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| '''Particle decay''' is the [[spontaneous process]] of one [[elementary particle]] transforming into other elementary particles. During this process, an elementary particle becomes a different particle with less mass and an [[intermediate particle]] such as [[W boson]] in [[Muon#Muon decay|muon decay]]. The intermediate particle then transforms into other particles. If the particles created are not stable, the decay process can continue.
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| ''Particle decay'' is also used to refer to the decay of [[hadrons]]. However, the term is not typically used to describe [[radioactive decay]], in which an unstable [[atomic nucleus]] is transformed into a lighter nucleus accompanied by the emission of particles or radiation, although the two are conceptually similar.
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| Note that this article uses [[natural units]], where <math>c=\hbar=1. \,</math>
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| ==Probability of survival and particle lifetime==
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| Particle decay is a [[Poisson process]], and hence the probability that a particle survives for time ''t'' before decaying is given by an [[exponential distribution]] whose time constant depends on the particle's velocity:
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| ::<math>P(t) = e^{-t/(\gamma \tau)} \,</math>
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| :where
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| ::<math>\tau</math> is the mean lifetime of the particle (when at rest), and
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| ::<math>\gamma = \frac{1}{\sqrt{1-v^2/c^2}}</math> is the [[Lorentz factor]] of the particle.
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| ===Table of some elementary and composite particle lifetimes===
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| All data is from the [[Particle Data Group]].
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| :{| class=wikitable style="text-align: center;"
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| !Type
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| !Name
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| !Symbol
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| ![[Energy]] ([[MeV]])
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| !Mean lifetime
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| |-
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| |rowspan="3" | Lepton
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| |[[Electron]] / [[Positron]]
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| |<math>e^- \, / \, e^+</math>
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| |0.511
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| |<math>> 4.6 \times 10^{26} \ \mathrm{years} \,</math>
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| |-
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| |[[Muon]] / Antimuon
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| |<math>\mu^- \, / \, \mu^+ </math>
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| |105.7
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| |<math>2.2\times 10^{-6} \ \mathrm{seconds} \,</math>
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| |-
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| |[[Tau lepton]] / Antitau
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| |<math>\tau^- \, / \, \tau^+</math>
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| |1777
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| |<math>2.9 \times 10^{-13} \ \mathrm{seconds} \,</math>
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| |-
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| |rowspan="2" | Meson
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| |Neutral [[Pion]]
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| |<math> \pi^0\,</math>
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| |135
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| |<math>8.4 \times 10^{-17} \ \mathrm{seconds} \,</math>
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| |-
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| |Charged [[Pion]]
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| |<math> \pi^+ \, / \, \pi^-</math>
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| |139.6
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| |<math>2.6 \times 10^{-8} \ \mathrm{seconds} \,</math>
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| |-
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| |rowspan="2" | Baryon
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| |[[Proton]] / [[Antiproton]]
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| |<math> p^+ \, / \, p^-</math>
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| |938.2
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| |<math>> 10^{29} \ \mathrm{years} \,</math>
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| |-
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| |[[Neutron]] / [[Antineutron]]
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| |<math> n \, / \, \bar{n} </math>
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| |939.6
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| |<math> 885.7 \ \mathrm{seconds} \,</math>
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| |-
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| |rowspan="2" | Boson
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| |[[W and Z bosons|W boson]]
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| |<math> W^+ \, / \, W^-</math>
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| |80,400
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| |<math>10^{-25} \ \mathrm{seconds} \,</math>
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| |-
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| |[[W and Z bosons|Z boson]]
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| |<math>Z^0 \,</math>
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| |91,000
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| |<math>10^{-25} \ \mathrm{seconds} \,</math>
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| |}
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| ==Decay rate==
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| The lifetime of a particle is given by the inverse of its decay rate, <math>\Gamma</math>, the probability per unit time that the particle will decay. For a particle of a mass ''M'' and [[four-momentum]] ''P'' decaying into particles with momenta <math>p_i</math>, the differential decay rate is given by the general formula (see also [[Fermi's golden rule]])
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| ::<math>d \Gamma_n = \frac{S \left|\mathcal{M} \right|^2}{2M} d \Phi_n (P; p_1, p_2,\dots, p_n) \,</math>
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| :where
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| ::''n'' is the number of particles created by the decay of the original,
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| ::''S'' is a combinatorial factor to account for indistinguishable final states (see below),
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| ::<math>\mathcal{M}\,</math> is the ''invariant matrix element'' or [[Probability amplitude|amplitude]] connecting the initial state to the final state (usually calculated using [[Feynman diagrams]]),
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| ::<math>d\Phi_n \,</math> is an element of the [[phase space]], and
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| ::<math>p_i \,</math> is the [[four-momentum]] of particle ''i''.
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| The factor ''S'' is given by
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| ::<math>S = \prod_{j=1}^m \frac{1}{k_j!}\,</math>
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| :where
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| ::''m'' is the number of sets of indistinguishable particles in the final state, and
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| ::<math>k_j \,</math> is the number of particles of type ''j'', so that <math>\sum_{j=1}^m k_j = n \,</math>.
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| The phase space can be determined from
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| ::<math>d \Phi_n (P; p_1, p_2,\dots, p_n) = (2\pi)^4 \delta^4\left(P - \sum_{i=1}^n p_i\right) \prod_{i=1}^n \frac{d^3 \vec{p}_i}{2(2\pi)^3 E_i}</math>
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| :where
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| ::<math>\delta^4 \,</math> is a four-dimensional [[Dirac delta function]],
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| ::<math>\vec{p}_i \,</math> is the (three-)momentum of particle ''i'', and
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| ::<math>E_i \,</math> is the energy of particle ''i''.
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| One may integrate over the phase space to obtain the total decay rate for the specified final state.
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| If a particle has multiple decay branches or ''modes'' with different final states, its full decay rate is obtained by summing the decay rates for all branches. The [[branching ratio]] for each mode is given by its decay rate divided by the full decay rate.
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| ==Two-body decay==
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| {{double image|right|2-body Particle Decay-CoM.svg|140|2-body Particle Decay-Lab.svg|160|In the '''Center of Momentum Frame''', the decay of a particle into two equal mass particles results in them being emitted with an angle of 180° between them.|...while in the '''Lab Frame''' the parent particle is probably moving at a speed close to the [[speed of light]] so the two emitted particles would come out at angles different than that of in the center of momentum frame.}}
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| ===Decay rate===
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| Say a parent particle of mass ''M'' decays into two particles, labeled '''1''' and '''2'''. In the rest frame of the parent particle,
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| :<math>|\vec{p}_1| = |\vec{p_2}| = \frac{[(M^2 - (m_1 + m_2)^2)(M^2 - (m_1 - m_2)^2)]^{1/2}}{2M}, \,</math>
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| which is obtained by requiring that [[four-momentum]] be conserved in the decay, i.e.
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| :<math>(M, \vec{0}) = (E_1, \vec{p}_1) + (E_2, \vec{p}_2).\,</math>
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| Also, in spherical coordinates,
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| :<math>d^3 \vec{p} = |\vec{p}\,|^2\, d|\vec{p}\,|\, d\phi\, d\left(\cos \theta \right). \,</math>
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| Using the delta function to perform the <math>d^3 \vec{p}_2</math> and <math>d|\vec{p}_1|\,</math> integrals in the phase-space for a two-body final state, one finds that the decay rate in the rest frame of the parent particle is
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| :<math>d\Gamma = \frac{ \left| \mathcal{M} \right|^2}{32 \pi^2} \frac{|\vec{p}_1|}{M^2}\, d\phi_1\, d\left( \cos \theta_1 \right). \,</math>
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| ===From two different frames===
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| The angle of an emitted particle in the lab frame is related to the angle it has emitted in the center of momentum frame by the equation
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| ::<math>\tan{\theta'} = \frac{\sin{\theta}}{\gamma \left(\beta / \beta' + \cos{\theta} \right)}</math>
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| ==3-body decay==
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| The phase space element of one particle decaying into three is
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| ::<math>d\Phi_3 = \frac{1}{(2\pi)^5} \delta^4(P - p_1 - p_2 - p_3) \frac{d^3 \vec{p}_1}{2 E_1} \frac{d^3 \vec{p}_2}{2 E_2} \frac{d^3 \vec{p}_3}{2 E_3} \,</math>
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| ==Complex mass and decay rate==
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| {{see|Resonance#Resonances in quantum mechanics}}
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| The mass of an unstable particle is formally a [[complex number]], with the real part being its mass in the usual sense, and the imaginary part being its decay rate in [[natural units]]. When the imaginary part is large compared to the real part, the particle is usually thought of as a [[resonance]] more than a particle. This is because in [[quantum field theory]] a particle of mass M (a [[real number]]) is often exchanged between two other particles when there is not enough energy to create it, if the time to travel between these other particles is short enough, of order 1/M, according to the [[uncertainty principle]]. For a particle of mass <math>\scriptstyle M+i\Gamma</math>, the particle can travel for time 1/M, but decays after time of order of <math>\scriptstyle 1/\Gamma</math>. If <math>\scriptstyle \Gamma > M</math> then the particle usually decays before it completes its travel.
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| ==See also==
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| *[[Relativistic Breit-Wigner distribution]]
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| *[[Particle physics]]
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| *[[List of particles]]
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| *[[Weak interaction]]
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| ==References==
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| *{{cite journal|author=J.D. Jackson|authorlink=J. D. Jackson|title=Kinematics|journal=[[Particle Data Group]]|year=2004|volume=|pages=|url=http://pdg.lbl.gov/2005/reviews/kinemarpp.pdf}} - See page 2.
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| *[http://pdg.lbl.gov/ Particle Data Group].
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| *"[http://particleadventure.org/ The Particle Adventure]" [[Particle Data Group]], Lawrence Berkeley National Laboratory.
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| [[Category:Particle physics]]
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