Renormalization Group Theory of Effective Field Theory Models in Low Dimensions

Abstract

We discuss the renormalization group approach to fundamental field theoretic models in low dimensions. We consider the models that are universal and frequently appear in physics, both in high-energy physics and condensed matter physics. They are the nonlinear sigma model, the φ 4 model and the sine-Gordon model. We use the dimensional regularization method to regularize the divergence and derive renormalization group equations called the beta functions. The dimensional method is described in detail.

Perturbative versus non-perturbative renormalization

“…This leads to the determination of critical frequency which separates the phases of the model. (A different strategy is when one uses an auxiliary mass term [23,30,31] where RG flow equations can be derived for the wave function renormalization but this is not discussed in this section. )…”

“…The perturbative renormalization of the SG model has been investigated by using the inclusion of an auxiliary mass term, see e.g. [23,30,31,57]. Thus, the perturbative renormalization of the SG model is done in the massive case when one adds an exilic mass term m 0 to the original periodic SG theory and RG flow equations of the massless SG model are given by considering the limit m 0 → 0.…”

“…(The Einstein summation convention is used.) First, we consider its perturbative renormalization around the Gaussian fixed point (m 2 = g 4 = 0) where the pRG flow equation for the quartic coupling in the minimal subtraction (MS) scheme, at 1-loop order has been givne in equation (10.52) of [23] or equation (45) of [24] for ò ≡ 0) with k ≡ μ. Its reads as follows…”

“…In addition, the perturbative RG treatment is possible for the SG model, but not in a traditional way, when the potential is expanded in Taylor series and only a finite number of terms are kept. In the case of the SG model, either one follows the scenario of [26,29] with all terms of the Taylor expansion summed up, or one uses the idea of an auxiliary mass term presented in [23,30,31]. The two-dimensional SG model is suitable for a detailed study and comparison of the scheme-and regulator-dependence.…”

Renormalization group analysis of the hyperbolic sine-Gordon model: Asymptotic freedom from cosh interaction

“…We consider tadpole diagrams to take account of the renormalization of α up to the lowest order of α ( Fig. 1) [6,[31][32][33]. Using the expansion cosh φ = 1 + (1/2)φ 2 + (1/4!…”