In muon decay

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

Symmetry and Flavour, Ch From amplitude to rate: Lorentz-invariant phase space, widths and cross sections

Problem

In muon decay

\mu^-\to e^-\bar\nu_e\nu_\mu

, mediated by a point-like weak interaction, the electron energy spectrum in the muon rest frame (electron mass neglected in the dynamics) is

\frac{\mathrm{d}\Gamma}{\mathrm{d}E_e}=\frac{G_F^2m_\mu^2}{12\pi^3}E_e^2\left(3-\frac{4E_e}{m_\mu}\right).

((a)) Treat the two neutrinos together as a single recoiling system

X

of invariant mass

m_X

, so that the decay is kinematically a two-body decay

\mu\to e+X

. Using the two-body decay kinematics (the energy formula that accompanies the Kallen-function momentum

p^*

), write the electron's rest-frame energy

E_e^*

as a function of

m_\mu

m_e

and

m_X

. ((b)) The two neutrinos are massless, so

m_X

can range down to zero (when they move collinearly). Use this to derive the maximum electron energy

E_e^{\max}

allowed by kinematics, in terms of

m_\mu

and

m_e

alone. ((c)) Evaluate

E_e^{\max}

numerically for

m_\mu=105.658\,\mathrm{MeV}

and

m_e=0.511\,\mathrm{MeV}

, and compare it with the massless-electron approximation

m_\mu/2

used implicitly in the spectrum formula above, quoting the difference in

\mathrm{keV}

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