Scope: This page is a compact, reviewer-friendly reference for the Z boson within the $\chi = 24$ / K3 framework. It defines standard Z-pole observables, shows how they tie to $\alpha(m_Z)$ and $\sin^2\theta_W$, and lists the numerical anchors we use when cross-checking our predictions ($\alpha$ page, Generations page, and any $Z'$ discussions).

1. Standard-Model Essentials (Neutral Current)

$Z$—“$f\bar{f}$ Couplings

For a fermion $f$ with weak isospin $T_3^f$ and charge $Q_f$, the $Z$—“$f\bar{f}$ couplings are:

g_A^f = T_3^f, \quad g_V^f = T_3^f - 2Q_f \sin^2\theta_W^{\text{eff}}

where $\sin^2\theta_W^{\text{eff}}$ is the effective weak mixing angle at the Z pole (scheme-dependent)

Partial Width (Born level):

\Gamma_f = N_c^f \frac{G_F m_Z^3}{6\sqrt{2} \pi} \left[(g_V^f)^2 + (g_A^f)^2\right] (1 + \delta_{\text{QCD}} + \delta_{\text{EW}})

with $N_c^f = 3$ for quarks and $1$ for leptons, and $\delta$ denoting known radiative corrections.

Foundation

Total/visible/invisible widths, peak cross sections, and asymmetries derive from these quantities.

2. Z-pole Observables (Definitions You Actually Use)

Lineshape (LEP scheme):

\{m_Z, \Gamma_Z, \sigma_h^0, R_\ell, A_{FB}^{0,\ell}\}

with $\sigma_h^0 \equiv \frac{12\pi}{m_Z^2}\frac{\Gamma_e \Gamma_{\text{had}}}{\Gamma_Z^2}$ and $R_\ell \equiv \frac{\Gamma_{\text{had}}}{\Gamma_\ell}$

Asymmetries:

A_f = \frac{2g_V^f g_A^f}{(g_V^f)^2 + (g_A^f)^2}, \quad A_{FB}^{0,f} = \frac{3}{4}A_e A_f

These pin down $\sin^2\theta_W^{\text{eff}}$

Invisible width:

\Gamma_{\text{inv}} = \Gamma_Z - \sum_{\text{vis}} \Gamma_f

Interpreting $\Gamma_{\text{inv}}$ as neutrinos yields the number of light active neutrinos $N_\nu \approx 3$

3. Relations Among $\alpha$, $G_F$, $m_Z$, $m_W$

On-Shell Scheme

In the on-shell scheme:

\sin^2\theta_W \equiv 1 - \frac{m_W^2}{m_Z^2}

\sin^2\theta_W \cos^2\theta_W = \frac{\pi\alpha(m_Z)}{\sqrt{2} G_F m_Z^2} \times \frac{1}{1-\Delta r}

where $\Delta r$ encodes loop corrections (notably $m_t$, $m_H$)

Scheme Dependence

When you change scheme (e.g., $\overline{\text{MS}}$, effective angle), the numerical value of $\sin^2\theta_W$ shifts slightly but is unambiguously related by known corrections. This matters when comparing your $\alpha$-page output (at $m_Z$) to Z-pole fits.

4. Data Anchors (What Numbers We Adopt)

PDG-Consolidated Z-pole Set

We use the PDG-consolidated Z-pole set (LEP/SLD-era fits, updated where appropriate):

✓“ Mass and Width

$m_Z \simeq 91.1876$ GeV (Breit—“Wigner with $s$-dependent width)

$\Gamma_Z \simeq 2.495$ GeV

✓“ Effective Leptonic Mixing Angle

$\sin^2\theta_W^{\text{eff},\ell} \approx 0.2315$ (scheme-specific; use PDG table for exact values)

✓“ Number of Light Neutrinos

$N_\nu \approx 3$ from the invisible Z width (LEP)

This anchors the Generations page argument against a sequential light fourth family

Important: If you quote any of these numerically on the page, copy the current PDG central values and uncertainties verbatim.

5. Cross-checks You Should Pass (and What Would Break)

A. $\alpha$-page Consistency

If you run $\alpha(0) \to \alpha(m_Z)$ and plug into the on-shell relation above with the PDG $m_Z$, $m_W$, $G_F$, your predicted $\sin^2\theta_W$ should land inside the global-fit envelope after including $\Delta r$. Disagreement beyond quoted uncertainties signals a normalization/loop-treatment error on the $\alpha$ page.

B. Generations Page Hook

Your quoted $N_\nu$ must match the PDG-derived value from $\Gamma_{\text{inv}}$. Any "extra light neutrino" claim would contradict LEP directly.

C. $Z'$ / Dark Sector Notes (if present)

Small kinetic or mass mixing shifts Z-pole observables; any $Z'$ claim must respect the PDG electroweak precision constraints and updated $\sin^2\theta_W$ determinations.

6. Minimal Formulas You Can Cite on the Page

Peak hadronic cross section:

\sigma_h^0 = \frac{12\pi}{m_Z^2}\frac{\Gamma_e \Gamma_{\text{had}}}{\Gamma_Z^2}

Partial width (fermion $f$):

\Gamma_f = N_c^f \frac{G_F m_Z^3}{6\sqrt{2} \pi} \left[(g_V^f)^2 + (g_A^f)^2\right] (1 + \delta_{\text{QCD}} + \delta_{\text{EW}})

Forward—“backward at the pole:

A_{FB}^{0,f} = \frac{3}{4} A_e A_f, \quad A_f = \frac{2g_V^f g_A^f}{(g_V^f)^2 + (g_A^f)^2}

Experimental Compatibility

These are the exact objects experimental fits use; they're scheme-aware through the effective angle and radiative factors.