General mass formula in broken supersymmetry

Abstract

The mass formula $\Sigma J^{}{\mathrm{}(-1\mathrm{})\mathrm{}}^{2J}\mathrm{}(2J+1)\mathrm{}{m_J}^2=0\mathrm{}$ is derived for a very general class of interactions with spontaneously broken supersymmetry. It shows the vanishing of the graded trace of the square of the mass operator, with $m_J$ the mass associated with a (real) field of spin $J$. This mass relation is shown to be true even in the presence of explicit breaking, provided it fulfills suitable requirements.