For the general reader interested in particle physics or astronomy:
Most of the Standard Model’s particles have a mass [a rest mass, to be precise], excepting only the photon (the particle of light) and the gluon (found in protons and neutrons.) For reasons not understood at all, these masses stretch out over a range of a trillion or more.
If it weren’t for the three types of neutrinos, the range would be a mere 400,000, from the top quark’s mass (172 GeV/c2) to the electron’s (0.000511 GeV/c2), still puzzling large. But neutrinos make the puzzle extreme! The universe’s properties strongly suggest that the largest mass among the neutrinos can’t be more than 0.0000000001 GeV/c2 , while other experiments tell us it can’t be too much less. The masses of the other two may be similar, or possibly much smaller.
This striking situation is illustrated in Figure 1, in which
- I’ve used a “logarithmic plot”, which compresses the vertical scale; if I used a regular “linear” plot, you’d see only the heaviest few masses, with the rest crushed to the bottom;
- For later use, I’ve divided the particles into two classes: “fermions” and “bosons”.
- Also, though some of these particles have separate anti-particles, I haven’t shown them; it wouldn’t add anything, since the anti-particle of any particle type has exactly the same mass.
As you can see, the neutrinos are way down at the bottom, far from everyone else? What’s up with that? The answer isn’t known; it’s part of ongoing research. But today I’ll tell you why
- once upon a time it was thought that the Standard Model solved this puzzle;
- today we know of two simple solutions to it, but don’t know which one is right;
- each of these requires a minor modification of the Standard Model: in one case a new type of particle, in another case a new phenomenon.
How the Majority of Standard Model Particles Get Mass
The story begins with Steven Weinberg in 1967 and Abdus Salam in 1968, who first introduced the basic concept of how the Standard Model’s “fermions” (often referred to, to my dismay, as “matter particles”) get their masses. This is illustrated in Figure 2.
It’s a weird, awkward idea, certainly at first glance, so much so that if experiment didn’t confirm it, it would be hard to believe. The idea is that particles like the electron are really put together from two particles, not one. These half-electrons are not half-particles, though; they are particles in and of their own right, except that a particle like this (called a “Weyl fermion“) must have zero mass.
Without the Higgs field, there is a fundamental obstruction to the electron having any mass at all. Although both halves of the electron have “electric charge” (meaning they are affected by electric and magnetic forces), only one half of the electron interacts with the particle known as the W boson, a crucial component of the weak nuclear force. You can’t “marry” two half-particles into one if they behave in fundamentally different ways. They have to behave the same way with respect to all the elementary forces of nature. So on the face of it, these two half-electrons must remain unmarried, and with zero mass, forever.
But when the Higgs field switches on, it changes the rules, giving the W boson its mass. Along the way, as Weinberg and Salam pointed out, it allows these two discrepant electron-halves to be married into one. The resulting electron is a “Dirac fermion“, with a mass.
If the Higgs field’s interaction with the electron’s halves were very strong, then this would be a strong marriage and the newly formed electron would have a very large mass, like the top quark. But instead this interaction is very weak, and the marriage is a loose one, resulting in an electron whose mass is much smaller than the top quark’s.
The same logic applies for the heavier cousins of the electron (the muon and the tau), as well as for the six types of quarks. Each one is really made from two half-particles — two Weyl fermions — of which only one half interacts with W bosons, and each of which would have zero mass were it not for the marriage engineered by the Higgs field.
Massless Neutrinos? Standard Model 1.0
But the logic for neutrinos has a twist. As far as experiment was able to tell, for several decades after the discovery of the first neutrino in 1956, each of the three neutrinos is a sort of half-neutrino… a Weyl fermion that interacts with the W boson and thus experiences the weak nuclear force (but is unaffected by the electromagnetic and strong nuclear forces.) By the logic I just gave you, this Weyl fermion can only have zero mass; it has nothing to marry.
Well, for decades that seemed fine; there wasn’t any experimental sign that neutrinos have any mass. It seemed that the Standard Model gave a simple explanation as to why neutrinos were (apparently) massless: they alone among the fermions of the Standard Model (version 1.0) lack their other half, which is needed for a mass-making marriage.
But gradually, evidence accumulated that neutrinos can change their type while in flight. This “neutrino mixing”, as it is called, is a long story (here’s my article about it), and today it is a major area of particle physics research. The mixings are most easily explained by at least two of the three types of neutrinos having mass. Well, if any of them do, then the logic that predicted all neutrinos have zero mass must be wrong; and if two of them do, it may well be all three.
Meanwhile, because neutrinos are easily made during the early era of the universe, they are abundant in the universe even today. Were their masses large, this would have had an impact on how galaxies and clusters of galaxies form, and more generally on why the universe isn’t uniform and instead has lots of structures in it. Careful study of these structures indicates that neutrino masses must all be more than several million times smaller than the electron’s mass, and more than a trillion times smaller than the top quark’s.
So that poses a puzzle. The neutrinos aren’t “Weyl fermions” with zero mass; they have mass just like all the other fermions do. But if that’s so, why are all their masses so much smaller than those of all the other fermions?
For that matter, can the Standard Model even accommodate these experimental discoveries? This risks getting us into a semantic argument about what is and isn’t “the Standard Model” when it comes to neutrinos, an argument I will take pains to avoid. Let’s just ask: What can we do to the Standard Model 1.0 to make it consistent with experiment? Theoretical physics offers two very different possible stories, and either one (or a combination of the two) is consistent with all experimental data.
Let me start with the easy one: let’s just make the neutrinos like the electron. We will imagine that there is an as-yet-undiscovered other half to each type of neutrino, a Weyl fermion with initially zero mass, until the Higgs field switches on. But there’s something unique about neutrinos: the second half of each neutrino is “sterile,” in the sense that its interactions with ordinary matter are staggeringly weak. This is because:
- neither half of the neutrino is affected by the electromagnetic force (it is electrically neutral, hence its name)
- neither half of the neutrino is affected by the strong nuclear force
- unlike the known half of the neutrino, the added half does not interact with the W boson — and more generally it is unaffected by the weak nuclear force too.
No electromagnetic force, no strong nuclear force, no weak nuclear force. That means the only known forces that affect it are gravity and the Higgs force. Such a particle would leave no direct traces in any current experiments.
These sterile half-neutrinos would then allow the neutrinos to be married by the Higgs field to form “Dirac fermions” with a mass, as shown in Figure 4; this is the same as Figure 2 for the electron. But I haven’t addressed why these masses are much smaller than an electron’s. In fact, the small masses for these Dirac neutrinos would have to arise from very tiny interactions with the Higgs fields. Why should those interactions be so tiny?
No one knows. Yet the very fact that these half-neutrinos are sterile suggests a direction for speculation. Their lack of interaction with the known particles and forces offers these particles possibilities to interact easily with as-yet-unknown strong forces and as-yet-unknown particles. (Such possibilities are highly restricted for non-sterile particles, like electrons, quarks and W bosons.) These additional interactions, in turn, could potentially suppress the sterile half-neutrinos’ interactions with the Higgs field. So if neutrinos are Dirac fermions, there may be ways to explain their small masses, though it will likely require a big addition to the Standard Model.
The second possible origin of the neutrino masses is quite different. It uses a unique feature of the known half-neutrino; by marrying itself, it can become a “Majorana neutrino”, with a mass. It does this by visiting the Higgs field twice, as shown in Figure 5. [This all may seem socially awkward to some readers, but that is completely appropriate given its namesake.] As this self-marriage is only possible for an electrically neutral fermion, it’s not an option for the electron or the quarks.
There’s really nothing wrong with this idea, so why wasn’t it introduced very early on in the Standard Model’s history? The problem is that visiting the Higgs field twice comes at a price: it inevitably makes the Standard Model incomplete. Though the Standard Model may still work fine at current experiments, giving neutrinos a mass by two visits to the Higgs field assures that there must be a maximum energy Emax beyond which the Standard Model’s equations cannot do their entire job. If you do experiments above that energy, there will be measurements for which the Standard Model makes no predictions at all. That means that someday, something else will have to be added to the Standard Model to fix this problem.
Well, so what? There was a time when this bothered people; they felt that incomplete theories weren’t really consistent. That time is past; with better understanding of quantum field theory (the math that underlies the Standard Model,) we no longer view this with concern. After all, we already know the Standard Model is incomplete: gravity is not a part of it, because we aren’t sure how best to combine Einstein’s theory of gravity with the other forces. The incompleteness for gravity may only show up at an Emax a million trillion times higher than we currently access at our particle accelerators. For Majorana neutrinos it won’t be so extreme; Emax may be no more than one trillion times higher than we currently can reach. But the conceptual issue of incompleteness isn’t necessarily so different for the two examples.
So yes, neutrinos can be Majorana fermions. It comes at the price of an incomplete theory, but conversely it avoids the need to add three new half-neutrinos to the theory. More importantly, it provides the seeds of an explanation here as to why neutrino masses are so small! The larger the energy Emax at which the Standard Model ceases to work, the smaller the neutrino masses mneutrino have to be.
This is known as the “generalized see-saw mechanism”, and the fulcrum of the see-saw is the Higgs field’s value: about 250 GeV, and usually called “v“. However small v is relative to Emax, the neutrino masses mneutrino must be even smaller compared to v.
For instance, if v is a trillion times smaller than Emax, then the largest mneutrino must be at least a trillion times smaller than v. If you like equations, you can say this more briefly:
- mneutrino < v2/Emax .
So if the Standard Model (or at least its neutrino portions) were to continue to be valid to energies a trillion times higher than the Large Hadron Collider, and the neutrinos are Majorana fermions, then we would be guaranteed that the neutrinos’ masses must be comparable to or smaller than what is shown in Figure 1.
But don’t misread the logic. If neutrino masses are someday measured to be a trillion times smaller than the Higgs field’s value, then although Emax cannot be larger than a trillion times the Higgs field’s value, it could be much smaller. It could, for example, be just ten times larger than v, and within reach of the Large Hadron Collider or its successor. If that were the case, we might see experimental evidence of the Standard Model’s breakdown in the relatively near future.
In short, what the see-saw mechanism does is promise us that the Standard Model will break down at or before the energy scale v2/mneutrino, the square of the Higgs field’s value divided by the largest mass among the neutrinos. [Remember, though, that this assumes that neutrinos are Majorana. If they’re Dirac, then this isn’t true!]
The Original See-Saw
If you’re curious (otherwise you can skip the section), let me describe to you the original see-saw mechanism. Introduced in the mid-1970s, it involves adding the same sterile neutrinos needed for Dirac fermion neutrinos in Figure 4, and repurposing them for the see-saw mechanism. What happens is that our familiar but hapless neutrino marries a sterile neutrino as in Figure 4, unaware that this sterile neutrino has an enormous mass Msterile. (Wait; wasn’t the sterile neutrino a massless Weyl neutrino? Ah, in Figure 4 it was; but because it is sterile it doesn’t have to be massless! Being sterile, it can have a Majorana mass all on its own, without any Higgs field!)
The presence of this sterile neutrino with a large mass causes the Standard Model’s equations to break down; what we generically called Emax above is simply Msterile. And since this sterile neutrino’s mass is so big, we cannot see it in any ongoing experiments; as far as our experiments are concerned, our familiar neutrino is lonely, essentially married to itself after all — a Majorana neutrino, despite its efforts to be Dirac. The larger is Msterile, the smaller is the familiar Majorana neutrino’s mass — hence the name “see-saw”.
The Many Open Questions
There’s a huge question looming over us that I haven’t even addressed yet: what are the actual neutrino masses? We know some relations between them; we know a maximum for them; and we know a minimum mass for at least one of them; but beyond that, the issue remains open. Experiments are ongoing. But that’s a story for another day.
Once we know their masses, will we be able to tell whether the neutrinos are of Dirac-type or of Majorana-type? Not unless we’re lucky. It could be that particularly prominent clues are accessible to current technology; but if not, we may not know the answer for decades. Why is it so difficult to figure out? It’s not hard in principle; if you’re an impractical sort of theoretical physicist, you can imagine all sorts of of methods. Here’s a fun one: if you could increase the Higgs field’s value v inside a box by one percent, the mass of an electron in the box would increase by one percent; similarly a Dirac neutrino’s mass would increase by one percent; but because the Majorana neutrino visits the Higgs field twice, its mass would increase by two percent. Very easy, conceptually. But unfortunately, neither this nor any related method will work; principle is not practice, especially when it comes to neutrinos, whose interactions with ordinary matter are too few and far between for anything about them to be easy.
Clearly, when it comes to neutrinos, there’s a lot more research to be done. But meanwhile, I hope this post gives you some insight into how the very structure of the Standard Model plays into these issues. The fact that neutrinos are affected only by the weak nuclear force, and that their other half, if it exists, would be a sterile particle, gives them unique character, allowing them potentially to be either Dirac fermions or Majorana fermions. This distinctiveness makes them a breed apart within the Standard Model, and many physicists suspect that this is in some way responsible for their remarkably small masses.