The Science of Everything Podcast - Episode 164: Gauge Theory and the Higgs Boson

Episode Date: August 30, 2026

We conclude our series on the standard model with an introduction to gauge theory. We discuss how the requirement of local gauge invariance generates interaction terms in the Lagrangian, and introduce the U(1) x SU(2) x SU(3) group structure of the standard model. We also consider the problem of massive W and Z bosons, and how this was resolved by the introduction of the Higgs field and the mechanism of spontaneous symmetry breaking. Recommended pre-listening is Episode 163: The Standard Model of Particle Physics. If you enjoyed the podcast please consider supporting the show by making a PayPal donation or becoming a Patreon supporter. https://www.patreon.com/jamesfodor https://www.paypal.me/ScienceofEverything

You're listening to The Science of Everything podcast, episode 164, gauge theory and the Higgs boson. I'm your host, James Fodor. So this episode picks up directly from where we left off in the previous episode, where we talked about the different particles in the standard model. And in that episode, we made a distinction between the fermions, which are kind of like matter particles, and the bosons, which are kind of like energy particles in a sense, and that they mediate the, fundamental forces that affects the fermions. So in this episode, we're going to talk in more detail about the different gauge bosons and how they work and the mathematics behind that. And we're also going to discuss the Higgs boson, which is involved in the origin of mass for the fermions and some of the gauge bosons. So unsurprisingly, recommended pre-listing is episode 163,

the standard model and in turn its various prerequisites. And warning in advance, this will be a bit more technical episodes, so those prerequisites will be assumed and bear with me if it gets a little technical at times, but I'm doing my best to explain it as clearly as I can. So let's start by talking about the term gauge. The term gauge refers to any mathematical formalism that's used to regulate redundant degrees of freedom in a physical system. Essentially, what happens when we try to explain or describe a, well, anything really, but here we're talking about fundamental physics, when we try to describe how they work mathematically, often the mathematics that we use will either deliberately or inadvertently introduce extra degrees

of freedom that don't correspond to anything physically meaningful. Let me give a simple example of this. It's not really a physics example, but hopefully it will illustrate the intuition. So when people invented numbers, they started with the integers, one, two, three, four, and so forth. You can add physical objects together and you get higher numbers. Two plus three is five, right? And you can have two rocks, bring them together for three rocks, you have five rocks. We can also imagine taking rocks away. So if you have five rocks and take away three, you have two rocks left over.

Interestingly, when we think about taking things away, that introduces this notion of subtraction, which when you apply it to then taking a bigger number away from a smaller number, it leads to this notion of negative numbers. Negative numbers are very useful because they can be used to describe, for example, someone who owes money. If they have a negative amount of money, that kind of means they owe someone else money. They can also be used in a wide range of other applications. But if you try to ask, like, what does it mean to have a negative rock? Well, I mean, it doesn't really mean anything in that sense. It's sort of like, it's not exactly the same, but it's sort of like an extra degree

freedom in that what we've introduced with the mathematics goes beyond what we can directly apply in a physical sense. You can't literally have a negative number of things. A negative number is more like exists conceptually as like the difference between a big and a number and a smaller number, but it can't be directly instantiated physically. So that's just an example, that's not a gauge theory or anything, but it's just an example of how often the mathematics used to describe things in the physical world can lead to us describing things that don't have a direct physical correlate. And in the case of a gauge theory, that amounts to having extra degrees of freedom that don't correspond to anything physical. That's fine to have that in the theory, but what it does is it introduces

extra parameters that we need to introduce to get the theory to work, but don't actually make any difference physically. So I've talked previously about the Lagrangian. That's a way to describe the energy of a system, and it's what we use in quantum field theory to describe particles and their interactions. In gauge theories, Lagrangians have internal degrees of freedom, which exist in the mathematics, but don't correspond to anything physical. So when we actually use the theory to compute something that's experimentally observable, we have to, as it's called, choose the gauge. Really, all it means is that we have to make a choice about fixing these degrees of freedom, degree or degrees of freedom, so that we can get a specific measurable result out of it.

You have to make a choice. What we want is a theory where anything that's observable from the theory, any experimentally testable results, do not depend on the gauge. Obviously, that's the whole point, right? If the gauge does not correspond to anything physical, that means that the results shouldn't depend on what gauge we choose. That should be just an arbitrary choice that we choose for computational convenience. An analogy to this is choosing our coordinate system. You may be familiar with Cartesian coordinates versus spherical coordinates or radial coordinates. They're just different ways of describing three-dimensional space. They can be useful because certain coordinate systems are more convenient for conducting certain types of calculations in. But the answer shouldn't

depend on the coordinate system that you pick, because that's just arbitrary. You can pick whatever you like. Choice of the gauge is like this. It's not a coordinate system as such, but it's sort of similar in the idea that it's something that's fixed by the scientist's computing a calculation, but we want to make sure that the theory is constructed so that the results don't depend on the gauge because it's just a degree of freedom that you can just sort of pick. Now, there's something very interesting about this notion of the gauge, because you might be wondering, well, if we have internal degrees of freedom in our Lagrangians specifically, which means that we have to like choose one in order to get a computational, like an experimentally verifiable number out of it.

Why don't we just reformulate the theory so that they don't have these internal degrees of freedom and they don't have to worry about this? Well, it turns out that it's actually good to keep these internal degrees of freedom in because it makes the calculations much easier and it has, there's actually quite a few desirable properties of these. As long as we have a way to get rid of them in the end, then it's fine. It doesn't matter if they're there during the calculation. kind of like negative numbers. It's good to have negative numbers during the calculations. It's just that as long as we don't then try to say that there's a negative rock that exists in the real world, that doesn't really make sense. Likewise, having the internal degrees of freedom in the Lagrangean is fine, as long as we have a way to ensure that they don't affect the final experimentally verifiable results, right?

And there's an interesting property of these internal degrees of freedom that correspond to this gauge choice, that whenever there is a reasonable, redundant degree of freedom in a Lagrangian, there will always be a corresponding symmetry of that theory, where the theory gives the same answers, even after you transform all of the terms in a certain way. This is called a gauge transform, which gives rise to a certain type of symmetry. Let me give an example to make this a bit more concrete. So when physicists were developing a theory of quantum electrodynamics, so electromagnetism but quantized, they realized that The way we describe the photon field, the electromagnetic field, I'll just call it the photon fields for short, is we use complex numbers, right? So that's when you have a real and an imaginary component.

Complex numbers are a very convenient way of describing the phase of a field, which essentially is like where you are on the oscillation scale from like peaks to troughs. The position sort of relative to one cycle is your phase. So we have those numbers in the theory, in the Lagrangian, as terms describing the phase of a given electromagnetic field. But what they discovered is that to get an experimentally verifiable result, you always multiply multiple fields together. You can never observe a single field. You always observe interactions of fields. And so what that comes down to is that the phase of a photon is never measurable. We never observe just the phase.

All we ever observe is phase differences. and that has a very specific measurable outcome, like when we see interference of light, like constructive and destructive interference, which we've talked about that in previous episodes. Like you have with water waves, for example, you can also see this with light waves when you have patterns of light and dark bands.

That's an interference pattern. That is caused by phase differences between photons as they interfere constructively or destructively depending on their distance and different frequencies and things like that. So we can observe phase differences, but a phase difference is precisely one phase minus another phase, or like mathematically that's what it is.

The point there is that in the formalism, we have two numbers, phase of photon one, phase of photon two, but we can only experimentally measure one number, which is their difference. So this gives rise to an extra degree of freedom in the formalism, which is essentially like one of the phases. We can only measure the phase difference, but we have two numbers from the phase of each of the photons, which is an extra degree of freedom. In order to get rid of that, we need to choose a gauge, which then we do the calculations in. But looking at the Lagrangians they were writing, physicists realized that this internal degree of freedom, this extra parameter here corresponding to unmeasurable phase, gives rise to a symmetry under a gauge transformation.

And specifically, that is, if you take a field in your Lagrangian in that equation that describes energy, different types of energy effectively, remember Lagrangians have like your kinetic energy, interaction energies and potential energies. as well as mass energy terms. So there's different terms in your Lagrange. We've talked about that before. I'm not going to go over that in detail here. A gauge transformation in this case is when we take our Fermion field. So think of this is just like your electron field for simplicity, but you can do this for other fields too.

You take your electron field and you transform it mathematically using a particular rule. Basically, you just shift the phase. If I shift the phase of the field everywhere in space and time, because remember these theories describe space and time like four-dimensionally, If you shift the field everywhere in space and time by a certain amount, then that doesn't change the results. That makes sense intuitively, right? We can't measure the absolute phase, only phase differences. So if you shift all the phases by the same amount, then there's not going to be any difference in the results.

So this extra degree of freedom in the Lagrangian gives rise to a gauge symmetry, which is the symmetry produced by shifting everything over all of the phases over by a certain amount. That's a global symmetry. and it corresponds to something that is physically conserved, which is conservation of charge. So the symmetry, the global symmetry of shifting the phase of the electromagnetic field corresponds to, or is a way of describing the fact that electric charge is conserved in interactions involving electromagnetism. So that's pretty cool.

There's this connection between symmetries and conservation. So there's already a kind of a reason to keep the redundancy in the theory because there has this sort of nice property that we can read off a conservation law off of the theory. It also just makes the theory easier if we can give each photon field its own phase instead of having to somehow subtract that off, which would make the theory more complicated to work with mathematically. So there's a bunch of reasons to maintain the gauge invariants and even more as we'll see going forward.

But here's the really interesting part that we need to understand. And this is something that took me many years to sort of get, because the way this is taught is often quite poor, I think. So let me try to develop the idea. When physicists were developing quantum electrodynamics, they already had a classical template to use, which is Maxwell's theories of electromagnetism, right? We already had those equations.

And so they knew what they should expect to find, particularly when we're talking about interactions between electrons and photons, like light and matter. They knew what those equations should look like, And so they wrote the Lagrangian to reproduce those equations of motion just within the quantized version, right? We talked about how quantization is done in previous episodes again. I won't get into that. But basically, you can take your equations of motion, work out what the Lagrangian should be from those equations of motion,

and then just quantize the resulting fields, and then you have a quantum field theory. This meant that in constructing the Lagrangian for the quantum field theory of electromagnetism, physicists started with the equations of motion of classical electromagnetism, worked backwards to figure out, which you can do mathematically, to figure out the corresponding Lagrangian, and then they look at the fields there and quantize those and work forward. So then we had a quantum field theory of electromagnetism. But they had this problem that they found where there was this extra internal degree of freedom. So the physicists tried to work out what are the conditions that we have to impose, like mathematically, in order to make it so that this gauge symmetry. doesn't change the experimental results. There was an extra degree of freedom, so we need to impose extra structure in order to get

the result not being dependent on the extra degree of freedom. And they worked out that a very specific mathematical choice actually meant that any dependency on this internal degree of freedom is cancelled out, and the results are always independent of the choice of gauge. So you can choose different gauges and get the same answer. So what was this mathematical trick that they needed to use? Well, it turns out that's very interesting. Remember I said earlier that an internal degree of freedom gives rise to a symmetry.

In this case, the extra phase that we have that's not measurable, the phase of the photon, form that by a global transformation. So the same everywhere, like just add the same amount to the field everywhere throughout space and time, then that doesn't change the theory, and that gives rise to the symmetry, which corresponds to the conservation of charge. Well, it turns out that if we modify this symmetry to make it a local symmetry instead of a global symmetry, the result is actually that this gauge term is effectively cancelled out.

And so we can pick any gauge we like. How does this work? Well, so-called upgrading the symmetry from a global symmetry to a local symmetry means that instead of changing the field by a constant at every point in space and time, just like adding five to it. Instead, now the transformation is a function of our coordinates, of our X coordinates. So the transformation of the field is a function of X, where X is just space-time coordinate. So this means that in one point in space-time, we could increase the field by one,

and in another point we could decrease it by two, and another we could increase it by seven. I'm just using arbitrary unity to illustrate the point, right? So it's not that we're shifting the field by the phase, rather, by a constant amount, is that we're shifting it by different amounts. In fact, an arbitrary function. function. Like we don't specify specifically. We just write it as an arbitrary function. Now, that might sound a bit crazy. Like, well, surely that will create differences between the phase at different points. But actually, it turns out that it doesn't. And the reason is because

there are other terms in the Lagrangian that cancel out the changes that are brought about by the transformation. So when we make this transformation, this local transformation, that allows the phase to be shifted by different amounts at each point, there is now a, this gets a bit technical, but I just want to illustrate why this happens. There's a bunch of derivative terms in the Lagrangian, which mean that we have to take the derivative of the field, how the field changes in space time, and then that gives rise to certain terms,

like the kinetic terms, for example. When we shift the phase by a constant amount, if you know anything about calculus, constants come out of the derivative. When we differentiate a constant, it doesn't change it, right? Like the derivative constant is just zero. So constant global transformation is unaffected by derivatives. But local transformations are a function of X, and so they are affected by derivatives.

And this means we pick up an extra term when we take the derivative, which then adds an extra term in our Lagrangian, which turns out to cancel out just the other terms that we had in there to mean that it's now gauging. The theory is now gauge invariant. We still need to choose a gauge, but it doesn't matter which gauge we choose. Whereas previously, the choice of gauge affected the result, which was bad. We don't want that to happen. So this was great, right? This introduction of this local gauge transformation

meant that we were able to solve this problem of the results depending on the gauge. And now we have a quantum field theory of electromagnetism. All we had to do was read the form of the interactions off of our Maxwell's equations and then add this extra constraint to fix the gauge or to ensure that the gauge choice didn't matter specifically by turning the global invariance into a local invariance and then applying the transformation

and then we work out, oh, now the theory works. Now there's an extra key thing that we need to understand about this gauge theory sort of solution to this problem and that it requires that the Fermion field, like the electron field in this case, is coupled in a certain way with a gauge field. In this case, that's the photon field. So there's a direct mathematical connection between requiring this local phase invariance to these transformations of the phase that can be different in every place, and coupling between the electron field and the electromagnetic field.

Coupling meaning there's an interaction between them. They interact with each other. Like in the fireman diagrams, you think about the lines coming in and lines coming out, that's an interaction we're talking about a coupling. And that's exactly why we call the photon a gauge boson, because it's the particle that interacts with Fermion field to generate or to facilitate this local gauge invariance. Or to put it another way, every local gauge invariance

will necessitate the existence of a gauge field, which will then describe the field that mediates interactions between fermions. We can loosely think about this as if the virtual photons that are constantly being emitted and absorbed and are fluctuating in and out of existence by, say, different electrons as interacting with, each other. This sea of virtual photons that are being exchanged between them is carrying information

about the phase of each electron. And the exchange of photons effectively ensures that there's this local invariance to the phase, that information is constantly being moved about between points in the electron field by photons to maintain that local gauge invariance. You shouldn't take that story too literally, but I think it's a helpful way of picturing the connection between this mathematical property of local phase invariance and the existence of this boson field that mediates interactions between the electrons. You can think of the photon field as consisting of these virtual particles that are exchanging phase information, constantly moving it about to ensure that there's this invariance between the different electrons. Yet another way to

describe this mathematically is that local phase invariance equals coupling with a gauge field. So if we want to get our particles to interact with each other through an intermediary force particle, we need to couple them to a gauge field, and we can do that by introducing or requiring mathematically this invariance of the field to certain types of local phase transformations. So this was all discovered initially in the context of quantum electrodynamics, where we already knew the form of the Lagrangian, including the interaction terms between electrons and photons. So we already had that in the Lagrangian. It just turned out that there was this undesirable property that the results depended on the gauge. So in order to make that not the case, the results invariant to gauge, we figured we could do that by introducing this local phase invariance.

And then we get this nice relationship between the gauge field and the local transformation. This might not be that interesting if it was just quantum electrodynamics, because we already knew what the Lagrangian should look like. But it becomes much more interesting and much more important when we come to the weak and strong nuclear forces. Because when we were trying to develop quantum field theories of these forces, we didn't have a classical analog. There is no Maxwell's equations for the strong or weak nuclear forces.

We didn't have anything to go by. So that means we didn't know what the interaction forms should look like in the Lagrangian. We had to figure it out. So, you know, there's different ways of doing that and people try different methods. But the technique that turned out to be very successful is taking the logic that was developed in quantum electrodynamics and reversing it. Remember in quantum electrodynamics, we started with the Lagrangian, which we inferred from the equations, from Maxwell's equations of motion. And then from that, we got the Lagrangian. And then we saw in the Lagrangian that there was this extra degree of freedom corresponding to phase, which gave rise to this dependence of our calculations on this internal degree of freedom.

which was not measurable. It depends on the gauge that we chose. That was not desirable. And so then we worked out that we could get rid of that dependency by introducing this local phase invariance, or specifically by promoting the global phase invariance, which already existed to a local phase invariance, which is an extra requirement that we make,

but then it turned out to cancel out the problematic terms and resolve the problem for us. So the logic in QED, electromagneticism, was that interaction between electron and photon, gives rise to local phase invariance. But remember, for the weak nuclear force and the strong nuclear force, we didn't know what the Lagrangian should look like, including the interaction terms. I mean, we had an idea about what the kinetic terms and the mass terms should look like,

because they're pretty standard. But we didn't know what the interaction terms should look like. So we reversed the logic. We said, well, let's impose a local gauge invariance to the relevant fields, just like we did in electromagnetism, except with some differences which we're talking about. So we'll require this local gauge invariance, and that will then give us an interaction term. And we'll assume that that interaction term is the correct one, and we'll test experimentally. So there's no guarantee that this would work.

This was a methodological assumption, but it turned out to be successful in both cases. It's been incredibly successful. Now, the reason I think this is often poorly explained is because many, many lecture notes and even textbooks will say that requiring local gauge invariants, in your Lagrantian, like, generates or produces a gauge boson, which in the case of the strong nucleophores is your gluons, right, or the weak nucleicorosis is your W&Z bosons. Now, I think that this is sort of mathematically true, but it's not really physically true.

Writing down an equation doesn't generate a physical field. That doesn't make sense, despite the fact that you often see this language used. I think that we just have to understand this speaking sort of mathematically. When we require local gauge invariants, That means that they're, by necessity, will be an extra field that we introduce into the mathematics that then couples to, like, interacts with the initial fermion field. Remember, we're always requiring the gauge invariance of the fermion field. So that's like your electron, or for chromodynamics, it's your clock.

So that's the sort of field we start with. We then introduce the phase invariance because of the mathematics of how the derivative works. That then necessitates mathematically. There must be an extra field that cancels out. certain terms and that gives rise to the invariance. And that extra field is our gauge boson field. But the mathematics of it doesn't determine the physics.

It's kind of the other way around. This was a methodological assumption that was made on the basis of the success of this strategy for quantum electrodynamics, where we already knew the interaction form. We took a guess, an informed guess, that something like this would also apply to the weak and the strong forces, and it turns out that it did. So that's why gauge theory is so important to understanding the standard model. because all of the forces are described using gauge theories in which we require this local phase invariance, and that then gives rise to exactly the types of interaction forces, coupling between the Fermion field

and the force mediator field, the gauge field, that explains the interactions that actually exists physically. So the mathematics doesn't determine the physics, but we took a guess based on mathematics in one force, and it turned out to apply to the other forces as well. So that's why people sort of say that the local phase invariance gives rise to the boson. But yeah, you have to understand the sense in which they mean that. Now, there are some complexities. It's not exactly the same as electromagnetism. And so I need to explain those now. And this is where we get into a little bit of group theory, which I'm by no means an expert on. I have studied a bit of group theory. But a group theory is a mathematical theory that describes certain mathematical objects called

groups. And a group is, it's not that complicated. Fundamentally, a group is a set of mathematical objects which obey certain transformation properties. Objects within the group transform into each other and you can't transform out of the group. So just to use a very simple example that illustrates the vague idea, if I have, if I start with positive numbers and I multiply them by other positive numbers, I can only get positive numbers. You can never get a negative number by multiplying a positive number by another positive number. But if I start with negative numbers and multiply them by negative numbers, I can get positive numbers. If I'm multiplying by, you're negative numbers, I can get positive or negative numbers. Whereas if I'm multiplying only by

positive numbers, I can only get positive numbers. So these would be kind of like two different groups that are closed under different operations of like multiplying by a positive versus a negative number. There's different sets of numbers in this case that you get depending on what operation you're talking about. Groups don't have to be comprised just of numbers. I said mathematical objects. They can be comprised of what kind of anything, including other sets, or in the case it'll be relevant here, matrices. Okay, so why are we talking about all this group theory? And don't worry too much if you don't get the mathematics, because it's not that important to understand the mathematics. It's just important to understand some of the terminology. It is said that the standard model consists of three gauge groups, or like a gauge structure of the standard model called SU3, SU2, and U1. And you'll often see it written as like the product of these. So SU3 times SU2 times U1. These terms here are referring to the types of groups that correspond to each of the gauge symmetries.

This is all, we'll be confusing, but hopefully it'll become a little bit clearer as I pass some of this terminology for you. So let's start with the simplest case, which is electromagnetism. Electromagnetism only has a single field, a single gauge field, it's the photon. As such, there's only a single extra degree of freedom in the theory. That's the phase of the Fermion phase. We only measure phase differences, but there's two phases in the formalism. We need to reduce the degrees of freedom by one, and so that field there is what we perform

the transformation on when we do the local phase invariance. So one extra internal degree of freedom means one gauge field. That corresponds to transformations of quite a simple group. It's called the unitary group of dimension 1, or U1. The U here for unitary, without getting into too much detail, is basically just referring to the the fact that probability has to be conserved. So particles can be created or destroyed, but you have to have a conservation of things like momentum and other quantum numbers. So loosely, that's where Unitary is referring to. I don't want to get too much into detail of that, but the U stands for Unitary.

Okay, so electromagnetism is fairly simple. What about the weak nuclear force? Well, the weak nuclear force is a bit more complicated because instead of there being only one extra degree of freedom, there's actually three extra degrees of freedom that we have. And The reason for the reason for this is because the weak nuclear force describes not just a single type of particle like electrons or quarks, but it describes what are called weak doublets. And this just means a pair of particles that kind of go together in the interactions. I'll explain a little bit what I mean by that. So what are the weak doublets exactly? Well, we've actually already met them.

So remember I said that there's this distinction between quarks and leptons, and there's three generations of matter. but within a generation, both of the leptons form a weak doublet. So the electron and the electron neutrino form a weak doublet. The mion and the neuron neutrino, likewise the tau and the tau and the tau neutrino. And as for the quarks, well, same thing there. The up and the down quark form a weak doublet. Same with the charm and the strange, same with the top and the bottom quark. So there's actually a more specific logic to why we group like the quarks and the leptons together,

particularly the leptons, is that they form a weak doublet. And that means that they transform and interact in certain ways in the Lagrangian so that we have to kind of write them together. And this means that when we write the Lagrangian, we end up with even more redundant degrees of freedom. Remember, the redundant degrees of freedom in the electromagnetic case came from the fact that when we have two electron fields, we can only measure the phase difference between them,

but mathematically each has its own phase that's in the equation. So there's an extra term there. But in the case of the weak nuclear force, we actually have kind of four fields that multiply together or kind of interact with each other, right? That's the, in the lepton case, that's like your electron and your electron neutrino, and then there's two of those, there's two-week doublers that appear in the form of the Lagrangian. So instead of giving rise to one redundant degree of freedom,

now we have three redundant degrees of freedoms. Technically, what this refers to is the rotation of the phase between the spaces of the electron versus the electron neutrino. There's not just a phase shift, but now a phase rotation. because there's multiple dimensions. Don't worry too much if that's a little confusing. Just understand that because there's a weak doublet that appears in the equations,

not just one field, but a doublet, we now have more kind of mathematical interactions and that gives rise to more redundant degrees of freedom. So we now have three instead of just the one. That means there are now going to be three boson fields that are generated by our gauge symmetries when we require these transforms, that these transformations that preserves invariance of the fields, now in multiple dimensions

instead of just the one, we now get three gauge fields instead of the one gauge field. That's why there are three bosons that mediate the weak force, W plus W minus and the Z bosons. It's because of the weak doublets that the particles exist in, that there's this sort of pairing between them. And this relates to the fact that they share the same weak flavor. Each doublet has a corresponding weak flavor that they have. And that means they sort of interact together with the weak nuclear force. I'll explain that a little more later when we get to interactions. But basically, there's a pairing that happens.

It doesn't mean that you can only find the particles together. Understand this. It means that when they interact, they interact in pairs. Electrons with an electron neutrino or a muon with a muon neutrino, an up with a down, and so forth. So these pairings that happen with at least many types of weak interactions it gives rise to this property that we call weak flavor. I mentioned this before.

Electrons and electron neutrinos have the flavor of essentially like the electron type weak flavor. This is why they're paired together in these doublets that share the same flavor. They interact in those doublets. And this gives rise to the extra degrees of freedom in the Lagrangian, which gives rise to the three bosons that then couple with and interact with the corresponding Fermion field. So we have three of them instead of just the one because of the weak doublet structure. And just as the invariance of the electron phase gives rise to conservation of electric charge, so too does this invariance with respect to rotations in the sort of weak space, weak doublet space, gives rise to the conservation of weak flavor.

So that means that whenever, for example, we have an electron electron neutron neutrino, those can interact in certain ways, but we always need to preserve the total amount of weak flavor, that version of weak flavor that we have before and afterwards. So that's the corresponding conservation and also the quantum properties that relate to the force. It's this weak flavor. The group that describes the symmetries and transformations that we've just been talking about is called SU2, the two being in brackets. And this is the group that describes the set of matrices.

that all transform between each other, which have a dimension of 2 by 2 and a determinant of 1. Now, the S stands for special, and there's a mathematical distinction between SU2 and then just U2, which is important, but I'm just not going to talk about that here just to simplify things a little bit. So basically, just in the U1 case, basically that just means it's a group of one-by-one matrices, essentially, that describes transforms of a very simple type. In this case, we have a group of two by two matrices which describe this rotation in weak phase space and this invariance of all of these transformations within that space. Because again, it doesn't matter what the phase is for each of our two fields in the weak double,

like the electron electron neutrino. It's only phase differences that matter. But now we've got two fields and so there's more interactions. So we describe that with a two by two matrix instead of just like essentially a scalar or a one-by-one matrix, if you like. Because there are three generators of this group, in other words, you can describe all of the matrices in this group as a combination of three of these three basic ones, the generators. So that's why we have three gauge bosons. It's because essentially the space of redundancies of the number of internal degrees of freedom is described by three two by two matrices. And so there's going to be one for each of our gauge bosons.

So there's this nice symmetry between the mathematics and the physics. There's three generators that describe invariances of our group of the 2x2 matrices that describe the invariances of the internal degrees of freedom, of the extra phase terms that we have for our weak doublets, and that corresponds to these three gauge bosons that mediate interactions of the weak nuclear force. So that's the SU2 part of the standard model. Now let's go to the SU3 part, the most complicated part. So just as the U1 describes electromagnetism, the SU2 describes the weak, nuclear. force, the SU3 of course, describes the strong nuclear force. And so here, instead of using two by two matrices to describe the invariances of the gauge transformations in our Lagrangian, we now use three by three matrices. So there's even more redundancies. The difference is that

the redundancies here come not from the weak doublet. Instead, it comes from the color charges, as you might have imagined. So remember that there's three different color charges and also the corresponding anti-charges. Each of them carries its own field, which has its own phase term. So now we're going to have transformation terms between all three of these types of fields. And theoretically, that would give rise to actually that group, the U3 group is described by nine different generators, but we lose one because it's the special unitary group. So we get only eight instead of nine. And those eight different generators of this SU3 group that describes these transformations corresponds to our eight gluon.

I mentioned before that there's multiple different types of gluons, but I didn't exactly explain how many. That's because to understand it, you kind of need to understand the gauge theory that describes the mathematics of it, but there's actually eight different gluons. We often group them together and talk about it as the gluon, which is a bit misleading because these diagrams also show it as one particle, but there's actually eight of them. They're all pretty similar, I mean, that they have similar properties otherwise. And for example, they don't carry an electric chart, so there's not like a w plus and a w minus version. They're also all massless, so there's no difference in mass. And maybe that's why they're written differently, because W and Z bosons have mass, and their difference mass.

They also have different charges, so they're written separately. But the gluons all have the same mass, zero, and the same charge, zero. And so they're sort of written together. But in terms of understanding the symmetries of the gauge theory, you should really think of there being eight different gluons. And basically, these eight gluons correspond to eight different ways of combining or like transferring charge between, between quarks or between other gluons.

So this means that each of these eight different types of gluons is kind of comprised of different combinations of color charge. So for example, one of the eight gluons is red anti-blue plus blue anti-red. And they essentially correspond to all of the different ways you can transfer, if you think of mathematical transformations, what we're doing is rotating within this three-dimensional space of charge phases. Each of the colour charge fields has its phase associated with it, just like every other field. But only phase differences are observable, so we've got all of these extra degrees of freedom.

We need it to be invariant between any of these rotations in this space. And so it turns out that that gives rise to eight degrees of freedom that can be described with the eight different gluons that have these different combinations of these color charges within them. And just like we had the conservation laws for the weak and the electromagnetic force, likewise that the global conservation law that relates to the invariance of the phase of the charge, part of the equation, the color charge, gives rise to conservation of charge. So we have conservation of weak flavor, conservation of electromagnetic charge, and then conservation of color charge, each of the three forces.

That's gauge theory in how it relates to the standard model. Each of the three forces has its own gauge group that describes the extra degrees of freedom we have in our Lagrangian. It turns out mathematically that we can generate the form of the interaction between the thermions and the gauge fields by requiring this local phase invariance by shifting the phase of our fermions at each point in space, and that will then essentially remove the extra degrees of freedom at the expense of or remove the degrees of freedom by introducing this extra field, the gauge field, that then determines the form of our interaction potential.

So essentially, requiring gauge invariants in this way is a nice way to get the form of the interactions to be the way that it should be, or the way that it's the way that experimentally we've found that the interactions are. Again, as far as I know, there's no deep reason as to why this is the way that nature works, but it just kind of is. And so this gauge theory formalism has been very success. And that's why the standard model is often described in terms of these groups and the symmetries that they describe because it's a very convenient mathematical formalism that relates directly

to the number of gauge bosons and the conserved property that they mediate. So at this point, we understand the four different forces, well, three that we're talking about here, and we understand the difference between fermions and bosons, and we understand how the bosons mediate interactions of a particular type of force between the fermions and how gauge theory allows us to describe that in a very eloquent mathematical formalism, and that gives rise to this term of gauge invariance and gauge bosons and so forth. But, and there's a big but, it turns out that this doesn't work, or specifically it doesn't work for the weak nuclear force.

This was discovered in the 1960s. that the gauge formalism works very well for the strong nuclear force, no problems there. But when you apply it to the weak nuclear force, it turns out, and I won't try to describe the mathematics behind this, but it turns out that gauge theory can only describe massless-gauge bosons. That's fine for the electromagnetic force because the photon is massless. And it's fine for the strong nuclear force because gluons are massless, all eight of them. But for the weak nuclear force, it was known that the W and Z bosons would have to be massless,

They hadn't actually been discovered at the time, but it was known from various inferential means from other experimental results that they had substantial mass. And that's a problem because trias they might, they could not figure out any way to get this gauge formalism to work with massive gauge bosons. Because of the properties required by these gauge fields to like couple with the Fermion field in the right way and to be consistent with this invariance of the phase and so forth, they have to be massless. mass terms, if you can write the mass terms in the Lagrangian, you can add them there, which they tried to do, but it turns out that the resulting theories are unrenormalizable, which means they give implausible infinities when you try to compute experimentally observable results from them, like the cross sections or the decay rates. And so the theories don't really work.

So this gave rise to a problem. We needed to have some way of describing the mass of W and Z bosons, but without giving up gauge invariable. Because gauge invariance was so successful and it was critical to describe the interactions of all of the theories and giving up gauge that they tried giving up gauge invariance and that made them unremormalizable. So it seemed like that the right theory should be gauge invariant, should have these internal degrees of freedom canceled out by these requiring these invariance of the of the fermion fields to certain types of phase transformations. Those transformations are described by these symmetry groups, which then gives rise to a certain number of gauge bosons, the same number as the same number as. the number of generators in the group, right? So that's what I mean by this gauge invariance here. But for a while, no one could figure out a way of doing this until they did. And that's what I'm going

to now talk about. This is where we introduce the Higgs boson, or really more generally, the Higgs mechanism. The Higgs boson is sort of a part of that. So this was very important work, and it's kind of the most important work that's happened to finish out the standard model, really, and was largely developed in the 1960s and into the 1970s. This relates to the Higgs mechanism and the Higgs mechanism and associated electro-week force, the unification of the weak and electromagnetic forces. So, what's all this about?

Let's introduce the players here. I did say earlier that there was another type of boson, the Higgs boson, which is not a gauge boson. The Higgs boson is actually just a part of this picture, and so we need to take a step back and talk about the Higgs field. which has a new field that's introduced in order to resolve this conundrum of how to describe massive WNZ bosons whilst retaining gauge invariants. That's the puzzle. And the answer is you can do that by introducing the Higgs field. But it's going to take a little bit of explanation to explain what this is and how it does that.

So the Higgs field is a new field. It has a spin zero, which is interesting, that differentiates it from the other gauge bosons, which will have a spin of 1, as well as the thermonnes, which have a half integer spin. So the Higgs field has a spin of zero, and it has four components initially. So how does this Higgs field solve our problem of describing how W&Z bosons get their mass, whilst retaining the Gagian variance? Well, to understand that, let's take a few steps back and talk about what mass actually is in quantum electrodynamics. Mass is just another type of energy that goes into the Lagrangian. I said before, we have our kinetic energy terms that have derivatives, and then we have

have our interaction energy terms, which involve like multiplying our gauge fields with our fermion fields. And the form of that is given by the requirement of this gauge invariance, right? That interaction terms then directly fall out of that. So we have our kinetic terms, our interaction terms. There's separate kinetic terms for the fermions as well as the gauge bosons. And there's also mass terms. So for a particle to be massive, it needs to have the right kind of energy term in its Lagrangian. Specifically, that just consists a constant and then two fields multiplied together, like two electron fields multiplied together and then a constant in front of that. The constant is the mass. That's a very simple term, but you need that specific form of term that

exists in the Lagrangian in order for a particle to be massive. And the point was that if you just write those terms into the Lagrangian for the W&Z bosons, then they have a mass term, but it turns out the theory is unremormalizable. So that didn't work. So physicists tried to work out, how can we get these mass terms to exist for the W&Z bosons? without breaking gauge invariance. Well, it turns out that there is a way to do this, that there's sort of a sneaky way that you can get a mass term that would not necessarily, if you, you know, arrange it correctly,

would not necessarily involve breaking gauge invariance. But it requires a new field, but not just that. It requires a new field that has a very special property. And that property is called a non-zero vacuum expectation value. It's written VEV for short, seen that vacuum expectation value. What does that mean and why is it relevant? A vacuum expectation value is just the value that you expect the field to be, like, or that it is on average, in a vacuum, so where there are no particles. You may recall that Fermion fields, photon fields,

all of the fields, as far as you know, exist like everywhere in space and time, and their value fluctuates, like the fields can be, have high or low values, depending on what's going on. If you have a particle that's propagating through free space, obviously the field value will be non-zero there. The expectation will not be zero, but that's because you have a particle that's propagating through there. If you have empty space with no particles that are moving about there or like no matter that exists there, the quantum fields will still be there and they'll still have energy. They'll still be fluctuating, but the expected value will be zero.

So this is important to understand. Vacuum expectation value of zero doesn't mean that there's no field there or like that there's nothing happening. It means there is a field there. It's fluctuating. There's like virtual particles, if you want to think of it that way, or just fluctuations in the field. But the expected value of those fluctuations is zero. So they cancel each other out on average. That's the key point. Every field that we've talked about so far, apart from the Higgs field, but before the Higgs field, all of the gauge bosons and the fermions, they, all of those fields have a vacuum expectation value of zero. But it turns out that if you introduce a new field that has a

non-zero vacuum expectation value, then you can get some very interesting results in your Lagrangian. And it turns out that if you couple a field with a non-zero vacuum expectation value with, say, a massless gauge field, and you rearrange the terms in the Lagrangian, you actually get your gauge field to have mass. You get this mass term for the gauge field. So that the mass-less gauge field, when it couples with the Higgs field, turns into a massive gauge field, which is exactly what we want to happen. And it turns out. you can do that while preserving gauge invariants. And the reason that this is possible is sort of fairly simple mathematically.

It's just because the problem is normally if you interact fields or with other fields, you get interaction terms. You don't get a mass term. Why is it different here? It's because of this non-zero vacuum expectation value. The non-zero vacuum expectation value is a constant. It's not a field term. It's just like a number.

And so when we interact that number with another field, you actually get a mass term. you don't get an interaction term. So that's really cool. It's basically this trick of introducing a special type of, a field with a special type of property that allows us to get our mass term for, well, for any particle, actually, not just to gauge boson. So just to summarize that,

normally what would happen if we just introduced a new field and interacted it with the field that we wanted to become massive, it would just make new interaction terms. It wouldn't give us any mass terms. However, if that field has a non-zero vacuum expectation value, that basically introduces a constant term which then when we interact it with our other field gives a mass term for that new field. So this is sort of like a special type of interaction, which instead of giving an interaction term,

it gives you a mass term, which is very cool, not something we've seen before in the standard model. And it's dependent on this special property of a non-zero vacuum expectation value. But a non-zero vacuum expectation value is pretty weird property to have. What that means is that even in empty space where there's no particles, the field still has a positive value, a positive expected value, not just fluctuating values. Every field has that, but it has an expected value that is positive. So kind of like there's sort of extra energy there would be sort of one way of thinking about this. That's weird, but there's nothing inconsistent about that or impossible about that as far as we know.

So the hypothesis was, well, there must be this extra field out there. They call it the Higgs field after one of the scientists who worked on this. In fact, there are many, and you can see longer versions of the name that have many hyphenations. I'm just going to call it the Higgs field. But if we introduce this Higgs field, which has this non-zero vacuum expectation value, then we can get our gauge bosons to become massive while preserving gauge invariance, and that fixes all of the problems that we had with the weak nuclear force. But, of course, that comes at the expense of a new problem.

How is it that we get this non-zero vacuum expectation value? Like, where does it come from? Another way to ask this question is we're going to add in an energy term into our Lagrangian, right? It's going to be a new potential, which corresponds to this Higgs field. So we're going to have to add this in. What does the form of that potential have to look like in order to get a non-zero vacuum expectation value? Because it's going to have to be something different to normal fields. Most of the potentials and interactions that we put into our Lagrangian have a quadratic structure,

which means it's two fields multiplied together. So you're kind of a squared term, and so the potential looks like a quadratic. It's like a parabola. like it goes down and then it goes up again. It turns out that, however, we can get this non-zero vacuum expectation value by introducing a potential with a different structure. Instead of this quadratic term, which is typical, it has a quartic function. So that's to the power of four.

And if you sketch a quatic function, what it looks like, at least the sort of positive version, is that it looks a bit like a W, except the middle part is smaller. So like you have a, it starts high, goes down, has a trough, goes up again, and there's a little hill in the middle, down again, and then up again at the end. So like a W, but with a small bit in the middle. That's what the potential looks like. And that is very important, because just to explain what this potential means, the vertical axis corresponds to the potential energy of the field.

The horizontal axis corresponds to the value of the field. So normally, let's think about what happens when we have our quadratic case. Normally, the higher the value of the field, the higher the potential energy. That's what a quadratic is. it goes up either way. Negative numbers or positive numbers, the higher the value of the field, the higher the potential energy.

And that kind of is sort of more intuitive, potentially. But with our quatic function, the higher the value of the field does not necessarily mean that there's a higher potential. So let's walk this through. Suppose that we start with a zero field,

like zero, no field there, zero value of the field, it has no potential. But now let's imagine increasing the field value a little bit. Well, what happens with this special structure is that the potential actually goes down. There's a lower potential energy with a positive field.

And at some point it reaches a minimum. And then if you increase the field value even further, then the potential starts going up. So this is weird. It's a different type of potential structure that the potential actually goes down for a bit with when you increase the field before going up. Now this is important because those minima on either side of, like you think of it as a hill and there's like a valley that surrounds the hill. The minimum energy point of the field is actually in that valley.

So the minimum energy value of the field corresponds to. to a non-zero field value. What happens in nature is that physical states tend towards a lowest energy state, at least in expectation, like it'll fluctuate around that lowest energy state. For every other field, the lowest energy state is where the expected value of the field is zero. That's our parabola, right? What I'm saying is that for the Higgs field, the lowest energy state is actually not where the field is zero on average, not where the field has an expectation value of zero. It's actually where the field has a positive expected value. And that's what is the non-zero VEV, the non-zero vacuum expectation value. It's because it arises because of this weird

potential shape. So what happens is that the expected value of the field is positive because that's where it's at its lowest energy, just because of the value, the shape of the potential. This arises initially because there's something called spontaneous symmetry breaking. When the universe was very hot and very young, you know, fractions of a second after the Big Bang, the Higgs field was in, it did have a vacuum expectation value of zero, it was in that sort of central point of zero field, zero potential. But then, but what happened is very soon afterwards, the symmetry broke and the field adopted a positive value. It was sort of randomly fluctuated out. So now it has a positive value in expectation, which means that it has a lower potential than it

used to, it went from a zero potential to like a negative potential. So the potential went down by the field becoming positive. So this is only possible because of the special quartic shape. The normal quadratic shape doesn't allow for this. So most fields, this won't happen for. It's only the special properties of the potential associated with the Higgs field that allow this spontaneous symmetry breaking to happen, which gives us our non-zero vacuum expectation value, which then when we couple the Higgs field to our gauge bosons, causes those gauge bosons to acquire mass. Now, you may recall, I said earlier that the Higgs field initially, before spontaneous symmetry

breaking, had four degrees of freedom. After spontaneous symmetry breaking, three of those degrees of freedom kind of combine in, mathematically, they sort of mix up together and combine with the three gauge bosons for the weak nuclear force, so the W plus W minus and Z bosons, and give each of those particles mass. And it turns out that a vector bowons, which has zero mass has two degrees of freedom. So like photon, each of the gluons, they have two degrees of freedom. A massive one has three degrees of freedom.

So each of those weak force gauge bosons acquired an extra degree of freedom from the Higgs field, which then lost that degree of freedom after the spontaneous symmetry breaking. The way that you'll sometimes see this described is that after spontaneous symmetry breaking, when the Higgs field acquires a non-zero vacuum expectation value, this generates three different gold stone bosons, which are then eaten by the gauge bosons of the weak nuclear force, causing him to have mass. That phrase of eaten is a bit confusing. I don't really know where that comes from, but it's just effectively what's happening is

that we are introducing a new potential term into the Lagrangian, which results in rearranging and mathematical redefining of things to remove extra degrees of freedom, essentially. After that all plays out, what happens is we now have three gauge bosons that are massive instead of three massless ones, and that's where those degrees of freedoms went. But if you see this language of like eating the goldstone bosons and giving mass, that's what that's referring to. It's a confusing term. You may wonder, what about the other gauge bosons? What about our gluons and our photon? Well, the answer there is that these, the gluons do not couple with the Higgs field, and so they don't have mass. That's why they don't have mass,

and it's similar with the case of the photon. So only the weak gauge bowels. bosons get mass, the other ones don't. It also turns out that the Higgs mechanism also explains why any of the particles have mass, not just the gauge bosons, but actually the fermions too. They also couple with the Higgs field and because of the vacuum expectation value, they gain mass too. The mechanism is slightly different in terms of exactly how the mathematics works, but it comes from the same ultimate source, which is the non-zero vacuum expectation value and the

coupling with the Higgs field. So that's pretty cool. We actually, by introducing this Higgs field, have now an explanation where all of the particles gain their mass. The Higgs field itself, as I said, starts with four degrees of freedom. After spontaneous symmetry breaking, it kind of loses three of those, which are absorbed by the weak gauge bosons, and it's left with only one degree of freedom left. And this gives rise to the Higgs boson. So when you hear people talk about the Higgs boson and the Higgs boson, and the Higgs boson being discovered, which happened in 2012, that's what they're referring to.

It's this kind of leftover of the rest of the Higgs field. Now, here it's maybe you'll sort of realize why it's misleading when sometimes, and in most of the popular accounts and even some more scholarly accounts, you will say that the Higgs boson accounts for the origin of mass, or it gives rise to mass. This is really not correct. The Higgs field and the Higgs mechanism gives rise to mass. The Higgs boson is a part of that process, but it's not like a Higgs boson is interacting with the other particles to give them mass. That's not correct. The other particles have mass because of a coupling with other degrees of freedom of the Higgs field, and specifically the fact that the Higgs field has a non-zero vacuum expectation value.

That doesn't directly have anything to do with the Higgs boson. The Higgs boson is kind of a residual leftover of that Higgs field. It's still important because it needs to exist for the theory to be correct, to have the right numbers of degrees of freedom, for example. And so the existence of this Higgs boson was predicted way back in the 1960s or 1970s, and at the time we didn't have the particle accelerators needed to generate it because it's quite short-lived and needs high energies to produce one. And so it took until 2012 for that to happen.

But it was predicted many decades before and was final sort of proof that the standard model was indeed correct that this was able to be predicted. But the Higgs boson itself does not generate mass. It's part of the process by which mass is generated through the Higgs mechanism and the non-zero vacuum expectation value. That also means, by the way, that prior to the spontaneous symmetry breaking, so fractions of a second after the Big Bang,

before the Higgs field acquired non-zero vacuum expectation value, all of the particles were massless, not just the Gage bosons, but all of the fermions as well, and the Higgs field itself was also massless. So there was no mass prior to the spontaneous symmetry breaking. All right. So now we have resolved this conundrum of how we can account for massive gauge bosons,

in the case of the weak nuclear force, whilst retaining gauge symmetry, we introduced this Higgs mechanism, which has a special property of the non-zero vacuum expectation value, which when coupling with a gauge boson allows us to generate a mass term. And in turn, we can get that non-zero vacuum expectation value as a result of this spontaneous symmetry breaking when we introduced this special quatic potential of the Higgs field into our Lagrangian. And so all of the rest then follows from this sort of single, very simple alteration.

And again, so this was a hypothesis when this was introduced. It was a way to resolve the problem. It wasn't guaranteed to be true. It was a hypothesis which then led to this prediction, which was subsequently verified when we discovered the Higgs boson. So that's very nice. But there's still a problem. And those of you familiar with this may have noticed that there's something that I've skipped

over. And again, I did that deliberately because you can't introduce everything at once. It's an important bit in some ways minor detail, but it does need to be explained. And this relates to the unification of the electromagnetic and the weak nuclear force to together into the electro-weak force. When this theory of the Higgs mechanism was being developed in order to resolve this problem with the weak bosons having mass and reconciling that with the gauge invariance, the way I initially described it in terms of like you have your massless gauge bosons, then they acquire mass by interacting with the Higgs mechanism, whereas the photon

doesn't interact with the Higgs field so it doesn't get a mass. It turns out that that doesn't work. In fact, prior to the Higgs field gaining its non-zero vacuum expectation value, prior to the spontaneous symmetry breaking, we didn't even have W bosons or Z bosons or photon at all. We had something different. The terms that are used for this is W1, W2, W3, and B. The way you can think about this is like W1, W2, and W3 are kind of like our proto-W and Z bosons, whereas B is like our proto-photon. They're not the same, but they're kind of related to the fields that we now have. But I'll explain the way they're related. Basically, they're related by a transformation. So, for example, let's take our photon, our electromagnetic field. The electromagnetic field

that we currently have and that exists in the universe is actually formed by a combination of two of the fields that previously existed prior to spontaneous symmetry breaking. The B field and the W3 field. Those two combined together using a particular mathematical equation, fairly simple equation, which then gives rise to our electromagnetic field. And that's the same with the W and Z fields as well. So W plus is formed from the W1 and W2 field. W minus is also formed from the W1 and W2 field. Z is formed from W3 and B,

and then our electromagnetic field is formed from W3 and B. So they kind of recombine with each other to form four new fields from four old fields. Before spontaneous symmetry breaking, we had W1, W2, W3, and B. and all of them were massless. After spontaneous symmetry breaking, we had W plus, W minus, and Z, all of which are massive, and then the photon, which is massless.

So the reason why the photon is massless is because it doesn't couple with the Higgs field, whereas the other three did in that transformation process, and so they acquire mass. This also means that the way I've been talking about gauge symmetries also needs to be modified slightly, because there's actually sort of two versions of the gauge symmetry. There's the gauge symmetries prior to spontaneous symmetry breaking and after spontaneous symmetry breaking.

So prior to spontaneous symmetry breaking, we didn't actually have electric charge. The color charge is unaffected because that doesn't couple to the Higgs at all. It doesn't, there's no symmetry that's broken. It's just the weak and the electromagnetic forces. So prior to spontaneous symmetry breaking, there was these two other properties called weak isospin and weak hypercharge. And the sort of proto-week force, if you like, the, the, the, the, the, the, the, parts of the gauge invariance that most closely relate to the weak force today, so the W1, W2, and W3 fields. Their invariances correspond to conservation of weak iso spin, whereas those of the B field,

which is closely related to the electromagnetic field today, their conserved charge was called weak hypercharge. So weak isosperchase spin and weak hypercharge still exists, but now they've sort of been rearranged a bit. And so this sort of coupling and interactions between the weak and the electromagnetic forces is why those are called electro-week in terms of when they're unified, which happens at very high energies prior to the spontaneous symmetry breaking. They're essentially treated as a single force, which has different manifestations at lower energy levels after the spontaneous symmetry breaking. And that draws our journey through the standard model to a conclusion.

So hopefully you found this series of episodes interesting. I know it's a bit technical, but if you need to re-listen, feel free to do that. It might be handy to sort of consolidate some of the ideas. But gauge theory is very interesting, and I think it really does unify the structure of the standard model and helps us to understand how all of the different particles and forces relate to each other. In the future, I may do further discussion of additional complications that I haven't talked about, such as neutrino oscillation and further details of the strong nuclear force.

We'll leave it there for the moment. So thanks again for everyone's support with listening to the podcast. If you would like to make a financial contribution, you can become a Patreon supporter on my Patreon, or you can make a one-off donation via PayPal to my email address. You can also leave questions, suggestions, or other feedback to my email. That's Fods12 at gmail.com. F-O-D-S-1-2 at g-gmail.com. Another way you can help the show is to go to our YouTube channel, just type Science of Everything

podcast into YouTube, and you can like and comment on any of the video. you've seen or listened to there that really helps to bring the show to a new audience on that platform so thanks very much for listening i'll talk to you next time