Determine the Neutrino-Hadron Deep Inelastic Cross Section

Get Tracer and set up:

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

VectorDimension[4]

Define the muon projection operator:

muon  =  G[lmu,  kp  +  mmu U]

And the neutrino projection operator:

munu  =  G[lmu,  k]

Calculate the lepton current:

\(muontensor\  = \ 
      muon\  ** \ G[lmu, {mu}]\  ** \ 
        G[lmu, U\  + \ G5] ** 
			munu\  ** \ G[lmu, {nu}] ** 
        G[lmu, U\  + \ G5]\)

Spur[1]

mt  =  muontensor /. lmu -> 1

Write down the most general hadron tensor:

\(hadrontensor\  = \ 
      V1\ S[{mu}, {nu}]\  - \ 
        S[P, {mu}]\ S[
            P, {nu}]\ V2\  + 
		\((S[P, {mu}]\ S[q, {nu}] + 
              S[P, {nu}]\ S[q, {mu}])\) 
          V3\  + 
		S[q, {mu}]\ S[q, {nu}]\ V4\  + \ 
        I\ Eps[{mu}, {nu}, P, q]\ V5\)

Contract with the lepton tensor:

matrix  = 
      Collect[ContractEpsGamma[mt hadrontensor], {V1, V2, V3, V4, 
          V5}]

Show that V3 and V4 terms are proportional to muon mass:

Simplify[
      Simplify[matrix[[5]] /. q -> k - kp] /. {kp . kp -> mmu^2, 
          k . k -> 0}]

Simplify[
      Simplify[matrix[[4]] /. q -> k - kp] /. {kp . kp -> mmu^2, 
          k . k -> 0}]

Drop these terms from further consideration:

matrix1  =  
      matrix[[1]] + matrix[[2]] + matrix[[3]]

Change to deep inelstic scattering variables:

matrix2  =  
      Collect[Simplify[(matrix1 /. {k . q -> -2 k . kp, 
                  kp . q -> 2  k . kp, kp . P -> k . P - q . P}) /. {k . 
                  kp  ->  Mn Enu x y, P . P -> Mn^2, k . P -> Enu Mn, 
              q . P -> Mn Enu y}], {V1, V2, V5}]

Change to "normal" structure functions:

matrix3  =  
      Collect[Simplify[ 
          matrix2 /. {V1 -> -F1/Mn,  V2 -> -F2/Enu/y/Mn^2, 
              V5 -> F3/Mn^2/y/Enu/4}], {F1, F2, F3}]

Calculate cross section:

\(Xsect\  = \ 
      Collect[Simplify[
          y\ \(Mn\ /\((2\ Pi)\)\)\ /2\ \((g/2)\)^2\ g^2\ /Mw^4  
            matrix3], 
		{F1, F2, F3}]\)

Xsect1 = (Gf^2  Mn Enu/Pi) 
        Collect[Simplify[(Xsect /. g -> Sqrt[Gf] Mw)/(Gf^2  
                  Mn Enu/Pi)], {F1, F2, F3}]