Field can interact with itself, also interact with different fields. How does that happen? it follows a recipe based on gauge invariance principle:

step1 begin from a free Dirac Lagrangian with a global symmetry, step2 promote the global symmetry to a local gauge symmetry with the addition of a compensating gauge field that itself must transform in a specific way, which introduces interactions, step3 then allow the gauge field to propagate by adding a field strength term to the Lagrangian.

Illustrate with an abelian case first, step1:

Step2 is to promote to local gauge :

If the Amiu in above only contains potential energy, no kinetic part, it only constitutes a background, nothing interesting happen, so we need to add back the kinetic term to EM A field.

The recipe for a gauge theory has been applied to an abelian symmetry. Later it will be extended to
non-abelian symmetries, but before that a few clarifications are necessary, because the weak and the
strong interactions are based on a non-abelian symmetry. To simplify the non-abelian case, the abelian
case can be formulated differently.