λϕ4 theory — Part I. The symmetric phase beyond NNNNNNNNLO

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 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.

Renormalization scheme dependence, RG flow, and Borel summability in ϕ4 theories in d<4

“…The Borel summability of the fixed dimension perturbative series in the φ 4 theory at parametrically small coupling and in the unbroken phase m 2 > 0 has been established long ago [4,5] (see also ref. [6]) and recently extended for finite values of the coupling and to a more general class of scalar field theories, including for instance d = 3 N -component φ 4 theories with both m 2 > 0 and m 2 < 0, using steepest descent arguments [7,8]. The Borel summability of the -expansion in the φ 4 theory remains instead to be proven.…”

“…As such, there are no RS-independent logs to be resummed and the β-function of the theory is entirely given by RS-dependent terms. Ironically, the results of both the early works [4,5] and the recent ones [7,8] are based on a "minimal" RS, equivalent to an operatorial normal ordering prescription in d = 2, where the β-function is trivial to all orders in perturbation theory! We start in section 2 by reviewing known facts about RS-dependence of relevant (and irrelevant) couplings.…”

“…[3], based on a properly defined β-function, and recall in subsection 2.2 how the critical regime can be analyzed in minimal RSs (already used in ref. [7] for the study of the d = 2 φ 4 theory) with no need of RG techniques.…”

“…[4,5] and ref. [7], emphasizing their RS-dependence. In particular, we show that the arguments of ref.…”

“…In particular, we show that the arguments of ref. [7] do not straightforwardly apply in RSs such as the one used in ref. [3] based on momentum subtraction, essentially because they lead to counterterms that have an infinite number of contributions in perturbation theory and involve quartic φ 4 terms that dominate the path integral for large field configurations.…”