# Special Relativity: Pion Decay - Two Photons

In summary, the problem involves a (pi)0 meson decaying into two photons in flight. By using the invariant and conservation of energy and momentum equations in the lab frame and introducing gamma, the energies of the two photons can be found. The relative direction of travel of the photons must also be taken into consideration. By finding the energies in the ZMF and using the doppler shift, the energies in the lab frame can be determined.

## Homework Statement

A (pi)0 meson whose rest mass is 135 MeV/c2 is moving with a kinetic energy of 1 GeV. It decays in flight into two photons whose paths are along the direction of motion of the meson. Find the energies of the two photons.

## Homework Equations

Lab Frame:

The invariant: $$E^2 = P^2c^2 + m^2c^4$$
Conservation of Energy: $$E = E_1 + E_2 = hf_1 + hf_2$$
Conservation of Momentum: $$P = P_1 + P_2 = hf_1/c + hf_2/c$$

ZMF Frame:

The invariant: $$E^2 = P'^2c^2 + m^2c^4$$
Conservation of Energy(ZMF frame): $$m_0c^2 = E'_1 + E'_2 = hf'_1 + hf'_2$$
Conservation of Momentum(ZMF frame): $$P' = P'_1 + P'_2 = 0$$

## The Attempt at a Solution

Using invariant and $$E = E_1 + E_2$$, $$P = P_1 + P_2$$ (lab frame)

We have $$E_1^2 + E_2^2 + 2E_1E_2 = (P_1^2 + P_2^2 + 2P_1P_2)c^2 + m_o^2c^4$$

Not sure what other equations I can get from conditions given. Too many unknowns?? I get the feeling this should end up with a quadratic to give the two different energies...

I find it easier to treat problems of this kind by introducing gamma in the equations. Then, for a particle,

$$E=\gamma m_0c^2$$

$$K=(\gamma-1)m_0c^2$$

$$p=\sqrt{\gamma^2-1}m_0c$$

Use these in your energy and momentum conservation equations. Pay attention to the relative direction of travel of the two photons. Is the angle betwen the direction of the two photons zero or 180o?

Last edited:
Thanks a lot for the help :) I sorted it in the end by finding the energies in the ZMF after the decay and then using the doppler shift to transform them back into the lab frame.

## 1. What is special relativity?

Special relativity is a theory developed by Albert Einstein in 1905 that explains the behavior of objects moving at speeds close to the speed of light. It states that the laws of physics are the same for all observers in uniform motion and that the speed of light is constant in all inertial frames of reference.

## 2. What is pion decay?

Pion decay is a process in which a subatomic particle called a pion spontaneously breaks down into other particles. This process is governed by the weak nuclear force and is an important phenomenon in particle physics.

## 3. How does special relativity apply to pion decay?

Special relativity plays a crucial role in understanding pion decay because it explains the behavior of particles moving at high speeds. In the case of pion decay, the particles produced in the decay process are moving at speeds close to the speed of light, so special relativity must be taken into account in order to accurately describe and predict the outcome of the decay.

## 4. What are the two photons in pion decay?

In pion decay, the two photons refer to the two particles that are produced when the pion breaks down. These photons are a type of subatomic particle that carries electromagnetic energy and do not have any mass.

## 5. Why is the study of pion decay important?

Pion decay is an important process to study because it provides valuable insights into the fundamental forces and particles that make up our universe. It also has practical applications in fields such as medical imaging and nuclear energy. Studying pion decay can also help us further our understanding of special relativity and its implications for the behavior of particles at high speeds.

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