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 with (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 2Mδ(E2 − M2). 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 E 4/M2, etc.) when the resonance is wide. The form of the relativistic Breit–Wigner distribution arises from the propagator of an unstable particle, which has a denominator of the form p2 − M2 + iMΓ. (Here, p2 is the square of the four-momentum carried by that particle in the tree Feynman diagram involved.