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

Taylor expansion about any supersymmetric vacuum $v$ givesTherefore the fluctuation scalar potential isFor the nonzero vacuum $w=-2M/g$ in the shifted coordinate, $M_v=M+gw=-M$. Thus it is also $|-M\tilde{\phi }+g{\tilde{\phi }}^2/2{|}^2$. The cubic interaction changes sign compared with expansion at the other root, whereas the quadratic mass term is the same.