# Little confusion with see-saw mechanism

Neutinos are either Dirac particles or Majorana particles but can’t be both at the same time. Then how can we write a general mass term as the sum of a Dirac mass term and a Majorana mass term? When we write such a term what nature of neutrinos (Dirac or Mjorana) do we have in mind?

A massive Dirac field, has four independent degrees of freedom (DOF): $$\psi_L,\psi_R,(\psi_L)^c=(\psi^c)_R,(\psi_R)^c=(\psi^c)_L$$ In contrast with this, a Majorana fermion has only two independent DOF: $$\psi_L, (\psi_L)^c=(\psi^c)_R$$ Then, in particular, how is it possible to replace $\psi_L$ by $(\psi_R)^c$ in the Dirac mass term because $\psi_L=(\psi_R)^c$ is true only for Majorana particles and not for Dirac particles. This is the headache I have.

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I believe, though I'm not sure, that in theory you should write down all the "allowable" Lagrangian terms. By "allowable" I mean: (i) it is Lorentz invariant, (ii) it is gauge invariant, and (iii) it is renormalizable. Since both the Dirac mass and the Majorana mass satisfy this property, they both must be written down (if we assume that the neutrino is Majorana particle; if it is not a Majorana particle, then we can not write down the Majorana mass term). –  Hunter Mar 29 '14 at 7:15
@Hunter - But the reverse is true, right? Namely if we take neutrinos to be Majorana particles then we can write down a legitimate Dirac mass for it. Right? Then this basically implies we only have 2 independent degrees of freedom to play with. This is the impression I have. –  SRS Mar 29 '14 at 7:28
I'm not sure about the two degrees of freedom, because I have never studies this in a lot of detail. But if a neutrino is a Dirac particle, then you are not allowed to write down the Majorana term; whereas if the neutrino is a Majorana particle, then there is nothing that prevents you from writing down a Dirac term (but this doesn't mean that the neutrino is a Dirac term). –  Hunter Mar 29 '14 at 7:32

## 1 Answer

There is a lot to be said about Majorana and Dirac neutrino masses, but I will try to address only the specific point you raise. When you write a Majorana mass term, for example, $$\mathcal{L}_{\text{Majorana}} = -\frac{m_M}{2} \left[\bar\nu_L(\nu_L)^C + \overline{\nu^C_L}\nu_L\right]$$ you are not assuming that $(\nu_L)^C = \nu_R$, but you can write that $$\eta = (\nu_L)^C + \nu_L = \eta^C$$ and rewrite the mass term as $$\mathcal{L}_{\text{Majorana}} = -\frac{m_M}{2} \bar\eta \eta$$ This construction is forbidden in the Standard Model, because it violates gauge symmetries.

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Where have you read that? –  innisfree Mar 31 '14 at 8:36