 Explicit twofold anisotropy of a complex scalar field

Adding $-({\varphi }^2+{\varphi }^{\ast 2})/2$ to the exponential complex-scalar potential gives $V=-2x^2+e^{\lambda (x^2+y^2)}$, where $\varphi =x+iy$. Phase rotations retain only $\varphi \mapsto -\varphi$; the full real-field internal symmetry also permits $\varphi \mapsto {\varphi }^{\ast }$, giving two independent sign reflections. For $0<\lambda <2$ the minima are $\varphi =\pm v$, $v^2=\ln (2/\lambda )/\lambda$. Around either minimum, $\varphi =v+(h+i\pi )/\sqrt{2}$ gives squared masses $4\ln (2/\lambda )$ and $2$. For $\lambda \ge 2$ the unique minimum is zero. Explicit anisotropy removes the continuous vacuum circle, so discrete symmetry breaking supplies no mandatory Goldstone boson.