2 Calculation of the dynamical critical exponent

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. Introduction The method of the renormalization group (RG) nowadays is the main tool of calculation of critical exponents in the theory of critical phenomena. The basis of that method is the technique of the ultraviolet renormalization of a model. Critical exponents are found from the calculated renormalization constants by means of the standard RG rules. Thus calculation of the renormalization constants is the primary concern of the method. The most consistent scheme of applying the RG method is combining it with the ε-expansion (with ε = 4 − d, where d is spatial dimension) and that is the way it is used throughout the paper. Creation of subtle analytical methods of calculations allowed to attain 5-loop accuracy (fifth order of the ε-expansion) in the theory of static critical phenomena [1] and maximum 3-loop accuracy (third order of the ε-expansion) in the theory of critical dynamics [2]. These record results have held since 1991 and 1984 respectively. Attempts of analytical calculations in higher orders met fundamental difficulties [3]. For numerical calculation of the renormalization constants it is essential to be able to extract residues at ε = 0 from the graphs. We developed such technique based on well-known Roperation of the theory of renormalizations. It reduces calculation of renormalization constants to the evaluation of multiple UV-convergent integrals which can be performed by standard programs for numerical integration. We used the technique for calculation of dynamical exponent of the model A in 4-loop approximation (fourth order of the ε-expansion). Brief formulation of the model and the results obtained are given below. Department of Theoretical Physics, St. Petersburg University, Ulyanovskaya 1, St. Petersburg, Petrodvorets, 198504, Russia. E-mail: [email protected], leo [email protected] Google St. Petersburg, Alia Tempora, ul. Mayakovskogo, Bldg 3B, Floors 8-9, St. Petersburg, 191025, Russia. E-mail: [email protected]