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The relativistic Breit–Wigner distribution (after the 1936 nuclear resonance formula of
Gregory Breit Gregory Breit (russian: Григорий Альфредович Брейт-Шнайдер, ''Grigory Alfredovich Breit-Shneider''; July 14, 1899, Mykolaiv, Kherson Governorate – September 13, 1981, Salem, Oregon) was a Russian-born Jewish Am ...
and
Eugene Wigner Eugene Paul "E. P." Wigner ( hu, Wigner Jenő Pál, ; November 17, 1902 – January 1, 1995) was a Hungarian-American theoretical physicist who also contributed to mathematical physics. He received the Nobel Prize in Physics in 1963 "for his con ...
) is a continuous
probability distribution In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon ...
with the following
probability density function In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can ...
, Se
Pythia 6.4 Physics and Manual
(page 98 onwards) for a discussion of the widths of particles in the
PYTHIA Pythia (; grc, Πυθία ) was the name of the high priestess of the Temple of Apollo at Delphi. She specifically served as its oracle and was known as the Oracle of Delphi. Her title was also historically glossed in English as the Pythoness ...
manual. Note that this distribution is usually represented as a function of the squared energy.
: f(E) = \frac~, where is a constant of proportionality, equal to : k = \frac ~~~~   with   ~~~~ \gamma=\sqrt ~. (This equation is written using
natural units In physics, natural units are physical units of measurement in which only universal physical constants are used as defining constants, such that each of these constants acts as a coherent unit of a quantity. For example, the elementary charge may ...
, .) It is most often used to model
resonances Resonance describes the phenomenon of increased amplitude that occurs when the frequency of an applied periodic force (or a Fourier component of it) is equal or close to a natural frequency of the system on which it acts. When an oscillat ...
(unstable particles) in
high-energy physics Particle physics or high energy physics is the study of fundamental particles and forces that constitute matter and radiation. The fundamental particles in the universe are classified in the Standard Model as fermions (matter particles) and b ...
. In this case, is the center-of-mass
energy In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of heat ...
that produces the resonance, is the
mass Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different elementa ...
of the resonance, and Γ is the resonance width (or '' decay width''), related to its
mean lifetime A quantity is subject to exponential decay if it decreases at a rate proportional to its current value. Symbolically, this process can be expressed by the following differential equation, where is the quantity and (lambda) is a positive rate ...
according to . (With units included, the formula is .)


Usage

The probability of producing the resonance at a given energy is proportional to , 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 off the maximum at such that , (hence for ), the distribution 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 sharpens infinitely to . In general, Γ can also be a function of ; this dependence is typically only important when Γ is not small compared to and the
phase space In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. For mechanical systems, the phase space usuall ...
-dependence of the width needs to be taken into account. (For example, in the decay of the
rho meson Rho (uppercase Ρ, lowercase ρ or ; el, ρο or el, ρω, label=none) is the 17th letter of the Greek alphabet. In the system of Greek numerals it has a value of 100. It is derived from Phoenician letter res . Its uppercase form uses the sa ...
into a pair of
pion In particle physics, a pion (or a pi meson, denoted with the Greek letter pi: ) is any of three subatomic particles: , , and . Each pion consists of a quark and an antiquark and is therefore a meson. Pions are the lightest mesons and, more gen ...
s.) The factor of 2 that multiplies Γ2 should also be replaced with 2 (or 4/2, etc.) when the resonance is wide. The form of the relativistic Breit–Wigner distribution arises from the
propagator In quantum mechanics and quantum field theory, the propagator is a function that specifies the probability amplitude for a particle to travel from one place to another in a given period of time, or to travel with a certain energy and momentum. In ...
of an unstable particle, which has a denominator of the form . (Here, 2 is the square of the
four-momentum In special relativity, four-momentum (also called momentum-energy or momenergy ) is the generalization of the classical three-dimensional momentum to four-dimensional spacetime. Momentum is a vector in three dimensions; similarly four-momentum is ...
carried by that particle in the tree Feynman diagram involved.) The propagator in its rest frame then is proportional to the quantum-mechanical amplitude for the decay utilized to reconstruct that resonance, :\frac~. The resulting probability distribution is proportional to the absolute square of the amplitude, so then the above relativistic Breit–Wigner distribution for the probability density function. The form of this distribution is similar to the amplitude of the solution to the classical equation of motion for a driven harmonic oscillator damped and driven by a
sinusoidal A sine wave, sinusoidal wave, or just sinusoid is a mathematical curve defined in terms of the ''sine'' trigonometric function, of which it is the graph. It is a type of continuous wave and also a smooth periodic function. It occurs often in ...
external force. It has the standard
resonance Resonance describes the phenomenon of increased amplitude that occurs when the frequency of an applied periodic force (or a Fourier component of it) is equal or close to a natural frequency of the system on which it acts. When an oscillati ...
form of the Lorentz, or
Cauchy distribution The Cauchy distribution, named after Augustin Cauchy, is a continuous probability distribution. It is also known, especially among physicists, as the Lorentz distribution (after Hendrik Lorentz), Cauchy–Lorentz distribution, Lorentz(ian) func ...
, but involves relativistic variables  = 2, here = 2. The distribution is the solution of the differential equation for the amplitude squared w.r.t. the energy energy (frequency), in such a classical forced oscillator, : f'(\text) \left(\left(\text^2-M^2\right)^2+\Gamma^2 M^2\right)-4 \text f(\text) (M-\text) (\text+M)=0 , with : f(M)=\frac.~


Gaussian broadening

In experiment, the incident beam that produces resonance always has some spread of energy around a central value. Usually, that is a Gaussian/normal distribution. The resulting resonance shape in this case is given by the
convolution In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions ( and ) that produces a third function (f*g) that expresses how the shape of one is modified by the other. The term ''convolution'' ...
of the Breit–Wigner and the Gaussian distribution, :V_(E; M, \Gamma, k, \sigma)= \int_^ \frac\frace^dE'. This function can be simplified by introducing new variables, :t=\frac, \quad u_1=\frac, \quad u_2=\frac, \quad a=\frac, to obtain :V_(E;M,\Gamma, k, \sigma)=\frac, where the relativistic line broadening function has the following definition, :H_2(a,u_1,u_2)=\frac\int_^\fracdt. H_2 is the relativistic counterpart of the similar line-broadening function for the
Voigt profile The Voigt profile (named after Woldemar Voigt) is a probability distribution given by a convolution of a Cauchy-Lorentz distribution and a Gaussian distribution. It is often used in analyzing data from spectroscopy or diffraction. Definition Wit ...
used in spectroscopy (see also Section 7.19 of ).


References

{{DEFAULTSORT:Relativistic Breit-Wigner Distribution Continuous distributions Particle physics