--- title: The W and Z Bosons module: The Weak Interaction moduleNumber: 7 lessonNumber: 2 order: 702 summary: > The contact theory hides a massive mediator. The charged $W^\pm$ carries the current that changes flavour; the neutral $Z^0$ carries a current that does not. Both were found at CERN's proton–antiproton collider in 1983 at the masses the electroweak theory demanded. Their decay widths partition into leptonic and hadronic channels, and the $Z$ carries a decisive extra: an invisible width from decays to neutrinos that counts the number of light generations at exactly three. Beta decay and muon decay are re-read at the parton level as $W$ exchange. topics: [The Weak Interaction] draft: false sources: - book: Thomson ref: "Ch. 15–16 — The W and Z bosons; electroweak measurements" - book: Griffiths ref: "Ch. 10 — Weak Interactions, §10.5–10.7" - book: Tong ref: "The Standard Model (Cambridge Part III), §5.2 Electroweak Symmetry Breaking, §5.3 Weak Decays" - book: PDG ref: "Gauge-boson masses, widths, and branching fractions — pdg.lbl.gov" --- The previous lesson ended with a mediator whose mass, hidden in the denominator of its propagator, is the real reason the weak interaction looks feeble at low energy. This lesson gives that mediator flesh. The charged-current $W^\pm$ and the neutral-current $Z^0$ are the massive gauge bosons of the electroweak theory; their masses were predicted before they were seen, their 1983 discovery at CERN confirmed the framework, and their decays — especially the $Z$'s invisible width — measure some of the sharpest numbers in the Standard Model, including the count of light neutrino generations. We work at energies comparable to the boson masses, where the full propagator, not the contact approximation, is in play. ## Charged and neutral currents The electroweak interaction has two kinds of current. The **charged current**, carried by the $W^\pm$, changes a fermion into its doublet partner and so changes electric charge by one unit: an electron becomes a neutrino, an up quark becomes a down quark. The **neutral current**, carried by the $Z^0$, couples a fermion to itself without changing its identity — like the photon, but reaching neutrinos and with a chirality-dependent strength. Reading off the electroweak Lagrangian in terms of the physical fields, the couplings organize as $$ \mathcal{L}_{\rm int} \supset -\frac{e}{\sqrt 2\,\sin\theta_W}\big(W^+_\mu J^{+\mu} + W^-_\mu J^{-\mu}\big) -\frac{e}{\sin\theta_W\cos\theta_W}\,Z_\mu J^\mu_Z - e\,A_\mu J^\mu_{\rm EM} , $$ where $\theta_W$ is the Weinberg angle mixing the $W^3$ and hypercharge fields into the $Z$ and the photon, and $e = g\sin\theta_W$ is the electric charge.[^tong-currents] The charged current $J^\pm$ is purely left-chiral — the $V\!-\!A$ structure of the previous lesson — while the neutral current $J_Z$ carries both a left-chiral piece and a piece proportional to the electromagnetic current, so its coupling to each fermion depends on the fermion's weak isospin and charge through $\sin^2\theta_W$. $$ % caption: The two weak vertices. The charged-current W vertex changes a fermion % into its partner, altering charge by one unit (up to down, neutrino to electron). % The neutral-current Z vertex returns the same fermion, coupling even to % neutrinos. The photon vertex is shown for comparison and ignores neutrinos. \begin{tikzpicture}[>=stealth, font=\scriptsize, scale=1.0] \definecolor{acc}{HTML}{4A6FA5} % W vertex at x=-4.6 \draw[->, black, thick] (-5.5,1.2) -- (-4.6,0.4); \node[black, above left, font=\scriptsize] at (-5.5,1.2) {up}; \draw[->, black, thick] (-4.6,0.4) -- (-3.7,1.2); \node[black, above right, font=\scriptsize] at (-3.7,1.2) {down}; \draw[acc, very thick, variable=\y] plot[domain=-0.7:0.4, samples=40] ({-4.6 + 0.08*sin((\y+0.7)/1.1*1440)}, \y); \node[acc, right, font=\scriptsize] at (-4.45,-0.15) {W}; \node[font=\scriptsize] at (-4.6,-1.4) {charged current}; % Z vertex at x=0 \draw[->, black, thick] (-0.9,1.2) -- (0,0.4); \node[black, above left, font=\scriptsize] at (-0.9,1.2) {neutrino}; \draw[->, black, thick] (0,0.4) -- (0.9,1.2); \node[black, above right, font=\scriptsize] at (0.9,1.2) {neutrino}; \draw[acc, very thick, variable=\y] plot[domain=-0.7:0.4, samples=40] ({0 + 0.08*sin((\y+0.7)/1.1*1440)}, \y); \node[acc, right, font=\scriptsize] at (0.15,-0.15) {Z}; \node[font=\scriptsize] at (0,-1.4) {neutral current}; % photon vertex at x=4.6 \draw[->, black, thick] (3.7,1.2) -- (4.6,0.4); \node[black, above left, font=\scriptsize] at (3.7,1.2) {electron}; \draw[->, black, thick] (4.6,0.4) -- (5.5,1.2); \node[black, above right, font=\scriptsize] at (5.5,1.2) {electron}; \draw[black, thick, variable=\y] plot[domain=-0.7:0.4, samples=40] ({4.6 + 0.06*sin((\y+0.7)/1.1*1800)}, \y); \node[black, right, font=\scriptsize] at (4.75,-0.15) {photon}; \node[font=\scriptsize] at (4.6,-1.4) {electromagnetic}; \end{tikzpicture} $$ ## Masses and the discovery at the SppS The electroweak theory ties the boson masses to the Higgs vacuum expectation value $v$, the gauge couplings, and the Weinberg angle: $$ M_W = \tfrac12 g v , \qquad M_Z = \frac{M_W}{\cos\theta_W} , $$ so the theory predicts $M_W < M_Z$ and fixes their ratio by $\cos\theta_W$ alone. The measured values are $$ M_W = 80.3692 \pm 0.0133\ \text{GeV}, \qquad M_Z = 91.1880 \pm 0.0020\ \text{GeV}, $$ consistent with $\sin^2\theta_W \approx 0.231$.[^pdg-masses] These are enormous masses for elementary particles — the $W$ weighs as much as a rubidium atom — and producing them required a dedicated machine. In 1983 the UA1 and UA2 experiments at CERN's Super Proton–Antiproton Synchrotron ($\mathrm{Sp\bar p S}$) collided protons and antiprotons at $\sqrt s = 540$ GeV, and Carlo Rubbia's team identified the $W$ through its leptonic decay $W\to e\nu$ — a high-energy electron recoiling against missing momentum carried off by the unseen neutrino — and the $Z$ through the cleaner $Z\to e^+e^-$ and $\mu^+\mu^-$, a sharp pair of opposite leptons peaking at the $Z$ mass. The discovery, at the masses the theory demanded, earned Rubbia and van der Meer the 1984 Nobel Prize.[^thomson-discovery] ## Decay widths and branching ratios A heavy gauge boson is unstable: it decays to every fermion–antifermion pair light enough to be produced. The total widths are large, $$ \Gamma_W = 2.085 \pm 0.042\ \text{GeV}, \qquad \Gamma_Z = 2.4955 \pm 0.0023\ \text{GeV}, $$ corresponding to lifetimes of order $3\times10^{-25}$ s.[^pdg-widths] The $W$ decays either to a charged lepton and its neutrino or to a quark pair; each of the three lepton channels takes about $10.9\%$, and the quark channels together about $67\%$, enhanced by the three colours available to each quark pair. The $Z$ partitions similarly but includes a channel the $W$ cannot have — decay to a neutrino pair $\nu\bar\nu$, which escapes every detector: $$ Z \to \text{hadrons}\ (69.9\%), \quad Z \to \ell^+\ell^-\ (3.37\%\ \text{each}), \quad Z \to \nu\bar\nu\ (20.0\%,\ \text{invisible}) . $$ The colour factor of three is directly visible here: the hadronic width is enhanced over a single leptonic channel by roughly the number of accessible quark flavours times three colours, and measuring the ratio $\Gamma_{\rm had}/\Gamma_{\rm lep}$ is one confirmation that quarks come in three colours. $$ % caption: Branching fractions of the Z boson as a bar partition. Hadronic channels % dominate at about seventy percent, the three charged-lepton channels take about % ten percent between them, and the invisible neutrino channels take twenty percent. % The invisible slice counts the light neutrino species. \begin{tikzpicture}[font=\footnotesize, scale=1.0] \definecolor{acc}{HTML}{4A6FA5} % total bar length 8 units = 100% \draw[black, thick, fill=black!12] (0,0) rectangle (5.59,0.7); \node[black, font=\scriptsize] at (2.8,0.35) {hadrons 70}; \draw[black, thick, fill=black!22] (5.59,0) rectangle (6.4,0.7); \node[black, font=\scriptsize] at (6.0,1.05) {leptons 10}; \draw[acc, very thick, fill=acc!14] (6.4,0) rectangle (8.0,0.7); \node[acc, font=\scriptsize] at (7.2,0.35) {invisible 20}; \draw[black] (0,-0.15) -- (0,-0.35) node[below, font=\scriptsize] {0}; \draw[black] (8.0,-0.15) -- (8.0,-0.35) node[below, font=\scriptsize] {100 percent}; \end{tikzpicture} $$ ## The invisible width and three generations The invisible width is the $Z$'s most consequential number. Each light neutrino species — one that the $Z$ can decay into, meaning $m_\nu < M_Z/2$ — adds one identical partial width $\Gamma_\nu$ to the total. The measured invisible width, divided by the theoretical width per neutrino, counts the number of light generations: $$ N_\nu = \frac{\Gamma_{\rm inv}}{\Gamma_{\nu}^{\rm SM}} = 2.9840 \pm 0.0082 . $$ The result is three, to better than one percent.[^pdg-nnu] This is measured at LEP not by catching the invisible decays but by their effect on the visible $Z$ resonance: the total width sets both the height and the breadth of the peak in $e^+e^-\to Z\to\text{anything visible}$. A fourth light neutrino would widen the resonance and lower its peak by adding an unseen decay path. The data sit squarely on the three-neutrino curve, excluding a fourth generation with a light neutrino. This is among the most economical profound measurements in physics: the number of matter generations, read from the shape of a single resonance. $$ % caption: The Z resonance line shape measured at LEP. The cross section for % electron-positron annihilation into visible final states peaks at the Z mass. The % peak height and width depend on the number of light neutrino species: two (tall, % dashed), three (solid, the measured curve with data points), or four (low, % dotted). The data select three. \begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0] \definecolor{acc}{HTML}{4A6FA5} \draw[->, black] (0,0) -- (6.0,0) node[right, font=\scriptsize] {energy}; \draw[->, black] (0,0) -- (0,3.6) node[above, font=\scriptsize] {cross section}; \node[black, font=\scriptsize, below] at (3.0,-0.05) {Z mass}; % N=2 (tall dashed) \draw[black, thick, dashed] (0.6,0.5) .. controls (2.3,0.5) and (2.75,3.25) .. (3.0,3.25) .. controls (3.25,3.25) and (3.7,0.5) .. (5.6,0.5); \node[black, font=\scriptsize, right] at (3.35,3.05) {two}; % N=3 (solid acc) \draw[acc, very thick] (0.6,0.5) .. controls (2.3,0.5) and (2.75,2.55) .. (3.0,2.55) .. controls (3.25,2.55) and (3.7,0.5) .. (5.6,0.5); \node[acc, font=\scriptsize, right] at (3.45,2.35) {three}; % N=4 (low dotted) \draw[black, thick, densely dotted] (0.6,0.5) .. controls (2.3,0.5) and (2.75,2.0) .. (3.0,2.0) .. controls (3.25,2.0) and (3.7,0.5) .. (5.6,0.5); \node[black, font=\scriptsize, right] at (4.05,1.5) {four}; % data points on the three curve \foreach \x/\y in {2.4/0.95, 2.7/1.7, 3.0/2.55, 3.3/1.7, 3.6/0.95} \fill[acc] (\x,\y) circle (1.8pt); \end{tikzpicture} $$ ## Beta decay at the parton level With the $W$ in hand, the oldest weak process becomes transparent. Nuclear beta decay is, at the level of quarks, a single down quark inside a neutron emitting a $W^-$ and turning into an up quark: $$ d \to u + W^-, \qquad W^- \to e^- + \bar\nu_e , $$ so the neutron ($udd$) becomes a proton ($uud$) while the $W$ materializes the electron and antineutrino. The neutron lifetime is about $879$ s — ten minutes, extraordinarily long for a decay, precisely because the tiny energy release $m_n - m_p - m_e \approx 0.78$ MeV sits so far below $M_W$ that the process is doubly suppressed: by the small phase space and by the $1/M_W^4$ propagator factor. The same diagram with a muon in place of the electron is muon decay; the same diagram at the hadron level, with the spectator quarks drawn in, is beta decay. One vertex structure, dressed differently, runs the entire zoo of weak decays.[^tong-beta] $$ % caption: Beta decay at the parton level. A down quark in the neutron emits a W % boson and becomes an up quark, turning the neutron into a proton. The W then % decays to an electron and an electron anti-neutrino. The two spectator quarks % carry through unchanged. \begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0] \definecolor{acc}{HTML}{4A6FA5} % neutron bracket \node[black, font=\scriptsize, left] at (-3.4,0.9) {neutron}; \draw[black] (-3.0,0.2) -- (-3.2,0.2) -- (-3.2,1.6) -- (-3.0,1.6); % spectator up \draw[->, black, thick] (-3.0,1.4) -- (1.8,1.4); \node[black, right, font=\scriptsize] at (1.85,1.4) {up}; % spectator down \draw[->, black, thick] (-3.0,0.9) -- (1.8,0.9); \node[black, right, font=\scriptsize] at (1.85,0.9) {down}; % active down -> up \draw[->, black, thick] (-3.0,0.3) -- (-1.0,0.3); \node[black, font=\scriptsize, below] at (-2.0,0.25) {down}; \draw[->, black, thick] (-1.0,0.3) -- (1.8,0.5); \node[black, right, font=\scriptsize] at (1.85,0.5) {up}; % proton bracket \node[black, font=\scriptsize, right] at (3.1,0.9) {proton}; \draw[black] (2.7,0.2) -- (2.9,0.2) -- (2.9,1.6) -- (2.7,1.6); % W propagator (vertical wave) \draw[acc, very thick, variable=\y] plot[domain=-1.3:0.3, samples=48] ({-1.0 + 0.08*sin((\y+1.3)/1.6*1620)}, \y); \node[acc, left, font=\scriptsize] at (-1.15,-0.5) {W}; % electron and antineutrino \draw[->, black, thick] (-1.0,-1.3) -- (1.0,-0.9); \node[black, right, font=\scriptsize] at (1.05,-0.9) {electron}; \draw[->, black, thick] (-1.0,-1.3) -- (0.4,-2.1); \node[black, right, font=\scriptsize] at (0.45,-2.1) {anti-nu-e}; \end{tikzpicture} $$ ## Summary The weak interaction is carried by two massive gauge bosons: the charged $W^\pm$, which changes a fermion into its doublet partner, and the neutral $Z^0$, which does not. Their masses $M_W = 80.37$ GeV and $M_Z = 91.19$ GeV obey $M_W = M_Z\cos\theta_W$ and were confirmed by the 1983 $\mathrm{Sp\bar p S}$ discovery. Their large widths partition into leptonic and colour-enhanced hadronic channels; the $Z$ additionally decays invisibly to neutrinos, and the invisible width measures $N_\nu = 2.984$, fixing three light generations. Beta decay, muon decay, and every heavy-flavour decay are one $W$-exchange diagram in different costumes. What we have not yet explained is why the $W$ couples up quarks not only to down quarks but, more weakly, to strange and bottom — the quark mixing that the next lesson organizes into the CKM matrix. [^tong-currents]: Tong, _The Standard Model_ (Cambridge Part III), §5.3.1 — the electroweak currents, the $W/Z$/photon couplings written via the Weinberg angle, and the chiral structure of the neutral current. [damtp.cam.ac.uk/user/tong/standardmodel.html](http://www.damtp.cam.ac.uk/user/tong/standardmodel.html) [^pdg-masses]: Boson masses $M_W$, $M_Z$ and $\sin^2\theta_W$: Particle Data Group, _Review of Particle Physics_, gauge-boson listings, [pdg.lbl.gov](https://pdg.lbl.gov). The mass relation $M_W = M_Z\cos\theta_W$ is derived in Tong §5.2. [^thomson-discovery]: Thomson, _Modern Particle Physics_, Ch. 16 — the UA1/UA2 discovery of the $W$ and $Z$ at the $\mathrm{Sp\bar p S}$; original reports UA1 (Arnison et al.) and UA2 (Banner et al.), _Phys. Lett. B_ (1983). [^pdg-widths]: Total and partial widths $\Gamma_W$, $\Gamma_Z$ and the branching fractions: Particle Data Group, [pdg.lbl.gov](https://pdg.lbl.gov). [^pdg-nnu]: The invisible-width determination $N_\nu = 2.9840 \pm 0.0082$ from the combined LEP line-shape measurements is reported by the Particle Data Group, [pdg.lbl.gov](https://pdg.lbl.gov); see also Thomson, Ch. 16. [^tong-beta]: Tong, _The Standard Model_ (Cambridge Part III), §5.3.3 — beta decay as $d\to u W^-$ with the neutron ($udd$) becoming a proton ($uud$), and the common vertex structure of beta, muon, and heavy-flavour decays. The neutron lifetime value follows the Particle Data Group, [pdg.lbl.gov](https://pdg.lbl.gov).