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

<math> f(E) \sim \frac{1}{\left(E^2-M^2\right)^2+M^2\Gamma^2}. \!</math>

It is most often used to model resonances (i.e., 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 <math>\Gamma</math> is the resonance's width, related to its lifetime according to <math>\tau=1/\Gamma</math>. 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, <math>\Gamma</math> can also be a function of E; this dependence is typically only important when <math>\Gamma</math> is not small compared to M (i.e., when the particle has a large width relative to its mass) and the phase-space dependence of the width needs to be taken into account. This occurs, for example, for the decay of the rho meson into a pair of pions. The factor of M² that multiples <math>\Gamma^2</math> should also be replaced with E² 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 <math>(p^2-M^2+iM\Gamma)</math>. 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.

[edit] See also

• Cauchy distribution

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