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253a/SchwartzDue September 28, 2010Problem Set 3Problems marked with a * are optional, but strongly recommended.1. We found fore+e-→μ+μ-that the cross section was proportional to1+cos2θ.a) Work out the Lorentz invariant quantitiess= (pe++pe-)2,t= (pμ--pe-)2andu= (pμ+-pe-)2in terms ofEcmand cosθ(still assumingmμ=me=0)b) Derive a relationship betweens, tandu.c) Rewrite the differential cross section in terms ofs, tandu.d) Now assumemμandmeare nonzero. Derive a relationship betweens, tanduand themasses (hint: do NOT choose coordinates). We will use thisa lot.2. One of the most important scattering experiments ever was Rutherford’s gold foil experiment.a) What is the formula for theclassicalRutherford scattering cross section (α-particle/nucleusscattering:αN→αN)? What are the assumptions go into its derviation?b) Calculate Rutherford scattering using quantum field theory.i. Write down a Feynman diagram for this process. What is the momentum of the vir-tual photonkμ