Abstract
We study masses, lifetimes and weak decays of the triply heavy tetraquarks \({b{\bar{c}}}{b{\bar{q}}}\). The masses of tetraquark are explored under two different models. Further more, our calculations show a possibility of existence for the stable triply heavy tetraquark \(b{\bar{c}} b{\bar{q}} \) with \(J^P=1^+\) which should be verified in experiment. Following the heavy quark expanding(HQE), the lifetimes of tetraquarks \({b{\bar{c}}}{b{\bar{q}}}\) can be expressed as the summation of different dimension operators. Particularly, we obtain the lifetimes of \({b{\bar{c}}}{b{\bar{q}}}\) at the next-to-leading order(NLO) given as \(\tau (T^{\{bb\}}_{\{{\bar{c}}{\bar{q}}\}})=0.70\times 10^{-12}s\). Besides, we construct the weak decays Hamiltonian of \({b{\bar{c}}}{b{\bar{q}}}\) in hadronic level under the SU(3) flavor symmetry. The discussion of the Hamiltonian can deduce the decay amplitudes and width relations of the tetraquarks. Following the choosing rules, we collect some golden channels for the mesonic decays of tetraquarks, which will be helpful to search for triply heavy tetraquarks \({b{\bar{c}}}{b{\bar{q}}}\) in future experiments.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
The quark model could well explain the hadronic states for a long time, in which mesons are handled as the bound states of a constituent quark and anti-quark, and baryons are bound by three constituent quarks. Nevertheless, the Belle collaboration changes the situation in 2003 for the discovery of the X(3872) [1], in \(B^{\pm }\rightarrow K^{\pm } X(X\rightarrow \pi ^+\pi ^- J/\psi )\) decays. The new state which contains the \(c{\bar{c}}\) pair is not match with the ordinary quarkonium state, consequently expected to be the candidate of four quark exotic state [2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30]. Subsequently, many more exotic states are observed in processes, for instance, BES-III and Belle collaborations observe charged heavy quarkoniumlike states \(Z_c(3900)^{\pm }\) [31,32,33,34], \(Z_c(4020)^{\pm }\) [35, 36], \(Z_b(10610)^{\pm }\) [37] and \(Z_b(10650)^{\pm }\) [37]. Until now, the X(3872) is one of the best studied four quark exotic state candidates, whose mass right at the \(D^0 {\bar{D}}^{*0}\) threshold, and spin-parity quantum numbers \(J^{PC}=1^{++}\) [38]. However, there is still a open issue whether the state is a loosely bound molecule or a compact tetraquark.
The four-quark state \(b{\bar{c}}b{\bar{q}}\) with three heavy quark is apparently different from the discovered quarkoniumlike state, in this respect, it might offer a new platform to study the internal structure of the exotic states. In addition, it can also be an ideal source to understand the hadronic dynamic and QCD factorization approach. Some theoretical and experimental studies have been carried out to explore the properties of the exotic state at present, especially the mass spectrum by the simple quark model [39], QCD sum rule [40]. In this paper, we will concern on the masses, lifetimes and decays channels of the state. The non-relativistic quark model [41] which has successfully used in the mass spectrum of S-wave hadron can be applied in multiquark system. While the quark state might be one stable tetraquark, its dominant decay modes would be induced by the weak interaction. In the diquark-antidiquark model [42], the lifetimes of lowest lying tetraquarks with the spin-parity \(0^+\) and \(1^+\) can be achieved by the operator product expansion(OPE) technique [43, 44]. Further more, the flavor SU(3) symmetry analysis which have been successfully applied into heavy mesons or baryons [45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61] can be used to study the weak decays of \(b{\bar{c}}b{\bar{q}}\).
Generally, the lifetimes of the tetraquarks can be expressed into several matrix elements of effective operators in the OPE technique. In this case, we investigate the lifetimes at the leading and next-to-leading order in the heavy quark expansion. In the other case, the weak decays of the tetraquarks can be considered at the hadronic level Hamiltonian in the SU(3) symmetry. The triply heavy tetraquarks \(b{\bar{c}}b{\bar{q}}\) and the heavy quark transition can be classified in the SU(3) light quark symmetry. Thus one can construct the Hamiltonian in the hadron level, and the non-perturbative effect can be absorbed into some parameters \(a_i,b_i,c_i,\ldots \). The discussion of the Hamiltonian can tell us the decay amplitudes and relations between different channels. Such analyses are benefit to the searching for the triply heavy tetraquarks in future experiment.
The paper is organized as follows. In Sect. 2, we will study the mass spectrum of tetraquark \(b{\bar{c}} b{\bar{q}}\) with two different models. Section 3 concerns on the lifetimes of the tetraqurak \({b{\bar{c}}}{b{\bar{q}}}\) with the approach of OPE. The SU(3) flavor symmetry analysis including particle multiplets and possible weak decays will be discussed in Sects. 4 and 5, in which the study of weak decays includes mesonic two-body non-leptonic decays and the three- or four-body semi-leptonic decays. In Sect. 6, we present a collection of golden decay channels. The summary is given in the end.
2 Masses
We will study the mass of the triply heavy tetraquark \(bb{\bar{c}}{\bar{q}}\) signed as \(T_{\{{\bar{c}}{\bar{q}}\}}^{\{bb\}}\) in the section. Since the large ambiguity still be indicated in the quark masses and effective couplings which are actually determined by experiments, the study will no tell the mass of the tetraquark with higher precision. Even though, the calculation may show the possibility of the existence of the stable tetraquark against strong interaction, which is worthwhile to search in future experiment.
The study will be carried on with two models for the tetraquark state \(J^P=0^+,1^+(L=0)\). Model-I is the diquark-antidiquark picture described by the effective Hamiltonian [3], given as
where \(m_{ij}\) is the mass of constituent diquark, \(S_i\) and \(\kappa _{ij}\) indicate the spin and spin-spin coupling of quarks. Following the idea of “good” and “bad” diquark from Jaffe [45], we introduce the allowed tetraquark states \(bb{\bar{c}}{\bar{q}}\) with the orbital angular moment \(L=0\) in the flavor(f) \(\otimes \) spin(s) \(\otimes \) color(c) spcaes.
Here, the quantum numbers of diquark (e.g. \(|\{bb,1_f,1_s,{\bar{3}}_c\},\ldots \rangle \)) and tetraquark (\(|\ldots ,3_f,0_s,1_c\rangle \)) are displayed respectively. With the help of Hamiltonian in Eq.(1), we write the corresponding splitting mass matrices for the \(J^P=0^+,1^+\) tetraquark
and
The spin-spin couplings of quark-quark \(\kappa _{ij}=(\kappa _{ij})_{{\bar{3}}}\) and quark-antiquark \(\kappa _{ij}=\frac{1}{4}(\kappa _{ij})_0\) are chosen respectively as [62]: \((\kappa _{cq})_{{\bar{3}}}=22\)MeV, \((\kappa _{cs})_{{\bar{3}}}=24\)MeV, \((\kappa _{b{\bar{c}}})_0=20\)MeV, \((\kappa _{b{\bar{q}}})_0=23\)MeV, \((\kappa _{b{\bar{s}}})_0=23\)MeV and \((\kappa _{b{\bar{b}}})_0=36\)MeV. The diquark masses are adopted as [62]: \(m_{bb}=10.016\)GeV, \(m_{cq}=1.975\)GeV. One diagonalizes the mass matrix and collect the results given in Table 1.
Model-II is the simple nonrelativistic quark model which includes the hyperfine (e.g. color-spin) interaction [41, 63]. The Hamiltonian of color-magnetic interaction is given as
where the coupling constant \(C_{ij}/(m_im_j)\) can be extracted from the hadron mass differences. One gets the corresponding parameters [64]: \(C_{cq}/{m_cm_q}=23\)MeV, \(C_{b{\bar{q}}}/(m_bm_q)=23\)MeV, \(C_{c s}/(m_cm_s)=47\)MeV, \(C_{b{\bar{s}}}/(m_bm_s)=24\)MeV and \(C_{b{\bar{b}}}/{m_b^2}=31\)MeV. The \(\vec {\lambda }_i\) is the Gell-Mann matrix for color SU(3) group, and \(\vec {s}_i=\vec {\sigma }_i/2\) is the spin matrix for spin SU(2) group. We use three different schemes [64] for the effective quark mass, which respectively are SET-1 fitted from the mesons, SET-2 fitted from the baryons and SET-3 widely used in quark potential model. Consistently, we compare the results from different models and schemes in Table 1.
There is small gap prediction between Model-I and Model-II according to the Table 1. Actually, most of the states are above their \(B_cB(B_cB_s)\) thresholds, except for the \(bb{\bar{c}}{\bar{u}}/bb{\bar{c}}{\bar{d}}\) with \(J^P=1^+\) in the Model-II in the scheme of SET-3, which below the \(B_cB^*\) threshold about 13MeV. It indicates that the state may be one stable particle under the strong interaction. In fact, our results are widely affected by the theoretical uncertainties, such as the effective bottom quark mass and the coupling constant of \(C_{bb}/(m_b^2)\). However, it offers a possibility of existence for the stable triply heavy tetraquark which should be verified in experiment. To hunt for the states, we will study their weak decays properties in the next section, which may offer useful help for the future experimental search for these states.
3 Lifetimes
In the section, we will study the lifetimes of the tetraquark \(T_{\{{\bar{c}} {\bar{q}}\}}^{\{bb\}}\) under the OPE. We proceed as follows, the decays width of \(T^{\{bb\}}_{\{{\bar{c}}{\bar{q}}\}}\rightarrow X\) with the spin-parity \(J^P=(0^+,1^+)\) can be expressed respectively as
here, \(m_T\), \(\lambda \) and \(p_T^{\mu }\) are the mass, spin and four-momentum of tetraquark \(T_{\{{\bar{c}} {\bar{q}}\}}^{\{bb\}}\) respectively. The matrix in full theory Hamiltonian \({\mathcal {H}}\) can match with that of electro-weak effective Hamiltonian \({\mathcal {H}}_{eff}^{ew}\), whose Hamiltonian given as
where \(C_i\) and \(O_i\) are Wilson coefficient and effective operator. Besides, Vs are the combinations of Cabibbo-Kobayashi-Maskawa(CKM) elements. The total decay width of \(\Gamma (T^{\{bb\}}_{\{{\bar{c}}{\bar{q}}\}}\rightarrow X)\) can be deduced in the optical theorem, rewritten as
Further more, the transition operator can be expanded in the heavy quark expanding(HQE), which given as
here, \(G_F\) is the Fermi constant and \(V_{CKM}\) is the CKM element. The coefficients \(c_{i,Q}\) which coming from the heavy quark decays are the perturbative coefficients. Therefore, the total decay width of the tetraquark \(T_{\{{\bar{c}} {\bar{q}}\}}^{\{bb\}}\) can be given in the leading dimension contribution as
where the heavy quark matrix elements are corresponding with bottom and charm number in the tetraquark state.
The perturbative short-distance coefficients \(c_{3,Q}\)s have been determined as \(c_{3,b}=5.29\pm 0.35,\ c_{3,c}=6.29\pm 0.72\) at the leading order(LO) and \(c_{3,b}=6.88\pm 0.74,\ c_{3,c}=11.61\pm 1.55\) at the next-to-leading order(NLO) [43]. Accordingly, we expect the lifetimes of the tetraquark given as
where the heavy quark masses \(m_c=1.4\ \mathrm{GeV}\) and \(m_b=4.8\ \mathrm{GeV}\).
4 Particle multiplets
The tetraquark with the quark constituents \({bb}{{\bar{c}}{\bar{q}}}\) is a flavor SU(3) anti-triplet. In the following, we will concern on the mesonic decays of \(T_{\{{\bar{c}} {\bar{q}}\}}^{\{bb\}}\). Therefore, the particle multiplets are given firstly. Following the SU(3) analysis, we give the SU(3) representation of the tetraquark \(T_{\{{\bar{c}} {\bar{q}}\}}^{\{bb\}}\),
The light pseudoscalar mesons can form an octet, which represented as
Besides, we give the SU(3) representations of bottom and charmed mesons,
For completeness, we show the weight diagrams of the states above given in Fig. 1.
5 Weak decays
In this section, we will study the possible weak decay of the tetraquark \(T_{\{{\bar{c}}{\bar{q}}\}}^{\{bb\}}\). Generally, we can classify the decays modes by the quantities of CKM matrix elements.
For the b/c quark semi-leptonic decays, we adopt the following groups.
$$\begin{aligned} b\rightarrow c/u \ell ^- {\bar{\nu }}_{\ell },\ \ \ {\bar{c}}\rightarrow {\bar{d}}/{\bar{s}} \ell ^- {\bar{\nu }}_{\ell }. \end{aligned}$$(21)The general electro-weak Hamiltonian can be expressed as
$$\begin{aligned} \mathcal{H}_{eff}= & {} \frac{G_F}{\sqrt{2}} \left[ V_{q'b} {\bar{q}}' \gamma ^\mu (1-\gamma _5)b {\bar{\ell }}\gamma _\mu (1-\gamma _5) \nu _{\ell } \right. \nonumber \\&\left. +V_{cq} {\bar{c}} \gamma ^\mu (1-\gamma _5)q {\bar{\ell }} \gamma _\mu (1-\gamma _5) \nu _{\ell }\right] +h.c.,\nonumber \\ \end{aligned}$$(22)where \(q'=(u,c)\), \(q=(d,s)\). The operators of \(b\rightarrow u/c \ell ^- \bar{\nu }_{\ell }\) transition form an SU(3) flavor triplet \(H_{3}'\) or singlet. In addition, the transition \({\bar{c}}\rightarrow {\bar{q}} \ell ^-\bar{\nu }\) can form a triplet \(H_{ 3}\).
For the \({\bar{c}}\) quark non-leptonic decays, the groups are given as
$$\begin{aligned} {\bar{c}}\rightarrow {\bar{s}} d {\bar{u}}, \; {\bar{c}}\rightarrow {\bar{u}} d {\bar{d}}/s {\bar{s}}, \; {\bar{c}}\rightarrow {\bar{d}} s {\bar{u}}, \; \end{aligned}$$(23)which are Cabibbo allowed, singly Cabibbo suppressed and doubly Cabibbo suppressed respectively. The operator \({\bar{c}} q_1 {\bar{q}}_2 q_3\) of the transition transforms as \({\bar{\mathbf{3}}}\otimes \mathbf{3}\otimes {\bar{\mathbf{3}}}={\bar{\mathbf{3}}}\oplus {\bar{\mathbf{3}}}\oplus \mathbf{6}\oplus {\overline{\mathbf{15}}}\), marked as \(H_{ 3},H_{ 6}\) and \(H_{ \overline{15}}\).
For the b quark non-leptonic decays, the groups are classified as
$$\begin{aligned} b\rightarrow c{\bar{c}} d/s, \; b\rightarrow c {\bar{u}} d/s, \; b\rightarrow u {\bar{c}} d/s, \; b\rightarrow q_1 {\bar{q}}_2 q_3,\nonumber \\ \end{aligned}$$(24)\(q_{1,2,3}\) represent the light quark(d/s). The operator of the transition \(b\rightarrow c{\bar{c}} d/s\) form an triplet \(H_{ 3}\), and the operator of the transition \(b\rightarrow c {\bar{u}} d/s\) can form an octet \(H_{{8}}\). In addition, the operator of transition \(b\rightarrow u{\bar{c}}s/d\) can form an anti-symmetric \({\bar{\mathbf{3}}}\) plus a symmetric \(\mathbf{6}\) tensors. While the charmless tree level operator \(({\bar{q}}_1 b)({\bar{q}}_2 q_3)\) (\(q_i=d,s\)) can be decomposed as \(\mathbf{3}\otimes \bar{\mathbf{3}}\otimes \mathbf{3}=\mathbf{3}\oplus \mathbf{3}\oplus {\bar{\mathbf{6}}}\oplus \mathbf{15}\), they are signed as \(H_{ 3},H_{ {\overline{6}}}\) and \(H_{ 15}\).
The parameters of the tensors above can be found in Ref. [59, 64,65,66], such as \((H_{ 6})_{31}^2=-(H_{6})_{13}^2=1\) for the transition of \({\bar{c}}\rightarrow {\bar{s}} d {\bar{u}}\). In the following, we will study the possible weak decay modes of \(T^{\{bb\}}_{\{{\bar{c}}{\bar{q}}\}}\) in order.
5.1 Semi-leptonic decays
Following the SU(3) analysis, we respectively construct the Hamiltonian of hadronic level for the semi-leptonic decays \(b\rightarrow c/u \ell ^- {\overline{\nu }}_{\ell }\),
The corresponding Feynman diagrams are shown in Fig. 2a–d. Further more, the diagram Fig. 2a is relative to the \(a_1\) and \(a_5\) terms. The diagrams Fig. 2b–d of four-body semi-leptonic decays are respectively corresponding with the remanding terms in the Hamiltonian Eqs. (25) and (26). Expanding the Hamiltonian, we can obtain the decay amplitudes of different decay channels, which are collected in Table 2. The relations of decay widths can be further deduced, when the phase space effect is neglected.
For the semi-leptonic transition of \({\bar{c}}\rightarrow {\bar{d}}/{\bar{s}} \ell ^- {\overline{\nu }}_{\ell }\), the Hamiltonian of mesonic decays is easy to construct, given as follows,
The corresponding Feynman diagram is drawn in Fig. 2e. More technically, we can expand the Hamiltonian above and collect the decay amplitudes,
The relations of decay widths given as follows.
5.2 Non-leptonic two-body mesonic decays
In the SU(3) flavor symmetry, the two-body mesonic decays Hamiltonian of the transition \(b\rightarrow c{\bar{c}} d/s\) is constructed as
We draw the corresponding Feynman diagrams shown in Fig. 3a–e. Particularly, the terms \(a_1\) and \(a_2\) with the final states B plus D are related to the diagrams Fig. 3c, e. The terms \(a_4\) is corresponding with the diagrams Fig. 3a, b. In addition, the \(a_3\) term with the light meson and \(B_c\) meson in the final states is relevant with the Fig. 3d. Based on the Hamiltonian above, we get the decay amplitudes collected in Table 3. Consistently, the relations between different decay widths given as follows.
The hadronic level Hamiltonian of the transition of \(b\rightarrow c{\bar{u}} d/s\) can be constructed as
The corresponding Feynman diagrams are shown in Fig. 3a–e. Further more, we expand the Hamiltonian and obtain the decay amplitudes collected in Table 4. Besides, the relation between different decay channels can be deduced directly, given as follows.
We can easily construct the hadronic level Hamiltonian for the transition of \(b\rightarrow u{\bar{c}} d/s\) in the SU(3) symmetry,
The corresponding Feynman diagrams are shown in Fig. 3a–d. In addition, the amplitudes obtained from the Hamiltonian are collected in Table 4, and the relations between different decay channels can be deduced as
Following the SU(3) analysis, the hadronic level Hamiltonian for the charmless transition of \(b\rightarrow q_1{\bar{q}}_2 q_3\) can be constructed as
We expand the Hamiltonian and collect the decay amplitudes given in Table 5. In particular, there is no relation between different decay channels.
For the charm quark decay \({\bar{c}}\rightarrow {\bar{q}}_1 q_2 {\bar{q}}_3\), the Hamiltonian of hadronic level can be easily constructed as
Expanding the Hamiltonian above, one can obtain the decay amplitudes given as follows,
Consistently, the relation between different decay channels is deduced as
6 Golden decay channels
In order to reconstruct the tetraquark \(T_{\{{\bar{c}} {\bar{q}}\}}^{\{bb\}}\) experimentally, we will discuss the golden channels in the section. In the discussion given in the previous sections, the final meson can be replaced by its corresponding counterpart, which has the same quark constituent but different quantum numbers. For instance, we can replace \(\pi ^0\) by \(\rho ^0\), \(K^+\) by \(K^{*+}\).
Generally, the weak decays are closely related with the CKM element. In addition, the charged particles are easy to detect than neutral one at hadron colliders like LHC. Therefore, one choose the following criteria for the Golden decay channels.
Branching fractions: One should choose the cabibbo allowed decay modes \(c\rightarrow s \ell ^+ \nu _{\ell }\) for the charm quark semi-leptonic decays and \(c\rightarrow s {\bar{d}} u\) for the non-leptonic decays. For bottom quark decays, the semi-leptonic decays \(b\rightarrow c \ell ^- {\overline{\nu }}_{\ell }\) and non-leptonic decays \(b\rightarrow c{\bar{u}}d\) or \(b\rightarrow c{\bar{c}}s\) give the largest branching fractions.
Detection efficiency: Since it is easy to detect the charged particles, one remove the channels with \(\pi ^0\), \(\eta \), \(\phi \), \(\rho ^{\pm }(\rightarrow \pi ^{\pm }\pi ^0\)), \(K^{*\pm }(\rightarrow K^{\pm }\pi ^0\)) and \(\omega \), but keep the modes with \(\pi ^\pm , K^0(\rightarrow \pi ^+\pi ^-), \rho ^0(\rightarrow \pi ^+\pi ^-)\).
We collect the golden channels with the two-body non-leptonic decays and three- or four-body semi-leptonic decays for the triply heavy tetraquark \(T_{\{{\bar{c}} {\bar{q}}\}}^{\{bb\}}\), given in Table 6.
Some comments are appropriate in the following. For the charm quark decays, the typical branching fraction is at a few percent level. In addition, the reconstruction of final states, such as bottom mesons, is at the order \(10^{-3}\) or even smaller on the experiment side. Therefore the branching fraction for the charm quark decays chains to reconstruct the \(T_{\{{\bar{c}} {\bar{q}}\}}^{\{bb\}}\) might be at the order \(10^{-5}\), or smaller. For the bottom quark decays, the typical branching fraction is at the order \(10^{-3}\). The reconstruction of final states \(J/\psi \) or D meson would introduce another factor \(10^{-3}\). Therefore the branching fraction of these channels could be at the order \(10^{-5}\).
7 Conclusions
In the paper, we have studied the masses, lifetimes and weak decays of the heavy tetraquarks \({b{\bar{c}}}{b{\bar{q}}}\). The mass spectrums of tetraquark \(bb{\bar{c}}{\bar{q}}\) were calculated in non-relativistic model, we found the triply heavy tetraquark with \(J^P=1^+\) could be the stable state which should be verified by further experiments. With the help of heavy quark expanding(HQE), we calculated the lifetimes of \({b{\bar{c}}}{b{\bar{q}}}\) at the leading order and next-to-leading order. Particularly, the lifetime at NLO is \(\tau (T^{\{bb\}}_{\{{\bar{c}}{\bar{q}}\}})=(0.70\pm 0.09)\times 10^{-12} \ s\). In addition , we discussed the weak decays of \({b{\bar{c}}}{b{\bar{q}}}\) under the SU(3) analysis which had been successfully applied into heavy meson and baryon processes. Following the building blocks in SU(3) analysis, we constructed the hadronic level Hamiltonian. More technically, the non-perturbative effects are collected into parameters, such as \(a_{i},b_{j},\,\ldots \). According to the Hamiltonian, one can derive the relations between different decay channels. Finally, we analyzed the semi-leptonic and non-leptonic mesonic decays of tetraquarks \({b{\bar{c}}}{b{\bar{q}}}\), and made a collection of the golden channels for the searching of triply heavy tetraquarks \({b{\bar{c}}}{b{\bar{q}}}\) in further experiments.
Data Availability Statement
This manuscript has no associated data or the data will not be deposited. [Authors’ comment: This is a theoretical study and no experimental data has been listed.]
References
S.K. Choi et al., [Belle Collaboration], Phys. Rev. Lett. 91, 262001 (2003). https://doi.org/10.1103/PhysRevLett.91.262001hep-ex/0309032
J.P. Ader, J.M. Richard, P. Taxil, Phys. Rev. D 25, 2370 (1982). https://doi.org/10.1103/PhysRevD.25.2370
L. Maiani, F. Piccinini, A.D. Polosa, V. Riquer, Phys. Rev. D 71, 014028 (2005). https://doi.org/10.1103/PhysRevD.71.014028. arXiv:hep-ph/0412098
L. Maiani, V. Riquer, W. Wang, Eur. Phys. J. C 78(12), 1011 (2018). https://doi.org/10.1140/epjc/s10052-018-6486-5. arXiv:1810.07848 [hep-ph]
S.S. Agaev, K. Azizi, B. Barsbay, H. Sundu, Eur. Phys. J. A 53(1), 11 (2017). https://doi.org/10.1140/epja/i2017-12201-2. arXiv:1608.04785 [hep-ph]
S.S. Agaev, K. Azizi, B. Barsbay, H. Sundu, Nucl. Phys. B 939, 130 (2019). https://doi.org/10.1016/j.nuclphysb.2018.12.021. arXiv:1806.04447 [hep-ph]
F.K. Guo, C. Hanhart, U.G. Meißner, Q. Wang, Q. Zhao, B.S. Zou, Rev. Mod. Phys. 90(1), 015004 (2018). https://doi.org/10.1103/RevModPhys.90.015004. arXiv:1705.00141 [hep-ph]
F.K. Guo, C. Hidalgo-Duque, J. Nieves, M.P. Valderrama, Phys. Rev. D 88, 054007 (2013). https://doi.org/10.1103/PhysRevD.88.054007. arXiv:1303.6608 [hep-ph]
M. Cleven, Q. Wang, F.K. Guo, C. Hanhart, U.G. Meissner, Q. Zhao, Phys. Rev. D 87(7), 074006 (2013). https://doi.org/10.1103/PhysRevD.87.074006. arXiv:1301.6461 [hep-ph]
F.K. Guo, U.G. Meißner, W. Wang, Commun. Theor. Phys. 61, 354 (2014). https://doi.org/10.1088/0253-6102/61/3/14. arXiv:1308.0193 [hep-ph]
X.H. Liu, G. Li, Eur. Phys. J. C 76(8), 455 (2016). https://doi.org/10.1140/epjc/s10052-016-4308-1. arXiv:1603.00708 [hep-ph]
G. Li, Eur. Phys. J. C 73(11), 2621 (2013). https://doi.org/10.1140/epjc/s10052-013-2621-5. arXiv:1304.4458 [hep-ph]
F.K. Guo, U.G. Meißner, W. Wang, Z. Yang, JHEP 1405, 138 (2014). https://doi.org/10.1007/JHEP05(2014)138. arXiv:1403.4032 [hep-ph]
F.K. Guo, C. Hanhart, Q. Wang, Q. Zhao, Phys. Rev. D 91(5), 051504 (2015). https://doi.org/10.1103/PhysRevD.91.051504. arXiv:1411.5584 [hep-ph]
Y.H. Chen, M. Cleven, J.T. Daub, F.K. Guo, C. Hanhart, B. Kubis, U.G. Meißner, B.S. Zou, Phys. Rev. D 95(3), 034022 (2017). https://doi.org/10.1103/PhysRevD.95.034022. arXiv:1611.00913 [hep-ph]
Q. Wang, M. Cleven, F.K. Guo, C. Hanhart, U.G. Meißner, X.G. Wu, Q. Zhao, Phys. Rev. D 89(3), 034001 (2014). https://doi.org/10.1103/PhysRevD.89.034001. arXiv:1309.4303 [hep-ph]
G. Li, W. Wang, Phys. Lett. B 733, 100 (2014). https://doi.org/10.1016/j.physletb.2014.04.029. arXiv:1402.6463 [hep-ph]
M. Albaladejo, F.K. Guo, C. Hanhart, U.G. Meißner, J. Nieves, A. Nogga, Z. Yang, Chin. Phys. C 41(12), 121001 (2017). https://doi.org/10.1088/1674-1137/41/12/121001. arXiv:1709.09101 [hep-ph]
F.K. Guo, C. Hanhart, U.G. Meißner, Q. Wang, Q. Zhao, Phys. Lett. B 725, 127 (2013). https://doi.org/10.1016/j.physletb.2013.06.053. arXiv:1306.3096 [hep-ph]
M.B. Voloshin, Phys. Rev. D 87(7), 074011 (2013). https://doi.org/10.1103/PhysRevD.87.074011. arXiv:1301.5068 [hep-ph]
D.Y. Chen, X. Liu, Phys. Rev. D 84, 094003 (2011). https://doi.org/10.1103/PhysRevD.84.094003. arXiv:1106.3798 [hep-ph]
D.Y. Chen, X. Liu, T. Matsuki, Phys. Rev. D 84, 074032 (2011). https://doi.org/10.1103/PhysRevD.84.074032. arXiv:1108.4458 [hep-ph]
A.E. Bondar, A. Garmash, A.I. Milstein, R. Mizuk, M.B. Voloshin, Phys. Rev. D 84, 054010 (2011). https://doi.org/10.1103/PhysRevD.84.054010. arXiv:1105.4473 [hep-ph]
G. Li, Z. Zhou, Phys. Rev. D 91(3), 034020 (2015). https://doi.org/10.1103/PhysRevD.91.034020. arXiv:1502.02936 [hep-ph]
D.Y. Chen, X. Liu, T. Matsuki, Phys. Rev. D 88(1), 014034 (2013). https://doi.org/10.1103/PhysRevD.88.014034. arXiv:1306.2080 [hep-ph]
Z.G. Wang, Z.Y. Di, Acta Phys. Polon. B 50, 1335 (2019). https://doi.org/10.5506/APhysPolB.50.1335. arXiv:1807.08520 [hep-ph]
N. Brambilla, S. Eidelman, C. Hanhart, A. Nefediev, C.P. Shen, C.E. Thomas, A. Vairo, C.Z. Yuan,. arXiv:1907.07583 [hep-ex]
A. Esposito, M. Papinutto, A. Pilloni, A.D. Polosa, N. Tantalo, Phys. Rev. D 88(5), 054029 (2013). https://doi.org/10.1103/PhysRevD.88.054029. arXiv:1307.2873 [hep-ph]
A. Esposito, A.L. Guerrieri, F. Piccinini, A. Pilloni, A.D. Polosa, Int. J. Mod. Phys. A 30, 1530002 (2015). https://doi.org/10.1142/S0217751X15300021. arXiv:1411.5997 [hep-ph]
E.J. Eichten, C. Quigg, Phys. Rev. Lett. 119(20), 202002 (2017). https://doi.org/10.1103/PhysRevLett.119.202002. arXiv:1707.09575 [hep-ph]
M. Ablikim et al., [BESIII Collaboration], Phys. Rev. Lett. 110, 252001 (2013). https://doi.org/10.1103/PhysRevLett.110.252001. arXiv:1303.5949 [hep-ex]
Z.Q. Liu et al., [Belle Collaboration] Phys. Rev. Lett. 110, 252002 (2013). https://doi.org/10.1103/PhysRevLett.110.252002. arXiv:1304.0121 [hep-ex]
T. Xiao, S. Dobbs, A. Tomaradze, K.K. Seth, Phys. Lett. B 727, 366 (2013). https://doi.org/10.1016/j.physletb.2013.10.041. arXiv:1304.3036 [hep-ex]
M. Ablikim et al., [BESIII Collaboration], Phys. Rev. Lett. 115, no.11, 112003 (2015) https://doi.org/10.1103/PhysRevLett.115.112003 arXiv:1506.06018 [hep-ex]
M. Ablikim et al. [BESIII Collaboration], Phys. Rev. Lett. 111 (2013) no.24, 242001 https://doi.org/10.1103/PhysRevLett.111.242001 arXiv:1309.1896 [hep-ex]
M. Ablikim et al. [BESIII Collaboration], Phys. Rev. Lett. 113 (2014) no.21, 212002 https://doi.org/10.1103/PhysRevLett.113.212002 arXiv:1409.6577 [hep-ex]
A. Bondar et al., [Belle Collaboration], Phys. Rev. Lett. 108, 122001 (2012). https://doi.org/10.1103/PhysRevLett.108.122001. arXiv:1110.2251 [hep-ex]
R. Aaij et al. [LHCb Collaboration], Phys. Rev. D 92 (2015) no.1, 011102 https://doi.org/10.1103/PhysRevD.92.011102 arXiv:1504.06339 [hep-ex]
K. Chen, X. Liu, J. Wu, Y.R. Liu, S.L. Zhu, Eur. Phys. J. A 53(1), 5 (2017). https://doi.org/10.1140/epja/i2017-12199-3. arXiv:1609.06117 [hep-ph]
J.F. Jiang, W. Chen, S.L. Zhu, Phys. Rev. D 96(9), 094022 (2017). https://doi.org/10.1103/PhysRevD.96.094022. arXiv:1708.00142 [hep-ph]
A. De Rujula, H. Georgi, S.L. Glashow, Phys. Rev. D 12, 147 (1975). https://doi.org/10.1103/PhysRevD.12.147
R.L. Jaffe, F. Wilczek, Phys. Rev. Lett. 91, 232003 (2003). https://doi.org/10.1103/PhysRevLett.91.232003. arXiv:hep-ph/0307341
A. Lenz, Int. J. Mod. Phys. A 30(10), 1543005 (2015). https://doi.org/10.1142/S0217751X15430058. arXiv:1405.3601 [hep-ph]
A. Ali, A.Y. Parkhomenko, Q. Qin, W. Wang, Phys. Lett. B 782, 412 (2018). https://doi.org/10.1016/j.physletb.2018.05.055. arXiv:1805.02535 [hep-ph]
R.L. Jaffe, Phys. Rep. 409, 1 (2005). https://doi.org/10.1016/j.physrep.2004.11.005. arXiv:hep-ph/0409065
M. J. Savage and M. B. Wise, Phys. Rev. D 39 (1989) 3346 Erratum: [Phys. Rev. D 40 (1989) 3127]. https://doi.org/10.1103/PhysRevD.39.3346, https://doi.org/10.1103/PhysRevD.40.3127
M. Gronau, O.F. Hernandez, D. London, J.L. Rosner, Phys. Rev. D 52, 6356 (1995). https://doi.org/10.1103/PhysRevD.52.6356. arXiv:hep-ph/9504326
X.G. He, Eur. Phys. J. C 9, 443 (1999). https://doi.org/10.1007/s100529900064. arXiv:hep-ph/9810397
C.W. Chiang, M. Gronau, J.L. Rosner, D.A. Suprun, Phys. Rev. D 70, 034020 (2004). https://doi.org/10.1103/PhysRevD.70.034020. arXiv:hep-ph/0404073
Y. Li, C.D. Lu, W. Wang, Phys. Rev. D 77, 054001 (2008). https://doi.org/10.1103/PhysRevD.77.054001. arXiv:0711.0497 [hep-ph]
W. Wang, C.D. Lu, Phys. Rev. D 82, 034016 (2010). https://doi.org/10.1103/PhysRevD.82.034016. arXiv:0910.0613 [hep-ph]
H.Y. Cheng, S. Oh, JHEP 1109, 024 (2011). https://doi.org/10.1007/JHEP09(2011)024. arXiv:1104.4144 [hep-ph]
Y.K. Hsiao, C.F. Chang, X.G. He, Phys. Rev. D 93(11), 114002 (2016). https://doi.org/10.1103/PhysRevD.93.114002. arXiv:1512.09223 [hep-ph]
C. D. L, W. Wang and F. S. Yu, Phys. Rev. D 93 (2016) no.5, 056008 https://doi.org/10.1103/PhysRevD.93.056008 arXiv:1601.04241 [hep-ph]
X. G. He, W. Wang and R. L. Zhu, J. Phys. G 44 (2017) no.1, 014003 https://doi.org/10.1088/0954-3899/44/1/014003, https://doi.org/10.1088/0022-3727/44/27/274003 arXiv:1606.00097 [hep-ph]
W. Wang, R.L. Zhu, Phys. Rev. D 96(1), 014024 (2017). https://doi.org/10.1103/PhysRevD.96.014024. arXiv:1704.00179 [hep-ph]
W. Wang, Z.P. Xing, J. Xu, Eur. Phys. J. C 77(11), 800 (2017). https://doi.org/10.1140/epjc/s10052-017-5363-y. arXiv:1707.06570 [hep-ph]
W. Wang, F.S. Yu, Z.X. Zhao, Eur. Phys. J. C 77(11), 781 (2017). https://doi.org/10.1140/epjc/s10052-017-5360-1. arXiv:1707.02834 [hep-ph]
Y.J. Shi, W. Wang, Y. Xing, J. Xu, Eur. Phys. J. C 78(1), 56 (2018). https://doi.org/10.1140/epjc/s10052-018-5532-7. arXiv:1712.03830 [hep-ph]
X.G. He, W. Wang, Chin. Phys. C 42(10), 103108 (2018). https://doi.org/10.1088/1674-1137/42/10/103108. arXiv:1803.04227 [hep-ph]
W. Wang, J. Xu, Phys. Rev. D 97(9), 093007 (2018). https://doi.org/10.1103/PhysRevD.97.093007. arXiv:1803.01476 [hep-ph]
A. Ali, C. Hambrock, I. Ahmed, M.J. Aslam, Phys. Lett. B 684, 28 (2010). https://doi.org/10.1016/j.physletb.2009.12.053. arXiv:0911.2787 [hep-ph]
M. Karliner, J.L. Rosner, Phys. Rev. Lett. 119(20), 202001 (2017). https://doi.org/10.1103/PhysRevLett.119.202001. arXiv:1707.07666 [hep-ph]
Y. Xing, R. Zhu, Phys. Rev. D 98(5), 053005 (2018). https://doi.org/10.1103/PhysRevD.98.053005. arXiv:1806.01659 [hep-ph]
G. Li, X.F. Wang, Y. Xing, Eur. Phys. J. C 79(8), 645 (2019). https://doi.org/10.1140/epjc/s10052-019-7150-4. arXiv:1902.05805 [hep-ph]
G. Li, X.F. Wang, Y. Xing, Eur. Phys. J. C 79(3), 210 (2019). https://doi.org/10.1140/epjc/s10052-019-6729-0. arXiv:1811.03849 [hep-ph]
Acknowledgements
We thank Prof. Wei Wang for invaluable advice and helpful discussions throughout the project. This work was supported by the Research Start-up Funding and Sailing Plan (Ye Xing) of China University of Mining and Technology under Grant No. 102519108, 102519092 and No. 102519147, by the National Natural Science Foundation of China under Grant No. 11774417.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
Funded by SCOAP3
About this article
Cite this article
Xing, Y. Weak decays of triply heavy tetraquarks \({b{\bar{c}}}{b{\bar{q}}}\) . Eur. Phys. J. C 80, 57 (2020). https://doi.org/10.1140/epjc/s10052-020-7625-3
Received:
Accepted:
Published:
DOI: https://doi.org/10.1140/epjc/s10052-020-7625-3