# Why is gravitino a spin 3/2 particle?

## Main Question or Discussion Point

Graviton is a spin 2 particle. Why is its superpartner gravitino a spin 3/2 (2-1/2) particle and not a spin 5/2 (2+1/2) particle?

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blechman
one point is that there is no known way to write down an interacting theory with massless fields of spin higher than 2. you might not be impressed by that argument, but it matters if we want to actually calculate anything!

another (perhaps more robust) point is that the gravitino field plays the role of a "gauge connection" for the "gauged supersymmetry" that is supergravity. That is: the gravitino couples to the supercurrent, which is a vector-spinor, and must therefore be a vector-spinor itself if it's to couple in a Poincare-invariant way. Such an object is a combination of spin-1/2 and spin-3/2 fields, no spin-5/2 (and we project out the spin-1/2 in supergravity for physical reasons).

Graviton is a spin 2 particle. Why is its superpartner gravitino a spin 3/2 (2-1/2) particle and not a spin 5/2 (2+1/2) particle?
Well we can ask the same question about fermions (1/2-1/2 and not 1/2+1/2) or electroweak gauge bosons..
the point is that it is "simpler" to construct supermultiplets with particles and sparticles that respect this scheme (s-1/2 and not s+1/2).

Think about the Chiral supermultiplet. we have a fermion with two degrees of freedom, so the simplest bosonic thing to add is a complex scalar field..

blechman
Well we can ask the same question about fermions (1/2-1/2 and not 1/2+1/2) or electroweak gauge bosons..
the point is that it is "simpler" to construct supermultiplets with particles and sparticles that respect this scheme (s-1/2 and not s+1/2).

Think about the Chiral supermultiplet. we have a fermion with two degrees of freedom, so the simplest bosonic thing to add is a complex scalar field..
I don't agree with this logic. Nowhere does it say nature has to be "simple"! And what about the Higgs, that goes against this rule?

The reason why we don't have spin-1 partners for the fermions is that spin-1 particles have a gauge symmetry associated with them (they HAVE to if you want to satisfy unitary conditions) and gauge particles must be in the adjoint (real) representation. However, the fermions (like the top quark) are in complex representations, so this is a contradiction. This is why the standard model fermion SUSY partners are scalars and not vectors. It has nothing to do with the "easiness" of the model.

Similarly, the only way to couple spin-3/2 is as the "gauge field of local supersymmetry" which requires a graviton, as hinted to above. that's why the gauginos are not spin-3/2 - such a representation cannot have interactions!

That being said: there are ways to generalize the MSSM to have vector partners for the fermions, but they are quite complicated. For example, take a look at http://arxiv.org/abs/0806.0386