Muon
Theoretical decay rate

The muon decay width that follows from Fermi's golden rule has dimension of energy, and must be proportional to the square of the amplitude, and thus the square of Fermi's coupling constant (G F {\displaystyle G_{\text{F}}}), with over-all dimension of inverse fourth power of energy. By dimensional analysis, this leads to Sargent's rule of fifth-power dependence on m,

where I (x) = 1 − 8 x − 12 x 2 ln x + 8 x 3 − x 4 {\displaystyle I(x)=1-8x-12x^{2}\ln x+8x^{3}-x^{4}}, and:

The decay distributions of the electron in muon decays have been parameterised using the so-called Michel parameters. The values of these four parameters are predicted unambiguously in the Standard Model of particle physics, thus muon decays represent a good test of the spacetime structure of the weak interaction. No deviation from the Standard Model predictions has yet been found.

For the decay of the muon, the expected decay distribution for the Standard Model values of Michel parameters is

where θ {\displaystyle \theta } is the angle between the muon's polarization vector P μ {\displaystyle \mathbf {P} _{\mu }} and the decay-electron momentum vector, and P μ = | P μ | {\displaystyle P_{\mu }=|\mathbf {P} _{\mu }|} is the fraction of muons that are forward-polarized. Integrating this expression over electron energy gives the angular distribution of the daughter electrons:

The electron energy distribution integrated over the polar angle (valid for x < 1 {\displaystyle x<1}) is

Because the direction the electron is emitted in (a polar vector) is preferentially aligned opposite the muon spin (an axial vector), the decay is an example of non-conservation of parity by the weak interaction. This is essentially the same experimental signature as used by the original demonstration. More generally in the Standard Model, all charged leptons decay via the weak interaction and likewise violate parity symmetry.
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