Weak decays of doubly heavy baryons: the $1/2\to3/2$ case

As a continuation of our previous works, we investigate the weak decays of doubly heavy baryons into a spin-$3/2$ singly or doubly heavy baryon. Light-front approach is adopted to handle the dynamics in the transitions, in which the two spectator quarks are approximated as a diquark. Results for form factors are then used to calculate decay widths of semi-leptonic and nonleptonic processes. The flavor SU(3) symmetry and symmetry breaking effects in semi-leptonic decays modes are explored, and we point out that in charm sector, there are sizable symmetry breaking effects. For nonleptonic decay modes, we study only the factorizable channels induced by the external W-emission. We find that branching fractions for most 1/2 to 3/2 transitions are approximately one order of magnitude smaller than the corresponding ones for the 1/2 to 1/2 transitions. Parametric uncertainties are also investigated in detail. This work, together with our previous works, are beneficial to the experimental studies of doubly heavy baryons at LHC and other experiments.

A doubly heavy baryon is composed of two heavy quarks and one light quark. Light flavor SU (3) symmetry arranges the doubly heavy baryons into the presentation 3. If it is symmetric under interchange of b and c quarks, this set is denoted by Ξ +,0 bc and Ω 0 bc , while for the asymmetric case, the corresponding set is denoted by Ξ ′+,′0 bc and Ω ′0 bc . 1 In reality these two sets probably mix with each other, which is not taken into account in this work. Spin-3/2 doubly heavy baryons have the same flavor wave functions with but different spin structures compared to the spin-1/2 counterparts. The quantum numbers of low-lying doubly heavy baryons can be found in Table I.  To be explicit, we will investigate the following decay modes of doubly heavy baryons.
The authors of Ref. [7] have investigated the doubly heavy baryon decays with the help of flavor SU(3) symmetry. Based on the available data, a great number of decay modes ranging from semi-leptonic decays to multi-body nonleptonic decays can be predicted. However, in the c quark decay, SU(3) symmetry breaking effects may be sizable and can not be omitted. A quantitative study of SU(3) symmetry breaking effects will be conducted within the light-front approach.
The rest of the paper is arranged as follows. In Sec. II, we will present briefly the framework of light-front approach under the diquark picture, and flavor-spin wave functions will also be discussed.
Numerical results are shown in Sec. III, including the results for form factors, predictions on semileptonic and nonleptonic decay widths, detailed discussions on the SU(3) symmetry, the error estimates and a comparison with the previous 1/2 to 1/2 results. A brief summary and discussions on future improvements are given in the last section.

II. THEORETICAL FRAMEWORK
Theoretical framework for 1/2 → 3/2 transition will be briefly introduced in the first subsection, including the definitions of the states for spin-1/2 and spin-3/2 baryons, and the extraction of form factors. More details can be found in [42,46]. Flavor-spin wave functions will be given in the second subsection.
1 is the initial (final) quark momentum, p 2 is the diquark momentum and the cross mark denotes the weak interaction.

A. Light-front approach
In the framework of light-front approach, the wave functions of 1/2 + baryon with an axial-vector diquark is expressed as Here with and with q = u, d or s for Ξ ++ cc , Ξ + cc or Ω + cc , respectively. It is similar for the doubly bottom baryons. For the bottom-charm baryons, there are two sets of states, with bc as a scalar or an axial-vector diquark. The wave functions of bottom-charm baryons with an axial-vector bc diquark are while those with a scalar bc diquark are given as with q = u, d or s for Ξ For the final state, the spin-3/2 baryon with quark contents of Qqq ′ has while for the Qqq baryon, an additional factor √ 2 should be added. For the spin-3/2 baryon with quark contents of QQ ′ q, we have The masses of the axial-vector diquarks are approximated by m {Qq} = m Q + m q . The shape parameters β in Eq. (5) are given as (in units of GeV) [28] β u{cq} = β d{cq} = 0.470, β s{cq} = 0.535, β c{cq} = 0.753, β b{cq} = 0.886, where q = u, d, s.
The masses and lifetimes of the parent baryons are collected in Table III [1,3,23,[48][49][50]. Note that, in the Table III, the masses and lifetimes of B bc and B ′ bc are taken the same. Also note that we have taken a new value for the lifetime of Ω + cc compared with our previous work [4]. Because according to Ref. [48], lifetimes of doubly charmed baryons should satisfy the following pattern:  [49] 370 [49] 800 [50] The masses of the final state baryons are given in Table IV [ In the calculation of nonleptonic decays, these mesons will present in the final states: π, ρ, a 1 , K, K * , D, D * , D s , D * s . Their masses can be found in Ref. [51], while their decay constants are given as follows [28,41,52]: Wilson coefficients a 1 = C 1 (µ c ) + C 2 (µ c )/3 = 1.07 [53], will be used.  [23,51]. 13.600

B. Results for form factors
To access the q 2 -distribution, the following single pole structure is assumed for form factors: F (0) is the value of the form factors at q 2 = 0, the corresponding numerical results predicted by the light-front approach are collected in Tables V to VIII. For c → d/s decays, m pole is taken as 1.87 GeV, while for b → u/c decays, m pole is taken as 5.28 GeV and 6.28 GeV, respectively. In practice, these quantities are taken as the masses of D, B and B c mesons. The discussion for the validity of this assumption can be found in our previous work [41].
The physical form factor can be obtained by multiplying Eq. (27) by the corresponding overlapping factor.

C. Results for semi-leptonic decays
Helicity amplitudes are defined by  (27) will be adopted, and m pole is taken as follows: for b → q process, m pole = 5.28 GeV while for b → c process, m pole = 6.28 GeV.
amplitudes are related to the form factors by the following expressions.
where the upper (lower) sign corresponds to is the mass of the baryon in the initial (final) state. The remaining helicity amplitudes can be obtained by Partial differential decay widths are obtained as  Numerical results are collected in Tables IX to XIV. Some comments are given in subsection III F.

D. Results for nonleptonic decays
For the nonleptonic processes, we are constrained to consider only those of a W boson emitting outward. For the process with a pseudoscalar meson in the final state, the decay width is obtained Here a 1 ≡ C 1 + C 2 /3.
For the process with a vector meson in the final state, the decay width is obtained as Note that in the above equations, q 2 = m 2 is understood, where m is the mass of the meson.     All the corresponding results are collected in Tables XV to XX. Some comments are given in subsection III F.

E. SU(3) symmetry for semi-leptonic decays
According to the flavor SU(3) symmetry, there exist the following relations among these semileptonic decay widths [7], which can also be readily rederived using the overlapping factors given in Table II: • bb sector • bc sector with the c quark decay • bc sector with the b quark decay • bc ′ sector with the c quark decay • bc ′ sector with the b quark decay Also, we have compared the predictions of the light-front approach with those of SU(3) symmetry method taking cc and bb sectors as examples, which can be seen in Tables XXI and XXII. Some comments are given in order.
• Note that, 1/2 to 3/2 process has completely the same SU(3) relations as the corresponding 1/2 to 1/2 case. This can be expected, because spin-3/2 baryon shares the same flavor wave function as the corresponding spin-1/2 baryon.
• SU(3) predictions for the corresponding two channels in bc and bc ′ sectors have completely the same form, as can be explained by the facts that they have the same final states and the formally fixed initial states as in Eqs. (15) and (16).  • We can see from where Q = c/b. And also, note that, SU(3) symmetry breaking in c quark decay is usually larger than that in b quark decay.
For a comparison, we also list the results in Ref. [41]. Some comments will be given on the results of semi-leptonic and nonleptonic decays.
• Since there exist large uncertainties in the lifetimes, we have also presented the results for decay widths.
• We find that the result for 1/2 to 3/2 process is roughly one order of magnitude smaller than the corresponding 1/2 to 1/2 case except for the B ′ bc decays. Both the c quark and b quark decays of B ′ bc baryons are comparable to the corresponding 1/2 to 1/2 cases.
Some comments are given in order.
• Eq. (27) is also adopted for Ξ ++ cc → Σ + c transitions for this time. In our previous work Ref. [4], the following fit formulas were adopted However, only a few percent is changed in Eqs. (47) compared with our previous results in Ref. [4].
• It can be seen that, the variation in these parameters may cause a sizable change in the decay width, but the order of magnitude will not change.

IV. CONCLUSIONS
In our previous work, we have performed the calculation of doubly heavy baryon weak decays for 1/2 to 1/2 case. As a continuation, we investigate the 1/2 to 3/2 case in this work. Lightfront approach under the diquark picture is once again adopted to extract the form factors. In Ref. [54], the same method was used to study the bottom and charm baryon decays and reasonable results were obtained. The extracted form factors are then applied to predict the decay widths of simi-leptonic and nonleptonic decays. We find that the result for 1/2 to 3/2 case is roughly one order of magnitude smaller than the corresponding 1/2 to 1/2 case except for the B ′ bc decays. For B ′ bc baryons, both the c quark and b quark decays are comparable to the corresponding 1/2 to 1/2 cases. SU(3) symmetry and sources of SU(3) symmetry breaking for semi-leptonic decays are discussed. The error estimates are also performed.
It should be noted that the decay branching ratio is proportional to the lifetime of the initial baryon. However, as we have pointed out in Ref. [4], there exist large uncertainties in the lifetimes of these doubly heavy baryons. Our future work will aim to fix this problem.