Gauge-Boson Decays and Precision Electroweak

The worked-calculation companion to the electroweak derivation: how the $SU(2{)}_L\times U(1{)}_Y$ Lagrangian predicts the decay widths of the $W$ and $Z$, the invisible width that counts neutrino species, and the $\rho$ parameter and oblique corrections that turn the $Z$-pole into a precision probe of new physics. This is the electroweak parallel of the QED Compton/hydrogen calculations.

Conventions: $\hslash =c=1$, mostly-minus metric; ${\theta }_W$ = weak mixing angle.

$Z$ partial widths

The $Z$ couples to each fermion through a vector/axial mixture set by its quantum numbers, $g_V^f=T_f^3-2Q_f{\sin }^2{\theta }_W$, $g_A^f=T_f^3$. The partial width to a fermion pair follows from the decay master formula:

$$\boxed{\Gamma (Z\to f\bar{f})=\frac{N_c^f{G}_F{m}_Z^3}{6\sqrt{2}\pi }[(g_V^f{)}^2+(g_A^f{)}^2],}$$

with the color factor $N_c^f=3$ for quarks, $1$ for leptons. Summing over all kinematically accessible fermions gives the total width ${\Gamma }_Z=2.4952\mathrm{GeV}$, in agreement with the LEP line-shape measurement. Each partial width is a distinct observable; their ratios test lepton universality at the $10^{-3}$ level.

The invisible width and $N_{\nu }=3$

The decays $Z\to \nu \bar{\nu }$ produce no detectable particles, contributing an invisible width ${\Gamma }_{\text{inv}}={\Gamma }_Z-{\Gamma }_{\text{visible}}$. Since each neutrino species contributes an identical, calculable $\Gamma (Z\to \nu \bar{\nu })=\frac{G_F{m}_Z^3}{12\sqrt{2}\pi }$ (from the formula above with $g_V=g_A=\frac{1}{2}$), the invisible width counts light neutrino flavors:

$$N_{\nu }=\frac{{\Gamma }_{\text{inv}}}{\Gamma (Z\to \nu \bar{\nu }{)}_{\text{SM}}}=2.984\pm \mathrm{0.008.}$$

This single LEP number establishes there are exactly three light ($m_{\nu }<m_Z/2$) neutrino generations, ruling out a fourth chiral SM generation — a striking example of a total-width measurement constraining the particle content of the Standard Model.

$W$ partial widths

The $W$ decays via the charged current, a pure left-handed (V−A) coupling:

$$\Gamma (W\to \ell \nu )=\frac{G_F{m}_W^3}{6\sqrt{2}\pi },\Gamma (W\to q_i{\bar{q}}_j)=3|V_{ij}{|}^2\frac{G_F{m}_W^3}{6\sqrt{2}\pi },$$

the quark channel carrying the color factor 3 and the CKM element $|V_{ij}{|}^2$. Summing gives branching ratios $\approx \frac{1}{9}$ per lepton and $\approx \frac{6}{9}$ to hadrons, and total ${\Gamma }_W=2.085\mathrm{GeV}$ — matching data and again testing lepton universality.

The $\rho$ parameter and custodial symmetry

A key structural prediction relates the $W$, $Z$ masses and the mixing angle. At tree level the Higgs mechanism gives $m_W=\frac{1}{2}gv$, $m_Z=\frac{1}{2}\sqrt{g^2+g^{\prime 2}}v$, so

$$\boxed{\rho \equiv \frac{m_W^2}{m_Z^2{\cos }^2{\theta }_W}=1}$$

exactly, at tree level. This is not an accident: it follows from an approximate custodial $SU(2)$ symmetry of the Higgs potential. The measured $\rho =1$ to $\sim 10^{-3}$ confirms that electroweak symmetry is broken by a scalar doublet (a triplet or other representation would give $\rho \ne 1$) — indirect evidence for the Higgs sector's structure long before the boson was found.

Oblique corrections: S, T, U

Loop corrections shift $\rho$ and the gauge-boson self-energies. The dominant new- physics effects are oblique (entering only through the gauge-boson propagators), captured by the Peskin–Takeuchi parameters $S,T,U$:

$$\alpha T=\Delta \rho ,S\sim \text{(non-degenerate new doublets)},U\sim \text{(}SU(2)\text{-breaking)}.$$

• The top quark contributes $\Delta \rho \propto m_t^2$ — a quadratic sensitivity that let the $Z$-pole data predict $m_t\approx 170\mathrm{GeV}$ before its 1995 discovery.
• The Higgs contributes $\propto \ln m_h$ — a weaker, logarithmic sensitivity that nonetheless predicted a light $m_h\lesssim 200\mathrm{GeV}$ before 2012.
• BSM physics (extra doublets, technicolor, heavy fourth generation) generically shifts $S,T$ by amounts the data already exclude — the $Z$-pole is a stringent new-physics filter.

This is the payoff of renormalizability: because the electroweak theory is a consistent quantum theory, its loop corrections are finite and predictive, so precision measurements probe particles too heavy to produce directly.

Summary

Where this fits

• The observables: electroweak observables.
• The mechanism: the Higgs mechanism, renormalizability / BRST.
• The flavor companion: GIM Mechanism and FCNC Suppression.

References

• Peskin & Schroeder, An Introduction to Quantum Field Theory, Ch. 20–21.
• Peskin & Takeuchi, Phys. Rev. D 46, 381 (1992).
• Particle Data Group, Review of Particle Physics, "Electroweak model".