Color-allowed Bottom Baryon to Charmed Baryon non-leptonic Decays

We study color allowed $\Lambda_b\to \Lambda^{(*,**)}_c M^-$, $\Xi_b\to\Xi_c^{(**)} M^-$ and $\Omega_b\to\Omega^{(*)}_c M^-$ decays with $M=\pi, K,\rho, K^*, D, D_s, D^*, D^*_s, a_1$, $\Lambda^{(*,**)}_c=\Lambda_c$, $\Lambda_c(2595)$, $\Lambda_c(2765)$, $\Lambda_c(2940)$, $\Xi_c^{(**)}=\Xi_c, \Xi_c(2790)$ and $\Omega^{(*)}_c=\Omega_c, \Omega_c(3090)$, in this work. There are three types of transitions, namely ${\cal B}_b({\bf \bar 3_f},1/2^+)$ to ${\cal B}_c({\bf \bar 3_f},1/2^+)$, ${\cal B}_b({\bf 6_f},1/2^+)$ to ${\cal B}_c({\bf 6_f},1/2^+)$ and ${\cal B}_b({\bf \bar 3_f},1/2^+)$ to ${\cal B}_c({\bf \bar 3_f},1/2^-)$ transitions. The bottom baryon to charmed baryon form factors are calculated using the light-front quark model. Decay rates and up-down asymmetries are predicted using na\"{i}ve factorization and can be checked experimentally. We find that in ${\cal B}_b\to {\cal B}_cP$ decays, rates in type~(ii) transition are smaller than those in type (i) transition, but similar to those in type (iii) transition, while in ${\cal B}_b\to{\cal B}_c V, {\cal B}_c A$ decays, rates in type~(ii) transition are much smaller than those in type (i) transition and are also smaller than those in type (iii) transition. For the up-down asymmetries, the signs are mostly negative, except for those in the type (ii) transition. All of these asymmetries are large in sizes. The study on these decay modes may shed light on the quantum numbers of some of the charmed baryons as the decays depend on the bottom baryon to charmed baryon form factors, which are sensitive to the configurations of the final state charmed baryons.

Among low lying singly bottom baryons, only Λ b , Ξ b and Ω b decay weakly [1]. Several color allowed Λ b → Λ c P decay rates with P = π, K, D, D s were reported by LHCb in year 2014 [12][13][14]. We expect more to come in the near future. It will be interesting and timely to study weak decays of singly bottom baryons to final states involving singly charmed baryons. In general, baryon decays are complicate processes. Nevertheless, when the transition only involve the heavy quarks, namely b → c transition, while the light quarks are spectating, the decay processes are easier. Accordingly we will study color allowed Λ

II. SPECTROSCOPY OF SINGLY CHARMED AND BOTTOM BARYONS
In this section we briefly review the spectroscopy of singly charmed and bottom baryons. Our discussion follows closely to those in [31,32]. The singly charmed or bottom baryon is composed of a charmed quark or a bottom quark and two light quarks. We will discuss the allowed quantum numbers for the light quark system before the brief review.
A. Allowed quantum numbers for the light quark system From Fermi statistics the wave function of the light quarks needs to be antisymmetry under permutation. As the charm or bottom quark is a color triplet 3 c , the diquark system, consists of the two light quarks, can only be an anti-color triplet3 c state, which is anti-symmetric (denoted as (3 c ) A ) under permutation of the two light quarks. The remaining part of the diquark wave function consists of must be symmetry under permutation. The spin of the light quarks can be in a symmetric triplet state (3 sp ) S (S l = 1) or an antisymmetric singlet state (1 sp ) A (S l = 0). Under permutation, the spin wave function picks up an phase factor ψ(spin) → (−) S l +1 ψ(spin). ( Given that each light quark is a triplet of the flavor SU (3) with N f = 3, 6 for3 f , 6 f , respectively. In the quark model, the orbital angular momentum of the light diquark can be decomposed into L = L k + L K , where L k is the orbital angular momentum between the two light quarks and L K the orbital angular momentum between the diquark (the light quark pair) and the heavy quark. Roughly speaking, we have where R n is the radial wave function, Y lm is the spherical harmonics, k is basically the relative momentum of the two light quarks and K is the relative momentum of the heavy quark and the diquark system. 3 In the above equation, we do not show explicitly the Clebsch-Gordan coefficient 2 We have followed the Particle Data Group's convention [1] to use a prime to distinguish the iso-doublet in the 6 f from the one in the3 f . 3 Explicitly, we have k µ ≡ (p 1 − p 2 ) µ − (p 1 + p 2 ) µ [(p 1 + p 2 ) · (p 1 − p 2 )/(p 1 + p 2 ) 2 ] and K µ ≡ (p 1 + p 2 − p 3 ) µ − P µ [P · (p 1 + p 2 − p 3 )/P 2 ] with p 1 , p 2 and p 3 the momenta of the light quarks and the heavy quark, respectively, and P ≡ p 1 + p 2 + p 3 . Note the above constructions in K and k are to make sure that in the rest frame of the light-quark system and in the whole baryon system, we have k = (0, k) and K = (0, K), respectively. I: Allowed quantum number of the diquark system satisfying Fermi statistics. S [qq] = L k + S qq is the angular momentum of the diquark system, without taking into account the orbital momentum of the Q − [qq] system ( L K ). The parity eigenvalues of the diquark [qq] system and the Qqq system are given by (−) L k and (−) L k +L K , respectively. ψ(color) L K L k ψ(flavor) ψ(spin) S [qq] parity([qq]) parity(Qqq) (3 c ) A even even (3 f ) A (1 sp ) A L k + + (3 c ) A even even (6 f ) S (3 sp ) S |L k − 1|, · · · , L k + 1 (3 c ) A odd even (6 f ) S (3 sp ) S |L k − 1|, · · · , L k + 1 + − and the Wigner rotation (see later discussion), as the rest frame of the whole system and the rest frame of the diquark system are not identical. Nevertheless the above wave function can still be used as an book keeping devise for working out the allowed quantum numbers. The angular momentum of the diquark system, without taking into account the orbital momentum of the Q − [qq] system, is with S [qq] given by |L k − S qq |, . . . , L k + S qq . Note that S qq is the spin of the light quark pair without taking into account the orbital momentum between them. The combination of S [qq] is better when viewing the diquark as a sub-system, i.e. one may have scalar diquark, axial-vector diquark and so on. The angular momentum of the diquark system, with the orbital momentum of the Q − [qq] system, is with J l given by |S [qq] − L k |, . . . , S [qq] + L k . Consequently, the total angular momentum is Under permutation of the light quark momenta, p 1 ↔ p 2 , we have k → − k and K → K, while under party we have k → − k and K → − K. Consequently, using the well known symmetry property of Y lm , under the permutation, the space part wave function, see Eq. (4), transforms as while under parity, it transforms as The parity eigenvalues of the [qq] diquark and the whole Qqq systems are given by (−) L k and (−) L k +L K , respectively.
Putting all of these together, under permutation of the light quarks, we have Fermi statistics requires the wave function to be antisymmetric giving the following constraint: The quantum numbers of all possible allowed configurations of the diquark system satisfying the Fermi statistic are shown in Table I. The corresponding parity eigenvalues of the diquark and the heavy baryons are also shown.

B. Charmed Baryons
The observed mass spectra and decay widths of charmed baryons are summarized in Table II. The J P quantum numbers of Λ + c , Λ c (2595) + , Λ c (2860) + , Λ c (2880) + , Λ c (2940) + and Σ c (2455), are determined up to different levels of certainty, while the J P quantum numbers given in Table II other states are either from quark model predictions or totally undetermined. In fact, there are 16 states out of 40 states in Table II having unknown quantum numbers.
In Table III configurations with L k +L K = 0, 1, 2 for charmed baryons are shown. The quantum number assignments are from Tables I and II, while those with ( †) are taken from ref. [4]. Only several multiplets are well established. These include the , respectively. Ref. [4] makes further suggestions on the classification on some other states. As noted previously PDG and LHCb assign Λ c (2940) + as a 3 2 − state [1,2], while the authors of ref. [4] take it as a 1 2 − state. We follow the suggestions of ref. [4] on the quantum numbers of Λ c (2940) + and some other states. Note that other quantum number assignments on the newly observed Ω ( * , * * ) c states, such as those avocated in refs. [5][6][7][8][9][10], are not shown in the table.
From Table III we see that there are plenty of states in the L k + L K = 0, 1, 2 sector to be discovered.

C. Bottom Baryons
The observed mass spectra and decay widths of bottom baryons are summarized in Table IV. Note that except Ξ b (5935) − and Ξ b (5955) − other J P quantum numbers given in Table IV are unmeasured. One has to rely on the quark model to determine the J P assignments.
In Table V configurations with L k + L K = 0, 1, 2 for charmed baryons are shown. The quantum number assignments are from Tables I and IV. Only the J P = 1 2 +3 f multiplet with states: , is established. Several miltiplets are to be completed with the yet to be discovered states, such as Σ 0 b , Σ * 0 b , Ξ b (5935) 0 and so on. From Table V we see that there are plenty of states in the L k + L K = 0, 1, 2 sector to be discovered.
As shown in Table IV and Ω b are the few singly bottom baryons that decay weakly. We will study their decay modes in this work. In particular, Λ 0 b → Λ     ≡ L k + S qq , J l ≡ S [qq] + L K and J ≡ J l + S Q , which are the angular momenta of the diquark system, the light-degree of freedom and the whole baryon, respectively. The quantum number assignments are from Tables I and II, while those with ( †) are taken from [4]. There are different assignments of the quantum number of Λ c (2940), see text for more details. There are plenty of states to be discovered.  [1], except those of Ξ b (6227) − , which are from [33].
excited s-wave state, Λ c (2940) a radial excited p-wave state and Ω c (3090) a radial excited s-wave state. The study on these B b → B c transitions may shed light on the quantum numbers of these charmed baryons.

III. FORM FACTORS IN THE LIGHT-FRONT APPROACH
We consider a heavy baryon consisting a heavy quark Q and a scalar isosinglet diquark [qq] or an axial-vector isovector diquark [qq]. In the light-front approach, the baryon bound state with the total momentum P and spin J can be written as (see, for example [34,35]) where S [qq] is the spin of the diquark, L K is the orbital angular momentum of the Q − [qq] system, J l is the total angular momentum of the light degree of freedom, n is the quantum number of the wave-function (see later), α, β, γ and b, c are color and flavor indices, respectively, λ i denotes helicity, p 1 and p 2 are the on-mass-shell light-front momenta, and  Tables I and IV. There are plenty of states to be discovered.
with λ 2 = S 2 = 0 for scalar diquark and λ 2 = 0, ±1 and S 2 = 1 for axial vector diquark . The coefficient C αβγ is a normalized color factor and F bc is a normalized flavor coefficient, obeying the Note that type (i) and (iii) transitions involve scalar diquarks, while type (ii) transitions involve axial-vector diquarks. Type (iii) has odd parity baryons in the final states. The quantum number assignments are from Tables III and V, while those with ( †) are taken from ref. [4]. The asterisks indicate that the baryons in the final states are radial excited.
Tthe momenta can be defined in terms of the light-front relative momentum variables, (x i , k i⊥ ) for i = 1, 2, The momentum-space wave-function Ψ JJz nL K S [qq] J l can be expressed as where φ nL K Lz (x 1 , x 2 , k 1⊥ , k 2⊥ ) describes the momentum distribution of the constituents in the bound state, J J ; m m |J J ; Jm is the Clebsch-Gordan coefficients and λ i |R † M (p + 1 , p 1⊥ , m i )|s i is the well normalized Melosh transform matrix element. We will return to these quantities later.
We normalize the state as consequently, φ nLLz (x, p ⊥ ) satisfies the following orthonormal condition, The wave function is defined as with where are proportional to the spherical harmonics Y 1Lz in momentum space, and ϕ ns and ϕ np are the distribution amplitudes of s-wave and p-wave states, respectively. For a Gaussian-like wave function, one has (the first two are from refs. [34,35]) Under the constraint of Now we turn to the Melosh transform. For the heavy quark part, we have [36,37], with u (D) , a Dirac spinor in the light-front (instant) form. For the diquark part, if it is a scalar diquark the Melosh transform is a trivial one, i.e.
but if it is a axial vector diquark, the Melosh transform is more interesting, where ε LF and ε I are polarization vectors in light-front and instant forms, respectively. Note that we have u D (k, s) = u(k, λ) λ|R † M |s and ε I (k, s) = ε LF (k, λ) λ|R † M |s . Consequently, the state |Q(k, λ) λ|R † M |s and |[qq](k, λ) λ|R † M |s transforms like |Q(k, s) and |[qq](k, s) , respectively, under rotation, i.e. their transformation do not depend on their momentum. A crucial feature of the light-front formulation of a bound state, such as the one shown in Eq. (12), is the frameindependence of the light-front wave function [36,38]. Namely, the hadron can be boosted to any (physical) (P + , P ⊥ ) without affecting the internal variables (x i , k ⊥i ) of the wave function, which is certainly not the case in the instant-form formulation.
In practice it is more convenient to use the covariant form for Ψ 1/2Jz with Γ s00 = 1, for baryon states with a S 2 = 0 or S 2 = 1 diquark. The derivation of the above results can be found in Appendix A. Note that Γ s00 agrees with the one in ref. [25], while Γ s11 and Γ p01 are new results and Γ s11 is different from those in ref. [26,29,39], which have Γ s11 proportional to γ 5 ε * LF (p 2 , λ 2 ), instead.
It should be remarked that in the conventional LF approachP = p 1 + p 2 is not equal to the baryon's four-momentum as all constituents are on-shell and consequently u(P , S z ) is not equal to u(P, S z ); they satisfy different equations of motions ( P −M 0 )u(P , S z ) = 0 and ( P −M )u(P, S z ) = 0. This is similar to the case of a vector meson bound state where the polarization vectors ε(P , S z ) and ε(P, S z ) are different and satisfy different equations ε(P , S z ) ·P = 0 and ε(P, S z ) · P = 0 [40]. Although u(P , S z ) is different than u(P, S z ), they satisfy the relation followed from γ + γ + = 0,P + = P + ,P ⊥ = P ⊥ . This is again in analogy with the case of ε(P , ±1) = ε(P, ±1).
Note that the normalization of state, Eq. (18), implies To verify it we note that the right-hand-side of Eq. (31) is a matrix element of a 2 × 2 hermitian matrix. Hence, it's value can be extracted by taking traces with unit and sigma matrices, giving and where we have made use of the following identities in the above equations, Eqs. (32) and (33) are non-trivial requirements and we check that using Γ s00 , Γ s11 and Γ p01 in Eq. (29) and φ nL K in Eqs. (21) and (22), the above relations are indeed satisfied. 4 The Feynman diagram for a typical B b → B c transition, is shown in Fig. 1. For the B b (1/2 + ) → B c (1/2 + ) transition, the matrix element can be parameterized as Armed with the light-front quark model description of |B b (P, J z ) in the previous subsection, we are ready to calculate the weak transition matrix element of heavy baryons. For where the scalar or axial-vector diquark is denoted by a dashed line and the corresponding V − A current vertex by X. (37) where the diquark acts as an spectator and with Γ L K S [qq] J l given in Eq. (29). As in [15,34,41], we consider the q + = 0, q ⊥ = 0 case. We follow [15,41] to project out various form factors from the above transition matrix elements (see Appendix B for details). The results are given below.

B. Form factors for
where we follow ref. [4] to take Λ c (2765) + as a radial excited s-wave state. In these transitions the scalar diquarks are spectators.
We obtain the following transition form factors for type (i) transition: For the transition with low laying final state (n = 1), the above equations are similar to those obtained in ref. [15] and are identical to those in ref. [25]. [4] to consider Ω c (3090) 0 as a radial excited s-wave state. In these transitions the axial-vector diquarks are spectators.
We obtain the following transition form factors for type (ii) transition: where we have

D. Form factors for
where we follow ref. [4] to consider Λ c (2940) + as a radial excited p-wave state. In these transitions, the scalar diquarks are spectators.
We obtain the following transition form factors for type (iii) transition: where we have and Note that we have k 1⊥ − k 1⊥ = x 2 q ⊥ and q 2 = −q 2 ⊥ . The above formulas of the form factors are new results.

IV. NUMERICAL RESULTS
In this section we will show the numerical results of various B b → B c transition form factors using formulas obtained in Sec. III. We then proceed to estimate the decay rates and up-down , m q and β's (in units of GeV) appearing in the Gaussiantype wave function (22). (The superscript S and A mean scalar and axial vector, repestively.) The constituent quark and diquark masses are taken from ref. [42].

A. B b → B c form factors
The input parameters m [qq ] , m q , β are summarized in Table VII. The constituent quark and diquark masses are taken from ref. [42]. For the diquark masses, we use m S for B b → B c transitions. The parameters a, b and F (0) are first determined in the spacelike region. We then employ this parametrization to determine the physical form factors at q 2 ≥ 0. The parameters a, b are expected to be of order O(1). As we shall see this is usually true in our numerical results. Occasionally some as and bs are larger than O(1), but in most of these cases the corresponding form factors are small and do not have much impact on decay rates. The B b (3 f , 1/2 + ) → B c (3 f , 1/2 + ) transition form factors f V 1,2 (q 2 ) and g A 1,2 (q 2 ) are given in Table VIII and are plotted in Fig. 2. These include the form factors for Λ b → Λ c , Λ c (2765) and Ξ b → Ξ c transitions. In this case, we have f V 1 , g A 1 > 0 and f V 2 , g A 2 < 0. We see that |f V 1 | and |g A 1 | are larger than |f V 2 | and |g A 2 | in these transitions. Note that except |f V 2 |, the Λ b → Λ c (2765) transition form factors have smaller sizes comparing to those in the other two transitions. This is reasonable, since Λ c (2765) is a radial excited state. The configurations of the final states in excited state differ from those in the low lying states and larger mis-match between initial and final state configurations, usually lead to smaller form factors.
The B b (6 f , 1/2 + ) → B c (6 f , 1/2 + ) transition form factors f V 1,2 (q 2 ) and g A 1,2 (q 2 ) are given in Table IX and are plotted in Fig. 3. These includes the form factors for Ω b → Ω c and Ω c (3090) transitions. In this case, we have f V 1 , f V 2 > 0, g A 1 and g A 2 < 0. We see that |f V 1 | and |f V 2 | are larger than |g A 1 | and |g A 2 | in these transitions. Note that except g A 2 , the Ω b → Ω c (2940) transition form factors have smaller sizes comparing to those in the Ω b → Ω c transition. This is reasonable, since we take Ω c (2940) as a radial excited state. Larger mis-match between initial and final state configurations, usually lead to smaller form factors. The transition form factors f A 1,2 (q 2 ) and g V 1,2 (q 2 ) are given in Table X and are plotted in Fig. 4. These includes the form factors for Λ b → Λ c (2595), Λ c (2940) and Ξ b → Ξ c (2790) transitions. In this case, we have f A 1 , g V 1 > 0 and f A 2 , g V 2 < 0. The signs of the form factors are identical to those in the B b (3 f , 1/2 + ) → B c (3 f , 1/2 + ) case. The transitions in this case have p-wave final state baryons. In the previous two cases, the initial and final state baryons belong to the same categories [B b (3 f , 1/2 + ) or B c (6 f , 1/2 + )], while in this case they are in different categories, the initial state is a s-wave baryon, but the final state is a p-wave baryon. We see that some of these form factors behavior rather differently from the previous ones. For example, as shown in Fig. 4, f A 1 (q 2 ) are almost independent of q 2 , which are different from the f V 1 (q 2 ) in the previous cases. Furthermore, all four form factors are of similar sizes in this case, while in the previous cases either one or two form factors are much smaller than the others. Note that the transition form factors of Λ b → Λ c (2940) are similar to those in Λ b → Λ c (2595), even though Λ c (2940) is a radial excited p-wave state. This feature is also different from the two previous cases, where form factors involving radial excited states are usually smaller in sizes.

B. B b → B c M decay rates and up-down asymmetries
Under the factorization approximation, the decay amplitudes for color-allowed B b → B c M − decays are given by where V cb,ij are the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements and a 1 is the effective color-allowed Wilson coefficient. In naïve factorization a 1 is given by c 1 + c 2 /N c with c 1 = 1.081 and c 2 = −0.190 at the scale of µ = 4.2 GeV [45]. The matrix element B c |V µ − A µ |B b is given by Eqs. (35) and (36), while M − (q i q j )|V µ − A µ |0 for M = P, V, A (with P , V and A stand for pseudoscalar, vector and axial vector mesons, respectively) are given by where f P,V,A are the corresponding decay constants. In type (i) and (ii) transitions [B b (3 f , 1/2 + ) → B c (3 f , 1/2 + ) and B b (6 f , 1/2 + ) → B c (6 f , 1/2 + ) transitions], the decay amplitudes are given by [17] with , one simply replaces f V i and g A i in the above equations by −f A i and −g V i , respectively. The decay rates and asymmetries read [17,47] with κ ≡ p c /(E + M ), where p c is the momentum in the center of mass frame. All hadron masses and life-times are taken from PDG [1]. The CKM matrix elements are taken from the latest results of the CKM fitter group [48]. Type The branching ratios are given in unit of 10 −3 . The asterisks in the first column indicate that the baryons in the final states are radial excited. work the decay rates are estimated using the naïve factorization approach. Note that in ref. [46] using QCD factorization the authors obtained |a 1 (B → DP )| = 1.055 +0.019 −0.017 − (0.013 +0.011 −0.006 )α P 1 with α π 1 = 0 and |α K 1 | < 1 [see Eq. (230) in [46]]. The |a 1 (DP )| agrees with the naïve factorization value (ref. [46] used a LO 1 = 1.025) within few %. For estimations, we assign 10% uncertainty in the effective Wilson coefficient a 1 and 10% uncertainty in form factors. Note that in B b → B c P decays, in principle one needs f 3 and g 3 contributions, see Eq. (60). Since these contributions are suppressed by a m 2 P /(M + M ) 2 factor compared to the f 1 and g 1 terms and f 3 , g 3 are expected to be vanishing in the heavy quark limit [25], we shall neglect them, but enlarge the form factor uncertainties to 15% in B b → B c D and B b → B c D s decays.
Note that as shown in refs. [18,19] non-factorizable contributions to B b → B c P non-leptonic decay amplitudes can contribute as large as 30% comparing to the factorized ones. A precise estimation of non-factorization contributions is beyond the scope of the present work. 5 If needed, one can scale up the uncertainties of our numerical results on rates.
The branching ratios for B b → B c P , B c V and B c A decays, with P = π, K, D, D s , and are summarized in Tables XI and XII. As shown in Table XI the Λ b → Λ c P rates can reasonably reproduce the data within errors. We see that the Λ b → Λ c π and Λ b → Λ c K rates prefer lower values, while the Λ b → Λ c D and Λ b → Λ c D s rates prefer higher values. Branching ratios for other modes are predictions. We find that for Λ b decays, we have the following pattern in the decay rates: The first two decays are of type (i) while Λ c (2765) is a radial excited state, and the last two decays are of type (iii) transitions is a radial excited state. We see that rates of type (i) transitions are greater than those of type (iii) transitions, and decay rates involving excited states are smaller within the same type. These are reasonable as the configurations of the final states in excited s-wave B c (3 f , 1/2 + ) state and low lying or excited p-wave B c (3 f , 1/2 − ) states differ from those in the low lying s-wave B b,c (3 f , 1/2 + ) states. Larger mis-match between initial and final state configurations, usually lead to smaller form factors, and, consequently, smaller rates.
The Ξ b → Ξ c P modes are of type (i) decays, while Ξ b → Ξ c (2790)P decays are of type (iii) decays, where Ξ c (2790) is a p-wave baryon. From Table XI we have We see again that rates of type (i) transitions are greater than those of type (iii) transitions. Note that the Ξ b → Ξ c P rate is slightly smaller than the Λ b → Λ c P rate.
For Ω b decays, we have These decays are type (ii) decays [B b (6 f , 1/2 + ) → B c (6 f , 1/2 + ) transitions] and Ω c (3090) is a radial excited state. Again the decay rate involving an excite state is smaller. Note that in B b → B c P decays, rates in type (ii) transition are smaller than those in type (i) transition, but similar to those in type (iii) transition. The branching ratios for the weak decays Table XII. We find that for Λ b decays, except for V = ρ − , we have the following pattern in the decay rates: For the case of V = ρ − , we have Finally for Ω b decays, we have and Ω b → Ω c M decays are compared. The branching ratios are given in the unit of 10 −3 . These are to be compared to the experimental branching ratios for Λ b → Λ c π − , Λ c K − , Λ c D − , Λ c D − s decays, which are 4.9 ± 0.4, 0.359 ± 0.030, 0.46 ± 0.06 and 11.0 ± 1.0 in unit of 10 −3 , respectively. See text for the results in ref. [17].

Mode
This     ) and Ω c (3090) are radial excited states. Larger mis-match between initial and final state configurations, usually lead to smaller rates. Note that in B b → B c V, B c A decays, rates in type (ii) transition are much smaller than those in type (i) transition and are also smaller than those in type (iii) transition.
In Tables XIII, we compare our results on the branching ratios of Λ b → Λ c M , Ξ b → Ξ c M and Ω b → Ω c M decays to those obtained in other works. Note that in the table the results of ref. [17] are obtained by using Table II in [17] with a 1 1, while for the B b → B c V rates the numerics are corrected by a factor of two, see footnote 7 in [47]. Overall speaking our results agree reasonably well with most of the results obtained in other works. Note that in Ω b → Ω c M − decays, the predicted rates are in general smaller than those obtained in other works, except that the predicted Br(Ω b → Ω c π − ) is close to the one in ref. [30].
In Tables XIV and XV,

Type
Mode     In Tables XVI, we compare our results on the up-down asymmetries of Λ b → Λ c M , Ξ b → Ξ c M and Ω b → Ω c M decays to those obtained in other works. Our results agree well in signs and magnitudes of the asymmetries with those in other works, except in Ω b → Ω c D − s , Ω c ρ − , Ω c D * − s decays the predicted asymmetries are larger than those in ref. [17], but nevertheless the signs agree. The predictions on rates and asymmetries presented in Tables XI, XII, XIV and XV can be verified experimentally. These information may shed light on the quantum numbers of Λ c (2765), Λ c (2940) and Ω c (3090).
We began with a brief overview of the charmed and bottom baryon spectroscopy and discussed their possible structure and J P assignment in the quark model. As a working assumption we follow ref. [4] to assign the quantum numbers of some singled charmed states. It is known that among low lying singly bottom baryons, only Λ b , Ξ b and Ω b decay weakly. Consequently, we study Λ b → Λ For the up-down asymmetries, the signs are mostly negative, except for those in the B b (6 f , 1/2 + ) → B c (6 f , 1/2 + ) [type (ii)] transition. These asymmetries are large in sizes. Note that in type (i) and (ii) cases, the asymmetries |α(B b → B c D * (s) )| are smaller than |α(B b → B c ρ)|, |α(B b → B c K * )| and |α(B b → B c a − )|.
We compare our results of rates and asymmetries of Λ b → Λ c M , Ξ b → Ξ c M and Ω b → Ω c M decays to existing results in other works. In most cases the agreements are reasonably well.
Predictions on rates and asymmetries can be checked experimentally. The study on these decay modes may shed light on the quantum numbers of the charmed baryons, as the decays depend on bottom baryon to charmed baryon form factors, which are sensitive to the configurations of the final state charmed baryons. This work can be further extended by including transitions having spin-3/2 baryons in the final states. The result will be reported elsewhere.