**Renormalization Group Theory of Effective Field Theory Models in Low Dimensions**

Takashi Yanagisawa

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.68214

## 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<sup>4</sup> 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.

Keywords: renormalization group theory, dimensional regularization, scalar model, non-linear sigma model, sine-Gordon model

## 1. Introduction

The renormalization group is a fundamental and powerful tool to investigate the property of quantum systems [1–15]. The physics of a many-body system is sometimes captured by the analysis of an effective field theory model [16–19]. Typically, effective field theory models are the φ<sup>4</sup> model, the non-linear sigma model and the sine-Gordon model. Each of these models represents universality as a representative of a universal class.

The φ<sup>4</sup> model is the model of a phase transition, which is often referred to as the Ginzburg-Landau model. The renormalization of the φ<sup>4</sup> model gives a prototype of renormalization group procedures in field theory [20–24].

The non-linear sigma model appears in various fields of physics [15, 25–27] and is the effective model of Quantum chromodynamics (QCD) [28] and also that of magnets (ferromagnetic and anti-ferromagnetic materials) [29–32]. This model exhibits an important property called the

asymptotic freedom. The non-linear sigma model is generalized to a model with fields that take values in a compact Lie group G [33–42]. This is called the chiral model.

The sine-Gordon model also has universality [43–49]. The two-dimensional (2D) sine-Gordon model describes the Kosterlitz-Thouless transition of the 2D classical XY model [50, 51]. The 2D sine-Gordon model is mapped to the Coulomb gas model where particles interact with each other through a logarithmic interaction. The Kondo problem [52, 53] also belongs to the same universality class where the scaling equations are just given by those for the 2D sine-Gordon model, i.e. the equations for the Kosterlitz-Thouless transition [53–57]. The onedimensional Hubbard model is also mapped onto the 2D sine-Gordon model on the basis of a bosonization method [58, 59]. The Hubbard model is an important model of strongly correlated electrons [60–65]. The Nambu-Goldstone (NG) modes in a multi-gap superconductor become massive due to the cosine potential, and thus the dynamical property of the NG mode can be understood by using the sine-Gordon model [66–71]. The sine-Gordon model will play an important role in layered high-temperature superconductors because the Josephson plasma oscillation is analysed on the basis of this model [72–75].

In this paper, we discuss the renormalization group theory for the φ4 theory, the non-linear sigma model and the sine-Gordon model. We use the dimensional regularization procedure to regularize the divergence [76].

## 2. φ<sup>4</sup> model

#### 2.1. Lagrangian

The φ<sup>4</sup> model is given by the Lagrangian

$$\mathcal{L} = \frac{1}{2} (\partial\_{\mu} \phi)^{2} - \frac{1}{2} m^{2} \phi^{2} - \frac{g}{4!} \phi^{4} \, , \tag{1}$$

where φ is a scalar field and g is the coupling constant. In the unit of the momentum μ, the dimension of <sup>L</sup> is given by <sup>d</sup>, where <sup>d</sup> is the dimension of the space-time: <sup>½</sup>L� ¼ <sup>μ</sup><sup>d</sup>. The dimension of the field <sup>φ</sup> is <sup>ð</sup><sup>d</sup> � <sup>2</sup>Þ=2: <sup>½</sup>φ� ¼ <sup>μ</sup>ðd�2Þ<sup>=</sup>2. Because <sup>g</sup>φ<sup>4</sup> has the dimension <sup>d</sup>, the dimension of g is given by 4 – d: [g] = μ<sup>4</sup> – <sup>d</sup> . Let us adopt that φ has N components as φ = (φ1, φ2, …, φN). The interaction term φ<sup>4</sup> is defined as

$$
\phi^4 = \left(\sum\_{i=1}^N \phi\_i^2\right)^2. \tag{2}
$$

The Green's function is defined as

$$\mathbf{G}\_i(\mathbf{x} - \mathbf{y}) = -i \langle \mathbf{0} | T \phi\_i(\mathbf{x}) \phi\_i(\mathbf{y}) | \mathbf{0} \rangle \,\tag{3}$$

where T is the time-ordering operator and |0〉 is the ground state. The Fourier transform of the Green's function is

Renormalization Group Theory of Effective Field Theory Models in Low Dimensions http://dx.doi.org/10.5772/intechopen.68214 99

$$G\_i(p) = \int d^d \mathbf{x} e^{ip \cdot \mathbf{x}} G\_i(\mathbf{x}). \tag{4}$$

In the non-interacting case with g = 0, the Green's function is given by

$$G\_i^{(0)}(p) = \frac{1}{p^2 - m^2},\tag{5}$$

where <sup>p</sup><sup>2</sup> ¼ ðp0<sup>Þ</sup> <sup>2</sup> � <sup>p</sup> !2 for p ¼ ðp0, p !Þ.

asymptotic freedom. The non-linear sigma model is generalized to a model with fields that

The sine-Gordon model also has universality [43–49]. The two-dimensional (2D) sine-Gordon model describes the Kosterlitz-Thouless transition of the 2D classical XY model [50, 51]. The 2D sine-Gordon model is mapped to the Coulomb gas model where particles interact with each other through a logarithmic interaction. The Kondo problem [52, 53] also belongs to the same universality class where the scaling equations are just given by those for the 2D sine-Gordon model, i.e. the equations for the Kosterlitz-Thouless transition [53–57]. The onedimensional Hubbard model is also mapped onto the 2D sine-Gordon model on the basis of a bosonization method [58, 59]. The Hubbard model is an important model of strongly correlated electrons [60–65]. The Nambu-Goldstone (NG) modes in a multi-gap superconductor become massive due to the cosine potential, and thus the dynamical property of the NG mode can be understood by using the sine-Gordon model [66–71]. The sine-Gordon model will play an important role in layered high-temperature superconductors because the Josephson plasma

In this paper, we discuss the renormalization group theory for the φ4 theory, the non-linear sigma model and the sine-Gordon model. We use the dimensional regularization procedure to

> <sup>2</sup> � <sup>1</sup> 2 m2 <sup>φ</sup><sup>2</sup> � <sup>g</sup> 4! φ4

where φ is a scalar field and g is the coupling constant. In the unit of the momentum μ, the dimension of <sup>L</sup> is given by <sup>d</sup>, where <sup>d</sup> is the dimension of the space-time: <sup>½</sup>L� ¼ <sup>μ</sup><sup>d</sup>. The dimension of the field <sup>φ</sup> is <sup>ð</sup><sup>d</sup> � <sup>2</sup>Þ=2: <sup>½</sup>φ� ¼ <sup>μ</sup>ðd�2Þ<sup>=</sup>2. Because <sup>g</sup>φ<sup>4</sup> has the dimension <sup>d</sup>, the

<sup>φ</sup><sup>4</sup> <sup>¼</sup> <sup>X</sup><sup>N</sup>

Giðx � y޼�i〈0jTφ<sup>i</sup>

i¼1 φ2 i � �<sup>2</sup>

where T is the time-ordering operator and |0〉 is the ground state. The Fourier transform of the

ðxÞφ<sup>i</sup>

, ð1Þ

: ð2Þ

ðyÞj0〉, ð3Þ

. Let us adopt that φ has N components as φ = (φ1,

take values in a compact Lie group G [33–42]. This is called the chiral model.

oscillation is analysed on the basis of this model [72–75].

<sup>L</sup> <sup>¼</sup> <sup>1</sup> 2 ð∂μφÞ

regularize the divergence [76].

98 Recent Studies in Perturbation Theory

The φ<sup>4</sup> model is given by the Lagrangian

dimension of g is given by 4 – d: [g] = μ<sup>4</sup> – <sup>d</sup>

The Green's function is defined as

Green's function is

φ2, …, φN). The interaction term φ<sup>4</sup> is defined as

2. φ<sup>4</sup> model

2.1. Lagrangian

Let us consider the correction to the Green's function by means of the perturbation theory in terms of the interaction term gφ<sup>4</sup> . A diagram that appears in perturbative expansion contains, in general, L loops, I internal lines and V vertices. They are related by

$$L = I - V + 1.\tag{6}$$

There are L degrees of freedom for momentum integration. The degree of divergence D is given by

$$D = d \cdot L - 2I.\tag{7}$$

We have a logarithmic divergence when D = 0. Let E be the number of external lines. We obtain

$$4V = E + 2I.\tag{8}$$

Then, the degree of divergence is written as

$$D = d \cdot L - 2I = d + (d - 4)V + \left(1 - \frac{d}{2}\right)E. \tag{9}$$

In four dimensions d = 4, the degree of divergence D is independent of the numbers of internal lines and vertices

$$D = 4 - E \tag{10}$$

When the diagram has four external lines, E = 4, we obtain D = 0 which indicates that we have a logarithmic (zero-order) divergence. This divergence can be renormalized.

Let us consider the Lagrangian with bare quantities

$$\mathcal{L} = \frac{1}{2} (\partial\_{\mu} \phi\_0)^2 - \frac{1}{2} m\_0^2 \phi\_0^2 - \frac{1}{4!} g\_0 \phi\_{0'}^4 \tag{11}$$

where φ<sup>0</sup> denotes the bare field, g<sup>0</sup> denotes the bare coupling constant and m<sup>0</sup> is the bare mass. We introduce the renormalized field φ, the renormalized coupling constant g and the renormalized mass m. They are defined by

$$
\phi\_0 = \sqrt{\mathbf{Z}\_\phi} \phi,\tag{12}
$$

$$\mathbf{g}\_0 = \mathbf{Z}\_\mathbf{g} \mathbf{g}\_\prime \tag{13}$$

$$\mathfrak{m}\_0^2 = \mathfrak{m}^2 \mathbf{Z}\_2 / \mathbf{Z}\_{\phi\nu} \tag{14}$$

where Zφ, Zg and Z<sup>2</sup> are renormalization constants. When we write Zg as

$$\mathcal{Z}\_{\mathfrak{Z}} = \mathcal{Z}\_{4} / \mathcal{Z}\_{\phi'}^{2} \tag{15}$$

we have <sup>g</sup>0Z<sup>2</sup> <sup>φ</sup> ¼ gZ4. Then, the Lagrangian is written by means of renormalized field and constants

$$\mathcal{L} = \frac{1}{2} Z\_{\phi} (\partial\_{\mu} \phi)^{2} - \frac{1}{2} m^{2} Z\_{2} \phi^{2} - \frac{1}{4!} \text{g} \mathbf{Z}\_{4} \phi^{4}. \tag{16}$$

#### 2.2. Regularization of divergences

#### 2.2.1. Two-point function

We use the perturbation theory in terms of the interaction gφ<sup>4</sup> . For a multi-component scalar field theory, it is convenient to express the interaction φ<sup>4</sup> as in Figure 1, where the dashed line indicates the coupling g. We first examine the massless case with m ! 0. Let us consider the renormalization of the two-point function Γð2<sup>Þ</sup> ðpÞ ¼ iGðpÞ �1 . The contributions to Γ(2) are shown in Figure 1. The first term indicates p<sup>2</sup> Z<sup>φ</sup> and the contribution in the second term is represented by the integral

$$I = \int \frac{d^d q}{(2\pi)^d} \frac{1}{q^2 - m^2}. \tag{17}$$

Using the Euclidean co-ordinate q<sup>4</sup> = –iq0, this integral is evaluated as

$$I = -i\frac{\Omega\_d}{(2\pi)^d} m^{d-2} \frac{1}{2} \Gamma\left(\frac{d}{2}\right) \Gamma\left(1 - \frac{d}{2}\right),\tag{18}$$

where Ω<sup>d</sup> is the solid angle in d dimensions. For d > 2, the integral I vanishes in the limit m ! 0. Thus, the mass remains zero in the massless case. We do not consider mass renormalization in the massless case. Let us examine the third term in Figure 2.

There are 4<sup>2</sup> � <sup>2</sup><sup>N</sup> <sup>þ</sup> 42 � 22 <sup>¼</sup> <sup>32</sup><sup>N</sup> <sup>þ</sup> 64 ways to connect lines for an <sup>N</sup>-component scalar field to form the third diagram in Figure 2. This is seen by noticing that this diagram is represented as a sum of two terms in Figure 3.

The number of ways to connect lines is 32N for (a) and 64 for (b). Then we have the factor from these contributions as

Renormalization Group Theory of Effective Field Theory Models in Low Dimensions http://dx.doi.org/10.5772/intechopen.68214 101

Figure 1. φ<sup>4</sup> interaction with the coupling constant g.

φ<sup>0</sup> ¼

Zg <sup>¼</sup> <sup>Z</sup>4=Z<sup>2</sup>

<sup>2</sup> � <sup>1</sup> 2 m2

field theory, it is convenient to express the interaction φ<sup>4</sup> as in Figure 1, where the dashed line indicates the coupling g. We first examine the massless case with m ! 0. Let us consider the

<sup>φ</sup> ¼ gZ4. Then, the Lagrangian is written by means of renormalized field and

<sup>Z</sup>2φ<sup>2</sup> � <sup>1</sup> 4! gZ4φ<sup>4</sup>

ðpÞ ¼ iGðpÞ

1

�1

<sup>Γ</sup> <sup>1</sup> � <sup>d</sup> 2 � �

m2 <sup>0</sup> <sup>¼</sup> <sup>m</sup><sup>2</sup>

where Zφ, Zg and Z<sup>2</sup> are renormalization constants. When we write Zg as

Zφð∂μφÞ

I ¼

Using the Euclidean co-ordinate q<sup>4</sup> = –iq0, this integral is evaluated as

<sup>I</sup> ¼ �<sup>i</sup> <sup>Ω</sup><sup>d</sup> ð2πÞ

the massless case. Let us examine the third term in Figure 2.

ð d <sup>d</sup> q ð2πÞ d

<sup>d</sup> md�<sup>2</sup> <sup>1</sup> 2 <sup>Γ</sup> <sup>d</sup> 2 � �

where Ω<sup>d</sup> is the solid angle in d dimensions. For d > 2, the integral I vanishes in the limit m ! 0. Thus, the mass remains zero in the massless case. We do not consider mass renormalization in

There are 4<sup>2</sup> � <sup>2</sup><sup>N</sup> <sup>þ</sup> 42 � 22 <sup>¼</sup> <sup>32</sup><sup>N</sup> <sup>þ</sup> 64 ways to connect lines for an <sup>N</sup>-component scalar field to form the third diagram in Figure 2. This is seen by noticing that this diagram is represented as

The number of ways to connect lines is 32N for (a) and 64 for (b). Then we have the factor from

<sup>L</sup> <sup>¼</sup> <sup>1</sup> 2

We use the perturbation theory in terms of the interaction gφ<sup>4</sup>

renormalization of the two-point function Γð2<sup>Þ</sup>

shown in Figure 1. The first term indicates p<sup>2</sup>

we have <sup>g</sup>0Z<sup>2</sup>

100 Recent Studies in Perturbation Theory

2.2. Regularization of divergences

2.2.1. Two-point function

represented by the integral

a sum of two terms in Figure 3.

these contributions as

constants

ffiffiffiffiffiffi Z<sup>φ</sup> q

φ, ð12Þ

g<sup>0</sup> ¼ Zgg, ð13Þ

Z2=Zφ, ð14Þ

<sup>φ</sup>, ð15Þ

: ð16Þ

. For a multi-component scalar

. The contributions to Γ(2) are

, ð18Þ

Z<sup>φ</sup> and the contribution in the second term is

<sup>q</sup><sup>2</sup> � <sup>m</sup><sup>2</sup> : <sup>ð</sup>17<sup>Þ</sup>

Figure 2. The contributions to the two-point function Γð2<sup>Þ</sup> <sup>ð</sup>p<sup>Þ</sup> up to the order of <sup>g</sup><sup>2</sup> .

Figure 3. The third term in Figure 2 is a sum of two configurations (a) and (b).

$$
\left(\frac{1}{4!}g\right)^2(32N+64) = \frac{N+2}{18}g^2.\tag{19}
$$

The momentum integral of this term is given as

$$J(k) := \int \frac{d^d p}{\left(2\pi\right)^d} \frac{d^d q}{\left(2\pi\right)^d} \frac{1}{p^2 q^2 \left(p + q + k\right)^2}. \tag{20}$$

The integral J exhibits a divergence in four dimensions d = 4. We separate the divergence as 1/E by adopting d = 4 – E. The divergent part is regularized as

$$J = -\left(\frac{1}{8\pi^2}\right)^2 \frac{1}{8\epsilon} + \text{regular terms} \tag{21}$$

To obtain this, we first perform the integral with respect to q by using

$$\frac{1}{q^2(p+q+k)^2} = \int\_0^1 d\mathbf{x} \frac{1}{\left[q^2\mathbf{x} + (p+q+k)^2(1-\mathbf{x})\right]^2}.\tag{22}$$

For q<sup>0</sup> = q + (1 – x)(p + k), we have

$$\begin{split} &\int \frac{d^d q}{(2\pi)^d} \frac{1}{q^2 (p+q+k)^2} = \int \frac{d^d q'}{(2\pi)^d} \int\_0^1 d\mathbf{x} \frac{1}{[q'^2 + \mathbf{x}(1-\mathbf{x})(p+k)^2]^2} \\ &= \frac{\Omega\_d}{(2\pi)^d} \int\_0^1 d\mathbf{x} \left(\mathbf{x}(1-\mathbf{x})\right)^{\frac{d}{2}-2} \left((p+k)^2\right)^{\frac{d}{2}-2} \int\_0^\infty dr r^{d-1} \frac{1}{\left(r^2+1\right)^2} \\ &= \frac{\Omega\_d}{\left(2\pi\right)^d} \frac{1}{2} \,\Gamma\left(\frac{d}{2}\right) \Gamma\left(2-\frac{d}{2}\right) \Gamma\left(\frac{d}{2}-1\right)^2 \,\frac{1}{\Gamma(d-2)} \left((p+k)^2\right)^{\frac{d}{2}-2} .\end{split} \tag{23}$$

Here, the following parameter formula was used

$$\frac{1}{A^n B^m} = \frac{\Gamma(n+m)}{\Gamma(n)\Gamma(m)} \int\_0^1 d\mathbf{x} \frac{\mathbf{x}^{n-1} (1-\mathbf{x})^{m-1}}{[\mathbf{x}A + (1-\mathbf{x})B]^{n+m}}.\tag{24}$$

Then, we obtain

$$\begin{split} \int \frac{d^d p}{\left(2\pi\right)^d} \frac{1}{p^2 \left(\left(p+k\right)^2\right)^{2-d/2}} &= \frac{\Gamma(3-d/2)}{\Gamma(2-d/2)} \int\_0^1 dx (1-x)^{1-d/2} \int \frac{d^d p'}{\left(2\pi\right)^d} \frac{1}{\left[p'^2 + x(1-x)k^2\right]^{3-d/2}} \\ &= \frac{\Omega\_d}{\left(2\pi\right)^d} \frac{\Gamma(3-d/2)}{\Gamma(2-d/2)} B\left(d-2, \frac{d}{2}-1\right) \frac{1}{2} B\left(\frac{d}{2}, 3-d\right) (k^2)^{d-3}. \end{split} \tag{25}$$

Here B(p, q) = Γ(p)Γ(q)/Γ(p+q). We use the formula

$$
\Gamma(\epsilon) = \frac{1}{\epsilon} + \text{finite terms} \tag{26}
$$

for E ! 0. This results in

$$\int \frac{d^d p}{(2\pi)^d} \frac{d^d q}{(2\pi)^d} \frac{1}{p^2 q^2 (p+q+k)^2} = -\left(\frac{1}{8\pi^2}\right)^2 \frac{1}{8\epsilon} \, k^2 + \text{ regular terms} \tag{27}$$

Therefore, the two-point function is evaluated as

$$
\Gamma^{(2)}(p) = Z\_{\phi}p^2 + \frac{1}{8\epsilon} \frac{N+2}{18} \left(\frac{g}{8\pi^2}\right)^2 p^2,\tag{28}
$$

up to the order of O(g<sup>2</sup> ). In order to cancel the divergence, we choose Z<sup>φ</sup> as

$$Z\_{\phi} = 1 - \frac{1}{8\epsilon} \frac{N+2}{18} \left(\frac{1}{8\pi^2}\right)^2 g^2. \tag{29}$$

#### 2.2.2. Four-point function

1 q<sup>2</sup>ðp þ q þ kÞ

> 1 q<sup>2</sup>ðp þ q þ kÞ

For q<sup>0</sup> = q + (1 – x)(p + k), we have

102 Recent Studies in Perturbation Theory

ð dd q ð2πÞ d

> <sup>¼</sup> <sup>Ω</sup><sup>d</sup> ð2πÞ d ð1 0 dx �

<sup>¼</sup> <sup>Ω</sup><sup>d</sup> ð2πÞ d 1 <sup>2</sup> <sup>Γ</sup> <sup>d</sup> 2 � �

1

<sup>¼</sup> <sup>Ω</sup><sup>d</sup> ð2πÞ d

Then, we obtain

p2 � ðp þ kÞ 2

for E ! 0. This results in

up to the order of O(g<sup>2</sup>

ð dd p ð2πÞ d dd q ð2πÞ d

ð dd p ð2πÞ d

Here, the following parameter formula was used

1 An

�<sup>2</sup>�d=<sup>2</sup> <sup>¼</sup> <sup>Γ</sup>ð<sup>3</sup> � <sup>d</sup>=2<sup>Þ</sup>

Γð3 � d=2Þ

Here B(p, q) = Γ(p)Γ(q)/Γ(p+q). We use the formula

Therefore, the two-point function is evaluated as

Γð2<sup>Þ</sup>

<sup>2</sup> ¼ ð1 0

<sup>2</sup> ¼

xð1 � xÞ

<sup>Γ</sup> <sup>2</sup> � <sup>d</sup> 2 � �

<sup>B</sup><sup>m</sup> <sup>¼</sup> <sup>Γ</sup>ð<sup>n</sup> <sup>þ</sup> <sup>m</sup><sup>Þ</sup> ΓðnÞΓðmÞ

Γð2 � d=2Þ

�

<sup>Γ</sup>ð<sup>2</sup> � <sup>d</sup>=2<sup>Þ</sup> B d � <sup>2</sup>,

<sup>Γ</sup>ðEÞ ¼ <sup>1</sup> E

1 p<sup>2</sup>q<sup>2</sup>ðp þ q þ kÞ

<sup>ð</sup>pÞ ¼ <sup>Z</sup>φp<sup>2</sup> <sup>þ</sup>

<sup>Z</sup><sup>φ</sup> <sup>¼</sup> <sup>1</sup> � <sup>1</sup>

8E

ð dd q0 ð2πÞ d ð1 0

> <sup>Γ</sup> <sup>d</sup> <sup>2</sup> � <sup>1</sup> � �<sup>2</sup> 1

ð1 0

ð1 0

dxð1 � xÞ

d <sup>2</sup> � <sup>1</sup> � � 1

<sup>2</sup> ¼ � <sup>1</sup>

1 8E

). In order to cancel the divergence, we choose Z<sup>φ</sup> as

N þ 2 18

8π<sup>2</sup> � �<sup>2</sup> 1

N þ 2 18

8E k

g 8π<sup>2</sup> � �<sup>2</sup>

g2

1 8π<sup>2</sup> � �<sup>2</sup> p2

�d <sup>2</sup>�<sup>2</sup> �

dx <sup>1</sup> ½q<sup>2</sup>x þ ðp þ q þ kÞ

> ðp þ kÞ 2 �d <sup>2</sup>�<sup>2</sup> ð<sup>∞</sup> 0

2

dx <sup>1</sup>

Γðd � 2Þ

<sup>0</sup> <sup>2</sup> þ xð1 � xÞðp þ kÞ

� ðp þ kÞ 2 �d <sup>2</sup>�2 :

m�1

drr<sup>d</sup>�<sup>1</sup> <sup>1</sup>

ðr<sup>2</sup> þ 1Þ 2

½q

dx xn�<sup>1</sup>ð<sup>1</sup> � <sup>x</sup><sup>Þ</sup>

½xA þ ð1 � xÞB�

1�d=2

2 <sup>B</sup> <sup>d</sup> 2 , 3 � d � �

ð dd p0 ð2πÞ d

> ðk2 Þ d�3 :

þ finite terms ð26Þ

<sup>ð</sup><sup>1</sup> � <sup>x</sup>Þ�<sup>2</sup> : <sup>ð</sup>22<sup>Þ</sup>

2 � 2

<sup>n</sup>þ<sup>m</sup> : <sup>ð</sup>24<sup>Þ</sup>

1 <sup>½</sup>p0<sup>2</sup> <sup>þ</sup> <sup>x</sup>ð<sup>1</sup> � <sup>x</sup>Þk<sup>2</sup>

<sup>2</sup> <sup>þ</sup> regular terms <sup>ð</sup>27<sup>Þ</sup>

, ð28Þ

: ð29Þ

� 3�d=2 ð23Þ

ð25Þ

Let us turn to the renormalization of the interaction term g<sup>4</sup> . The perturbative expansion of the four-point function is shown in Figure 4. The diagram (b) in Figure 4, denoted as ΔΓ<sup>ð</sup>4<sup>Þ</sup> <sup>b</sup> , is given by for N = 1:

$$
\Delta\Gamma\_b^{(4)}(p) = \,\_3\mathrm{g}^2\frac{1}{2}\int \frac{d^d q}{(2\pi)^d} \frac{1}{(q^2 - m^2)\left(\left(\left(p+q\right)^2 - m^2\right)\right)}.\tag{30}
$$

As in the calculation of the two-point function, this is regularized as

$$
\Delta\Gamma\_b^{(4)}(p) = \ i\frac{1}{8\pi^2}\frac{1}{2\epsilon}\ g^2,\tag{31}
$$

for d = 4 – E. Let us evaluate the multiplicity of this contribution for N > 1. For N = 1, we have a factor 42 32 2/4!4!=1/2 as shown in Eq. (30). Figure 4c and d gives the same contribution as in Eq. (31), giving the factor 3/2. For N > 1, there is a summation with respect to the components of φ. We have the multiplicity factor for the diagram in Figure 4b as

$$2\left(\frac{1}{4!}\right)^2 2^2 2^2 2N = \frac{N}{18}.\tag{32}$$

Since we obtain the same factor for diagrams in Figure 4c and d, we have N/6 in total. We subtract 1/6 for N = 1 from 3/2 to have 8/6. Finally, the multiplicity factor is given by (N + 8)/6. Then, the four-point function is regularized as

$$
\Delta\Gamma^{(4)}(p) = \operatorname\*{i}\frac{1}{8\pi^2}\frac{N+8}{6}\frac{1}{\epsilon}g^2. \tag{33}
$$

Because g has the dimension 4 – d such as [g] = μ4–<sup>d</sup> , we write g as gμ4–<sup>d</sup> so that g is the dimensionless coupling constant. Now, we have

$$\Gamma^{(4)}(p) = \ -igZ\_4\mu^\epsilon + i\frac{1}{8\pi^2}\frac{N+8}{6}\frac{1}{\epsilon}\ g^2. \tag{34}$$

for d = 4 – E where we neglect μ<sup>E</sup> in the second term. The renormalization constant is determined as

Figure 4. Diagrams for four-point function.

$$Z\_4 = 1 + \frac{N+8}{6\epsilon} \frac{1}{8\pi^2} \text{g.} \tag{35}$$

As a result, the four-point function Γ(4) becomes finite.

#### 2.3. Beta function β(g)

The bare coupling constant is written as <sup>g</sup><sup>0</sup> <sup>¼</sup> Zggμ<sup>4</sup>�<sup>d</sup> ¼ ðZ4=Z<sup>2</sup> <sup>φ</sup>Þgμ<sup>4</sup>�<sup>d</sup>. Since <sup>g</sup><sup>0</sup> is independent of the energy scale, μ, we have μ∂g0=∂μ ¼ 0. This results in

$$
\mu \frac{\partial \mathbf{g}}{\partial \mu} = (d - 4)\mathbf{g} - \mathbf{g}\mu \frac{\partial \mathbf{g}}{\partial \mu} \frac{\partial \ln Z\_{\mathbf{g}}}{\partial \mathbf{g}},\tag{36}
$$

where Zg <sup>¼</sup> <sup>Z</sup>4=Z<sup>2</sup> <sup>φ</sup>. We define the beta function for g as

$$
\beta(\mathbf{g}) = \mu \frac{\partial \mathbf{g}}{\partial \mu'} \tag{37}
$$

where the derivative is evaluated under the condition that the bare g<sup>0</sup> is fixed. Because

$$Z\_{\S} = 1 + \frac{N+8}{6\epsilon} \frac{1}{8\pi^2} \text{g} + O(\text{g}^2). \tag{38}$$

the beta function is given as

$$\beta(\mathbf{g}) = \frac{-\epsilon \mathbf{g}}{1 + \mathbf{g}\frac{\partial \ln Z\_{\mathbf{g}}}{\partial \mathbf{g}}} = -\epsilon \mathbf{g} + \frac{N+8}{6} \frac{1}{8\pi^2} \mathbf{g}^2 + O(\mathbf{g}^3). \tag{39}$$

β(g) up to the order of g<sup>2</sup> is shown as a function of g for d < 4 in Figure 5. For d < 4, there is a non-trivial fixed point at

$$g\_c = \epsilon \frac{48\pi^2}{N+8}.\tag{40}$$

For d = 4, we have only a trivial fixed point at g = 0.

For d = 4 and N = 1, the beta function is given by

$$\beta(\mathbf{g}) = \frac{3}{16\pi^2} \mathbf{g}^2 + \dotsb \,\tag{41}$$

In this case, the β(g) has been calculated up to the fifth order of g [77]:

$$\beta(\mathbf{g}) = \frac{3}{16\pi^2} \mathbf{g}^2 - \frac{17}{3} \frac{1}{\left(16\pi^2\right)^2} \mathbf{g}^3 + \left(\frac{145}{8} + 12\zeta(3)\right) \frac{1}{\left(16\pi^2\right)^3} \mathbf{g}^4 + A\_5 \frac{1}{\left(16\pi^2\right)^4} \mathbf{g}^5,\qquad(42)$$

where

Renormalization Group Theory of Effective Field Theory Models in Low Dimensions http://dx.doi.org/10.5772/intechopen.68214 105

Figure 5. The beta function of g for d < 4. There is a finite fixed point gc.

$$A\_5 = -\left(\frac{3499}{48} + 78\zeta(3) - 18\zeta(4) + 120\zeta(5)\right),\tag{43}$$

and ζ(n) is the Riemann zeta function. The renormalization constant Zg and the beta function β(g) are obtained as a power series of g. We express Zg as

$$Z\_{\mathcal{S}} = 1 + \frac{N+8}{6\epsilon}g + \left(\frac{b\_1}{\epsilon^2} + \frac{b\_2}{\epsilon}\right)g^2 + \left(\frac{c\_1}{\epsilon^3} + \frac{c\_2}{\epsilon^2} + \frac{c\_3}{\epsilon}\right)g^3 + \dotsb,\tag{44}$$

and then β(g) is written as

Z<sup>4</sup> ¼ 1 þ

<sup>∂</sup><sup>μ</sup> ¼ ð<sup>d</sup> � <sup>4</sup>Þ<sup>g</sup> � <sup>g</sup><sup>μ</sup>

βðgÞ ¼ μ

where the derivative is evaluated under the condition that the bare g<sup>0</sup> is fixed. Because

N þ 8 6E

¼ �Eg þ

β(g) up to the order of g<sup>2</sup> is shown as a function of g for d < 4 in Figure 5. For d < 4, there is a

48π<sup>2</sup> N þ 8

gc ¼ E

<sup>β</sup>ðgÞ ¼ <sup>3</sup>

145

<sup>8</sup> <sup>þ</sup> <sup>12</sup>ζð3<sup>Þ</sup> 1

In this case, the β(g) has been calculated up to the fifth order of g [77]:

<sup>2</sup> <sup>g</sup><sup>3</sup> <sup>þ</sup>

1 ð16π<sup>2</sup>Þ

As a result, the four-point function Γ(4) becomes finite.

The bare coupling constant is written as <sup>g</sup><sup>0</sup> <sup>¼</sup> Zggμ<sup>4</sup>�<sup>d</sup> ¼ ðZ4=Z<sup>2</sup>

μ ∂g

<sup>φ</sup>. We define the beta function for g as

Zg ¼ 1 þ

∂ ln Zg ∂g

<sup>β</sup>ðgÞ ¼ �E<sup>g</sup> 1 þ g

For d = 4, we have only a trivial fixed point at g = 0.

For d = 4 and N = 1, the beta function is given by

<sup>16</sup>π<sup>2</sup> <sup>g</sup><sup>2</sup> � <sup>17</sup>

3

of the energy scale, μ, we have μ∂g0=∂μ ¼ 0. This results in

2.3. Beta function β(g)

104 Recent Studies in Perturbation Theory

where Zg <sup>¼</sup> <sup>Z</sup>4=Z<sup>2</sup>

the beta function is given as

non-trivial fixed point at

<sup>β</sup>ðgÞ ¼ <sup>3</sup>

where

N þ 8 6E

1

∂g ∂μ

∂g

1

<sup>8</sup>π<sup>2</sup> <sup>g</sup> <sup>þ</sup> <sup>O</sup>ðg<sup>2</sup>

N þ 8 6

1

<sup>8</sup>π<sup>2</sup> <sup>g</sup><sup>2</sup> <sup>þ</sup> <sup>O</sup>ðg<sup>3</sup>

∂lnZg

<sup>8</sup>π<sup>2</sup> <sup>g</sup>: <sup>ð</sup>35<sup>Þ</sup>

<sup>φ</sup>Þgμ<sup>4</sup>�<sup>d</sup>. Since <sup>g</sup><sup>0</sup> is independent

<sup>∂</sup><sup>g</sup> , <sup>ð</sup>36<sup>Þ</sup>

Þ, ð38Þ

: ð40Þ

<sup>16</sup>π<sup>2</sup> <sup>g</sup><sup>2</sup> <sup>þ</sup> <sup>⋯</sup>: <sup>ð</sup>41<sup>Þ</sup>

<sup>3</sup> <sup>g</sup><sup>4</sup> <sup>þ</sup> <sup>A</sup><sup>5</sup>

1 ð16π<sup>2</sup>Þ

4 g5

, ð42Þ

ð16π<sup>2</sup>Þ

Þ: ð39Þ

<sup>∂</sup><sup>μ</sup> , <sup>ð</sup>37<sup>Þ</sup>

$$\begin{split} \boldsymbol{\beta}(\mathbf{g}) &= -\epsilon \mathbf{g} + \epsilon \mathbf{g}^2 \left[ \frac{N+8}{6\epsilon} + 2 \left( \frac{b\_1}{\epsilon^2} + \frac{b\_2}{\epsilon} \right) \mathbf{g} + \frac{(N+8)^2}{36\epsilon^2} \mathbf{g} + \cdots \right] \\ &= -\epsilon \mathbf{g} + \frac{N+8}{6} \mathbf{g}^2 - \frac{9N+42}{36} \mathbf{g}^3 + \cdots \end{split} \tag{45}$$

Here, the factor 1/8π<sup>2</sup> is included in g. The terms of order 1/E <sup>2</sup> are cancelled because of

$$b\_1 = \quad -\frac{\left(N+8\right)^2}{72}.\tag{46}$$

In general, the nth order term in β(g) is given by n!gn . The function β(g) is expected to have the form

$$\beta(\mathbf{g}) = \ -\epsilon \mathbf{g} + \frac{N+8}{6} \mathbf{g}^2 + \ \cdots + n! a^n n^b c \mathbf{g}^n + \ \cdots \tag{47}$$

where a, b and c are constants.

#### 2.4. n-point function and anomalous dimension

Let us consider the n-point function Γ(n) . The bare and renormalized n-point functions are denoted as Γ<sup>ð</sup>n<sup>Þ</sup> <sup>B</sup> ðpi , g0, m0, <sup>μ</sup><sup>Þ</sup> and <sup>Γ</sup><sup>ð</sup>n<sup>Þ</sup> <sup>R</sup> ðpi , g, m, μÞ, respectively, where pi (i = 1,…, n) indicate momenta. The energy scale μ indicates the renormalization point. Γ<sup>ð</sup>n<sup>Þ</sup> <sup>R</sup> has the mass dimension <sup>n</sup> <sup>+</sup> <sup>d</sup> – nd/2: <sup>½</sup>Γ<sup>ð</sup>n<sup>Þ</sup> <sup>R</sup> � ¼ <sup>μ</sup><sup>n</sup>þd�nd=2. These quantities are related by the renormalization constant <sup>Z</sup><sup>φ</sup> as

$$
\Gamma\_{\mathbb{R}}^{(n)}(p\_{i^\prime} \mathbb{g}, m^2, \mu) \;= \, \, \mathcal{Z}\_{\phi}^{n/2} \Gamma\_{\mathbb{B}}^{(n)}(p\_{i^\prime} \mathbb{g}\_{0^\prime} m\_{0^\prime}^2 \mu). \tag{48}
$$

Here, we consider the massless case and omit the mass. Because the bare quantity Γ<sup>ð</sup>n<sup>Þ</sup> <sup>B</sup> is independent of μ, we have

$$\frac{d}{d\mu}\Gamma\_B^{(n)} = 0.\tag{49}$$

This leads to

$$
\mu \frac{d}{d\mu} \left( Z\_{\phi}^{-n/2} \Gamma\_{\mathbb{R}}^{(n)} \right) = 0. \tag{50}
$$

Then we obtain the equation for Γ<sup>ð</sup>n<sup>Þ</sup> R :

$$
\lambda \left( \mu \frac{\partial}{\partial \mu} + \mu \frac{\partial \mathbf{g}}{\partial \mu} \frac{\partial}{\partial \mathbf{g}} - \frac{n}{2} \boldsymbol{\gamma}\_{\phi} \right) \Gamma\_{\mathcal{R}}^{(n)} (\boldsymbol{p}\_{i}, \mathbf{g}, \mu) = 0,\tag{51}
$$

where γφ is defined as

$$
\gamma\_{\phi} = \, \mu \frac{\partial}{\partial \mu} \text{lnZ}\_{\phi}.\tag{52}
$$

A general solution of the renormalization equation is written as

$$\Gamma\_R^{(n)}(p\_{i'}\mathbf{g},\mu) = \exp\left(\frac{n}{2}\left[\frac{\mathcal{V}\_\phi(\mathbf{g'})}{\beta(\mathbf{g'})}d\mathbf{g'}\right]f^{(n)}(p\_{i'}\mathbf{g},\mu),\tag{53}$$

where

$$f^{(n)}(p\_{i'} \mathbf{g}, \mu) = F\left(p\_{i'} \ln \mu - \int\_{\mathcal{g}\_1}^{\mathcal{S}} \frac{1}{\beta(\mathbf{g'})} d\mathbf{g'}\right),\tag{54}$$

for a function F and a constant g1. We suppose that β(g) has a zero at g = gc. Near the fixed point gc, by approximating γφðg<sup>0</sup> <sup>Þ</sup> by γφðgcÞ, <sup>Γ</sup><sup>ð</sup>n<sup>Þ</sup> <sup>R</sup> is expressed as

Renormalization Group Theory of Effective Field Theory Models in Low Dimensions http://dx.doi.org/10.5772/intechopen.68214 107

$$
\Gamma\_{\mathbb{R}}^{(n)}(p\_{i'} \mathbb{g}\_{c'} \mu) = \ \mu^{\#\prime\_{\phi}(\mathbb{g}\_{c})} f^{(n)}(p\_{i'} \mathbb{g}\_{c'} \mu). \tag{55}
$$

In general, we define γ(g) as

$$\gamma(\mathbf{g})\ln\mu = \int\_{\mathbf{g}\_1}^{\mathbf{g}} \frac{\gamma\_\phi(\mathbf{g}')}{\beta(\mathbf{g}')}d\mathbf{g}',\tag{56}$$

Then, we obtain

2.4. n-point function and anomalous dimension

, g0, m0, <sup>μ</sup><sup>Þ</sup> and <sup>Γ</sup><sup>ð</sup>n<sup>Þ</sup>

Γ<sup>ð</sup>n<sup>Þ</sup> <sup>R</sup> ðpi <sup>R</sup> ðpi

momenta. The energy scale μ indicates the renormalization point. Γ<sup>ð</sup>n<sup>Þ</sup>

, g, m<sup>2</sup>

μ d dμ � Z�n=<sup>2</sup> <sup>φ</sup> <sup>Γ</sup><sup>ð</sup>n<sup>Þ</sup> R �

R :

� �

γφ ¼ μ

n 2 ð g

0 B@

g1

γφðg<sup>0</sup> Þ βðg<sup>0</sup> <sup>Þ</sup> dg<sup>0</sup>

, lnμ �

for a function F and a constant g1. We suppose that β(g) has a zero at g = gc. Near the fixed point

<sup>R</sup> is expressed as

ðg g1

!

1 βðg<sup>0</sup> Þ dg<sup>0</sup>

1 CAf ðnÞ ðpi

∂ ∂μ

μ ∂ ∂μ þ μ ∂g ∂μ ∂ <sup>∂</sup><sup>g</sup> � <sup>n</sup> <sup>2</sup> γφ

A general solution of the renormalization equation is written as

, g, μÞ ¼ exp

<sup>Þ</sup> by γφðgcÞ, <sup>Γ</sup><sup>ð</sup>n<sup>Þ</sup>

, g, μÞ ¼ F pi

Γ<sup>ð</sup>n<sup>Þ</sup> <sup>R</sup> ðpi

> f ðnÞ ðpi

, <sup>μ</sup>Þ ¼ <sup>Z</sup><sup>n</sup>=<sup>2</sup>

Here, we consider the massless case and omit the mass. Because the bare quantity Γ<sup>ð</sup>n<sup>Þ</sup>

d dμ Γ<sup>ð</sup>n<sup>Þ</sup>

. The bare and renormalized n-point functions are

, g, m, μÞ, respectively, where pi (i = 1,…, n) indicate

<sup>B</sup> ¼ 0: ð49Þ

¼ 0: ð50Þ

, g, μÞ ¼ 0; ð51Þ

, g, μÞ, ð53Þ

, ð54Þ

lnZφ: ð52Þ

<sup>R</sup> � ¼ <sup>μ</sup><sup>n</sup>þd�nd=2. These quantities are related by the renormalization constant <sup>Z</sup><sup>φ</sup> as

, g0, m<sup>2</sup>

<sup>φ</sup> <sup>Γ</sup><sup>ð</sup>n<sup>Þ</sup> <sup>B</sup> ðpi

> Γ<sup>ð</sup>n<sup>Þ</sup> <sup>R</sup> ðpi

<sup>R</sup> has the mass dimension

<sup>B</sup> is

<sup>0</sup>, μÞ: ð48Þ

Let us consider the n-point function Γ(n)

<sup>B</sup> ðpi

106 Recent Studies in Perturbation Theory

independent of μ, we have

Then we obtain the equation for Γ<sup>ð</sup>n<sup>Þ</sup>

where γφ is defined as

gc, by approximating γφðg<sup>0</sup>

where

denoted as Γ<sup>ð</sup>n<sup>Þ</sup>

<sup>n</sup> <sup>+</sup> <sup>d</sup> – nd/2: <sup>½</sup>Γ<sup>ð</sup>n<sup>Þ</sup>

This leads to

$$
\Gamma\_{\mathbb{R}}^{(n)}(p\_{i'} \mathbb{g}, \mu) = \ \mu^{\#\langle \mathfrak{g} \rangle} f^{(n)}(p\_{i'} \mathbb{g}, \mu). \tag{57}
$$

Under a scaling pi ! ρpi , Γ<sup>ð</sup>n<sup>Þ</sup> <sup>R</sup> is expected to behave as

$$
\Gamma\_{\mathbb{R}}^{(n)}(\rho p\_{i'} \text{ g}\_{c'} \ \mu) = \ \rho^{n+d-nd/2} \Gamma\_{\mathbb{R}}^{(n)}(p\_{i'} \text{ g}\_{c'} \ \mu/\rho),
\tag{58}
$$

because Γ<sup>ð</sup>n<sup>Þ</sup> <sup>R</sup> has the mass dimension n þ d � nd=2. In fact, Figure 4b gives a contribution being proportional to

$$\begin{split} \left[g^2(\mu^{4-d})^2\right] \int d^d q \frac{1}{q^2(\rho p + q)^2} &= \left. g^2(\mu^{4-d})^2 \rho^{d-4} \right\| d^d q \frac{1}{q^2(p+q)^2} \\ &= \left. \rho^{4-d} g^2 \left(\frac{\mu}{\rho}\right)^{2(4-d)} \right\| d^d q \frac{1}{q^2(p+q)^2} \end{split} \tag{59}$$

after the scaling pi ! ρpi for n = 4. We employ Eq. (58) for n = 2

$$\begin{split} \Gamma\_{\mathbb{R}}^{(2)}(\rho p\_{i^{\prime}} \text{ g}\_{\varepsilon^{\prime}} \, \mu) &= \, \rho^{2} \Gamma\_{\mathbb{R}}^{(2)}(p\_{i^{\prime}} \text{ g}\_{\varepsilon^{\prime}} \, \mu/\rho) = \, \rho^{2} \binom{\mu}{\rho}^{\mathbb{V}} f^{(2)}(p\_{i^{\prime}} \text{ g}\_{\varepsilon^{\prime}} \, \mu/\rho) \\ &= \, \rho^{2-\gamma} \mu^{\gamma} f^{(2)}(p\_{i^{\prime}} \text{ g}\_{\varepsilon^{\prime}} \, \mu/\rho) = \, \rho^{2-\gamma} \Gamma\_{\mathbb{R}}^{(2)}(p\_{i^{\prime}} \text{ g}\_{\varepsilon^{\prime}} \, \mu/\rho) . \end{split} \tag{60}$$

This indicates

$$
\Gamma^{(2)}(p) \, = \, p^{2-\eta} \, = \, p^{2-\gamma} \, = \, (p^2)^{1-\gamma/2} \,. \tag{61}
$$

Thus, the anomalous dimension η is given by η = γ. From the definition of γ(g) in Eq. (56), we have

$$
\gamma\_{\phi}(\mathbf{g}) = \left. \gamma(\mathbf{g}) + \beta(\mathbf{g}) \frac{\partial \gamma(\mathbf{g})}{\partial \mathbf{g}} \ln \mu. \tag{62}
$$

At the fixed point g = gc, this leads to

$$
\eta = \underline{\gamma} = \underline{\gamma}(\underline{\mathfrak{g}}\_c) = \underline{\gamma}\_\phi(\underline{\mathfrak{g}}\_c). \tag{63}
$$

The exponent η shows the fluctuation effect near the critical point.

The Green's function <sup>G</sup>ðpÞ ¼ <sup>Γ</sup>ð2<sup>Þ</sup> ðpÞ �<sup>1</sup> is given by

$$G(p) = \frac{1}{p^{2-\eta}}.\tag{64}$$

The Fourier transform of G(p) in d dimensions is evaluated as

$$G(r) = \int \frac{1}{p^{2-\eta}} e^{ip\cdot r} d^d p = \,\,\Omega\_d \frac{1}{r^{d-2+\eta}} \frac{\pi}{2\Gamma(4-\eta-d)\sin\left((4-\eta-d)\pi/2\right)}.\tag{65}$$

When 4 – η – d is small near four dimensions, G(r) is approximated as

$$G(r) \approx \Omega\_d \frac{1}{r^{d-2+\eta}}.\tag{66}$$

The definition of γφ in Eq. (52) results in

$$
\gamma\_{\phi}(\mathbf{g}) = \mu \frac{\partial \mathbf{g}}{\partial \mu} \frac{\partial}{\partial \mathbf{g}} \ln \mathbf{Z}\_{\phi} = \beta(\mathbf{g}) \frac{\partial}{\partial \mathbf{g}} \ln \mathbf{Z}\_{\phi}.\tag{67}
$$

Up to the lowest order of g, γφ is given by

$$\begin{split} \mathcal{V}\_{\phi} &= \left( -\frac{1}{8\epsilon} \frac{N+1}{9} \frac{1}{\left( 8\pi^{2} \right)^{2}} g \right) \beta(g) + \mathcal{O}(g^{3}) \\ &= \frac{N+2}{72} \frac{1}{\left( 8\pi^{2} \right)^{2}} g^{2} + \mathcal{O}(g^{3}). \end{split} \tag{68}$$

At the critical point g = gc, where

$$\frac{1}{8\pi^2}\mathbf{g}\_c = \frac{6\mathbf{\bar{e}}}{N+8} \tag{69}$$

the anomalous dimension is given as

$$
\eta = \mathcal{V}\_{\phi}(\mathbf{g}\_c) = \frac{N+2}{2(N+8)} \epsilon^2 + \mathcal{O}(\epsilon^3). \tag{70}
$$

For N = 1 and E = 1, we have η = 1/54.

#### 2.5. Mass renormalization

Let us consider the massive case m 6¼ 0. This corresponds to the case with T > Tc in a phase transition. The bare mass m<sup>0</sup> m and renormalized mass m are related through the relation <sup>m</sup><sup>2</sup> <sup>¼</sup> <sup>m</sup><sup>2</sup> <sup>0</sup>Zφ=Z2. The condition μ ∂ m0=∂μ ¼ 0 leads to

Renormalization Group Theory of Effective Field Theory Models in Low Dimensions http://dx.doi.org/10.5772/intechopen.68214 109

$$
\mu \frac{\partial \ln m}{\partial \mu} = \mu \frac{\partial}{\partial \mu} \ln \frac{Z\_{\phi}}{Z\_{2}}.\tag{71}
$$

From Eq. (50), the equation for Γ<sup>ð</sup>n<sup>Þ</sup> <sup>R</sup> is

$$
\hbar \left[ \mu \frac{\partial}{\partial \mu} + \beta(g) \frac{\partial}{\partial g} - \frac{n}{2} \gamma\_{\phi} + \mu \frac{\partial}{\partial \mu} \ln \left( \frac{Z\_{\phi}}{Z\_{2}} \right) \cdot m^{2} \frac{\partial}{\partial m^{2}} \right] \Gamma\_{\mathbb{R}}^{(n)} (p\_{\nu} \text{ g. } \mu \text{ } m^{2}) = 0. \tag{72}$$

We define the exponent ν by

$$\frac{1}{\nu} - 2 = \mu \frac{\partial}{\partial \mu} \ln \left( \frac{Z\_2}{Z\_\phi} \right) \tag{73}$$

then

<sup>G</sup>ðpÞ ¼ <sup>1</sup>

1 rd�2þ<sup>η</sup>

GðrÞ ≈ Ω<sup>d</sup>

∂g ∂μ ∂ ∂g

N þ 1 9

1

<sup>η</sup> <sup>¼</sup> γφðgcÞ ¼ <sup>N</sup> <sup>þ</sup> <sup>2</sup>

!

1 ð8π<sup>2</sup>Þ 2 g

1 ð8π<sup>2</sup>Þ

<sup>8</sup>π<sup>2</sup> gc <sup>¼</sup> <sup>6</sup><sup>∈</sup>

N þ 8

2 E <sup>2</sup> <sup>þ</sup> <sup>O</sup>ð<sup>E</sup> 3

2ðN þ 8Þ

Let us consider the massive case m 6¼ 0. This corresponds to the case with T > Tc in a phase transition. The bare mass m<sup>0</sup> m and renormalized mass m are related through the relation

The Fourier transform of G(p) in d dimensions is evaluated as

p ¼ Ω<sup>d</sup>

When 4 – η – d is small near four dimensions, G(r) is approximated as

γφðgÞ ¼ μ

γφ ¼ � <sup>1</sup>

8E

<sup>¼</sup> <sup>N</sup> <sup>þ</sup> <sup>2</sup> 72

GðrÞ ¼

108 Recent Studies in Perturbation Theory

ð 1 p<sup>2</sup>�<sup>η</sup> e ip�r dd

The definition of γφ in Eq. (52) results in

Up to the lowest order of g, γφ is given by

At the critical point g = gc, where

the anomalous dimension is given as

For N = 1 and E = 1, we have η = 1/54.

<sup>0</sup>Zφ=Z2. The condition μ ∂ m0=∂μ ¼ 0 leads to

2.5. Mass renormalization

<sup>m</sup><sup>2</sup> <sup>¼</sup> <sup>m</sup><sup>2</sup>

<sup>p</sup><sup>2</sup>�<sup>η</sup>: <sup>ð</sup>64<sup>Þ</sup>

ð4 � η � dÞπ=2

rd�2þ<sup>η</sup> : <sup>ð</sup>66<sup>Þ</sup>

lnZφ: ð67Þ

, ð69Þ

Þ: ð70Þ

� : <sup>ð</sup>65<sup>Þ</sup>

ð68Þ

π

�

2Γð4 � η � dÞ sin

1

lnZ<sup>φ</sup> <sup>¼</sup> <sup>β</sup>ðg<sup>Þ</sup> <sup>∂</sup>

<sup>2</sup> <sup>g</sup><sup>2</sup> <sup>þ</sup> <sup>O</sup>ðg<sup>3</sup>

∂g

<sup>β</sup>ðgÞ þ <sup>O</sup>ðg<sup>3</sup><sup>Þ</sup>

Þ:

$$
\hbar \left[ \mu \frac{\partial}{\partial \mu} + \beta(\mathbf{g}) \frac{\partial}{\partial \mathbf{g}} - \frac{n}{2} \gamma\_{\phi} - \left( \frac{1}{\nu} - 2 \right) m^2 \frac{\partial}{\partial m^2} \right] \Gamma\_R^{(n)}(p\_{\nu} \text{ g. } \mu \text{ } m^2) = 0. \tag{74}$$

At the critical point g = gc, we obtain

$$
\left[\mu \frac{\partial}{\partial \mu} - \frac{n}{2} \eta - \zeta m^2 \frac{\partial}{\partial m^2} \right] \Gamma\_R^{(n)} (p\_{i'} \text{ g}\_{c'} \ \mu \ \ \_\prime m^2) = 0,\tag{75}
$$

where γφ = η and we set

$$
\zeta = \frac{1}{\nu} - 2.\tag{76}
$$

At g = gc, Γ<sup>ð</sup>n<sup>Þ</sup> <sup>R</sup> has the form

$$
\Gamma\_{\mathbb{R}}^{(n)}(p\_{i'} \ g\_{c'} \ \mu\_{\prime} \ m^2) = \ \mu^{\frac{n}{2}} F^{(n)}(p\_{i'} \ \mu m^{2/\zeta}) . \tag{77}
$$

because this satisfies Eq. (75).

In the scaling pi ! ρpi , we adopt

$$
\Gamma\_{\mathbb{R}}^{(n)}(\rho p\_{i'} \ g\_{c'} \ \mu\_{\prime} \ m^2) = \ \rho^{n+d-nd/2} \Gamma\_{\mathbb{R}}^{(n)}(p\_{i'} \ g\_{c'} \ \mu/\rho, \ m^2/\rho^2). \tag{78}
$$

From Eq. (77), we have

$$
\Gamma\_{\mathbb{R}}^{(n)}(\mathbf{k}\_{i\prime} \ g\_{c\prime}, \mu, m^2) = \rho^{n+d-nd/2-m\eta/2} \mu^{\#\eta} \mathcal{F}^{(n)}\Big(\rho^{-1} \mathbf{k}\_{i\prime} \ \rho^{-1} \mu (\rho^{-2} m^2)^{1/\zeta}\Big), \tag{79}
$$

where we put <sup>ρ</sup>pi <sup>¼</sup> ki. We assume that <sup>F</sup>(n) depends only on <sup>ρ</sup>�1ki. We choose <sup>ρ</sup> as

$$
\rho = (\mu m^{2/\zeta})^{\zeta/(\zeta+2)} = \left. \mu \left(\frac{m^2}{\mu^2}\right)^{1/(\zeta+2)} \right. \tag{80}
$$

This satisfies <sup>ρ</sup>�<sup>1</sup>μðρ�<sup>2</sup>m<sup>2</sup><sup>Þ</sup> <sup>1</sup>=<sup>ς</sup> <sup>¼</sup> 1 and results in

$$\Gamma\_{\mathbb{R}}^{(n)}(\mathbf{k}\_{\prime} \ g\_{c}, \ \mu, \ m^{2}) = \mu^{d + \frac{\mu}{2}(2 - d - \eta)} \left(\frac{m^{2}}{\mu^{2}}\right)^{\left\{d + \frac{\mu}{2}(2 - d - \eta)\right\} \frac{1}{\kappa^{2}}} \mu^{\frac{\mu}{2}\eta} F^{(n)}\left(\mu^{-1} \left(\frac{m^{2}}{\mu^{2}}\right)^{-\frac{1}{\kappa + 2}} k\_{l}\right). \tag{81}$$

We take μ as a unit by setting μ = 1, so that Γ<sup>ð</sup>n<sup>Þ</sup> <sup>R</sup> is written as

$$
\Gamma\_{\mathbb{R}}^{(n)}(k\_{i\prime} \text{ g}\_{\mathcal{L}'} \mathbf{1}, \; m^2) = \; m^{2\upsilon \left\{ d + \frac{\mu}{2} (2 - d + \eta) \right\}} F^{(n)}(k\_i m^{-2\upsilon}), \tag{82}
$$

because ς þ 2 ¼ 1=ν. We can define the correlation length ξ by

$$(m^2)^{-\nu} = \ \xi. \tag{83}$$

The two-point function is written as

$$
\Gamma\_R^{(2)}(k, m^2) = \ m^{2\upsilon(2-\eta)} F^{(2)}(km^{-2\upsilon}).\tag{84}
$$

Now let us turn to the evaluation of ν. Since γφ ¼ μ ∂ lnZφ=∂μ, from Eq. (73) ν is given by

$$\frac{1}{\nu} = 2 + \mu \frac{\partial}{\partial \mu} \ln \left( \frac{Z\_2}{Z\_\phi} \right) = 2 + \beta(g) \frac{\partial}{\partial g} \ln Z\_2 - \gamma\_\phi(g). \tag{85}$$

The renormalization constant Z<sup>2</sup> is determined from the corrections to the bare mass m0. The one-loop correction, shown in Figure 6, is given by

$$i\Sigma(p^2) = i\frac{N+2}{6}g\left\{\frac{d^d k}{(2\pi)^d}\frac{1}{k^2 - m\_0^2},\tag{86}$$

where the multiplicity factor is (8 + 4N)/4!. This is regularized as

$$\Sigma(p^2) = \frac{N+2}{6}g \left\{ \frac{d^d k}{(2\pi)^d} \frac{1}{k\_E^2 + m\_0^2} = -\frac{N+2}{6}g \frac{1}{8\pi^2} m\_0^2 \frac{1}{\epsilon'} \right. \tag{87}$$

for d = 4–E. Therefore the renormalized mass is

Figure 6. Corrections to the mass term. Multiplicity weights are 8 for (a) and 2N for (b).

Renormalization Group Theory of Effective Field Theory Models in Low Dimensions http://dx.doi.org/10.5772/intechopen.68214 111

$$m^2 = m\_0^2 + \Sigma(p^2) = m\_0^2 \left( 1 - \frac{N+2}{6\epsilon} \frac{1}{8\pi^2} g \right) \tag{88}$$

Z<sup>2</sup> is determined to cancel the divergence in the form m<sup>2</sup> Z2/Zφ. The result is

$$Z\_2 = 1 + \frac{N+2}{6\epsilon} \frac{1}{8\pi^2} \text{g.} \tag{89}$$

Then, we have

Γ<sup>ð</sup>n<sup>Þ</sup>

110 Recent Studies in Perturbation Theory

<sup>R</sup> <sup>ð</sup>ki, gc, <sup>μ</sup>, m<sup>2</sup>

The two-point function is written as

We take μ as a unit by setting μ = 1, so that Γ<sup>ð</sup>n<sup>Þ</sup>

Γ<sup>ð</sup>n<sup>Þ</sup>

1 <sup>ν</sup> <sup>¼</sup> <sup>2</sup> <sup>þ</sup> <sup>μ</sup>

one-loop correction, shown in Figure 6, is given by

<sup>Σ</sup>ðp<sup>2</sup>

for d = 4–E. Therefore the renormalized mass is

Þ ¼ <sup>μ</sup><sup>d</sup>þ<sup>n</sup>

<sup>R</sup> <sup>ð</sup>ki, gc, <sup>1</sup>, m<sup>2</sup>

because ς þ 2 ¼ 1=ν. We can define the correlation length ξ by

Γ<sup>ð</sup>2<sup>Þ</sup> <sup>R</sup> <sup>ð</sup>k, m<sup>2</sup>

> ∂ ∂μ

<sup>Σ</sup>ðp<sup>2</sup> Þ ¼ i

where the multiplicity factor is (8 + 4N)/4!. This is regularized as

Figure 6. Corrections to the mass term. Multiplicity weights are 8 for (a) and 2N for (b).

Þ ¼ <sup>N</sup> <sup>þ</sup> <sup>2</sup> <sup>6</sup> <sup>g</sup>

ln <sup>Z</sup><sup>2</sup> Z<sup>φ</sup> � �

ð d<sup>d</sup> k ð2πÞ d

<sup>2</sup>ð2�d�η<sup>Þ</sup> <sup>m</sup><sup>2</sup>

μ2 � �{dþ<sup>n</sup>

Þ ¼ <sup>m</sup><sup>2</sup><sup>ν</sup> <sup>d</sup>þ<sup>n</sup>

Þ ¼ <sup>m</sup><sup>2</sup>νð2�η<sup>Þ</sup>

Now let us turn to the evaluation of ν. Since γφ ¼ μ ∂ lnZφ=∂μ, from Eq. (73) ν is given by

ðm2 Þ <sup>2</sup>ð2�d�ηÞ} <sup>1</sup> ζþ2 μ n

<sup>R</sup> is written as

<sup>2</sup> f g <sup>ð</sup>2�dþη<sup>Þ</sup> Fðn<sup>Þ</sup>

Fð2<sup>Þ</sup>

<sup>¼</sup> <sup>2</sup> <sup>þ</sup> <sup>β</sup>ðg<sup>Þ</sup> <sup>∂</sup>

The renormalization constant Z<sup>2</sup> is determined from the corrections to the bare mass m0. The

ð dd k ð2πÞ d

1 k 2 <sup>E</sup> <sup>þ</sup> <sup>m</sup><sup>2</sup> 0

= +

N þ 2 <sup>6</sup> <sup>g</sup> <sup>ð</sup>km�2<sup>ν</sup>

∂g

1 k <sup>2</sup> � <sup>m</sup><sup>2</sup> 0

¼ � <sup>N</sup> <sup>þ</sup> <sup>2</sup> <sup>6</sup> <sup>g</sup>

1 <sup>8</sup>π<sup>2</sup> <sup>m</sup><sup>2</sup> 0 1 E

(*a*) (*b*)

<sup>2</sup><sup>η</sup>Fðn<sup>Þ</sup> <sup>μ</sup>�<sup>1</sup> <sup>m</sup><sup>2</sup>

<sup>ð</sup>kim�2<sup>ν</sup>

�<sup>ν</sup> <sup>¼</sup> <sup>ξ</sup>: <sup>ð</sup>83<sup>Þ</sup>

μ2 � �� <sup>1</sup> ζþ2 ki

!

: ð81Þ

Þ, ð82Þ

Þ: ð84Þ

lnZ<sup>2</sup> � γφðgÞ: ð85Þ

, ð86Þ

, ð87Þ

$$
\beta(\mathbf{g}) \frac{\partial}{\partial \mathbf{g}} \ln \mathbf{Z}\_2 = -\frac{N+2}{6} \frac{1}{8\pi^2} \mathbf{g} + \mathcal{O}(\mathbf{g}^2). \tag{90}
$$

Eq. (85) is written as

$$\frac{1}{\nu} = 2 - \frac{N+2}{6} \frac{1}{8\pi^2} \text{g}\_c - \ \eta = 2 - \frac{N+2}{N+8} \epsilon + \mathcal{O}(\epsilon^2),\tag{91}$$

where we put g = gc and used η ¼ γφðgÞ¼ðN þ 2Þ= 2ðN þ 8Þ 2 � E. Now the exponent ν is

$$\nu = \frac{1}{2} \left( 1 + \frac{N+2}{2(N+8)} \epsilon \right) + \mathcal{O}(\epsilon^2). \tag{92}$$

In the mean-field approximation, ν = 1/2. This formula of ν contains the fluctuation effect near the critical point. For N = 1 and E = 1, we have ν = 1/2 + 1/12 = 7/12.

## 3. Non-linear sigma model

#### 3.1. Lagrangian

The Lagrangian of the non-linear sigma model is

$$\mathcal{L} = \frac{1}{2g} (\partial\_{\mu}\phi)^{2} \, , \tag{93}$$

where φ is a real N-component field φ = (φ1,…,φN) with the constraint φ<sup>2</sup> = 1. This model has an O(N) invariance. The field φ is represented as

$$
\phi = \begin{pmatrix} \sigma \ \pi\_1 \ \pi\_2 \ \cdots \ \pi\_{N-1} \end{pmatrix} \tag{94}
$$

with the condition <sup>ο</sup><sup>2</sup> <sup>þ</sup> <sup>π</sup><sup>2</sup> <sup>1</sup> <sup>þ</sup> <sup>⋯</sup> <sup>þ</sup> <sup>π</sup><sup>2</sup> <sup>N</sup>�<sup>1</sup> <sup>¼</sup> 1. The fields <sup>π</sup><sup>i</sup> (<sup>i</sup> = 1, …, <sup>N</sup> – 1) are regarded as representing fluctuations. The Lagrangian is given by

$$\mathcal{L} = \frac{1}{2g} \{ (\partial\_{\mu} \sigma)^{2} + (\partial\_{\mu} \pi\_{i})^{2} \}, \tag{95}$$

where summation is assumed for index i. In this Section we consider the Euclidean Lagrangian from the beginning. Using the constraint <sup>σ</sup><sup>2</sup> <sup>þ</sup> <sup>π</sup><sup>2</sup> <sup>i</sup> ¼ 1, the Lagrangian is written in the form

$$\mathcal{L} = \frac{1}{2g} (\partial\_{\mu} \pi\_i)^2 + \frac{1}{2g} \frac{1}{1 - \pi\_i^2} (\pi\_i \partial\_{\mu} \pi\_i)^2 \tag{96}$$

$$=\frac{1}{2g}(\eth\_{\mu}\pi\_{i})^{2}+\frac{1}{2g}(\pi\_{i}\eth\_{\mu}\pi\_{i})^{2}+\cdots \tag{97}$$

The second term in the right-hand side indicates the interaction between π<sup>i</sup> fields. The diagram for this interaction is shown in Figure 7.

Here, let us check the dimension of the field and coupling constant. Since <sup>½</sup>L� ¼ <sup>μ</sup><sup>d</sup>, we obtain <sup>½</sup>π� ¼ <sup>μ</sup><sup>0</sup> (dimensionless) and <sup>½</sup>g� ¼ <sup>μ</sup><sup>2</sup>�<sup>d</sup>. <sup>g</sup><sup>0</sup> and <sup>g</sup> are used to denote the bare coupling constant and renormalized coupling constant, respectively. The bare and renormalized fields are indicated by πBi and πRi, respectively. We define the renormalization constants Zg and Z by

$$\mathbf{g}\_0 = \mathbf{g}\mu^{2-d}\mathbf{Z}\_{\mathbb{S}'} \tag{98}$$

$$
\pi\_{\text{Bi}} = \sqrt{\underline{Z}} \,\, \pi\_{\text{Bi}} \tag{99}
$$

where g is the dimensionless coupling constant. Then, the Lagrangian is expressed in terms of renormalized quantities:

$$\mathcal{L} = \frac{\mu^{d-2} Z}{2g Z\_{\mathcal{g}}} \left\{ (\partial\_{\mu} \pi\_{\text{Ri}})^2 + \frac{1}{4} (\partial\_{\mu} \pi\_{\text{Ri}}^2)^2 + \cdots \right\}. \tag{100}$$

In order to avoid the infrared divergence at d = 2, we add the Zeeman term to the Lagrangian which is written as

Figure 7. Lowest order interaction for πi.

Renormalization Group Theory of Effective Field Theory Models in Low Dimensions http://dx.doi.org/10.5772/intechopen.68214 113

$$\mathcal{L}\_{\mathbf{Z}} = \frac{H\_{\rm B}}{\mathcal{g}\_{0}} \boldsymbol{\sigma} = \frac{H\_{\rm B}}{\mathcal{g}\_{0}} \left( 1 - \frac{\mathbf{Z}}{2} \pi\_{\rm Ri}^{2} - \frac{\mathbf{Z}^{2}}{8} \pi\_{\rm Ri}^{4} + \dotsb \right) \tag{101}$$

$$\mathbf{k} = \text{const.} \, - \, H\_B \frac{\mathbf{Z}}{2 \mathbf{g} \mathbf{Z}\_3} \boldsymbol{\mu}^{d-2} \pi\_{\text{Ri}}^2 - \, H\_B \frac{\mathbf{Z}^2}{8 \mathbf{g} \mathbf{Z}\_3} \boldsymbol{\mu}^{d-2} (\pi\_{\text{Ri}}^2)^2. \tag{102}$$

Here, HB is the bare magnetic field and the renormalized magnetic field H is defined as

$$H = \frac{\sqrt{Z}}{Z\_{\ $}} H\_{\$ } \tag{103}$$

Then, the Zeeman term is given by

$$\mathcal{L}\_z = \text{const.} \ -\frac{\sqrt{Z}}{2g} H \mu^{d-2} \pi\_{Ri}^2 - \frac{Z^2}{8g} H \mu^{d-2} (\pi\_{Ri}^2)^2 + \dotsb \tag{104}$$

#### 3.2. Two-point function

<sup>L</sup> <sup>¼</sup> <sup>1</sup> 2g

<sup>L</sup> <sup>¼</sup> <sup>1</sup> 2g ð∂μπiÞ 2 þ 1 2g

<sup>¼</sup> <sup>1</sup> 2g ð∂μπiÞ 2 þ 1 2g

<sup>L</sup> <sup>¼</sup> <sup>μ</sup><sup>d</sup>�<sup>2</sup><sup>Z</sup> 2gZg

*p*

*− p* + *q*

from the beginning. Using the constraint <sup>σ</sup><sup>2</sup> <sup>þ</sup> <sup>π</sup><sup>2</sup>

112 Recent Studies in Perturbation Theory

for this interaction is shown in Figure 7.

renormalized quantities:

which is written as

Figure 7. Lowest order interaction for πi.

fð∂μσÞ

where summation is assumed for index i. In this Section we consider the Euclidean Lagrangian

The second term in the right-hand side indicates the interaction between π<sup>i</sup> fields. The diagram

Here, let us check the dimension of the field and coupling constant. Since <sup>½</sup>L� ¼ <sup>μ</sup><sup>d</sup>, we obtain <sup>½</sup>π� ¼ <sup>μ</sup><sup>0</sup> (dimensionless) and <sup>½</sup>g� ¼ <sup>μ</sup><sup>2</sup>�<sup>d</sup>. <sup>g</sup><sup>0</sup> and <sup>g</sup> are used to denote the bare coupling constant and renormalized coupling constant, respectively. The bare and renormalized fields are indicated by πBi and πRi, respectively. We define the renormalization constants Zg and Z by

<sup>g</sup><sup>0</sup> <sup>¼</sup> <sup>g</sup>μ<sup>2</sup>�<sup>d</sup>

<sup>π</sup>Bi <sup>¼</sup> ffiffiffiffiffi Z

ð∂μπRiÞ

where g is the dimensionless coupling constant. Then, the Lagrangian is expressed in terms of

2 þ 1 4 <sup>ð</sup>∂μπ<sup>2</sup> RiÞ <sup>2</sup> <sup>þ</sup> <sup>⋯</sup>

In order to avoid the infrared divergence at d = 2, we add the Zeeman term to the Lagrangian

*q<sup>µ</sup> − q<sup>µ</sup>*

� �

<sup>2</sup> þ ð∂μπi<sup>Þ</sup>

1 <sup>1</sup> � <sup>π</sup><sup>2</sup> i

ðπi∂μπiÞ

2

ðπi∂μπiÞ

g, ð95Þ

<sup>2</sup> <sup>ð</sup>96<sup>Þ</sup>

<sup>2</sup> <sup>þ</sup> <sup>⋯</sup> <sup>ð</sup>97<sup>Þ</sup>

Zg, ð98Þ

: ð100Þ

<sup>p</sup> <sup>π</sup>Ri <sup>ð</sup>99<sup>Þ</sup>

*p*'

*− p*' *− q*

<sup>i</sup> ¼ 1, the Lagrangian is written in the form

The diagrams for the two-point function Γð2<sup>Þ</sup> <sup>ð</sup>pÞ ¼ <sup>G</sup>ð2<sup>Þ</sup> ðpÞ �<sup>1</sup> are shown in Figure 8. The contributions in Figure 8c and d come from the magnetic field. Figure 8b presents

$$I\_b = \int \frac{d^d k}{(2\pi)^d} \frac{(k+p)^2}{k^2 + H} = \left(p^2 - H\right) \int \frac{d^d k}{(2\pi)^d} \frac{1}{k^2 + H'} \tag{105}$$

where we used the formula in the dimensional regularization given as

$$\int d^d k = 0.\tag{106}$$

Near two dimensions, d =2+ E, the integral is regularized as

$$I\_b = (p^2 - H) \frac{\Omega\_d}{\left(2\pi\right)^d} H^{\frac{d}{2}-1} \Gamma\left(\frac{d}{2}\right) \Gamma\left(1 - \frac{d}{2}\right) = \ -(p^2 - H) \frac{\Omega\_d}{\left(2\pi\right)^d \epsilon} \frac{1}{\epsilon}.\tag{107}$$

The H-term Ic in Figure 8c just cancels with –H in Ib. The contribution Id in Figure 8d has the multiplicity 2 � 2 � ðN � 1Þ because (πi) has N – 1 components. Id is evaluated as

$$I\_c = \frac{1}{8} \cdot 4(N-1) \int \frac{d^d k}{(2\pi)^d} \frac{1}{k^2 + H} = -\frac{\Omega\_d}{(2\pi)^d} \frac{N-1}{2} \frac{1}{\epsilon}. \tag{108}$$

As a result, up to the one-loop-order the two-point function is

Figure 8. Diagrams for the two-point function. The diagrams (c) and (d) come from the Zeeman term.

$$
\Gamma^{(2)}(p) = \frac{Z}{Z\_3g}p^2 + \frac{\sqrt{Z}}{g}H - \frac{1}{\epsilon} \left(p^2 + \frac{N-1}{2}H\right),
\tag{109}
$$

where the factor Ωd=ð2πÞ <sup>d</sup> is included in g for simplicity. To remove the divergence, we choose

$$\frac{Z}{Z\_{\otimes}} = 1 + \frac{\mathcal{g}}{\epsilon'} \tag{110}$$

$$
\sqrt{Z} = 1 + \frac{N-1}{2\epsilon} \text{g.}\tag{111}
$$

This set of equations indicates

$$Z\_{\mathfrak{F}} = 1 + \frac{N-1}{\epsilon} \mathfrak{g} + \mathcal{O}(\mathfrak{g}^2),\tag{112}$$

$$Z = 1 \, + \, \frac{N-1}{\epsilon} \, \mathbf{g} + O(\mathbf{g}^2). \tag{113}$$

The case N = 2 is s special case, where we have Zg = 1. This will hold even when including higher order corrections. For N = 2, we have one π field satisfying

$$
\sigma\_2 + \pi\_2 = 1\tag{114}
$$

When we represent σ and π as σ = cos θ and π = sin θ, the Lagrangian is

$$\mathcal{L} = \frac{1}{2g} \{ \left( \partial\_{\mu} \sigma \right)^{2} + \left( \partial\_{\mu} \pi \right)^{2} \} = \frac{1}{2g} (\partial\_{\mu} \theta)^{2} . \tag{115}$$

If we disregard the region of θ, 0 ≤ θ ≤ 2π, the field θ is a free field suggesting that Zg = 1.

#### 3.3. Renormalization group equations

The beta function β(g) of the coupling constant g is defined by

$$
\beta(\mathbf{g}) = \mu \frac{\partial \mathbf{g}}{\partial \mu'} \tag{116}
$$

where the bare quantities are fixed in calculating the derivative. Since μ ∂ g0=∂μ ¼ 0, the beta function is derived as

Renormalization Group Theory of Effective Field Theory Models in Low Dimensions http://dx.doi.org/10.5772/intechopen.68214 115

$$\beta(\mathbf{g}) = \frac{\epsilon \mathbf{g}}{1 + \mathbf{g}\frac{\partial}{\partial \mathbf{g}} \ln \mathbf{Z}\_{\mathbf{g}}} = \epsilon \mathbf{g} - (\mathbf{N} - \mathbf{2})\mathbf{g}^2 + O(\mathbf{g}^3), \tag{117}$$

for d =2+ E. The beta function is shown in Figure 9 as a function of g. We mention here that the coefficient N – 2 of g<sup>2</sup> term is related with the Casimir invariant of the symmetry group O(N) [34, 49].

In the case of N = 2 and d = 2, β(g) vanishes. This case corresponds to the classical XY model as mentioned above and there may be a Kosterlitz-Thouless transition. The Kosterlitz-Thouless transition point cannot be obtained by a perturbation expansion in g.

In two dimensions d = 2, β(g) shows asymptotic freedom for N > 2. The coupling constant g approaches zero in high-energy limit μ ! ∞ in a similar way to QCD. For N = 1, g increases as μ ! ∞ as in the case of QED. When d > 2, there is a fixed point gc:

$$\mathbf{g}\_{\varepsilon} = \frac{\epsilon}{N - 2},\tag{118}$$

for N > 2. There is a phase transition for N > 2 and d > 2.

Let us consider the n-point function Γðn<sup>Þ</sup> ðki, g, μ, HÞ. The bare and renormalized n-point functions are introduced similarly and they are related by the renormalization constant Z

$$
\Gamma\_{\mathbb{R}}^{(n)}(\mathbf{k}\_{i\prime} \text{ g. } \mu \text{ } H) = Z^{n/2} \Gamma\_{\mathbb{B}}^{(n)}(\mathbf{k}\_{i\prime} \text{ g. } \mu \text{ } H). \tag{119}
$$

From the condition that the bare function Γ<sup>ð</sup>n<sup>Þ</sup> <sup>B</sup> is independent of <sup>μ</sup>, <sup>μ</sup> <sup>d</sup> <sup>Γ</sup><sup>ð</sup>n<sup>Þ</sup> <sup>B</sup> =dμ ¼ 0, the renormalization group equation is followed

$$
\hbar \left[ \mu \frac{\partial}{\partial \mu} + \mu \frac{\partial \mathbf{g}}{\partial \mu} \frac{\partial}{\partial \mathbf{g}} - \frac{n}{2} \zeta(\mathbf{g}) + \left( \frac{1}{2} \zeta(\mathbf{g}) + \frac{1}{g} \beta(\mathbf{g}) - (d-2) \right) H \frac{\partial}{\partial H} \right] \Gamma\_{\mathbf{R}}^{(n)}(\mathbf{k}\_{\nu} \text{ g, } \mu, H) = 0,\tag{120}
$$

where we defined

Γð2<sup>Þ</sup>

where the factor Ωd=ð2πÞ

114 Recent Studies in Perturbation Theory

This set of equations indicates

<sup>ð</sup>pÞ ¼ <sup>Z</sup> Zgg <sup>p</sup><sup>2</sup> <sup>þ</sup>

ffiffiffi Z p g

+ + +

(*a*) (*b*) (*c*) (*d*)

Z Zg

ffiffiffi Z <sup>p</sup> <sup>¼</sup> <sup>1</sup> <sup>þ</sup>

Figure 8. Diagrams for the two-point function. The diagrams (c) and (d) come from the Zeeman term.

Zg ¼ 1 þ

Z ¼ 1 þ

higher order corrections. For N = 2, we have one π field satisfying

<sup>L</sup> <sup>¼</sup> <sup>1</sup> 2g

The beta function β(g) of the coupling constant g is defined by

3.3. Renormalization group equations

function is derived as

When we represent σ and π as σ = cos θ and π = sin θ, the Lagrangian is

fð∂μσÞ

<sup>H</sup> � <sup>1</sup>

<sup>¼</sup> <sup>1</sup> <sup>þ</sup> <sup>g</sup> E

N � 1

N � 1

The case N = 2 is s special case, where we have Zg = 1. This will hold even when including

<sup>2</sup> þ ð∂μπ<sup>Þ</sup>

If we disregard the region of θ, 0 ≤ θ ≤ 2π, the field θ is a free field suggesting that Zg = 1.

βðgÞ ¼ μ

where the bare quantities are fixed in calculating the derivative. Since μ ∂ g0=∂μ ¼ 0, the beta

2 g ¼ <sup>1</sup> 2g ð∂μθÞ 2

∂g

N � 1

<sup>E</sup> <sup>g</sup> <sup>þ</sup> <sup>O</sup>ðg<sup>2</sup>

<sup>E</sup> <sup>g</sup> <sup>þ</sup> <sup>O</sup>ðg<sup>2</sup>

<sup>E</sup> <sup>p</sup><sup>2</sup> <sup>þ</sup>

N � 1 2 H

, ð109Þ

, ð110Þ

*H H*

<sup>2</sup><sup>E</sup> <sup>g</sup>: <sup>ð</sup>111<sup>Þ</sup>

σ<sup>2</sup> þ π<sup>2</sup> ¼ 1 ð114Þ

<sup>∂</sup><sup>μ</sup> , <sup>ð</sup>116<sup>Þ</sup>

Þ, ð112Þ

Þ: ð113Þ

: ð115Þ

� �

<sup>d</sup> is included in g for simplicity. To remove the divergence, we choose

Figure 9. The beta function β(g) as a function of g for d = 2 (a) and d > 2 (b). There is a fixed point for N > 2 and d > 2. β(g) is negative for d = 2 and N > 2, which indicates that the model exhibits an asymptotic freedom.

$$\mathcal{L}(\mathbf{g}) = \, \mu \frac{\partial}{\partial \mu} \ln Z = \, \beta(\mathbf{g}) \frac{\partial}{\partial \mathbf{g}} \ln Z. \tag{121}$$

From Eq. (113), ζ(g) is given by

$$
\mathbb{Q}(\mathbf{g}) = (N - 1)\mathbf{g} + \mathcal{O}(\mathbf{g}^2). \tag{122}
$$

Let us define the correlation length ξ ¼ ξðg, μÞ. Because the correlation length near the transition point will not depend on the energy scale, it should satisfy

$$
\mu \frac{d}{d\mu} \xi(\mathbf{g}, \mu) = \left(\mu \frac{\partial}{\partial \mu} + \beta(\mathbf{g}) \frac{\partial}{\partial \mathbf{g}}\right) \xi(\mathbf{g}, \mu) = 0. \tag{123}
$$

We adopt the form <sup>ξ</sup> <sup>¼</sup> <sup>μ</sup>�<sup>1</sup>fðg<sup>Þ</sup> for a function <sup>f</sup>(g), so that we have

$$
\beta(\mathbf{g}) \frac{df(\mathbf{g})}{d\mathbf{g}} = f(\mathbf{g}). \tag{124}
$$

This indicates

$$f(\mathbf{g}) = \mathbb{C} \exp\left(\int\_{\mathcal{S}\_\*}^{\mathcal{S}} \frac{1}{\beta(\mathbf{g'})} d\mathbf{g'}\right),\tag{125}$$

where C and g\* are constants. In two dimensions (E = 0), the beta function in Eq. (117) gives

$$\xi = \mathbb{C}\mu^{-1} \exp\left(\frac{1}{N-2}\left(\frac{1}{g} - \frac{1}{g\_\*}\right)\right). \tag{126}$$

When <sup>N</sup> > 2, <sup>ξ</sup> diverges as <sup>g</sup> ! 0, namely, the mass proportional to <sup>ξ</sup>�<sup>1</sup> vanishes in this limit. When d >2(E > 0), there is a finite-fixed point gc. We approximate β(g) near g = gc as

$$
\beta(\mathbf{g}) \approx \mathfrak{a}(\mathbf{g} - \mathbf{g}\_c),
\tag{127}
$$

with a < 0, ξ is

$$\xi = \mu^{-1} \exp\left(\frac{1}{a} \ln \left| \frac{\mathbf{g} - \mathbf{g}\_c}{\mathbf{g}\_\* - \mathbf{g}\_c} \right|\right). \tag{128}$$

Near the critical point g ≈ gc, ξ is approximated as

$$\|\xi^{-1} \approx \mu\|\mathbf{g} - \mathbf{g}\_c\|^{1/\|\mathbf{a}\|}. \tag{129}$$

This means that ξ ! ∞ as g ! gc. We define the exponent v by

Renormalization Group Theory of Effective Field Theory Models in Low Dimensions http://dx.doi.org/10.5772/intechopen.68214 117

$$\|\xi^{-1} \approx \|\mathbf{g} - \mathbf{g}\_c\|^\nu\_\nu\tag{130}$$

then we have

ζðgÞ ¼ μ

tion point will not depend on the energy scale, it should satisfy

We adopt the form <sup>ξ</sup> <sup>¼</sup> <sup>μ</sup>�<sup>1</sup>fðg<sup>Þ</sup> for a function <sup>f</sup>(g), so that we have

ξðg, μÞ ¼ μ

βðgÞ

fðgÞ ¼ C exp

<sup>ξ</sup> <sup>¼</sup> <sup>C</sup>μ�<sup>1</sup>

<sup>ξ</sup> <sup>¼</sup> <sup>μ</sup>�<sup>1</sup>

Near the critical point g ≈ gc, ξ is approximated as

This means that ξ ! ∞ as g ! gc. We define the exponent v by

μ d dμ

From Eq. (113), ζ(g) is given by

116 Recent Studies in Perturbation Theory

This indicates

with a < 0, ξ is

∂ ∂μ ln<sup>Z</sup> <sup>¼</sup> <sup>β</sup>ðg<sup>Þ</sup> <sup>∂</sup>

<sup>ζ</sup>ðgÞ¼ ð<sup>N</sup> � <sup>1</sup>Þ<sup>g</sup> <sup>þ</sup> <sup>O</sup>ðg<sup>2</sup>

Let us define the correlation length ξ ¼ ξðg, μÞ. Because the correlation length near the transi-

<sup>þ</sup> <sup>β</sup>ðg<sup>Þ</sup> <sup>∂</sup> ∂g

� �

∂ ∂μ

dfðgÞ

ðg g�

1 N � 2

When <sup>N</sup> > 2, <sup>ξ</sup> diverges as <sup>g</sup> ! 0, namely, the mass proportional to <sup>ξ</sup>�<sup>1</sup> vanishes in this limit.

where C and g\* are constants. In two dimensions (E = 0), the beta function in Eq. (117) gives

exp

When d >2(E > 0), there is a finite-fixed point gc. We approximate β(g) near g = gc as

exp 1 a

<sup>ξ</sup>�<sup>1</sup> <sup>≈</sup> <sup>μ</sup>⌊<sup>g</sup> � gc⌋<sup>1</sup>=⌊a⌋

1 βðg<sup>0</sup> Þ dg<sup>0</sup>

> 1 <sup>g</sup> � <sup>1</sup> g�

� � � �

ln <sup>g</sup> � gc <sup>g</sup>� � gc � � � �

!

∂g

ln Z: ð121Þ

Þ: ð122Þ

ξðg, μÞ ¼ 0: ð123Þ

, ð125Þ

: ð126Þ

: ð128Þ

: ð129Þ

dg <sup>¼</sup> <sup>f</sup>ðgÞ: <sup>ð</sup>124<sup>Þ</sup>

βðgÞ ≈ aðg � gcÞ, ð127Þ

$$\nu = -\frac{1}{\beta'(\mathfrak{g}\_c)}.\tag{131}$$

Since β 0 ðgcÞ ¼ E � 2ðN � 2Þgc ¼ �E, this gives

$$\frac{1}{\nu} = \epsilon + \mathcal{O}(\epsilon^2) = d - 2 + \mathcal{O}(\epsilon^2). \tag{132}$$

Including the higher-order terms, ν is given as

$$\frac{1}{N} = d - 2 + \frac{\left(d - 2\right)^2}{N - 2} + \frac{\left(d - 2\right)^3}{2\left(N - 2\right)} + \mathcal{O}(\epsilon^4). \tag{133}$$

#### 3.4. 2D quantum gravity

A similar renormalization group equation is derived for the two-dimensional quantum gravity. The space structure is written by the metric tensor gμν and the curvature R. The quantum gravity Lagrangian is

$$\mathcal{L} = -\frac{1}{16\pi G} \sqrt{g} R \tag{134}$$

where g is the determinant of the matrix ðgμνÞ and G is the coupling constant. The beta function for G was calculated as [78–81]

$$
\beta(G) = \,\,\epsilon G - bG^2,\tag{135}
$$

for d ¼ 2 þ E with a constant b. This has the same structure as that for the non-linear sigma model.

## 4. Sine-Gordon model

#### 4.1. Lagrangian

The two-dimensional sine-Gordon model has attracted a lot of attention [43–49, 82–91]. The Lagrangian of the sine-Gordon model is given by

$$\mathcal{L} = \frac{1}{2t\_0} \left( \partial\_{\mu} \phi \right)^2 + \frac{\alpha\_0}{t\_0} \cos \phi,\tag{136}$$

where φ is a real scalar field, and t<sup>0</sup> and α<sup>0</sup> are bare coupling constants. We also use the Euclidean notation in this section. The second term is the potential energy of the scalar field. We adopt that t and α are positive. The renormalized coupling constants are denoted as t and <sup>α</sup>, respectively. The dimensions of <sup>t</sup> and <sup>α</sup> are <sup>½</sup>t� ¼ <sup>μ</sup><sup>2</sup>�<sup>d</sup> and <sup>½</sup>α� ¼ <sup>μ</sup>2. The scalar field <sup>φ</sup> is dimensionless in this representation. The renormalization constants Zt and Z<sup>α</sup> are defined as follows

$$
\hbar \mathfrak{t}\_0 = \mathfrak{t} \mu^{2-d} \mathcal{Z}\_{t\prime} \,\, \alpha\_0 = \,\, \alpha \mu^2 \mathcal{Z}\_a. \tag{137}
$$

Here, the energy scale μ is introduced so that t and α are dimensionless. The Lagrangian is written as

$$\mathcal{L} = \frac{\mu^{d-2}}{2tZ\_t} (\partial\_\mu \phi)^2 + \frac{\mu^d \alpha Z\_a}{tZ\_t} \cos \phi. \tag{138}$$

We can introduce the renormalized field <sup>φ</sup><sup>B</sup> <sup>¼</sup> ffiffiffiffiffiffi Z<sup>φ</sup> <sup>p</sup> <sup>φ</sup><sup>R</sup> where <sup>Z</sup><sup>φ</sup> is the renormalization constant. Then the Lagrangian is

$$\mathcal{L} = \frac{\mu^{d-2} Z\_{\phi}}{2t Z\_{t}} (\mathfrak{d}\_{\mu} \phi)^{2} + \frac{\mu^{d} a Z\_{a}}{t Z\_{t}} \cos \phi. \tag{139}$$

where φ denotes the renormalized field φR.

#### 4.2. Renormalization of α

We investigate the renormalization group procedure for the sine-Gordon model on the basis of the dimensional regularization method. First consider the renormalization of the potential term. The lowest-order contributions are given by diagrams with tadpole contributions. We use the expansion cos <sup>φ</sup> <sup>¼</sup> <sup>1</sup> � <sup>1</sup> <sup>2</sup> <sup>φ</sup><sup>2</sup> <sup>þ</sup> <sup>1</sup> <sup>4</sup>! <sup>φ</sup><sup>4</sup> � <sup>⋯</sup> . Then the corrections to the cosine term are evaluated as follows. The constant term is renormalized as

$$1 - \frac{1}{2} \langle \phi^2 \rangle + \frac{1}{4!} \langle \phi^4 \rangle - \dots = 1 - \frac{1}{2} \langle \phi^2 \rangle + \frac{1}{2} \left( \frac{1}{2} \langle \phi^2 \rangle \right)^2 - \dots = \exp\left( -\frac{1}{2} \langle \phi^2 \rangle \right). \tag{140}$$

Similarly, the φ<sup>2</sup> is renormalized as

$$-\frac{1}{2}\phi^2 + \frac{1}{4!}\theta\langle\phi^2\rangle\phi^2 - \frac{1}{6!}15\cdot 3\langle\phi^2\rangle^2\phi^2 + \cdots = \exp\left(-\frac{1}{2}\langle\phi^2\rangle\right)\left(-\frac{1}{2}\phi^2\right). \tag{141}$$

Hence the <sup>α</sup>Z<sup>α</sup> cos <sup>ð</sup> ffiffiffiffiffiffi Z<sup>φ</sup> <sup>p</sup> <sup>φ</sup><sup>Þ</sup> is renormalized to

$$aZ\_{\rm at} \exp\left(-\frac{1}{2}Z\_{\phi}\langle\phi^2\rangle\right)\cos\left(\sqrt{Z\_{\phi}}\phi\right) \approx aZ\_{\rm at}\left(1-\frac{1}{2}Z\_{\phi}\langle\phi^2\rangle+\cdots\right)\cos\left(\sqrt{Z\_{\phi}}\phi\right).\tag{142}$$

The expectation value 〈φ<sup>2</sup> 〉 is regularized as Renormalization Group Theory of Effective Field Theory Models in Low Dimensions http://dx.doi.org/10.5772/intechopen.68214 119

$$Z\_{\phi} \langle \phi^2 \rangle = t \mu^{2-d} Z\_t \int \frac{d^d k}{(2\pi)^d} \frac{1}{k^2 + m\_0^2} = -\frac{t}{\epsilon} \frac{\Omega\_d}{(2\pi)^d} \tag{143}$$

where d ¼ 2 þ E and we included a mass m<sup>0</sup> to avoid the infrared divergence and Zt=1 to this order. The constant Z<sup>α</sup> is determined to cancel the divergence:

$$Z\_{\alpha} = 1 - \frac{t}{2} \frac{1}{\epsilon} \frac{\Omega\_d}{(2\pi)^d}. \tag{144}$$

From the equations μ ∂t0=∂μ ¼ 0 and μ ∂α0=∂μ ¼ 0, we obtain

$$
\mu \frac{\partial t}{\partial \mu} = (d - 2)t - t\mu \frac{\partial \ln Z\_t}{\partial \mu},
\tag{145}
$$

$$
\mu \frac{\partial \alpha}{\partial \mu} = -2\alpha - \alpha \mu \frac{\partial \ln Z\_a}{\partial \mu} \tag{146}
$$

The beta function for α reads

We adopt that t and α are positive. The renormalized coupling constants are denoted as t and <sup>α</sup>, respectively. The dimensions of <sup>t</sup> and <sup>α</sup> are <sup>½</sup>t� ¼ <sup>μ</sup><sup>2</sup>�<sup>d</sup> and <sup>½</sup>α� ¼ <sup>μ</sup>2. The scalar field <sup>φ</sup> is dimensionless in this representation. The renormalization constants Zt and Z<sup>α</sup> are defined as

Zt, <sup>α</sup><sup>0</sup> <sup>¼</sup> αμ<sup>2</sup>

<sup>2</sup> <sup>þ</sup> <sup>μ</sup><sup>d</sup>αZ<sup>α</sup> tZt

Z<sup>φ</sup>

<sup>2</sup> <sup>þ</sup> <sup>μ</sup><sup>d</sup>αZ<sup>α</sup> tZt

Here, the energy scale μ is introduced so that t and α are dimensionless. The Lagrangian is

ð∂μφÞ

ð∂μφÞ

We investigate the renormalization group procedure for the sine-Gordon model on the basis of the dimensional regularization method. First consider the renormalization of the potential term. The lowest-order contributions are given by diagrams with tadpole contributions. We

Zα: ð137Þ

cos φ: ð138Þ

cos φ: ð139Þ

<sup>p</sup> <sup>φ</sup><sup>R</sup> where <sup>Z</sup><sup>φ</sup> is the renormalization con-

<sup>4</sup>! <sup>φ</sup><sup>4</sup> � <sup>⋯</sup> . Then the corrections to the cosine term are

� <sup>⋯</sup> <sup>¼</sup> exp � <sup>1</sup>

2 〈φ<sup>2</sup> 〉 � �

〉 þ ⋯

cos

<sup>φ</sup><sup>2</sup> <sup>þ</sup> <sup>⋯</sup> <sup>¼</sup> exp � <sup>1</sup>

� �

2 〈φ<sup>2</sup> 〉 � �

� 1 2 φ2 � �

ffiffiffiffiffiffi Z<sup>φ</sup> q φ � � : ð140Þ

: ð141Þ

: ð142Þ

<sup>t</sup><sup>0</sup> <sup>¼</sup> <sup>t</sup>μ<sup>2</sup>�<sup>d</sup>

<sup>L</sup> <sup>¼</sup> <sup>μ</sup><sup>d</sup>�<sup>2</sup> 2tZt

<sup>L</sup> <sup>¼</sup> <sup>μ</sup><sup>d</sup>�<sup>2</sup>Z<sup>φ</sup> 2tZt

<sup>2</sup> <sup>φ</sup><sup>2</sup> <sup>þ</sup> <sup>1</sup>

2 〈φ<sup>2</sup> 〉 þ 1 2 1 2 〈φ<sup>2</sup> 〉 � �<sup>2</sup>

<sup>15</sup> � <sup>3</sup>〈φ<sup>2</sup> 〉 2

<sup>≈</sup> <sup>α</sup>Z<sup>α</sup> <sup>1</sup> � <sup>1</sup>

2 Zφ〈φ<sup>2</sup>

evaluated as follows. The constant term is renormalized as

〉 � <sup>⋯</sup> <sup>¼</sup> <sup>1</sup> � <sup>1</sup>

〉φ<sup>2</sup> � <sup>1</sup> 6!

<sup>p</sup> <sup>φ</sup><sup>Þ</sup> is renormalized to

ffiffiffiffiffiffi Z<sup>φ</sup> q φ � �

〉 is regularized as

cos

We can introduce the renormalized field <sup>φ</sup><sup>B</sup> <sup>¼</sup> ffiffiffiffiffiffi

where φ denotes the renormalized field φR.

stant. Then the Lagrangian is

4.2. Renormalization of α

<sup>1</sup> � <sup>1</sup> 2 〈φ<sup>2</sup> 〉 þ 1 4! 〈φ<sup>4</sup>

use the expansion cos <sup>φ</sup> <sup>¼</sup> <sup>1</sup> � <sup>1</sup>

Similarly, the φ<sup>2</sup> is renormalized as

Z<sup>φ</sup>

� �

� 1 2 <sup>φ</sup><sup>2</sup> <sup>þ</sup> 1 4! 6〈φ<sup>2</sup>

Hence the <sup>α</sup>Z<sup>α</sup> cos <sup>ð</sup> ffiffiffiffiffiffi

<sup>α</sup>Zαexp � <sup>1</sup>

The expectation value 〈φ<sup>2</sup>

2 Zφ〈φ<sup>2</sup> 〉

follows

118 Recent Studies in Perturbation Theory

written as

$$\beta(\alpha) \equiv \begin{array}{c} \frac{\partial \alpha}{\partial \mu} = \ -2\alpha + \ t\alpha \frac{1}{2} \frac{\Omega\_d}{\left(2\pi\right)^{d'}} \end{array} \tag{147}$$

where we set μ ∂ t=∂μ ¼ ðd � 2Þt with Zt ¼ 1 up to the lowest order of α. The function β(α) has a zero at t ¼ tc ¼ 8π.

#### 4.3. Renormalization of the two-point function

Let us turn to the renormalization of the coupling constant t. The renormalization of t comes from the correction to p<sup>2</sup> term. The lowest-order two-point function is

$$
\Gamma\_B^{(2)(0)}(p) = \frac{1}{t\_0} p^2 = \frac{1}{t\mu^{2-d}Z\_t} p^2. \tag{148}
$$

The diagrams that contribute to the two-point function are shown in Figure 10 [88]. These diagrams are obtained by expanding the cosine function as cos <sup>φ</sup> <sup>¼</sup> <sup>1</sup> � ð1=2Þ<sup>φ</sup> <sup>2</sup> <sup>þ</sup> <sup>⋯</sup>. First, we consider the Green's function,

$$\mathcal{G}\_0(\mathbf{x}) = \mathcal{Z}\_\phi \, \phi(\mathbf{x})\phi(\mathbf{0}) > = t\mu^{2-d}\mathcal{Z}\_t \left[\frac{d^d p}{(2\pi)^d} p \frac{e^{i\mathbf{p}\cdot\mathbf{x}}}{p^2 + m\_0^2} = t\mu^{2-d}\mathcal{Z}\_t \frac{\Omega\_d}{\left(2\pi\right)^d} \mathcal{K}\_0(m\_0|\mathbf{x}|), \tag{149}$$

where K<sup>0</sup> is the zeroth modified Bessel function and m<sup>0</sup> is introduced to avoid the infrared singularity. Because sinh I � I ¼ I 3 =3! þ ⋯, the diagrams in Figure 10 are summed up to give

Figure 10. Diagrams that contribute to the two-point function.

$$\Sigma(p) = \int d^d \mathbf{x} [e^{ip \cdot \mathbf{x}} (\sinh I - I) - (\cosh I - 1)] \,\tag{150}$$

Where <sup>I</sup> <sup>¼</sup> <sup>G</sup>0ðxÞ. Since sinh <sup>I</sup> � <sup>I</sup> <sup>≈</sup> eI =2 and cosh I ≈ e<sup>I</sup> =2, the diagrams in Figure 10 lead to

$$\Gamma\_{\mathcal{B}}^{(2)c}(p) = -\frac{1}{2} \left( \frac{a\mu^d Z\_{\alpha}}{tZ\_t} \right)^2 \int d^d x (e^{ip \cdot x} - 1) e^{G\_0(\mathbf{x})}.\tag{151}$$

We use the expansion <sup>e</sup>ip�<sup>x</sup> <sup>¼</sup> <sup>1</sup> <sup>þ</sup> ip � <sup>x</sup> � ð1=2Þð<sup>p</sup> � <sup>x</sup><sup>Þ</sup> <sup>2</sup> <sup>þ</sup> <sup>⋯</sup>, and keep the <sup>p</sup><sup>2</sup> term. We denote the derivation of t from the fixed point tc ¼ 8π as ν:

$$\frac{t}{8\pi} = 1 + \nu\_{\prime} \tag{152}$$

for <sup>d</sup> = 2. Using the asymptotic formula <sup>K</sup>0ðxÞ<sup>e</sup> � <sup>γ</sup> � lnðx=2<sup>Þ</sup> for small <sup>x</sup>, we obtain

$$\Gamma\_{B}^{(2)c}(p) = \frac{1}{8} \left(\frac{a\mu^{d}}{tZ\_{t}}\right)^{2} p^{2} (c\_{0}m\_{0}^{2})^{-2-2\nu} \Omega\_{d} \int\_{0}^{\varkappa} d\mathbf{x} \mathbf{x}^{d+1} \frac{1}{(\mathbf{x}^{2} + a^{2})^{2+2\nu}}$$

$$= -\frac{1}{8} p^{2} \left(\frac{a\mu^{d}}{tZ\_{t}}\right)^{2} (c\_{0}m\_{0}^{2})^{-2} \Omega\_{d} \frac{1}{\epsilon} + \mathcal{O}(\nu) \tag{153}$$

$$\approx -\frac{1}{t\mu^{2-d}Z\_{t}} p^{2} \frac{1}{32} a^{2} \mu^{d+2} (c\_{0}m\_{0}^{2})^{-2} \frac{1}{\epsilon} + O(\nu)$$

where c<sup>0</sup> is a constant and a ¼ 1=μ is a small cut-off. The divergence of α was absorbed by Zα. Now the two-point function up to this order is

$$\Gamma\_{\rm B}^{(2)}(p) = \frac{1}{t\mu^{2-d}\mathcal{Z}\_t} \left[p^2 - \frac{1}{32}\alpha^2\mu^{d+2}(c\_0m\_0^2)^{-2}\frac{1}{\epsilon}\right] \tag{154}$$

The renormalized two-point function is Γ<sup>ð</sup>2<sup>Þ</sup> <sup>R</sup> <sup>¼</sup> <sup>Z</sup>φΓ<sup>ð</sup>2<sup>Þ</sup> <sup>B</sup> . This indicates that Renormalization Group Theory of Effective Field Theory Models in Low Dimensions http://dx.doi.org/10.5772/intechopen.68214 121

$$\frac{Z\_{\phi}}{Z\_{t}} = 1 + \frac{1}{32} \alpha^{2} \mu^{d+2} (c\_{0} m\_{0}^{2})^{-2} \frac{1}{\epsilon}. \tag{155}$$

Then, we can choose Z<sup>φ</sup> = 1 and

ΣðpÞ ¼

Figure 10. Diagrams that contribute to the two-point function.

Γ<sup>ð</sup>2Þ<sup>c</sup>

We use the expansion <sup>e</sup>ip�<sup>x</sup> <sup>¼</sup> <sup>1</sup> <sup>þ</sup> ip � <sup>x</sup> � ð1=2Þð<sup>p</sup> � <sup>x</sup><sup>Þ</sup>

8

¼ � <sup>1</sup> 8

<sup>≈</sup> � <sup>1</sup> tμ<sup>2</sup>�dZt

<sup>B</sup> <sup>ð</sup>pÞ ¼ <sup>1</sup>

αμ<sup>d</sup> tZt � �<sup>2</sup>

derivation of t from the fixed point tc ¼ 8π as ν:

Γ<sup>ð</sup>2Þ<sup>c</sup> <sup>B</sup> <sup>ð</sup>pÞ ¼ <sup>1</sup>

Now the two-point function up to this order is

The renormalized two-point function is Γ<sup>ð</sup>2<sup>Þ</sup>

Γ<sup>ð</sup>2<sup>Þ</sup>

Where <sup>I</sup> <sup>¼</sup> <sup>G</sup>0ðxÞ. Since sinh <sup>I</sup> � <sup>I</sup> <sup>≈</sup> eI

120 Recent Studies in Perturbation Theory

ð dd x½e ip�x

<sup>B</sup> <sup>ð</sup>pÞ¼ � <sup>1</sup>

2

=2 and cosh I ≈ e<sup>I</sup>

αμdZ<sup>α</sup> tZt � �<sup>2</sup>ð

t

for <sup>d</sup> = 2. Using the asymptotic formula <sup>K</sup>0ðxÞ<sup>e</sup> � <sup>γ</sup> � lnðx=2<sup>Þ</sup> for small <sup>x</sup>, we obtain

p2 <sup>ð</sup>c0m<sup>2</sup> 0Þ �2�2ν Ω<sup>d</sup> ð∞ 0

<sup>p</sup><sup>2</sup> αμ<sup>d</sup> tZt � �<sup>2</sup>

tμ<sup>2</sup>�dZt

<sup>p</sup><sup>2</sup> <sup>1</sup> <sup>32</sup> <sup>α</sup><sup>2</sup> μ<sup>d</sup>þ<sup>2</sup>

<sup>ð</sup>c0m<sup>2</sup> 0Þ �2 Ω<sup>d</sup> 1 E

where c<sup>0</sup> is a constant and a ¼ 1=μ is a small cut-off. The divergence of α was absorbed by Zα.

<sup>p</sup><sup>2</sup> � <sup>1</sup> <sup>32</sup> <sup>α</sup><sup>2</sup> μ<sup>d</sup>þ<sup>2</sup>

<sup>R</sup> <sup>¼</sup> <sup>Z</sup>φΓ<sup>ð</sup>2<sup>Þ</sup>

<sup>ð</sup>c0m<sup>2</sup> 0Þ �<sup>2</sup> 1 E þ OðνÞ

dd xðe

ip�<sup>x</sup> � <sup>1</sup>Þ<sup>e</sup>

ðsinh I � IÞ�ðcosh I � 1Þ�, ð150Þ

G0ðxÞ

<sup>8</sup><sup>π</sup> <sup>¼</sup> <sup>1</sup> <sup>þ</sup> <sup>ν</sup>, <sup>ð</sup>152<sup>Þ</sup>

ðx<sup>2</sup> þ a<sup>2</sup>Þ

2þ2ν

dxxdþ<sup>1</sup> <sup>1</sup>

þ OðνÞ

<sup>ð</sup>c0m<sup>2</sup> 0Þ �<sup>2</sup> 1 E

<sup>B</sup> . This indicates that

� �

=2, the diagrams in Figure 10 lead to

<sup>2</sup> <sup>þ</sup> <sup>⋯</sup>, and keep the <sup>p</sup><sup>2</sup> term. We denote the

: ð151Þ

ð153Þ

ð154Þ

$$Z\_t = 1 - \frac{1}{32} \alpha^2 \mu^{d+2} (c\_0 m\_0^2)^{-2} \frac{1}{\epsilon}. \tag{156}$$

Zt=Z<sup>φ</sup> can be regarded as the renormalization constant of t up to the order of α2, and thus we do not need the renormalization constant Z<sup>φ</sup> of the field φ. This means that we can adopt the bare coupling constant as <sup>t</sup><sup>0</sup> <sup>¼</sup> <sup>t</sup>μ<sup>2</sup>�dZ~<sup>t</sup> with <sup>Z</sup>~<sup>t</sup> <sup>¼</sup> Zt=Zφ.

The renormalization function of <sup>t</sup> is obtained from the equation <sup>μ</sup> <sup>∂</sup> <sup>t</sup>0=∂<sup>μ</sup> <sup>¼</sup> 0 for <sup>t</sup><sup>0</sup> <sup>¼</sup> <sup>t</sup>μ<sup>2</sup>�dZt:

$$\begin{split} \beta(t) & \equiv \mu \frac{\partial t}{\partial \mu} = (d-2)t + \frac{1}{32} (c\_0 m\_0^2)^{-2} \frac{1}{\epsilon} \Big( 2a\mu^{d+2} \mu \frac{\partial \alpha}{\partial \mu} + (d+2)a^2 \mu^{d+2} \Big) t \\ & = (d-2)t + \frac{1}{32} \mu^{d+2} (c\_0 m\_0^2)^{-2} t a^2 \end{split} \tag{157}$$

Because the finite part of <sup>G</sup>0ð<sup>x</sup> ! <sup>0</sup><sup>Þ</sup> is given by <sup>G</sup>0ð<sup>x</sup> ! <sup>0</sup>Þ ¼ �ð1=2πÞlnðe<sup>γ</sup>m0=2μÞ, we perform the finite renormalization of <sup>α</sup> as <sup>α</sup> ! <sup>α</sup>c0m<sup>2</sup> <sup>0</sup>a<sup>2</sup> <sup>¼</sup> <sup>α</sup>c0m<sup>2</sup> <sup>0</sup>μ�2. This results in

$$
\beta(t) = \ (d-2)t + \frac{1}{32}t\alpha^2. \tag{158}
$$

As a result, we obtain a set of renormalization group equations for the sine-Gordon model

$$\beta(\alpha) = \ \mu \frac{\partial \alpha}{\partial \mu} = \ -\alpha \left(2 - \frac{1}{4\pi}t\right),\tag{159}$$

$$
\beta(t) = \mu \frac{\partial t}{\partial \mu} = (d-2)t + \frac{1}{32}t\alpha^2,\tag{160}
$$

Since the equation for α is homogeneous in α, we can change the scale of α arbitrarily. Thus, the numerical coefficient of tα<sup>2</sup> in β(t) is not important.

#### 4.4. Renormalization group flow

Let us investigate the renormalization group flow in two dimensions. This set of equations reduces to that of the Kosterlitz-Thouless (K-T) transition. We write t ¼ 8πð1 þ νÞ, and set x ¼ 2ν and y ¼ α=4. Then, the equations are

$$
\mu \frac{\partial \mathbf{x}}{\partial \mu} = \mathbf{y}^2,\tag{161}
$$

$$
\mu \frac{\partial y}{\partial \mu} = \text{ xy},
\tag{162}
$$

These are the equations of K-T transition. We have

$$x^2 - y^2 = \text{const.}\tag{163}$$

The renormalization flow is shown in Figure 11. The Kosterlitz-Thouless transition is a beautiful transition that occurs in two dimensions. It was proposed that the transition was associated with the unbinding of vortices, that is, the K-T transition is a transition of the bindingunbinding transition of vortices.

The Kondo problem is also described by the same equations. In the s-d model, we put

$$\mathbf{x} = \pi \beta \mathbf{J}\_z - \mathbf{2}, \ y = \mathbf{2} |\mathbf{J}\_\perp| \mathbf{\tau}. \tag{164}$$

where Jz and J⊥ð¼ Jx ¼ JyÞ are exchange coupling constants between the conduction electrons and the localized spin, and β is the inverse temperature. τ is a small cut-off with τ∝1=μ. The scaling equations for the s-d model are [53, 57]

$$
\tau \frac{\partial \mathbf{x}}{\partial \tau} = -\frac{1}{2} y^2,\tag{165}
$$

$$
\tau \frac{\partial y}{\partial \tau} = -\frac{1}{2}xy. \tag{166}
$$

The Kondo effect occurs as a crossover from weakly correlated region to strongly correlated region. A crossover from weakly to strongly coupled systems is a universal and ubiquitous

Figure 11. The renormalization group flow for the sine-Gordon model as μ ! ∞.

phenomenon in the world. There appears a universal logarithmic anomaly as a result of the crossover.

## 5. Scalar quantum electrodynamics

μ ∂x <sup>∂</sup><sup>μ</sup> <sup>¼</sup> <sup>y</sup><sup>2</sup>

μ ∂y

The renormalization flow is shown in Figure 11. The Kosterlitz-Thouless transition is a beautiful transition that occurs in two dimensions. It was proposed that the transition was associated with the unbinding of vortices, that is, the K-T transition is a transition of the binding-

where Jz and J⊥ð¼ Jx ¼ JyÞ are exchange coupling constants between the conduction electrons and the localized spin, and β is the inverse temperature. τ is a small cut-off with τ∝1=μ. The

The Kondo effect occurs as a crossover from weakly correlated region to strongly correlated region. A crossover from weakly to strongly coupled systems is a universal and ubiquitous

*a*

*t t*

*c*

Figure 11. The renormalization group flow for the sine-Gordon model as μ ! ∞.

The Kondo problem is also described by the same equations. In the s-d model, we put

τ ∂x <sup>∂</sup><sup>τ</sup> ¼ � <sup>1</sup> 2 y2

τ ∂y <sup>∂</sup><sup>τ</sup> ¼ � <sup>1</sup> 2

These are the equations of K-T transition. We have

scaling equations for the s-d model are [53, 57]

unbinding transition of vortices.

122 Recent Studies in Perturbation Theory

, ð161Þ

<sup>∂</sup><sup>μ</sup> <sup>¼</sup> xy, <sup>ð</sup>162<sup>Þ</sup>

<sup>x</sup><sup>2</sup> � <sup>y</sup><sup>2</sup> <sup>¼</sup> const: <sup>ð</sup>163<sup>Þ</sup>

x ¼ πβJz � 2, y ¼ 2jJ⊥jτ: ð164Þ

, ð165Þ

xy: ð166Þ

We have examined the φ<sup>4</sup> theory and showed that there is a phase transition. This is a secondorder transition. What will happen when a scalar field couples with the electromagnetic field? This issue concerns the theory of a complex scalar field φ interacting with the electromagnetic field Aμ, called the scalar quantum electrodynamics (QED). The Lagrangian is

$$\mathcal{L} = \frac{1}{2} |(D\_{\mu}\phi)|^{2} - \frac{1}{4}g(|\phi|^{2})^{2} - \frac{1}{4}F\_{\mu\nu}^{2},\tag{167}$$

where g is the coupling constant and Fμν ¼ ∂μA<sup>ν</sup> � ∂νAμ. D<sup>μ</sup> is the covariant derivative given as

$$D\_{\mu} = \left. \partial\_{\mu} - \mathrm{ie}A\_{\mu} \right. \tag{168}$$

with the charge e. The scalar field φ is an N component complex scalar field such as φ ¼ ðφ1, ⋯, φNÞ. This model is actually a model of a superconductor. The renormalization group analysis shows that this model exhibits a first-order transition near four dimensions d ¼ 4 � E when 2N < 365 [92–96]. Coleman and Weinberg first considered the scalar QED model in the case N = 1. They called this transition the dimensional transmutation. The result based on the E-expansion predicts that a superconducting transition in a magnetic field is a first-order transition. This transition may be related to a first-order transition in a high magnetic field [97].

The bare and renormalized fields and coupling constants are defined as

$$
\phi\_0 = \sqrt{Z\_{\phi}} \phi,\tag{169}
$$

$$\mathbf{g}\_0 = \frac{\mathbf{Z\_4}}{Z\_\phi^2} \mathbf{g} \mu^{4-d},\tag{170}$$

$$\mathbf{e}\_0 = \frac{\mathbf{Z}\_\mathbf{e}}{\sqrt{\mathbf{Z}\_A \mathbf{Z}\_\phi}} \mathbf{e}\_\prime \tag{171}$$

$$A\_{\mu 0} = \sqrt{Z\_A} A\_{\mu} \tag{172}$$

where φ, g, e and A<sup>μ</sup> are renormalized quantities. We have four renormalization constants. Thanks to the Ward identity

$$\mathbf{Z}\_{\mathbf{t}} = \mathbf{Z}\_{\mathbf{A}\prime} \tag{173}$$

three renormalization constants should be determined. We show the results:

$$Z\_{\phi} = 1 + \frac{3}{8\pi^2 \epsilon} e^2,\tag{174}$$

$$Z\_A = 1 - \frac{2N}{48\pi^2\epsilon}e^2,\tag{175}$$

$$Z\_{\S} = 1 + \frac{2\mathcal{N} + 8}{8\pi^2 \epsilon} \mathcal{g} + \frac{3}{8\pi^2 \epsilon} \frac{1}{\mathcal{g}} e^4. \tag{176}$$

The renormalization group equations are given by

$$
\mu \frac{\partial e^2}{\partial \mu} = \ -\epsilon e^2 + \frac{N}{24\pi^2} e^4,\tag{177}
$$

$$
\mu \frac{\partial \mathbf{g}}{\partial \mu} = -\epsilon g + \frac{N+4}{4\pi^2} \mathbf{g}^2 + \frac{3}{8\pi^2} e^4 - \frac{3}{4\pi^2} e^2 \mathbf{g}. \tag{178}
$$

The fixed point is given by

$$
\epsilon\_{\mathfrak{c}} = \frac{24}{N} \pi^2 \epsilon\_{\mathfrak{c}} \tag{179}
$$

$$g\_c = \epsilon \frac{2\pi^2}{N+4} \left\{ 1 + \frac{18}{N} \pm \frac{(n^2 - 360n - 2160)^{1/2}}{n} \right\},\tag{180}$$

where <sup>n</sup> <sup>¼</sup> <sup>2</sup>N. The square root <sup>δ</sup> � ðn<sup>2</sup> � <sup>360</sup><sup>n</sup> � <sup>2160</sup><sup>Þ</sup> <sup>1</sup>=<sup>2</sup> is real when 2N > 365. This indicates that the zero of a set of beta functions exists when N is sufficiently large as long as 2N > 365. Hence there is no continuous transition when N is small, 2N ≤ 365, and the phase transition is first-order.

There are also calculations up to two-loop-order for scalar QED [98, 99]. This model is also closely related with the phase transition from a smectic-A to a nematic liquid crystal for which a second-order transition was reported [100]. When N is large as far as 2N > 365, the transition becomes second-order. Does the renormalization group result for the scalar QED contradict with second-order transition in superconductors? This subject has not been solved yet. A possibility of second-order transition was investigated in three dimensions by using the renormalization group theory [101]. An extra parameter c was introduced in [101] to impose a relation between the external momentum p and the momentum q of the gauge field as q ¼ p=c. It was shown that when c > 5:7, we have a second-order transition. We do not think that it is clear whether the introduction of c is justified or not.

## 6. Summary

We presented the renormalization group procedure for several important models in field theory on the basis of the dimensional regularization method. The dimensional method is very useful and the divergence is separated from an integral without ambiguity. We invested three fundamental models in field theory: φ4 theory, non-linear sigma model and sine-Gordon model. These models are often regarded as an effective model in understanding physical phenomena. The renormalization group equations were derived in a standard way by regularizing the ultraviolet divergence. The renormalization group theory is useful in the study of various quantum systems.

The renormalization means that the divergences, appearing in the evaluation of physical quantities, are removed by introducing the finite number of renormalization constants. If we need infinite number of constants to cancel the divergences for some model, that model is called unrenormalizable. There are many renormalizeable field theoretic models. We considered three typical models among them. The idea of renormalization group theory arises naturally from renormalization. The dependence of physical quantities on the renormalization energy scale easily leads us to the idea of renormalization group.

## Author details

Z<sup>φ</sup> ¼ 1 þ

Zg ¼ 1 þ

μ ∂e<sup>2</sup> <sup>∂</sup><sup>μ</sup> ¼ �E<sup>e</sup>

<sup>∂</sup><sup>μ</sup> ¼ �E<sup>g</sup> <sup>þ</sup>

2π<sup>2</sup> <sup>N</sup> <sup>þ</sup> <sup>4</sup> <sup>1</sup> <sup>þ</sup>

The renormalization group equations are given by

μ ∂g

gc ¼ E

where <sup>n</sup> <sup>¼</sup> <sup>2</sup>N. The square root <sup>δ</sup> � ðn<sup>2</sup> � <sup>360</sup><sup>n</sup> � <sup>2160</sup><sup>Þ</sup>

clear whether the introduction of c is justified or not.

The fixed point is given by

124 Recent Studies in Perturbation Theory

first-order.

6. Summary

ZA <sup>¼</sup> <sup>1</sup> � <sup>2</sup><sup>N</sup>

2N þ 8 <sup>8</sup>π<sup>2</sup><sup>E</sup> <sup>g</sup> <sup>þ</sup>

N þ 4 <sup>4</sup>π<sup>2</sup> <sup>g</sup><sup>2</sup> <sup>þ</sup>

ec <sup>¼</sup> <sup>24</sup> <sup>N</sup> <sup>π</sup><sup>2</sup>

that the zero of a set of beta functions exists when N is sufficiently large as long as 2N > 365. Hence there is no continuous transition when N is small, 2N ≤ 365, and the phase transition is

There are also calculations up to two-loop-order for scalar QED [98, 99]. This model is also closely related with the phase transition from a smectic-A to a nematic liquid crystal for which a second-order transition was reported [100]. When N is large as far as 2N > 365, the transition becomes second-order. Does the renormalization group result for the scalar QED contradict with second-order transition in superconductors? This subject has not been solved yet. A possibility of second-order transition was investigated in three dimensions by using the renormalization group theory [101]. An extra parameter c was introduced in [101] to impose a relation between the external momentum p and the momentum q of the gauge field as q ¼ p=c. It was shown that when c > 5:7, we have a second-order transition. We do not think that it is

We presented the renormalization group procedure for several important models in field theory on the basis of the dimensional regularization method. The dimensional method is very

18

3 8π<sup>2</sup>E e 2

48π<sup>2</sup>E e 2

2 þ N <sup>24</sup>π<sup>2</sup> <sup>e</sup> 4

> 3 <sup>8</sup>π<sup>2</sup> <sup>e</sup>

<sup>N</sup> � <sup>ð</sup>n<sup>2</sup> � <sup>360</sup><sup>n</sup> � <sup>2160</sup><sup>Þ</sup>

( )

n

<sup>4</sup> � <sup>3</sup> <sup>4</sup>π<sup>2</sup> <sup>e</sup> 2

3 8π<sup>2</sup>E 1 g e 4

, ð174Þ

, ð175Þ

: ð176Þ

, ð177Þ

E, ð179Þ

<sup>1</sup>=<sup>2</sup> is real when 2N > 365. This indicates

1=2

g: ð178Þ

, ð180Þ

Takashi Yanagisawa

Address all correspondence to: t-yanagisawa@aist.go.jp

National Institute of Advanced Industrial Science and Technology, Tsukuba, Ibaraki, Japan

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