  From 't Hooft's argument, one expects that the analyticity domain of an
asymptotically free quantum field theory is horned shaped. In the usual Borel
summation, the function is obtained through a Laplace transform and thus has a
much larger analyticity domain. However, if the summation process goes through
the process called acceleration by Ecalle, one obtains such a horn shaped
analyticity domain. We therefore argue that acceleration, which allows to go
beyond standard Borel summation, must be an integral part of the toolkit for
the study of exactly renormalisable quantum field theories. We sketch how this
procedure is working and what are its consequences.
