253a/Schwartz Due October 5, 2010 Problem Set 4 1. Consider the Lagrangian for φ 3 theory L = − 1 2 φ ( s + m 2 ) φ + g 3! φ 3 (1) a) Draw a tree level feynman diagram for the decay φ → φφ . Write down the corre-sponding amplitude using the Feynman rules. b) Now consider the one loop correction, given by Write down the corresponding amplitude using the Feynman rules. c) Now start over and write down the diagram from part b in position space, in terms of integrals over the intermediate points and Wick contractions, represented with D F ′ s . d) Show that after you apply LSZ, what you got in c ) reduces to what you got in b ) , by integrating the phases into δ-functions, and integrating over those δ-functions. 2. Non-relatavistic Moller scattering: e − e − → e − e − . If the electron and photon were spinless, we can write the Lagrangian as L = − 1 2 φ e ( s + m e 2 ) φ e − 1 2 A s A + e 2 m e Aφ e φ e (2) the factor of m e comes from the non-relativistic limit (or by dimensional analysis!). a) Draw the three tree-level
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