Open access peer-reviewed chapter

Renormalization Group Theory of Effective Field Theory Models in Low Dimensions

By Takashi Yanagisawa

Submitted: October 31st 2016Reviewed: February 28th 2017Published: June 14th 2017

DOI: 10.5772/intechopen.68214

Downloaded: 532

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 [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

L=12(μϕ)212m2ϕ2g4!ϕ4,E1

where ϕ is a scalar field and g is the coupling constant. In the unit of the momentum μ, the dimension of Lis given by d, where d is the dimension of the space-time: [L]=μd. The dimension of the field ϕis (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

 ϕ4=(i=1Nϕi2)2.E2

The Green’s function is defined as

Gi(xy)=i0|Tϕi(x)ϕi(y)|0,E3

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

Gi(p)=ddxeipxGi(x).E4

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

 Gi(0)(p)=1p2m2,E5

where p2=(p0)2p2for 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

L=IV+1.E6

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

D=dL2I.E7

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

4V=E+2I.E8

Then, the degree of divergence is written as

D=dL2I=d+(d4)V+(1d2)E.E9

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

D=4EE10

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

L=12(μϕ0)212m02ϕ0214!g0ϕ04,E11

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

ϕ0=Zϕϕ,E12
g0=Zgg,E13
m02=m2Z2/Zϕ,E14

where Zϕ, Zg and Z2 are renormalization constants. When we write Zg as

Zg=Z4/Zϕ2,E15

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

L=12Zϕ(μϕ)212m2Z2ϕ214!gZ4ϕ4.E16

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.

I=ddq(2π)d1q2m2.E17

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

I=iΩd(2π)dmd212Γ(d2)Γ(1d2),E18

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+64ways 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

(14!g)2(32N+64)=N+218g2.E19

The momentum integral of this term is given as

J(k):= ddp(2π)dddq(2π)d 1p2q2(p+q+k)2.E20

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+regulartermsE21

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.E22

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.E23

Here, the following parameter formula was used

1AnBm= Γ(n+m)Γ(n)Γ(m) 01dxxn1(1x)m1[xA+(1x)B]n+m.E24

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.E25

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

Γ(ϵ)= 1ϵ+finitetermsE26

for ϵ → 0. This results in

ddp(2π)dddq(2π)d 1p2q2(p+q+k)2= (18π2)218ϵ k2+ regular termsE27

Therefore, the two-point function is evaluated as

Γ(2)(p)= Zϕp2+18ϵ N+218 (g8π2)2 p2,E28

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.E29

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).E30

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

ΔΓb(4)(p)= i18π2 12ϵ g2,E31

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.E32

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.E33

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.E34

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

Z4=1+N+86ϵ 18π2g.E35

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,E36

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

β(g)= μgμ,E37

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

Zg=1+N+86ϵ18π2g+O(g2),E38

the beta function is given as

β(g)= ϵg1+glnZgg= ϵg+N+8618π2g2+O(g3).E39

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.E40

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+ .E41

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,E42

where

A5= (349948+78ζ(3)18ζ(4)+120ζ(5)),E43

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

Zg=1+N+86ϵg+(b1ϵ2+b2ϵ)g2+(c1ϵ3+c2ϵ2+c3ϵ)g3+,E44

and then β(g) is written as

β(g)= ϵg+ϵg2[N+86ϵ+2(b1ϵ2+b2ϵ)g+(N+8)236ϵ2g+]= ϵg+N+86g29N+4236g3+ E45

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

b1= (N+8)272.E46

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+ ,E47

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).E48

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

ddμΓB(n)=0.E49

This leads to

μddμ(Zϕn/2ΓR(n))=0.E50

Then we obtain the equation for ΓR(n):

(μμ+μgμgn2γϕ)ΓR(n)(pi,g,μ)=0,E51

where γϕ is defined as

γϕ= μμlnZϕ.E52

A general solution of the renormalization equation is written as

ΓR(n)(pi,g,μ)=exp(n2g1gγϕ(g)β(g)dg)f(n)(pi,g,μ),E53

where

f(n)(pi,g,μ)=F(pi,lnμg1g1β(g')dg'),E54

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).E55

In general, we define γ(g) as

γ(g)lnμ= g1gγϕ(g)β(g)dg,E56

Then, we obtain

ΓR(n)(pi,g,μ)= μn2γ(g)f(n)(pi,g,μ).E57

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

ΓR(n)(ρpi, gc, μ)= ρn+dnd/2ΓR(n)(pi, gc, μ/ρ),E58

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,E59

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).E60

This indicates

Γ(2)(p) = p2η = p2γ = (p2)1γ/2.E61

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.E62

At the fixed point g = gc, this leads to

η = γ = γ(gc) = γϕ(gc).E63

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

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

G(p) = 1p2η.E64

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

G(r) = 1p2ηeiprddp = Ωd1rd2+ηπ2Γ(4ηd)sin((4ηd)π/2).E65

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

G(r)  Ωd 1rd2+η.E66

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

γϕ(g) = μgμglnZϕ = β(g)glnZϕ.E67

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).E68

At the critical point g = gc, where

18π2gc= 6N+8,E69

the anomalous dimension is given as

η= γϕ(gc)= N+22(N+8)2ϵ2+O(ϵ3).E70

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=0leads to

μlnmμ=μμlnZϕZ2.E71

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

[μμ+β(g)g n2γϕ+μμln(ZϕZ2)m2m2]ΓR(n)(pi, g, μ, m2)=0.E72

We define the exponent ν by

1ν2= μμln(Z2Zϕ),E73

then

[μμ+β(g)g n2γϕ(1ν2)m2m2]ΓR(n)(pi, g, μ, m2)=0.E74

At the critical point g = gc, we obtain

[μμn2ηζm2m2]ΓR(n)(pi, gc, μ, m2)=0,E75

where γϕ = η and we set

ζ= 1ν2.E76

At g = gc, ΓR(n)has the form

ΓR(n)(pi, gc, μ, m2)= μn2F(n)(pi, μm2/ζ).E77

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).E78

From Eq. (77), we have

ΓR(n)(ki, gc, μ, m2)= ρn+dnd/2nη/2μn2ηF(n)(ρ1ki, ρ1μ(ρ2m2)1/ζ),E79

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).E80

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

ΓR(n)(ki, gc, μ, m2)=μd+n2(2dη)(m2μ2){d+n2(2dη)}1ζ+2μn2ηF(n)(μ1(m2μ2)1ζ+2ki). E81

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),E82

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

(m2)ν = ξ.E83

The two-point function is written as

ΓR(2)(k,m2)= m2ν(2η)F(2)(km2ν).E84

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

1ν=2+μμln(Z2Zϕ)=2+β(g)glnZ2γϕ(g).E85

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).

Σ(p2)=iN+26gddk(2π)d1k2m02,E86

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

Σ(p2)= N+26gddk(2π)d1kE2+m02= N+26g18π2m021ϵ,E87

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

m2= m02+ Σ(p2)= m02(1N+26ϵ18π2g)E88

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

Z2=1+N+26ϵ18π2g.E89

Then, we have

β(g)glnZ2= N+2618π2g+O(g2).E90

Eq. (85) is written as

1ν =2N+2618π2gc η=2N+2N+8ϵ+O(ϵ2),E91

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).E92

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,E93

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)E94

with the condition ο2+π12++πN12=1. The fields πi (i = 1, …, N – 1) are regarded as representing fluctuations. The Lagrangian is given by

L=12g{(μσ)2+(μπi)2},E95

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)2E96
= 12g(μπi)2+ 12g(πiμπi)2+ E97

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

g0=gμ2dZg,E98
πBi= Z πRiE99

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

L= μd2Z2gZg{(μπRi)2+ 14(μπRi2)2+ }.E100

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+ )E101
=const.  HBZ2gZgμd2πRi2 HBZ28gZgμd2(πRi2)2.E102

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

H= ZZgHBE103

Then, the Zeeman term is given by

Lz=const.  Z2gHμd2πRi2 Z328gHμd2(πRi2)2+ .E104

3.2. Two-point function

The diagrams for the two-point function Γ(2)(p)=G(2)(p)1are 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+HE105

where we used the formula in the dimensional regularization given as

ddk=0.E106

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

Ib= (p2H)Ωd(2π)dHd21Γ(d2)Γ(1d2)= (p2H)Ωd(2π)d1ϵ.E107

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.E108

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

Γ(2)(p)= ZZggp2+ ZgH 1ϵ(p2+N12H),E109

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

ZZg=1+gϵ,E110
Z=1+N12ϵg.E111

This set of equations indicates

Zg=1+N1ϵg+O(g2),E112
Z=1 + N1ϵg+O(g2).E113

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

σ2+π2=1E114

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.E115

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,E116

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),E117

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:

gc= ϵN2,E118

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).E119

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,E120

where we defined

ζ(g)= μμlnZ= β(g)glnZ.E121

From Eq. (113), ζ(g) is given by

ζ(g)= (N1)g+O(g2).E122

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.E123

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

β(g)df(g)dg=f(g).E124

This indicates

f(g)=C exp(g*g1β(g)dg),E125

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

ξ=Cμ1exp(1N2(1g1g*)).E126

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),E127

with a < 0, ξ is

ξ= μ1exp(1alnggcg*gc). E128

Near the critical point ggc, ξ is approximated as

ξ1  μggc1/a.E129

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

ξ1  ggcν,E130

then we have

ν= 1β'(gc).E131

Since β'(gc)=ϵ2(N2)gc=ϵ, this gives

1ν= ϵ + O(ϵ2) = d2+O(ϵ2).E132

Including the higher-order terms, ν is given as

1ν=d2+ (d2)2N2+ (d2)32(N2)+O(ϵ4).E133

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 gμνand the curvature R. The quantum gravity Lagrangian is

L= 116πGgRE134

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,E135

for d=2+ϵwith 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,E136

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]=μ2dand [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.E137

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

L= μd22tZt(μϕ)2+ μdαZαtZtcosϕ.E138

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

L= μd2Zϕ2tZt(μϕ)2+ μdαZαtZtcosϕ.E139

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).E140

Similarly, the ϕ2 is renormalized as

12ϕ2+ 14!6ϕ2ϕ2 16!153ϕ22ϕ2+ =exp(12ϕ2)(12ϕ2).E141

Hence the αZαcos(Zϕϕ)is renormalized to

αZαexp(12Zϕϕ2)cos(Zϕϕ)αZα(112Zϕϕ2+)cos(Zϕϕ).E142

The expectation value ϕ2is regularized as

Zϕϕ2=tμ2d Ztddk(2π)d1k2+m02= tϵΩd(2π)d,E143

where d=2+ϵand 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:

Zα=1t21ϵΩd(2π)d.E144

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

μtμ= (d2)ttμlnZtμ,E145
μαμ= 2α αμlnZαμE146

The beta function for α reads

β(α) μαμ= 2α+ tα12Ωd(2π)d,E147

where we set μt/μ=(d2)twith Zt=1up 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.E148

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|),E149

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)], E150

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

ΓB(2)c(p)= 12(αμdZαtZt)2ddx(eipx1)eG0(x).E151

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=8πas ν:

t8π=1+ν,E152

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)E153

where c0 is a constant and a=1/μis 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]E154

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

ZϕZt=1+ 132α2μd+2(c0m02)21ϵ.E155

Then, we can choose Zϕ = 1 and

Zt=1 132α2μd+2(c0m02)21ϵ.E156

Zt/Zϕcan 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=0for t0=tμ2dZt:

β(t)  μtμ= (d2)t+132(c0m02)21ϵ(2αμd+2μαμ+(d+2)α2μd+2)t= (d2)t+132μd+2(c0m02)2tα2E157

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.E158

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

β(α)= μαμ= α(214πt),E159
β(t)= μtμ= (d2)t+132tα2,E160

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=2νand y=α/4. Then, the equations are

μxμ= y2,E161
μyμ= xy,E162

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

x2y2=const.E163

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.E164

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,E165
τyτ= 12xy.E166

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,E167

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

Dμ= μieAμ,E168

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=4ϵwhen 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

ϕ0= Zϕϕ,E169
g0= Z4Zϕ2gμ4d,E170
e0= ZeZAZϕe,E171
Aμ0= ZAAμ,E172

where ϕ, g, e and Aμ are renormalized quantities. We have four renormalization constants. Thanks to the Ward identity

Ze= ZA,E173

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

Zϕ=1+38π2ϵe2,E174
ZA=12N48π2ϵe2,E175
Zg=1+2N+88π2ϵg+38π2ϵ1ge4.E176

The renormalization group equations are given by

μe2μ= ϵe2+N24π2e4,E177
μgμ= ϵg+N+44π2g2+38π2e434π2e2g.E178

The fixed point is given by

ec= 24Nπ2ϵ,E179
gc= ϵ2π2N+4{1+18N±(n2360n2160)1/2n},E180

where n=2N. The square root δ(n2360n2160)1/2is 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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Takashi Yanagisawa (June 14th 2017). Renormalization Group Theory of Effective Field Theory Models in Low Dimensions, Recent Studies in Perturbation Theory, Dimo I. Uzunov, IntechOpen, DOI: 10.5772/intechopen.68214. Available from:

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