Electroweak Theory

The electroweak theory is the relativistic quantum field theory of leptons, quarks, the photon, the $W^{\pm }$ and $Z^0$ gauge bosons, and the Higgs boson. It unifies the electromagnetic interaction (QED) with the weak interaction in a single gauge theory based on the group

$$G_{\text{EW}}=SU(2{)}_L\times U(1{)}_Y,$$

spontaneously broken by the vacuum expectation value of a Higgs scalar doublet down to the residual electromagnetic $U(1{)}_{\text{EM}}$. The full unbroken theory has four massless gauge bosons; the broken theory has one massless gauge boson (photon) and three massive ones ($W^{\pm },Z^0$), with masses generated by the Higgs mechanism.

It is the second pillar of the Standard Model (alongside QCD), the canonical example of:

• spontaneous symmetry breaking (SSB) in a gauge theory (the Higgs mechanism),
• chiral fermions (left-handed and right-handed components transform differently under the gauge group),
• explicit parity violation ($P$, $C$ separately violated; $CP$ violated through the CKM phase),
• flavor mixing (the CKM and PMNS matrices),

and historically the first instance of a unification of two of the four fundamental interactions.

What Electroweak Theory Describes Physically: Electromagnetism and the Weak Force

Electroweak theory subsumes two of the four fundamental forces of nature. The electromagnetic force is treated in detail in QED (the unbroken $U(1{)}_{\text{EM}}$ that remains after symmetry breaking — see §3.1). This section unpacks the second, less familiar half: the weak force.

What the weak force is

The weak force (weak interaction) is the fundamental force responsible for processes that change one type of fermion into another. Unlike electromagnetism (which preserves species and only shifts energies/momenta) or the strong force (which binds quarks of fixed flavor), the weak force can turn a down quark into an up quark, an electron into a neutrino, a muon into an electron, and so on.

Canonical processes:

• Nuclear $\beta$-decay: $n\to p+e^{-}+{\bar{\nu }}_e$, the historical entry point (radioactivity, discovered 1896; understood as a weak process in the 1930s). Microscopically, a down quark inside the neutron emits a virtual $W^{-}$ and turns into an up quark; the $W^{-}$ then decays into an electron and an electron antineutrino.
• Muon decay: ${\mu }^{-}\to e^{-}+{\bar{\nu }}_e+{\nu }_{\mu }$ — the cleanest weak process, used to define the Fermi constant $G_F$.
• Pion decay ${\pi }^{-}\to {\mu }^{-}+{\bar{\nu }}_{\mu }$, kaon decay, hyperon decay, and the entire menagerie of "slow" hadron decays whose lifetimes ($10^{-10}\mathrm{s}$ or longer) signal a weak-force origin.
• Hydrogen fusion in the Sun: $p+p\to d+e^{+}+{\nu }_e$. A proton must turn into a neutron for the deuteron to form — only the weak force allows that. The slowness of this process (${\tau }_{pp}\sim 10^{10}\mathrm{yr}$ per proton) is why the Sun burns for billions of years rather than seconds.
• Neutrino interactions: neutrinos feel only the weak force (and gravity), which is why a typical solar neutrino traverses a light-year of lead before scattering.

Distinguishing features

The name "weak" is a low-energy accident: at energies $\gtrsim m_W$ the weak coupling is comparable to the electromagnetic one. The apparent weakness at everyday energies comes from the propagator suppression $1/(q^2-m_W^2)\approx -1/m_W^2$ for $|q^2|\ll m_W^2$ — pulling a factor of $1/m_W^2$ into every weak amplitude. Strip away that propagator factor and the underlying coupling $g$ is stronger than the electromagnetic $e$. This is exactly the hint that led to electroweak unification: the two forces look qualitatively different at low energy, but their underlying gauge couplings are of the same order.

Two types of weak interactions

Empirically the weak force splits into two distinct classes:

• Charged-current (CC) interactions, mediated by $W^{\pm }$. They change electric charge by one unit and the flavor of one fermion. Vertex (in the doublet $({\nu }_e,e^{-})$, similarly for $(u,d)$): $$W_{\mu }^{+}\to {\bar{\nu }}_e{\gamma }^{\mu }{P}_L{e}^{-}+\bar{u}{\gamma }^{\mu }{P}_L{V}_{\text{CKM}}d.$$ All known $\beta$-decays, hyperon decays, and most flavor physics happen via CC.
• Neutral-current (NC) interactions, mediated by $Z^0$. They preserve flavor and electric charge but provide a new weak force between like and unlike species, including neutrino–electron scattering. The $Z$ couples to a mixture of vector and axial-vector currents and was predicted by electroweak unification before direct observation (Gargamelle, 1973 — the first hard confirmation of the GSW theory).

V−A structure and parity violation

The single most surprising experimental fact about the weak force is maximal parity violation: only left-handed fermions ${\psi }_L=P_L\psi =\frac{1}{2}(1-{\gamma }^5)\psi$ couple to the $W^{\pm }$. A right-handed electron does not feel the charged-current weak force at all. This was discovered in the Wu experiment (1956–57): polarized ${}^{60}\mathrm{Co}$ nuclei were found to emit $\beta$ electrons preferentially in one direction relative to the nuclear spin, a result that cannot happen in a parity-conserving theory. Lee and Yang shared the Nobel Prize in 1957 for predicting it.

The compact statement is that the charged-current interaction has V−A (vector-minus-axial) structure: it couples through $\bar{\psi }{\gamma }^{\mu }(1-{\gamma }^5)\psi$ rather than the parity-symmetric $\bar{\psi }{\gamma }^{\mu }\psi$ of QED. This is encoded in the modern theory by putting ${\psi }_L$ in $SU(2{)}_L$ doublets and ${\psi }_R$ in singlets — a structural choice (Step 1 below), but the empirical input it formalizes is the V−A structure that Wu's experiment exposed.

The terminology mismatch: is there a "QFD"? By analogy with QED (the standalone gauge theory of EM, $U(1{)}_{\text{EM}}$) and QCD (the standalone gauge theory of the strong force, $SU(3{)}_C$), one might expect a standalone theory of the weak force — sometimes called QFD (Quantum Flavordynamics) in older literature. There is no such theory in nature. The weak interaction is inseparably unified with electromagnetism in electroweak theory: the $W^{\pm },Z^0,\gamma$ are four mixtures (rotated by the Weinberg angle ${\theta }_W$) of the original $SU(2{)}_L\times U(1{)}_Y$ gauge bosons, and they cannot be cleanly disentangled into a "weak-only" and "EM-only" sector at the Lagrangian level. The historical predecessor — Fermi's effective theory of $\beta$-decay (1933) with a four-fermion contact interaction $G_F(\bar{p}{\gamma }^{\mu }n)(\bar{e}{\gamma }_{\mu }{\nu }_e)$ — was a standalone theory of the weak force, but it is non-renormalizable and breaks down at energies $\sqrt{s}\gtrsim 300\mathrm{GeV}$. EW theory replaces it with a renormalizable gauge theory in which Fermi's $G_F$ emerges as the low-energy limit, $G_F/\sqrt{2}=g^2/(8m_W^2)$. So "QFD" never caught on, and the modern view is that the weak force is one half of electroweak.

Short historical lineage

• Becquerel 1896 — discovery of radioactivity; later identified as $\beta$-decay = weak.
• Fermi 1933 — first quantitative theory: a four-fermion contact interaction ${\mathcal{L}}_F=G_F(\bar{p}{\gamma }^{\mu }n)(\bar{e}{\gamma }_{\mu }{\nu }_e)+\text{h.c.}$, with $G_F\approx 1.17\times 10^{-5}{\mathrm{GeV}}^{-2}$. Predicts $\beta$ spectra correctly. Non-renormalizable.
• Lee–Yang 1956, Wu 1957 — parity violation. The weak interaction is the only one in nature that distinguishes left from right at the fundamental level.
• Feynman–Gell-Mann / Sudarshan–Marshak 1958 — V−A structure: the weak interaction couples only to left-handed currents.
• Glashow 1961, Weinberg 1967, Salam 1968 — electroweak unification: $SU(2{)}_L\times U(1{)}_Y$ with the Higgs mechanism. Predicts $W^{\pm },Z^0$, neutral currents, and the photon all from the same gauge structure. Nobel Prize 1979.
• 't Hooft–Veltman 1971–72 — proof that the spontaneously broken non-abelian gauge theory is renormalizable. Settles EW as a viable QFT. Nobel Prize 1999.
• Gargamelle 1973 — discovery of weak neutral currents (the first confirmation of $Z^0$ as predicted by EW).
• UA1/UA2 1983 — direct discovery of $W^{\pm }$ and $Z^0$ bosons at CERN. Nobel Prize 1984.
• LHC ATLAS/CMS 2012 — discovery of the Higgs boson at $m_h\approx 125\mathrm{GeV}$. Nobel Prize 2013.

The "weak interaction" label predates the field-theoretic understanding by 40+ years; the modern view is that the weak force and electromagnetism are two low-energy faces of a single $SU(2{)}_L\times U(1{)}_Y$ gauge interaction, broken by the Higgs VEV to expose the difference between them.

Companion presentations.

• Modern Foundations — Wigner–Weinberg derivation shows in §5.2 that Lorentz consistency of any set of massless spin-1 species forces a Yang–Mills structure (the "Generalization" callout). It also flags (in the "What this story does not derive" callout) that massive gauge bosons are not derivable from M1+M2+M3 alone — their longitudinal polarizations break perturbative unitarity unless the gauge symmetry is spontaneously broken by a scalar VEV. This document is where that gap is filled: the Higgs mechanism is the empirical input that makes massive $W^{\pm },Z^0$ consistent with Lorentz + unitarity.
• QED and QCD are the two simpler gauge theories built on the same template (gauge symmetry → covariant derivative → renormalizable Lagrangian). Electroweak adds chirality, symmetry breaking, and mixing. See §Comparison with QED and QCD for a side-by-side.
• Standard Model combines this document with QCD into the full $SU(3{)}_C\times SU(2{)}_L\times U(1{)}_Y$ theory, with the additional cross-sector ingredients (anomaly cancellation, three generations, Yukawa hierarchy, CP problems).

Following the tag convention of foundations-modern.md, each step below is labelled (Empirical input), (Postulate), (Theorem), or (Standard machinery) so the assumption budget on top of the QFT postulates is transparent.

Derivation of Electroweak Theory from QFT

Electroweak inherits all postulates of QFT (see Postulates of QFT). The chain of choices parallels the QED / QCD derivations but with three new structural ingredients: chirality in the field content (Step 1), spontaneous symmetry breaking in the Lagrangian (Step 3), and mass mixing at the quantization stage (Step 4).

Step 1 — Specify the Field Content (Empirical input; specializes QFT Postulate 4)

The electroweak field content has four irreducible pieces, each playing a distinct structural role.

Gauge fields. Four real vector fields, one for each generator of $SU(2{)}_L\times U(1{)}_Y$:

• $W_{\mu }^a(x)$, $a=1,2,3$ — three real vector fields in the adjoint of $SU(2{)}_L$.
• $B_{\mu }(x)$ — one real vector field, neutral under $SU(2{)}_L$, carrying hypercharge $Y$ via $U(1{)}_Y$.

At the level of the unbroken Lagrangian all four are massless. The physical $W^{\pm },Z^0,\gamma$ emerge after symmetry breaking (Step 3) as linear combinations.

Matter fields — chiral fermions. The genuinely novel feature relative to QED/QCD is that left-handed and right-handed components transform differently: they belong to inequivalent representations of $SU(2{)}_L$. Writing ${\psi }_L\equiv P_L\psi =\frac{1}{2}(1-{\gamma }^5)\psi$ and ${\psi }_R\equiv P_R\psi =\frac{1}{2}(1+{\gamma }^5)\psi$:

(Right-handed neutrinos ${\nu }_R$ are not present in the original SM but are added in extensions to give neutrinos Dirac masses; see Caveats.) The full theory has three generations of this pattern (electron / muon / tau, with up/down, charm/strange, top/bottom); a single generation is shown above for clarity. Hypercharge assignments are not free: they are fixed by anomaly cancellation (see standard-model/from-postulates.md).

Why chirality? The empirical input is the V−A (vector-minus-axial) structure of charged-current weak interactions, first systematized by Feynman–Gell-Mann (1958) and Sudarshan–Marshak (1958) and traced to parity violation in $\beta$-decay (Wu, 1957). Putting ${\psi }_L$ and ${\psi }_R$ in inequivalent gauge multiplets is the modern way of encoding this: a Dirac mass term $\bar{\psi }\psi ={\bar{\psi }}_L{\psi }_R+{\bar{\psi }}_R{\psi }_L$ mixes the two chiralities and is therefore forbidden by gauge invariance until the Higgs gives the symmetry-breaking VEV that supplies the missing quantum numbers.

Scalar field — the Higgs. A complex scalar doublet under $SU(2{)}_L$ with hypercharge $Y=+\frac{1}{2}$:

$$\Phi (x)=\left(\begin{array}{c}{\varphi }^{+}(x)\\ {\varphi }^0(x)\end{array}\right),\Phi \in 2\text{ of }SU(2{)}_L,Y(\Phi )=+\frac{1}{2}.$$

This is the single field whose presence is novel relative to QED/QCD. The four real degrees of freedom in $\Phi$ become, after symmetry breaking, three longitudinal modes of $W^{\pm },Z^0$ (eaten by the gauge bosons, Goldstone-style) and one physical scalar — the Higgs boson $h$, mass $m_h\approx 125\mathrm{GeV}$ (discovered at the LHC, 2012).

By the spin–statistics theorem (QFT Postulate 7), all fermions are quantized with anticommutators, all bosons with commutators.

Step 2 — Postulate Local $SU(2{)}_L\times U(1{)}_Y$ Gauge Symmetry (Postulate)

The next ingredient is local non-abelian × abelian gauge invariance. With ${\tau }^a/2={\sigma }^a/2$ the $SU(2)$ generators (half the Pauli matrices) and $Y$ the hypercharge generator:

• A field $\psi$ in a representation $R$ of $SU(2{)}_L$ with hypercharge $Y(\psi )$ transforms as $$\psi (x)\to e^{i{\alpha }^a(x)T_R^a+i\beta (x)Y(\psi )}\psi (x).$$
• The gauge fields transform inhomogeneously: $$W_{\mu }^a\to W_{\mu }^a+\frac{1}{g}{\partial }_{\mu }{\alpha }^a+{ϵ}^{abc}{\alpha }^b{W}_{\mu }^c,B_{\mu }\to B_{\mu }+\frac{1}{g^{\prime }}{\partial }_{\mu }\beta .$$

The covariant derivative acting on $\psi$ is

$$D_{\mu }\psi =({\partial }_{\mu }-ig{W}_{\mu }^a{T}_R^a-i{g}^{\prime }Y(\psi )B_{\mu })\psi ,$$

with two independent gauge couplings: $g$ for $SU(2{)}_L$, $g^{\prime }$ for $U(1{)}_Y$. The non-abelian field strengths are

$$W_{\mu \nu }^a={\partial }_{\mu }{W}_{\nu }^a-{\partial }_{\nu }{W}_{\mu }^a+g{ϵ}^{abc}{W}_{\mu }^b{W}_{\nu }^c,B_{\mu \nu }={\partial }_{\mu }{B}_{\nu }-{\partial }_{\nu }{B}_{\mu }.$$

As in QCD, $W_{\mu \nu }^a$ contains 3- and 4-gauge-boson self-couplings; $B_{\mu \nu }$ is abelian like the photon and has no self-couplings of its own.

Local $SU(2{)}_L\times U(1{)}_Y$ has the following immediate consequences:

• All four gauge bosons ($W^a$, $B$) must be massless at the Lagrangian level.
• A Dirac fermion mass term $m\bar{\psi }\psi =m({\bar{\psi }}_L{\psi }_R+{\bar{\psi }}_R{\psi }_L)$ is forbidden because ${\psi }_L$ and ${\psi }_R$ live in different representations.
• The fermion–gauge-boson couplings are uniquely fixed by the covariant derivative; the gauge boson self-couplings are uniquely fixed by the ${ϵ}^{abc}$ structure constants.

Equivalent framings. As in QED and QCD, postulating local $SU(2{)}_L\times U(1{)}_Y$ gauge invariance and deriving the masslessness and self-couplings of the gauge bosons is equivalent to postulating four massless spin-1 species (with the right Lorentz quantum numbers and an internal $SU(2{)}_L\times U(1{)}_Y$ multiplet structure) and deriving the Yang–Mills + abelian gauge structure via foundations-modern.md §5.2. The gauge-first framing is taken here, as in QCD, because it makes the symmetry-breaking pattern (Step 3) and the residual unbroken $U(1{)}_{\text{EM}}$ transparent.

Step 3 — Build the Lagrangian: Higgs Mechanism (Theorem + Postulate; specializes QFT Postulate 9)

The most general Lagrangian density that is

• Lorentz-invariant,
• gauge-invariant under local $SU(2{)}_L\times U(1{)}_Y$,
• $CPT$-invariant (with $C$, $P$ separately and $CP$ all violated, as we will see),
• renormalizable (operators of mass dimension $\le 4$),

built from the fields of Step 1, splits naturally into four pieces:

$$\boxed{{\mathcal{L}}_{\text{EW}}={\mathcal{L}}_{\text{gauge}}+{\mathcal{L}}_{\text{fermion}}+{\mathcal{L}}_{\text{Higgs}}+{\mathcal{L}}_{\text{Yukawa}}}$$

with:

$${\mathcal{L}}_{\text{gauge}}=-\frac{1}{4}{W}_{\mu \nu }^a{W}^{a\mu \nu }-\frac{1}{4}{B}_{\mu \nu }{B}^{\mu \nu }.$$

$${\mathcal{L}}_{\text{fermion}}=\sum _f{\bar{\psi }}_{f}i{\gamma }^{\mu }{D}_{\mu }{\psi }_f,$$

where the sum runs over all chiral multiplets ${\psi }_f\in \{L_L,e_R,Q_L,u_R,d_R\}$ (per generation), each with its own covariant derivative determined by its $SU(2{)}_L$ representation and hypercharge.

$${\mathcal{L}}_{\text{Higgs}}=(D_{\mu }\Phi {)}^{†}(D^{\mu }\Phi )-V(\Phi ),V(\Phi )=-{\mu }^2{\Phi }^{†}\Phi +\lambda ({\Phi }^{†}\Phi {)}^2.$$

$${\mathcal{L}}_{\text{Yukawa}}=-\sum _{ij}[y_{ij}^e{\bar{L}}_L^i\Phi e_R^j+y_{ij}^d{\bar{Q}}_L^i\Phi d_R^j+y_{ij}^u{\bar{Q}}_L^i\tilde{\Phi }{u}_R^j+\text{h.c.}],$$

with $\tilde{\Phi }\equiv i{\sigma }^2{\Phi }^{\ast }$ the charge-conjugate doublet (needed to give up-type quarks mass while preserving hypercharge), $y_{ij}^{e,d,u}$ generic complex Yukawa matrices in generation space, and h.c. denoting Hermitian conjugate.

The Higgs mechanism is what makes this Lagrangian describe massive gauge bosons and massive fermions consistently with gauge invariance.

3.1 Spontaneous symmetry breaking (Theorem)

When ${\mu }^2>0$ in the Higgs potential, the minimum is not at $\Phi =0$ but on a sphere ${\Phi }^{†}\Phi =v^2/2$ with

$$v\equiv \sqrt{{\mu }^2/\lambda }\approx 246\mathrm{GeV},$$

the electroweak scale, fixed empirically by measured $W$ and $Z$ masses. Picking the gauge

$$⟨\Phi ⟩=\frac{1}{\sqrt{2}}\left(\begin{array}{c}0\\ v\end{array}\right),$$

three of the four generators of $SU(2{)}_L\times U(1{)}_Y$ are broken by this VEV; one combination remains unbroken:

$$Q\equiv T^3+Y,Q⟨\Phi ⟩=0.$$

This is the electric charge generator. The unbroken subgroup is $U(1{)}_{\text{EM}}$ — exactly the gauge group of QED. So electroweak theory is the unique $SU(2{)}_L\times U(1{)}_Y$ extension of QED that admits a renormalizable scalar potential breaking it down to $U(1{)}_{\text{EM}}$.

3.2 Gauge-boson masses (Theorem)

Expanding $\Phi (x)=⟨\Phi ⟩+\cdots$ and reading off the quadratic terms in $(D_{\mu }\Phi {)}^{†}(D^{\mu }\Phi )$ produces gauge-boson mass terms. Diagonalizing them defines the physical gauge bosons:

$$\begin{array}{rlrl}{W}_{\mu }^{\pm }& =\frac{1}{\sqrt{2}}(W_{\mu }^1\mp i{W}_{\mu }^2),& m_W^2& =\frac{1}{4}{g}^2{v}^2,\\ Z_{\mu }& =\cos {\theta }_W{W}_{\mu }^3-\sin {\theta }_W{B}_{\mu },& m_Z^2& =\frac{1}{4}(g^2+g^{\prime 2})v^2,\\ A_{\mu }& =\sin {\theta }_W{W}_{\mu }^3+\cos {\theta }_W{B}_{\mu },& m_A& =0,\end{array}$$

where the weak mixing angle ${\theta }_W$ (also called the Weinberg angle) is fixed by

$$\tan {\theta }_W=g^{\prime }/g,\cos {\theta }_W=m_W/m_Z.$$

So the $W^{\pm }$ bosons get mass from $SU(2{)}_L$ alone; the $Z^0$ is a $g,g^{\prime }$-mixed combination; and the photon $A_{\mu }$ is the unbroken combination of $W^3$ and $B$, remaining massless to enforce $U(1{)}_{\text{EM}}$ on the physical states. The electric charge is

$$e=g\sin {\theta }_W=g^{\prime }\cos {\theta }_W,$$

connecting the electroweak couplings $(g,g^{\prime })$ to the QED coupling $e$.

The three "eaten" Goldstone bosons — the $T^1,T^2$, and broken-$T^3-Y$ generators acting on $\Phi$ — become the longitudinal polarizations of $W^{\pm }$ and $Z^0$, restoring the count of degrees of freedom: $\Phi$ has 4 real components, of which 3 are eaten and 1 (the physical Higgs $h$) remains.

3.3 Higgs boson mass (Empirical input)

The fluctuation $h(x)$ around the VEV, $\Phi =\frac{1}{\sqrt{2}}\left(\genfrac{}{}{0ex}{}{0}{v+h(x)}\right)$ (in unitary gauge), is a physical real scalar field with mass

$$m_h^2=2{\mu }^2=2\lambda v^2.$$

Empirically $m_h\approx 125\mathrm{GeV}$ (LHC, 2012), which fixes $\lambda \approx 0.13$ for the measured $v$. The Higgs self-couplings (cubic $h^3$ and quartic $h^4$) are then predictions, currently being measured at the LHC.

3.4 Fermion masses and CKM mixing (Theorem)

The Yukawa terms become fermion mass matrices after EW symmetry breaking. For example,

$$y_{ij}^e{\bar{L}}_L^i\Phi e_R^j\stackrel{⟨\Phi ⟩}{\to }\frac{v}{\sqrt{2}}{y}_{ij}^e{\bar{e}}_L^i{e}_R^j\equiv M_{ij}^e{\bar{e}}_L^i{e}_R^j,$$

a general complex mass matrix $M^e=(v/\sqrt{2})y^e$. Singular-value decomposition $M^e=V_L^e{\hat{M}}^e{V}_R^{e†}$ produces three positive eigenvalues — the physical electron, muon, and tau masses — and rotates the flavor fields into mass eigenstates.

In the quark sector, doing this independently for up-type and down-type yields the Cabibbo–Kobayashi–Maskawa (CKM) matrix

$$V_{\text{CKM}}=V_L^u{V}_L^{d†},$$

a $3\times 3$ unitary matrix that cannot be removed by field redefinitions and appears explicitly in the charged-current interactions $W_{\mu }^{+}{\bar{u}}_L^i{\gamma }^{\mu }{V}_{\text{CKM}ij}{d}_L^j$. The CKM matrix has three real angles and one physical CP-violating phase — the sole source of CP violation in the Standard Model (modulo the QCD $\theta$-angle).

For leptons, an analogous Pontecorvo–Maki–Nakagawa–Sakata (PMNS) matrix governs neutrino oscillations, but it requires neutrino masses (and hence either right-handed neutrinos ${\nu }_R$ or a Majorana mass term) — see Caveats.

Where is the postulate vs. theorem boundary? Steps 1 and 2 (field content + gauge group) are empirical / postulate; the existence of a Higgs mechanism solving the mass problem is a theorem (it is the unique renormalizable way to give masses to the gauge bosons of a spontaneously broken gauge theory while preserving unitarity). What is postulated in §3 is (a) the specific scalar representation (an $SU(2{)}_L$ doublet with $Y=+1/2$ — the minimal choice consistent with $W$/$Z$ mass relations) and (b) the Yukawa couplings $y_{ij}^{e,d,u}$ (whose values are empirical and unexplained). The hierarchical pattern of fermion masses ($m_t\sim 173\mathrm{GeV}$ down to $m_e\sim 0.5\mathrm{MeV}$) is one of the major open puzzles — see Caveats.

Step 4 — Quantize (Standard machinery)

The fields are promoted to operators following the standard QFT path-integral machinery. Two wrinkles relative to QED:

• Non-abelian gauge fixing requires Faddeev–Popov ghosts in covariant gauges, as in QCD.
• $R_{\xi }$ gauges for spontaneously broken theories ('t Hooft, 1971) mix the gauge fields with the would-be Goldstone bosons in a way that produces a manifestly renormalizable Lagrangian. The would-be Goldstones acquire $\xi$-dependent masses and ghosts appear for both $W^{\pm },Z^0$.

In unitary gauge ($\xi \to \infty$), the would-be Goldstones decouple and the propagators of $W^{\pm },Z^0$ become

$$\frac{-i}{k^2-m_V^2}({\eta }_{\mu \nu }-\frac{k_{\mu }{k}_{\nu }}{m_V^2}),$$

manifestly massive. In Feynman–'t Hooft gauge ($\xi =1$) the propagators are simpler:

$$\frac{-i{\eta }_{\mu \nu }}{k^2-m_V^2},$$

at the cost of carrying around explicit Goldstone bosons and ghost fields.

The renormalizability of the spontaneously broken non-abelian gauge theory was proved by 't Hooft and Veltman (1971–72), Nobel Prize 1999. The proof requires the BRST formalism and the unitarity of the $S$-matrix on the physical Hilbert space (Goldstones + ghosts cancel against unphysical gauge-boson polarizations).

Step 5 — Renormalize (Standard machinery)

The electroweak theory is renormalizable in any $R_{\xi }$ gauge. Its $\beta$-functions for the two gauge couplings are, at one loop:

$${\beta }_g=-\frac{g^3}{(4\pi {)}^2}(\frac{22}{3}-\frac{4}{3}{n}_g-\frac{1}{6}{n}_h),{\beta }_{g^{\prime }}=+\frac{g^{\prime 3}}{(4\pi {)}^2}(\frac{20}{9}{n}_g+\frac{1}{6}{n}_h),$$

with $n_g$ the number of generations and $n_h$ the number of Higgs doublets. For the SM ($n_g=3$, $n_h=1$), ${\beta }_g<0$ (asymptotic freedom for $SU(2{)}_L$, like QCD) and ${\beta }_{g^{\prime }}>0$ (a Landau pole at very high energies for $U(1{)}_Y$, like QED — though far above the Planck scale and so phenomenologically irrelevant).

The renormalized parameters are usually traded for measured observables: the most common modern scheme is

$$\{\alpha (M_Z),G_F,m_Z\},$$

with the Fermi constant $G_F$ defined from muon decay and $\alpha (M_Z)$ the running fine-structure constant at the $Z$-pole.

Step 6 — Predictions (Standard machinery; specializes QFT Postulate 10)

The Feynman rules follow from ${\mathcal{L}}_{\text{EW}}+{\mathcal{L}}_{\text{gauge-fix}}+{\mathcal{L}}_{\text{ghost}}$ in any $R_{\xi }$ gauge. The new ingredients relative to QED/QCD:

Cross sections and decay rates follow from the standard machinery of cross-sections.md and decay-rates.md.

Comparison with QED and QCD

So electroweak adds qualitatively new content to the QED/QCD template:

• Chiral fermions — the first place left and right components live in inequivalent gauge representations.
• Spontaneous symmetry breaking — the empirical input that foundations-modern §5.2 flagged as not derivable from M1+M2+M3.
• Mass mixing (CKM, PMNS) — the source of all flavor-physics phenomenology.
• Discrete-symmetry violation — $P,C,CP$ all violated.

Successes and Tested Predictions

Full inventory of electroweak observables — what is measured, where, and how it overlaps with QED/QCD/SM — lives in observables/electroweak.md. This section lists historical and structural highlights only.

• Predicted the $W^{\pm }$ and $Z^0$ bosons with masses $m_W\approx 80$ GeV, $m_Z\approx 91$ GeV before they were directly discovered (UA1/UA2 at CERN, 1983; Nobel Prize 1984).
• Predicted neutral-current weak interactions (Gargamelle, CERN, 1973) before any direct detection of weak neutral currents.
• Predicted the Higgs boson as a necessary consequence of the renormalizable Higgs mechanism (Higgs / Englert / Brout, 1964); discovered at the LHC ATLAS/CMS in 2012 at $m_h\approx 125\mathrm{GeV}$, Nobel Prize 2013.
• Precision tests at LEP, SLC, and the Tevatron: $Z$-pole observables (line-shape, asymmetries, partial widths) measured to per-mille accuracy and consistent with electroweak global fits, severely constraining new-physics scenarios.
• CKM unitarity tests ($|V_{ud}{|}^2+|V_{us}{|}^2+|V_{ub}{|}^2=1$ etc.) all consistent to $\sim 10^{-4}$.
• Anomalous magnetic moment of the muon $a_{\mu }$ receives an electroweak contribution from $W,Z$ loops, computed to high order and required for agreement with experiment.
• The CP-violating phase in the CKM matrix correctly accounts for the observed CP violation in $K$- and $B$-meson systems.

Caveats and Open Issues

• No rigorous mathematical construction in $d=4$. As with QED and QCD, the renormalizable perturbative theory is well-defined formally but lacks a rigorous Wightman / Haag–Kastler construction.
• The hierarchy / naturalness problem. The Higgs mass receives quadratically divergent radiative corrections, $\delta m_h^2\sim {\Lambda }^2$. Why $m_h^2\sim (125\mathrm{GeV}{)}^2$ remains so small compared to any plausible UV cutoff $\Lambda$ (Planck scale, GUT scale) is unexplained. The standard proposals (supersymmetry, composite Higgs, large extra dimensions, anthropic / multiverse) are all so far disfavored or unconfirmed by LHC data.
• Neutrino masses. The SM as stated above has massless neutrinos — but solar / atmospheric / reactor oscillation experiments (Super-K, SNO, KamLAND, T2K, NOvA, ...) show neutrinos have small but non-zero masses ($\Delta m^2\sim 10^{-3}-10^{-5}{\mathrm{eV}}^2$). Generating them requires either adding right-handed neutrinos ${\nu }_R$ (Dirac masses, with anomalously small Yukawas) or a dim-5 Weinberg operator $\frac{1}{\Lambda }({\bar{L}}_L^c{\Phi }^{\ast })({\Phi }^{†}{L}_L)$ (Majorana masses, signalling new physics at scale $\Lambda \sim 10^{14}\mathrm{GeV}$). The SM as renormalizable theory does not incorporate either; this is the cleanest evidence for physics beyond the SM at any scale.
• The flavor / Yukawa hierarchy. The 13 fermion masses + 4 CKM parameters span six orders of magnitude with no SM explanation. Proposals include Froggatt–Nielsen mechanisms, family symmetries, and extra dimensions — none confirmed.
• The strong CP problem in QCD (mentioned in QCD/from-postulates.md) is not solved by electroweak unification — it remains an outstanding puzzle.
• Anomaly cancellation appears miraculous within EW alone. The hypercharge assignments are tuned so that $SU(2{)}_L^2\cdot U(1{)}_Y$, $U(1{)}_Y^3$, and mixed gravitational-gauge anomalies all cancel per generation. This is a strong hint that quarks and leptons should be unified in larger representations — the motivation for grand unified theories. The full anomaly bookkeeping is collected in the Standard Model doc.

See Also

• QED — the abelian $U(1{)}_{\text{EM}}$ that survives after electroweak symmetry breaking.
• QCD — the non-abelian gauge theory of the strong interaction, the third pillar of the SM.
• Standard Model — combines EW with QCD into the full $SU(3{)}_C\times SU(2{)}_L\times U(1{)}_Y$ theory.
• Modern Foundations — for the derivation of the QFT framework EW specializes, including the §5.2 Generalization callout pointing here for the Higgs mechanism.
• Group Theory — for $SU(2)$, $SU(2)\times U(1)$, Lie algebras, and spontaneous symmetry breaking.