Charmed Ωc\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega _c$$\end{document} weak decays into Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega $$\end{document} in the light-front quark model

More than ten Ωc0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega _c^0$$\end{document} weak decay modes have been measured with the branching fractions relative to that of Ωc0→Ω-π+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega ^0_c\rightarrow \Omega ^-\pi ^+$$\end{document}. In order to extract the absolute branching fractions, the study of Ωc0→Ω-π+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega ^0_c\rightarrow \Omega ^-\pi ^+$$\end{document} is needed. In this work, we predict Bπ≡B(Ωc0→Ω-π+)=(5.1±0.7)×10-3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {B}}}_\pi \equiv {{\mathcal {B}}}(\Omega _c^0\rightarrow \Omega ^-\pi ^+)=(5.1\pm 0.7)\times 10^{-3}$$\end{document} with the Ωc0→Ω-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega _c^0\rightarrow \Omega ^-$$\end{document} transition form factors calculated in the light-front quark model. We also predict Bρ≡B(Ωc0→Ω-ρ+)=(14.4±0.4)×10-3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {B}}}_\rho \equiv {{\mathcal {B}}}(\Omega _c^0\rightarrow \Omega ^-\rho ^+)=(14.4\pm 0.4)\times 10^{-3}$$\end{document} and Be≡B(Ωc0→Ω-e+νe)=(5.4±0.2)×10-3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {B}}}_e\equiv {{\mathcal {B}}}(\Omega _c^0\rightarrow \Omega ^-e^+\nu _e)=(5.4\pm 0.2)\times 10^{-3}$$\end{document}. The previous values for Bρ/Bπ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {B}}}_\rho /{{\mathcal {B}}}_\pi $$\end{document} have been found to deviate from the most recent observation. Nonetheless, our Bρ/Bπ=2.8±0.4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {B}}}_\rho /{{\mathcal {B}}}_\pi =2.8\pm 0.4$$\end{document} is able to alleviate the deviation. Moreover, we obtain Be/Bπ=1.1±0.2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {B}}}_e/{{\mathcal {B}}}_\pi =1.1\pm 0.2$$\end{document}, which is consistent with the current data.

One still manages to measure more than ten 0 c decays, such as 0 c → − ρ + , 0K ( * )0 and − + ν , but with the branching fractions relative to B( 0 c → − π + ) [5]. To extract the absolute branching fractions, the study of a e-mail: yukuohsiao@gmail.com b e-mail: yangling@ihep.ac.cn c e-mail: cclih@phys.nthu.edu.tw d e-mail: shangyuu@gmail.com (corresponding author) 0 c → − π + is crucial. Fortunately, the 0 c → − π + decay involves a simple topology, which benefits its theoretical exploration. In Fig. 1a, 0 c → − π + is depicted to proceed through the 0 c → − transition, while π + is produced from the external W -boson emission. Since it is a Cabibboallowed process with V * cs V ud 1, a larger branching fraction is promising for measurements. Furthermore, it can be seen that 0 c → − π + has a similar configuration to those of 0 c → − ρ + and 0 c → − + ν , as drawn in Fig. 1, indicating that the three 0 c decays are all associated with the 0 c → − transition. While is a decuplet baryon that consists of the totally symmetric identical quarks sss, behaving as a spin-3/2 particle, the form factors of the 0 c → − transition can be more complicated, which hinders the calculation for the decays. As a result, a careful investigation that relates 0 c → − π + , − ρ + and 0 c → − + ν has not been given yet, despite the fact that the topology associates them together.

The light-front quark model
The baryon bound state B (c) contains three quarks q 1 , q 2 and q 3 , with the subscript c for q 1 = c. Moreover, q 2 and q 3 are combined as a diquark state q [2,3] , behaving as a scalar or axial-vector. Subsequently, the baryon bound state |B (c) (P, S, S z ) in the light-front quark model can be written as [31] where SS z is the momentum-space wave function, and ( p i , λ i ) stand for momentum and helicity of the constituent (di)quark, with i = 1, 2 for q 1 and q [2,3] , respectively. The tilde notations represent that the quantities are in the light-front frame, and one defines P = (P − , P + , P ⊥ ) and P = (P + , P ⊥ ), with P ± = P 0 ± P 3 and P ⊥ = (P 1 , P 2 ).
Besides,p i are given bỹ with where x and k ⊥ are the light-front relative momentum variables with k ⊥ from k = (k ⊥ , k z ), ensuring that P + = p + 1 + p + 2 and P ⊥ = p 1⊥ + p 2⊥ . According to e i ≡ m 2 i + k 2 and M 0 ≡ e 1 + e 2 in the Melosh transformation [30], we obtain Consequently, SS z can be given in the following representation [41][42][43][44][45]: where the vertex function S(A) is for the scalar (axial-vector) diquark in B c , and α A for the axial-vector diquark in B . We have used the variableP ≡ p 1 + p 2 to describe the internal motions of the constituent quarks in the baryon [32], which leads to (P μ γ μ − M 0 )u(P, S z ) = 0, different from (P μ γ μ − M)u(P, S z ) = 0. For the momentum distribution, φ(x, k ⊥ ) is presented as the Gaussian-type wave function, given by where β shapes the distribution.
Using |B c (P, S, S z ) and |B (P, S , S z ) from Eq. (6) and their components in Eqs. (10), (11) and (12), we derive the matrix elements of the B c → B transition in Eq. (4) as with Then, we multiply J 5 j (J 5 j ) by T ( T ) as F 5 j ≡ J 5 j · T andF 5 j ≡J 5 j · T with T and T in Eqs. (4) and (13), respectively, resulting in [45] In the connection of F 5 j =F 5 j , we construct four equations. By solving the four equations, the four form factors F V 1 , F V 2 , F V 3 and F V 4 can be extracted. The form factors F A i can be obtained in the same way.

Numerical analysis
In the Wolfenstein parameterization, the CKM matrix elements are adopted as V cs = V ud = 1 − λ 2 /2 with λ = 0.22453 ± 0.00044 [5]. We take the lifetime and mass of the 0 c baryon and the decay constants ( f π , f ρ ) = (132, 216) MeV from the PDG [5]. With (c 1 , c 2 ) = (1.26, −0.51) at the m c scale [47], we determine a 1 . In the generalized factorization, N c is taken as an effective color number with N c = (2, 3, ∞) [28,29,46,50], in order to estimate the non-factorizable effects. For the + c (css) → − (sss) transition form factors, the theoretical inputs of the quark masses and parameter β in Eq. (15) are given by [34,40] where Table 1. For the momentum dependence, we have used the double-pole parameterization: with m F = 1.86 GeV.
Using the theoretical inputs, we calculate the branching fractions, whose results are given in Table 2.

Discussions and conclusions
In Table 2, we present B π and B ρ with N c = (2, 3, ∞). The errors come from the form factors in Table 1, of which the uncertainties are correlated with the charm quark mass. By comparison, B π and B ρ are compatible with the values in Ref. [28]; however, an order of magnitude smaller than those in Refs. [20,22], whose values are obtained with the total decay widths π(ρ) = 2.09a 2 1 (11.34a 2 1 ) × 10 11 s −1 Table 2 Branching fractions of (non-)leptonic 0 c decays and their ratios, where The three numbers in the parenthesis correspond to N c = (2, 3, ∞), and the errors come from the uncertainties of the form factors in Table 1 B(R)
The helicity amplitudes can be used to better understand how the form factors contribute to the branching fractions. 0.05, which shows that H A 1 20 dominates B π , instead of H V 1 20 .
More specifically, it is the F A 4 term in H A 1 20 that gives the main contribution to the branching fraction. By contrast, the F A 1,3 terms in H A