Continuity and Resurgence: towards a continuum definition of the CP(N-1) model

Abstract: We introduce a non-perturbative continuum framework to study the dynamics of quantum field theory (QFT), applied here to the CP(N-1) model, using Ecalle's theory of resurgent trans-series, combined with the physical principle of continuity, in which spatial compactification and a Born-Oppenheimer approximation reduce QFT to quantum mechanics, while preventing all intervening rapid cross-overs or phase transitions. The reduced quantum mechanics contains the germ of all non-perturbative data, e.g., mass gap, of the QFT, all of which are calculable. For CP(N-1), the results obtained at arbitrary N are consistent with lattice and large-N results. These theories are perturbatively non-Borel summable and possess the elusive IR-renormalon singularities. The trans-series expansion, in which perturbative and non-perturbative effects are intertwined, encapsulates the multi-length-scale nature of the theory, and eliminates all perturbative and non-perturbative ambiguities under consistent analytic continuation of the coupling. We demonstrate the cancellation of the leading non-perturbative ambiguity in perturbation theory against the ambiguity in neutral bion amplitudes. This provides a weak-coupling interpretation of the IR-renormalon, and a theorem by Pham et al implies that the mass gap is a resurgent function, for which resummation of the semi-classical expansion yields finite exact results.