Perturbation theory of a large class of scalar field theories in d < 4 can be shown to be Borel resummable using arguments based on Lefschetz thimbles. As an example we study in detail the λφ 4 theory in two dimensions in the Z 2 symmetric phase. We extend the results for the perturbative expansion of several quantities up to N 8 LO and show how the behavior of the theory at strong coupling can be recovered successfully using known resummation techniques. In particular, we compute the vacuum energy and the mass gap for values of the coupling up to the critical point, where the theory becomes gapless and lies in the same universality class of the 2d Ising model. Several properties of the critical point are determined and agree with known exact expressions. The results are in very good agreement (and with comparable precision) with those obtained by other non-perturbative approaches, such as lattice simulations and Hamiltonian truncation methods.

Abstract:We study quantum mechanical systems with a discrete spectrum. We show that the asymptotic series associated to certain paths of steepest-descent (Lefschetz thimbles) are Borel resummable to the full result. Using a geometrical approach based on the PicardLefschetz theory we characterize the conditions under which perturbative expansions lead to exact results. Even when such conditions are not met, we explain how to define a different perturbative expansion that reproduces the full answer without the need of transseries, i.e. non-perturbative effects, such as real (or complex) instantons. Applications to several quantum mechanical systems are presented.

In quantum mechanics and quantum field theory perturbation theory generically requires the inclusion of extra contributions non-perturbative in the coupling, such as instantons, to reproduce exact results. We show how full non-perturbative results can be encoded in a suitable modified perturbative series in a class of quantum mechanical problems. We illustrate this explicitly in examples which are known to contain non-perturbative effects, such as the (supersymmetric) double-well potential, the pure anharmonic oscillator, and the perturbative expansion around a false vacuum.The coefficients of saddle-point expansions of (path) integrals generically grow factorially and produce nonconvergent asymptotic series. In special cases exact results can be obtained from such series by "Borel resummation", consisting in taking the Laplace transform of the function obtained by resumming the original series after dividing their terms by a factorially growing coefficient. However this procedure generically fails, because exact results take the form of trans-series, i.e. series in powers of the coupling λ, e −1/λ and log(−λ), which capture behaviours non-perturbative in the coupling λ.There are several well understood tools which determine the properties of perturbative series of finitedimensional integrals. For path integrals some results are available in specific cases (mostly within quantum mechanics or supersymmetric theories using localization techniques) but a systematic characterization of the behavior of their perturbation theories is still lacking. Quantum mechanics (QM) represents the simplest playground where to test our understanding of the interplay between perturbative and non-perturbative effects in path integrals, for exact results can be extracted efficiently by other means. In this letter we present a method to recover the full non-perturbative answer from a deformation of the perturbative series in a certain class of QM problems that contain non-perturbative effects. Such technique could be used to improve numerical computations based on perturbation theory in QM and might, in principle, be extended to higher dimensional quantum field theories (QFT).Consider one-dimensional quantum mechanical systems described by the Hamiltonianwhere the potential V (x; λ) is a regular function of x describing a bounded system (i.e. lim |x|→∞ V (x) = ∞). If the dependence on the coupling λ is such that V (x; λ) = V (x √ λ; 1)/λ then the perturbative expansion in λ coincides with the expansion. We call such potential classical. If V 0 (x; λ) and V 1 (x; λ) are two such potentials then the combination V (x; λ) = V 0 (x; λ) + λV 1 (x; λ) is a sum of a classical contribution (V 0 ) and a quantum one (V 1 ). Consider now the classical anharmonic oscillatorwhose energy eigenvalues at small λ are close to those of the harmonic oscillator E ao n = n + 1 2 + O(λ). By studying the analytic properties of the eigenvalues, their perturbative series has been shown to be Borel resummable to the exact result [1,2]. The conditions under whic...