The Cluster's Breit-Wigner Distribution
A page about the Breit-Wigner distribution — the characteristic line shape of a resonant scattering process.
The Breit-Wigner formula
The Breit-Wigner distribution describes the cross section of a resonant process: sigma(E) = sigma_0 * (Gamma^2 / 4) / ((E - E_R)^2 + Gamma^2 / 4) where E_R is the resonance energy (mass), Gamma is the width (decay rate), and sigma_0 is the peak cross section. In terms of the dimensionless variable x = (E - E_R) / (Gamma / 2): sigma(E) = sigma_0 / (1 + x^2)
The full width at half maximum (FWHM) is Gamma. In the cluster, the edit Breit-Wigner formula gives an edit resonant cross section.
The physical origin
A resonance is a quasi-bound state with a finite lifetime tau = hbar / Gamma. The energy-time uncertainty relation Delta E Delta t ~ hbar gives the width. The resonance appears as a peak in the scattering cross section because the intermediate state has an enhanced probability of being formed.
The scattering amplitude near resonance is: f(E) ~ 1 / (E_R - E - i Gamma / 2) and the cross section is sigma ~ |f(E)|^2, giving the Breit-Wigner form.
In the cluster, the edit scattering amplitude gives an edit cross section.
The relativistic Breit-Wigner
For a relativistic particle with mass m and width Gamma: sigma(s) = sigma_0 * (m^2 Gamma^2) / ((s - m^2)^2 + m^2 Gamma^2) where s = E^2 is the center-of-mass energy squared. The propagator denominator is: D(s) = s - m^2 + i m Gamma
The branching ratio to channel i is: B_i = Gamma_i / Gamma_total where Gamma_i is the partial width for the i-th decay channel.
In the cluster, the edit relativistic Breit-Wigner gives an edit propagator denominator.
Examples
- Delta baryon: E_R = 1232 MeV, Gamma = 117 MeV (strong decay)
- Z boson: E_R = 91.187 GeV, Gamma = 2.495 GeV
- Higgs boson: E_R = 125.25 GeV, Gamma = 4.07 MeV (tiny — electromagnetic/weak decay)
- J/psi: E_R = 3097 MeV, Gamma = 92.9 keV (narrow — OZI suppression)
In the cluster, the edit examples give an edit mass and an edit width.
Applications
- Particle physics: Identifying new resonances in collider data
- Nuclear physics: Nuclear level densities and scattering cross sections
- Condensed matter: Kondo resonance, Fano resonance
- Atomic physics: Autoionizing states, Rydberg series
- Medical physics: Positron emission tomography (PET) uses Breit-Wigner line shapes
In the cluster, edit applications include:
- edit Particle physics
- edit Nuclear physics
- edit Condensed matter
- edit Atomic physics
- edit Medical physics
The Fano profile
Fano (1961) generalized the Breit-Wigner to include interference between a resonance and a continuum: sigma(E) = sigma_0 * (q + x)^2 / (1 + x^2) where q is the Fano parameter. This asymmetric line shape is common in atomic and condensed matter physics.
In the cluster, the edit Fano profile gives an edit asymmetric line shape.
This distribution
This page is about the Breit-Wigner distribution. sigma(E) = sigma_0 / (1 + ((E-E_R)/(Gamma/2))^2). Gamma = hbar / tau. Examples: Delta(117 MeV), Z(2.5 GeV), H(4.07 MeV). The distribution is real.