A Wilsonian Energy-Momentum Tensor

For local conformal field theories, it is shown how to construct an expression for the energy-momentum tensor in terms of a Wilsonian effective Lagrangian. Tracelessness implies a single, unintegrated equation which enforces both the Exact Renormalization Group equation and its partner encoding invariance under special conformal transformations.


I. INTRODUCTION
In the context of quantum field theory, consider the action for a free, scalar field in d-dimensional Euclidean space: What is the scaling dimension of the field? This is, of course, a trivial question to answer: since the theory is non-interacting, we can just use naïve power counting to obtain (d − 2)/2.
However, this method does not carry over to the interacting case; consequently, we seek other, more elaborate ways of answering this question that are more generalizable.
Our next approach-which, at least in its preliminary incarnation, is in a sense just a somewhat more formal means of power-counting-involves constructing a functional repre-sentation of the dilatation generator. First define D (δ) = x · ∂ + δ, (1.2) where δ is an a priori unknown scaling dimension and, in this context, the dot denotes a contraction of indices. Next introduce where here the dot denotes an integral over the position of the field. Demanding dilatation invariance of the action recovers the previous result: As it stands, this method is also of no use in the interacting case, but the approach has an appropriate generalization.
If we wish to retain a representation of the dilatation generator for interacting quantum field theories which is linear in the functional derivative, then we must take it to act on the correlation functions, which are non-local. To preserve a local description, we are forced instead consider representations of the dilatation generator which are (at least) quadratic in derivatives. The Exact Renormalization Group (ERG) provides a particular realization of this: where G and G incorporate an ultraviolet cutoff function, K, in a manner to be specified later. Scale invariance is enforced by the fixed-point condition: A particularly interesting feature of this equation is that, in general, not only is the scaling dimension a priori unknown but, so too, is the action. The correct understanding is to recognize the ERG equation as a non-linear eigenvalue equation [1]. In principle, then, the ERG equation allows the self-consistent determination of the spectrum of local fixed-points.
However, the equation itself is fiendishly difficult to attack and, in general, various rather brutal approximation schemes must be employed.
Within the context of approaches that are intrinsically local, are there any options to improve upon the ERG? A clue comes from the fact that (1.6) is only a statement of scale invariance; it does not automatically incorporate full conformal invariance (for a detailed discussion of the relationship between scale and conformal invariance, see [2]). Whilst it is true that for many theories of interest scale invariance in fact enhances to conformal invariance, this is not a general property of all solutions of (1.6). Indeed, just as (1.5) provides a representation of the dilatation generator, so too is there an associated representation of the special conformal generator [3][4][5]. The form we use is equivalent to that derived in [4]: where G µ is related to G (again, with the details deferred until later), and η is defined via (1.9) A conformally invariant theory thus satisfies not just (1.6) but also K µ e −S = 0. (1.10) However rather than attempting to solve the pair of equations (1.6) and (1.10) as they stand, in this paper we shall instead seek a single equation which incorporates full conformal invariance. To understand how such an equation might arise, let us consider a third answer to the question posed at the start of this paper: we will obtain the scaling dimension of our Gaussian theory via consideration of the energy-momentum tensor. This is significantly more involved than either of the above approaches, but it has the merit of providing an interesting and potentially powerful generalization.
For the Gaussian theory, we can take the energy-momentum tensor to be defined by the following three equations: In position space, the × just represents the product of two quantities at the same location; it can be omitted and is generally used to emphasise the lack of an integral (which would be denoted by a dot). These three equations encode, respectively, translational, rotational and scale-invariance of the action; however, these three invariances can only be expressed in terms of a single object-the energy momentum tensor-if the theory is in fact conformally invariant.
It is instructive to see how these equations can be used to determine δ. Substituting (1.1) into (1.11a) and rearranging gives: Therefore, where w is determined by the condition for scale invariance (1.11c): A bit of simple algebra reveals that: as before.
The generalizations of (1.11a), (1.11b) and (1.11c) appropriate to interacting theories within the framework of the ERG were derived in [4] and subsequently explored in [6]. As emphasised in [4], if one is to consider various representations of the conformal algebra, then one must be prepared to consider associated representations of the energy-momentum tensor. In this section we work in a local but otherwise arbitrary representation and, to indicate this, use the symbol T αβ . As in the introduction, throughout the rest of this paper we work in d-dimensional Euclidean space.
To motivate an appropriate generalization of (1.11a), (1.11b) and (1.11c), note that the first and last equations both involve the two separate objects, ϕ and δS/δϕ. The restriction to the Gaussian fixed-point has been lifted since henceforth our interest is in general conformal field theories. Intuitively, the first object, ϕ, should have a scaling dimension of δ; from this and a straightforward dimensional analysis, it follows that the second object has Now, in an arbitrary local representation of the conformal algebra, while we can reasonably expect the quasi-primary field of scaling dimension δ to be intimately related to ϕ, it may have a more complicated form. For example, in an ERG representation, it contains additional terms which depend on the cutoff, but which vanish when the cutoff is removed.
Thus, instead of explicitly working with ϕ, we prefer the more general O (δ) , which is defined to be a quasi-primary field of scaling dimension δ viz.
Similarly, we trade δS/δϕ for O (d−δ) , where the latter satsfies With this in mind, the generalization of (1.11a), (1.11b) and (1.11c) which we seek-and which is fully justified in [4]-reads: The origin of the commutator may be seen by acting on (2.3c) with D, bearing in mind that T αα is expected to be of scaling dimension d.
Integrating (2.5) and using (2.3b) it is apparent that, for local solutions, T αβ is symmetric up to total derivatives. Therefore, for some F λαβ and some symmetric T sym αβ we may write whereupon it follows that Combining with (2.3b) and (2.5) yields The goal now is to solve the conditions (2.6) and (2.9) for B λαβ in terms of F λαβ . To this end, it is useful to follow Belinfante and Rosenfeld [7,8]. Defining the solution is given by where Y αλβµ has the following symmetries: Substituting (2.10) into (2.11), it follows immediately that the desired constraints (2.6) and (2.9) are satisfied. Therefore, substituting (2.11) into (2.5) yields a tensor which is both conserved and symmetric: where, for convenience, we have defined which inherits from (2.12) the following: The first term on the right-hand side is just an expression of dilatation invariance. This is readily seen for a classical representation, where it reduces to D (δ) ϕ · δS/δϕ. That it holds more generally is discussed fully in [4]. The point is that this term vanishes, revealing an integrated form of (2.3c).
Therefore, given an assumption of dilatation invariance, a solution to (2.3a) and (2.3b) solves (2.3c) up to a total derivative. Temporarily supposing that W λαβ = 0, this implies that, for some H λ , Inspecting our explicit solution (2.13), the question as to whether the energy-momentum tensor can be improved amounts to asking whether it is possible to find a suitable W λαβ to remove the unwanted total derivative term [4]. Therefore, we seek a solution to (2.15) and (2.17) together with This may be accomplished by introducing H τ λ and taking in terms of which we recover Polchinski's solution [9] 1 : However, the right-hand side of (2.19a) is readily seen to be independent of a and so this ambiguity in H τ λ has no effect on the energy-momentum tensor.
Having discussed these generalities, let us now return to our explicit solution (2.13) and take the trace: Comparing this with (2.3c) gives a consistency condition: which, if satisfied, amounts to the energy-momentum tensor being improvable, so that it is not only conserved and symmetric, but also traceless. Substituting (2.17) into (2.21) yields which will be central to our analysis below and is the basis for the conformal fixed-point equation. Note that any terms containing two or more total derivatives on the right-hand side can be absorbed by redefining H τ λ ; this will be exploited in subsequent sections.
An important subtlety is that, for (2.22) to be soluble, it seems there must be an interesting conspiracy, since the right-hand side is not manifestly O ∂ 2 . The resolution is as follows: integrating this equation yields whereas first multiplying by x µ gives Therefore, existence of a solution to (2.22) implies one scalar and one vector condition on the action, both independent of H τ λ ; respectively, these are naturally associated with dilatation and special conformal invariance.
There is an additional, interesting subtlety in d = 2: it is possible to have a theory for which the action satisfies both dilatation and special conformal invariance but, nevertheless, the quantum theory is not conformal! This can arise if it is not possible to express H τ λ = δ τ λ H; we shall encounter this later when examining the higher derivative ϕ ∂ 4 ϕ theory, which suffers from particularly bad infrared behaviour in d = 2.
B. Classical Theories

Analysis
In this section we apply the general methodology of section II A in a classical context.
On the one hand, this will provide some experience with the advocated approach; on the other the results of this section will form part of the full quantum field theoretic result.
Classically, the appropriate form for the defining equations of the energy-momentum tensor, (2.3a), (2.3b) and (2.3c) are: where t αβ denotes the classical energy-momentum tensor. Note that these equations are exactly of the form as in the introduction, (1.11a), (1.11b) and (1.11c) and, as with the earlier equations, respectively encode translational, rotational and scale-invariance. Throughout this section, when we talk of an energy-momentum tensor which is conserved/traceless, we mean that it is conserved/traceless up to terms which vanish on the equations of motion.
To proceed, let us takeL to be an arbitrary element of the equivalence class of objects that integrate to the Wilsonian effective action, viz.
but for which any total-derivative contributions have been discarded. For example, the with a = 0 singled out as the Lagrangian. We henceforth demand quasi-locality, meaning that we restrict our attention to Lagrangians (and hence actions) which exhibit a derivative expansion, viz.
where V and Z do not contain any derivatives, and the ellipsis denotes terms higher order in derivatives.
To aid the analysis, define: (2.28) Using this notation, we have where it is understood that after expanding out using (2.28), the repeated indices σ 1 , . . . σ n are summed over. Defining it is apparent that Courtesy of the chain rule, . (2.32) In order to generate a contribution to the energy-momentum tensor, we must convert the whole of the right-hand side into a total derivative. As preparation for manipulating the middle term, observe that for some A, B and j < i Feeding this result back into the final term and iterating yields, for j < i: We now wish to apply this result, with j = 0, to the middle term of (2.33). Since this may only be done for i ≥ 1, we first separate off the i = 0 term before proceeding. This yields: The first term cancels the final one of (2.33) leaving, after relabelling the dummy indices This equation is s a consequence of translation invariance. Using (2.30) to substitute for S α , the resulting term can be absorbed into the final one above by replacing both lower limits of the sums over j and i with zero. Recalling (2.5), we deduce the following contribution to the energy-momentum tensor: .
(2.38) plus a total derivative, as in (2.7). Utilizing (2.35) gives: The key point is that the first two terms are both symmetric under α ↔ β. This is manifest for δ αβL . For the second term, we argue as follows. First, observe that . (2.40) With this in mind, we utilize rotational invariance of the action, together with (2.29) and integration by parts: (2.41) The first term vanishes after using the chain rule and integrating by parts and so we conclude that rotational invariance alone is sufficient to ensure that the integrand of the final piece vanishes, at least up to total derivative terms. However, we can go further by exploiting quasi-locality ofL. Recall that this is a statement thatL has a derivative expansion; of course, sinceL is a scalar, all partial derivatives must be paired up. In the integrand under analysis, the effect onL of the [D σ i 1 , x α ∂ β ] term is to split all such pairs up into ∂ α and ∂ β . Since all possible splittings are summed over, the result is symmetric under interchange of indices and so we conclude that Returning to (2.39), we use the recipe (2.7) to extract where, in the final expression, the multiple summations have been simplified. From (2.43a) and (2.43b) we can construct the full energy-momentum tensor according to (2.13), at least up to the conserved, symmetric ∂ λ w λαβ . To determine the latter, we take the tracefor which we can use (2.20)-and compare with (2.25c). Mimicking section II A, we take . the game now is to simplify this expression by absorbing various O ∂ 2 terms into the first term on the right-hand side. We begin by noting that: .

(2.45)
To see this, start by shifting i → i − 1 on the left-hand side, so that the sum over i now starts from 2. Permutation of the dummy indices on the left-hand side means that each term arising from the sum over k (which now runs to i − 1) is identical and therefore there are i(i − 1)/2 such terms. It is easy to see that this matches the right-hand side. To exploit this result in (2.44) we must first re-express the final term using a variant of (2.35): whereupon we obtain . (2.48) In accord with the discussion under (2.22) we have ignored any additional transverse contributions. Comparing (2.47) with (2.25c) we arrive at the following constraint for CFTs: . (2.49) As mentioned at the end of section II A, the existence of solutions to this equation implies two conditions that must be satisfied by the action. This will allow us to check the consistency of (2.49). First of all, we integrate it directly. The total derivative terms vanish. To process the surviving terms, observe that . (2.50) The first term can be processed by the chain rule to give −dS. It is thus apparent that, upon integration, (2.49) reduces to which is of course nothing but the statement of dilatation invariance.
Returning to (2.49), we now multiply by 2x µ and then integrate. To see what this gives, observe that . (2.52) The final term can be processed by manipulating the commutator: From this it follows that When substituted into (2.52), the first term on the right-hand side vanishes as a consequence of (2.42); returning to (2.49), we thus conclude that multiplying by x µ and integrating implies: From (2.48) we see that h τ λ =h τ λ , with the latter determined by (2.49) therefore implying that δ = δ 0 , as expected, with To construct the full energy-momentum tensor we employ (2.19a) and (2.19b); conveniently, for the present case where h τ λ ∼ δ τ λ , the former reduces to the latter and so we find, using (2.13) as expected. It is easy to check that takingL = − 1 2 ϕ ∂ 2 ϕ gives the same result and so, reassuringly, t αβ is independent of which of the representatives ofL we use. b. Higher Derivative Theory As a slightly more involved example, we will explore the free theory with a kinetic term quartic in derivatives. To begin with, we shall considerL = 1 2 ∂ µ ∂ ν ϕ ∂ µ ∂ ν ϕ. Power counting informs us that the scaling dimension of ϕ is (d − 4)/2 < δ 0 and so, in Minkowski space, the theory is non-unitary. Nevertheless, at least for d > 2, we can construct the energy-momentum tensor. Referring back to (2.43a) and (2.43b), we have: corresponding, according to (2.7), to the conserved but not symmetric tensor Following the recipe (2.13) yields the conserved, symmetric tensor where ∂ λ w λαβ is determined by (2.19a) in terms of h τ λ . From (2.48), in this particular case h τ λ =h τ λ and from (2.49) Components of the first two terms can be transferred to the final term by writing The final term can be massaged to give and so we conclude that η = −2, as expected, and, for d > 2, simplifying yields the full (if rather unwieldy) energy-momentum tensor: (2.68) The expression is manifestly symmetric and it is easy to check that it is conserved and traceless (as usual up to terms which vanish on the equations of motion).
A variant of the above analysis which exercises all terms involved in the construction of the energy-momentum tensor is achieved by taking insteadL = − 1 2 ∂ µ ϕ ∂ µ ∂ 2 ϕ. In this case we find: it can be checked that the sum of these terms, t αβ is conserved. According to the recipe (2.13), we construct the conserved, symmetric tensor Contrary to the previous analysis, h τ λ is non-trivially related toh τ λ . From (2.48), and from (2.49) and so, as before, η = −2. After a bit of rearrangement (and utilizing the result for η) we find, for d > 2:h As before, it is relatively easy to confirm (2.67) but somewhat involved to reconstruct the full energy-momentum tensor (2.68).
c. Interacting Theories A crucial feature of the equations which define the classical energy momentum tensor, (2.25a), (2.25b) and (2.25c), is that contributions to the action depending on powers of the field decouple from one another. However, consistency between these various terms is enforced by the final condition encoding dilatation invariance. For example, consider adding a potential term to the gaussian theory: The energy-momentum tensor for this theory is given by: with dilatation invariance requiring that, for d > 2, This is very different from the quantum case, where the right-hand side of the analogues of (2.25a) and (2.25c) are quadratic both in the action and also functional derivatives, making the problem of finding explicit solutions very much more difficult.

Notation and Conventions
To formulate the ERG equation, we introduce an ultraviolet cutoff function which, as in the introduction, we denote by K(x, y). As with all ingredients of a good ERG equation this function must be quasi-local (cf. the discussion below (2.26)). Concretely, for coefficients k i , we may write where G 0 is Green's function, so that −∂ 2 G 0 = 1l.
From the cutoff function we construct an object G satisfying This is perhaps more intuitive in momentum space 2 where it translates to p · ∂ p K = p 2 G or just G(p 2 ) = 2 dK(p 2 )/dp 2 . From G it is helpful to construct G µ (x, y) ≡ (x + y) µ G(x, y).

(2.81)
It is useful to define an ultraviolet regulated version of Green's function: where, as alluded to in the introduction, we use the following shorthand for integrals: Now we have all the ingredients we need; the ERG equation and its partner encoding special conformal invariance read, up to vacuum terms (which are neglected throughout this paper): The ERG equation is slightly different from that usually appearing in the literature-which is the variant of the Wilson/Polchinski equations [11,12] proposed in [13]-on account of it using the full Wilsonian effective action; the relationship to the more common form is given in appendix A. The special conformal equation (2.84b) is a similar re-expression of the equation written down in [4].
The analysis of the following section will utilize some new notation. To motivate this, let us anticipate that in the ERG treatment of the energy-momentum tensor we will encounter a term like where given A(x, y) and B(x, y) we understand The first term on the right-hand side of (2.85) we recognize from the classical analysis and so our task will be to process the second term. In particular, we would like to re-write it as a total derivative. To this end observe that, for some quasi-local F (x − y) 2 , where F α (y, z; x) is also quasi-local and the partial derivative on the right-hand side is understood to be with respect to x. This is easy to see by making the coordinate dependence explicit and integrating: and, since F is quasi-local, (2.87) follows. Note that, courtesy of translation invariance, F can be rewritten as a function of coordinate differences: F α y − x, z − x . Here we are overloading notation so that the two argument form of F is considered separate from the three argument form.
Were we to directly utilize (2.87) in (2.85), our notation would be potentially confusing since up until now a single object sandwiched between dots-such as ·G· in the ERG equation-is such that all coordinates are integrated over. To avoid possible confusion as to whether or not expressions are fully integrated, without having to pollute all of our equations with explicit coordinate dependence, we develop some new notation which retains the compactness of the dot notation. Given test functions Ψ and Φ, we hijack the symbols ⋊ and ⋉ as follows: The left-hand side could be written as Ψ · F (x) · Φ, where the coordinates of F which are integrated over have been suppressed. However, whereas we should retain the x in this case to avoid ambiguity, we will be able to use the new notation with the explicit coordinate dependence dropped. Amongst other things, this allows such terms such as Ψ ⋊ F ⋉ Φ to cleanly appear in the same expression as e.g. Ψ · K × Φ.
Overloading notation so that the same symbol is used for both a function of position and its Fourier transform, and using k, p to denote momenta, we have: (2.89) The notation of (2.88) naturally extends to the case where Φ and Ψ depend on two arguments, viz.
clearly it can also be applied to the mixed case where Φ has one argument but Ψ has two (or vice-versa).
To gain some experience with the new notation, let us record several useful properties: The first equation is a trivial consequence of (2.88), given the convention stated under (2.87) that (∂ α F )(y, z; x) = ∂F (y, z; x)/∂x α . Equation (2.91b) exploits translation invariance of F : Integrating by parts, the desired result follows. There are several ways to see (2.91c). Most directly, one could simply substitute δ-functions for the 1ls and apply (2.90): after applying (2.88) again, (2.91c) follows. At a more heuristic level, one could swap the ⋉ and ⋊ for dots-mindful of the ambiguity above-and then convert them back again after eliding the 1ls.
The notation of (2.88) allows us to neatly express F α in terms of derivatives of a scalar: The backward-pointing arrow indicates that the associated operator-in this case a partial derivative-acts on the last argument of the object to its left, which here would correspond to ∂δ (d) (y, u)/∂u α , with δ (d) (y, u) = δ (d) (y − u). To justify (2.93), we establish a relationship between F and F . Employing (2.91b), observe that: This can be compared with (2.87); it is particularly transparent to do so in momentum space, which yields: The expansion in k will play a key role later on; in its most useful form it is: where the prime denotes a derivative with respect to the argument. Equivalently, where the second line defines the notation {·, ·} and, in position space, we understand F ′ to be the Fourier transform of dF (p 2 )/dp 2 , and so forth. Equivalently, recalling (1.2), we can define F ′ via: where

Analysis
To specialize the general analysis of the energy-momentum tensor of section II A to the ERG requires expressions for the O (δ) andÔ (d−δ) appearing in (2.3a) and (2.3c). First of all, we note the existence of a pair of primary fields [4,[14][15][16] O (δ) where the subscript ERG is a reminder that we are in the (quasi-local) ERG representation and, in momentum space, for η < 2 R(p 2 ) = p 2(η/2−1) K(p 2 ) . (2.103) Given that the cutoff function is normalized such that K(p 2 ) = 1 + O p 2 it follows that, as anticipated earlier, O ERG is, up to cutoff-dependent terms, just ϕ, whereas O ERG is similarly related to δS/δϕ. These equations may be verified by checking that (2.1) and (2.2) hold with D and K µ given, respectively, by (1.5) and (1.7), so long as we multiply O as may be checked using (2.4), with the same qualifications as above.
Recall that, in the classical case, by a conserved and traceless energy-momentum tensor we mean that the right-hand sides of (2.3a) and (2.3c) vanish on the equations of motion.
The equivalent statement in the ERG treatment is that the right-hand sides of the equations are 'redundant'-with a redundant field defined such that it is generated by quasi-local field redefinition [17,18]. (The term 'inessential' is also used in the literature.) We are now ready to attempt to solve (2.3a), (2.3b) and (2.3c) in the ERG representation.
The first step is to split the energy-momentum tensor up into a 'classical' piece and 'quantum' piece: Conservation of the energy-momentum tensor implies: where equality in the final equation is strictly only up to a vacuum contribution. Since the classical part has been treated in section II B, we focus on the quantum contribution, the first term of which can be readily re-expressed by exploiting (2.87) and (2.91a): (2.108) To process the double derivative term, we again exploit (2.87) to pull out a total derivative piece: from expanding out the right-hand side using (2.91a) and (2.87) and noting that K ·R = R·K (which is particularly obvious in momentum space). Thus we can construct the following contribution to the energy-momentum tensor: To form the symmetric energy-momentum tensor, according to the recipe which ultimately produces (2.13), we use (2.101b) and (2.101c) to re-express in terms of symmetric pieces plus total derivative terms: Utilizing the manifest symmetry under α ↔ β of the final terms allows us to decompose Q αβ along the lines of (2.7), with: (2.113) As before, the next step is to construct the trace using (2.20) and, after adding the classical piece, to compare with the ERG version of (2.3c). The first step gives: where we have utilized (2.91a) and, recalling (2.17), have split H τ λ into classical and quantum pieces: Again, the strategy is to simplify by absorbing O ∂ 2 pieces into the first term on the right-hand side of (2.114). To proceed we exploit (2.101c) to re-express where we recall that δ 0 ≡ (d − 2)/2. Exploiting the notation of (2.100) gives, for some I (1) τ λ : where, in the final term on the right-hand side, we have used (2.82) to set To treat the double functional derivative terms in (2.114) note that, for quasi-local A 1 This can be seen by integrating: the integrand of the left-hand side is odd and so vanishes. 3 This result implies that various quantities of interest are O ∂ 2 . (2.120c) The first equation is a trivial consequence of (2.119).
To treat the final double derivative term in (2.114) we utilize (2.101b) rather than (2.101c) to re-express: Noting from (2.103) that dR dp 2 = the double derivative term under consideration can now be processed similarly to the previous case, yielding: For the full quantum field theoretic case, the trace of the energy-momentum tensor is given by: (2.126) To extract the 'quantum' part, we mirror the decomposition of (2.106) into (2.107a) and (2.107b): Putting the classical ϕ × δ/δϕ term to one side, we compare the remainder of (2.126) to (2.125). To facilitate this, define That the right-hand side is O ∂ 2 follows by using (2.101c), followed by (2.97). The comparison of quantum terms now yields ∂ λ ∂ τĨτ λ e −S + . . . = 1 2 ϕ · G −1 · G, 1l · δ δϕ where (2.28) is employed.
We can check consistency of (2.131) just as we did in the classical case: first, we integrate the equation as it stands. The total derivative terms vanish; the first three terms on the righthand side have already been dealt with in the reduction of (2.49) to (2.51) and it is easy to see that they combine with the remaining terms combine to give the ERG equation (2.84a).
Returning to (2.131), now we multiply by 2x µ and then integrate. In this case, all four 'classical' terms have been processed in the reduction of (2.49) to (2.55 (2.132) The various numbered contributions on the right-hand side of (2.130) may be extracted where, consistent with the rest of this paper, we have ignored a vacuum term on the righthand side. It is thus apparent that (A2a) transforms into (2.84a).