\rightarrow \pi ^- \overline{K}^0 \right)= & {} {2}\Gamma \left( T^{Bc}_{\eta _1}\rightarrow \pi ^0 K^- \right) \\= & {} {6}\Gamma \left( T^{Bc}_{\eta _1}\rightarrow K^- \eta _8 \right) ,\\ \Gamma \left( T^{Bc}_{\pi ^+}\rightarrow \pi ^0 \eta _1 \right)= & {} \Gamma \left( T^{Bc}_{\pi ^0}\rightarrow \pi ^- \eta _1 \right) ,\\ \Gamma \left( T^{Bc}_{\pi ^+}\rightarrow \eta _8\eta _1 \right)= & {} \Gamma \left( T^{Bc}_{\eta _8}\rightarrow \pi ^- \eta _1 \right) ,\\ \Gamma \left( T^{Bc}_{K^0}\rightarrow \pi ^- \eta _1 \right)= & {} {2}\Gamma \left( T^{Bc}_{K^+}\rightarrow \pi ^0 \eta _1 \right) ,\\ \Gamma \left( T^{Bc}_{\eta _1}\rightarrow K^0 K^- \right)= & {} {\frac{3}{2}}\Gamma \left( T^{Bc}_{\eta _1}\rightarrow \pi ^- \eta _8\right) ,\\ \Gamma \left( T^{Bc}_{\pi ^+}\rightarrow \overline{K}^0 \eta _1\right)= & {} {2}\Gamma \left( T^{Bc}_{\pi ^0}\rightarrow K^- \eta _1 \right) . \end{aligned}$$
Table 5 Tetraquark \(T_{b\bar{c}8}\)/\(T_{b\bar{c}1}\) decays into two anti-charmed mesons
Full size table

4.3 \(b\rightarrow u \bar{c} d/s\) transition: two-body decays into mesons

The operator \((\bar{u}b)(\bar{q}c)\) can form an anti-symmetric \(\mathbf{\bar{3}}\) and a symmetric \(\mathbf{6}\) representations. In the transition \(b\rightarrow u\bar{c}s\), the nonzero components of the anti-symmetric tensor \(H_{\bar{3}}''\) and the symmetric tensor \(H_{ 6}\) are given respectively as \( (H_{\bar{3}}'')^{13} =- (H_{\bar{3}}'')^{31} =V_{cs}^*\), \( (H_{\bar{6}})^{13}=(H_{\bar{6}})^{31} =V_{cs}^*. \) In the transition \(b\rightarrow u\bar{c}d\), the nonzero components can be obtained by interchanging the subscripts \(2\leftrightarrow 3\), and replacing \(V_{cs}\) by \(V_{cd}\). Therefore, the effective Hamiltonian at the hadron level for \(T_{b\bar{c}8}\)/\(T_{b\bar{c}1}\) producing two mesons is constructed as

$$\begin{aligned} \mathcal {H}_{eff}= & {} a_{16} (T_{b\bar{c}8})^i_j (H_{\bar{3}} )^{[jk]}D_i D_k +a_{17} (T_{b\bar{c}8})^i_j (H_{6} )^{\{jk\}}D_i D_k\nonumber \\&+\,a'_{16} (T_{b\bar{c}1})^i_i (H_{6} )^{\{jk\}}D_j D_k. \end{aligned}$$
(16)

Also the topological diagrams corresponding with the Hamiltonian above are given in Fig. 2a–d, k, o. One then deduces the decay amplitudes for different channels shown in Table 5, which leads to the relations for decay widths as

$$\begin{aligned} \Gamma \left( T^{Bc}_{\pi ^-}\rightarrow D^- D^-_s\right)= & {} \frac{1}{2}\Gamma \left( T^{Bc}_{K^-}\rightarrow D^-_s D^-_s\right) \\= & {} 2\Gamma \left( T^{Bc}_{\pi ^0}\rightarrow \overline{D}^0 D^-_s\right) ,\\ \Gamma \left( T^{Bc}_{\pi ^-}\rightarrow D^-D^-\right)= & {} 2\Gamma \left( T^{Bc}_{K^-}\rightarrow D^-_sD^-\right) ,\\ \Gamma \left( T^{Bc}_{\pi ^+}\rightarrow \overline{D}^0\overline{D}^0\right)= & {} 2\Gamma \left( T^{Bc}_{\overline{K}^0}\rightarrow \overline{D}^0 D^-_s\right) ,\\ \Gamma \left( T^{Bc}_{K^+}\rightarrow \overline{D}^0\overline{D}^0\right)= & {} 2\Gamma \left( T^{Bc}_{K^0}\rightarrow \overline{D}^0D^-\right) . \end{aligned}$$
Table 6 Tetraquark \(T_{b\bar{c}8}\)/\(T_{b\bar{c}1}\) decays into anti-charmed and light mesons induced by the charmless \(b\rightarrow d\) transition
Full size table
Table 7 Tetraquark \(T_{b\bar{c}8}\)/\(T_{b\bar{c}1}\) decays into anti-charmed and light mesons induced by the charmless \(b\rightarrow s\) transition
Full size table

4.4 Charmless \(b\rightarrow q_1 \bar{q}_2 q_3\) transition: two body decays into mesons

The charmless tree level operator \((\bar{q}_1 b)(\bar{q}_2 q_3)\) can be decomposed into a triple \(H_\mathbf{3}\), an antisymmetric sextet \(H_\mathbf{\overline{6}}\) and a traceless symmetric \(H_{\mathbf{15}}\) in upper indices, while the charmless penguin level operator behaves as the triplet \(H_\mathbf{3}\) . For the \(\Delta S=0 (b\rightarrow d)\) decays, the nonzero components of these irreducible tensors are given as

$$\begin{aligned} (H_3)^2= & {} 1,\;\;\;(H_{\overline{6}})^{12}_1=-(H_{\overline{6}})^{21}_1=(H_{\overline{6}})^{23}_3=-(H_{\overline{6}})^{32}_3=1,\nonumber \\ 2(H_{15})^{12}_1= & {} 2(H_{15})^{21}_1=-3(H_{15})^{22}_2= -6(H_{15})^{23}_3\nonumber \\= & {} -6(H_{15})^{32}_3=6. \end{aligned}$$
(17)

For the \(\Delta S=1 (b\rightarrow s)\) decays, the nonzero entries in the irreducible tensor \(H_{\mathbf{3}}\), \(H_\mathbf{\overline{6}}\), \(H_{\mathbf{15}}\) can be obtained from Eq. (17) with the exchange \(2\leftrightarrow 3\). Accordingly, the hadron-level effective Hamiltonian for \(T_{b\bar{c}8}\)/\(T_{b\bar{c}1}\) decays into mesons is constructed as

$$\begin{aligned} \mathcal {H}_{eff}= & {} b_4(T_{b\bar{c}8})^i_j (H_3 )^{k}D_i M^j_k +b_5(T_{b\bar{c}8})^i_j (H_3 )^{j}D_k M^k_i \nonumber \\&+\,b_6 (T_{b\bar{c}8})^i_j (H_3 )^{k}D_k M^j_i +b_7(T_{b\bar{c}8})^i_j (H_{\bar{6}} )^{[jk]}_i D_l M^l_k \nonumber \\&+\, b_8(T_{b\bar{c}8})^i_j (H_{\bar{6}} )^{[kl]}_i D_k M^j_l + b_9(T_{b\bar{c}8})^i_j (H_{\bar{6}} )^{[jk]}_l D_i M^l_k \nonumber \\&+\, b_{10}(T_{b\bar{c}8})^i_j (H_{\bar{6}} )^{[jk]}_l D_k M^l_i +b_{11} (T_{b\bar{c}8})^i_j (H_{15} )^{\{jk\}}_i D_l M^l_k\nonumber \\&+\, b_{12}(T_{b\bar{c}8})^i_j (H_{15} )^{\{kl\}}_i D_k M^j_l + b_{13}(T_{b\bar{c}8})^i_j (H_{15} )^{\{jk\}}_l D_i M^l_k \nonumber \\&+\, b_{14}(T_{b\bar{c}8})^i_j (H_{15} )^{\{jk\}}_l D_k M^l_i\nonumber \\&+\,b'_4(T_{b\bar{c}1})^i_i (H_3 )^{k}D_j M^j_k +b'_5 (T_{b\bar{c}1})^i_i (H_3 )^{k}D_k M^j_j\nonumber \\&+\, b'_6 (T_{b\bar{c}8})^i_j (H_3 )^{j}D_i M^k_k+ b'_7(T_{b\bar{c}1})^i_i (H_{\bar{6}} )^{[kl]}_j D_k M^j_l \nonumber \\&+\, b'_{8}(T_{b\bar{c}8})^i_j (H_{\bar{6}} )^{[jk]}_i D_k M^l_l +b'_{9} (T_{b\bar{c}1})^i_i (H_{15} )^{\{kl\}}_j D_k M^j_l\nonumber \\&+\, b'_{10}(T_{b\bar{c}8})^i_j (H_{15} )^{\{jk\}}_i D_k M^l_l. \end{aligned}$$
(18)

In the two-body decays of the transition, the decay amplitudes are given in Table 6 for the transition \(b\rightarrow d\) and Table 7 for the transition \(b\rightarrow s\). We obtain no direct relation of these decay widths.

Fig. 3
figure 3

Topological diagrams for the \(\bar{c}\)-quark non-leptonic decays of tetraquark \(T_{b\bar{c}8}\)/\(T_{b\bar{c}1}\). The two-body processes are given in a–l. The \(\bar{c}\)-quark decays have the similar structures with the b-quark decays. The a–l contribute to the process of \(T_{b\bar{c}8}\)/\(T_{b\bar{c}1}\) decays into B plus a light meson

Full size image
Table 8 Tetraquark \(T_{b\bar{c}8}\)/\(T_{b\bar{c}1}\) decays into a B meson and light meson
Full size table
Table 9 Cabibbo allowed \(T_{b\bar{c}8}\)/\(T_{b\bar{c}1}\) \(\bar{c}\)-quark decays
Full size table
Table 10 Golden channels for \(T_{b\bar{c}8}\)/\(T_{b\bar{c}1}\) b-quark decays
Full size table

4.5 \(\bar{c}\rightarrow \bar{q}_1 q_2 \bar{q}_3\) transition: two body decays into mesons

Under the flavor SU(3) symmetry, the operator \(\bar{c} q_1 \bar{q}_2 q_3\) transforms as \({\bar{\mathbf{3}}}\otimes \mathbf{3}\otimes {\bar{\mathbf{3}}}={\bar{\mathbf{3}}}\oplus {\bar{\mathbf{3}}}\oplus \mathbf{6}\oplus \mathbf{\overline{15}}\). Following the classifications mentioned before, the Cabibbo allowed transition to be \(\bar{c}\rightarrow \bar{s} d \bar{u}\), and the nonzero tensor components are given as

$$\begin{aligned} (H_{ 6})_{31}^2=-(H_{6})_{13}^2=1,\;\;\; (H_{\overline{15}})_{31}^2= (H_{\overline{15}})_{13}^2=1. \end{aligned}$$
(19)

In the singly Cabibbo suppressed transition \(\bar{c}\rightarrow \bar{u} d\bar{d}\) and \(\bar{c}\rightarrow \bar{u} s\bar{s}\) , the combination of tensor components are given as

$$\begin{aligned}&(H_{6})_{31}^3 =-(H_{6})_{13}^3 =(H_{ 6})_{12}^2 =-(H_{ 6})_{21}^2 =\sin (\theta _C),\nonumber \\&(H_{\overline{15}})_{31}^3= (H_{\overline{15}})_{13}^3=-(H_{\overline{15}})_{12}^2=-(H_{\overline{15}})_{21}^2= \sin (\theta _C).\nonumber \\ \end{aligned}$$
(20)

while for the doubly Cabibbo suppressed transition \(\bar{c}\rightarrow \bar{d} s \bar{u}\), we have

$$\begin{aligned}&(H_{ 6})_{21}^3=-(H_{ 6})_{12}^3=\sin ^2\theta _C,\nonumber \\&(H_{\overline{15}})_{21}^3= (H_{\overline{15}})_{12}^3=\sin ^2\theta _C. \end{aligned}$$
(21)

Therefore, it is convenient to construct the hadron-level effective Hamiltonian for \(T_{b\bar{c}8}\)/\(T_{b\bar{c}1}\) decays into mesons as

$$\begin{aligned} \mathcal {H}_{eff}= & {} f_3(T_{b\bar{c}8})^i_j (H_6)^j_{[ik]} \overline{B}^l M^k_l + f_4(T_{b\bar{c}8})^i_j (H_6)^l_{[ik]} \overline{B}^j M^k_l \nonumber \\&+f_5 (T_{b\bar{c}8})^i_j (H_6)^l_{[ik]} \overline{B}^k M^j_l + f_6(T_{b\bar{c}8})^i_j (H_6)^j_{[kl]} \overline{B}^k M^l_i \nonumber \\&+f_7(T_{b\bar{c}8})^i_j (H_{\overline{15}})^j_{\{ik\}} \overline{B}^l M^k_l +f_8 (T_{b\bar{c}8})^i_j (H_{\overline{15}})^l_{\{ik\}} \overline{B}^j M^k_l \nonumber \\&+ f_9(T_{b\bar{c}8})^i_j (H_{\overline{15}})^l_{\{ik\}} \overline{B}^k M^j_l +f_{10} (T_{b\bar{c}8})^i_j (H_{\overline{15}})^j_{\{kl\}} \overline{B}^k M^l_i\nonumber \\&+f'_3(T_{b\bar{c}1})^i_i (H_6)^l_{[jk]} \overline{B}^j M^k_l+f'_4(T_{b\bar{c}8})^i_j (H_6)^j_{[ik]} \overline{B}^k M^l_l\nonumber \\&+f'_5(T_{b\bar{c}1})^i_i (H_{\overline{15}})^l_{\{jk\}} \overline{B}^j M^k_l +f'_6(T_{b\bar{c}8})^i_j (H_{\overline{15}})^j_{\{ik\}} \overline{B}^k M^l_l .\nonumber \\ \end{aligned}$$
(22)

As usual, the corresponding topological diagrams are given in Fig. 3. Expanding the Hamiltonian above to obtain decay amplitudes, and listed in Table 8. One deduces the relations between different decay widths given as

$$\begin{aligned} \Gamma \left( T^{Bc}_{\pi ^-}\rightarrow B^-\pi ^- \right)= & {} { }\Gamma \left( T^{Bc}_{K^-}\rightarrow B^-K^- \right) ,\\ \Gamma \left( T^{Bc}_{\pi ^0}\rightarrow B^-\pi ^0 \right)= & {} { }\Gamma \left( T^{Bc}_{\eta _8}\rightarrow B^-\eta _8\right) ,\\ \Gamma \left( T^{Bc}_{\pi ^+}\rightarrow B^-\pi ^+ \right)= & {} { }\Gamma \left( T^{Bc}_{K^+}\rightarrow B^-K^+ \right) ,\\ \Gamma \left( T^{Bc}_{\pi ^+}\rightarrow \overline{B}^0_s\pi ^0 \right)= & {} { }\Gamma \left( T^{Bc}_{\pi ^0}\rightarrow \overline{B}^0_s\pi ^- \right) ,\\ \Gamma \left( T^{Bc}_{K^0}\rightarrow B^-K^0 \right)= & {} { }\Gamma \left( T^{Bc}_{\overline{K}^0}\rightarrow B^-\overline{K}^0 \right) ,\\ \Gamma \left( T^{Bc}_{K^0}\rightarrow \overline{B}^0_s\pi ^- \right)= & {} { }\Gamma \left( T^{Bc}_{\overline{K}^0}\rightarrow \overline{B}^0K^- \right) ,\\ \Gamma \left( T^{Bc}_{K^+}\rightarrow \overline{B}^0K^0 \right)= & {} { }\Gamma \left( T^{Bc}_{\pi ^+}\rightarrow \overline{B}^0_s\overline{K}^0 \right) ,\\ \Gamma \left( T^{Bc}_{\pi ^+}\rightarrow \overline{B}^0 \eta _1 \right)= & {} \Gamma \left( T^{Bc}_{K^+}\rightarrow \overline{B}_s^0 \eta _1 \right) ,\\ \Gamma \left( T^{Bc}_{\eta _8}\rightarrow B^- \eta _1 \right)= & {} {3} \Gamma \left( T^{Bc}_{\pi ^0}\rightarrow B^- \eta _1 \right) ,\\ \Gamma \left( T^{Bc}_{\eta _1}\rightarrow B^- \eta _8 \right)= & {} {3}\Gamma \left( T^{Bc}_{\eta _1}\rightarrow B^- \pi ^0 \right) ,\\ \Gamma \left( T^{Bc}_{\eta _1}\rightarrow \overline{B}^0 \pi ^- \right)= & {} \Gamma \left( T^{Bc}_{\eta _1}\rightarrow \overline{B}_s^0 K^- \right) . \end{aligned}$$

5 Golden decay channels

In this section, we will discuss the golden channels to reconstruct the \(T_{b\bar{c}8}\)/\(T_{b\bar{c}1}\) and give an estimate of the decay branching fractions. In our analysis given in the previous sections, the final meson can be replaced by its corresponding counterpart with the same quark constituent but different quantum numbers. For instance, one may replace \(\overline{K}^0\) by \(\overline{K}^{*0}\).

Golden decay channels must satisfy the following criteria:

  • Large branching fractions  For charm quark decays, one should use the Cabibbo allowed decay modes, while for bottom quark, the quark level transition \(b\rightarrow c\bar{u}d\) or \(b\rightarrow c\bar{c}s\) gives the largest branching fractions.

  • Large detection efficiency  At hadron colliders like LHC, charged particle has better chance to be detected than neutral state. So we will remove the channels with \(\pi ^0\), \(\eta \), \(\phi \), \(\rho ^{\pm }(\rightarrow \pi ^{\pm }\pi ^0\)), \(K^{*\pm }(\rightarrow K^{\pm }\pi ^0\)) and \(\omega \) in final states, but keep the modes with \(\pi ^\pm , K^0(\rightarrow \pi ^+\pi ^-), \rho ^0(\rightarrow \pi ^+\pi ^-)\).

The two-body decay modes that can be used to reconstruct the \(T_{b\bar{c}8}\)/\(T_{b\bar{c}1}\) are collected in Tables 9 and  10. Some comments are given in order.

From the data on the charmed meson decays, we can infer that the typical branching fraction of these Cabibbo allowed decay channels for charm quark decay in Table 9 is at a few percent level. On the experimental side, to construct the bottom meson in final states, another factor, at the order \(10^{-3}\) or even smaller, due to the weak decay of bottom meson is needed. So the branching fraction for the decay chains to reconstruct the \(T_{b\bar{c}8}\)/\(T_{b\bar{c}1}\) might reach the order \(10^{-5}\), or smaller.

If the b quark in \(T_{b\bar{c}8}\)/\(T_{b\bar{c}1}\) decay first, from the data on the bottom mesons decays, one may deduce that the typical branching fraction is at the order \(10^{-3}\). The final states which include the \(J/\psi \) or D meson such as \(T^{Bc}_{\pi ^-}\rightarrow K^- D^- \) would introduce a factor \(10^{-3}\) to reconstruct in the experiment. So the branching fraction of these channels may also reach the order \(10^{-5}\).

The channels with two light mesons such as \(T^{Bc}_{\pi ^-}\rightarrow \pi ^- \pi ^- \) require the annihilation of the two heavy quarks. But since the CKM matrix element \(V_{cb}\), these channels might have sizable decay branching fractions.

6 Conclusions

Tetraquarks with the quark content \([b q_i] [\bar{c}\bar{q_j}]\) are of great research interest but have not been discovered yet. In this paper, we have systematically studied the weak decays of the doubly-heavy tetraquark \(T_{b\bar{c}8}\)/\(T_{b\bar{c}1}\) under the SU(3) flavor symmetry, which include the semileptonic and nonleptonic b and \(\bar{c}\) quark decays. The \(\bar{c}\) quark decays are dominant, of which the typical branching fraction is at a few percents level.

Using the building blocks in SU(3), we construct the effective Hamiltonian at the hadron level for their weak decays. The nonperturbative effects are parametrized into a few quantities (\(a_{i},b_{j},...\)). Therefore, one can easily derive the decay amplitudes, based on which relations between different channels can be obtained. Finally, we give a list of the golden channels which is useful to search for the \(T_{b\bar{c}8}\)/\(T_{b\bar{c}1}\) state tetraquark in future experiments.