Weak Decays of Heavy Baryons in Light-Front Approach

In this work, we perform a analysis of semi-leptonic and nonleptonic weak decays of heavy baryons: $\Lambda_{b},\Xi_{b},\Omega_{b}$ and $\Lambda_{c},\Xi_{c},\Omega_{c}$. For nonleptonic decay modes, we study only the factorizable channels induced by the external W-emission. The two spectator quarks in baryonic transitions are treated as a diquark and form factors are calculated in the light-front approach. Using the results for form factors, we also calculate some corresponding semi-leptonic and nonleptonic decay widths. We find that our results are comparable with the available experimental data and other theoretical predictions. Decay branching fractions for many channels are found to reach the level $10^{-3}\sim10^{-2}$, which are promising to be discovered in the future measurements at BESIII, LHCb and BelleII. The SU(3) symmetry in semi-leptonic decays is examined and sources of symmetry breaking are discussed.

Inspired by this discovery, we also expect a renaissance in the study of singly bottom or charm baryons. Particularly there are rapid progresses in the study of Λ c decays at BESIII [22][23][24][25][26][27] and some recent studies on Λ b and Λ c decays by LHCb can be found in Refs. [28][29][30][31][32][33]. It is anticipated that many more decay modes will be established in future. Thus an up-to-date theoretical analysis is highly demanded, and this work aims to do so. Here the "mixed symmetric (antisymmetric)" means the state is symmetric (antisymmetric) under interchange of the first two quarks. The wave functions for baryons in the initial and final states are collected in the Appendix A.
On the theoretical side, the singly heavy baryon decays have been investigated by various theoretical methods, and some of them can be found in Refs. . In this work, we will adopt the light-front approach. This method has been widely used to study the properties of mesons [57][58][59][60][61][62][63][64][65][66][67][68][69][70][71][72][73][74]. Its application to baryons can be found in Refs. [75][76][77][78]. In the transition form factors, the two spectator quarks do not change and can be viewed as a diquark. In this diquark scheme, the two quarks are treated as a whole system, and thus its role is similar to that of the antiquark in the meson case, see Fig. 2. In the process like Λ b → Λ c , where the light quarks u and d are considered to form a scalar diquark, which is denoted by [ud], while in the process like Ω b → Ω c , the light s quarks are believed to form an axial-vector diquark, which is denoted by {ss}.
Some recent works have been devoted to investigate the singly heavy baryon decays with the help of flavor SU(3) symmetry [79][80][81][82]. Based on the available data, the SU(3) analysis can give predictions on a great number of decay modes ranging from semi-leptonic decays to multi-body 1 is the initial (final) quark momentum, p 2 is the diquark momentum and the cross mark denotes the corresponding vertex of weak interaction. nonleptonic decays. However, as we know, in the case of c quark decay, SU(3) symmetry breaking effects are 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 the wave function overlapping factors are also given. Our results are shown in Sec. III, including the results for form factors, predictions on semi-leptonic and nonleptonic decay widths, and detailed discussions on the SU(3) symmetry and sources of symmetry breaking. A brief summary will be given in the last section.

II. THEORETICAL FRAMEWORK
In this section, we will briefly overview the theoretical framework for form factors: the lightfront approach. More details can be found in Refs. [75] and [4]. It is necessary to point out that the physical form factor should be multiplied by a factor due to the overlap of wave funtions in the initial and final states.
for a scalar diquark involved in the initial and final baryons, or if an axial-vector diquark is involved.

B. Spin and flavor wave functions
In the last subsection, we have presented the explicit expressions of form factors. It should be noted that, the physical transition form factor should be multiplied by the corresponding overlapping factor: From the discussions in the Appendix A, we can obtain these factors and the corresponding results are collected in Tab. I.
The shape parameters are given as Some remarks on the above parameters are given in order.
• m [ud] = 0.50GeV and m {ud} = 0.77GeV are taken from Refs. [75,77]. Other diquark masses are taken as the above values since the s quark mass is expected to be 0.1 GeV higher than that of u or d quark.

B. Form factors
Results for form factors are collected in Tab. V for charmed baryons and Tab. VI for bottomed baryons. The following expressions have been used to access the q 2 distribution: where the F (0) is the form factor at q 2 = 0. The m fit and δ are two parameters to be fitted from numerical results. For the form factor g 2 , a plus sign is adopted in Eq. (8) otherwise the fitted parameter m fit becomes purely imaginary. The minus sign is adopted for all the other situations.
Some comments are given in order.
• Only the scalar diquark contributes to the Λ Q and Ξ Q decays and only the axial-vector diquark contributes to the Ω Q decays, where Q = c/b.
• It should be noted that, in Tabs. V and VI, the overlapping factors are not taken into account.
The physical transition form factor should be multiplied by the corresponding overlapping factor, see Eq. (6).
• An advantage of the results in Tab. V is that, they can be directly be used to explore SU (3) symmetry and its breaking effects. In fact, if we take the approximations all the form factors will be the same. From the results in Tab. V, we can see that the SU (3) symmetry is not severely broken.
In Tab. IV, we compare our results with other theoretical predictions in Refs. [39,84,85].
Some comments are given as follows.
• It can be seen that, our results are comparable to other predictions. However, there still exists an uncertainty about the sign of g 2 (0). The sign of g 2 (0) in this work is same as that derived by LCSR method in Ref. [85] but different from those obtained by other quark models. More careful analysis should be devoted to fixing this problem.  • The form factors f 3 and g 3 are not obtained in this work because we have taken the q + = 0 frame, while another method adopted in Refs. [86,87] may be applied to extract these form factors.
• Also note that, in Refs. [39,84,85], only few channels are investigated but this work aims to give a comprehensive investigation to the heavy baryon decays. Only in this way, SU (3) symmetry and sources of SU(3) symmetry breaking can be seen clearly.

C. Semi-leptonic results
The differential decay width for semi-leptonic process reads with the polarized decay widths are given as Here the q 2 is the lepton pair invariant mass, p = is the mass of the parent (daughter) baryon.
The helicity amplitudes are related to the form factors through the following expressions: the ones with asterisk, and the minus sign is adopted for all the others.
The negative helicity amplitudes are given as The helicity amplitudes for the left-handed current are obtained as Numerical results are given in Tabs. VII and IX. Comparisons with some recent works [51-53, 80, 81, 84, 88] and the experimental results [83] can be found in Tabs. VIII and X.

D. Non-leptonic results
For nonleptonic decays, we are constrained to consider only the processes of a W boson emitting outward. The naive factorization assumption is employed [89,90]. The decay width for the B → B ′ P (P denotes a pseudoscalar meson) is given as where E (E ′ ) is the energy of the meson (daughter baryon) in the final state, and A, B, A 1,2 and B 1,2 are given as Here λ = G F √ 2 V CKM V * q 1 q 2 a 1 with a 1 = C 1 (µ c ) + C 2 (µ c )/3 = 1.07 [91], the first CKM matrix element corresponds to the process of B → B ′ and the second comes from the production of the meson.
M (M ′ ) is the mass of the parent (daughter) baryon and m is the mass of the emitted meson.
For the decay mode with an axial-vector meson involved, f V should be replaced by −f A in the expressions of A 1,2 and B 1,2 in Eqs. (17).
Note that the P-wave meson a 1 emission case is included. The naive factorization can still work for these processes [92].
The masses of the mesons in the final states can be taken from Ref. [83]. The decay constants are adopted as [61,74,93] The numerical results are given in Tabs. XI, XIII and XIV. Comparisons with some recent works [54,81] and the experimental results [83] can be found in Tab. XII and Tab. XV.

E. SU(3) analysis for semi-leptonic decays
From the overlapping factors above, we would expect the following relations for c-baryon sector and for b-baryon sector, if the flavor SU(3) symmetry is respected. These relations for the charmed baryons are consistent with those in Refs. [79,80], while the ones for the bottomed baryons, as far as we know, are first derived by this work.
In the following, we will investigate the SU (3) • If we have considered the differences of CKM and the overlapping factors between these two channels but take all the other parameters as the same, we get the precise SU(3) symmetry . This prediction is also obtained in Refs. [79,80].
• If we consider only the difference of daughter baryon mass but take all the other parameters as the same, we get a ratio 0.538. It means that SU(3) symmetry is broken by about 50% between these two modes. The more accurate number is 35% (see Tab. XVI), when all the other relevant impacts are taken into account.
We can see from Tabs. XVI and XVII: • The SU(3) symmetry breaking is sizable for c-baryon decays while it is small for the b-baryon decays. This can be understood due to a much smaller phase space in c-baryon decays, and thus the decay width significantly depends on the mass differences of the baryons in the initial and final states.
• SU(3) symmetry is broken more severely in the c → s processes than in the c → d processes because of the larger mass of s quark than u and d quarks. The typical value of SU (3) symmetry breaking for c → s processes is 35% while that for c → d processes is 15%.
• All the form factors are not very sensitive to the diquark mass m di .
• g 2 is one order of magnitude smaller than the other form factors, and it is sensitive to β i and β f , while f 1 , f 2 and g 1 are still not very sensitive to these shape parameters.
• It can be seen from Eq. (22)  where (q 1 , q 2 ) = (u, d), (u, s), (d, s) for Λ + c and Ξ +,0 c , respectively. The wave functions of the baryon octet in the final states in the standard flavor-spin basis are given as follows.

Wave functions in the diquark basis
From the coupling of two angular momenta j 1 = 1 and j 2 = 1 2 , we know that So, it is natural to define the baryon state with an axial-vector diquark as follows.
Equipped with above expressions, the baryon wave functions in the initial and final states in the diquark basis can be derived as follows.
For Σ 0 and Λ, we have