 Decay Width
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Decay Width

The relativistic Breit-Wigner distribution (after the 1936 nuclear resonance formula of Gregory Breit and Eugene Wigner) is a continuous probability distribution with the following probability density function,

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

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 ? is the resonance width (or decay width), related to its mean lifetime according to . (With units included, the formula is .)

## Usage

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 |E2 - M2| = M?, (hence |E - M| = ?/2 for M >> ?), the distribution f has attenuated to half its maximum value, which justifies the name for ?, width at half-maximum.

In the limit of vanishing width, ? -> 0, the particle becomes stable as the Lorentzian distribution f sharpens infinitely to 2M?(E2 - M2).

In general, ? can also be a function of E; this dependence is typically only important when ? 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 M2 that multiplies ?2 should also be replaced with E2 (or E 4/M2, etc.) when the resonance is wide.

The form of the relativistic Breit-Wigner distribution arises from the propagator of an unstable particle, which has a denominator of the form p2 - M2 + iM?. (Here, p2 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 amplitude of the solution to 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 = p2, here = E2. 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,

$f'({\text{E}})\left(\left({\text{E}}^{2}-M^{2}\right)^{2}+\Gamma ^{2}M^{2}\right)-4{\text{E}}f({\text{E}})(M-{\text{E}})({\text{E}}+M)=0,$ with

$f(M)={\frac {k}{\Gamma ^{2}M^{2}}}.~$ 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 of the Breit-Wigner and the Gaussian distribution,

$V_{2}(E;M,\Gamma ,k,\sigma )=\int _{-\infty }^{\infty }{\frac {k}{(E'^{2}-M^{2})^{2}+(M\Gamma )^{2}}}{\frac {1}{\sigma {\sqrt {2\pi }}}}e^{-{\frac {(E'-E)^{2}}{2\sigma ^{2}}}}dE'.$ This function can be simplified  by introducing new variables,

$t={\frac {E-E'}{{\sqrt {2}}\sigma }},\quad u_{1}={\frac {E-M}{{\sqrt {2}}\sigma }},\quad u_{2}={\frac {E+M}{{\sqrt {2}}\sigma }},\quad a={\frac {k\pi }{2\sigma ^{2}}},$ to obtain

$V_{2}(E;M,\Gamma ,k,\sigma )={\frac {H_{2}(a,u_{1},u_{2})}{\sigma ^{2}2{\sqrt {\pi }}}},$ where the relativistic line broadening function  has the following definition,

$H_{2}(a,u_{1},u_{2})={\frac {a}{\pi }}\int _{-\infty }^{\infty }{\frac {e^{-t^{2}}}{(u_{1}-t)^{2}(u_{2}-t)^{2}+a^{2}}}dt.$ $H_{2}$ is the relativistic counterpart of the similar line-broadening function  for the Voigt profile used in spectroscopy (see also Section 7.19 of ).