Particle decay

Decay rate

This section uses natural 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 mass M and four-momentum P decaying into particles with momenta p_i, the differential decay rate is given by the general formula (expressing Fermi'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 the invariant matrix element or amplitude connecting the initial state to the final state (usually calculated using Feynman diagrams),

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

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

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

where

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

${\displaystyle {\overrightarrow{p}}_i}$ 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

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

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, $${\displaystyle |{\overrightarrow{p}}_1|=|{\overrightarrow{p}}_2|=\frac{\sqrt{[M^2-(m_1+m_2{)}^2][M^2-(m_1-m_2{)}^2]}}{2M},}$$ which is obtained by requiring that four-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

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

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

Complex mass and decay rate

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

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 ${\displaystyle \frac{1}{M},}$ according to the uncertainty 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.