Resurgence and the meaning of perturbation series

4 problems

• Does QCD perturbation theory plus condensates resum to the exact answer

  Open

  QCD predictions combine a divergent power series with small corrections from vacuum condensates. Whether this combination, properly resummed, is complete and unambiguous is unknown.

• Semiclassical origin of the O(N) model trans-series sectors

  Partially resolved

  For solvable two-dimensional theories, the full set of exponentially small corrections to the divergent series has been extracted from the exact solution. Which field configurations of the theory produce each of these corrections is unknown.

• Semiclassical origin of renormalons in asymptotically free theories

  Contested

  The factorial growth of perturbation series in QCD-like theories is linked to singularities called renormalons, which no known classical solution explains. What nonperturbative object cancels their ambiguity is unknown.

• Nonperturbative completion of the string genus expansion

  Open

  String theory computes processes as a sum over surfaces with more and more holes, and the sum diverges like factorials. Which nonperturbative effects complete it, and whether the result is unique, is known only in toy models and special settings.