Products of composite operators in the exact renormalization group formalism

We discuss a general method of constructing the products of composite operators using the exact renormalization group formalism. Considering mainly the Wilson action at a generic fixed point of the renormalization group, we give an argument for the validity of short distance expansions of operator products. We show how to compute the expansion coefficients by solving differential equations, and test our method with some simple examples.


I. INTRODUCTION
In the framework of the Wilsonian renormalization group (RG), the physics of a system is completely characterized by a Wilson  The purpose of this paper is to improve our understanding of composite operators using the formalism of the exact renormalization group (ERG) or functional RG. The importance of composite operators in ERG was emphasized early by Becchi [1], and his results have been extended in some later works such as [2][3][4].
In this work we use ERG to construct products of composite operators and study their properties. We discuss the insertion of two (or more) composite operators in correlation functions of the elementary fields. Particular attention is paid to the ERG differential equation satisfied by composite operators and their products at the fixed point.
When considering the product of two composite operators, it is natural to ask about its short distance behavior. At short distances the operator product expansion (OPE) of K. Wilson [5] is expected to be valid. In the past ERG has been used to provide an alternative proof of the existence of the OPE in perturbation theory [6][7][8][9][10][11][12][13]; the original perturbative proof goes back to Zimmermann [14]. The purpose of examining the OPE within ERG is to fill the gap between ERG and other nonperturbative approaches to quantum field theory where the OPE forms the backbone structure of the theory. Particularly relevant is the case of conformal field theories, especially in the two dimensional case. Using ERG we argue the plausibility (if not a proof) of the existence of OPE. In particular, we derive ERG differential equations (a.k.a. flow equations) satisfied by the Wilson coefficients and solve them for some simple examples.
The paper is organized as follows. In section II we define composite operators at a fixed point of the RG transformation. We introduce three equivalent approaches using the Wilson action [15,16], the generating functional of connected correlation functions with an infrared cutoff [17], and its Legendre transform (called the effective average action) [18,19], respectively. The three approaches differ in the natural choice of field variables: φ, J, Φ. In section III we generalize our construction to the product of two composite operators and consider how the OPE arises. In section IV some working examples are presented. In section V we explain how to generalize the ERG differential equations to consider the insertion of an arbitrary number of composite operators, and in section VI we discuss the ERG differential equations for composite operators away from the fixed point. We summarize our findings in section VII. We confine some technical parts in three appendices. In Appendix A we review the basics of the ERG formalism that this paper is based on. A best pedagogical effort has been made for those readers familiar with [16] but not with [15]. In Appendix B we explain how to construct local composite operators in the massive free scalar theories. In Appendix C we derive the asymptotic behavior of a short-range function necessary for the examples of Sec. IV.
We shall work in the dimensionless convention, where all dimensionful quantities have been rescaled via a suitable power of the cutoff. We also adopt the following notation; (1)

II. COMPOSITE OPERATORS AT A FIXED POINT
At a fixed point of the exact renormalization group, the Wilson action satisfies the ERG equation [15,20] where γ is the anomalous dimension. (We have prepared Appendix A for the readers who are familiar with [16] but not with [15].) We have introduced two positive cutoff functions: 1. K(p) approaches 1 as p 2 → 0, and decays rapidly for p 2 ≫ 1.
2. R(p) must be nonvanishing at p = 0 and decays rapidly for p 2 ≫ 1. The inverse transform of R(p) is a function in space that is nonvanishing only over a region of unit size.
Eq. (2) implies that the modified correlation functions defined by satisfy the scaling law for arbitrary momenta. [21] In our discussion of composite operators we find it more convenient to deal with a func- where In fact it is even more convenient to deal with the Legendre transform of W [J] [18,23]: is often called the effective average action. We can interpret W [J] as the generating functional of connected correlation functions [17,22] and Γ[Φ] as the effective action [18,23], both in the presence of an infrared cutoff. (The same cutoff is called an ultraviolet cutoff for S and an infrared cutoff for W and Γ. This is because we regard S as the weight of functional integration over low momenta to be done, but we regard W and Γ as consequences of functional integration over high momenta already done. It has recently been shown that the high momentum limit of W and Γ gives the corresponding functionals without the infrared cutoff. [24]) The ERG equations satisfied by W and Γ are given by (see [20] and reference therein) where Now, composite operators can be thought of as infinitesimal changes of S, W , or Γ.
Correspondingly, we can regard composite operators as functionals of φ, J, or Φ. Let O(p) be a composite operator of scale dimension −y and momentum p. Regarding it as a functional of J, we obtain the following ERG equation: Similarly, regarding O(p) as a functional of Φ, we can rewrite the above as where G is defined by (9).
The above two ERG equations are equivalent, and they imply that the modified correlation functions defined by satisfy the scaling law O(pe t ) φ(p 1 e t ) · · · φ(p n e t ) = exp t −y − n D + 2 2 O(p) φ(p 1 ) · · · φ(p n ) . (14)

III. PRODUCTS OF COMPOSITE OPERATORS
Given two composite operators O 1 (p), O 2 (p) of scale dimensions −y 1 , −y 2 , we wish to define their product as a composite operator of scale dimension −(y 1 + y 2 ). The naive product O 1 (p)O 2 (q) will not do because it does not satisfy (11) or (12). We must define the product by adding a local counterterm: Otherwise the product will not satisfy the scaling law: In the ERG formalism we have a dimensionless cutoff of order 1 (either in momentum space or in coordinate space). In coordinate space the inverse Fourier transform is expected to have a support of unit size around the coordinate r. We expect the same property for the product of two composite operators. Given O 1 (p) and O 2 (q), we denote their inverse Fourier transforms using the same symbol: Both have a distribution of unit size in coordinate space. The limit is well defined. If there are short-distance singularities, we cannot find them in O 1 (p)O 2 (q): we must look for them in the counterterm P 12 (p, q), which is required by ERG (or equivalently scaling). Even without the help of ERG, we expect the need for the counterterm in defining the Fourier transform; the integration over the case where the two operators are dangerously close together requires special attention, resulting in a local counterterm.
Let us further analyze the nature of P 12 (p, q). Regarding composite operators as functionals of Φ, we obtain where (20c) is equivalent to the scaling (16).
From (20), we obtain the following ERG equation for the counterterm: where we have used Since R is the Fourier transform of a function nonvanishing only over a region of unit size, (22) is local in space. That means that the inverse Fourier transform is nonvanishing only when the distance |x − x ′ | is of order 1 or less.
Therefore, we can expand the counterterm P 12 (p, q) using a basis of local composite operators: where O i is a composite operator of scale dimension −y i , satisfying The coefficient c 12,i depends only on p − q; we have absorbed all the dependence on p + q into O i . Similarly, we can expand the right-hand side of (22) as Substituting (24) and (26) into (22), we obtain or equivalently which determines c 12,i (p) in terms of d 12,i (p).
Before discussing the short-distance behavior of c 12,i (p − q) for large |p − q|, we would like to consider the solvability of (27) and uniqueness of its solution. We assume analyticity: both c 12,i (p − q) and d 12,i (p) are regular functions of p at p = 0. (27) can be solved uniquely If (28)  If (28) holds, we need to modify (20c) so that where the constant d is determined so that has no term proportional to p n . Then, we need to solve instead of (27). This can be solved, but the solution is not unique. To fix the coefficient of p n , we must introduce a convention such as the absence of the p n term in c 12,i (p): All this implies that the scale dimension y 1 + y 2 is extended to a matrix: the product at the Gaussian fixed point in D = 4, the product (y 1 = y 2 = 2) mixes with δ(p + q). See Example 1 of Sec. IV for more details. So much for the discussion of (28). Now, we consider the short-distance limit of the product. We consider taking |p − q| large while fixing p + q. As has been explained, a singular behavior is expected We can regard P 12 (p, q) as a functional of J with momentum p + q. Since we keep the momenta p 1 , · · · , p n finite in the above, we can assume δO 1 (p) δJ(r) and δO 2 (q) δJ(−r) of (22) to depend only on J's with finite momenta. Hence, r in (22) must be of order p by momentum conservation. Therefore, R(r) becomes extremely small. Hence, from (26), we expect Thus, we obtain This implies where C 12,i is a constant.
We thus obtain a short-distance expansion (a.k.a. operator product expansion) To be able to neglect the contribution of O 1 (p)O 2 (q), we must restrict the sum over i to corresponding to singularities in space. This condition can be rewritten as where D − y i is the scale dimension of the inverse Fourier transform of O i (operator in coordinate space). The operator O i with the lowest scale dimension provides the highest short-distance singularity.

IV. EXAMPLES
We would like to provide concrete applications of the general theory we have developed.
In the first subsection we consider a generic fixed point action, and in the second we take examples from the Gaussian fixed point in D dimensions (2 < D ≤ 4).
Let O(p) be an arbitrary composite operator of scale dimension −y. Its product with Φ(q) must satisfy This gives [3] [ This implies the counterterm For the simplest case of O = Φ, we use (6b) and (42) to obtain This generalizes to [25] [ which can be checked to satisfy (41): Using (9) we can rewrite (43) as Hence, for p + q fixed, we obtain From the scaling law we obtain the coefficient of the identity operator as Further coefficients can be computed by employing some approximation scheme. For instance, we have checked in the φ 4 theory in dimension D = 4 that the order λ correction is given by In coordinate space the order λ correction is proportional to the logarithm of the distance.

B. Examples from the Gaussian fixed point in D dimensions
We now consider the composite operators at the Gaussian fixed point: There is no anomalous dimension: γ = 0. The high-momentum propagator is given by For convenience we introduce which vanishes rapidly for p ≫ 1. Now, the "differential" operator D defined by (21) can be written as In the remaining part of this section we consider the products of the composite operators where the constant κ 2 is defined by The scale dimensions are −2 and D − 4, respectively. See Appendix B for the construction of composite operators in the free theory.
Example 1: 1 2 φ 2 (p) 1 2 φ 2 (q) The scale dimension of the product is −y = −4. Hence, in D = 4, we expect mixing with the unit operator δ(p + q). Let The counterterm must satisfy To solve this, let us expand Substituting this into (59), we obtain and The homogeneous solution of (61a) is excluded on account of analyticity at zero momentum.
Hence, we obtain Substituting this into (61b), we obtain For 2 < D < 4, this is uniquely solved by (See Fig. 1.) Now, for D = 4, (63) does not admit a solution analytic at p = 0. Since the left-hand side vanishes at p = 0, we must modify it to by subtracting a constant from the right-hand side. The constant can be evaluated as (65) determines u 0 (p) up to an additive constant. We define F (p) by As a convention we adopt the choice u 0 (p) = F (p). The subtraction in (67a) implies the mixing of the product with the identity operator, and the product satisfies the ERG equation Let us find the asymptotic behavior of the product as p → ∞ for a fixed p + q. We find From Appendix C, we obtain where c F is given by (C6). Hence, we obtain for 2 < D < 4, and for D = 4.
Example 2: The scale dimension of the product is −y = D − 6. Hence, in D = 4, the product mixes with 1 2 [φ 2 (p + q)]. Let we obtain, for 2 < D < 4, (See Fig. 2.) The above expression is valid also for D = 4 except that the ERG equation for the product is modified to Using (70), we obtain for 2 < D < 4, and for D = 4.
where the counterterm P 12 is the same counterterm that makes a composite operator. The extra counterterm P 123 (p, q, r) has to do with the three operators close to each other simultaneously. We obtain the ERG equation If there is a local composite operator O whose scale dimension −y satisfies the product [O 1 (p)O 2 (q)O 3 (r)] may mix with O(p + q + r), and we obtain where d(p − q, q − r) is a degree n polynomial of p − q, q − r.
In the absence of mixing the last counterterm satisfies This can be generalized to higher order products of composite operators.

VI. AWAY FROM A FIXED POINT
Let g be a parameter with scale dimension y E > 0. The Wilson action is parametrized by g. Assuming the anomalous dimension is independent of γ, we obtain the ERG equation of the action as The modified correlation functions defined by satisfy the scaling law: Rewriting the ERG equation for we obtain Comparing this with (87), we find that for the constant source the sum satisfies (97). Therefore, is the n-th order product of the zero momentum composite operator of scale dimension −y E , defined at the fixed point.
Considering operator products, it is even more convenient to introduce the effective action with an infrared cutoff: Then, a composite operator of scale dimension −y satisfies where D is given by (21) except that G is now defined with the g-dependent W as The discussion of Sec. III goes through as long as we use the g-dependent D. We introduce g-dependent coefficients c 12,i (g; p) and d 12,i (g; p) via (24) and (26), respectively. Eq. (27) is replaced by p · ∂ p + y E g ∂ ∂g + y 1 + y 2 − y i c 12,i (g; p) = d 12,i (g; p) .
Since the locality of the cutoff function R gives we obtain the asymptotic behavior assuming the analyticity of c 12,i (g; p) in g.
The simplest example is given by the massive Gaussian theory where the squared mass m 2 plays the role of g. (See Appendix B for the construction of composite operators.) We can show, for 3 < D < 4 (the term proportional to m 2 is not singular for 2 < D ≤ 3) and for D = 4

VII. CONCLUSIONS
In this work we have studied the OPE (operator product expansions) in the framework of ERG (the exact renormalization group). The key concepts underlying our analysis are the composite operators and their products, which we define respectively in Sec. II and Sec. III. We have argued that the ERG differential equation associated with the product of two operators can be expanded in a local basis of composite operators, leading to the ERG differential equations for the Wilson  To make the content of Sec. II easier to understand for the readers already familiar with [16] but not with [15] (and the subsequent extensions done more recently), we would like to summarize the basics of the ERG formalism adopted in this paper. We rely on perturbation theory for intuition.
In [16] the Wilson action is given in the form where the first term gives the propagator K(p/Λ) p 2 that damps rapidly for p > Λ, and the second term gives interactions. To preserve physics below the momentum scale Λ, the interaction part must obey the following differential equation [16]: We can rewrite this equation for the whole action as .
The correlation functions calculated with S Λ are not entirely independent of Λ. Take the sum of diagrams contributing to the connected part of the n-point function for n > 2. (Fig. 4) Polchinski's equation (A2) guarantees that the shaded blob (denoted G n ) of Fig. 4 is independent of Λ. But the external propagators are multiplied by K, and we obtain φ(p 1 ) · · · φ(p n ) connected Λ = G n (p 1 , · · · , p n ) For small momenta p i , the cutoff function K(p i /Λ) is almost 1. We only need to divide the correlation function by a product of K's to make this strictly Λ-independent: We are left with the two-point function which is given by where G(p) is independent of Λ. (Fig. 5) Again, this is almost independent of Λ for p < Λ.
To make the two-point function strictly Λ-independent, we first modify it by subtracting a high momentum propagator K(p/Λ) (1 − K(p/Λ)) p 2 (A8) This is independent of Λ.
We have thus explained that Λ-independent connected correlation functions are given by where n > 2. Including the disconnected parts, the Λ-independent four-point correlation function is given by φ(p 1 ) · · · φ(p 4 ) ≡ φ(p 1 ) · · · φ(p 4 ) connected + φ(p 1 )φ(p 2 ) φ(p 3 )φ(p 4 ) + (t, u-channels) This structure generalizes to higher-point functions. Using a formal but more convenient notation, we can express Λ-independent correlation functions by These Λ-independent correlation functions were first introduced in [21] and termed modified correlation functions. Two Wilson actions are called there equivalent if their modified correlation functions are the same. We find the word "modified" somewhat misleading since these are the proper correlation functions given by the Wilson action.
The ERG differential equation (A3) still differs from the ERG equations of the main text. In order to discuss a fixed point of the renormalization group, we need to introduce an anomalous dimension and adopt the dimensionless convention to fix the momentum cutoff.
Let us explain this one by one.
1. Anomalous dimension -To keep the kinetic term of S Λ independent of Λ, we must introduce an appropriate Λ-dependence to the normalization of φ. This introduces an anomalous dimension γ Λ so that To obtain this we must change (A3) to (There are other ways of introducing γ Λ such as the one given in [28]. Here we have followed [20,21].) Rewriting the second integral of the right-hand side, we obtain 2. Dimensionless convention -Finally, to obtain a fixed point we must adopt the dimensionless convention by measuring physical quantities in powers of appropriate The high-momentum propagator is given by where h(m 2 , p) ≡ 1 p 2 + m 2 + R(p) .
We define A composite operator of scale dimension (−y) satisfies where D is given by .
Appendix C: Asymptotic behavior of F (p) For 2 < D < 4, F (p) ≡ 1 2 q h(q)h(q + p) (C1) satisfies the differential equation we obtain where c F is a constant. From we obtain For D = 4, F (p) is determined by and F (0) = 0 .
For large p, the differential equation becomes which gives The constant is determined by the initial condition F (0) = 0, but it depends on the choice of a cutoff function R.
[1] C. Becchi, "On the construction of renormalized gauge theories using renormalization group