Large-$N$ solution and effective action of a twisted-mass deformed $\mathrm{CP}(N-1)$ model

We study effective dynamics of the nonsupersymmetric two-dimensional $\mathbb{CP}(N-1)$ model in the large $N$ limit. This model is deformed by a mass term $m$ preserving ${\mathbb{Z}}_N$ symmetry of the Lagrangian. At small $m$ the theory is strongly coupled and resembles the undeformed $\mathbb{CP}(N-1)$ model, while at large $m$ it is in a weakly coupled Higgs phase with spontaneously broken ${\mathbb{Z}}_N$. We find the phase transition point and discuss the fate of the kink-antikink “mesons” at strong coupling. We also resolve an issue of instability that arose in previous studies of this model.

Figure 1

Behavior of $\sigma$ and $|n{|}^2$ near the phase transition point $m=\sqrt{c}\mathrm{\Lambda }$. (For this plot $c=1$.) Black solid lines are the exact numerical solution, red dashed lines show the asymptotic expressions (3.26) and (3.27) respectively.

Figure 2

Vacuum energy and phase transition. Solid black line is the ground state (true vacuum) energy. Dashed blue line shows the would-be Coulomb phase energy analytically continued beyond the phase transition point $m=\sqrt{c}\mathrm{\Lambda }$. (For this plot $c=1$.).

Figure 3

Loop corrections to the $A_{\alpha }$ and $\sigma$ propagators.