Relativistic Breit–Wigner distribution
Where k is the 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.
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.
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 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.
See also 
- Cauchy distribution, also known as the (non-relativistic) Breit–Wigner distribution or Lorentz distribution.
- See  for a discussion of the widths of particles in the PYTHIA manual. Note that this distribution is usually represented as a function of the squared energy.
- See the treatment of the Z-boson cross-section in, for example, G. Giacomelli, B. Poli (2002). "Results from high-energy accelerators". arXiv:hep-ex/0202023 [hep-ex].
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