QED calculation: Peskin Sec.	5

(SuperMinus[e])(SuperMinus[μ])->(SuperMinus[e])(SuperMinus[μ]) (QED): Peskin P.153-54

Initialization

Exit[];

$Language="English";

<<FeynArts`
<<FormCalc`

Create Topology

topologies=CreateTopologies[0,2->2];
(*Paint[topologies];*)

Insert Particles

Appendix B of the FeynArts manual contains a list of the particles in the (default) SM model.

diagrams=InsertFields[topologies,{F[2,{1}],F[2,{2}]}->{F[2,{1}],F[2,{2}]},InsertionLevel->{Classes}];

Paint[diagrams];

diagrams=InsertFields[topologies,{F[2,{1}],F[2,{2}]}->{F[2,{1}],F[2,{2}]},InsertionLevel->{Classes},
	ExcludeParticles->{V[2],S[1],S[2]}];
Paint[diagrams];

Pass the diagram to FormCalc

ClearProcess[];
amplitude=CalcFeynAmp[CreateFeynAmp[diagrams],FermionChains->VA]

Remember that SquaredME returns a list of {squared, squaredRules}.
In Lecture 1-1 we applied the rules right now to avoid confusion.
Here we follow the official recommendation: applying the rules as late as possible to improve the calculation speed.

squared=SquaredME[amplitude]

helSquared=squared//.HelicityME[amplitude]

matrixElement=helSquared//.{Hel[_]->0}

We have obtained unpolarized amplitude by Hel[_]->0. Instead, setting _Hel=0 in advance gives the same result (and actually faster).

_Hel=0;
squared = SquaredME[amplitude] 
helAveragedSquared=squared//.HelicityME[amplitude]//Simplify

This is not the result. We should

multiply it by four (Do you remember why?), and

apply the second part (we above called “squaredRules” ) to the first part.

So,

result=4*(helAveragedSquared[[1]]//.helAveragedSquared[[2]])//Simplify

ourResult=result//.{ME->0,ME2->0,Den[a_,b_]:>(a-b)^(-1),S->(((kk+EE))^2),T->-2(kk^2)(1-CosTheta),U->2ME2+2MM2-S-T}

PeskinResult=((2(4π Alfa)^2)/((kk^2)(((1-CosTheta))^2)))((((EE+kk))^2)+(((EE+kk CosTheta))^2)-MM2(1-CosTheta))
FullSimplify[ourResult==PeskinResult,Assumptions->(EE^2)==(kk^2)+MM2]

OK.
Now let us review the procedure:

topologies=CreateTopologies[0,2->2];
diagrams=InsertFields[topologies,{F[2,{1}],F[2,{2}]}->{F[2,{1}],F[2,{2}]},InsertionLevel->{Classes},
	ExcludeParticles->{V[2],S[1],S[2]}];
Paint[diagrams];

ClearProcess[];
amplitude=CalcFeynAmp[CreateFeynAmp[diagrams],FermionChains->VA];
_Hel=0;
helSquared=SquaredME[amplitude]//.HelicityME[amplitude];
result=4*helSquared[[1]]//.helSquared[[2]]//Simplify

result//.{ME->0,ME2->0,Den[a_,b_]:>(a-b)^(-1),S->(((kk+EE))^2),T->-2(kk^2)(1-CosTheta),U->2ME2+2MM2-S-T}
FullSimplify[%==PeskinResult,Assumptions->(EE^2)==(kk^2)+MM2]