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+(u2+s2)t2)
Using the fine structure constant (with c=ℏ=1)
α≡e24π
dσdΩ=α22s((t2+s2)u2−2s2tu+(u2+s2)t2)
In the center of mass frame the Mandelstam variables are given by:
s≡4E∗2
Using the relationship
cosθ=−1+cosθ2
In the ultra-relativistic limit, the electron mass is small enough compared to the energy such that it can be neglected when compared to the momentum
E2≡m2+p2→E2≈p2
t≡−2E∗2(1−cosθ)=−2E∗2(1−2cos2θ2+1)=−4E∗2(1−2cos2θ2)=−4E∗2sin2θ2
u≡−2E∗2(1+cosθ)=−2E∗2(1+2cos2θ2−1)=−4E∗2cos2θ2
dσdΩ=α28E∗2(16E∗4sin4θ2+16E∗416E∗4cos4θ2−32E∗44E∗2sin2θ24E∗2cos2θ2+16E∗4cos4θ2+16E∗416E∗4sin4θ2)
dσdΩ=α28E∗2(sin4θ2+1cos4θ2−2sin2θ2cos2θ2+cos4θ2+1sin4θ2)