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Theoretical and Mathematical Physics

Representation of Renormalization Group Functions By Nonsingular Integrals in a Model of the Critical Dynamics of Ferromagnets: The Fourth Order of The ε-Expansion

L. Ts. Adzhemyan, Svetlana Evgen'evna Vorobyeva, Ella V. Ivanova, Mikhail Vladimirovich Kompaniets

Published Apr 1, 2018• 3 citations
Publisher

Abstract

Using the representation for renormalization group functions in terms of nonsingular integrals, we calculate the dynamical critical exponents in the model of critical dynamics of ferromagnets in the fourth order of the ε-expansion. We calculate the Feynman diagrams using the sector decomposition technique generalized to critical dynamics problems.

DOI
10.1134/s0040577918040104

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Using the representation for renormalization group functions in terms of nonsingular integrals, we calculate the dynamical critical exponents in the model of critical dynamics of ferromagnets in the fourth order of the ε-expansion. We calculate the Feynman diagrams using the sector decomposition technique generalized to critical dynamics problems.
Abstract Within the framework of the renormalization group approach to the models of critical dynamics, we propose a method for a considerable reduction of the number of integrals needed to calculate the critical exponents. With this method we perform a calculation of the critical exponent z of model A at 4-loop level, where our method allows one to reduce the number of integrals from 66 to 17. The way of constructing the integrand in a Feynman representation of such diagrams is discussed. Integrals were estimated numerically with a sector decomposition technique.
A new method based on the R'-operation of the renormalization theory is proposed for the numerical calculation of the renormalization constants in the theory of critical behaviour. The problem of finding residues of the poles of the Green's functions at ε= 0, where ε= 4 - d, is reduced to the evaluation of multiple UV-finite integrals, which can be performed by means of standard integration programs. The method is used to calculate the renormalization group functions of the model A of critical dynamics in four-loop approximation. Dynamical exponent z of the model A is calculated in the fourth order of the ε-expansion.
The Renormalization group method (RG) is applied to the investigation of the E model of critical dynamics, which describes the transition from the normal to the superfluid phase in He4. The technique “Sector decomposition” with R’ operation is used for the calculation of the Feynman diagrams. The RG functions, critical exponents and critical dynamical exponent z, which determines the growth of the relaxation time near the critical point, have been calculated in the two-loop approximation in the framework of ε-expansion. The relevance of a fixed point for helium, where the dynamic scaling is weakly violated, is briefly discussed.
The critical dynamics of a model isotropic ferromagnet is studied in $6\ensuremath{-}\ensuremath{\epsilon}$ dimensions both above and below Tc{T}_{c}Tc​. The renormalization group and the characteristic frequency exponent zzz are determined to order $\ensuremath{\epsilon}$. Implications of the renormalization group are discussed. A direct calculation of the dynamic response function to order $\ensuremath{\epsilon}$ shows many interesting features, especially for T<TcT<{T}_{c}T<Tc​.