Semileptonic $\Xi_c$ baryon decays in the light cone QCD sum rules

Form factors of the weak $\Xi_c \to \Xi(\Lambda)$ transitions are calculated within the light cone QCD sum rules. The pollutions coming from the contribution of the negative parity $\Xi_c^*$ baryon is eliminated by considering the combinations of sum rules corresponding to the different Lorentz structures. Having obtained the form factors, the branching ratios of the $\Xi_c \to \Xi (\Lambda) \ell \nu$ decays are also calculated, and our predictions are compared with the results of other approaches as well as the measurements done by BELLE and ALICE Collaborations.


I. INTRODUCTION
The electroweak decays of the heavy flavored hadrons provide useful information about the helicity structure of the effective Hamiltonian and the matrix elements of the Cabibbo- Moreover, ALICE Collaboration has also announced the result for the branching ratio of the Ξ 0 c → Ξ − ℓν transition (B = (1.8 ± 0.2)%) [2], which agrees with BELLE's measurement within error ranges.
The semileptonic decays of the Ξ c baryon have been comprehensively studied in the framework of different approaches, such as the light-front formalism [3,4], relativistic quark model [5], lattice QCD [6], 3-point QCD sum rules [7], and light cone QCD sum rules method [8].
In this study, we calculated the form factors and branching ratios of the semileptonic decay of Ξ c within the LCSR framework. It should be noted that the same channel was already studied in [8]. However, the prediction of the branching ratio obtained in that study is considerably larger than the results of the other approaches as well as experimental measurements. This study also aims to understand the source of this discrepancy. In our opinion, the reason for the discrepancy can be attributed to the fact that the interpolating current for the given heavy baryon couples not only to the ground state baryon with positive parity J P = 1 2 + but also to a heavier baryon with negative parity J P = 1 2 − . Hence, the dispersion relation of the Ξ c baryon gets modified when the contribution of the negative parity Ξ c baryon is taken into account, which is 300 MeV heavier compared to the ground state Ξ c baryon. In the light of new experimental data, we reanalyze the semileptonic decays of Ξ b → Ξ(Λ)lν within light-cone sum rules in detail by taking into account the contributions of J P = 1 2 − heavy baryon.
So far, the light cone sum rules (LCSR) have successfully been applied to the wide range of problems of the hadronic physics, such as nucleon electromagnetic form factor [9], form factors and strong coupling constants of the heavy baryons [10], rare Λ b → N(N * )ℓ + ℓ − decays [11], etc.
The form factors responsible for the Ξ * c → B transition can be obtained from Eqs. (1) and (2) with the replacements f i → f i , g i → g i , inserting the Dirac matrix γ 5 after the Ξ c baryon bispinor, and replacing Ξ c with Ξ * c .
In order to derive the LCSR for the form factors, we start by considering the following correlation function(s), where η Ξc is the interpolating current of the Ξ c baryon, and J V (A) µ =cγ µ q(cγ µ γ 5 q) are the transition currents. In further calculations, we use the general form of the interpolating current Ξ c , [12].
Here q is the light quark, C is the charge conjugation operator, a, b and c are the color indices, and β is an arbitrary parameter, and β = −1 corresponds to the Ioffe current.
To derive the LCSR for the transition form factors, we first calculate the hadronic part of the correlation function, which is achieved by inserting the full set of charmed-baryon states between the interpolating current η Ξc and the transition current J µ in Eq. (3). Thus, the hadronic part contains the contributions of the lowest positive-parity Ξ c , as well as its negative-parity partner Ξ * c , i.e., where · · · denote the contributions of all excited and continuum states with the quantum numbers of Ξ c , and summation is performed over the ground and first orbital excited states.
The first term in the right hand side of the Eq. (5) describes the coupling of the Ξ c (Ξ * c ) baryon with the interpolating current η Ξc which is defined as, where λ Ξc (λ Ξ * c ) is the residue of the corresponding baryon.
Using the definitions of the transition form factors for the vector transition current, and using the Dirac equation pu B (p) = m B u B (p) we get, , and multiplying γ 5 matrix to the right end, i.e., We now turn our attention to the calculation of the correlation function (3) for the Ξ c (Ξ * c ) → B transition. We take (p − q) 2 , q 2 ≪ m 2 Ξc to justify the expansion of the product of the two currents in the correlation function (3) near the light cone x 2 ≈ 0,hence, the matrix element ε abc 0|q a α (0)s b β (0)s c γ (0)|B(p) is obtained. This matrix element is parametrized in terms of the Ξ and Λ baryon distribution amplitudes (DAs) of a different twist. The explicit expressions of the Ξ and Λ baryon DAs can be found in [13][14][15]. The operator product expansion is obtained by convolution of the hard-scattering amplitudes formed by the virtual c-quark propagator and the Ξ(Λ) baryon DAs with increasing twists. In our calculations, we take into account all three particle B-baryon DAs up to twist-6. However, we neglect the contributions of the four-particle (quark and gluon) DAs.
Matching the coefficient of the relevant Lorentz structures in both representations of the correlation function, we get the sum rules for the transition form factors. Finally, we perform Borel transformation over −(p − q) 2 in order to suppress the higher state and continuum contributions and obtain the following sum rules for the form factors of thecγ µ s transition current, , and Π B 6 (p, q) are the invariant functions for the Lorentz structures, p µ , p µ q,γ µ q, γ µ , q µ , and q µ q structures, respectively.
Note that, the equations for the axial vector current, γ µ γ 5 , can be obtained from Eq. (8) by making the following replacements Solving the six equations given in (8) we obtain the LCSR for the transition form factors f i , f i for vector current and g i , and g i for axial vector which read as: ) , ) , ) .
Few words about the theoretical calculations are in order. The correlation function with the γ µ and γ µ γ 5 transition currents can be transformed to the following form, and x 0 is the solution of the equation The expressions of the form factors involve residues of Ξ c and Ξ * c baryons. These residues can be calculated using the two-point correlation function, Following the standard sum rules methodology, namely, saturating the correlation function with Ξ c and Ξ * c , and performing the Borel transformation and continuum subtraction, we obtain, where λ Ξc (λ Ξ * c ) and m Ξc (m Ξ * c ) are the residues and the masses of the Ξ c (Ξ * c ) baryons, re-spectively. Solving these equations, for the residue of the Ξ c baryon we get, The invariant functions Π B 1 and Π B 2 are calculated in [11], which we will use in our numerical analysis.

III. NUMERICAL ANALYSIS
This section is devoted to the numerical analysis of the form factors derived in the previous section. The main input parameters of LCSR are the DAs of the Ξ baryon, which are calculated in [13,14].
The normalization parameters of these DAs are obtained from the analysis of the twopoint sum rules (see for example [13][14][15]), whose values are, f Ξ = (9.9 ± 0.4) × 10 −3 GeV 2 , The numerical values of other input parameters used in the calculations are presented in   Finally, to find the working region of β, where β = tan θ, we study the dependency of mass on cos θ at several fixed values of M 2 and s 0 . We observe that the mass exhibits good stability to the variation of cos θ in the interval −1.0 < cos θ < −0.6.
It should be emphasized that, the LCSR predictions, unfortunately, are not applicable The LCSR for the form factors are reliable only up to q 2 ≤ 0.5 GeV 2 . To extend this restricted domain to the full physical domain given above, we use the z-series parametrization of the form factors [18], which is given as, The following parametrization The values of the fit parameters a f 0 , a f 1 and a f 2 for the Ξ c → Ξ and Ξ c → Λ form factors are presented in Tables II and III, respectively.    Having obtained the results for the form factors, we estimate the decay widths of the Ξ c → Bℓν decays. The width of these decays can be calculated using helicity formalism [19]. We choose the rest frame of Ξ c baryon, where the z-axis points along the W of f −shell to calculate the helicity amplitudes, and we obtain where m ± = m Ξc ± m B , and Q ± = m 2 ± − q 2 .
In these expressions, the first and second subindices describe the helicities of the B baryon and virtual W , correspondingly. The amplitudes for the negative values of the helicities can be obtained from the parity consideration, i.e., The total helicity amplitude is given by, Using the expressions of above-given helicity amplitudes for the differential decay widths, we obtain where G F is the Fermi constant, V cq is the CKM matrix element (q = s or d), and The differential decay width for the Ξ * c → Bℓν decay can be obtained from Ξ c → Bℓν decay by making the following replacements, Using the values of the CKM matrix elements |V cd | = 0.2211 ± 0.0700 and |V cs | = 0.987 ± 0.011 [16] and the Ξ c life time τ (Ξ 0 c ) = (1.53 ± 0.06) × 10 −13 s, and τ (Ξ + c ) = (4.56 ± 0.05) × 10 −13 s we can predict the branching ratios of the corresponding semileptonic decays. Our results are presented in Table IV. In this table, we also present the values of the branching ratios of the semileptonic Ξ c → Bℓν decays obtained from other theoretical approaches, as well as the latest announced experimental results. From a comparison of the predictions of the different approaches, we see that our results are close to that of the ones given in [20] as well as the experimental measurements [1,2]. On the other hand, the obtained branching ratios are slightly smaller than the results presented in [4][5][6] but larger than the values obtained in [7,21]. However, our results are considerably different for the results obtained in [8] for the Ξ c → Ξℓν decay, although they applied the same method as used in this work. This discrepancy can be explained as follows. The interpolating current of Ξ c baryon interacts not only with ground state positive parity baryons J P = ( 1 2 ) + but also with J P = 1 2 − negative parity baryon which was neglected in [8]. Thus, the dispersion relation of Ξ c baryon is modified, and since the mass difference between these states is around 300 MeV, the results change considerably.
Our predictions on the branching ratios of Ξ + c → Λℓν ℓ are also quite in agreement with  the results of [5] within the error. The predictions on the branching ratios can further be improved by more precise determination of the input parameters appearing in DAs of the Ξ and Λ baryons, as well as taking into account O(α s ) corrections.

IV. CONCLUSION
The form factors of the semileptonic Ξ c → Ξ(Λ)ℓν decays are studied in the framework of the light cone QCD sum rules method. In order to eliminate the contamination of the negative parity Ξ * c baryon, the combination of the sum rules obtained from different Lorentz structures is used.
Using the obtained results on the form factors and applying the helicity formalism, we also estimated the corresponding branching ratios of the considered decays. Moreover, our results on the branching ratios are compared with the predictions of the other approaches as well as with the experimental measurements.
The branching ratios of Ξ c → Ξℓν decays has already been studied in various models like Relativistic Quark Model [5], LATTICE QCD [6], 3-point sum rules [7], Light Front Quark Models [20,21]. Our calculations within the light cone sum rule showed that the resutls are in good agreement with the experimental measurements done by BELLE [1] and ALICE [2] Collaborations.
The discrepancy between our finding and the results of [8] in which the same method was used can be explained by taking into account the contributions of the Ξ * c baryon that was neglected in [8].
Moreover, we also estimated the decay width of the CKM suppressed semileptonic Ξ → Λlν decay within the light-cone sum rules. The obtained branching ratios are close the predictions of [5] and the magnitude of the obtained value shows that it has potential to be measured in the future experiments