Electroweak Theory and the Higgs

Choose an electroweak route by the object that must be made consistent: representations and charges, the Higgs orbit, mass or current eigenstates, fermion masses, a low-energy limit, a longitudinal high-energy limit, a renormalized input map, or a pole/fiducial Higgs observable. The chapter connects those objects through one $SU(2)_L\times U(1)_Y$ Lagrangian while keeping gauge choices, tree parameters, renormalized inputs, and measured observables conceptually distinct.

Check readiness, then choose a route

The required-background notes on individual leaves are hard dependencies for their derivations. The order below is a suggested complete pass, not a claim that every reader must begin at the first page. A reader who can already diagonalize the neutral mass matrix may enter at weak currents; a precision calculation should begin with its renormalized input and pole conventions.

This diagnostic is informal and unscored. Each symptom can be observed in a draft calculation, and each repair supplies the missing capability.

If the draft currently…	Missing capability	Exact repair
assigns hypercharges but cannot recover every electric charge	weights, representations, and the declared $Q=T_3+Y$ normalization	Compact Lie groups, roots, and weights
treats $\langle H\rangle$ as a gauge-invariant observable	the distinction between gauge fixing and the Higgs phase	Elitzur’s theorem and the gauge-invariant Higgs mechanism
rotates $W^3$ and $B$ without checking the zero eigenvector	spectral decomposition of a symmetric mass matrix	Normal forms, spectra, and projectors
writes a fermion mass before checking its chiral gauge charges	Weyl chirality and Lorentz-invariant bilinears	Weyl fields and chirality
replaces a $W$ propagator by a constant near its pole	controlled heavy-field matching	Tree-level matching by classical elimination
uses $\epsilon_L^\mu\simeq p^\mu/m$ as the complete equivalence theorem	Ward identities, partial waves, and high-energy hypotheses	Partial-wave unitarity
mixes an on-shell mass, a running weak angle, and a $G_F$-scheme correction	renormalized perturbation theory and finite scheme translation	Renormalized perturbation theory
calls a Breit–Wigner fit parameter, complex pole, and fiducial rate the same observable	unstable-particle observable layers	Unstable-particle observables and resonance approximations

The chapter guide lists every leaf once, in manifest order.

Route	Use it when the central question is	Main output
Electroweak gauge and matter structure	Which chiral multiplets and hypercharges define one generation?	A charge- and gauge-invariance-checked field table
The Higgs doublet and electroweak symmetry breaking	What phase and physical scalar content follow from the doublet potential?	A gauge-qualified vacuum orbit and mode count
Gauge-boson masses and electroweak mixing	How do $W^\pm$, $Z$, and $\gamma$ emerge?	The charged and neutral mass matrices, photon zero mode, and tree relations
Charged and neutral weak currents	How do physical vector fields couple to chiral fermions?	$J_\mathrm{em}$, $J_\pm$, and $J_Z$ with vector/axial checks
Yukawa couplings and fermion masses	How do gauge-invariant matrices become masses, Higgs vertices, and flavor misalignment?	A basis-aware mass and coupling map
Higgs self-interactions and the scalar potential	What do cubic and quartic Higgs couplings mean at tree and loop level?	Tree vertices plus a renormalized observable contract
The Fermi limit of weak interactions	What remains when momentum transfer is far below $m_W$?	A normalized left-handed four-fermion interaction and power remainder
Longitudinal vector bosons and the equivalence theorem	How is high-energy growth cancelled and when may Goldstones replace longitudinal vectors?	A channel-level cancellation and theorem-domain test
Electroweak renormalization and input schemes	How are measured inputs converted into consistent loop predictions?	An input/output map with counterterm, tadpole, and complex-pole conventions
Higgs interactions, production, decay, and pole observables	How do couplings reach production, decay, pole, pseudo-observable, and fiducial layers?	A durable mechanism and covariance map with a dated-evidence boundary

The model-wide consistency chain

The reusable logic is

$$\begin{gathered} \{G,\text{ representations},Y,g,g'\} \longrightarrow \{H,V(H),\text{ vacuum orbit}\} \longrightarrow \{W^\pm,Z,A,h\},\\ \longrightarrow \{J_\pm,J_Z,J_{\mathrm{em}},M_f,Y_f\} \longrightarrow \{G_F,\text{ high-energy cancellations}\} \longrightarrow \{\text{renormalized inputs, poles, fiducial observables}\}. \end{gathered}$$

Every arrow has a distinct check:

1. Representation check: each kinetic and Yukawa term is invariant, electric charges follow from $Q=T_3+Y$, and anomaly cancellation is not confused with classical gauge invariance.
2. Phase check: the minimum is described by a gauge orbit; physical claims use masses, poles, and gauge-invariant matrix elements rather than a gauge-variant field expectation value.
3. Mass check: the neutral matrix has one exact zero mode and $\rho=1$ at tree level for one doublet.
4. Current check: the photon coupling is vectorlike and universal, the charged current is left-handed, and the neutral current has the correct $T_3-s_W^2Q$ structure.
5. Limit check: the Fermi interaction is the $q^2/m_W^2\to0$ expansion, while the equivalence theorem is an $E/m_V\to\infty$ statement at fixed scattering geometry. They are opposite limits.
6. Renormalization check: an observable is ultraviolet finite and gauge independent after one input set, tadpole prescription, mass/width definition, and perturbative order are used throughout.

These tree-level field, mass, current, and low-energy relations are developed coherently in Schwartz 2014, §§29.1–29.4, pp. 584–604. Loop calculations require the additional counterterm and input-scheme organization reviewed in Denner 1993, §§3–4, pp. 334–364.

Chapter conventions and boundaries

This chapter fixes

$$Q=T_3+Y, \qquad D_\mu=\partial_\mu-i gT^aW^a_\mu-i g'YB_\mu,$$

in addition to the site’s Hermitian-generator and $(+---)$ conventions. Thus the Higgs doublet has $Y_H=\tfrac12$. Define

$$s_W=\sin\theta_W,\qquad c_W=\cos\theta_W,$$

but attach a qualifier beyond tree level: $s_W^2=1-M_W^2/M_Z^2$ is an on-shell definition, whereas a hatted or running angle is a different renormalized parameter. Similarly, $v$ and the minimum condition are tree-level parameters until an input and tadpole scheme defines them perturbatively.

For an unstable particle, the preferred invariant mass/width datum is the complex pole

$$s_p=\mu_p^2-i\mu_p\gamma_p.$$

Do not substitute a running-width line-shape mass into a complex-pole formula without translation. Production, partial widths, branching fractions, pseudo-observables, and fiducial rates are separate layers even when a narrow-width approximation relates them.

The chapter applies the Standard Model doublet. General spontaneous-symmetry-breaking theory, generic renormalization and matching, flavor phenomenology, additional scalar representations, and live experimental combinations remain with their dedicated owners.

Informal synthesis check

This is an informal, unscored work-product check. Starting from one generation plus a Higgs doublet, construct a symbolic prediction for a process with a charged-current production stage and a resonant Higgs decay. A satisfactory response should:

• list every representation and hypercharge and verify the relevant Yukawa term;
• minimize the potential without treating the chosen Higgs direction as an observable;
• diagonalize the neutral mass matrix and recover $e=gs_W=g'c_W$;
• derive the charged and neutral currents, then obtain the Fermi coefficient by expanding the $W$ propagator;
• show a representative cancellation of longitudinal energy growth;
• choose one renormalized input set, tadpole convention, and complex-pole convention; and
• separate the pole residue, partial width, total-width correlation, fiducial measurement, and any dated evidence.

Criterion	A minimally complete answer contains	Repair if absent
Charges	explicit $(SU(3)_c,SU(2)_L)_Y$ assignments and $Q=T_3+Y$ checks	Multiplets, invariants, and selection rules
Higgs phase	the orbit, unbroken generator, four-to-one scalar count, and gauge-invariant interpretation	Elitzur’s theorem and the gauge-invariant Higgs mechanism
Masses and currents	the exact neutral zero mode plus electromagnetic, charged, and neutral current normalizations	Quantum currents, improvements, and conservation
Low/high limits	$O(q^2/m_W^2)$ for Fermi matching and $O(m_V/E)$ for equivalence	Controlled effective-field-theory expansion and partial-wave unitarity
Loop prediction	one coherent input, tadpole, gauge, pole, and truncation record	Renormalization conditions, schemes, and finite parts
Evidence interface	release, dataset period, covariance/likelihood, theory version, corrections, and evidence date	Unstable-particle observables

Purpose-keyed exits

When the next task is…	Continue to…
derive CKM misalignment and weak flavor amplitudes	Quark Flavor and CP
add neutrino masses, mixing, and propagation	Neutrino and Lepton Physics
combine electroweak, QCD, flavor, anomalies, and global form	Standard Model Assembly and Consistency
define pseudo-observables, likelihoods, and correlated precision tests	Precision Standard Model
replace the single-doublet sector by a controlled extension	Consistent Extensions and Portals
inspect dated Higgs or electroweak status and open problems	EFT and Standard Model Tests

References

• Denner, Ansgar. “Techniques for the Calculation of Electroweak Radiative Corrections at the One-Loop Level and Results for W-Physics at LEP 200.” Fortschritte der Physik 41, no. 4 (1993): 307–420. DOI. Open PDF.
• Schwartz, Matthew D. Quantum Field Theory and the Standard Model. Cambridge University Press, 2014, §§29.1–29.4, pp. 584–604. DOI.