Stability of a two-scalar quartic potential ($\lambda \ge 0,\mu \ge -\lambda$)

For real scalar fields with quartic potential energy density $V_4=\lambda ({\varphi }_1^4+{\varphi }_2^4)+2\mu {\varphi }_1^2{\varphi }_2^2$, nonnegativity in every direction is equivalent to the displayed conditions. Necessity follows on an axis and on ${\varphi }_1={\varphi }_2$. Sufficiency follows by writing $V_4=\lambda ({\varphi }_1^2-{\varphi }_2^2{)}^2+2(\lambda +\mu ){\varphi }_1^2{\varphi }_2^2$. With a positive quadratic mass term, equality cases still give a stable isolated classical vacuum at the origin. The quartic alone is strictly positive away from the origin precisely when $\lambda >0$ and $\mu >-\lambda$. Rotation symmetry additionally requires $\mu =\lambda$.