The renormalization group and ultraviolet asymptotics

Vladimirov, Alexey; Ширков, Дмитрий Васильевич

doi:10.1070/pu1979v022n11abeh005644

Cited by 64 publications

(98 citation statements)

References 3 publications

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“…Such a comparison is useful for orientation, but full coincidence of the results should not be expected. It is known that the renormalization group functions depend on the choice of the renormalization scheme [15] and only observables (critical exponents) are invariant. Result (20) is valid for field theoretical model (1) with the isotropic momentum cutoff |k| < Λ, whereas hightemperature series are constructed for lattice models, where the effective momentum cutoff is anisotropic.…”

Section: Function β(G)mentioning

confidence: 99%

“…The coefficients U N for N ≤ N av ≈ 20 are directly calculated by Eq. (14) and are then continued by power law (15) in order to avoid the catastrophic increase of errors [7,12]. Thus, the consistent implementation of the algorithm necessarily requires the determination of the strong-coupling asymptotic behavior for W(g) [see Eq.…”

Section: Initial Information and Summation Proceduresmentioning

confidence: 99%

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Renormalization group functions for two-dimensional phase transitions: To the problem of singular contributions

Pogorelov

Suslov

2007

J. Exp. Theor. Phys.

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-According to the available publications, the field theoretical renormalization group approach in the two-dimensional case gives the critical exponents that differ from the known exact values. This property is associated with the existence of nonanalytic contributions in the renormalization group functions. The situation is analysed in this work using a new algorithm for summing divergent series that makes it possible to determine the dependence of the results for the critical exponents on the expansion coefficients for the renormalization group functions. It has been shown that the exact values of all the exponents can be obtained with a reasonable form of the coefficient functions. These functions have small nonmonotonities or inflections, which are poorly reproduced in natural interpolations. It is not necessary to assume the existence of singular contributions in the renormalization group functions.

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Section: Function β(G)mentioning

confidence: 99%

Section: Initial Information and Summation Proceduresmentioning

confidence: 99%

Renormalization group functions for two-dimensional phase transitions: To the problem of singular contributions

Pogorelov

Suslov

2007

J. Exp. Theor. Phys.

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“…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]]. …”

Section: Introductionmentioning

confidence: 99%

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

2018

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We develop a resummation approach based on Meijer-G functions and apply it to approximate the Borel sum of divergent series and the Borel-Écalle sum of resurgent transseries in quantum mechanics and quantum field theory (QFT). The proposed method is shown to vastly outperform the conventional Borel-Padé and Borel-Padé-Écalle summation methods. The resulting Meijer-G approximants are easily parametrized by means of a hypergeometric ansatz and can be thought of as a generalization to arbitrary order of the Borel-hypergeometric method [Mera et al., Phys. Rev. Lett. 115, 143001 (2015)]. Here we demonstrate the accuracy of this technique in various examples from quantum mechanics and QFT, traditionally employed as benchmark models for resummation, such as zero-dimensional ϕ 4 theory; the quartic anharmonic oscillator; the calculation of critical exponents for the N-vector model; ϕ 4 with degenerate minima; self-interacting QFT in zero dimensions; and the summation of one-and two-instanton contributions in the quantum-mechanical double-well problem.

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“…The main physical result of this study consists in reconstructing the Gell-Mann-Low function of the ϕ 4 theory (Section 8). The task is solved proceeding from the same information as that used in [13], namely, the first four coefficients of expansion of the β(g) function in the subtraction scheme [15,18] (5) and their asymptotics according to Lipatov, taking into account the first-order correction [19]:…”

Section: Introductionmentioning

confidence: 99%

Summing divergent perturbative series in a strong coupling limit. The Gell-Mann-Low function of the ϕ4 theory

Suslov

2001

J. Exp. Theor. Phys.

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-An algorithm is proposed for determining asymptotics of the sum of a perturbative series in the strong coupling limit using given values of the expansion coefficients. Application of the algorithm is illustrated, methods for estimating errors are developed, and an optimization procedure is described. Applied to the ϕ 4 theory, the algorithm yields the Gell-Mann-Low function asymptotics of the type β ( g ) ≈ 7.4 g 0.96 for large g . The fact that the exponent is close to unity can be interpreted as a manifestation of the logarithmic branching of the type β ( g ) ~ g (ln g ) -γ (with γ ≈ 0.14), which is confirmed by independent evidence. In any case, the ϕ 4 theory is self-consistent. The procedure of summing perturbative series with arbitrary values of the expansion parameter is discussed.

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The renormalization group and ultraviolet asymptotics

Cited by 64 publications

References 3 publications

Renormalization group functions for two-dimensional phase transitions: To the problem of singular contributions

Renormalization group functions for two-dimensional phase transitions: To the problem of singular contributions

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

Summing divergent perturbative series in a strong coupling limit. The Gell-Mann-Low function of the ϕ4 theory

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