Semiclassical origin of renormalons in asymptotically free theories
In plain words
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.
Precise statement
In asymptotically free theories (4D Yang-Mills and QCD, 2D $O(N)$, $CP^{N-1}$ and principal chiral models), the Borel transform of perturbation series has renormalon singularities at positions set by the one-loop beta-function coefficient, not matched by any finite-action classical solution. Identify the nonperturbative sectors (saddles, fractional instantons, bions on R^{d-1} x S^1, or OPE condensates) whose imaginary ambiguities cancel the renormalon ambiguities, and demonstrate the cancellation.
What would settle it
An explicit computation in at least one asymptotically free theory showing order-by-order cancellation between the renormalon ambiguity and an identified nonperturbative sector.
Status in the literature
Unverified note
Bion proposals (2012) were questioned when their ambiguities did not match renormalon positions in compactified $\mathrm{CP}^{N-1}$ and $\mathrm{O}(N)$ models (2019-2024).
Related problems
- More general than Does QCD perturbation theory plus condensates resum to the exact answer
- More general than Semiclassical origin of the O(N) model trans-series sectors