Modifying a renormalisable theory$.$ Apparently, if we take a certain renormalisable theory, then any modification consistent with the symmetries must render the theory non-renormalisable. Is this claim true? Has it been discussed rigorously in the literature?
 A: "Any modification" is pretty vague. The point is the following. Given a number of fields $\phi_I$ (that can be bosons, fermions, gauge fields, transforming under some internal symmetries etc.) you can write down a finite number of local, gauge-invariant operators of dimension $\leq 4$. Call these $\mathcal{O}_\alpha$. Then the most general renormalizable action reads
$$L = \text{kinetic terms} + \sum_{\alpha} g_\alpha \mathcal{O}_\alpha$$
where the couplings $g_\alpha$ have mass dimension $4 - \text{dimension of } \mathcal{O}_\alpha$.
You can of course write down many more operators, of dimension $5,6,\ldots$. In principle we can add such operators, with couplings $g'_\beta$, to the action as well. So a point in theory space is parametrized by an infinite vector $(g_\alpha, g'_\beta)$. The point is that the submanifold with $g'_\beta = 0$ corresponds to the set of renormalizable trajectories. This is what that David Bar Moshe was aiming at. In the case of Yang-Mills, there is only one coupling $g_\alpha = g_{YM}$, so any other operator you add to the action will destroy renormalizability.
This whole story is well-known to anyone in the field but may not be properly discussed in your university QFT class. The standard reference is "Renormalization and Effective Lagrangians" [NPB 213, 1984] by Polchinski.
This entire discussion is of course modulo field redefinitions, scheme choices etc. - there are various trivial ways to write an action in a different form, without changing physical predictions.
