Difference between two varieties

[qraal]

How do Majorana and Dirac neutrinos differ?

 

[Bill_K]

qraal, Basically a Majorana neutrino is its own antiparticle. That means all of its quantum numbers have to be 0. You already mentioned it must be a type of neutrino (electric charge 0) Also lepton number, muon number, hypercharge, and so on. No charge means that it cannot couple to the photon. Also it cannot couple to the W or Z, and hence does not participate in the weak interactions. This makes it what we call a "sterile" neutrino. Sterile neutrinos are one of the possible candidates for a dark matter particle.

 

[Parlyne]

Also it cannot couple to the W or Z, and hence does not participate in the weak interactions. This makes it what we call a "sterile" neutrino. Sterile neutrinos are one of the possible candidates for a dark matter particle.

This is not actually correct. A Majorana particle can, in fact, have purely axial couplings to gauge fields (at least to massive ones). This would simply necessitate that the Majorana mass of the particle is generated by the same symmetry breaking mechanism that gives the gauge fields mass. So, for instance, one can have heavy neutrinos that interact under a new Z'.

A more down to Earth point is that the neutrinos in the standard model may actually have Majorana masses; but, this doesn't prevent them from interacting with the W or Z. Generally, models that do this generate the Majorana masses as part of electroweak symmetry breaking, either by having EWSB create terms that mix the light neutrinos with heavy neutrinos that already have Majorana masses or by adding new scalars that contribute to EWSB and give the neutrinos Majorana mass in a similar way to how the ordinary Higgs gives the other particles Dirac mass.

The reason that this can work is that there are no conserved charges associated with interactions with the W or Z. (That is, in fact, what EWSB gets rid of.)

 

[Bill_K]

Ok, what I've picked up on this subject differs from what you're saying in several respects, so I'd appreciate further explanation if you can provide it

First, that a Dirac particle can have a Majorana mass, but not the other way around.
Second, that the seesaw mechanism is a result of mixing between a standard model Dirac particle and a very heavy Majorana particle.
Third, that although this Majorana particle acquires its mass from some interaction, it must be a scalar under the electroweak symmetry, and sterile in that sense.

Am I wrong?

 

[ParticleGrl]

It might help to think about things in terms of Weyl spinors. If we work in a chiral basis for the $\gamma$ matrices, then we can think of a Dirac spinor as two different objects, a right-chiral spinor and a left -chiral spinor-

$$\psi_{d} = \left(\begin{array}{c}\psi_L\\ \psi_R\end{array}\right)$$

So the Dirac mass term connects right handed with left handed spinors.

$$\mathcal{L}_{md} = m\left(\bar{\psi}_R \psi_L + \bar{\psi}_L \psi_R \right)$$

The goal of a majorana mass is to build a mass term with only one of these spinors.

$$\mathcal{L}_{mm} = \frac{1}{2} m\left(\bar{\psi}_L C \psi^*_L + H.C. \right)$$

Here, C is the charge conjugation matrix.

From this it should be clear that a Majorana spinor requires fewer degrees of freedom than a Dirac spinor. You can build 3 mass terms with a Dirac spinor, the Dirac mass, a left Majorana mass and a right Majorana mass. You can only build one Majorana mass from a Majorana spinor.

As to the idea that a Majorana particle has to be sterile under electroweak symmetry, this depends on your model. There are GUTs (like SO(10)) that naturally admit a particle that is a singlet under the standard model interactions. In these models, you can give the right hand neutrino a mass related to the GUT breaking scale, which then mixes with the standard model dirac masses, as you suggest.

However- these GUTs are speculative. The majorana mass term for left-handed neutrinos in the standard model does not require such particles. Consider that we can make an SU(2) singlet out of any particle in an SU(2) doublet (call the doublet $\phi$) as follows-

$$singlet = \phi^T\epsilon \phi$$

where $\epsilon$ is a 2d levi-civita. Under an SU(2) transformation T, this transforms to det(T) = 1. So you can add a term to the standard model like

$$\phi^T\epsilon \phi (E_L^T\epsilon E_L)^*$$

We can only do this with the left-handed lepton doublet, because of the hypercharge.

Here $\phi$ is the SM higgs and E the SM neutrino/lepton SU(2) doublet. This will give majorana masses to the neutrinos.

Its worth noting that this term has a mass dimension 5 (which breaks renormalization), and so the coupling can be written as $\frac{g}{\Lambda}$, where $\Lambda$ is a mass scale. The smallness of neutrino masses may well be related to the fact that this mass scale is large. In your see-saw models it would be the mass of the right handed neutrino.