Fast-convergent resummation algorithm and critical exponents of φ4-theory in three dimensions

Abstract

We develop an efficient algorithm for evaluating divergent perturbation expansions of field theories in the bare coupling constant g B for which we possess a finite number L of expansion coefficients plus two more informations: The knowledge of the large-order behavior proportional to (−α) k k!k β g k B , with a known growth parameter α, and the knowledge of the approach to scaling being of the type c + c ′ /g ω B , with constants c, c ′ and a critical exponent of approach ω. The latter information leads to an increase in the speed of convergence and a high accuracy of the results. The algorithm is applied to the six-and seven-loop expansions for the critical exponents of O(N )-symmetric φ 4 -theories, and the result for the critical exponent α is compared with the recent satellite experiment.

Critical exponents and equation of state of the three-dimensional Heisenberg universality class

“…In particular, for small L min , we see a clear dependence of the result for ν on λ. For instance, for L min = 8 the difference between the result for λ = 4.0 and λ = 5.0 is 0.00176 (46).…”

“…However, the reliability of the error bars reported in Ref. [46] is unclear: indeed, Ref. [47] analyzes the same perturbative series and reports much more cautious error estimates.…”

“…The FT results of Refs. [46][47][48][49][50][51] were derived by analyzing perturbative expansions in different frameworks: fixed-dimension expansion (6th-and 7th-order series, see Refs. [58,48]), ǫ-expansion (to O(ǫ 5 ), see Refs.…”

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

“…[41,[54][55][56], where the authors used it as part of an algorithm to find convergent strong coupling expansions (from divergent weak coupling ones). Similarly, in Refs.…”

Order of superconductive phase transition

“…This theory has been developed and discussed in detail in [8,[26][27][28][29][30] and it has proven to be a powerful tool for determining critical exponents in three [28] as well as in 4 − ε dimensions [30] of pure φ 4 −theories. Variational perturbation theory is a procedure which allows us to determine resummed quantities from a strong-coupling limit of divergent expansions in powers of the bare coupling constant.…”