Consider the decay at rest \(\tau^{-} \rightarrow \pi^{-} v_{\tau}\), where the spin of the tau is in the positive \(z\)-direction and the \(v_{\tau}\) and \(\pi^{-}\)travel in the \(\pm z\)-directions. Sketch the allowed spin configurations assuming that the form of the weak charged-current interaction is (i) \(V-A\) and (ii) \(V+A\).

Weak interactions are a type of fundamental force that is responsible for processes like radioactive decay. They are denoted by the exchange of W and Z bosons. Unlike the strong and electromagnetic forces, weak interactions can change the type (or 'flavor') of particles. For instance, in the decay \tau^{-} \rightarrow \pi^{-} v_{\tau}\, a tau lepton \(\tau^{-}\) decays into a pion \(\pi^{-}\) and a tau neutrino \(v_{\tau}\). Weak interactions are unique because they are the only type of fundamental interaction that violates parity symmetry, meaning they have different behaviors for left-handed and right-handed particles.

• Roles: Weak interactions are primarily involved in processes such as beta decay and the fusion reactions in the Sun.
• Exchange Particles: The W and Z bosons mediate these interactions, with the W boson being charged and the Z boson neutral.

Helicity refers to the projection of a particle's spin onto the direction of its momentum. For a given particle, if the spin is aligned with the momentum, the helicity is right-handed. If the spin is opposite to the momentum, the helicity is left-handed. In weak interactions, helicity plays a significant role because particles interact differently depending on their helicity.
For example: In the decay \tau^{-} \rightarrow \pi^{-} v_{\tau}\, the resulting tau neutrino \(v_{\tau}\) involving the V-A interaction has left-handed helicity, while for the V+A interaction, it would have right-handed helicity. Understanding the helicity configurations helps in comprehending the conservation laws and interaction dynamics in particle physics.

Angular momentum conservation is a crucial principle in physics, stating that the total angular momentum of a closed system remains constant if no external torque acts on it. This principle is vital in analyzing particle decays. In the context of tau decay: \(\tau^{-} \rightarrow \pi^{-} v_{\tau}\), it's necessary to ensure the initial and final states' angular momentum is conserved.

• Tau Lepton: Initially, the tau lepton has a spin aligned positively along the z-axis.
• Pion: Since the pion is a spin-0 particle, its contribution to the system's total spin is zero.
• Neutrino: The tau neutrino's spin and momentum direction must be aligned such that the total angular momentum is conserved, respecting the initial spin direction of the tau lepton.

The V-A interaction stands for Vector minus Axial vector interaction. In weak decay processes like \(\tau^{-} \rightarrow \pi^{-} v_{\tau}\), V-A indicates that left-handed components dominate, making left-handed particles more likely to interact. For a tau lepton with spin in the positive z-direction, its left-handed component has spin opposite to its momentum.

• Spin Configuration: The tau lepton decays in such a way that the tau neutrino \((v_{\tau})\) has left-handed helicity.
• Directional Analysis: The tau lepton's spin is in the positive z-direction. The pion \(\pi^{-}\) travels in the negative z-direction, while the tau neutrino travels in the positive z-direction with spin opposite to its movement.

These configurations must respect angular momentum conservation laws.

The V+A interaction, or Vector plus Axial vector interaction, implies that right-handed components dominate the weak interaction. In the decay \(\tau^{-} \rightarrow \pi^{-} v_{\tau}\), this results in the right-handed particles being more likely to interact. For a tau lepton with spin in the positive z-direction, its right-handed component's spin aligns with its momentum.

• Spin Configuration: The tau lepton decays such that the resulting tau neutrino \(v_{\tau}\) has right-handed helicity.
• Directional Analysis: The tau lepton's spin is in the positive z-direction. Hence, in the positive z-direction, the tau neutrino travels with its spin also in the positive z-direction, which aligns with its movement direction.

This respects the conservation of angular momentum, showing a different helicity outcome based on the interaction type.