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PHYS 633: Elementary Particle Physics Spring 2012 Homework #4 Due Date: 10/30/12 1. In the center-of-mass frame for the process AB CD , show that dQ pf 1 4 4 s 2 d F 4 pi s And hence that the differential cross section is d 1 pf 2 M d cm 64 2 s pi 2. Show that for very high-energy “spinless” electron-muon scattering, d 2 3 cos d cm 4s 1 cos 2 Where θ is the scattering angle and α=e2/4π. Neglect the particle masses. 3. Show that in the reaction A B C D , s t u m A2 mB2 mC2 mD2 4. Taking e e e e to be the s channel process, verify that s 4( k 2 m 2 ) t 2k 2 (1 cos ) u 2k 2 (1 cos ) Where θ is the center-of-mass scattering angle and k k i k f , where k i and k f are, respectively, the momenta of the incident and scattered electrons in the center-of-mass frame. Show that the process is physically allowed provided s 4m 2 , t 0, u 0 . The physical region is shown shaded in the figure drawn in class. Note that t 0(u 0) correspond to forward (backward) scattering. 5. Show that the invariant amplitude for “spinless” electron-electron scattering, iΜe e 2 p A pC pD pB p A pB pD pC 2 i e e 2 pC pD 2 p p D B , can be written as s u t u Μe e ( s, t , u ) e 2 . s t Comment on the symmetry of M under s ↔ t. 6. In e e near threshold, one can obviously not neglect the mass of the . Working from d 1 pf 2 M and the exact spin d 64 2 s pi averaged amplitude M 2 8e 4 p pk k k p k p m 2 k p M 2 k p 2m 2 M 2 4 q k , p : incoming e , e four vecto rs k , p : scattered , four vecto rs m, M : masses of e, Show that the total cross-section for production is given by 4 2 3s 3 2 v where . c 2