Copyright © 2009 University of Cambridge. Not to be quoted or reproduced without permission.Quantum Field Theory: Example Sheet 4Prof A C Davis, Michaelmas Term 20091.A real scalar field withφ4interaction has the LagrangianL=12∂μφ∂μφ−12m2φ2−λ4!φ4(1)Use Dyson’s formula and Wick’s theorem to show that the leading order contributionto 3-particle→3-particle scattering includes the amplitudep32pp1p12pp3///= (−iλ)2i(p1+p2+p3)2−m2(2)Check that this result is consistent with the Feynman rules for the theory. What otherdiagrams also contribute to this process?2.Examine(0|S|0)to orderλ2inφ4theory. Identify the different diagrams withthe different contributions arising from an application of Wick’s theorem. Confirmthat to orderλ2, the combinatoric factors work out so that the the vacuum to vacuumamplitude is given by the exponential of the sum of distinct vacuum bubble types,(0|S|0)= exp(+++...)(3)3.Consider the Lagrangian for 3 scalar fieldsφi,i= 1,2,3, given byL=3summationdisplayi=112(∂μφi)(∂μφi)−12m2(3summationdisplayi=1φ2i)−λ8(3summationdisplayi=1φ2i)2(4)Show that the Feynman propagator for the free field theory (i.e.λ= 0) is of the form(0|Tφi(x)φj(y)|0)=δijDF(x−y)(5)whereDF(x−y) is the usual scalar propagator. Write down the Feynman rules of thetheory. Compute the amplitude for the scatteringφiφj→φkφlto lowest order inλ.