QED calculation: Peskin Sec.	5

(SuperMinus[e])(SuperPlus[e])->γ γ : Peskin P.153-54

Initialization

Exit[];

$Language="English";

<<FeynArts`
<<FormCalc`

Create Topology & Insert Particles

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

Pass the diagram to FormCalc

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

Take an average for initial fermions

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

What are these abbreviations?

Abbr[]
Subexpr[]

helSquared //. Subexpr[] //. Abbr[]

e[1], ec[1], etc. are the photon polarization, which we have to sum up by PolarizationSum.
As the name says, PolarizationSum is expected to return the sum over polarizations.

polSummed=PolarizationSum[helSquared]

Note that, with PolarizationSum, we need not do the replacement “sq[[1]] //. sq[[2]]” that we did in the previous lectures.

polSummed//.Subexpr[]//.Abbr[]

eta[1] etc. are parameters that undertake the gauge-dependence (see manual for details).
A gauge-independent quantity does not depend on them.
We know this amplitude is gauge-independent, so PolSummedME is actually independent of eta.
Let’s check it before proceeding, approximating (Subscript[m, e])=0.

polSummed//.Join[Subexpr[],Abbr[],{
ME2->0,
S->-T-U,
Den[a_,b_]:>(a-b)^(-1),
Pair[eta[a_],k[1]]:>-Pair[eta[a],k[2]]+Pair[eta[a],k[3]]+Pair[eta[a],k[4]]
}]//FullSimplify

Gauge independent as expected!
In practice we are perfectly sure that we are calculating gauge-independent quantity.
So we neglect the gauge-dependent parameters by GaugeTerms->False.

polSummedNoEta=PolarizationSum[helSquared,GaugeTerms->False]//.Join[Subexpr[],Abbr[]]

You can check this agrees with Peskin Eq. (5.105).

Collect[polSummedNoEta//.{
Den[a_,b_]:>(a-b)^(-1),
Alfa2->(((((e^2)/(4π))))^2),
S->-T-U+2ME2,
T->ME2-2"[p1.k1]",
U->ME2-2"[p1.k2]"
},ME2,Simplify]