Journal of High Energy Physics

, 2012:170 | Cite as

Resurgence and trans-series in Quantum Field Theory: the \(\mathbb{C}{{\mathbb{P}}^{N-1 }}\) model



This work is a step towards a non-perturbative continuum definition of quantum field theory (QFT), beginning with asymptotically free two dimensional non-linear sigma-models, using recent ideas from mathematics and QFT. The ideas from mathematics are resurgence theory, the trans-series framework, and Borel-Écalle resummation. The ideas from QFT use continuity on \({{\mathbb{R}}^1}\times \mathbb{S}_L^1\), i.e., the absence of any phase transition as N → ∞ or rapid-crossovers for finite-N, and the small-L weak coupling limit to render the semi-classical sector well-defined and calculable. We classify semi-classical configurations with actions 1/N (kink-instantons), 2/N (bions and bi-kinks), in units where the 2d instanton action is normalized to one. Perturbation theory possesses the IR-renormalon ambiguity that arises due to non-Borel summability of the large-orders perturbation series (of Gevrey-1 type), for which a microscopic cancellation mechanism was unknown. This divergence must be present because the corresponding expansion is on a singular Stokes ray in the complexified coupling constant plane, and the sum exhibits the Stokes phenomenon crossing the ray. We show that there is also a non-perturbative ambiguity inherent to certain neutral topological molecules (neutral bions and bion-anti-bions) in the semiclassical expansion. We find a set of “confluence equations” that encode the exact cancellation of the two different type of ambiguities. There exists a resurgent behavior in the semi-classical trans-series analysis of the QFT, whereby subleading orders of exponential terms mix in a systematic way, canceling all ambiguities. We show that a new notion of “graded resurgence triangle” is necessary to capture the path integral approach to resurgence, and that graded resurgence underlies a potentially rigorous definition of general QFTs. The mass gap and the Θ angle dependence of vacuum energy are calculated from first principles, and are in accord with large-N and lattice results.


Field Theories in Lower Dimensions Solitons Monopoles and Instantons Nonperturbative Effects Renormalization Regularization and Renormalons 


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Copyright information

© SISSA, Trieste, Italy 2012

Authors and Affiliations

  1. 1.Department of PhysicsUniversity of ConnecticutStorrsU.S.A.
  2. 2.Department of Physics and Astronomy, SFSUSan FranciscoU.S.A.

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