Issue with γ-matrix simplifications after FermionSpinSum in FeynCalc

[impermast]

Hello,

I am experiencing an issue with FeynCalc (version 9.3) in Mathematica while computing the amplitude for the process tt → Z → ZZ in the 2HDM model.

Previously, I successfully computed the amplitude for ZZZ, and γ-matrix expressions were correctly simplified. However, after introducing quarks in the initial state and applying FermionSpinSum, along with SUNSimplify, γ-matrix expressions no longer simplify as expected.

MWE

<< FeynArts`
<< FeynCalc`

(*only one diagram for simplicity*)
diagsAll = InsertFields[
    CreateTopologies[1, 2 -> 2, ExcludeTopologies -> {Tadpoles}], 
    {F[9], -F[9]} -> {V[2], V[2]}, 
    InsertionLevel -> {Classes}, 
    Model -> THDMCPV, 
    ExcludeParticles -> {F[_], V[1], V[3], S[2], S[3], U[_]}, 
    LastSelections -> {V[2]}
];

diagHHH = DiagramExtract[diagsAll, {19}];

ampHHH[0] = FCFAConvert[
    CreateFeynAmp[diagHHH, Truncated -> True], 
    IncomingMomenta -> {k1, k2}, 
    OutgoingMomenta -> {p1, p2}, 
    LoopMomenta -> {q}, 
    TransversePolarizationVectors -> {p1, p2}, 
    UndoChiralSplittings -> True, 
    ChangeDimension -> D, 
    List -> False, 
    SMP -> True, 
    Contract -> False, 
    DropSumOver -> True
];

ampHHH[1] = ampHHH[0] // FCTraceFactor // SUNSimplify;

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

ampHHH[2] = ampHHH[1] // TID[#, q, ToPaVe -> True, UsePaVeBasis -> True] &;

ampHHH[3] = (ampHHH[2] /. M$FACouplings) // FCReplaceD[#, D -> 4 - 2 Epsilon] & //
    Series[#, {Epsilon, 0, 0}] & // Normal;

ampHHH[4] = ampHHH[3] // ChangeDimension[#, 4] &;

(*just contract amp, no summing*)
(ampHHH[4] ComplexConjugate[ampHHH[4]]) // Contract

(*fermion and vector sum*)
ampSquared[0] = 1/2 (ampHHH[4] (ComplexConjugate[ampHHH[4]])) //  
                FermionSpinSum // Contract//
        DiracSimplify //
        SUNSimplify  // 
        DoPolarizationSums[#, k1] & // 
        DoPolarizationSums[#, k2] &//
        TrickMandelstam[#, {s, t, u, 2 SMP["m_t"]^2 + 2 SMP["m_Z"]^2}] &;

ampSquared[0]

Expected Behavior

The squared amplitude should not explicitly depend on γ-matrices. After applying the necessary simplifications, terms involving DiracGamma should be completely eliminated, leaving a scalar result.

Observed Behavior

In ampSquared[0], I still have terms of the form:

DiracGamma[Momentum[p1]] \cdot DiracGamma[7] \cdot DiracGamma[Momentum[p2]]

These terms exist even after applying FermionSpinSum, DiracSimplify, SUNSimplify, and other simplifications.

Questions:

1. What is the correct order of operations to ensure proper γ-matrix simplification?

• Contract
• TID (Tensor Integral Decomposition)
• FermionSpinSum
• DiracSimplify
• SUNSimplify
• DoPolarizationSums[#, k1] & // DoPolarizationSums[#, k2] & //

2. Should DiracSimplify be applied before or after FermionSpinSum to ensure correct simplifications?

I would be grateful for any clarification on whether I am missing a step or misapplying a function. Thank you in advance for your help!

[vsht]

I can't run your MWE, since the model is missing. Could you post the expression for ampHHH[0]?

From looking at the code, I guess the issue might be that you have traces with an odd number of $\gamma^5$, which cannot be evaluated in the NDR scheme.

https://feyncalc.github.io/FeynCalcBookDev/Extra/Gamma5.html

You can check it by setting

FCSetDiracGammaScheme["BMHV"]

and rerunning the code of ampSquared[0].

Of course, the $\gamma^5$-issue always requires some thinking for every given process, so using BMHV doesn't magically solve those issues you might have.

[impermast]

Thanks for the suggestion!

I ended up solving the issue — it turned out that the leftover DiracGamma terms (especially involving gamma6 and gamma7) were due to an incorrect order of simplifications.

Using this sequence of operations gave the expected result (no leftover gamma matrices and free tensor indexes in the squared amplitude):

ampSquared = 1/2 (SubjAmp (ComplexConjugate[SubjAmp]));
ampSquared = DiracSimplify[ampSquared];
ampSquared = DoPolarizationSums[ampSquared, k1, NumberOfPolarizations -> 3];
ampSquared = DoPolarizationSums[ampSquared, k2, NumberOfPolarizations -> 3];
ampSquared = FermionSpinSum[ampSquared];
ampSquared = DiracSimplify[Contract[ampSquared]];
ampSquared = SUNSimplify[ampSquared];

I think, the issue was due to the γ₅ scheme in the end — but i solve it with better control over the γ₆/γ₇ terms during simplification.

Thanks again for the help!

[vsht]

Thanks for the feedback.