Astroparticle Physics

40 3 Kinematics and Cross Sectionse –e + e − pair productionin the Coulomb fieldof a nucleuse − p scattering e –Example 4: Photoproduction of an electron–positron pairin the Coulomb field of a nucleusIn this example the incoming photon γ is real, while thephoton γ ∗ exchanged between the electron and the nucleusis virtual (Fig. 3.1).Example 5: Electron–proton scattering (Fig. 3.2)The virtuality of the exchanged photon γ ∗ can easilybe determined from the kinematics based on the fourmomentumvectors of the electron and proton. The fourmomentumvectors are defined in the following way: incomingelectron q e = ( E e)p e , final-state electron q′e =( E ′ )ep , incoming proton ′ qp = ( E p)ep p , final-state protonq p ′ = ( E p′ )p . Since energy and momentum are conserved,′ palso four-momentum conservation holds:q e + q p = q ′ e + q′ p . (3.29)ppThe four-momentum squared of the exchanged virtualphoton qγ 2 ∗ is determined to beFig. 3.2The process e − + p → e − + p qγ 2 ∗ = (q e − q′ e )2(Ee − E=e′ ) 2p e − p ′ = (E e − E e ′ )2 − (p e − p ′ e )2e= Ee 2 − p2 e + E′2 e − p′2 e − 2E e E′ e + 2p e · p′ ee + += 2m 2 e − 2E e E′ e (1 − β e β′ e cos θ) , (3.30)e – –(3.30) is simplified to *where β e and β e ′ are the velocities of the incoming andoutgoing electron and θ is the angle between p e and p ′ e .For high energies and not too small scattering anglesFig. 3.3The process e + e − → µ + µ − q2 γ ∗ =−2E e E′ e (1 − cos θ)=−4E e E ′ e sin2 θ 2 . (3.31)If sin θ 2 can be approximated by θ 2, one gets for not toosmall anglesq 2 γ ∗ =−E e E′ e θ 2 . (3.32)space-like photonsThe mass squared of the exchanged photon in this caseis negative! This means that the mass of γ ∗ is purelyimaginary. Such photons are called space-like.