The free Lagrangian is the part of the Lagrangian density that is quadratic in the fields — it describes particles that propagate without interacting. It fixes the propagator and the mass, and it is the starting point onto which interactions are added (in QCD, via the gauge principle).

For a single quark flavour, ignoring color coupling, the free Lagrangian is the Dirac Lagrangian

\[ \mathcal{L}_0 = \overline\psi\,(i\gamma^\mu\partial_\mu - m)\,\psi , \]

a spinor bilinear that is Lorentz invariant and yields the free Dirac equation \((i\gamma^\mu\partial_\mu-m)\psi=0\).

Because the quark is a three-component object in color space, \(\psi=(\psi_r,\psi_g,\psi_b)^T\), the free Lagrangian is a sum over colors (and over the six flavours \(f\)),

\[ \mathcal{L}_0 = \sum_{f}\ \sum_{c=r,g,b}\ \overline{\psi_c}^{\,f}\big(i\gamma^\mu\partial_\mu - m\big)\psi_c^{\,f} = \overline\psi_{\substack{\alpha\\a}}\big(i\gamma^\mu_{\alpha\beta}\delta_{ab}\partial_\mu - m\,\delta_{ab}\delta_{\alpha\beta}\big)\psi_{\substack{\beta\\b}} , \]

which is diagonal in color: the free theory has no interaction mixing the colors. Note it is invariant under a global \(SU(3)\) rotation \(\psi\to e^{i\alpha^aT^a}\psi\) with constant \(\alpha^a\), but not under a local one — the derivative spoils local invariance.

Demanding local gauge invariance forces the replacement \(\partial_\mu\to D_\mu=\partial_\mu-igA_\mu\). The free Lagrangian then becomes

\[ \mathcal{L} = \overline\psi\,(i\gamma^\mu D_\mu - m)\,\psi = \mathcal{L}_0 + g\,\overline\psi\gamma^\mu A_\mu\psi , \]

exposing the quark–gluon coupling as the unique minimal interaction. Adding the gluon kinetic term completes \(\mathcal{L}_{QCD}\).