scholarly journals Dimensional Regularization Approach to the Renormalization Group Theory of the Generalized Sine-Gordon Model

2018 ◽  
Vol 2018 ◽  
pp. 1-7 ◽  
Author(s):  
Takashi Yanagisawa
Keyword(s):  
Renormalization Group ◽  
Group Theory ◽  
High Frequency ◽  
Dimensional Regularization ◽  
Renormalization Group Theory ◽  
Generalized Model ◽  
New Class ◽  
Scaling Property ◽  
Renormalization Group Equations ◽  
Sine Gordon Model

We present the dimensional regularization approach to the renormalization group theory of the generalized sine-Gordon model. The generalized sine-Gordon model means the sine-Gordon model with high frequency cosine modes. We derive renormalization group equations for the generalized sine-Gordon model by regularizing the divergence based on the dimensional method. We discuss the scaling property of renormalization group equations. The generalized model would present a new class of scaling property.

scholarly journals Renormalization group theory of the generalized multi-vertex sine-Gordon model

2021 ◽  
Vol 2021 (3) ◽  
Author(s):  
Takashi Yanagisawa
Keyword(s):  
Renormalization Group ◽  
Group Theory ◽  
Regularization Method ◽  
Dimensional Regularization ◽  
Group Method ◽  
Regular Tetrahedron ◽  
Renormalization Group Theory ◽  
Renormalization Group Method ◽  
Triangle Condition ◽  
Sine Gordon Model

Abstract We investigate the renormalization group theory of the 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 $k_j$ ($j=1,2,\ldots,M$) are momentum vectors and $\phi$ is an $N$-component scalar field. The beta functions are calculated for the sine-Gordon model with multiple 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 the 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. A 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.

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