### Neutrino Dark Matter

#### Posted by John Baez

I talked to Neil Turok at a café today. He used to be the head of the Perimeter Institute, but now he’s at the University of Edinburgh.

He coauthored a paper arguing that dark matter is very heavy right-handed neutrinos:

- Latham Boyle, Kieran Finn and Neil Turok, The Big Bang, CPT, and neutrino dark matter.

It’s very natural to add right-handed neutrinos to the Standard Model, and if they’re heavy they can make the observed left-handed neutrinos light via the ‘see-saw mechanism’. The problem is to keep them from decaying too fast!

For a heavy neutrino to serve as dark matter, it needs to be quite stable. Apparently this is tough if it interacts with the Higgs—how true is that, exactly? But neutrino that’s its own antiparticle can have a mass without interacting with the Higgs: a so-called ‘Majorana mass’.

In Turok’s theory all the neutrinos have Majorana masses, described by a mass matrix. To make the heaviest right-handed neutrino stable, a bunch of matrix entries must vanish—and this makes the lightest left-handed neutrino massless!

So, out of this theorizing Turok gets a testable prediction: one of the observed neutrinos is massless!

This is interesting, because we only know that *two* of the three observed neutrinos have a nonzero mass. Neutrino oscillation experiments don’t let us measure neutrino masses: they only tell us differences between squares of neutrino masses!

In 1998, research results at the Super-Kamiokande neutrino detector determined that neutrinos can oscillate from one flavor to another, which requires that they must have a nonzero mass. While this shows that neutrinos have mass, the absolute neutrino mass scale is still not known. This is because neutrino oscillations are sensitive only to the difference in the squares of the masses. As of 2020, the best-fit value of the difference of the squares of the masses of mass eigenstates 1 and 2 is 0.00007 $\mathrm{eV}^2$ , while for eigenstates 2 and 3 it is 0.00251 $\mathrm{eV}^2$. Since this is the difference of two squared masses, at least one of them must have a value which is at least the square root of this value. Thus, there exists at least one neutrino mass eigenstate with a mass of at least 0.05 eV

We need other experiments to measure, or bound, the actual masses of neutrinos! Luckily, some good experiments are in the works now.

For example, Euclid is a space telescope that will study the large-scale structure of the Universe in exquisite detail.

It should be able to measure the sum of all 3 light neutrino masses to within 0.03 eV:

- Anton Chudaykin and Mikhail M. Ivanov, Measuring neutrino masses with large-scale structure: Euclid forecast with controlled theoretical error.

Right now we just know this sum is $\lt$ 0.1 eV. With Euclid, and other forthcoming experiments, we may be able to constrain it much better. Combining this with our knowledge of the differences of squared masses, we should either be able to prove the lightest neutrino isn’t massless, disproving the theory I’m talking about… or get evidence that it’s almost massless, supporting it.

Turok explained a lot more to me, but in case you’re not used to this stuff, here’s a summary of what I just said: if neutrinos have Majorana masses and one is massless, that’s indirect evidence that dark matter is a heavy right-handed neutrino!

Finally, let me state a few of my own prejudices.

I want leptons to be very similar to quarks, since they’re already quite similar in many ways, and the mathematics of theories like the SO(10) GUT make these similarities seem very natural.

This makes me *want* right-handed neutrinos. So I *like* the idea of dark matter being heavy right-handed neutrinos - especially since the see-saw mechanism says this is compatible with the left-handed ones being very light.

But I *don’t like* Majorana masses for neutrinos, since the quarks don’t have those.

So, I want to know if, in the theories I prefer, where neutrinos gets masses in the same way that quarks do, one very heavy and stable (or almost stable) right-handed neutrino is compatible with—or implies—an extremely light or massless left-handed neutrino.

This must have been studied; I just need to read some more papers. Actually I once read some papers by Mohapatra about this, but I need to reread them with a closer eye to the stability of the most massive neutrino.

Finally, I should add that it was very invigorating to talk to someone who is really excited about fundamental physics, with some concrete testable ideas about it. Turok thinks a lot of the big remaining problems in physics can be solved using the Standard Model with very slight tweaks, no new particles. I don’t know if I believe that, but it was great talking to him. I haven’t thought seriously about particle physics for years, because it seems so depressing.

## Re: Neutrino Dark Matter

While we’re waiting for someone to answer your question, I – a mathematician with limited knowledge of physics – would like to ask something more basic.

I think I understand the field content of the theory you want to consider: we have the fields from the 1970s version of the Standard Model (SM), plus a three-generation right-handed Weyl spinor field $\nu^R$. I.e., the field $\nu^R$ is a section in a complex vector bundle over spacetime whose typical fiber is $\mathbb{C}^2\otimes\mathbb{C}^3$: spinor part $\mathbb{C}^2$ tensor generation part $\mathbb{C}^3$. Let $G$ be the SM gauge group. The physically relevant group $\SL(2,\mathbb{C})\times G$ acts on $\mathbb{C}^2\otimes\mathbb{C}^3$ by $(A,g)(v\otimes w) = A(v)\otimes w$. I agree that this field content has a much prettier description (that I don’t have to spell out here) than the 1970s version in which $\nu^R$ is omitted.

According to the usual philosophy, the SM is an effective field theory. At length scales much larger than the fundamental scale, we expect to see the effects of

allrenormalizable Lagrangian terms that we can build from the given field content (because we don’t believe in coincidences that make the value of some physical constant experimentally indistinguishable from $0$ without good reason). In your preferred theory, we can build all terms we had in the 1970s version of the SM, plus a few obvious lepton analogues of quark terms we know from the 1970s version. They include Yukawa-type terms that give rise to neutrino masses, analogously to how the quark masses arise in the SM. So far, so good.But we can also build an additional (super-)renormalizable term, involving only $\nu^R$, that has no quark analogue. This works as follows. We fix an antisymmetric complex-bilinear form $M\colon \mathbb{C}^3\times\mathbb{C}^3 \to \mathbb{C}$ on the generation tensor factor. (We can think of $M$ as a bunch of new physical constants.) The antisymmetric complex-bilinear form $B\colon \mathbb{C}^2\times\mathbb{C}^2 \to \mathbb{C}$ given by

(where the determinant $\det$ is interpreted as a map $\bigwedge^2\mathbb{C}^2 \to \mathbb{C}$) is $\SL(2,\mathbb{C})$-invariant. Hence $B\otimes M$ is a symmetric complex-bilinear form on $\mathbb{C}^2\otimes\mathbb{C}^3$, invariant under our action of $\SL(2,\mathbb{C})\times G$. We can therefore build the Lagrangian

where $\text{Re}$ denotes the real part.

So, in your beautified Standard Model, we have automatically neutrino terms without quark analogue. (There is no quark analogue because the nontrivial action of $G$ on the quark parts spoils the invariance.)

What is the relation (if any) between these terms and Majorana mass terms? Majorana mass terms as usually discussed by physicists refer to a different field content than we have in your theory, don’t they?

Which observable consequences do the additional terms have? Which experimental constraints are there on $M$?