In David Tong's lecture notes on quantum field theory, at the bottom of page 138 he states that, regarding the coupling of the electromagnetic field to a real scalar field, there exists "no suitable conserved current... This means we can't couple a real scalar field to a gauge field".
I don't understand what this means (why would the lack of a conserved current be a deal-breaker), and consequently why we cannot couple the electromagnetic field to a real scalar field.
I will just explain what David Tong must have meant by that comment. Any coupling between electromagnetism and some other field needs to be gauge invariant. The standard coupling is of the form
$$\int d^4x J^{\mu}A_{\mu}$$
Where $J^{\mu}$ is a conserved current. This is gauge invariant because after we make a gauge transformation $A_{\mu}\to A_{\mu}+\partial_{\mu}\alpha$, we have
$$\int d^4x J^{\mu}A_{\mu}+\int d^4x J^{\mu}\partial_{\mu}\alpha$$ $$=\int d^4x J^{\mu}A_{\mu}-\int d^4x \partial_{\mu}J^{\mu}\alpha$$
The last term now vanishes because of the conservation of the current. But a real scalar field has no conserved current, so one cannot construct this interaction term.
Now, it is possible to construct other interaction terms say multiplying a gauge and Lorentz invariant quantity like $E\cdot B$ with a real scalar field. This has been studied and is called the Peccei-Quinn theory.
David Tong is only talking about the canonical minimal coupling however.
