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Field Theory Renormalization Group: The Tomonaga-Luttinger Model Revisited

We apply field theoretical renormalization group (RG) methods to describe the Tomonaga-Luttinger model as an important test ground to deal with spin-charge separation effects in higher spatial dimensions. We compute the anomalous dimension associated with the one-particle Green's functions $G_{(a)R}$. We calculate the RG equations for the renormalized forward couplings $g_{2R}$ and $g_{4R}$ up to two-loop order and demonstrate that they don't flow in the vicinities of the Fermi points (FPs). We solve the Callan-Symanzik equation for $G_{(a)R}$ in the neighborhood of the FPs. We calculate the related spectral function and the momentum distribution function at $p=k_{F},p_{0}=ω$. We compute the renormalized one-particle irreducible function $Γ_{(+)R}^{(2)}(p,p_{0}=0;Λ)$ and show it carries important spin-charge separation effects in agreement with well known results. Finally we discuss the implementation of the RG scheme taking into account the important simplifications produced by the Ward identities.