Conformal invariance for Wilson actions

We discuss the realization of conformal invariance for Wilson actions using the formalism of the exact renormalization group. This subject has been studied extensively in the recent works of O. J. Rosten. The main purpose of this paper is to reformulate Rosten's formulas for conformal transformations using a method developed earlier for the realization of any continuous symmetry in the exact renormalization group formalism. The merit of the reformulation is simplicity and transparency via the consistent use of equation-of-motion operators. We derive equations that imply the invariance of the Wilson action under infinitesimal conformal transformations which are non-linearly realized but form a closed conformal algebra. The best effort has been made to make the paper self-contained; ample background on the formalism is provided.


I. INTRODUCTION
The study of conformally invariant field theories (in dimensions D > 2) was initiated long ago by J. Wess [1] with a hope that conformal invariance constrains a theory more than scale invariance, since the latter is implied by the former. Requirement of conformal invariance seemed much stronger than that of scale invariance at first sight, but the difference turned out to be subtle. In the seminal work [2], J. Polchinski showed the equivalence of conformal invariance to the vanishing of the trace of the energy-momentum tensor; scale invariance requires the vanishing of only its integral. The question of whether scale invariance implies conformal invariance has attracted much attention lately, and we would like to refer the reader to a recent review by Y. Nakayama [3] and references therein.
The subject of this paper is realization of conformal symmetry using Wilson actions. [4] This was recently taken up by O. J. Rosten [5] and also by Delamotte, Tissier, and Wschebor [6]. Rosten has extended his work further in [7,8]. It is the recent works of Rosten (especially [5] and [8]) that we wish to improve upon by using the method of symmetry realization developed and reviewed in [9]. We aim to add simplicity and transparency to the structure of conformal transformations in the exact renormalization group formalism.
Wilson actions come with a finite momentum cutoff, and it is generally accepted that only the physics at scale below the cutoff is effectively described by Wilson actions. This is indeed the case with a generic Wilson action, but there are exceptions. Those Wilson actions flowing out of a fixed point under the renormalization group transformations correspond to a continuum limit, and the physics at all momentum scales are described by the Wilson actions. (In [4] these Wilson actions form a finite dimensional space S(∞).) Hence, if the continuum limit of a theory has symmetry, we can realize the symmetry using its Wilson action. Now, a fixed point of the renormalization group transformation is a continuum limit.
If the limit possesses conformal symmetry, its Wilson action must realize the symmetry, too.
The method of [9] has recently been applied to the construction of the energy-momentum tensor in [10]. Our expression of special conformal transformation (15d) was in fact first derived there from the assumption of the vanishing trace. We summarize this derivation in Appendix C.
We organize the paper as follows. In Sect. II we introduce infinitesimal conformal transformations of the elementary scalar field in D-dimensional Euclidean space. In Sect. III, we review quickly how to express continuous symmetry of a Wilson action in terms of equationof-motion composite operators. Then, in Sect. IV, we construct equation-of-motion composite operators for the conformal symmetry, and subsequently in Sect. V we construct the products of the infinitesimal transformations to show the closure of the algebra. Sects. IV and V constitute the main part of this paper. In Sect. VI we rewrite the invariance of the Wilson action as that of the associated generating functional and 1PI action. In Sect. VII we construct the 1PI action of a Wilson-Fisher fixed point in D = 4 − ǫ dimensions to first order in ǫ. We extend the conformal transformation to the scalar composite operators in Sect. VIII before we conclude the paper in Sect. IX.
We have kept the main text reasonably short by relegating the technicalities to five appendices. The effort has been made to make this technical paper an easy read; the first reading of the main text had better be done without referring to the appendices. We have adopted the following notation to simplify the formulas.

II. CONFORMAL ALGEBRA
We consider a real scalar field theory in D dimensional Euclidean space. We first consider the field in coordinate space. Infinitesimal conformal transformations act on the field as follows [1]: where D−2 2 + γ is the full scale dimension of the scalar field including the anomalous dimension γ. We have chosen the superscript T for translation, R for rotation, S for scale transformation, and K for the special conformal transformation that results from the succession of inversion, translation, and inversion. The algebra of the differential operators is closed, and is called the conformal algebra [1]: We formulate the Wilson action in momentum space; it is more convenient to rewrite the above transformations in momentum space. Denoting the Fourier transform of the scalar field by we obtain The above D(p)'s obey the same conformal algebra as (3): for example, we obtain

III. INVARIANCE OF A WILSON ACTION
The infinitesimal conformal transformations are linear transformations of the scalar field.
There is no guarantee, however, that they are realized as linear transformations for the Wilson where the second term comes from the Jacobian. This can be written as In the ERG formalism we choose K(p) is a positive momentum cutoff function: it depends only on p 2 , is nearly 1 for momenta low compared with the cutoff p = 1, and decreases rapidly for p ≫ 1. O(p) is a composite operator (i.e., a functional of φ) with momentum p. Using the above ∆φ, we obtain the invariance as This is the general form of the equation of motion in the ERG formalism. The equation of motion implies the Ward-Takahashi identity for the correlation functions: where the i-th φ is replaced by O. Note that we use · · · for the continuum limit of correlation functions. We refer the reader to Appendices A & B, where we give technical details on the ERG formalism such as modified correlation functions and equations-of-motion composite operators.

IV. CONFORMAL INVARIANCE
For infinitesimal conformal transformations, we choose O(p) of the previous section as where S is the Wilson action, and K, k are cutoff functions. Φ(p) has the same correlation functions as the elementary field φ(p): See Appendix A for the precise definition of both sides. Hence, We now introduce the following the equation-of-motion composite operators: These carry no momentum. The conformal invariance amounts to the vanishing of the above operators: Substituting these into the correlation functions, we obtain the following Ward-Takahashi identities: Note that the scale invariance, given by Σ S = 0, is nothing but the ERG differential equation for a fixed point Wilson action, which is usually given in the form [4] This rewriting has been explained in Appendix B of [10].
As for the special conformal invariance Σ K µ = 0, an equivalent formula was first derived by Rosten as (3.25) in [5]. The particular form Σ K µ given by (15d) was first obtained in [10]. (This is briefly explained in Appendix C.) Rosten has rewritten his result as (2.79b) in [7].
We will explain how to derive (a formula similar to) his (2.79b) by rewriting our Σ K µ = 0 in Appendix E. For realization of the algebra, we need the product of two infinitesimal transformations.
We construct the product as where is a composite operator corresponding to the product of Φ(p) and Σ 2 . (See Appendix A.) (where the hat above φ(p i ) implies omission), we obtain Therefore, we obtain This implies Hence, the algebra of D's translates into the algebra of Σ's.
The higher products of Σ's can be defined recursively as so that Similarly, the formulas we obtain the following results: where the integrals with R have been simplified by partial integration.
where q is set equal to p only after the derivative is taken. The integrals with R have been simplified by partial integration. This step is explained in Appendix D.
The first two types of invariance are free of the cutoff function R. In fact, the invariance of the Wilson action under translation and rotation can also be written without R [10]: On the other hand, the invariance under the scale and special conformal transformations depends non-trivially on the cutoff function R.
As for the special conformal invariance, Eq. (34b) for Γ has been obtained by Rosten as (4.16) in [8]. A similar expression has also been derived as (10) in [6].

VII. WILSON-FISHER FIXED POINT TO ORDER ǫ
As a concrete example, we consider the Wilson-Fisher fixed point in D = 4−ǫ dimensions, and construct a conformally invariant 1PI action Γ to first order in ǫ. Assuming γ = 0 at this order, we obtain the following equations from (33) and (34):

Special conformal invariance
We will solve these equations with the ansatz where m 2 , λ are both of order ǫ. Note this is automatically invariant under translation and rotation.
The high momentum propagator G −p,q [Φ] is now defined by and it is obtained as up to first order in ǫ.

A. Scale invariance
Substituting (38) into (36), we obtain two equations, one quadratic in Φ, and the other quartic in Φ. The latter is given by This gives which is trivially satisfied to order ǫ. We are now left with This is solved by

B. Special conformal invariance
Substituting (38) into (37), we obtain two equations, one quadratic in Φ, and the other quartic in Φ. The latter is given by Using (independent of i), we obtain which gives (42) again. The equation quadratic in Φ is given by Integration by parts reduces this to Hence, we obtain (44) again. We have thus seen that scale invariance automatically leads to conformal invariance.
We need a second order calculation to fix λ to order ǫ. the same way as on φ(p); we only need to generalize D S (p) and D K µ (p) as where the product of composite operators is defined by Eqs. (51) imply Alternatively, regarding O(p) as a functional of Φ, we obtain It is the easiest to obtain the above results by varying either W or Γ infinitesimally by O(p) in (33) and (34).
at the Gaussian fixed point in D > 2. With y = 2, both of (51) are satisfied if the constant κ 2 is chosen as

IX. CONCLUSION
The main purpose of this paper is to reformulate the recent results of Rosten [5,7,8] using the method of equation-of-motion composite operators advocated in [9]. The Wilson action of the continuum limit of a theory has all the symmetry intact despite the presence of a finite momentum cutoff. We hope that we have convinced the reader that a finite UV cutoff does not stand in the way of making a Wilson action invariant under conformal transformations.
Note added: Rosten extends his work further in a recent article [11].
Appendix A: Quick summary of the ERG formalism The purpose of this and next appendices is to give the reader (without the working knowledge of ERG) just enough to follow the flow of the present paper. For further details we recommend [12] and references cited therein.
As in the main text, we use the dimensionless notation in which dimensionful quantities are measured in units of an appropriate power of the momentum cutoff. Hence, the momentum cutoff becomes 1 in this convention.
The renormalization group flow of the Wilson action S t [φ] is given by the exact renormalization group equation [4] ∂ where t is the logarithmic scale factor. This is a generalized version with two cutoff functions K(p), k(p) [13]: K(p) approaches 1 as p → 0, and decreases rapidly for p ≫ 1, and k(p) vanishes at p = 0. In the popular adaptation by Polchinski [14], k(p) is taken as To obtain S t+∆t from S t , we first integrate over the field with momenta between 1 and e −∆t .
We then rescale the momentum by the factor e ∆t to restore the cutoff at 1, and renormalize the field so that, for example, the kinetic term is canonically normalized. It is remarkable that this whole procedure can be expressed as a functional differential equation.
In this paper we are not interested in t-dependent actions, but only interested in a fixed where γ is a constant anomalous dimension. This S has a UV cutoff p = 1, just like a generic bare action with the same cutoff p = 1, but it corresponds to a massless continuum theory. The field with momenta p > 1 have already been integrated, and the Wilson action can provide the continuum limit of correlation functions only with a little modification [13]: k(p) modifies the two-point functions trivially at high momenta, and K(p) corrects the normalization of the field. As befits the continuum limit, the modified correlation functions are defined for arbitrary momenta, and satisfy the scaling law φ(p 1 e t ) · · · φ(p n e t ) = exp n − D + 2 2 Hence, the two-point function is given by We next introduce the concept of composite operators. (For more details than given here, see Sect. 4 of [9].) A composite operator O(p) is a functional of φ, and it can be regarded as an infinitesimal variation of the action. We define its modified correlation functions by operators playing important roles in this paper. One is which has the correlation functions where O(p) is a composite operator. E O has the correlation functions (Derivation) Using (A7), we obtain Functionally integrating this by part, we obtain where the hat above φ implies the omission. (End of derivation) Given two composite operators O 1 (p), O 2 (q), their product O 1 (p)O 2 (q) is not necessarily a composite operator. When one of them is Φ(p), however, its product with an arbitrary O(q) is easy to construct: The product has the correlation functions Now, in [10] we have assumed the invariance of the Wilson action under translations and rotations Σ T µ ≡ −e −S p K(p) δ δφ(p) D T µ (p)φ(p) e S = 0 , where D T µ , D R µν are defined in (5). We have then shown the existence of the energymomentum tensor Θ µν (p) satisfying Θ µν (p) = Θ νµ (p) .
where D S , D K µ are defined in (5). We wish to show how to obtain these from the trace condition: In [10] it is shown that a fixed point Wilson action, satisfying (C3a), also satisfies (C4) at p = 0. Conversely, to obtain (C3a) from (C2) and (C4), we differentiate (C2a) with respect to p ν , sum over ν, and then set p = 0 to obtain Θ(0) = q K(q)e −S δ δφ(q) (D + q · ∂ q )Φ(q) e S .
Getting (C3b) from (C2) & (C4) is a little more involved. (This has been done in Sect. VI of [10], where (C4) is assumed up to a two-derivative term p µ p ν L µν (p). For simplicity, we have removed the two-derivative term by redefining Θ µν (p).) We apply ∂ 2 ∂p α ∂p ν − 1 2 δ αν ∂ 2 ∂p β ∂p β on (C2a) and set p = 0. Using (C4), we can write the left side as The right side gives Equating the two sides, we obtain Σ K α = 0. In a recent work [7] Rosten regards (C2) and (C4) as fundamental equations from which he attempts to construct a conformally invariant Wilson action.
which is valid for any symmetric F (−p, q) satisfying