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Engineering » Electrical and Electronic Engineering » "Recent Studies in Perturbation Theory", book edited by Dimo I. Uzunov, ISBN 978-953-51-3262-2, Print ISBN 978-953-51-3261-5, Published: June 14, 2017 under CC BY 3.0 license. © The Author(s).

Chapter 4

Renormalization Group Theory of Effective Field Theory Models in Low Dimensions

By Takashi Yanagisawa
DOI: 10.5772/intechopen.68214

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ϕ4 interaction with the coupling constant g.
Figure 1. ϕ4 interaction with the coupling constant g.
The contributions to the two-point function Γ(2)(p) up to the order of g2.
Figure 2. The contributions to the two-point function Γ(2)(p) up to the order of g2.
The third term in Figure 2 is a sum of two configurations (a) and (b).
Figure 3. The third term in Figure 2 is a sum of two configurations (a) and (b).
Diagrams for four-point function.
Figure 4. Diagrams for four-point function.
The beta function of g for d<4. There is a finite fixed point gc.
Figure 5. The beta function of g for d<4. There is a finite fixed point gc.
Corrections to the mass term. Multiplicity weights are 8 for (a) and 2N for (b).
Figure 6. Corrections to the mass term. Multiplicity weights are 8 for (a) and 2N for (b).
Lowest order interaction for πi.
Figure 7. Lowest order interaction for πi.
Diagrams for the two-point function. The diagrams (c) and (d) come from the Zeeman term.
Figure 8. Diagrams for the two-point function. The diagrams (c) and (d) come from the Zeeman term.
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.
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.
Diagrams that contribute to the two-point function.
Figure 10. Diagrams that contribute to the two-point function.
The renormalization group flow for the sine-Gordon model as μ → ∞.
Figure 11. The renormalization group flow for the sine-Gordon model as μ → ∞.

Renormalization Group Theory of Effective Field Theory Models in Low Dimensions

Takashi Yanagisawa
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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 [115]. The physics of a many-body system is sometimes captured by the analysis of an effective field theory model [1619]. Typically, effective field theory models are the ϕ4 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 ϕ4 model is the model of a phase transition, which is often referred to as the Ginzburg-Landau model. The renormalization of the ϕ4 model gives a prototype of renormalization group procedures in field theory [2024].

The non-linear sigma model appears in various fields of physics [15, 2527] and is the effective model of Quantum chromodynamics (QCD) [28] and also that of magnets (ferromagnetic and anti-ferromagnetic materials) [2932]. 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 [3342]. This is called the chiral model.

The sine-Gordon model also has universality [4349]. 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 [5357]. 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 [6065]. 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 [6671]. 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 [7275].

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. ϕ4 model

2.1. Lagrangian

The ϕ4 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 L is given by d, where d is the dimension of the space-time: [L]=μd. The dimension of the field ϕ is (d2)/2: [ϕ]=μ(d2)/2. Because 4 has the dimension d, the dimension of g is given by 4 – d: [g] = μ4 – d. Let us adopt that ϕ has N components as ϕ = (ϕ1, ϕ2, …, ϕN). The interaction term ϕ4 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 p2=(p0)2p2 for p=(p0,p).

Let us consider the correction to the Green’s function by means of the perturbation theory in terms of the interaction term 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 ϕ0 denotes the bare field, g0 denotes the bare coupling constant and m0 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 Zϕ, Zg and Z2 are renormalization constants. When we write Zg as

we have g0Zϕ2=gZ4. Then, the Lagrangian is written by means of renormalized field and constants


2.2. Regularization of divergences

2.2.1. Two-point function

We use the perturbation theory in terms of the interaction 4. For a multi-component scalar field theory, it is convenient to express the interaction ϕ4 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 Γ(2)(p)=iG(p)1. The contributions to Γ(2) are shown in Figure 1. The first term indicates p2Zϕ and the contribution in the second term is represented by the integral


Figure 1.

ϕ4 interaction with the coupling constant g.


Using the Euclidean co-ordinate q4 = –iq0, this integral is evaluated as


where Ωd 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.


Figure 2.

The contributions to the two-point function Γ(2)(p) up to the order of g2.

There are 422N+4222=32N+64 ways to connect lines for an 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.


Figure 3.

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

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


The momentum integral of this term is given as

J(k):= ddp(2π)dddq(2π)d 1p2q2(p+q+k)2.

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

J= (18π2)218ϵ+regularterms

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

1q2(p+q+k)2= 01dx1[q2x+(p+q+k)2(1x)]2.

For q′ = q + (1 – x)(p + k), we have

ddq(2π)d1q2(p+q+k)2= ddq'(2π)d 01dx1[q'2+x(1x)(p+k)2]2= Ωd(2π)d 01dx(x(1x))d22 ((p+k)2)d22 0drrd11(r2+1)2= Ωd(2π)d 12 Γ(d2)Γ(2d2)Γ(d21)21Γ(d2) ((p+k)2)d22.

Here, the following parameter formula was used

1AnBm= Γ(n+m)Γ(n)Γ(m) 01dxxn1(1x)m1[xA+(1x)B]n+m.

Then, we obtain

ddp(2π)d 1p2((p+k)2)2d/2= Γ(3d/2)Γ(2d/2) 01dx(1x)1d/2 ddp'(2π)d 1[p'2+x(1x)k2]3d/2= Ωd(2π)d Γ(3d/2))Γ(2d/2)B(d2,d21)12B(d2,3d) (k2)d3.

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

Γ(ϵ)= 1ϵ+finiteterms

for ϵ → 0. This results in

ddp(2π)dddq(2π)d 1p2q2(p+q+k)2= (18π2)218ϵ k2+ regular terms

Therefore, the two-point function is evaluated as

Γ(2)(p)= Zϕp2+18ϵ N+218 (g8π2)2 p2,

up to the order of O(g2). In order to cancel the divergence, we choose Zϕ as

Zϕ=1 18ϵ N+218 (18π2)2 g2.

2.2.2. Four-point function

Let us turn to the renormalization of the interaction term g4. The perturbative expansion of the four-point function is shown in Figure 4. The diagram (b) in Figure 4, denoted as ΔΓb(4), is given by for N = 1:


Figure 4.

Diagrams for four-point function.

ΔΓb(4)(p)= g212ddq(2π)d 1(q2m2)(((p+q)2m2).

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

ΔΓb(4)(p)= i18π2 12ϵ g2,

for d = 4 – ϵ. Let us evaluate the multiplicity of this contribution for N > 1. For N = 1, we have a factor 42322/4!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

(14!)222222N= N18.

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

ΔΓ(4)(p)= i18π2 N+86 1ϵg2.

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

Γ(4)(p)= igZ4μϵ+i18π2 N+86 1ϵ g2.

for d = 4 – ϵ where we neglect μϵ in the second term. The renormalization constant is determined as

Z4=1+N+86ϵ 18π2g.

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

2.3. Beta function β(g)

The bare coupling constant is written as g0=Zggμ4d=(Z4/Zϕ2)gμ4d. Since g0 is independent of the energy scale, μ, we have μg0/μ=0. This results in

μgμ= (d4)ggμgμlnZgg,

where Zg=Z4/Zϕ2. We define the beta function for g as

β(g)= μgμ,

where the derivative is evaluated under the condition that the bare g0 is fixed. Because


the beta function is given as

β(g)= ϵg1+glnZgg= ϵg+N+8618π2g2+O(g3).

β(g) up to the order of g2 is shown as a function of g for d < 4 in Figure 5. For d < 4, there is a non-trivial fixed point at


Figure 5.

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

gc= ϵ48π2N+8.

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

β(g)= 316π2g2+ .

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

β(g)= 316π2g2 1731(16π2)2g3+ (1458+12ζ(3))1(16π2)3g4+ A51(16π2)4g5,


A5= (349948+78ζ(3)18ζ(4)+120ζ(5)),

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


and then β(g) is written as

β(g)= ϵg+ϵg2[N+86ϵ+2(b1ϵ2+b2ϵ)g+(N+8)236ϵ2g+]= ϵg+N+86g29N+4236g3+ 

Here, the factor 1/8π2 is included in g. The terms of order 1/ϵ2 are cancelled because of

b1= (N+8)272.

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

β(g)= ϵg+N+86g2+ +n!annbcgn+ ,

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 ΓB(n)(pi,g0,m0,μ) and ΓR(n)(pi,g,m,μ), respectively, where pi (i = 1,…, n) indicate momenta. The energy scale μ indicates the renormalization point. ΓR(n) has the mass dimension n + dnd/2: [ΓR(n)]=μn+dnd/2. These quantities are related by the renormalization constant Zϕ as

ΓR(n)(pi,g,m2,μ) = Zϕn/2ΓB(n)(pi,g0,m02,μ).

Here, we consider the massless case and omit the mass. Because the bare quantity ΓB(n) is independent of μ, we have

This leads to


Then we obtain the equation for ΓR(n):


where γϕ is defined as

γϕ= μμlnZϕ.

A general solution of the renormalization equation is written as




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 γϕ(g') by  γϕ(gc), ΓR(n) is expressed as

ΓR(n)(pi,gc,μ)= μn2γϕ(gc)f(n)(pi,gc,μ).

In general, we define γ(g) as

γ(g)lnμ= g1gγϕ(g)β(g)dg,

Then, we obtain

ΓR(n)(pi,g,μ)= μn2γ(g)f(n)(pi,g,μ).

Under a scaling piρpi, ΓR(n) is expected to behave as

ΓR(n)(ρpi, gc, μ)= ρn+dnd/2ΓR(n)(pi, gc, μ/ρ),

because ΓR(n) has the mass dimension n+dnd/2. In fact, Figure 4b gives a contribution being proportional to

g2(μ4d)2ddq1q2(ρp+q)2= g2(μ4d)2ρd4ddq1q2(p+q)2= ρ4dg2(μρ)2(4d)ddq1q2(p+q)2,

after the scaling piρpi for n = 4. We employ Eq. (58) for n = 2

ΓR(2)(ρpi, gc, μ)= ρ2ΓR(2)(pi, gc, μ/ρ)= ρ2(μρ)γf(2)(pi, gc, μ/ρ)= ρ2γμγf(2)(pi, gc, μ/ρ)= ρ2γΓR(2)(pi, gc, μ/ρ).

This indicates

Γ(2)(p) = p2η = p2γ = (p2)1γ/2.

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

γϕ(g) = γ(g)+β(g)γ(g)glnμ.

At the fixed point g = gc, this leads to

η = γ = γ(gc) = γϕ(gc).

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

The Green’s function G(p)= Γ(2)(p)1 is given by

G(p) = 1p2η.

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

G(r) = 1p2ηeiprddp = Ωd1rd2+ηπ2Γ(4ηd)sin((4ηd)π/2).

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

G(r)  Ωd 1rd2+η.

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

γϕ(g) = μgμglnZϕ = β(g)glnZϕ.

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

γϕ=(18ϵN+191(8π2)2g)β(g)+ O(g3)= N+272 1(8π2)2g2 + O(g3).

At the critical point g = gc, where

18π2gc= 6N+8,

the anomalous dimension is given as

η= γϕ(gc)= N+22(N+8)2ϵ2+O(ϵ3).

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 > Tc in a phase transition. The bare mass m0 m and renormalized mass m are related through the relation m2=m02Zϕ/Z2. The condition μm0/μ=0 leads to


From Eq. (50), the equation for ΓR(n) is

[μμ+β(g)g n2γϕ+μμln(ZϕZ2)m2m2]ΓR(n)(pi, g, μ, m2)=0.

We define the exponent ν by

1ν2= μμln(Z2Zϕ),


[μμ+β(g)g n2γϕ(1ν2)m2m2]ΓR(n)(pi, g, μ, m2)=0.

At the critical point g = gc, we obtain

[μμn2ηζm2m2]ΓR(n)(pi, gc, μ, m2)=0,

where γϕ = η and we set

At g = gc, ΓR(n) has the form

ΓR(n)(pi, gc, μ, m2)= μn2F(n)(pi, μm2/ζ).

because this satisfies Eq. (75).

In the scaling piρpi, we adopt

ΓR(n)(ρpi, gc, μ, m2)= ρn+dnd/2ΓR(n)(pi, gc, μ/ρ, m2/ρ2).

From Eq. (77), we have

ΓR(n)(ki, gc, μ, m2)= ρn+dnd/2nη/2μn2ηF(n)(ρ1ki, ρ1μ(ρ2m2)1/ζ),

where we put ρpi=ki. We assume that F(n) depends only on ρ−1ki. We choose ρ as

ρ= (μm2/ζ)ζ/(ζ+2)= μ(m2μ2)1/(ζ+2).

This satisfies ρ1μ(ρ2m2)1/ς=1 and results in

ΓR(n)(ki, gc, μ, m2)=μd+n2(2dη)(m2μ2){d+n2(2dη)}1ζ+2μn2ηF(n)(μ1(m2μ2)1ζ+2ki). 

We take μ as a unit by setting μ = 1, so that ΓR(n) is written as

ΓR(n)(ki, gc, 1, m2)= m2ν{d+n2(2d+η)}F(n)(kim2ν),

because ς+2=1/ν. We can define the correlation length ξ by

(m2)ν = ξ.

The two-point function is written as

ΓR(2)(k,m2)= m2ν(2η)F(2)(km2ν).

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


The renormalization constant Z2 is determined from the corrections to the bare mass m0. The one-loop correction, shown in Figure 6, is given by


Figure 6.

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


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

Σ(p2)= N+26gddk(2π)d1kE2+m02= N+26g18π2m021ϵ,

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

m2= m02+ Σ(p2)= m02(1N+26ϵ18π2g)

Z2 is determined to cancel the divergence in the form m2Z2/Zϕ. The result is


Then, we have

β(g)glnZ2= N+2618π2g+O(g2).

Eq. (85) is written as

1ν =2N+2618π2gc η=2N+2N+8ϵ+O(ϵ2),

where we put g = gc and used η=γϕ(g)=(N+2)/(2(N+8)2)ϵ. Now the exponent ν is

ν= 12(1+N+22(N+8)ϵ)+O(ϵ2).

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

L= 12g(μϕ)2,

where ϕ is a real N-component field ϕ = (ϕ1,…,ϕN) with the constraint ϕ2 = 1. This model has an O(N) invariance. The field ϕ is represented as

ϕ= (σ, π1, π2, , πN1)

with the condition ο2+π12++πN12=1. The fields πi (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 σ2+πi2=1, the Lagrangian is written in the form

L= 12g(μπi)2+12g11πi2(πiμπi)2
= 12g(μπi)2+ 12g(πiμπi)2+ 

The second term in the right-hand side indicates the interaction between πi fields. The diagram for this interaction is shown in Figure 7.


Figure 7.

Lowest order interaction for πi.

Here, let us check the dimension of the field and coupling constant. Since [L]=μd, we obtain [π]=μ0 (dimensionless) and [g]=μ2d. g0 and g 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

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

L= μd2Z2gZg{(μπRi)2+ 14(μπRi2)2+ }.

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

LZ=HBg0σ=HBg0(1Z2πRi2Z28πRi4+ )
=const.  HBZ2gZgμd2πRi2 HBZ28gZgμd2(πRi2)2.

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

Then, the Zeeman term is given by

Lz=const.  Z2gHμd2πRi2 Z328gHμd2(πRi2)2+ .

3.2. Two-point function

The diagrams for the two-point function Γ(2)(p)=G(2)(p)1 are shown in Figure 8. The contributions in Figure 8c and d come from the magnetic field. Figure 8b presents


Figure 8.

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

Ib= ddk(2π)d (k+p)2k2+H= (p2H)ddk(2π)d 1k2+H

where we used the formula in the dimensional regularization given as

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

Ib= (p2H)Ωd(2π)dHd21Γ(d2)Γ(1d2)= (p2H)Ωd(2π)d1ϵ.

The H-term Ic in Figure 8c just cancels with –H in Ib. The contribution Id in Figure 8d has the multiplicity 22(N1) because (πi) has N – 1 components. Id is evaluated as

Ic= 184(N1)ddk(2π)d 1k2+H= Ωd(2π)dN121ϵ.

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

Γ(2)(p)= ZZggp2+ ZgH 1ϵ(p2+N12H),

where the factor Ωd/(2π)d is included in g for simplicity. To remove the divergence, we choose


This set of equations indicates

Z=1 + N1ϵg+O(g2).

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

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

L= 12g{(μσ)2+(μπ)2}= 12g(μθ)2.

If we disregard the region of θ, 0θ2π, 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

β(g)= μgμ,

where the bare quantities are fixed in calculating the derivative. Since μg0/μ=0, the beta function is derived as

β(g)= ϵg1+gglnZg= ϵg(N2)g2+O(g3),

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 g2 term is related with the Casimir invariant of the symmetry group O(N) [34, 49].


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.

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:

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

Let us consider the n-point function Γ(n)(ki, g, μ, H). The bare and renormalized n-point functions are introduced similarly and they are related by the renormalization constant Z

ΓR(n)(ki, g, μ, H)= Zn/2ΓB(n)(ki, g, μ, H).

From the condition that the bare function ΓB(n) is independent of μ, μdΓB(n)/dμ=0, the renormalization group equation is followed

[μμ+μgμgn2ζ(g)+(12ζ(g)+1gβ(g)(d2))HH]ΓR(n)(ki, g, μ, H)=0,

where we defined

ζ(g)= μμlnZ= β(g)glnZ.

From Eq. (113), ζ(g) is given by

ζ(g)= (N1)g+O(g2).

Let us define the correlation length ξ=ξ(g,μ). Because the correlation length near the transition point will not depend on the energy scale, it should satisfy

μddμξ(g,μ)= (μμ+β(g)g)ξ(g,μ)=0.

We adopt the form ξ=μ1f(g) for a function f(g), so that we have


This indicates

f(g)=C exp(g*g1β(g)dg),

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 gc. We approximate β(g) near g = gc as

β(g) a(ggc),

with a < 0, ξ is

ξ= μ1exp(1alnggcg*gc). 

Near the critical point ggc, ξ is approximated as

ξ1  μggc1/a.

This means that ξ → ∞ as ggc. We define the exponent v by

ξ1  ggcν,

then we have

ν= 1β'(gc).

Since β'(gc)=ϵ2(N2)gc=ϵ, this gives

1ν= ϵ + O(ϵ2) = d2+O(ϵ2).

Including the higher-order terms, ν is given as

1ν=d2+ (d2)2N2+ (d2)32(N2)+O(ϵ4).

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

L= 116πGgR

where g is the determinant of the matrix (gμν) and G is the coupling constant. The beta function for G was calculated as [7881]

β(G)= ϵGbG2,

for d=2+ϵ 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 [4349, 8291]. The Lagrangian of the sine-Gordon model is given by

L= 12t0(μϕ)2+ α0t0cosϕ,

where ϕ is a real scalar field, and t0 and α0 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 [t]=μ2d and [α]=μ2. The scalar field ϕ is dimensionless in this representation. The renormalization constants Zt and Zα are defined as follows

t0=tμ2dZt, α0= αμ2Zα.

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

L= μd22tZt(μϕ)2+ μdαZαtZtcosϕ.

We can introduce the renormalized field ϕB=ZϕϕR where Zϕ is the renormalization constant. Then the Lagrangian is

L= μd2Zϕ2tZt(μϕ)2+ μdαZαtZtcosϕ.

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 cosϕ=112ϕ2+14!ϕ4 . Then the corrections to the cosine term are evaluated as follows. The constant term is renormalized as

1 12ϕ2+14!ϕ4 =1 12ϕ2+ 12(12ϕ2)2 =exp(12ϕ2).

Similarly, the ϕ2 is renormalized as

12ϕ2+ 14!6ϕ2ϕ2 16!153ϕ22ϕ2+ =exp(12ϕ2)(12ϕ2).

Hence the αZαcos(Zϕϕ) is renormalized to


The expectation value ϕ2 is regularized as

Zϕϕ2=tμ2d Ztddk(2π)d1k2+m02= tϵΩd(2π)d,

where d=2+ϵ and we included a mass m0 to avoid the infrared divergence and Zt=1 to this order. The constant Zα is determined to cancel the divergence:


From the equations μt0/μ=0 and μα0/μ=0, we obtain

μtμ= (d2)ttμlnZtμ,
μαμ= 2α αμlnZαμ

The beta function for α reads

β(α) μαμ= 2α+ tα12Ωd(2π)d,

where we set μt/μ=(d2)t with Zt=1 up to the lowest order of α. The function β(α) has a zero at t=tc=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 p2 term. The lowest-order two-point function is

ΓB(2)(0)(p)= 1t0p2= 1tμ2dZtp2.

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 cosϕ=1(1/2)ϕ2+. First, we consider the Green’s function,


Figure 10.

Diagrams that contribute to the two-point function.

G0(x)= Zϕ<ϕ(x)ϕ(0)>=tμ2dZtddp(2π)dpeipxp2+m02 =tμ2dZtΩd(2π)dK0(m0|x|),

where K0 is the zeroth modified Bessel function and m0 is introduced to avoid the infrared singularity. Because sinhII=I3/3!+, the diagrams in Figure 10 are summed up to give

Σ(p)= ddx[eipx(sinhII)(coshI1)], 

Where I=G0(x). Since sinhIIeI/2 and coshIeI/2, the diagrams in Figure 10 lead to

ΓB(2)c(p)= 12(αμdZαtZt)2ddx(eipx1)eG0(x).

We use the expansion eipx=1+ipx(1/2)(px)2+, and keep the p2 term. We denote the derivation of t from the fixed point tc=8π as ν:

for d = 2. Using the asymptotic formula K0(x)~γln(x/2) for small x, we obtain

ΓB(2)c(p)= 18(αμdtZt)2p2(c0m02)22νΩd0dxxd+11(x2+a2)2+2ν= 18p2(αμdtZt)2(c0m02)2Ωd1ϵ+ O(ν) 1tμ2dZtp2132α2μd+2(c0m02)21ϵ+O(ν)

where c0 is a constant and a=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

ΓB(2)(p)= 1tμ2dZt[p2132α2μd+2(c0m02)21ϵ]

The renormalized two-point function is ΓR(2)=ZϕΓB(2). This indicates that

ZϕZt=1+ 132α2μd+2(c0m02)21ϵ.

Then, we can choose Zϕ = 1 and

Zt=1 132α2μd+2(c0m02)21ϵ.

Zt/Zϕ can be regarded as the renormalization constant of t up to the order of α2, and thus we do not need the renormalization constant Zϕ of the field ϕ. This means that we can adopt the bare coupling constant as t0=tμ2dZt˜ with Zt˜=Zt/Zϕ.

The renormalization function of t is obtained from the equation μt0/μ=0 for t0=tμ2dZt:

β(t)  μtμ= (d2)t+132(c0m02)21ϵ(2αμd+2μαμ+(d+2)α2μd+2)t= (d2)t+132μd+2(c0m02)2tα2

Because the finite part of G0(x0) is given by G0(x0)=(1/2π)ln(eγm0/2μ), we perform the finite renormalization of α as ααc0m02a2=αc0m02μ2. This results in

β(t)= (d2)t+132tα2.

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

β(α)= μαμ= α(214πt),
β(t)= μtμ= (d2)t+132tα2,

Since the equation for α is homogeneous in α, we can change the scale of α arbitrarily. Thus, the numerical coefficient of 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 t=8π(1+ν), and set x=2ν and y=α/4. Then, the equations are

μxμ= y2,
μyμ= xy,

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.


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

x=πβJz2, y=2|J|τ.

where Jz and J(=Jx=Jy) 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 τ1/μ. The scaling equations for the s-d model are [53, 57]

τxτ= 12y2,
τyτ= 12xy.

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 ϕ4 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 Aμ, called the scalar quantum electrodynamics (QED). The Lagrangian is

L= 12|(Dμϕ)|2 14g(|ϕ|2)2 14Fμν2,

where g is the coupling constant and Fμν=μAννAμ. Dμ is the covariant derivative given as

Dμ= μieAμ,

with the charge e. The scalar field ϕ is an N component complex scalar field such as ϕ=(ϕ1, , ϕN). 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=4ϵ when 2N<365 [9296]. 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 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

g0= Z4Zϕ2gμ4d,
e0= ZeZAZϕe,

where ϕ, g, e and Aμ 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

μe2μ= ϵe2+N24π2e4,
μgμ= ϵg+N+44π2g2+38π2e434π2e2g.

The fixed point is given by

ec= 24Nπ2ϵ,
gc= ϵ2π2N+4{1+18N±(n2360n2160)1/2n},

where n=2N. The square root δ(n2360n2160)1/2 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, 2N365, 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=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.


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