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

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?

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.