Quantum Chromodynamics

Quantum Chromodynamics (QCD) is the relativistic quantum field theory of quarks and gluons — the gauge theory of the strong interaction. Like QED, it is obtained by specializing the general postulates of Quantum Field Theory with three additional inputs: a choice of field content, a gauge symmetry, and the renormalizable Lagrangian uniquely determined by these together with Lorentz and discrete-symmetry invariance.

The single structural change relative to QED — replacing the abelian gauge group $U(1)$ by the non-abelian $SU(3{)}_{\text{color}}$ — has profound consequences:

• Gluons self-interact (3- and 4-gluon vertices appear automatically from gauge invariance), unlike photons.
• The running coupling decreases at high energy: asymptotic freedom.
• At low energy the coupling becomes strong and quarks/gluons are not asymptotic states: color confinement.
• Faddeev–Popov ghosts do not decouple — they are required for unitarity in covariant gauges.

These three facts make QCD qualitatively different from QED at both the perturbative and non-perturbative level, despite the formal similarity of the Lagrangians.

What QCD Describes Physically: The Strong Force

QCD is the modern theory of the strong force (strong interaction), one of the four fundamental forces alongside electromagnetism, the weak interaction, and gravity. The strong force is responsible for two phenomena that look quite different at first glance:

1. Fundamental level — binding quarks into hadrons. Quarks carry color charge (3 values: red, green, blue — labels, no relation to optics) and interact by exchanging gluons, the strong-force analogue of the photon. Unlike photons, gluons themselves carry color (8 of them, in the adjoint of $SU(3{)}_C$) and so interact with each other. The only physical states are color-singlet combinations: mesons ($\bar{q}q$), baryons ($qqq$), glueballs, etc. — isolated quarks and gluons have never been observed, a non-perturbative phenomenon called confinement (see §Confinement).
2. Residual level — binding nucleons into nuclei. Protons and neutrons are themselves color-singlets, so the force holding them together inside a nucleus is a residual effect of QCD, analogous to van der Waals forces between neutral atoms. At the nucleon level it is well-described by the Yukawa exchange of mesons (Yukawa, 1935): primarily pions ($\pi$) and heavier mesons ($\rho ,\omega ,\sigma ,\dots$). The resulting nucleon–nucleon potential is short-range (range $\sim 1/m_{\pi }\sim 1.4\mathrm{fm}$), attractive at moderate distances, repulsive at very short ones. This is the original "nuclear force" of pre-quark-era nuclear physics, now understood as an emergent low-energy consequence of the deeper QCD dynamics below.

Why "strong"?

The four fundamental couplings, at low energy:

So at hadronic energies the strong force is genuinely the strongest — by an order of magnitude over electromagnetism, six orders over the weak interaction, and an astronomical 38 orders over gravity between elementary particles. Asymptotic freedom (Step 5) is what tames it at short distances: ${\alpha }_s(\mu )$ decreases with energy, so the force gets weaker at higher energies — the opposite of the everyday intuition that pulling things apart costs more energy.

Why most mass is QCD

One of QCD's most striking qualitative predictions, often missed in introductory treatments: only $\sim 1\%$ of the proton's mass comes from the Higgs mechanism. The up-quark and down-quark masses (Higgs-generated) total roughly $m_u+m_u+m_d\approx 9\mathrm{MeV}$ — about $1\%$ of the proton mass $m_p\approx 938\mathrm{MeV}$. The remaining $\sim 99\%$ is dynamically generated by QCD: it is the energy stored in the gluon field and in quark kinetic motion confined inside the proton, related to the dimensional-transmutation scale ${\Lambda }_{\text{QCD}}$ (Step 5).

This is why almost all the mass of ordinary matter (atoms, you, this document) is QCD-binding energy, not Higgs-generated rest mass. The Higgs is essential for most particle masses (electron, muon, $W,Z$, individual quarks), but the bulk of baryonic matter mass is the strong force at work. Lattice QCD computations confirm this ab initio: the entire light-hadron spectrum (proton, neutron, pion, kaon, ...) emerges from a Lagrangian whose only inputs are ${\alpha }_s$ and the small quark masses, with no further mass parameter.

Short historical lineage

The strong force was named long before QCD existed:

• Yukawa (1935) — postulated a massive scalar mediator of the nuclear force, predicting the pion ($m_{\pi }\sim 140\mathrm{MeV}$, discovered 1947).
• Particle zoo (1950s–60s) — proliferation of hadrons (mesons, baryons, resonances) at accelerators suggested they were composite.
• Quark model (Gell-Mann, Zweig, 1964) — hadrons organized into multiplets of an approximate $SU(3)$ flavor symmetry, postulated to arise from three quark "building blocks" $(u,d,s)$.
• Color (Han–Nambu, Greenberg, 1965) — extra quantum number needed for the ${\Delta }^{++}$ baryon to be antisymmetric under fermion exchange.
• Asymptotic freedom (Gross–Wilczek, Politzer, 1973) — gauge $SU(3{)}_{\text{color}}$ with quarks in the fundamental representation produces a negative $\beta$-function. Settled QCD as the theory of the strong force, Nobel Prize 2004.
• Confirmation (1970s–present) — deep inelastic scattering at SLAC (partons, late 1960s); the 3-jet event at PETRA (gluons, 1979); precision QCD at LEP, Tevatron, and the LHC.

The "strong interaction" label predates the field-theoretic understanding by 40 years; the modern view is that the strong force is the $SU(3{)}_{\text{color}}$ gauge interaction of QCD, and everything else (nuclear binding, mesons, hadron spectroscopy) is an emergent consequence.

Companion presentation. This document picks up where Modern Foundations — Wigner–Weinberg derivation ends: the latter shows in §5.2 (and its Generalization callout) that any set of massless spin-1 particles forming a multiplet under an internal symmetry $G$ forces the Yang–Mills structure as a theorem of Lorentz consistency. Here we go the opposite direction — postulate $G=SU(3{)}_{\text{color}}$ as the gauge group and derive the resulting massless adjoint multiplet of gluons — and then proceed to the full Lagrangian, quantization, renormalization, and qualitative consequences (asymptotic freedom, confinement). See §Equivalent framings in Step 2 for the precise correspondence.

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 QCD from QFT

QCD inherits all postulates of QFT (see Postulates of QFT). The chain of choices that turns the generic QFT framework into QCD parallels the QED derivation.

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

QCD postulates two fundamental field types:

• Quark fields $q_f^a(x)$ — Dirac spinors carrying a color index $a=1,2,3$ (transforming in the fundamental representation $3$ of $SU(3{)}_{\text{color}}$) and a flavor index $f\in \{u,d,s,c,b,t\}$. Each quark flavor has its own mass $m_f$.
• Gluon fields $A_{\mu }^A(x)$, $A=1,\dots ,8$ — eight real vector fields, one for each generator of $SU(3)$, transforming in the adjoint representation $8$.

By the spin–statistics theorem (QFT Postulate 7), quarks are quantized with anticommutators and gluons with commutators.

Color vs. flavor. Color $SU(3)$ is the gauge symmetry — it is local and is the dynamical content of the theory. Flavor (which distinguishes up/down/strange/...) is a global approximate symmetry, broken explicitly by the different quark masses. The quark flavors are not related by any gauge transformation; they are different species, each with its own mass parameter.

Step 2 — Postulate a Local $SU(3{)}_{\text{color}}$ Gauge Symmetry (Postulate)

The genuinely new ingredient is local non-abelian gauge invariance:

$$q(x)\to U(x)q(x),A_{\mu }(x)\to U(x)A_{\mu }(x)U^{†}(x)+\frac{i}{g_s}U(x){\partial }_{\mu }{U}^{†}(x),$$

where $U(x)=\exp (i{\alpha }^A(x)T^A)\in SU(3)$, the $T^A$ are the eight Hermitian generators of $SU(3)$ in the fundamental representation (often written $T^A={\lambda }^A/2$ with ${\lambda }^A$ the Gell-Mann matrices), and $A_{\mu }\equiv A_{\mu }^A{T}^A$ is the matrix-valued gauge field. The generators satisfy the $\mathfrak{su}(3)$ Lie algebra (see group-theory.md):

$$[T^A,T^B]=i{f}^{ABC}{T}^C,\mathrm{tr}(T^A{T}^B)=\frac{1}{2}{\delta }^{AB},$$

with totally antisymmetric structure constants $f^{ABC}$.

The covariant derivative on a quark field is

$$D_{\mu }q=({\partial }_{\mu }-i{g}_s{A}_{\mu }^A{T}^A)q,$$

and the non-abelian field strength is

$$F_{\mu \nu }^A={\partial }_{\mu }{A}_{\nu }^A-{\partial }_{\nu }{A}_{\mu }^A+g_s{f}^{ABC}{A}_{\mu }^B{A}_{\nu }^C.$$

The last term is the crucial difference from QED: $F_{\mu \nu }^A$ is not gauge-invariant by itself — it transforms covariantly, $F_{\mu \nu }\to U{F}_{\mu \nu }{U}^{†}$ — and the gauge-invariant kinetic term $F_{\mu \nu }^A{F}^{A\mu \nu }$ contains $A^3$ and $A^4$ pieces, giving gluon self-interactions.

Local $SU(3{)}_{\text{color}}$ has two immediate consequences:

• Gluons must be massless ($m_g^2{A}_{\mu }^A{A}^{A\mu }$ is not gauge-invariant).
• The way quarks couple to gluons is uniquely fixed by the covariant derivative; in addition, the gauge group itself dictates 3-gluon and 4-gluon vertices.

Equivalent framings. Postulating local $SU(3{)}_{\text{color}}$ gauge invariance and deriving an octet of massless gluons (this doc) is logically equivalent to postulating eight massless spin-1 species transforming in the adjoint of $SU(3)$ and deriving the Yang–Mills self-coupling structure from Lorentz consistency — the route sketched in the Generalization callout of foundations-modern.md §5.2. The two are the same theory viewed from opposite ends. As with QED, this document picks the gauge-symmetry-first framing because it generalizes uniformly to any compact Lie group $G$ (electroweak $SU(2{)}_L\times U(1{)}_Y$, GUTs, etc.) and exposes the non-abelian structure constants $f^{ABC}$ as the cause of gluon self-interaction rather than as an accident.

Step 3 — Build the Lagrangian (Theorem; specializes QFT Postulate 9)

The most general Lagrangian density that is

• Lorentz-invariant,
• gauge-invariant under local $SU(3{)}_{\text{color}}$,
• $C$-, $P$-, and $T$-invariant (the $\theta$-term below is a parity-odd exception, see caveats),
• renormalizable (operators of mass dimension $\le 4$),
• built only from $q_f$, ${\bar{q}}_f$, $A_{\mu }^A$, and their derivatives,

is uniquely

$$\boxed{{\mathcal{L}}_{\text{QCD}}=\sum _f{\bar{q}}_f(i{\gamma }^{\mu }{D}_{\mu }-m_f)q_f-\frac{1}{4}{F}_{\mu \nu }^A{F}^{A\mu \nu }}$$

with the covariant derivative and field strength given in Step 2. The free parameters are the strong coupling $g_s$ (equivalently ${\alpha }_s\equiv g_s^2/(4\pi )$) and the quark masses $\{m_f\}$.

Expanding $-\frac{1}{4}{F}_{\mu \nu }^A{F}^{A\mu \nu }$ in powers of $A$ exposes the new vertices absent in QED:

$$-\frac{1}{4}{F}_{\mu \nu }^A{F}^{A\mu \nu }=\underset{\text{kinetic, like photon}}{\underbrace{-\frac{1}{4}({\partial }_{\mu }{A}_{\nu }^A-{\partial }_{\nu }{A}_{\mu }^A{)}^2}}-\underset{\text{3-gluon vertex}}{\underbrace{\frac{g_s}{2}{f}^{ABC}({\partial }_{\mu }{A}_{\nu }^A-{\partial }_{\nu }{A}_{\mu }^A)A^{B\mu }{A}^{C\nu }}}-\underset{\text{4-gluon vertex}}{\underbrace{\frac{g_s^2}{4}{f}^{ABC}{f}^{ADE}{A}_{\mu }^B{A}_{\nu }^C{A}^{D\mu }{A}^{E\nu }}}.$$

Renormalizability rules out higher-dimension operators such as $(\bar{q}q{)}^2$ (dim 6) and $\bar{q}{\sigma }^{\mu \nu }{G}_{\mu \nu }^A{T}^{A}q$ (chromomagnetic Pauli term, dim 5). The hypothetical $CP$-violating term

$${\mathcal{L}}_{\theta }=\theta \frac{g_s^2}{32{\pi }^2}{ϵ}^{\mu \nu \rho \sigma }{F}_{\mu \nu }^A{F}_{\rho \sigma }^A$$

is allowed by all symmetries and renormalizability and — unlike the analogous $F\tilde{F}$ term in QED — is not a total derivative in the non-abelian case (it contributes via instantons). The experimental smallness of the neutron electric dipole moment forces $|\theta |\lesssim 10^{-10}$, the unexplained strong CP problem (see Caveats).

The classical equations of motion are the Dirac equation in a color background and the non-abelian Maxwell (Yang–Mills) equations with a color current:

$$(i{\gamma }^{\mu }{D}_{\mu }-m_f)q_f=0,D_{\mu }{F}^{A\mu \nu }=g_s\sum _f{\bar{q}}_f{\gamma }^{\nu }{T}^A{q}_f\equiv j^{A\nu },$$

where $D_{\mu }{F}^{A\mu \nu }={\partial }_{\mu }{F}^{A\mu \nu }+g_s{f}^{ABC}{A}_{\mu }^B{F}^{C\mu \nu }$ is the gauge-covariant divergence.

Step 4 — Quantize (Standard machinery)

The fields are promoted to operators following the standard QFT machinery, with one important wrinkle absent in QED.

Canonical quantization proceeds as for QED, but resolving the primary constraint ${\pi }^{A0}=0$ in covariant gauges requires the Faddeev–Popov procedure. The most useful presentation is the path integral:

$$Z[J]=\int \mathcal{D}A\mathcal{D}\bar{q}\mathcal{D}q\mathcal{D}\bar{c}\mathcal{D}c\exp \left(i\int d^{4}x[{\mathcal{L}}_{\text{QCD}}+{\mathcal{L}}_{\text{gf}}+{\mathcal{L}}_{\text{ghost}}+\text{sources}]\right),$$

with covariant gauge-fixing and ghost terms

$${\mathcal{L}}_{\text{gf}}=-\frac{1}{2\xi }({\partial }_{\mu }{A}^{A\mu }{)}^2,{\mathcal{L}}_{\text{ghost}}={\partial }_{\mu }{\bar{c}}^A(D^{\mu }c{)}^A={\partial }_{\mu }{\bar{c}}^A({\partial }^{\mu }{c}^A+g_s{f}^{ABC}{A}^{B\mu }{c}^C).$$

The Grassmann-valued scalar fields $c^A,{\bar{c}}^A$ are the Faddeev–Popov ghosts. The non-abelian piece $g_s{f}^{ABC}{A}^{B\mu }{c}^C$ of the ghost-gluon coupling means ghosts run in loops — they do not decouple as they do in QED, and are required to cancel unphysical gluon polarizations and preserve unitarity. (Ghosts only appear inside loops; they are never external states.)

The full quantum theory has a residual BRST symmetry that replaces the broken classical gauge symmetry; this is what guarantees gauge-independence of physical $S$-matrix elements after gauge-fixing.

Step 5 — Renormalize: Asymptotic Freedom (Standard machinery + key theorem)

QCD is renormalizable (proved by 't Hooft, 1971): all UV divergences can be absorbed into multiplicative redefinitions of the fields, masses, and the coupling $g_s$.

The defining feature of QCD is the sign of the beta function. To one loop,

$$\mu \frac{d{g}_s}{d\mu }=\beta (g_s)=-\frac{g_s^3}{(4\pi {)}^2}{\beta }_0+O(g_s^5),{\beta }_0=\frac{11}{3}{C}_A-\frac{2}{3}{n}_f,$$

with $C_A=N=3$ for $SU(3)$ and $n_f$ the number of active quark flavors. For $n_f\le 16$ (in particular the physical $n_f=6$), ${\beta }_0>0$, so $\beta <0$ and the coupling decreases as $\mu$ increases. Equivalently,

$${\alpha }_s({\mu }^2)=\frac{4\pi }{{\beta }_0\ln ({\mu }^2/{\Lambda }_{\text{QCD}}^2)}+O(1/{\ln }^2),$$

where ${\Lambda }_{\text{QCD}}\sim 200\mathrm{MeV}$ is the dimensional-transmutation scale at which ${\alpha }_s$ formally diverges. This is asymptotic freedom (Gross–Wilczek and Politzer, 1973), the historical justification for taking QCD seriously as the theory of the strong interaction:

• High energy / short distance ($\mu \gg {\Lambda }_{\text{QCD}}$): ${\alpha }_s$ is small, perturbation theory works, quarks behave as nearly free particles — the regime probed by deep inelastic scattering ($Q^2\gg {\Lambda }^2$) and high-energy jets.
• Low energy / long distance ($\mu \sim {\Lambda }_{\text{QCD}}$): ${\alpha }_s$ is large, perturbation theory breaks down, and the spectrum is dominated by hadrons (mesons, baryons), not quarks and gluons. This is the regime of confinement and chiral symmetry breaking.

The crossover scale ${\Lambda }_{\text{QCD}}$ is generated dynamically; it is the only intrinsic mass scale of the massless-quark theory. The hadron mass spectrum (proton, pion, ...) is set by ${\Lambda }_{\text{QCD}}$, with quark masses providing only quantitative corrections.

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

Time-ordered correlators are computed using Feynman rules read off from ${\mathcal{L}}_{\text{QCD}}+{\mathcal{L}}_{\text{gf}}+{\mathcal{L}}_{\text{ghost}}$:

LSZ reduction and the cross-section / decay-rate master formulas of cross-sections.md / decay-rates.md apply unchanged for partonic processes (where quarks and gluons can be treated as asymptotic states at high energy via the factorization theorems below). For low-energy observables, the partonic $S$-matrix is not directly measurable — see Caveats.

Confinement and the Asymptotic-State Problem

QCD violates the version of QFT Postulate 10 that identifies asymptotic states with single-particle quark/gluon excitations: free quarks and gluons have never been observed. Instead the asymptotic Hilbert space is built from color-singlet hadrons — mesons $\bar{q}q$, baryons $qqq$, glueballs, etc.

Confinement is a non-perturbative phenomenon — invisible to any finite order in ${\alpha }_s$. The standard pieces of evidence and tools:

• Lattice QCD: Wilson's discretization of the Euclidean path integral on a spacetime lattice, evaluated by Monte Carlo. Confirms a linearly rising quark–antiquark potential $V(r)\sim \sigma r$ at large $r$ (string tension $\sigma \sim (440\mathrm{MeV}{)}^2$), reproduces hadron masses ab initio, and is currently the only first-principles method for QCD at low energy.
• 't Hooft large-$N$ limit: at $N\to \infty$ with $g_s^{2}N$ fixed, planar diagrams dominate; hadronic resonance physics organizes into a $1/N$ expansion that explains qualitative facts (OZI suppression, narrow widths, meson dominance).
• Wilson loops: $⟨W(\mathcal{C})⟩\sim e^{-\sigma \text{Area}(\mathcal{C})}$ (area law) signals confinement, vs. $e^{-\sigma \text{Perimeter}}$ for an unconfined Coulomb phase.
• The Yang–Mills mass gap (existence of a non-zero lightest glueball mass in pure $SU(3)$) is supported numerically by lattice computations and is one of the Clay Millennium Prize Problems.

Chiral Symmetry and Its Spontaneous Breaking

In the limit $m_u=m_d=m_s=0$, the QCD Lagrangian has a global chiral symmetry $SU(3{)}_L\times SU(3{)}_R$ acting on left- and right-handed quark flavors independently. The QCD vacuum spontaneously breaks this to the diagonal $SU(3{)}_V$ via the chiral quark condensate

$$⟨\bar{q}q⟩\ne 0.$$

Goldstone's theorem then predicts 8 massless pseudoscalar bosons, identified with the pseudoscalar meson octet $(\pi ,K,\eta )$. Small explicit quark masses ($m_u,m_d,m_s\ne 0$) lift these to small but non-zero pseudo-Goldstone masses — the pions are unusually light because chiral symmetry is only weakly broken in nature. The low-energy effective theory built around this picture is chiral perturbation theory ($\chi$PT).

The $U(1{)}_A$ piece of the classical chiral symmetry is broken by the chiral anomaly (not spontaneously): this resolves the historical ${\eta }^{\prime }$ puzzle and is intimately tied to instantons and the $\theta$-term.

Factorization and Practical Calculations

High-energy hadron-collider observables (e.g. proton–proton scattering at the LHC) cannot be computed directly because the asymptotic states (protons) are non-perturbative bound states. The bridge is factorization theorems: schematically,

$$d{\sigma }_{pp\to X}(s)=\sum _{i,j}\int d{x}_{1}d{x}_2\underset{\text{PDFs, non-perturbative}}{\underbrace{f_i(x_1,{\mu }_F)f_j(x_2,{\mu }_F)}}\underset{\text{partonic, perturbative in }{\alpha }_s}{\underbrace{d{\hat{\sigma }}_{ij\to X}(x_1{x}_{2}s,{\mu }_F,{\mu }_R)}},$$

separating the calculation into:

• Parton distribution functions $f_i(x,{\mu }_F)$: probability of finding parton $i$ carrying momentum fraction $x$ inside a proton; non-perturbative, extracted from data.
• Partonic cross section $d\hat{\sigma }$: computed perturbatively in ${\alpha }_s$ using the Feynman rules above and the standard cross-section machinery.
• DGLAP evolution: the ${\mu }_F$-dependence of $f_i$ is governed by perturbative QCD via the Dokshitzer–Gribov–Lipatov–Altarelli–Parisi equations.

This is what makes QCD predictive at colliders despite confinement.

Summary: What QCD Adds to QFT and Differs from QED

So the "derivation" of QCD from QFT amounts to the same three choices as QED — field content, gauge group, renormalizability — with $U(1)\to SU(3)$. The dynamics that follows is qualitatively different.

Successes and Tested Predictions

• Deep inelastic scattering (DIS) and Bjorken scaling: high-energy electron–proton scattering revealed point-like constituents (partons) inside the proton, and the logarithmic violations of exact Bjorken scaling match the DGLAP evolution predicted by perturbative QCD.
• Asymptotic freedom: the measured running of ${\alpha }_s$ from $\mu \sim 1$ GeV (where ${\alpha }_s\sim 0.5$) to $\mu =M_Z$ (where ${\alpha }_s(M_Z)\approx 0.118$) agrees with QCD predictions across two decades in energy.
• Jet physics at colliders: 2-jet, 3-jet, and multi-jet rates at $e^{+}{e}^{-}$ machines (LEP, SLC) and hadron colliders (Tevatron, LHC) confirm both the gluon's existence (3-jet events at PETRA, 1979) and the QCD prediction for jet cross sections.
• Lattice QCD ab initio computations of the hadron spectrum reproduce the proton, neutron, pion, kaon, ... masses to a few percent using only ${\alpha }_s$ and the quark masses as inputs.
• Heavy-quark physics: the spectroscopy and decays of charmonium ($J/\psi$), bottomonium (${\rm Y}$), and $B$-mesons are computed with NRQCD and HQET, effective theories built from QCD by integrating out the heavy-quark scale.

Caveats and Open Issues

• No rigorous construction of $SU(3)$ Yang–Mills in $d=4$ with a mass gap — one of the Clay Millennium Prize Problems. Confinement is universally believed but has never been proved analytically.
• The strong CP problem. The $\theta$-term is allowed by all symmetries and would generate a neutron EDM at the level $d_n\sim 10^{-16}\theta e\cdot \mathrm{cm}$; the experimental bound $d_n\lesssim 10^{-26}e\cdot \mathrm{cm}$ forces $|\theta |\lesssim 10^{-10}$. Why this parameter is so tiny is unexplained within QCD; the most popular resolution is the Peccei–Quinn mechanism, which predicts a new pseudoscalar particle, the axion.
• Sign problem in lattice QCD at finite density. Monte Carlo importance sampling fails when the Euclidean action becomes complex (finite chemical potential, real-time dynamics); this obstructs first-principles study of dense QCD (neutron-star interiors, the QCD phase diagram).
• Confinement mechanism. Several pictures (dual superconductor / monopole condensation, center-vortex condensation, Gribov–Zwanziger horizon) capture aspects of confinement, but no single mechanism is established as the answer.
• Quark masses are inputs. The 6 quark masses + the strong coupling are free parameters of QCD; their hierarchical pattern (over six orders of magnitude from $m_u\sim 2$ MeV to $m_t\sim 173$ GeV) is unexplained within QCD and must be supplied by physics beyond.

See Also

• QED — the abelian gauge theory, structurally identical at the Lagrangian level but qualitatively different in dynamics.
• Modern Foundations — the Wigner–Weinberg derivation framework into which QCD fits as a specialization (with the asymptotic-completeness caveat above).
• Postulates of QFT — the general framework QCD specializes.
• Group Theory — for $SU(3)$, its Lie algebra $\mathfrak{su}(3)$, structure constants, and representation theory.
• Remarks and Open Issues — for the Yang–Mills existence problem and related foundational gaps.