The irreducible representations of supersymmetry combined withSU N

Abstract

Supersymmetry introduced by W~ss and ZUMINO (1,2) s e e m s to offer an interesting possibility to classify hadronic states because the multiplets of supersymmetry (supermultiplets) contain fermions and bosons. No-go theorems like that proven by COL~MAZ~ and MANDULA (3) do not apply due to the fact that the supersymmetry algebra involves anticommutators besides commutators. On the other hand, the supersymmetry algebra implies that particles within the same multiplct have equal mass. Therefore supersymmetry is (at best) a broken symmetry. In order to classify the hadrons into supermultiplets one has to find the irreducible representations of the supersymmetry algebra. SALAME and STRATHDEE (4) have constructed a set of irreducible representations and we have shown in earlier note (5) tha t ~his set contains already all irreducible representations. The supermultiplets contain in general four particles. Two of them have spin j, say, and different parity, one has spin j § 89 and the last one has spin j 89 However since these multiplets are so small and supersymmetry is broken considerably it seems hard to classify the hadrons into these multiplets in any convincing manner. Therefore it seems tempting to combine supersymmetry with an internal symmetry in a nontrivial way to obtain larger multiplets. There are several possibilities for doing this. A natural choice of the internal symmetry would be S U2, S U 3 or S U4 if one believes in charm. The coupling of supersymmetry can be performed in the way indicated in Sect. 7 of (2). In this note we show that one obtains all irreducible representations of this algebra (internal symmetry S U ~ ) by applying the construction of SALA~ and STRATHD~ t~ all irreducible representations of S U 2 • S U ~ . The original supersymmetry algebra (without an additional internal symmetry) is generated by the generators of the Poincard group J ~ , Pa and a l~lajorana spinors S~