Inparticle physics,particle decay is thespontaneous process of one unstablesubatomic particle transforming into multiple other particles. The particles created in this process (thefinal state) must each be less massive than the original, although thetotal mass of the system must be conserved. A particle is unstable if there is at least oneallowed final state that it can decay into. Unstable particles will often have multiple ways of decaying, each with its ownassociated probability. Decays are mediated by one or severalfundamental forces. The particles in the final state may themselves be unstable and subject to further decay.

The term is typically distinct fromradioactive decay, in which an unstableatomic nucleus is transformed into a lighter nucleus accompanied by the emission of particles orradiation, although the two are conceptually similar and are often described using the same terminology.

Particle decay is aPoisson process, and hence the probability that a particle survives for timet before decaying (thesurvival function) is given by anexponential distribution whosetime constant depends on the particle's velocity:

$${\displaystyle P(t)=\exp \left(-\frac{t}{\gamma \tau }\right)}$$

where

${\displaystyle \tau }$ is the mean lifetime of the particle (when at rest), and

${\displaystyle \gamma =\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}}$ is theLorentz factor of the particle.

All data are from theParticle Data Group.

Type	Name	Symbol	Mass (MeV)	Mean lifetime
Lepton	Electron /Positron[1]	${\displaystyle {\mathrm{e}}^{-}/{\mathrm{e}}^{+}}$	0.511	>6.6×1028 years
	Muon / Antimuon	${\displaystyle {\mu }^{-}/{\mu }^{+}}$	105.7	2.2×10−6 seconds
	Tau lepton / Antitau	${\displaystyle {\tau }^{-}/{\tau }^{+}}$	1777	2.9×10−13 seconds
Meson	NeutralPion	${\displaystyle {\pi }^0}$	135	8.4×10−17 seconds
	ChargedPion	${\displaystyle {\pi }^{+}/{\pi }^{-}}$	139.6	2.6×10−8 seconds
Baryon	Proton /Antiproton[2][3]	${\displaystyle {\mathrm{p}}^{+}/{\mathrm{p}}^{-}}$	938.2	1.67×1034 years
	Neutron /Antineutron	${\displaystyle \mathrm{n}/\overline{\mathrm{n}}}$	939.6	885.7 seconds
Boson	W boson	${\displaystyle {\mathrm{W}}^{+}/{\mathrm{W}}^{-}}$	80400	10−26 seconds
	Z boson	${\displaystyle {\mathrm{Z}}^0}$	91000	10−26 seconds

This section usesnatural units, where${\displaystyle c=\hslash =1.}$

The lifetime of a particle is given by the inverse of its decay rate,Γ, the probability per unit time that the particle will decay. For a particle of a massM andfour-momentumP decaying into particles with momentap_i, the differential decay rate is given by the general formula (expressingFermi's golden rule)$${\displaystyle d{\mathrm{\Gamma }}_n=\frac{S{\left|\mathcal{M}\right|}^2}{2M}d{\mathrm{\Phi }}_n(P;p_1,p_2,\dots ,p_n)}$$

where

n is the number of particles created by the decay of the original,

S is a combinatorial factor to account for indistinguishable final states (see below),

${\displaystyle \mathcal{M}}$ is theinvariant matrix element oramplitude connecting the initial state to the final state (usually calculated usingFeynman diagrams),

${\displaystyle d{\mathrm{\Phi }}_n}$ is an element of thephase space, and

p_i is thefour-momentum of particlei.

The factorS is given by$${\displaystyle S=\prod _{j=1}^m\frac{1}{k_j!}}$$

where

m is the number of sets of indistinguishable particles in the final state, and

k_j is the number of particles of typej, so that${\displaystyle \sum _{j=1}^m{k}_j=n.}$

The phase space can be determined from$${\displaystyle d{\mathrm{\Phi }}_n(P;p_1,p_2,\dots ,p_n)=(2\pi {)}^4{\delta }^4\left(P-\sum _{i=1}^n{p}_i\right)\prod _{i=1}^n\frac{d^3{\overrightarrow{p}}_i}{2(2\pi {)}^3{E}_i}}$$

where

${\displaystyle {\delta }^4}$ is a four-dimensionalDirac delta function,

${\displaystyle {\overrightarrow{p}}_i}$ is the (three-)momentum of particlei, and

${\displaystyle E_i}$ is the energy of particlei.

One may integrate over the phase space to obtain the total decay rate for the specified final state.

If a particle has multiple decay branches ormodes with different final states, its full decay rate is obtained by summing the decay rates for all branches. Thebranching ratio for each mode is given by its decay rate divided by the full decay rate.

This section usesnatural units, where${\displaystyle c=\hslash =1.}$

In theCenter 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 theLab Frame the parent particle is probably moving at a speed close to thespeed of light so the two emitted particles would come out at angles different from those in the center of momentum frame.

Say a parent particle of massM decays into two particles, labeled1 and2. In the rest frame of the parent particle,$${\displaystyle |{\overrightarrow{p}}_1|=|{\overrightarrow{p}}_2|=\frac{\sqrt{[M^2-(m_1+m_2{)}^2][M^2-(m_1-m_2{)}^2]}}{2M},}$$which is obtained by requiring thatfour-momentum be conserved in the decay, i.e.$${\displaystyle (M,\overrightarrow{0})=(E_1,{\overrightarrow{p}}_1)+(E_2,{\overrightarrow{p}}_2).}$$

Also, in spherical coordinates,$${\displaystyle d^3\overrightarrow{p}=|\overrightarrow{p}{|}^{2}d|\overrightarrow{p}|d\varphi d\left(\cos \theta \right).}$$

Using the delta function to perform the${\displaystyle d^3{\overrightarrow{p}}_2}$ and${\displaystyle d|{\overrightarrow{p}}_1|}$ 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

$${\displaystyle d\mathrm{\Gamma }=\frac{{\left|\mathcal{M}\right|}^2}{32{\pi }^2}\frac{|{\overrightarrow{p}}_1|}{M^2}d{\varphi }_{1}d\left(\cos {\theta }_1\right).}$$

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$${\displaystyle \tan {\theta }^{\prime }=\frac{\sin \theta }{\gamma \left(\beta /{\beta }^{\prime }+\cos \theta \right)}}$$

This section usesnatural units, where${\displaystyle c=\hslash =1.}$

The mass of an unstable particle is formally acomplex number, with the real part being its mass in the usual sense, and the imaginary part being its decay rate innatural units. When the imaginary part is large compared to the real part, the particle is usually thought of as aresonance more than a particle. This is because inquantum field theory a particle of massM (areal 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${\displaystyle \frac{1}{M},}$ according to theuncertainty principle. For a particle of mass${\displaystyle M+i\mathrm{\Gamma }}$, the particle can travel for time${\displaystyle \frac{1}{M},}$ but decays after time of order of${\displaystyle \frac{1}{\mathrm{\Gamma }}.}$ If${\displaystyle \mathrm{\Gamma }>M}$ then the particle usually decays before it completes its travel.^[4]

• Relativistic Breit-Wigner distribution
• Particle physics
• Particle radiation
• List of particles
• Weak interaction

• J. D. Jackson (2004)."Kinematics"(PDF).Particle Data Group. Archived fromthe original(PDF) on 2014-11-21. Retrieved2006-11-26. (See page 2).
• Particle Data Group.
• "The Particle Adventure"Particle Data Group, Lawrence Berkeley National Laboratory.

• Particle physics

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