Gluon mass

[BobekJosef]

Hello,
apologies if the answer is obvious - I could not find it.

Is it possible to add gluon mass to the propagator? I guess I could try adjusting the amplitude and then proceed with the calculation. However, I would rather have some more automatic solution for all the processes.

Since I'm beginner in Wolfram (and FeynCalc) I'm also sending short script for easier communication if needed.

Many thanks in advance.

Josef

MakeBoxes[p1, TraditionalForm] := "\!\(\*SubscriptBox[\(p\), \(1\)]\)";
MakeBoxes[p2, TraditionalForm] := "\!\(\*SubscriptBox[\(p\), \(2\)]\)";
MakeBoxes[k1, TraditionalForm] := "\!\(\*SubscriptBox[\(k\), \(1\)]\)";
MakeBoxes[k2, TraditionalForm] := "\!\(\*SubscriptBox[\(k\), \(2\)]\)";

diags = InsertFields[
   CreateTopologies[0, 2 -> 2], {F[4, {1}], F[3, {1}]} ->
    	{F[4, {1}], F[3, {1}]}, InsertionLevel -> {Classes}, 
   Model -> "SMQCD",
   	ExcludeParticles -> {S[_], V[2], V[1], V[3], -V[3]}];

Paint[diags, ColumnsXRows -> {2, 1}, Numbering -> Simple,
  	SheetHeader -> None, ImageSize -> {512, 256}];

amp[0] = FCFAConvert[CreateFeynAmp[diags], IncomingMomenta -> {p1, p2},
  	OutgoingMomenta -> {k1, k2}, UndoChiralSplittings -> True, 
  ChangeDimension -> 4,
  	List -> False, SMP -> True, Contract -> True, DropSumOver -> True]

FCClearScalarProducts[];
SetMandelstam[s, t, u, p1, p2, -k1, -k2, SMP["m_d"], SMP["m_u"], 
  SMP["m_d"], SMP["m_u"]];

ampSq[0] = 1/(SUNN^2) (amp[0] (ComplexConjugate[amp[0]])) //
       	FeynAmpDenominatorExplicit // SUNSimplify[#, Explicit -> True,
        	SUNNToCACF -> True] & // 
     FermionSpinSum[#, ExtraFactor -> 1/2^2] & //
    	DiracSimplify // 
   TrickMandelstam[#, {s, t, u, 2 SMP["m_u"]^2 + 2 SMP["m_d"]^2}] & //
  	Simplify

[vsht]

Hi,

to add the gluon mass to all processes you need to create a new FeynArts model with massive gluons. Usually it goes
by first creating a FeynRules model and then exporting that to FeynArts. FeynRules takes some time to master, but is definitely a useful skill.

If it's too much work, then "hacking" the amplitudes manually is still the simplest workaround.

Cheers,
Vlad

[BobekJosef]

Many thanks for very quick the reply! Just follow up...
If I'd create FeynRules model with gluon mass term. Then exported it into FeynCalc/FeynArts/Models/...mod, then generate diagrams, followed by FeynCalc calculation.

Since right now I'm interested only into QCD LO 2->2 processes (so only gluon propagator is adjusted), can't I just go to FeynCalc/FeynArts/Models/ create copies of SMQCD.mod SM.mod and edit gluon mass

  V[5] == {
	SelfConjugate -> True,
	Indices -> {Index[Gluon]},
	Mass -> FCGV["MG"],
	QuantumNumbers -> {Sqrt[3] ColorCharge},
	PropagatorLabel -> "g",
	PropagatorType -> Cycles,
	PropagatorArrow -> None },

Now this produces seemingly "correct" amplitude

amp[0] = FCFAConvert[CreateFeynAmp[diags], IncomingMomenta -> {p1, p2},
  	OutgoingMomenta -> {k1, k2}, UndoChiralSplittings -> True, 
  ChangeDimension -> 4,
  	List -> False, SMP -> True, Contract -> True, DropSumOver -> True]

However there is
(FCGV(MG)(Glu5))^2

I added FCGV["MG"] -> SMP["m_G"] to FCPrepareFAAmp.m
and

SMP /:
	MakeBoxes[SMP["m_G"], TraditionalForm] :=
		SubscriptBox["m", "G"];

to SMP.m

After that I get this:

So there is still (Glu5)... But this seems like what I would want from adjusting amplitudes manually.

Proceeding with:

FCClearScalarProducts[];
SetMandelstam[s, t, u, p1, p2, -k1, -k2, SMP["m_d"], SMP["m_u"], 
  SMP["m_d"], SMP["m_u"]];

and

ampSq[0] = 1/(SUNN^2) (amp[0] (ComplexConjugate[amp[0]])) //
       	FeynAmpDenominatorExplicit // SUNSimplify[#, Explicit -> True,
        	SUNNToCACF -> True] & // 
     FermionSpinSum[#, ExtraFactor -> 1/2^2] & //
    	DiracSimplify // 
   TrickMandelstam[#, {s, t, u, 2 SMP["m_u"]^2 + 2 SMP["m_d"]^2}] & //
  	Simplify

I get this error:
FCRenameDummyIndices: Error! FCRenameDummyIndices has encountered a fatal problem and must
abort the computation. The problem reads: The input expression violates Einstein summation. Some dummy indices appear more than twice.

Any idea how to proceed? I suspect this Glu5 but I cannot pinpoint solutions.

Again, many thanks for your time.

Josef

[vsht]

It seems like for some reason your gluon mass is getting an adjoint color index, which confuses SUNSimplify because you can't have three identical dummy indices in a term.

Since right now I'm interested only into QCD LO 2->2 processes (so only gluon propagator is adjusted), can't I just go to FeynCalc/FeynArts/Models/ create copies of SMQCD.mod SM.mod and edit gluon mass

You can do that too, but the price to pay are glitches like you're experiencing here. You can still fix them by hand with the right replacement rules, though.

[BobekJosef]

I see, could you give me a hint how to replace it? I think doing replacement manually is still the easiest option for me.

I was able to replace the m_G but index is still there I don't know how to replace it whole with the index.

amp[0] = FCFAConvert[CreateFeynAmp[diags], IncomingMomenta -> {p1, p2},
  	OutgoingMomenta -> {k1, k2}, UndoChiralSplittings -> True, 
  ChangeDimension -> 4,
  	List -> False, SMP -> True, Contract -> True, DropSumOver -> True]

`ampClean[0] = amp[0] //. SMP["m_G", ___] -> m;`

Many thanks, again.

Josef

[vsht]

Just apply StandardForm or FullForm to your expression to see its structure. This should give you an idea what replacement rule is needed.

[BobekJosef]

Many thanks for your guide! I'll post my full solution in case somebody else would need it and found it here. So no further questions from my side. Thank you.

---- SOLUTION ----
Edited gluon mass in FeynCalc/FeynArts/Models/SMQCD.mod

  V[5] == {
	SelfConjugate -> True,
	Indices -> {Index[Gluon]},
	Mass -> FCGV["MG"],
	QuantumNumbers -> {Sqrt[3] ColorCharge},
	PropagatorLabel -> "g",
	PropagatorType -> Cycles,
	PropagatorArrow -> None },

add FCGV["MG"] -> SMP["m_G"] to FCPrepareFAAmp.m
and

SMP /:
	MakeBoxes[SMP["m_G"], TraditionalForm] :=
		SubscriptBox["m", "G"];

to SMP.m

After converting FeynArt to FeynCalc

amp[0] = FCFAConvert[CreateFeynAmp[diags], IncomingMomenta -> {p1, p2},
  	OutgoingMomenta -> {k1, k2}, UndoChiralSplittings -> True, 
  ChangeDimension -> 4,
  	List -> False, SMP -> True, Contract -> True, DropSumOver -> True]

I got rid of Glu5 index: `ampClean[0] = amp[0] //. SMP["m_G"][SUNIndex[Glu5]] -> SMP["m_G"];`