Trying to get 2-loop amplitude/form factor in GLI format using ApartFF and FCLoopTensorReduce

[mashhoudtt]

I am attempting to get my two-loop form factor in with 5 propagators in terms of GLIs. I get my amplitude using FCFAConvert and then use DiracSimplify. I then apply
ampForm = DiracSimplify[DiracTrace[P[1, \[Mu]] . ampraw], FCParallelize -> True];
with P being my projection operator. This gives me products such as Pair[Momentum[k2,D],Momentum[p1,D]]Pair[Momentum[k2,D],Momentum[p1,D]]
To which I then proceed to implement as an example
Pair[Momentum[k2, D], Momentum[p1, D]] -> 1/2 ((Pair[Momentum[k2 + p1, D], Momentum[k2 + p1, D]] - m^2) - Pair[Momentum[k2, D], Momentum[k2, D]])
ampFF = ApartFF[ampForm, {k1, k2}, FDS -> False];
I am having two issues with this.

1. I get terms with products again as before.
2. ApartFF creates a new propagator GenericPropagatorDenominator[Pair[Momentum[k2, D], Momentum[k2, D]] - 2 Pair[Momentum[k2, D], Momentum[p2, D]] + Pair[Momentum[p2, D], Momentum[p2, D]], {-1, 1}] inside FeynAmpDenominator, to which it becomes 1/1/propagator.
   A good representation of my issues can be seen in the image attached

I would appreciate any and all help, as I am a beginner to all of this.

Edit: The diagram is given by
diags = InsertFields[CreateTopologies[2, 2 -> 1, ExcludeTopologies -> {Tadpoles}], {F[2, {1}], -F[2, {1}]} -> {V[1]},InsertionLevel -> {Particles}, ExcludeParticles -> {V[2 | 3], S[_], U[_]}] ;

The amplitude is
ampRaw = FCFAConvert[CreateFeynAmp[DiagramExtract[diags, {88}], Truncated -> True, GaugeRules -> {}, PreFactor -> 1], IncomingMomenta -> {p1, p2}, OutgoingMomenta -> {q}, LorentzIndexNames -> {\[Mu], \[Nu], \[Rho], \[Theta], \[Eta]}, LoopMomenta -> {k1, k2}, UndoChiralSplittings -> True, ChangeDimension -> D, List -> True, SMP -> True, DropSumOver -> True] /. {q -> p1 + p2, GaugeXi[A] -> 1} /. {k1 ->k1 + p1} // SMPToSymbol

I then used DiracSimplify and set the masses to zero. The projection is given by
P[j_, \[Mu]_] := ((GSD[p1] + m)/m ) . (I*g1[j]* GAD[\[Mu]] + (I/2 m)*g2[j] (FVD[p2, \[Mu]] - FVD[p1, \[Mu]])) . ((GSD[p2] - m)/m) where in the case i've given we're dealing with j=1 so g1[1] := -1/(e * Nc) * 1/(4 (1 - \[Epsilon]))* 1/(s/m^2 - 4); and g2[1] := 1/(e * Nc) * (3 - 2 \[Epsilon])/(1 - \[Epsilon])* 1/(s/m^2 - 4)^2;
Then I do ampForm = DiracSimplify[DiracTrace[P[1, \[Mu]] . ampRaw], FCParallelize -> True]; followed by replacing it with this mess
/. {Pair[Momentum[k1, D], Momentum[p1, D]] -> 1/2 ((Pair[Momentum[k1 + p1, D], Momentum[k1 + p1, D]] - m^2) - Pair[Momentum[k1, D], Momentum[k1, D]]), Pair[Momentum[k1, D], Momentum[p2, D]] -> -1/ 2 ((Pair[Momentum[k1 - p2, D], Momentum[k1 - p2, D]] - m^2) - Pair[Momentum[k1, D], Momentum[k1, D]]), Pair[Momentum[k1, D], Momentum[k2, D]] -> -1/ 2 (Pair[Momentum[k1 - k2, D], Momentum[k1 - k2, D]] - Pair[Momentum[k1, D], Momentum[k1, D]] - Pair[Momentum[k2, D], Momentum[k2, D]]), Pair[Momentum[k2, D], Momentum[p1, D]] -> 1/2 ((Pair[Momentum[k2 + p1, D], Momentum[k2 + p1, D]] - m^2) - Pair[Momentum[k2, D], Momentum[k2, D]]), Pair[Momentum[k2, D], Momentum[p2, D]] -> -1/ 2 ((Pair[Momentum[k2 - p2, D], Momentum[k2 - p2, D]] - m^2) - Pair[Momentum[k2, D], Momentum[k2, D]]), Pair[Momentum[p1, D], Momentum[p2, D]] -> 1/2 (s - 2 m^2), Pair[Momentum[p1, D], Momentum[p1, D]] -> m^2 , Pair[Momentum[p2, D], Momentum[p2, D]] -> m^2 }
and I end with:
ampFF = ApartFF[ampForm, {k1, k2}, FDS -> False];

[vsht]

Can you please post explicit Mathematica code and the original expression so that one can reproduce it?

Images are not helpful here because you obviously can't run them in Mathematica and reconstructing expressions from images
is a waste of everyone's time.

[mashhoudtt]

Hello, yes I will edit it now!

[vsht]

This already looks better. But still ampRaw is not provided, so it is not reproducible. Can you add this one too?

[mashhoudtt]

I provided ampRaw=FCFAConvert[...] in my edit. The only thing I did on top of that was DiracSimplify[ampRaw /. {me->m, mu->0}, FCParallelize->True];].
If the confusion is with my ampFormRep term in the image provided, that is just my ampForm which I also gave instructions to in the edit + the kinematic substitutions I provided, i.e. ampFormRep = ampForm /. {Pair[Momentum[k1, D], Momentum[p1, D]] -> 1/2 ((Pair[Momentum[k1 + p1, D], Momentum[k1 + p1, D]] - m^2) - Pair[Momentum[k1, D]...

[vsht]

I definitely cannot reproduce the appearance of GenericPropagatorDenominator

$FAVerbose = 0;
diags = InsertFields[
   CreateTopologies[2, 2 -> 1, 
    ExcludeTopologies -> {Tadpoles}], {F[2, {1}], -F[2, {1}]} -> {V[1]},
    InsertionLevel -> {Particles}, 
   ExcludeParticles -> {V[2 | 3], S[_], U[_]}];
ampRaw = 
 FCFAConvert[
     CreateFeynAmp[DiagramExtract[diags, {88}], Truncated -> True, 
      GaugeRules -> {}, PreFactor -> 1], IncomingMomenta -> {p1, p2}, 
     OutgoingMomenta -> {q}, 
     LorentzIndexNames -> {\[Mu], \[Nu], \[Rho], \[Theta], \[Eta]}, 
     LoopMomenta -> {k1, k2}, UndoChiralSplittings -> True, 
     ChangeDimension -> D, List -> False, SMP -> True, 
     DropSumOver -> True] /. {q -> p1 + p2, GaugeXi[A] -> 1} /. {k1 ->
      k1 + p1} // SMPToSymbol
P[j_, \[Mu]_] := ((GSD[p1] + m)/
    m) . (I*g1[j]*GAD[\[Mu]] + (I/2 m)*
     g2[j] (FVD[p2, \[Mu]] - FVD[p1, \[Mu]])) . ((GSD[p2] - m)/m) 
g1[1] := -1/(e*Nc)*1/(4 (1 - \[Epsilon]))*1/(s/m^2 - 4); 
g2[1] := 1/(e*Nc)*(3 - 2 \[Epsilon])/(1 - \[Epsilon])*1/(s/m^2 - 4)^2;
ampForm = 
  DiracSimplify[DiracTrace[P[1, \[Mu]] . ampRaw], 
   FCParallelize -> True];

amfFormRep = 
  ampForm /. {Pair[Momentum[k1, D], Momentum[p1, D]] -> 
     1/2 ((Pair[Momentum[k1 + p1, D], Momentum[k1 + p1, D]] - m^2) - 
        Pair[Momentum[k1, D], Momentum[k1, D]]), 
    Pair[Momentum[k1, D], 
      Momentum[p2, D]] -> -1/
       2 ((Pair[Momentum[k1 - p2, D], Momentum[k1 - p2, D]] - m^2) - 
        Pair[Momentum[k1, D], Momentum[k1, D]]), 
    Pair[Momentum[k1, D], 
      Momentum[k2, D]] -> -1/
       2 (Pair[Momentum[k1 - k2, D], Momentum[k1 - k2, D]] - 
        Pair[Momentum[k1, D], Momentum[k1, D]] - 
        Pair[Momentum[k2, D], Momentum[k2, D]]), 
    Pair[Momentum[k2, D], Momentum[p1, D]] -> 
     1/2 ((Pair[Momentum[k2 + p1, D], Momentum[k2 + p1, D]] - m^2) - 
        Pair[Momentum[k2, D], Momentum[k2, D]]), 
    Pair[Momentum[k2, D], 
      Momentum[p2, D]] -> -1/
       2 ((Pair[Momentum[k2 - p2, D], Momentum[k2 - p2, D]] - m^2) - 
        Pair[Momentum[k2, D], Momentum[k2, D]]), 
    Pair[Momentum[p1, D], Momentum[p2, D]] -> 1/2 (s - 2 m^2), 
    Pair[Momentum[p1, D], Momentum[p1, D]] -> m^2, 
    Pair[Momentum[p2, D], Momentum[p2, D]] -> m^2};

ampFF = ApartFF[ampForm, {k1, k2}, FDS -> False];

FreeQ[ampFF, GenericPropagatorDenominator]
(*True*)

Regarding

I get terms with products again as before.

I'm not sure that I understand what you're trying to achieve by those replacements. ApartFF doesn't remove scalar products
in the numerators so you're expecting something different than what it is supposed to do.

Perhaps you should have a look at the examples for doing multiloop calculations with FeynCalc, e.g.

https://github.com/FeynCalc/feyncalc/blob/master/FeynCalc/Examples/QCD/TwoLoops/Mathematica/GlGl-H.m
https://github.com/FeynCalc/feyncalc/blob/master/FeynCalc/Examples/QCD/TwoLoops/Mathematica/Gh-Gh.m

[mashhoudtt]

I see where the issue is coming from in terms of the GenericPropagatorDenominator. When I include the scalar product replacements, FreeQ becomes false, i.e. ampFF = ApartFF[ampFormRep, {k1, k2}, FDS -> False]; instead of ampForm.
With regards to what I'm trying to achieve with the replacements; I'm trying to rewrite the scalar products in terms of the propagators only, so I can have my result be in the format of (scalar terms)*G[n1,n2,n3,n4,n5] +...and so on.
The way I made it this far was by looking at the examples but there have been some differences that I couldn't make sense of/wrap my head around which is why I'm asking for help.
Can I ask how you'd suggest the best way I can get the scalar products in terms of the propagators so as to then be able to get it in the format I would like?
Also, thank you so much for your help thus far.

[vsht]

Can I ask how you'd suggest the best way I can get the scalar products in terms of the propagators so as to then be able to get it in the format I would like?

That's exactly how the examples I quoted work. The amplitude just before running IBPs is always linear combination of GLIs.

What you're probably looking for is

https://feyncalc.github.io/FeynCalcBookDev/FCLoopCreateRulesToGLI.html

But you cannot apply it to the amplitude in the form you have it now. Instead, you must follow all the steps starting with FCLoopFindTopologies because this is how the functions are intended to work. This is why I linked the examples because
they precisely show the template to use.