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
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 .
The Green’s function is defined as
In the non-interacting case with
where for .
Let us consider the correction to the Green’s function by means of the perturbation theory in terms of the interaction term
We have a logarithmic divergence when
Then, the degree of divergence is written as
In four dimensions
When the diagram has four external lines,
Let us consider the Lagrangian with bare quantities
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
Using the Euclidean co-ordinate
The number of ways to connect lines is 32
The momentum integral of this term is given as
To obtain this, we first perform the integral with respect to
Here, the following parameter formula was used
Then, we obtain
Therefore, the two-point function is evaluated as
up to the order of
2.2.2. Four-point function
Let us turn to the renormalization of the interaction term
As in the calculation of the two-point function, this is regularized as
Since we obtain the same factor for diagrams in Figure 4
As a result, the four-point function Γ(4) becomes finite.
2.3. Beta function
The bare coupling constant is written as . Since
where . We define the beta function for
where the derivative is evaluated under the condition that the bare
the beta function is given as
In this case, the
Here, the factor 1/8
In general, the
n-point function and anomalous dimension
Let us consider the
Here, we consider the massless case and omit the mass. Because the bare quantity is independent of
This leads to
Then we obtain the equation for :
A general solution of the renormalization equation is written as
for a function
In general, we define
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
Thus, the anomalous dimension
At the fixed point
The Green’s function is given by
The Fourier transform of
When 4 –
The definition of
Up to the lowest order of
At the critical point
the anomalous dimension is given as
2.5. Mass renormalization
Let us consider the massive case
From Eq. (50), the equation for is
We define the exponent ν by
At the critical point
because this satisfies Eq. (75).
In the scaling , we adopt
From Eq. (77), we have
where we put . We assume that
This satisfies and results in
because . We can define the correlation length
The two-point function is written as
Now let us turn to the evaluation of
The renormalization constant
where the multiplicity factor is (8 + 4
Then, we have
Eq. (85) is written as
where we put
In the mean-field approximation,
3. Non-linear sigma model
The Lagrangian of the non-linear sigma model is
with the condition . The fields
where summation is assumed for index
The second term in the right-hand side indicates the interaction between
Here, let us check the dimension of the field and coupling constant. Since , we obtain (dimensionless) and .
In order to avoid the infrared divergence at
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,
As a result, up to the one-loop-order the two-point function is
where the factor is included in
This set of equations indicates
When we represent
If we disregard the region of
3.3. Renormalization group equations
The beta function
where the bare quantities are fixed in calculating the derivative. Since , the beta function is derived as
In the case of
In two dimensions
Let us consider the
From the condition that the bare function is independent of
where we defined
From Eq. (113),
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
Near the critical point ,
This means that
then we have
Since , this gives
Including the higher-order terms,
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
for with a constant
4. Sine-Gordon model
Here, the energy scale
We can introduce the renormalized field where
4.2. Renormalization of
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
Hence the is renormalized to
The expectation value is regularized as
where and we included a mass
From the equations and , we obtain
The beta function for
where we set with up to the lowest order of
4.3. Renormalization of the two-point function
Let us turn to the renormalization of the coupling constant
Where . Since and , the diagrams in Figure 10 lead to
We use the expansion , and keep the
The renormalized two-point function is . This indicates that
Then, we can choose
can be regarded as the renormalization constant of
The renormalization function of
Because the finite part of is given by , we perform the finite renormalization of
As a result, we obtain a set of renormalization group equations for the sine-Gordon model
Since the equation for
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
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
with the charge
The bare and renormalized fields and coupling constants are defined as
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
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
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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