# Perturbative series in physics: why are coeffcieints of Gevrey-1 type (i.e. bounded by $\alpha C^n(n!)^1$

I have only been able to find this explicitly mentioned in this paper on resurgence techniques in physics. And have chased up the hints it gives, but they are not very explanatory.

Essentially, the question at hand is: why are are pertubative series in physics (always?) of Gevrey-1 type, meaning that the growth of the coefficients, $$a_n$$ is bounded as $$a_n \leq \alpha C^n(n!)^1$$, using the notation from the ref. The paper refers to 'arguments of Dyson and Lipatov', which essentially amount to considering the source of divergences in standard perturbation theory (instantons and renormalons) from the combinatoric growth of Feynman diagrams (giving an n! component) and IR/UV divergences.

While I agree that these are the mpst common sources of divergences in standard QFT, I am yet to find a convincing reason why the coefficients in an impotant theory might not go as $$~ (n!)^2$$ say, in which case the Borel transformed series would still have a vanishing radius of convergence. It is not nbelievable to me that sources of factorial divergences might not be multip[licative, making the coefficients of Gevrey-2 type!

• Minor comment to the post (v1): In the future please link to abstract pages rather than pdf files. – Qmechanic Mar 27 at 17:39