Longitudinal and Transverse Correlation Functions in the φ4 Model below and near the Critical Point

We have extended our method of grouping Feynman diagrams (GFD theory) to study the transverse and longitudinal correlation functions G(perpendicular to)(k) and G(parallel to)(k) in phi(4) model below the critical point (T < T-c) in the presence of an infinitesimal external field. Our method allows a qualitative analysis without cutting the perturbation series. The long-wave limit k -> 0 has been studied at T < T-c, showing that G(perpendicular to)(k) similar or equal to ak(-lambda perpendicular to) and G(parallel to)(k) similar or equal to bk(-lambda parallel to) with exponents d/2 < lambda(perpendicular to) < 2 and lambda(parallel to) = 2 lambda(perpendicular to) - d are the physical solutions of our equations at the spatial dimensionality 2 < d < 4, which coincides with the asymptotic solution at T -> T-c as well as with a nonperturbative renormalization group (RG) analysis provided in our paper. This has been confirmed also by recent Monte Carlo simulations. The exponents as well as the ratio bM(2)/a(2) (where M is magnetization) are universal. The results of the perturbative RG method are reproduced by formally setting lambda(perpendicular to) = 2, although our analysis yields lambda(perpendicular to) < 2.