Wilson’s Renormalization Group
and Its Applications in
Perturbation Theory111Lectures given at the APCTP Field
Theory Winter School, Feb. 2-6, 2006 in Pohang, Korea; preprint
The general prescription for constructing the continuum limit of a field theory is explained using Wilson’s renormalization group. We then formulate the renormalization group in perturbation theory and apply it to dimensional and QED.
- 0 Introduction
- 1 Lecture 1 – Continuum Limits
- 2 Lecture 2 – Wilson’s RG
- 3 Lecture 3 – Perturbative Exact RG
- 4 Lecture 4 – Application to QED
- A non-linear sigma model in dimensions
- B Large approximation for the lattice theory
Quantum Field Theory has a very long history. It started with the quantization of the electromagnetic field by Dirac in 1927. This is almost 80 years ago. Currently every student of particle theory (and string theory) takes a course on quantum field theory which covers such important topics as perturbative renormalization, gauge theories, spontaneous symmetry breaking, and the Higgs mechanism. I assume that you know at least perturbative renormalization.
The problem with perturbative renormalization theory is the lack of physical insights. It consists of procedures for subtracting UV divergences to get finite results. Unfortunately, many if not most people still regard renormalization this way. It is the purpose of the next four lectures to introduce the physics of renormalization.
The essence of modern renormalization theory has been known for a long time. It was initiated mainly by Ken Wilson in the late 60’s. You must have heard the expressions such as renormalization group (RG), RG flows, fixed points, relevant and irrelevant parameters. Unfortunately there is no textbook introducing these ideas of Wilson’s. After 40 years, the best reference is still the two lecture notes [2, 3] by himself. I believe Wilson’s renormalization theory is best studied in a second course on field theory, and my four lectures will give merely an outline.
The four lectures are organized as follows. In lecture 1, we start with concrete examples of renormalization to introduce the relation between criticality and renormalizability. We will give hardly any derivation, but the examples will illuminate the meaning of renormalization. In lecture 2, we introduce the exact renormalization group (ERG) as a tool to understand the nature of renormalization. The emphasis is on the ideas of fixed points, relevance/irrelevance of parameters, and universality. In lectures 3 & 4, we apply Wilson’s exact renormalization group to perturbation theory. This was initiated by J. Polchinski in 1984. We mainly discuss the theory in dimensions in lecture 3, and QED in lecture 4. I think that it is a reflection of the depth of the renormalization group that after almost 40 years of conception it is still under active research.
1 Lecture 1 – Continuum Limits
The purpose of the first lecture is to familiarize ourselves with the concept of renormalization through concrete examples. Before we start, we should agree on the use of the euclidean metric as opposed to the Minkowski metric.
Given an -point Green function of a scalar field
we obtain an -point correlation function
by the analytic continuation
For example, the free scalar propagator
Problem 1-1: Derive this.
Hence, the free propagator in Minkowski space
in euclidean space.
1.1 The idea of a continuum limit
The idea of a continuum limit is very simple and must be already familiar to you. We consider a theory with a momentum cutoff , meaning that the theory is defined only up to the scale . For example, a lattice theory defined on a cubic lattice of a lattice unit
has the momentum cutoff . (Figure 1)
The continuum limit is the limit
and renormalization is the specific way of taking the continuum limit so that the physical mass scale , say the mass of an elementary particle, remains finite.
From the viewpoint of a lattice theory, it is more natural to measure distances in lattice units. Hence, the lattice unit becomes simply .
In this convention, the Compton length or equivalently the inverse of the physical mass is a dimensionless number called the correlation length. Therefore, we obtain
Clearly, as we take while keeping finite. Thus, as we take the continuum limit, the lattice theory must obtain an infinite correlation length.
A lattice theory with an infinite correlation length is called a critical theory. Therefore, to obtain a continuum limit the corresponding lattice theory must be critical. Let us look at examples.333 See Appendix A for an example of asymptotic free theories. In lecture 2 we simply quote the results without any derivation. See Appendix B for explicit calculations for in & dimensions.
1.2 Ising model in dimensions
The Ising model on a square lattice is defined by the action
At each site of the lattice, we introduce a classical spin variable . The parameter is a dimensionless positive constant, which we can regard as the inverse of a reduced (i.e., dimensionless) temperature.
The partition function is defined by
and the correlation functions are defined by
The action is invariant under the global transformation
With respect to this symmetry, the model has two phases:
High temperature phase : the symmetry is exact, and
Low temperature phase : the symmetry is spontaneously broken, and
For large , the two-point function behaves exponentially as
This defines the correlation length . At the theory is critical with . Two critical exponents
characterize the theory near criticality as follows:
As , the correlation length behaves as
As , the VEV behaves as
The correlation functions near the critical point obeys the scaling law:
where for . For , the scaling law is valid only for large separation of lattice sites:
For , the scaling law simply boils down to the exponential behavior of near criticality. For , the scaling law gives
where . For the limit to exist as , the function must behave like
for . Hence, at the critical point , the two-point function is given by the exponential
In fact this is another way of introducing the critical exponent .
The scaling law introduced above implies that we can renormalize the Ising model to construct a scalar field theory as follows:
where both and have mass dimension .444We chose a sign convention so that the is spontaneously broken for .
Let us explain this formula in several steps:
— Hence, as , the theory approaches criticality. The particular dependence was chosen so that .
Necessity of — was introduced so that the physical length of a lattice unit is . Hence, has mass dimension , and has mass dimension .
Given an arbitrary coordinate , is not necessarily a vector with integer components. Since , however, we can always find an integral vector which approximates to the accuracy .
Applying the scaling law, we can compute the limit as
where we used
Thus, the limit exists. The limit depends not only on the mass parameter but also on the arbitrary mass scale .
RG equation — The correlation function satisfies
This implies that the scale change of the coordinates
can be compensated by the change of the mass parameter :
and renormalization of the field:
Hence, is the scale dimension of , and is that of . The general solution of the RG equation is given by the scaling formula with as arbitrary functions.
dependence (alternative RG equation)
This is obtained from the previous RG equation by dimensional analysis. The change of is compensated by renormalization of .
Elimination of — If we want, we can eliminate the arbitrary scale from the continuum limit by giving the mass dimension to the scalar field. By writing as the new scalar field , we obtain
Before ending, let us examine the short distance behavior using RG. The RG equation can be rewritten as
Hence, in the short distance limit, the correlation functions are given by those at the critical point :
Especially for the two-point function, we obtain
1.3 Ising model in dimensions
We can construct the continuum limit of the dimensional Ising model the same way as for the dimensional one. The only difference is the value of the critical exponents and .
These are known only approximately. gives the difference of the scale dimension of the scalar field from the free field value, and is called the anomalous dimension.
In defining the continuum limit, we can give any engineering dimension to the scalar field. Here, let us pick , the same as the free scalar field. The two-point function can be defined as
Problem 1-2: Define the -point function.
We have arbitrarily given the engineering to the parameter .
The two-point function obeys the following RG equation:
This implies the short-distance behavior
1.4 Alternative: on a cubic lattice
The same continuum limit can be obtained from a different model. Let be a real variable taking a value from to . We consider the theory on a cubic lattice:
where , but can be negative.555See Appendix B for explicit calculations.
For fixed, the theory has two phases depending on the value of :
symmetric phase : is intact.
broken phase : is spontaneously broken, and .
Note that the critical value depends on .
The continuum limit is obtained as
where are the same critical exponents as in the Ising model. This limit is not necessarily independent of . For independence, we need to do rescale both and :
Universality consists of two statements:
are the same as in the Ising model.
The continuum limit is the same as in the Ising model. (We only have to choose and appropriately.)
We have defined the continuum limit using the lattice units for the lattice theory. How do we take the continuum limit if we use the physical units instead? To go to the physical units, we assign the length
to the lattice unit. The action is now given by
Then, to obtain the continuum limit we must choose
and we obtain
Note that is a finite arbitrary constant. The bare squared mass has not only a quadratic divergence but also a divergence of power . The bare coupling is linearly divergent. We see clearly that the UV divergences of parameters are due to the use of physical units. If we use the lattice units, there is no divergence.666Except for the divergences in the lattice distance and the normalization constant .
1.5 Ising model in dimensions
The dimensional case is very different from the lower dimensional cases in that we cannot take the continuum limit without obtaining a free theory. This is called triviality. For example, we obtain
and are appropriate constants.777See Appendix B for explicit calculations using the dimensional theory.
To keep the theory interacting, we must choose
large but finite so that we can define a coupling constant by888This is what we usually call
where999To follow the rest, we don’t lose much by assuming .
is a function of satisfying the differential equation
Problem 1-3: Derive this.
Note that for , we find
Hence, a large cutoff implies a small coupling.
We define the theory by
To derive the RG equation, we scale infinitesimally to . To keep invariant, we must change to . This changes by
To keep invariant, we must change by
Hence, we obtain101010There is no anomalous dimension for the field . In general it is of order , and it can be eliminated by field redefinition.
In perturbation theory we compute the correlation functions in powers of . Since the cutoff is given by
it is infinite in perturbation theory. Hence, as long as we use perturbation theory, we can take the continuum limit of the theory.
2 Lecture 2 – Wilson’s RG
There are three important issues with renormalization:
how to renormalize a theory — in the previous lecture we have seen how to take the continuum limit. We would like to understand why the limit exists.
universality — the continuum limit does not depend on the specific model we use. For example, the three dimensional theory can be defined using either the Ising model or the theory on a cubic lattice.
finite number of parameters — the continuum limit depends only on a finite number of parameters such as for the three dimensional and for the four dimensional . We would like to understand why.
Ken Wilson clarified all these by introducing his RG.[2, 3] We may add the adjective exact or Wilson’s or Wilsonian to distinguish it from the RG acting on a finite number of parameters. Exact RG can be abbreviated as ERG.
2.1 Definition of ERG
It is difficult to formulate ERG precisely. (We will omit E from ERG from now on.) In the next lecture we will see how to formulate it within perturbation theory. Here, we simply assume the existence of a non-perturbative formulation, and use it to illustrate the expected properties of RG.111111For a more rigorous approach than presented here, please consult with Wegner’s article . It is important to note that we don’t need to use RG to define continuum limits as we saw in the previous lecture. The most important role of RG is to give us an insight into the actual procedure of renormalization. Without RG it is hard to understand the meaning of renormalization.
To simplify our discussion, we introduce only one real scalar field. Given a momentum cutoff , the scalar field can be expanded as
where for -dimensional space. We would like to define RG so that the cutoff does not change under renormalization. This calls for the use of “lattice units”: we make everything dimensionless by multiplying appropriate powers of . Hence, the scalar field can be expanded as
where is a dimensionless spatial vector in units of , and is a dimensionless momentum in units of .
A theory is defined by the action which is a functional of .121212Precisely speaking this is not quite correct. Two different actions related to each other by a change of variables correspond to the same theory. We will ignore this subtlety in the rest. Again consult with  for more serious discussions. Besides the usual invariance under translation and rotation, there are two constraints on the choice of :
positivity — must be bounded from below so that the functional integral
is well defined.131313Of course, what is well defined is by unit volume (free energy/volume).
locality — If we expand in powers of , the interaction kernel
is non-vanishing only within a separation of order . This condition is hard to incorporate, though, unless we express as a power series of .
These two properties are very important physically, but they do not play any major roles in the following discussion of RG. Hence, you may happily ignore them ;-)
RG is introduced as a transformation from one to another. It consists of three steps:
integration over high momentum modes — we integrate over only for
where is infinitesimal. We obtain
where depends only on with
rescaling of space or equivalently momentum — we are left with the scalar field
We now define
so that its Fourier mode is given by
and that the momentum of ranges over the entire domain
renormalization of — we change the normalization of the field
and denote the action by . Like the original , the transformed action is defined for the field with momentum . We determine the renormalization constant , for example, by the condition that the kinetic term141414The coefficient of the quadratic term can be expanded in powers of . The kinetic term is the linear term in this expansion. of is normalized as
The value of depends on , and we may write for clarity. This last step is necessary for the RG transformation to obtain fixed points.
The transformation from to is the infinitesimal RG transformation so that we write . Then, under the , we obtain
This is valid for large separations
for which the modes do not contribute. By integrating over , we obtain a finite RG transformation , under which
The RG transformation gives a flow, called an RG flow, in the space of all permissible actions.
Along an RG flow, the correlation length changes trivially as
as a consequence of rescaling of space. This simple equation implies the following:
If a theory is critical with , it remains critical under RG.
If a theory is non-critical, its correlation length becomes less and less along its RG flow.
2.2 Fixed points and relevant & irrelevant parameters
A fixed point is an action which is invariant under the RG transformation. At the fixed point we find
Especially for , we obtain151515We can extend this equation to define the correlation function for arbitrary .
This implies that the correlation length is infinite:
The scale transformation is compensated by renormalization of the field, and the physics at the fixed point is scale invariant with no characteristic length scale.
The scale dimension of the scalar field, , is not the only important quantity that characterizes the fixed point. To explain this, let us consider the RG flow in a small (but finite) neighborhood of , where we expect to be able to linearize the RG transformation. Let
The RG transformation is given by
where is a linear transformation. We introduce eigenvectors of by161616We assume all the eigenvalues are real.
Using we can expand the difference :
Since the RG transformation is linearized in this basis, we obtain
We can use the parameters as the coordinates of the theory space in the neighborhood of . We call with positive eigenvalues relevant, and those with negative eigenvalues irrelevant.171717Those with zero eigenvalues are called marginal. We assume the absence of marginal parameters to simplify the discussion below. Like the scale dimension of the scalar field, the scale dimensions of the relevant and irrelevant parameters characterize the fixed point .
2.3 Critical subspace and renormalized trajectory
As an example, let us consider a fixed point with one relevant parameter with scale dimension . All the other (an infinite number of them) parameters are irrelevant. Let be the least irrelevant, meaning that its scale dimension is the largest. In an infinitesimal neighborhood of , it is a good approximation to keep just and satisfying the RG equations
The solution is easily obtained as
As grows, goes to while grows unless to begin with. All the other irrelevant parameters approach zero faster than .
In the neighborhood of , where is well defined, we can define a subspace by a single condition
This is called a critical subspace . The critical subspace can be defined in a larger neighborhood of as the subspace which flows into the fixed point as :
This implies that the long-distance behavior of any theory in is the same as the fixed point . Especially, for any theory on , we find the correlation length infinite
and the theory is critical. Given an arbitrary theory with parameters, what is the likelihood that the theory is critical? Since is defined by a single condition , we expect that we need to fine tune only one of the parameters to make the theory critical.
Another subspace, called a renormalized trajectory , is defined by the vanishing of all the irrelevant parameters:
This is a one-dimensional subspace with a single parameter . Alternatively, can be defined as the solution of the equation
implying that if is the coordinate of an arbitrary theory on , then also belongs to . can be also defined as the RG flow coming out of the fixed point . Note that along the renormalized trajectory, the correlation length is given by
where is a constant, and is for . is a one-dimensional subspace in the infinite dimensional space. Hence, a theory with a finite number of parameters has no chance of lying in .
2.4 Example: dimensional theory
Let us now consider the dimensional theory defined by the action
For simplicity, we assume that the theory lies in the subspace where we can linearize the RG transformation. Then, the coordinates are obtained as functions of and :
For any , we only need to tune to make the theory critical. This implies that there is only one relevant parameter , and are all irrelevant. Hence, we obtain
The critical exponents introduced in the previous lecture are given by the scale dimensions of the fixed point :
For a given , let us consider a theory near criticality:
Expanding with respect to , we obtain
Applying for large , we obtain
Hence, by choosing
for , we obtain
Thus, RG gives