The Z resonance & LEP

Every process that Quantum Electrodynamics builds from photon exchange has a hidden twin. Because the neutral Z boson couples to every flavour of fermion, the photon in any diagram can be replaced by a Z — and for $e^+e^-\to\mu^+\mu^-$ this gives a second lowest-order s-channel graph sitting beside the QED one, with propagator $\propto g_Z^2/(q^2-m_Z^2)$ instead of $e^2/q^2$ . At low energy the massive denominator smothers the Z and QED rules; near $\sqrt{s}=m_Z\approx 91$ GeV the Z denominator collapses and its rate overwhelms pure QED by three orders of magnitude. The annihilation cross section soars into a sharp resonance — the Z resonance. Taken literally the propagator would diverge at $\sqrt{s}=m_Z$ , which is nonsense: it forgets that the Z is unstable. Promoting the rest-frame phase $e^{-imt}$ of a decaying state to $e^{-imt}e^{-\Gamma t/2}$ makes the replacement $m_Z\to m_Z-i\Gamma_Z/2$ , and the pole slides off the real axis. What was a divergence becomes a finite Breit–Wigner peak whose height is set by $1/(m_Z^2\Gamma_Z^2)$ , whose width is the total decay rate $\Gamma_Z$ , and whose position is the mass. Between 1989 and 2000 the four LEP experiments at CERN scanned this peak with ~17 million Z decays and measured its shape to parts in $10^5$ — pinning $m_Z=91.1876$ GeV, $\Gamma_Z=2.4952$ GeV, and, from the peak's height, the number of light neutrino generations: three. This article is where a QED cross section, an unstable particle's lifetime, and the most precise test of the electroweak Standard Model become one curve.

Key takeaways

Replacing the photon by a Z in $e^+e^-\to f\bar f$ adds an s-channel diagram with propagator $\propto g_Z^2/(s-m_Z^2)$ that dominates near $\sqrt{s}=m_Z$ .
The Z's finite lifetime enters as $m_Z\to m_Z-i\Gamma_Z/2$ , giving the propagator $1/(s-m_Z^2+im_Z\Gamma_Z)$ and a finite peak instead of a divergence.
The cross section traces a Breit–Wigner in $\sqrt{s}$ : $\sigma_{f\bar f}=\dfrac{12\pi s}{m_Z^2}\dfrac{\Gamma_{ee}\Gamma_{f\bar f}}{(s-m_Z^2)^2+m_Z^2\Gamma_Z^2}$ , peaking at $\sigma^0_{f\bar f}=\dfrac{12\pi}{m_Z^2}\dfrac{\Gamma_{ee}\Gamma_{f\bar f}}{\Gamma_Z^2}$ .
The peak position gives $m_Z$ , its width gives $\Gamma_Z$ , and its height gives the partial widths and hence the Z couplings $c_V^f=T_3^f-2Q_f\sin^2\theta_W$ , $c_A^f=T_3^f$ .
The "missing" height of the peak measures the invisible width $\Gamma_{\text{inv}}$ , and dividing by $\Gamma_{\nu\nu}$ counts light neutrino species: $N_\nu=2.984\pm0.008$ — exactly three.
Key idea: an unstable particle's finite lifetime regularises its propagator pole, turning what would be a divergence at $\sqrt{s}=m_Z$ into a finite, measurable Breit–Wigner peak whose position, width, and height read off the Z's mass, total decay rate, and couplings.

The core insight

An unstable particle's finite lifetime regularises its propagator pole, turning what would be a divergence at $\sqrt{s}=m_Z$ into a finite, measurable Breit–Wigner peak whose position, width, and height directly read off the Z's mass, total decay rate, and its vector and axial couplings to every fermion.

Where this lives today

The Z line shape is the anchor of the electroweak precision program, and its measurements are still, decades later, among the most accurate numbers in physics. The LEP value $m_Z=91.1876\pm0.0021$ GeV is a defining input to the global electroweak fit, and its precision demanded beam-energy calibration by resonant depolarization at the $10^{-5}$ level — sensitive enough that the LEP team had to correct for the tidal deformation of the 27 km ring by the Moon, the water level in Lake Geneva, and even stray currents from passing TGV trains leaking into the beam pipe. That same fit, using $m_Z$ , $\sin^2\theta_W^{\text{eff}}$ , and the W mass, predicted the top-quark mass before the Tevatron found it and boxed in the Higgs mass to below ~160 GeV years before the LHC discovery at 125 GeV — a triumph of loop-level consistency. Today the frontier is to do it all again a hundred times better: proposed circular electron–positron colliders, the FCC-ee at CERN and the CEPC in China, would run as "Tera-Z" factories producing $10^{12}$ Z bosons — five orders of magnitude more than LEP — shrinking the uncertainties on $m_Z$ , $\Gamma_Z$ , and the couplings to the point where they could reveal new heavy particles indirectly through their virtual effects. The invisible width also remains a live probe of physics beyond the Standard Model: any new light state the Z can decay into would widen $\Gamma_Z$ and lower the peak, so the resonance that counted three neutrinos still stands guard over what else might be hiding in the dark.

The physical picture

Start from a process you already know — $e^+e^-\to\mu^+\mu^-$ through a virtual photon — and add its electroweak twin. The incoming electron and positron can annihilate not only into a photon but into a Z, because the Z, like the photon, is a neutral spin-1 boson that couples to charged leptons. The two amplitudes carry the same Dirac spinor structure; they differ only in their coupling and their propagator. The photon's propagator is $1/s$ ; the Z's is $1/(s-m_Z^2)$ . Far below the peak the $m_Z^2$ in the Z denominator dwarfs $s$ and the Z contribution is suppressed by $s/m_Z^2$ . As $\sqrt{s}$ climbs toward $m_Z$ the denominator shrinks toward zero and the Z amplitude explodes — until its own decay width cuts it off. The first figure shows the two diagrams that compete for the same final state.

Figure

The two lowest-order s-channel diagrams for e⁺e⁻ → μ⁺μ⁻: photon exchange with propagator ∝ e²/s and Z exchange with propagator ∝ g_Z²/(s − m_Z²). Far below the peak the massive Z denominator suppresses it; at √s = m_Z the Z propagator 1/(s − m_Z² + i m_Z Γ_Z) is enormous and the Z rate beats pure QED by about a factor of 1000

Now sweep the collision energy and watch the Z amplitude. Because the propagator would blow up at $\sqrt{s}=m_Z$ , the theory only makes sense once the Z's own instability is put in: the pole is displaced into the complex plane by $-i\Gamma_Z/2$ , and the cross section becomes a finite bump. Its width in energy is precisely the total decay rate $\Gamma_Z$ , so the peak is narrow — $\Gamma_Z/m_Z\approx 2.7\%$ — and tall. The height of the peak is not arbitrary: it is fixed by the partial widths into the initial ( $\Gamma_{ee}$ ) and final ( $\Gamma_{f\bar f}$ ) states divided by $\Gamma_Z^2$ . That last fact is the entire precision program in one line — because $\Gamma_Z$ is the sum over every decay channel including invisible ones, the peak height weighs the neutrinos you cannot see. The second figure is the actual measured line shape, and the plot that announced three generations.

Figure

The measured Z line shape: hadronic cross section σ(e⁺e⁻ → Z → hadrons) versus √s, a Breit-Wigner peaking at σ⁰ = 41.5 nb at m_Z = 91.19 GeV with full width Γ_Z = 2.495 GeV. The three curves are the Standard Model prediction for 2, 3, and 4 light neutrino species; the LEP data (red) sit squarely on the N = 3 curve, because more neutrinos widen Γ_Z and lower the peak as 1/Γ_Z²

The core equations

The whole story begins with a single edit to the Z propagator. A stable-particle propagator has a real pole at $s=m_Z^2$ ; an unstable particle carries a phase that also decays, $e^{-imt}\to e^{-imt}e^{-\Gamma t/2}$ , which is the statement $m_Z\to m_Z-i\Gamma_Z/2$ . Squaring $m_Z-i\Gamma_Z/2$ and dropping the tiny $\tfrac14\Gamma_Z^2$ term ( $\Gamma_Z\ll m_Z$ ) moves the pole off the real axis.

\frac{1}{s-m_Z^2}\;\longrightarrow\;\frac{1}{s-m_Z^2+im_Z\Gamma_Z} \qquad (1)

where: $s=q^2$ — squared centre-of-mass energy (the s-channel invariant); $m_Z$ — Z mass; $\Gamma_Z$ — total Z decay width; $i m_Z\Gamma_Z$ — the imaginary displacement that keeps the amplitude finite. Reading it: the pole never touches the real $s$ axis, so the cross section peaks at a finite height instead of diverging. Squaring the amplitude that contains propagator (1) and dressing the vertices with the neutral-current couplings gives the Breit–Wigner cross section — the master formula LEP measured point by point.

\sigma_{f\bar f}(s)=\frac{12\pi\,s}{m_Z^2}\,\frac{\Gamma_{ee}\,\Gamma_{f\bar f}}{(s-m_Z^2)^2+m_Z^2\Gamma_Z^2} \qquad (2)

where: $\Gamma_{ee}$ — partial width for $Z\to e^+e^-$ (the production vertex, fixed by the $e^+e^-$ beams); $\Gamma_{f\bar f}$ — partial width into the observed final state (the decay vertex); $\Gamma_Z$ — total width, setting the peak's breadth; $12\pi$ — spin and colour bookkeeping. The three measurable features are the position ( $m_Z$ ), the width ( $\Gamma_Z$ ), and the height. Setting $s=m_Z^2$ collapses the denominator to $m_Z^2\Gamma_Z^2$ and gives the peak cross section — the single most information-dense number on the resonance.

\sigma^0_{f\bar f}=\frac{12\pi}{m_Z^2}\,\frac{\Gamma_{ee}\,\Gamma_{f\bar f}}{\Gamma_Z^2} \qquad (3)

where: $\sigma^0_{f\bar f}$ — pole (Born-level) peak cross section into channel $f\bar f$ . The height falls as $1/\Gamma_Z^2$ , so anything that broadens the Z — including decays to invisible neutrinos — pushes the peak down. This is the lever that counts generations. The partial widths are computed from the Z's chiral couplings to each fermion, which in the electroweak Standard Model are fixed by weak isospin, charge, and one mixing angle.

c_V^f=T_3^f-2Q_f\sin^2\theta_W,\qquad c_A^f=T_3^f \qquad (4)

where: $T_3^f$ — third component of weak isospin ( $+\tfrac12$ for $u,\nu$ ; $-\tfrac12$ for $d,e$ ); $Q_f$ — electric charge in units of $e$ ; $\sin^2\theta_W\approx0.231$ — the weak mixing (Weinberg) angle. For charged leptons $c_A=-\tfrac12$ and $c_V=-\tfrac12+2\sin^2\theta_W\approx-0.038$ — tiny, which is exactly why leptonic asymmetries are razor-sharp probes of $\sin^2\theta_W$ . Each partial width follows from those couplings, with a colour factor $N_c$ ( $=3$ for quarks, $1$ for leptons and neutrinos).

\Gamma_{f\bar f}=\frac{\sqrt{2}\,G_F\,m_Z^3}{12\pi}\,N_c^f\big[(c_V^f)^2+(c_A^f)^2\big] \qquad (5)

where: $G_F=1.166\times10^{-5}\ \mathrm{GeV^{-2}}$ — the Fermi constant; the bracket $(c_V^f)^2+(c_A^f)^2$ — the squared coupling strength that also governs the resonance height. Summing (5) over all kinematically open channels builds the total width $\Gamma_Z=\sum_f\Gamma_{f\bar f}$ . Finally the invisible width — the part of $\Gamma_Z$ not seen as charged leptons or hadrons — counts the light neutrino species, the flagship LEP result.

N_\nu=\frac{\Gamma_{\text{inv}}}{\Gamma_{\nu\nu}}=\frac{\Gamma_Z-\Gamma_{\text{had}}-3\,\Gamma_{\ell\ell}}{\Gamma_{\nu\nu}} \qquad (6)

where: $\Gamma_{\text{inv}}$ — invisible width (neutrinos); $\Gamma_{\text{had}}$ — sum over all quark channels; $3\,\Gamma_{\ell\ell}$ — the three charged-lepton channels; $\Gamma_{\nu\nu}$ — the Standard-Model width per neutrino species, $\approx166$ MeV. Everything on the right is read off the line shape; the ratio comes out $2.984\pm0.008$ .

Derivation AHow an unstable particle's lifetime turns a pole into a peak

The goal is to show, without hand-waving, why the naive Z propagator $1/(s-m_Z^2)$ must be replaced by $1/(s-m_Z^2+im_Z\Gamma_Z)$ , and why that single change converts an unphysical infinity at $\sqrt{s}=m_Z$ into the finite Breit–Wigner resonance LEP actually measured. We build the width in from the decay law, insert it into the propagator, square the amplitude, and identify the peak height and full width so that every feature of the line shape is traced to the Z's lifetime.

Step 1. A stable state of mass $m$ evolves in its rest frame as $\psi(t)\propto e^{-imt}$ , so $|\psi|^2$ is constant — it never decays. An unstable state with lifetime $\tau=1/\Gamma$ must instead obey $|\psi|^2\propto e^{-\Gamma t}=e^{-t/\tau}$ , which forces the amplitude itself to carry $\psi(t)\propto e^{-imt}e^{-\Gamma t/2}=e^{-i(m-i\Gamma/2)t}$ . Why: the only way to make the probability decay exponentially while keeping a definite phase is to give the mass a small negative imaginary part, $m\to m-i\Gamma/2$ .

Step 2. In the propagator the mass appears as $m_Z^2$ , so make the same substitution: $m_Z^2\to(m_Z-i\Gamma_Z/2)^2=m_Z^2-im_Z\Gamma_Z-\tfrac14\Gamma_Z^2$ . Since $\Gamma_Z/m_Z\approx0.027\ll1$ , the last term is a $10^{-3}$ correction and is dropped, leaving the effective replacement $m_Z^2\to m_Z^2-im_Z\Gamma_Z$ and the propagator $1/(s-m_Z^2+im_Z\Gamma_Z)$ . Why: keeping the leading imaginary piece is what shifts the pole off the real axis; the quadratic piece is negligible for a narrow resonance.

Step 3. The cross section is proportional to the modulus squared of the amplitude, so the complex propagator becomes $\big|\,1/(s-m_Z^2+im_Z\Gamma_Z)\,\big|^2=1/[(s-m_Z^2)^2+m_Z^2\Gamma_Z^2]$ . This is the Breit–Wigner denominator: real, positive, and never zero. Why: squaring a complex number adds the real and imaginary parts in quadrature, so the once-divergent denominator now bottoms out at $m_Z^2\Gamma_Z^2$ instead of at zero.

Step 4. Read off the shape. The maximum sits at $s=m_Z^2$ , i.e. $\sqrt{s}=m_Z$ — the peak locates the mass. The denominator doubles when $(s-m_Z^2)^2=m_Z^2\Gamma_Z^2$ , i.e. $s=m_Z^2\pm m_Z\Gamma_Z$ ; converting to energy gives a full width at half maximum of $\Gamma_Z$ in $\sqrt{s}$ — the breadth measures the total decay rate. And the peak value scales as $1/(m_Z^2\Gamma_Z^2)$ — the height measures the couplings and the width together. Why: the three independent features of one curve carry three independent physics numbers, which is what makes the line shape such an efficient measuring instrument. *As a consistency check: $\Gamma_Z/m_Z=2.4952/91.19=0.0274$ , so the dropped $\tfrac14\Gamma_Z^2$ term is only $7\times10^{-4}$ of $m_Z^2$ — confirming the narrow-resonance approximation is excellent. The peak sits at $\sqrt{s}=m_Z$ , the FWHM equals $\Gamma_Z$ , and the height scales as $1/(m_Z\Gamma_Z)^2$ .*

Go deeper · Derivation B · 2 minFrom the neutral-current amplitude to the peak cross section

Derivation BFrom the neutral-current amplitude to the peak cross section

The aim is to connect the microscopic Z vertex to the macroscopic peak height $\sigma^0_{f\bar f}=(12\pi/m_Z^2)(\Gamma_{ee}\Gamma_{f\bar f}/\Gamma_Z^2)$ , showing that the same coupling combination $(c_V)^2+(c_A)^2$ appears both in the production of the Z from $e^+e^-$ and in its decay to $f\bar f$ , so that the resonance height factorises cleanly into an entrance width and an exit width divided by the total width squared. We start from the chiral vertex, reduce the helicity sum, spin-average, and repackage into partial widths.

Step 1. The neutral-current vertex is $-i\tfrac{g_Z}{2}\gamma^\mu(c_V^f-c_A^f\gamma^5)$ . Writing it in the chiral basis $c_V=c_L+c_R$ , $c_A=c_L-c_R$ separates left- and right-handed couplings, and on the resonance only four of the sixteen helicity combinations survive — exactly the same four that appear in QED $e^+e^-\to\mu^+\mu^-$ , because a massless fermion keeps its chirality at the vertex. Why: chirality conservation collapses the spin sum to the same angular structure as the photon process, so the Z calculation is the QED one with new couplings.

Step 2. Squaring and spin-averaging the amplitude on the peak gives a result proportional to the product of the initial- and final-state coupling factors, $[(c_V^e)^2+(c_A^e)^2]\,[(c_V^f)^2+(c_A^f)^2]$ , times the Breit–Wigner denominator from Derivation 1. Why: the electron couplings enter where the Z is made and the fermion couplings where it decays, so the two vertices multiply — the amplitude does not know the beginning and end are correlated.

Step 3. Recognise each coupling factor as a partial width. From equation (5), $\Gamma_{f\bar f}\propto N_c^f[(c_V^f)^2+(c_A^f)^2]$ , so $[(c_V^e)^2+(c_A^e)^2]\propto\Gamma_{ee}$ and $[(c_V^f)^2+(c_A^f)^2]N_c^f\propto\Gamma_{f\bar f}$ . Substituting converts the coupling product into $\Gamma_{ee}\Gamma_{f\bar f}$ . Why: partial widths are the physical, radiatively-correctable repackaging of the couplings, so the cross section written in widths is what one actually fits to data.

Step 4. Assemble the pieces at $s=m_Z^2$ : the flux and phase-space factors combine with the spin-averaging $\tfrac14$ and the colour and $\pi$ factors into the universal prefactor $12\pi/m_Z^2$ , leaving $\sigma^0_{f\bar f}=(12\pi/m_Z^2)(\Gamma_{ee}\Gamma_{f\bar f}/\Gamma_Z^2)$ . Why: the peak height is manifestly a ratio of a "way in" times a "way out" over the total decay rate squared — the defining signature of a resonance formed and destroyed through definite channels. *As a numerical check: with $\Gamma_{ee}=83.9$ MeV, $\Gamma_{\text{had}}=1744$ MeV, and $\Gamma_Z=2495$ MeV, equation (3) gives $\sigma^0_{\text{had}}=41.5$ nb and $\sigma^0_{\mu\mu}=2.0$ nb, in agreement with the LEP measurements.*

Go deeper · Derivation C · 2 minCounting light neutrino generations from the invisible width

Derivation CCounting light neutrino generations from the invisible width

The purpose is to derive, from quantities that are all read off the measured line shape, the number of light neutrino species $N_\nu$ , and to explain why the Z peak height — a laboratory cross section — settles a question about how many kinds of neutrino exist in nature, including any that couple only weakly and are otherwise invisible. We isolate the invisible width, express it through the peak observables, and divide by the Standard-Model prediction per species.

Step 1. Split the total width into what is seen and what is not: $\Gamma_Z=3\,\Gamma_{\ell\ell}+\Gamma_{\text{had}}+\Gamma_{\text{inv}}$ , where $\Gamma_{\text{inv}}=N_\nu\,\Gamma_{\nu\nu}$ is the sum over neutrino channels that leave the detector empty-handed. Why: neutrinos escape undetected, so they never appear as a final state but do contribute to the total decay rate that sets the peak's width and height.

Step 2. Measure the three "visible" line-shape observables — the peak hadronic cross section $\sigma^0_{\text{had}}$ , the total width $\Gamma_Z$ , and the ratio $R_\ell=\Gamma_{\text{had}}/\Gamma_{\ell\ell}$ . From equation (3) at the hadronic peak, $\Gamma_{ee}\Gamma_{\text{had}}=\sigma^0_{\text{had}}\,m_Z^2\,\Gamma_Z^2/12\pi$ , which fixes $\Gamma_{ee}$ and $\Gamma_{\text{had}}$ once $R_\ell$ and lepton universality are used. Why: the height and width together over-determine the partial widths, so the visible sector is fully pinned by the scan.

Step 3. Form the invisible width by subtraction, $\Gamma_{\text{inv}}=\Gamma_Z-\Gamma_{\text{had}}-3\,\Gamma_{\ell\ell}$ , and divide by the Standard-Model per-species width $\Gamma_{\nu\nu}$ (equation (5) with $c_V=c_A=\tfrac12$ , giving $\approx166$ MeV). To cancel common theory uncertainties one actually uses the ratio $\Gamma_{\text{inv}}/\Gamma_{\ell\ell}$ compared to $(\Gamma_{\nu\nu}/\Gamma_{\ell\ell})_{\text{SM}}$ . Why: taking a ratio to a well-predicted Standard-Model quantity removes the leading radiative-correction errors, sharpening the count.

Step 4. Evaluate: LEP finds $\Gamma_{\text{inv}}=499$ MeV, and $(\Gamma_{\nu\nu}/\Gamma_{\ell\ell})_{\text{SM}}=1.991$ , giving $N_\nu=2.984\pm0.008$ . The result is three standard deviations below three — not because there are fewer than three neutrinos, but a mild statistical fluctuation; the count is unambiguously three light species and no fourth. Why: a fourth Standard-Model neutrino lighter than $m_Z/2$ would add $\approx166$ MeV to $\Gamma_Z$ , lowering the peak by ~15% — an effect LEP would have seen a thousand times over. *As a numerical check: $\Gamma_{\text{inv}}=2495-1744-3(83.9)=499$ MeV, and $499/166=3.01$ ; the full LEP fit gives $N_\nu=2.984\pm0.008$ .*

Limiting cases

Case	Result	Meaning
$\sqrt{s}\ll m_Z$	Z suppressed by $s/m_Z^2$ ; QED dominates	at PETRA/PEP energies the photon rules; Z shows only as a tiny forward–backward asymmetry
$\sqrt{s}\gg m_Z$	$\sigma\propto1/s$ ; $\gamma$ and Z comparable	above the peak both fall as $1/s$ and interfere; the Z tail probes electroweak radiative effects
$\sqrt{s}=m_Z$ (peak)	$\sigma^0=\dfrac{12\pi}{m_Z^2}\dfrac{\Gamma_{ee}\Gamma_{f\bar f}}{\Gamma_Z^2}$	pure resonance; interference with the photon nearly vanishes at the pole
$\Gamma_Z\to0$ (stable Z)	pole returns to real axis; $\sigma\to\infty$	the unphysical divergence — recovered only if the Z could not decay
extra light $\nu$ (or exotic invisible decay)	$\Gamma_Z\uparrow$ , peak $\downarrow$ as $1/\Gamma_Z^2$	the height is a total-width meter; caps $N_\nu$ and any invisible Z decay

Common trapThe cross section really blows up to infinity at √s = m_Z

It does not, and believing so mistakes an artefact for physics. The divergence you would get from the bare propagator $1/(s-m_Z^2)$ is an error of neglecting the Z's instability, not a real prediction. Because the Z decays with rate $\Gamma_Z$ , its propagator carries $+im_Z\Gamma_Z$ in the denominator, and after squaring the amplitude the denominator is $(s-m_Z^2)^2+m_Z^2\Gamma_Z^2$ , which bottoms out at the finite value $m_Z^2\Gamma_Z^2$ . The peak height is therefore proportional to $1/(m_Z^2\Gamma_Z^2)$ — large, about 41.5 nb into hadrons and a thousand times the pure-QED rate, but perfectly finite. A truly stable particle would have $\Gamma_Z=0$ and a genuine pole, but a particle with zero width has infinite lifetime and no decays to detect in the first place — you could never scan across such a "resonance". Finiteness and observability are the same statement: it is precisely because the Z decays that the peak is measurable.

Worked examples

1 · The peak hadronic cross section (confidence builder)

Using the Z partial widths $\Gamma_{ee}=83.9$ MeV and $\Gamma_{\text{had}}=1744$ MeV, the total width $\Gamma_Z=2495$ MeV, and $m_Z=91.19$ GeV, compute the pole hadronic cross section $\sigma^0_{\text{had}}$ and compare to the leptonic peak $\sigma^0_{\mu\mu}$ .

Step 1. Write the peak formula (3) in convenient units. With everything in GeV, $\sigma$ comes out in $\mathrm{GeV^{-2}}$ ; convert at the end using $1\ \mathrm{GeV^{-2}}=0.3894\ \mathrm{mb}=3.894\times10^{5}\ \mathrm{nb}$ . Assemble the prefactor: $12\pi/m_Z^2=37.70/8316=4.534\times10^{-3}\ \mathrm{GeV^{-2}}$ .

Step 2. Form the width ratio for hadrons: $\Gamma_{ee}\Gamma_{\text{had}}/\Gamma_Z^2=(0.0839)(1.744)/(2.495)^2=0.14636/6.225=0.02351$ . Multiply: $\sigma^0_{\text{had}}=4.534\times10^{-3}\times0.02351=1.066\times10^{-4}\ \mathrm{GeV^{-2}}$ .

Step 3. Convert to nanobarns: $\sigma^0_{\text{had}}=1.066\times10^{-4}\times3.894\times10^{5}=\mathbf{41.5\ nb}$ , matching the measured pole cross section $41.540\pm0.037$ nb.

Step 4. Repeat for muons, replacing $\Gamma_{\text{had}}\to\Gamma_{\mu\mu}=\Gamma_{ee}=0.0839$ GeV: $\Gamma_{ee}\Gamma_{\mu\mu}/\Gamma_Z^2=(0.0839)^2/6.225=1.131\times10^{-3}$ , so $\sigma^0_{\mu\mu}=4.534\times10^{-3}\times1.131\times10^{-3}\times3.894\times10^{5}=\mathbf{2.0\ nb}$ . The hadronic peak is about $\Gamma_{\text{had}}/\Gamma_{\mu\mu}\approx21$ times taller — the Z decays to quarks far more often than to any single lepton, which is why the line shape is measured in the hadronic channel. σ⁰_had = 41.5 nb, σ⁰_μμ = 2.0 nb.

2 · Counting the neutrino generations

Given the LEP-measured widths $\Gamma_Z=2495$ MeV, $\Gamma_{\text{had}}=1744$ MeV, and $\Gamma_{\ell\ell}=83.9$ MeV per charged lepton, and the Standard-Model per-species neutrino width $\Gamma_{\nu\nu}=166$ MeV, find the number of light neutrino species.

Step 1. Add up everything visible. Charged leptons: $3\Gamma_{\ell\ell}=3\times83.9=251.7$ MeV. Hadrons: $\Gamma_{\text{had}}=1744$ MeV. Visible total $=1995.7$ MeV.

Step 2. Subtract from the measured total width to isolate the invisible part: $\Gamma_{\text{inv}}=\Gamma_Z-3\Gamma_{\ell\ell}-\Gamma_{\text{had}}=2495-251.7-1744=\mathbf{499\ MeV}$ . This is energy carried off by particles the detectors never saw.

Step 3. Divide by the width one Standard-Model neutrino would contribute: $N_\nu=\Gamma_{\text{inv}}/\Gamma_{\nu\nu}=499/166=\mathbf{3.01}$ . The full LEP fit, using ratios to cancel theory error, gives $N_\nu=2.984\pm0.008$ .

Step 4. Interpret the lever. A fourth light neutrino would raise $\Gamma_Z$ to $\approx2661$ MeV; by equation (3) the hadronic peak would drop as $1/\Gamma_Z^2$ to $41.5\times(2495/2661)^2=36.5$ nb — a 12% collapse of a peak measured to $10^{-3}$ . LEP therefore excludes a fourth light, weakly-coupled neutrino at overwhelming significance, and with it a fourth Standard-Model generation with a light neutrino. N_ν = 2.984 ± 0.008 — exactly three.

3 · The leptonic couplings from the mixing angle

Using $\sin^2\theta_W=0.231$ , compute the Z's vector and axial couplings to a charged lepton and the resulting asymmetry parameter $A_\ell=2c_Vc_A/(c_V^2+c_A^2)$ .

Step 1. For a charged lepton $T_3=-\tfrac12$ , $Q=-1$ , so the axial coupling is $c_A=T_3=-0.5$ and the vector coupling is $c_V=T_3-2Q\sin^2\theta_W=-0.5+2(0.231)=\mathbf{-0.038}$ . The vector coupling is nearly zero — an accident of $\sin^2\theta_W$ sitting close to $\tfrac14$ .

Step 2. Form the asymmetry parameter: $A_\ell=2c_Vc_A/(c_V^2+c_A^2)=2(-0.038)(-0.5)/[(0.038)^2+(0.5)^2]=0.0376/0.2514=\mathbf{0.150}$ . The forward–backward asymmetry on the peak is then $A_{FB}^{0,\ell}=\tfrac34A_e A_\ell=\tfrac34(0.150)^2=\mathbf{0.017}$ .

Step 3. See the leverage: because $c_V\approx-0.038$ is small, $A_\ell$ is roughly linear in $c_V$ and hence extremely sensitive to $\sin^2\theta_W$ — a 1% change in $\sin^2\theta_W$ shifts $c_V$ by ~25% and $A_\ell$ almost as much. This is why the asymmetries, not the peak height, give the sharpest single measurement of the weak mixing angle: $\sin^2\theta_W^{\text{eff}}=0.23153\pm0.00016$ . c_V = −0.038, c_A = −0.5, A_ℓ = 0.150.

Physics of the portrait

The Z resonance is where a theoretical prediction, a technological gamble, and a precision measurement met. Sheldon Glashow, Steven Weinberg, and Abdus Salam predicted a neutral, massive partner to the W in the 1960s; Gargamelle saw its fingerprint in 1973; and then one experimentalist bet a collider on making the particle itself — a bet that made the LEP precision program possible a decade later.

Origins · 1 minCarlo Rubbia and how this was discovered

Carlo Rubbia

1934– · the discovery of the W and Z and the road to LEP precision

Carlo Rubbia is the Italian physicist who turned the electroweak Z from a prediction into an object you could produce and count. In the late 1970s the Z's mass was expected near 90 GeV — far beyond any existing machine — and Rubbia's audacious idea was to convert CERN's existing Super Proton Synchrotron into a proton–antiproton collider, so that valence quarks and antiquarks could annihilate at high enough energy to make real W and Z bosons. The scheme only worked because Simon van der Meer had invented stochastic cooling, a feedback technique that squeezed a diffuse antiproton beam into a dense, usable bunch. In 1983 Rubbia's UA1 experiment, alongside UA2, found the W and then the Z, and he and van der Meer shared the 1984 Nobel Prize barely a year later — one of the fastest recognitions in the prize's history. But a hadron collider is a blunt instrument for precision: the Z appeared over a large QCD background at poorly known parton energies. The decisive precision would come from doing the opposite — colliding point-like electrons and positrons at an exquisitely known energy. That machine was LEP, the 27 km Large Electron–Positron collider, whose four experiments (ALEPH, DELPHI, L3, OPAL) scanned the Z line shape from 1989 to 1995 and measured $m_Z$ to $2\times10^{-5}$ , counted the neutrino generations, and tested the Standard Model to the loop level. Rubbia made the Z; LEP measured it to death. Their combined legacy is that the electroweak sector is the most precisely verified theory in particle physics.

History

1967

— Weinberg and Salam predict a massive neutral Z boson from electroweak unification

1973

— the Gargamelle bubble chamber at CERN observes weak neutral currents

1983

— UA1 and UA2 discover the W and Z at the SppS collider, Rubbia and van der Meer

1989

— LEP and the SLC begin operation; the first Z scans already limit the light neutrinos to three

2000

— LEP is shut down after roughly 17 million Z decays to make way for the LHC

Going further

Texts. Thomson, Modern Particle Physics, §16.1 (the Z resonance and Z production cross section) and §15–16 (electroweak unification) · Halzen & Martin, Quarks and Leptons, Ch. 13 (electroweak interactions of leptons) · Peskin & Schroeder, An Introduction to Quantum Field Theory, §20.3 (the Standard Model at the Z pole) · Griffiths, Introduction to Elementary Particles, 2nd ed., §9–10 · Schwartz, Quantum Field Theory and the Standard Model, Ch. 29 (electroweak precision tests). The measured line-shape parameters ( $m_Z=91.1876$ GeV, $\Gamma_Z=2.4952$ GeV, $\sigma^0_{\text{had}}=41.540$ nb, $N_\nu=2.984$ ) are the combined LEP results (ALEPH, DELPHI, L3, OPAL and the SLD/LEP electroweak working group, 2006). Key numerical cross-checks are: (1) $\sigma^0_{\text{had}}=41.5$ nb and $\sigma^0_{\mu\mu}=2.0$ nb from equation (3); (2) $c_V=-0.038$ , $c_A=-0.5$ , $A_\ell=0.150$ ; (3) $\Gamma_{\text{inv}}=499$ MeV and $N_\nu=3.0$ ; (4) FWHM $=\Gamma_Z$ and peak at $\sqrt{s}=m_Z$ .

Prerequisites. Breit–Wigner Resonances · $e^+e^-\to\mu^+\mu^-$ in QED · The Weak Neutral Current · Electroweak Unification · Chirality in QED. Leads to. Z Decays and Partial Widths · The Number of Generations · Electroweak Radiative Corrections · The Weinberg Angle.

Ask the tutor: Ask me to walk through why the Z peak height counts three neutrino generations, or to work any Breit–Wigner line-shape number from the partial widths.

Why replace the photon with a Z at all — aren't they completely different forces?

They are two faces of one unified electroweak interaction, which is exactly why the substitution works. In the Standard Model the photon and the Z are orthogonal mixtures of the same underlying $SU(2)\times U(1)$ gauge fields, rotated by the weak mixing angle $\theta_W$ . Both are neutral, spin-1, and couple to fermion currents with the same $\gamma^\mu$ Dirac structure; they differ only in their coupling strengths and in the fact that the Z is heavy while the photon is massless. So any QED annihilation diagram has a Z twin obtained by swapping the coupling $e\to g_Z(c_V-c_A\gamma^5)/2$ and the propagator $1/s\to1/(s-m_Z^2)$ . At low energy the Z is invisible because of its mass; near $\sqrt{s}=m_Z$ it takes over. Unification is the statement that these are not two separate calculations but one, split by a mass.

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

The total width $\Gamma_Z$ is the full decay rate of the Z, the inverse of its lifetime ( $\tau=\hbar/\Gamma_Z\approx3\times10^{-25}$ s). A partial width $\Gamma_{f\bar f}$ is the rate into one specific final state — $e^+e^-$ , $\mu^+\mu^-$ , a quark pair, or a neutrino pair. By definition $\Gamma_Z=\sum_f\Gamma_{f\bar f}$ , summed over every channel the Z can reach. The branching ratio into a channel is $\Gamma_{f\bar f}/\Gamma_Z$ : about 3.4% per charged lepton, 70% into hadrons, and 20% into invisible neutrinos. In the resonance formula the total width sets the width of the peak (its breadth in energy), while the product of two partial widths sets its height. Measuring both breadth and height therefore separates "how fast does the Z decay overall" from "how strongly does it couple to this particular pair".

Why measure the line shape in the hadronic channel rather than in muons?

Statistics. The Z decays to quark–antiquark pairs about 70% of the time but to any single charged-lepton species only ~3.4% of the time, so the hadronic peak is roughly twenty times taller — 41.5 nb versus 2.0 nb — and delivers twenty times the event rate for the same running time. Since the line-shape parameters ( $m_Z$ , $\Gamma_Z$ , $\sigma^0$ ) are ultimately limited by how many Z decays you collect, the hadronic channel gives the smallest statistical errors on the mass, width, and peak height. The leptonic channels are still indispensable — they test lepton universality and, through the forward–backward asymmetries, give the sharpest handle on $\sin^2\theta_W$ — but for the shape itself, hadrons win on sheer numbers.

How can a cross section measure something invisible like the neutrino count?

Through the total width, which the peak height depends on even for channels you never detect. The Z decays into three light neutrino species that leave the apparatus without a trace, yet each such decay still counts toward $\Gamma_Z$ — it removes a Z from the world just as surely as a visible decay does. Because the peak cross section scales as $1/\Gamma_Z^2$ , adding invisible decay channels broadens the resonance and lowers its top. So you measure everything visible (the peak height, the total width, the hadron-to-lepton ratio), reconstruct the visible widths, subtract them from the total, and whatever is left over is the invisible width. Dividing by the calculable width of a single neutrino species gives the count: $N_\nu=2.984\pm0.008$ . The neutrinos announce their number by the shadow they cast on the peak.

Why is the vector coupling c_V of the Z to leptons so small, and does it matter?

It is small because of an arithmetic near-coincidence: $c_V=T_3-2Q\sin^2\theta_W=-\tfrac12+2\sin^2\theta_W$ , and the measured $\sin^2\theta_W\approx0.231$ is close to $\tfrac14$ , which would make $c_V$ vanish. The actual value is $c_V\approx-0.038$ , about thirteen times smaller than $c_A=-\tfrac12$ . Far from being a nuisance, this smallness is a gift: quantities like the asymmetry parameter $A_\ell=2c_Vc_A/(c_V^2+c_A^2)$ are nearly linear in $c_V$ near zero, so they amplify any change in $\sin^2\theta_W$ . That is why the forward–backward and polarization asymmetries at the Z pole give the world's most precise value of the weak mixing angle, and why a tiny coupling turns out to be the sharpest tool on the resonance.

What made LEP's energy calibration so extraordinarily precise?

LEP measured the beam energy by resonant depolarization: transversely polarized electrons precess at a rate set by their energy, and driving that precession with a small oscillating field to depolarize the beam pins the energy to a few parts per million. At that sensitivity the machine became a giant seismometer and voltmeter. The circumference of the 27 km ring changes by about a millimetre as the tidal pull of the Moon and Sun deforms the Earth's crust, shifting the beam energy measurably; the LEP team observed and corrected for it. They also traced slow drifts to the water level in Lake Geneva and, most famously, to stray leakage currents from the electric TGV trains running nearby, which returned to ground through the beam pipe. Controlling these effects is what let $m_Z$ be quoted to $\pm0.0021$ GeV — a relative precision of $2\times10^{-5}$ .