Feynman diagram of muon to electron decay

The Michel parameters, usually denoted by

\rho, \eta, \xi

and

\delta

, are four parameters used in describing the phase space distribution of leptonic decays of charged

lepton In particle physics, a lepton is an elementary particle of half-integer spin ( spin ) that does not undergo strong interactions. Two main classes of leptons exist: charged leptons (also known as the electron-like leptons or muons), and neutr ...

l_^-\rightarrow l_^\nu_\bar

. They are named after the physicist

Louis Michel Louis Michel (born 2 September 1947) is a Belgian politician. He served in the government of Belgium as Minister of Foreign Affairs from 1999 to 2004 and was European Commissioner for Development and Humanitarian Aid from 2004 to 2009. From 20 ...

. Sometimes instead of

\delta

, the product

\xi\delta

is quoted. Within the

Standard Model The Standard Model of particle physics is the theory describing three of the four known fundamental forces (electromagnetism, electromagnetic, weak interaction, weak and strong interactions - excluding gravity) in the universe and classifying a ...

electroweak interaction In particle physics, the electroweak interaction or electroweak force is the unified description of two of the four known fundamental interactions of nature: electromagnetism and the weak interaction. Although these two forces appear very differe ...

s, these parameters are expected to be :

\rho=, \quad \eta=0, \quad \xi=1, \quad \xi\delta=.

Precise measurements of energy and angular distributions of the daughter leptons in decays of polarized

muon A muon ( ; from the Greek letter mu (μ) used to represent it) is an elementary particle similar to the electron, with an electric charge of −1 '' e'' and a spin of , but with a much greater mass. It is classified as a lepton. As wi ...

s and

tau lepton The tau (), also called the tau lepton, tau particle, tauon or tau electron, is an elementary particle similar to the electron, with negative electric charge and a spin of . Like the electron, the muon, and the three neutrinos, the tau is a l ...

s are so far in good agreement with these predictions of the Standard Model.

Muon decay

Consider the decay of the positive muon: :

\mu^+\to e^+ + \nu_e + \bar\nu_\mu.

In the muon

rest frame In special relativity, the rest frame of a particle is the frame of reference (a coordinate system attached to physical markers) in which the particle is at rest. The rest frame of compound objects (such as a fluid, or a solid made of many vibratin ...

, energy and angular distributions of the

positron The positron or antielectron is the antiparticle or the antimatter counterpart of the electron. It has an electric charge of +1 '' e'', a spin of 1/2 (the same as the electron), and the same mass as an electron. When a positron collides ...

s emitted in the decay of a polarised muon expressed in terms of Michel parameters are the following, neglecting electron and

neutrino A neutrino ( ; denoted by the Greek letter ) is a fermion (an elementary particle with spin of ) that interacts only via the weak interaction and gravity. The neutrino is so named because it is electrically neutral and because its rest mass ...

masses and the radiative corrections: :

\frac \sim (3-3x) + \frac\rho (4x-3) + P_\xi\cos\theta 1-x)+\frac\delta(4x-3)

where

P_

is muon polarisation,

x=E_e/E_e^

, and

\theta

is the angle between muon

spin Spin or spinning most often refers to: * Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning * Spin, the rotation of an object around a central axis * Spin (propaganda), an intentionally b ...

direction and positron momentum direction. For the decay of the negative muon, the sign of the term containing

\cos \theta

should be inverted. For the decay of the positive muon, the expected decay distribution for the

values of Michel parameters is :

\frac \sim x^2 3-2x) - P_\cos\theta(1-2x)

Integration of this expression over electron energy gives the angular distribution of the daughter positrons: :

\frac \sim 1 + \fracP_\cos\theta.

The positron energy distribution integrated over the polar angle is :

\frac \sim (3x^2-2x^3).

References

Lecture on Lepton Universality
by Michel Davier at the 1997 SLAC Summer Institute.
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. Electroweak theory {{particle-stub