Understand
Perturbation series in quantum field theory are generally divergent asymptotic series which are also typically not Borel resummable in the sense that the resummed series is ambiguous.
- The ambiguity is associated with singularities in the Borel plane on the positive real axis.
- In quantum mechanics there are cases in which the ambiguity that arises in perturbation theory cancels against a similarly ambiguous contribution from instanton--anti-instanton events.
- In asymptotically free gauge theories this mechanism does not suffice because perturbation theory develops ambiguities associated with singularities in the Borel plane which are closer to the origin by a factor of about N (the rank of the gauge group) compared to the singularities realized by instanton events.
Reading the bibliography…