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

Yanagisawa, Takashi

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

Advances in Mathematical Physics

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Research Article | Open Access

Volume 2018 | Article ID 9238280 | https://doi.org/10.1155/2018/9238280

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

Takashi Yanagisawa¹

Academic Editor: Emmanuel Lorin

Received07 Feb 2018

Accepted18 Jul 2018

Published06 Sept 2018

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 .

4. Renormalization of

There is an effect of renormalization on the coupling constant that is the correction to the kinetic term. Let us consider the two-point function . The bare lowest order two-point function is given byThis corresponds to the kinetic part of the bare Lagrangian:

4.1. Real Space Formulation

The lowest order correction to the two-point function is given by a second-order term for () such as . From the formula , the correction to the action comes fromwhere we consider connected contributions. By taking into account the contribution of tadpole diagrams, this reduces toThe expectation value is given bywhere is the 0th modified Bessel function. Because increases divergently as approaches zero, is approximated aswhere we put . The cosine function would oscillate as a function of , the contribution for will be small. Thus, we consider only the contributions with :We extract the divergent term in . There may be two ways to do this. We discuss these methods in the following.

In the first method, we regularize by introducing a cutoff in the real space:by replacing with . By using the asymptotic relation with the Euler constant , the integral with respect to is performed as follows [49]:near where we set . We consider the case where is close to the critical value :where represents the deviation from the critical point. In the lowest order of , we haveThen we obtainThe constant was absorbed for the renormalization of . Then, by taking the sum from each term, the kinetic part is renormalized toThis indicates that we choose and appear as a ratio in this order, and then the coupling constant is renormalized as or we can choose . The equation results inLastly, we put to obtainThe numerical coefficient is not important and this depends on the choice of the cutoff .

In the second way, the divergence comes from where we adopt that the integral with respect to is finite. This treatment is similar to that in [31] where the Wilson renormalization group method was used. The correction is written asIn order to let the integral for be dimensionless, we change the variable and putby introducing a cutoff in the integral. Then, we have This results in the same beta function with the numerical factor being slightly different:

4.2. Momentum Space Formulation

In the momentum space, we evaluate the two-point function by calculating the diagrams in Figure 2 [6]. This set of diagrams gives the self-energy . is written as a sum of that comes from the interaction term . The diagrams in Figure 2 are summed up to givewhere we putSince is divergently large as , is approximated asUsing the expansion , we keep the term. By using the formula for small , is written asThe integral diverges when and . Then we consider the case , which gives the correction to the two-point function when as follows:where we set . This term mainly comes from the region where . The two-point function up to this order isThe renormalized two-point function is given as . The renormalization constants are determined as shown above and thus we obtain the same renormalization group equation.

5. Renormalization Group Flow

Let us consider the case with two parameters and . The renormalization group equations areWe have the critical value for and for . The parameter is an increasing function of in two dimensions . The space may be divided into four regions which are classified by the values to which the pair is renormalized as increases. We call them regions I, II, III, and IV:In region , we put , , and . The equations readWhen is small, the equations reduce to those of the conventional sine-Gordon model (Kosterlitz-Thouless transition).

When , we put and to obtainIn this region, acts as a perturbation to the scaling equation of the conventional sine-Gordon model. We show the renormalization group flow as increases in Figure 3.

6. Summary

We have discussed the dimensional regularization approach to the renormalization group theory of the generalized sine-Gordon model. There are multiple critical points for the coupling constant given as . In the case where is close to for some , the renormalization group equations are approximated by those for the sine-Gordon model with single-cosine potential (conventional sine-Gordon model). A nontrivial simple generalized model is the sine-Gordon model with and . When is , acts as a perturbation for and . The renormalization flow as or depends on an initial set of parameters , , and . This can be viewed as a competition between two interactions and . This may lead to a generalization of the Kosterlitz-Thouless transition, the crossover phenomenon in the Kondo effect, and other phenomena. In the Kondo problem, the appearance of logarithmic singularity [22, 56, 57] suggested the renormalizability of the model. In materials with many magnetic impurities, the interaction between magnetic impurities, called the RKKY interaction [58–60], should be considered. In this case the renormalization group flow is drawn on a two-dimensional plane of two parameters. There may be a relation to the generalized sine-Gordon model.

Data Availability

No data were used to support this study.

Conflicts of Interest

The author declares that there are no conflicts of interest.

Acknowledgments

The author expresses his sincere thanks to K. Odagiri for useful discussions. This work was supported in part by Grant-in-Aid from the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan (no. 17K05559).

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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.

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