Spring Problem Set 1
Problem 4

Get Tracer program:

<< "dirac/Tracer.m"

VectorDimension[4]

Spur[1]

Define spin projection operator:

spin[ll_, ss_] = (1/2) (G[ll, U] + G[ll, G5]**G[ll, ss])

Electron/Postron Current with definite spin:

electroncurrent = 
      G[le, p1 + me U]**spin[le, s1]**G[le, {mu}]**G[le, p2 - me U]**
        spin[le, s2]**G[le, {nu}]

Trace it:

ecurrent = electroncurrent /. le -> 1

Mu+, Mu- current without spin:

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

Trace it:

mcurrent = muoncurrent /. lmu -> 1

Contract the two tensors:

msq1 = ContractEpsGamma[ecurrent mcurrent]/(((2 energy))^4)

Start defining vectors.
e- in +z direction, e+ in -z:

pp1 = {energy, 0, 0, pe}pp2 = {energy, 0, 0, -pe}

Mu- at + theta, Mu- 180o opposite:

pp3 = {energy, pmu Sin[theta], 0, pmu Cos[theta]}pp4 = {energy, -pmu Sin[theta], 0, -pmu Cos[theta]}

Metric tensor:

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

Substitute momentum dot products:

msq2 = 
      Simplify[msq1 /. {p1 . p2 -> pp1 . gmunu . pp2, 
            p1 . p3 -> pp1 . gmunu . pp3, 
            p1 . p4 -> pp1 . gmunu . pp4, 
            p2 . p3 -> pp2 . gmunu . pp3, 
            p2 . p4 -> pp2 . gmunu . pp4, 
            p3 . p4 -> pp3 . gmunu . pp4}]

Define helicity spinor states:

msq3 = msq2 /. {p1 . s1 -> 0, p2 . s2 -> 0}

Define spin states for frame (take particle 1 to be always right handed):

ss1 = {(pe/me), 0, 0, (energy/me)}

ss1 . gmunu . pp1

Particle 2 right handed state:

ss2R = {(pe/me), 0, 0, -(energy/me)}

Particle 2 left handed state:

ss2L = {-(pe/me), 0, 0, (energy/me)}

ss2R . gmunu . pp2

ss2L . gmunu . pp2

Substitute particle 1 dot products:

same = {p2 . s1 -> pp2 . gmunu . ss1, 
        p3 . s1 -> pp3 . gmunu . ss1, p4 . s1 -> pp4 . gmunu . ss1}

msq4 = Simplify[msq3 /. same]

Define right handed particle 2 dot products:

right = {p1 . s2 -> pp1 . gmunu . ss2R, 
        p3 . s2 -> pp3 . gmunu . ss2R, 
        p4 . s2 -> pp4 . gmunu . ss2R, 
        s1 . s2 -> ss1 . gmunu . ss2R}

Calculate matrix elecment sq with right handed states for both spinors (total spin = 0):

msq5R = Simplify[msq4 /. right]

msq6R = 
      Simplify[PowerExpand[
          msq5R /. {pe -> (Sqrt[(energy^2) - (me^2)]), 
              pmu -> (Sqrt[(energy^2) - (mmu^2)]), 
              Cos[2 theta] -> 2 Cos[theta]^2 - 1}], 
        Trig -> False]

msq7R = Collect[msq6R, energy]

Now do state with left handed particle 2 (spin 1 state):

left = {p1 . s2 -> pp1 . gmunu . ss2L, 
        p3 . s2 -> pp3 . gmunu . ss2L, 
        p4 . s2 -> pp4 . gmunu . ss2L, 
        s1 . s2 -> ss1 . gmunu . ss2L}

msq5L = Simplify[msq4 /. left]

msq6L = 
      Simplify[PowerExpand[
          msq5L /. {pe -> (Sqrt[(energy^2) - (me^2)]), 
              pmu -> (Sqrt[(energy^2) - (mmu^2)]), 
              Cos[2 theta] -> 2 Cos[theta]^2 - 1}], 
        Trig -> False]

msq7L = Collect[msq6L, energy]

Check total matrix element squared for unpolarized beams:

msqtotal = 
      Simplify[Expand[((msq7L + msq7R)/2)], Trig -> False]