The Standard Model

The Standard Model (SM) of particle physics is the relativistic quantum field theory that combines the three gauge theories of the previous documents — QCD, and electroweak theory (the latter unifying QED with the weak interaction) — into a single $S U (3)_{C} \times S U (2)_{L} \times U (1)_{Y}$ gauge theory. It is the most precisely tested theory of nature ever constructed, accounting for every laboratory-scale particle-physics observation made to date.

The SM is the union of QCD and EW plus a small number of genuinely new ingredients that only make sense across the two sectors:

Three generations of fermions with identical gauge quantum numbers.
Anomaly cancellation across quark and lepton sectors — the central reason hypercharge assignments take the specific values they do.
Asymmetric coupling of the same Higgs to all charged fermions via the Yukawa sector, with masses spanning six orders of magnitude.
Two distinct CP problems (CKM phase vs. QCD $θ$ -angle) that the unification does not relate.

This document collects the cross-sector content. For the individual gauge factors, see the parent docs.

Companion presentations.

Modern Foundations — derives the QFT framework all SM constituents specialize.

QED, QCD, Electroweak — the three constituent gauge theories.

Following the tag convention of foundations-modern.md, each step below is labelled (Empirical input), (Postulate), (Theorem), or (Standard machinery).

Derivation of the SM from QFT

The SM inherits all QFT postulates and is obtained by combining the field content and gauge symmetries of QCD and EW. The "new" ingredients are constraints that link the two sectors.

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

The SM postulates a single Higgs doublet and three generations of chiral fermions, each generation a copy of the EW pattern with QCD color attached. Writing $(S U (3)_{C}, S U (2)_{L})_{Y}$ representations:

Field (one generation)	Multiplet	Generations
Left-handed quark doublet $Q_{L} = (d _{L} u _{L})$	$(3, 2)_{+ 1/6}$	$(u, d), (c, s), (t, b)$
Right-handed up-quark $u_{R}$	$(3, 1)_{+ 2/3}$	$u, c, t$
Right-handed down-quark $d_{R}$	$(3, 1)_{- 1/3}$	$d, s, b$
Left-handed lepton doublet $L_{L} = (e _{L} ν _{L})$	$(1, 2)_{- 1/2}$	$(ν_{e}, e), (ν_{μ}, μ), (ν_{τ}, τ)$
Right-handed charged lepton $e_{R}$	$(1, 1)_{- 1}$	$e, μ, τ$
Higgs doublet $Φ$	$(1, 2)_{+ 1/2}$	one
Gluons $G_{μ}^{A}$ , $A = 1 \dots 8$	$(8, 1)_{0}$	one set
EW gauge bosons $W_{μ}^{a}$ , $a = 1, 2, 3$	$(1, 3)_{0}$	one set
$U (1)_{Y}$ gauge boson $B_{μ}$	$(1, 1)_{0}$	one set

(Right-handed neutrinos $ν_{R}$ are not part of the original SM; they are added in extensions to give neutrinos Dirac masses — see Caveats.)

The three-generation pattern is an empirical input. The SM neither predicts the number of generations nor explains why it is exactly three; the constraint $N_{ν} = 2.984 \pm 0.008$ from the $Z$ -boson invisible width at LEP rules out a fourth light generation, but says nothing about why there should be three rather than one.

Step 2 — Gauge Symmetry (Postulate)

The SM gauge group is

$G_{SM} = S U (3)_{C} \times S U (2)_{L} \times U (1)_{Y},$

with three independent gauge couplings $(g_{s}, g, g^{'})$ . The covariant derivative on any field $ψ$ in representation $(R_{3}, R_{2})_{Y}$ is

$D_{μ} ψ = (\partial_{μ} - i g_{s} G_{μ}^{A} T_{R_{3}}^{A} - i g W_{μ}^{a} T_{R_{2}}^{a} - i g^{'} Y B_{μ}) ψ .$

After electroweak symmetry breaking the residual gauge group is $S U (3)_{C} \times U (1)_{EM}$ .

Step 3 — The SM Lagrangian (Theorem; specializes QFT Postulate 9)

The most general renormalizable, $G_{SM}$ -gauge-invariant, Lorentz-invariant, $CPT$ -invariant Lagrangian built from the Step 1 fields is

$L_{SM} = L_{gauge} + L_{fermion} + L_{Higgs} + L_{Yukawa} + L_{θ}$

where

$L_{gauge} = - \frac{1}{4} (G_{μν}^{A} G^{A μν} + W_{μν}^{a} W^{a μν} + B_{μν} B^{μν})$ ,
$L_{fermion} = \sum_{f} \overset{ˉ}{ψ}_{f} i γ^{μ} D_{μ} ψ_{f}$ , summed over all chiral multiplets and all three generations,
$L_{Higgs} = (D_{μ} Φ)^{†} (D^{μ} Φ) - V (Φ)$ with the EW potential,
$L_{Yukawa}$ is the generation-mixing Yukawa structure from EW Step 3.4,
$L_{θ} = θ \frac{g _{s}^{2}}{32 π ^{2}} ϵ^{μν ρ σ} G_{μν}^{A} G_{ρ σ}^{A}$ (the QCD $θ$ -term — see QCD Step 3).

A direct $U (1)_{Y}$ counterpart $θ_{Y} ϵ B_{μν} B_{ρ σ}$ exists but is a total derivative for an abelian theory in 4D and so does not contribute perturbatively. The non-abelian $Tr (W \tilde{W})$ term would be physical, but the $S U (2)_{L}$ vacuum angle is unobservable because of the chiral nature of the SM fermions (it can be rotated away by a baryon-number redefinition — see Reviews).

The SM has 19 free parameters that must be measured (assuming massless neutrinos): 3 gauge couplings, 2 Higgs parameters $(μ^{2}, λ)$ , 9 fermion masses, 4 CKM parameters (3 angles + 1 phase), and the QCD $θ$ -angle. With non-zero neutrino masses one adds at least 7 more (3 masses + 4 PMNS parameters, with possible extra Majorana phases).

Step 4 — Quantize (Standard machinery)

The quantization machinery is the union of QCD's and EW's: path integral with Faddeev–Popov ghosts for both $S U (3)_{C}$ and $S U (2)_{L}$ , $R_{ξ}$ gauges for the spontaneously broken EW sector. Renormalizability of the combined theory was established by the same 't Hooft–Veltman analysis that handled EW alone.

Step 5 — Renormalize (Standard machinery)

All three gauge couplings run; their measured low-energy values and the SM $β$ -functions extrapolate to nearly meet near $μ \sim 1 0^{15} GeV$ but not exactly — the almost-unification is one of the main historical motivations for grand unified theories and for supersymmetry (which makes them meet much more precisely).

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

The Feynman rules are the union of QED/QCD/EW; all cross-sector predictions (e.g. flavor-changing neutral currents constrained by the GIM mechanism, electroweak corrections to QCD observables and vice versa) follow from the standard machinery.

Cross-Sector Content (What Combining QCD + EW Buys Us)

The substantive content of the SM as a separate document, beyond its constituents, is the following.

A. Anomaly cancellation (Theorem)

Gauge invariance must survive quantization. A chiral gauge theory generically suffers triangle anomalies: fermion-loop diagrams with three gauge currents on the legs produce $\partial_{μ} j^{μ} \neq = 0$ , violating the gauge symmetry at the quantum level. For the SM to be consistent the anomaly contribution must cancel summed over all chiral fermions in the theory.

The relevant anomalies are:

Anomaly	Contribution from one generation	Comment
$S U (3)_{C}^{3}$	0	Vector-like in color (left and right quarks both in $3$ )
$S U (3)_{C}^{2} \cdot U (1)_{Y}$	$2 Y_{Q_{L}} - Y_{u_{R}} - Y_{d_{R}} = \frac{1}{3} - \frac{2}{3} + \frac{1}{3} = 0$	Cancels
$S U (2)_{L}^{3}$	0	$S U (2)$ reps are self-conjugate
$S U (2)_{L}^{2} \cdot U (1)_{Y}$	$3 Y_{Q_{L}} + Y_{L_{L}} = \frac{1}{2} - \frac{1}{2} = 0$	Quark contribution exactly cancels lepton contribution
$U (1)_{Y}^{3}$	$\sum 2 Y_{L}^{3} - \sum Y_{R}^{3} = 0$	Cancels (long computation; depends on factor of 3 from color)
$grav^{2} \cdot U (1)_{Y}$	$\sum 2 Y_{L} - \sum Y_{R} = 0$	Mixed gauge-gravitational anomaly

The factor of 3 in $S U (2)_{L}^{2} \cdot U (1)_{Y}$ from color combined with the matching hypercharges is what makes quarks and leptons "fit together". This is the strongest hint inside the SM that quarks and leptons are not independent — they belong in larger multiplets of some unified gauge group (the original motivation for $S U (5)$ and $SO (10)$ GUTs, where one generation fits into a $5^{*} \oplus 10$ of $S U (5)$ or a single $16$ of $SO (10)$ ).

Why this matters. Anomaly cancellation is what forces the hypercharge assignments listed in EW Step 1 — they are not free phenomenological inputs. The deeper question — why exactly this charge assignment, and the apparent quark-lepton complementarity — is the central pre-existing-mystery the SM hands forward to BSM physics.

B. Generations and GIM (Theorem)

With three generations and a single Higgs doublet, the Yukawa matrices $y^{e, d, u}$ are $3 \times 3$ complex. After diagonalization (EW Step 3.4) the charged-current weak interactions carry the CKM matrix; the neutral-current weak interactions and the Higgs couplings, in contrast, remain diagonal in flavor at tree level. This is the GIM mechanism (Glashow–Iliopoulos–Maiani, 1970), which is why flavor-changing neutral currents (FCNCs) — e.g. $K^{0} \to μ^{+} μ^{-}$ — are dramatically suppressed in nature.

The GIM mechanism required the existence of the charm quark to cancel the FCNC contribution from up; this was a successful pre-discovery prediction (charm found at SLAC and Brookhaven, 1974). The same logic applied to the down sector + measured CP violation in $K^{0}$ decays led Kobayashi and Maskawa (1973) to predict a third generation; bottom (1977) and top (1995) quarks confirmed it.

C. Two CP problems, one solved, one not (Postulate)

The SM has two independent sources of $CP$ violation:

Source	Status
CKM phase $δ_{CKM}$	$δ_{CKM} \approx 70°$ , large. Accounts for the observed $CP$ violation in $K$ and $B$ mesons.
QCD $θ$ -angle	$∣ θ ∣ ≲ 1 0^{- 10}$ from neutron EDM bound. Unexplained smallness — the strong CP problem (see QCD Caveats).

The two are independent — the CKM phase does not "solve" the strong CP problem, and they cannot be rotated into each other without violating gauge invariance. This is one of the cleanest indications that the SM is incomplete.

D. Accidental symmetries (Theorem)

Renormalizable $G_{SM}$ -invariance happens to automatically preserve four global $U (1)$ symmetries:

$U (1)_{B}$ — baryon number (each quark $+ 1/3$ , leptons $0$ ).
$U (1)_{L_{e}}$ , $U (1)_{L_{μ}}$ , $U (1)_{L_{τ}}$ — three separate lepton-flavor numbers.

These are accidental: they were not postulated; they emerge as consequences of $G_{SM}$ -invariance + renormalizability + the absence of $ν_{R}$ . They are broken by:

Non-perturbative electroweak instantons ('t Hooft, 1976): $B + L$ is anomalously broken; $B - L$ is exactly conserved. Practically irrelevant at low energy (rate $\sim e^{- 2 π / α_{W}}$ ), but important at very high temperatures and central to baryogenesis scenarios.
Neutrino masses (Majorana): break $L$ explicitly; lepton flavor mixed via PMNS.
Beyond-SM physics (proton decay): predicted by GUTs at rate $\sim 1 0^{- 34} / year$ , not observed.

So conservation of baryon number is not a SM postulate — it is a derived approximate symmetry. Lepton-flavor universality (electrons / muons / taus interact with the same gauge couplings) is a postulate, embedded in the choice of identical gauge multiplets across generations.

Comparison: SM as a Whole vs. its Constituents

Aspect	Combined SM (this doc)	Sum of QED + QCD + EW alone
Gauge group	$S U (3)_{C} \times S U (2)_{L} \times U (1)_{Y}$	Same
Free parameters (massless $ν$ )	19	Same
Hypercharge assignments	Fixed by anomaly cancellation	Would appear free
FCNCs	Suppressed by GIM	Would be unconstrained
3rd generation	Required for $CP$ violation by KM	Two would suffice for masses
$B, L_{e}, L_{μ}, L_{τ}$ conservation	Accidental at tree level; $B + L$ non-perturbatively broken	Would need to be postulated
CP-violation sources	CKM phase ✓; strong $θ$ ✗ (unexplained)	Same; not unified

So the cross-sector content the SM doc captures, beyond its parts, is exactly the four items in §Cross-Sector Content above: anomaly cancellation, GIM/generations, the two CP problems, and accidental symmetries.

Successes and Tested Predictions

Full inventory of SM cross-sector observables — CKM unitarity triangle, global EW fit, GIM mechanism, lepton-universality tests, cosmological constraints — lives in observables/standard-model.md. Sector-specific observables live in observables/electroweak.md, with QED/QCD per-theory inventories noted as a future gap in observables/README.md. This section lists historical and structural highlights only.

The SM is the most precisely tested theory in physics. Highlights:

Anomalous magnetic moment of the electron — agrees with theory to better than $1 0^{- 12}$ (when combined with the most precise $α$ measurement). Includes electroweak loop corrections.
$Z$ -pole observables at LEP/SLC — $Z$ line-shape, partial widths, forward-backward asymmetries all match SM predictions at the per-mille level.
CKM-matrix unitarity tests — $∣ V_{u d} ∣^{2} + ∣ V_{u s} ∣^{2} + ∣ V_{u b} ∣^{2} = 1$ to $\sim 1 0^{- 4}$ ; CP violation in $B$ -mesons consistent with a single KM phase.
Higgs discovery at $m_{h} \approx 125 GeV$ (LHC, 2012); production and decay rates match SM predictions in every measured channel to $\sim 10%$ .
Asymptotic freedom of QCD confirmed across $α_{s} (μ)$ measurements spanning $μ \in [1, 1000] GeV$ .
Lattice QCD computations of the light hadron spectrum agree with experiment at the few-percent level.
Neutral-current discovery (Gargamelle, 1973) — predicted by EW before observation.
Numerous discovered particles in advance of measurement: $W^{\pm}, Z^{0}, c, b, t, h, τ, ν_{τ}$ , the gluon (from 3-jet events at PETRA).

Caveats and Open Issues

The SM, despite its empirical success, leaves known gaps:

Gravity. The SM contains no graviton and no quantization of general relativity. The fundamental obstruction is non-renormalizability of perturbatively quantized Einstein gravity in $d = 4$ . The SM is an effective theory below the Planck scale $M_{Pl} \sim 1 0^{19} GeV$ .
Neutrino masses. Already covered in EW Caveats. The cleanest evidence for BSM physics at any scale.
Dark matter. Astronomical evidence (rotation curves, cluster dynamics, CMB anisotropies, large-scale structure) requires $\sim 27%$ of cosmic energy density to be in a non-luminous, non-baryonic, cold component. No SM particle fits; candidates (WIMPs, axions, sterile neutrinos, primordial black holes) are all BSM.
Dark energy. $\sim 68%$ of cosmic energy density behaves like a cosmological constant $Λ$ . The SM vacuum energy is many orders of magnitude larger than the observed $Λ$ (the cosmological constant problem), one of the largest fine-tuning puzzles in physics.
Baryogenesis. The universe is matter-dominated; the SM has the three Sakharov conditions ( $B$ violation, $C$ and $CP$ violation, out-of-equilibrium dynamics) only in principle — the CKM CP violation is too small and the SM electroweak phase transition is too smooth to generate the observed baryon asymmetry. BSM CP-violation sources are required.
Hierarchy problem. Why $m_{h} ≪ M_{Pl}$ ? See EW Caveats.
Strong CP problem. Why $∣ θ_{QCD} ∣ ≲ 1 0^{- 10}$ ? See QCD Caveats. The most popular resolution (Peccei–Quinn axion) is BSM.
Flavor puzzle. Why three generations? Why the hierarchical Yukawa pattern? Why CKM mixing angles small but PMNS mixing angles large?
No rigorous construction in $d = 4$ . As with QED/QCD/EW individually, a Wightman / Haag–Kastler construction of the SM is an open problem.

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