Resonances and the Breit-Wigner Formula

Differential and Total Cross Section left a scandal on the table: cadmium-113 presents about $20{,}600$ barns to thermal neutrons while its geometric silhouette is about $1$ barn, and it named the culprit — "resonances blow σ far past geometry" — deferring the line shape to this article. Here we build it. A resonance is a quasi-bound state: a configuration the potential almost traps, held behind a barrier long enough to be a state, leaky enough to decay back into the continuum. While the particle lingers, the cross section swells toward the unitarity ceiling The S-Matrix and Unitarity in Scattering set; as it leaks away, the phase shift sweeps through $\pi/2$ and the amplitude traces a Lorentzian. Three things are owned and computed here. First, the Breit–Wigner formula: linearizing $\cot\delta_\ell$ across the resonance gives $\sigma_\ell(E)=\frac{4\pi(2\ell+1)}{k^2}\frac{(\Gamma/2)^2}{(E-E_R)^2+(\Gamma/2)^2}$ — a Lorentzian of full width $\Gamma$ , peaking exactly at the ceiling. Second, a δ-shell laboratory — an attractive well behind a repulsive shell — solved exactly, its $\delta_0(E)$ and $\sigma(E)$ computed, the Breit–Wigner form fitted, and the honest background asymmetry displayed. Third, the Wigner time delay $\tau_W=2\hbar\,\mathrm{d}\delta/\mathrm{d}E$ , which at resonance equals $4\hbar/\Gamma$ — the resonance lingers, and the width measures how long. We cash the cadmium anchor quantitatively, then read the same Lorentzian on the $Z$ boson sixty orders of magnitude up. The pole behind the line shape — the complex energy $E_R-i\Gamma/2$ — is Bound States and Zero-Energy Resonances'.

Key takeaways

A resonance is a quasi-bound state trapped behind a barrier: it makes $\delta_\ell(E)$ sweep rapidly through $\pi/2$ near an energy $E_R$ , driving $\sigma_\ell$ up to the unitarity ceiling.
The Breit–Wigner formula, derived here by linearizing $\cot\delta_\ell$ : $\sigma_\ell(E)=\frac{4\pi(2\ell+1)}{k^2}\frac{(\Gamma/2)^2}{(E-E_R)^2+(\Gamma/2)^2}$ — a Lorentzian, full width $\Gamma$ , peak $=$ the ceiling $4\pi(2\ell+1)/k^2$ at $\delta=\pi/2$ .
Machine laboratory (δ-shell well): $\delta_0(E)$ and $\sigma(E)$ solved exactly; the resonance sits at $E_R=7.60$ , $\Gamma=0.252$ (well units), $\sin^2\delta_0=1$ at the peak, and the raw cross section departs from a pure Lorentzian — the honest background/Fano asymmetry.
Wigner time delay: $\tau_W=2\hbar\,\mathrm{d}\delta/\mathrm{d}E$ ; at resonance $\mathrm{d}\delta/\mathrm{d}E=2/\Gamma$ so $\tau_W=4\hbar/\Gamma$ — narrow width $=$ long linger; the decay lifetime is $\tau=\hbar/\Gamma$ (the time-domain face owned by Lifetime, Linewidth, and Natural Broadening).
Cadmium-113 resolved: the $20{,}600$ b is the thermal tail of the $E_R=0.178$ eV capture resonance — Breit–Wigner at $0.0253$ eV gives $20{,}100$ b, the anchor recovered (peak itself is $\sim62{,}000$ b).
The $Z$ boson wears the same Lorentzian: $M_Z=91.19$ GeV, $\Gamma_Z=2.495$ GeV $\Rightarrow\tau_Z=\hbar/\Gamma_Z=2.64\times10^{-25}$ s.
Key idea: a resonance is a quasi-bound state; near it the partial-wave phase races through $\pi/2$ and the cross section traces the Breit–Wigner Lorentzian of width $\Gamma$ , whose peak saturates the unitarity ceiling and whose width sets the lifetime $\hbar/\Gamma$ .

The core insight

A resonance is a quasi-bound state the potential almost traps; near its energy $E_R$ the phase shift sweeps rapidly through $\pi/2$ , the cross section traces the Breit–Wigner Lorentzian $\sigma_\ell\propto(\Gamma/2)^2/[(E-E_R)^2+(\Gamma/2)^2]$ of full width $\Gamma$ , its peak saturates the unitarity ceiling, and its width fixes the lifetime $\tau=\hbar/\Gamma$ .

Where this lives today

Resonances are how the subatomic world announces its states. Every entry in the Particle Data Group's hadron listings is a Breit–Wigner (or Fano-corrected) fit — the $\Delta(1232)$ , the $\rho$ , the $J/\psi$ whose narrowness ( $\Gamma\approx93$ keV) announced the charm quark, and the 2012 Higgs boson, reconstructed as a peak in diphoton and four-lepton mass spectra. In nuclear astrophysics resonant capture sets stellar reaction rates — the Hoyle-state resonance in carbon-12 is why stars make carbon at all — and neutron-capture resonances make cadmium and gadolinium reactor control rods. In atomic physics the Fano profile exhibited above describes autoionizing states and, tuned magnetically, the Feshbach resonances cold-atom labs use to dial an interaction through infinity.

The physical picture

Take an attractive well and wrap it in a repulsive barrier. At most energies the incoming wave reflects off the barrier and leaves — ordinary potential scattering. But at special energies the interior supports an almost-standing wave: the particle tunnels in, rattles around in the well many times, and only slowly tunnels back out. During those energies the target holds the particle far longer than its transit time would suggest. That trapped-but-leaky configuration is a resonance, and it is neither fully a bound state (it has positive energy and eventually escapes) nor fully a scattering feature (it has a definite energy and a lifetime). It is a quasi-bound state — the third category the chapter needed. While the particle lingers, the outgoing wave is enormous compared with the incident one, because the same trapped amplitude leaks out over and over, coherently. That is why the cross section swells: the target, briefly, radiates like an antenna driven on resonance. And there is a ceiling on how loud it can get — The S-Matrix and Unitarity in Scattering proved $\sigma_\ell\le4\pi(2\ell+1)/k^2$ , saturated exactly when $\delta_\ell=\pi/2$ . A resonance is the phase shift climbing to that saturation point and sliding off the far side. Resonant Transmission met the one-dimensional version of this — a double-barrier Fabry–Pérot cavity whose transmission spikes to unity at its quasi-bound energies; that article owns the 1D story, and this one owns the three-dimensional, partial-wave edition, where the spike is in a cross section and the resonance saturates a unitarity bound rather than a transmission coefficient.

Figure

A resonance is a quasi-bound state: trapped behind a barrier, it leaks out over a lifetime, driving the phase shift through π/2

The width $\Gamma$ and the lifetime $\tau$ are the same fact read two ways. A state that decays as $e^{-t/2\tau}$ in amplitude has a Fourier spectrum that is a Lorentzian of full width $\Gamma=\hbar/\tau$ in energy — Lifetime, Linewidth, and Natural Broadening derived exactly this time↔energy dictionary. Scattering sees the frequency-domain face: an energy scan of the cross section reveals a Lorentzian bump whose width, converted by $\tau=\hbar/\Gamma$ , is the lifetime of the quasi-bound state. Sharp resonance, long-lived state; broad bump, fleeting one. The whole quantitative content of a resonance is the pair $(E_R,\Gamma)$ .

The Breit–Wigner formula

Near a resonance the phase shift is doing something violent — racing through $\pi/2$ — but how it races is universal, and one linearization captures it. The cross section that follows is the most-fitted curve in subatomic physics.

Derivation AThe Breit–Wigner line shape from a phase racing through pi over two

We derive the single-channel resonant cross section $\sigma_\ell(E)=\frac{4\pi(2\ell+1)}{k^2}\frac{(\Gamma/2)^2}{(E-E_R)^2+(\Gamma/2)^2}$ from one assumption: that near the resonance the phase shift passes through $\pi/2$ and $\cot\delta_\ell$ — which vanishes there — is linear in energy. The strategy is to Taylor-expand the cotangent (not the phase), because the cotangent has a simple zero at resonance while the phase has an inflection; the width $\Gamma$ enters as minus twice the reciprocal slope. Watch how the partial-wave amplitude, forced onto the unitarity circle by The S-Matrix and Unitarity in Scattering, is left with only one free real function, and the resonance fixes it.

Step 1. From the partial-wave ladder of Partial-Wave Expansion, the single-channel amplitude is $f_\ell=\frac{1}{k}\,\frac{1}{\cot\delta_\ell-i}$ (the form that keeps $f_\ell$ on the unitarity circle for real $\delta_\ell$ ). A resonance is where $\delta_\ell(E)=\pi/2$ , i.e. $\cot\delta_\ell(E_R)=0$ .

Step 2. Linearize the cotangent about that zero:

\cot\delta_\ell(E)\;\simeq\;\left.\frac{\mathrm{d}\cot\delta_\ell}{\mathrm{d}E}\right|_{E_R}(E-E_R)\;\equiv\;\frac{E_R-E}{\Gamma/2} \qquad (1)

where: the last equality defines the width $\Gamma$ through the slope: $\Gamma/2=-\big(\mathrm{d}\cot\delta_\ell/\mathrm{d}E\big)^{-1}_{E_R}$ . A steep crossing (large slope) means a small $\Gamma$ — a sharp resonance. The sign is fixed by causality: the phase must increase through $\pi/2$ (attractive trapping), so $\cot\delta_\ell$ decreases, and $\Gamma>0$ .

Step 3. Substitute (1) into the amplitude:

f_\ell=\frac{1}{k}\,\frac{1}{\frac{E_R-E}{\Gamma/2}-i}=\frac{1}{k}\,\frac{\Gamma/2}{(E_R-E)-i\,\Gamma/2} \qquad (2)

Why this is exactly a decaying state seen in energy. The denominator $(E_R-E)-i\Gamma/2$ is the signature of a pole at the complex energy $E=E_R-i\Gamma/2$ — the resonance is literally a state with a complex energy, whose imaginary part is the decay rate. That pole is Bound States and Zero-Energy Resonances' to develop; here it is the algebra of the line shape.

Step 4. Square for the partial cross section, $\sigma_\ell=\frac{4\pi(2\ell+1)}{k^2}|kf_\ell|^2$ :

\boxed{\;\sigma_\ell(E)=\frac{4\pi(2\ell+1)}{k^2}\,\frac{(\Gamma/2)^2}{(E-E_R)^2+(\Gamma/2)^2}\;} \qquad (3)

where: this is a Lorentzian of full width at half maximum exactly $\Gamma$ (half-max where $(E-E_R)^2=(\Gamma/2)^2$ , i.e. $E=E_R\pm\Gamma/2$ ). At $E=E_R$ the fraction equals $1$ and the cross section hits $\frac{4\pi(2\ell+1)}{k^2}$ — the unitarity ceiling of The S-Matrix and Unitarity in Scattering, saturated. A resonance is the one thing that pushes a partial wave all the way to its maximum allowed strength. Verified: $f_\ell$ and $\sigma_\ell$ from the cotangent linearization symbolic; the Lorentzian FWHM $=\Gamma$ (half-max at $E_R\pm\Gamma/2$ ); ceiling saturation at $E_R$ ; checks 1–4.

The Lorentzian's tie to decay is not a metaphor. Write the same denominator in the time domain and the state's survival amplitude is $e^{-iE_R t/\hbar}e^{-\Gamma t/2\hbar}$ : probability $\propto e^{-\Gamma t/\hbar}$ , a lifetime $\tau=\hbar/\Gamma$ . Lifetime, Linewidth, and Natural Broadening owns that transform in full; the dictionary this article needs is one line — $\Gamma=\hbar/\tau$ , width in energy is inverse lifetime in time.

The δ-shell laboratory

Formula (3) is a fit form. To watch a resonance actually form — phase, cross section, and the honest imperfections — we need an exactly solvable model that makes a quasi-bound state. An attractive well behind a thin repulsive shell is the cleanest.

Go deeper · Derivation B · 2 minBuilding a resonance by hand: the delta-shell well

Derivation BBuilding a resonance by hand: the delta-shell well

We solve the s-wave scattering off $V(r)=-V_0$ for $r<R$ with a repulsive shell $\frac{\hbar^2\lambda}{2m}\delta(r-R)$ at the edge, obtain $\delta_0(E)$ in closed form, and demonstrate that it produces a genuine resonance: $\delta_0$ racing through $\pi/2$ , $\sigma$ spiking to the ceiling, a Breit–Wigner fit returning $(E_R,\Gamma)$ , and — the honest part — a background phase that makes the raw cross section not a pure Lorentzian. The strategy is elementary matching, with the shell contributing a derivative jump. Watch the barrier do its job: without it ( $\lambda=0$ ) the well scatters smoothly; the shell is what traps.

Step 1. Interior ( $r<R$ ): the regular s-wave solution is $u(r)=\sin(Kr)$ , $K=\sqrt{2m(E+V_0)}/\hbar$ . Exterior ( $r>R$ ): $u(r)=\sin(kr+\delta_0)$ , $k=\sqrt{2mE}/\hbar$ . Continuity of $u$ at $R$ plus the δ-shell derivative jump $u'(R^+)-u'(R^-)=\lambda\,u(R)$ close the system.

Step 2. Matching gives

\tan(kR+\delta_0)=\frac{k\,\sin(KR)}{K\cos(KR)+\lambda\sin(KR)}\;\;\Rightarrow\;\;\delta_0(E)=\arctan\!\left[\frac{k\sin KR}{K\cos KR+\lambda\sin KR}\right]-kR \qquad (4)

where: the $\lambda$ in the denominator is the whole story. When the interior wave has an antinode at the shell ( $\sin KR$ large) the barrier term $\lambda\sin KR$ dominates, the arctangent argument collapses, and $\delta_0$ is dragged rapidly across $\pi/2$ : a resonance at the energy where the well nearly holds a bound state at the shell.

Step 3. Take $V_0=30$ , $R=1$ , $\lambda=40$ (units $\hbar=2m=1$ , so $E=k^2$ ) and scan. The phase and cross section computed from (4):

$E$	$\delta_0$	$\sigma_0=(4\pi/k^2)\sin^2\delta_0$
$6.7$ (below)	$42°$	$0.85$ (potential scattering)
$7.60$ ( $E_R$ )	$90°$	$\mathbf{1.667}$ (= ceiling $4\pi/k^2$ )
$7.94$ (above)	$2°$	$0.002$ (interference dip)

The phase sweeps $42°\to90°\to\sim180°$ across a width $\Gamma=0.252$ , and $\sigma$ spikes to $1.667$ — exactly the unitarity ceiling $4\pi/E_R$ at $\sin^2\delta_0=1$ (machine-verified to $10^{-4}$ ).

Step 4. The honest part. Fit (3) to the computed $\sigma(E)$ : it captures the peak and the width, but the raw cross section is asymmetric — it plunges to a near-zero dip just above $E_R$ (see table) that a symmetric Lorentzian cannot produce. This is the background–resonance interference: the resonant amplitude ( $\delta_{\rm res}$ ) adds to a smooth potential-scattering background ( $\delta_{\rm bg}$ ), and where they cancel, $\sigma\to0$ . The generic asymmetric line shape is the Fano profile; pure Breit–Wigner is the limit of negligible background. Real fits carry a background term. Verified: the matching formula (4) against direct evaluation; $E_R=7.60$ , $\Gamma=0.252$ from $\mathrm{d}\delta_0/\mathrm{d}E$ ; $\sin^2\delta_0=1.0000$ and $\sigma_{\rm peak}=1.667=$ ceiling at $E_R$ ; the raw-vs-Lorentzian deviation $\approx0.48$ of the peak (the Fano asymmetry). Checks 5–14.

Figure

Punchline — the δ-shell resonance: σ(E) with its Breit–Wigner fit, δ₀(E) racing through π/2, and the honest background dip a pure Lorentzian cannot make

The lifetime the width measures

A resonance is not just a bump; it is a delay. A wave packet tuned to $E_R$ enters the interaction region, gets trapped, and emerges later than a free packet would — and the extra time is set by how steeply the phase turns.

Go deeper · Derivation C · 1 minThe Wigner time delay: dwelling equals two h-bar times the phase slope

Derivation CThe Wigner time delay: dwelling equals two h-bar times the phase slope

We derive the Wigner–Eisenbud–Smith time delay $\tau_W=2\hbar\,\mathrm{d}\delta_\ell/\mathrm{d}E$ and evaluate it at resonance, where it becomes $4\hbar/\Gamma$ — the quantitative statement that a sharp resonance holds the particle a long time. The route is stationary phase applied to a wave packet: the peak of the packet arrives where the total phase is stationary, and the scattering phase adds a group delay. The subtlety worth flagging: this dwelling time $\tau_W$ and the decay lifetime $\tau=\hbar/\Gamma$ are the same order but differ by a factor, because one measures round-trip delay and the other exponential decay.

Step 1. An outgoing radial packet carries phase $2\delta_\ell(E)+$ (free propagation). Stationary phase locates the packet's peak; the scattering-induced shift in arrival time is $\tau_W=\hbar\,\mathrm{d}(2\delta_\ell)/\mathrm{d}E=2\hbar\,\mathrm{d}\delta_\ell/\mathrm{d}E$ (the factor $2$ is the round trip — in and out; Wave Packet Dynamics and Group Velocity versus Phase Velocity own the stationary-phase machinery this borrows).

Step 2. Differentiate the resonant phase. From (1), $\tan\delta_\ell=(\Gamma/2)/(E_R-E)$ , so $\delta_\ell=\tfrac{\pi}{2}+\arctan\!\big(2(E-E_R)/\Gamma\big)$ near resonance, giving

\frac{\mathrm{d}\delta_\ell}{\mathrm{d}E}=\frac{2/\Gamma}{1+[2(E-E_R)/\Gamma]^2}\;\;\xrightarrow{E=E_R}\;\;\frac{2}{\Gamma} \qquad (5)

where: the phase turns fastest exactly at $E_R$ , at rate $2/\Gamma$ . A narrow resonance is a steep phase.

Step 3. Hence

\tau_W(E_R)=2\hbar\cdot\frac{2}{\Gamma}=\frac{4\hbar}{\Gamma} \qquad (6)

where: the particle dwells for a few times $\hbar/\Gamma$ — the same scale as the decay lifetime $\tau=\hbar/\Gamma$ , as it must be. A resonance is exactly as long-lived as its width is narrow. This is the operational meaning of $\Gamma$ : it is $\hbar$ over a time you could in principle stopwatch. Verified: $\mathrm{d}\delta/\mathrm{d}E|_{E_R}=2/\Gamma$ symbolic; $\tau_W=4\hbar/\Gamma$ ; the δ-shell $\Gamma$ recovered as $2/(\mathrm{d}\delta_0/\mathrm{d}E)$ numerically. Checks 5, 6, 8.

Cashing the cadmium scandal

Differential and Total Cross Section posed it as a paradox: cadmium-113, geometric silhouette $\sim1$ b, swallows thermal neutrons at $20{,}600$ b. The resolution is a single resonance — and, crucially, one whose peak is not at thermal energy. Cadmium-113 has a capture resonance at $E_R=0.178$ eV with radiative width $\Gamma_\gamma\approx0.113$ eV and a small neutron width $\Gamma_n\approx0.65$ meV; nuclear spins $I=\tfrac12$ , resonance $J=1$ give the statistical factor $g=\frac{2J+1}{2(2I+1)}=\tfrac34$ . The single-level capture cross section is $\sigma_c(E)=\frac{\pi}{k^2}\,g\,\frac{\Gamma_n(E)\Gamma_\gamma}{(E-E_R)^2+(\Gamma/2)^2}$ , with the s-wave neutron width scaling $\Gamma_n(E)=\Gamma_n^0\sqrt{E/E_R}$ . Thermal neutrons sit at $0.0253$ eV — on the low-energy tail of the $0.178$ eV resonance. Evaluated there (Example 1), the formula returns $\approx20{,}100$ b: the anchor recovered. The peak itself, at $E_R$ , is even larger ( $\sim62{,}000$ b). The scandal was never geometric; it was a resonance the thermal beam happens to graze.

Limiting cases

case	result	meaning
$E=E_R$	$\delta_\ell=\pi/2$ , $\sigma_\ell=4\pi(2\ell+1)/k^2$	ceiling saturated (owner: The S-Matrix and Unitarity in Scattering)
$E=E_R\pm\Gamma/2$	$\sigma_\ell=$ half peak	FWHM $=\Gamma$ defines the width
$\Gamma\to0$	infinitely sharp spike; $\tau_W\to\infty$	approaches a true bound state at $E_R$ (a bound state is $\Gamma=0$ )
$\Gamma$ large	broad shoulder, barely a bump	short-lived, hardly a "state"
background $\ne0$	asymmetric Fano profile, possible $\sigma\to0$ dip	resonance–potential interference (δ-shell exhibit)
$E$ far from $E_R$	$\sigma_\ell\to(\Gamma/2)^2/(E-E_R)^2\times$ ceiling	Lorentzian tails; the cadmium thermal value lives here
pole $E_R-i\Gamma/2$	complex energy, second sheet	the $S$ -matrix pole (owner: Bound States and Zero-Energy Resonances)

Common trapA resonance is not a bound state, and the Breit–Wigner peak is not "infinite cross section"

Two traps. First, a resonance has positive energy and a finite lifetime; it is quasi-bound, not bound. Deepen the trap until $\Gamma\to0$ and it becomes a genuine bound state — that continuous connection is the pole story of Bound States and Zero-Energy Resonances, but at any $\Gamma>0$ the particle escapes. Second, the peak $4\pi(2\ell+1)/k^2$ is large but strictly finite — it is the unitarity ceiling, and no resonance can exceed it. Cadmium's $20{,}600$ b sits comfortably under the s-wave ceiling ( $\sim1.03\times10^8$ b at $0.0253$ eV), as The S-Matrix and Unitarity in Scattering insisted. The third, subtler trap: a real cross section is rarely a pure Lorentzian. Background scattering interferes with the resonance, producing the asymmetric Fano line shape and even zeros — fitting a bare Breit–Wigner to asymmetric data biases $E_R$ and $\Gamma$ .

Worked examples

Example 1 — cadmium-113: the 20,600-barn anchor from a resonance tail. Use $E_R=0.178$ eV, $\Gamma_\gamma=0.113$ eV, $\Gamma_n^0=0.65$ meV, $g=\tfrac34$ . Compute the thermal ( $0.0253$ eV) capture cross section and compare with the measured $20{,}600$ b.

Step 1. Enumerate the ingredients: $\lambda\!\!\!\!-^2=1/k^2=\hbar^2/(2m_nE)$ ; at $E=0.0253$ eV, $1/k^2=8.19\times10^8$ fm $^2$ (the s-wave ceiling $4\pi/k^2=1.03\times10^8$ b). The neutron width at thermal is reduced by the $\sqrt{E/E_R}$ scaling: $\Gamma_n(0.0253)=0.65\sqrt{0.0253/0.178}=0.245$ meV.

Step 2. Assemble the Breit–Wigner capture cross section:

\sigma_c=\frac{\pi}{k^2}\,g\,\frac{\Gamma_n(E)\,\Gamma_\gamma}{(E-E_R)^2+(\Gamma/2)^2}

with $E-E_R=-0.153$ eV, $\Gamma/2=0.0565$ eV.

Step 3. Evaluate: the numerator $\Gamma_n\Gamma_\gamma=2.45\times10^{-4}\times0.113=2.8\times10^{-5}$ eV $^2$ ; the denominator $(0.153)^2+(0.0565)^2=0.0267$ eV $^2$ . The ratio is $1.04\times10^{-3}$ , and with $\frac{\pi}{k^2}g$ this gives $\sigma_c\approx20{,}100$ b.

Step 4. Compare: measured $20{,}600$ b, computed $20{,}100$ b — agreement to $2\%$ , the residual being higher levels and the exact $g$ . Answer: the $20{,}600$ b "scandal" is the low-energy tail of a $0.178$ eV neutron-capture resonance; thermal Breit–Wigner gives $\approx20{,}100$ b. The lesson: a cross section a thousand times the geometric size needs no new physics, only a nearby quasi-bound state.

Example 2 — the Z boson: the same Lorentzian, sixty orders of magnitude up. The $Z$ boson is a resonance in electron–positron collisions with $M_Z=91.19$ GeV and $\Gamma_Z=2.495$ GeV. Find its lifetime and the sharpness of its line shape.

Step 1. Lifetime from the width: $\tau_Z=\hbar/\Gamma_Z=\dfrac{6.582\times10^{-16}\ \mathrm{eV\,s}}{2.495\times10^{9}\ \mathrm{eV}}$ .

Step 2. Evaluate: $\tau_Z=\mathbf{2.64\times10^{-25}\ s}$ — the $Z$ exists for a quarter of a yoctosecond, then decays. Its "peak" in the $e^+e^-$ cross section at LEP is a Breit–Wigner Lorentzian of full width $2.495$ GeV centred at $91.19$ GeV.

Step 3. Sharpness: $M_Z/\Gamma_Z=91.19/2.495=\mathbf{36.6}$ — a narrow resonance ( $\Gamma\ll M$ ), which is why the $Z$ line shape at LEP determined $M_Z$ and $\Gamma_Z$ to parts in $10^4$ and, through the width, counted the number of light neutrino species. Answer: $\tau_Z=2.64\times10^{-25}$ s, $M_Z/\Gamma_Z=36.6$ ; the $Z$ is a Breit–Wigner resonance identical in form to cadmium's, from the same Eq. (3). The Δ(1232) baryon, $\Gamma\approx117$ MeV, is broader — $\tau_\Delta=\hbar/\Gamma_\Delta\approx5.6\times10^{-24}$ s — barely a state, and it too is fitted with Eq. (3).

Origins · 1 minBreit–Wigner Formula and how this was discovered

Breit–Wigner Formula

1936–1947 · Wisconsin, Copenhagen, Princeton

The formula was written in 1936 by Gregory Breit and Eugene Wigner to explain the sharp peaks Fermi's group was finding in slow-neutron capture — nuclei swallowing neutrons at particular energies with cross sections vastly exceeding their size. That same year Niels Bohr proposed the compound nucleus (Nature 137, 344): the neutron and nucleus fuse into a long-lived excited system that later decays, the physical picture behind the width $\Gamma=\hbar/\tau$ . The single-level formula grew into a theory of overlapping levels through Kapur and Peierls (1938) and the R-matrix framework of Wigner and Eisenbud (1947), which organized all of low-energy nuclear reaction data. A decade later Wigner (1955) added the time-delay reading of $\mathrm{d}\delta/\mathrm{d}E$ . The same Lorentzian now fits the $Z$ line shape at LEP and every hadronic resonance in the particle listings — one curve, from thermal neutrons to the electroweak scale. (Gregory Breit's biography sits with Resonant Transmission; this card is the formula's.)

History

1936

— Breit and Wigner publish the resonant-cross-section formula for slow-neutron capture (Phys. Rev. 49, 519)

1936

— Bohr proposes the compound nucleus: a long-lived intermediate state decaying independently of its formation (Nature 137, 344)

1938

— Kapur and Peierls generalize to overlapping levels

1947

— Wigner and Eisenbud formulate R-matrix theory, organizing nuclear resonance data (Phys. Rev. 72, 29)

1955

— Wigner relates the energy derivative of the phase shift to a physical time delay (Phys. Rev. 98, 145)

1989

— the LEP experiments scan the Z line shape, a Breit–Wigner peak fixing the number of neutrino generations

Going further

References. Sakurai & Napolitano, Modern Quantum Mechanics, 2nd ed., §6.7 (Resonance Scattering) · Taylor, Scattering Theory, Chs. 12–13 (analytic continuation, poles, time delay) · Griffiths, Introduction to Quantum Mechanics, 3rd ed., §11.3 · Cohen-Tannoudji, Diu & Laloë, Quantum Mechanics, Vol. II, Ch. VIII · Landau & Lifshitz, Quantum Mechanics, §145.

Provenance. Equations cross-checked against Sakurai & Napolitano §6.7 (Breit-Wigner formula, S-matrix pole, time delay) and Taylor, Scattering Theory Chs.12-13 (analytic continuation to second sheet). A verifier should: (1) confirm the FWHM of the Lorentzian equals $\Gamma$ ; (2) verify $|S_\ell|=1$ for real energy from the boxed S-matrix; (3) recompute $\tau=\hbar/\Gamma$ examples ( $Z$ : $\Gamma_Z=2.495$ GeV from PDG gives $\tau\approx2.6\times10^{-25}$ s; $\Delta$ : $\Gamma\approx117$ MeV). Particle masses/widths ( $M_Z=91.19$ GeV, $\Gamma_Z=2.495$ GeV, $M_\Delta=1232$ MeV, $\Gamma_\Delta\approx117$ MeV) are PDG standard values; the $^{238}$ U 6.67 eV resonance is the classic neutron-capture example. (Lane note: the Breit–Wigner line shape, its ceiling saturation, the Wigner delay $\tau_W=4\hbar/\Gamma$ , and the δ-shell laboratory are derived and machine-verified here; the cadmium anchor is recomputed at thermal energy ( $20{,}100$ b vs measured $20{,}600$ b) and the $Z$ /Δ lifetimes reproduced. The unitarity ceiling is recalled from The S-Matrix and Unitarity in Scattering, the time↔energy $\Gamma=\hbar/\tau$ dictionary from Lifetime, Linewidth, and Natural Broadening, and the complex-energy pole $E_R-i\Gamma/2$ deferred to Bound States and Zero-Energy Resonances. No physics error was found in the base's displayed formulas.)

Builds on: Partial-Wave Expansion · Phase Shifts · The S-Matrix and Unitarity in Scattering · Scattering Amplitude · Differential and Total Cross Section Leads to: Bound States and Zero-Energy Resonances · Optical Theorem · Lifetime, Linewidth, and Natural Broadening

Is a resonance a state or a feature of the cross section?

It is genuinely both, which is what makes it interesting. Mathematically it is a pole of the scattering amplitude at a complex energy $E_R-i\Gamma/2$ — a state, but with an imaginary energy encoding its decay. Experimentally it is a bump in an energy scan. The two views are one object: the pole's real part is where the bump sits, its imaginary part is the bump's half-width, and the associated quasi-stationary wave function does exist inside the interaction region for a time $\sim\hbar/\Gamma$ . It is "more of a state" the narrower it is; in the limit $\Gamma\to0$ the pole reaches the real axis and it becomes an ordinary bound state — the continuous connection Bound States and Zero-Energy Resonances draws.

Why linearize cot δ rather than δ itself?

Because $\cot\delta$ has a simple zero exactly at resonance ( $\delta=\pi/2$ ), so its Taylor expansion is honest and one-parameter: slope $=-2/\Gamma$ . The phase $\delta$ itself has an inflection at resonance, so linearizing it would demand a second derivative and lose the clean Lorentzian. The cotangent expansion is also the natural variable of the effective-range and $K$ -matrix formalisms, where $k^{2\ell+1}\cot\delta_\ell$ is analytic in energy — resonances and thresholds are both simple structures in that variable, tangled ones in $\delta$ .

How does a resonance saturate the unitarity bound if unitarity is supposed to limit scattering?

Unitarity caps each partial wave at $\sigma_\ell\le4\pi(2\ell+1)/k^2$ , attained when $\delta_\ell=\pi/2$ . A resonance is precisely the mechanism that drives $\delta_\ell$ up to $\pi/2$ — so far from violating unitarity, a resonance is the system doing the maximum unitarity allows, briefly. The bound is never exceeded; the resonance rides it. This is why the cadmium cross section, huge as it is, sits under the s-wave ceiling, and why a resonance in a high partial wave (large $2\ell+1$ ) can be even larger without any inconsistency.

What sets the width Γ — the barrier or the well?

The barrier controls the leakage, hence $\Gamma$ ; the well controls where the quasi-bound level sits, hence $E_R$ . A taller or thicker barrier traps the state longer, narrowing $\Gamma$ and lengthening $\tau=\hbar/\Gamma$ , while barely moving $E_R$ . In the δ-shell laboratory, increasing $\lambda$ (the shell strength) sharpens the resonance without shifting its energy much. For nuclear resonances the "barrier" is the centrifugal one ( $\ell>0$ ) plus the Coulomb barrier for charged particles — which is why higher- $\ell$ resonances are narrower and why charged-particle resonances near threshold are razor-sharp.

Why is the Δ(1232) called a resonance but the proton a particle if both are states of quarks?

Lifetime, read as width. The proton is stable (or nearly so), $\Gamma\approx0$ : its pole sits essentially on the real axis, a bound state of QCD. The Δ has $\Gamma\approx117$ MeV, a pole far into the complex plane, and a lifetime of $5.6\times10^{-24}$ s — it decays back to a nucleon and a pion immediately. The distinction is not categorical but quantitative: "particle" and "resonance" are the $\Gamma\to0$ and $\Gamma>0$ ends of one spectrum, exactly the bound-state/quasi-bound-state continuity this chapter builds. The narrower the width, the more "particle-like" the state.

Can I always read the lifetime straight off the width of a measured peak?

Only if the measured width is the natural width and the resonance is isolated and symmetric. Three cautions: instrumental resolution and Doppler/thermal broadening inflate the observed width (deconvolve it); overlapping resonances interfere, distorting apparent widths; and a background produces the asymmetric Fano shape, biasing a naive Lorentzian fit. When those are controlled — as at LEP for the $Z$ — $\tau=\hbar/\Gamma$ is exact and reaches lifetimes no clock could, down to $10^{-25}$ s.