Question

In which of the following decays are the three lepton numbers conserved? In each case, explain your reasoning. (a) \(\mu^-\rightarrow e^- + \nu_e + \overline{\nu}_\mu\); (b) \(\tau^-\rightarrow e^- + \overline{\nu}_e + \overline {\nu} _\tau\); (c) \(\pi^+ \rightarrow e^+ + \gamma\); (d) \(n \rightarrow p + e^- + \overline{\nu}_e\).

Short answer

Decays (a), (b), and (d) conserve lepton numbers. Decay (c) does not.

Step-by-step solution

Understand Lepton Number Conservation

Lepton number conservation states that in any nuclear or particle reaction, the sum of the lepton numbers before the reaction must equal the sum after the reaction. There are three types of lepton numbers: electron lepton number \(L_e\), muon lepton number \(L_\mu\), and tau lepton number \(L_\tau\). These numbers must be independently conserved in a reaction.

Evaluate Decay (a) \(\mu^-\rightarrow e^- + \nu_e + \overline{\nu}_\mu\)

Assign lepton numbers to each particle: \(\mu^-, L_e = 0, L_\mu = 1, L_\tau = 0\); \(e^-, L_e = 1, L_\mu = 0, L_\tau = 0\); \(u_e, L_e = 1, L_\mu = 0, L_\tau = 0\); \(\overline{u}_\mu, L_e = 0, L_\mu = -1, L_\tau = 0\). Before decay: \(L_e = 0, L_\mu = 1, L_\tau = 0\). After decay: \(L_e = 1 + 1, L_\mu = 0 - 1, L_\tau = 0\). All lepton numbers are conserved.

Evaluate Decay (b) \(\tau^-\rightarrow e^- + \overline{\nu}_e + \overline{\nu}_\tau\)

Assign lepton numbers to each particle: \(\tau^-, L_e = 0, L_\mu = 0, L_\tau = 1\); \(e^-, L_e = 1, L_\mu = 0, L_\tau = 0\); \(\overline{u}_e, L_e = -1, L_\mu = 0, L_\tau = 0\); \(\overline{u}_\tau, L_e = 0, L_\mu = 0, L_\tau = -1\). Before decay: \(L_e = 0, L_\mu = 0, L_\tau = 1\). After decay: \(L_e = 1 - 1, L_\mu = 0, L_\tau = 0 - 1\). All lepton numbers are conserved.

Evaluate Decay (c) \(\pi^+ \rightarrow e^+ + \gamma\)

Assign lepton numbers to each particle: \(\pi^+, L_e = 0, L_\mu = 0, L_\tau = 0\); \(e^+, L_e = -1, L_\mu = 0, L_\tau = 0\); \(\gamma, L_e = 0, L_\mu = 0, L_\tau = 0\). Before decay: \(L_e = 0, L_\mu = 0, L_\tau = 0\). After decay: \(L_e = -1, L_\mu = 0, L_\tau = 0\). Lepton numbers are not conserved.

Evaluate Decay (d) \(n \rightarrow p + e^- + \overline{\nu}_e\)

Assign lepton numbers to each particle: \(n, L_e = 0, L_\mu = 0, L_\tau = 0\); \(p, L_e = 0, L_\mu = 0, L_\tau = 0\); \(e^-, L_e = 1, L_\mu = 0, L_\tau = 0\); \(\overline{u}_e, L_e = -1, L_\mu = 0, L_\tau = 0\). Before decay: \(L_e = 0, L_\mu = 0, L_\tau = 0\). After decay: \(L_e = 1 - 1, L_\mu = 0, L_\tau = 0\). All lepton numbers are conserved.

Key concepts

Muon Decay

The muon, denoted as \(\mu^-\), is a type of lepton, which is similar to an electron but much heavier. In the decay process, a muon can transform into other particles.
In the decay \(\mu^-\rightarrow e^- + u_e + \overline{u}_\mu\), the muon decays into:

• an electron \((e^-)\),
• an electron neutrino \((u_e)\), and
• a muon antineutrino \((\overline{u}_\mu)\).

Lepton number conservation must be obeyed, meaning the sum of lepton numbers must be the same before and after the decay. For this reaction:

• Electron lepton number \( (L_e) \) starts at 0 and ends at 1 (electron) + 1 (electron neutrino) = 2.
• Muon lepton number \( (L_\mu) \) starts at 1 (muon) and ends at 0 - 1 (muon antineutrino) = 0.
• Tau lepton number \( (L_\tau) \) remains unchanged at 0 throughout the process.

In this decay, all lepton numbers are conserved, confirming it aligns with lepton number conservation laws.

Tau Decay

A tau lepton, symbolized by \(\tau^-\), is another heavier cousin of the electron. It's part of the lepton family and can undergo a decay process.

In the decay \(\tau^-\rightarrow e^- + \overline{u}_e + \overline{u}_\tau\), the tau transforms into:

• an electron \((e^-)\),
• an electron antineutrino \((\overline{u}_e)\), and
• a tau antineutrino \((\overline{u}_\tau)\).

Lepton number conservation applies again. Let's consider:

• Electron lepton number \( (L_e) \) begins at 0 and changes to 1 (electron) - 1 (electron antineutrino) = 0.
• Muon lepton number \( (L_\mu) \) starts and remains at 0 both before and after decay.
• Tau lepton number \( (L_\tau) \) begins at 1 (tau) and results in 0 - 1 (tau antineutrino) = 0.

Thus, all lepton numbers are conserved in this tau decay, adhering to the rules of lepton number conservation.

Leptons in Particle Physics

Leptons are fundamental particles in physics. They include electrons, muons, and tau particles, each having a corresponding neutrino.

There are three types of lepton numbers in particle physics:

• Electron lepton number \((L_e)\)
• Muon lepton number \((L_\mu)\)
• Tau lepton number \((L_\tau)\)

Each lepton family has a specific lepton number, and these numbers must be conserved in all nuclear and particle reactions.
Conservation of lepton number ensures that for each type of lepton participating in a reaction:

• The number of these leptons minus the number of their corresponding antineutrinos remains constant.
• This conservation maintains the balance and predictability in particle interactions.
• It plays a crucial role in understanding processes such as particle decay and interactions.

Understanding this principle is essential in predicting and analyzing outcomes in particle physics, ensuring that reactions follow the physical laws of the universe.