# Decay of a pion into a muon and a neutrino

References: Griffiths, David J. (2007), Introduction to Electrodynamics, 3rd Edition; Pearson Education – Chapter 12, Post 31.

[Griffiths’s approach to the relativistic four-velocity is similar to that of Moore, although rather confusingly, he uses different notation (as well as keeping factors of ${c}$ in the equations rather than setting ${c=1}$). To keep the notation consistent with Griffiths, I’ll use his notation here, but anyone attempting to follow both books should beware\ldots ]

We can use the conservation of relativistic energy and momentum to analyze the interaction of elementary particles. For example, a pion at rest can decay into a muon and a neutrino. Conservation of energy and 3-momentum require

 $\displaystyle E_{\pi}=m_{\pi}c^{2}$ $\displaystyle =$ $\displaystyle E_{\mu}+E_{\nu}\ \ \ \ \ (1)$ $\displaystyle \mathbf{p}_{\pi}=0$ $\displaystyle =$ $\displaystyle \mathbf{p}_{\mu}+\mathbf{p}_{\nu} \ \ \ \ \ (2)$

We can use the relation

$\displaystyle E^{2}-p^{2}c^{2}=m^{2}c^{4} \ \ \ \ \ (3)$

to relate energy and momentum. Assuming the neutrino is massless (it isn’t quite, but it’s close) we have

$\displaystyle E_{\nu}=cp_{\nu} \ \ \ \ \ (4)$

while for the muon

$\displaystyle E_{\mu}=c\sqrt{p_{\mu}^{2}+m_{\mu}^{2}c^{2}} \ \ \ \ \ (5)$

so

$\displaystyle m_{\pi}c=\sqrt{p_{\mu}^{2}+m_{\mu}^{2}c^{2}}+p_{\nu} \ \ \ \ \ (6)$

But ${p_{\nu}=-p_{\mu}}$ from 2 so

 $\displaystyle m_{\pi}c$ $\displaystyle =$ $\displaystyle \sqrt{p_{\mu}^{2}+m_{\mu}^{2}c^{2}}-p_{\mu}\ \ \ \ \ (7)$ $\displaystyle \sqrt{p_{\mu}^{2}+m_{\mu}^{2}c^{2}}$ $\displaystyle =$ $\displaystyle m_{\pi}c+p_{\mu}\ \ \ \ \ (8)$ $\displaystyle p_{\mu}$ $\displaystyle =$ $\displaystyle \frac{m_{\mu}^{2}-m_{\pi}^{2}}{2m_{\pi}}c\ \ \ \ \ (9)$ $\displaystyle E_{\mu}$ $\displaystyle =$ $\displaystyle \frac{m_{\mu}^{2}+m_{\pi}^{2}}{2m_{\pi}}c^{2} \ \ \ \ \ (10)$

where the last line follows from 5.

The velocity of the muon can be found from

 $\displaystyle E_{\mu}$ $\displaystyle =$ $\displaystyle p^{0}c\ \ \ \ \ (11)$ $\displaystyle$ $\displaystyle =$ $\displaystyle \frac{m_{\mu}c^{2}}{\sqrt{1-u^{2}/c^{2}}}\ \ \ \ \ (12)$ $\displaystyle \frac{m_{\mu}^{2}+m_{\pi}^{2}}{2m_{\pi}}c^{2}$ $\displaystyle =$ $\displaystyle \frac{m_{\mu}c^{2}}{\sqrt{1-u^{2}/c^{2}}}\ \ \ \ \ (13)$ $\displaystyle u$ $\displaystyle =$ $\displaystyle c\sqrt{1-\frac{4m_{\pi}^{2}m_{\mu}^{2}}{\left(m_{\pi}^{2}+m_{\mu}^{2}\right)}}\ \ \ \ \ (14)$ $\displaystyle$ $\displaystyle =$ $\displaystyle \frac{m_{\pi}^{2}-m_{\mu}^{2}}{m_{\pi}^{2}+m_{\mu}^{2}}c \ \ \ \ \ (15)$