Continuity and Resurgence: Towards a continuum definition of theCP(N−1)model

Abstract

We introduce a nonperturbative continuum framework to study the dynamics of quantum field theory (QFT), applied here to the $CP(N\ensuremath{-}1)$ model. We show that the ambiguities in perturbation theory due to infrared renormalons are exactly canceled by corresponding ambiguities in the nonperturbative sector coming from amplitudes of certain nonperturbative objects: neutral bions and bion-antibions. This provides an explicit weak-coupling interpretation of the IR-renormalons. We use \'Ecalle's theory of resurgent trans-series and the physical principle of continuity to continuously connect QFT to quantum mechanics, while preventing all intervening rapid crossovers or phase transitions. The quantum mechanics contains the germ of all nonperturbative data, e.g., mass gap, of the QFT, all of which are calculable. For $CP(N\ensuremath{-}1)$, the results obtained at arbitrary $N$ are consistent with lattice and large-$N$ results. The trans-series expansion, in which perturbative and nonperturbative effects are intertwined, encapsulates the multilength-scale nature of the theory, and eliminates all perturbative and nonperturbative ambiguities under consistent analytic continuation of the coupling. A theorem by Pham et al. implies that the mass gap is a resurgent function, for which resummation of the semiclassical expansion yields finite exact results in the weakly coupled domain.