So far we have seen the two noble pieces, Yang-Mills and Dirac. Their noblesse has even convinced mathematicians, Donaldson has used a non-Abelian Yang-Mills theory to discover exotic differential structures on R4 and the Dirac operator has been elected differential operator of the decade by Atiyah & Singer. I feel that these two actions deserve the comparison with the circles of planetary motion and we are ready for the epicycles, the other three pieces are indeed cheap copies of the circles with the gauge boson A replaced by a scalar φ. We need these three epicycles to cure only one problem, give masses to some gauge bosons and to some fermions. These masses are forbidden by gauge invariance and parity violation. To simplify the notation we will work from now on in units with c = ħ = 1.
The Yang-Mills action contains the kinetic term for the gauge boson. This is simply the quadratic term, (dA, dA) that by Euler-Lagrange produces linear field equations. Again we need this minimal coupling ψ*Aψ for gauge invariance. The non-Abelian Yang-Mills action contains interaction terms for the gauge bosons, a bounded, invariant, forth order polynomial, 2(dA, [A, A])+([A, A], [A,A]). The two circles, Yang-Mills and Dirac, contain three types of couplings, a trilinear self coupling AAA, a quadrilinear self coupling AAAA and a the trilinear minimal coupling ψ*Aψ. The gauge self couplings are absent if the group G is Abelian, the photon has no electric charge, Maxwell's equations are linear.