Weak Decays of Doubly Heavy Baryons: Multi-body Decay Channels

The newly-discovered $\Xi_{cc}^{++}$ decays into the $ \Lambda_{c}^+ K^-\pi^+\pi^+$, but the experimental data has indicated that this decay is not saturated by any two-body intermediate state. In this work, we analyze the multi-body weak decays of doubly heavy baryons $\Xi_{cc}$, $\Omega_{cc}$, $\Xi_{bc}$, $\Omega_{bc}$, $\Xi_{bb}$ and $\Omega_{bb}$, in particular the three-body nonleptonic decays and four-body semileptonic decays. We classify various decay modes according to the quark-level transitions and present an estimate of the typical branching fractions for a few golden decay channels. Decay amplitudes are then parametrized in terms of a few SU(3) irreducible amplitudes. With these amplitudes, we find a number of relations for decay widths, which can be examined in future.


I. INTRODUCTION
Nowadays Lattice QCD is the sole approach that can study nonperturbative strong interactions from first principle. Despite the fact that there have been great progresses on Lattice QCD, hadron structures are still often encoded by phenomenological approaches like quark models or QCD sum rules. The quark model can be used to classify the hadrons, in which a baryon is assigned as a three-quark system. Among various baryonic states, doubly heavy baryons are of particular interest since they provide a platform to study the nonperturbative dynamics in the presence of heavy quarks. These states have been searched for a long time [1][2][3][4][5][6], and in 2017 the LHCb collaboration has announced an observation of the Ξ ++ cc (ccu) with the mass m Ξ ++ cc = (3621.40 ± 0.72 ± 0.27 ± 0.14)MeV [7]. This analysis is based on the 1.7 f b −1 data accumulated at 13 TeV, and confirmed in the additional sample of data collected at 8 TeV.
To handle weak decays of heavy mesons, factorization approach is widely adopted in order to separate high-energy and low-energy degrees of freedoms. High-energy contributions are calculable using the ordinary perturbation theory. The low-energy degrees, or equivalently the long-distance contributions, are usually parameterized as low energy inputs such as light-cone distribution amplitudes. In terms of heavy baryon decays, neither the low-energy inputs nor the short-distance coefficients are available in the literature. Only recently the "decay constants" were studied in QCD sum rules [26].
This work is an extension of a series of previous works [10,11,14,24,26]. In Ref. [14], instead of factorization, we have adopted the flavor SU(3) symmetry and classified various decays of doubly heavy baryons. In that work, however, we have limited ourselves to two-body nonleptonic decay modes. The Ξ ++ cc baryon has been firstly observed in the mode Ξ ++ cc → Λ + c K − π + π + [7], and experimental data has indicated that this mode is not saturated by two-body intermediate state.
This motivates us to study the multi-body decays. The main objective of this work is to do so, and we will focus on the cases where the final states contain one additional light meson, namely three-body nonleptonic decay and four-body semileptonic decays.
The rest of this paper is organized as follows. In Sec. II, we will collect representations for the particle multiplets in the SU(3) symmetry. In Sec. III, we will give a list of golden channels that can be used to reconstruct the doubly heavy baryons, and we present an estimate of their branching fractions. In Sec. IV, we will analyze the semileptonic decays of the doubly-heavy baryons, in which the final state contains two hadrons. The three-body nonleptonic decays of doubly-charmed baryons, doubly-bottom baryons and the baryons with b, c quarks are investigated in Sec. V, VI and Sec. VII, respectively. The last section contains a brief summary.

II. PARTICLE MULTIPLETS
In this section, we start with the representations for the multiplets of the flavor SU(3) group.
Quantum numbers of the doubly heavy baryons are derived from the quark model. These baryons can form an SU(3) triplet: (1) The light baryons form an SU(3) octet and a decuplet. The octet has the expression: and the light decuplet is given as The above classification is also applicable to bottom mesons.

III. GOLDEN DECAY CHANNELS
Before presenting the decay amplitudes for various channels, we will make a list of the golden channels and give an estimate of the decay branching fractions in this section [10]. In the following list we give, a hadron is generic and can be replaced by the states with the same quark structure, for instance one can replace K 0 by K * 0 which decays into K − π + . Since a neutral meson is very difficult to reconstruct at LHC, we have removed the modes involving the π 0 , η, ρ + , ω.

I. Ξ cc and Ω cc
The Feynman diagrams for the Cabibbo-allowed decays are given in Fig. 1. We only show one type of penguin diagrams. The C, C ′ , B, E diagrams are suppressed by 1/N c compared to the tree amplitude T . For the Ξ cc and Ω cc decays, we collect Cabibbo allowed decays in Tab. I. From the D and Λ c decay data, we infer that these Cabibbo allowed decay channels have typical branching fractions at a few percent level.

II. Ξ bc and Ω bc
A list of possible modes to reconstruct the bcq baryons is given in Tab. II. For the charm quark decay, the typical branching fractions might be a few percents. The final state contains either a bottom meson, or a bottom baryon, whose decay branching fraction is then at the order 10 −3 . So the branching fraction to reconstruct the Ξ bc and Ω bc is very likely at the order 10 −5 .
If the bottom quark decay first in the bcq baryons, the branching fraction might be even smaller than 10 −3 , since the total width of Ξ bc and Ω bc is dominated by charm quark decay. In this case, the branching fraction to reconstruct Ξ bc and Ω bc might be even smaller than 10 −5 .

III. Ξ bb and Ω bb
The channels that can be used to reconstruct the Ξ bb and Ω bb are collected in Tab. III. Their typical branching fractions are at the order 10 −3 . However in order to reconstruct the bottom meson and bottom baryons in the final state, the price to pay is another factor of 10 −3 . Including the fraction for J/ψ or D or charmed baryons, we have the largest decay branching fraction for Ξ bb and Ω bb at the order of 10 −8 .  The c → qlν transition is induced by the effective Hamiltonian: where q = d, s and the V cd and V cs are CKM matrix elements. The heavy-to-light quark operators will form an SU(3) triplet, denoted as H 3 with the components ( At the hadron level, the effective Hamiltonian for decays of Ξ cc and Ω cc into a singly charmed baryon and a light meson is constructed as: Here the a i are SU (3)  From these amplitudes, we can find the relations for decay widths in the SU(3) symmetry limit.
For decays into a singly charmed baryon (anti-triplet), we have Relations for decays into a singly charmed baryon (sextet) are given as: II. Semileptonic Ξ bb and Ω bb decays The b quark decay is governed by the electro-weak Hamiltonian The hadron level Hamiltonian for semileptonic Ξ bb and Ω bb decays is constructed as The decay amplitudes can be deduced from this Hamiltonian, and the results are given in Tab. V.
For decays into a bcq, we have the relations for decay widths For Ξ bb and Ω bb decays into a singly bottom baryon, we have the relations for decay widths:

III. Semileptonic Ξ bc and Ω bc decays
For the charm quark decays in Ξ bc and Ω bc , one can obtain the decay amplitudes from those for Ξ cc and Ω cc decays with the replacement of T cc → T bc , T c → T b and D → B. For the bottom quark decay, one can obtain them from those for Ξ bb and Ω bb decays with T bb → T bc , T b → T c and B → D. Thus we do not repeat the tedious results here.
So the hadron-level Hamiltonian can be decomposed in terms of a vector (H 3 ), a traceless tensor antisymmetric in upper indices, H 6 , and a traceless tensor symmetric in upper indices, H 15 . As we will show in the following, the representation H 3 will vanishes from the unitarity of CKM matrix.
For the c → sud transition, we have the nonzero matrix element: while for the doubly Cabibbo suppressed transition c → dus, we have For the transition c → udd, we have with all other remaining entries zero. The overall CKM factor is V * cd V ud ≃ − sin(θ C ). While for the transition c → uss, we have with all other remaining entries zero. The overall CKM factor is V * cs V us ≃ sin(θ C ). Since the CKM factors for c → udd and c → uss are almost equal in magnitudes, we combine the two transitions.
Thus the singly Cabibbo-suppressed channel has the following hadron-level Hamiltonian:
So we can remove b 1 and b 9 term in the expanded amplitude.
It should be mentioned that the dynamical mechanisms of these terms are not all the same. For the production of final two light mesons, some terms contain one QCD coupling while the others contain two QCD couplings. Expanding the above equations, we will obtain the decay amplitudes given in Tab. VI, Tab. VII for the anti-triplet baryon and Tab. VIII, Tab. IX, Tab. X for the sextet. Based on the expanded amplitudes, we derive the relations for decay widths collected in Appendix A 1.

II. Decays into a light baryon, a charmed meson and a light meson
The hadron-level Hamiltonian for the decays of T cc into a light octet baryon, a charmed meson and a light meson is given as (c 1 + 2c 2 + c 4 + 2c 5 + c 6 + c 8 + 2c 9 − 4c 10 − 4c 11 + 2c 12 Expanding the above equations, we will obtain the decay amplitudes given in Tab. XI, Tab. XII and Tab. XIII. This leads to the relations for decay widths: in Appendix A 2.
For a light decuplet in the final state, the Hamiltonian is given as The corresponding decay amplitudes are given in Tab. XIV, and it leads to the relations for decay widths also collected in Appendix A 2.

VI. NON-LEPTONIC Ξ bb AND Ω bb DECAYS
For the bottom quark decay, there are generically four kinds of quark-level transitions: with q 1,2,3 as a light quark. Each of them can induce more than one types of decay modes at hadron level, which will be discussed in order.

bottom baryon and a light meson
These decays have the same topology with semileptonic b → sℓ + ℓ − decays, and thus the SU (3) relations derived in this subsection are also applicable to semileptonic b → sℓ + ℓ − decays. The transition operator b → ccd/s can form an SU(3) triplet, which leads to the effective Hamiltonian: Decay amplitudes are given in Tab. XV, from which we derive the relations for decay widths: Appendix A 3.
This Hamiltonian denotes the decays into doubly heavy baryon bcq plus an anti-charmed meson.
Decay amplitudes are given in Tab. XVI. Thus we obtain the following relations for decay widths: III. b → cūd/s transition: Decays into a doubly heavy baryon bcq plus two light mesons The operator to produce a charm quark from the b-quark decay,cbqu, is given by The Hamiltonian is then given as Decay amplitudes are expanded in Tab. XVII, which leads to the relations: Appendix A 5.

IV. b → cūd/s transition: Decays into a bottom baryon bqq, a charmed meson and a light meson
The effective Hamiltonian from the operatorcbqu gives Results are given in Tab. XVIII for anti-triplet and Tab. XIX for sextet, thus we have the relations for decay amplitudes: Appendix A 6 for sextet. Actually, for the anti-triplet case there's no definite relations between the decay withs.

V. b → ucd/s: Decays into a bottom baryon bqq plus anti-charmed meson and a light meson
For the anti-charm production, the operator having the quark contents (ūb)(qc) is given by The two light anti-quarks form the3 and 6 representations. The anti-symmetric tensor H ′′ 3 and the symmetric tensor H 6 have nonzero components (H ′′ 3 ) 13 = −(H ′′ 3 ) 31 = V * cs , (H6) 13 = (H6) 31 = V * cs , for the b → ucs transition. For the transition b → ucd one requests the interchange of 2 ↔ 3 in the subscripts, and V cs replaced by V cd .
The effective Hamiltonian is constructed as Decay amplitudes for different channels are given in Tab. XX and Tab. XXI. We derive relations for decay amplitudes given in Appendix A 7.
The Ξ bb can decay into both D 0 and D 0 . The D 0 and D 0 can form the CP eigenstates D + and D − . Thus using the Ξ bb decays into the D ± , one may construct the interference between the b → cūs and b → ucs. The CKM angle γ can then be extracted from measuring decay widths of these channels, as in the case of B → DK [27][28][29][30][31][32], B → DK * 0,2 [33,34] and others. This is also similar for the Ω bb → D ± decays and the following Ξ bc → D ± and Ω bc → D ± channels.

VI. Charmless b → q 1q2 q 3 Decays: Decays into a bottom baryon and two light mesons
The charmless b → q (q = d, s) transition is controlled by the weak Hamiltonian H ef f : where O i is a four-quark operator or a moment type operator. At the hadron level, penguin
VII. NON-LEPTONIC Ξ bc AND Ω bc DECAYS Decays of Ξ bc and Ω bc can proceed via the b quark decay or the c quark decay. As we have shown in the semileptonic channels, for the charm quark decays, one can obtain the decay amplitudes from those for Ξ cc and Ω cc decays with the replacement of T cc → T bc , T c → T b and D → B. For the bottom quark decay, one can obtain them from those for Ξ bb and Ω bb decays with T bb → T bc , T b → T c and B → D. Thus we do not present the tedious results again.

VIII. CONCLUSIONS
Quite recently, the LHCb collaboration has observed the Ξ ++ cc in the final state Λ c K − π + π + . Such an important observation will undoubtedly promote the research on both hadron spectroscopy and weak decays of doubly heavy baryons.
In this paper, we have analyzed weak decays of doubly heavy baryons Ξ cc , Ω cc , Ξ and Ω bb under the flavor SU(3) symmetry, where the final states involve one or two light mesons. This is inspired by the experimental fact that the Ξ ++ cc → Λ c K − π + π + is not dominated by any two-body intermediate state. Decay amplitudes for various semileptonic and nonleptonic decays have been parametrized in terms of a few SU(3) irreducible amplitudes. We have found a number of relations or sum rules between decay widths, which can be examined in future measurements at experimental facilities like LHC [8], Belle II [38] and CEPC [39]. On the one hand, at first sight the number of relations is desperately large. On the other hand, once a few decay branching fractions were measured in future, these relations can provide richful important clues for the exploration of other decay modes.
It should be stressed that our analysis in this work using the flavor SU(3) symmetry is only applicable to non-resonant contributions. For a complete exploration of three-body decays, one should also take into account resonant contributions from two-body states and this has been given in Ref. [14]. Relative phases between them can be obtained in a Dalitz plot analysis or measurements of invariant mass distributions. In addition, SU(3) symmetry breaking effects might also be relevant. Such effects in the phase space can be incorporated once masses of all involved hadrons are known. This will remedy the relations for decay widths we derived. Actually, we have removed the channels kinematically prohibited. Further deviations, if found by experimentalists in future, would have the indications on decay dynamics in the doubly heavy baryon system. We hope this analysis together with experimental measurements in future will help to establish a QCD-rooted