= Flat direction of a scalar potential
= Flat directions of a scalar potential
{synonym}
In vacuum analysis, an exact <flat direction of a scalar potential> is a nonconstant continuous family of degenerate stationary vacua. The <scalar potential> stays constant along this family and its <gradient> vanishes there. An ordinary level set does not suffice: $V(x,y)=x^2+y^2$ is constant on every circle, but its vacuum at the origin has no flat direction. A vanishing <eigenvalue> of the quadratic <Hessian matrix> also does not suffice: $V(x)=x^4$ has zero quadratic mass at zero but no family of nearby vacua. Exact flat directions can be lifted by corrections absent from the classical <scalar potential>.
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