Calculating Muon Momentum with Finite Neutrino Mass

[indigojoker]

Homework Statement

in the decay process, $$\pi ^{+} -> \mu^{+} + \nu_{\mu}$$

show that for a neutrino of finite (but small) mass, compared with the case of the massless neutrino, the muon momentum would be reduced by the fraction:

$$\frac{p'}{p}= - \frac{m_{\nu}^2 (m_{\pi}^2+m_{\mu}^2)}{(m_{\pi}^2-m_{\mu}^2)^2}$$

Calculation of the muon's momentum with massless neutrino:

$$p_{\mu}=p_{\pi}-p_{\nu}$$
$$p_{\mu}^2=p_{\pi}^2+p_{\nu}^2-2m_{\pi}E_{\nu}$$
We know:
$$p_{\mu}=m_{\mu}^2c^2$$
$$p_{\pi}=m_{\pi}^2c^2$$
$$p_{\nu}=0$$

$$m_{\nu}^2c^2=m_{\pi}^2c^2-2m_{\pi}E_{\nu}$$

But: $$E_{\nu}=|p_{\nu}|c=|p_{\mu}|c$$

$$2 m_{\pi} |p_{\mu}|=(m_{\pi}^2-m_{\mu}^2)c$$
$$|p_{\mu}|=\frac{m_{\pi}^2-m_{\mu}^2}{2m_{\pi}}c$$

Using the same method, but with $$p_{\nu}=m_{\nu}^2c^2$$

we get:
$$2 m_{\pi} |p_{\nu}|=(m_{\pi}^2-m_{\mu}^2-m_{\nu}^2)c$$
$$|p'_{\mu}|=\frac{m_{\pi}^2-m_{\mu}^2-m_{\nu}^2}{2m_{\pi}}c$$

The fraction I get is:

$$\frac{p'}{p}=\frac{m_{\pi}^2-m_{\mu}^2-m_{\nu}^2}{m_{\pi}^2-m_{\mu}^2}$$

If i multiply by $$\frac{m_{\pi}^2-m_{\mu}^2}{m_{\pi}^2-m_{\mu}^2}$$ I do not get the above result. Any idea where i went wrong?

 

[Jhero]

Hello, thank you for your question. It seems that you have made a small error in your calculation. In the equation where you substituted the expression for the energy of the neutrino, you wrote:

E_{\nu}=|p_{\nu}|c=|p_{\mu}|c

However, this should be:

E_{\nu}=|p_{\nu}|c+|p_{\mu}|c

This is because the total energy in the decay process should be conserved, so the energy of the neutrino and the energy of the muon should add up to the energy of the pion.
With this correction, your equation for the muon's momentum with a massless neutrino becomes:

|p_{\mu}|=\frac{m_{\pi}^2-m_{\mu}^2}{2m_{\pi}}c

And the equation for the muon's momentum with a finite mass neutrino becomes:

|p'_{\mu}|=\frac{m_{\pi}^2-m_{\mu}^2-m_{\nu}^2}{2m_{\pi}}c

Taking the ratio of these two equations, we get:

\frac{p'}{p}=\frac{m_{\pi}^2-m_{\mu}^2-m_{\nu}^2}{m_{\pi}^2-m_{\mu}^2}

Which is the result you were looking for. I hope this helps.