Relativistic Breit-Wigner distribution
Encyclopedia
The relativistic Breit–Wigner distribution (after Gregory Breit
and Eugene Wigner) is a continuous probability distribution
with the following probability density function
:
Where k is the constant of proportionality, equal to
with 
(This equation is written using natural units
, .) 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 . (With units included, the formula is .) 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 pion
s.) 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 . 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.
Gregory Breit
Gregory Breit was a Russian-born American physicist and professor at universities in New York, Wisconsin, Yale, and Buffalo...
and Eugene Wigner) is a continuous probability distribution
Probability distribution
In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....
with the following probability density function
Probability density function
In probability theory, a probability density function , or density of a continuous random variable is a function that describes the relative likelihood for this random variable to occur at a given point. The probability for the random variable to fall within a particular region is given by the...
:
Where k is the constant of proportionality, equal to
with (This equation is written using natural units
Natural units
In physics, natural units are physical units of measurement based only on universal physical constants. For example the elementary charge e is a natural unit of electric charge, or the speed of light c is a natural unit of speed...
, .) It is most often used to model resonances
Resonance (particle physics)
In particle physics, a resonance is the peak located around a certain energy found in differential cross sections of scattering experiments. These peaks are associated with subatomic particles and their excitations...
(unstable particles) in high-energy physics. In this case E is the center-of-mass
Center of mass
In physics, the center of mass or barycenter of a system is the average location of all of its mass. In the case of a rigid body, the position of the center of mass is fixed in relation to the body...
energy
Energy
In physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...
that produces the resonance, M is the mass
Mass
Mass can be defined as a quantitive measure of the resistance an object has to change in its velocity.In physics, mass commonly refers to any of the following three properties of matter, which have been shown experimentally to be equivalent:...
of the resonance, and Γ is the resonance width (or decay width), related to its mean lifetime according to . (With units included, the formula is .) 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
Phase space
In mathematics and physics, a phase space, introduced by Willard Gibbs in 1901, is a space in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in the phase space...
-dependence of the width needs to be taken into account. (For example, in the decay of the rho meson
Rho meson
In particle physics, a rho meson is a short-lived hadronic particle that is an isospin triplet whose three states are denoted as , and . After the pions and kaons, the rho mesons are the lightest strongly interacting particle with a mass of roughly for all three states...
into a pair of pion
Pion
In particle physics, a pion is any of three subatomic particles: , , and . Pions are the lightest mesons and they play an important role in explaining the low-energy properties of the strong nuclear force....
s.) 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
Propagator
In quantum mechanics and quantum field theory, the propagator gives the probability amplitude for a particle to travel from one place to another in a given time, or to travel with a certain energy and momentum. Propagators are used to represent the contribution of virtual particles on the internal...
of an unstable particle, which has a denominator of the form . Here p2 is the square of the four-momentum
Four-momentum
In special relativity, four-momentum is the generalization of the classical three-dimensional momentum to four-dimensional spacetime. Momentum is a vector in three dimensions; similarly four-momentum is a four-vector in spacetime...
carried by the particle. The propagator appears in the quantum mechanical amplitude
Probability amplitude
In quantum mechanics, a probability amplitude is a complex number whose modulus squared represents a probability or probability density.For example, if the probability amplitude of a quantum state is \alpha, the probability of measuring that state is |\alpha|^2...
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.
See also
- Cauchy distributionCauchy distributionThe Cauchy–Lorentz distribution, named after Augustin Cauchy and Hendrik Lorentz, is a continuous probability distribution. As a probability distribution, it is known as the Cauchy distribution, while among physicists, it is known as the Lorentz distribution, Lorentz function, or Breit–Wigner...
, also known as the (non-relativistic) Breit–Wigner distribution or Lorentz curve