The power of perturbation theory

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

Hyperasymptotics for integrals with saddles

Integrals involving exp { –kf(z)}, where |k| is a large parameter and the contour passes through a saddle of f(z), are approximated by refining the method of steepest descent to include exponentially small contributions from the other saddles, through which the contour does not pass. These contributions are responsible for the divergence of the asymptotic expansion generated by the method of steepest descent. The refinement is achieved by means of an exact ‘resurgence relation', expressing the original integral as its truncated saddle-point asymptotic expansion plus a remainder involving the integrals through certain ‘adjacent’ saddles, determined by a topological rule. Iteration of the resurgence relation, and choice of truncation near the least term of the original series, leads to a representation of the integral as a sum of contributions associated with ‘multiple scattering paths’ among the saddles. No resummation of divergent series is involved. Each path gives a ‘hyperseries’, depending on the terms in the asymptotic expansions for each saddle (these depend on the particular integral being studied and so are non-universal), and certain ‘hyperterminant’ functions defined by integrals (these are always the same and hence universal). Successive hyperseries get shorter, so the scheme naturally halts. For two saddles, the ultimate error is approximately ∊2.386, where ∊ (proportional to exp (—A│k│) where A is a positive constant), is the error in optimal truncation of the original series. As a numerical example, an integral with three saddles is computed hyperasymptotically.<br/

lambda phi(4) theory - Part I. The symmetric phase beyond NNNNNNNNLO

Perturbation theory of a large class of scalar field theories in d &lt; 4 can be shown to be Borel resummable using arguments based on Lefschetz thimbles. As an example we study in detail the \u3bb\u3d54 theory in two dimensions in the Z2 symmetric phase. We extend the results for the perturbative expansion of several quantities up to N8LO 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