Ajaltouni, Ziad,

Pakhlova, G.

Robbe, P.

Ajaltouni, Ziad, Zj

HAL-IN2P3

Weak radiative decays of beauty baryons - PedagogicalapproachZiad Ajaltouni, G. Pakhlova, P. RobbeTo cite this version:Ziad Ajaltouni, G. Pakhlova, P. Robbe. Weak radiative decays of beauty baryons - Pedagogicalapproach. 2004, pp.1-8. <in2p3-00020447>HAL Id: in2p3-00020447http://hal.in2p3.fr/in2p3-00020447Submitted on 25 Feb 2004HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.LHCb 2004-009PHYSICSFebruary 2004Weak Radiative Decays of BeautyBaryonsPedagogical ApproachZ.J.Ajaltouni1, G.Pakhlova2, P.Robbe31 Laboratoire de Physique Corpusculaire de Clermont-FerrandIN2P3/CNRS Universite´ Blaise PascalF-63177 Aubie`re Cedex FRANCE2 Institute of Theoretical and Experimental Physics (ITEP), Moscow, RUSSIA3 Universite´ Paris-Sud, LAL, Orsay, FRANCEAbstractWeak decays of beauty baryons Bb = Λb ,Ωb oﬀer the opportunity to performsome tests of CP symmetry and time-reversal invariance TR. In this note, theradiative two-body decay channel is studied and emphasis is put on the angulardistributions of ﬁnal particles and on the importance of the polarization of the Bbbaryon.11 IntroductionWeak decays of beauty baryons represent a wide investigation ﬁeld for checking symmetrylaws like CP or TR (Time -Reversal) symmetry [1]. In the case of weak radiative decay(WRD) like Λb → Λ + γ , interesting tests of Λ polarization can be performed in orderto cross-check the Standard Model predictions. Same kind of tests can be done in thecase of three body decays like Λb → Λ + − where the lepton pairs could come eitherfrom a vector-meson like J/ψ or a virtual photon. Furthermore, because of the Λ ﬁnalpolarization, TR invariance can be checked with observables built from the kinematicsparameters of the ﬁnal particles.On the phenomenological side, the WRD of any baryon B1 → B2 + γ is described bythe phenomenological lagrangian [2] :L = ψ¯1(C + Dγ5)σµνF µνψ2 + h.c.where coeﬃcients C and D are respectively the Parity Conserving (PC) and the ParityViolating (PV) parts. From this lagrangian, the matrix element describing the radiativetransition can be computed and the corresponding radiative width is given by :Γ(B1 → B2 + γ) = 18π(M12 −M22)M1(|C|2 + |D|2)Mi being the mass of the baryon Bi.Furthermore, coeﬃcients C and D are also related to asymmetries arising from theﬁnal state decays and help to probe Parity and CP symmetries in baryonic weak decays.[2]2 Angular Momentum AnalysisPerforming angular momentum analysis is an important task to cross-check the validityof symmetry laws, especially CP symmetry, in baryonic weak decays. We will focus onthe channel Λb → Λ + γ followed by the decay Λ → p + π− which, at the quark level,corresponds to the penguin electroweak process b→ s + γ .Let J be the spin of the decaying baryon, L is the orbital angular momentum of theγΛ system and S is their total spin. Due to the massless nature of the photon, it couldnot state that S = s1 + s2 , s1 and s2 being respectively the spin of the photon and theﬁnal baryon. Indeed, the spin angular momentum of the photon cannot be disentagledfrom its orbital one and, according to the Landau-Lifschitz approach [3], it can be setthat the parity of the ﬁnal γΛ system is given by :P (γΛ) = (−1)(+1)P (Λ)where  is an L eigenvalue with two possible values  = 0, 1.While P (Λb) = P (Λ) = +1, the following relations can be inferred : = 0 ⇒ P (γΛ) = −1 =⇒ Parity Violation2 = 1 ⇒ P (γΛ) = +1 =⇒ Parity ConservationIn the remaining study, the Λb rest-frame will be used :• The z-axis is given by the Λb momentum in the laboratory frame, z || p0 = pΛb/pΛbIt is chosen as the quantization axis in the helicity formalism.• The y-axis is given by the direction of the vector product p0× pΛ• Finally the x-axis is obtained as x = y×zpΛ being the Λ momentum in the Λb rest-frame.Helicity configurationsThe spin of the initial decaying particle being 1/2 and the photon helicity having only twovalues, λγ = ±1, the helicity of the ﬁnal state according to the (∆)-axis where (∆) || pΛis well constrained :λf = λΛ − λγ = ± 1/2So, two conﬁgurations may arise :(λΛ, λγ) = (1/2, 1) =⇒ λf = −1/2(λΛ, λγ) = (−1/2,−1) =⇒ λf = +1/2Departing from these two helicity states and from the initial spin component of theΛb, m = sz(Λb), the decay amplitude can be computed. The diﬀerent computationalsteps are given below :• Let m,λΛ, λγ and pΛ = (p, θ, φ) be ﬁxed. According to the Jackson convention [4]for the Wigner rotation matrices, the decay amplitude can be written as :Am(pˆ, λΛ, λγ) ∝ < jm;λΛλγ|S|jm > Djmλ(φ, θ, 0)where Djmλ(φ, θ, 0) is the representation of the rotation matrix given byDjmλ(φ, θ, 0) = exp(−imφ) djmλ(θ) and λ = λfIn a more compact form, the above amlitude can be written as :Am ∝ M(λΛ, λγ)Djmλ(φ, θ, 0)•• In a second step, we must take into account the polarization of the initial baryonΛb when this latter is produced in the proton-proton collisions. The most suitable way isto introduce the polarization density-matrix of the Λb whose expression is :3(ρ++ ρ+−ρ−+ ρ−−)where indice + is related to the positive value of the spin projection m = +1/2 andindice − indicates the negative one, m = −1/2 ; ρ++ = ρ 12, 12, etc..The density-matrix (DM) is hermitean , ρij = ρji and its trace veriﬁes Trρ = 1.So, a general expression of the ﬁnal angular distributions W (θ, φ) must take accountof the following physical requirements :1) Instead of the ordinary squared matrix element |Am|2, a summation over the DMelements must be performed :Σm,m′ρmm′AmAm′ with m,m′ = ±1/22) The helicities of the ﬁnal particles γ and Λ being not measured, a summation overλ1 = λγ and λ2 = λΛ must be done too. These last ones appear in the ﬁnal helicityaccording to the ∆-axis which takes only two values, λ = λ1 − λ2 = ±1/2.Thus the ﬁnal angular distribution will be given by the expression :W (θ, φ) ∝ Σλ1,λ2 (Σm,m′ρmm′AmAm′)∝ Σm,m′,λρmm′M(λ1, λ2)M(λ1, λ2)DjmλDjm′λThe above relation contains only 8 terms instead of 16 and if we replace Djmλ by theirstandard expression, the ﬁnal angular distribution will be :W (θ, φ) = N Σm,m′,λ|M(λ)|2exp(i(m−m′)φ)djmλ(θ)djm′λ(θ) (1)N being a normalization factor. Among the 8 terms, only two diﬀerent values ofM arise, M(++) = M(1/2, 1) and M(−−) = M(−1/2,−1). Expressing this relationaccording to the DM elements ρij , we get :W (θ, φ) = |M(++)|2(ρ++sin2(θ/2) + ρ−−cos2(θ/2)−(ρ+−exp(iφ))sin θ)+ |M(−−)|2(ρ++cos2(θ/2) + ρ−−sin2(θ/2) + (ρ+−exp(iφ))sin θ)Due to the algebraic properties of the DM, there are three unknown parameters :ρ++ , (ρ+−) and (ρ+−)It is worth noticing that the matrix elements M(++) and M(−−) correspondrespectively to the right-handed photon and to the left-handed one and they will bewritten as MR and ML respectively.43 Final Angular DistributionsThe ﬁnal step consists in averaging over the azimuthal angle φ around the z-axis.Consequently, the interference terms represented by the non-diagonal matrix elementρ+− disappear and a simple expression is obtained for the polar angle distribution θ :R(θ) =dNd cos θ∝ |MR|2(ρ++sin2(θ/2) + ρ−−cos2(θ/2))+ |ML|2(ρ++cos2(θ/2) + ρ−−sin2(θ/2))At this stage, new physical parameters can be introduced :i) The initial polarization of the Λb baryon, P = PΛb = ρ++ − ρ−−ii) The asymmetry parameter of the photon (due to its own polarization),αγ =|ML|2 − |MR|2|ML|2 + |MR|2The ﬁnal expression of R(θ) becomes :R(θ) =|MR|2 + |ML|22(1 + αγPcos θ) (2)Interesting remarks can be drawn from the expression above :• This angular distribution is identical to the standard one describing the weakhyperon two-body decay like Ξ , Σ→ Λ+ γ and, as it is expected, this distributionviolates parity symmetry because R(π − θ) = R(θ) [5]• To detect any parity violation in Λb decay, the photon must have diﬀerent helicityweights, |MR|2 = |ML|2 =⇒ αγ = 0• To get evidence for a photon asymmetry even when this one is polarized, the initialbaryon Λb must be polarized, ρ++ = ρ−−So, knowing the polarization density-matrix of the Λb is an essential ingredient toperform any test of symmetry violation like CP or TR with the beauty baryon decays.Simulation MethodThe main point in simulating the decay process Λb → Λ+γ is the polar angle generation.By normalizing the R(θ) distribution, one can get the probability density function :f(x) =12(1 + Ax) where x = cos θwith A = αγP and |A| ≤ 1By taking a random variable Y uniformly generated in the interval [0, 1], x is obtainedfrom the following relation :x = −1 +√1 + A2 + A(4Y − 2)54 How to generate Λb → Λγ decay with EvtGenSuch decays Λb → Λγ can be generated with EvtGen[6], the event generator usedin the LHCb simulation software (Gauss)[7]. This section explains how to write thecorresponding decay ﬁle and shows some results obtained with the generator.4.1 Decay fileThe decay model to use to generate this decay mode is HELAMP. In this case it needs 4parameters that have to be given as arguments to the model. The parameters are themagnitudes and phases of the amplitudes M(++) and M(−−) deﬁned in Eq. (1).The syntax of the decay ﬁle will be:Decay Lambda b0sig1.000 Lambda0 gamma HELAMP |M(++)| arg[M(++)] |M(--)| arg[M(--)];EnddecayCDecay anti-Lambda b0sigEndwhere Lambda b0sig is the alias to the signal Λb in the event.It is important to note that Λb will be produced unpolarized by Pythia in the currentGauss software versions.4.2 Some results obtained with EvtGenFig.1 shows the cosine of the decay angle θ as deﬁned in section 2 obtained with a specialproduction where the Λb were fully polarized, with parameters:M(++) = 1M(−−) = 0ρ++ = 1ρ−− = 0As expected, the distribution follows Eq. (2) with αγ = −1 and P = +1.6References[1] Hyperon decays and CP nonconservationJ.F.Donoghue et al, Phys.Rev. D34(1986), 833[2] Theory of Weak Interactions in Particle PhysicsR.E.Marshak et, Wiley-Interscience 1969[3] Quantum ElectrodynamicsLandau-Lifschitz, Course in Theoretical Physics, Vol. IV, (Pergamon)[4] Resonance DecaysJ.D.Jackson, Lectures in Les Houches Summer School (1965)[5] General Theory of Angular Correlations of Decaying ParticlesY.Ueda, S.Okubo Nucl.Phys.49 (1963), 345[6] http://www.slac.stanford.edu/ lange/EvtGen/[7] Gauss User Guide and Reference Manual7Figure 1: Helicity angle8

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