Earlier this week I explained how neutrinos can get their mass within the Standard Model of particle physics, either by engaging with the Higgs field once, the way the other particles do, or by engaging with it twice. In the first case, the neutrinos would be “Dirac fermions”, just like electrons and quarks. In the second, they’d be “Majorana fermions”. Decades ago, in the original Standard Model, neutrinos were thought not to have any mass at all, and were “Weyl fermions.” Although I explained in my last post what these three types of fermions are, today I want go a little deeper, and provide you with a diagrammatic way of understanding the differences among them, as well as a more complete view of the workings of the “see-saw mechanism”, which may well be the cause of the neutrinos’ exceptionally small masses.
[N.B. On this website, mass means “rest mass” except when otherwise indicated.]
The Three Types of Fermions
What’s a fermion? All particles in our world are either fermions or bosons. Bosons are highly social and are happy to all do the same thing, as when huge numbers of photons are all locked in synch to make a laser. Fermions are loners; they refuse to do the same thing, and the “Pauli exclusion principle” that plays a huge role in atomic physics, creating the famous shell structure of atoms, arises from the fact that electrons are fermions. The Standard Model fermions and their masses are shown below.
The simplest and most basic type of fermion is a Weyl fermion, though we are not aware of any such particles in nature. A particle which is a Weyl fermion must have zero mass; this is what neutrinos were once thought to be, before evidence of their masses was uncovered. I’ll draw a Weyl fermion as a single line, indicating a particle traveling from, say, left to right.
The second type of fermion, a Dirac fermion, is common in the Standard Model; the electron is a Dirac fermion, as are its its cousins the tau and the muon, and all the quarks. A Dirac fermion is made from the marriage of two Weyl fermions
A Dirac fermion is made from the marriage of two Weyl fermions. It is sometimes convenient mathematically to view the mass as the mechanism that makes the particle switch from one Weyl fermion to the other. (But do not take this too seriously, because it is only strictly true in processes where the fermion is “virtual”. [For mathy folks: this is a generalization of the statement that 1/(e-m)=1/e+m/e2+m2/e3+… to a matrix calculation known as a Neumann series; this works for m<<e, but a real particle would have the equivalent of m=e, where the expansion is ill-defined.])
This viewpoint is illustrated in Figure 3 below, where a Dirac fermion moves from left to right; one of its Weyl fermions is shown in blue and the other in green, flipping back and forth at a rate set by the mass m. If the mass were zero, the flip would never happen, and the particle would be two separate Weyl fermions, one blue, one green.
For this mass to be possible, the two Weyl fermions must have the same properties; they must interact with all the forces of nature in the same way. For instance, they must have the same electric charge.
The third type, a Majorana fermion, marries a Weyl fermion to itself; even more precisely, it marries a Weyl fermion to its anti-particle. This is shown below, where the antiparticle is shown as a dashed line of the same color. [Physicists often use arrows to show the difference between the particle and its anti-particle, but this can get confusing, because the anti-particle traveling to the right is often depicted with an arrow pointing to the left.] Just as for the Dirac fermion, this can only work if the mass connects objects with the same properties. That means that a Weyl fermion can only be a Majorana fermion if it and its anti-particle have the same properties. For instance, since an anti-particle has the opposite electric charge as the corresponding particle, both particle and anti-particle must have electric charge zero. This isn’t true for the electron, muon, tau or the quarks, but it is true for the neutrinos, which is why they will eventually have the option of being Majorana fermions. But not quite yet.
The Standard Model, the Higgs Field, and Dirac Fermions
One thing that makes the Standard Model truly remarkable is that in the absence of the Higgs field, all of its fermions would be massless Weyl fermions. What we call “an electron” would actually be two different massless particles with different properties; they wouldn’t even deserve the same name. Here, looking ahead to the effect of the Higgs field, we will call them both “electron”, but I’ll color them differently to remind you that they are really different. One half of the electron is affected by the weak nuclear force (in particular, the W field and its corresponding particle, the W boson), while the other half is not. Because of this, they cannot marry… not until the Higgs field comes along and, by switching on and taking on a non-zero value, hides the effect of the weak nuclear force. (In fact this hiding, which involves giving the W and Z bosons a big mass, is what makes this force weak, in the sense discussed here and here.)
By hiding the weak nuclear force, and making it an effect that is small for processes at low energy (hence the enhanced lifetimes of particles that decay via the weak nuclear force), the Higgs field makes the two halves of the electron compatible. The two Weyl fermion halves were unable to marry because of the weak nuclear force, but with that issue hidden away, they are now free to marry, and a Dirac fermion results. This is indicated in Figure 4.
You see that we begin with an interaction at left between the two halves of the electron (blue and green) and the Higgs field. Once the Higgs field has a constant non-zero value across the universe, the Higgs field acts just as a Dirac mass term would, flipping the two halves of the electron from one to the other. The mass of the electron is then the product of two quantities:
- the strength of the Higgs field’s interaction with the electron, called ye (the “electron Yukawa coupling”), which was present in the original diagram at left, and
- the magnitude of the value of the Higgs field, v, of about 250 GeV .
(There is also a technical square root of 2, not worth discussing here.) As promised, if v were zero the electron would be massless. Also, since the electron’s mass is only 0.0005 GeV, the quantity ye is only 0.000003. The reason for its small size is unknown.
If neutrinos are Dirac fermions, the same mechanism applies as for electrons. Figure 6 reminds us that one of the two half-neutrinos is “sterile”, meaning it responds neither to the electromagnetic, strong nuclear, or weak nuclear force; but other than that, Figure 6 looks the same as Figure 5. In fact, in this case, all the Standard Model’s fermions get their masses in the same way.
However, to explain the fact that the neutrinos have mass no more than a millionth of the electron’s, their Yukawa couplings must also be at least a million times smaller, i.e. yν < 10-12. There’s no obvious reason why this should be the case. (Perhaps I’ll tell you at a later time how new forces impacting the sterile neutrinos might be able to cause this, but that’s too advanced a subject for today.) With Dirac neutrinos, there’s no obvious principle to make their masses small.
The Original See-Saw
But with Majorana neutrinos, there is a possible principle, and this involves the “see-saw” mechanism that I sketched last time. This is part of what makes it appealing (but remember, that’s far from saying it’s true, even though some physicists occasionally seem to get mixed up about that.)
In the mid-to-late 1970s, a number of people pointed out the basic idea, to different degrees of detail. They observed that
- although, without the Higgs field being switched on, an ordinary neutrino cannot have a Majorana mass, as it is not the same as its anti-neutrino [technically, they have different “hypercharges”, the name give for the U(1) in SU(3)xSU(2)xU(1)],
- a sterile neutrino is the same as its anti-particle, so it can have a Majorana mass, as in Figure 4, even without the Higgs field, and
- a sterile neutrino with a very large mass and an ordinary interaction with the Higgs field (no tiny Yukawa coupling needed) would lead to an ordinary neutrino with a very small mass.
This is illustrated in Figure 7. There are now several steps to the logic.
- First, we take the same Higgs/neutrino/sterile-neutrino interaction shown in Figure 6 above; but now we use it twice, and in between, we add a huge Majorana mass M for the sterile neutrino.
- Second, with such a huge mass, this sterile neutrino is irrelevant for any current experiments; all it does is create an interaction between a neutrino, its anti-particle, and two instances of the Higgs field; and this interaction (which would disappear altogether if M were infinite) is suppressed by 1/M.
- Third, the Higgs field switches on, and gives the neutrino a mass that contains two factors of yν, two factors of v, and one factor of 1/M .
A key question you should ask: if you look at the discussion of how the weak nuclear force works in this post, (see its Figure 6), you’ll see there is a similar suppression that comes from a W boson because of its large mass mW — except that there the suppression is 1/mW2. Well, this is a key difference between bosons and fermions: where a boson suppresses effects by the square of its mass, a fermion does so by only one factor of its mass.
What we see in Figure 7 tells us that both yν and M enter into the neutrinos’ masses. This means we could get neutrino masses a million times smaller than the electron’s in different ways. For instance, we could have yν = 1 and have M be up around 1015 GeV. Or we could have yν = ye = 0.000003, and M somewhere around or just above 1000 GeV, even possibly within range of the Large Hadron Collider [LHC]. Or anything in between.
So you see, even if we measure the neutrino masses, we won’t actually know M. We’ll only know a maximum for it: Yukawa couplings much, much bigger than 1 don’t really make sense (this is a long story) so if the largest neutrino mass is about a trillion times smaller than v, as experiment currently suggests, we have an estimate that M can’t ever be much larger than a trillion times bigger than v. It could be much smaller.
But remember this estimate only holds if the see-saw mechanism is in operation!
The Generalized See-Saw
A sterile neutrino with a very large mass is a particularly simple idea, but reality might be a lot more complex than this. Because that reality may be hidden up at very high energy, we may not learn anything about it anytime soon. Other high-energy phenomena, of an unknown source, could potentially mimic the effect of a sterile neutrino, and we wouldn’t easily know. All we can potentially do in current experiments is probe the neutrino/anti-neutrino interaction with two Higgs fields, as shown in Figure 8, without knowing how it was generated. Any such interaction must be suppressed by a high-energy scale, which we can call Emax.
In my previous post I wrote that Emax becomes M in the original see-saw. But now you can see that this was a bit of cheat. Comparing Figure 7 to Figure 8, you can see that Emax is really M/yν2 (ignoring factors of 2), and could, in the original see-saw, be much, much larger than the actual sterile neutrino mass M.
Or, said differently: if we measure the largest neutrino mass mν and try to infer Emax by using the neutrino masses, namely by using the estimate Emax < v2/mν, we have to remember that the scale at which new phenomena appear might be much, much lower than this estimate.
This is why physicists at the LHC actively look for sterile Majorana neutrinos under the rationale of the see-saw. They can do this because sterile neutrinos, despite being impervious to the electromagnetic, strong nuclear and weak nuclear forces, still interact with ordinary matter a little bit. This can occur via the interaction with the Higgs field seen in Figure 6 and 7, and moreover, after the dust settles, it’s always the case that ordinary and sterile neutrinos mix; sterile neutrinos will be a little bit ordinary, and the ordinary neutrinos a little bit sterile. These effects potentially allow a sterile neutrino to be produced at the LHC and to decay to observable particles, as long as M is small enough. (See for example https://arxiv.org/abs/1506.06020 .) I think most theoretical physicists view this effort as a long shot, but theorists might well be wrong here, since we are really quite confused about the fermions’ masses… and if you don’t check via an experiment, you don’t know for sure.
But it could easily be the case that sterile neutrinos or whatever mimics them is far out of reach of current experiments. In that case, we are limited to detailed study of the neutrinos we know — and there is still a lot more to learn there. That, however, is for another day.
Technical Epilogue: Three Neutrinos are Better than One
Let me just finish with a comment about the fact that there are, of course, three neutrinos, and everything I’ve said so far has focused on one at a time. Doing three completely is more complicated, of course, but if you learned linear algebra in school then you can follow it. For Dirac neutrinos, the Yukawa couplings yν become a 3×3 matrix Yν, and the 3×3 mass matrix for the neutrinos is just
- Mν = Yν v .
The actual masses of the neutrinos are the three eigenvalues of the matrix Mν .
In the Majorana case, the Yukawa couplings are still a 3×3 matrix Yν, but now we also have the sterile neutrino masses, which also form a 3×3 matrix M . Define the inverse of M, which we call M-1, as the matrix such that M M-1 = 1, where 1 is the 3×3 identity matrix. Then the ordinary neutrino masses are
- Mν = v2 Yν M-1 (Yν)T
where (Yν)T is the conjugate of the Yukawa matrix. Again, the actual masses of the neutrinos are the three eigenvalues of the matrix Mν ; but this time they involve the combinations of the two matrices Yν and M, so unfortunately neither matrix could be uniquely determined even if the matrix Mν were fully known. This does represent a significant limitation on what we can hope to learn about neutrinos in the near future, if they are indeed of Majorana type.