Four-loop Approximation Research Articles

On loop corrections to integrable 2D sigma model backgrounds

We study regularization scheme dependence of β-function for sigma models with two-dimensional target space. Working within four-loop approximation, we conjecture the scheme in which the β-function retains only two tensor structures up to certain terms containing ζ3. Using this scheme, we provide explicit solutions to RG flow equation corresponding to Yang-Baxter- and λ-deformed SU(2)/U(l) sigma models, for which these terms disappear.

Critical behaviour of a O(n)-symmetric model with antisymmetric tensor order parameter: Four-loop approximation

Critical behaviour of a O(n)-symmetric model with antisymmetric tensor order parameter: Four-loop approximation

Erratum to: On the relation between pole and running heavy quark masses beyond the four-loop approximation

Citations in the original article are corrupted due to error made by Editors QUARKS-2018. See the Full text with correct citations in PDF file.

On the relation between pole and running heavy quark masses beyond the four-loop approximation

The effective charges motivated method is applied to the relation between pole and M̅S̅-scheme heavy quark masses to study high order perturbative QCD corrections in the observable quantities proportional to the running quark masses. The non-calculated five- and six-loop perturbative QCD coefficients are estimated. This approach predicts for these terms the sign-alternating expansion in powers of number of lighter flavors nl, while the analyzed recently infrared renormalon asymptotic expressions do not reproduce the same behavior. We emphasize that coefficients of the quark mass relation contain proportional to π2 effects, which result from analytical continuation from the Euclidean region, where the scales of the running masses and QCD coupling constant are initially fixed, to the Minkowskian region, where the pole masses and the running QCD parameters are determined. For the t-quark the asymptotic nature of the non-resummed PT mass relation does not manifest itself at six-loops, while for the b-quark the minimal PT term appears at the probed by direct calculations four-loop level. The recent infrared renormalon based studies support these conclusions.

Calculation of critical index η of the φ3-theory in four-loop approximation by the conformal bootstrap technique

The method of calculation of [Formula: see text]-expansion in model of scalar field with [Formula: see text]-interaction based on conformal bootstrap equations is proposed. This technique is based on self-consistent skeleton equations involving full propagator and full triple vertex. Analytical computations of the Fisher’s index [Formula: see text] are performed in four-loop approximation. The three-loop result coincides with one obtained previously by the renormalization group equations technique based on calculation of a larger number of Feynman diagrams. The four-loop result agrees with its numerical value obtained by other authors.

Representation of the β-function and anomalous dimensions by nonsingular integrals: Proof of the main relation

A method for calculating the β-function and anomalous dimensions, convenient for numerical calculations in the ɛ-expansion framework, was previously proposed, and the relation underlying the method was verified up to the four-loop approximation. We prove this relation in all orders of the perturbation theory.

Heavy quark masses from sum rules in four-loop approximation

New data for the total cross section σ ( e + e − → hadrons ) in the charm and bottom threshold region are combined with an improved theoretical analysis, which includes recent four-loop calculations, to determine the short distance MS ¯ charm and bottom quark masses. A detailed discussion of the theoretical and experimental uncertainties is presented. The final result for the MS ¯ -masses, m c ( 3 GeV ) = 0.986 ( 13 ) GeV and m b ( 10 GeV ) = 3.609 ( 25 ) GeV , can be translated into m c ( m c ) = 1.286 ( 13 ) GeV and m b ( m b ) = 4.164 ( 25 ) GeV . This analysis is consistent with but significantly more precise than a similar previous study.

Analytic Invariant Charge in QCD with Suppression of Nonperturbative Contributions at Large Q 2

Based on the analytic invariant charge obtained from the results of the standard perturbation theory up to the four-loop approximation, we construct a “synthetic” model of the invariant charge in quantum chromodynamics. In the suggested model, the perturbative discontinuity on the timelike semiaxis in the complex Q2 plane is preserved, and nonperturbative contributions not only cancel nonphysical perturbation theory singularities in the infrared region but also rapidly decrease in the ultraviolet region. On one hand, the effective coupling function in this model is enhanced at zero (the dual superconductivity property of the quantum chromodynamics vacuum); on the other hand, a dynamical gluon mass appears. In our approach, fixing the parameter corresponding to the string tension parameter and normalizing (for example, at the point Mτ) entirely fix the synthetic invariant-charge model. The dynamical gluon mass mg is then fixed and is stable as the number of loops of the original perturbative approximation increases.

Critical thermodynamics of two-dimensional N-vector cubic model in the five-loop approximation

The critical behavior of the two-dimensional N-vector cubic model is studied within the field-theoretical renormalization-group (RG) approach. The beta-functions and critical exponents are calculated in the five-loop approximation, RG series obtained are resummed using Pade-Borel-Leroy and conformal mapping techniques. It is found that for N = 2 the continuous line of fixed points is well reproduced by the resummed RG series and an account for the five-loop terms makes the lines of zeros of both beta-functions closer to each another. For N > 2 the five-loop contributions are shown to shift the cubic fixed point, given by the four-loop approximation, towards the Ising fixed point. This confirms the idea that the existence of the cubic fixed point in two dimensions under N > 2 is an artifact of the perturbative analysis. In the case N = 0 the results obtained are compatible with the conclusion that the impure critical behavior is controlled by the Ising fixed point.

Critical behavior of two-dimensional cubic andMNmodels in the five-loop renormalization group approximation

The critical thermodynamics of the two-dimensional N-vector cubic and MN models is studied within the field-theoretical renormalization-group (RG) approach. The beta functions and critical exponents are calculated in the five-loop approximation and the RG series obtained are resummed using the Borel-Leroy transformation combined with the generalized Pad\'e approximant and conformal mapping techniques. For the cubic model, the RG flows for various N are investigated. For N=2 it is found that the continuous line of fixed points running from the XY fixed point to the Ising one is well reproduced by the resummed RG series and an account for the five-loop terms makes the lines of zeros of both beta functions closer to each another. For the cubic model with N\geq 3, the five-loop contributions are shown to shift the cubic fixed point, given by the four-loop approximation, towards the Ising fixed point. This confirms the idea that the existence of the cubic fixed point in two dimensions under N>2 is an artifact of the perturbative analysis. For the quenched dilute O(M) models ($MN$ models with N=0) the results are compatible with a stable pure fixed point for M\geq1. For the MN model with M,N\geq2 all the non-perturbative results are reproduced. In addition a new stable fixed point is found for moderate values of M and N.

Renormalization group and nonlinear susceptibilities of cubic ferromagnets at criticality

For the three-dimensional cubic model, the nonlinear susceptibilities of the fourth, sixth, and eighth orders are analyzed and the parameters \delta^(i) characterizing their reduced anisotropy are evaluated at the cubic fixed point. In the course of this study, the renormalized sextic coupling constants entering the small-field equation of state are calculated in the four-loop approximation and the universal values of these couplings are estimated by means of the Pade-Borel-Leroy resummation of the series obtained. The anisotropy parameters are found to be: \delta^(4) = 0.054 +/- 0.012, \delta^(6) = 0.102 +/- 0.02, and \delta^(8) = 0.144 +/- 0.04, indicating that the anisotropic (cubic) critical behavior predicted by the advanced higher-order renormalization-group analysis should be, in principle, visible in physical and computer experiments.

Critical dynamics of slightly disordered spin systems

This paper is a field theoretic description of the critical dynamics of spin systems with frozen nonmagnetic impurities. For three-dimensional systems the dynamical critical exponent is found directly by employing the three-loop approximation with the Pade-Borel summation technique. The results are compared with those obtained by calculating the dynamical exponent for homogeneous systems in the four-loop approximation, and with the values obtained by computer simulation of the critical dynamics by Monte Carlo methods. Calculations of the dynamical exponent for the two-dimensional Ising model in the four-loop approximation are also presented.

Critical dynamics of disordered magnets in the three-loop approximation

A field-theoretic description of the critical dynamics of magnetic systems with frozen nonmagnetic impurities is given. The values of the dynamical critical exponent in the three-loop approximation are obtained directly for three-dimensional systems using the Pade-Borel summation technique. A comparison is made with the values of the dynamical exponent for homogeneous systems calculated in the four-loop approximation as well as with the values obtained by Monte Carlo methods.

Critical dynamics of spin systems in the four-loop approximation

A field-theoretic description of the critical dynamics of spin systems is constructed. Using the Pade-Borel summation technique, the dynamic critical exponent for two-and three-dimensional systems is evaluated directly in the four-loop approximation. The results are compared with the values obtained for the dynamic exponent by Monte Carlo methods.

VARIATIONAL PERTURBATION THEORY

A nonperturbative method — variational perturbation theory (VPT) — is discussed. A quantity we are interested in is represented by a series, a finite number of terms of which not only describe the region of small coupling constant but reproduce well the strong coupling limit. The method is formulated only in terms of the Gaussian quadratures, and diagrams of the conventional perturbation theory are used. Its efficiency is demonstrated for the quantum-mechanical anharmonic oscillator. The properties of convergence are studied for series in VPT for the [Formula: see text] model. It is shown that it is possible to choose variational additions such that they lead to convergent series for any values of the coupling constant. Upper and lower estimates for the quantities under investigation are considered. The nonperturbative Gaussian effective potential is derived from a more general approach, VPT. Various versions of the variational procedure are explored and the preference for the anharmonic variational procedure in view of convergence of the obtained series is argued. We investigate the renormalization procedure in the φ4 model in VPT. The nonperturbative β function is derived in the framework of the proposed approach. The obtained result is in agreement with four-loop approximation and has the asymptotic behavior as g3/2 for a large coupling constant. We construct the VPT series for Yang-Mills theory and study its convergence properties. We introduce coupling to spinor fields and demonstrate that they do not influence the VPT series convergence properties.

ß-function for the ϕ4-model in variational perturbation theory

We consider the renormalization procedure for the ϕ 4-model in the variational perturbation theory. The nonperturbative ß-function is derived in the framework of the proposed approach. The obtained result is in agreement with the four-loop approximation and has an asymptotic behaviour as g 3 2 for a large coupling constant.

Four-loop approximation in the? 4 model

Four-loop approximation in the? 4 model