# FeynCalc: FermionSpinSum problems with interference of diagrams

I encountered a problem in FeynCalc, where i wanted to calculate the $$\sum_\text{spins}|\mathcal{M}|^2$$ of a process and therefore need the function FermionSpinSum[ ] of FeynCalc.

I run:

 TESTM = SpinorUBar[l, m].ChiralityProjector[1].DiracSlash[l].ChiralityProjector[-1].SpinorV[k, m];

STESTM = ComplexConjugate[TESTM].TESTM;

STESTM2 = FermionSpinSum[STESTM] /. DiracTrace -> Tr //StandardForm


and the output is

2 Pair[Momentum[k], Momentum[l]] Pair[Momentum[l], Momentum[l]]


which is perfectly fine. The problem arises when i add another diagram of the same order to the matrix element

TESTM = SpinorUBar[l, m].ChiralityProjector[1].DiracSlash[l].ChiralityProjector[-1].SpinorV[k, m] +
SpinorUBar[k, m].ChiralityProjector[1].DiracSlash[l].ChiralityProjector[-1].SpinorV[l, m];

STESTM = ComplexConjugate[TESTM].TESTM;

STESTM2 = FermionSpinSum[STESTM] /. DiracTrace -> Tr // StandardForm


which generates the following output

(Spinor[-Momentum[k], m, 1].DiracGamma[6].DiracGamma[
Momentum[l]].DiracGamma[7].Spinor[Momentum[l], m, 1] +
Spinor[-Momentum[l], m, 1].DiracGamma[6].DiracGamma[
Momentum[l]].DiracGamma[7].Spinor[Momentum[k], m, 1]).(Spinor[
Momentum[k], m, 1].DiracGamma[6].DiracGamma[
Momentum[l]].DiracGamma[7].Spinor[-Momentum[l], m, 1] +
Spinor[Momentum[l], m, 1].DiracGamma[6].DiracGamma[
Momentum[l]].DiracGamma[7].Spinor[-Momentum[k], m, 1])


This clearly cannot be correct, since it still contains $$\gamma$$-matrices and spinors, which should've been gone after FermionSpinSum[ ].

I'm confused, can anybody help me at this stage?

• May I suggest directly contacting the developer and FeynCalc community by posting your question on the FeynCalc forum? You'll very likely get a response from them. Feb 8 '19 at 4:28
• I hope i didn't miss anything trivial here. I wrote a mail to them; thanks for encouraging me to do so. I will post any updates as answers here if i get any. Feb 8 '19 at 9:36

Majorana fermions are not supported in the stable version, so FermionSpinSum cannot automatically reorder the chains to apply the summation formulas.

I've just added this functionality to the development version today, so there

TESTM = SpinorUBar[k, m].GA[6].GS[l].GA[7].SpinorV[l, m] +
SpinorUBar[l, m].GA[6].GS[l].GA[7].SpinorV[k, m];
STESTM = ComplexConjugate[TESTM] TESTM;
FermionSpinSum[STESTM] // DiracSimplify


just works yielding

-4 m^2 Pair[Momentum[l], Momentum[l]] +
4 Pair[Momentum[k], Momentum[l]] Pair[Momentum[l], Momentum[l]]


Since it's a "brand new feautre", there might be some bugs lurking around, so sensible cross-checks of the results are a must.