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Differential Cross-Section
dσdΩ=164π2spfinalpinitial|M|2
Working in the center of mass frame
pfinal=pinitial
Determining the scattering amplitude in the center of mass frame
M=e2(u−st+t−su)
M2=e4(u−st+t−su)(u−st+t−su)
M2=e4((u−s)2t2+(t−s)2u2+2(u−s)t(t−s)u)
M2=e4((u2−2us+s2)t2+(t2−2ts+s2)u2+2(ut−st+s2−us)tu)
M2=e4((t2+s2)u2+2s2tu+2−2ust2−2st−2tsu2−2su+(u2+s2)t2)
Using the fine structure constant (with c=ℏ=ϵ0=1)
α≡e24π
dσdΩ=α24s((t2+s2)u2+2s2tu+2−2ust2−2st−2tsu2−2su+(u2+s2)t2)
In the center of mass frame the Mandelstam variables are given by:
s≡4E∗2
t≡−2p∗2(1−cosθ)
u≡−2p∗2(1+cosθ)
Calculating the parts to have common denominators:
(1)2=2p∗4sin4θp∗4sin4θ=2p∗4(1−cos2θ)2p∗4sin4θ=2p∗4(1−2cos2θ+cos4θ)p∗4sin4θ
(2)2s2tu=32E∗44p∗4(1+cosθ)(1−cosθ)=8E∗4p∗4sin2θ=8E∗4sin2θp∗4sin4=8E∗4(1−cos2θ)p∗4sin4θ
(3)t2u2=4p∗2(1−cosθ)24p∗2(1+cosθ)2=tan4θ2=p∗4(1−cosθ)4p∗4sin4θ=p∗4(cos4θ−4cos3θ+6cos2θ−4cosθ+1)p∗4sin4θ
(4)u2t2=4p∗2(1+cosθ)24p∗2(1−cosθ)2=cot4θ2=p∗4(1+cosθ)4p∗4sin4θ=p∗4(cos4θ+4cos3θ+6cos2θ+4cosθ+1)p∗4sin4θ
(5)s2u2=16E∗44p∗4(1+cosθ)2=E∗4sec4θ2p∗4=4E∗4p∗4(1+cosθ)2=4E∗4(1−cosθ)2p∗4(1+cosθ)2(1−cosθ)2=4E∗4(cos2θ−2cosθ+1)p∗4sin4θ
(6)s2t2=16E∗44p∗4(1−cosθ)2=E∗4csc4θ2p∗4=4E∗4p∗4(1−cosθ)2=4E∗4(1+cosθ)2p∗4(1−cosθ)2(1+cosθ)2=4E∗4(cos2θ+2cosθ+1)p∗4sin4θ
(7)−2st=8E∗22p∗2(1−cosθ)=4E∗2p∗2(1−cosθ)=4E∗2(1+cosθ)p∗2(1−cosθ)(1+cosθ)=4E∗2(1+cosθ)p∗2sin2θ=4E∗2p∗2sin2θ(1+cosθ)p∗4sin4θ=4E∗2p∗2(1−cos2θ)(1+cosθ)p∗4sin4θ=4E∗2p∗2(1+cosθ−cos2θ−cos3θ)p∗4sin4θ
(8)−2su=8E∗22p∗2(1+cosθ)=4E∗2p∗2(1+cosθ)=4E∗2(1−cosθ)p∗2(1+cosθ)(1−cosθ)=4E∗2(1−cosθ)p∗2sin2θ=4E∗2p∗2sin2θ(1−cosθ)p∗4sin4θ=4E∗2p∗2(1−cos2θ)(1−cosθ)p∗4sin4θ=4E∗2p∗2(1−cosθ−cos2θ+cos3θ)p∗4sin4θ
(9)−2tsu2=4p∗2(1−cosθ)4E∗24p∗2(1+cosθ)2=4E∗2(1−cosθ)sec4θ2p∗2=4E∗2p∗2(1−cosθ)p∗4(1+cosθ)2=4E∗2p∗2(1−cosθ)(1−cosθ)2p∗4(1+cosθ)2(1−cosθ)2=4E∗2p∗2(−cos3θ+3cos2θ−3cosθ+1)p∗4sin4θ
(10)−2ust2=4p∗2(1+cosθ)4E∗24p∗2(1−cosθ)2=4E∗2(1+cosθ)csc4θ2p∗2=4E∗2p∗2(1+cosθ)p∗4(1−cosθ)2=4E∗2p∗2(1+cosθ)(1+cosθ)2p∗4(1−cosθ)2(1+cosθ)2=4E∗2p∗2(cos3θ+3cos2θ+3cosθ+1)p∗4sin4θ
Combing like terms further,
(1,3/4)→4p∗4cos4θ+8p∗4cos2θ+4p∗4
(2,6/5)→16E∗4
(8/7,10/9)→16E∗2p∗2+16E∗2p∗2cos2θ
Expressing this as the differential cross-section
dσdΩ=α24sp∗4sin4θ(4p∗4cos4θ+8p∗4cos2θ+4p∗4+16E∗2p∗2+16E∗2p∗2cos2θ+16E∗4)
dσdΩ=α2sp∗4sin4θ(p∗4cos4θ+2p∗4cos2θ+p∗4+4E∗2p∗2+4E∗2p∗2cos2θ+4E∗4)
In the Ultra-relativistic limit as E≈p
dσdΩ=α24E∗2sin4θ(cos4θ+6cos2θ+9)=α2(3+cos2θ)24E∗2sin4θ
In the non-relativistic limit as E≈m
dσdΩ=α24m∗2p∗4sin4θ(p∗4cos4θ+2p∗4cos2θ+p∗4+4m∗2p∗2+4m∗2p∗2cos2θ+4m∗4)
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