Decay Width
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Decay Width

The relativistic Breit-Wigner distribution (after the 1936 nuclear resonance formula[1] of Gregory Breit and Eugene Wigner) is a continuous probability distribution with the following probability density function,[2]

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

where k is a constant of proportionality, equal to

${\displaystyle k={\frac {2{\sqrt {2}}M\Gamma \gamma }{\pi {\sqrt {M^{2}+\gamma }}}}~~~~}$   with   ${\displaystyle ~~~~\gamma ={\sqrt {M^{2}\left(M^{2}+\Gamma ^{2}\right)}}~.}$

(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.[3]

The form of the relativistic Breit-Wigner distribution arises from the propagator of an unstable particle,[4] 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,

${\displaystyle {\frac {\sqrt {k}}{\left(E^{2}-M^{2}\right)+iM\Gamma }}~.}$

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,

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

with

${\displaystyle f(M)={\frac {k}{\Gamma ^{2}M^{2}}}.~}$

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,

${\displaystyle V_{2}(E;M,\Gamma ,k,\sigma )=\int _{-\infty }^{\infty }{\frac {k}{(E'^{2}-M^{2})^{2}+(M\Gamma )^{2}}}{\frac {1}{\sigma {\sqrt {2\pi }}}}e^{-{\frac {(E'-E)^{2}}{2\sigma ^{2}}}}dE'.}$

This function can be simplified [5] by introducing new variables,

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

to obtain

${\displaystyle V_{2}(E;M,\Gamma ,k,\sigma )={\frac {H_{2}(a,u_{1},u_{2})}{\sigma ^{2}2{\sqrt {\pi }}}},}$

where the relativistic line broadening function [5] has the following definition,

${\displaystyle H_{2}(a,u_{1},u_{2})={\frac {a}{\pi }}\int _{-\infty }^{\infty }{\frac {e^{-t^{2}}}{(u_{1}-t)^{2}(u_{2}-t)^{2}+a^{2}}}dt.}$

${\displaystyle H_{2}}$ is the relativistic counterpart of the similar line-broadening function [6] for the Voigt profile used in spectroscopy (see also Section 7.19 of [7]).

References

1. ^ Breit, G.; Wigner, E. (1936). "Capture of Slow Neutrons". Physical Review. 49 (7): 519. Bibcode:1936PhRv...49..519B. doi:10.1103/PhysRev.49.519.
2. ^ See Pythia 6.4 Physics and Manual (page 98 onwards) 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.
3. ^ Bohm, A.; Sato, Y. (2005). "Relativistic resonances: Their masses, widths, lifetimes, superposition, and causal evolution". Physical Review D. 71 (8). arXiv:hep-ph/0412106. Bibcode:2005PhRvD..71h5018B. doi:10.1103/PhysRevD.71.085018.
4. ^ Brown, L S (1994). Quantum Field Theory, Cambridge University press, ISBN 978-0521469463 , Chapter 6.3.
5. ^ a b Kycia, Rados?aw A.; Jadach, Stanis?aw (2018-07-15). "Relativistic Voigt profile for unstable particles in high energy physics". Journal of Mathematical Analysis and Applications. 463 (2): 1040-1051. arXiv:1711.09304. doi:10.1016/j.jmaa.2018.03.065. ISSN 0022-247X.
6. ^ Finn, G. D.; Mugglestone, D. (1965-02-01). "Tables of the Line Broadening Function H ( a, ? )". Monthly Notices of the Royal Astronomical Society. 129 (2): 221-235. doi:10.1093/mnras/129.2.221. ISSN 0035-8711.
7. ^ NIST handbook of mathematical functions. Olver, Frank W. J., 1924-, National Institute of Standards and Technology (U.S.). Cambridge: Cambridge University Press. 2010. ISBN 978-0-521-19225-5. OCLC 502037224.CS1 maint: others (link)

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