QCD calculation: Webber et al. Sec. 7

We have learned how to sum/average helicities and polarizations.
Now it’s time to handle color; you will use ColourME[].

Two-jet cross sections (Only QCD; mass ignored) : Ellis-Stirling-Webber Sec.7.5

Initialization

Exit[];

$Language="English";
<<FeynArts`
<<FormCalc`

Create Topology (and also ignore helicities)

topologies=CreateTopologies[0,2->2];
_Hel=0;

q q' -> q q'

diag=InsertFields[topologies,{F[3,{1}],F[3,{2}]}->{F[3,{1}],F[3,{2}]},InsertionLevel->{Classes},
Model->"SMQCD",
ExcludeParticles->{V[1],V[2],V[3],V[4],S[1],S[2]}];
Paint[diag];
ClearProcess[];
amp=CalcFeynAmp[CreateFeynAmp[diag],FermionChains->VA];

squared = SquaredME[amp];
helSquared=squared//.HelicityME[amp]//.{MU->0,MU2->0,MC->0,MC2->0}//Simplify

ColourME is used like HelicityME.

ColourME[amp]

helSquared//.ColourME[amp]//Simplify

%[[1]]//.%[[2]]//.Join[Abbr[],Subexpr[],{MC2->0,MU2->0,Den[a_,b_]:>(1/(a-b))}]
FullSimplify[%/.T->-S-U]

FullSimplify[(gs^4)(4/9)(((S^2)+(U^2))/(T^2))//.{gs->(Sqrt[4π Alfas])}]

Now you notice that, while HelicityME takes AVARAGE, ColourME takes SUM!!!
This behavior can be summarized into two “lessons”:

LESSON

×2 for each FINAL fermion

DIVIDE by INITIAL colour factor

So, in summary, we should calculate the amplitude as follows:

sq=SquaredME[amp]//.HelicityME[amp]//.ColourME[amp];
result=(sq[[1]]//.sq[[2]])*((2^2)*((((1/3)))^2))//Simplify
FullSimplify[result//.{Alfas2->(((((gs^2)/(4π))))^2),MC->0,MU->0,MC2->0,MU2->0,Den[a_,b_]:>(1/(a-b))}//.U->-S-T]
FullSimplify[(gs^4)(4/9)(((S^2)+(U^2))/(T^2))//.U->-S-T]

q q' -> q q' (Again)

Let’s do it again at a stretch.

diag=InsertFields[topologies,{F[3,{1}],F[3,{2}]}->{F[3,{1}],F[3,{2}]},InsertionLevel->{Classes},
Model->"SMQCD",
ExcludeParticles->{V[1],V[2],V[3],V[4],S[1],S[2]}];
Paint[diag];
ClearProcess[];
amp=CalcFeynAmp[CreateFeynAmp[diag],FermionChains->VA];
sq=SquaredME[amp]//.HelicityME[amp]//.ColourME[amp];
FullSimplify[(sq[[1]]//.sq[[2]])*((2^2)/(3^2))];
FullSimplify[%//.{Alfas2->(((((gs^2)/(4π))))^2),MC->0,MU->0,MC2->0,MU2->0,Den[a_,b_]:>(1/(a-b))}//.U->-S-T]
FullSimplify[(gs^4)(4/9)(((S^2)+(U^2))/(T^2))//.U->-S-T]

g q -> g q

Now you can calculate all the QCD tree-level diagrams.

diag=InsertFields[topologies,{V[5],F[3,{1}]}->{V[5],F[3,{1}]},InsertionLevel->{Classes},
Model->"SMQCD",
ExcludeParticles->{V[1],V[2],V[3],V[4],S[1],S[2]}];
Paint[diag];
ClearProcess[];
amp=CalcFeynAmp[CreateFeynAmp[diag],FermionChains->VA];
sqHelCol=SquaredME[amp]//.HelicityME[amp]//.ColourME[amp]

Now do PolarizationSum. As we want to take an average (the sum) over initial (final) photon, we divide the result by 2.
With the helicity factor 2 and color factor (1/3)×(1/8), the result is given by

MyResult=(1/2)*2*(1/24)*PolarizationSum[sqHelCol,GaugeTerms->False]//.Subexpr[]//.{Den[p2_,m2_]:>(1/(p2-m2)),MU->0,MU2->0,Alfas2->(((((gs^2)/(4π))))^2)}//FullSimplify
WebberResult=(gs^4)(-(4/9)(((S^2)+(U^2))/(S U))+(((U^2)+(S^2))/(T^2)))//FullSimplify

FullSimplify[MyResult==WebberResult,Assumptions->{S+T+U==0}]

We used GaugeTerms->False, because we know the diagram set is gauge-independent.
We can calculate with GaugeTerms and check that the result is independent of eta.
Let’s do it.

MyResultWithGaugeTerms=(1/2)*2*(1/24)*PolarizationSum[sqHelCol,GaugeTerms->True]//.Subexpr[]//.Abbr[]//.Den[p2_,m2_]:>(1/(p2-m2))

MyResultWithGaugeTerms//.MU2->((S+T+U)/2)//Simplify

It seems that this amplitude depends on “eta” (i.e., gauge-dependent). However, note that, in this 2->2 process, k[1]+k[2]==k[3]+k[4]. Therefore...

%//.{Pair[eta_,k[2]]:>Pair[eta,k[3]]+Pair[eta,k[4]]-Pair[eta,k[1]]}//Simplify

And we just have confirmed that this amplitude is gauge independent.
Needless to say, this agrees with the result in the canon.

FullSimplify[%==WebberResult,Assumptions->{S+T+U==2MU2,MU2==0,Alfas2==(((((gs^2)/(4π))))^2)}]

Summary of Calculation

Exit[];

(* SET UP *)
$Language="English";
<<FeynArts`
<<FormCalc`

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

(* Diagram *)
diag=InsertFields[topologies,
{V[5],F[3,{1}]}->{V[5],F[3,{1}]},
InsertionLevel->{Classes},
Model->"SMQCD",
ExcludeParticles->{V[1],V[2],V[3],V[4],S[1],S[2]}];

Paint[diag];

(* Amplitude *)
ClearProcess[];
amp=CalcFeynAmp[CreateFeynAmp[diag],FermionChains->VA];
squ=SquaredME[amp];

(* If contains Fermion *)
squ=squ//.HelicityME[amp]//.{Hel[_]->0}; (* or simply execute "_Hel=0" before this evaluation.*)

(* If contains Colored Particles *)
squ=squ//.ColourME[amp];

(* If contains Vector *)
rawResult = PolarizationSum[squ,GaugeTerms->False]; (* Asserting gauge invariance! *)
(* otherwise:
rawResult = squ[[1]]//.squ[[2]];
*)

(* Appropreate factors depending on the process *)
result=2*(1/(3*8))*(1/2)*rawResult;
FullSimplify[result]

MyResult=FullSimplify[%//.Subexpr[]//.{Den[p2_,m2_]:>(1/(p2-m2)),MU->0,MU2->0,Alfas2->(((((gs^2)/(4π))))^2)}]
WebberResult=(gs^4)(-(4/9)(((S^2)+(U^2))/(S U))+(((U^2)+(S^2))/(T^2)))

FullSimplify[MyResult==WebberResult,Assumptions->{S+T+U==0}]

LESSON

×2 for each FINAL fermion

DIVIDE by INITIAL colour factor

DIVIDE by INITIAL vector particles.