Why FormCalc gives zero amplitude for this process

Question

Why when I generate self energy diagram of for neutrinos, FormCalc gives zero amplitude ? here is the code

t12 := CreateTopologies[1, 1 -> 1, ExcludeTopologies -> Internal]
ins := InsertFields[t12, F[1, {3}] -> F[1, {3}]];

amp = CalcFeynAmp[CreateFeynAmp[ins]]

which gives zero amplitude:

Amp[{{F[1, {3}], k[1], 0, {0, LeptonNumber}}} -> {{F[1, {3}], k[2], 
     0, {0, LeptonNumber}}}][0]

It's known that neutrino is massless in SM (the default choice of FA) but painting [ins] gives diagrams mediated by W and Z bosons as in the next figure, so why their amplitudes have not evaluated ?

Answer 1

Although I have nothing to do with FormCalc, I think that I understand what is the problem here: Since you want to compute self-energy, you need to use the option Truncated when generating the amplitudes. Otherwise the amplitude is generated with external spinors and vanishes because of the equation of motion (Dirac equation). The correct code reads:

<< FormCalc`FormCalc`
<< FeynArts`FeynArts`

$FAVerbose = 0;
t12 = CreateTopologies[1, 1 -> 1, ExcludeTopologies -> Internal];
ins = InsertFields[t12, F[1, {3}] -> F[1, {3}], 
   InsertionLevel -> {Particles}];
Paint[ins, ColumnsXRows -> {3, 1}, ImageSize -> {768, 256}];

CalcFeynAmp[CreateFeynAmp[ins, Truncated -> True], OnShell -> False]

and the result is

Amp[{{F[1, {3}], k[1], 0, {0, LeptonNumber}}} -> {{F[1, {3}], k[2], 
     0, {0, LeptonNumber}}}][-(1/(\[Pi] SW2))
  Alfa (1/16 (2 + 1/CW2) Finite + 1/8 (B0i[bb1, Pair[k[1], k[1]], 0, MZ2]/
        CW2 + (2 + ML2/MW2) B0i[bb1, Pair[k[1], k[1]], ML2, MW2])) GA[6, k[1]]]

To see why the amplitude is zero without Truncated->True one can use the following FeynCalc code

$LoadFeynArts = True;
<< FeynCalc`

$FAVerbose = 0;
t12 = CreateTopologies[1, 1 -> 1, ExcludeTopologies -> Internal];
ins = InsertFields[t12, F[1, {3}] -> F[1, {3}], InsertionLevel -> {Particles}];
amp1 = FCFAConvert[CreateFeynAmp[ins], IncomingMomenta -> {p}, 
    OutgoingMomenta -> {p}, LoopMomenta -> {q}, SMP -> True, List -> False] // Contract;
amp2 = amp1 // TID[#, q] & // ToPaVe[#, q] & //Collect2[#, DOT, IsolateNames -> KK] &

Here we simplify the Dirac algebra and perform 1-loop tensor decomposition. After that we collect terms w.r.t to the remaining Dirac structures and introduce abbreviations to make the result better readable. What we get is

$$ \frac{1}{128} \text{KK}(103) (\varphi (p)).(\gamma \cdot p).\bar{\gamma }^7.(\varphi (p)), $$

where $\bar{\gamma }^7$ is the usual left handed projector. The resulting expression is zero by Dirac equation, since neutrinos are massless in the SM model of FeynArts. We can also see this in FeynCalc by evaluating

amp2 // DiracSimplify

which gives zero.