\(Samara\ State\ University

      Department\ of\ General\ and\ Theoretical\ Physics


      Program\ for\ calculation\ \ of\ 

      cross\ section\ dependence\ σ = σ \((s)\)\ \ \ 

      for\ \ γγ -> \(e\^-\) \(e\^+\)\ process\ 

      with\ using\ FeynCalc\ package


      author : \ Yu . \ P . \ Philippov


      Samara\ 2007\)

First step - FeynArts package load for Feynman diagrams drawing (if you have other address please put it in next line)

<< "C:\\Program Files\\Wolfram \
Research\\Mathematica\\4.1\\AddOns\\Applications\\FeynArts32\\FeynArts.m"

a1 = CreateVFTopologies[0, 2 -> 2];

a2 = 
        InsertFields[a1, {V, V} -> {F, -F}, Model -> "QED", 
          InsertionLevel -> {Particles}];

Paint[a2]

Second step - FeynCalc package load for process calculation (if you have other address please put it in next line)

<< HighEnergyPhysics`FeynCalc`

FeynCalc 4.1.1    For help, type ?FeynCalc,  use the built-in help system  or visit www.feyncalc.org

Third step - Introduction of process constants

m1 = m;
m2 = m;
mγ = 0;

Fourth step - Introduction of  Mandelshtam variations

ScalarProduct[p1, p1] = (m1^2);
\(\(\(ScalarProduct[p2, p2] = m2\^2;\)\(
\)
    \)\)
ScalarProduct[k1, k1] = m(γ^2);
\(\(\(ScalarProduct[k2, k2] = mγ\^2;\)\(
\)
    \)\)
ScalarProduct[k1, k2] = (1/2) s;
ScalarProduct[p1, k1] = (1/2)*((m^2) - t);
ScalarProduct[p1, k2] = 
        (1/2)*((m^2) - u);

    
gs[p_] := DiracSlash[p];
ga[mu_] := DiracMatrix[mu]

Fifth step - Introduction of Spinor Amplitudes for the process

\(A1 = 
      Contract[
I*\((SpinorUBar[p1, 
                m1] . \((I*eс*
                  GA[ν])\) . \(\(gs[k1] - gs[p1] + m\)\/\(t - 
                    m\^2\)\) . \((I*eс*GA[μ])\) . \ 
              SpinorU[\(-p2\), m2])\)]\)

AC1 = ComplexConjugate[A1]

A11 = FermionSpinSum[
              A1*AC1*PolarizationSum[ComplexIndex[μ], μ]*
                PolarizationSum[ComplexIndex[ν], ν]] /. 
            DiracTrace -> Tr /. 
          p2 -> (k1 + k2 - p1);
AMP11 = Contract[A11] // Simplify

\(A2 = 
      Contract[I*\((SpinorUBar[p2, 
                m2] . \((I*eс*
                  GA[ν])\) . \(\(gs[k2] - gs[p1] + m\)\/\(u - 
                    m\^2\)\) . \((I*eс*GA[μ])\) . \ 
              SpinorU[\(-p1\), m1])\)]\)

AC2 = ComplexConjugate[A2]

A22 = FermionSpinSum[
              AC2*A2*PolarizationSum[ComplexIndex[μ], μ]*
                PolarizationSum[ComplexIndex[ν], ν]] /. 
            DiracTrace -> Tr /. 
          p2 -> (k1 + k2 - p1);
AMP22 = Contract[A22] // Simplify

AC1*A2*PolarizationSum[ComplexIndex[μ], μ]*
      PolarizationSum[ComplexIndex[ν], ν]

A12 = FermionSpinSum[
              AC1*A2*PolarizationSum[ComplexIndex[μ], μ]*
                PolarizationSum[ComplexIndex[ν], ν]] /. 
            p2 -> (k1 + k2 - p1) /. 
          DiracTrace -> Tr;
AMP12 = Contract[A12] // Simplify

Sixth step - Building of the process resulting amplitude

AMP = Simplify[AMP11 + AMP22 + 2*AMP12]

Seventh step - Cross section definition

u = 2*m^2 - t - s;
dsigma = Simplify[AMP/(16*Pi*s^2)]

tmax = 
        m^2 - s/2 + Sqrt[s*(s - 4*m^2)]/2;
tmin = m^2 - s/2 - Sqrt[s*(s - 4*m^2)]/2;

CrSection0 = Integrate[dsigma, {t, tmin, tmax}]

CrSection = Simplify[CrSection0]

m = 0.000511;
e = Sqrt[4*Pi/137];

CrSection1[s_]:=-(1/(2 π (s^3))) ((e^4) (2 (m^2) + 
                  s) (-4 Log[(Sqrt[s (s - 4 (m^2))]) - 
                        s] (m^2) + 
                  4 Log[s + (Sqrt[s (s - 4 (m^2))])] (m^2) + (4 (m^2) - 
                        s) Log[-s - (Sqrt[s (s - 4 (m^2))])] + (s - 4 (m^2)) Log[s - (Sqrt[s (s - 4 (m^2))])] + 
                  s log((Sqrt[s (s - 4 (m^2))]) - s) - 
                  s Log[s + (Sqrt[s (s - 4 (m^2))])] + 
                  4 (Sqrt[s (s - 4 (m^2))])))

CrSection2[ξ_] := CrSection1[(ξ^2)];

\(Plot[CrSection2[ξ], {ξ, 0.002, 100}, 
      AxesLabel -> {\*""(Sqrt[s]), [(GeV^2)]"", \*""(Subscript[σ, Total]), [(1/(GeV^2))]""}]\)

\(Plot[Re[CrSection2[ξ]], {ξ, 0.0018, 0.05}, 
      AxesLabel -> {\*""(Sqrt[s]), [(GeV^2)]"", \*""(Subscript[σ, Total]), [(1/(GeV^2))]""}]\)

\(Plot[Re[CrSection2[ξ]], {ξ, 20, 100}, 
      AxesLabel -> {\*""(Sqrt[s]), [(GeV^2)]"", \*""(Subscript[σ, Total]), [(1/(GeV^2))]""}]\)