Why must gluinos be spin 1/2 instead of 3/2?

Question

Is there some condition in the N=1 SUSY algebra telling that the spin of the superpartners of gauge bosons (either for colour or for electroweak) must be less than the spin of the gauge boson? I am particularly puzzled because sometimes a supermultiplet is got from sugra that contains one spin 2 particle, four spin 3/2, and then some spin 1, 1/2 and 0. If this supermultiplet is to be broken to N=1 it seems clear that the graviton will pair with the gravitino and the rest of spin 3/2 should pair with spin 1, so in this case it seems that a superpair (3/2, 1) is feasible. Why not in gauge supermultiplets?

Answer 1

Fermionic spin 3/2 fields, much like bosonic fields with spin 1 and higher, contain negative-norm polarizations. Roughly speaking, a spin 3/2 field is $R_{\mu a}$ where $\mu$ is a vector index and $a$ is a spinor index. If $\mu$ is chosen to be 0, the timelike direction, one gets components of the spintensor $R$ that creates negative-norm excitations.

This is not allowed to be a part of the physical spectrum because probabilities can't be negative. It follows that there must be a gauge symmetry that removes the $R_{0a}$ components - a spinor of them. The generator of this symmetry clearly has to transform as a spinor, too. There must be a spinor worth of gauge symmetry generators. The generators are fermionic because the original field $R$ is also fermionic, by the spin-statistics relation.

It follows that the conserved spinor generators are local supersymmetry generators and their anticommutator inevitably includes a vector-like bosonic symmetry which has to be the energy-momentum density. This completes the proof that in any consistent theory, spin 3/2 fields have to be gravitinos. The number of "minimal spinors" - the size of the gravitinos - has to be equal to the number of supercharges spinors which counts how much the local supersymmetry algebra is extended. In particular, it can't be linked to another quantity such as the dimension of a Yang-Mills group.

So while both 3/2 and 1/2 differ by 1/2 from $j=1$, more detailed physical considerations show that it is inevitable for the superpartner of a gauge boson, a gaugino, to have spin equal to 1/2 and not 3/2. Similarly, one can show that the superpartner of the graviton can't have spin 5/2 because that would require too many conserved spin-3/2 fermionic generators which would make the S-matrix essentially trivial, in analogy with the Coleman-Mandula theorem. Gravitinos can only have spin 3/2, not 5/2.