Complex sine-Gordon theory ($\mathcal{L}=|\partial \psi {|}^2/(1-{\lambda }^2|\psi {|}^2)-M^2|\psi {|}^2$)

A nonlinear complex scalar field theory with a global phase symmetry and a curved target-space kinetic coefficient. In physical coordinates its classical density is displayed above; dimensionless coordinates measured in $M^{-1}$ put a common $M^2$ outside the reduced density. Its regular coordinate domain is ${\lambda }^2|\psi {|}^2<1$. Rotating localized solutions are charged complex sine-Gordon solitons. The local classical density does not by itself specify the treatment of the singular coordinate boundary or a global quantum completion.

The rest-frame field $\psi =\cos \alpha e^{i\sin \alpha t}/[\lambda \cosh (\cos \alpha x)]$ has mass $4M\cos \alpha /{\lambda }^2$ and charge $4[\mathrm{sgn}(\alpha )\pi /2-\alpha ]/{\lambda }^2$, for the generator $\delta \psi =i\psi$ and $\alpha \ne 0$. The conserved phase angle is a collective coordinate with momentum $Q$. Quantization of a periodic soliton coordinate gives integer $Q$ in units $\hslash =1$ and the displayed leading mass formula. On the regular branch $0<|Q|<2\pi /{\lambda }^2$, its concave increasing sine law prevents fragmentation into smaller like-charge states.

At zero rotation the field reaches ${\lambda }^2|\psi {|}^2=1$ at its center. Its energy remains finite, but its charge expression is singular and has the two displayed one-sided limits. The regular local branch excludes this endpoint. A global completion must say whether and how these two limiting charge labels represent a physical state.

An additional integer-level completion identifies the charge labels modulo $k$ and has $k-1$ nonzero sectors. For even $k$, the two maximal labels describe one self-conjugate sector. This interpretation is specified in Dorey and Hollowood, section 2; it is additional to the local classical density.