Why are all the interactions particle of a gauge theory bosons. Are fermionic gauge particle fields somehow forbidden by the theory ?
Answer
The reason that the gauge particle must be a spin 1 gauge boson is because there aren't any renormalizable alternatives. To see this consider the Dirac Lagrangian:
ˉψiγμ∂μψ
The question is what to add to Dμ. We can potentially add spin 0,12,1,32, and 2 particles to fix this. We go case by case.
There is no combination of spin zero fields that transform as a vector without adding derivatives (adding derivative to fix the derivative covariance would take you in circles), thus we can't have a spin zero gauge boson.
Next consider adding a spin 1/2 gauge boson we could write (ψa is a gauge particle, not ψ), Dμ=∂μ+∑aTa(gˉψaγμψa+g′ˉψaγμγ5ψa)
The spin 1 field works well and exists in the SM. I'm not sure about the spin 3/2 field as I have no experience with working with such fields however, I presume it won't work for similar reasons. I also know that spin 2 fields must mediate gravitational fields and thus would give a nonsensible result.
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