# PS3-solutions - Physics 253a 1 Problem Set 3 Solutions...

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Physics 253a1Problem Set 3 SolutionsOctober 6, 20101.(a) In the center of mass frame, thee+ande-approach each other with oppositemomenta and total energyEcm:pe+=Ecm2(1,1,0,0) andpe-=Ecm2(1,-1,0,0).The muon and anti-muon look the same, but with their momenta rotated byθ:pμ+=Ecm2(1,cosθ,sinθ,0) andpμ-=Ecm2(1,-cosθ,-sinθ,0).Now,s=(pe++pe-)2=(Ecm,0)2=E2cmt=(pμ--pe-)2=-2pμ-·pe-=-E2cm2(1-cosθ)u=(pμ+-pe-)=-2pμ+·pe-=-Ecm2(1 + cosθ)(b) Note thats+t+u= 0.(c) In class we founddΩ=e464π2E2cm(1 + cos2θ)Note that4tus2= 1-cos2θ. So we can rewrite the above asdΩ=e432π2s1-2tus2=e432π2s3(t2+u2)where we’ve useds2= (t+u)2=t2+ 2tu+u2.(d) We haves+t+u=(2m2e+ 2pe-·pe+) + (m2e+m2μ-2pe-·pμ-)+(m2e+m2μ-2pe-·pμ+)=4m2e+ 2m2μ-2pe-·(pμ++pμ--pe+)=4m2e+ 2m2μ-2pe-·pe-=2(m2e+m2μ)Where we’ve used conservation of momentum in the last line. In general, we’dhaves+t+u=im2i.
Physics 253a22.

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