Quarks are the spin-1/2 fermion constituents of hadrons and is the only matter fields of QCD. Quarks and gluons are elementary particles that carry color charge and therefore participate in the strong interaction, and they also carry electric charge, weak isospin and hypercharge, so they feel all four fundamental interactions.

A quark field carries three independent labels:

• Flavour \(f\): there are six flavours grouped in three generations, \[ (u,d),\quad (c,s),\quad (t,b) \]

with electric charges \(+\tfrac{2}{3}e\) for the up-type (\(u,c,t\)) and \(-\tfrac{1}{3}e\) for the down-type (\(d,s,b\)).

• Color \(a=1,2,3\) (red, green, blue): the quark transforms in the fundamental representation \(\mathbf{3}\) of the gauge group \(SU(3)_c\).
• Dirac index \(\alpha=1,2,3,4\): it is a 4-component spinor obeying the Dirac equation.

Hence a single quark field \(\psi_{\substack{\alpha \\ a}}^{f}\) carries \(4\times 3 = 12\) components per flavour, in contrast to the 4-component electron field of QED. This is exactly why the free Lagrangian in Continuum QCD carries both a Dirac index \(\alpha,\beta\) and a color index \(a,b\).

The current-quark masses span many orders of magnitude:

\[ \begin{align*} m_u &\approx 2.2\ \text{MeV}, & m_d &\approx 4.7\ \text{MeV}, & m_s &\approx 95\ \text{MeV},\\ m_c &\approx 1.27\ \text{GeV}, & m_b &\approx 4.18\ \text{GeV}, & m_t &\approx 173\ \text{GeV}. \end{align*} \]

The \(u,d,s\) are the "light quarks" relevant for the low-energy dynamics of nuclear matter; the \(t\) quark is so heavy that it decays before it can hadronize.

Two defining non-perturbative features distinguish quarks from QED leptons:

• Confinement: quarks are never observed in isolation. The potential between a static quark–antiquark pair rises linearly at large separation, \[ V(r) \sim \sigma r , \]

so that pulling them apart costs an energy that grows without bound (with string tension \(\sigma\)). Only color-singlet combinations propagate as physical states.

• Asymptotic freedom: the running coupling decreases at short distance, \[ \alpha_s(\mu) = \frac{g^2(\mu)}{4\pi}\ \xrightarrow{\ \mu\to\infty\ }\ 0 , \]

so that at high energy quarks behave almost as free particles. This is a consequence of the non-abelian structure of \(SU(3)_c\).

Quarks combine into color-singlet bound states also known as hadrons (for example neutron, protons, pions, kaons, etc.). We are made up of these hadrons.

• Mesons \(q\bar q\) (e.g. the pion \(\pi\)),
• Baryons \(qqq\) (e.g. the proton \(uud\), the \(\Omega^-=sss\), the \(\Delta^{++}=uuu\)).

The existence of the spin-\(\tfrac{3}{2}\) baryons \(\Omega^-\) and \(\Delta^{++}\), made of three identical quarks in a symmetric state, was the historical motivation for introducing color so as to preserve the Pauli exclusion principle.