2 problems

Relevant-background conjecture. These are all the backgrounds that are relevant at low energies.

Let $\phi_\alpha(x):=\phi(x)e^{i\alpha(x)}$, $\phi^*_\alpha(x):=\phi^*(x)e^{-i\alpha(x)}$, and, for a local functional $F$, let $F_\alpha:=F(\phi_\alpha,\phi^*_\alpha,A^\mu)$, wher…