Semileptonic decays of $\Lambda_c^+$ in dynamical approaches

We study the semileptonic decays of $\Lambda_c^+ \to \Lambda(n)\ell^+ \nu_{\ell}$ in two relativistic dynamical approaches of the light-front constituent quark model (LFCQM) and MIT bag model (MBM). By considering the Fermi statistic between quarks and determining spin-flavor structures in baryons along with the helicity formalism in the two different dynamical models, we calculate the branching ratios (${\cal B}$s) and averaged asymmetry parameters ($\alpha$s) in the decays. Explicitly, we find that ${\cal B}( \Lambda_c^+ \to \Lambda e^+ \nu_{e})=(3.43\pm0.57,3.48)\%$ and ${\alpha}( \Lambda_c^+ \to \Lambda e^+ \nu_{e})=(-0.96\pm0.03,-0.83)$ in (LFCQM, MBM), in comparison with the data of ${\cal B}( \Lambda_c^+ \to \Lambda e^+ \nu_{e})=(3.6\pm0.4)\%$ and ${\alpha}( \Lambda_c^+ \to \Lambda e^+ \nu_{e})=-0.86\pm 0.04$ given in the Particle Data Group, respectively. We also predict that ${\cal B}( \Lambda_c^+ \to n e^+ \nu_{e})=(2.15\pm0.41, 2.55)\times 10^{-3}$ and ${\alpha}( \Lambda_c^+ \to n e^+ \nu_{e})=(-0.97\pm0.01,-0.85)$ in (LFCQM, MBM), which could be observed by the ongoing experiments at BESIII, LHCb and BELLEII.

There have been recently many works discussing the anti-triplet charm baryon decays.
Because of the complicated structures of these baryons with large non-perturbative effects of the quantum chromodynamic (QCD), it is very hard to calculate the decay amplitudes from first principles. In the literature, people use the flavor symmetry of SU(3) f to analyze various charmed baryon decay processes, such as semi-leptonic, two-body and three-body non-leptonic decays, to obtain reliable results . However, the SU(3) f symmetry is an approximate symmetry, resulting in about 10% error for the predictions naturally. In order to more precision calculations, we need a dynamical QCD model to understand each process. To avoid other complicated problem like the non-factorizable effect, we only discuss the semi-leptonic processes, which are purely factorizable ones. In particular, we focus on the Λ + c semi-leptonic decays in this work. There are several theoretical analyses and lattice QCD calculations on the charmed baryon semi-leptonic decays with different models in the literature [26][27][28][29][30][31]. In this paper, we will mainly use the light-front (LF) formalism to study the decays and check the results in the MIT bag model (MBM) as comparisons.
The LF formalism is considered as a consistent relativistic approach, which has been very successful in the mesonic and light quark sectors [32,33]. Due to this success, it has been extended to other systems, such as those involving the heavy mesons, pentaquarks and so on [34][35][36][37][38][39][40][41][42][43][44][45][46][47][48]. In addition, the bottom baryon to charmed baryon nonleptonic decays in the LF approach have been done in Refs. [49,50]. For a review on the non-perturbative nature in the equation of motion and QCD vacuum structure for the LF constituent quark model (LFCQM), one can refer to the article in Ref. [32]. The advantage of LFCQM is that the commutativity of the LF Hamiltonian and boost generators provide us with a good convenience to calculate the wave-function in different inertial frames because of the recoil effect. In addition, since the AdS/CFT correspondence [51] was proposed by Juan Maldacena in the late of 1997, the LF holography as a feature of the AdS/CFT duality has brought the LF QCD from a phenomenological theory to a more fundamental one [52]. This paper is organized as follows. We present our formal calculations of the baryonic transition form factors for LFCQM and MBM in Secs. II and III, respectively. We show our numerical results of the form factors, branching ratios and averaged asymmetry parameters in Sec. IV. We also compare our results with those in the literature. In Sec. V, we give our discussions and conclusions.

A. Vertex function of baryon
In LFCQM, a baryon with its momentum P and spin S as well the z-direction projection of S z are considered as a bound state of three constitute quarks. As a result, the baryon state can be expressed by [32,33,42,[53][54][55] |B, P, S, where Ψ SSz (p 1 ,p 2 ,p 3 , λ 1 , λ 2 , λ 3 ) is the vertex function, which can be formally solved from Bethe-Salpeter equations by the Faddeev decomposition method, C αβγ and F abc are the color and flavor factors, λ i andp i with i = 1, 2, 3 are the LF helicities and 3-momentum of the on-mass-shell constituent quarks, defined as and To describe the internal motion of the constituent quarks, we introduce the kinematic variables (q ⊥ , ξ) and (Q ⊥ , η) and P tot , given by where (q ⊥ , ξ) characterize the relative motion between the first and second quarks, while (Q ⊥ , η) the third quark and other two quarks. The invariant masses of (q ⊥ , ξ) and (Q ⊥ , η) systems are represented by [33] respectively. Unlike Refs. [54,55] or Ref. [53], which treat the diquark as a point like object or spectator, we consider the three constituent quarks in the baryon independently with suitable quantum numbers satisfying Fermi statistics to have a correct baryon bound state system. The vertex function of Ψ SSz (p 1 ,p 2 ,p 3 , λ 1 , λ 2 , λ 3 ) in Eq. (2) can be written as [32,33,56] Ψ SSz (p 1 ,p 2 ,p 3 , λ 1 , λ 2 , where Φ(q ⊥ , ξ, Q ⊥ , η) is the momentum distribution of constituent quarks and Ξ SSz (λ 1 , λ 2 , λ 3 ) represents the momentum-depended spin wave function, given by with 1 2 s 1 , 1 2 s 2 , 1 2 s 3 SS z the usual SU(2) Clebsch-Gordan coefficient, and R i the well-known Melosh transformation, which corresponds to the ith constituent quark and can be expressed by and where σ i is the Pauli matrix and n = (0, 0, 1). This is the generalization of the Melosh transformation from two-particle systems, which can be derived from the transformation property of angular momentum operators [33,57]. We further represent the LF kinematic variables (ξ, q ⊥ ) and (η, Q ⊥ ) in the forms of the ordinary 3-momenta q and Q: to get more clear physical pictures of the momentum distribution wave functions.
It is known that the exact momentum wave function cannot be solved from the QCD first principle currently due to the lake of knowledge in the QCD effective potential in the threebody system. Hence, we choose the phenomenological Gaussian type wave function with suitable shape parameters to including the diquark clustering effects in Λ + c and Λ baryons [33,53]. The baryon spin-flavor-momentum wave function F abc Ψ SSz (p 1 ,p 2 ,p 3 , λ 1 , λ 2 , λ 3 ) should be totally symmetric under any permutations of quarks to keep the Fermi statistics. The spin-flavor-momentum wave functions of Λ + c , Λ and the neutron are given by respectively, where and φ 1(2) has the form by replacing (q, Q) with (q 1(2) , Q 1(2) ) in φ 3 , with N = 2(2π) 3 (β q β Q π) −3/2 and β q,Q being the normalized constant and shape parameters, respectively. Explicitly, q 1 (2) and Q 1(2) are given by Here, the baryon state is normalized as resulting in the normalization of the momentum wave function, given by We emphasize that the momentum wave functions of φ i with the different shape parameters of β q and β Q describe the scalar diquark effect in Λ (c) . For the neutrons, the momentum distribution functions are the same, i.e. φ = φ 3 (β q = β Q ), for any spin-flavor state due to the isospin symmetry. Note that there is no SU(6) spin-flavor symmetry in Λ (c) even though the forms of these states are similar to those with the SU(6) spin-flavor wave functions.

B. Transition form factors
The baryonic transition form factors of the V − A weak current are defined by where σ µν = i 2 [γ µ , γ ν ] and P ′ − P = k. We choose the frame such that P + is conserved (k + = 0, k 2 = −k 2 ⊥ ) to calculate the form factors to avoid other x + -ordered diagrams in the LF formalism [33]. The Matrix elements of the vector and axial-vector currents at quark level correspond to three different lowest-order Feynman diagrams as shown in Fig. 1.
Since the spin-flavor-momentum wave functions of baryons are totally symmetric under the permutation of quarks, we have that (a) + (b) + (c) = 3(a) = 3(b) = 3(c) [33]. As an illustration, we only present the calculations for the diagram (c), which contains simpler Feynman diagrams for the baryonic weak transitions at the lowest order, where the sign and cleaner forms with the notation (q ⊥ , Q ⊥ , ξ, η). We can extract the form factors from the matrix elements through the relations Note that f 3 and g 3 cannot be obtained when k + = 0, but they are negligible because of the suppressions of the k 2 factors . With the help of the momentum distribution functions and the Melosh transformation matrix, the transition matrix elements can be expressed as Using Eqs. (18), (19) and (20), we find that

III. BARYONIC TRANSITION FORM FACTORS IN MBM
The formalism and other details for MBM can be found in Ref. [31]. In the calculation of MBM, we take the same notations as those in Ref. [31]. In this approach, the current quark masses are used, given by where R corresponds to the bag size. Note that the form factors can be only evaluated at k = 0 (k 2 = ∆M 2 ) due to the assumption of the static bag. The form factors are decomposed as follows: where A is the normalized factor for the baryon, corresponding to the baryon spin-flavor structures as given by Table. II in Ref. [31], W q ± are associated with the normalized factors for quarks, given by with ω q representing the quark energy, and I and J stand for the overlapping factors for the quark wave functions, defined by with j n the Bessel function and x q 0 the lowest root of the transcendental equation of

IV. NUMERICAL RESULTS
In section II, we have derived the baryonic transition form factors in LFCQM. The form factors can be evaluated only in the space-like region (k 2 = −k 2 ⊥ ) because of the condition k + = 0. Thus, we follow the standard procedures in Refs. [41,42,54] to extract the information of the form factors in the time-like region. These procedures have widely been tested and discussed in the mesonic sector [58,59]. We fit f 1(2) (k 2 ) and g 1(2) (k 2 ) with some analytic functions in the space-like region, which are analytically continued to the physical time-like region (k 2 > 0). We employ the numerical values of the constituent quark masses and shape parameters in Table. I. The values of the shape parameters can be determined approximately by the calculations in the mesonic sectors [53,60]. By assuming that the Coulomb-like potential is dominant in the quark-quark strong interaction, one can deduce the shape parameter of quark pairs to be √ 2 greater than those fro the mesonic sectors because the interaction is about twice stronger between the quark-quark pair than quarkanti-quark one [53]. Since the reciprocals of the shape parameters are related to the sizes of systems, we adopt β qΛc ≃ 2( √ 2β ud ) and β qΛ ≃ 1.2( √ 2β ud ), where the factors of 2 and 1.2 come from the effects of the diquark clusterings, respectively, which make the light quark pairs to be more compact. By using Eqs.
In the Λ + c → Λ transition, we choose p 1 = 0, p 2 = 0 for f 1,2 and g 1 , but only c q 1 = 0 for g 2 . On the other hand, in the Λ + c → n transition, we take p 1 = p 2 = 0 for all form factors to fit the numerical values in the space-like region. We present our fitting results in Table. II.
For MBM, we use the Lorentzian type functions for the k 2 dependences of the form factors, given by where  Table. III.  In order to calculate the decay branching ratios and other physical quantities, we introduce the the helicity amplitudes of H V (A) λ 2 λ W , which give more intuitive physical pictures and simpler expressions when discussing the asymmetries of the decay processes, such as the integrated (averaged) asymmetry, also known as the longitudinal polarization of the daughter baryon. Relations between the helicity amplitudes and form factors are given by where We note that both f 3 and g 3 have been set to be 0 in LFCQM.
The differential decay width and asymmetries can be expressed in the analytic forms in terms of the helicity amplitudes, which can be found in our previous work of Ref. [22]. In our numerical calculations, we use the center value of τ Λ + c = 203.5 × 10 −15 s in Eq. (1) [1]. Our predictions of the decay branching ratios (Brs) and asymmetries (αs) are listed in Table. IV. In Table. V, we compare our results with the experimental data and those in various calculations in the literature. In LF [30] and HQET [26], the authors use a specific spin-flavor structure of c(ud−du)χ ρ 3 sz for the charmed baryon state, in which only the permutation relation is considered between light quarks. In addition, they assume that the diquarks from the light quark pairs are spectator and structureless. These simplifications in Refs.
[30] and [26] make their results to be not good compared with the experimental data as shown in Table. V. Based on the Fermi statistics, the overall spin-flavor-momentum structures are determined, from which the parameters like quark masses, baryon masses and shape parameters can recover the spin-flavor symmetry. In LFCQM, we consider the different diquark clustering effect in different baryons. We expect that this effect is stronger if the mass of the third quark is greater than others, which is encoded in the shape parameter of β qB . There is an interesting observation that the shape parameters β QB and β qB in our study are almost the same in each baryon, which implies the totally symmetric momentum distribution of three constituent quarks in the baryon. In addition, the flavor symmetry breaking effect due to the quark masses seems to get canceled due to the clustering effect of the shape parameters in the momentum distribution functions. Our numerical results indicate that the form factors follow the Lorentzian functions of F (k 2 ) = F (0)/(1 − q 1 k 2 + q 2 k 4 ) except g 2 (k 2 ) in the Λ + c → Λ processes. Our results of f i (k 2 ) = g i (k 2 ) show that the lowest order of the heavy quark symmetry is failure because the constituent charm quark mass is not heavy enough.
For MBM, Although the semi-leptonic processes have been fully studied in Ref. [31], their results are mismatched with the current data. By using the same formalism with the same input parameters, we are able to get the same values of the form factors at the zero recoil point. By taking the Lorentzian k 2 dependences for the form factors, inspired from our LF calculations, we obtain much better results as shown in Table. V. It is interesting to see that our results for Λ + c → ne + ν e are consistent with other calculations except those from SU(3) F and LQCD.

V. CONCLUSIONS
We have studied the semi-leptonic decays of Λ + c → Λ(n)ℓ + ν ℓ in the two dynamical approaches of LFCQM and MBM. We have used the Fermi statistics to determine the overall spin-flavor-momentum structures and recover the spin-flavor symmetry with the quark and baryon masses and shape parameters. We have found that B(Λ + c → Λe + ν e ) = (3.43±0.57)% and 3.48% in LFCQM and MBM, which are consistent with the experimental data of (3.6 ± 0.4) × 10 −2 [2] as well as the values predicted by SU(3) F [22], LQCD [27,28] and RQM [29], but about a factor of two larger than those in HQET [26] and LF [30]. We have also obtained that α(Λ + c → Λe + ν e ) = (−0.96 ± 0.03) and −0.83 in LFCQM and MBM, which are lower and higher than but acceptable by the experimental data of −0.86 ±0.04 [2], respectively. We have also predicted that B(Λ + c → ne + ν e ) = (2.15 ± 0.41, 2.55) × 10 −3 and α(Λ + c → ne + ν e ) = (−0.97 ± 0.01, −0.85) in (LFCQM, MBM), in which our results of B(Λ + c → ne + ν e ) in LFCQM and MBM as well as that in RQM [29] are consistent with each other, but about two times smaller than those in SU(3) F [22] and LQCD [27,28]. It is clear that our predicted values for the decay branching ratio and asymmetry in Λ + c → ne + ν e could be tested in the ongoing experiments at BESIII, LHCb and BELLEII. Finally, we re-mark that our calculations in LFCQM and MBM can be also extended to the other charmed baryons, such as Ξ + c , Ξ 0 c , and even b baryons.