To get to the mass thing, we need to cover some ground.
- How to give a particle mass the easy way.
- How to introduce fundamental forces.
- Why doing these together is a problem.
- How to fix the problem via the Higgs mechanism (boson case).
- How to fix the problem via the Higgs mechanism (fermion case).
How to give a particle its mass the easy way.
Imagine a boring universe with one type of particle that interacts with nothing, not even its own species. The mathematical description of such a theory is pretty lightweight. There is a piece that describes the energy and momentum side, and there is a piece that describes the mass side. The latter looks sort of like a self interaction, but it has a very specific mathematical form to interface with the energy/momentum piece in the right way and to allow well-behaved propagation of particles. We can write these piece out notionally as (kinetic piece) + (mass piece). But keep in mind that these pieces have very specific mathematical forms.
This works fine. You can make a particle theory that has particles with mass in it. You just add that mass piece, and set the value of the mass to match observation.
How to introduce fundamental forces.
Separately, we need interactions (or “forces”) in the theory. I noted in the other thread that symmetries are often the source of physical laws. An example from that thread: if physics should work the same at different places (a type of spatial symmetry), then momentum must be conserved. Many “laws” of physics are just consequences of symmetries of nature.
This is true for all the fundamental forces in the Standard Model. The electromagnetic, weak, and strong forces are not put into the theory, but rather they emerge from it in an elegant fashion by requiring certain symmetries of nature to be present. The relevant symmetries are somewhat subtle and mathematical in their description, but in their own way they are simple. These are known as “gauge symmetries”. These gauge symmetries insist that the laws of nature should be unchanged if you do a certain type of mathematical operation to the fields.
(Technical aside: the freedom that this symmetry imposes is the freedom to rotate the quantum mechanical phases of the fields independently at all points in spacetime while also introducing a “covariant derivative” to ensure the kinetic piece of the theory is aware of the changing phases.)
When this symmetry – this freedom to make phase changes everywhere with no consequence – is enforced on the theory, the fundamental forces and their corresponding bosons just… emerge. By adding a few bells and whistles to the structure of the fields, and by choosing a few different gauge symmetries to enforce, you can get all the fundamental forces that we see in nature. Almost.
Why doing these together is a problem.
This way of getting forces to emerge only works if there are no masses for any particles. You can’t have both the gauge symmetry that yields the forces and also the mass piece(s) from earlier in this post. Those mass pieces are fundamentally not gauge invariant (at least not for the specific symmetries we need for our universe and not without something to help restore invariance).
But we do have massive particles and gauge interactions in nature. The solution: insert no masses by hand into the model, and instead have all masses emerge in a well-behaved way.
(You might ask: do I need all this prologue? Well, it’s a little unexpected that we need some trick to get masses into the theory at all. And, if there weren’t forces in the theory – or if there were only certain types of forces – we could do it just fine. But in our universe, it is not allowed to introduce the masses “bare”.)
How to fix the problem via the Higgs mechanism (boson case).
Start with the mass-free theory with just your fermions in it. Then, postulate a new field – the Higgs – that has the following two key properties. (1) It has some of the same bells and whistles mentioned above so that its complexity aligns with the forthcoming gauge symmetries. In short, it doesn’t just have one value everywhere in spacetime but a small, structured set of values. (2) Give it a self-interaction that serves as a source of its own potential energy, with a form like that blue curve from the prior post so that it will acquire a non-zero vacuum expectation value.
With those features in place, we now:
- Impose gauge symmetry as above to make the fundamental forces appear.
- Everywhere that the Higgs field appears in the theory, “re-center” it to the now-relevant reference value (the vacuum value), which leads to substantial rearrangement of all the pieces in the theory.
- After this rearrangement, some of the residual pieces look (and act!) precisely like the original mass pieces that we wanted but couldn’t have. Since they’ve come about now through this “Higgs mechanism”, the theory as a whole is well-behaved and respects gauge invariance end-to-end.
So, while you can’t add boson masses directly, you can make them pop into the theory through a combination of enforcing gauge invariance (which gets you the forces in the first place) and having this “offset” Higgs field that twists all the pieces up when it shifts to its off-center value.
(Note: The “shifting” part is the narrow piece of all this that the original pendulum analogy has anything to do with. The rest is not conveyed through that analogy.)
Fermion case
For fermions, it works a little differently. Starting with massless fermions, you can safely add parts in the theory that make the massless fermions interact with the original raw Higgs field. This doesn’t mess with any gauge invariance requirements. But then, when the Higgs field shifts to its vacuum value, those interaction pieces get split into two chunks: one that acts like a fermion-to-(new)-Higgs interaction piece and another that looks (and acts!) precisely like a fermion mass piece, of the sort that was forbidden when jammed in by hand. In a sense, an aspect of the Higgs field is “spun off” into a genuine fermion mass. It’s just a mass, not a force or drag or anything else. It looks just like a mass would look if we put it in by hand. The only issue is that putting it in by hand while also imposing gauge invariance to generate the forces makes things break. Recovering fermion masses via the Higgs mechanism gets us around that breakage.
Confirmed?
When the Higgs gets re-centered around its new vacuum value, many parts of the theory get entwined in the process, as seen above. This means that a lot of observable quantities should have detailed relationships with one another. For example, the mass of each particle can be measured, the interaction strength of each particle with the Higgs can be measured, and the vacuum expectation value of the Higgs can be measured. If all this holds water, these measurable values should have very precise algebraic relationships with each other. And they are all observed to do so.
Summary
- You can’t simultaneously put (some) forces and masses into the theory by hand without breaking things.
- So, start with no masses.
- Then put in a Higgs that picks up a non-zero vacuum value.
- When you re-center the Higgs field around its new vacuum value instead of the original unstable value, the “shifted away” parts end up looking like masses for bosons and fermions, if through slightly different mechanisms.
These emergent masses in the theory look and act precisely like regular ol’ masses, even though we had to use the Higgs to ensure that the theory as a whole maintained compatibility with gauge invariance. What this rigmarole really gets us is various counteracting pieces that restore gauge invariance when those masses are present.
Side notes
As @Chronos notes, this provides masses for fundamental particles, but most of the mass we experience everyday is due to something else entirely, namely the energy of quarks and gluons bound together in protons and neutrons. You could have that sort of mass even if none of the fundamental particles had any mass at all.
I haven’t iterated on this post really at all, so do pick out / pick on any pieces that are not cutting it for you.