•The differential cross section

Squaring the amplitude, transforming the spin sums into traces and evaluating the traces:

squaredamp = FermionSpinSum[ampFC ComplexConjugate[ampFC /. {μ1 -> ν1, μ2 -> ν2}] /. D -> Sequence[] // Expand] /. DiracTrace -> Tr // DiracSimplify // Simplify ;

squaredampfinal = squaredamp // Contract // Simplify

32 (e^(0 ))^4 (2 1/(-p _ 2 - p _ 3)^2^2 (m _ ψ^(ó  0 ))^4 + 2 1/(-p _ 2 - p _ 4)^2^2 (m _ ψ^(ó  0 ))^4 + 2 1/(-p _ 2 - p _ 3)^2 1/(-p _ 2 - p _ 4)^2 (m _ ψ^(ó  0 ))^4 + p _ 1 · p _ 4 1/(-p _ 2 - p _ 3)^2^2 (m _ ψ^(ó  0 ))^2 + p _ 2 · p _ 3 1/(-p _ 2 - p _ 3)^2^2 (m _ ψ^(ó  0 ))^2 + p _ 2 · p _ 4 1/(-p _ 2 - p _ 4)^2^2 (m _ ψ^(ó  0 ))^2 + p _ 1 · p _ 4 1/(-p _ 2 - p _ 3)^2 1/(-p _ 2 - p _ 4)^2 (m _ ψ^(ó  0 ))^2 + p _ 2 · p _ 3 1/(-p _ 2 - p _ 3)^2 1/(-p _ 2 - p _ 4)^2 (m _ ψ^(ó  0 ))^2 + p _ 2 · p _ 4 1/(-p _ 2 - p _ 3)^2 1/(-p _ 2 - p _ 4)^2 (m _ ψ^(ó  0 ))^2 - p _ 3 · p _ 4 1/(-p _ 2 - p _ 3)^2 1/(-p _ 2 - p _ 4)^2 (m _ ψ^(ó  0 ))^2 + p _ 1 · p _ 4 p _ 2 · p _ 3 1/(-p _ 2 - p _ 4)^2^2 + p _ 1 · p _ 3 (1/(-p _ 2 - p _ 4)^2 (1/(-p _ 2 - p _ 3)^2 + 1/(-p _ 2 - p _ 4)^2) (m _ ψ^(ó  0 ))^2 + p _ 2 · p _ 4 1/(-p _ 2 - p _ 3)^2^2) + p _ 1 · p _ 2 (p _ 3 · p _ 4 (1/(-p _ 2 - p _ 3)^2 + 1/(-p _ 2 - p _ 4)^2)^2 - (m _ ψ^(ó  0 ))^2 1/(-p _ 2 - p _ 3)^2 1/(-p _ 2 - p _ 4)^2))

samp = squaredampfinal // PropagatorDenominatorExplicit // MandelstamReduce[#, Masses -> {ParticleMass[Electron, RenormalizationState[0]], ParticleMass[Electron, RenormalizationState[0]], ParticleMass[Electron, RenormalizationState[0]], ParticleMass[Electron, RenormalizationState[0]]}, MandelstamCancel -> MandelstamU] & // Simplify

(16 (e^(0 ))^4 (64 (m _ ψ^(ó  0 ))^8 + 16 (t - 6 s) (m _ ψ^(ó  0 ))^6 + 4 (13 s^2 + 3 t s + 3 t^2) (m _ ψ^(ó  0 ))^4 - 4 (3 s^3 + 3 t s^2 + 3 t^2 s + 2 t^3) (m _ ψ^(ó  0 ))^2 + (s^2 + t s + t^2)^2))/(t^2 (-4 (m _ ψ^(ó  0 ))^2 + s + t)^2)

The kinematical factor in the center of mass frame:

kinfac = 1/(64 π^2 s) ;

The full differential cross section in the center of mass frame expressed in terms of Mandelstam variables:

dcrosssection[s_, t_, u_] := 1/4 kinfac * samp /. {MandelstamS -> s, MandelstamU -> u, MandelstamT -> t} ;

dcrosssection[s, t, u]

((e^(0 ))^4 (64 (m _ ψ^(ó  0 ))^8 + 16 (t - 6 s) (m _ ψ^(ó  0 ))^6 + 4 (13 s^2 + 3 t s + 3 t^2) (m _ ψ^(ó  0 ))^4 - 4 (3 s^3 + 3 t s^2 + 3 t^2 s + 2 t^3) (m _ ψ^(ó  0 ))^2 + (s^2 + t s + t^2)^2))/(16 π^2 s t^2 (-4 (m _ ψ^(ó  0 ))^2 + s + t)^2)

The formula usually quoted:

MoellerFormula = dcrosssection[s, t, u] /. {s -> 4 ω^2, t -> -2 q2 (1 - (1 - sin[θ]^2)^(1/2))} /. q2 -> ω^2 - ParticleMass[Fermion[7], RenormalizationState[0]]^2 // Simplify

((e^(0 ))^4 (((sin (θ))^2 - 4)^2 ω^4 - 2 (m _ ψ^(ó  0 ))^2 ((sin (θ))^4 - 2 (sin (θ))^2 + 8) ω^2 + (m _ ψ^(ó  0 ))^4 ((sin (θ))^4 + (sin (θ))^2 + 4)))/(64 π^2 ω^2 (ω^2 - (m _ ψ^(ó  0 ))^2)^2 (sin (θ))^4)

Converted by Mathematica  (July 10, 2003)