Mass Spectrum of the Two-Dimensional O(3) Sigma Model with a $\theta$ Term

Form factor perturbation theory is applied to study the spectrum of the $\mathrm{O}(3)$ nonlinear sigma model with the topological term in the vicinity of $\theta =\pi$. Its effective action near this value is given by the nonintegrable double sine-Gordon model. Using previous results by Affleck and the explicit expressions of the form factors of the exponential operators $e^{\pm i\sqrt{8\pi }\phi (x)}$, we show that the spectrum consists of a stable triplet of massive particles for all values of $\theta$ and a singlet state of higher mass. The singlet is a stable particle only in an interval of values of $\theta$ close to $\pi$, whereas it becomes a resonance below a critical value ${\theta }_c$.

Figure 1

Unperturbed and first order correction of the masses of the triplet and the singlet states.

Figure 2

Qualitative behavior of the masses of the triplet (continuum line) and the singlet (long-dashed line) as functions of $\theta$ in the interval $(0,\pi )$. The singlet particle is stable in the interval above ${\theta }_c$.