|
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,〔 See (Pythia3 ) 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.〕 : where is a constant of proportionality, equal to : (This equation is written using natural units, .) It is most often used to model resonances (unstable particles) in high-energy physics. In this case, is the center-of-mass energy that produces the resonance, 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 .) The probability of producing the resonance at a given energy is proportional to , 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 off the maximum at such that , (hence for ), the distribution 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 sharpens infinitely to . In general, can also be a function of ; this dependence is typically only important when is not small compared to 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 2 that multiplies 2 should also be replaced with 2 (or 4/ 2, etc.) when the resonance is wide. The form of the relativistic Breit–Wigner distribution arises from the propagator of an unstable particle,〔Brown, L S (1994). ''Quantum Field Theory'', Cambridge University press, ISBN 978-0521469463 , Chapter 6.3.〕 which has a denominator of the form . (Here, 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, : 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 = ², here = 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, :. == References == 〔 de:Breit-Wigner-Verteilung 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Relativistic Breit–Wigner distribution」の詳細全文を読む スポンサード リンク
|