A. I. Sokolov, V. A. Ul'kov, E. V. Orlov
Department of Physical Electronics, State Electrotechnical University,
5 Prof. Popov Str., St.Petersburg 197376, Russia
Higher-order vertices at zero external momenta for the scalar field theory, describing the critical behaviour of the Ising model, are studied within the field--theoretical renormalization group (RG) approach in three dimensions. Dimensionless six-point $g_6$ and eight-point $g_8$ effective coupling constants are calculated in the three-loop approximation. Their numerical values, universal at criticality, are estimated by means of the Pad\'e and Pad\'e-Borel summation of the RG expansions found and by putting the renormalized quartic coupling constant equal to its universal fixed-point value known from six-loop RG calculations. The values of $g_6^*$ obtained are compared with their analogs resulting from the $\epsilon$-expansion, Monte Carlo simulations, the Wegner-Houghton equations and the linked cluster expansion series. The field-theoretical estimates for $g_6^*$ are shown to be in a good agreement with each other, differing considerably from the values given by other methods.