Electroweak Observables

The set of experimentally measured quantities that test the electroweak theory. This page is the inventory; the theoretical machinery used to compute each entry lives in cross-sections.md, decay-rates.md, and the general observables map, specialized to the $S U (2)_{L} \times U (1)_{Y}$ field content of electroweak theory.

Each observable is tagged with overlap status with QED, QCD, and the Standard Model:

Tag	Meaning
[EW-exclusive]	No analogue in QED or QCD alone — relies on $W, Z, h$ or chiral structure
[QED-shared]	QED contribution at low energy; EW contribution required at the $Z$ -pole or for $sin^{2} θ_{W}$ sensitivity
[QCD-shared]	EW vertex but hadronic input from QCD (form factors, PDFs, decay constants)
[SM-cross-sector]	Tests the combined QCD + EW structure; lives in standard-model.md

1. Gauge-Boson Properties [EW-exclusive]

The masses and widths of $W^{\pm}$ and $Z^{0}$ are the headline EW observables — they directly fix the gauge couplings $g, g^{'}$ and the EW scale $v$ .

1.1 Boson masses

Observable	Value	Where measured
$m_{Z}$	$91.1876 \pm 0.0021 GeV$	LEP1 (1989–95, $e^{+} e^{-} \to Z$ line-shape scan)
$m_{W}$	$80.369 \pm 0.013 GeV$ (PDG 2024 world avg.)	LEP2, Tevatron (CDF/D0), LHC (ATLAS, CMS, LHCb)
$Γ_{Z}$	$2.4952 \pm 0.0023 GeV$	LEP1 line-shape
$Γ_{W}$	$2.085 \pm 0.042 GeV$	LEP2, Tevatron, LHC

How these masses are actually measured. $m_{Z}$ comes from a Breit–Wigner line-shape scan at LEP1 ( $e^{+} e^{-}$ beam energy stepped through the resonance, beam energy calibrated to $1 0^{- 5}$ via resonant depolarization). $m_{W}$ is much harder because every $W$ decay involves a neutrino — it is extracted from the transverse-mass $m_{T}$ distribution at hadron colliders, using $Z \to ℓℓ$ events as a lepton-momentum calibration anchor. Full pipelines, including the CDF $m_{W}$ anomaly, are walked through in collider-measurements.md §3.1–3.2.

CDF $m_{W}$ tension. In 2022 the CDF collaboration reported $m_{W} = 80.434 \pm 0.009 GeV$ , $\sim 7 σ$ above the world average. Subsequent ATLAS and CMS measurements have agreed with the world average; the CDF result remains an unresolved outlier as of 2026. Either CDF is wrong, or one of the experiments handling the same data is wrong, or this is the cleanest hint of BSM physics in the EW sector.

1.2 $Z$ -pole line shape and partial widths

At the $Z$ pole the $e^{+} e^{-}$ cross section is a pure resonance:

$σ_{e^{+} e^{-} \to f \overset{ˉ}{f}} (s) \propto \frac{12 π}{m _{Z}^{2}} \frac{Γ _{ee} Γ _{ff}}{( s - m _{Z}^{2} ) ^{2} + m _{Z}^{2} Γ _{Z}^{2}} .$

Here $s = (p_{e^{-}} + p_{e^{+}})^{2} = E_{cm}^{2}$ is the Mandelstam invariant — the squared total CM energy, equal to $(2 E_{beam})^{2}$ at a symmetric collider like LEP. The resonance peaks at $s = m_{Z}^{2}$ , i.e. $s = m_{Z}$ . (See cross-sections.md § Mandelstam variables for the general $s, t, u$ kinematics.)

Fitting the line shape across $s \approx (88 - 94) GeV$ at LEP1 pinned down:

All hadronic and leptonic partial widths $Γ (Z \to ℓ \overset{ˉ}{ℓ}), Γ (Z \to q \overset{q}{ˉ})$ to $\sim 0.1%$ .
The invisible width $Γ_{inv}^{Z} = Γ_{Z} - \sum_{visible} Γ_{i}$ — equals $N_{ν} Γ_{ν \overset{ν}{ˉ}}^{SM}$ — giving

$N_{ν} = 2.984 \pm 0.008.$

This is the measurement that pins down the number of light ( $m_{ν} < m_{Z} /2$ ) neutrino flavors. Rules out a fourth chiral SM generation.

1.3 $W$ branching ratios

Channel	BR (SM tree-level)	Comment
$W \to e ν$	$\sim 1/9 \approx 10.8%$	$O (α_{s})$ corrections small
$W \to μν$	same	Lepton universality test: ratios = 1 to $\sim 1 0^{- 3}$
$W \to τν$	same	Same
$W \to hadrons$	$\sim 6/9 \approx 67.5%$	$u \overset{ˉ}{d} + c \overset{s}{ˉ}$ via CKM; the factor of 3 from color

Lepton universality is enforced by construction in the SM (all three generations have identical EW couplings); the measured BR ratios are universality tests at $\sim 1 0^{- 3}$ .

2. $Z$ -Pole Asymmetries and $sin^{2} θ_{W}$ [EW-exclusive]

Parity-violating asymmetries at the $Z$ pole are the cleanest way to measure the Weinberg angle. They expose the V−A structure of the $Z$ coupling.

Observable	Definition	Best measurement
Forward–backward $A_{FB}^{f}$	$(σ_{F} - σ_{B}) / (σ_{F} + σ_{B})$ for $e^{+} e^{-} \to f \overset{ˉ}{f}$ near $Z$ -pole	LEP1, all $f = μ, τ, b, c$
Left–right $A_{L R}$	$(σ_{L} - σ_{R}) / (σ_{L} + σ_{R})$ with polarized $e^{-}$ beam	SLC (SLAC, $\sim 1 0^{- 4}$ precision)
$τ$ polarization $P_{τ}$	helicity asymmetry of $τ$ produced in $Z \to ττ$	LEP1
$A_{FB}^{b}, A_{FB}^{c}$	quark-flavor-tagged FB asymmetries	LEP1

These all reduce to one underlying parameter, the effective leptonic weak mixing angle:

$sin^{2} θ_{W}^{eff,lep} = 0.23153 \pm 0.00016 (world average from Z -pole asymmetries) .$

This single number, combined with $m_{Z}$ and $G_{F}$ , constrains the entire EW radiative-correction structure — including (historically) predicting $m_{t}$ before its 1995 discovery and $m_{h}$ before 2012.

2.1 The $ρ$ parameter

$ρ \equiv \frac{m _{W}^{2}}{m _{Z}^{2} cos ^{2} θ _{W}} = 1 + Δ ρ .$

At tree level $ρ = 1$ (custodial $S U (2)$ symmetry of the Higgs sector). Measured $∣Δ ρ ∣ < 1 0^{- 3}$ . BSM physics with isospin-breaking would shift $Δ ρ$ — encoded in the Peskin–Takeuchi oblique parameters $S, T, U$ :

Parameter	Probes	Current bound
$T \propto Δ ρ$	Custodial-isospin breaking	$∣ T ∣ < 0.1$
$S$	Non-universal new physics in gauge-boson self-energies	$∣ S ∣ < 0.1$
$U$	$S U (2)_{L}$ doublet symmetry	$∣ U ∣ < 0.1$

Most BSM extensions (Two-Higgs-Doublet Models, technicolor, extra $W^{'} / Z^{'}$ , heavy fourth-generation fermions) shift one or more of $(S, T, U)$ by an amount the data already excludes.

3. Fermi Constant and Low-Energy Charged Currents [Mostly QCD-shared]

The most precisely measured electroweak quantity:

$G_{F} = (1.1663787 \pm 0.0000006) \times 1 0^{- 5} GeV^{- 2},$

extracted from the muon lifetime $τ_{μ} = 2.1969811 (22) μ s$ via the purely leptonic decay $μ^{-} \to e^{-} \overset{ν}{ˉ}_{e} ν_{μ}$ :

$\frac{1}{τ _{μ}} = \frac{G _{F}^{2} m _{μ}^{5}}{192 π ^{3}} [1 + QED + EW corrections] .$

[EW-exclusive] — no hadronic input needed.

Other low-energy CC observables that do require QCD form factors:

Observable	EW input	QCD input
Neutron lifetime $τ_{n} = 878.4 \pm 0.5 s$	$V_{u d}$ , $G_{F}$	$g_{A} = 1.2754 \pm 0.0013$ (axial coupling, lattice or low-energy fit)
$0^{+} \to 0^{+}$ super-allowed nuclear $β$ -decays	$V_{u d}$	Nuclear structure corrections (Vud-dominant)
$π \to μν, π \to e ν$	$G_{F}, V_{u d}$	$f_{π} = 130.5 \pm 0.1 MeV$ (lattice)
$K \to μν, K \to e ν$	$V_{u s}$	$f_{K} = 156.1 \pm 0.8 MeV$ (lattice)
$B \to τν, D_{s} \to μν$	$V_{u b}, V_{cs}$	$f_{B}, f_{D_{s}}$ (lattice)

The CKM matrix elements (Section 4) are extracted by combining these EW + QCD ingredients.

3.1 Lepton-universality ratios

Ratios cancel hadronic form factors and isolate pure EW structure:

$R_{e / μ}^{π} \equiv \frac{Γ ( π \to e ν )}{Γ ( π \to μν )} = (1.2352 \pm 0.0002) \times 1 0^{- 4},$

agrees with SM to $\sim 1 0^{- 3}$ — a stringent universality test.

3.2 Parity-violating atomic / electron physics [QED-shared]

The $Z$ -exchange weak charge $Q_{W}$ of a nucleus produces a parity-violating energy shift in heavy atoms:

System	Probes	Best result
Cesium-133 ( $^{133} Cs$ )	$Q_{W} (Cs) = - 72.82 \pm 0.42$	SM: $- 73.16 \pm 0.05$ ; consistent at $\sim 1 σ$
Møller scattering $e^{-} e^{-} \to e^{-} e^{-}$ at SLAC (E158) / JLab (MOLLER)	$sin^{2} θ_{W} (Q^{2} \to 0)$	Tests running of $sin^{2} θ_{W}$ from $m_{Z}$ down
Proton weak charge ( $Q_{weak}$ at JLab)	$Q_{W} (p)$	Tests SM at the $\sim 3%$ level

These low-energy parity-violation measurements probe BSM scales of $\sim 1 - 10 TeV$ , complementing direct LHC searches.

4. Flavor Physics: CKM and CP Violation [Heavily QCD-shared]

The CKM matrix has 4 physical parameters (3 angles + 1 phase). They are overdetermined by many independent measurements that all must agree.

4.1 Magnitudes

Element	Best determination	Value	QCD input
$∥ V_{u d} ∥$	Super-allowed nuclear $β$ -decay	$0.97373 (31)$	Nuclear structure
$∥ V_{u s} ∥$	$K_{ℓ 3}$ form factor, $Γ (K_{ℓ 2}) /Γ (π_{ℓ 2})$	$0.22431 (85)$	$f_{K} / f_{π}$ , $K_{ℓ 3}$ form factor (lattice)
$∥ V_{c d} ∥$	$D \to π ℓ ν$ , neutrino DIS	$0.221 (4)$	Form factors (lattice)
$∥ V_{cs} ∥$	$D \to K ℓ ν$ , $W \to cs$	$0.975 (6)$	Form factors (lattice)
$∥ V_{u b} ∥$	$B \to π ℓ ν$ (excl.), $B \to X_{u} ℓ ν$ (incl.)	$(3.82 \pm 0.20) \times 1 0^{- 3}$	Form factors / OPE matrix elements
$∥ V_{c b} ∥$	$B \to D^{*} ℓ ν$ (excl.), $B \to X_{c} ℓ ν$ (incl.)	$(41.0 \pm 1.4) \times 1 0^{- 3}$	Same
$∥ V_{t d} ∥$	$B^{0}$ – $\overset{ˉ}{B}^{0}$ oscillation $Δ m_{d}$	$(8.6 \pm 0.2) \times 1 0^{- 3}$	Bag parameter $B_{B_{d}}$
$∥ V_{t s} ∥$	$B_{s}$ – $\overset{ˉ}{B}_{s}$ oscillation $Δ m_{s}$	$(41.5 \pm 0.9) \times 1 0^{- 3}$	Bag parameter $B_{B_{s}}$
$∥ V_{t b} ∥$	$t \to Wb$ branching	$> 0.92$ at 95% CL	(clean — single-top production)

A long-standing $∣ V_{u b} ∣$ tension between exclusive and inclusive determinations ( $\sim 3 σ$ ) is one of the unresolved puzzles of flavor physics; it remains as of 2026.

4.2 CP-violating observables

Observable	Physical meaning	SM prediction
$ϵ_{K} = (2.228 \pm 0.011) \times 1 0^{- 3}$	Indirect CP violation in $K^{0}$ mixing	Consistent within $\sim 10%$
$ϵ^{'} / ϵ = (1.66 \pm 0.23) \times 1 0^{- 3}$	Direct CP violation in $K \to ππ$	Consistent (large lattice uncertainty)
$sin 2 β$ from $B^{0} \to J / ψ K_{S}$	Direct CP via interference of mixing + decay	$0.699 \pm 0.017$ — consistent with global UT fit
$γ = (66. 4_{- 3.6}^{+ 3.5}) °$ from $B \to DK$	CKM angle, theoretically cleanest	Consistent
$ϕ_{s}$ from $B_{s} \to J / ψ ϕ$	CP phase in $B_{s}$ mixing	Consistent with tiny SM value

The unitarity triangle $V_{u d} V_{u b}^{*} + V_{c d} V_{c b}^{*} + V_{t d} V_{t b}^{*} = 0$ — when plotted in the complex plane — must close. All current measurements close it to $\sim 1 0^{- 4}$ accuracy. This is the central SM flavor test, sensitive to a wide range of BSM models. See standard-model.md § Unitarity-Triangle Fit for the consolidated cross-sector view.

4.3 Rare FCNC decays [Pure quantum loop probes]

In the SM, flavor-changing neutral currents are forbidden at tree level by the GIM mechanism (see standard-model.md § GIM). The measured rates probe loop physics directly.

Decay	SM BR	Measured
$B_{s} \to μ^{+} μ^{-}$	$(3.66 \pm 0.14) \times 1 0^{- 9}$	$(3.4 5_{- 0.27}^{+ 0.29}) \times 1 0^{- 9}$ (LHCb 2022) — ✅
$B^{0} \to μ^{+} μ^{-}$	$(1.03 \pm 0.05) \times 1 0^{- 10}$	$1.2 1_{- 0.41}^{+ 0.46} \times 1 0^{- 10}$ — consistent
$B \to K^{} μ^{+} μ^{-}$ , $B \to K^{} e^{+} e^{-}$	computed	LFU ratios $R_{K^{(*)}}$ — once anomalous (2014–22), now consistent with SM (LHCb 2023)
$K^{+} \to π^{+} ν \overset{ν}{ˉ}$	$(8.4 \pm 1.0) \times 1 0^{- 11}$	$(10. 6_{- 3.4}^{+ 4.0}) \times 1 0^{- 11}$ (NA62) — consistent
$μ \to e γ$	$\sim 1 0^{- 54}$ in SM with $m_{ν} \neq = 0$	$< 3.1 \times 1 0^{- 13}$ (MEG) — any observation = BSM
$μ^{-} \to e^{-}$ conversion	$\sim 1 0^{- 54}$ in SM	$< 7 \times 1 0^{- 13}$ (SINDRUM-II); Mu2e/COMET aim for $1 0^{- 17}$

Lepton-flavor violation in charged leptons would be a smoking gun for BSM physics. SM predicts essentially zero; current experimental sensitivity is $\sim 1 0^{- 13}$ , future $\sim 1 0^{- 17}$ — enormous discovery potential.

5. Higgs Sector Observables [Mixed: QCD-shared production, mostly EW-exclusive couplings]

After 2012 the Higgs sector is its own subfield.

5.1 Mass

$m_{h} = 125.20 \pm 0.11 GeV (ATLAS+CMS combined) .$

Combined with $G_{F}, m_{W}$ this fixes $λ$ in the Higgs potential to $\approx 0.13$ .

5.2 Production cross sections [QCD-shared]

ATLAS/CMS resolve five production modes, each probing different couplings:

Mode	Diagram	Fraction at $s = 13 TeV$	Probes
Gluon fusion (ggH)	top loop $\to h$	$\sim 87%$	top Yukawa $y_{t}$ (and any BSM colored states)
Vector-boson fusion (VBF)	$qq \to qq + h$ via $W, Z$	$\sim 7%$	$hVV$ couplings, tagging signature
Associated $V H$	$q \overset{q}{ˉ} \to W^{*} \to Wh$ etc.	$\sim 4%$	$hWW, h ZZ$ couplings
$t \overset{ˉ}{t} H$	$gg \to t \overset{ˉ}{t} h$	$\sim 1%$	Direct $y_{t}$ measurement
$b \overset{ˉ}{b} H$	$gg \to b \overset{ˉ}{b} h$	$\sim 1%$	$y_{b}$ (small, hard)

Heavy QCD K-factors ( $σ_{NLO} / σ_{LO} \sim 2$ for ggH) — Higgs production calculations require $\sim α_{s}^{4}$ matching to be percent-level.

5.3 Branching ratios

Channel	BR (SM)	Measured
$h \to b \overset{ˉ}{b}$	$58.4%$	✓
$h \to W W^{*}$	$21.4%$	✓
$h \to gg$	$8.2%$	inferred
$h \to τ^{+} τ^{-}$	$6.3%$	✓
$h \to c \overset{c}{ˉ}$	$2.9%$	first evidence 2022, still poorly measured
$h \to Z Z^{*}$	$2.6%$	"golden channel": $4 ℓ$ final state
$h \to γγ$	$0.23%$	"diphoton" — original 2012 discovery channel
$h \to Z γ$	$0.15%$	first evidence 2023, currently $\sim 2 σ$ above SM
$h \to μ^{+} μ^{-}$	$0.022%$	$3 σ$ evidence (2020); the smallest measured Higgs coupling

5.4 Coupling modifiers $κ_{i}$

A common parameterization defines $κ_{i} \equiv g_{hi}^{obs} / g_{hi}^{SM}$ for each Higgs vertex. Current LHC measurements:

$κ_{W}, κ_{Z}, κ_{t}, κ_{b}, κ_{τ}, κ_{μ}, κ_{g}, κ_{γ} = 1 \pm few-to-ten percent .$

All consistent with $κ = 1$ (pure SM). Total Higgs width $Γ_{h}^{SM} = 4.07 MeV$ — also consistent with measurements.

5.5 CP and spin

Angular analyses of $h \to Z Z^{*} \to 4 ℓ$ and $h \to ττ$ confirm $J^{PC} = 0^{++}$ to high confidence; large CP-violating Higgs–top coupling already excluded.

5.6 Higgs self-couplings

$λ$ enters quadratically into di-Higgs $pp \to hh$ production:

Current bounds: $- 0.6 < κ_{λ} < 6.6$ at 95% CL (HL-LHC projection: $\sim 50%$ precision on $κ_{λ}$ ).
FCC-hh would reach $\sim 5%$ precision on $λ$ .

This is the least well-tested part of the SM Higgs sector and the cleanest probe of the shape of the Higgs potential — relevant to the cosmological electroweak phase transition and the universe's matter–antimatter asymmetry.

6. Anomalous Moments and Precision Tests [QED-shared]

The anomalous magnetic moments $a_{e}, a_{μ} = (g - 2) /2$ are predominantly QED observables but receive small required EW contributions.

Observable	Contribution from EW	Significance
$a_{e}^{EW}$	$\sim 3 \times 1 0^{- 14}$	Far below experimental sensitivity
$a_{μ}^{EW}$	$\sim 1.54 \times 1 0^{- 9}$	Required for SM consistency at $\sim 1 0^{- 10}$
$a_{τ}^{EW}$	$\sim 4 \times 1 0^{- 8}$	Not yet measured

The muon $g - 2$ anomaly (Fermilab E989, results 2021–2023) shows a $\sim 4.2 σ$ excess over SM theory — if the data-driven hadronic vacuum polarization is correct. Lattice-QCD (BMW 2020 and subsequent reproductions) gives a different hadronic value that brings SM closer to experiment. As of 2026 the situation remains contested; resolution requires either (a) the SM hadronic VP is right and there's BSM physics, or (b) lattice is right and the dispersive data-driven analysis has a subtle issue.

6.1 Electric dipole moments [BSM-sensitive]

$d_{e}^{SM} \sim 1 0^{- 38} e \cdot cm, d_{e}^{exp} < 4.1 \times 1 0^{- 30} e \cdot cm (ACME II), < 2 \times 1 0^{- 30} e \cdot cm (JILA HfF^{+}, 2023) .$

SM-CKM contribution is many orders of magnitude below current experimental sensitivity. Any positive electron-EDM measurement = BSM CP violation.

7. Neutrino Sector Observables [EW-exclusive, BSM-driven]

Neutrinos feel only the weak force (and gravity), so neutrino observables are 100% EW. Most current results require physics beyond the renormalizable SM (neutrino masses).

Observable	Source	Status
Atmospheric mass-squared diff. $Δ m_{32}^{2} \approx 2.5 \times 1 0^{- 3} eV^{2}$	Super-K, T2K, NOvA, IceCube	Measured; sign (NH vs IH) still unresolved
Solar mass-squared diff. $Δ m_{21}^{2} \approx 7.4 \times 1 0^{- 5} eV^{2}$	SNO, KamLAND, Super-K	Measured
PMNS mixing angles $θ_{12}, θ_{23}, θ_{13}$	Combined oscillation data	All large (unlike CKM); $θ_{13} \approx 8.6°$ measured 2012
Leptonic CP phase $δ_{CP}$	T2K + NOvA combined	Hints of $δ_{CP} \neq = 0$ ; not yet $5 σ$
Sum of neutrino masses $\sum m_{ν}$	Cosmology (CMB + LSS)	$< 0.12 eV$ (Planck 2018 + BAO)
Neutrinoless double- $β$ decay $⟨ m_{ββ} ⟩$	KamLAND-Zen, GERDA, LEGEND	$< 36 - 156 meV$ (depending on nuclear matrix element); positive observation would prove Majorana nature

Within the renormalizable SM neutrinos are massless. All of the above are evidence for BSM physics — either right-handed neutrinos (Dirac masses) or the Weinberg dim-5 operator (Majorana masses). See electroweak Caveats.

8. Computational Pipelines

The structural map of how each observable connects back to QFT machinery:

Observable type	Method
Gauge-boson masses & widths	Tree-level + radiative corrections in $R_{ξ}$ gauge → cross-sections.md, decay-rates.md
$Z$ -pole asymmetries	Differential cross sections at $s \approx m_{Z}$ ; ratios cancel normalizations → observables/README.md § 2.4
$G_{F}$ from $τ_{μ}$	Decay rate of three-body final state → decay-rates.md
CKM extraction	EW vertex + QCD form factor (lattice) — pair-wise → master formulas of cross-sections.md, decay-rates.md
Higgs cross sections	Multi-loop QCD + EW; production matched to NNLO+NNLL accuracy → general observables/README.md § 2.8
Higgs branching ratios	Decay-rate master formula × spectrum of channels → decay-rates.md
Anomalous moments	Vertex function $F_{2} (0)$ → observables/README.md § 2.7
Neutrino oscillations	Quantum-mechanical superposition of mass eigenstates; requires BSM neutrino mass term

9. See Also

Electroweak Theory — the underlying theory whose observables are listed here.
Standard Model Observables — cross-sector observables (CKM unitarity, GIM, anomaly fits, lepton universality, global EW fit).
Observables — General Map — the structural classification (Class A/B/C) and inventory of all QFT observables.
Cross Sections, Decay Rates — master formulas behind most observables above.
The Standard Model — how the SM combines EW + QCD; defines the global parameter space.

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