Resonances & the Breit-Wigner shape

A resonance is the way an unstable particle announces itself in data: not as a track that starts somewhere, but as a bump — a peak in a cross section or in an invariant-mass spectrum — sitting on top of a smooth background. Scan the collision energy of an $e^+e^-$ machine through $91$ GeV, or plot the invariant mass of every proton–pion pair you record, and where a short-lived state can be formed the count rate swells and then subsides. That bump has a definite centre, the resonance mass $M$ , and a definite full width at half maximum, the total decay width $\Gamma$ . The two numbers are not independent of the particle's life story: through $\tau=\hbar/\Gamma$ the width fixes the lifetime, and through $\Gamma=\sum_j\Gamma_j$ it is the sum of the partial widths into every open channel, each computed on its own from Fermi's golden rule. The universal shape of the bump is the Breit-Wigner lineshape, a Lorentzian $\sigma\propto[(E-M)^2+\Gamma^2/4]^{-1}$ . It is universal for a deep reason: it is the exact spectral fingerprint of exponential decay. A state that survives only $\sim1/\Gamma$ cannot have a sharp energy — Fourier duality smears it by $\sim\Gamma$ — so every fleeting particle, from the broad $\Delta(1232)$ baryon to the exquisitely sharp $Z$ boson, wears the same functional dress, only stretched or squeezed by its lifetime. This article builds the Breit-Wigner shape from three angles at once — the golden-rule bookkeeping of widths and branching ratios, the Fourier transform of a decaying amplitude, and the complex-mass pole of a field-theory propagator. They all give the same peak, and reading its height and width is how we weigh and time particles too short-lived to ever see directly.

Key takeaways

A resonance is a peak in a cross section or invariant-mass spectrum, centred at the mass $M$ with full width at half maximum equal to the total decay width $\Gamma$ .
The lineshape is the Breit-Wigner Lorentzian $\sigma(E)\propto1/[(E-E_R)^2+\Gamma^2/4]$ — the Fourier transform of exponential decay, not a detector artefact.
Width and lifetime are one fact: $\tau=\hbar/\Gamma$ ; a broad resonance is short-lived, a narrow one long-lived.
The total width is the sum of partial widths, $\Gamma=\sum_j\Gamma_j$ , and the branching ratio of a channel is $\mathrm{BR}(j)=\Gamma_j/\Gamma$ , with $\sum_j\mathrm{BR}(j)=1$ .
The peak height is $\sigma_{\text{peak}}=(4\pi/k^2)\,g\,\Gamma_i\Gamma_f/\Gamma^2$ , set by spins and branching ratios — so a lineshape scan weighs, times, and counts the decay modes at once.
Key idea: a resonance is exponential decay seen in the energy domain — its Breit-Wigner peak sits at the mass $M$ , its full width equals the total decay width $\Gamma$ , and through $\tau=\hbar/\Gamma$ and $\Gamma=\sum_j\Gamma_j$ that one curve reports the particle's lifetime and every branching ratio at once.

The core insight

A resonance is exponential decay seen in the energy domain: its Breit-Wigner peak sits at the mass $M$ , its full width at half maximum equals the total decay width $\Gamma$ , and through $\tau=\hbar/\Gamma$ and $\Gamma=\sum_j\Gamma_j$ that single curve encodes the particle's lifetime and the branching ratio of every decay channel.

Where this lives today

The Breit-Wigner shape is the working tool of every particle-mass measurement and every search for something new. At LEP, four experiments scanned the $Z$ resonance and fitted its lineshape to fix $M_Z$ and $\Gamma_Z$ to parts in $10^4$ ; the "invisible" slice of the width — unaccounted for by charged leptons and hadrons — gave $N_\nu=2.984\pm0.008$ , definitive proof of exactly three light neutrino families. The Higgs boson has a predicted width of only $\Gamma_H\approx4.1$ MeV, far narrower than any detector's resolution, so its width is extracted indirectly from off-shell production and interference rather than a visible peak. The top quark sits at the opposite extreme: its width $\Gamma_t\approx1.4$ GeV means a lifetime $\sim5\times10^{-25}$ s, shorter than the time to form a bound state, so the top decays before it can hadronise and no "toponium" resonance exists. Hadron spectroscopy still lives on Breit-Wigner fits — the broad $\rho$ and $\Delta$ , the razor-sharp $J/\psi$ ( $\Gamma=93$ keV), and the exotic $X(3872)$ whose sub-MeV width hints at a molecular structure. And every direct search for physics beyond the Standard Model — a heavy $Z'$ , a dijet resonance, a leptoquark — is, at bottom, a hunt for one more Breit-Wigner bump above a smooth background.

The physical picture

Picture a scattering experiment tuned across a range of energies. Most of the time the incoming particles glance off each other and the cross section drifts slowly with energy. But if, at some special energy, the two particles can briefly bind into a short-lived intermediate state — a resonance — then near that energy the reaction rate surges: the system "rings". The ringing is sharp in energy precisely to the degree that the intermediate state is long-lived, because a state that persists for a time $\tau$ can only be built from a band of energies of width $\sim\hbar/\tau$ . The first figure isolates this shape — the Breit-Wigner Lorentzian — and marks the one relation that defines the width: the peak falls to half its height a distance $\Gamma/2$ on either side of the mass, so the full width at half maximum is the total decay width $\Gamma$ .

Figure

The Breit-Wigner lineshape is a Lorentzian σ(E) ∝ 1/((E − E_R)² + Γ²/4): it peaks at the resonance energy E_R and drops to half its maximum a distance Γ/2 to each side, so the full width at half maximum equals the total decay width Γ — a short-lived state gives a broad peak, a long-lived one a sharp peak

The same shape describes states with lifetimes spanning twenty orders of magnitude — one only rescales $\Gamma$ . The broadest resonances, like the $\Delta(1232)$ baryon that dominates low-energy pion–proton scattering, decay by the strong force in $\sim10^{-24}$ s and are $\sim100$ MeV wide; the sharpest, like the $Z$ boson produced at $e^+e^-$ colliders, still live only $\sim10^{-25}$ s yet are measured to a few parts per thousand. The second figure is the crown jewel: the $Z$ resonance as scanned at LEP, a Breit-Wigner peak whose height and width, read off the real data, weigh the $Z$ , time it, and — through the missing "invisible" width — count the number of light neutrino species in the universe at three.

Figure

The Z resonance measured at LEP: the e⁺e⁻ → hadrons cross section traces a Breit-Wigner peak of σ = 41.5 nb at √s = M_Z = 91.19 GeV, with full width Γ_Z = 2.495 GeV fixing the lifetime τ = ħ/Γ_Z ≈ 2.6×10⁻²⁵ s; the peak height and width together pin the number of light neutrino families at 3

The core equations

The heart of the subject is a single lineshape. In its non-relativistic form — the shape Breit and Wigner first wrote down for slow-neutron capture — the cross section for forming a resonance in channel $i$ and observing it decay to channel $f$ is a Lorentzian in the total energy $E$ .

\sigma_{\text{BW}}(E)=\frac{\pi}{k^2}\,g\,\frac{\Gamma_i\,\Gamma_f}{(E-E_R)^2+\Gamma^2/4} \qquad (1)

where: $k$ — centre-of-mass momentum of the incoming particles; $g=\dfrac{2J+1}{(2s_a+1)(2s_b+1)}$ — spin-counting factor for a resonance of spin $J$ formed from spins $s_a,s_b$ ; $\Gamma_i,\Gamma_f$ — partial widths into the entrance and exit channels; $\Gamma$ — total width; $E_R\approx M$ — resonance energy. Reading it: the peak sits at $E=E_R$ , the height scales as $\Gamma_i\Gamma_f/\Gamma^2$ , and the denominator makes the curve a Lorentzian of full width $\Gamma$ . At collider energies the resonance is a genuine relativistic particle, and the lineshape descends from its Feynman propagator. Dressing the propagator with the decay it must undergo shifts its pole off the real axis to a complex mass, and the squared amplitude gives the relativistic Breit-Wigner.

\sigma(s)=\frac{16\pi}{s}\,g\,\frac{M^2\,\Gamma_i\,\Gamma_f}{(s-M^2)^2+M^2\Gamma^2},\qquad \mathcal{A}\propto\frac{1}{s-M^2+iM\Gamma} \qquad (2)

where: $s=E_{\text{cm}}^2$ — the Mandelstam invariant (squared centre-of-mass energy); $M$ — resonance mass; the amplitude $\mathcal{A}$ carries the resonant pole at $s=M^2-iM\Gamma$ . Reading it: the term $iM\Gamma$ is the decay width made visible in the propagator; squaring turns it into the real, symmetric denominator $(s-M^2)^2+M^2\Gamma^2$ , a Lorentzian of full width $\Gamma$ in $\sqrt{s}$ . Width, lifetime and branching ratio are three readings of the same total rate. Each open channel contributes a partial width from Fermi's golden rule; the total is their sum, its inverse is the lifetime, and the fraction going to any channel is the branching ratio.

\Gamma=\sum_j\Gamma_j,\qquad \tau=\frac{\hbar}{\Gamma},\qquad \mathrm{BR}(j)=\frac{\Gamma_j}{\Gamma},\qquad \sum_j\mathrm{BR}(j)=1 \qquad (3)

where: $\Gamma_j$ — partial width (rate) into channel $j$ ; $\tau$ — proper lifetime in the rest frame; $\mathrm{BR}(j)$ — branching ratio. Reading it: the lifetime is set by the total width, never a single partial width, while the peak's channel-dependence lives entirely in the branching ratios. Evaluating the lineshape on resonance gives the peak cross section — the number an experiment actually maximises when it tunes to the mass.

\sigma_{\text{peak}}=\frac{4\pi}{k^2}\,g\,\frac{\Gamma_i\,\Gamma_f}{\Gamma^2}=\frac{16\pi}{M^2}\,g\,\mathrm{BR}_i\,\mathrm{BR}_f \qquad (4)

where: the two forms are equal because on resonance $k^2=M^2/4$ for light initial particles. Reading it: the peak height depends only on the mass, the spins, and the two branching ratios — so a measured peak height returns $\mathrm{BR}_i\,\mathrm{BR}_f$ once the spins are known. Finally, each partial width is computed once and for all from the two-body phase-space integral, the result that grounds every $\Gamma_j$ in equation (3).

\Gamma_{fi}=\frac{p^*}{32\pi^2\,m_a^2}\int|M_{fi}|^2\,d\Omega \qquad (5)

where: $m_a$ — parent mass; $p^*$ — daughter momentum in the parent rest frame; $M_{fi}$ — Lorentz-invariant matrix element; $d\Omega$ — solid angle of one daughter. Reading it: all the kinematics collapses into the single number $p^*$ , and all the dynamics sits in $|M_{fi}|^2$ .

Derivation AHow exponential decay produces the Lorentzian lineshape

The resonance peak is not a quirk of any one reaction — it is the direct spectral fingerprint of exponential decay, and this derivation makes that identity exact. An unstable state whose probability drains as $e^{-\Gamma t}$ cannot carry a single sharp energy: by the Fourier duality between time and energy, a signal that lasts only $\sim1/\Gamma$ must be spread in energy by $\sim\Gamma$ . We write the amplitude of a decaying state, transform it to the energy domain, square it, and read off both the Breit-Wigner Lorentzian and the exact statement that its full width at half maximum equals $\Gamma$ .

Step 1. Model the state's amplitude in its rest frame as an oscillation at the rest energy $E_R$ that also decays, $\psi(t)=\psi(0)\,e^{-iE_R t}\,e^{-\Gamma t/2}$ for $t\ge0$ , so that the survival probability $|\psi(t)|^2=|\psi(0)|^2e^{-\Gamma t}$ reproduces the decay law with mean life $\tau=1/\Gamma$ . Why: the amplitude decays at half the rate of the probability, since probability is amplitude squared — this factor of two is exactly what makes the width $\Gamma$ rather than $\Gamma/2$ .

Step 2. The energy content of the signal is its Fourier transform, $\chi(E)=\int_0^\infty\psi(t)\,e^{iEt}\,dt=\dfrac{\psi(0)}{\Gamma/2-i(E-E_R)}$ . Why: a state with a definite energy would be a pure phase $e^{-iE_R t}$ and transform to a delta function; the decay envelope $e^{-\Gamma t/2}$ broadens that spike into a finite peak.

Step 3. The probability of finding the state at energy $E$ is the modulus squared, $|\chi(E)|^2\propto\dfrac{1}{(E-E_R)^2+\Gamma^2/4}$ — the Breit-Wigner (Lorentzian) lineshape, a symmetric peak centred on $E_R$ . Why: this is the same functional form as the resonant cross section, because the cross section is proportional to the squared amplitude and the resonant amplitude carries precisely this energy denominator.

Step 4. Read off the width: the Lorentzian equals half its peak value when $(E-E_R)^2=\Gamma^2/4$ , that is at $E=E_R\pm\Gamma/2$ , so the full width at half maximum is exactly $\Gamma$ . A short-lived state (large $\Gamma$ ) is broad; a long-lived state (small $\Gamma$ ) is sharp. Why: this cements $\Gamma\tau=\hbar$ as an energy–time uncertainty relation, not a measurement error.

Go deeper · Derivation B · 1 minThe unstable-particle propagator and the relativistic Breit-Wigner

Derivation BThe unstable-particle propagator and the relativistic Breit-Wigner

A stable particle of mass $M$ enters Feynman amplitudes through a propagator that diverges at $s=M^2$ — its mass shell. An unstable particle can never sit exactly on shell, and the field theory knows it: summing the self-energy corrections that let the particle dissolve into its decay products and reform shifts the pole off the real axis to a complex mass. This derivation traces that shift, shows it amounts to $M^2\to M^2-iM\Gamma$ in the propagator, and squares the amplitude to obtain the relativistic Breit-Wigner cross section whose peak sits at $s=M^2$ and whose width in $\sqrt{s}$ is $\Gamma$ .

Step 1. The tree-level propagator of a scalar of mass $M$ is $1/(s-M^2)$ ; dressing it with the one-particle-irreducible self-energy $\Pi(s)$ and resumming the geometric series gives $1/[\,s-M^2-\Pi(s)\,]$ . Why: the particle spends part of its existence as its own virtual decay products, and that dissociation is exactly what the self-energy $\Pi$ encodes.

Step 2. The optical theorem ties the imaginary part of the self-energy to the total decay rate, $\operatorname{Im}\Pi(M^2)=-M\Gamma$ . Absorbing $\operatorname{Re}\Pi$ into the physical mass and keeping the imaginary part, the propagator near the peak becomes $1/(s-M^2+iM\Gamma)$ . Why: a purely real pole would describe a stable state; the imaginary piece is the decay width made manifest, and its coefficient is fixed — not free — by unitarity.

Step 3. The cross section for $a\,b\to R\to f$ is proportional to the squared modulus of this amplitude, so $\sigma(s)\propto1/[(s-M^2)^2+M^2\Gamma^2]$ ; restoring the flux, spin and phase-space factors gives $\sigma(s)=\dfrac{16\pi}{s}\,g\,\dfrac{M^2\Gamma_i\Gamma_f}{(s-M^2)^2+M^2\Gamma^2}$ . Why: squaring the complex denominator produces the real, symmetric resonance denominator, while the numerator $\Gamma_i\Gamma_f$ records production through channel $i$ and decay through channel $f$ .

Step 4. In the narrow-width limit $\Gamma\ll M$ this is a sharp peak at $s=M^2$ of height $\propto1/\Gamma^2$ and full width $M\Gamma$ in $s$ , i.e. width $\Gamma$ in $\sqrt{s}$ , recovering the Lorentzian of the previous derivation. Why: this connects the field-theory pole directly to the observed lineshape — the same $\Gamma$ appears in the propagator, the peak, and the lifetime.

Go deeper · Derivation C · 1 minPartial widths, the total width, and the resonance peak

Derivation CPartial widths, the total width, and the resonance peak

A resonance rarely decays a single way, and the Breit-Wigner shape only makes sense once the bookkeeping of channels is clear. Each open channel contributes a partial width $\Gamma_j$ from Fermi's golden rule; the total width that sets both the lifetime and the lineshape is their sum; and the branching ratios control the peak height. This derivation assembles those pieces — partial widths, the two-body phase-space formula, branching ratios — and shows how they enter the peak so that measuring a height and a width returns the decay fractions.

Step 1. Treat the channels as independent: Fermi's golden rule gives a partial rate $\Gamma_j$ for $a\to(\text{channel }j)$ , and because the channels are mutually exclusive the total rate is the sum $\Gamma=\sum_j\Gamma_j$ , fixing the lifetime $\tau=\hbar/\Gamma$ . Why: independent, additive processes add their rates, just as several parallel leaks drain a tank at the sum of their flow rates.

Step 2. For a two-body channel $a\to1+2$ in the rest frame of $a$ , the daughters emerge back-to-back with momentum $p^*$ ; the momentum- and energy-conserving delta functions collapse the phase-space integral to $\Gamma_{fi}=\dfrac{p^*}{32\pi^2 m_a^2}\displaystyle\int|M_{fi}|^2\,d\Omega$ . Why: the kinematics reduces to the single number $p^*$ , so every channel's width is a clean product of a phase-space factor and a matrix element.

Step 3. Define the branching ratio $\mathrm{BR}_j=\Gamma_j/\Gamma$ , whence $\sum_j\mathrm{BR}_j=1$ . The Breit-Wigner numerator for entrance channel $i$ and exit channel $f$ is $\Gamma_i\Gamma_f=\Gamma^2\,\mathrm{BR}_i\mathrm{BR}_f$ , so the peak cross section $\sigma_{\text{peak}}=(16\pi/M^2)\,g\,\mathrm{BR}_i\mathrm{BR}_f$ depends only on the mass, the spins and the branching ratios. Why: this is what makes a lineshape scan so powerful — the width returns the lifetime, and the peak height returns the product of branching ratios.

Limiting cases

Case	Result	Meaning
Stable particle	$\Gamma\to0,\ \tau\to\infty$	the lineshape shrinks to $\delta(E-M)$ : a sharp mass, no width
Narrow resonance ( $\Gamma\ll M$ )	BW $\to$ Lorentzian; $\dfrac{1}{(s-M^2)^2+M^2\Gamma^2}\to\dfrac{\pi}{M\Gamma}\delta(s-M^2)$	production and decay factorise: $\sigma=\sigma_{\text{prod}}\times\mathrm{BR}_f$ (narrow-width approximation)
On peak ( $E=E_R$ )	$\sigma_{\text{peak}}=(4\pi/k^2)\,g\,\Gamma_i\Gamma_f/\Gamma^2$	maximal cross section, fixed by spins and branching ratios
Far tail ( $\|E-E_R\|\gg\Gamma$ )	$\sigma\propto\Gamma^2/(E-E_R)^2$	Lorentzian wings fall only as an inverse square — resonance tails are long
Broad or overlapping ( $\Gamma\sim M$ )	single-BW form fails	resonances interfere; one needs coupled-channel or $K$ -matrix treatments

Common trap“The width of a resonance is just the resolution of my apparatus.”

It is not. The natural width $\Gamma$ is an intrinsic property of the state, mandated by its finite lifetime through the energy–time uncertainty relation $\Delta E\,\Delta t\gtrsim\hbar$ : a particle that lives only $\tau=\hbar/\Gamma$ simply cannot possess an energy sharper than $\sim\Gamma$ . Detector resolution and beam-energy spread do broaden the observed peak, but they add on top of $\Gamma$ (in quadrature for Gaussian smearing), and precision machines such as LEP deconvolve them to recover the true $\Gamma$ . The decisive proof that $\Gamma$ is physical: the very same number, read as $\hbar/\Gamma$ , predicts the lifetime measured by a completely independent method. The $Z$ 's $2.5$ GeV width and its $2.6\times10^{-25}$ s lifetime are two faces of one fact. A companion trap: the lifetime is set by the total width $\Gamma=\sum_j\Gamma_j$ , never by any single partial width $\Gamma_j$ — a mode with a tiny branching ratio still contributes to $\Gamma$ .

Worked examples

1 · Weighing, timing and counting the Z boson (fully worked)

The $Z$ boson has mass $M_Z=91.1876$ GeV and total width $\Gamma_Z=2.4952$ GeV, with partial widths $\Gamma_{ee}=83.9$ MeV to $e^+e^-$ and $\Gamma_{\text{had}}=1744$ MeV to hadrons. Find its lifetime, and predict the peak hadronic cross section $e^+e^-\to Z\to\text{hadrons}$ .

Step 1 — lifetime from the width. Using $\tau=\hbar/\Gamma_Z$ with $\hbar=6.582\times10^{-25}$ GeV·s,

\tau=\frac{6.582\times10^{-25}\ \text{GeV}\,\cdot\,\text{s}}{2.4952\ \text{GeV}}=\mathbf{2.64\times10^{-25}\ s}.

Even at the speed of light such a $Z$ would travel only $c\tau\approx7.9\times10^{-17}$ m — a thousandth of a proton radius — which is why the $Z$ is never seen as a track but only as a resonance in the energy scan.

Step 2 — the spin factor. The $Z$ has spin $J=1$ , formed from two spin- $\tfrac12$ leptons, so

g=\frac{2J+1}{(2s_a+1)(2s_b+1)}=\frac{3}{2\times2}=\frac{3}{4}.

Then $16\pi\,g=16\pi\times\tfrac34=12\pi$ , recovering the familiar prefactor of the $Z$ lineshape.

Step 3 — the peak cross section. On resonance equation (4) gives $\sigma^0_{\text{had}}=\dfrac{12\pi}{M_Z^2}\dfrac{\Gamma_{ee}\Gamma_{\text{had}}}{\Gamma_Z^2}$ . Numerically, $\dfrac{12\pi}{M_Z^2}=\dfrac{37.70}{8315\ \text{GeV}^2}=4.53\times10^{-3}\ \text{GeV}^{-2}$ , and $\dfrac{\Gamma_{ee}\Gamma_{\text{had}}}{\Gamma_Z^2}=\dfrac{0.0839\times1.744}{2.4952^2}=0.02350$ , so

\sigma^0_{\text{had}}=4.53\times10^{-3}\times0.02350=1.066\times10^{-4}\ \text{GeV}^{-2}.

Step 4 — convert to physical units. With $1\ \text{GeV}^{-2}=0.3894\ \text{mb}=3.894\times10^{5}\ \text{nb}$ ,

\sigma^0_{\text{had}}=1.066\times10^{-4}\times3.894\times10^{5}\ \text{nb}=\mathbf{41.5\ nb},

in striking agreement with the LEP measurement of $41.54$ nb. So one Breit-Wigner peak returns the mass ( $91.19$ GeV), the lifetime ( $2.6\times10^{-25}$ s) and — through the invisible width $\Gamma_{\text{inv}}=\Gamma_Z-\Gamma_{\text{had}}-3\Gamma_{\ell\ell}$ — the count of three light neutrino families.

2 · The Δ(1232): a broad hadronic resonance (fully worked)

The $\Delta(1232)$ baryon dominates $\pi^+p$ scattering with mass $M=1232$ MeV and width $\Gamma=117$ MeV. It has spin $J=\tfrac32$ and decays essentially $100\%$ to $\pi N$ . Find its lifetime and its peak elastic cross section.

Step 1 — lifetime. $\tau=\hbar/\Gamma=(6.582\times10^{-22}\ \text{MeV}\,\cdot\,\text{s})/117\ \text{MeV}=\mathbf{5.6\times10^{-24}\ s}$ — the hallmark timescale of a strong decay. The $\Delta$ is broad precisely because it decays so fast.

Step 2 — the CM momentum on resonance. With $m_p=938.9$ MeV, $m_\pi=139.6$ MeV and $\sqrt{s}=1232$ MeV,

k=\frac{1}{2\sqrt{s}}\sqrt{[s-(m_p+m_\pi)^2]\,[s-(m_p-m_\pi)^2]}=\mathbf{227\ MeV}.

Step 3 — the spin factor. The pion is spinless and the proton is spin- $\tfrac12$ , so

g=\frac{2J+1}{(2s_\pi+1)(2s_p+1)}=\frac{4}{1\times2}=2.

Step 4 — the peak cross section. Since $\pi^+p\to\Delta\to\pi^+p$ is elastic, $\mathrm{BR}_i=\mathrm{BR}_f=1$ , and equation (4) gives $\sigma_{\text{peak}}=\dfrac{4\pi}{k^2}g=\dfrac{8\pi}{k^2}$ . With $k=0.227$ GeV, $k^2=0.0514\ \text{GeV}^2$ ,

\sigma_{\text{peak}}=\frac{8\pi}{0.0514\ \text{GeV}^2}=489\ \text{GeV}^{-2}=489\times0.3894\ \text{mb}=\mathbf{190\ mb},

close to the measured $\pi^+p$ peak of $\approx195$ mb (the small shortfall comes from the pole approximation and relativistic kinematics). A single, broad Breit-Wigner accounts quantitatively for the giant bump that dominates low-energy pion–nucleon scattering.

Physics of the portrait

The resonance formula was born in nuclear physics, in the golden years when quantum mechanics was being turned on the nucleus, and it migrated outward to become the universal language of every unstable particle.

Origins · 1 minGregory Breit and how this was discovered

Gregory Breit

1899–1981 · nuclear resonance theory and the Breit-Wigner formula

Gregory Breit, born in Nikolaev in the Russian Empire and educated at Johns Hopkins, was one of the quiet architects of American theoretical physics — a co-designer of the first large Van de Graaff accelerator and the wartime coordinator who first organised the theoretical work that became the Manhattan Project before handing it to Oppenheimer. In 1936, with the Hungarian-American theorist Eugene Wigner, he published Capture of Slow Neutrons (Phys. Rev. 49, 519), which asked why certain nuclei absorb neutrons ferociously at sharply defined energies. Their answer was that the neutron and nucleus form a short-lived compound state, and that the absorption cross section as a function of energy must trace a resonance curve — the Lorentzian $\sigma\propto[(E-E_R)^2+\Gamma^2/4]^{-1}$ now called the Breit-Wigner shape, with $\Gamma$ the width tied to the compound state's lifetime by $\Gamma\tau=\hbar$ . Wigner, who would win the 1963 Nobel Prize for the symmetry principles underlying such states, supplied the group-theoretic backbone; Breit supplied the scattering theory. What began as an account of neutron capture proved to be completely general: the same formula, upgraded to a relativistic propagator with a complex mass, describes the $\rho$ and $\Delta$ of the strong interaction, the $J/\psi$ and $\Upsilon$ of the quark model, and the $W$ and $Z$ of the electroweak theory. Every bump in every modern collider spectrum is read through the equation Breit and Wigner wrote for slow neutrons.

History

1936

— Breit and Wigner publish the resonance-capture formula for slow neutrons

1952

— Fermi's team at Chicago resolves the first Δ(1232) baryon resonance in pion–proton scattering

1961

— the ρ meson and a spectrum of hadronic resonances emerge in bubble-chamber experiments

1974

— the J/ψ appears as a resonance a thousand times narrower than the Δ, opening the charm era

1989

— LEP and SLC begin the precision Z-lineshape scan that fixes three light neutrino families

Going further

Texts. Thomson, Modern Particle Physics, §3.3 (particle decays) and §16.2 (the $Z$ resonance and its lineshape) · Halzen & Martin, Quarks and Leptons, §2.10 (resonances) · Griffiths, Introduction to Elementary Particles, 2nd ed., §6.3 · Peskin & Schroeder, An Introduction to Quantum Field Theory, §7.3 (the unstable-particle propagator and the optical theorem) · Perkins, Introduction to High Energy Physics, Ch. 4. Key numerical checks: the Fourier transform of $e^{-iE_R t-\Gamma t/2}$ gives $|\chi|^2\propto[(E-E_R)^2+\Gamma^2/4]^{-1}$ with FWHM $=\Gamma$ ; $\tau_Z=\hbar/\Gamma_Z=2.64\times10^{-25}$ s; $\sigma^0_{\text{had}}=12\pi\,\Gamma_{ee}\Gamma_{\text{had}}/(M_Z^2\Gamma_Z^2)=41.5$ nb; and the $\Delta(1232)$ peak $\sigma=8\pi/k^2=190$ mb with $k=227$ MeV. Standard values ( $M_Z,\Gamma_Z,N_\nu$ , the $\Delta$ and $J/\psi$ parameters) follow the Particle Data Group.

Prerequisites. Fermi's Golden Rule · Lorentz-Invariant Phase Space · Centre-of-Mass Frame · Particle Decay Rates. Leads to. The Z Resonance at LEP · Interaction Cross Sections · Quarkonium Bound States.

Ask the tutor: Ask me to walk through fitting a real Breit-Wigner peak — from a mass and width to a lifetime, branching ratios, and even a neutrino count.

What is the difference between a partial width and the total width?

A partial width $\Gamma_j$ is the decay rate into one specific channel; the total width $\Gamma=\sum_j\Gamma_j$ is the sum over every open channel. The distinction matters because different quantities depend on different combinations. The lifetime is set by the total width alone, $\tau=\hbar/\Gamma$ , so even a channel with a minuscule branching ratio shortens the lifetime a little. The Breit-Wigner peak height, by contrast, depends on the product of the entrance and exit partial widths, $\Gamma_i\Gamma_f$ . Never use a single partial width to infer a lifetime — the most common slip in the whole subject.

Why is the lineshape a Lorentzian and not a Gaussian?

Because it is the Fourier transform of an exponential decay, and the Fourier transform of $e^{-\Gamma t/2}$ (times a phase) is a Lorentzian, full stop. A Gaussian would arise from a Gaussian time-domain envelope, which is not how quantum states decay — they decay exponentially. The practical signature is in the tails: a Lorentzian falls only as $(E-E_R)^{-2}$ , far more slowly than a Gaussian, so resonance wings extend surprisingly far. When an observed peak looks Gaussian it is because detector resolution or beam spread — which really are Gaussian — dominates the intrinsic Lorentzian, giving a convolution of the two (a Voigt profile).

How can a particle have a “width” in its mass — isn't mass a fixed number?

For a stable particle, yes: its mass is a single sharp value. But an unstable particle has no exactly defined mass, because it does not persist long enough to define one. The energy–time uncertainty relation $\Delta E\,\Delta t\gtrsim\hbar$ says a state that lives for a time $\tau$ has an unavoidable energy spread $\Delta E\sim\hbar/\tau=\Gamma$ . So when you plot the invariant mass of a resonance's decay products, you get a distribution of width $\Gamma$ , not a spike. "The $Z$ has mass $91.19$ GeV and width $2.5$ GeV" means the reconstructed mass scatters around $91.19$ with a Breit-Wigner spread of $2.5$ GeV — a genuine quantum blur, not a measurement error.

When do I need the relativistic Breit-Wigner rather than the non-relativistic one?

The non-relativistic form, a Lorentzian in energy $E$ with denominator $(E-E_R)^2+\Gamma^2/4$ , is accurate whenever the width is small compared with the mass and you are working near the peak — most nuclear and atomic resonances qualify. The relativistic form, with denominator $(s-M^2)^2+M^2\Gamma^2$ in the invariant $s$ , is needed when the resonance is a relativistic particle produced at a collider, when $\Gamma$ is not tiny compared with $M$ , or when you fit the tails far from the peak. The two agree in the narrow-width limit near resonance; they differ in the wings and in the precise peak normalisation, which matters for parts-per-thousand measurements like the $Z$ lineshape.

How did the Z lineshape at LEP count the neutrinos?

The total width $\Gamma_Z$ is the sum of the widths into every final state the $Z$ can reach: charged leptons, hadrons, and neutrinos. The charged and hadronic pieces are measured directly from their event rates. Whatever width is left over — the "invisible" width — must be the $Z$ decaying to neutrino pairs, which escape undetected. Since each neutrino species contributes a calculable partial width $\Gamma_\nu\approx167$ MeV, the number of light species is $N_\nu=\Gamma_{\text{inv}}/\Gamma_\nu$ . Fitting the peak height and width of the $Z$ resonance gave $N_\nu=2.984\pm0.008$ — decisively three, and a landmark constraint on the particle content of the universe.

Why do broad, overlapping resonances need more than a single Breit-Wigner?

A single Breit-Wigner assumes one isolated pole with a fixed width. When two resonances sit close together, or a single state is broad enough to feel the opening of new decay channels, that picture breaks. Amplitudes, not cross sections, are what add, so nearby resonances interfere — producing peaks, dips and asymmetric shapes that a lone Lorentzian cannot reproduce. A width that changes with energy as thresholds open (a "running" width) further distorts the shape. The proper tools are coupled-channel formalisms and the $K$ -matrix, which enforce unitarity across all channels simultaneously and reduce to independent Breit-Wigners only in the isolated, narrow limit.