QED calculation: Peskin Sec. 5

(SuperPlus[e])(SuperMinus[e])->(SuperPlus[μ])(SuperMinus[μ]) (QED): Peskin P.131-136

Initialization

You should choose English to avoid unnecessary confusions.

Exit[];

$Language="English";

<<FeynArts`
<<FormCalc`

Create Topology

topologies=CreateTopologies[0,2->2]
Paint[topologies]

Insert Particles

FeynArts Manual Appendix B has a list of the particles in the (default) SM model.

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

Paint[diagrams]

Insert Particles (again...)

(* We concentrate on QED for simplicity. *)
diagrams=InsertFields[topologies,{F[2,{1}],-F[2,{1}]}->{F[2,{2}],-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];
matrix=SquaredME[amplitude]//.HelicityME[amplitude]

(* Assumes unpolarized fermions *)
matrix = matrix//.Hel[_]->0

matrix//.Abbr[]//.Subexpr[]//.{Den[p2_,m2_]:>(1/(p2-m2)),Alfa->((e^2)/(4π)),Alfa2->(((((e^2)/(4π))))^2),ME2->0}

%//.{S->4(EE^2),T->(MM^2)-2((EE^2)-EE(Sqrt[(EE^2)-(MM^2)])CosTheta),U->-S-T+2MM2}

%[[1]]//.%[[2]]

Collect[FullSimplify[%],{e,CosTheta}]

This is different from the result in Peskin’s book.
We should be very careful to use tools for automatic calculation.

Doing again with patience.

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

Here we used “Weyl” as an option, which is the default according to the manual.
Because this process is vector-like (i.e. not chiral), we use “VA” as the FermionChains option.

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

Here you see the incoming and outgoing particles with, e.g., momentum “k[1]”, mass “ME”, etc.
The “amplitude” is the last object, which is

amplitude[[1]]

with some abbreviations. You can expand the abbreviations by

amplitude[[1]] //. Join[Abbr[],Subexpr[]]

Here,  “Abbr[]” and “Subexpr[]” give the abbreviations used in this calculation.

Abbr[]

Subexpr[]

Then we square the matrix element.

squared = SquaredME[amplitude]

In FormCalc v9.6 (and later), SquaredME returns a list of {squared, squaredRules} to improve calculation speed.
Theoretically we should apply these rules as late as possible, but in this first lecture we apply them right now so that you are not confused by the compressed notation.

squaredWithRules=SquaredME[amplitude]
squared=squaredWithRules[[1]]//.squaredWithRules[[2]]

Then we apply HelicityME.

squared//.HelicityME[amplitude]

Hmm... anyway we want to remove the helicity....

squared//.HelicityME[amplitude]//.{Hel[1]->0,Hel[2]->0,Hel[3]->0,Hel[4]->0}

This is actually THE AVERAGE over the helicity! Because we want to average for the incoming states, while to sum over the outgoing, we have to multiply by four.

result=4*squared//.HelicityME[amplitude]//.{Hel[1]->0,Hel[2]->0,Hel[3]->0,Hel[4]->0}
result//.{Den[p2_,m2_]:>(1/(p2-m2)),Alfa->((e^2)/(4π)),Alfa2->(((((e^2)/(4π))))^2),ME2->0}//.{S->4(EE^2),T->(MM^2)-2((EE^2)-EE(Sqrt[(EE^2)-(MM^2)])CosTheta),U->-T-S+2MM2}

Collect[FullSimplify[%],{e,CosTheta},Expand]

Perfect.