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)

2 thoughts on “Decay of a pion into a muon and a neutrino

  1. Pingback: Collision of a pion and a proton | Physics pages

  2. Pingback: Elastic collision of two identical particles | Physics pages

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