# 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.