Longitudinal and Transverse Greens Functions in Φ Model below and near the Critical Point

We have extended our method of grouping of Feynman diagrams (GFD theory) to study the transverse and longitudinal Greens functions G ⊥ (k) and G (k) in ϕ 4 model below the critical point (T < T c) in presence of an infinitesimal external field. Our method allows a qualitative analysis not cutting the perturbation series. We have shown that the critical behavior of the Greens functions is consistent with a general scaling hypothesis, where the same critical exponents, found within the GFD theory, are valid both at T < T c and T > T c. The long–wave limit k → 0 has been studied at T < T c , showing that G ⊥ (k) ≃ a k −λ ⊥ and G (k) ≃ b k −λ with exponents d/2 < λ ⊥ < 2 and λ = 2λ ⊥ − d is the physical solution 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 non–perturbative renormalization group (RG) analysis provided in our paper. It is confirmed also by Monte Carlo simulation. 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 λ ⊥ = 2. Nevertheless, we disprove the conventional statement that λ ⊥ = 2 is the exact result.