Quantum Chromodynamics

Electroweak theory describes beta decay, electromagnetic interactions, and the masses generated by the Higgs field. It does not explain what binds quarks into protons and neutrons. Quantum chromodynamics, or QCD, describes that strong interaction.

We will first identify the particles and the extra charge involved, called colour. Then we repeat a construction used in QED and WI: introduce a global symmetry, make it local, and read the resulting interactions. The important new feature is that the gauge fields themselves carry colour. Their interactions contribute to very different behaviour at short and long distances.

We use $\hslash =c=1$, metric $(+,-,-,-)$, and the plus-sign covariant derivative of the preceding chapters. We write $g_s$ for the strong coupling, reserving $g$ and $g^{\prime }$ for the electroweak couplings.

Introducing Quarks

A proton is not an elementary particle like the electron. Scattering experiments reveal internal constituents, and the pattern of strongly interacting particles is described using spin-$1/2$ particles called quarks. The different quark species are called flavours.

The up quark $u$ has electric charge $+2e/3$ and the down quark $d$ has charge $-e/3$. A proton has valence content $uud$ and a neutron $udd$, giving

$$Q_p=\frac{2}{3}+\frac{2}{3}-\frac{1}{3}=1,Q_n=\frac{2}{3}-\frac{1}{3}-\frac{1}{3}=0.$$

Here the charges $Q$ are in units of $e>0$. Valence content specifies the net quark flavour content. It does not mean that a proton contains only three particles at all times: its quantum state also includes gluons and quark–antiquark pairs.

A composite particle held together by the strong interaction is a hadron. The simplest families are baryons, with three valence quarks, and mesons, with a valence quark and antiquark. Protons and neutrons are baryons. The positively charged pion, with valence content $u\bar{d}$, is a meson; the antiquark contributes charge $+e/3$, giving total charge $+e$. More complicated hadrons exist, but these examples are enough for the construction below.

There are six quark flavours. Their electric charges fall into two sets:

The additional names do not introduce new kinds of strong charge. Each flavour has the same three colour components, which we introduce next. Their masses differ. The top quark decays through the weak interaction before forming ordinary hadrons.

Quarks therefore participate in both strong and electroweak interactions. In beta decay, the weak interaction changes a down quark into an up quark. QCD instead binds the quarks and, at a quark–gluon vertex, preserves the quark flavour.

Confinement and Color Quantum Number

Quarks carry a charge different from electric charge, called colour. The three basis labels are conventionally red, green, and blue. They have no connection to visible colours. An up quark can have any of these colour components while keeping the same mass, electric charge, and flavour.

For one flavour, collect the components into a column:

$$q(x)=\left(\begin{array}{c}{q}_r(x)\\ q_g(x)\\ q_b(x)\end{array}\right).$$

Here $q$ denotes a quark field, not an electric-charge number. Each entry is a Dirac spinor. The column adds a colour index to the spinor components already introduced in the Dirac chapter. An antiquark transforms with the corresponding anticolour representation.

Why introduce this additional label? One motivation comes from hadron states containing identical quarks. For example, the ground-state ${\Delta }^{++}$ baryon has valence content $uuu$ and spin $3/2$. In its fully aligned spin state, its spatial, spin, and flavour factors are symmetric under exchanging quarks. The Pauli principle requires the full state to be antisymmetric. An additional antisymmetric colour factor supplies the missing part. This motivates colour; it does not alone prove that the correct gauge theory must be $SU(3)$.

In ordinary vacuum experiments we observe isolated hadrons, not isolated quarks or gluons. This is confinement. Hadrons are colour singlets: their complete colour state is unchanged by a common colour rotation. A colour singlet need not be electrically neutral; the proton is an example.

A meson's colour-singlet state is a superposition,

$$|1{⟩}_{q\bar{q}}=\frac{1}{\sqrt{3}}\left(|r\bar{r}⟩+|g\bar{g}⟩+|b\bar{b}⟩\right).$$

It is the whole combination that is invariant. A single named colour–anticolour term is generally not a singlet. A baryon uses an antisymmetric combination containing all three colours.

Confinement is a statement about the dynamics, not a rule proved by listing singlet states. A useful picture is that separating a quark and antiquark concentrates colour-field energy between them. With dynamical light quarks, enough energy can create another quark–antiquark pair, producing separate colour-singlet hadrons rather than a free quark. This picture is not an analytic proof of confinement. We return below to what can and cannot be calculated with a small-coupling expansion.

SU(3)C Global Symmetry

The three colour components of a given flavour have the same mass and free kinetic form. We can mix them without changing their total free Lagrangian. For one flavour, write

$${\mathcal{L}}_0=\bar{q}(i{\gamma }^{\mu }{\partial }_{\mu }-m_q)q=\sum _{i=r,g,b}{\bar{q}}_i(i{\gamma }^{\mu }{\partial }_{\mu }-m_q)q_i.$$

Try $q\to Uq$, where $U$ is a constant unitary three-by-three matrix with determinant one. These matrices form $SU(3{)}_C$; the subscript specifies that they act on colour. They do not mix up quarks with down quarks.

The adjoint transforms as $\bar{q}\to \bar{q}{U}^{†}$. Since $U$ is constant, it passes through the derivative, and $U^{†}U=I$ cancels in both terms. Gamma matrices act on spinor indices, while $U$ acts on colour, so their actions commute. The free Lagrangian is invariant.

As in WI, write transformations using generators:

$$U=\exp (i{\alpha }^a{T}^a),a=1,\dots ,8.$$

The parameters ${\alpha }^a$ specify the transformation. The eight matrices $T^a$ are fixed, Hermitian, and traceless. Why eight? A Hermitian three-by-three matrix has three real diagonal entries and three complex off-diagonal entries, giving $3+6=9$ real parameters. Requiring its trace to vanish removes one. The remaining eight independent matrix directions generate $SU(3)$.

For example, one generator mixes red and green:

$$T^1=\frac{1}{2}\left(\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ 0& 0& 0\end{array}\right),(I+i\alpha T^1)q=\left(\begin{array}{c}{q}_r+i\alpha q_g/2\\ q_g+i\alpha q_r/2\\ q_b\end{array}\right).$$

This changes the colour components of one flavour, without changing its electric charge. Other generators mix different pairs or change their relative phases. We need not list all eight to construct the theory. We use the normalization and commutator

$$\mathrm{tr}(T^a{T}^b)=\frac{1}{2}{\delta }^{ab},[T^a,T^b]=i{f}^{abc}{T}^c.$$

The structure constants $f^{abc}$ are numbers specifying these commutators. Their nonzero values mean that colour rotations generally depend on order: $SU(3)$ is non-Abelian, just like the $SU(2)$ used in WI.

The free colour components also admit a common phase rotation. Choosing to gauge the traceless $SU(3)$ transformations, without adding a ninth strong gauge field for that phase, is part of the QCD model. The free-field symmetry alone does not uniquely select the interaction theory.

SU(3)C Local and QCD

Now let $U$ depend on spacetime. Differentiating gives

$${\partial }_{\mu }(Uq)=U{\partial }_{\mu }q+({\partial }_{\mu }U)q.$$

The second term prevents the earlier cancellation. Introduce a compensating gauge field for each generator, exactly as in WI. There are eight gluon fields $G_{\mu }^a$, with one shared strong coupling $g_s$:

$$D_{\mu }={\partial }_{\mu }+i{g}_s{G}_{\mu },G_{\mu }=G_{\mu }^a{T}^a.$$

The index $a$ labels the eight gluon directions; $\mu$ labels spacetime components. Requiring $D_{\mu }^{\prime }{q}^{\prime }=U{D}_{\mu }q$ fixes the gauge-field transformation. It does not fix the numerical value of $g_s$.

Replacing the derivative produces the quark–gluon interaction:

$$\bar{q}i{\gamma }^{\mu }{D}_{\mu }q=\bar{q}i{\gamma }^{\mu }{\partial }_{\mu }q-g_s\bar{q}{\gamma }^{\mu }{T}^{a}q{G}_{\mu }^a.$$

Written with colour indices, the interaction is $-g_s{\bar{q}}_i{\gamma }^{\mu }(T^a{)}_{ij}{q}_j{G}_{\mu }^a$. The matrix specifies which colour components connect. It acts on neither the flavour nor the spinor index. In particular, a gluon interaction can change colour but cannot turn a down quark into an up quark. Its vertex factor is $-i{g}_s{\gamma }^{\mu }{T}^a$, the QED fermion vertex with an additional colour matrix.

Gluons also interact with gluons

We still need a kinetic term so gluon disturbances can propagate. The Yang–Mills construction supplies it. Define $G_{\mu \nu }$ by $[D_{\mu },D_{\nu }]=i{g}_s{G}_{\mu \nu }$. Then

$$G_{\mu \nu }={\partial }_{\mu }{G}_{\nu }-{\partial }_{\nu }{G}_{\mu }+i{g}_s[G_{\mu },G_{\nu }],$$

or, in components,

$$G_{\mu \nu }^a={\partial }_{\mu }{G}_{\nu }^a-{\partial }_{\nu }{G}_{\mu }^a-g_s{f}^{abc}{G}_{\mu }^b{G}_{\nu }^c.$$

The minus sign follows from our plus-sign convention for $D_{\mu }$: the factor $i$ in the matrix commutator multiplies the explicit $i$. The full derivation is the same as for $SU(2)$ on the linked page, with different generators and structure constants.

The QCD Lagrangian is therefore

$$\boxed{{\mathcal{L}}_{\mathrm{QCD}}=-\frac{1}{4}{G}_{\mu \nu }^a{G}^{a\mu \nu }+\sum _f{\bar{q}}_f(i{\gamma }^{\mu }{D}_{\mu }-m_f)q_f.}$$

The sum runs over quark flavours. The $m_f$ are the mass parameters generated through the Higgs interactions; we are writing the strong sector after electroweak symmetry breaking. For this chapter we omit the possible additional strong-CP term, which concerns a separate vacuum-symmetry question.^[1]

Unlike the photon, a gluon itself transforms nontrivially under its gauge group: it carries colour. Squaring the field strength gives terms with two, three, and four gluon fields. The quadratic terms describe free propagation; the cubic and quartic terms give three-gluon and four-gluon interactions. Thus the local symmetry fixes both quark–gluon and gluon self-interactions.

There is no Higgs vacuum value that breaks colour in the Standard Model: the Higgs is a colour singlet. The Lagrangian has no gluon mass term. This does not imply that a free massless gluon can be detected in isolation; confinement concerns the observable states of the interacting theory.

Short distances and confinement

The coupling used in a quantum calculation depends on the momentum scale at which we resolve the interaction. Write ${\alpha }_s=g_s^2/(4\pi )$ for the strong coupling in a convenient normalization. At large momentum transfer, corresponding to short distances, it becomes smaller. This is asymptotic freedom. At hadronic distances the coupling grows, and a calculation retaining only a few Feynman diagrams is no longer reliable.^[2]

At high energies, quarks and gluons produced in a collision ultimately form hadrons, a process called hadronization. The particles often arrive in collimated groups called jets. A jet can therefore carry evidence of an energetic quark or gluon without being a free quark or gluon in the detector.

Confinement and hadron masses require methods beyond a short perturbation series. Lattice QCD approximates spacetime by a grid and evaluates the quantum theory numerically. It provides nonperturbative evidence and calculations; we do not derive confinement here. The distinction matters: gluon self-interactions are explicit in the Lagrangian, while long-distance bound states require solving its quantum dynamics.

Most of the proton's mass arises from this strong-interaction dynamics, not from adding the Higgs-generated masses of its light valence quarks. The residual strong interaction between colour-singlet protons and neutrons also binds nuclei. QCD therefore accounts for both the constituents of nuclear matter and the underlying origin of nuclear forces, although calculating nuclei is a further many-body problem.

SU(3)C × SU(2)L × U(1)Y — Intuitive Explanation

The product means that a field can carry several kinds of charge at once. Colour transformations act on its colour index, weak-isospin transformations on its doublet index, and hypercharge transformations multiply it by a phase. These actions commute because they act on different indices, with a common hypercharge within each weak multiplet.

For example, arrange a left-chiral quark doublet as

$$Q_L={\left(\begin{array}{ccc}{u}_r& u_g& u_b\\ d_r& d_g& d_b\end{array}\right)}_L.$$

Each displayed entry is a spinor field. The two rows distinguish weak-isospin components; the three columns distinguish colour components. A colour transformation mixes columns within each flavour, while a weak transformation mixes the up and down entries for each colour. This is one set of quark fields participating in several interactions, not separate strong and weak copies of the quarks.

Its covariant derivative contains all three gauge contributions:

$$D_{\mu }{Q}_L=\left({\partial }_{\mu }+i{g}_s{T}_C^a{G}_{\mu }^a+ig{T}_L^i{W}_{\mu }^i+i{g}^{\prime }\frac{Y}{2}{B}_{\mu }\right)Q_L,Y=\frac{1}{3}.$$

Here the formula uses index notation implicitly: $T_C^a$ acts on colour, $T_L^i$ on the weak index. They are not matrices acting on the same two- or three-dimensional space. There are eight colour generators and three weak generators.

A triplet has three colour components; a colour singlet has no gluon coupling. Left and right quarks both carry colour, so both couple to gluons. Leptons and the Higgs do not. The hypercharges are those already chosen in EW to reproduce the electric charges.

The gauge fields are eight gluons, three weak fields, and one hypercharge field. After the Higgs mechanism, we describe the electroweak fields as $W^{+},W^{-},Z,A$; the gluons remain the colour gauge fields. The three constants $g_s,g,g^{\prime }$ are independent inputs. The product group combines the interactions in one theory without asserting that their strengths are equal.

We began with the need to bind quarks into hadrons. Colour and local $SU(3)$ specify the gluon interactions, while their quantum dynamics explains why short-distance calculations and long-distance bound states behave so differently. The next Standard Model chapter collects this strong sector with the electroweak and Higgs sectors.