Particle decay

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 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 some composite particles, such as 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.

Note that this article uses natural units, where ${\displaystyle c=\hbar =1.\,}$

Probability of survival and particle lifetime

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:

${\displaystyle P(t)=e^{-t/(\gamma \tau )}\,}$
where
${\displaystyle \tau }$ is the mean lifetime of the particle (when at rest), and
${\displaystyle \gamma ={\frac {1}{\sqrt {1-v^{2}/c^{2}}}}}$ is the Lorentz factor of the particle.

Table of some elementary and composite particle lifetimes

All data is from the Particle Data Group.

Type Name Symbol Energy (MeV) Mean lifetime
Lepton Electron / Positron ${\displaystyle e^{-}\,/\,e^{+}}$ 0.511 ${\displaystyle >4.6\times 10^{26}\ \mathrm {years} \,}$
Muon / Antimuon ${\displaystyle \mu ^{-}\,/\,\mu ^{+}}$ 105.7 ${\displaystyle 2.2\times 10^{-6}\ \mathrm {seconds} \,}$
Tau lepton / Antitau ${\displaystyle \tau ^{-}\,/\,\tau ^{+}}$ 1777 ${\displaystyle 2.9\times 10^{-13}\ \mathrm {seconds} \,}$
Meson Neutral Pion ${\displaystyle \pi ^{0}\,}$ 135 ${\displaystyle 8.4\times 10^{-17}\ \mathrm {seconds} \,}$
Charged Pion ${\displaystyle \pi ^{+}\,/\,\pi ^{-}}$ 139.6 ${\displaystyle 2.6\times 10^{-8}\ \mathrm {seconds} \,}$
Baryon Proton / Antiproton ${\displaystyle p^{+}\,/\,p^{-}}$ 938.2 ${\displaystyle >10^{29}\ \mathrm {years} \,}$
Neutron / Antineutron ${\displaystyle n\,/\,{\bar {n}}}$ 939.6 ${\displaystyle 885.7\ \mathrm {seconds} \,}$
Boson W boson ${\displaystyle W^{+}\,/\,W^{-}}$ 80,400 ${\displaystyle 10^{-25}\ \mathrm {seconds} \,}$
Z boson ${\displaystyle Z^{0}\,}$ 91,000 ${\displaystyle 10^{-25}\ \mathrm {seconds} \,}$

Decay rate

The lifetime of a particle is given by the inverse of its decay rate, ${\displaystyle \Gamma }$, 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 ${\displaystyle p_{i}}$, the differential decay rate is given by the general formula (expressing Fermi's golden rule)

${\displaystyle d\Gamma _{n}={\frac {S\left|{\mathcal {M}}\right|^{2}}{2M}}d\Phi _{n}(P;p_{1},p_{2},\dots ,p_{n})\,}$
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 the invariant matrix element or amplitude connecting the initial state to the final state (usually calculated using Feynman diagrams),
${\displaystyle d\Phi _{n}\,}$ is an element of the phase space, and
${\displaystyle p_{i}\,}$ is the four-momentum of particle i.

The factor S 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
${\displaystyle k_{j}\,}$ is the number of particles of type j, so that ${\displaystyle \sum _{j=1}^{m}k_{j}=n\,}$.

The phase space can be determined from

${\displaystyle d\Phi _{n}(P;p_{1},p_{2},\dots ,p_{n})=(2\pi )^{4}\delta ^{4}\left(P-\sum _{i=1}^{n}p_{i}\right)\prod _{i=1}^{n}{\frac {d^{3}{\vec {p}}_{i}}{2(2\pi )^{3}E_{i}}}}$
where
${\displaystyle \delta ^{4}\,}$ is a four-dimensional Dirac delta function,
${\displaystyle {\vec {p}}_{i}\,}$ is the (three-)momentum of particle i, and
${\displaystyle E_{i}\,}$ 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 for each mode is given by its decay rate divided by the full decay rate.

Two-body decay

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 from those in the center of momentum frame.

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,

${\displaystyle |{\vec {p}}_{1}|=|{\vec {p_{2}}}|={\frac {[(M^{2}-(m_{1}+m_{2})^{2})(M^{2}-(m_{1}-m_{2})^{2})]^{1/2}}{2M}},\,}$

which is obtained by requiring that four-momentum be conserved in the decay, i.e.

${\displaystyle (M,{\vec {0}})=(E_{1},{\vec {p}}_{1})+(E_{2},{\vec {p}}_{2}).\,}$

Also, in spherical coordinates,

${\displaystyle d^{3}{\vec {p}}=|{\vec {p}}\,|^{2}\,d|{\vec {p}}\,|\,d\phi \,d\left(\cos \theta \right).\,}$

Using the delta function to perform the ${\displaystyle d^{3}{\vec {p}}_{2}}$ and ${\displaystyle d|{\vec {p}}_{1}|\,}$ 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

${\displaystyle d\Gamma ={\frac {\left|{\mathcal {M}}\right|^{2}}{32\pi ^{2}}}{\frac {|{\vec {p}}_{1}|}{M^{2}}}\,d\phi _{1}\,d\left(\cos \theta _{1}\right).\,}$

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

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

3-body decay

The phase space element of one particle decaying into three is

${\displaystyle d\Phi _{3}={\frac {1}{(2\pi )^{5}}}\delta ^{4}(P-p_{1}-p_{2}-p_{3}){\frac {d^{3}{\vec {p}}_{1}}{2E_{1}}}{\frac {d^{3}{\vec {p}}_{2}}{2E_{2}}}{\frac {d^{3}{\vec {p}}_{3}}{2E_{3}}}\,}$

Complex mass and decay rate

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 ${\displaystyle \scriptstyle M+i\Gamma }$, the particle can travel for time 1/M, but decays after time of order of ${\displaystyle \scriptstyle 1/\Gamma }$. If ${\displaystyle \scriptstyle \Gamma >M}$ then the particle usually decays before it completes its travel.[1]