Electroweak Theory and the Origin of the Fundamental Forces (script) (Patreon)

Our universe seems pretty complicated. We have a weird zoo of elementary particles, which interact through very different fundamental forces. But some extremely subtle clues in nature have led us to believe that the forces of nature were once unified, ruled by a single, grand symmetry. But how does one force separate into multiple? And how do the forces of nature arise from mathematical symmetries in the first place?

The best way to understand how the universe went from a simpler, more symmetric state to its current complicated condition - is to look at the unification that we understand the best. Today we're going to begin an exploration of how the electromagnetic and weak forces were once a single thing. And this dive into electroweak unification will lead us inevitably to the Higgs field and an understanding of how particles gain mass.

Let’s start with the mysterious and often misunderstood weak interaction. You may have heard that it’s the force responsible for some types of radioactive decay. Not very satisfying - but it’s how the weak force was first identified.  Beta decay is when a neutron turns into a proton by emitting an electron and neutrino. The electron was called a beta particle by Ernest Rutherford back in 1899 back before we knew that these things were electrons. It’s one of the main ways radioactive nuclei decay - the other being alpha decay, where the emitted “alpha particle” is really a helium-4 nucleus.

Fast forward to the early 30s. While the brand new field of quantum mechanics could describe the behaviour of electrons, nuclear processes remained mysterious. Enrico Fermi made the first attempt at a full quantum description of beta decay with his ‘four fermion’ interaction. Basically, he tried to model this as a direction interaction - in which all four “fermion” particles literally touch. So an ingoing neutron is directly converted into the outgoing proton, electron and neutrino, with all the conservation laws satisfied. Fermi was motivated by the apparent extreme short range of the interaction - and that short range is what earned it the name “weak” interaction. But Fermi’s model only worked at low energies, nor could it or its successors explain why the weak interaction violates charge-parity symmetry.

Meanwhile, there was a very different effort to explain electromagnetism that was enjoying much more success. That effort was quantum electrodynamics, in which charged particles interact not by actually touching - but via a mediating “force particle” - in this case a photon. By the way, force-mediating particles are bosons, as opposed to the fermions that make up matter. QED is what we call a gauge theory - its force-carrying fields and particles arise from the symmetries of the quantum equations of motion. We’ll come back to that.

In 1957, Julian Schwinger proposed a set of force mediating gauge bosons for the weak interaction. Given that the weak interaction could change a neutral particle into a pair of charged ones this mediating particle must itself be charged. This was an early hint that somehow the electromagnetic force, which acts on charged particles, was playing a role here. Although the involvement of the neutron and neutrino meant it couldn’t be entirely electromagnetism.

Not only that, but experiments at the time indicated that, if they existed, these new ‘weak bosons’, W bosons, had to have mass due to the short range nature of the interaction, quite a lot of it in fact. The short range nature of forces with massive force carriers is usually attributed to the energy-time uncertainty relation, and we’ve presented it that way previously. Though as we’ll discuss in a future episode, that’s not the full story. Anyway, this new gauge theory of the weak interaction seemed to be okay with parity violation, and wasn’t only accurate at low energies. Problem solved. Easy peasy, right? Uh, not even close. That requirement of the W bosons having mass was an enormous theoretical headache. Gauge bosons should just not have mass. Ever! At least assuming that the symmetries of gauge theory remain intact. The simple requirement that the weak force was mediated by massive particles ultimately unified the weak force with  electromagnetism, and revealed the existence of the Higgs particle.

To get all of this we need to do a quick recap of what a gauge theory actually is. Let’s try an example:

In quantum mechanics, the wavefunction determines the probabilities of certain outcomes being measured for observables like particle position and momentum. Quantum mechanical equations of motion like the Schrodinger equation describe how the wavefunction evolves through space and time. Like all waves, the wavefunction has a phase - the current position of the peaks and troughs. But that phase should be totally unobservable by any experiment. Phase is what we call a degree of freedom, because it should be possible to change it however you like at any point in space and still get the same physical observables. For that to be the case, we need to add some stuff to the Schrodinger equation. That stuff turns out to describe the electromagnetic field. When we quantize that field - when we let it oscillate with discrete packets of energy - we get the photon.

So there we go - we insisted on a symmetry - that the Schrodinger equation is invariant to changes in the local phase of the wavefunction. That resulted in a new quantum field and a corresponding particle. So that’s a gauge theory in a nutshell: it explains a force by imposing symmetries on the equations of motion.

Actually, a visual representation would really help here. We said that at each point in space, the physics shouldn’t change when we shift both the real and complex parts of the wavefunction by some amount, a process which leaves the magnitude of the wavefunction unchanged. This is equivalent to considering a complex 2D vector, where the axes are the real and imaginary components, which we can rotate but not stretch or shrink: the length of the vector is the magnitude of the wavefunction, and the rotation amount is our local phase shift.

The collection of transformations which are just rotations of 2D vectors are members of something called the Unitary group of size 1, written U(1). The ‘U’ stands for unitary and tells us that elements of this group preserve vector length, or in our case quantum probability. The ‘1’ just means the symmetry transformations act on states which can be completely  represented with just one function, like our single wavefunction from the Schrodinger equation.

Imposing this U(1) symmetry on our equations of motion led us to something very real and physical - electromagnetism. So it makes sense to see what other symmetries do.   There, the single 2D vector could only change in its angle - its phase. It had 1 degree of freedom - corresponding to a gauge field with a single mode - the electromagnetic field and its photon.

In SU(2), each individual vector can change length, rotate, and be reflected - overall 8 degrees of freedom. But if we require this transformation to now respect the SU(2) symmetry - for example requiring unitarity - we knock out 5 degrees of freedom leaving us with 3.  If we require our equations of motion to respect SU(2) we have to introduce a new field with 3 degrees of freedom. Quantizing this field gives 3 bosons which are *almost* exactly what we need to make a gauge theory of the weak field work.

Almost. The fields and corresponding particles produced by the pure symmetries we described are fundamentally massless. The bosons of the version of SU(2) that I just described are simple light-speed oscillations in their fields, just like photons. But the bosons of the weak force have to have mass. So is this a bust? Maybe, but the whole gauge field thing seemed so promising, so it's worth asking: how can we give mass to something that seems fundamentally massless?

The perfect masslessness of these gauge fields and bosons is a direct consequence of the perfect symmetries from which they come. Adding in mass screws up those symmetries. For example, adding mass to a photon means adding an extra term to the electromagnetic field stuff in the Schrodinger equation so that it would no longer be invariant to local phase shifts. In order for the weak force bosons to have mass, we have to willingly break the symmetry that gave us them in the first place.

But wouldn't that mean we're throwing away the core idea that gave us our weak field? Not necessarily. It's possible for the equations of motion describing a system to have a particular symmetry - to not care about changes in some property - But at the same time, for the state of the system those equations describe to NOT have the same symmetry. That sounds counter-intuitive, so let me give you an example.

Consider a magnetic material, made up of many little magnetic dipoles, like little bar magnets that can point in different directions. The equations of motion governing the overall magnetisation of this material aren’t encoded with some special direction - the whole system could be rotated and nothing should change. This represents the symmetry in our equations. But the individual magnetic particles interact with each other, they want to align with their neighbours. At high enough temperatures - above the Curie temperature - thermal energy causes the particles to rotate randomly so the overall magnetization of the material is random.

However, if we cool the material down, the interactions between magnetic particles can start to come into play. They align with each other. If we cool the system enough all particles get frozen into one aligned state. This is a state of broken symmetry. Even though the equations of motion say one direction is not preferred to the other, the ground state has randomly picked out a single direction. We call this spontaneous symmetry breaking.

So what if something similar is happening with the field that gives us the weak force? Perhaps the equations of motion respect the symmetries that give us the necessary gauge field, but the physical system - the field itself - evolves into a state which no longer respects those symmetries. That could lead to our force carriers having mass.

This has been an analogy-heavy episode. Spinning vectors and magnetic materials. But what specifically is happening with the weak field? What is this symmetry, really? And what causes it to break? Well remember that I said that the weak interaction somehow seems to involve electromagnetism? That’s another clue. It turns out that both the weak force AND electromagnetism come from the breaking of a larger symmetry group.

And actually, that larger symmetry group is just the combination of our U(1) and SU(2) symmetries. The combined SU(2)xU(1) symmetry is the electroweak field, and it has 4 massless bosons, like a well behaved symmetry should. But at low temperatures, this symmetry is spontaneously broken - leaving an independent, massless U(1) field for the photon and a massive, broken SU(2) field that gives the massive weak force bosons.

By the way - by “low temperature” I mean anything below around 10^15 Kelvin. The entire universe was that hot when it was less than a trillionth of a second old. Back then, electromagnetism and photons and the weak force didn’t exist. This was the electroweak era, and we can also produce these conditions in our particle accelerators - and in fact we’ve verified this whole electroweak thing with fantastic precision.

Let’s take a second to remember how we got here. We just asked what would happen if the equations of motion governing nature had certain symmetries. The very existence of those symmetries requires a family of fields and particles that we now observe in nature. But to really make this work - to give the weak force bosons their mass - we have to conclude that really these fields, these symmetries, are ultimately unified. And then broken. As we’ll see soon, by the Higgs field. But it doesn’t stop there - the electroweak unification set us on this grand quest to find deeper symmetries that could bring the strong nuclear force into the fold, allowing us to collect the full Pokemon-evolution set of the standard model: U(1)xSU(2)xSU(3). And from which arises the fantastically rich palette of particles and the complexity they enable, inevitable consequences of the broken symmetries of spacetime.