Longitudinal and transverse Greens functions in phi⁴ model below and near the critical point

We have extended our method of grouping of Feynman diagrams (GFD theory) to study the transverse (G_t) and longitudinal (G_l) Greens functions in phi^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 (correlation) 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 the transverse and the longitudinal correlation functions diverge as 1/k in the power of lambda_t and lambda_l, respectively, where d/2< lambda_t < 2 and lambda_l = 2 lambda_t - d holds at the spatial dimensionality 2<d<4. It is the physical solution of our equations, 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, a and b are the amplitudes of G_t and G_l at k->0) are universal. The results of the perturbative RG method are reproduced by formally setting lambda_t=2. Nevertheless, we disprove the conventional statement that lambda_t=2 is the exact result.