Breit-Wigner distribution

The Breit-Wigner distribution is a Probability distribution that models particle resonance:

$E$ is the center-of-mass energy of the process, $E_{n}$ is the rest energy of the resonant particle and $\Gamma$ is the resonance width or decay width, measured in MeV or GeV, and equal to

where $\tau$ the mean lifetime of the resonant particle. This is a consequence of the energy-time uncertainty principle $\Delta E\Delta t\gtrsim\hbar$: if the duration $\Delta t$ is the life of the particle $\tau$, there is an uncertainty on its rest energy $\Delta E\gtrsim \hbar/\tau$. If we take this to be lowest-uncertainty and call $\Delta E=\Gamma$ then $\Gamma\sim \hbar/\tau$. The resonance width is then the inherent quantum uncertainty on the rest energy of the resonant particle, which increases are particles become more short-lived. This is why it's also called decay width, as large $\Gamma$ mean faster, more likely decay.

Written in terms of invariant mass we have $E=m_{0}c^{2}$ and $E_{n}=m_{X}c^{2}$ and so

which is the relativistic Breit-Wigner distribution.

The cross section of the particle scattering process in which the resonance occurs is locally proportional to the Breit-Wigner distribution around the mass peak. When $E=E_{n}$ (equivalently: $m_{0}=m_{X}$), the scattering resonates and maximizes the cross section (and therefore Probability) of the resonant particle being formed. This is useful to more reliably produce resonant particles by manipulating the center-of-mass energy to be just right for the resonance.

Mathematically, it is the distribution of the energy of a resonant state which decays exponentially in time. It is a Cauchy distribution centered on $E_{n}$. $\Gamma$ is the full width at half maximum of the distribution.