253a/SchwartzDue October 5, 2010Problem Set 41. Consider the Lagrangian forφ3theoryL=−12φ(square+m2)φ+g3!φ3(1)a) Draw a tree level feynman diagram for the decayφ→φφ. Write down the corre-sponding amplitude using the Feynman rules.b) Now consider the one loop correction, given byWrite down the corresponding amplitude using the Feynman rules.c) Now start over and write down the diagram from partbin position space, in termsof integrals over the intermediate points and Wick contractions, represented withDF′s.d) Show that after you apply LSZ, what you got inc)reduces to what you got inb),by integrating the phases intoδ-functions, and integrating over thoseδ-functions.2. Non-relatavistic Moller scattering:e−e−→e−e−. If the electron and photon were spinless,we can write the Lagrangian asL=−12φe(square+me2)φe−12AsquareA+e2meAφeφe(2)the factor ofmecomes from the non-relativistic limit (or by dimensional analysis!).a) Draw the three tree-levele−e−→e−e−diagrams following from this Lagrangian.