Astroparticle Physics

3.2 Four-Vectors 41Example 6: Muon pair production in e + e − interactions(Fig. 3.3)Assuming that electrons and positrons have the sametotal energy E and opposite momentum (p e + =−p e −),one has()qγ 2 ∗ = (q e + + q E + E 2e −)2 =p e + + (−p e +)= 4E 2 . (3.33)In this case the mass of the exchanged photon is 2E,which is positive. Such a photon is called time-like. Themuon pair in the final state can be created if 2E ≥ 2m µ .The elegant formalism of four-momentum vectors forthe calculation of kinematical relations can be also extendedto decays of elementary particles. In a two-body decay ofan elementary particle at rest the two decay particles getwell-defined discrete energies because of momentum conservation.Example 7: The decay π + → µ + + ν µFour-momentum conservation yieldsµ pair productiontime-like photonstwo-body decayq 2 π = (q µ + q ν ) 2 = m 2 π . (3.34)In the rest frame of the pion the muon and neutrino areemitted in opposite directions, p µ =−p νµ ,(Eµ + E νp µ + p νµ) 2= (E µ + E ν ) 2 = m 2 π . (3.35)Neglecting a possible non-zero neutrino mass for thisconsideration, one hasE ν = p νµwith the resultE µ + p µ = m π .Rearranging this equation and squaring it givesEµ 2 + m2 π − 2E µ m π = p2 µ ,2E µ m π = m 2 π + m2 µ ,E µ = m2 π + m2 µ2m π. (3.36)