Semileptonic Decay of $\Omega_c^0 \to \Xi^- l^+ \nu_l$ From Light-Cone Sum Rules

We calculate the form factors of the weak decay of $\Omega_c$ to $\Xi$ in the method of QCD light-cone sum rule. With the form factors obtained, we also calculate the decay width of $\Omega_c^0 \to \Xi^- l^+ \nu_l$ and its decay branching ratio. To the twist-6 distribution amplitudes, we give the form factors $f_1=-0.168, f_2=0.175, g_1=-0.0078$ and $g_2=0.176$ at zero recoil point. The result of the semileptonic decay width of $\Omega_c^0 \to \Xi^-l^+\nu_l$ is $\Gamma=(1.64\pm0.05)\times10^{-16} GeV$ , and the prediction of the decay branching ratio $Br(\Omega_c^0\to\Xi^-l^+\nu_l)=6.68\times10^{-5}$.

We calculate the form factors of the weak decay of Ωc to Ξ in the method of QCD light-cone sum rule. With the form factors obtained, we also calculate the decay width of Ω 0 c → Ξ − l + ν l and its decay branching ratio. To the twist-6 distribution amplitudes, we give the form factors f1 = −0.168, f2 = 0.175, g1 = −0.0078 and g2 = 0.176 at zero recoil point. The result of the semileptonic decay width of Ω 0 c → Ξ − l + ν l is Γ = (1.64 ± 0.05) × 10 −16 GeV , and the prediction of the decay branching ratio Br(Ω 0 c → Ξ − l + ν l ) = 6.68 × 10 −5 . In recent years, many new experimental results of Ω 0 c baryon have been developed. The lifetime has been updated by LHCb and a new value τ (Ω 0 c ) = (268±24±10± 2) × 10 −15 s [1] is given, about five times larger than the old measurements [2][3][4]. And also, five new narrow Ω 0 c states are reported by LHCb in 2017 [5], and confirmed by e + e − collisions on Belle [6]. These new discoveries enrich the nature of Ω 0 c baryon greatly. Two of the important properties of Ω 0 c baryon are its decay properties, strong and weak decays, but there is no strong decay observed in experiment until now. The weak decay channels are the main decay channels of Ω 0 c . The PDG listed fourteen Cabibbo-favored weak decay channels up to now [7], among them there are thirteen non-leptonic and only one semileptonic weak decay channel observed. For the investigation of the transition from Ω 0 c baryon to other lighter baryons, the simplicity object is the semileptonic weak decay.
One of the established semileptonic weak decay mode is the channel Ω 0 c → Ω − e + ν e [8].In this channel, the decay mode is from charm quark decay to strange quark and radiative positron and neutrino. For charm quark, the other possible decay channel c → dl + ν l is not forbidden in the standard model, which was observed in D + → ηµ + ν µ semileptonic decay by BESIII collaboration recently and also gives a new value of CKM element |V cd | = 0.242 [9]. Therefore, the study of semileptonic weak decay of charm baryon provides an additional source of information for determining the CKM matrix elements of charm quark as well as exploring the internal dynamics of systems containing heavy-light quarks. In the Ω 0 c baryon case, the Ω 0 c → Ξ − l + ν l decay channel gives the way to research.
The transition from Ω 0 c baryon to Ξ baryon has been studied with many theoretical approaches, such as non-relativistic quark model [10], heavy quark effective theory [11,12], light-front quark model [13], and MIT bag model [14], in these articles the form factors were calculated. With the development of experimental results of the mass, lifetime, and other parameters, the new calculation and prediction are necessary. In our work, we will study this transition with the method of light-cone sum rules [15][16][17][18][19][20], which is based on QCD sum rules [21][22][23]. We will evaluate our work from baryon and weak currents and estimate the semileptonic weak decay width with the new result of Ω 0 c baryon lifetime. This method has been used to study the strong decay properties of excited Ω 0 c baryons [24,25].
After the introduction in Sec. I, we give the details of the light-cone sum rules derivation of the semileptonic weak decay of Ω 0 c → Ξ − l + ν l , and the sum rules of the form factors are given in Sec. II. Sec. III is the numerical analysis of the four form factors of transition matrix element of Ω 0 c → Ξ. Conclusions are given in Sec. IV. In the Appendix, we give the explicit expressions of the Ξ baryon distribution amplitudes.
Weak decay dynamics can be investigated by the weak decay effective Hamiltonian [26], and the semileptonic decay can be processed as the rare decay of B mesons [27]. For the c → dl + ν l decay mode, we can write the Hamiltonian with the form [28]: where G F is fermi constant. |V ud | and |V cd | are the CKM matrix elements.
is quark and lepton current respectively. The light-cone sum rule starts from the current algebra structure of hadrons, and is evaluated with analytical methods, then gives the hadron transition matrix element with form factors in theoretical expressions. This method uses two kinds of representations, hadrons and QCD theoretical representations. On the one hand, one can write the hadrons transition correlation function with the hadrons quantum numbers, and on the other hand, one can calculate the hadrons properties with QCD quark currents on the light-cone by hadrons light-cone distribution amplitudes (LCDAs).
Decay properties of heavy baryons calculations need the decay matrix element of heavy baryon decay to light baryon. The decay matrix element of Ω c → Ξ can be parameterized to six form factors as follows.
In the equation above, the relation s u Ωc (p−q)ū Ωc = [( / p− / q)+ M Ωc ] is used, also set q ·z = 0 because the small value on the light cone of the transfer momentum. If we only consider the light leptons e ± and µ ± . Their masses are very small in our system. In the zero masses limit, with the relation q µl γ µ (1 − γ 5 ) = 0, the f 3 and g 3 terms do not contribute. We omit them and the corresponding terms in our following calculations [19].
Our next work is to derive the other way to describe the correlation function. In our work, we use the lightcone sum rule to express the QCD theoretical formalism. For this purpose, we expand the correlation function on the light-cone with the light-cone distribution amplitudes of Ξ baryon [20]. Substituted the heavy baryon current (4) and the weak decay current (5) into the correlation function (6), after contract the heavy quark, we obtain the correlation function In these calculations, we use the LCDAs transformation relations, and the LCDAs of Ξ has been given in [20]. We will also give the LCDAs and other formulas we need in our calculations in Appendix A, where we only display the parameters we need, and the completeness form can be found in [20,29]. After the standard procedure of light-cone sum rule calculations and performing Borel transformations both on the two sides of hadron and theoretical representation, we get the final light-cone sum rules of these form factors f i (g i )(i = 1, 2) as follows: −M Ωc f Ωc g 1 e −M 2 where Q 2 = −q 2 and the α 30 relates to the threshold s 0 is The Borel transform parameters is The signs B i (α 3 ) and C i (α 3 ) we used above are defined in the following expressions

III. NUMERICAL ANALYSIS.
In order to know the form factors and give the value properties of the semileptonic decay of Ω c → Ξ − l + ν l ,we should set the numerical value of the parameters in the formulas of form factors. In our analysis, we adopt the standard center values of charm quark mass and baryons masses from PDG [7], which gives the numbers m c = 1.27GeV, M Ξ − = 1.3217GeV , and the Ω c baryon mass M Ω 0 c = 2.6952GeV . The physical region of momentum transfer square q 2 varies in the region For the numerical analysis, we also need to know the nonperturbative parameters of the Ω c baryon decay constants and the Ξ baryon decay constants f Ξ and λ 1 . The nonpertubative parameters distribution amplitudes V i (i = 1, ..., 8) are dependent on the Ω c baryon decay constants, and their forms have been derived in [20] where we list them in the appendix. In our calculations, we adopt the decay constant f Ωc from reference [24] M Ωc f Ωc = √ 2 × 0.0438GeV 3 , and the decay constant of Ξ can be seen in [20], the values f Ξ = (9.9 ± 0.4) × 10 −3 GeV 2 , The other parameters we should determine are the Borel region. The chosen principle of Borel parameters is that we should promise to suppress both the higher resonance and twist contributions. So the threshold s 0 = 11.56GeV 2 which we choose is available in our calculations, and the stable window of the Borel parameter M 2 B working we analyse is 8 < M 2 B < 10GeV 2 . We analyse the varying of M 2 B at the point s 0 = 11.56GeV 2 and q 2 = 0GeV 2 . The pictures of curve are given in Fig. 1, and the pictures of form factors f i (g i )(i = 1, 2) are plotted in Fig. 2.
Because the arriving of the decay width of the semileptonic process should be investigated on the whole physical region, we extrapolate these form factors by the threeparameter dipole formula Eq. (22) on the whole kinematical region. Another benefit of using this fitting formula is that we can simplify the procedure of our calculation.
The form factors at point q 2 = 0GeV 2 are given in Table I, and the results obtained from other approaches are also listed in this table. The coefficients a and b we fitted are listed in Table II Due to the smallness of g 1 we do not consider the g 1 term. We use the decay width of semileptonic weak decay as in [19,20]. The differential decay rate dependent on q 2 is Where m ± = M Ωc ± M Ξ and q 2 ± = q 2 − m 2 ± . The fermi constant is G F = 1.166 × 10 −5 GeV −2 , and CKM matrix element is |V cd | = 0.221. Because the form factor g 1 is more than one order smaller compared with other form factors, we omit the term which contains g 1 . Substituting these constants into the differential decay formula and integrating it in the dynamical region 0 < q 2 < (M Ωc − M Ξ ) 2 , we obtain the decay width of the weak semileptonic decay Ω 0 c → Ξ − l + ν l , that is Γ = (1.64 ± 0.05) × 10 −16 GeV . The errors come from the Borel region which we choose. The picture of the differential decay width is plotted in Fig 3. With the mean lifetime of Ω 0 c in PDG [1,7], which gives the value The form factor fi(i = 1, 2) and gi(i = 1, 2) varies from zero momentum transfer square to q 2 = 2GeV 2 τ = 268 × 10 −15 s, the branching ratio of decay which we estimate by the dipole formula is Br(Ω 0 c → Ξ − l + ν l ) = 6.68 × 10 −5 . We also list the results calculated by other methods compared with ours in Table III.

IV. CONCLUSIONS
In this work we calculate the form factors of the semileptonic weak decay from heavy baryon Ω c to light baryon Ξ and two leptons with light-cone sum rule approach. The explicit expressions of the sum rules of form factors are given. By using the numerical value in PDG the form factors are calculated and are given in Table I. The form foctors obtained by other approaches are compatible with ours in Table I too. The decay width of Ω 0 c → Ξ − l + ν l is evaluated to the number Γ = (1.64 ± 0.05) × 10 −16 GeV .The branching ratio of it is given by Br(Ω 0 c → Ξ − l + ν l ) = 6.68 × 10 −5 . But there is no absolute branching of Ω 0 c decay which has been discovered in experiments, and the only reference of it is to set the Ω 0 c → Ω − π + decay channel to one, and the other decay channels are all relative to it [30]. Our calculation results may be tested in the future experiments.
In this appendix we give the distribution amplitudes of Ξ baryon [20]. 0|ǫ ijk s iT α (0)s j β (0)d k γ (x)|Ξ(p) = V 1 ( / pC) αβ (γ 5 Ξ) γ + V 2 M Ξ ( / pC) αβ (/ xγ 5 Ξ) γ + V 3 M Ξ (γ µ C) αβ (γ µ γ 5 Ξ) γ + The distribution amplitudes V i , (i = 1, ..., 6) did not have definite twist, in ordr to get the distribution amplitudes with the definite twist we should make the following transforming to get the amplitudes with definite twist These distribution amplitudes V i are functions of x · p, but we need the variable relevant to the longitude momentum fraction of quarks in the baryon, so we should make the following transformation formula.
The integration measure Dx is The distribution amplitudes of Ξ with twist-3 is Twist-4 distribution amplitudes Twist-5 distribution amplitudes And twist-6 distribution amplitudes