relativistic Breit–Wigner distribution

The probability of producing the resonance at a given energy {{mvar|E}} is proportional to {{math|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 {{mvar|E}} off the maximum at {{mvar|M}} such that {{nobr|{{math|{{!}}E{{sup|2}} − M{{sup|2}}{{!}} {{=}} MΓ}},}} (hence {{nobr|{{math|{{!}}E − M{{!}} {{=}} Γ/2}}}} for {{nobr|{{math|M ≫ Γ}}),}} the distribution {{mvar|f}} has attenuated to half its maximum value, which justifies the name width at half-maximum for {{math|Γ}}.

In the limit of vanishing width, {{nobr|{{math|Γ → 0}},}} the particle becomes stable as the Lorentzian distribution {{mvar|f}} sharpens infinitely to {{nobr|{{math|2Mδ(E{{sup|2}} − M{{sup|2}})}},}} where {{mvar|δ}} is the Dirac delta function (point impulse).

In general, {{math|Γ}} can also be a function of {{mvar|E}}; this dependence is typically only important when {{math|Γ}} is not small compared to {{mvar|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 {{math|M{{sup|2}}}} that multiplies {{math|Γ{{sup|2}}}} should also be replaced with {{math|E{{sup|2}}}}

(or {{nobr|{{math|E{{sup|4}}/M{{sup|2}}}},}} etc.) when the resonance is wide.{{cite journal |last1 = Bohm |first1 = A. |last2 = Sato |first2 = Y. |year = 2005 |title = Relativistic resonances: Their masses, widths, lifetimes, superposition, and causal evolution |journal = Physical Review D |volume = 71 |issue = 8 |page = 085018 |arxiv = hep-ph/0412106 |bibcode = 2005PhRvD..71h5018B |s2cid = 119417992 |doi = 10.1103/PhysRevD.71.085018 }}

The form of the relativistic Breit–Wigner distribution arises from the propagator of an unstable particle,{{cite book |last=Brown |first=L. S. |year=1994 |title=Quantum Field Theory |publisher=Cambridge University Press |ISBN=978-0521469463 |at=§ 6.3 }} which has a denominator of the form {{nobr|{{math|p{{sup|2}} − M{{sup|2}} + iMΓ}}.}} (Here, {{math|p{{sup|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,

$$$$

\frac{\sqrt{k}}{(E^2 - M^2) + 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 {{nobr|{{math|s {{=}} p{{sup|2}}}},}} here {{nobr|{{math|{{=}} E{{sup|2}}}}.}} 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'(\mathrm{E})\big[(\mathrm{E}^2 - M^2)^2 + \Gamma^2 M^2\big]

- 4 \mathrm{E} (M^2 - \mathrm{E}^2) f(\mathrm{E}) = 0,

or rather

$$$$

\frac{f'(\mathrm{E})}{f(\mathrm{E})} =

\frac{4 (M^2 - \mathrm{E}^2) \mathrm{E}}{(\mathrm{E}^2 - M^2)^2 + \Gamma^2 M^2},

with

$$$$

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

{{further|Cauchy distribution}}

The cross-section for resonant production of a spin-$J$ particle of mass $M$ by the collision of two particles with spins $S\_1$ and $S\_2$ is generally described by the relativistic Breit–Wigner formula:{{cite journal

|last1=Navas |first1=S.

|collaboration=Particle Data Group

|title=Review of Particle Physics: 51. Cross-Section Formulae for Specific Processes

|journal=Physical Review D

|year=2024

|volume=110 |issue=3 |page=030001

|bibcode=2018PhRvD..98c0001T

|doi=10.1103/PhysRevD.110.030001 |doi-access=free

|url= https://pdg.lbl.gov/2024/web/viewer.html?file=../reviews/rpp2024-rev-cross-section-formulae.pdf

|hdl=20.500.11850/695340

|hdl-access=free

}}

$$$$

\sigma(E_\text{cm}) = \frac{2J + 1}{(2S_1 + 1)(2S_2 + 1)} \frac{4\pi}{p_\text{cm}^2}\left[\frac{\Gamma^2/4}{(E_\text{cm} - E_0)^2 + \Gamma^2/4}\right] B_\text{in},

where $E\_\backslash text\{cm\}$ is the centre-of-mass energy of the collision, $E\_0\; =\; Mc^2$, $p\_\backslash text\{cm\}$ is the centre-of-mass momentum of each of the two colliding particles, $\backslash Gamma$ is the resonance's full width at half maximum, and $B\_\backslash text\{in\}$ is the branching fraction for the resonance's decay into particles $S\_1$ and $S\_2$.

If the resonance is only being detected in a specific output channel, then the observed cross-section will be reduced by the branching fraction ($B\_\backslash text\{out\}$) for that decay channel.

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{{cite journal |last1=Kycia |first1=Radosław A. |last2=Jadach |first2=Stanisław |date=15 July 2018 |title=Relativistic Voigt profile for unstable particles in high energy physics |journal=Journal of Mathematical Analysis and Applications |volume=463 |issue=2 |pages=1040–1051 |arxiv=1711.09304 |doi=10.1016/j.jmaa.2018.03.065 |s2cid=78086748 |issn=0022-247X |url=http://www.sciencedirect.com/science/article/pii/S0022247X18302774 |language=en }} 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{{Cite journal |last1=Finn |first1=G. D. |last2=Mugglestone |first2=D. |date=1965-02-01 |title=Tables of the line broadening function {{math|H(a, v)}} |journal=Monthly Notices of the Royal Astronomical Society |volume=129 |issue=2 |pages=221–235 |issn=0035-8711 |doi=10.1093/mnras/129.2.221 |doi-access=free |url=https://academic.oup.com/mnras/article/129/2/221/2604178 |language=en |url-access=subscription }} for the Voigt profile used in spectroscopy (see also § 7.19 of {{cite book |title=NIST Handbook of Mathematical Functions |year=2010 |publisher=Cambridge University Press |editor1=Olver, Frank W. J. |editor2=Lozier, Daniel W. |editor3=Boisvert, Ronald F. |editor4=Clark, Charles W. |series=U.S. National Institute of Standards and Technology |isbn=978-0-521-19225-5 |location=Cambridge, UK |oclc=502037224 }}).

{{reflist|25em}}

{{ProbDistributions|continuous-semi-infinite}}

{{DEFAULTSORT:Relativistic Breit-Wigner Distribution}}

Category:Continuous distributions

Category:Particle physics