{'fraction_non_alphanumeric': 0.044937821881673154, 'fraction_numerical': 0.01953272201984675, 'mean_word_length': 3.820617620345141, 'pattern_counts': {'":': 0, '<': 4, '<?xml version=': 0, '>': 15, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 22, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Polarization effects in the cascade decay Lambda_b ->Lambda(->p pi-) + J/psi(->l+ l-) in the covariant confined quark model

We calculate the invariant and helicity amplitudes for the nonleptonic decay Lambda_b ->Lambda + J/psi, psi(2S) in the covariant confined quark model. We discuss joint angular decay distributions in the cascade decay Lambda_b ->Lambda(->p pi-) + J/psi, psi(2S) (->l+ l-) and calculate some of the asymmetry parameters that characterize the joint angular decay distribution. We confirm expectations from the naive quark model that the transitions into the lambda_Lambda=1/2 helicity states of the daughter baryon Lambda are strongly suppressed leading to a near maximal negative polarization of the Lambda. For the same reason the azimuthal correlation between the two decay planes spanned by (p pi-) and (l+ l-) is negligibly small. We provide form factor results for the whole accessible q2-range. Our results are close to lattice results at minimum recoil and light-cone sum rule results at maximum recoil. A new feature of our analysis is that we include lepton mass effects in the calculation which allows us to also describe the cascade decay Lambda_b ->Lambda(->p pi-) + psi(2S)(->tau+ tau-).


I Introduction
Recently the LHCb Collaboration has performed an angular analysis of the decay Λ b → Λ + J/ψ where the Λ b 's are produced in pp collisions at √ s = 7 TeV at the LHC (CERN) [1]. They reported on the measurement of the relative magnitude of the helicity amplitudes in the decay Λ b → Λ + J/ψ by a fit to several asymmetry parameters in the cascade decay distribution Λ b → Λ(→ pπ − ) + J/ψ(→ ℓ + ℓ − ). In the fit they were also able to measure the transverse polarization of the Λ b relative to the production plane. From a theoretical point of view the nonleptonic decay Λ b → Λ + J/ψ is quite attractive in as much as the factorizable tree diagram is the only contribution to the decay; i.e. there are no W -exchange contributions [color compensation (C), exchange (E) and bow-tie (B) in the terminology of [2]] as e.g. in Λ b → Λ + ρ 0 . There have been a number of theoretical quark model calculations for the decay Λ b → Λ + J/ψ that are based on the factorization hypothesis [3]- [10]. The results of some of these calculations have been compared to the new experimental results by the LHCb Collaboration. We mention that the LHCb Collaboration has not given a result on the branching fraction B(Λ b → Λ + J/ψ) for which the PDG quotes an average value of (5.8 ± 0.8) × 10 −4 [11]. The latter was deduced from the measurements by the CDF [12] and D0 Collaborations [13].
In this paper we present a detailed analysis of the decay process Λ b → Λ + J/ψ in the framework of the covariant quark model proposed and developed in Refs. [14]- [25] for the study of mesons and baryons that are treated as bound states of their constituent quarks. Particle transitions are calculated from multiloop Feynman diagrams in which freely propagating constituent quark fields connect the different nonlocal particle-quark vertices. We mention that the covariant quark model has recently been also applied to exotic tetraquark states [26,27] and their decays. Quark confinement has been incorporated into the covariant quark model in an effective way [28]- [32] through an infrared regularization of the relevant quark-loop diagrams that removes quark thresholds in the loop diagrams (see details in Refs. [28]- [32]).
Our paper is structured as follows. In Sec. II, we review the phenomenological aspects of the decay Λ b → Λ + V where V = J/ψ or ψ(2S). This includes a discussion of kinematics, matrix elements, and invariant and helicity amplitudes. In Sec. III we write down joint angular decay distributions for the cascade decay Λ b → Λ(→ pπ − ) + V (→ ℓ + ℓ − ) where V = J/ψ or ψ(2S). We also define some pertinent decay asymmetry parameters that characterize the angular decay distributions. In Sec. IV we review the salient features of the covariant confined quark model and present our form factor results, which we compare with the results of other model calculations. In Sec. V we carefully discuss the heavy quark limit (HQL) of our Λ b → Λ form factor expressions. In Sec. VI we present our numerical results on helicity amplitudes, on the rate and on the asymmetry parameters in the decay processes Λ b → Λ + J/ψ and Λ b → Λ + ψ(2S). We have included the latter decay since it allows us to discuss nonzero lepton mass effects in the kinematically allowed decay Λ b → Λ + ψ(2S) (→ τ + τ − ). Finally, in Sec. VII, we summarize our results.
II Λ b → Λ + J/ψ decay: matrix element and observables The effective Lagrangian [33] for the b → scc transition is given by where the Q i are the set of effective four-quark flavor-changing b → s operators and where V cb = 0.0406 and V * cs = 0.974642 are Cabibbo-Kabayashi-Maskawa (CKM) matrix elements; O µ = γ µ (1 − γ 5 ) andÕ µ = γ µ (1 + γ 5 ). The C i are the set of Wilson coefficients [33]: The quark-level matrix element contributing to the Λ b → Λ + J/ψ decay is given by As in Ref. [32] we introduce the following combinations of helicity amplitudes: The partial helicity width corresponding to the two specific combinations of helicity amplitudes in (10) is defined by For the Λ b → Λ + V decay width one finds III Joint angular decay distributions in the cascade decay As in the case of the rare meson decays B → K ( * ) + ℓ + ℓ − (ℓ = e, µ, τ ) treated in [21] one can exploit the cascade nature of the decay Λ b (↑) → Λ(→ pπ − ) + V (→ ℓ + ℓ − ) of polarized Λ b (↑) decays to write down a fivefold angular decay distribution involving the polar angles θ 1 , θ 2 and θ, and the two azimuthal angles φ 1 and φ 2 . V stands for J/ψ or ψ(2S). Since the decay ψ(2S) → τ + τ − is kinematically allowed, we include lepton mass effects in our decay formulas. The angular decay distribution involves the helicity amplitudes h V λ ℓ + λ ℓ − for the decay V → ℓ + ℓ − , H λΛλV for the decay Λ b → Λ + V and h B λp0 for the decay Λ → p + π − . We do not write out the full fivefold angular decay distribution that can be found in [35], or that can be adapted from the corresponding fivefold decay distributions for the semileptonic baryon decays Ξ 0 → Σ + + ℓ −ν ℓ and Λ c → Λ + ℓ + ν ℓ written down in [34] and [36], respectively. Instead we discuss a threefold polar angle distribution for polarized Λ b decay and a threefold joint decay distribution for unpolarized Λ b decay. These can be obtained from the full fivefold decay distributions written down in [34][35][36] by the appropriate angular integrations or by setting the polarization of the Λ b to zero.

A.
Polar angle distribution in polarized Λb decay Let us first consider the polar angle distribution W (θ, θ 1 , θ 2 ) for polarized Λ b decays which has been discussed before in [1,37] in the zero lepton mass approximation (see Fig. 1).
The angular decay distribution can be derived from the master formula where the summation extends over all possible helicities λ ℓ + , λ ℓ − , λ Λ b , λ Λ , λ p = ± 1 2 and λ V = 0, ±1. For the diagonal terms of the density matrix ρ λΛ b ,λΛ b (θ) appearing in Eq. (13) one has The vector current lepton helicity amplitudes are given by (see [32]) Finally, factorizing out the combination |h B such that the angular factors h i (cos θ, cos θ 1 , cos θ 2 ) (i = 1, . . . , 7) in the second row of Eq. (19) integrate to zero after polar integration. The functions f i , g i , h i and ℓ i (i = 0, . . . , 7) that describe the normalized angular distribution (19) are listed in Table I. Setting ε = m 2 ℓ /M 2 V to zero and the velocity parameter v = √ 1 − 4ε to 1 as is appropriate in the zero lepton mass approximation one recovers Table II of Ref. [1]. It is clear that one can determine the four parameters (P b , α b , r 0 and r 1 ) from a global fit to the polar angle distribution as has been done in [1].
i fi(α b , r0, r1) gi(P b , αΛ) hi(cos θ, cos θ1, cos θ2) ℓi(ε) Let us briefly dwell on the powers of the velocity factor v in Table I. The common factor v in the fifth column of Table I has its origin in the phase space factor v in the decay V → ℓ + ℓ − . The remaining v dependence results from a dominant S-wave contribution in the factor (1 + 2ε) and a dominant S − D-interference contribution in the factor v 2 = (1 − 4ε), respectively, as can be seen by performing an LS analysis of the decay V → ℓ + ℓ − . The LS amplitudes M LS are given by M 01 = 2/3 (h V ). One then finds By integrating over two respective angles of the three polar angles one obtains the single angle distributions W (θ), W (θ 1 ) and W (θ 2 ). In their normalized forms they read where the factor defines the longitudinal polarization of the daughter baryon Λ and the factor is a measure of the longitudinal/transverse polarization composition of the vector charmonium state. Since the decay V → ℓ + ℓ − is electromagnetic and therefore parity conserving, the decay is not sensitive to the difference of the transverse-plus and transverse-minus helicity contributions For the same reason there is no linear cos θ 2 contribution in Eq. (23).
The distributions W (θ) and W (θ 1 ) are asymmetric in cos θ and cos θ 1 such that they can be characterized by the forward-backward (FB) asymmetries The distribution W (θ 2 ) is symmetric in cos θ 2 with its convexity parameter c f given by In Table II we have listed the numerical values of the two velocity factors v(1 + 2ε) and v 3 for the cases involving the muon and the tau lepton. It is quite apparent that the zero lepton mass approximation is quite good for J/ψ → µ + µ − and ψ(2S) → µ + µ − . For the case ψ(2S) → τ + τ − the factor v(1 + 2ε) provides a reduction of about 60% relative to the e + e − and µ + µ − cases while the factor v 3 becomes negligibly small. This means that for ψ(2S) → τ + τ − one loses the analyzing power of the lepton-side decay, i.e. of the last four rows of Table I. For the same reason, one will have an almost flat decay distribution W (cos θ 2 ) for ψ(2S) → τ + τ − because of the factor v 2 = 1 − 4ε in the expression Eq. (28) for the convexity parameter.

B. Azimuthal angle distribution
Consider the angular decay distribution of an unpolarized Λ b that is characterized by the two polar angles (θ 1 ) and (θ 2 ), and an azimuthal angle χ defined by the azimuth of the two decay planes defined by the decays V → ℓ + ℓ − and Λ → pπ − (see Fig. 2). The angular decay distribution can then be calculated from the master formula Let us first present a qualitative argument that the azimuthal correlation between the two decay planes is small. Azimuthal correlations result from the configurations which again follows from the fact that Λ b is treated as unpolarized. The azimuthal correlations are therefore determined by bilinear forms such as H 1 | (as also expected from the naive quark model) such that one concludes that the azimuthal correlations between the two decay planes are quite small. Let us cast this reasoning into a more quantitative form. The threefold angular decay distribution resulting from Eq. (29) reads Integrating over the hadron-side polar angle θ 1 one obtains If one wants to define a measure of the azimuthal correlation, one cannot integrate Eq. (31) over the whole range of the lepton-side polar angle θ 2 because 1 −1 d cos θ 2 sin 2θ 2 = 0. However, one can recover a nonzero azimuthal correlation measure by defining a FB asemmtry with respect to the lepton-side polar angle θ 2 by writing On reintroducing the normalized helicity amplitudes one obtains

IV The Λ b → Λ form factors in the covariant confined quark model
For the description of the couplings of the baryons Λ Q (Q = b, s) and the charmonium vector meson states V = J/ψ, ψ(2S) to their three and two constituent quarks, respectively, we employ a generic Lagrangian that reads The color index is denoted by a and C = γ 0 γ 2 is the charge conjugation matrix. In the baryon case we take the u and d quarks to be in a [ud] diquark configuration antisymmetric in spin and isospin. We emphasize, however, that we treat the u and d quarks as separate dynamical entities and not as a dynamical diquark state. Vertex functions in momentum space are obtained from the Fourier transformations of the vertex functions F H in Eqs. (34) and (35). In the numerical calculations we choose a simple Gaussian form for the vertex functions (for both mesons and baryons): where Λ H is a size parameter describing the distribution of the quarks inside a given hadron H. We use the values of these parameters fixed before in [29,30,32]. We would like to stress that the Minkowskian momentum variable P 2 turns into the Euclidean form − P 2 E needed for the appropriate falloff behavior of the correlation function (36) in the Euclidean region. We emphasize that any choice for the correlation functionΦ H is acceptable as long as it falls off sufficiently fast in the ultraviolet region of Euclidean space. The choice of a Gaussian form forΦ H has obvious calculational advantages.
For given values of the size parameters Λ H the coupling constants g ΛQ and g V are determined by the compositeness condition suggested by Weinberg [38] and Salam [39] (for a review, see [40]) and extensively used in our approach (for details, see [41]). The compositeness condition implies that the renormalization constant of the hadron wave function is set equal to zero where Σ ′ H is the on-shell derivative of the hadron mass function Σ H with respect to its momentum. The compositeness condition can be seen to provide for the correct charge normalization for a charged bound state (see e.g. [29]).
Next we discuss the calculation of the matrix element of the Λ b → Λ + V transition. We work in the so-called factorization approximation in which the matrix element for Λ b → Λ + V factorizes into a (b → s) current-induced matrix element < Λ|J µ |Λ b > and a (c → c) current-induced vacuum to vector meson matrix element < V |J µ |0 >. In our approach the Λ b → Λ transition is described by a two-loop Feynman-type diagram and the current-induced vacuum to vector meson transition is described by a one-loop Feynman-type diagram. The latter diagram is proportional to the leptonic decay constant of the vector meson denoted by f V . We have calculated f J/ψ before in Ref. [29] and have found f J = 415 MeV in almost perfect agreement with the measured value. In the calculation of quark-loop diagrams we use the set of model parameters fixed in our previous studies. The model parameters are the constituent quark masses m q and the infrared cutoff parameter λ responsible for quark confinement. They are taken from a fit done in the papers [29,30] The dimensional size parameters of the Λ b and Λ baryons have been determined in [32] by a fit to the semileptonic The resulting values are Λ Λ = 0.490 GeV, and Λ Λ b = 0.569 GeV. For the size parameter of the J/ψ we take Λ J/ψ = 1.482 GeV resulting from the fit in [29].
As of yet we cannot treat radial excitations in our approach. We therefore take the experimental value f ψ(2S) = 286.7 MeV for the ψ(2S).
It should be quite clear that the evaluation of the form factors is technically quite involved since it involves the calculation of a two-loop Feynman diagram with a complex spin structure resulting from the quark propagators and the vertex functions, which leads to a number of two-loop tensor integrals. To tackle this difficult task we have automated the calculation in the form of FORM [42] and FORTRAN packages written for this purpose. The q 2 behavior of the form factors are shown in Fig. 3.
is not shown since it would not be visible at the scale of the figure. The values of q 2 = 0 and q 2 = q 2 max correspond to the maximal and minimal recoil points, respectively.
The results of our numerical calculations are well represented by a double-pole parametrization Using such a parametrization facilitates further treatment such as the q 2 integrations without having to do a numerical evaluation for each q 2 value separately. The values of f (0), a and b are listed in Table III. It is quite noteworthy that the numerical values of a and b for each form factor in Table III are Table III. It is quite gratifying that for each case the effective dipole masses come out to be close to the average weighted mass (m Bs + 3m B * s /4) = 5.31 GeV of the ground state (bs) mesons, which would set the scale for the q 2 behavior of the form factors in a generalized vector dominance picture.
In Table IV we list our form factor results for three different values of q 2 and compare them to the results of the light-front diquark model calculation of [9] and the potential quark model calculation of [10]. At q 2 = m 2 J/ψ all three model form factors agree for the large form factors f V 1 and f A 1 , while the small form factors f V 2 and f A 2 of [9] differ from those of the other two models. There are larger discrepancies of the three sets of form factors for the other two q 2 values. In particular, at q 2 max the form factor values of [9] are much smaller than those of the other two models, while at q 2 = 0 the large form factors f V 1 and f A 1 of [10] come out much smaller than in the two other models. The q 2 = 0 values of our form factors in Table III slightly differ from those in Table IV because     Comparison of our form factor values at q 2 = 0, q 2 = m 2 J/ψ and q 2 = q 2 max with those obtained in [9,10]. At maximum recoil q 2 = 0 we can compare our results with the light-cone sum rules (LCSR) results of [43] on the Λ b → p transition form factors if we assume SU (3) to hold. In the limit of SU (3) the Λ b → Λ and Λ b → p form factors are related by F (Λ b → Λ) = 2/3 F (Λ b → p). This can be seen by using the3 ⊗ 3 → 8 Clebsch-Gordan table listed in [44]. Based on the observation that the [ud] diquark is the (Y = 2/3, I = 0) member of the3 multiplet one needs the Clebsch-Gordan (C.G.) coefficients The labeling in (40) proceeds according to the sequence | R, Y, I, I z > where R denotes the relevant SU (3) representation.
The LCSR results of [43] have been summarized in Table 1 of [45]. We take central values of the results listed in [45] and average over the two options of Λ b currents. We finally multiply these numbers by 2/3 and obtain f V 1 = 0.11 (0.10), f V 2 = 0.041 (0.039), f A 1 = 0.11 (0.099) and f A 2 = 0.018 (−0.0036) where we have added our model predictions in brackets. The agreement is quite satisfactory except for the small form factor f A 2 where we obtain a smaller value that differs in sign from that in [45].

V The heavy quark limit for the Λ b → Λ form factors
It is instructive to explore the HQL for the heavy-to-light transition Λ b → Λ in our form factor expressions. The HQL corresponds to the limit m Λ b , m b → ∞ while keeping the difference m Λ b − m b =Λ and the size parameter Λ Λ b fixed. The limit has to be taken in the relevant expressions for the coupling constants and the form factors.
First consider the local b−quark propagator that reduces to the static form where Φ ΛQ is the correlation function of the three constituent quarks with the coordinates x 1 , x 2 , x 3 and the masses m 1 , m 2 , m 3 , respectively. The variable w i is defined by In the present application m 1 = m b , m 2 = m u and m 3 = m d . In the limit m 1 → ∞ one has As it turns out, the next-to-leading order corrections m2 m1 and m3 m1 contribute significantly to the HQL and cannot be neglected in the numerical calculations. On the other hand, we must keep such terms because the terms p 1 w 2 ∼ m 2 and p 1 w 3 ∼ m 3 occurring in the vertex function do not vanish in the heavy quark limit (where p 1 is the momenta of the Λ b baryon).
In the HQL the coupling constant g ΛQ does not depend on the b−quark mass and the Λ b mass [there is a dependence on the O(m 0 Q ) parameters -Λ and Λ]. This is specific to the three-quark system. For example, in the meson case the meson-quark coupling constant scales as √ m b when m b → ∞. The constancy of g ΛQ in the HQL will be used to demonstrate the validity of the HQL for the transition form factors. First, we stress that the Ward identity relating the derivative of the mass operator and the electromagnetic vertex function at p 1 = p 2 for the heavy baryon with charge ±1 is still valid. To show the validity of the Ward identity we consider the charged baryon Λ c (the heavy quark symmetry partner of the Λ b baryon), which has the charge e Λc = 1. Using a Ward identity one can rewrite the compositeness condition Z Λc = 0 for the heavy Λ c baryon, in the form (see Ref. [32]) where the electromagnetic vertex function Λ µ Λc (p, p) obtains contributions from the electromagnetic current coupling to the quark lines (triangle contributions) and the vertices (bubble contributions) (see Ref. [32] for details). To make the HQL more transparent we perform a shift of the loop momenta k 1 → k 1 + (w 2 + w 3 )p and k 2 → k 2 + w 2 p. One has Λ µ Λc (p, p) = 6 g 2 Λc where the double bracket notation << . . . >> stands for the two integrations over the loop momenta (see Ref. [32] ). Also we use the definitions and Here s ΛQ ≡ 1/Λ 2 ΛQ and the argument of the vertex function is The calculational techniques of the matrix elements in the heavy quark limit can easily be demonstrated for the example of the structure integral occurring in the heavy meson case (the extension to the heavy baryon case is straightforward) where s = 1/Λ 2 . For simplicity we only keep the product of the heavy quark propagator and the denominator of the light quark propagator. We start with the Schwinger representation for the quark propagators assuming that both loop and external momenta are Euclidean. In the Euclidean region the denominator of the quark propagator is positive and the integral over the Schwinger parameter is absolutely convergent. However, to use the Schwinger representation for the heavy quark propagator in Eq. (41) in a straightforward way is not quite correct because the HQL has to be taken in Minkowski space where the denominator is not necessarily positive. We will use the heavy quark propagator in the form assuming again that all momenta in the exponential are in the Euclidean region. For the numerator of the heavy quark propagator we take the HQL: m Q + k + p → m Q (1+ v). Next we demonstrate how to proceed with the HQL for such a representation. As mentioned above, we start in the Euclidean region where k 2 ≤ 0 and (k + p) 2 ≤ 0. By using Schwinger's representation for the heavy quark propagator and the denominator of the light quark propagator, scaling the Schwinger parameters α i → tα i and imposing an infrared cutoff, we arrive at We will use this representation for the analytical continuation to the physical region p 2 = (m Q +Λ) 2 with m Q → ∞. Note that in a theory without a cutoff (λ → 0) the integral I 2 (Λ) has a branch point atΛ = m. The confinement ansatz allows one to remove this singularity. Then we scale the integration variable α → α/m Q with m Q → ∞.
Finally, one has The calculation of the HQL for the coupling constant g Λ b and the transition form factors Λ µ Λ b →Λ (p 1 , p 2 ) proceed in the described way. All analytical calculations are done by FORM [42] and the numerical calculations are done using FORTRAN. One finds The coupling constant in the HQL is smaller than the exact coupling constant only by about 10 % which shows that one is quite close to the HQL for the Λ b → Λ transitions. When calculating the HQL for the coupling constant it is important to keep the numerical value of the parameterΛ b fixed at its physical valueΛ b = m Λ b − m b = 0.53 GeV. In fact, the value of g HQL Λ b depends very sensitively on the choice of the parameterΛ b . For example, if one putsΛ b = 0 then one calculates g HQL Λ b = 185.36 GeV −2 which differs significantly from the exact result g Λ b = 65.23 GeV −2 . This demonstrates how important it is to keep the physical value ofΛ b . It is interesting to compare our results for the coupling constants g Λ b and g Λc . For g Λc one finds One can see that there is a small difference between the coupling constants in the exact case, while in the HQL they are practically degenerate. This happens because, as was emphasized before, the coupling constant g ΛQ in the HQL does not depend on the heavy quark mass. A small difference of the coupling constants g Λ b and g Λc in HQL is due to dependence on O(m 0 Q ) parameters -Λ and Λ). The vertex function Λ µ Λ b →Λ (p 1 , p 2 ) that describes the heavy-to-light Λ b → Λ transition reads where In the present application we have to consider two cases of Γ µ : γ µ and γ µ γ 5 . The loop calculation contains the variable v 1 p 2 that, for a given q 2 , is fixed by the kinematics of the process through where p 2 and m Λ are the momentum and mass of the Λ.
In the heavy quark limit we reproduce the form factor structure derived previously from heavy quark effective theory [46][47][48], which is usually written in the form Again, for the present application, Γ µ is γ µ or γ µ γ 5 . From Eq. (58) one finds i.e. there are only two independent form factors in the HQL. We emphasize that the form factor relations Eq. (59) are valid in the full kinematical region 4m 2 Fig. 4 we plot the q 2 dependence of the form factors f V 1 and f V 2 and compare them to the corresponding form factors calculated in the HQL. For the large form factor f V 1 the HQL form factor exceeds the full form factor by O(10 %) while the small form factor f V 2 is lowered by O(50 %).
It is interesting to compare our HQL form factors with the corresponding static lattice results presented in [49]. Although the main concern of [49]   We mention that the static results given in [49] approximately satisfy the SU At the large recoil end of the q 2 spectrum soft collinear effective theory predicts that the form factor F 2 vanishes in the heavy quark limit at the leading order of α s [45,50]. The plot of f V, HQL 2 (q 2 ) = −F 2 (q 2 ) in Fig. 4 shows that our prediction for F 2 reaches a very small albeit nonzero value at large recoil.

VI Numerical results
We begin by presenting our numerical results for the decay Λ b → Λ + J/ψ(→ e + e − , µ + µ − ) for which one can safely use the zero lepton mass approximation (see Table II). Our results are presented in Tables V-VII where we compare them with the available data [1,11] and predictions of other theoretical approaches [3]- [10]. In Table V we present our result on the branching fraction B(Λ b → ΛJ/ψ) and compare it with data and the results of other theoretical models.
given in [11] and a value of B(b → Λ b ) = 0.1 as used in previous editions of the PDG. Our result on the branching fraction is based on the lifetime measurement 1.429 × 10 −12 s as listed in the 2013 update of the PDG [11]. If one would instead take the value of 1.482 × 10 −12 s reported in [51], one would have to scale our result on the branching fraction upward by 3.7 %. For easy comparison we have taken the freedom to present the results of [9, 10] using our parameters (Λ b life time, CKM matrix elements and Wilson coefficients). Our branching fraction is slightly larger than those of [9,10]. All three branching fractions are somewhat larger than the experimental PDG average value. To judge on the significance of this discrepancy one would have to wait for an absolute measurement of the branching fraction B(Λ b → ΛJ/ψ). We mention that the remaining theoretical branching fractions in Table V would have to be readjusted upward by (4 − 7)% (depending on the year of publication) if the new 2013 lifetime measurement of the LHCb Collaboration [51] is used.
In Table VI we present our result for the asymmetry parameter α b and compare it with the data and the results of other theoretical models. In agreement with the measurement the theoretical results on the asymmetry parameter come out to be quite small. Since the measurement carries large error bars, one cannot really draw any conclusions on the quality of the agreement between the experiment and the model predictions.
In Table VII we compare our predictions for the asymmetry parameters and the moduli squared of the normalized helicity amplitudes with the corresponding measurements of [1]. The fourth column of Table VII contains our predictions for the corresponding quantities in the decay Λ b → Λ + ψ(2S), which differ notably from the corresponding quantities in the Λ b → Λ + J/ψ mode. The numerical results clearly show the dominance of the λ Λ = −1/2 helicity configurations in both cases where the dominance is more pronounced for the Λ b → ΛJ/ψ mode. For the Λ b → ΛJ/ψ mode the agreement between our results and the data is quite satisfactory.
Using our results in Table VII we can calculate the three measures Eqs. (26)-(28) characterizing the single angle decay distributions Eqs. (21)(22)(23). For the polarization related FB asymmetry A F B θ calculated in Eq. (26) one obtains The analyzing power related to the measurement of the Λ b -polarization P b is quite small due to the fact that in our model | H + 1 2 0 | 2 ≈ | H + 1 2 +1 | 2 ≈ 0 and | H − 1 2 0 | 2 ≈ | H − 1 2 −1 | 2 leading to a very small value of the parameter α b . This is reflected in the poor precision of the polarization measurement P b = 0.05 ± 0.07 ± 0.02 reported by the LHCb Collaboration [1].
Negative convexities correspond to a downward open parabola for the decay distribution as is also evident from Fig. 5. Figure 5 also contains a plot of the cos θ 2 distribution for the case Λ b → Λ ψ(2S)(→ τ + τ − ) for which the convexity parameter is given by c f = −0.016 implying an almost flat cos θ 2 distribution. Because of the small factor v 2 = 1 − 4ε = 0.071 in the cos θ 2 distribution Eq. (23) the lepton side has lost all of its analyzing power in this case. As has already been argued after Eq. (29) one expects a minimal azimuthal correlation of the two decay planes spanned by (pπ − ) and (ℓ + ℓ − ). In the zero lepton mass approximation one obtains the azimuthal decay distribution [see Eq.
to evaluate the other modes. The results for the branching ratios of the decays Λ b → Λ + ψ(2S)(→ ℓ + ℓ − ) are given in Table VIII. Again, one can see that the τ -lepton mass plays an essential role in reducing the value of the decay width as compared to the electron and muon modes. Our prediction for the branching fraction of the Λ b → Λ + ψ(2S) transition is (based on the lifetime value τ Λ b = 1.429 × 10 −12 s) B(Λ b → Λ + ψ(2S)) = 7.25 × 10 −4 . (65)

VII Summary
We have performed a detailed analysis of the decay process Λ b → Λ + J/ψ in the framework of the covariant quark model. We have worked out two variants of threefold joint angular decay distributions in the cascade decay Λ b → Λ(→ pπ − ) + J/ψ(→ ℓ + ℓ − ) for polarized and unpolarized Λ b decays. We have reported our numerical results on helicity amplitudes, on the rate and on the asymmetry parameters in the decay processes Λ b → Λ + J/ψ and Λ b → Λ + ψ(2S). We have included the decay Λ b → Λ + ψ(2S) in our analysis since this decay allows one to discuss nonzero lepton mass effects in the kinematically allowed decay Λ b → Λ + ψ(2S)(→ τ + τ − ). We confirm expectations from the naive quark model that the transitions into the λ Λ = 1/2 helicity states of the daughter baryon Λ are strongly suppressed leading to a near maximal negative polarization of the Λ. For the same reason the azimuthal correlation between the two decay planes spanned by (pπ − ) and (ℓ + ℓ − ) is negligibly small. We have compared our results with the available experimental data and with the results of other theoretical approaches. In a separate section we have presented form factor results over the whole accessible range of q 2 values. These results are close to lattice results at minimum recoil and to LCSR results at maximum recoil.