Question

How do we know for certain that the scattering process \(e^{+}+\nu_{\mu} \rightarrow e^{+}+\nu_{\mu}\) proceeds through an intermediate \(Z\) boson and cannot proceed through an intermediate charged \(W\) boson, while both options are possible for \(e^{+}+\nu_{e} \rightarrow e^{+}+\nu_{e} ?\)

Short answer

Answer: The scattering process \(e^{+}+\nu_{\mu} \rightarrow e^{+}+\nu_{\mu}\) cannot proceed through an intermediate charged \(W\) boson due to violation of electric charge conservation and lepton flavor conservation. However, it can proceed through an intermediate neutral \(Z\) boson, which conserves both electric charge and lepton flavor.

Step-by-step solution

Analyze the given scattering processes and identify the particles involved

In both processes, we have a lepton (positron) and a neutrino involved. In the first process, we have a positron (\(e^{+}\)) and a muon neutrino (\(\nu_{\mu}\)) as the initial particles and the same particles in the final state. In the second process, we have a positron (\(e^{+}\)) and an electron neutrino (\(\nu_{e}\)) as initial particles, and the same particles in the final state.

Apply the conservation laws - Electric charge conservation

Let's analyze the initial and final states of the processes. For the first process: Initial state: \(e^{+}+\nu_{\mu}\). A positron has charge +1, while neutrinos are neutral. Therefore, the initial state has an electric charge of +1. Final state: Again, \(e^{+}+\nu_{\mu}\). The final state also has a charge of +1. For the process to be allowed, the intermediate particle must conserve the electric charge. \(W\) bosons are charged particles: \(W^+\)(charge +1) and \(W^-\)(charge -1). For the process \(e^{+}+\nu_{\mu} \rightarrow e^{+}+\nu_{\mu}\), the intermediate charged \(W\) would lead to an inconsistency in charge conservation. However, the \(Z\) boson is neutral, and the charge is conserved.

Apply the conservation laws - Lepton flavor conservation

Lepton flavor conservation forbids transitions between different flavors of leptons. In the case of the process \(e^{+}+\nu_{\mu} \rightarrow e^{+}+\nu_{\mu}\), the initial state involves an electron antilepton (positron) and a muon neutrino. An intermediate charged \(W\) boson would change the lepton flavors, which would violate lepton flavor conservation. However, an intermediate \(Z\) boson does not change lepton flavors and thus conserves lepton flavor.

Discuss the possibility of \(W\) boson and \(Z\) boson for the process \(e^{+}+\nu_{e} \rightarrow e^{+}+\nu_{e}\)

In this case, the initial state is \(e^{+}+\nu_{e}\), and the final state is also \(e^{+}+\nu_{e}\). Both \(Z\) boson (neutral) and \(W\) boson (charged) can mediate this process without violating electric charge conservation or lepton flavor conservation. The process can proceed through a \(Z\) boson via neutral current interaction, or through a \(W\) boson via charged current interaction, where the initial state and the final state have the same lepton flavor, and the electric charge is conserved. In conclusion, the scattering process \(e^{+}+\nu_{\mu} \rightarrow e^{+}+\nu_{\mu}\) proceeds through an intermediate \(Z\) boson and cannot proceed through an intermediate charged \(W\) boson, while both options are possible for \(e^{+}+\nu_{e} \rightarrow e^{+}+\nu_{e}\) due to conservation laws involving electric charge and lepton flavor.