Relativistic Breit–Wigner distribution

The relativistic Breit–Wigner distribution (after the 1936 nuclear resonance formula of Gregory Breit and Eugene Wigner) is a continuous probability distribution with the following probability density function,

${\displaystyle f(E)={\frac {k}{\left(E^{2}-M^{2}\right)^{2}+M^{2}\Gamma ^{2}}}~,}$

where k is a constant of proportionality, equal to

${\displaystyle k={\frac {2{\sqrt {2}}M\Gamma \gamma }{\pi {\sqrt {M^{2}+\gamma }}}}~~~~}$   with   ${\displaystyle ~~~~\gamma ={\sqrt {M^{2}\left(M^{2}+\Gamma ^{2}\right)}}~.}$

(This equation is written using natural units, ħ = c = 1.)

It is most often used to model resonances (unstable particles) in high-energy physics. In this case, E is the center-of-mass energy that produces the resonance, M is the mass of the resonance, and Γ is the resonance width (or decay width), related to its mean lifetime according to τ = 1/Γ. (With units included, the formula is τ = ħ/Γ.)