Editing Muon (section)

=== Theoretical decay rate ===
{{See also|Michel parameters}}
{{More citations needed section|date=June 2021}}

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 constant|Fermi's coupling constant]] (<math>G_\text{F} </math>), with over-all dimension of inverse fourth power of energy. By [[dimensional analysis]], this leads to [[Q value (nuclear science)#Applications|Sargent's rule]] of fifth-power dependence on {{math|''m''<sub>''μ''</sub>}},<ref>{{cite thesis |title=Muon Decay Width and Lifetime in the Standard Model |first=Mahgoub Abbaker |last=Kabbashi |type=MSc |publisher=Sudan University of Science and Technology, Khartoum |date=August 2015 |url=http://repository.sustech.edu/bitstream/handle/123456789/11856/Research.pdf?sequence=2&isAllowed=y |access-date=May 21, 2021}}</ref><ref name=DecayFormula>{{cite web |url=https://www.uni-muenster.de/imperia/md/content/physik_tp/lectures/ss2017/standard_model/sheet10.pdf |title=Einführung in das Standardmodell der Teilchenphysik – Sheet 10 |date=2017 |first1=M. |last1=Klasen |first2=D. |last2=Frekers |first3=K. |last3=Kovařík |first4=P. |last4=Scior |first5=S. |last5=Schmiemann |access-date=May 21, 2021 |language=English }}</ref>
: <math>\Gamma=\frac{G_\text{F}^2 m_\mu^5}{192\pi^3}~ I\left(\frac{m_\text{e}^2}{m_\mu^2}\right),</math>
where <math>I(x)=1-8x-12x^2\ln x+8x^3-x^4</math>,<ref name=DecayFormula/> and:
: <math> x=\frac{2\,E_\text{e}}{m_\mu\,c^2}</math> is the fraction of the maximum energy transmitted to the electron.

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
: <math>\frac{\partial^2\Gamma}{\partial x\,\partial{\cos\theta}} \sim x^2[(3-2x) + P_\mu\cos\theta\,(1-2x)]</math>
where <math>\theta</math> is the angle between the muon's polarization vector <math>\mathbf P_\mu</math> and the decay-electron momentum vector, and <math>P_\mu = |\mathbf P_\mu|</math> is the fraction of muons that are forward-polarized. Integrating this expression over electron energy gives the angular distribution of the daughter electrons:
: <math>\frac{\mathrm{d}\Gamma}{\mathrm{d}{\cos\theta}} \sim 1 - \frac{1}{3}P_\mu\cos\theta.</math>
The electron energy distribution integrated over the polar angle (valid for <math> x < 1 </math>) is
: <math>\frac{\mathrm{d}\Gamma}{\mathrm{d}x} \sim (3x^2-2x^3).</math>

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