= Solution
Taylor expansion about any <supersymmetric vacuum> $v$ gives
$$
W'(v+\widetilde\varphi)
=W'(v)+W''(v)\widetilde\varphi+\frac g2\widetilde\varphi^2
=M_v\widetilde\varphi+\frac g2\widetilde\varphi^2,
\qquad M_v=m+gv.
$$
Therefore the fluctuation <scalar potential> is
$$
\boxed{V(\widetilde\varphi)
=|M_v|^2|\widetilde\varphi|^2
+\frac12\left(M_v^*g\,\widetilde\varphi^*\widetilde\varphi^2
+M_vg^*\,\widetilde\varphi(\widetilde\varphi^*)^2\right)
+\frac{|g|^2}{4}|\widetilde\varphi|^4.}
$$
For the nonzero vacuum $w=-2M/g$ in the shifted coordinate, $M_v=M+gw=-M$. Thus it is also $|-M\widetilde\varphi+g\widetilde\varphi^2/2|^2$. The cubic interaction changes sign compared with expansion at the other root, whereas the quadratic mass term is the same.
Back to article page
