Semileptonic weak decays of anti-triplet charmed baryons in the light-front formalism

We systematically study the semileptonic decays of ${\bf B_c} \to {\bf B_n}\ell^+ \nu_{\ell}$ in the light-front constituent quark model, where ${\bf B_c}$ represent the anti-triplet charmed baryons of $(\Xi_c^0,\Xi_c^+,\Lambda_c^+)$ and ${\bf B_n}$ correspond to the octet ones. We determine the spin-flavor structures of the constituents in the baryons with the Fermi statistics and calculate the decay branching ratios (${\cal B}$s) and averaged asymmetry parameters ($\alpha$s) with the helicity formalism. In particular, we find that ${\cal B}( \Lambda_c^+ \to \Lambda e^+ \nu_{e}, ne^+ \nu_{e})=(3.55\pm1.04, 0.36\pm0.15)\%$, ${\cal B}( \Xi_c^+ \to \Xi^0 e^+ \nu_{e},\Sigma^0 e^+ \nu_{e},\Lambda e^+ \nu_{e})=(11.3\pm3.35), 0.33\pm0.09,0.12\pm0.04\%$ and ${\cal B}( \Xi_c^0 \to \Xi^- e^+ \nu_{e},\Sigma^- e^+ \nu_{e})=(3.49\pm0.95,0.22\pm0.06)\%$. Our results agree with the current experimental data. Our prediction for ${\cal B}( \Lambda_c^+ \to n e^+ \nu_{e})$ is consistent with those in the literature, which can be measured by the charm-facilities, such as BESIII and BELLE. Some of our results for the $\Xi_c^{+(0)}$ semileptonic channels can be tested by the experiments at BELLE as well as the ongoing ones at LHCb and BELLEII.


I. INTRODUCTION
In the recent few years, there have been two important experiments in charmed baryon physics. One is the measurement of the absolute branching ratios of B(Ξ 0 c → Ξ − π + ) = (1.8± 0.5)% [1] and B(Ξ + c → Ξ − π + π + ) = (2.86±1.21±0.38)% [2] from the Belle Collaboration, and the other is the most precise measurement of the Ξ 0 c 's lifetime of τ Ξ 0 c = 154.5±1.7±1.6±1.0 fs from the LHCb Collaboration [3], which significantly deviates from the past world averaged value of τ Ξ 0 c = 112 +13 −10 fs in PDG [4]. Both of them bring us new hints as well as new problems in charm physics. We are now in a precision-era of charm physics. It is expected that as more high quality data will be accumulated in the future, stronger constraints on various baryonic QCD models as well as physics beyond the standard model can be given.
Recently, the anti-triplet charm baryon decays have been extensively discussed in the literature. However, due to the large non-perturbative effects from the quantum chromodynamics (QCD), the decay amplitudes and observables of hadrons are very hard to obtain from the QCD first principle. To avoid the difficulties, various charmed baryon decay processes have been studied based on the flavor symmetry of SU(3) f , such as semi-leptonic, two-body and three-body non-leptonic decays, to get reliable results [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24]. It is known that SU(3) f is an approximate symmetry, resulting in about 10% of uncertainties for the predictions inherently. Moreover, the SU(3) f symmetry itself does not directly reveal any clue about the QCD dynamics. In fact, its applications heavily rely on the experimental data as inputs. In order to have precise calculations, we need a specific dynamical QCD model to understand each decay process. For simplicity, we only discuss the semi-leptonic processes of the anti-triplet charmed baryons in this study, which involve purely factorizable contributions. There are several theoretical calculations on these decay processes with different QCD frameworks in the literature [25][26][27][28][29][30][31][32][33][34].
The light front (LF) QCD formalism in the quark model is a consistent relativistic approach, which has been tested successfully in the mesonic and light quark sectors in early times [35,36]. Because of these successes, it has been used in other generalized systems, such as those containing the heavy mesons, pentaquarks and so on [37][38][39][40][41][42][43][44]. Apart from the charm system, the bottom to charmed baryon nonleptonic decays have been recently analyzed in the LF formalism [45]. For a review on the comprehensive introduction of the LF QCD and its vacuum structure, one can refer to Ref. [35]. For the LF constituent quark model (LFCQM), we recommend Ref. [36] and references therein. The advantage of LFCQM is that we can boost the reference frame without changing the equation of motion because of the commutativity of the LF Hamiltonian and boost generators. It provides us with a great convenience to calculate the wave-function in different inertial frames because of the recoil effects in the transition form factors. This paper is organized as follows. We first present our formal calculations of the branching ratios and averaged decay asymmetries in terms of the helicity amplitudes, the baryonic states in LFCQM, and the baryonic transition form factors in Sec. II. In Sec. III, we show our numerical results and compare them with those in the literature. In Sec. IV, we give our conclusions.

A. Helicity Amplitudes and Observables
The effective Hamiltonian for the anti-triplet charmed baryon semi-leptonic weak decays can be written as where G F is the Fermi constant, V cq is the CKM matrix element, and (lν ℓ ) V −A and (qc) The weak transition amplitudes of the anti-triplet charmed baryons are given by time-like form factors, we use LFCQM, in which a baryon is treated as a bound state of three constituent quarks quantized in the LF formalism and its state is denoted by the momentum P , canonical spin S and the z-direction projection of spin S z , respectively. As a result, the baryon state can be expressed by [35,36,39,[46][47][48] |B, P, S, S z = {d 3p }2(2π) 3 where Ψ SSz (p 1 ,p 2 ,p 3 , λ 1 , λ 2 , λ 3 ) is the vertex function describing the overlapping between the baryon and its constituents, which can be formally solved from the three-body Bethe-Salpeter equations, C αβγ (F abc ) are the color (flavor) factors, λ i andp i with i = 1, 2, 3 are the LF helicities and 3-momenta of the on-mass-shell constituent quarks, defined as and |q a α (p, λ) and {d 3p } correspond to the light front constituent quark states and the integral measure, given by respectively, with the quark field operators satisfied the following anti-commutation relations To separate the internal motion of the constituents from the bulk motion, we use the kinematic variables of (q ⊥ , ξ), (Q ⊥ , η) and P tot , given by where (q ⊥ , ξ) and (Q ⊥ , η) capture the relative motions between the first and second quarks, and the third and other two quarks, respectively. We consider the three constituent quarks in the baryon independently with suitable spin-flavor structures satisfying the 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. (7) can be further written into two parts [35,36,49], 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-dependent spin wave function, given by with the SU(2) Clebsch-Gordan coefficients of 1 2 s 1 , 1 2 s 2 , 1 2 s 3 SS z , and R i is the Melosh transformation matrix [36,50], which corresponds to the ith constituent quark, expressed by Here, σ stands for the Pauli matrix, n = (0, 0, 1), and M and M 3 are invariant masses of (q ⊥ , ξ) and (Q ⊥ , η) systems, represented by [36] respectively.
The spin-flavor structures of B c and B n are given by where is the momentum distribution of the constituents with the corresponding spin-flavor configuration, and In principle, one could solve φ (i) from the Bethe-Salpeter equation with an explicit QCDinspired potential but it is beyond the scope of this paper. Nonetheless, we use a Gaussian type distribution with the phenomenological shape parameters β Q and β q to describe the relative motions of constituents. Consequently, we represent the LF kinematic variables (ξ, q ⊥ ) and (η, Q ⊥ ) in the forms of ordinary 3-momenta q = (q ⊥ , q z ) and Q = (Q ⊥ , Q z ): where N is the normalization constant. Since the one-particle baryonic state is normalized the normalization condition of the momentum wave function is given by In this paper, we take different shape parameters of β q and β Q in the momentum wave functions φ i to describe the scalar diquark effects in B c . On the other hand, we assume the momentum-distribution function φ of octet baryons B n is flavor symmetric for all constituents. In other words, the SU(3) f flavor symmetry is hold in the momentum wave function of B n . As a result, the shape parameters of φ are equal, i.e. β QBn = β qBn = β Bn .
Note that there is no SU(6) spin-flavor symmetry in B c and B n because of the momentumdependent Melosh transformation even though the forms of these states are similar to those with the SU(6) spin-flavor wave functions.

C. Transition form factors
We pick theqγ + (1 − γ 5 )c current or so-called "good component" of the baryon transition amplitudes and choose the frame such that p + i(f ) is conserved (k + = 0, k 2 = −k 2 ⊥ ) to calculate the form factors to avoid zero-mode contributions and other x + -ordered diagrams in the LF formalism [35,36]. 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 constituents, we have that (a) + (b) + (c) = 3(a) = 3(b) = 3(c) [36]. We only present the calculation for the diagram (c), which contains simpler and cleaner forms with our notation (q ⊥ , Q ⊥ , ξ, η) as a demonstration. We can extract the form factors from the matrix elements through the relations which do not contribute to the semileptonic decays in the massless lepton limit [19]. As a result, we can safely set both f 3 and g 3 to be 0 in this study. With the help of the momentum distribution functions and Melosh transformation matrix in Eq. (7), the transition matrix elements can be written as where q ′ ⊥ = q ⊥ and Q ′ ⊥ = Q ⊥ + k ⊥ . Using Eqs. (22) and (23), we find that

III. NUMERICAL RESULTS
To find out the decay branching ratios and averaged asymmetries in the helicity formalism,  We fit f 1(2) (k 2 ) and g 1(2) (k 2 ) with the analytic functions in the space-like region with the following form 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 cases [46,53]. Because the strength of the quark-quark pairs is a half of the quark-anti-quark one [46], we will get the shape parameters of the quark pairs, which are approximately √ 2 smaller than those in the mesonic cases.
We adopt β qΛc ≃ 2(β ud / √ 2) and β qΞc ≃ 2(β sū(d) / √ 2), where the factor of 2 comes from the effects of the diquark clustering, making the light quark pairs to be more compact in B c baryons. For the octet baryons of B n , we assume that the SU(3) f flavor symmetry is hold, and therefore, the shape parameters is flavor symmetric for each constituent, i.e. β Q = β q . As a result, we approximate the shape parameters of the octet baryons equal to the mesonic ones by effectively treating any pair of two constituents as a heavier anti-quark.  Tables II, III and IV, and our predictions of the branching ratios and asymmetry parameters in Table V. The comparisons with different   theoretical models are presented in Tables VI, VII [1,2]. Our branching ratios are also consistent with the predictions of the relativistic quark model (RQM) [28,29], the covariant constituent quark model (CCQM) [30] and the SU(3) F approach [19]. The results given by the LF formalism [33] and heavy quark effective theory (HQET) [25] are half of ours because they choose the spin-flavor wave function of B c to be c(q 1 q 2 − q 2 q 1 )χ ρ 3 sz instead of the totally symmetric one. The averaged asymmetry parameter predicted by LFCQM in Λ + c → Λe + ν e is 10% lower than the data. Since we do not consider any parameters about spin interactions  [28,29] have considered a comprehensive QCD-inspired potential including the chromomagnetic effect, their results are more close to the experimental values than ours. Meanwhile, the SU(3) f approach is a model independent way to analyze B c → B n ℓ + ν ℓ decays, which automatically includes the information related to the spin interaction when the asymmetry parameters are used as the fitting inputs. Clearly, our results of the asymmetry parameters could be improved by considering the full QCD potential and its solutions.

IV. CONCLUSIONS
We have systematically studied the semi-leptonic decays of B c → B n ℓ + ν ℓ in LFCQM.
By requiring the constituents in the baryonic states to obey the Fermi statistics, we are able to determine the overall spin-flavor-momentum structures of the baryons. We assume that the momentum distribution of B n is symmetric in flavor indices and any pair of two quarks can be effectively treated as a heavier anti-quark. We have found that B(Λ + c → Λe + ν e ) = (3.55 ± 1.04)%, B(Ξ + c → Ξ 0 e + ν e ) = (11.3 ± 3.35)% and B(Ξ 0 c → Ξ − e + ν e ) = (3.49 ± 0.95)% in LFCQM, which are consistent with the experimental data of (3.6±0.4)×10 −2 [4], (6.6 +3.7 −3.5 )× 10 −2 [2,4] and (1.8 ± 1.2) × 10 −2 [1,4] as well as the values predicted by SU(3) F [19], LQCD [26,27], RQM [28], and CCQM [30], but twice larger than those in HQET [25] and LF [33]. We have also obtained that α(Λ + c → Λe + ν e ) = (−0.97 ± 0.03) in LFCQM, which is 10% lower than the experimental data of −0.86 ± 0.04 [4]. The reason of this deviation may arise from the QCD spin-spin interacting effects, which are not included in our phenomenological wave functions. Our results of the averaged decay asymmetries could be improved if we consider the wave function solved from the full QCD potential. It is clear that our predicted values for the decay branching ratios and asymmetries in Ξ +(0) c → B n e + ν e could be tested in the ongoing experiments at LHCb and BELLEII.