Semileptonic $\Xi_c$ baryon decays in the relativistic quark model

The form factors of the weak $\Xi_c\to \Xi(\Lambda)$ transitions are calculated in the framework of the relativistic quark model based on the quasipotential approach. All relativistic effects including transformations of the baryon wave functions from the rest to moving reference frame and contributions of the intermediate negative energy states are systematically taken into account. The explicit analytic expressions which reliably approximate the momentum transfer $q^2$ dependence of the form factors in the whole accessible kinematical range are given. The calculated form factors are applied for the evaluation of the semileptonic $\Xi_c\to \Xi \ell\nu_\ell$ and $\Xi_c\to \Lambda \ell\nu_\ell$ ($\ell=e,\mu$) decay rates, different asymmetry and polarization parameters within helicity formalism. The obtained results are compared with available experimental data and previous calculations.


I. INTRODUCTION
This year significant experimental progress has been achieved in studying weak decays of the charmed Ξ c baryons. Until now the absolute branching fractions of both neutral Ξ 0 c and charged Ξ + c baryons were not measured. All decay modes were only measured relative to Ξ 0 c → Ξ − π + and Ξ + c → Ξ − π + π + modes [1]. This fact significantly complicated comparison of theoretical predictions with experimental data. However, recently the Belle Collaboration presented the first measurement of absolute branching fractions of the neutral Ξ 0 c baryon in three decay modes including Ξ 0 c → Ξ − π + one [2]. Its branching fraction is Br(Ξ 0 c → Ξ − π + ) = (1.80 ± 0.50 ± 0.14)%. Then the absolute branching fractions of its charged partner Ξ c were also reported [3] for three decay modes including Ξ + c → Ξ − π + π + with Br(Ξ + c → Ξ − π + π + ) = (2.86 ± 1.21 ± 0.38)%. These results can be combined with Ξ c branching fractions measured relative to corresponding modes to get other absolute Ξ c branching fractions. Thus the experimental values of Ξ c semileptonic branching fractions can be determined.
In this paper we calculate the weak Ξ c → Ξ(Λ) transition form factors in the framework of the relativistic quark model based on the quasipotential approach and use them to evaluate the semileptonic branching fractions of the Ξ c baryon. This model was successfully applied for studying semileptonic decays of bottom Λ b [4] and Ξ b [5] and charmed Λ c [6] baryons. The form factors are expressed as the overlap integrals of the baryon wave functions. The important advantage of the employed model is the comprehensive inclusion of the relativistic effects which allows us to explicitly determine the q 2 dependence of the form factors in the whole kinematical range, thus increasing reliability of the results. The calculated form factors can be then used for the determination of branching fractions and other important observables which can be measured experimentally. The results can be confronted with previous theoretical predictions and new experimental data.
where M B and u B (p, s) are masses and Dirac spinors of the initial and final baryons (B = Ξ c , Ξ, Λ), q = p ′ − p.
In the relativistic quark model these form factors are expressed through the overlap integrals of the baryon wave functions [4] which are known from the calculations of the baryon mass spectra [8,9]. The explicit expressions are given in Ref. [4]. They systematically take into account all relativistic effects including transformation of baryon wave functions from the rest to moving reference frame and contributions of the intermediate negative energy states.
Following Ref. [5] we fit the numerically calculated form factors by the following analytic expression where the variable  Tables I, II. The difference of the fitted form factors from the calculated ones does not exceed 0.5%. The form factors are plotted in Figs. 1, 2. We roughly estimate the total uncertainty of our form factor calculation to be about 5%.
In Table III we compare our results for the values of form factors at q 2 = 0 with previous calculations. Results [10] are based on the light-front approach. 1 In this paper form factors f V,A 3 (0) were not evaluated. We find reasonable agreement for the form factors parametrizing the Ξ c → Ξ weak transitions, while in the case of the Ξ c → Λ transition our form factors are FIG. 2: Form factors of the weak Ξ c → Λ transitions. 1 In Table III we present the "physical transition form factors" from Ref. [10] which include overlapping factors.  about a factor of 2 larger. In Ref. [11] the light cone QCD sum rules were employed for the calculation of the Ξ c → Ξ transition form factors. Their form factors f V 1,2,3 (0) parametrizing the vector current significantly differ from our results. The central values [11] of the form factor f V 1 (0) is about 2 times smaller while those for f V 2,3 (0) are a factor of about 3 larger than other predictions.
The differential and total semileptonic decay rates, branching fractions and different asymmetry and polarization parameters can be calculated using the decay form factors and the helicity formalism [7]. The relation between helicity amplitudes and decay form factors are the following.
and the amplitudes for negative values of the helicities are obtained from the relations The total helicity amplitude for the V − A current is given by The differential decay rates and angular distributions are expressed in terms of the helicity structures which are the following combinations of the total helicity amplitudes (5) The expression for the differential decay rate is given by [7] dΓ where G F is the Substituting in these expressions the Ξ c decay form factors calculated in the previous section we obtain the differential decay rates. We plot them for the Ξ c → Ξℓν ℓ (left) and Ξ c → Λℓν ℓ (right) semileptonic decays in Fig. 3. Many important observables are expressed in terms of the helicity combinations (6) (see [7] for details): • The forward-backward asymmetry of the charged lepton • The convexity parameter  Decay • The longitudinal polarization of the final baryon Ξ c (Λ) ; (11) • The longitudinal polarization of the charged lepton ℓ These observables are plotted for Ξ c → Ξℓ + ν ℓ and Ξ c → Λℓ + ν ℓ semileptonic decays in Figs. 4-7. The predictions for the decay branching fractions and asymmetry parameters are presented in Table IV. We calculated the decay rates using the CKM values |V cs | = 0.995 ± 0.016, |V cd | = 0.220 ± 0.005 [1]. The average values of the A F B , C F , P L and P ℓ were obtained by separately integrating the numerators and denominators in Eqs. (9)-(12) over q 2 . We roughly estimate uncertainties of our predictions for the branching fractions to be about 10%.

IV. CONCLUSION
The relativistic quark model was used for the calculation of form factors of the semileptonic Ξ c transitions, both for the CKM favored Ξ c → Ξℓ + ν ℓ and CKM suppressed Ξ + c → Λℓ + ν ℓ decays. All relativistic effects, including baryon wave function transformations from the rest to moving reference frame and contributions of the intermediate negative-energy states, were comprehensively taken into account. This allowed us to explicitly determine the momentum transfer q 2 dependence of the weak form factors in the whole kinematical range without additional model assumptions or extrapolations. We give the values of parameters in the analytic expression (2) which accurately approximates the numerically calculated form factors in Tables I, II and show the form factor q 2 dependence in Figs. 1, 2.
Using the calculated form factors and helicity formalism we estimated important observables for the CKM favored and CKM suppressed Ξ c semileptonic decays: differential decay rates, branching fractions, different asymmetry and polarization parameters, which are given in Table IV and plotted in Figs. 4-7. Our results for the branching fractions of the Ξ c → Ξℓν ℓ decays are in reasonable agreement with previous calculations based on the light-front quark model [10] and application of the SU(3) flavor symmetry [13,14], but significantly lower than the light-cone QCD sum rule predictions [11]. They agree reasonably with experimental values which can be obtained combining the corresponding decay ratios from PDG [1] and recent measurements of absolute branching fractions by the Belle Collaboration [2,3]. For the suppressed Ξ + c → Λℓ + ν ℓ decays available theoretical predictions differ significantly. Our model predicts larger values of the decay branching fractions than other calculations [10,14]. Thus their direct experimental measurement can help to discriminate between theoretical approaches.