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

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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
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.

This post imported from StackExchange Physics at 2016-05-31 07:24 (UTC), posted by SE-user CuriousOne
arxiv.org/abs/hep-ph/0510142

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

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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.

This post imported from StackExchange Physics at 2016-05-31 07:24 (UTC), posted by SE-user Thomas
answered Nov 1, 2015 by (720 points)
In the case of double well problems, the splitting is actually resultant of some new vacuum that have large structure in space. If we don't resolve small energy we have a degenerate vacuum, but if we resolve the small energy we can see that the vacuum is actually formed by a small tunneling through the barriers. In the case of Kondo problem, is a big screening cloud around the impurity that is not resolved by high energy (short distance physics) but for exponentially small energy (Kondo temperature), long distance physics is important and the true vacuum make the appearance.

This post imported from StackExchange Physics at 2016-05-31 07:24 (UTC), posted by SE-user Nogueira
What I'm saying is that this term is related to small structures in energy spectra near the vacuum. Very far the vacuum this term gives small error (in terms of percentage over the value), (instanton corrections?).

This post imported from StackExchange Physics at 2016-05-31 07:24 (UTC), posted by SE-user Nogueira
I'm giving an up vote ;)

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

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