# Special relativity: pion decay

• abeboparebop

#### abeboparebop

I'm not sure whether this should go in this forum or the Advanced forum, but here goes.

## Homework Statement

Given:
A pion+ decays into muon+ and neutrino,
$$\pi^+ \rightarrow \mu^+ + \nu$$
neutrino mass approaches zero, and
the pion is initially at rest.

Problem statement: Use conservation of momentum and energy to show that the speed of the muon is given by:

$$\frac{u}{c} = \frac{(m_\pi/m_\mu)^2 - 1}{(m_\pi/m_\mu)^2 + 1}$$

## Homework Equations

E_total = KE + rest energy = (gamma-1)m*c^2 + m*c^2 = gamma*m*c^2

$E_t = (\gamma-1)mc^2 + mc^2 = \gamma mc^2$

$p = \gamma mv$

## The Attempt at a Solution

I assume that the fact that pion is initially at rest, in combination with conservation of momentum, means that total momentum of the muon and neutrino is zero. Given that the neutrino mass ~ zero (as stated in the problem), it would seem that it's momentum is zero, therefore the momentum of the muon is zero, therefore the velocity of the muon is zero. This is obviously not the case, as the problem gives me an equation to solve towards.

I tried setting the mass energy of the pion equal to the total energy of the muon (mass energy plus kinetic energy), and solving for v/c, but I got an answer quite different from the stated solution. Is the problem here an algebra error or do I need to account somehow for the momentum of the neutrino?

Thanks for the help.

Last edited:
Trying to figure out LaTeX... sorry it's transitionally ugly.

after setting c=1 (for convenience) use the following to help establish the result
$$E_\nu = p_\nu$$
$$E^2_\mu = p^2_\mu+m^2_\mu$$
$$m_\pi=E_\pi = E_\mu+E_\nu$$
$$p_\mu + p_\nu=0$$

write everything in terms of $$E_\mu, p_\mu, m_\pi, m_\nu$$ and note that $$u=p_\mu/E_\mu$$.