\u03bb\u03d54 theory \u2014 Part I. The symmetric phase beyond NNNNNNNNLO

Marco Serone,Giovanni Villadoro,Gabriele Spada

doi:10.1007/jhep08(2018)148

Abstract

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 λϕ4 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.

Highlights

Expansions that are Borel resummable to the exact result in theories, such as the quantum mechanical symmetric double well, where ordinary perturbation theory is not
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
In this paper we have shown, using arguments based on Lefschetz thimbles in the spirit of refs. [3, 4], that the Schwinger functions in a large class of Euclidean scalar field theories in d < 4 are Borel reconstructable from their loopwise expansion

Summary

Introduction

Expansions that are Borel resummable to the exact result in theories, such as the quantum mechanical symmetric double well, where ordinary perturbation theory is not. The Borel summability of Schwinger functions in this class of theories is deduced by a change of variables in the path integral and the absence of non-trivial positive finite actions solutions to the classical equations of motion. This is equivalent to establishing that the original path integral over real field configurations coincides with a single Lefschetz thimble. Using certain bounds and analytic properties of the Schwinger functions, the perturbation series of arbitrary correlation functions had already been rigorously proved to be Borel resummable, though only for parametrically small coupling constant and large positive squared mass, i.e. for g 1 [6]

Methods

Results

Conclusion