= Stability of a two-scalar quartic potential
{title2=$\lambda\ge0,\quad\mu\ge-\lambda$}
For <real scalar fields> with quartic <potential energy> density $V_4=\lambda(\phi_1^4+\phi_2^4)+2\mu\phi_1^2\phi_2^2$, nonnegativity in every direction is equivalent to the displayed conditions. Necessity follows on an axis and on $\phi_1=\phi_2$. Sufficiency follows by writing $V_4=\lambda(\phi_1^2-\phi_2^2)^2+2(\lambda+\mu)\phi_1^2\phi_2^2$. 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 $\mu=\lambda$.
Back to article page
