If you were to ask most physicists about the success of the standard model of particle physics, you would get a very strange answer: that it is frustratingly successful. Odd indeed, that’s like calling a painting frustratingly beautiful. Further prodding would reveal that in fact most physicists not only find the model frustrating, but also think of it as ugly! I’m going to do my best to explain what physicists mean by this, not to convince you one way or the other that the model is ugly or beautiful, but to help you as a reader become a well informed art critic of the standard model.

First and foremost, the standard model is not a theory of particles, it is a theory of fields. Instead of there being an electron, or several in the universe, there is simply one electron field. This field permeates all space, and when the field gets excited, an electron appears, similar to how an excitation in water creates a wave. If, in this analogy, we were to define sea level as zero height, waves in the water could have positive values at their crests, and negative values at their troughs. The electron field is expected to be “zero height” far away from any other matter, and “splashing” in the electron field amounts to having very energetic electric particles move around in space (other electrons, photons, positrons etc.). The value of the electron field, its “height” at any point in space, gives us information about the probability of there being an electron located at that point in the field.

To match what we’ve seen in laboratories, the standard model needs to declare the existence of at least 24 fields, all of which are very much like the electron field: 18 quark fields, the building blocks of the atomic nucleus, and 6 lepton fields (these are the electron, its heavier cousins, and the electrically neutral neutrinos). All of these fields are fermion fields, which is a fancy way of saying excitations (particles) in these fields cannot overlap each other (the fields themselves have to overlap of course, they exist throughout all of spacetime); this is opposed to something like photons, particles of light, which can overlap with each other. Speaking of, photons are pretty pivotal to how we understand the universe, why aren’t they in our model yet? This is inarguably the most beautiful part of the standard model: how the so called “gauge bosons” (particles which mediate forces) enter our theory.

So far we have all of our fermions (if this word scares you, just think “electrons”) and they’re not doing much. They can have masses and momentum, but they can’t interact! Experiments show us that these fields do interact, so a good theory should account for this. The way this is done, is by enforcing certain symmetries on our theory of physics. Some symmetries are natural to us, like translation symmetry, the idea that if everything in the universe happened to be 5 meters to the left of where it is now, the universe would appear identical. If your theory of physics describes the same universe after rotating the whole universe, your theory is symmetric under rotation. If your theory of physics looks the same in both the future and the past, it’s symmetric through time. One more example of this is the theory of classical electromagnetism: if we swapped every positive charge for a negative charge, and every negative charge for a positive charge, the theory would describe an indistinguishable universe from our own. The theory of classical electromagnetism is symmetric under swapping electric charges, or “charge conjugation”. The take away from this is that if the universe your theory of physics describes looks the same even after you make some change to it, then that change is known as a symmetry of your theory.

The symmetries outlined above are known as global symmetries, they are changes to everything everywhere in the universe in unison. We swap all the charges, or we move everything over by a kilometer, etc. Much less common are local symmetries, where we preform the same type of action to the whole universe, but not exactly the same action everywhere. An example of this would be going through the universe and moving everything over, but by varying amounts and/or in different directions depending on where you are. If you had the power to do this, you would inevitably throw planets out of orbit, or tear objects apart, and so this local spacial translation is not a symmetry of our universe, it won’t leave everything looking the same afterwards. (Interestingly, the universe our model describes thus far, with no interactions and only fermions, does exhibit this local translation symmetry, can you think of why?)

Suppose you dear reader are reckless and wish to do this: shift everything in the universe over, but differently in different places; how would you keep track of where and how you move what? One simple way would be to have a mapping that takes in a coordinate of where something is, and spits out another coordinate of where you ought to put it. This is exactly what a field is! It is something that, for each and every point of spacetime, spits out a value. For the electron field, if we input a coordinate, it tells us information about the probability of finding an electron at that coordinate. If you choose to implement local spacial translation (i.e. move all the stuff in the universe differently), and you wanted to know how to move Jupiter; you would enter the coordinates of Jupiter into your local translation mapping, or “field”, and out would pop the coordinate to place it at, and so on until you have moved everything in the universe around, based on this handy mapping. The take away here is that while acting on a system via some global symmetry can be characterized by only a few numbers, specifying the effect of some local symmetry requires you to have a value for every single point in spacetime. If we want to keep track of local symmetries, we require a field (a value for every point in spacetime).

Remarkably the act of gaining the interacting bits of our theory (photons, gluons, W and Z, the particles which mediate interactions) is nothing more than changing our theory of physics until it has some local symmetry we want. So far we have our fermions (electrons, quarks), and we have a basic theory of what they do when left alone. With perfect hindsight, we can then demand our theory to be locally U(1) symmetric. What this means is that we want a theory, where nothing in the universe will look any different if we multiply our electron (and quark, and muon…) field by any number that looks like e^iθ (any complex number which has size of 1), and the number we choose to do this can be different at different places in spacetime. Remember, the electron field tells us about the probability of finding electrons, so changing the value of these fields by multiplying them everywhere, will change these probabilities: this means we should expect our universe to look different after taking this action, and thus we will have to adjust the theory until this multiplication has no effect (if we want this strange U(1) symmetry). Much like with local translation, the best way to keep track of what we are multiplying the electron field by at every point in spacetime, is with another field, i.e. in order to track the action that we preform on the electron field, we need to bring in another field (which you may have heard called the gauge field). If you’re feeling a little lost at this point, this symmetry that we’re demanding is strange and abstract, not something that is easy to understand intuitively; the overarching ideas are more important than the details here, so don’t abandon hope yet.

Bringing this all back together, the theory we started with, of all the non-interacting fermions is not invariant under this action, i.e. our naive theory of physics is not U(1) symmetric: if we go through and multiply the different fermion fields by different numbers everywhere, we don’t get back the same universe. In order to make our theory of physics invariant under this weird and abstract local symmetry, we actually need to put this so called gauge field into our equations! This is where the magic happens: not only do we need a (gauge) field to keep track of how we act on the fermion fields, when we demand our theory to look the same no matter how we choose to do this strange act of multiplying differently everywhere, the gauge field enters our equations just as physically as the electron field would. This implies the gauge field is as physical as the fermion fields, in a universe with the symmetry. In fact, for this strange U(1) symmetry, the field we get ends up acting identically to how the vector potential does in electromagnetism, and can be interpreted as the photon field. Simply by having charged fermions, and demanding U(1) symmetry, the entire theory of quantum electromagnetism falls out of our equation. While to some extent mysterious, this is undeniably beautiful. As a professor once told me “Any student who evades being impressed by the mysterious appearance of the gauge bosons born of the ether of mathematical abstraction simply does not understand the theory.”

Roughly speaking, we do this again with fields that have certain properties (demand the universe be identical under local SU(2) actions on weak-isospin, and local SU(3) actions of quark color charge), and we get the equations of the weak and strong nuclear forces! This is what people mean when they say the standard model is a U(1) ✕ SU(2) ✕ SU(3) theory. Really take a moment to let this settle in your brain. By stating some basic properties of our fermions, and then demanding symmetries, we get all of physics (aside from gravity). Abstract and esoteric? Certainly; but ugly? Given the relative simplicity in order to obtain the correct theory of physics for our universe how could anyone call it ugly? Oh, of course: the Higgs Field.

As it turns out, there is one major problem with our nearly perfect theory. In order for the weak force (the SU(2) piece) to work, which accounts for how particles decay: our particles can not have mass. The reasoning behind this is beyond the scope of this article, but if you start with your basic (non-interacting) theory and try to make your universe look the same under “local SU(2) actions”, you have to throw away any parts of your theory corresponding to the mass of particles! This means everything in the universe which our theory describes, flies around at the speed of light! Our theory is so close to perfection. It gets us all the forces of nature for free, as a consequence of symmetry, but at the cost of everything being massless. In order to get mass back, physicists added one more field, unlike they rest.

The Higgs field just as all the others, permeates all of space and takes a value at every point in space. However this is not a fermion field, or a gauge field (it is not a new force). The value of this field does not drop off like one would expect the electromagnetic field, or electron field to. Instead this field has an expected value that is not zero, even very far away from any matter. This is one reason people find the Higgs field unnatural, it doesn’t act like any of the other fields in nature; and if any of them behaved like the Higgs, the Universe would look very different than it does now. In order for the Higgs to deliver mass-like-terms back into our theory, it also needs to be an infinite source of what is known as “weak hyper-charge”. It is a property of particles which determines how they interact with the weak force, much like how electric charge determines how things interact with the electromagnetic force. The Higgs field, because it has a non-zero value nearly everywhere, can then constantly give particles weak hyper-charge, then take it away, and then do it again. This causes the particle to oscillate between states of having this weak-charge, and not having it. The frequency with which this oscillation happens will pop up in the equations of physics, to function as a mass (This can be understood through a combination of Mass-Energy equivalence, and De Broglie wave relations)!

Whew! We got our masses back. Unfortunately this time the things we want don’t simply fall out of the theory, the frequency with which the Higgs causes a particle to oscillate between these “weak” and “weakless” states (i.e. how strongly the Higgs couples to each field), has to be put in by hand for each field. The final unfortunate piece of the mechanism behind the Higgs giving mass to particles, is that in order to get something as light as even the mass of the electron, the Higgs field needs to make an electron oscillate between the two states at a rate of 300,000,000,000,000,000,000, (or 300 quintillion) times a second. This is so enormously fast that there is essentially no hope that any experiment will ever be able to verify this is happening. Fortunately the Higgs does predict its own mass, and a particle which matches the description of the Higgs was detected at the Large Hadron Collider in 2012 (and other times subsequently), however many people are happy to ignore this and theorize it is not the Higgs, or that it is some “prettier” variant of the Higgs.

All in all the standard model of particle physics starts off arguably beautiful. We have some non-interacting fields, and we demand that our theory should have some local symmetries. By enforcing these symmetries we get a universe much like our own, with all the same fundamental interactions, but it is a universe where nothing has mass. To solve this we add one final field which has a non-zero expected value everywhere, so that it may act as an infinite source and sink of this weak hyper-charge, in order to oscillate each field at some enormous frequency, so that these massless versions of all our particles appear massive.

The Standard Model is frustratingly successful because theorists in physics desire a more beautiful theory. One that has the stunning simplicity of gauge theory, but maybe predicts the masses of particles, or explains the patterns we see in the groups and families of the current set of particles. And so long as the standard model continues being experimentally verified, we have nothing concrete to base a next new theory on. It is up to the reader now to decide whether the standard model is an ugly, massive failure (pun intended), and the Higgs field is one complicated band aid, or if physicists are ruffling their own feathers for nothing and should simply be amazed that they can understand any part of the Universe at all.

(If you wan’t to think more about this, I have another post which discusses very similar ideas)

Perhaps this goes without saying, but there is an incredible amount of nuance being over-looked here, and this post should be read as an introductory piece to get the reader familiar with the ideas of fields, local symmetry, gauge theory and the Higgs. Below I link a few blogs which are a bit more technically involved for those interested, but in truth no amount of blog posts will ever be a substitute for digging into the math in order to understand what is really going on. This is not a substitute for a physics text book, but it does allow the avid science enthusiast, or novice scientist, to see what lies ahead in their studies, without being bogged down by jargon and mathematical detail.

https://profmattstrassler.com

https://www.quantumdiaries.org/2011/06/19/helicity-chirality-mass-and-the-higgs/

https://coherence.wordpress.com/2012/07/08/the-higgs-boson-simply-explained/

Image Source: https://physics.info/standard/