b-baryon semi-tauonic decays in the Standard Model

Within the framework of HQET, $\Lambda_{b}\rightarrow\Lambda_{c}\tau\bar{\nu}_{\tau}$ and $\Omega_{b}\rightarrow\Omega_{c}^{(*)}\tau\bar{\nu}_{\tau}$ weak decays are studied to the order of $1/m_c$ and $1/m_b$. Helicity amplitudes are written down. Relevant Isgur-Wise functions given by QCD sum rule and large $N_c$ methods are applied in the numerical analysis. The baryonic R-ratios $R(\Lambda_c)$ and $R(\Omega_c^{(*)})$ are obtained.

The outline of the paper is as follows. In section II, general description of the decays in terms of helicity amplitudes is given. In section III, form factors are expressed by Isgur-Wise functions which were obtained from the large N c and QCD sum rules. In section IV, numerical results are presented. Section V gives the summary.
where q = p − p ′ and σ µν = i [γ u , γ ν ] /2, form factors f i and g i are functions of q 2 . It is convenient to reexpress the form factors as functions of velocities of baryons, where v and v ′ denote four-velocities of Λ b and Λ c , respectively, ω = v · v ′ , F i and G i are functions of ω.
Similarily, for the decays of Ω b → Ω , where the u Ω * c λ is the Rarita-Schwinger spinor for a spin-3/2 particle.

B. Helicity amplitudes
In analyzing decays, polarization gives detailed physics information. The decay Λ b → Λ c τν τ can be thought of as two sub-processes Λ b → Λ c + W off-shell and W off-shell → τ +ν τ .
Consider the decay Λ b → Λ c + W off-shell in the rest system of Λ b . W off-shell moves in the +z direction, and Λ c moves in the −z direction. The momentums of Λ b , Λ c and W off-shell are respectively. The current is composed of a spin-1 and a spin-0 components. The relevant expression of polarization 4-vectors of the current is that [48] ε µ (t) = 1 The t-label stands for the time-component of the corresponding current. Notice that in our case, the tauon mass will be taken into consideration, so the time-component of the W off− shell should be included. The helicity amplitudes are defined in the following, where M V,A µ stand for the matrix elements of vector and axial vector currents, λ 2 and λ W are the helicities of the daughter baryon Λ c and the W off−shell , respectively. Helicity amplitudes are then expressed in terms of the form factors.
For Λ b → Λ c transition, Similarly for Ω b → Ω c transition, and for Ω b → Ω * c transition, Other relations can be obtained by relations: Decay rates can be given in terms of these helicity amplitudes.

C. Decay rates
The differential decay rate dΓ/dω is obtained as following [25,26], where G F is the Fermi coupling constant, V cb is the CKM matrix element, and and dΓ t ± dω are defined as the transverse, longitudinal and time-component contribution to the decay rate with ± denoting the final baryon helicity.
Following the same method, we get that for where h (Q) v denotes the heavy quark field defined in the HQET with velocity v, and Γ stands for any gamma matrices. ξ(ω) is normalized at the zero recoil, ξ(1) = 1.
When 1/m Q correction is taken into consideration, another Isgur-Wise function χ and a mass parameterΛ appear. The subleading Isgur-Wise function χ(ω) is defined by whereΛ is the heavy baryon mass in HQET,Λ = m Λ Q − m Q .
Including α s and Λ QCD /m c,b corrections, the form factors are given as following [30][31][32], where the perturbative QCD coefficient in the leading logarithmic approximation is and For Ω ( * ) b(c) cases, similarly, based on the standard tensor method [30,33], we denote Ω Q and Ω * Q as B 1 µ and B 2 µ respectively, In the leading order of heavy quark expansion, the fourteen form factors are reduced to two Isgur-Wise functions which are defined as, The form factors are expressed as [49], B. QCD sum rule and Large N c Isgur-Wise functions and the mass parameters should be calculated by nonperturbative methods. In this work, we make use of results from QCD sum rule [11,13] and large N c methods [12,18,20,50].

QCD sum rule
Within HQET, the QCD sum rule method gives the following results [11,13],

Large N c
In the large N c limit, the leading Isgur-Wise function ξ(ω) and the mass parameterΛ are given as [50] ξ This ξ is actually a reallization of δ function [12]. Ref. [12] further showed that χ(ω) = 0 in the large N c limit.
We finally obtain the form factors as

IV. NUMERICAL RESULTS
Numerical results for Λ b → Λ c lν l and Ω b → Ω ( * ) c lν l (l = e, µ, τ ) can be obtained now. In the calculation it takes m Λ b = 5.62 GeV, m Λc = 2.23 GeV, m Ω b = 6.07 GeV, m Ωc = 2.70 GeV, m Ω * c = 2.77 GeV, |V cb | = 0.04 and G F = 1.166 × 10 −5 GeV −2 [53]. And m c = 1.44 GeV, m b = 4.83 GeV, µ = 0.47 GeV [31,32]. ω is in the range 1 ≤ ω ≤ Tauonic decay distributions are plotted in Figs. 1-7. Fig. 1 presents the Λ b → Λ c τν τ differential decay rate, both QCD sum rule and large N c results are given for comparison, with the uncertainty of the QCD sum rule considered. The two results are close to each other, especially in the low recoil region. In Figs. 2 and 3, we display the ω dependence of Λ b → Λ c τν τ partial differential rates T, L, t and the total differential rate. The transverse rate T dominates in the low recoil region while the longitudinal rate L dominates in the large recoil region. Fig. 2 is for the QCD sum rule method. And Fig. 3 is that from the large N c method. Figs. 4-7 show the corresponding plots for Ω b → Ω ( * ) c τν τ decays for the large N c limit. For the partial decay distribution of Ω b → Ω * c τν τ (Fig. 7), what should be discussed is that the t + channel is almost 0, and the L ± channels are almost the same. As for the tauonic decay, time-components should be considered specifically, because they are absent in the massless charged lepton case. In the Λ b case, time-component is still small. However, in the Ω b → Ω c case, time-component gets comparatively larger. In the Ω b → Ω * c case, time-component gets to be even much larger and begins to dominate in the large recoil region.
The decay rates are obtained by ω integration. For the Λ b → Λ c τν τ decay, we obtain the total decay rate, the branching ratio, and the R-ratio in the following from the QCD sum rule, where uncertainties are due to the error of QCD sum rules in Eq. (20). For the large N c case, The error of the large N c result is estimated to be 1/N c ∼ 30% in general. However, the uncertainty of R which is what we are really interested in, is supposed to be smaller because of the cancellation in the ratios [12]. Thus, the uncertainty in R (Λ c ) is estimated as small as ∼ 10%. Table I lists results for the Λ b → Λ c ℓν ℓ semileptonic decay. Experimental data [53] and results from the quark model [37], HQET [44], and lattice QCD [21]  Via the same procedure, the result of the Ω b → Ω ( * ) c τν τ decay is obtained by using the large N c method in the following, And Like in the Λ b decay, the 1/N c uncertainty for R(Ω ( * ) c ) is expected to be ∼ 10%.

ACKNOWLEDGMENTS
We are very grateful to Jürgen G. Körner for valuable comments. We acknowledge support from the National Natural Science Foundation of China (No. 11875306).