Simulation of the Polarized Fermion Decay
aa r X i v : . [ h e p - ph ] A p r Simulation of the Polarized Fermion De aysA.A. Ashimova 1Mos ow State University,Mos ow 119992, RussiaS.R. Slabospitsky 2State Resear h CenterInstitute for High Energy Physi s,Protvino, Mos ow Region 142281, RussiaAbstra tIn this paper the modi(cid:28) ation of the method onventionally used forthe modeling of the massive fermions produ tion and de ays is pro-posed. The step by step algorithm is presented. Under the stri t onditions the proposed method of modeling allow distin tly raise thee(cid:30) ien y of the omputations.1E-mail: ashimova_aa @ mail.ru2E-mail: Sergey.Slabospitsky @ ihep.ruimulation of massive fermion produ tion (let us denote it F ) with sub-sequent de ay frequently o urs in High Energy Physi s. Indeed, the angulardistributions of the fermion de ay produ ts are very sensitive to its polariza-tion.The most obvious way is the al ulation of the squared amplitude of thewhole pro ess in luding the fermion de ays. However su h way may be a - ompanied by some di(cid:30) ulties. First, the al ulation of the amplitude of thewhole pro ess (in luding the de ay hain) ould be rather umbersome. Moreover the orresponding (cid:28)nal state an appear without the massive fermion ontribution. In other words the (cid:28)nal state of the pro ess ould be the resultof the additional diagrams not ontaining F at all (see Fig. 1). bd WF Wbu a) b bd uWg á) Figure 1: The example of the diagrams des ribing the pro ess with the same (cid:28)nalstate. Diagram a) orresponds to the massive fermion F resonan e ontribution.Here F de ays into bW . Diagram b) has the same (cid:28)nal state, but not ontainsfermion F .However, one an assume that the invariant mass of the F de ay produ tsis lose to fermion mass m F . In this ase the ontribution of non-resonan ediagrams (see Fig.1b) will be notably suppressed.Fairly often the problem ould be simpli(cid:28)ed with the narrow resonan eapproximation, i.e. the intermediate fermion is onsidered to be ¾on theshell¿ ( p F = m F , p F is 4-momentum of the fermion F ). Su h approa hallows to simplify the al ulations and split the simulation into two steps:1. the produ tion of the massive polarized fermion;2. the subsequent de ay of the polarized fermionThereby the squared amplitude of the pro ess an be written in the ¾fa tor-ized¿ form: | A | = | A P rod ( s ) | · | A Dec ( s ) | ⊗ Φ( s ) . (1)1ere A P rod ( s ) and A Dec ( s ) are the amplitudes des ribing the produ tion andde ay of the polarized fermion, s denotes the spin of the fermion ( ( sp F ) = 0 )and Φ( s ) is the fa tor of ¾spin transfer¿ from produ tion to de ay.The problem posed in su h way used to be solved within the methodof spiral amplitudes [1℄. In this method the amplitudes of the fermion F produ tion and de ay al ulated depending on the di(cid:27)erent heli ities ( λ i ).Than the total amplitude of the pro ess an be performed as the followingsum: | A | = X λ i ,λ j A P rod ( λ i ) A Dec ( λ i ) · ( A P rod ( λ j )) + ( A Dec ( λ j )) + , (2)The equivalent approa h was proposed by Jada h and Was [2℄. Theyshowed that expression (2) an be rewritten as follows: | A | = | A P rod | · | A Dec | (1 + ~H ~V ) . (3)Here A P rod and A Dec are the amplitudes of unpolarized fermion produ tionand de ay. ~H and ~V are so- alled polarization ve tors. They are determinedin the fermion rest frame and ontain information about fermion spin. | A P rod ( s ) | = | A P rod | (1 + ( Hs )) (4) | A Dec ( s ) | = | A Dec | (1 + ( V s )) (5)One example is given below. A ordingly to (5) the matrix elementsquared for the t quark de ay to bW + with the following de ay of W + bosonto ℓν an be written as follows: | M | = 2 ( p b p ν )( p t p ℓ )( p w − M W ) + Γ W M W × (cid:20) − m t ( p ℓ s )( p t p ℓ ) (cid:21) , (6)where p t , p b , p l , p n u are the momenta of the t and b quarks and (cid:28)nal leptons, Γ W and M W are the total de ay width and the mass of the W -boson. Hen eone an derive the polarization ve tor for this de ay V µ = − m t p µℓ ( p t p ℓ ) (7)As an example for expression (3) the pro ess u ¯ d → t ¯ b , t → bℓν is onsidered (see Fig.2).Using (6) the expli it expression for the matrix element squared of thispro ess an be written in the form ( ompare to (3)) as follows: | M | ∝ (cid:18) g ( p u p ¯ b )( p t p ¯ d )( p w − M W ) + Γ W M W (cid:19) × (cid:18) g ( p b p ν )( p t p ℓ )( p w − M W ) + Γ W M W (cid:19) × (1 + ~n ℓ ~n ¯ d ) , (8)2 u ¯ d W + ¯ b νℓ + t t Figure 2: Feynman diagram for the pro ess u ¯ d → bℓν ¯ b where ~n ℓ and ~n ¯ d are dire tions of ℓ + and ¯ d -quark momenta in t -quark restframe.Fairly often the expression (3) is preferable in the numeri al al ulationssin e it redu es the quantity of logi al operation. Su h onventional methodof simulation uses dis arding te hnique (the reje t-and a ept method) [3℄and supposes realization of the following algorithm (see details in [2℄):1) to al ulate the (cid:28)lal momenta of the parti les from the fermion F pro-du tion pro ess;2) a ording to a given kinemati s to evaluate the polarization ve tor ~H in F rest frame;3) to perform simulation of the F -fermion de ay and to (cid:28)x the de ayprodu ts kinemati s;4) to evaluate the de ay polarization ve tor ~V (in F rest frame);5) to al ulate the additional weight W = (1+ ~H ~V ) and using a dis ardingte hnique to reje t or a ept an event;6) if the event is dis arded than return to step 3).However using this algorithm one an expe t the in rease of omputation timein ase of ompli ated expressions for the massive fermion matrix elements.In this Note the modi(cid:28) ation of this algorithm is proposed. Very oftensu h modi(cid:28) ation allows to raise an e(cid:30) ien y of the numeri al al ulations.The basi ondition of the modi(cid:28)ed algorithm appli ation is | ~V | = const ,i.e. the ve tor ~V absolute value must be independent of the fermion de aysimulation results.One should noti e that the value of the additional weight W = (1 + ~H ~V ) (see step 5 of the algorithm) depends on the dire tion of ~V only. Besides thedi(cid:27)erent kinemati s of the de ay an lead to the same polarization ve tor ~V .This ideas are the basi for our modi(cid:28)ed algorithm.Thus we o(cid:27)er to hose orre tly the polarization ve tor ~V (a tually thedire tion of ~V ) before the fermion de ay simulation. Corre tly means inorder to a ept an event. Hen e the algorithms steps starting from step 3)modi(cid:28)es: 3as) hose the ~V dire tion using an dis arding te hnique with the weight W = 1 + ~H ~V ;4as) to simulate the massive fermion de ay ;5as) to evaluate the polarization ve tor ~v orresponding to the F de aykinemati s (mote, that in general ~v = ~V sin e the ve tors may havedi(cid:27)erent dire tions);6as) to rotate the referen e system so that ve tor ~v oin ides with ~V in F rest frame .Let us point out that for ea h a epted event the simulation of fermionde ay made just on e. This fa t is for sure an advantage of the proposedalgorithm. Note, that the method des ribed above an be laso used in ase oftwo fermions produ tion with a subsequent de ays. In this ase the additionalweight takes the form: W = 1 + ~h ~v + ~h ~v + h ik v i v k . (9)where (like in (3)) two ve tors h , and tensor h ikik