electron+mu+->electron + mu+

Input diarc matrix package:

Needs["Tracer`", 
      ToString[ToFileName[{$TopDirectory, "Addons", "Applications", "\
dirac"}, "Tracer.m"]]]

Define vectors in 4 dimensions:

VectorDimension[4]

Gamma matrix contraction switch:

Spur[1]

Electron Current:

currentE = 
      G[le, p1 + me U] ** G[le, {mu}] ** G[le, p2 + me U] ** 
        G[le, {nu}]

Expand electron current:

cE = currentE /. le -> 1

Muon Current:

currentmu = 
      G[lmu, p3 - mmu U] ** G[lmu, {mu}] ** G[lmu, p4 - mmu U] ** 
        G[lmu, {nu}]

Expand muon current:

cmu = currentmu /. lmu -> 1

Calculate the square of the matrix element:

matrixsq = 
      Simplify[(((ge^4) ContractEpsGamma[cE cmu])/(4 (q^4)))]

This is Griffiths 7.126 except my definition of p2 and p3 are reversed fromhis.

Construct laboratory 4 momenta:

gmunu = {{1, 0, 0, 0}, {0, -1, 0, 0}, {0, 0, -1, 0}, {0, 0, 
          0, -1}}

pp1 = {wi, 0, 0, ki}

pp2 = {wf, kf Sin[theta], 0, kf Cos[theta]}

pp3 = {mmu, 0, 0, 0}

pp4 = pp1 + pp3 - pp2

m1sq = 
      Simplify[matrixsq /. {p1 . p2 -> pp1 . gmunu . pp2, 
            p3 . p4 -> pp3 . gmunu . pp4, 
            p1 . p3 -> pp1 . gmunu . pp3, 
            p1 . p4 -> pp1 . gmunu . pp4, 
            p2 . p3 -> pp2 . gmunu . pp3, 
            p2 . p4 -> pp2 . gmunu . pp4}]

Assume electron has zero mass:

m2sq = 
      Simplify[PowerExpand[
          m1sq /. {wf -> kf, wi -> ki, me -> 0}]]

Bj&D matrix element (page 115):
	ge^4 -> (4 Pi)^2 alpha^2
	q^4 -> 16 ki^2 kf^2 Sin[theta/2]^4
	All must be divided by 16 mu^2 me^2 (because of difference in wave function normalization)
	ki-kf->q^2/2/mmu

m2sqBD  =  
      m2sq/(16 mmu^2  me^2) /. {ge^4 -> (4 Pi)^2 alpha^2, 
          q^-4 -> 1/(16 ki^2 kf^2 Sin[theta/2]^4)}

m3sqBD = 
      Simplify[Numerator[m2sqBD /. ki -> kf  + q^2/mmu/2]/
          Denominator[m2sqBD]]

Now put in terms of DIS variables:

matrixsq

m3sq = 
      (matrixsq /. p4 -> p3 + q) /. {me -> 0, p3 . p3 -> mmu^2, 
          p1 . p1 -> 0}

m4sq = (m3sq /. p2 -> p1 - q) /. p1 . p1 -> 0

m5sq = 
      Simplify[m4sq /. {p3 . q ->  mmu nu,  p3 . p1 ->  mmu nu/y, 
            p1 . q -> -mmu nu,  q . q -> -2 nu mmu, 
            ge^4 -> (4 Pi)^2 alpha^2}]

Comparing this to the Deep Inelastic results gives (x=1 for elastic reactions):
	2 x F1 =4 mu nu/4 Pi
	F2 = 4 mu nu /4 Pi
	
Thus for elastic scattering 2 x F1 = F2 (Callen Gross relationship).