Editing Muon (section)

== Muon decay ==
[[Image:Muon Decay.svg|right|thumb|The most common decay of the muon]]
Muons are unstable elementary particles and are heavier than electrons and neutrinos but lighter than all other matter particles. They decay via the [[weak interaction]]. Because [[lepton number|leptonic family numbers]] are conserved in the absence of an extremely unlikely immediate [[neutrino oscillation]], one of the product neutrinos of muon decay must be a muon-type neutrino and the other an electron-type antineutrino (antimuon decay produces the corresponding antiparticles, as detailed below).

Because charge must be conserved, one of the products of muon decay is always an electron of the same charge as the muon (a positron if it is a positive muon). Thus all muons decay to at least an electron, and two neutrinos. Sometimes, besides these necessary products, additional other particles that have no net charge and spin of zero (e.g., a pair of photons, or an electron-positron pair), are produced.

The dominant muon decay mode (sometimes called the Michel decay after [[Louis Michel (physicist)|Louis Michel]]) is the simplest possible: the muon decays to an electron, an electron antineutrino, and a muon neutrino. Antimuons, in mirror fashion, most often decay to the corresponding antiparticles: a [[positron]], an electron neutrino, and a muon antineutrino. In formulaic terms, these two decays are:
<!-- anybody who will make this from <math> will be cursed -->
: {{subatomic particle|muon}} → {{subatomic particle|electron}} + {{math|{{subatomic particle|electron antineutrino|link=yes}} + {{subatomic particle|muon neutrino|link=yes}}}} 
: {{subatomic particle|antimuon}} → {{subatomic particle|positron|link=yes}} + {{math|{{subatomic particle|electron neutrino}} + {{subatomic particle|muon antineutrino}}}}

The mean lifetime, {{math|{{var|τ}} {{=}} {{var|ħ}}/{{var|Γ}}}}, of the (positive) muon is {{val|2.1969811|0.0000022|ul=us}}.<ref name="PDG2012" /> The equality of the muon and antimuon lifetimes has been established to better than one part in 10<sup>4</sup>.<ref name="ChargeRatio">{{cite journal | last1 = Bardin | first1 = G. | last2 = Duclos | first2 = J. | 
 last3 = Magnon | first3 = A. | last4 = Martino | first4 = J. | last5 = Zavattini| first5 = E. | title = A New Measurement of the Positive Muon Lifetime | journal = Phys Lett B | volume = 137 | pages = 135–140 | year = 1984 | issue = 1–2 | doi=10.1016/0370-2693(84)91121-3| bibcode = 1984PhLB..137..135B | url = https://cds.cern.ch/record/149657 }}</ref>

=== Prohibited decays ===
Certain neutrino-less decay modes are kinematically allowed but are, for all practical purposes, forbidden in the [[Standard Model]], even given that neutrinos have mass and oscillate. Examples forbidden by lepton flavour conservation are:
: {{subatomic particle|muon}} → {{subatomic particle|electron}} + {{subatomic particle|photon}}
and
: {{subatomic particle|muon}} → {{subatomic particle|electron}} + {{subatomic particle|positron}} + {{subatomic particle|electron}} .

Taking into account neutrino mass, a decay like {{subatomic particle|muon}} → {{subatomic particle|electron}} + {{subatomic particle|photon}} is technically possible in the Standard Model (for example by [[neutrino oscillation]] of a virtual muon neutrino into an electron neutrino), but such a decay is extremely unlikely and therefore should be experimentally unobservable. Fewer than one in 10<sup>50</sup> muon decays should produce such a decay.

Observation of such decay modes would constitute clear evidence for theories [[beyond the Standard Model]]. Upper limits for the branching fractions of such decay modes were measured in many experiments starting more than {{rounddown|{{age|format=raw|1964|1|1}}|-1}} years ago. The current upper limit for the {{subatomic particle|antimuon}} → {{subatomic particle|positron}} + {{subatomic particle|photon}} branching fraction was measured 2009–2013 in the [[Mu to E Gamma|MEG]] experiment and is {{val|4.2e-13}}.<ref name=Baldini>
{{cite arXiv
 | last1 = Baldini | first1 =A.M.
 | collaboration=MEG collaboration
 | display-authors=etal
 | title =Search for the lepton flavour violating decay μ<sup>+</sup> → e<sup>+</sup>γ with the full dataset of the MEG experiment
 | date =May 2016
 | eprint=1605.05081
 | class =hep-ex
}}</ref>

=== 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.