Tau-leptons are produced in the process \(\mathrm{e}^{+} \mathrm{e}^{-} \rightarrow \tau^{+} \tau^{-}\)at a centre-of-mass energy of \(91.2 \mathrm{GeV}\). The angular distribution of the \(\pi^{-}\)from the decay \(\tau^{-} \rightarrow \pi^{-} v_{\tau}\) is $$ \frac{\mathrm{d} N}{\mathrm{~d}\left(\cos \theta^{*}\right)} \propto 1+\cos \theta^{*} $$ where \(\theta^{*}\) is the polar angle of the \(\pi^{-}\)in the tau-lepton rest frame, relative to the direction defined by the \(\tau(\) tau \()\) spin. Determine the laboratory frame energy distribution of the \(\pi^{-}\)for the cases where the tau-lepton spin is (i) aligned with or (ii) opposite to its direction of flight.

The polar angle distribution describes how particles are scattered in various directions after a decay or collision.
The given distribution for the \(\pi^{-}\) in the tau-lepton rest frame is represented by the equation $$ \frac{dN}{d(\text{cos} \theta^{*})} \propto 1 + \text{cos} \theta^{*} $$
Where \(\theta^{*}\) is the angle of \(\pi^{-}\) relative to the spin direction of the tau-lepton.
This formula tells us that more \(\pi^{-}\) particles will be emitted in the direction of the tau's spin (low \(\theta^{*}\)) than in the opposite direction (high \(\theta^{*}\)).
This asymmetric distribution is due to the nature of weak interactions that violate parity symmetry.
Practical applications of polar angle distribution include analyzing particle collisions in accelerators and understanding the behavior of subatomic particles.

In all particle interactions and decays, the law of conservation of energy and momentum must be satisfied.
When a tau-lepton decays, its total energy and momentum are shared between the resulting particles, such as \(\pi^{-}\) and neutrinos.
Conservation of energy can be expressed as: \ E_{\tau} = E_{\pi} + E_{\u}
For momentum conservation: \ \vec{P}_{\tau} = \vec{P}_{\pi} + \vec{P}_{\u}
These conservation laws help determine the energy distribution of the decay products.
For instance, if tau surface decays produces \(\pi^{-}\) and neutrino, their combined energy and momentum in each frame should match the initial tau-lepton's.
This is crucial for the further steps of the solution where we need these principles to transform and understand the decay products' behavior in different reference frames.

To understand the behavior of decay products in different reference frames, we use Lorentz transformations.
These are mathematical equations derived from special relativity that relate the coordinates and time of events between different inertial frames moving at constant velocities relative to each other.
The Lorentz transformation for energy and momentum is: \ E' = \gamma(E - vP\text{cos}\theta) \ \
And for momentum: \ P' = \gamma(P - \frac{vE\text{cos}\theta}{c^2} ).
Where \(\gamma\) is the Lorentz factor: \ \gamma = \frac{1}{\text{√}(1 - \frac{v^{2}}{c^{2}} )}.
Applying these transformations allows us to convert the energy distribution from the tau-rest frame to the laboratory frame.
This is critical for understanding how the tau-lepton’s spin orientation affects the \(\pi^{-}\)’s energy distribution relative to the tau-lepton’s direction of motion.