# Michel parameters

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{{Short description|Phase space parameters in leptonic decays}}
[[File:Feynman diagram of muon to electron decay.svg|right|frame|[Feynman diagram](/source/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](/source/phase_space) distribution of leptonic decays of charged [lepton](/source/lepton)s, <math>l_{i}^-\rightarrow l_{j}^{-}\nu_{i}\bar{\nu_{j}}</math>. They are named after the physicist [Louis Michel](/source/Louis_Michel_(physicist)). Sometimes instead of <math>\delta</math>, the product <math>\xi\delta</math> is quoted. Within the [Standard Model](/source/Standard_Model) of [electroweak interaction](/source/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](/source/muon)s and [tau lepton](/source/tau_lepton)s are so far in good agreement with these predictions of the Standard Model.

==Muon decay==
{{See also|Muon#Muon decay}}
Consider the decay of the positive muon:
:<math>\mu^+\to e^+ + \nu_e + \bar\nu_\mu.</math>
In the muon [rest frame](/source/rest_frame), energy and angular distributions of the [positron](/source/positron)s emitted in the decay of a polarised muon expressed in terms of Michel parameters are the following, neglecting [electron](/source/electron) and [neutrino](/source/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](/source/Spin_(physics)) direction and positron momentum direction.<ref>
{{cite journal
 |author=R. Bayes ''et al.'' ([TWIST collaboration](/source/TWIST_experiment))
 |year=2011
 |title=Experimental Constraints on Left-Right Symmetric Models from Muon Decay
 |journal=[Physical Review Letters](/source/Physical_Review_Letters)
 |volume=106|issue= 4|article-number=041804
 |doi=10.1103/PhysRevLett.106.041804|bibcode = 2011PhRvL.106d1804B
 |pmid=21405321
 |doi-access=free
 }}</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](/source/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](/source/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

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Adapted from the Wikipedia article [Michel parameters](https://en.wikipedia.org/wiki/Michel_parameters) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Michel_parameters?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.