Weak Decays of Triply Heavy Baryons

After the experimental establishment of doubly heavy baryons, baryons with three quarks are the last missing pieces of the lowest-lying baryon multiplets in quark model. In this work we study semileptonic and nonleptonic weak decays of triply heavy baryons, $\Omega_{ccc}^{++}, \Omega_{ccb}^{+}, \Omega_{cbb}^{0}, \Omega_{bbb}^{-}$. Decay amplitudes for various channels are parametrized in terms of a few SU(3) irreducible amplitudes. We point out that branching fractions for Cabibbo allowed processes, $\Omega_{ccc}\to (\Xi_{cc}^{++} \overline K^0, \Xi_{cc}^{++}K^-\pi^+, \Omega_{cc}^{+}\pi^+, \Xi_{c}^+ D^+, \Xi_{c}^{\prime} D^+, \Lambda_c D^+\overline K^0, \Xi_{c}^+ D^0 \pi^+, \Xi_{c}^0 D^+\pi^+)$ may reach a few percents. We suggest our experimental colleagues to perform a search at hadron colliders and the electron and positron collisions in future, which will presumably lead to discoveries of triply heavy baryons and complete the baryon multiplets. Using the expanded amplitudes, we derive a number of relations for the partial widths which can be examined in future.

Ω ++ ccc , Ω + ccb , Ω 0 cbb , Ω − bbb . Decay amplitudes for various channels are parametrized in terms of a few SU(3) irreducible amplitudes. We point out that branching fractions for Cabibbo allowed processes, Ω ccc → (Ξ ++ cc K 0 , Ξ ++ cc K − π + , Ω + cc π + , Ξ + c D + , Ξ ′ c D + , Λ c D + K 0 , Ξ + c D 0 π + , Ξ 0 c D + π + ) may reach a few percents. We suggest our experimental colleagues to perform a search at hadron colliders and the electron and positron collisions in future, which will presumably lead to discoveries of triply heavy baryons and complete the baryon multiplets. Using the expanded amplitudes, we derive a number of relations for the partial widths which can be examined in future.

I. INTRODUCTION
In the past decades, hadron spectroscopy has experienced a rapid renaissance, predominantly propelled by discoveries of a number of hadron resonances that defy the standard quark model interpretations. These resonant states are generically classified as hadron exotics, and for reviews on recent progresses, please see Refs. [1][2][3][4]. Among all exotic hadrons, the X(3872) plays a most important role due to its special properties. Aside from these unexpected discoveries, there are also gradual progresses in the traditional sector of the charmonium and bottomonium spectroscopy.
This observed state fills well in the quark model as the lowest-lying ccu baryon [6]. After the tentative establishment of Ξ ++ cc , it is plausible to fill the baryon family with the last missing members, namely, baryons made of three heavy quarks. Charm and bottom quarks are much heavier than the u, d, s, thus baryons with three heavy quarks will refrain from light quark contaminations and the study of triply heavy baryons can help us to better understand the dynamics of strong interactions.
Previous studies of triply heavy baryons concentrated on three facets: spectroscopy, production and decays. Most theoretical investigations in the literature, especially in recent years, have focused on the masses and magnetic moments [7][8][9][10][11][12][13][14][15][16][17][18][19][20], while less attentions have been paid to the production and decay properties. The only available estimate of the production is conducted in Refs. [21,22], where the cross sections at the LHC with √ s = 7 TeV are found to reach the 0.1 nb level depending on different kinematics cuts. In the b → c transitions among triply heavy baryons, Ref. [23] has discussed the implications of heavy quark spin symmetry. Some decay modes of the Ω ccc are analyzed recently in Ref. [24].
The main focus of this paper is to provide a systematic analysis of weak decays of the lowestlying triply heavy baryons, Ω ccc,ccb,cbb,bbb . The Ω ccc and Ω bbb have spin 3/2, while the Ω ccb and Ω cbb can be the J P = 1/2 + or J P = 3/2 + state. As we will show, various types of weak decays of triply heavy baryons occur, but unfortunately, a universal dynamical (factorization) approach can not be established yet. This gives a barrier for us to predict their decay widths. Instead we will use an optional theoretical tools to analyze heavy quark decays, the flavor SU(3) symmetry .
One advantage of the SU(3) analysis is that it is independent of the factorization details. In this work, we consider semileptonic decay channels with one or two hadrons in the final state, while for nonleptonic decays, the two-body and three-body modes will be analyzed.
The rest of this paper is organized as follows. In Sec. II, we will collect the representation matrices for various particle multiplets in the SU(3) group. In Sec. III, semileptonic decay modes with one or two hadrons in the final state are analyzed. In Sec. IV, Sec. V and Sec. VI, nonleptonic decays of Ω ccc , Ω bbb , Ω ccb and Ω cbb will be studied in order. In Sec. VII, we shall present a collection of golden modes which are most likely to discover the triply heavy baryons. A short summary is given in the last section.

II. PARTICLE MULTIPLETS
In this section, we will collect the representations for hadron multiplets under the flavor SU (3) group. We start with the baryon sector.
The above two SU(3) triplets are also applicable to the bottom mesons.
In the following we will construct the hadron-level effective Hamiltonian for various decay modes.
It is necessary to point out that a hadron in the final state must be created by its anti-particle field. For instance, in order to produce a Ξ ++ ccu , one must use the Ξ −− ccu in the Hamiltonian, and the doubly heavy baryon anti-triplet is abbreviated as T cc .
The c → qlν transition is induced by the effective electro-weak Hamiltonian: where q = d, s and ℓ = e, µ. The V cd and V cs are CKM matrix elements. Heavy-to-light quark operators are an SU(3) triplet, denoted as H 3 with the nonzero components ( cs . At hadron level, the effective Hamiltonian for three-body and four-body semileptonic Ω ccc decays can be constructed as: where the a i s are SU(3) irreducible amplitudes. Decay amplitudes for different channels can be deduced from the Hamiltonian in Eq. (10), and collected in Tab. I. A few remarks are given in order.
• In this table and following ones, the light pseudoscalar mesons can be replaced by their light counterparts. For instance the K 0 can be replaced by a K * 0 , which is reconstructed by the K − π + final state. The η meson is difficult to reconstruct at hadron colliders, while the vector φ meson can be reconstructed in the K + K − final state with a high efficiency.
• The c → s transition is proportional to the V cs ∼ 1, and the c → d transition has a smaller CKM matrix element V cd ∼ 0.2. Inspired by the experimental data on D meson decays [6], we can infer that branching fractions for the c → s channels are about a few percents, and the ones for the c → d transitions are at the order 10 −3 .
• A number of relations for decay widths can be easily read off from Tab. I. For instance for the c → s decays into a doubly charmed baryons, we have However it is necessary to point out that the above relations will be modified due to the different masses of the final hadrons. Once the mass of Ω ccc is experimentally measured in future, kinematical corrections can be included, and these relations can be refined.

B. Semileptonic Ω bbb decays
The b quark decay is controlled by the electro-weak Hamiltonian with q ′ = u, c, and here ℓ = e, µ, τ . The b → c transition is an SU(3) singlet and thus the transition is simply a singlet. The b → u transition forms an SU(3) triplet H ′ 3 with (H ′ 3 ) 1 = 1 and (H ′ 3 ) 2,3 = 0. The hadron level Hamiltonian is given as The b i s are the SU(3) independent amplitudes like the a i s in Eq. (10). The decay amplitudes can be deduced from this Hamiltonian, and the results are given in Tab. II.
A few remarks are given in order.
• The Ω cbb in the final state can be 1/2 + or 3/2 + . It is similar for the T bb , T bc and others.
• The b → c transition has a larger CKM matrix element V cb ∼ 0.04, and the typical branching fractions might reach the order 10 −3 to 10 −2 . However such decay modes still contain a triply heavy baryon which must be detected through its subsequent weak decays.
• The b → u transition is suppressed due to V ub ∼ 10 −3 . Typical branching fractions are at the order 10 −4 .

C. Semileptonic Ω ccb decays
Both charm quark and bottom quark in Ω ccb can weakly decay. Thus the hadron level Hamiltonian for semileptonic Ω ccb decays is given as Decay amplitudes for different channels are obtained by expanding the above Hamiltonian and are collected in Tab. III.

D. Semileptonic Ω cbb decays
Similarly the hadron level Hamiltonian for semileptonic Ω cbb decay is given as Expanding the above equations, we will obtain the decay amplitudes given in Tab. IV.

IV. NON-LEPTONIC Ω ccc DECAYS
Nonleptonic charm quark decays into light quarks are classified into three groups: Feynman diagrams for two-body decays induced by the c → sdu are given in Fig For charm quark decays, the vector representation H 3 will vanishes as an approximation. For the c → sud transition, we have while for the doubly Cabibbo suppressed c → dus transition, we have CKM matrix elements for c → udd and c → uss transitions are approximately equal in magnitude but different in sign. With both contributions, one has the nonzero components: In the following we will use s C to abbreviate the sine of Cabibbo angle θ.

A. Decays into a doubly-charmed baryon and one (two) light meson(s)
For decays into a doubly-charmed baryon and a light meson, one may derive the effective Hamiltonian: whose Feynman diagrams are given in Fig. 1. It is necessary to stress that the above SU (3) independent amplitudes a i s are different with the ones in Eq. (10).
Expanding the above equations, we will obtain the decay amplitudes given in Tab For the reactions with one additional light meson in the final state, one has the Hamiltonian: Expanding the above equation, we will obtain the decay amplitudes given in Tab. VI. The following remarks are in order.
• From the expanded Hamiltonian, one can find that the amplitudes b 1 and b 2 always appear in the combination b 1 − b 2 . Thus we have removed the amplitude b 2 in Tab. VI • For channels with two identical particles, there is a factor 1/2 in the decay width.

B. Decays into a charmed baryon and a charmed meson
For the two-body decays into a charmed baryon and a charmed meson, the effective Hamiltonian for the decays of Ω ccc into a singly charmed baryon and a charmed meson is given as: The Feynman diagram is shown in the last panel of Fig. 1, and decay amplitudes are collected in Tab. VII.
The three-body decays of Ξ ++ ccc can involve an additional light meson in the final state. For the modes with an anti-triplet baryon, we have while the effective Hamiltonian for a sextet baryon is constructed as: Expanding the above equations, we will obtain the decay amplitudes given in Tab. VIII for anti-triplet baryon and in Tab. IX for sextet baryon.
which will be studied in order.
A. b → ccd/s

Decays into a J/ψ
Such decays will have the same topology with the b → sℓ + ℓ − decays. The transition operator b → ccd/s can form an SU(3) triplet: with (H 3 ) 2 = V * cd and (H 3 ) 3 = V * cs . Decay amplitudes for different channels are obtained by expanding the above Hamiltonian and are collected in Tab. X.

Decays into a triply heavy baryon cbb 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. Decay amplitudes for different channels are obtained by expanding the above Hamiltonian and are collected in Tab. XI.

Decays into a triply heavy baryon cbb plus light mesons
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 cb V * us for the b → cūs transition.
Decay amplitudes for different channels are obtained by expanding the above Hamiltonian and are collected in Tab. XII.
Decay amplitudes for different channels are obtained by expanding the above Hamiltonian and are collected in Tab. XIII.

C. The CKM suppressed b → ucd/s transition
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 derived as: Decay amplitudes for different channels are obtained by expanding the above Hamiltonian and are collected in Tab. XIV.

Decays into a doubly bottom baryon bbq 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. In the SU (3)  If the final state contains one light meson, the effective Hamiltonian is given as: Decay amplitudes for different channels are obtained by expanding the above Hamiltonian and are collected in Tab. XV.
With one additional light meson, we have Decay amplitudes for different channels are obtained by expanding the above Hamiltonian and are collected in Tab. XVI. A few remarks are given in order.
• In Tab • Expanding Eq. (38), one can find the amplitudes l 6 and l 7 are not independent and they always appear in the product l 6 − l 7 . So in the two tables, we did not show the l 7 .
• Inspired from the B meson decay data, we can infer that the typical branching fractions are at the order 10 −6 . Thus these channels are rare decays, and can be studied with a large amount of data. However, the direct CP asymmetries in these channels are typically sizable.
• For the b → q 1q2 q 3 decays, there are two amplitudes with different CKM factors. One can consider the U-spin connected decays with the decay amplitudes where r is a constant factor, and A T and A P are the amplitudes without the CKM factors.

Such channel pairs include [Ω
and etc. As pointed out in Refs. [29,32,55], there exists a relation for the CP violating quantity ∆ = Γ −Γ: The future experiment data will be valuable to test flavor SU(3) symmetry and the CKM mechanism for CP violation.

Decays into a bottom meson and a bottom baryon bqq
If the bottom baryon is an anti-triplet, we have the effective Hamiltonian for two-body and three-body decays: Decay amplitudes for different channels are obtained by expanding the above Hamiltonian and are collected in Tab. XVII.
In the case of a sextet, we have Decay amplitudes for different channels are obtained by expanding the above Hamiltonian and are collected in Tab. XIX.
The final state can be a bottom baryon (sextet) which has: • The light pseudoscalar meson in these two tables can be replaced by its vector counterpart.
For instance a K 0 can be replaced by a K * 0 decaying into K − π + .
• Branching fractions for semileptonic Ω ccc decay channels in Tab. XXIII can reach a few percents, but there is a neutrino in the final state, reducing somewhat experimental efficiency.
• Nonleptonic Ω ccc such as Ω +++ ccc → Ξ ++ cc K − π + might be used to search for Ω ccc especially at LHC, since their branching fractions are sizable, and the final state can be easily to identify.
This will make use of the doubly heavy baryon Ξ ++ cc which has been just discovered by LHCb.
• For nonleptonic decays of Ω − bbb , the largest branching fraction might reach 10 −3 . Taking into account its daughter decays, we expect the branching fraction for Ω − bbb decaying into charmless final state is at most 10 −9 . Thus the triply bottom baryon can be only observed with a large amount of data in future, such as the high luminosity LHC.

VIII. CONCLUSIONS
Up to date, quark model is a most successful theoretical tools to describe the hadron spectrum especially the lowest lying hadrons. Since the charm and bottom quarks are much heavier than the lighter ones, hadrons with a different number of heavy quarks will have distinct dynamics. On experimental side, light hadrons with no heavy quark, singly heavy baryons, and doubly heavy baryons have been established, but triply heavy baryons are still missing. Thus it deserves more theoretical and experimental efforts to study various properties of triply heavy baryons from both theoretical and experimental sides.
In this work, we have systematically analyzed weak decays of triply heavy baryons for the first time in the literature. Decay amplitudes for various transitions have been parametrized in terms of the SU(3) independent amplitudes. Using these results, we find a number of relations for the partial decay widths. We also give a list of decay channels with sizable branching fractions. We suggest our experimental colleagues to perform a search at hadron colliders and the electron and positron collisions in future.