Color SU(3), Gluons, and the QCD Lagrangian

Color entered the quark model as a bookkeeping charge that rescued Fermi statistics for the ${\Delta }^{++}$ and, through the color-singlet rule, 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, $\hslash =c=1$ and ${\eta }^{\mu \nu }=\mathrm{diag}(+,-,-,-)$. The strong coupling is written $g_s$, with ${\alpha }_s=g_s^2/4\pi$ the analogue of the fine-structure constant.¹

Color as an SU(3) gauge symmetry

Each quark flavor comes in three colors, assembled into a complex three-component vector in color space,

which transforms in the fundamental representation $3$ of $SU(3)$. A color rotation is

with eight real parameters ${\theta }^A$ because $SU(3)$ has $N^2-1=8$ generators. The generators are $T^A=\frac{1}{2}{\lambda }^A$, where the ${\lambda }^A$ are the Gell-Mann matrices — the $SU(3)$ counterpart of the Pauli matrices — normalized by

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.

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 $3$ — and the eight gluons occupy the adjoint $8$: six at the outer hexagon and two at the center.

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 $G_{\mu }^A$, one per generator, packaged into the matrix $G_{\mu }=G_{\mu }^A{T}^A$, and define

The gluon field is postulated to transform so that $D_{\mu }q\to U{D}_{\mu }q$, which requires

The inhomogeneous second term is the non-abelian generalization of the QED gauge shift $A_{\mu }\to A_{\mu }-\frac{1}{e}{\partial }_{\mu }\chi$; 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

The interaction term $g_s\bar{q}{\gamma }^{\mu }{T}^{A}q{G}_{\mu }^A$ 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 $g_s$.

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, $G_{\mu \nu }=\frac{i}{g_s}[D_{\mu },D_{\nu }]$, giving

The last term is quadratic in the gluon field and has no counterpart in QED, where $f^{ABC}=0$. The full QCD Lagrangian is

summed over the six quark flavors $f$. Squaring the field strength produces the gluon propagator from the $\partial G$ pieces, but also — through the $f^{ABC}GG$ term — cubic and quartic gluon vertices:

• The three-gluon vertex, from the cross term $\partial G\cdot (fGG)$, couples three gluon lines with strength $g_s$ and one factor of $f^{ABC}$.
• The four-gluon vertex, from $(fGG{)}^2$, 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.

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:

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:

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, $\mathrm{Tr}(T^A{T}^B)=T_F{\delta }^{AB}$ with $T_F=\frac{1}{2}$.

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 $-\frac{4}{3}$ (attractive), while the same exchange in a color-octet configuration carries $+\frac{1}{6}$ (repulsive); the attraction of the singlet channel is why color-neutral bound states form.²

The QCD Feynman rules

The rules mirror QED with color inserted. Read a diagram against the fermion arrows, as in the QED case, and carry the color indices along.

• Quark-gluon vertex: $i{g}_s{\gamma }^{\mu }{T}_{ij}^A$, 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: $\frac{i({\gamma }^{\mu }{q}_{\mu }+m)}{q^2-m^2+i\epsilon }{\delta }_{ij}$, diagonal in color.
• Gluon propagator (Feynman gauge): $\frac{-i{\eta }_{\mu \nu }{\delta }^{AB}}{q^2+i\epsilon }$, 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 with ${\alpha }^2\to {\alpha }_s^2$ and a color average.

The following table sets the two theories side by side.

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

roughly sixteen times the electromagnetic $\alpha (M_Z)\approx 1/128$ at the same scale.³ 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.