Samara State University
Department of General and Theoretical Physics

Program for calculation of
Total cross section  σ=σ(s)
for \(\(e\^-\) \(e\^+\) -> γγ\) process with using FeynCalc
(in the framework of Mathematica 4.1)

author: Yu.P. Philippov

Samara 2007

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

\(<< "C:\Program Files\Wolfram Research\Mathematica\4.1\AddOns\Applications\FeynArts-3.2\FeynArts.m"\)

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

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

Paint[a2]

Second step -  FeynCalc package loading (if you have other address please  put it in next line)

<< "C:\\Program Files\\Wolfram \
Research\\Mathematica\\4.1\\AddOns\\Applications\\HighEnergyPhysics\\fc.m"

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

mp1 = m;
mp2 = m;
mγ = 0;

Fourth step -  Introduction of Mandelshtam variables

ScalarProduct[p1, p1] = (m^2);
\(\(\(ScalarProduct[p2, p2] = m\^2;\)\(
\)
    \)\)
ScalarProduct[k1, k1] = m(γ^2);
\(\(\(ScalarProduct[k2, k2] = mγ\^2;\)\(
\)
    \)\)
ScalarProduct[k1, k2] = (1/2) s;
ScalarProduct[p1, k1] = (1/2)*((mp1^2) - t);
ScalarProduct[p1, k2] = (1/2)*((mp1^2) - u);

Fifth step - Introduction of Spinor Amplitudes of the process

SA1 = 
Contract[(SpinorUBar[-p2, 
              mp2] . (I*e*
                GA[ν]) . (I*(((GS[p1 - k1] + m)/(t - (m^2))))) . (I*e*GA[μ]) .  
            SpinorU[p1, mp1])]

SAC1 = ComplexConjugate[SA1]

A11 = FermionSpinSum[
              SA1*SAC1*PolarizationSum[ComplexIndex[μ], μ]*
                PolarizationSum[ComplexIndex[ν], ν]] /. 
            DiracTrace -> Tr /. 
          p2 -> (k1 + k2 - p1);
AMP11 = Simplify[Contract[A11] /. u -> 2*(m^2) - t - s]

SA2 = 
      Contract[(SpinorUBar[-p2, 
              mp2] . (I*e*
                GA[ν]) . (I*(((GS[p1] - GS[k2] + m)/(u - (m^2))))) . (I*e*GA[μ]) .  
            SpinorU[p1, mp1])]

SAC2 = ComplexConjugate[SA2]

A22 = FermionSpinSum[
              SA2*SAC2*PolarizationSum[ComplexIndex[μ], μ]*
                PolarizationSum[ComplexIndex[ν], ν]] /. 
            DiracTrace -> Tr /. 
          p2 -> (k1 + k2 - p1);
AMP22 = Simplify[Contract[A22] /. u -> 2*(m^2) - t - s]

A12 = FermionSpinSum[
              SAC1*SA2*PolarizationSum[ComplexIndex[μ], μ]*
                PolarizationSum[ComplexIndex[ν], ν]] /. 
            p2 -> (k1 + k2 - p1) /. 
          DiracTrace -> Tr;
AMP12 = Simplify[Contract[A12] /. u -> 2*(m^2) - t - s]

Sixth step - Building of the process total amplitude

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

dsigma = 
      Simplify[Expand[AMP/(16*Pi*s*(s - 4*(m^2)))]]

tmax = (m^2) - (s/2) + (1/2)*(Sqrt[s*(s - 4*(m^2))]);
tmin = (m^2) - (s/2) - (1/2)*(Sqrt[s*(s - 4*(m^2))]);

sigma = 
      Simplify[Integrate[Expand[dsigma], {t, tmin, tmax}]] // 
        StandardForm

e := (Sqrt[4*π/137])

CS[m_, s_] := 
      Re[(1/(18769 (4 (m^2) - 
                    s) (s^2))) (8 π (8 (m^2) (Sqrt[s (-4 (m^2) + s)]) - (8 (m^4) + 6 (m^2) s + 
                        (s^2)) Log[-s - (Sqrt[s (-4 (m^2) + s)])] + (8 (m^4) + 6 (m^2) s + (s^2)) Log[
                      s - (Sqrt[s (-4 (m^2) + s)])] + 
                  8 (m^4) Log[-s + (Sqrt[s (-4 (m^2) + s)])] + 
                  6 (m^2) s Log[-s + (Sqrt[s (-4 (m^2) + s)])] + 
                  (s^2) Log[-s + (Sqrt[s (-4 (m^2) + s)])] - 
                  8 (m^4) Log[s + (Sqrt[s (-4 (m^2) + s)])] - 
                  6 (m^2) s Log[s + (Sqrt[s (-4 (m^2) + s)])] - 
                  (s^2) Log[
                      s + (Sqrt[s (-4 (m^2) + s)])]))]

CS[0.5, 1.1]
(* For hypothetical particle with mass m=0.01 MeV annihilation *)
CS[0.01, 5]

Plot[{CS[0.5, (s^2)], CS[0.01, (s^2)]},{s, 0, 7}, AxesLabel->{"(Sqrt[s]), MeV","(Subscript[σ, tot]),(1/(MeV^2))"}]