Introduction of the vector potential $A_{\mu}$ for the local gauge invariance of the complex scalar field lagrangian [duplicate]

Question

In Ryder, when trying to restore the local $U(1)$ gauge symmetry of the complex scalar field $\phi=\phi_1+i\phi_2$, the final Lagrangian consists of the following four parts: $$L_0=(\partial_{\mu}\phi)(\partial^{\mu}\phi^*)-m^2\phi^*\phi$$ which is the free particle Lagrangian, $$L_1=-eJ^{\mu}A_{\mu}$$ which is where the $A_{\mu}$ is introduced, $$L_2=e^2A_{\mu}A^{\mu}\phi^*\phi$$ and $$L_3=-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}$$ with $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$, which is gauge invariant. On top of $L_0$, the additional $L_1$ and $L_2$ seems to be quite natural. However, for the $L_3$ part, it doesn't seem to be necessary to me, while Ryder's argument is that "the field $A_{\mu}$ must presumably contribute by itself to the Lagrangian". If we now look back, we know that the $L_3$ term actually gives us the Maxwell's equations. but I'm still not convinced by Ryder's argument. So the question is how do we know that $A_{\mu}$ must contribute to the Lagrangian?

Answer 1

An intuitive way to see why a term proportional to $F_{\mu\nu}F^{\mu\nu}$ is natural is recognizing the fact that this term introduces dynamics for the gauge field. If you make the dependence on the gauge field explicit, you will see that it contains a properly normalized kinetic term, allowing us to treat it as a field propagating in spacetime. This, as you suggested, makes sense since the term is needed to recover the full set of Maxwell equations.

Furthermore, the term is consistent with all symmetries one would want the theory to possess, namely gauge and Poincaré invariance.