$SU(3)$ Flavor Symmetry for Weak Hadronic Decays of ${\bf B}_{bc}$ Baryons

Baryons with a heavy c-quark or a heavy b-quark and also two c-quarks have been discovered. These states are expected in QCD and therefore provide a test for the theory. There should be double beauty baryons, and also an intriguing possibility that baryons ${\bf B}_{bc}$ with a c-quark, a b-quark. These states are yet to be discovered. The main decay modes of ${\bf B}_{bc}$ are expected to be weak processes from theoretical understanding of their mass spectrum. These decay modes can provide crucial information about these heavy baryons ${\bf B}_{bc}$. We analyze two body hadronic weak decays for ${\bf B}_{bc}$ using $SU(3)$ flavor symmetry. Any one of the $c$ and $b$ decays will induce ${\bf B}_{bc}$ to decay. We find that the Cabibbo allowed decays ${\bf B}_{bc} \to {\bf B}_b + M$ due to $c \to s u \bar d$ can be crucial for exploration. The LHC may have the sensitivity to discover such decays. Other ${\bf B}_{bc}$ decays due to $b \to c q' \bar q$ are sub-leading. Several relations among branching ratios are obtained which can be used to test $SU(3)$ flavor symmetry.

These three states form a fundamental representation 3 of SU (3). The subscript "i" taking the values 1,2,3 in B bc i is the SU (3) representation index. Changing it to a superscript, it becomes the anti-representation index. The subscript bc is related to the naming of the particle. We will use the same notions for other particles and also Hamiltonian in our later discussions.
The weak decays of B bc baryons can be induced by the constituent b-quark or c-quark decays. The dominant c-quark decay is induced by c → uq ′ q with q ′ and q taking the values d and s. These decays are proportional to λ q ′ q = V uq ′ V * cq with Wilson coefficients of order one. These decays are classified as Cabibbo allowed (λ ds = V ud V * cs ), Cabibbo suppressed (λ dd = V ud V * cd or λ ss = V us V * cs ), and doubly Cabibbo suppressed (λ sd = V us V * cd ) decays. The lifetimes of the three B bc states are estimated to be an order of a few times 10 −13 s [18][19][20].
The b can decay in several ways at tree level, a) b → ccq, b) b → cūq, c) b → ucq, d) b → uūq. Among them a) and b) are dominating ones which are proportional to λ c cq = V cb V * cq and λ c uq = V cb V * uq with the Wilson coefficients of order one. c) and d) are proportional to λ c bq = V ub V * cq and λ u bq = V ub V * uq with Wilson coefficients of order one. At one loop level, there are also the penguin decay modes b → q q ′ =u,d,s,cq ′ q ′ . These decays are, however, suppressed by loop induced small Wilson coefficients. In our later discussions we will only keep decays with q=d for a) and b), respectively, and neglect the suppressed parts [21]. Note that even the two largest classes of b-quark induced decays are suppressed by a factor of |V cb /V cs | 2 ∼ 1.7 × 10 −3 compared with Cabibbo allowed c-quark decays. But they are different from c-quark decay induced processes and we hope experiments may find some different favorable search strategies. For instance, the b-quark induced B bc decays can have a displaced baryon for subsequent decays, which is able to travel a 10 3 longer distance before decaying than that from the c-quark decay induced ones. One can hence consider a displaced vertex as a signal of the b-decay mode since the baryon's decay vertex is displaced from the prompt production vertex, which has been used as a technique of searching for doubly-bottom hadrons [22]. For the main B bc baryon weak decays described above, the effective Hamiltonians are given by [23] H c ef f = , for the c and b-quark decays, respectively. G F is the Fermi constant. a takes the values c and u. In the above (q 1 q 2 ) =q 1 γ µ (1 − γ 5 )q 2 , and the subscripts (α, β) denote the color indices. The Wilson coefficients c i are scale We now specify the notation of SU (3) group tensor properties for the effective Hamiltonians, omitting Lorentz structure, for c → q jqi q k and b → cq i q j . We will denote them by H jk i for H c ef f , H i j for H b ef f with a = u, and H i for H b ef f with a = c, respectively. Their nonzero entries are given by [15,24] H jk i contains a3 H , a 6 H and a 15 H [13]. Here the subscript for the representations indicates where they come from, and the H shows that the given representation comes from the effective Hamiltonian. When calculating various decays, the results for3 H will be proportional to λ dd + λ ss . Using the unitarity property of the CKM matrix, this combination is equal to −V ub V * cb which is much smaller than any of the λ q ′ q and can be safely neglected. In other words, λ ss = −λ dd will be a good approximation for our purpose. H i j and H i transform as a 8 H and3 H , respectively. The above effective Hamiltonians can induce various hadronic B bc decays, including two and multi-body channels [5]. We will concentrate on the two-body hadronic decays which may provide the most promising chances for experimental measurements.

II. THE TWO BODY B bc DECAY MODES
We now list some of the interesting two body decay modes of B bc . The effective Hamiltonian H jk i can induce B bc to decay into an octet-8 B (decuplet-10 B ′ ) baryon B (B ′ ) and a3 M b b-meson M b and also into a3 where B (′) stands for the octet (decuplet) baryon and B (′) Q the anti-triplet (sextet) Q-baryon with Q = (b, c). H i j can induce B bc to decay into a triplet-3 Bcc double charmed baryon B cc and a meson M , and also into a3 Bc H i can induce the following decay modes where Mc denotes the anti-particle of M c , and M cc = (η c , J/ψ).

III. SU(3) INVARIANT AMPLITUDES
We now provide some details for the SU (3) invariant decay amplitudes for the decays mentioned in previous sections.
To obtain SU (3) amplitudes, one just needs to contract all upper and lower indices of the hadrons and the Hamiltonian to form all possible SU (3) singlets and associate each with a parameter which lumps up the Wilson coefficients and unknown hadronization effects. These parameters can be determined theoretically and experimentally. Our emphasis will not be on how to determine these hadronic parameters, but to identify the dominant modes and some relations for experimental search. We will normalize the decay amplitudes as A ≡ (G F / √ 2)M and leave the spinor and Lorentz structure out.
The H jk i induced B bc decays are given by We show the corresponding topological diagrams for these two classes of decays in Fig. 1.
In the above, we have neglected terms needing to contract two indices of the Hamiltonian H ij i . Because this will result in the combination of λ dd + λ ss , leading to −V ub V * cb , whose absolute value ∼ 10 −4 is very small compared to |λ dd,ss | ≃ 0.22.
One needs to make sure if the above amplitudes are all independent. This can be checked by group theoretical considerations. The number of independent amplitudes for formed which correspond to 2 invariant amplitudes. However, to get a3 requires to contract two indices and to have H ij i whose contributions are small, proportional to λ dd + λ ss and can be neglected. For 6 H and 15 H , each produces 2 singlets. Therefore there are total 4 independent invariant amplitudes. Naively there are 6 terms as can be seen in the above equation. However using the identity The corresponding topological diagrams for these decays are shown in Fig. 2.
The corresponding topological diagrams are shown in Fig. 3. Similar group theoretical analysis shows that the above invariant amplitudes are all independent ones.
The corresponding topological diagrams are shown in Fig. 4.

IV. DECAY MODES FOR EXPERIMENTAL ANALYSIS
For experimental discovery of B bc , the most favored decay modes will certainly be those with large branching ratios, and at the same time particles in the final state can be easily identified and analyzed. For the decay modes discussed in previous sections, there are two classes of decays, one is decay induced by c decays and another induced by b decays.
We discuss c induced decays below, Among   Tables I and II, we can identify the following Cabibbo allowed decays as We expect the above decay modes to be the dominant ones which are some of the most likely to-be-discovered decay modes. Since the lifetime of B bc is mainly determined by c → sud, the lifetime of Ξ + bc would be similar to Ξ ++ cc and the decay phase spaces and the particle masses are different and therefore may deviate from each other. This expectation is in agreement with some theoretical estimates. For example, in Ref. [20] it is estimated to be a few times of 10 −13 s. A similar argument would lead to the expectation that the decay width of Ξ + bc → Ξ 0 b +π + is similar to Ξ ++ cc → Ξ + c +π + up to phase space modifications. Taking factorizable contributions as the example for an order of magnitude estimate, we derive that M(Ξ Qc → Ξ Q π + ) = iλ ds a 1 f π q µ Ξ Q |(sc)|Ξ Qc , where the decay constant f π is from π + |(ūd)|0 = if π q µ , and a 1 = c 1 + c 2 /N c with N c the color number. With Ξ Q |(sc)|Ξ Qc ≃ū ΞQ (f 1 γ µ − g 1 γ µ γ 5 )u ΞQc [25], we obtain where m π ≃ 0 has been used. We can hence have Γ( 1.93, 0.83, 0.68) [1,7]. This leads to B(Ξ + bc → Ξ 0 b + π + ) an order of a few times 10 −2 [25], in agreement with the calculations in Refs. [26][27][28][29]. One expects other Cabibbo-allowed B bc → B (′) b + M and also B bc → B (′) + M b branching fractions to have similar order of magnitudes [26][27][28][29].
According to the gluon-gluon fusion mechanism that dominantly produces the baryons with two heavy quarks [3,30,31], the cross section (X) for the Ξ bc production is estimated to be (17 ± 3) and (35 ± 7) nb at LHC for the centerof-mass (c.m.) energy √ S = 7 GeV and √ S = 14 GeV, respectively, where the theoretical errors mainly consider the uncertainties from the quark masses m b and m c , together with the non-perturbative effects and the factorization scale.
One also sees from the Tables I and II that there are several relations among decay amplitudes induced by c → uq ′ q.
Those Cabibbo allowed decays offer good chances to be tested experimentally, given by and triangle relations There are also some relations between Cabibbo allowed, Cabibbo suppressed and doubly Cabibbo suppressed decay modes which can be read off from Tables I and II. With more and more data being collected, these relations can also serve further examintations.
For b-decay induced decays, we have As mentioned before that b-decay induced modes are suppressed by a factor of |V cb /V cs | ≃ 0.04 compared with the Cabibbo allowed ones in c-decay induced modes. Nonetheless, their branching fractions are not necessarily small.
Using the B bc → B c , B cc transition form factors obtained in Refs. [34], we estimate that Therefore, we expect that the branching fractions of B bc → B (′) c +(M c , M cc ) and B bc → B cc +(M, Mc) can be as large as (10 −6 , 10 −5 , 10 −4 , 10 −3 ), respectively. According to the current luminosity at LHCb, they are much more difficult to be measured experimentally.
The decayed meson particles are all easy to be identified. In particular, by adopting the situation in Ref. [22], where the weakly decaying double beauty hadrons have been discussed, we take B (′) c and B cc as the displaced baryons. They travel sizeable distances before decaying, which helps to distinguish the signal from the prompt background that might come from the c-quark induced B bc → B b M (BM b ) decays [22]. There may be some chances when more and more data are collected. For example, with the subsequent Ξ ++ cc decay, the needed Ξ bc events to discovery Ξ + bc → Ξ ++ cc π − are estimated to be around 10 9 . We list the SU (3) invariant decay amplitudes in Tables III and IV for completeness. There are also several relations among different decay modes.
In conclusion, we have studied the two-body B bc weak decays using the SU (3) flavor symmetry aiming to provide the most promising decay channels to discover B bc . With the branching fractions estimated as a few times 10 −2 for the Cabibbo-allowed c-quark weak decays, the LHC may be able to discover B bc given their excellent capabilities of identifying B ( They may have some chance to be eventually detected at the LHCb. We strongly urge our experimental colleagues to search for B bc using two-body weak decays.  Decay modes Amplitudes Decay modes Amplitudes