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It is common wisdom that a gluon propagator (Gribov-)like $$ G(p^2)=\frac{a+bp^2}{cp^4+dp^2+e} $$ should give rise to a linear rising potential. So far, I have not seen a proof of this and I would like to get a mathematical derivation, or a reference, displaying how a potential emerges from it.


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As far as I know it doesn't. When people say that a Gribov-type propagator is "confining", they normally mean that a gluon with such a propagator cannot be an asymptotic state (because there is no real pole in the propagator), which is then interpreted as the gluon being removed from the spectrum (by being bound in glueballs for example).

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