Do we need an alternative to local gauge coupling to electromagnetic fields?

The local gauge coupling through the recipe Formula: see text, that works so well with Dirac spinors in QED and in the gauge theories of the Standard Model, has a peculiarity when applied to scalar fields: it generates in the Lagrangian a coupling term Formula: see text in which Formula: see text does not coincide with the conserved Nöther current associated to the global gauge symmetry. This is not an inconsistency, just a feature that appears when working out the locally gauge invariant action, and which ensures that the correct conserved current is the source of the gauge field. What would happen then if we were to assume for the scalar field the same coupling Formula: see text through a conserved current which holds for spinor QED and classical electrodynamics? The consequence is that one is forced in that case to renounce to the principle of local gauge symmetry and must thus consider the electromagnetic (e.m.) field to be described by electrodynamic theories compatible with that lack of invariance, like the extended electrodynamics by Aharonov-Bohm. No differences with the usual theory appear for fermion systems when strict local charge conservation applies. In particular, if we consider the non-relativistic quantum theory as the low-energy limit of the relativistic theory, we would expect no modifications of Schrödinger equation when applied to fermion systems. However, when scalar boson systems are considered, like Cooper pairs quasi-particles in superconductors, in the new formulation the e.m. fields include a source, additional to the usual conserved four-current, and, besides, the corresponding Schrödinger equation acquires a new term, proportional to Formula: see text, which can lead to observable consequences, like a sizable change in the estimate of the magnetic penetration depth in certain superconductors, compatible with the experimental data. In conclusion, the alternative coupling considered yields a viable effective model for bosonic condensed matter systems, while for Dirac fermions it reduces to standard QED. Soft photon factorization and KLN cancellations in scalar QED fail in this framework, therefore particle physics scattering is outside the scope.