Astroparticle Physics

17.2 Chapter 2 3412. For any unstable elementary particle like, e.g., a muon, the quantity ‘lifetime’ shouldbe considered in the average sense only. In other words, it does not mean that a particlewith lifetime τ will decay exactly the time τ after it was produced. Its actual lifetime tis a random number distributed with a probability density function,f(t; τ)dt = 1 τ e− t τ dt,giving a probability that the lifetime t lies between t and t + dt. It can easily be checkedthat the mean value of t equals τ. For an unstable relativistic particle, a mean rangebefore it decays is given by the product of its velocity βc and lifetime γτ to βγcτ.Formuons cτ = 658.653 m. Therefore, to survive to sea level from an altitude of 20 km, theaverage√range should equal l = 20 km or βγcτ µ = l = 20 km and βγ = l/cτ µ .Fromβγ = γ 2 − 1 one gets γ 2 = (l/cτ µ ) 2 + 1.Since l/cτ ≫ 1, one finally obtainsγ ≈ l/cτ µ = 20 × 103 m658.653 m ≈ 30.4 .Then the total energy isE µ = γm µ c 2 ≈ 3.2GeVand the kinetic energy isE kinµ = E µ − m µ c 2 ≈ 3.1GeV.3. The Coulomb force isF Coulomb = 1 q 1 q 21(1.602 × 10 −19 As) 24πε 0 r 2 ≈4π × 8.854 × 10 −12 Fm −1 (10 −15 m) 2≈ 230.7Nand the gravitational force isF gravitation = G m 1m 2r 2−11 m3 (2.176 × 10 −8 kg) 2≈ 6.674 × 10kg s 2 (10 −15 m) 2 ≈ 31 600 N .4. Energy–momentum conservation requiresq e + + q e − = q f ,where q e +,q e −,andq f are the four-momenta of the positron, electron, and final state,respectively. To produce a Z, the invariant mass of the initial state squared should be notless than the invariant mass of the required final state squared or(q e + + q e −) 2 ≥ m 2 Z .Since q e + = (E e +, p e +) and q e − = (m e , 0), one obtains