# Relativistic Breit–Wigner distribution

The relativistic Breit–Wigner distribution (after Gregory Breit and Eugene Wigner) is a continuous probability distribution with the following probability density function:[1]

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

Where k is the constant of proportionality, equal to

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

(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.) The probability of producing the resonance at a given energy E is proportional to f(E), so that a plot of the production rate of the unstable particle as a function of energy traces out the shape of the relativistic Breit–Wigner distribution.

In general, Γ can also be a function of E; this dependence is typically only important when Γ is not small compared to M and the phase space-dependence of the width needs to be taken into account. (For example, in the decay of the rho meson into a pair of pions.) The factor of M2 that multiplies Γ2 should also be replaced with E2 (or E4/M2, etc.) when the resonance is wide.[2]

The form of the relativistic Breit–Wigner distribution arises from the propagator of an unstable particle, which has a denominator of the form p2M2 + i. Here p2 is the square of the four-momentum carried by the particle. The propagator appears in the quantum mechanical amplitude for the process that produces the resonance; the resulting probability distribution is proportional to the absolute square of the amplitude, yielding the relativistic Breit–Wigner distribution for the probability density function as given above.

The form of this distribution is similar to the solution of the classical equation of motion for a damped harmonic oscillator driven by a sinusoidal external force.