Mathematical proof that $\exp(-1/|g|)$ is always related with formation of bound states through scales?

I know that this function ($g$ means coupling) is non-analytical in $g=0$, so this function is only appreciable under non-perturbative calculations, so is a non-perturbative phenomena. This function is present on many critical/cross temperatures like in Kondo problem and Superconductors. This functions happens in QCD, when we fix physical coupling equal one. Is always that: $$E=E_0\,e^{-\frac{1}{\rho |g|}}$$ or, replacing $|g|$, $g^2$ and $\rho$ is some density of state.

When we perceive (physically) that the perturbative series don't converge (like Dyson argument), we treat our series as assymptotic. If the series diverges as $n!$, we can use Borel summation and come up with some integration over a meromorphic function in $(0,\,\infty)$. After some calculation, the poles of this meromorphic function gives contributions like $e^{-\frac{1}{\rho |g|}}$.

From this site, seems to me that only instanton made this contribution (instanton corrections). But renormalons could give the same contribution (no?). Bound states, nearly-bound states and tunneling mechanisms that connect different nearly-bound states seems to me the reason of the appearance of this terms and the divergence of perturbative calculation. But is very interesting that this corrections added in perturbative calculations are very tiny, exponentially tiny,...a far scale,... the typical scale of the bound state or the width of a tunneling barrier that holds nearly-bound state.

In the physical examples that I gave, the Kondo temperature tells us the size of the cloud around the impurity, the QCD energy gives us the size of the proton, Cooper instability gives the size of the electron-electron pair, a QM double well problem gives the distance of the wells,...so on, so on. Always a formation of bound state through scales. Short distance plus small interactions giving long distance bounded states.

I came with this by physical intuition. Can someone can give a mathematical proof of that?

• I can't give a mathematical answer, but the physical answer is completely trivial: whenever something in a model diverges badly or has very difficult convergence problems like in this case that have to be mended, then the model is simply wrong. – CuriousOne Oct 27 '15 at 2:08
• arxiv.org/abs/hep-ph/0510142 – Count Iblis Oct 27 '15 at 4:42

1) $\exp(-1/g)$ is not necessarily related to bound states. In the standard QM double well problem it is the splitting, not the binding energy, that is $O(\exp(-1/g))$. In conformal field theories instantons can give $\exp(-1/g)$ effects even though there are no bound states at all.
2) Instantons are one source of $\exp(-1/g)$ effects, but there are others. You already mentioned renormalons in gauge theories. Also, the $\exp(-1/g)$ in BCS or the Kondo problem is not in any obvious way an instanton effect.
3) There is a folklore that $\exp(-1/g)$ is always related to some semi-classical configuration (like the instanton). There is no proof of this. For example, it is not known what classical field corresponds to the renormalon, anlthough there are some recent ideas.