Past exam of the mathematics course of the University of Cambridge / 2007 / ib / Paper 4 / 17B / b

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- Solution b

Solution

In the pion rest frame, conservation of three-momentum makes the electron and massless neutrino momenta equal and opposite, of common magnitude

p

. Their energies obey

E_{e} + p c = m_{π} c^{2}, E_{e}^{2} = p^{2} c^{2} + m_{e}^{2} c^{4} .

Squaring

E_{e} = m_{π} c^{2} - p c

and cancelling

p^{2} c^{2}

gives

p c = \frac{m _{π}^{2} - m _{e}^{2}}{2 m _{π}} c^{2}, E_{e} = \frac{m _{π}^{2} + m _{e}^{2}}{2 m _{π}} c^{2} .

The electron four-momentum satisfies

p = E_{e} v / c^{2}

. Thus, for the physically allowed decay

m_{π} > m_{e}

,

v = c \frac{m _{π}^{2} - m _{e}^{2}}{m _{π}^{2} + m _{e}^{2}} = c \frac{1 - ( m _{e} / m _{π} ) ^{2}}{1 + ( m _{e} / m _{π} ) ^{2}} .

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