Name: Philipp Date: 08/31/17-09:28:04 PM Z

Hi,

I was decomposing the following expressions in the 3 loop momenta {p1,p2,p3} with FCMultiLoopTID

exp = CA^3 FeynAmpDenominator[PropagatorDenominator[Momentum[p1,D],0],PropagatorDenominator[Momentum[p2,D],I mE],PropagatorDenominator[Momentum[p3,D],I mE],PropagatorDenominator[Momentum[p1,D],0],PropagatorDenominator[Momentum[p1,D],0],PropagatorDenominator[Momentum[p1,D],0],PropagatorDenominator[Momentum[-p1+p2,D],I mE],PropagatorDenominator[Momentum[-p1+p3,D],I mE]] Pair[LorentzIndex[Lor1,D],Momentum[p1,D]] Pair[LorentzIndex[Lor2,D],Momentum[p1,D]] (Pair[Momentum[p1,D],Momentum[p1,D]]-2 (Pair[Momentum[p1,D],Momentum[p2,D]]+Pair[Momentum[p1,D],Momentum[p3,D]]-2 Pair[Momentum[p2,D],Momentum[p3,D]]))^2 SMP[g_s]^6 SUNDelta[SUNIndex[a],SUNIndex[b]]

However after the decomposition I am still left with an expression that is not scalarised and has open Lorentz indices lor1,lor2 and only depends on two loop momenta even:

(1/D)2 CA^3 FeynAmpDenominator[PropagatorDenominator[Momentum[p1,D],0],PropagatorDenominator[Momentum[p1,D],0],PropagatorDenominator[Momentum[p1,D],0],PropagatorDenominator[Momentum[p1,D]-Momentum[p2,D],I mE],PropagatorDenominator[Momentum[p1,D]-Momentum[p3,D],I mE]] Pair[LorentzIndex[Lor1,D],LorentzIndex[Lor2,D]] SMP[g_s]^6 SUNDelta[SUNIndex[a],SUNIndex[b]]-(1/D)2 CA^3 FeynAmpDenominator[PropagatorDenominator[Momentum[p1,D],0],PropagatorDenominator[Momentum[p1,D],0],PropagatorDenominator[Momentum[p1,D],0],PropagatorDenominator[Momentum[p2,D],I mE],PropagatorDenominator[Momentum[p1,D]-Momentum[p3,D],I mE]] Pair[LorentzIndex[Lor1,D],LorentzIndex[Lor2,D]] SMP[g_s]^6 SUNDelta[SUNIndex[a],SUNIndex[b]]+(1/D)2 CA^3 FeynAmpDenominator[PropagatorDenominator[Momentum[p1,D],0],PropagatorDenominator[Momentum[p1,D],0],PropagatorDenominator[Momentum[p2,D],I mE],PropagatorDenominator[Momentum[p1,D]-Momentum[p2,D],I mE],PropagatorDenominator[Momentum[p1,D]-Momentum[p3,D],I mE]] Pair[Lorentz
 Index[Lor1,D],LorentzIndex[Lor2,D]] SMP[g_s]^6 SUNDelta[SUNIndex[a],SUNIndex[b]]-(1/D)2 CA^3 FeynAmpDenominator[PropagatorDenominator[Momentum[p3,D],I mE],PropagatorDenominator[Momentum[p1,D],0],PropagatorDenominator[Momentum[p1,D],0],PropagatorDenominator[Momentum[p1,D],0],PropagatorDenominator[-Momentum[p1,D]+Momentum[p2,D],I mE]] Pair[LorentzIndex[Lor1,D],LorentzIndex[Lor2,D]] SMP[g_s]^6 SUNDelta[SUNIndex[a],SUNIndex[b]]+(1/D)CA^3 FeynAmpDenominator[PropagatorDenominator[Momentum[p3,D],I mE],PropagatorDenominator[Momentum[p1,D],0],PropagatorDenominator[Momentum[p1,D],0],PropagatorDenominator[Momentum[p1,D]-Momentum[p2,D],I mE],PropagatorDenominator[Momentum[p1,D]-Momentum[p3,D],I mE]] Pair[LorentzIndex[Lor1,D],LorentzIndex[Lor2,D]] SMP[g_s]^6 SUNDelta[SUNIndex[a],SUNIndex[b]]-(1/D)CA^3 FeynAmpDenominator[PropagatorDenominator[Momentum[p3,D],I mE],PropagatorDenominator[Momentum[p1,D],0],PropagatorDenominator[Momentum[p1,D],0],PropagatorDenominator[Momentum[p2,D],I mE],Prop
 agatorDenominator[-Momentum[p1,D]+Momentum[p2,D],I mE]] Pair[LorentzIndex[Lor1,D],LorentzIndex[Lor2,D]] SMP[g_s]^6 SUNDelta[SUNIndex[a],SUNIndex[b]]-(1/D)CA^3 FeynAmpDenominator[PropagatorDenominator[Momentum[p3,D],I mE],PropagatorDenominator[Momentum[p1,D],0],PropagatorDenominator[Momentum[p1,D],0],PropagatorDenominator[Momentum[p2,D],I mE],PropagatorDenominator[Momentum[p1,D]-Momentum[p3,D],I mE]] Pair[LorentzIndex[Lor1,D],LorentzIndex[Lor2,D]] SMP[g_s]^6 SUNDelta[SUNIndex[a],SUNIndex[b]]+(1/D)CA^3 FeynAmpDenominator[PropagatorDenominator[Momentum[p3,D],I mE],PropagatorDenominator[Momentum[p1,D],0],PropagatorDenominator[Momentum[p2,D],I mE],PropagatorDenominator[Momentum[p1,D]-Momentum[p2,D],I mE],PropagatorDenominator[Momentum[p1,D]-Momentum[p3,D],I mE]] Pair[LorentzIndex[Lor1,D],LorentzIndex[Lor2,D]] SMP[g_s]^6 SUNDelta[SUNIndex[a],SUNIndex[b]]+(1/D)CA^3 mE^2 FeynAmpDenominator[PropagatorDenominator[Momentum[p1,D],0],PropagatorDenominator[Momentum[p1,D],0],PropagatorDeno
 minator[Momentum[p1,D],0],PropagatorDenominator[Momentum[p2,D],I mE],PropagatorDenominator[Momentum[p1,D]-Momentum[p2,D],I mE],PropagatorDenominator[Momentum[p1,D]-Momentum[p3,D],I mE]] Pair[LorentzIndex[Lor1,D],LorentzIndex[Lor2,D]] SMP[g_s]^6 SUNDelta[SUNIndex[a],SUNIndex[b]]-CA^3 FeynAmpDenominator[PropagatorDenominator[Momentum[p1,D],0],PropagatorDenominator[Momentum[p1,D],0],PropagatorDenominator[Momentum[p1,D],0],PropagatorDenominator[Momentum[p1,D],0],PropagatorDenominator[Momentum[p2,D],I mE],PropagatorDenominator[-Momentum[p1,D]+Momentum[p2,D],I mE]] Pair[LorentzIndex[Lor1,D],Momentum[p1,D]] Pair[LorentzIndex[Lor2,D],Momentum[p1,D]] SMP[g_s]^6 SUNDelta[SUNIndex[a],SUNIndex[b]]+(1/D)2 CA^3 FeynAmpDenominator[PropagatorDenominator[Momentum[p3,D],I mE],PropagatorDenominator[Momentum[p1,D],0],PropagatorDenominator[Momentum[p1,D],0],PropagatorDenominator[Momentum[p1,D],0],PropagatorDenominator[Momentum[p1,D]-Momentum[p2,D],I mE],PropagatorDenominator[Momentum[p1,D]-Momen
 tum[p3,D],I mE]] Pair[LorentzIndex[Lor1,D],LorentzIndex[Lor2,D]] Pair[Momentum[p1,D],Momentum[p2,D]] SMP[g_s]^6 SUNDelta[SUNIndex[a],SUNIndex[b]]-(1/D)8 CA^3 FeynAmpDenominator[PropagatorDenominator[Momentum[p1,D],I mE],PropagatorDenominator[Momentum[p2,D],0],PropagatorDenominator[Momentum[p2,D],0],PropagatorDenominator[Momentum[p2,D],0],PropagatorDenominator[Momentum[p1,D]-Momentum[p2,D],I mE],PropagatorDenominator[Momentum[p2,D]-Momentum[p3,D],I mE]] Pair[LorentzIndex[Lor1,D],LorentzIndex[Lor2,D]] Pair[Momentum[p1,D],Momentum[p3,D]] SMP[g_s]^6 SUNDelta[SUNIndex[a],SUNIndex[b]]-(1/D)2 CA^3 FeynAmpDenominator[PropagatorDenominator[Momentum[p3,D],I mE],PropagatorDenominator[Momentum[p1,D],0],PropagatorDenominator[Momentum[p1,D],0],PropagatorDenominator[Momentum[p1,D],0],PropagatorDenominator[Momentum[p2,D],I mE],PropagatorDenominator[-Momentum[p1,D]+Momentum[p2,D],I mE]] Pair[LorentzIndex[Lor1,D],LorentzIndex[Lor2,D]] Pair[Momentum[p1,D],Momentum[p3,D]] SMP[g_s]^6 SUNDelta[SU
 NIndex[a],SUNIndex[b]]-(1/D)8 CA^3 FeynAmpDenominator[PropagatorDenominator[Momentum[p3,D],I mE],PropagatorDenominator[Momentum[p1,D],I mE],PropagatorDenominator[Momentum[p2,D],0],PropagatorDenominator[Momentum[p2,D],0],PropagatorDenominator[Momentum[p1,D]-Momentum[p2,D],I mE],PropagatorDenominator[Momentum[p2,D]-Momentum[p3,D],I mE]] Pair[LorentzIndex[Lor1,D],LorentzIndex[Lor2,D]] Pair[Momentum[p1,D],Momentum[p3,D]] SMP[g_s]^6 SUNDelta[SUNIndex[a],SUNIndex[b]]+(1/D)8 CA^3 FeynAmpDenominator[PropagatorDenominator[Momentum[p3,D],I mE],PropagatorDenominator[Momentum[p1,D],I mE],PropagatorDenominator[Momentum[p2,D],0],PropagatorDenominator[Momentum[p2,D],0],PropagatorDenominator[Momentum[p2,D],0],PropagatorDenominator[-Momentum[p1,D]+Momentum[p2,D],I mE]] Pair[LorentzIndex[Lor1,D],LorentzIndex[Lor2,D]] Pair[Momentum[p1,D],Momentum[p3,D]] SMP[g_s]^6 SUNDelta[SUNIndex[a],SUNIndex[b]]-(1/D)8 CA^3 FeynAmpDenominator[PropagatorDenominator[Momentum[p3,D],I mE],PropagatorDenominator[M
 omentum[p2,D],0],PropagatorDenominator[Momentum[p2,D],0],PropagatorDenominator[Momentum[p2,D],0],PropagatorDenominator[Momentum[p1,D]-Momentum[p2,D],I mE],PropagatorDenominator[Momentum[p2,D]-Momentum[p3,D],I mE]] Pair[LorentzIndex[Lor1,D],LorentzIndex[Lor2,D]] Pair[Momentum[p1,D],Momentum[p3,D]] SMP[g_s]^6 SUNDelta[SUNIndex[a],SUNIndex[b]]+(1/D)16 CA^3 FeynAmpDenominator[PropagatorDenominator[Momentum[p3,D],I mE],PropagatorDenominator[Momentum[p1,D],I mE],PropagatorDenominator[Momentum[p2,D],0],PropagatorDenominator[Momentum[p2,D],0],PropagatorDenominator[Momentum[p2,D],0],PropagatorDenominator[Momentum[p1,D]-Momentum[p2,D],I mE],PropagatorDenominator[Momentum[p2,D]-Momentum[p3,D],I mE]] Pair[LorentzIndex[Lor1,D],LorentzIndex[Lor2,D]] Pair[Momentum[p1,D],Momentum[p3,D]]^2 SMP[g_s]^6 SUNDelta[SUNIndex[a],SUNIndex[b]]+(1/D)CA^3 FeynAmpDenominator[PropagatorDenominator[Momentum[p1,D],0],PropagatorDenominator[Momentum[p1,D],0],PropagatorDenominator[Momentum[p1,D],0],PropagatorD
 enominator[Momentum[p2,D],I mE],PropagatorDenominator[Momentum[p1,D]-Momentum[p2,D],I mE],PropagatorDenominator[Momentum[p1,D]-Momentum[p3,D],I mE]] Pair[LorentzIndex[Lor1,D],LorentzIndex[Lor2,D]] Pair[Momentum[p3,D],Momentum[p3,D]] SMP[g_s]^6 SUNDelta[SUNIndex[a],SUNIndex[b]]

Am I missing something here? It works, however, when performing the decomposition with ApartFF->False and then individually using ApartFF afterwards.

Also in some cases when using your FIRE interface it appears that after the tensor integral decomposition one is still left with integrals that have linearly dependent propagators (or at least an error is given:

exp2 = -(1/2) CA^3 FeynAmpDenominator[PropagatorDenominator[Momentum[p1,D],I mE],PropagatorDenominator[Momentum[p2,D],I mE],PropagatorDenominator[Momentum[p1+p2,D],0],PropagatorDenominator[Momentum[p1+p2,D],0],PropagatorDenominator[Momentum[p3,D],I mE],PropagatorDenominator[Momentum[-p1-p2+p3,D],I mE],PropagatorDenominator[Momentum[p2,D],I mE],PropagatorDenominator[Momentum[p1,D],I mE]] Pair[LorentzIndex[Lor1,D],Momentum[p1,D]] Pair[LorentzIndex[Lor2,D],Momentum[p2,D]] (Pair[Momentum[p1,D],Momentum[p1,D]]-2 Pair[Momentum[p1,D],Momentum[p3,D]]-Pair[Momentum[p2,D],Momentum[p2,D]]+2 Pair[Momentum[p2,D],Momentum[p3,D]])^2 SMP[g_s]^6 SUNDelta[SUNIndex[a],SUNIndex[b]];

FCMultiLoopTID[exp2,{p1,p2,p3}];
FIREBurn[%,{p1,p2,p3},{q}];

Here FIREBurn gives the error FIREBurn::lindep which shouldn’t appear given the fact that the decomposition was done automatically with ApartFF.

Thanks for your help.

Cheers,
Philipp