Advances in Mathematical Physics

Volume 2018, Article ID 9238280, 7 pages

https://doi.org/10.1155/2018/9238280

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

Electronics and Photonics Research Institute, National Institute of Advanced Industrial Science and Technology, 1-1-1 Umezono, Tsukuba, Ibaraki 305-8568, Japan

Correspondence should be addressed to Takashi Yanagisawa; pj.og.tsia@awasiganay-t

Received 7 February 2018; Accepted 18 July 2018; Published 6 September 2018

Academic Editor: Emmanuel Lorin

Copyright © 2018 Takashi Yanagisawa. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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.

#### 1. Introduction

The sine-Gordon model is an interesting model and plays an important role in physics [1–13]. There are many phenomena that are related to the sine-Gordon model. In this sense, the sine-Gordon model has universality. In the weak coupling phase the sine-Gordon model is perturbatively equivalent to the massive Thirring model [1, 14–16]. The two-dimensional (2D) sine-Gordon model describes a crossover between weak coupling region and strong coupling region. The renormalization equations are the same as those for the Kosterlitz-Thouless transition of the 2D classical XY model [17–19]. The 2D sine-Gordon model is mapped to the Coulomb gas model where particles interact with each other through logarithmic interaction [4, 20, 21]. The Kondo problem belongs to the same universality class where the renormalization group equations are given by the same equations for the 2D sine-Gordon model [20–27]. The renormalization group equations in the Kondo problem was derived before those by Kosterlitz and Thouless. The one-dimensional Hubbard model is mapped to the 2D sine-Gordon model by using a bosonization method [28–31], where the Hubbard model is an important model that describes the metal-insulator transition and high-temperature superconductivity [32–39]. The sine-Gordon model appears in a multiband superconductor where the Nambu-Goldstone modes become massive due to the Josephson couplings [40–47]. The Josephson plasma oscillation in layered high-temperature superconductors was analyzed based on the sine-Gordon model [48]. In a series of papers [41–43, 45, 46] we introduced the sine-Gordon model into the study of superconductivity and examined significant excitation modes in superconductors. A generalization from to a compact continuous group G for the sine-Gordon model was also investigated [49] where the sine-Gordon model considered in this paper and in references cited above is a model with group.

In this paper, we investigate the renormalization group theory for the 2D generalized sine-Gordon model by using the dimensional regularization method to regularize the divergence [50–52]. Here the generalized sine-Gordon model is a sine-Gordon model that includes high frequency cosine potential terms such as for an integer n. The renormalization of the generalized sine-Gordon model was investigated [53] by the Wegner-Houghton method [54] and by the functional renormalization group method [55]. We use the dimensional regularization method in deriving the renormalization group equation for the generalized sine-Gordon model. The divergence is regularized near two dimensions by putting the dimension . The divergent part of integral is evaluated as a pole in the form . This is called the minimal subtraction method. Then the beta function for the coupling constant is derived.

#### 2. Lagrangian

Let us consider a real scalar field . The Lagrangian of the generalized sine-Gordon model is given bywhere is a bare real scalar field and and are bare coupling constants. The second term indicates the potential energy of the scalar field . The generalized sine-Gordon model contains high frequency terms such as (n = 1, 2, ). We write the renormalized coupling constants as and , respectively. We adopt that and . for some may be zero, but at least one should be positive (nonzero). The dimensions of and are given as and where is a parameter representing the energy scale. The scalar field is dimensionless. The relations between bare and renormalized quantities are given bywhere and are renormalization constants. and are dimensionless constants by virtue of the energy scale . We define the renormalized field bywhere is the renormalization constant for the field . The Lagrangian with renormalized quantities is written aswhere denotes the renormalized field . The second term represents the interaction of the field as seen by expanding as a power series. There is the other representation of interaction parameters. We can absorb the parameter in the definition of the field and the parameter . In this case, field in the interaction term includes the parameter in the form where . We will obtain the same result since it does not depend on the representation.

#### 3. Renormalization of

We consider the renormalization of up to the lowest order of . By considering tadpole diagrams in Figure 1, the cosine function is renormalized to Since the expectation value diverges, we regularize it using the dimensional regularization method:for where is introduced to avoid infrared divergence, is the solid angle in dimensions and was put as 1. In order to remove the divergence, the constant is determined as follows:Since the bare coupling constant is independent of , we have . This results inWe set up to the lowest order of , so that we have . The beta function for at the lowest order in is given by has a zero at :for . There is a fixed point of for each .