Video: Lepton Number Conservation in Pair Production

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

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Video Transcript

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.

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