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<<FeynCalc`

Mandelstam Variables

ScalarProduct[k,p]=((s-(me^2))/2);
ScalarProduct[kp,pp]=((s-(me^2))/2);
ScalarProduct[k,pp]=-(((t-(me^2))/2));
ScalarProduct[p,kp]=-(((t-(me^2))/2));
ScalarProduct[k,kp]=-(u/2);
ScalarProduct[pp,p]=-(u/2);

Matrix element from lecture

iMCC=-SpinorUBar[pp,me].GA[μ].ChiralityProjector[-1].SpinorU[k,0]*SpinorUBar[kp,0].GA[ν].ChiralityProjector[-1].SpinorU[p,me]

Square the matrix element, contract indices

iMaC=ComplexConjugate[iMCC]/.{μ-> μC,ν-> νC}
Ma2=FermionSpinSum[iMCC*iMaC]*MetricTensor[μ,μC]*MetricTensor[ν,νC]//Contract
Masq=Ma2/.DiracTrace-> Tr//ExpandScalarProduct//Simplify
Msq=8*(GF^2)*Masq/.u-> -s-t//Simplify

Compute the cross-section, see Byckling & Katanje Chapter 2 final states interactions

(*Kallen function*)
λ[x_,y_,z_]:=(((x-y-z))^2)-4y*z;
dσdt=(1/4)(1/(16π))(1/(λ[s,(me^2),0]))Msq; (*factor of (1/4) for summing over final states and averaging over initial states*)
tplus[masq_,mbsq_,m1sq_,m2sq_,s_]:=masq+m1sq-(1/(2s))*((s+masq-mbsq)*(s+m1sq-m2sq)-((λ[s,masq,mbsq])^(1/2))*((λ[s,m1sq,m2sq])^(1/2)));
tminus[masq_,mbsq_,m1sq_,m2sq_,s_]:=masq+m1sq-(1/(2s))*((s+masq-mbsq)*(s+m1sq-m2sq)+((λ[s,masq,mbsq])^(1/2))*((λ[s,m1sq,m2sq])^(1/2)));
Simplify[(Integrate[dσdt,t]/.t-> tplus[0,(me^2),(me^2),0,s])-(Integrate[dσdt,t]/.t->tminus[0,(me^2),(me^2),0,s]),Assumptions->{s>0,me>0}]

Expand[(((GF^2) ((((me^2)-s))^2) ((me^2) s+(me^4)+(s^2)))/(12 π (s^3)))]