Patterns

While it’s nice to know about electrons, up quarks, and down quarks, there are a lot of weird numbers to remember about them. Imagine how scientists felt, then, when they found out that there are more than a hundred more particles to learn the numbers for! It’s too complicated to memorize all that stuff, and it doesn’t explain anything. What we need are easy-to-remember patterns. That’s what The Standard Model gives us.

Elementary Fermions

We’ll start with the patterns for elementary fermions. Elementary fermions are the particles that are good for making stuff. We’ll worry about the other particles, the elementary bosons, later.

The standard model says that we can describe fermions using a mathematical idea called U(1)×SU(2)×SU(3). The “U(1)” part tells us that every fermion has a hypercharge, the “SU(2)” part tells us that every fermion has an isospin, and the “SU(3)” part tells us that every fermion has a color. So now you know why we were talking about hypercharge, isospin, and color. But what’s more interesting is that U(1)×SU(2)×SU(3) is a subgroup of SU(5),^[1] which is a fancy math way of saying that you can pick a fermion and know nearly everything about it by answering five yes/no questions. Here are the five questions:

• It the fermion isospin up? (Worth +1 hypercharge and +1/2 isospin, which means +1 charge.)
• It the fermion isospin down? (Worth +1 hypercharge and -1/2 isospin, which means no change in charge.)
• It the fermion red? (Worth -2/3 hypercharge, which means -1/3 charge.)
• It the fermion green? (Worth -2/3 hypercharge, which means -1/3 charge.)
• It the fermion blue? (Worth -2/3 hypercharge, which means -1/3 charge.)

This is kind of like making a character in a video game; you answer the questions and see what it does to the particle’s stats. No matter how you answer, you will always get the stats for a real fermion.

For example, let’s say I design a particle that is isospin up but not down and has only the color red. My answers would be “yes”, “no”, “yes”, “no”, and “no”, or YNYNN when abbreviated. Then the particle’s hypercharge will be 1 - 2/3 = +1/3, its isospin will be +1/2, and its charge will be 1 - 1/3 = 2/3. That sounds exactly like a left-handed red up quark.

In fact, the answers to the five questions not only tell you stats, but also tell you other things:

• A fermion is right-handed if the number of “yes” answers is odd. It is left-handed if the number of “yes” answers is even.
• A fermion is matter if the number of “yes” answers to color questions is odd. It is antimatter if the number of “yes” answers to color questions is even.
• A fermion is a lepton if the color questions all have the same answer. It is a quark if the color questions have different answers.
• A lepton is a charged lepton if the “isospin up” question has a different answer than the color questions. It is a neutrino if the “isospin up” question and the color questions all have the same answers.
• A lepton is white if it is matter and black if it is antimatter.
• A quark is red, green, or blue if that is the only color with a “yes” answer. A quark is antired, antigreen, or antiblue if the opposite is the only color with a “no” answer.

The one thing these questions can’t easily tell you is how heavy the particle is. But you can sort-of tell by answering a sixth question:

• Is the fermion first-generation (light and unlikely to fall apart), second generation (heavy and quick to fall apart), or third generation (extremely heavy and extremely quick to fall apart)?

Again, you can make any choice you want here and still be talking about a real particle.

Elementary Bosons (Force Carriers)

Now imagine an experiment: We smash together a left-handed red down quark (NYYNN, isospin=-1/2) and a left-handed electron neutrino (YNYYY, isospin=+1/2). Because they are both have nonzero isospin, the weak force can affect them. One way that could happen is that they switch isospin answers (the first two yes/no answers) with each other. That means that the quark changes to YNYNN, a left-handed red up quark, and the neutrino changes to NYYYY, a left-handed electron. But other outcomes never happen. Why?

One important rule is that when two particles affect each other using some force, the changes to one particle have to be balanced out by changes to the other. If one particle loses something, the other has to gain it. In The Standard Model, a something that can be lost or gained because of a force is called a force carrier or an elementary boson.

In the example, the quark lost its isospin downness (worth +1 hypercharge and -1/2 isospin) and gained isospin upness (worth +1 hypercharge and +1/2 isospin), so the neutrino had to gain isospin downness and lose isospin upness. Or, said another way, the quark gave away 1 isospin downness and -1 isospin upness to the neutrino through the weak force. That transfer, 1 isospin downness and -1 isospin upness together, is called the W⁻ boson, which is a force carrier for the weak force. And we can compute its stats by knowing what its changes are worth: (+1 hypercharge and -1/2 isospin) - (+1 hypercharge and +1/2 isospin) = (0 hypercharge and -1 isospin).

The give and take doesn’t have to be giving and taking question answers though. For instance, the electromagnetic force’s boson is a photon. When particles exchange photons, they transfer momentum and energy, not anything to do with their types.

In some cases, force carriers can also exist on their own, without obviously being part of a trade. For example, sometimes an electron in an atom loses momentum and energy by shooting a photon off into empty space. We call photons like that light.

Inside The Standard Model, these are the fundamental force carriers:

• The B boson is the force carrier for hypercharge.
• The W⁻, W⁰, and W⁺ bosons are the force carriers for isospin.
• Eight gluons are force carriers for color. (There are different sets of eight gluons that you can pick to describe the effects of the strong force. Scientists don’t agree on which set of eight is the best one.)
• The Higgs boson is unlike all the other force carriers. It follows unusual rules, and it makes other particles follow unusual rules. The only way to overcome those effects is with an insane amount of energy.

However, you won’t hear many people talking about the B boson or the W⁰ boson. The reason is that the Higgs boson messes up how they work, so we don’t see them on their own. Instead, we usually see them in special combinations:

• The Z boson is the combination of a W⁰ boson minus half of a B boson, and
• The photon is the combination of a W⁰ boson plus half of a B boson.

So the weak force’s force carriers are usually listed as the W⁻, Z, and W⁺ bosons, while the electromagnetic force’s force carrier is the photon. (Remember how the B boson goes with hypercharge, and the W⁰ boson goes with isospin? Since the photon is half of a B boson plus a W⁰ boson means, that’s where we get the Gell-Mann–Nishijima formula from, the rule that electric charge is half a particle’s hypercharge plus its isospin.)

Everyday Matter, Hypercharge, Isospin, and the Three Forces

Just like the ingredients in a bakery have numbers on their nutrition labels to tell us how they will affect our health, in The Standard Model, elementary particles have quantum numbers that tell us how different forces affect them. For each particle, we will talk about two quantum numbers, hypercharge and isospin.^[1] Think of hypercharge and isospin as two different nutrients that can be found in the ingredients.

Also, just like different kinds of food are in different food groups, we can also classify elementary particles into groups using colors. These aren’t really how the particles’ look—when we say that a particle is “red”, what we mean is that it’s in the “red” group. There are three color groups, “red”, “green”, and “blue”. If we say that a particle is “black”, that means that it isn’t in any groups, and if we say that it is “white”, that means that it is in all of the groups. If we say that a particle is something like “antired”, that means it’s in all of the groups except red—in other words, the green and blue groups.

Now, according to The Standard Model, most of the stuff on earth is made from particles called electrons, up quarks, and down quarks. Before we get into patterns, let’s talk about these particles’ quantum numbers and colors (their “nutrition facts” and their “food groups”), starting with electrons. That will also give us a chance to talk about the three forces in The Standard Model.

Electrons

In The Standard Model, there are two kinds of electrons: right-handed electrons and left-handed electrons. You can tell them apart because, when a right-handed electron moves, it gives the illusion that it is spinning to the right like a corkscrew or drill bit, even though it is not actually spinning, whereas when a left-handed electron moves, it seems to be spinning to the left. A right-handed electron has a hypercharge of -2 but zero isospin. A left-handed electron has hypercharge of -1, and its isospin is -1/2. Both electrons are white, which is to say that they have all of the colors, that they’re in all of the food groups and make a balanced meal all by themselves.

What do those numbers mean? Well, the most important thing to know for an electron is the Gell-Mann–Nishijima formula, that you can add half a particle’s hypercharge to its isospin to get another quantum number called electric charge. Electric charge that tells us how the particle is affected by the electromagnetic force. For a left-handed electron, half of -2 plus 0 is -1, and for a right-handed electron, half of -1 plus -1/2 is also -1. So both kinds of electrons have the same negative electric charge. The electromagnetic force pushes negative charges away from negative charges and positive charges away from positive charges, but pulls negative and positive charges together. That means that electrons run away from each other and try to go towards positively charged particles. Because of something called “relativity”, the electromagnetic force also means that moving electric charges create magnetic fields and moving magnetic fields create electric forces. We use those effects to move electrons around—we call that electricity—and to change electricity back to motion with devices like motors.

The electrons’ isospin tells us another thing, this time about the weak force, a force that can change particles from one type to another. Since the right-handed electron has zero isospin, it isn’t affected by the weak force, so we can’t use it to make new elementary particles. But the left-handed electron does have some isospin, so if we smash it with a careful choice of other ingredients, we can make new elementary particles.

There’s also a third force in The Standard Model, the strong force, but it doesn’t affect black or white particles, so we have to wait to talk about it.

Up Quarks

How about up quarks? They too come in right- and left-handed forms. The hypercharge of a right-handed up quark is +4/3, but just like a right-handed electron, it has no isospin. A left-handed up quark has a hypercharge of +1/3 and an isospin of +1/2. Like we did before, we can compute the electric charge by adding half the hypercharge to the isospin. Half of +4/3 plus 0 is +2/3 for the right-handed up quark, and half of +1/3 plus +1/2 is also +2/3. So up quarks are positively charged and will be attracted to and attract electrons. And since only the left-handed up quark has isospin, only the left-handed up quark can be used to make new elementary particles with the weak force.

Up quarks are not color-balanced, so now we get to talk about the strong force. There are actually six kinds of up quarks: red, green, and blue right-handed up quarks and red, green, and blue left-handed up quarks. The Standard Model says that the strong force causes color confinement, which means that it pulls particles of different colors together to make a balanced meal so strongly that it’s impossible to have a combination that isn’t balanced. That means you can’t have an up quark all by itself—it would be like trying to make a healthy meal out of only desserts. But you could put a red up quark with a green quark and a blue quark since then you’d have all three colors. It doesn’t matter that the electromagnetic force is trying to push them apart when the strong force pulling them together is so much stronger. A balanced diet is important!

Down Quarks

What about down quark? Just like up quarks, they can be right- or left-handed, and they come in three colors: red, green, and blue, so there are six kinds of down quark in total. The right-handed down quarks’ hypercharge is -2/3, but they have no isospin. The left-handed down quarks have a hypercharge of +1/3, the same as the left-handed up quarks, but their isospin is -1/2, the opposite. If we do the math, down quarks’ electric charge is always -1/3, so they are negatively charged, like electrons. Only the left-handed down quarks are affected by the weak force because the right-handed down quarks have no isospin. (Notice a pattern?) The color confinement rules apply to down quarks the same as they do to up quarks.

Nucleons and Atoms

Now that we know those particles, we can make some recipes to put them together. The basic recipes for earthly matter are the recipes for protons, neutrons, and atoms.

Our first recipe will make a proton. Protons are very important for chemistry because you can control the kind of chemical you have by changing the number of protons in each bundle of quarks. The recipe for a proton is two up quarks and one down quark, all of different colors. (Actually, something interesting happens when you put these ingredients together. The “up” and “down” gets mixed all around so that it’s like each individual quark is actually two parts “up” and one part “down”.) A proton’s electric charge is 2/3 + 2/3 - 1/3 = +1, which means that it is the electromagnetic opposite of an electron.

Our second recipe will make a neutron. Having the right number of neutrons in the right places helps keep big balanced meals of quarks from splitting into smaller balanced meals, and without them, most chemicals would just fall apart. On the other hand, by purposely using the wrong number of neutrons, we can make a pile of quarks break into pieces and maybe get some energy out of the split. That’s how nuclear reactors work. The recipe for a neutron is one up quark and two down quarks, all different colors. A neutron’s electric charge is 2/3 - 1/3 - 1/3 = 0, which means that they aren’t easily affected by the electromagnetic force. (But the quarks themselves are! That’s how scientists first figured out that neutrons are made up of smaller ingredients.)

Since quarks have different colors, and different colors like to be near each other, if we can get them close enough for the strong force to start pulling, we can stick a bunch of protons and neutrons together to make a jumble of quarks called an atomic nucleus. Then, if we surround an atomic nucleus with electrons, we get an atom. Almost everything you see around is you is made from atoms.

I’ll lead with an apology: I got ambitious. You said you wanted to know about particles; I wanted to say a lot about them, so I guess I wrote a book.

Also a disclaimer: I only had physics through undergrad. My source is Sections 1 through 3.1 in Baez, J., & Huerta, J. (2010). The Algebra of Grand Unified Theories. Bulletin of the American Mathematical Society, 47(3), 483–552. if you want to consult something more reliable.

Introduction

If you look inside a bakery, you will find lots of different ingredients—flour, sugar, butter, milk, eggs, and so on. By combining these ingredients in different ways, the baker can make all kinds of different breads and desserts. To make the kind of food they want, the baker has to understand their ingredients, how much of each kind to use, and how to combine them.

In the same way, we know that everything in the world can be made out of different kinds of very tiny ingredients called elementary particles. By combining these particles in different ways, we can create all kinds of stuff. But to make the stuff we want, we have to understand the elementary particles and how they affect each other, just like the baker has to understand their ingredients and how they go together.

There are over one hundred different kinds of elementary particles that we know about and maybe more that we don’t. We could just memorize them all, but scientists think it’s better to look for patterns. Think about when you were learning addition: its easier to learn that zero plus anything is always the same thing than it is to memorize 0 + 0 = 0, 0 + 1 = 1, 0 + 2 = 2, 0 + 3 = 3, and so on. Scientists do the same thing; they find patterns. Eventually they wrote down all of the elementary particles that they had discovered and all of the patterns they were really sure about. They called this knowledge “The Standard Model”.

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