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

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We have extended our method of grouping of Feynman diagrams (GFD theory) to study the transverse (Gₜ) and longitudinal (Gₗ) Greens functions in phi⁴ model below the critical point (T<Tc) 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<Tc and T>Tc. The long-wave limit k->0 has been studied at T<Tc, showing that the transverse and the longitudinal correlation functions diverge as 1/k in the power of lambdaₜ and lambdaₗ, respectively, where d/2< lambdaₜ < 2 and lambdaₗ = 2 lambdaₜ - 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 -> Tc 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 bM²/a² (where M is magnetization, a and b are the amplitudes of Gₜ and Gₗ at k->0) are universal. The results of the perturbative RG method are reproduced by formally setting lambdaₜ=2. Nevertheless, we disprove the conventional statement that lambdaₜ=2 is the exact result.