Uniqueness of Yang-Mills theory

Question:

Is there any sense of uniqueness in Yang-Mills gauge field theories?

Details:

Let's say we are after the most general Lagrangian Quantum Field Theory of (possibly self-interacting) $N$ spin $j=1$ particles (and matter). Yang-Mills' construction is based on the following:

• 
  

  Pick a compact semi-simple Lie Group $G$ with $\dim G=N$, and introduce $N$ vector fields $A_\mu^a$, $a=1,\dots,N$. Then $$ F^a_{\mu\nu}\equiv 2\partial_{[\mu}A_{\nu]}^a+gf^{abc}A_\mu^b A_\nu^c $$

  
  
  


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  The Lagrangian is given by $$ \mathcal L=-\frac12\text{tr}(F^2)+\mathcal L_\mathrm{matter}(\psi,\nabla\psi)+\text{gauge-fixing} $$ where $\nabla\psi\equiv\partial\psi-ig T^a A^a$.

  
  

My question is about how unique this procedure is. For example, some questions that come to mind:

1. 
   

   Is $-\frac12\text{tr}(F^2)$ the most general Lagrangian $\mathcal L=\mathcal L(A^a_\mu)$ that leads to a consistent theory? or can we add new self interactions, and new free terms, without spoiling unitarity, covariance, or renormalisability?

   
   



2. 
   

   Is minimal coupling $\partial\to \nabla$ the most general introduction of interactions with the matter fields? or can we add non-minimal interactions without spoiling unitarity, covariance, or renormalisability?

   
   

In short: does Yang-Mills' construction lead to the most general Lagrangian that can accommodate the interactions of these spin $j=1$ particles consistently? This construction has many different ingredients, some of which can be motivated through geometric considerations, but I've never seen any claim about uniqueness.