Astroparticle Physics

}28 2 The Standard Model of Elementary ParticlesMuon decay can be described in a similar fashion (Fig. 2.8).The muon transfers its charge to a W − , thereby transform-ing itself into the neutral lepton of the second family, the ν µ .The W − in turn decays again into in e − ¯ν e .muon decayFig. 2.7Neutron decayFig. 2.8Muon decaynuu dd}pd uW –e – ee – – W – e }Finally, pion decay will be discussed (Fig. 2.9). In prin-ciple, the W + can also decay in this case, into an e + ν estate. Helicity reasons, however, strongly suppress this decay:as a spin-0 particle, the pion decays into two leptonsthat must have antiparallel spins due to angular-momentumconservation. The helicity is the projection of the spin ontothe momentum vector, and it is fixed for the neutrino (formassless particles the spin is either parallel or antiparallel tothe momentum). Particles normally carry negative helicity(spin ‖−p, left-handed) so that the positron, as an antiparticle(spin ‖ p, right-handed), must take on an unnaturalhelicity (Fig. 2.10). The probability of carrying an abnormalhelicity is proportional to 1 − v c(where v is velocityof the charged lepton). Owing to the relatively high mass ofthe muon (m µ ≫ m e ), it takes on a much smaller velocitycompared to the electron in pion decay, i.e., v(µ) ≪ v(e).The consequence of this is that the probability for the decaymuon to take on an unnatural helicity is much larger comparedto the positron. For this reason, the π + → e + ν e decayis strongly suppressed compared to the π + → µ + ν µ decay(the suppression factor is 1.23 × 10 −4 ).The various elementary particles are characterized byquantum numbers. In addition to the electric charge, themembership of a quark generation (quark flavour) or leptongeneration (lepton number) is introduced as a quantum num-ber. Leptons are assigned the lepton number +1 intheirrespectivegeneration, whereas antileptons are given the leptonnumber −1. Lepton numbers for the different lepton families(L e , L µ , L τ ) are separately conserved, as is shown inthe example of the muon decay:pion decay +uW + + dFig. 2.9Pion decay espin e + e +spin e +Fig. 2.10Helicity conservation in π + decayquantum numberslepton numberµ − → ν µ + e − + ¯ν eL µ 1 1 0 0 . (2.4)L e 0 0 1 −1