Longitudinal and Transverse Correlation Functions in the 4 Model below and near the Critical Point

Abstract

We have extended our method of grouping Feynman diagrams (GFD theory) to study the transverse and longitudinal correlation functions G⊥(k) and G∥(k) in ϕ4 model below the critical point (T < Tc) 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 < Tc, showing that G⊥(k) ≃ ak−λ⊥ and G∥(k) ≃ bk−λ∥ with exponents d/2 < λ⊥ < 2 and λ∥ = 2 λ⊥ − d are the physical solutions of our equations at the spatial dimensionality 2 < d < 4, which coincides with the asymptotic solution at T → Tc 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 bM2/a2 (where M is magnetization) are universal. The results of the perturbative RG method are reproduced by formally setting λ⊥ = 2, although our analysis yields λ⊥ < 2.