--- title: Color SU(3), Gluons, and the QCD Lagrangian module: Quantum Chromodynamics moduleNumber: 8 lessonNumber: 1 order: 801 summary: > Color is the exact gauged SU(3) charge of the strong force. Gauging it forces eight massless gluons in the adjoint representation and, because the gauge group is non-abelian, three- and four-gluon self-couplings absent from QED. This lesson builds the QCD Lagrangian from the covariant derivative and the non-abelian field strength, states the Feynman rules with their color factors, and computes the Casimir invariants that set the strength of quark-gluon and gluon-gluon coupling. topics: [Quantum Chromodynamics] draft: false sources: - book: Griffiths ref: "Ch. 9 — Quantum Chromodynamics, §9.1–9.2" - book: Halzen & Martin ref: "Ch. 14 (color, the QCD Lagrangian, and the Feynman rules)" - book: Thomson ref: "Ch. 10 — Quantum Chromodynamics" - book: Tong ref: "The Standard Model (Cambridge Part III), §3 (the QCD action, gluons, color)" --- Color entered the quark model as a bookkeeping charge that rescued Fermi statistics for the [$\Delta^{++}$](/particle-physics/quark-model/baryon-spectroscopy) and, through the [color-singlet rule](/particle-physics/quark-model/color-confinement-exotics), selected the observed hadrons. Quantum chromodynamics promotes color from a static label to a dynamical gauge charge. The recipe is the one that produced QED from the global phase symmetry of the electron: take the symmetry that rotates the color of a quark, demand that it hold independently at every spacetime point, and the demand generates a set of gauge fields — the gluons — with a fixed coupling to the quarks. The difference from QED is that the color symmetry is $SU(3)$, a non-abelian group, and non-commuting generators force the gauge fields to couple to themselves. That single fact is the origin of asymptotic freedom, confinement, and everything that separates the strong force from electromagnetism. Throughout, $\hbar = c = 1$ and $\eta^{\mu\nu} = \operatorname{diag}(+,-,-,-)$. The strong coupling is written $g_s$, with $\alpha_s = g_s^2/4\pi$ the analogue of the fine-structure constant.[^tong-action] ## Color as an SU(3) gauge symmetry Each quark flavor comes in three colors, assembled into a complex three-component vector in color space, $$ q = \begin{pmatrix} q_r \\ q_g \\ q_b \end{pmatrix}, $$ which transforms in the fundamental representation $\mathbf 3$ of $SU(3)$. A color rotation is $$ q \to U q, \qquad U = \exp\!\left(i\,\theta^A T^A\right)\in SU(3), \qquad A = 1,\dots,8, $$ with eight real parameters $\theta^A$ because $SU(3)$ has $N^2 - 1 = 8$ generators. The generators are $T^A = \tfrac12\lambda^A$, where the $\lambda^A$ are the Gell-Mann matrices — the $SU(3)$ counterpart of the Pauli matrices — normalized by $$ \operatorname{Tr}\!\left(T^A T^B\right) = \tfrac12\,\delta^{AB}, \qquad \left[T^A, T^B\right] = i f^{ABC}\,T^C . $$ The structure constants $f^{ABC}$ are totally antisymmetric and nonzero: $SU(3)$ is non-abelian, and this non-vanishing commutator is the technical seed of every feature below. Two of the eight generators, $\lambda^3$ and $\lambda^8$, are diagonal — the group has rank two — so a color state is labeled by two additive quantum numbers, the color isospin $I_3^c$ and the color hypercharge $Y^c$, exactly parallel to the flavor $SU(3)$ of the [eightfold way](/particle-physics/quark-model/eightfold-way-su3). > **Definition (Color gauge invariance).** Global color invariance requires the > Lagrangian to be unchanged under a single $U$ applied everywhere. Local (gauge) > invariance requires it to be unchanged when $U = U(x)$ varies from point to point. > The free Dirac Lagrangian $\bar q\,(i\gamma^\mu\partial_\mu - m)\,q$ is globally but > not locally invariant, because $\partial_\mu q$ picks up a term $\propto\partial_\mu U$. Because $SU(3)$ has rank two, color charge is a point in a two-dimensional plane spanned by the two diagonal generators. The three quark colors sit at the corners of a triangle — the fundamental $\mathbf 3$ — and the eight gluons occupy the adjoint $\mathbf 8$: six at the outer hexagon and two at the center. $$ % caption: Color charge in the plane of the two diagonal generators. The three quark % colors form the fundamental triplet at the corners of a triangle; the antiquarks % form the inverted antitriplet. The eight gluons fill the adjoint octet, a hexagon of % six color-changing states with two states stacked at the center. The pattern is the % color analogue of the flavor weight diagrams of the eightfold way. \begin{tikzpicture}[font=\footnotesize, scale=1.0] \definecolor{acc}{HTML}{4A6FA5} % --- triplet (left) --- \begin{scope}[shift={(-4.4,0)}] \draw[black, ->] (-1.5,0) -- (1.5,0); \draw[black, ->] (0,-1.5) -- (0,1.6); \fill[black] (-0.9,-0.5) circle (2.2pt) node[below left, font=\scriptsize] {G}; \fill[black] (0.9,-0.5) circle (2.2pt) node[below right, font=\scriptsize] {R}; \fill[black] (0,1.0) circle (2.2pt) node[above, font=\scriptsize] {B}; \node[black, font=\scriptsize] at (0,-2.0) {triplet}; \end{scope} % --- octet (right) --- \begin{scope}[shift={(3.4,0)}] \draw[black, ->] (-2.0,0) -- (2.0,0) node[right, black, font=\scriptsize] {color isospin}; \draw[black, ->] (0,-1.7) -- (0,1.8) node[above, black, font=\scriptsize] {hypercharge}; \foreach \x/\yy in {-1.6/0, 1.6/0, -0.8/0.9, 0.8/0.9, -0.8/-0.9, 0.8/-0.9} \fill[black] (\x,\yy) circle (2.2pt); \draw[black, thick] (0,0) circle (2.6pt); \fill[black] (0,0) circle (2.2pt); \node[black, font=\scriptsize] at (0,-2.0) {octet (two at center)}; \end{scope} \end{tikzpicture} $$ ## The covariant derivative and the gluon field Local invariance is restored by replacing the ordinary derivative with a covariant derivative that carries a compensating gauge field. Introduce eight gluon fields $G^A_\mu$, one per generator, packaged into the matrix $G_\mu = G^A_\mu T^A$, and define $$ D_\mu q = \left(\partial_\mu - i g_s\,G^A_\mu T^A\right) q . $$ The gluon field is postulated to transform so that $D_\mu q \to U D_\mu q$, which requires $$ G_\mu \to U G_\mu U^{-1} - \frac{i}{g_s}\,(\partial_\mu U)\,U^{-1}. $$ The inhomogeneous second term is the non-abelian generalization of the QED gauge shift $A_\mu \to A_\mu - \tfrac1e\partial_\mu\chi$; the first term, absent in QED, rotates the gluon among its eight components because the gauge field itself carries color. With the covariant derivative in hand, the quark part of the Lagrangian is $$ \mathcal L_{\text{quark}} = \bar q\,\left(i\gamma^\mu D_\mu - m\right) q = \bar q\,(i\gamma^\mu\partial_\mu - m)\,q + g_s\,\bar q\,\gamma^\mu T^A q\;G^A_\mu . $$ The interaction term $g_s\,\bar q\gamma^\mu T^A q\,G^A_\mu$ is the quark-gluon vertex: a color current $\bar q\gamma^\mu T^A q$ coupled to the gluon. It is the QED vertex $e\,\bar\psi\gamma^\mu\psi A_\mu$ with the color matrix $T^A$ inserted, so the gluon both changes the quark's color and couples with strength $g_s$. $$ % caption: The eight gluons carry a color and an anticolor. Six are the off-diagonal % color-changing states (red-antigreen and the like); the remaining two are neutral % mixtures built from the diagonal Gell-Mann generators. The would-be ninth, the fully % symmetric colorless singlet, is absent because the gauge group is SU(3) and not U(3). \begin{tikzpicture}[font=\footnotesize, scale=1.0] \definecolor{acc}{HTML}{4A6FA5} % six off-diagonal \foreach \lab/\x/\yy in {{$R\overline{G}$}/-4.2/1.2, {$R\overline{B}$}/-2.6/1.2, {$G\overline{R}$}/-1.0/1.2, {$G\overline{B}$}/-4.2/0.0, {$B\overline{R}$}/-2.6/0.0, {$B\overline{G}$}/-1.0/0.0} \node[draw=black, minimum size=0.85cm] at (\x,\yy) {\lab}; \node[black, font=\scriptsize] at (-2.6,-0.9) {six color-changing gluons}; % two diagonal \node[draw=black, minimum size=0.85cm] at (1.4,0.9) {mix 1}; \node[draw=black, minimum size=0.85cm] at (3.0,0.9) {mix 2}; \node[black, font=\scriptsize] at (2.2,0.1) {two neutral mixtures}; % excluded singlet \node[draw=acc, dashed, minimum size=0.85cm] at (2.2,-1.2) {singlet}; \node[black, font=\scriptsize, align=center] at (2.2,-2.15) {excluded: colorless\\ singlet is not a gluon}; \end{tikzpicture} $$ ## The gluon field strength and self-interaction The gluon kinetic term is built from the field strength. In QED the field strength $F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu$ is gauge invariant. For a non-abelian field it must be defined through the covariant derivative, $G_{\mu\nu} = \tfrac{i}{g_s}[D_\mu, D_\nu]$, giving $$ G^A_{\mu\nu} = \partial_\mu G^A_\nu - \partial_\nu G^A_\mu + g_s\,f^{ABC}\,G^B_\mu G^C_\nu . $$ The last term is quadratic in the gluon field and has no counterpart in QED, where $f^{ABC}=0$. The full QCD Lagrangian is $$ \mathcal L_{\text{QCD}} = -\tfrac14\,G^A_{\mu\nu}G^{A\,\mu\nu} + \sum_f \bar q_f\,\left(i\gamma^\mu D_\mu - m_f\right) q_f , $$ summed over the six quark flavors $f$. Squaring the field strength produces the gluon propagator from the $\partial G$ pieces, but also — through the $f^{ABC}G G$ term — cubic and quartic gluon vertices: - The **three-gluon vertex**, from the cross term $\partial G\cdot(f G G)$, couples three gluon lines with strength $g_s$ and one factor of $f^{ABC}$. - The **four-gluon vertex**, from $(f G G)^2$, couples four gluon lines with strength $g_s^2$. Gluons carry color, so they act as sources of the very field they are, and the color charge itself is not gauge invariant the way electric charge is. Photons are electrically neutral and never couple to one another; gluons do, and the difference governs the running of $\alpha_s$ in the [next lesson](/particle-physics/qcd/asymptotic-freedom-confinement). $$ % caption: The QCD vertices. On the left, the quark-gluon vertex carries the coupling % times a color matrix, the direct analogue of the QED vertex. The two vertices on the % right have no electromagnetic counterpart: three gluons meet with one power of the % coupling, and four gluons meet with two powers. The gluon self-interaction is what % makes QCD non-abelian. \begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0] \definecolor{acc}{HTML}{4A6FA5} % quark-gluon vertex \begin{scope}[shift={(-5.0,0)}] \draw[thick, ->] (-1.0,1.0) -- (-0.5,0.5); \draw[thick] (-0.5,0.5) -- (0,0); \draw[thick, ->] (0,0) -- (-0.5,-0.5); \draw[thick] (-0.5,-0.5) -- (-1.0,-1.0); \draw[thick] (0,0) sin (0.25,0.12) cos (0.5,0) sin (0.75,-0.12) cos (1.0,0); \fill (0,0) circle (2.0pt); \node at (0,-1.5) {quark-gluon}; \node[black, font=\scriptsize] at (0,-2.0) {order $g_s$}; \end{scope} % three-gluon \begin{scope}[shift={(-0.4,0)}] \draw[acc, thick] (0,0) sin (-0.3,0.35) cos (-0.6,0.6) sin (-0.75,0.75) cos (-0.9,0.9); \draw[acc, thick] (0,0) sin (-0.3,-0.35) cos (-0.6,-0.6) sin (-0.75,-0.75) cos (-0.9,-0.9); \draw[acc, thick] (0,0) sin (0.35,0.12) cos (0.7,0) sin (1.0,-0.12) cos (1.2,0); \fill[acc] (0,0) circle (2.0pt); \node[acc] at (0,-1.5) {three-gluon}; \node[black, font=\scriptsize] at (0,-2.0) {order $g_s$}; \end{scope} % four-gluon \begin{scope}[shift={(4.4,0)}] \draw[acc, thick] (-0.9,0.9) sin (-0.45,0.55) cos (0,0) sin (0.45,-0.55) cos (0.9,-0.9); \draw[acc, thick] (-0.9,-0.9) sin (-0.45,-0.55) cos (0,0) sin (0.45,0.55) cos (0.9,0.9); \fill[acc] (0,0) circle (2.0pt); \node[acc] at (0,-1.5) {four-gluon}; \node[black, font=\scriptsize] at (0,-2.0) {order $g_s^2$}; \end{scope} \end{tikzpicture} $$ ## Color factors and Casimir invariants Every QCD amplitude factorizes into a kinematic part, identical in form to the QED result, and a color part built from traces of the $T^A$. Two group invariants set the scale of these color factors. The **fundamental Casimir** $C_F$ governs a gluon emitted and reabsorbed by a single quark, which contracts $T^A T^A$ over the color indices: $$ \left(T^A T^A\right)_{ij} = C_F\,\delta_{ij}, \qquad C_F = \frac{N^2 - 1}{2N} = \frac{4}{3} \quad (N = 3). $$ It plays the role of the squared charge $e^2$ for a quark, setting the strength with which a quark radiates gluons. The **adjoint Casimir** $C_A$ governs a gluon, whose color lives in the adjoint representation, and the analogous contraction uses the structure constants: $$ f^{ACD} f^{BCD} = C_A\,\delta^{AB}, \qquad C_A = N = 3 . $$ It measures how strongly a gluon couples to gluons. The **trace normalization** $T_F$ governs a gluon splitting into a quark loop of a single flavor, $\operatorname{Tr}(T^A T^B) = T_F\,\delta^{AB}$ with $T_F = \tfrac12$. That $C_A = 3$ is nonzero while the photon's self-coupling vanishes is the quantitative statement of the difference between the two theories. The single-gluon exchange between a quark and an antiquark in a color singlet carries a color factor $-\tfrac43$ (attractive), while the same exchange in a color-octet configuration carries $+\tfrac16$ (repulsive); the attraction of the singlet channel is why color-neutral bound states form.[^grif-color] $$ % caption: Color flow in single-gluon exchange between two quarks. The gluon carries % a color and an anticolor, so it repaints the quark lines: a red quark emitting a % red-antiblue gluon turns blue, and the blue quark absorbing it turns red. Color is % conserved at every vertex, and the exchanged gluon transports the difference. \begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0] \definecolor{acc}{HTML}{4A6FA5} % top quark line \draw[thick, ->] (-3.4,1.4) -- (-1.7,1.4); \draw[thick] (-1.7,1.4) -- (0,1.4); \draw[thick, ->] (0,1.4) -- (1.7,1.4); \draw[thick] (1.7,1.4) -- (3.4,1.4); \node[left, font=\scriptsize] at (-3.4,1.4) {R}; \node[right, font=\scriptsize] at (3.4,1.4) {B}; \fill (0,1.4) circle (2.0pt); % bottom quark line \draw[thick, ->] (-3.4,-1.4) -- (-1.7,-1.4); \draw[thick] (-1.7,-1.4) -- (0,-1.4); \draw[thick, ->] (0,-1.4) -- (1.7,-1.4); \draw[thick] (1.7,-1.4) -- (3.4,-1.4); \node[left, font=\scriptsize] at (-3.4,-1.4) {B}; \node[right, font=\scriptsize] at (3.4,-1.4) {R}; \fill (0,-1.4) circle (2.0pt); % gluon \draw[acc, thick] (0,1.4) sin (0.12,1.05) cos (0,0.7) sin (-0.12,0.35) cos (0,0) sin (0.12,-0.35) cos (0,-0.7) sin (-0.12,-1.05) cos (0,-1.4); \node[acc, right, font=\scriptsize] at (0.28,0) {$R\overline{B}$ gluon}; \end{tikzpicture} $$ ## The QCD Feynman rules The rules mirror QED with color inserted. Read a diagram against the fermion arrows, as in the [QED case](/particle-physics/qed/feynman-rules-qed), and carry the color indices along. - **Quark-gluon vertex:** $i g_s\,\gamma^\mu\,T^A_{ij}$, where $i,j$ are the quark colors and $A$ the gluon color. - **Three-gluon vertex:** $-g_s f^{ABC}$ times a momentum-dependent Lorentz structure. - **Four-gluon vertex:** $-i g_s^2$ times products of $f^{ABE}f^{CDE}$ and metric tensors. - **Quark propagator:** $\dfrac{i(\gamma^\mu q_\mu + m)}{q^2 - m^2 + i\varepsilon}\,\delta_{ij}$, diagonal in color. - **Gluon propagator (Feynman gauge):** $\dfrac{-i\,\eta_{\mu\nu}\,\delta^{AB}}{q^2 + i\varepsilon}$, diagonal in the adjoint index. The Lorentz and spinor structure is identical to QED, so a quark-level cross section is the corresponding QED cross section with $e^2 \to g_s^2$ and an extra color factor. For example, the quark-quark scattering rate is the electron-muon rate of the [QED tree lesson](/particle-physics/qed/qed-tree-processes) with $\alpha^2 \to \alpha_s^2$ and a color average. The following table sets the two theories side by side. | Feature | QED | QCD | | --- | --- | --- | | Gauge group | $U(1)$, abelian | $SU(3)$, non-abelian | | Gauge bosons | one photon | eight gluons | | Boson charge | neutral | colored (adjoint $\mathbf 8$) | | Boson self-coupling | none | 3- and 4-gluon vertices | | Vertex factor | $ie\gamma^\mu$ | $ig_s\gamma^\mu T^A$ | | Matter Casimir | $Q^2$ | $C_F = \tfrac43$ | | Coupling at low $E$ | $\alpha \approx \tfrac{1}{137}$ | $\alpha_s \sim 1$ | | Running | grows with $E$ | falls with $E$ | The last two rows are the physically decisive ones. Because the gluon self-coupling $C_A = 3$ overwhelms the quark screening $T_F n_f$, the QCD coupling runs opposite to QED: it weakens at short distance and strengthens at long distance. Establishing that sign, and following it to asymptotic freedom and confinement, is the subject of the next lesson. ## The value of the coupling The strong coupling is not a single number because it runs, but at the reference scale of the $Z$ boson mass its value is $$ \alpha_s\!\left(M_Z^2\right) = 0.1180 \pm 0.0009, $$ roughly sixteen times the electromagnetic $\alpha(M_Z)\approx 1/128$ at the same scale.[^pdg-alphas] At the scale of light-hadron physics, $Q \sim 1$ GeV, the coupling is of order unity and perturbation theory in $\alpha_s$ breaks down — the regime where confinement takes over. The two faces of QCD, a weakly coupled theory of quarks and gluons at high energy and a strongly coupled theory of hadrons at low energy, are the same Lagrangian read at different scales. [^tong-action]: The QCD action, the covariant derivative $D_\mu q = \partial_\mu q - i g_s G_\mu q$, the non-abelian field strength $G_{\mu\nu} = \partial_\mu G_\nu - \partial_\nu G_\mu - i g_s[G_\mu, G_\nu]$, and the Gell-Mann basis $T^A = \tfrac12\lambda^A$ with $\operatorname{Tr}(T^A T^B) = \tfrac12\delta^{AB}$ are given in Tong, _The Standard Model_ (Cambridge Part III), §3, [damtp.cam.ac.uk/user/tong/standardmodel.html](http://www.damtp.cam.ac.uk/user/tong/standardmodel.html). [^grif-color]: The color factors of single-gluon exchange, $-\tfrac43$ for the singlet and $+\tfrac16$ for the octet, and the Casimirs $C_F = \tfrac43$, $C_A = 3$, are derived in Griffiths, _Introduction to Elementary Particles_, §9.2, and Halzen & Martin, _Quarks and Leptons_, Ch. 14. [^pdg-alphas]: The world-average $\alpha_s(M_Z^2) = 0.1180 \pm 0.0009$ and the running electromagnetic coupling $\alpha(M_Z)\approx 1/128$ are from the Particle Data Group, _Review of Particle Physics_, [pdg.lbl.gov](https://pdg.lbl.gov).