Lesson Video: The Conservation of Lepton Number Physics

Video Transcript

In this video, we’re talking about the conservation of lepton number. We’re going to learn what lepton number is, how it relates to particles called leptons as well as their antiparticles. And we’re also going to see how this number is conserved across nuclear reactions.

Before we talk about lepton number though, let’s remind ourselves about leptons. These are elementary subatomic particles that do not engage in the strong nuclear force. If we look at an atom, we know that atoms in general are comprised of a nucleus, their core, and electrons orbiting that nucleus. Protons and neutrons make up that core. And we know that neither of these particles is elementary. And furthermore, they are held together in this tightly compacted ball by the strong nuclear force.

All this to say, anything we find in the nucleus of an atom is not a lepton. But when we look at the particles outside the nucleus, the electrons, we find that these do satisfy the conditions of a lepton. And indeed, electrons are one of the six kinds of particles that are classified as leptons. To remind ourselves just what these elementary subatomic particles that do not engage in the strong force are, let’s clear a bit of space on screen and then write out the six kinds of leptons. These are the electron, the muon, the tauon, and then the three types of neutrinos, electron neutrinos, mu neutrinos, and tau neutrinos.

When we talk about the relative electric charge of leptons, the first three of these have a relative charge of negative one, while all the neutrinos, true to their name, which sounds like neutral, have a relative charge of zero. More importantly for our lesson though, we want to focus on the fact that every subatomic particle has what’s called a lepton number. The lepton number of any lepton, any particle on this list, is one. And the lepton number of any nonlepton particle, say a proton or a neutron, is zero. So that’s clear enough. But let’s not forget that all particles, leptons included, have corresponding antiparticles. The antiparticles of the leptons are the positron, the antimuon, the antitauon, and three antineutrinos.

The relative electric charge of the first of these three antileptons is positive one. And the antineutrinos, like the neutrinos, have no charge. Even more than this, though, it’s worth noting that the lepton number of antileptons is negative one. So, in general, for any elementary particle, the lepton number of that particle is either positive one, zero, or negative one. And on our screen right now, we see all those particles that have a lepton number of plus or minus one. By the way, if we’re able to remember the list of leptons and that each one of them has a lepton number of positive one, then we can be helped in identifying the antileptons by looking for this prefix anti- in a particle’s name. If we find it there and that antiparticle corresponds to a lepton, then it must have a lepton number of negative one.

Notice that the one exception to this is the first antilepton, the positron. But otherwise, this rule of looking for that prefix anti- holds true. Now, all this discussion of lepton number comes into play when we consider nuclear equations. As an example, say that we start out with a muon, represented by this symbol. And imagine further that this muon decays into other particles, in particular, an electron, an electron antineutrino, and a mu neutrino. If we consider this reaction from the perspective of lepton number, we can identify that a muon, according to our table, has a lepton number of positive one. So does an electron, while an electron antineutrino has a lepton number of negative one. And then the last particle, the mu neutrino, has a lepton number of positive one.

If we consider the reactant in the product side of this reaction as two sides of an equation, then notice that if we add together the lepton numbers of the particles on the right, that equals one overall. And so the total lepton number on the reactant side equals the total lepton number on the product side. We can say then that in this equation, lepton number is conserved. We’re just looking here at one single nuclear equation, but it turns out that this conservation of lepton number is very general. In fact, it’s a rule that applies to any nuclear equation. It must always be the case that the total lepton number on the reactant side equals that on the product side.

This is such a strictly held rule that scientists use it to determine which nuclear equations are possible and which are not. If a potential nuclear equation would violate the conservation of lepton number, then it’s considered to be not possible. Knowing all this about lepton number and its conservation, let’s get some practice now through an example exercise.

Which of the following particles have a lepton number of one? Muon, antitauon, tau neutrino, electron antineutrino, tauon, tau antineutrino.

Alright, the first thing we can realize is that any particle that has a lepton number of one is itself a lepton. A lepton is a subatomic elementary particle that does not participate in the strong nuclear force. There are six kinds of leptons, the electron, the muon, the tauon, and three types of neutrinos. And as we said, all six of these particles have a lepton number of one. Looking at our answer options then, we see that the muon, the tau neutrino, and the tauon are all leptons and therefore have a lepton number of one, while the rest of the answer options all have a word with this prefix anti- somewhere in it.

That tells us that these are antiparticles. In particular, they’re antileptons. The lepton number of any antilepton is negative one. So we won’t include any of these in our answer. We can say then that it’s the muon, the tau neutrino, and the tauon that all have a lepton number of one.

Let’s look now at a second example exercise.

The following equation shows a muon and an antimuon being produced via pair production from a gamma-ray photon. What is the total lepton number before the interaction takes place? What is the total lepton number after the interaction takes place?

Looking at this equation, we indeed see a gamma-ray photon here on the reactant side, producing a muon and an antimuon. This is called pair production because the muon and the antimuon are antiparticles one of another. The first part of our question asks, what is the total lepton number before the interaction takes place, that is, before the muon and antimuon have been generated? This is another way of asking, what is the total lepton number of this gamma-ray photon that’s all that existed before the interaction? When we consider the lepton number of a particle or group of particles, the rule for that goes like this. If a particle is a lepton, then its lepton number is positive one. If it’s an antilepton, then its lepton number is negative one. And if the particle is neither a lepton nor an antilepton, then its lepton number is zero.

When it comes to classifying this photon, we can recall that all leptons, and there are six of them, have mass; that is, none of them are massless particles. A photon, though, is massless. And from this, we can tell that it’s not a lepton. But then it’s also not an antilepton because these antiparticles have the same mass as their corresponding lepton. This tells us that a photon is neither a lepton nor an antilepton. And therefore, it has a lepton number of zero. Since that’s the only particle involved in our interaction before the interaction takes place, we know that the total lepton number before that happens is therefore zero.

Part two of our question asks, what is the total lepton number after the interaction takes place? There are two ways we could figure this out. One is to recall that lepton number is conserved in any nuclear equations. This means the total lepton number before an interaction must equal the total lepton number after an interaction. A second way we could get at the same answer is by recognizing that a muon is a lepton and therefore has a lepton number of positive one, while an antimuon is an antilepton and therefore has a lepton number of negative one. The total lepton number after the interaction then would be positive one minus one or zero. So both before and after this interaction, the total lepton number involved is zero.

Let’s look now at one last example exercise.

Which of the following equations shows a particle interaction that would violate the conservation of lepton number? (A) A gamma-ray photon produces a muon and an antimuon. (B) A muon decays into an electron, an electron antineutrino, and a mu neutrino. (C) Carbon-14 decays into nitrogen-14 and an electron. (D) A tauon and an antitauon combine to form a gamma-ray photon. And (E) oxygen-15 decays into nitrogen-15 plus a positron plus an electron neutrino.

Considering these five interactions, we want to identify which one would violate the conservation of lepton number. Lepton number, we can recall, is a property of elementary particles. If a given particle is classified as a lepton, then that means its lepton number is positive one. The particles that make up this class are the electron, the muon, the tauon, and three different types of neutrino. Now, on the other hand, any antiparticles of leptons, that is, antileptons, have a lepton number of negative one. These particles include the positron, the antimuon, the antitauon, and then the electron antineutrino, the mu antineutrino, and the tau antineutrino.

Of course, it is possible for a particle to be neither a lepton nor an antilepton. In that case, its lepton number is simply zero. This list includes particles such as photons, neutrons, and protons. All of these have a lepton number of zero. Any particle that we could come up with then has a lepton number of plus or minus one or zero. And this fits in to what’s called the conservation of lepton number. This conservation is a physical law that says that in any nuclear reaction, the total lepton number before the interaction must equal the total lepton number after it.

So to figure out which of our five answer options shows a particle interaction that would violate this conservation, what we’ll need to do is, for each interaction, figure out the total lepton number on the left- and on the right-hand sides. Let’s start with this first interaction, a gamma-ray photon producing a muon and an antimuon. Since a photon is neither a lepton nor an antilepton, it has a lepton number of zero, while on the product side, a muon, being a lepton, has a lepton number of positive one and an anti muon has a lepton number of negative one. One minus one is zero. So we can say the total lepton number on either side of this interaction is the same, and therefore it’s conserved.

This first interaction then would not violate the conservation of lepton number. Considering the next interaction, here we have a muon, which we see is a lepton and so it has a lepton number of positive one, decaying into an electron, which also has a lepton number of positive one, then an electron antineutrino, which as an antilepton has a lepton number of negative one, plus a mu neutrino, which we can see from our list is a lepton and therefore has a lepton number of positive one. Looking at all the numbers on the product side, we have one minus one plus one. That adds up to one. And so, once again, we have a conservation of lepton number in this interaction.

Now, let’s look at option (C) with carbon-14 decaying into nitrogen-14 and an electron. This interaction involves two atomic nuclei, carbon and nitrogen, without their electrons. And so, other than this electron here, all we’re considering are the protons and neutrons that make up carbon-14 and nitrogen-14. We’ve seen that both protons and neutrons are neither leptons nor antileptons and therefore have a lepton number of zero. So that means our entire carbon nucleus has a lepton number of zero, as does our nitrogen nucleus.

This is because the particles that make up these nuclei all have lepton numbers of zero. An electron, on the other hand, is a lepton and therefore has a lepton number of positive one. And this shows us that, in this potential interaction, we do not have a conservation of lepton number. Therefore, this interaction violates that law. Let’s continue on and see if any of the remaining interactions also violate this conservation law.

In our next interaction, we have a tauon combining with an antitauon to form a gamma-ray photon. And from our table, we see that a tauon, being a lepton, has a lepton number of positive one, while an antitauon has a lepton number of negative one. And then a photon has a lepton number of zero. One minus one equals zero. So this interaction does not violate the conservation of lepton number.

Lastly, considering interaction (E), here we have oxygen-15 decaying into nitrogen-15 plus a positron plus an electron neutrino. Once again, the atomic symbols we see, oxygen and nitrogen, refer to nuclei of those atoms; no electrons included. Therefore, they’re comprised entirely of protons and neutrons. And so each have a total lepton number of zero.

Our positron though, being an antilepton, has a lepton number of negative one, while the electron neutrino, being a lepton, has a lepton number of positive one. On the product side of this interaction then, we have zero minus one plus one, which is simply zero. So this interaction does not violate the conservation of lepton number. So the only interaction that does violate this conservation is carbon-14 decaying into nitrogen-14 plus an electron. And the fact that this interaction would violate the conservation of lepton number means that it’s not a possible interaction. That is, carbon-14 can’t decay into just a nitrogen-14 nucleus and an electron. As we’ve seen, that’s because it violates lepton number conservation.

Let’s summarize now what we’ve learned about the conservation of lepton number. In this lesson, we saw that lepton number is a property of all elementary particles. For all leptons, symbolized here, the lepton number is positive one. For all antileptons, the antiparticles of leptons, that lepton number is negative one. And for any particle that’s neither a lepton nor an antilepton, its lepton number is zero. And lastly, we saw that in any possible nuclear interaction, lepton number is conserved. This means that the total lepton number before the interaction must equal the total after the interaction. If this is not the case, then that given interaction is considered not possible. This is a summary of the conservation of lepton number.