Astroparticle Physics

44 3 Kinematics and Cross Sections( )( )qµ 2 + q2 e − 2q µ q e = m2 µ + m2 e − 2 mµ Ee0 p e= (q¯νe + q νµ ) 2 ,E e = m2 µ + m2 e − (q¯ν e+ q νµ ) 2. (3.46)2m µThe electron energy is largest, if (q¯νe + q νµ ) 2 takes on aminimum value. For vanishing neutrino masses this meansthat the electron gets a maximum energy, ifq¯νe q νµ = E¯νe E νµ − p ¯νe · p νµ = 0 . (3.47)Equation (3.47) is satisfied for p ¯νe ‖ p νµ . This yieldsE maxe= m2 µ + m2 e≈ m µ= 52.83 MeV . (3.48)2m µ 2In this configuration the electron momentum p e is antiparallelto both neutrino momenta which in turn are parallel toeach other.If the spins of all participating particles and the struc-ture of weak interactions are taken into consideration, oneobtains for the electron spectrum, using the shorthand x =2E e /m µ ≈ E e /Ee max ,electron spectrumin muon decayN(x) = const x 2 (1.5 − x) . (3.49)Just as in nuclear beta decay (n → p + e − +¯ν e )theavailable decay energy in a three-body decay is distributedcontinuously among the final-state particles (Fig. 3.4).Fig. 3.4Energy spectrum of electrons frommuon decayIf interaction or decay processes are treated, it is fully suffi-cient to consider the process in the center-of-mass system. Ina different system (for example, the laboratory system) thetransformationbetween laboratoryand center-of-mass system3.3 Lorentz Transformation“We have learned something about thelaws of nature, their invariance withrespect to the Lorentz transformation,and their validity for all inertial systemsmoving uniformly, relative to each other.We have the laws but do not know theframe to which to refer them.”Albert Einstein