Both and are weak decays of a charged meson to a muon and a (massless

PHYSICS · Multi-part · Fermi University Physics · Particle Physics & Quantum Fields · Particles, interactions and experiment · Particles, interactions and experiment

Particle Physics, Ch Decay rates, cross sections and phase space

Problem

Both

\pi^+\to\mu^+\nu_\mu

and

K^+\to\mu^+\nu_\mu

are weak decays of a charged meson to a muon and a (massless, for this purpose) neutrino, through vertices of the same general type. The measured lifetimes are

\tau_\pi = 2.603\times10^{-8}\,\mathrm{s}

(essentially all to

\mu^+\nu_\mu

) and

\tau_K = 1.238\times10^{-8}\,\mathrm{s}

, with

\mathcal{B}(K^+\to\mu^+\nu_\mu) = 63.56\%

. Take

m_\pi = 139.57\,\mathrm{MeV}/c^2

m_K = 493.68\,\mathrm{MeV}/c^2

m_\mu = 105.66\,\mathrm{MeV}/c^2

. ((a)) Find the momentum

p^*

of the muon in each parent's rest frame (the neutrino is taken massless). ((b)) Find the partial width

\Gamma(\to\mu^+\nu_\mu)

for both the pion and the kaon. ((c)) State whether the ratio of the two

p^*

values is reflected in the ratio of partial widths, and what this implies about the squared amplitudes

|\mathcal{M}|^2

of the two decays.

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