$\Lambda_b$ Decays into $\Lambda$-Vector

A complete study of the angular distributions of the processes, $\Lambda_b \to {\Lambda} V(1^-)$, with $\Lambda \to p {\pi}^-$ and $V (J/{\Psi}) \to {\ell}^+ {\ell}^-$ or $V ({\rho}^0,\omega) \to {\pi}^+ {\pi}^-,$ is performed. Emphasis is put on the initial $\Lambda_b$ polarization produced in the proton-proton collisions. The polarization density-matrices as well as angular distributions are derived and help to construct T-odd observables which allow us to perform tests of both Time-Reversal and CP violation.


Introduction
With the advent of B-factories at the proton-proton colliders, huge statistics of beauty hadrons are expected to be produced. This will allow a thorough study of CP violation processes with B mesons. Moreover, some specific phenomena related to either b-quark physics or CP violation can be performed to put limits on the validity of the Standard Model (SM). One of these processes concerns the validity of the Time Reversal (TR) symmetry. A promising method to look for TR violation is the three body Λ b decay [1,2] as it was initiated with the hyperons long time ago by R. Gatto [3].
T-odd operator is derived from Time Reversal and it keeps the initial and final states unchanged. It is well known that the time reversing state of a decay like Λ → pπ − or β nucleon decay cannot be realized in the physical world, thus we must be contented with the following transformations which are the main ingredients of TR operator: where ℓ and s are respectively the angular momentum and the spin of any particle with momentum p. Consequently the helicity of the particle defined by λ = s · p/p remains unchanged by TR transformations.
In the past, it was pointed out by many authors [4] the importance to look for T-odd effects in the hyperon decays like Λ, Σ and Ξ, as being a consequence of both CP T theorem and CP violation in weak |∆S| = 1 decays. As far as beauty hadrons Λ b , Σ b and Ξ b are concerned, because of their numerous decay channels and the strength of CP violation in the b-quark sector, opportunities to find T-odd observables will increase and interesting tests of both the SM and models beyond the SM can be performed successfully. Due to the initial polarization of the Λ b baryon, T-odd observables can be constructed from the decay products such as v 1 · ( v 2 × v 3 ) where v i is either the spin or the momentum of the particle i. These observables change sign under TR transformations and a non-vanishing mean value of their distribution could be a sign of TR violation. This paper is devoted to a study and simulations of Λ b decays into Λℓ + ℓ − and Λh + h − . Final leptons, ℓ = e, µ, or final hadron, h = π, can originate from intermediate resonances which quantum numbers are those of a vector meson 1 − like J/ψ, ρ 0 and ω. The reminder of this paper is organized as it follows. In section 2, we present an analysis of both the intermediate states and the final particles in some appropriate frames, the helicity frames. By stressing on the importance of the polarizations of the initial Λ b as well as the intermediate resonances, calculations based on the helicity formalism are performed and take into account the spin properties of the final decay products. Dynamical assumption is made through the factorization framework applied in baryon decays in section 3. The following section is devoted to results and discussions for angular distributions and polarization density matrices. Finally, in the last section, we draw some conclusions.

Λ b decay analysis
In the collisions, pp → Λ b +X, the Λ b is produced with a transverse polarization in a similar way than the ordinary hyperons. Its longitudinal polarization is suppressed because of parity conservation in strong interactions. Let us define, N P , the vector normal to the production plane by: where p 1 and p b are the vector-momenta of one incident proton beam and Λ b , respectively. The mean value of the Λ b spin along N P is the Λ b transverse polarization usually greater than 20% [5]. Let (Λ b xyz) be the rest frame (see Fig. 1) of the Λ b particle. The quantization axis (Λ b z) is chosen to be parallel to N P . The other orthogonal axis (Λ b x) and (Λ b y) are chosen arbitrarily in the production plane. In our analysis, the (Λ b x) axis is taken parallel to the momentum p 1 . The spin projection, M i , of the Λ b along the transverse axis (Λ b z) takes the values ±1/2. The polarization density matrix 4 , ρ Λ b , of the Λ b is a (2 × 2) hermitian matrix. Its elements 5 , ρ Λ b ii , are real and 2 i=1 ρ Λ b ii = 1. The probability of having Λ b produced with M i = ±1/2 is given by ρ Λ b 11 and ρ Λ b 22 , respectively. Finally, the initial Λ b polarization, P Λ b , is given by is obtained by applying the Wigner-Eckart theorem to the S-matrix element in the framework of the Jacob-Wick helicity formalism [6]: where p = (p, θ, φ) is the vector-momentum of the hyperon Λ in the Λ b frame (Fig. 1). .
, contains all the decay dynamics. Finally, the Wigner matrix element, is expressed according to the Jackson convention [6].
In case of two intermediate resonances such as those described in the next section, the Λ b -decay plane is defined by the momenta of the Λ and leptons (or hadrons). This decay plane does not coincide with that one defined by the momenta of the J/Ψ, proton and pion.

Decay of the intermediate resonances
By performing appropriate rotations and Lorentz boosts, we can study the decay of each resonance in its own helicity frame (see Fig. 1) such that the quantization axis is parallel to the resonance momentum in the Λ b frame i.e. 4 The polarization density matrix elements (PDM), ρ Λ b ij , do not need to be exactly known since the initial, [5] is only required in our analysis. 5 Note as well that the respective helicities of the final particles are (λ 3 , λ 4 ) = (±1/2, 0) and (λ 5 , λ 6 ) = (±1/2, ±1/2) in case of leptons or (λ 5 , λ 6 ) = (0, 0) in case of 0 − mesons.
In the Λ helicity frame, the projection of the total angular momentum, m i , along the proton momentum, p P , is given by m 1 = λ 3 − λ 4 = ±1/2. In the vector meson helicity frame, this projection is equal to m 2 = λ 5 − λ 6 = −1, 0, +1 if leptons and m 2 = 0 if π. The decay amplitude, A i (λ i ), of each resonance can be written similarly as in Eq. (2), requiring only that the kinematics of its decay products are fixed. We obtain, where θ 1 and φ 1 are respectively the polar and azimuthal angles of the proton momentum in the Λ rest frame while θ 2 and φ 2 are those of ℓ − (h − ) in the V rest frame.

Analytical form of the decay probability
The general decay amplitude 6 , A I , for the process 7 must include all the possible intermediate states so that a sum over the helicity states (λ 1 , λ 2 ) is performed: The decay probability, dσ, depending on the amplitude, A I , takes the form, where the polarization density matrix, , is used to take into account the unknown Λ b spin component, M i . Since the helicities of the final particles are not measured, a summation over the helicity values λ 3 , λ 4 , λ 5 and λ 6 is performed as well. Finally, the decay probability, dσ, written in a such way that only the intermediate resonance helicities appear, reads as, where F Λ describing the decay dynamics of the intermediate resonances Λ → P π − and V → ℓ + ℓ − , respectively, are given in Appendix. Because of parity violation in weak hadronic decays, it is assumed that

Factorization procedure
In tree approximation, the effective interaction 8 , H ef f , written as, gives the weak following amplitude factorized into, The CKM matrix elements, V qb V ⋆ qs , read as V ub V ⋆ us and V cb V ⋆ cs , in case of Λ b → Λρ and Λ b → ΛJ/Ψ, respectively. The Wilson Coefficients, c i , are equal to c 1 = −0.3 and c 2 = +1.15. The hadronic matrix element, Λ|sγ µ (1 − γ 5 )b|Λ b , can be derived respecting Lorentz decomposition. Working in HQET, it is more convenient to use [7], where the four-velocity of , depending on the helicity state, (λ Λ , λ V ), reads as, (11) The q 2 dependence of the transition form factors, F i (q 2 ), or (F ± (q 2 )), resulting from QCD sum rules and HQET [8] takes the form as it follows, where the following values (0.462, −0.0182, −1.76×10 −4 ) and (−0.077, −0.0685, 1.46×10 −3 ) correspond to (F (0), a, b) in case of F 1 (q 2 ) and F 2 (q 2 ), respectively. We refer to the PDG [9] for all the numerical values used in our analysis. 8 All the terms of the effective interaction are extensively defined in literature. 9 We define F ± (q 2 ) = F 1 (q 2 ) ± F 2 (q 2 ), for convenience. 10 In Eq. (11), the factor, is not written only for simplicity.
Owing to the spin 1/2 of the Λ b , the angular momentum projection along the helicity axis (which direction is given by the Λ vector-momentum) has only two values, M i = ±1/2, with respective weights generally different. The helicity asymmetry parameter, α Λ b As , defined in Eq. (14), takes the following values: From these results, the angular momentum projection, M i = 1/2, appears to be largely dominant in the analyzed decays. The Λ-polarization, 17), can be computed in both decay cases. After normalization of P Λ , we obtain the values, P Λ = +31%, and P Λ = −9%, for Λ b → Λρ 0 and Λ b → ΛJ/ψ, respectively. The other important parameter concerning the spin state of the intermediate resonances is the density matrix element, ρ V ij , defined in Eq. (20). Let us focus on the matrix element, ρ V 00 , which is related to the longitudinal polarization of the vector meson V. After calculation, 65.5% and 55.5% are the results for the density matrix element, ρ V 00 , in case of Λ b → Λρ 0 and Λ b → ΛJ/ψ, respectively. It is important to notice that these parameters, α Λ b AS and ρ V ij , (as well as ρ Λ ij ) govern entirely the angular distributions, W i (θ i , φ i ), of the final particles in each resonance frame.
In Fig. 2, are shown the polar angular distributions (which do not depend on Λ b initial polarization) for proton and l(h) coming respectively from Λ and V decays. In the same figure, the transverse momentum distributions, P P ⊥ and P π ⊥ (Λ daughters) given in the Λ b rest frame, are plotted. These distributions look to be discriminant in the investigation of Λ b decay observables.
Finally, the last step is the computation of the branching ratios, BR(Λ b → Λρ 0 ) and BR(Λ b → ΛJ/ψ), which requires the calculation of their corresponding widths. The standard expression of a decay width, Γ(Λ b → ΛV ), is given by, where E Λ and P V are respectively the energy and momentum of the Λ baryon and vector meson in the Λ b rest frame. Ω corresponds to the decay solid angle. Performing all the calculations and keeping the number of color, N ef f c , to vary between the values 2 and 3 as it is suggested by the factorization hypothesis, we obtain the following branching ratio results: respectively for N ef f c = 2, 2.5 and 3. These interesting results suggest that the effective number of color might be taken greater than 2.5 in the framework of the factorization hypothesis in case of Λ b decay. It is worth comparing the theoretical branching ratio, BR th (Λ b → ΛJ/ψ), with the experimental one [9], BR exp (Λ b → ΛJ/ψ) = (4.7 ± 2.1 ± 1.9) × 10 −4 .

Conclusion
Calculations of the angular distributions as well as branching ratios of the process Λ b → ΛV with Λ → P π − and V → ℓ + ℓ − or V → h + h − have been performed by using the helicity formalism and stressing on the correlations which arise among the final decay products. In all these calculations, particular role of the Λ b polarization has been put into evidence. The initial polarization, P Λ b , appears explicitly in the polar angle distribution of the Λ hyperon in the Λ b rest-frame. Similarly, the azimuthal angle distributions of both proton and ℓ − in the Λ and V frames, respectively, depend directly on the Λ b polarization. Furthermore, a first computation of the asymmetry parameter, α As , in Λ b decays into ΛV (1 − ) has been performed as well as the longitudinal polarization of the vector meson, ρ V 00 , which is shown to be dominant (≥ 56%). On the other hand, it is well known that the violation of CP symmetry via the CKM mechanism is one of the corner-stone of the Standard Model of particle physics. Looking for TR violation effects in baryon decays provides us a new field of research: firstly as a complementary test of CP violation by assuming the correctness of the CP T theorem and, secondly, as a possibility to search for processes beyond the Standard Model. In particular, triple product correlations, which are T -odd under time reversal, can be extensively investigated in Λ b decays. However, this latter aim requires both experimental and theoretical improvements in order to increase our knowledge of b-physics.
the final angular distribution, W (θ, φ), deduced 11 from Eq. (7) and expressed as puts into evidence the parity violation.