Curious case of an effective theory

We describe an effective theory of a scalar field, motivated by some features expected in the low energy theory of gluodynamics in $3+1$ dimensions. The theory describes two propagating massless particles in a certain limit, which we identify with the Abelian QED limit, and has classical string solutions in the general case. The string solutions are somewhat unusual as they are multiply degenerate due to the spontaneous breaking of diffeomorphism invariance. Nevertheless, all solutions yield an identical electric field and have the same string tension.