Quantum Electrodynamics
Dirac fermions, U(1) gauge invariance — the prototype of all gauge theories.
QED couples the Dirac electron field $\psi$ to the photon $A_\mu$ via local $U(1)$ gauge invariance:
$$\mathcal L_{\rm QED} = \bar\psi(i\gamma^\mu D_\mu - m)\psi - \tfrac{1}{4} F_{\mu\nu} F^{\mu\nu},$$with covariant derivative $D_\mu = \partial_\mu + ieA_\mu$. Demanding invariance under $\psi \to e^{i\alpha(x)}\psi$ forces the photon — gauge symmetry generates the interaction. The coupling is the dimensionless fine-structure constant
$$\alpha = \frac{e^2}{4\pi\hbar c} \approx \frac{1}{137.036}.$$An explicit photon mass term $m_\gamma^2 A_\mu A^\mu$ would not be gauge invariant, so gauge symmetry forces the photon to be massless — a deep structural prediction that is experimentally verified to extraordinary precision. QED itself has been tested to about 12 significant figures (electron $g{-}2$) — the most precisely tested theory in physics.