On the $a$-theorem in the Conformal Window

We show that for four dimensional gauge theories in the conformal window, the Euler anomaly, known as the $a$-function, can be computed from a $2$-point function of the trace of the energy momentum tensor making it more amenable to lattice simulations. Concretely, we derive an expression for the $a$-function as an integral over the renormalisation scale of quantities related to $2$- and $3$-point functions of the trace of the energy momentum tensor.The crucial ingredients are that the square of the field strength tensor is an exactly marginal operator at the Gaussian fixed point and that the relevant $3$-point correlation function is finite when resummed to all orders. This allows us to define a scheme for which the $3$-point contribution vanishes, thereby explicitly establishing the strong version of the $a$-theorem for this class of theories.


Introduction
The conformal anomaly, first known as the central charge c of the Virasoro algebra, is key to the physics of conformal field theories (CFTs) as it is a measure of the number of degrees of freedom. The Weyl anomalies discovered in the 70's (see [1] for a review of the topic) state that this central charge appears in the trace of the energy momentum tensor (TEMT) when there is a curved background, T ρ ρ CFT = −(β c /(24π))R, elevating the central charge to a β-function; β c ≡ c. The c-theorem [2] can be stated in terms of Cardy's formula [3] ∆β 2D where T ρ ρ | flat → Θ and . . . c stands for the connected component of the vacuum expectation value (VEV). It is assumed that the theory flows from an ultraviolet (UV) to an infrared (IR) fixed point (FP). The inequality ∆β 2D c ≥ 0, establishes the irreversibility of the renormalisation group (RG) flow, and follows from the positivity of the spectral representation and the finiteness of the correlator in (1).
In 4D the situation is more involved as there are further terms in the TEMT Above H ≡ R/(d − 1) and as opposed to [4] we have omitted a cosmological constant term for brevity. In particular, we denote the coefficients of the geometric invariants by β-functions except the R-term which is a Weyl variation of the local R 2 -term. In CFTs, β b = 0 and β a , β b &b are the true conformal anomalies. The a-theorem, ∆β a = β UV a − β IR a > 0, was conjectured early on [5] and a proof in flat space uses the 4-point function of the TEMT, anomaly matching and analyticity [6,7]. 1 A stronger version of the theorem requires an interpolating functionβ a (µ) that reduces to β UV,IR a at the respective FPs, satisfying monoticity, d ln µβa ≥ 0, along the RG flow. A perturbative argument was given in [9] by finding a function satisfying d ln µβa = (χ AB + . . . )β A β B where the Zamolodchikov metric χ AB is positive by unitarity at the Gaussian FP. 2 A representation similar to (1) has been proposed involving 2-and 3-point functions [11] ∆β a = 1 3 · 2 8 where x ≡ d 4 x. This expression is derived in appendix A using conformal anomaly matching. As alluded above, this expression does not lend itself to positivity because of the presence of the 3-point function. In this work, we will show that for gauge theories with gauge couplings only, the 3-point function term drops for theories in the conformal window. This establishes the positivity with Euclidean methods and makes the evaluation more amenable to lattice simulations.

Executive Summary
In the remainder of this introduction, we give an executive summary of our work leaving the derivation of equations and definitions of schemes to the main part of the paper. Our assumptions are: (i) that the TEMT assumes the form (summation over A implied) and (ii) that the beta functions β A ≡ d d ln µ g A vanish in the IR & UV. 3 Above operators with square brackets denote renormalised composite operators, e.g. O A ∼ G 2 in the case at hand, where G 2 ≡ (G a µν ) 2 , is the standard field strength tensor squared known from quantum chromodynamics (QCD). Using these assumptions allows us to derive from (3). The β-functions are µ-dependent through the couplings and the χ's are the analogues of the Zamolodchikov metric. 4 The expression (3) is derived in section 2, and is a new result of this paper. For our work the crucial input is that the leading order correction to the non-interacting FP is The main focus of this paper will be on asymptotically free QCD in the conformal window regime where (6) follows by using the conformal OPE (c.f. appendix B ) and is of course easily 2 This argument was generalised to conformal perturbations at interacting FPs in [10]. In both cases the positivity is controlled by the smallness of perturbative corrections encoded in the dots. In 2D the strong c-theorem was proven in the original paper [2] without reference to perturbation theory. 3 For gauge theories with chiral symmetry breaking, the assumption Θ ∼ β A OA breaks down since the goldstone bosons couple with a term Θ ∼ π 2 which cannot be improved since it is in conflict with chiral symmetry [12,13] leading to subtleties for flow theorems [4,14]. 4 Whereas the χ's are dependent on a generic scheme R, the two flow integrals themselves are scheme independent, cf. [4]. We will refer to χABC as the 3-metric throughout in a loose analogy to the Zamolodchikov metric in two dimensions.
established by direct computation as well. Using (6) and our previous work on finiteness of 2and 3-point functions we are able to define a scheme, referred to as the R 3χ -scheme, for which χ R 3χ ABC (µ) = 0 along the flow. This establishes the main result of our paper valid for the assumptions specified above and satisfying (6). On a side note this means that ∆β a = 2∆b ≡ 2(b UV −b IR ), for the same conditions, since the flow theorem forb can be expressed in terms of the same 2-point function [4]. This relation was conjectured to hold for general classically conformal QFTs in [15]. In this work we show under what conditions this relation holds. A case where it fails is when the theory contains scalar couplings and the 3-point function does contribute. The paper is organised as follows. The cornerstones, formula (5) and the scheme χ Rχ ABC (µ) = 0, are established in sections 2 and 3 respectively. The sum rule (3) is derived in appendix A and a more formal argument for the first equation in (6), underlying the above mentioned scheme, is given in appendix B.

The Flow of E 4 (or β a ) as an Integral over the RG-scale
It is the aim of this section to derive (5). The presentation below is similar to the one given for the R-flow in [4] where we have shown that 5 The starting point is formula (3), which is derived in appendix A. Using (8) one may write where Γ θθθ (p x , p y ) = x y e i(px·x+py·y) Θ(x)Θ(y)Θ(0) c , andP λ 3 is defined in (A.10). 6 The transformation of the equation above into an integral representation over the RG scale, necessitates the discussion of the renormalisation prescription. The latter is defined by a splitting of the bare function into a renormalised Γ R and a counterterm L R Above λ 3 (Källén function) and P 3 are the 3-and 2-point kinematic structures where momentum conservation, p z + p x + p y = 0, is implied. In order to pass to the coupling coordinates one uses (4) to write with In a generic subtraction scheme R, using dimensional regularisation d = 4 − 2 , the 3-tensor structure, is defined in analogy to the 2-tensor Zamolodchikov metric χ AB . Above L β is the Lie derivative in coupling space (e.g. [4,16]) and L 1,R ABC follows from the analogous split in (11) (see section 4.2 of [17] for more detailed definition of the relevant quantities).
In establishing the flow formula (5), the so-called MOM-scheme, proves useful defined in analogy to the 2-point function case [4]. The scheme is defined by the following split where we used the MOM-scheme relation χ MOM θθθ = β A β B β C χ MOM ABC , which in turn follows from substituting (4) into (16).
The final expression for ∆β a is obtained by taking p → 0 limit of (17) and inserting the result to (9) Although as it stands (18) is written in a specific scheme, just like for ∆b [4], schemeindependence follow by observing that a change from a scheme R 1 to R 2 is given by a cohomologically trivial term where ω = (β A β B β C ω ABC ) with ω ABC parametrising the change of scheme in analogy with ω AB in [4]. This establishes the representation (5) and completes the aim of this section. We have also checked that eq. (18) is consistent with the MS-scheme formulae of [16] (namely 3.17b and 3.23 in this reference).

The 3-metric χ ggg in Gauge Theories
In this section we restrict ourselves to QCD-like theories with one gauge coupling g and massless fermions. The generalisation of the following result to multiple coupling theories satisfying (6) is straight forward. Before we proceed, let us establish the notation. The trace anomaly for gauge-theories reads where β = d ln g d ln µ is the logarithmic beta function. The corresponding operator O g is the field strength squared with the somewhat non-standard treatment of the coupling constant (and G 2 has been defined previously). The mapping to the general expressions (5) and (7) is done by comparing (4) and (20), which gives O A → O g and β A → 1 2 β omitting the superscript g on the β-function for brevity.
In section 3.1 it is shown that χ ggg = O(g 2 ), from where it is deduced, in section 3.2, that the counterterm to the 3-point function L ggg is finite when summed to all orders for all points along the flow. Based on this, in section 3.3, a scheme is defined for which χ R 3χ ggg (µ) = 0, leading to the main result of the paper.

The vanishing of χ ggg at the UV Fixed Point
Technically, we will show that χ ggg = O(g 2 ). An explicit evaluation of the corresponding 3-point function at leading order gives where c is a constant and the 2-and 3-point kinematic structures P 3 and λ 3 are defined in (12). It is observed that at leading order in the coupling there is no contribution to the 3-point function kinematic structure λ 3 . The 2-point function structure P 3 is not relevant for our work. From (11) and the definition of χ ggg (15), χ ggg = O(g 2 ) follows. In principle this completes the task of this section but we think it is instructive to add a few more comments. First, as demonstrated in the appendix B this can also be understood from the fact that for an exactly marginal operator the λ 3 -structure vanishes in a CFT [18]. In the language of [18] the structure P 3 is referred to as semilocal, and that is at least one delta function in coordinate space; δ(x) 1 y 2d + permutations ↔ p 4−2 1 + permutations, in the case (22). For non-coincident points the correlation function is indeed proportional to (d − 4) cf. [19,20]. It is in particular instructive to consider a case where this fails. An example is a free conformally coupled scalar field for which φ 4 is an operator of scaling dimension four but since its perturbation δL ∼ λφ 4 induces an RG-flow, namely β λ = 0, it is not exactly marginal. In the explicit computation, one obtains in coordinate space which is clearly not semi-local and will contribute to the λ 3 structure upon Fourier transformation. On a side note it is a remarkable circumstance that from the evaluation (23), one can infer that the β λ = 0, cf. discussion in appendix B and [18]. Second, one might wonder whether something similar is possible for the 2-point function. The answer is no for the following reasons. If it were possible to set χ AB = 0 in some scheme then it would also imply that Θ = 0 by reflection positivity in (8) which is incompatible with a non-trivial flow. The explicit straightforward computation for the 2-point function at leading order gives, , a non-contact term contribution, unlike (22), whose Fourier transform gives rise to ln µ-dependent term. Moreover, the formal argument given in appendix B does not descend to 2-point functions.

Finiteness of the 3-point Function
Following the analysis in [17], we study the finiteness of L 1,R ggg after removing the regulator which serves as the basis for defining the R 3χ -scheme in the next section. The quantity L 1,R ggg is defined in analogy to L 1,R θθθ in (11). In dimensional regularisation, the RGE (15) reads where we used the d-dimensional logarithmic beta functionβ = − + β and a s ≡ g 2 /(4π) 2 . This equation allows for the integral solution This expression is well-defined for µ > 0, convergent for µ → ∞, as we will show shortly for the asymptotically free and asymptotically safe case. Anticipating the result and removing regulator ( → 0) the expression becomes where the second integral representation in (26) is useful for practical computation.
• In the asymptotically free case (Gaussian FP), the β-function and the 3-metric behave as in accordance with (6). Inserting this back into the integral (26) we see that the solution behaves regularly near the UVFP since the ln µ −3 from the integral is cancelled by the 1/β 3 prefactor.
• We can also apply similar arguments for RG flows in the vicinity of a non-trivial UVFP a UV s , which corresponds to the asymptotically safe scenario (see [21] for some recent discussion of this possibility). In this case the UV behaviour is where χ * ggg ≡ χ R ggg (a UV s ) and γ * = ∂ ln as β| as * > 0 is the anomalous dimension of [G 2 ] at the FP. One gets ∞ ln µ By inserting this back to (26), we find again finite UV behaviour L 1,R ggg µ→∞ ∼ O(1). In fact, it is straightforward to see that provided γ * = 0, for any FP a s * the equation (24) always allows for a finite solution Let us now turn to the issue of IR convergence. Clearly, the presence of 1 β 3 in the solution (26) indicates additional problems when the IR limit a s → a IR s is taken. Note however, that near the IRFP another solution to (24) can be found well-defined for µ < ∞. After taking → 0 (which is justified for the same reasons as (25)) one gets By repeating the analysis leading to (27) and (28) near the IRFP, we conclude that (32) is well-defined in the vicinity of a IR s . Assuming that the theory is free from any singularities in the coupling space, the solutions (25) and (32) should be compatible on overlapping domains. By subtracting (25) from (32) (and taking → 0 limit) we find a continuity condition This is consistent with the vanishing of the 3-point contribution in (5), which can therefore be seen as the direct consequence of the finiteness and coupling continuity of L 1,R ggg . Indeed, in the next section, we will show how the above results can be used to construct a scheme, where the 3-metric vanishes.

Constructing the R 3χ -scheme for the 3-metric χ ggg
A change of scheme is given by a finite shift ω ggg (a s ) in the counterterms Using (24) we can deduce that under such a shift the 3-metric transforms as χ R 2 ggg = χ R 1 ggg + (2β∂ ln as + 6(∂ ln as β)) ω ggg .
Since the → 0 limit of L 1,R 1 ggg is uniform as shown in the previous section we can choose to define a new scheme R 3χ for which is automatic. This scheme is new to this paper and not to be confused with the previously discussed MOM-scheme. By using (38) in the general scheme-independent expression (18) we finally arrive at the desired result We would like to end this section by demonstrating how to construct such a scheme in perturbation theory. Using the two-loop formulas of [16], we extract where β 0 is one-loop coefficient of the beta function β = −β 0 a s + O(a s 2 ), the gluons and N f fermions are assumed to be in the adjoint and fundamental representation of an SU(N c ) gauge group respectively (n g ≡ N 2 c − 1), and MS denotes the standard minimal subtraction scheme. By performing a scheme change (36) with we achieve as expected at this order in the perturbation theory. At the perturbative Banks-Zaks FP a s IR ∝ β 0 1. Since χ ggg is absent, ∆β a in this theory can be computed purely by substituting the known perturbative expressions for the beta function and χ gg in (8) as was done up to five loops in [4] (see Eq. 60 and the discussion bellow in this reference).
To find χ R 3χ ggg at higher orders one would need to use the two-loop beta function together with the three-loop expression of χ ggg , which is not presently available to the authors. Nevertheless, using the formulae of this paper some general predictions about the behaviour of these higher order corrections can be made (cf. appendix C).

Conclusions and Discussions
Our starting point was the derivation of a formula for the Euler anomaly or a-function as an integral over the RG-scale of a 2-and 3-point functions of the trace of the energy momentum tensor (5), valid for theories which are governed by β-functions at both fixed points (4).
For gauge theories in the conformal window the formula collapses to the 2-point function (7). Our main assumption for the proof is that the ultraviolet and infrared solution (25) and  (7), as a function of the IR cut-off Λ IR on the spacial integral (44). The scale Λ −1 g is related to the running of the β-function and can be expected to be of the order of the scale where the derivative of the β-function changes sign [25,26]. The proposed formula is given in eq. (45). Note that the asymptotic value 2∆b| mq>0 is a non-trivial quantity whose value is not yet understood [4]. The determination of the latter would therefore be an additional benefit of a lattice investigation (32), of the RGE (24), can be matched continuously. This allowed us to define an explicit prescription, the R 3χ -scheme (37), for which the 3-metric vanishes. In particular, our result means that for those theories, the Euler flow and the R-flow are identical (8). The reason this works for gauge couplings, and not for generic couplings, is that for the former the 3-point function collapses to a 2-point function near the trivial FP (22). This is a consequence of the vanishing of the leading order contribution to the 3-metric (6) and can also be understood from the fact that the field strength tensor squared is an exactly marginal operator at zero coupling cf. appendix B. An example where this fails is a scalar free field theory for which φ 4 is not an exactly marginal operator and its non-zero β-function induces a 3-point function structure at leading order, prohibiting the use of the R 3χ -scheme.
In the light of the above remarks one might wonder, whether the result can be applied to gauge theories with supersymmetry including scalar fields such as supersymmetric QCD (SQCD). The extension is possible owing the same form of the anomaly (4) in SQCD [22]. In supersymmetric gauge theories without superpotential the matter and gauge contributions are related through the Konishi anomaly [23], so that the trace anomaly can be expressed solely in terms of the field strength tensor squared up to equation of motion terms. We therefore expect that the main results of this paper should apply to N = 1 SQCD in the conformal window. 7 Another corollary of our analysis is that for the class of theories studied in this paper the strong a-theorem applies. One defines the off-critical quantitỹ which reduces to β IR a in the limit µ → 0 by (39) and gives a (scheme dependent) interpolating function between the fixed points. The monotonicity of this function follows from the positivity of χ MOM gg , established in [4]. Moreover, the perspective of implementing the a-theorem in the conformal window on the lattice have improved since it is related to a 2-point function. 8 Supersymmetric lattice gauge theories [30], could be a particularly interesting test ground as the Euler anomaly is exactly known [31]. In practice though, lattice Monte Carlo simulation are done at finite quark mass which does not fall into our class of theories. 9 A pragmatic way to deal with this problem is to choose an infrared cutoff Λ −1 IR , on the integral (7). The function A is expected to plateau to ∆β a | mq=0 for Λ −1 IR lower than the inverse quark mass m q for which the theory behaves like a massless theory. 10 More precisely the flat region corresponds to the near-conformal behaviour in the vicinity of the IRFP. Thus one would expect ∆β a to plateau to the massless case for the above mentioned range, cf. fig.1 for a schematic illustration and an explanation about the scale Λ g . It would be interesting to apply this procedure to the case where the IR phase is chirally broken and investigate the expected appearance of the ln m q -divergence induced by the goldstone bosons [4,14]. γm is the quark mass anomalous dimension. Thus unless γ IR m = −1 this does not correspond to a CFT in the IR. 10 None of this scales should be confused with the lattice size L. In particular, Λ −1 IR < L, holds strictly by construction. To avoid finite size effects one has to impose m −1 q L as for the 2-point function the quark mass correction ought to be exponential exp(−mH L) where mH ∼ (mq) η H with ηH ≡ 1/(1 + γ IR m ) is a mass of a hadron [32].

A.1. Renormalisation in Curved Space and β UV a
In an external gravitational field one needs to add counterterms to the action to renormalise the theory [33]. The bare couplings are defined as a 0 = µ d−4 (a R (µ)+ L R a (µ)) etc. and geometric quantities are the same as in (2) except for R which being a total derivative does not contribute to the action. The main idea is that the quantum action principle (differentiation with respect to sources) leads to finite quantities and thus to constraints on the counterterms. Concretely, a triple Weyl variation δs(x) (g µν → e −2s(x) g µν ) leads to x y e i(px·x+py·y) δ 3 δs(x)δs(y)δs(0) is introduced and we used (10) to include the dynamical contribution. By using, (11) we conclude that the finiteness of L 1,R θθθ ensures finiteness of the combination (2k a 0 − 8b 0 ) in (A.2). Since b 0 has been shown to be finite [17] it is to be concluded that the quantity k a 0 is finite. In particular this means that the → 0 limit k a 0 is meaningful In the last step we used that a UV is finite and that L UV where we have used (A.3). In the preceding argument, the finiteness of k a 0 (and b 0 ) was essential to ensure UV finiteness of the dilaton effective action and match the the bare coefficient of the Wess-Zumino term to the Euler anomaly β UV a . Similarly, the IR effective action contains the term 2β IR a S W Z which contributes at O(s 3 ) We are now ready to put all the pieces together. By Fourier transforming the third functional derivative with respect to s of (A.7), we see that at low momenta, the LHS of (A.2) behaves as where the dots stand for nonlocal contributions subleading in the momentum expansion. Assuming momentum conservation, p z = −(p x + p y ), λ 3 (12) assumes the form with the associated projectorP λ 3 λ 3 = 1 beinĝ for which the P 3 -structure automatically vanishes (P λ 3 P 3 = 0). ApplyingP λ 3 to the right-hand side of (A.2), one gets which in position space assumes the form The Euler flow formula (A.13) is invariant under the addition of the local δL ∼ ω 0 R 2 -term unlike the sum rule for ∆b (8) which needs to be amended. More concretely, the 2-and 3-point term containb UV with opposite signs and is therefore invariant underb UV →b UV + ω 0 , The reason this has to happen is that β a is well-defined at each FP and not only as a difference, like ∆b. At last, we would like to mention that eq. (A.13) itself has been derived by Anselmi [11] using different methods.

B. Vanishing of χ ggg at the UV Fixed Point -formal Argument
In this appendix we will demonstrate how χ ggg = 0 in the free theory can be derived by using standard OPE arguments [34]. We start the discussion by considering a general perturbation for some coupling constant λ that can be set to 1 without loss of generality. We now deform the constant λ → λ + δλ, the corrections to a generic correlator . . . in the perturbed theory read The δλ 2 -term in (B.2) can be obtained by using the OPE where dots encompass terms irrelevant for the calculation. By inserting this expression back into (B.2) and evaluating the y integral with a UV cutoff Λ, one finds that a logarithmic divergence appears This divergence can be removed by adding a counterterm of the form However, adding such a term amounts to introducing a β-function Hence the non-vanishing of the β λ -function and the OPE coefficient C λ λλ are directly related. We restrict our attention to the case at hand where O λ ≡ O g = G 2 in the free-field theory. Since the value of δg only affect the normalisation of the kinetic term, it is clear that the theory remains free (and therefore a CFT) for any value of δg. Thus β = 0 and in the free theory Using the fact that in a CFT the 3-point function of an operator O λ is proportional to C λ λλ we conclude that also the three-point function of O g has to vanish, which directly implies that Note that the above argument shows that O g is exactly marginal at the Gaussian fixed point. In general an operator is called exactly marginal if its beta function vanishes, which is equivalent to the vanishing of the corresponding 3-point function as shown in [18,34]. 11

C. The R 3χ -scheme in Perturbation Theory
In this appendix, we will construct explicitly the solutions (26) and (32) for a theory with a trivial UVFP and a Banks-Zaks FP in the IR. We start with the analysis near the trivial UVFP. The scheme (37) means that given a χ R ggg , we should be able to obtain ω ggg to any given order in a s through (26). We demonstrate how this works for the first two non-vanishing orders in perturbation theory. Introducing the notation χ R ggg = χ (1) ggg a s + χ (2) ggg a s 2 + O(a s 3 ) , (C.1) the scheme change ω ggg to O(a s 2 ) is given by, performing the integral on the right-hand side of (26) and expanding the result in a s , ω ggg UV = χ where the two-loop β-function is parameterised by β = −β 0 a s − β 1 a s 2 . It is easily verified that χ R 3χ ggg = χ R ggg + (2β∂ ln as + 6(∂ ln as β)) ω ggg = 0 + O(a s 3 ) .

(C.3)
Note that (C.2) is O(1) and therefore nonzero at the UVFP even though χ R ggg itself vanishes there.
In the IR we assume a (Banks-Zaks) FP at a IR s = − β 0 β 1 1, which exists for asymptotically free theory with β 0 > 0, β 1 < 0. The 3-metric (C.1) expands to χ * ggg ≡ χ R ggg (a IR s ) = Above we used that γ * = −∂ ln as β| as * = β 2 0 β 1 is the anomalous dimension of O g (field strength tensor squared). In the limit a s → a IR s limit we get the expected dependence (31). Direct computation shows that (C.5) is compatible with (C.3). The leading O(1) parts of the IR solution (C.5) and the UV solution (C.2) match, up to O(β 0 ) corrections, provided that The above should be regarded as the necessary condition for continuity of ω ggg , equivalent to (34).