Relativistic description of the $\Xi_b$ baryon semileptonic decays

Semileptonic decays of the $\Xi_b$ baryon are studied in the framework of the relativistic quark-diquark model based on the quasipotential approach. The weak decay form factors are calculated with the comprehensive account of all relativistic effects without employing nonrelativistic and heavy quark expansions. On this basis differential and total decay rates as well as different asymmetry parameters are calculated for the heavy-to-heavy $\Xi_b\to \Xi_c\ell\nu_\ell$ and heavy-to-light $\Xi_b\to \Lambda\ell\nu_\ell$ semileptonic decays. Predictions for the ratios of such decays involving $\tau$ lepton and muon are presented.


I. INTRODUCTION
The investigation of the semileptonic decays of bottom baryons represents a very interesting and important problem. Indeed, their study provides an independent determination of the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements |V cb | and |V ub | and thus can help to better understand the origin of the disagreement of their values determined from inclusive and exclusive B meson decays [1]. Such decays can be also used to verify the lepton flavour universality, indications of which violation were reported in the semileptonic B meson decays governed by the b → c quark transitions (for recent review see [2] and references therein). The discrepancy between predictions of the Standard Model (SM) and experimental data was observed for the ratios of the branching fractions R D and R D * of the semileptonic B meson decays to D and D * mesons, respectively, involving τ lepton and muon. The combined excess of the measured ratios over the SM prediction is about 3.6σ [2]. Recently the LHCb Collaboration [3] measured the ratio R J/Ψ of semileptonic B c meson decays to J/Ψ with τ lepton and muon which exceeds the SM prediction by more than 2σ [2,4].
In our paper [5] we comprehensively investigated semileptonic decays of Λ b baryons in the framework of the relativistic quark-diquark model based on the quasipotential approach. The explicit expressions for the decay form factors were obtained in terms of the overlap integrals over the baryon wave functions. Baryons were treated as quark-diquark composite systems. All relativistic effects including contributions of the intermediate negative-energy states and wave function transformations from the rest to the moving reference frame were systematically taken into account. To achieve this goal we do not use either nonrelativistic or heavy quark expansions. As a result the obtained formulas are valid in the whole range of the momentum transfer q 2 both for the heavy-to-heavy (b → c weak transitions) and heavy-tolight (b → u weak transitions) semileptonic decays of bottom baryons. The calculated form factors of the Λ b baryon transitions were used for obtaining predictions for the differential and total decay rates and different asymmetry parameters. Good agreement of theoretical results with experimental data was found.
In the present paper we extend the previous analysis to the semileptonic decays of the Ξ b baryon. Such decay are significantly less studied both theoretically and experimentally.
However there are good chances that they will be soon observed at LHC.

II. WEAK DECAY FORM FACTORS
The hadronic matrix elements of the vector and axial vector weak currents for the semileptonic decay Ξ b → Ξ c (Λ) are parametrized in terms of six invariant form factors [6] where M B and u B (p, s) are masses and Dirac spinors of the B baryons ( The expressions for these form factors as the overlap integrals of baryon wave functions with the systematic account of the relativistic effects are given in Ref. [5]. They were obtained without employing the heavy quark expansion both for initial and final baryons. Therefore we can apply them for the calculation of the heavy-to-heavy Ξ b → Ξ c ℓν ℓ and heavy-to-light Ξ b → Λℓν ℓ decay form factors. For the numerical calculations we use the baryon wave functions obtained while studying their spectroscopy [7,8].
We found that the numerically calculated form factors can be approximated with high accuracy by the following analytic expression where the variable The pole masses have the following values: 280 GeV for f A 3 . We take the masses of the excited B c and B mesons from our previous study of their spectroscopy [9,10]. The fitted values of the parameters a 0 , a 1 , a 2 as well as the values of form factors at maximum q 2 = 0 and zero recoil q 2 = q 2 max are given in Tables I, II. The difference of the fitted form factors from the calculated ones does not exceed 0.5%. Our model form factors are plotted in Figs. 1, 2. We roughly estimate the total uncertainty of our form factor calculation to be about 5%.
Now we can use the obtained form factors for the calculation of the differential and total semileptonic decay rates, different asymmetry parameters and other observables. To achieve this goal it is convenient to use the helicity formalism [6]. The helicity amplitudes are expressed through the decay form factors by the following relations: The amplitudes for negative values of the helicities can be obtained using the relation The total helicity amplitude for the V − A current is given by The helicity structures entering the differential decay rates and angular distributions are expressed in terms of the total helicity amplitudes (5) by Then the differential decay rate can be presented by [6] dΓ where , and m ℓ is the lepton mass (ℓ = e, µ, τ ), It is plotted in Fig. 3 for the Ξ b → Ξ c ℓν ℓ (left) and Ξ b → Λℓν ℓ (right) semileptonic decays. Many important observables are expressed in terms of the helicity structures (6) (see [6] for details): a) The forward-backward asymmetry of the charged lepton b) The convexity parameter c) The longitudinal polarization of the final baryon Ξ c (Λ) d) The longitudinal polarization of the charged lepton ℓ We plot these observables in Figs. 4-7 for heavy-to-heavy Ξ b → Ξ c ℓν ℓ and heavy-to-light Ξ b → Λℓν ℓ semileptonic decays. Our predictions for the decay rates, branching fractions and asymmetry parameters are given in Table III. The decay rates are calculated using the CKM values |V cb | = (3.90 ± 0.15) × 10 −2 , |V ub | = (4.05 ± 0.20) × 10 −3 extracted from our previous analysis of the heavy B and B s meson decays [11]. The average values of the forward-backward asymmetry of the charged lepton A F B , the convexity parameter C F and the longitudinal polarization of the final baryon P L and the charged lepton P ℓ are calculated by separately integrating the numerators and denominators over q 2 .
Since the discrepancy between predictions of the Standard Model and experimental data in heavy meson semileptonic decays is observed for the ratio of branching ratios of decays   involving τ and a muon or electron [1, 2], it is important to investigate similar decays of the heavy baryons. Using our results we get the following predictions for the ratios of the Ξ b baryon branching fractions (l = e, µ) Note that part of the theoretical uncertainties cancels in these ratios and we roughly estimate them to be about 3%. The values of these ratios are slightly larger than the corresponding values R Λc = 0.313 and R p = 0.649 calculated previously for the semileptonic Λ b decays [5]. Any significant deviations from these results, if observed, can be interpreted as a signal of the new physics contributions. For the semileptonic Λ b decays the LHCb Collaboration measured the ratio of the heavyto-heavy and heavy-to-light decays [12]. If we consider the similar ratio for the Ξ b semileptonic decays, we get the following results The measurement of such ratios can provide an additional determination of the ratio of the CKM matrix elements |V ub | and |V cb |. The final value was obtained for our preferred CKM values and the error bar includes combined theoretical uncertainties in the ratio of decay branching fractions as well as the uncertainties in the ratio of the CKM values.
In Table IV we compare our results for the semileptonic decay Ξ b → Ξ c ℓν ℓ with other calculations [13][14][15]. 1 The previous investigations [13][14][15] employed heavy quark effective theory to study this heavy-to-heavy baryon transition. Both the infinitely heavy quark limit and the first order 1/m Q corrections were considered. In our present calculations we perform all calculations without application of the nonrelativistic or heavy quark expansions. We find our total decay rates and branching ratios to be somewhat lower and slight deviations in other observables. Note that the central value of our ratio R Ξc is approximately 1.3 times larger than in Ref. [15]. This can be attributed to the completely relativistic treatment of the decays in our study.

IV. CONCLUSION
The form factors of the heavy-to-heavy Ξ b → Ξ c and heavy-to-light Ξ b → Λ weak transitions were calculated in the relativistic quark-diquark picture with the comprehensive account of the relativistic effects. The relativistic baryon wave functions were used for the evaluation of the corresponding form factors. The form factor momentum transfer q 2 dependence was explicitly determined in the whole accessible kinematical range without extrapolations or additional model assumptions. It was found that the analytic approximation for these form factors (2) accurately reproduces their q 2 behaviour with the parameters listed in Tables I, II. The helicity formalism was used for the calculation of the differential and total semileptonic decay rates and other useful observables which are given in Table III. The ratios R Ξc and R Λ (13) of the semileptonic Ξ b to Ξ c and Λ decay rates involving τ and electron are obtained. Their measurement can serve as an additional test of the lepton flavour universality. The measurement of the ratio (14) of heavy-to-light Ξ b → Ξ c ℓν ℓ to heavy-to-heavy Ξ b → Ξ c ℓν ℓ semileptonic decays can provide the independent determination of the CKM ratio |V ub |/|V cb |. To our knowledge this is the first detailed study of the CKM suppressed semileptonic Ξ b → Λℓν ℓ decays.