Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 45 / 2 / d / Solution

In the original coordinate, the possible zero-energy vacuum expectation values for $g\ne 0$ areThey coincide when the discriminant vanishes. With real parameters and a nonnegative discriminant these are real; otherwise the allowed complex scalar values must be retained.

In the coordinate $u=\phi -s$ used to remove the linear term, the same vacua areThe nonzero branch exists in this shifted coordinate when $M\ne 0$. Let $w=-2M/g$ be that branch and write $u=w+\tilde{\phi }$. Equivalently, in the original coordinate write $\phi =v+\tilde{\phi }$ with $v=s+w$. The physical fluctuation field is the same. For $g=0,m\ne 0$ there is the unique vacuum $v=-\kappa /m$; for the constant and pure-linear exceptional branches the scalar is a flat direction, with the supersymmetry status already distinguished in part (c).