Renormalization group theory of generalized multi-vertex sine-Gordon model

TL;DR

This paper develops a renormalization group framework for a generalized multi-vertex sine-Gordon model, revealing how multi-vertex interactions evolve and generate new interactions under RG flow, with specific geometric conditions on momentum vectors.

Contribution

It introduces a detailed RG analysis of multi-vertex sine-Gordon models, identifying conditions for interaction generation and deriving closed RG equations for specific geometric configurations.

Findings

New vertex interactions are generated when momentum vectors satisfy the triangle condition.

RG equations close for configurations where vectors form equilateral triangles or regular tetrahedra.

Wilsonian RG results agree qualitatively with dimensional regularization analysis.

Abstract

We investigate the renormalization group theory of generalized multi-vertex sine-Gordon model by employing the dimensional regularization method and also the Wilson renormalization group method. The vertex interaction is given by $\cos(k_j\cdot \phi)$ where $k_j$ ($j=1,2,\cdots,M$) are momentum vectors and $\phi$ is an $N$-component scalar field. The beta functions are calculated for the sine-Gordon model with multi cosine interactions. The second-order correction in the renormalization procedure is given by the two-point scattering amplitude for tachyon scattering. We show that new vertex interaction with momentum vector $k_{\ell}$ is generated from two vertex interactions with vectors $k_i$ and $k_j$ when $k_i$ and $k_j$ meet the condition $k_{\ell}=k_i\pm k_j$ called the triangle condition. Further condition $k_i\cdot k_j=\pm 1/2$ is required within the dimensional regularization method. The renormalization group equations form a set of closed equations when $\{k_j\}$ form an equilateral triangle for $N=2$ or a regular tetrahedron for $N=3$. The Wilsonian renormalization group method gives qualitatively the same result for beta functions.