Physics:Breit-Wigner distribution

Short description: Probability distribution describing resonance phenomena in particle physics

Breit–Wigner distribution

Parameters	$M>0$ (resonance mass) $\mathrm{\Gamma }>0$ (resonance width)
Support	$E\in (0,\mathrm{\infty })$ (relativistic form) or $x\in (-\mathrm{\infty },\mathrm{\infty })$ (non-relativistic form)
PDF	See below
CDF	(no simple closed form for the relativistic case)
Mean	$\simeq M$ (for $\mathrm{\Gamma }\ll M$)
Median	$M$ (non-relativistic)
Mode	$M$
Variance	Undefined (non-relativistic Cauchy form); finite for relativistic form with restricted support
MGF	Does not exist (non-relativistic form)

The Breit–Wigner distribution (also known as the Lorentzian distribution in the non-relativistic limit) is a probability distribution that arises in particle physics to describe the cross-section of a resonance. It is named after physicists Gregory Breit and Eugene Wigner, who first described the distribution in 1936 in the context of nuclear scattering theory.

The distribution characterises the energy (or invariant mass) profile of an unstable particle (resonance) and is intimately connected to the particle's mean lifetime through the uncertainty principle.

Overview

When an unstable particle of mass $M$ and total decay width $\mathrm{\Gamma }$ is produced in a scattering experiment, the probability of observing the particle at a given centre-of-mass energy $E$ (or equivalently at a given invariant mass $\sqrt{s}$) is not a delta function at $E=M$. Instead, the finite lifetime $\tau =\hslash /\mathrm{\Gamma }$ of the resonance leads to a spread in energy described by the Breit–Wigner distribution.

In the non-relativistic limit, the Breit–Wigner distribution reduces to the well-known Cauchy distribution (Lorentzian). In high-energy particle physics, however, the relativistic form is more commonly used.

Probability density function

Non-relativistic (Cauchy) form

The non-relativistic Breit–Wigner distribution is given by:

$f(E)=\frac{1}{2\pi }\frac{\mathrm{\Gamma }}{(E-M{)}^2+{\left(\frac{\mathrm{\Gamma }}{2}\right)}^2}$

where:

• $E$ is the energy (or mass),
• $M$ is the resonance energy (mass of the unstable particle),
• $\mathrm{\Gamma }$ is the full width at half maximum (FWHM), equal to the total decay width.

This is identical to the Cauchy distribution with location parameter $x_0=M$ and scale parameter $\gamma =\mathrm{\Gamma }/2$.

Relativistic form

In relativistic quantum field theory, the Breit–Wigner distribution is more properly written in terms of the Lorentz invariant Mandelstam variable $s=E_{\mathrm{c}\mathrm{m}}^2$ (the square of the centre-of-mass energy). The relativistic Breit–Wigner distribution is:

$f(s)=\frac{1}{N}\frac{k}{{(s-M^2)}^2+M^2{\mathrm{\Gamma }}^2}$

where:

• $s$ is the square of the centre-of-mass energy,
• $M$ is the rest mass of the resonance (in natural units),
• $\mathrm{\Gamma }$ is the total decay width of the resonance,
• $k$ is a constant of proportionality (often $k=\frac{2\sqrt{2}M\mathrm{\Gamma }{\gamma }_m}{\pi \sqrt{M^2+{\gamma }_m}}$ with ${\gamma }_m=\sqrt{M^2(M^2+{\mathrm{\Gamma }}^2)}$, depending on convention),
• $N$ is a normalisation factor.

A commonly used simplified form, expressed as a function of the invariant mass $m\equiv \sqrt{s}$, is:

$f(m)=\frac{A}{{(m^2-M^2)}^2+M^2{\mathrm{\Gamma }}^2}$

where $A$ is a normalisation constant.

Energy-dependent width

In many practical applications, the total width $\mathrm{\Gamma }$ is not constant but depends on the invariant mass. For a resonance decaying into two particles, the energy-dependent width (also called running width) is:

$\mathrm{\Gamma }(m)={\mathrm{\Gamma }}_0{\left(\frac{q}{q_0}\right)}^{2L+1}\frac{M}{m}\frac{B_L^2(q)}{B_L^2(q_0)}$

where:

• ${\mathrm{\Gamma }}_0\equiv \mathrm{\Gamma }(M)$ is the nominal width at the pole mass,
• $q$ is the magnitude of the momentum of either daughter particle in the rest frame of the resonance at mass $m$,
• $q_0$ is the same evaluated at $m=M$,
• $L$ is the orbital angular momentum of the decay,
• $B_L$ are the Blatt–Weisskopf barrier penetration factors.

The relativistic Breit–Wigner with energy-dependent width then becomes:

$f(m)\propto \frac{m^2\mathrm{\Gamma }(m{)}^2}{{(m^2-M^2)}^2+m^2\mathrm{\Gamma }(m{)}^2}$

This form is widely used in partial wave analysis and Dalitz plot analyses.

Connection to the propagator

The Breit–Wigner distribution arises naturally from the modulus squared of the propagator of an unstable particle. For a scalar resonance, the propagator in the vicinity of the pole is:

${𝒫}(s)=\frac{1}{s-M^2+iM\mathrm{\Gamma }}$

The cross-section (or rate) is proportional to $|{𝒫}(s){|}^2$:

$|{𝒫}(s){|}^2=\frac{1}{{(s-M^2)}^2+M^2{\mathrm{\Gamma }}^2}$

which is the relativistic Breit–Wigner line shape. The imaginary part of the self-energy of the particle gives rise to the finite width $\mathrm{\Gamma }$, and the real part shifts the mass.

Properties

Relation to the Cauchy distribution

In the non-relativistic limit, where $E\simeq M$ and $\mathrm{\Gamma }\ll M$, one can write $s-M^2=(E-M)(E+M)\simeq 2M(E-M)$, and the relativistic Breit–Wigner reduces to the non-relativistic (Cauchy) form up to normalisation.

Half-width at half-maximum

For the non-relativistic form, the full width at half maximum (FWHM) is exactly $\mathrm{\Gamma }$. For the relativistic form expressed in $\sqrt{s}$, the FWHM is approximately $\mathrm{\Gamma }$ when $\mathrm{\Gamma }\ll M$.

Relation to lifetime

The total width $\mathrm{\Gamma }$ is related to the mean lifetime $\tau$ of the resonance by:

$\tau =\frac{\hslash }{\mathrm{\Gamma }}$

This is a direct consequence of the energy–time uncertainty relation. A short-lived particle has a large width, and vice versa.

Moments

The non-relativistic Breit–Wigner (Cauchy) distribution has no finite moments—neither the mean nor the variance is defined. However, the median and the mode both equal $M$.

Applications in particle physics

The Breit–Wigner distribution is ubiquitous in particle physics. Some notable applications include:

• Z boson: The Z boson resonance at $M_Z\sim 91.2\text{GeV}/c^2$ with ${\mathrm{\Gamma }}_Z\sim 2.5\text{GeV}$ is one of the most precisely measured Breit–Wigner line shapes. Measurements at LEP and SLC used the Z line shape to determine the number of light neutrino generations.

• W boson: The W boson resonance at $M_W\sim 80.4\text{GeV}/c^2$ with ${\mathrm{\Gamma }}_W\simeq 2.1\text{GeV}$.

• Hadron resonances: Many hadronic resonances such as the $\rho (770)$, $\mathrm{\Delta }(1232)$, and $f0(500)/\sigma$ are described by Breit–Wigner (or modified Breit–Wigner) distributions. Very broad resonances may require more sophisticated parameterisations.

• Higgs boson: The Higgs boson at $M_H\simeq 125\text{GeV}/c^2$ has a predicted Standard Model width of ${\mathrm{\Gamma }}_H\simeq 4.1\text{MeV}$, which is far too narrow to be resolved directly by the LHC detectors. The observed line shape is dominated by detector resolution rather than the intrinsic Breit–Wigner width.

• Nuclear physics: The original application by Breit and Wigner was to neutron capture cross-sections of nuclei, where individual nuclear resonances appear as Breit–Wigner peaks.

Generalisations and related distributions

• Voigt profile: The convolution of a Breit–Wigner (Lorentzian) distribution with a Gaussian distribution. This is used when both the natural line width and instrumental (detector) resolution must be accounted for.

• Fano resonance: A generalisation that describes asymmetric line shapes arising from the interference between a resonant and a non-resonant (background) scattering amplitude. The Fano line shape reduces to the Breit–Wigner form in the limit of zero coupling to the continuum.

• Flatté distribution: A parameterisation used for resonances near a decay threshold, where the energy-dependent width has a specific threshold behaviour. It is commonly used for scalar mesons such as the $f0(980)$ and $a0(980)$.

• K-matrix formalism: A more general approach to parameterising overlapping resonances while maintaining unitarity.

• Cauchy distribution: The non-relativistic limit of the Breit–Wigner distribution, widely used in statistics and spectroscopy.

Interference effects

When multiple resonances overlap or when a resonance interferes with a non-resonant background amplitude, the simple Breit–Wigner form must be modified. The total amplitude is the coherent sum of individual amplitudes:

${𝒜}(s)=\sum _k\frac{a_k{e}^{i{\phi }_k}}{s-M_k^2+i{M}_k{\mathrm{\Gamma }}_k}+{{𝒜}}_{\text{bg}}(s)$

and the observable cross-section is proportional to $|{𝒜}(s){|}^2$. This can lead to characteristic interference patterns such as dips and asymmetric peaks, which are not described by an incoherent sum of Breit–Wigner distributions.

Pole position

In modern particle physics, the pole position of a resonance is considered a more fundamental quantity than the Breit–Wigner mass and width parameters, which can be scheme-dependent. The pole is located in the complex $s$-plane at:

$s_{\text{pole}}={(M_{\text{pole}}-\frac{i}{2}{\mathrm{\Gamma }}_{\text{pole}})}^2$

For narrow resonances, $M_{\text{pole}}\simeq M$ and ${\mathrm{\Gamma }}_{\text{pole}}\simeq \mathrm{\Gamma }$, but for broad resonances the pole parameters can differ significantly from the Breit–Wigner parameters.

The Particle Data Group now quotes pole parameters for many broad resonances in addition to (or instead of) Breit–Wigner parameters.

History

The distribution was introduced by Gregory Breit and Eugene Wigner in 1936 to describe the cross-section for slow neutron capture by atomic nuclei. They showed that near an isolated nuclear resonance, the capture cross-section takes the characteristic Lorentzian form. The formula was a landmark result in nuclear physics and laid the groundwork for the R-matrix theory of nuclear reactions.

With the development of particle physics in the mid-20th century, the Breit–Wigner distribution became the standard tool for describing short-lived particles (resonances) observed as peaks in invariant mass distributions and excitation curves.

See also

• Cauchy distribution
• Resonance (particle physics)
• Decay width
• Cross section
• Voigt profile
• Fano resonance
• Propagator (quantum field theory)
• Dalitz plot
• Partial wave analysis
• Particle Data Group

Further reading

• Particle Data Group (2022). "Resonances: kinematics, production and decay". Review of Particle Physics. American Physical Society. https://pdg.lbl.gov/. 
• Blatt, John M.; Weisskopf, Victor F. (1952). Theoretical Nuclear Physics. John Wiley & Sons. ISBN 0-486-66827-4. 
• Pilkuhn, Hartmut (1979). Relativistic Particle Physics. Springer-Verlag. ISBN 0-387-09348-6. 
• Byckling, Eero; Kajantie, Keijo (1973). Particle Kinematics. John Wiley & Sons. ISBN 0-471-12885-6. 

External links

• Particle Data Group
• Breit–Wigner Distribution at MathWorld