Astroparticle Physics

3.2 Four-Vectors 39Example 3: Consider the photoproductionof a neutral pion(mass m π 0 ≈ 135 MeV) on a target proton at rest (massm p ):π 0 productionγ + p → p + π 0 ; (3.24)√ s ={m2p + 2E γ m p } 1/2 ≥ m p + m π 0 ,m 2 p + 2E γ m p ≥ m2 p + m2 π 0 + 2m p m π 0 ,E γ ≥ 2m p m π 0 + m 2 π 02m p= m π 0 + m2 π 02m p(3.25)≥ m π 0 + 9.7MeV≈ 145 MeV .3.2 Four-Vectors“The physicist in preparing for his workneeds three things, mathematics, mathematics,and mathematics.”Wilhelm Conrad RöntgenFor calculations of this kind it is practical to introduceLorentz-invariant four-vectors. In the same way as time tand the position vector s = (x,y,z) can be combined toform a four-vector, also a four-momentum vector( Eq = with p = (p x ,p y ,p z ) (3.26)p)can be introduced. Because ofLorentz-invariantfour-vectorse +e – *( ) E 2q 2 = = E 2 − p 2 = m 2 0 (3.27)pthe square of the four-momentum is equal to the square ofthe rest mass. For photons one hasq 2 = E 2 − p 2 = 0 . (3.28)Those particles, which fulfill (3.27) are said to lie on themass shell. On-shell particles are also called real. Apart fromthat, particles can also borrow energy for a short time fromthe vacuum within the framework of Heisenberg’s uncertaintyprinciple. Such particles are called virtual. They arenot on the mass shell. In interaction processes virtual particlescan only occur as exchange particles.nucleusnucleus´Fig. 3.1The processγ + nucleus → e + + e − + nucleus ′mass shell