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?

This post imported from StackExchange Physics at 2016-05-31 07:24 (UTC), posted by SE-user Nogueira