Relativistic Breit–Wigner distribution

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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 ^[1]:

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

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 $Γ$ is the resonance's width, related to its lifetime according to $τ = 1 / Γ$ . 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 (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 $Γ 2$ 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 $(p 2 - M 2 + i M Γ)$ . 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, also known as the (non-relativistic) Breit–Wigner distribution

[edit] Sources

^ See [1] 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, Results from high-energy accelerators by G. Giacomelli, B. Poli, 2002.

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