Questions: 1. A neutral pion ( pi^0 ) with a total energy of 500 MeV in the laboratory reference frame travels along the ( x )-axis. The pion decays into two photons with the reaction [ pi^0 rightarrow gamma gamma ] In the laboratory reference frame: (a) Calculate the magnitude of the momentum of the pion. (b) Calculate the kinetic energy of the pion. (c) Draw a Feynman diagram for this decay. (d) What is the angle between the photons and the x-axis when they have minimum and maximum energy? (e) Derive the maximum and minimum energy that the two photons can have. (f) Calculate the angle between the two photons if they are emitted with the same energy.

1. A neutral pion ( pi^0 ) with a total energy of 500 MeV in the laboratory reference frame travels along the ( x )-axis. The pion decays into two photons with the reaction
[
pi^0 rightarrow gamma gamma
]

In the laboratory reference frame:
(a) Calculate the magnitude of the momentum of the pion.
(b) Calculate the kinetic energy of the pion.
(c) Draw a Feynman diagram for this decay.
(d) What is the angle between the photons and the x-axis when they have minimum and maximum energy?
(e) Derive the maximum and minimum energy that the two photons can have.
(f) Calculate the angle between the two photons if they are emitted with the same energy.

Transcript text: 1. A neutral pion $\pi^{0}$ with a total energy of 500 MeV in the laboratory reference frame travels along the $x$-axis. The pion decays into two photons with the reaction \[ \pi^{0} \rightarrow \gamma \gamma \] In the laboratory reference frame: (a) Calculate the magnitude of the momentum of the pion. (b) Calculate the kinetic energy of the pion. (c) Draw a Feynman diagram for this decay. (d) What is the angle between the photons and the x -axis when they have minimum and maximum energy? (e) Derive the maximum and minimum energy that the two photons can have. (f) Calculate the angle between the two photons if they are emitted with the same energy.

Solution

Solution Steps

Step 1: Calculate the Magnitude of the Momentum of the Pion

The total energy $ E $ of the pion is given as 500 MeV. The rest mass energy $ m_{\pi^0}c^2 $ of the neutral pion is approximately 135 MeV. The relationship between energy, momentum, and mass is given by the relativistic energy-momentum relation:

\[ E^2 = (pc)^2 + (m_{\pi^0}c^2)^2 \]

Solving for the momentum $ p $:

\[ p = \frac{1}{c} \sqrt{E^2 - (m_{\pi^0}c^2)^2} \]

Substituting the given values:

\[ p = \frac{1}{c} \sqrt{(500 \, \text{MeV})^2 - (135 \, \text{MeV})^2} \]

\[ p = \frac{1}{c} \sqrt{250000 \, \text{MeV}^2 - 18225 \, \text{MeV}^2} \]

\[ p = \frac{1}{c} \sqrt{231775 \, \text{MeV}^2} \]

\[ p \approx 481.5 \, \text{MeV}/c \]

Step 2: Calculate the Kinetic Energy of the Pion

The kinetic energy $ K $ is the total energy minus the rest mass energy:

\[ K = E - m_{\pi^0}c^2 \]

Substituting the given values:

\[ K = 500 \, \text{MeV} - 135 \, \text{MeV} = 365 \, \text{MeV} \]

Step 3: Draw a Feynman Diagram for the Decay

A Feynman diagram for the decay of a neutral pion into two photons would show the pion as an incoming line, which then splits into two outgoing lines representing the photons. The diagram is simple and typically looks like a "Y" shape, with the pion at the base and the two photons at the top ends.