PS6-solutions - Phys 253a 1 Problem Set 6 solutions November 7 2010 1(a There are many diagrams one can draw(I count 23 at order e4 Many of these are

# PS6-solutions - Phys 253a 1 Problem Set 6 solutions...

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Phys 253a 1 Problem Set 6 solutions November 7, 2010 1.(a) There are many diagrams one can draw (I count 23 at ordere4). Many of theseare corrections to the two and three point functions for the scalars or the photonand will cancel in small groups. We will begin with some of these. For example,consider the following diagram:p1p2p4p3To simplify subscripts we will refer to the initial particles ase’s and the finalparticles asμ’s.Let’s call the right moving momenta in the top and bottomphoton propagators in the looplandrrespectively. Then the numerator in theexpression for this diagram is given byN=(-ie)4(p1-p2)μ(-i)gμν-(1-ξ)(p1+p2)μ(p1+p2)ν(p1+p2)2(l-r)ν(l-r)λ×(-i)gμν-(1-ξ)(p3+p4)λ(p3+p4)ρ(p3+p4)2(p3-p4)ρ(2π)4δ4(p1+p2-l-r).Since we have (p1-p2)·(p1+p2) =m2e-m2e= 0 and (p3+p4)·(p3-p4) =m2μ-m2μ= 0 on shell, both (1-ξ) pieces vanish and this diagram is by itselfgauge invariant.Now let’s get our hands a little more dirty. Consider the following pair of dia-grams:  Phys 253a 3   • • • 