I have answered this question in this article. For reference, I copy the article below:
Quantum Chromodynamics is a Quantum Field Theory that describes Quarks, Gluons, and their interactions through the Strong Force. It is a strongly-coupled theory, which means that there is the need of Renormalisation (unless you believe in Vladimir Kalitvanski).
Free Quarks clearly should obey the Free (i.e. Potential-less) Dirac Equation:
Applying the Euler-Lagrange Equations, we see that the Lagrangian Density would then be:
Sidenote: This is the Lagrangian Density for a Free Quark. For the entire Quantum Chromodynamics Lagrangian Density, one also needs to find the Lagrangian Density \ for only Gluons, and the Lagrangian Density for Quark-Gluon interaction.
This Lagrangian Density ensures invariance under transformations. I.e. transforming the quark field as where does not change the Lagrangian Density.
In experiment, Quarks have never been observed as free, they are always interacting through the Strong Force, with Gluons and other Quarks. This phenomenon is known as Quark Confinement.
Gluons and the strong force
Let us now introduce 8 Gluon Potentials where goes from 1 to 8. The Quarks will no longer be free. The interaction between the Quarks and theGluons is known as the Strong Force. The Lagrangian Density due to the Strong Force is then given by the following expression:
The Gluons themselves are Yang-Mills Fields, and have the Yang-Mills Lagrangian Density. This is as follows:
With these Gluon Fields, the covariant derivative can be defined as follows:
Which invites correct comparisons with General Relativity, including bundle curvatures, etc.
Total Lagrangian Density
Adding up the three individual Lagrangian Densityies previously discussed, one obtains the total Lagrangian Density of Quantum Chrodynamics:
Note that we do not immediately observe the interaction term, but this is merely because we have quietly replaced with .
Quantum Chromodynamics is then invariant under transformations if the Gluon Potentials simultaneously transform as:
This is a Gauge Transformation, and therefore, Quantum Chromodynamics is a Gauge Theory of with a Gauge Group of .