Astroparticle Physics

3.1 Threshold Energies 373.1 Threshold Energies“Energy has mass and mass representsits energy.”Albert EinsteinIn astroparticle physics frequently the problem occurs to determinethe threshold energy for a certain process of particleproduction. This requires that in the center-of-mass systemof the collision at least the masses of all particles in thefinal state of the reaction have to be provided. In storagerings the center-of-mass system is frequently identical withthe laboratory system so that, for example, the creation of aparticle of mass M in an electron–positron head-on collision(e + and e − have the same total energy E) requiresthreshold energy2E ≥ M. (3.12)If, on the other hand, a particle of energy E interacts witha target at rest as it is characteristic for processes in cosmicrays, the center-of-mass energy for such a process must firstbe calculated.For the general case of a collision of two particles withtotal energy E 1 and E 2 and momenta p 1 and p 2 the Lorentzinvariantcenter-of-mass energy E CMS can be determined using(3.7) and (3.11) in the following way:E CMS = √ s={(E 1 + E 2 ) 2 − (p 1 + p 2 ) 2} 1/2{} 1/2= E1 2 − p2 1 + E2 2 − p2 2 + 2E 1 E 2 − 2p 1 · p 2{} 1/2= m 2 1 + m2 2 + 2E 1 E 2 (1 − β 1 β 2 cos θ) . (3.13)determinationof the center-of-mass energyIn this equation θ is the angle between p 1 and p 2 . For highenergies (β 1 ,β 2 → 1andm 1 ,m 2 ≪ E 1 ,E 2 ) and not toosmall angles θ (3.13) simplifies toE CMS = √ s ≈ {2E 1 E 2 (1 − cos θ)} 1/2 . (3.14)If one particle (for example, the particle of the mass m 2 )isat rest (laboratory system E 2 = m 2 , p 2 = 0), (3.13) leads to√ s ={m21 + m 2 2 + 2E 1 m 2 }1/2 . (3.15)Using the relativistic approximation (m 2 1 ,m2 2 ≪ 2E 1 m 2 )one gets