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[[Image:Muon-Electron-Decay.svg|right|frame|[[Feynman diagram]] of the muon decay]]
The '''Michel parameters''', usually denoted by <math>\rho, \eta, \xi</math> and <math>\delta</math>, are four parameters used in describing the phase space distribution of leptonic decays of charged [[lepton]]s, <math>l_{i}^-\rightarrow l_{j}^{-}\nu_{i}\bar{\nu_{j}}</math>. They are named after the physicist [[Louis Michel (physicist)|Louis Michel]]. Sometimes instead of <math>\delta</math>, the product <math>\xi\delta</math> is quoted. Within the [[Standard Model]] of [[electroweak interaction]]s, these parameters are expected to be
:<math> \rho={3\over4}, \quad \eta=0, \quad \xi=1, \quad \xi\delta={3\over4}.</math>
Precise measurements of energy and angular distributions of the daughter leptons in decays of polarized [[muon]]s and [[tau lepton]]s are so far in good agreement with these predictions of the Standard Model.

==Muon decay==
{{Seealso|Muon#Muon decay}}
Let us consider the decay of the positive muon:
:<math>\mu^+\to e^+ + \nu_e + \bar\nu_\mu.</math>
In the muon [[rest frame]], energy and angular distributions of the [[positron]]s emitted in the decay of a polarised muon expressed in terms of Michel parameters are the following, neglecting electron and [[neutrino]] masses and the radiative corrections:
:<math>\frac{d^2\Gamma}{x^2dxd\cos\theta} \sim (3-3x) + \frac{2}{3}\rho (4x-3) + P_{\mu}\xi\cos\theta
[(1-x)+\frac{2}{3}\delta(4x-3)],</math>
where <math>P_{\mu}</math> is muon polarisation, <math>x=E_e/E_e^{max}</math>, and <math>\theta</math> is the angle between muon [[Spin (physics)|spin]] direction and positron momentum direction.<ref>
{{cite journal
 |author=R. Bayes ''et al.'' ([[TWIST experiment|TWIST collaboration]])
 |year=2011
 |title=Experimental Constraints on Left-Right Symmetric Models from Muon Decay
 |journal=[[Physical Review Letters]]
 |volume=106|issue= 4|pages=041804
 |doi=10.1103/PhysRevLett.106.041804|bibcode = 2011PhRvL.106d1804B
 |unused_data=Open Access }}</ref> For the decay of the negative muon, the sign of the term containing <math>cos\theta</math> should be inverted.

For the decay of the positive muon, the expected decay distribution for the [[Standard Model]] values of Michel parameters is
:<math>\frac{d^2\Gamma}{dxd\cos\theta} \sim x^2[(3-2x) - P_{\mu}\cos\theta(1-2x)].</math>
Integration of this expression over electron energy gives the angular distribution of the daughter positrons:
:<math>\frac{d\Gamma}{d\cos\theta} \sim 1 + \frac{1}{3}P_{\mu}\cos\theta.</math>
The positron energy distribution integrated over the polar angle is
:<math>\frac{d\Gamma}{dx} \sim (3x^2-2x^3).</math>

==References==
{{reflist}}
* [http://www.slac.stanford.edu/pubs/confproc/ssi97/ssi97-002.html Lecture on Lepton Universality] by Michel Davier at the 1997 SLAC Summer Institute.
* [http://pos.sissa.it/archive/conferences/005/036/silafae-III_036.pdf Electroweak Couplings, Lepton Universality, and the Origin of Mass: An Experimental Perspective], article by John Swain, from the Proceedings of the Third Latin American Symposium on High Energy Physics.

[[Category:Electroweak theory]]
[[Category:Particle physics]]