Supersymmetry, parity and fermion-number conservation

Abstract

A general rule is formulated for the construction of renormalizable supersymmetric Lagrangians which admit a conserved quantum number (baryon or lepton number) associated with fermions. There are two unusual features. Firstly, there is a class of supermultiplets in which the spinor particle with unit fermion number is accompanied by a scalar particle which carries two units of fermion number. Only in a very restricted class of theories can the appearance of such difermions be avoided. Secondly, the fermion number content of a supermultiplet is governed by its chiral type. Thus, with the conventions adopted here, it is shown that, in scalar supermultiplets, right-handed spinor fermions are associated with ordinary scalar bosons while left-handed spinor fermions are associated with di-fermionic bosons. In gauge supermultiplets ordinary vector bosons are associated with left-handed spinor fermions. In consequence of this peculiar multiplet structure, the problem of defining a conserved parity is non-trivial. It is shown that parity doubling provides no solution unless either supersymmetry or fermion number is violated explicitly. In fact, the solution to the parity problem makes essential use of local symmetries. A class of renormalizable supersymmetric Lagrangians which conserve both fermion number and parity is given. For this class the interactions are governed by a gauge principle, and it turns out that they realize a complex extension of the original supersymmetry.