On free energy of three-dimensional Ising model at criticality

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 Pade and Pade-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.