Minimally subtracted six-loop renormalization of O(n) -symmetric ϕ4 theory and critical exponents

Abstract

We present the perturbative renormalization group functions of O(n)-symmetric φ 4 theory in 4 − 2ε dimensions to the sixth loop order in the minimal subtraction scheme. In addition, we estimate diagrams without subdivergences up to 11 loops and compare these results with the asymptotic behaviour of the beta function. Furthermore we perform a resummation to obtain estimates for critical exponents in three and two dimensions.

Fast summation of divergent series and resurgent transseries from Meijer- G approximants

“…In recent work, Kompaniets and Panzer (KP) [10] give the six-loop order β function, mass and field anomalous dimension, and critical exponents of the N-vector model in 4-2ϵ dimensions. Besides adding one extra coefficient to previously computed expansions, their work is an exercise in state-of-the-art practical resummation as well as an excellent compilation of reference results for this important model [34,66].…”

“…In this sense it is worth pointing out the work of Kompaniets and Panzer [10] that epitomizes state-of-theart resummation in the face of the incomplete information represented by the several coefficients that can actually be computed: these coefficients generate a vast approximant space; additional information (such as minimal sensitivity, known bounds on the function and its derivatives) needs to be used in order to narrow down the approximant space toward the physical result. We emphasize that conformal mapping technique and variational perturbation theory can reach high accuracy, but lack the algorithmic simplicity that make techniques such as Padé and Borel-Padé so popular and widely used.…”

“…However, the conventional Borel-Padé-Écalle resummation used for this purpose typically requires a large number of terms [23,29] from each sector in order to obtain reasonably accurate results beyond the weak coupling regime. This makes it useless for problems in QFT [4,10] or many-body perturbation theory in condensed matter physics [14] where ≲10 orders are available at best. In such cases the Borel-Padé (BP) technique can be used together with a conformal transformation in the Borel plane [1,10,[30][31][32][33][34][35][36][37]].…”

“…This is exemplified by the asymptotic series for the LoSurdo-Stark effect of atoms and molecules where first-and second-order terms match measurements well, but only for very weak electric fields [3], or by high precision calculations of multiloop Feynman diagrams in quantum electrodynamics [4]where the fine-structure constant is small. On the other hand, extracting physically relevant information from the asymptotic series at larger coupling constants calls, almost invariably, for resummation techniques as exemplified by the LoSurdo-Stark effect [5,6] and field assisted excitonic ionization in layered materials [7], anharmonic oscillators in quantum mechanics [8,9], ϕ 4 theory in QFT [10], quantum chromodynamics [11], string perturbation theory [12,13], and diagrammatic Monte Carlo techniques in condensed matter physics [14]. Given the ubiquity of divergent series in physics, research on summation techniques remains an active research area [15,16].…”

“…[29] needed tens of terms to find a good approximation at intermediate coupling strengths. In practice only a handful of expansion coefficients are typically available: for instance the ϵ expansion for the OðnÞ-symmetric ϕ 4 theory is only known to six-loop order [10], while the five-loop QCD beta function and anomalous dimension have been calculated only very recently [40]. It is then clear that, for a summation technique to be practical, it needs to be able to return accurate estimates of the sum of a divergent series with only a few coefficients.…”