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,
where k is a constant of proportionality, equal to
(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. Note that for values of E off the maximum at M such that |E2−M2| = MΓ, (hence |E−M| = Γ/2 for M≫Γ), the distribution f has attenuated to half its maximum value, which justifies the name for Γ, width at half-maximum.
In the limit of vanishing width, Γ→0, the particle becomes stable as the Lorentzian distribution f sharpens infinitely to 2M δ(E2−M2).
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