electron+postron->mu+ + 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 (((2 energy))^4)))]

Construct cm 4 momenta:

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

pp1 = {energy, 0, 0, pcme}

pp2 = {energy, 0, 0, -pcme}

pp3 = {energy, pcmmu Sin[theta], 0, pcmmu Cos[theta]}

pp4 = {energy, -pcmmu Sin[theta], 
        0, -pcmmu Cos[theta]}

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}, 
        Trig -> False]

m2sq = 
      Simplify[m1sq /. {(pcme^2) -> (energy^2) - (me^2), 
            (pcmmu^2) -> (energy^2) - (mmu^2), 
            Sin[theta]^2 -> 1 - Cos[theta]^2}, 
        Trig -> False]

m3sq = Collect[m2sq, {ge, energy}]

xtotal = ((((∫_0^π) m3sq Sin[theta] \[DifferentialD]theta) 2 π (Sqrt[(energy^2) - (mmu^2)]))/((((8 π))^2) (((2 energy))^2) (Sqrt[(energy^2) - (me^2)]))) /. 
        (ge^4) -> (((4 π alpha))^2)

Spin Average:

xtotalME = PowerExpand[xtotal /. {me -> 0}]

xtotalHE = PowerExpand[xtotalME /. {mmu -> 0}]