## 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 non-linear sigma model, the ϕ4 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*^{4} 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*^{4} model is the model of a phase transition, which is often referred to as the Ginzburg-Landau model. The renormalization of the *ϕ*^{4} 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 one-dimensional 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*^{4} model

### 2.1. Lagrangian

The *ϕ*^{4} model is given by the Lagrangian

where *ϕ* is a scalar field and *g* is the coupling constant. In the unit of the momentum *μ*, the dimension of *d*, where *d* is the dimension of the space-time: *gϕ*^{4} has the dimension *d*, the dimension of *g* is given by 4 – *d*: [*g*] = *μ*^{4 – d}. Let us adopt that *ϕ* has *N* components as *ϕ* = (*ϕ*_{1}, *ϕ*_{2}, …, *ϕ*_{N}). The interaction term *ϕ*^{4} is defined as

The Green’s function is defined as

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

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

where

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

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

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

Then, the degree of divergence is written as

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

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

where *ϕ*_{0} denotes the bare field, *g*_{0} denotes the bare coupling constant and *m*_{0} is the bare mass. We introduce the renormalized field *ϕ*, the renormalized coupling constant *g* and the renormalized mass *m*. They are defined by

where *Z*_{ϕ}, *Z*_{g} and *Z*_{2} are renormalization constants. When we write *Z*_{g} as

we have

### 2.2. Regularization of divergences

#### 2.2.1. Two-point function

We use the perturbation theory in terms of the interaction *gϕ*^{4}. For a multi-component scalar field theory, it is convenient to express the interaction *ϕ*^{4} 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 ^{(2)} are shown in Figure 1. The first term indicates *p*^{2}*Z*_{ϕ} and the contribution in the second term is represented by the integral

Using the Euclidean co-ordinate *q*_{4} = –*iq*_{0}, this integral is evaluated as

where Ω_{d} 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 *N*-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 32*N* for (a) and 64 for (b). Then we have the factor from these contributions as

The momentum integral of this term is given as

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

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

For *q*′ = *q* + (1 – *x*)(*p* + *k*), we have

Here, the following parameter formula was used

Then, we obtain

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

for *ϵ* → 0. This results in

Therefore, the two-point function is evaluated as

up to the order of *O*(*g*^{2}). In order to cancel the divergence, we choose *Z*_{ϕ} as

#### 2.2.2. Four-point function

Let us turn to the renormalization of the interaction term *g*^{4}. The perturbative expansion of the four-point function is shown in Figure 4. The diagram (*b*) in Figure 4, denoted as *N* = 1:

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

for *d* = 4 – *ϵ*. Let us evaluate the multiplicity of this contribution for *N* > 1. For *N* = 1, we have a factor 4^{2}3^{2}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

Since we obtain the same factor for diagrams in Figure 4**c** 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

Because *g* has the dimension 4 – *d* such as [*g*] = *μ*^{4–d}, we write *g* as *gμ*^{4–d} so that *g* is the dimensionless coupling constant. Now, we have

for *d* = 4 – *ϵ* where we neglect *μ*^{ϵ} in the second term. The renormalization constant is determined as

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

### 2.3. Beta function *β*(*g*)

The bare coupling constant is written as *g*_{0} is independent of the energy scale, *μ*, we have

where *g* as

where the derivative is evaluated under the condition that the bare *g*_{0} is fixed. Because

the beta function is given as

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

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

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

where

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

and then *β*(*g*) is written as

Here, the factor 1/8*π*^{2} is included in *g*. The terms of order 1/*ϵ*^{2} are cancelled because of

In general, the *n*th order term in *β*(*g*) is given by *n*!*g*^{n}. The function *β*(*g*) is expected to have the form

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 *p*_{i} (*i* = 1,…, *n*) indicate momenta. The energy scale *μ* indicates the renormalization point. *n* + *d* – *nd/2*: *Z*_{ϕ} as

Here, we consider the massless case and omit the mass. Because the bare quantity *μ*, we have

This leads to

Then we obtain the equation for

where *γ*_{ϕ} is defined as

A general solution of the renormalization equation is written as

where

for a function *F* and a constant *g*_{1}. We suppose that *β*(*g*) has a zero at *g* = *g*_{c}. Near the fixed point *g*_{c}, by approximating

In general, we define *γ*(*g*) as

Then, we obtain

Under a scaling

because

after the scaling *p*_{i}→ *ρp*_{i} for *n* = 4. We employ Eq. (58) for *n* = 2

This indicates

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

At the fixed point *g* = *g*_{c}, this leads to

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

The Green’s function

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

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

The definition of *γ*_{ϕ} in Eq. (52) results in

Up to the lowest order of *g, γ*_{ϕ} is given by

At the critical point *g* = *g*_{c}, where

the anomalous dimension is given as

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

### 2.5. Mass renormalization

Let us consider the massive case *m* ≠ 0. This corresponds to the case with *T* > *T*_{c} in a phase transition. The bare mass *m*_{0} *m* and renormalized mass *m* are related through the relation

From Eq. (50), the equation for

We define the exponent ν by

then

At the critical point *g* = *g*_{c}, we obtain

where *γ*_{ϕ} = *η* and we set

At *g* = *g*_{c},

because this satisfies Eq. (75).

In the scaling

From Eq. (77), we have

where we put *F*^{(n)} depends only on *ρ*−1*k*_{i}. We choose *ρ* as

This satisfies

We take *μ* as a unit by setting *μ* = 1, so that

because *ξ* by

The two-point function is written as

Now let us turn to the evaluation of *ν*. Since *ν* is given by

The renormalization constant *Z*_{2} is determined from the corrections to the bare mass *m*_{0}. The one-loop correction, shown in Figure 6, is given by

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

for *d* = 4–*ϵ*. Therefore the renormalized mass is

*Z*_{2} is determined to cancel the divergence in the form *m*^{2}*Z*_{2}/*Z*_{ϕ}. The result is

Then, we have

Eq. (85) is written as

where we put *g* = *g*_{c} and used

In the mean-field approximation, *ν* = 1/2. This formula of *ν* contains the fluctuation effect near the critical point. For *N* = 1 and *ϵ* = 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

where *ϕ* is a real *N*-component field *ϕ =* (*ϕ*_{1},…,*ϕ*_{N}) with the constraint *ϕ*^{2} = 1. This model has an O(N) invariance. The field *ϕ* is represented as

with the condition *π*_{i} (*i* = 1, …, *N* – 1) are regarded as representing fluctuations. The Lagrangian is given by

where summation is assumed for index *i*. In this Section we consider the Euclidean Lagrangian from the beginning. Using the constraint

The second term in the right-hand side indicates the interaction between *π*_{i} fields. The diagram for this interaction is shown in Figure 7.

Here, let us check the dimension of the field and coupling constant. Since *g*_{0} and *g* are used to denote the bare coupling constant and renormalized coupling constant, respectively. The bare and renormalized fields are indicated by *π*_{Bi} and *π*_{Ri}, respectively. We define the renormalization constants *Z*_{g} and *Z* by

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

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

Here, *H*_{B} is the bare magnetic field and the renormalized magnetic field *H* is defined as

Then, the Zeeman term is given by

### 3.2. Two-point function

The diagrams for the two-point function

where we used the formula in the dimensional regularization given as

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

The H-term *I*_{c} in Figure 8c just cancels with –*H* in *I*_{b}. The contribution *I*_{d} in Figure 8d has the multiplicity *π*_{i}) has *N* – 1 components. *I*_{d} is evaluated as

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

where the factor *g* for simplicity. To remove the divergence, we choose

This set of equations indicates

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

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

If we disregard the region of *θ*, *θ* is a free field suggesting that *Z*_{g} = 1.

### 3.3. Renormalization group equations

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

where the bare quantities are fixed in calculating the derivative. Since

for *d* = 2 + *ϵ*. The beta function is shown in Figure 9 as a function of *g*. We mention here that the coefficient *N* – 2 of *g*^{2} 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 *g*_{c}:

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

Let us consider the *n*-point function *n*-point functions are introduced similarly and they are related by the renormalization constant *Z*

From the condition that the bare function *μ*,

where we defined

From Eq. (113), *ζ*(*g*) is given by

Let us define the correlation length

We adopt the form *f*(*g*), so that we have

This indicates

where *C* and *g*_{*} are constants. In two dimensions (*ϵ* = 0), the beta function in Eq. (117) gives

When *N* > 2, *ξ* diverges as *g* → 0, namely, the mass proportional to *ξ*^{−1} vanishes in this limit. When *d* > 2 (*ϵ* > 0), there is a finite-fixed point *g*_{c}. We approximate *β*(*g*) near *g* = *g*_{c} as

with *a* < 0, *ξ* is

Near the critical point *ξ* is approximated as

This means that *ξ* → ∞ as *g* → *g*_{c}. We define the exponent *v* by

then we have

Since

Including the higher-order terms, *ν* is given as

### 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 *R*. The quantum gravity Lagrangian is

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

for *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

where *ϕ* is a real scalar field, and *t*_{0} and *α*_{0} 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 *α*, respectively. The dimensions of *t* and *α* are *ϕ* is dimensionless in this representation. The renormalization constants *Z*_{t} and *Z*_{α} are defined as follows

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

We can introduce the renormalized field *Z*_{ϕ} is the renormalization constant. Then the Lagrangian is

where *ϕ* denotes the renormalized field *ϕ*_{R}.

### 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

Similarly, the *ϕ*^{2} is renormalized as

Hence the

The expectation value

where *m*_{0} to avoid the infrared divergence and Z_{t}=1 to this order. The constant *Z*_{α} is determined to cancel the divergence:

From the equations

The beta function for *α* reads

where we set *α*. The function *β*(*α*) has a zero at

### 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*^{2} term. The lowest-order two-point function is

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

where *K*_{0} is the zeroth modified Bessel function and *m*_{0} is introduced to avoid the infrared singularity. Because

Where

We use the expansion *p*_{2} term. We denote the derivation of *t* from the fixed point *ν*:

for *d* = 2. Using the asymptotic formula *x*, we obtain

where *c*_{0} is a constant and *α* was absorbed by *Z*_{α}. Now the two-point function up to this order is

The renormalized two-point function is

Then, we can choose *Z*_{ϕ} = 1 and

*t* up to the order of α2, and thus we do not need the renormalization constant *Z*_{ϕ} of the field *ϕ*. This means that we can adopt the bare coupling constant as

The renormalization function of *t* is obtained from the equation

Because the finite part of *α* as

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

Since the equation for *α* is homogeneous in *α*, we can change the scale of *α* arbitrarily. Thus, the numerical coefficient of *tα*^{2} 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

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

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-unbinding transition of vortices.

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

where *J*_{z} and *β* is the inverse temperature. *τ* is a small cut-off with

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

## 5. Scalar quantum electrodynamics

We have examined the *ϕ*^{4} theory and showed that there is a phase transition. This is a second-order 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 *A*_{μ}, called the scalar quantum electrodynamics (QED). The Lagrangian is

where *g* is the coupling constant and *D*_{μ} is the covariant derivative given as

with the charge *e*. The scalar field *ϕ* is an *N* component complex scalar field such as *N* = 1. They called this transition the dimensional transmutation. The result based on the *ϵ*-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

where *ϕ, g, e* and *A*_{μ} are renormalized quantities. We have four renormalization constants. Thanks to the Ward identity

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

The renormalization group equations are given by

The fixed point is given by

where *N* is sufficiently large as long as *N* is small,

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 *c* was introduced in [101] to impose a relation between the external momentum *p* and the momentum *q* of the gauge field as *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.