Relativistic Breit–Wigner distribution

The relativistic Breit–Wigner distribution (after the 1936 nuclear resonance formula^[1] of Gregory Breit and Eugene Wigner) is a continuous probability distribution with the following probability density function,^[2]

f(E) = \frac{k}{\left(E^2-M^2\right)^2+M^2\Gamma^2}~,

where $k$ is a constant of proportionality, equal to

k = \frac{2 \sqrt{2} M \Gamma \gamma }{\pi \sqrt{M^2+\gamma}} ~~~~

with

~~~~ \gamma=\sqrt{M^2\left(M^2+\Gamma^2\right)} ~.

(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. Note that for values of $E$ off the maximum at $M$ such that $| E 2 - M 2 | = MΓ$ , (hence $| E - M | = Γ /2$ for $M≫Γ$ ), the distribution $f$ 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 $f$ sharpens infinitely to $2 M δ (E 2 - M 2)$ .

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 $M$ ² that multiplies $Γ$ ² should also be replaced with $E$ ² (or $E$ ⁴/ $M$ ², etc.) when the resonance is wide.^[3]

The form of the relativistic Breit–Wigner distribution arises from the propagator of an unstable particle,^[4] which has a denominator of the form $p 2 - M 2 + iMΓ$ . (Here, $p$ ² is the square of the four-momentum 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{\sqrt{k}}{\left(E^2-M^2\right)+iM\Gamma}~.

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 solution of the classical equation of motion for a driven harmonic oscillator damped and driven by a sinusoidal external force. It has the standard resonance form of the Lorentz, or Cauchy distribution, but involves relativistic variables $s$ = $p$ ², here = $E$ ².

The distribution is the solution of the differential equation, analogous to that for the time averaged input power of the above classical forced oscillator,

\left\{{\begin{array}{l}f'({\text{E}})\left(\left({\text{E}}^{2}-M^{2}\right)^{2}+\Gamma ^{2}M^{2}\right)-4{\text{E}}f({\text{E}})(M-{\text{E}})({\text{E}}+M)=0\\[10pt]f(M)={\frac {k}{\Gamma ^{2}M^{2}}}\end{array}}\right\}

Further information: Cauchy distribution

References

↑ Breit, G.; Wigner, E. (1936). "Capture of Slow Neutrons". Physical Review. 49 (7): 519. Bibcode:1936PhRv...49..519B. doi:10.1103/PhysRev.49.519.
↑ See Pythia 6.4 Physics and Manual (page 98 onwards) 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.
↑ Bohm, A.; Sato, Y. (2005). "Relativistic resonances: Their masses, widths, lifetimes, superposition, and causal evolution". Physical Review D. 71 (8). arXiv:hep-ph/0412106. Bibcode:2005PhRvD..71h5018B. doi:10.1103/PhysRevD.71.085018.
↑ Brown, L S (1994). Quantum Field Theory, Cambridge University press, ISBN 978-0521469463 , Chapter 6.3.

Probability distributions

List

Discrete univariate with finite support	Benford Bernoulli beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher discrete uniform Zipf Zipf–Mandelbrot

Discrete univariate with infinite support	beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta

Continuous univariate supported on a bounded interval	arcsine ARGUS Balding–Nichols Bates beta beta rectangular Irwin–Hall Kumaraswamy logit-normal noncentral beta raised cosine reciprocal triangular U-quadratic uniform Wigner semicircle

Continuous univariate supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind beta prime Burr chi-squared chi Dagum Davis exponential-logarithmic Erlang exponential F folded normal Flory–Schulz Fréchet gamma gamma/Gompertz generalized inverse Gaussian Gompertz half-logistic half-normal Hotelling's T-squared hyper-Erlang hyperexponential hypoexponential inverse chi-squared scaled inverse chi-squared inverse Gaussian inverse gamma Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal Lomax matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami noncentral chi-squared Pareto phase-type poly-Weibull Rayleigh relativistic Breit–Wigner Rice shifted Gompertz truncated normal type-2 Gumbel Weibull Discrete Weibull Wilks's lambda

Continuous univariate supported on the whole real line	Cauchy exponential power Fisher's z Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's S_U Landau Laplace asymmetric Laplace logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel Tracy–Widom variance-gamma Voigt

Continuous univariate with support whose type varies	generalized extreme value generalized Pareto Tukey lambda q-Gaussian q-exponential q-Weibull shifted log-logistic

Mixed continuous-discrete univariate	rectified Gaussian

Multivariate (joint)	Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet multivariate normal multivariate stable multivariate t normal-inverse-gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart

Directional	Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped asymmetric Laplace wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham

Degenerate and singular	Degenerate Dirac delta function Singular Cantor

Families	Circular compound Poisson elliptical exponential natural exponential location-scale maximum entropy mixture Pearson Tweedie wrapped

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