Find the maximum opening angle between the photons produced in the decay \(\pi^{0} \rightarrow \gamma \gamma\) if the energy of the neutral pion is \(10 \mathrm{GeV}\), given that \(m_{\pi^{a}}=135 \mathrm{MeV}\).

When discussing a relativistic particle, we mean a particle moving at speeds close to the speed of light. This is especially relevant in high-energy physics. The neutral pion in this problem has an energy of 10 GeV, far greater than its rest mass of 135 MeV. This indicates it is moving at a speed very close to the speed of light.

Key characteristics of relativistic particles include:

• Their total energy is dominated by their kinetic energy rather than their rest mass energy.
• Their momentum and energy are related by the famous equation \( E^2 = p^2 + m^2 \) where E is energy, p is momentum, and m is mass.
• Time and space behave differently for them; time dilation and length contraction become significant.

In practical terms, this means a highly energetic pion will behave differently from how we'd usually expect based on everyday experiences with slower-moving objects.

Photon emission occurs when a particle decays and releases energy in the form of light or other electromagnetic radiation. In this scenario, a neutral pion decays into two photons. These photons each carry away half the rest mass energy of the pion in the pion's rest frame.

• In the pion's rest frame, the photons are emitted directly opposite to each other.
• Each photon here would have an energy of \(67.5 \, \text{MeV}\).

However, things change in a frame where the pion is moving rapidly.
In such frames, the energy of each emitted photon is not simply half the rest mass energy and the photons no longer travel exactly back-to-back. Instead, the photons tend to move forward in a more collimated fashion due to the high momentum of the pion.

Understanding this emission pattern is crucial for accurate predictions in particle physics experiments.

Lorentz transformations are a set of equations used in relativity to transform between the measurements of observers in different inertial frames. They are essential when dealing with high-speed particles. In this problem, applying a Lorentz transformation helps us understand how the angles of photon emission change when moving from the pion's rest frame to the lab frame.

Key points:

• The angle \(\theta\) of the photons in the lab frame is different from their emission angle in the rest frame.
• For a particle moving close to the speed of light, the emission angle in the lab frame \(\theta \) is much smaller.
• The relationship used here is \(\theta \approx \frac{m_{\pi^{0}}}{E_{\pi^{0}}} \), which gives us the maximum opening angle.

In our case, this yields an angle of about 0.0135 radians, highlighting how a highly relativistic particle emits photons that appear to move almost in the same direction it was moving.
Understanding Lorentz transformations allows physicists to predict how particles behave in different reference frames, which is fundamental to modern physics.