Definition of neutrinos (Majorana vs. Weyl spinor)

[Socob]

In the included model files for the Standard Model (model “SM”), the Dirac spinor for the neutrinos is defined as

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DEFINITION[EWSB][DiracSpinors]={
…
 Fv ->{  vL, 0}};

In models with radiative neutrino masses (e. g. the scotogenic/radiative seesaw model), it should be possible to use a nonzero value for these masses (e. g. calculated at 1-loop level by SPheno). However, for SARAH’s (e. g.) micrOMEGAs output, using the definition above always seems to fix the masses to 0. On the other hand, if I define the neutrinos as Majorana spinors

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DEFINITION[EWSB][DiracSpinors]={
…
 Fv ->{  vL, conj[vL}};

parameters for the neutrino masses appear in the micrOMEGAs model files so that they can be set to nonzero values.

I’m wondering in general: Is one of these definitions of “Fv” “better”/which one should be used? (Physically, they should be equivalent, shouldn’t they?)

[FStaub]

Hi,

Majorana and Dirac neutrinos are not equivalent. Therefore, there are experimental efforts what the neutrino really is (neutrinoless double-beta decay). It's a model to model decision how to write the four-spinor.

As long as you have the SM without right-neutrino or Weinberg operator, the neutrinos are massless. For practical applications (as dark matter calculations) the neutrino masses shouldn't play any role anyway.

All in all, I'm not sure if I really got the question...

Cheers,
Florian

[Socob]

Of course, Dirac and Majorana neutrinos are different. But the spinor

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Fv -> {vL, 0}

doesn’t really describe a Dirac fermion – it’s just a Weyl spinor disguised in the four-component spinor formalism. Using the correct translation (gamma matrices instead of Pauli matrices etc.) to the four-component formalism, this spinor is basically equivalent to a left-handed two-component spinor, i. e. Fv ≅ vL. With this spinor, it shouldn’t be a problem to write down any term in the Lagrangian, including (Majorana) mass terms.

Essentially, I think that both versions of Fv could be used to describe Majorana neutrinos. My question is now which of these ways is the “correct” one to use for Majorana fermions in SARAH.

[FStaub]

If the neutrinos are Majorana, ie if there is a Majorana mass term, one must use

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Fv ->{  vL, conj[vL}};

The other definition can only be used for massless states.