Charmed baryon $\Omega_c^0 \rightarrow \Omega^- l^+ \nu_l$ and $\Omega_c^0 \rightarrow \Omega_c^- \pi^+ (\rho^+)$ decays in light cone sum rules

The semileptonic $\Omega_c^0 \rightarrow \Omega^- l \nu$ and non-leptonic $\Omega_c^0 \to \Omega^- \pi^+$, $\Omega_c^0 \to \Omega^- \rho^+$ decays of charmed $\Omega_c$ baryon are studied within the light cone sum rules. The form factors responsible for $\Omega_c \to \Omega$ transitions are calculated using the distribution amplitudes of $\Omega_c$ baryon. With the obtained form factors, the branching ratios of $\Omega_c^0 \to \Omega^- l^+ \nu_l$, $\Omega_c^0 \rightarrow \Omega^- \pi^+$, and $\Omega_c^0 \rightarrow \Omega^- \rho^+$ decays are estimated. The results are compared with Belle data as well as the findings of the other approaches.


I. INTRODUCTION
The semileptonic weak decays of hadrons represent a very promising class of decays. The study of semileptonic decays can provide us with useful information about the elements of the Cabibbo -Kobayashi-Maskawa (CKM) matrix. The investigation of these decays can play a crucial role in studying strong interaction, i.e., the form of the effective Hamiltonian. The decay amplitudes of semileptonic decays can be represented as a product of a well-understood leptonic current and a complicated hadronic current for describing the quark transitions. The hadronic part of the weak decays is usually parameterized in terms of form factors. The form factors belong to the nonperturbative region of QCD, hence some nonperturbative methods are needed to calculate them.
Among these methods, the QCD sum rules [1] occupies a special place. The advantage of this method is that it is based on the fundamental QCD Lagrangian.
The new measurement and the variation in the predictions of the branching fractions among different models need further attention. In the present work, we study the Ω 0 c → Ω − l + ν e decay within the light cone sum rules (LCSR) method (for more information about the LCSR method, see [9]). The paper is organized as follows. In Sec. II, the LCSR for the relevant form factors responsible for Ω 0 c → Ω − transitions are derived. Numerical analysis, including the results for the form factors and decay widths, is presented in Sec. III . The last section contains our conclusion.

II. FORM FACTORS FOR Ω c → Ω TRANSITION IN LCSR
To calculate the form factors for Ω 0 c → Ω − transition, we consider the following correlation function Here, the current J Ω µ = ǫ jkl s j T Cγ µ s k s l is the interpolating current of Ω baryon, )c is the current describing c → s transition and j, k and l are the color indices.
In LCSR, the correlation function is calculated both at hadronic and QCD level at the deep Euclidean region, i.e., p 2 << m 2 c , q 2 << m 2 c . Then, the results of the calculations for the two representations of the correlation function are matched by using the quark-hadron duality ansatz.
In result, the sum rules for the relevant form factors are derived.
Let us first calculate the correlation function from the hadronic side. Inserting the complete set of baryon states carrying the quantum numbers of Ω baryon and isolating the ground state for the correlation function, we obtain The first term of this matrix is defined where λ is the decay constant and u µ (p ′ ) is the Rarita-Schwinger spinor for spin-3/2 Ω baryon.
The second matrix element is parameterized in terms of eight transition form factor The summation over spins of Ω baryon is performed via the formula Before delving into the analysis, we would like to make the following two remarks.
• The current J Ω µ also couples with spin-1/2 negative parity baryons, i.e., where B denotes the negative parity baryon. Therefore, the structures with p ′ µ and γ µ terms also contain the contributions of spin-1/2 baryon. Using this fact, from Eq.(5) we find out that only the structure with g αβ term is free of spin-1/2 baryon contributions. In other words, the structure with g αβ term contains contributions coming only from spin-3/2 baryons.
• Note that not all the Lorentz structures are independent, and to overcome this problem, a specific order of Dirac matrices are chosen. In this work, we choose the structure γ µ/ qγ ν / p (γ µ/ qγ ν / pγ 5 ).
Taking into account these remarks, we obtain the correlation function from the hadronic part, where dots stand for the contributions of higher states and continuum, which denotes the contributions arising from quarks starting from some threshold s th value. In the following discussions, we will denote the momentum of Ω c baryon as p µ → m Ωc v µ where v µ is its velocity. Now let us turn our attention to the calculation of the correlation function from the QCD side with the help of the operator product expansion (OPE). Using the Wick theorem from Eq. (1) we get the correlation function where S(x) is the s-quark propagator, and T ν = γ ν (1 − γ 5 ). From Eq.(8), it follows that to calculate the correlation function from QCD side, we need the matrix element ǫ jkl 0|s j α (x)s k β (x)c l γ (0)|Ω c . This matrix element can be parametrized in terms of the heavy baryon distribution amplitudes (DAs). The DAs of the sextet baryons with quantum numbers J P = 1 2 + in the heavy quark mass limit is obtained in [10]. In this work, the DA's are classified by the total spin of two light quarks. If the polarization vector is parallel to the light cone plane, the matrix element ǫ jkl 0|q j 1α (t 1 )q k 2β (t 2 )Q l γ (0)|Ω Q (p) can be expressed in terms of the four DAs in the following way. where Here n µ = xµ vx ,n µ = 2v µ − 1 vx x µ ,v µ = xµ vx − v µ , and f (i) are the decay constants of Ω c baryon and ψ (i) are the distribution amplitudes. The Fourier transformation of the DAs are Ψ( where ω 1 and ω 2 are the momentum of two light quarks along the light-cone direction and their total momentum is ω = ω 1 + ω 2 with t 1 = x 1 n and t 2 = x 2 n. The DAs can be written as In our case since x 1 = x 2 , then we get Based on the heavy-quark symmetry, we can use the same DAs for the baryon containing charm quark, and b-quark. In [10], the DAs for Ω b baryon are obtained and we used the same DAs for Ω c in this work. where with C  Table I for completeness. For the numerical calculations we take A = 1/2. From Eq. (8), we obtain the following form using Eq. (9) After integrating over x, one can obtain the explicit expressions of the correlation function at QCD level. Separating the coefficients of the Lorentz structures (v µ/ qγ ν γ 5 , / qγ 5 v µ q ν , / qγ 5 v µ v ν , / qγ 5 g µν ,( / qγ ν v µ , / qv µ q ν , / qv µ v ν , / qg µν )) from both representations of the correlation function, we get the desired sum rules for the transition form factors F 1 (q 2 ), F 3 (q 2 ), F 2 (q 2 ) + F 3 (q 2 ), and F 4 (q 2 ) ((G 1 (q 2 ), G 3 (q 2 ), G 2 (q 2 ) + G 3 (q 2 )) and G 4 (q 2 )), respectively.
Here Π i are the invariant functions for the Lorentz structures mentioned above. In general, the invariant functions can be written in the following form, where ∆ = p ′ 2 −s(σ), σ = ω m Ωc , and i runs from 1 to 8 (each value of the i describes the corresponding structure). Since the explicit expressions of ρ i , and ρ (3) i are lengthy, we did not present them here. Applying Borel transformation with respect to the variable −(p − q) 2 , we get the sum rules for the form factors: Borel transformation and continuum subtraction is performed with the help of formula and Numerical analysis of the obtained sum rules for the form factors is carried out in the next section.
In these expressions Here m 1 and m 2 are the mass of Ω c and Ω baryon, respectively. The remaining helicity amplitudes can be obtained by symmetry relation.
From these helicity amplitudes the decay widths of the semileptonic and non-leptonic decays are calculated as and where Here, G F is the Fermi coupling constant and V ij are the elements of CKM. The factor a 1 = C 1 +C 2 /N c comes from the factorization [13], where N c is the color factor and C 1 = −0.25 and C 2 = 1.1 are the Wilson coefficients [11].

III. NUMERICAL ANALYSIS
In this section, we perform numerical analysis for the transition form factors obtained in the previous section. For this goal, first, we present the input parameters in our numerical analysis. f (1) = 0.093 ± 0.01 [14] f (2) = 0.093 ± 0.01 [14] (30) V cs = 0.97320 ± 0.00011 [2] V ud = 0.97401 ± 0.00011 [2]  Having determined the working intervals of the threshold and Borel mass parameters, the next problem is to find the best fitting for the form factors. The LCSR predictions for the form factors are not applicable for the whole physical region, m 2 l ≤ q 2 ≤ (m 2 Ωc − m Ω ) 2 but give reliable results for up to q 2 ≤ 0.5 GeV 2 region. Hence, we first obtained the form factors within QCD sum rules up to q 2 ≃ 0. Then, we extrapolated from the domain where LCSR predictions are reliable to the full physical region by applying the following z-expansion fit function [15,16]. where and t ± = (m Ωc ± m Ω ) 2 , t 0 = t + (1 − 1 − t − t + ). Here, m R,i are the corresponding masses of the resonances for the c → s transition in the spectrum, i.e., m Ds = 1.97 GeV.
The obtained parametrization that best reproduces the form factors predicted by the LCSR in the region q 2 ≤ 1.1 GeV 2 , is given in Table II. Note that a 0 corresponds to the form factor at q 2 = 0, i.e., a 0 = F i (q 2 = 0). To verify that the results of the form factors depend weakly on the chosen Exp. [2,8] Ref [4] Ref [18] Ref [19]  width expressions, we obtain the branching ratios that are presented in Table III.
Ω 0 c → Ω − transition had been studied in several models [4,7,11,18,19]. The obtained results are presented in Table III. Our results are close to the predictions of the light-front quark model [4] for the semileptonic part. However, there is a large discrepancy with other studies for these decays.
In addition, our theoretical predictions do not match with the recent BELLE II measurement [8], especially for the semileptonic Ω 0 c → Ω − l + ν l decay. A similar situation was also obtained in [4]. This discrepancy needs further discussion, and it may indicate the existence of new physics. On the other hand, an upper bound for R ρ/π > 1.3 is set by the experiment. This bound is not controversial with theoretical results; however, there is considerable discrepancy among the predictions of different theoretical approaches. Hopefully, more experimental and theoretical efforts on this transition will enlighten the discrepancy.

IV. CONCLUSION
In this work, the semileptonic and non-leptonic decays of charged decays of Ω 0 c , namely, Ω 0 c → Ω − l + ν l , Ω 0 c → Ω − π + and Ω 0 c → Ω − ρ + are studied within the framework of the light-cone sum rules by using the distribution amplitudes of Ω c baryon. In this study, DAs for Ω b baryons are used for their c-quark counterpart depending on the heavy-quark symmetry. We first calculated the transition form factors for Ω 0 c → Ω − decay in the LCSR method. Then, using the obtained results for the transition form factors, we predicted the branching ratios of the semileptonic Ω 0 c → Ω − l + ν l (where l = e, µ) and non-leptonic Ω 0 c → Ω − π + (ρ + ) decays. Finally, we compared our predictions with other approaches as well as recent BELLE II results. We obtained that our results, especially on branching ratio for semileptonic Ω 0 c → Ω − l + ν l decay normalized to Ω 0 c → Ω − π + is considerably smaller than existing experimental data. A similar discrepancy was also obtained in the light-front approach.
We obtained that there is a large deviation between the theoretical predictions and experimental results on the branching ratios of the semileptonic decays. Our results on branching ratios for the semileptonic decay (normalized to Ω → π) is compatible with the light-front quark model. However,