1 Copyright 2009 University of Cambridge Not to be quoted or reproduced without

# 1 copyright 2009 university of cambridge not to be

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Copyright © 2009 University of Cambridge. Not to be quoted or reproduced without permission.4.The Lagrangian for Yukawa theory is given byL=12(∂φ)212μ2φ2+¯ψ(i/m)ψ¯ψψ(6)a)Considerψψψψscattering, with the initial and final states given by,|i)=radicalbig4EvectorpEvectorqasvectorparvectorq|0)|f)=radicalbig4EvectorpEvectorqasvectorparvectorq|0)(7)Show using Dyson’s formula and Wick’s theorem that the scattering amplitude at orderg2is given by,M= (ig)2parenleftbiggus(vectorp)·us(vectorp)][¯ur(vectorq)·ur(vectorq)](pp)2μ2us(vectorp)·ur(vectorq)][¯ur(vectorq)·us(vectorp)](qp)2μ2parenrightbiggDraw the two Feynman diagrams that correspond to these two terms.b)Consider nowψ¯ψψ¯ψscattering, with initial and final states given by|i)=radicalbig4EvectorpEvectorqasvectorpbrvectorq|0)|f)=radicalbig4EvectorpEvectorqasvectorpbrvectorq|0)(8)Show that the amplitude is this time given byM=(ig)2parenleftbiggus(vectorp)·us(vectorp)][¯vr(vectorq)·vr(vectorq)](pp)2μ2vr(vectorq)·us(vectorp)][¯us(vectorp)·vr(vectorq)](p+q)2μ2parenrightbigg(Be careful with minus signs!!). What are the Feynman diagrams that now contribute?5.The Lagrangian for a pseudoscalar Yukawa interaction is given byL=12(∂φ)212μ2φ2+¯ψ(i/m)ψ¯ψγ5ψ(9)Write down the Feynman rules for this theory. Use this to write down the amplitudeat orderg2forψψψψscattering andψ¯ψψ¯ψscattering.6.Any vector functionf(x) has a decomposition into a sum of transverse (zero diver-gence) and longitudinal (zero curl) parts, namely