# Do perturbative renormalization groups help one understand when perturbation theory can be used in general?

If, as I asked in this question, a relevant operator in a renormalization group transformation can't be used in a perturbative expansion since it becomes large as the transformations are applied, does this mean that the operator can't be used in 'normal' perturbation theory?

I.e. Is using the renormalization group a way to determine whether or not perturbation theory can be used at all? Or is it only relevant within renormalization groups since without it, the operator remains small?

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Renormalization group is usually used because of some divergence your are avoiding in perturbation theory(such as the divergence of your greens function). In perturbation theory your still expanding in terms of the small parameter before you apply the renormalization group action. Therefore your expansion is still valid.

Also just because your parameter is relevant doesn't mean it diverges and invalidates renormalization group. Since RG still calculates the flow perturbativly, even if at one order your relevant parameter seems to diverge, higher order calculations may add bifurcations to your flow equations that have your parameter flow to a different fixed point as apposed to diverging to infinity.

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