Weak Decays of Doubly Heavy Baryons: SU(3) Analysis

Motivated by the recent LHCb observation of doubly-charmed baryon $\Xi_{cc}^{++}$ in the $\Lambda_c^+ K^-\pi^+\pi^+$ final state, we analyze the weak decays of doubly heavy baryons $\Xi_{cc}$, $\Omega_{cc}$, $\Xi_{bc}$, $\Omega_{bc}$, $\Xi_{bb}$ and $\Omega_{bb}$ under the flavor SU(3) symmetry. Decay amplitudes for various semileptonic and nonleptonic decays are parametrized in terms of a few SU(3) irreducible amplitudes. We find a number of relations or sum rules between decay widths and CP asymmetries, which can be examined in future measurements at experimental facilities like LHC, Belle II and CEPC. Moreover once a few decay branching fractions are measured in future, some of these relations may provide hints for exploration of new decay modes.

QCD as the fundamental theory for strong interactions shows two distinct facets. At high energy, the interaction strength is weak that allows the use of perturbation theory. At low energy, quarks and gluons are confined into hadrons. The large coupling constant prohibits a direct application of perturbative expansions. For a high energy process with generic hard scattering, one often uses the factorization to separate the high-energy and low-energy degrees of freedoms. The factorization approach has been widely applied to heavy meson decays [22][23][24][25][26][27][28][29][30], in which the long-distance contributions are parametrized in terms of the low energy inputs, mostly the light-cone distribution amplitudes. For heavy baryon decays, the factorization analysis is much more involved due to the lack of knowledge on low-energy inputs and the complicated hard-scattering kernels, and see Refs. [31][32][33][34][35][36] for some recent discussions.
In heavy quark decays, the flavor SU(3) symmetry is an useful tool . There are a few advantages to adopt the SU(3) symmetry. First once the branching fractions for a few decay channels are measured, the flavor SU(3) symmetry offers an opportunity to obtain the knowledge on the related channels. Secondly, the investigation of a few related decay channels can allow one to examine the CKM parameters with the help of SU(3) symmetry. Thirdly, when enough data is available, one may use the data to extract the SU(3) irreducible amplitudes. These amplitudes are expected to calculable in different factorization approaches, and can then be used to examine the factorization schemes themselves. Thus in this paper we will use the flavor SU(3) symmetry and analyze various decays of doubly heavy baryons.
The rest of this paper is organized as follows. In Sec. II, we will collect the representations for the particle multiplets in the SU(3) symmetry. In Sec. III, we will analyze the semileptonic decays of the doubly-heavy baryons. The nonleptonic decays of doubly-charmed baryons, doubly-bottom baryons and the baryons with b, c quarks are investigated in Sec. IV, V and Sec. VI, respectively.
The last section contains a brief summary.

II. PARTICLE MULTIPLETS
In this section, we will collect the representations for the multiplets of the flavor SU(3) group. Quantum numbers of the doubly heavy baryons are derived from the quark model [72] and are given in Table I The singly charmed baryons can form an anti-triplet or sextet as shown in Fig. 1. In the The above two SU(3) triplets are also applicable to the bottom mesons.

III. SEMI-LEPTONIC DECAYS
A. Ξ cc and Ω cc decays 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 is constructed as: Here the a 1 and a 2 are SU(3) irreducible nonperturbative amplitudes. Feynman diagrams for these decays are given in Fig. 2.
The decay amplitudes for different channels can be deduced from the Hamiltonian in Eq. (11), and given in Tab. II. From these amplitudes, we can find the relations for decay widths in the SU(3) symmetry limit: Recently, inspired by the LHCb observation of Ξ cc , the weak decays of doubly heavy baryons have been studied in Ref. [21], where the authors first derived the hadronic form factors for these transitions in the light-front approach and then applied the results to predict the partial widths for the semi-leptonic and non-leptonic decays of doubly heavy baryons. The SU(3) symmetry can be confronted with these results. We should note that the same comparison in semileptonic Ξ bb , Ω bb and Ξ bc , Ω bc decays and in non-leptonic decays of doubly heavy baryons can also be made.
Compared to these explicit model calculations, we found that the SU(3) symmetry works well for the bottom quark decays, while the symmetry breaking effects are sizable for the charm quark decays, largely due to the phase-space differences.
B. Semileptonic Ξ bb and Ω bb decays The b quark decay is controlled by the Hamiltonian Feynman diagrams for these decays are given in Fig. 3. The decay amplitudes can be deduced from this Hamiltonian, and the results are given in Tab. III. It leads to the relations for decay widths: C. Semileptonic Ξ bc and Ω bc decays The effective Hamiltonian for semileptonic Ξ bc and Ω bc decays is given as: In this equation, we have included both charm quark and bottom quark decays. Decay amplitudes for different channels are obtained by expanding the above Hamiltonian and are collected in Tab. IV.
Apparently, the Ξ bc and Ω bc decay amplitudes can be obtained by the ones for T cc and T bb decays. For the charm quark decays, one would derive the results with the replacement, T cc → T bc , Thus we have the following relations for decay widths: IV. NON-LEPTONIC Ξ cc AND Ω cc DECAYS Usually the charm quark decays into light quarks are categorized into three groups: Cabibbo allowed, singly Cabibbo suppressed, and doubly Cabibbo suppressed: while for the c → dus transition which is doubly Cabibbo suppressed, we have For the transition c → udd, we have with all other remaining entries zero. The overall 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 factor is V * cs V us ≃ sin(θ C ). With both the c → udd and c → uss, the singly Cabibbo-suppressed channel has the following effective Hamiltonian: A. Decays into a charmed baryon and a light meson With the above expressions, one may derive the effective Hamiltonian for decays involving the anti-triplet heavy baryons as For the sextet baryon, we have the Hamiltonian Feynman diagrams for these decays are given in Fig. 4.   Expanding the above equations, we will obtain the decay amplitudes given in Tab. V for the antitriplet baryon and Tab. VI for the sextet. Thus we have the following relations for decay widths: For the decays into the sextet, we have:

B. Decays into a light octet baryon and a charmed meson
The effective Hamiltonian for the decays of T cc into a light octet baryon and a charmed meson is given as In the above Hamiltonian we find the following relatios: Thus two of the reduced matrix elements are not independent. In the following, we will eliminate the c 4 and c 5 and use the effective Hamiltonian: Feynman diagrams for these decays are given in Fig. 5. Expanding the above equations, we will obtain the decay amplitudes given in Tab. VII, which leads to the relations for decay widths:

C. Decays into a light decuplet baryon and a charmed meson
The effective Hamiltonian for a light decuplet in the final state is given as Feynman diagrams for these decays are same as Fig. 5. The corresponding decay amplitudes are given in Tab. VIII and it leads to the relations for decay widths: In addition from the decay amplitudes, one can see that there are relations between the widths between Cabibbo-allowed, singly Cabibbo suppressed and doubly Cabibbo suppressed decay modes: These relations are given in Tab. IX, X and Tab. XI, respectively.
For the bottom quark decay, there are generically 4 kinds of quark-level transitions: with q 1,2,3 being the light quarks. Each of them will induce more than one types of decay modes at hadron level, which will be analyzed in order in the following.
A. b → ccd/s

Decays into J/ψ plus a bottom baryon
Such decays have the same topology with semileptonic b → sℓ + ℓ − decays. The transition operator b → ccd/s can form an SU(3) triplet, leadings to the effective Hamiltonian: with (H 3 ) 2 = V * cd and (H 3 ) 3 = V * cs . Feynman diagrams for these decays are given in Fig. 6. Decay amplitudes are given in Tab. XII. Thus we have the following relations for decay widths:

Decays into a doubly heavy baryon bcq plus a anti-charmed meson
The b → ccd/s transition can lead to another type of effective Hamiltonian: which corresponds to the decays into doubly heavy baryon bcq plus a anti-charmed meson. Feynman diagrams for these decays are given in Fig. 7. Decay amplitudes are given in Tab. XIII. Thus we  obtain the following relations for decay widths:

Decays into a doubly heavy baryon bcq plus a light meson
The operator to produce a charm quark from the b-quark decay,cbqu, is given by The light quarks in this effective Hamiltonian form an octet with the nonzero entry for the b → cūd transition, and (H 8 ) 3 1 = V * us for the b → cūs transition. The hadron-level effective Hamiltonian is then given as Feynman diagrams for these decays are given in Fig. 8. Decay amplitudes are given in Tab. XIV, which leads to:

Decays into a bottom baryon bqq plus a charmed meson
The effective Hamiltonian from the operatorcbqu gives Feynman diagrams for these decays are given in Fig. 9. Results are given in Tab. XV, thus we have the relations for decay amplitudes:   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 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 Feynman diagrams for these decays are given in Fig. 10. Decay amplitudes for different channels are given in Tab. XVI, thus we have the relations for decay amplitudes: As one can see, the Ξ bb can decay into both Ξ b D 0 and Ξ b D 0 . The D 0 and D 0 can form the CP eigenstate D + and D − . Thus using the Ξ bb decays into the Ξ b 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 [73][74][75][76][77][78], B → DK * 0,2 [79,80] and others. This is also similar for the Ω bb → Ω − D ± decays and the following Ξ bc → Ξ c D ± and Ω bc → Ω 0 D ± modes. D. Charmless b → q 1q2 q 3 Decays

Decays into a bottom baryon and a light meson
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 The effective hadron-level Hamiltonian for decays into the bottom anti-triplet is constructed as while for the sextet baryon, we have Feynman diagrams for these decays are given in Fig. 11. Decay amplitudes for different channels are given in Tab. XVII and Tab. XVIII for b → d transition and b → s transition respectively.
Thus, it leads to the relations for decay widths:

Decays into a bottom meson and a light baryon octet
The effective Hamiltonian is given as Similarly, we find the reduced matrix elements d 4 , d 7 and d 5 , d 6 are not independent. So we use the following effective Hamiltonian: Feynman diagrams for these decays are given in Fig. 12

Decays into a bottom meson and a light baryon decuplet
The effective Hamiltonian is given as Feynman diagrams for these decays are same as Fig. 12. Decay amplitudes for different channels are given in Tab. XIX and Tab. XX for b → d transition and b → s transition respectively.
We summarize the relations for decay widths for Ξ bb and Ω bb decay into a bottom meson and a light baryon, 4. U-spin for Ξ bb and Ω bb decays For Ξ bb , Ω bb decays induced by the b → q 1 q 2 q 3 , there are two amplitudes with different CKM factors. We consider the connected decays with the decay amplitudes As pointed out in Refs. [40,43,44], there exists a relation for the CP violating quantity ∆ = Γ −Γ.
The relation about decay widths Γ(∆S = i) and CP asymmetry A CP (∆S = i) is In Tab.XXI and Tab.XXII, we collect the Ξ bb , Ω bb decay pairs related by U-spin. The CP asymmetries and decay widths for these pairs satisfy relation in Eq. (97). The experiment data in future is important to test flavor SU(3) symmetry and also CKM mechanism for CP violation.

VI. NON-LEPTONIC Ξ bc AND Ω bc DECAYS
The decays of Ξ bc and Ω bc can proceed via the b quark decay or the c quark decay, which are induced by the following quark transitions: c → sdu, c → udd/ss, c → dsu, b →ccd/s, b → cūd/s, b → ucd/s, b → q 1q2 q 3 .
As we have shown in the semileptonic case, 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 it is not necessary to repeat the tedious results here.

VII. CONCLUSIONS
Quite recently, the LHCb collaboration has observed the doubly-charmed baryon Ξ ++ cc in the final state Λ c K − π + π + . Such an important observation will undoubtedly promote the research on the hadron spectroscopy and also on weak decays of doubly heavy baryons.
In this paper, we have analyzed the weak decays of doubly heavy baryons Ξ cc , Ω cc , Ξ Moreover once a few decay branching fractions were measured in future, some of these relations may provide hints for exploration of new decay modes.