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When testing a theory for its renormalizability, in practice one always calculates the mass dimension of the coupling constants $g_i$. If $[g_i]<0$ for any $i$ the theory is not renormalizable. I am wondering where this criterion/trick comes from? Is there an easy way to see that a coupling constant with negative mass dimension will yield a non-renormalizable theory?

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Yes, suppose $[g] = \delta$. By dimensional analysis only we can write that a loop diagram contributes $$ \sim g^{n} \int \frac{d^4 k}{k^{4-n\delta}} $$ If $\delta=0$, this diverges logarithmically, but can be re-normalized. If $\delta$ is less than zero, it diverges by simple power counting.

This is VERY informal. Technically, you should study the superficial degree of divergence of a diagram. But that's called superficial for a reason. So for the whole story I think you need Weinberg's theorem, which is a rule for telling exactly if a diagram diverges.

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