quantum field theory - Gauge fermions versus gauge bosons

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:

\begin{equation} \bar{\psi} i \gamma ^\mu \partial _\mu \psi \end{equation} This term is not gauge invariant under the transformation, $ \psi \rightarrow e ^{ i T ^a \theta ^a (x) } \psi $, because of the derivative spoils the desired transformation of $ \partial _\mu \psi $. To fix this we must add a contribution that transforms in the same way as the derivative, i.e., transforms as a vector. In other words we modify the derivative such that, \begin{equation} D _\mu \psi \rightarrow e ^{ i T _a \theta _a (x) } D _\mu \psi \end{equation}

The question is what to add to $ D _\mu $. We can potentially add spin $ 0, \frac{1}{2} , 1 , \frac{3}{2} , $ 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 ($ \psi _a $ is a gauge particle, not $ \psi $), \begin{equation} D _\mu = \partial _\mu + \sum _a T ^a \left( g\bar{\psi} ^a \gamma _\mu \psi ^a + g ' \bar{\psi} ^a \gamma _\mu \gamma ^5 \psi ^a \right) \end{equation} However, this would give an interaction \begin{equation} \sum _a i\left[ \bar{\psi} \gamma ^\mu\psi \right] \left[ \bar{\psi} _a \gamma _\mu \psi _a \right] \end{equation} and similarly for the $ \gamma ^5 $ term. These interactions are non-renormalizable as they involve four fermions. Non-renormalizable interactions arise from effective field theories and are suppressed by the scale at which they arise. This would make the gauge interactions non-fundamental but instead involve a massive vector particle integrated out. For the integrated out interaction to be renormalizable it must be between two fermions and a spin $1$ field. This brings us back to the usual case.

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.