# Neutral Pion Mass from Its Decay into Two Photons

• BOAS
In summary: The theta dependence comes from the dot product term, which is dependent on the angle between the two photon momenta. In summary, to find the mass of a ##\pi^0## meson based on the measured energies and angle between two emitted photons, one can use conservation of 4-momentum and invariant quantities to derive the expression ##M = 2 \sqrt{E_1 E_2} \sin \frac{\theta}{2}##, where ##E_1## and ##E_2## are the energies of the two photons and ##\theta## is the angle between their momenta.
BOAS

## Homework Statement

A ##\pi^0## meson decays predominantly to two photons. Suppose the energies (E1, E2) and angle (##\theta##) between the emitted photons are measured. Find an expression for the ##\pi^0## mass in terms of E1, E2, and ##\theta##.

## The Attempt at a Solution

[/B]
In the pion rest frame, by momentum conservation, I know that the two photons are emitted with equal but opposite momenta.

##E'_{\pi} = m_{\pi}##
##E'_{\gamma} = p'_{\gamma}##

##E'_{\gamma 1} = p'_{\gamma 1} = E'_{\gamma 2} = p'_{\gamma 2} = \frac{E'_{\pi}}{2} = \frac{m_{\pi}}{2}##

I know how to transform the photon energies into the lab frame, but I don't understand how to include the angular dependency.

You do not need to transform anything to the lab frame. You just need to apply conservation of 4-momentum and use invariant quantities.

Orodruin said:
You do not need to transform anything to the lab frame. You just need to apply conservation of 4-momentum and use invariant quantities.

Photon 1 has four-momentum ##P_{1 \mu} = (E_1, p_x, p_y, p_z)##

Photon 2 has four-momentum ##P_{2 \mu} = (E_2, -p_x, -p_y, -p_z)##

The pion has four-momentum ##P_{\pi^0 \mu} = (m_{\pi}, 0, 0, 0)##

I think that ##E_1 = E_2##

##M^2 = E_{tot}^2 - \vec{P^2}##

##M = 2E_1##

I don't think that can be right, but I don't see any reason for a theta dependance...

It is not right. It is not clear to me how you have accounted for 4-momentum conservation.

BOAS said:
I don't think that can be right, but I don't see any reason for a theta dependence...

You're not going to get one in the CM frame, but one will appear when you Lorentz transform back to the lab frame. But that's the longer way to solve this problem. Try analyzing the problem in the lab frame.

So I've done some research on using four vectors.

##M^2 = (P_{1 \mu} + P_{2 \mu})^2 = P^2_{1 \mu} + P^2_{2 \mu} + 2E_1E_2 - 2\vec{p_1} . \vec{p_2}##

The squared terms are the same as dot products which are zero for a massless particle and I am left with ##M^2 = 2E_1 E_2 - 2 \vec{p_1} . \vec{p_2}##

Since ##E' = p## for the photons I think I can write this as ##M^2 = 2 E_1 E_2 (1 - \cos \theta)##, and by employing a trig identity I can say that ##M = 2 \sqrt{E_1 E_2} \sin \frac{\theta}{2}##

I only used (i think) properties of the four vector but I have found a theta dependence.

BOAS said:
So I've done some research on using four vectors.

##M^2 = (P_{1 \mu} + P_{2 \mu})^2 = P^2_{1 \mu} + P^2_{2 \mu} + 2E_1E_2 - 2\vec{p_1} . \vec{p_2}##

The squared terms are the same as dot products which are zero for a massless particle and I am left with ##M^2 = 2E_1 E_2 - 2 \vec{p_1} . \vec{p_2}##

Since ##E' = p## for the photons I think I can write this as ##M^2 = 2 E_1 E_2 (1 - \cos \theta)##, and by employing a trig identity I can say that ##M = 2 \sqrt{E_1 E_2} \sin \frac{\theta}{2}##

I only used (i think) properties of the four vector but I have found a theta dependence.
This looks correct.

## 1. What is a neutral pion?

A neutral pion is a subatomic particle that is composed of a quark-antiquark pair. It is the lightest meson and is electrically neutral, meaning it has no charge.

## 2. How does a neutral pion decay into two photons?

A neutral pion decays into two photons through the strong nuclear force. This force causes the quark and antiquark within the pion to annihilate, producing two high-energy photons.

## 3. Why is the neutral pion mass important?

The neutral pion mass is important because it is a fundamental property of the particle that helps determine its interactions with other particles. It also has implications for the properties of the strong nuclear force.

## 4. How is the neutral pion mass measured?

The neutral pion mass is typically measured by studying its decay into two photons. By measuring the energy and momentum of the photons, scientists can calculate the mass of the pion using Einstein's famous equation, E=mc^2.

## 5. What is the current accepted value for the neutral pion mass?

The current accepted value for the neutral pion mass is 134.9770±0.0005 MeV/c^2. This value is based on numerous experimental measurements and is constantly being refined as new data becomes available.

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