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.
- renormalization group theory
- dimensional regularization
- scalar model
- non-linear sigma model
- sine-Gordon model
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 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 [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)  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 .
2. ϕ4 model
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 is given by d, where d is the dimension of the space-time: . The dimension of the field is : . Because gϕ4 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 for .
Let us consider the correction to the Green’s function by means of the perturbation theory in terms of the interaction term gϕ4. 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 . 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 gϕ4. 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 . 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
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.
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
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(g2). In order to cancel the divergence, we choose Zϕ as
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 , is given by for 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 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
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
Because g has the dimension 4 – d such as [g] = μ4–d, we write g as gμ4–d so that g is the dimensionless coupling constant. Now, we have
for d = 4 – ϵ where we neglect μϵ 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 . Since g0 is independent of the energy scale, μ, we have . This results in
where . We define the beta function for g as
where the derivative is evaluated under the condition that the bare g0 is fixed. Because
the beta function is given as
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
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 :
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
Here, the factor 1/8π2 is included in g. The terms of order 1/ϵ2 are cancelled because of
In general, the nth order term in β(g) is given by n!gn. 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 and , respectively, where pi (i = 1,…, n) indicate momenta. The energy scale μ indicates the renormalization point. has the mass dimension n + d – nd/2: . These quantities are related by the renormalization constant Zϕ as
Here, we consider the massless case and omit the mass. Because the bare quantity is independent of μ, we have
This leads to
Then we obtain the equation for :
where γϕ is defined as
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 by , is expressed as
In general, we define γ(g) as
Then, we obtain
Under a scaling , is expected to behave as
because has the mass dimension . In fact, Figure 4b gives a contribution being proportional to
after the scaling pi→ ρpi for n = 4. We employ Eq. (58) for n = 2
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 = gc, this leads to
The exponent η shows the fluctuation effect near the critical point.
The Green’s function is given by
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 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 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 . The condition leads to
From Eq. (50), the equation for is
We define the exponent ν by
At the critical point g = gc, we obtain
where γϕ = η and we set
At g = gc, has the form
because this satisfies Eq. (75).
In the scaling , we adopt
From Eq. (77), we have
where we put . We assume that F(n) depends only on ρ−1ki. We choose ρ as
This satisfies and results in
We take μ as a unit by setting μ = 1, so that is written as
because . We can define the correlation length ξ by
The two-point function is written as
Now let us turn to the evaluation of ν. Since , 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
where the multiplicity factor is (8 + 4N)/4!. This is regularized as
for d = 4–ϵ. Therefore the renormalized mass is
Z2 is determined to cancel the divergence in the form m2Z2/Zϕ. The result is
Then, we have
Eq. (85) is written as
where we put g = gc and used . Now the exponent ν is
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
The Lagrangian of the non-linear sigma model is
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
with the condition . 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 , the Lagrangian is written in the form
The second term in the right-hand side indicates the interaction between πi fields. The diagram for this interaction is shown in Figure 7.
Here, let us check the dimension of the field and coupling constant. Since , we obtain (dimensionless) and . 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:
In order to avoid the infrared divergence at d = 2, we add the Zeeman term to the Lagrangian which is written as
Here, HB 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
where we used the formula in the dimensional regularization given as
Near two dimensions, d = 2 + ϵ, the integral is regularized as
As a result, up to the one-loop-order the two-point function is
where the factor is included in 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 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
If we disregard the region of θ, , 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
where the bare quantities are fixed in calculating the derivative. Since , the beta function is derived as
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].
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 . The bare and renormalized n-point functions are introduced similarly and they are related by the renormalization constant Z
From the condition that the bare function is independent of μ, , the renormalization group equation is followed
where we defined
From Eq. (113), ζ(g) is given by
Let us define the correlation length . Because the correlation length near the transition point will not depend on the energy scale, it should satisfy
We adopt the form for a function f(g), so that we have
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
with a < 0, ξ is
Near the critical point , ξ is approximated as
This means that ξ → ∞ as g → gc. We define the exponent v by
then we have
Since , this gives
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 and the curvature R. The quantum gravity Lagrangian is
for with a constant b. This has the same structure as that for the non-linear sigma model.
4. Sine-Gordon model
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 and . The scalar field ϕ is dimensionless in this representation. The renormalization constants Zt and Zα 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 where Zϕ is the renormalization constant. Then the Lagrangian is
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 . Then the corrections to the cosine term are evaluated as follows. The constant term is renormalized as
Similarly, the ϕ2 is renormalized as
Hence the is renormalized to
The expectation value is regularized as
where 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 and , we obtain
The beta function for α reads
where we set with up to the lowest order of 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 p2 term. The lowest-order two-point function is
where K0 is the zeroth modified Bessel function and m0 is introduced to avoid the infrared singularity. Because , the diagrams in Figure 10 are summed up to give
Where . Since and , the diagrams in Figure 10 lead to
We use the expansion , and keep the p2 term. We denote the derivation of t from the fixed point as ν:
for d = 2. Using the asymptotic formula for small x, we obtain
where c0 is a constant and is a small cut-off. The divergence of α was absorbed by Zα. Now the two-point function up to this order is
The renormalized two-point function is . This indicates that
Then, we can choose Zϕ = 1 and
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 with .
The renormalization function of t is obtained from the equation for :
Because the finite part of is given by , we perform the finite renormalization of α as . This results in
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 tα2 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 , and set and . Then, the equations are
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 Jz and 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 . The scaling equations for the s-d model are [53, 57]
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
where g is the coupling constant and . Dμ is the covariant derivative given as
with the charge e. The scalar field ϕ is an N component complex scalar field such as . 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 when [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 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 .
The bare and renormalized fields and coupling constants are defined as
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
The fixed point is given by
where . The square root is real when . This indicates that the zero of a set of beta functions exists when N is sufficiently large as long as . Hence there is no continuous transition when N is small, , 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 . When N is large as far as , 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 . An extra parameter c was introduced in  to impose a relation between the external momentum p and the momentum q of the gauge field as . It was shown that when , we have a second-order transition. We do not think that it is clear whether the introduction of c is justified or not.
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.