Particle decay

# Particle decay

Discussion

Encyclopedia
Particle decay is the spontaneous process
Spontaneous process
A spontaneous process is the time-evolution of a system in which it releases free energy and moves to a lower, more thermodynamically stable energy state...

of one elementary particle
Elementary particle
In particle physics, an elementary particle or fundamental particle is a particle not known to have substructure; that is, it is not known to be made up of smaller particles. If an elementary particle truly has no substructure, then it is one of the basic building blocks of the universe from which...

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 decay. The intermediate particle then transforms into other particles. If the particles created are not stable, the decay process can continue.

Particle decay is also used to refer to the decay of hadrons. However, the term is not typically used to describe radioactive decay
Radioactive decay is the process by which an atomic nucleus of an unstable atom loses energy by emitting ionizing particles . The emission is spontaneous, in that the atom decays without any physical interaction with another particle from outside the atom...

, in which an unstable atomic nucleus
Atomic nucleus
The nucleus is the very dense region consisting of protons and neutrons at the center of an atom. It was discovered in 1911, as a result of Ernest Rutherford's interpretation of the famous 1909 Rutherford experiment performed by Hans Geiger and Ernest Marsden, under the direction of Rutherford. The...

is transformed into a lighter nucleus accompanied by the emission of particles or radiation, although the two are conceptually similar.

Natural units
In physics, natural units are physical units of measurement based only on universal physical constants. For example the elementary charge e is a natural unit of electric charge, or the speed of light c is a natural unit of speed...

, where

## Probability of survival and particle lifetime

Particle decay is a Poisson process
Poisson process
A Poisson process, named after the French mathematician Siméon-Denis Poisson , is a stochastic process in which events occur continuously and independently of one another...

, and hence the probability that a particle survives for time t before decaying is given by an exponential distribution
Exponential distribution
In probability theory and statistics, the exponential distribution is a family of continuous probability distributions. It describes the time between events in a Poisson process, i.e...

whose time constant depends on the particle's velocity:

where

is the mean lifetime of the particle (when at rest), and
is the Lorentz factor
Lorentz factor
The Lorentz factor or Lorentz term appears in several equations in special relativity, including time dilation, length contraction, and the relativistic mass formula. Because of its ubiquity, physicists generally represent it with the shorthand symbol γ . It gets its name from its earlier...

of the particle.

### Table of elementary particle lifetimes

All data is from the Particle Data Group
Particle Data Group
The Particle Data Group is an international collaboration of particle physicists that compiles and reanalyzes published results related to the properties of particles and fundamental interactions. It also publishes reviews of theoretical results that are phenomenologically relevant, including...

.
Type Name Symbol Energy
Energy
In physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...

(MeV
MEV
MeV and meV are multiples and submultiples of the electron volt unit referring to 1,000,000 eV and 0.001 eV, respectively.Mev or MEV may refer to:In entertainment:* Musica Elettronica Viva, an Italian musical group...

)
Lepton Electron
Electron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...

/ Positron
Positron
The positron or antielectron is the antiparticle or the antimatter counterpart of the electron. The positron has an electric charge of +1e, a spin of ½, and has the same mass as an electron...

0.511
Muon
Muon
The muon |mu]] used to represent it) is an elementary particle similar to the electron, with a unitary negative electric charge and a spin of ½. Together with the electron, the tau, and the three neutrinos, it is classified as a lepton...

/ Antimuon
105.6
Tau lepton
Tau lepton
The tau , also called the tau lepton, tau particle or tauon, is an elementary particle similar to the electron, with negative electric charge and a spin of . Together with the electron, the muon, and the three neutrinos, it is classified as a lepton...

/ Antitau
1777
Meson Neutral Pion
Pion
In particle physics, a pion is any of three subatomic particles: , , and . Pions are the lightest mesons and they play an important role in explaining the low-energy properties of the strong nuclear force....

135
Charged Pion
Pion
In particle physics, a pion is any of three subatomic particles: , , and . Pions are the lightest mesons and they play an important role in explaining the low-energy properties of the strong nuclear force....

139.6
Baryon Proton
Proton
The proton is a subatomic particle with the symbol or and a positive electric charge of 1 elementary charge. One or more protons are present in the nucleus of each atom, along with neutrons. The number of protons in each atom is its atomic number....

/ Antiproton
Antiproton
The antiproton is the antiparticle of the proton. Antiprotons are stable, but they are typically short-lived since any collision with a proton will cause both particles to be annihilated in a burst of energy....

938.2
Neutron
Neutron
The neutron is a subatomic hadron particle which has the symbol or , no net electric charge and a mass slightly larger than that of a proton. With the exception of hydrogen, nuclei of atoms consist of protons and neutrons, which are therefore collectively referred to as nucleons. The number of...

/ Antineutron
Antineutron
The antineutron is the antiparticle of the neutron with symbol . It differs from the neutron only in that some of its properties have equal magnitude but opposite sign. It has the same mass as the neutron, and no net electric charge, but has opposite baryon number...

939.6
Boson W boson
W and Z bosons
The W and Z bosons are the elementary particles that mediate the weak interaction; their symbols are , and . The W bosons have a positive and negative electric charge of 1 elementary charge respectively and are each other's antiparticle. The Z boson is electrically neutral and its own...

80,400
Z boson
W and Z bosons
The W and Z bosons are the elementary particles that mediate the weak interaction; their symbols are , and . The W bosons have a positive and negative electric charge of 1 elementary charge respectively and are each other's antiparticle. The Z boson is electrically neutral and its own...

91,000

## Decay rate

The lifetime of a particle is given by the inverse of its decay rate, , the probability per unit time that the particle will decay. For a particle of a mass M and four-momentum
Four-momentum
In special relativity, four-momentum is the generalization of the classical three-dimensional momentum to four-dimensional spacetime. Momentum is a vector in three dimensions; similarly four-momentum is a four-vector in spacetime...

P, the differential decay rate is given by the general formula

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),
is the invariant matrix element or amplitude
Probability amplitude
In quantum mechanics, a probability amplitude is a complex number whose modulus squared represents a probability or probability density.For example, if the probability amplitude of a quantum state is \alpha, the probability of measuring that state is |\alpha|^2...

connecting the initial state to the final state (usually calculated using Feynman diagrams),
is an element of the phase space
Phase space
In mathematics and physics, a phase space, introduced by Willard Gibbs in 1901, is a space in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in the phase space...

, and
is the four-momentum
Four-momentum
In special relativity, four-momentum is the generalization of the classical three-dimensional momentum to four-dimensional spacetime. Momentum is a vector in three dimensions; similarly four-momentum is a four-vector in spacetime...

of particle i.

The factor S is given by
where
m is the number of sets of indistinguishable particles in the final state, and
is the number of particles of type j, so that .

The phase space can be determined from
where
is a four-dimensional Dirac delta function
Dirac delta function
The Dirac delta function, or δ function, is a generalized function depending on a real parameter such that it is zero for all values of the parameter except when the parameter is zero, and its integral over the parameter from −∞ to ∞ is equal to one. It was introduced by theoretical...

,
is the (three-) momentum of particle i, and
is the energy of particle i.

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 or modes with different final states, its full decay rate is obtained by summing the decay rates for all branches. The branching ratio
Branching ratio
In particle physics and nuclear physics, the branching fraction for a decay is the fraction of particles which decay by an individual decay mode with respect to the total number of particles which decay. It is equal to the ratio of the partial decay constant to the overall decay constant...

for each mode is given by its decay rate divided by the full decay rate.

## Two-body decay

### Decay rate

Say a parent particle of mass M decays into two particles, labeled 1 and 2. In the rest frame of the parent particle,
which is obtained by requiring that four-momentum
Four-momentum
In special relativity, four-momentum is the generalization of the classical three-dimensional momentum to four-dimensional spacetime. Momentum is a vector in three dimensions; similarly four-momentum is a four-vector in spacetime...

be conserved in the decay, i.e.

Also, in spherical coordinates,

Using the delta function to perform the and 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

### From two different frames

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

## 3-body decay

The phase space element of one particle decaying into three is

## Complex mass and decay rate

The mass of an unstable particle is formally a complex number
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...

, with the real part being its mass in the usual sense, and the imaginary part being its decay rate in natural units
Natural units
In physics, natural units are physical units of measurement based only on universal physical constants. For example the elementary charge e is a natural unit of electric charge, or the speed of light c is a natural unit of speed...

. When the imaginary part is large compared to the real part, the particle is usually thought of as a resonance
Resonance
In physics, resonance is the tendency of a system to oscillate at a greater amplitude at some frequencies than at others. These are known as the system's resonant frequencies...

more than a particle. This is because in quantum field theory
Quantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...

a particle of mass M (a real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

) 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
Uncertainty principle
In quantum mechanics, the Heisenberg uncertainty principle states a fundamental limit on the accuracy with which certain pairs of physical properties of a particle, such as position and momentum, can be simultaneously known...

. For a particle of mass , the particle can travel for time 1/M, but decays after time of order of . If then the particle usually decays before it completes its travel.