Quantum U(1)-invariant theory from integrable classical models

A certain class of two-dimensional quantum field theories with solitonlike behavior and $\mathrm{O}\mathrm{}\left(N\right)\mathrm{}$ ($N\ge 1$) symmetry is considered. By imposing factorizability of the $S$ matrix for any value of the coupling constant, the classical theory turns to be uniquely defined. It reduces to sine-Gordon theory for $N=1\mathrm{}$, to complex sine-Gordon (CSG) theory for $N=2\mathrm{}$, and to a free theory for $N\ge 3$. The quantum CSG theory is investigated. The $\beta$ function is identically zero up to one loop, and moreover a first nontrivial classical conserved current is studied at this order. It is shown that it does not need proper renormalization and a corresponding quantum local conserved current is constructed. Finally, the semiclassical spectrum is explored.