= Explicit twofold anisotropy of a complex scalar field
Adding $-(\phi^2+\phi^{*2})/2$ to the <exponential complex-scalar potential> gives $V=-2x^2+e^{\lambda(x^2+y^2)}$, where $\phi=x+iy$. Phase rotations retain only $\phi\mapsto-\phi$; the full real-field internal symmetry also permits $\phi\mapsto\phi^*$, giving two independent sign reflections. For $0<\lambda<2$ the minima are $\phi=\pm v$, $v^2=\ln(2/\lambda)/\lambda$. Around either minimum, $\phi=v+(h+i\pi)/\sqrt2$ gives squared masses $4\ln(2/\lambda)$ and $2$. For $\lambda\geq2$ the unique minimum is zero. Explicit anisotropy removes the continuous vacuum circle, so discrete symmetry breaking supplies no mandatory <Goldstone boson>.
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