${\cal N} = 1$ Euler Anomaly Flow from Dilaton Effective Action

We consider ${\cal N} =1$ supersymmetric gauge theories in the conformal window. The running of the gauge coupling is absorbed into the metric by applying a suitable matter superfield- and Weyl-transformation. The computation becomes equivalent to one of a free theory in a curved background carrying the information of the renormalisation group flow. We use the techniques of conformal anomaly matching and dilaton effective action, by Komargodski and Schwimmer, to rederive the difference of the Euler anomaly coefficient $\Delta a \equiv a_{\rm UV} - a_{\rm IR} $ for the ${\cal N} =1$ theory. The structure of $\Delta a $ is therefore in one-to-one correspondence with the Wess-Zumino dilaton action.


Introduction
An exact expression for the difference of the ultraviolet (UV) and infrared (IR) Euler anomaly ∆a ≡ a UV − a IR was derived for N = 1 supersymmetric gauge theories by Anselmi, Freedman, Grisaru, and Johansen (AFGJ) [1]. Thereafter it has served as a fruitful laboratory for testing different techniques by rederiving the result. Examples include verification up to fourth loop order [2], the use of the local renormalisation group (RG) [3] and employing superspace techniques assuming a gradient flow equation [4]. In the latter case an expression valid outside the fixed point has been obtained [4] of a form conjectured earlier by a perturbative approach [5].
In this paper ∆a| N =1 is derived by using the techniques of conformal anomaly matching and dilaton effective action. The latter were used by Komargodski and Schwimmer (KS) [6,7] to derive the a-theorem ∆a ≥ 0 as conjectured in 1988 by Cardy [8]. A crucial ingredient is the introduction of an external field called the dilaton by coupling it to the renormalisation scale µ → µe τ (x) , thereby introducing a local scale interpretation analogous to the the local RG pioneered by Shore [9,10]. The locality of the approach is crucial and served Jack and Osborn to derive a proof the a-theorem at weak coupling (i.e. perturbation theory) by using it as a source term in a field theory in a generic curved background. KS and later Komargodski [7] focused on the four point dilaton function and were able to prove the a-theorem based on analyticity assumptions. In essence the dilaton serves as a compensator field to the Weyl-rescaling g µν → e −2α(x) g µν . (1) The transformation (1) corresponds to changing distances locally and implies that coordinate and momenta invariants change as x 2 → e −2α(x) x 2 and p 2 → e 2α(x) p 2 . Variation of the logarithm of the partion function with respect to the Weyl-parameter results in the vacuum expectation value (VEV) of the trace of the energy momentum tensor (EMT). For a theory on a curved space, with no explicit scale symmetry breaking, the EMT is parametrised by [11,12] where the abbreviations are used throughout. The quantities and R are the Euler density, the Weyl tensor squared and the Ricci-scalar; and R µναβ and R µν denote the Riemann and Ricci tensors. The Euler density E 4 is a topological quantity and the Weyl tensor squared W 2 vanishes on a conformally flat space. The absence of c, and therefore the H 2 -term, in a 4D conformal field theory (CFT) was shown in Ref. [13]. The R term can be removed by a finite H 2 -counterterm in the action [11] and will therefore not be discussed any further throughout. The constants a, b, c and c ′ depend on the dynamics of the theory. Their free field values for various spins were computed in [11]. Note, the non-vanishing of a and b therefore establish the conformal or Weyl anomaly in 4D [11,12].
This work is structured as follows. In section 2 the general framework is outlined by restating some of the results of [6] in a language appropriate for this work. The specific construction is presented and illustrated in sections 2.1 and 2.2 respectively. In section 3, in particular 3.1, the AFGJ Euler anomaly result is rederived within our framework using the Konishi anomaly. The paper ends with conclusions in section 4. Appendices A, B and C include a review of the derivation of the NSVZ beta function using the Konishi anomaly, the derivation of the the trace anomaly for a free theory for a dilaton background field without additional background curvature and renormalisation group equations for the generating functional.

General framework
Consider a massless theory with fields φ and a coupling g. The path integral is given by where the action S W is to be understood in a Wilsonian sense and W is proportional to the negative free energy. For the purposes of this work S W is interpreted to be on a renormalisation trajectory from the UV to an IR fixed point. In massless theories correlation functions depend on ratios of q 2 /µ 2 where q denotes an external momentum. Hence the renormalisation scale transforms as µ → e −α µ under the Weyl-rescaling. An external field, known as the dilaton τ , is introduced in the action transforming under Weyl-rescaling (1) as such that the product µe τ is Weyl-invariant. The dilaton therefore serves as a spurion (or compensator) formally restoring scale invariance. In this work no dynamic nature is attributed to the dilaton field which is in line with [7] but not the first paper [6] on the a-theorem in 2011. The dilaton serves as a source term for the EMT and when made a local field the, yet to-be-defined, Wess-Zumino term carries the information on the Euler anomaly. Promoting the dilaton to a local field τ → τ (x) requires local Weyl invariance and demands changes similar to passing from global to local gauge invariance. The specific implementation will be discussed in the the explicit examples. The space-dependance of τ augments the couplings to local objects g(µ) → g(µe τ (x) ). Note that the functional form µe τ renders local Weyl-rescaling equivalent to a local RG transformation. The path integral becomes τ -dependent, The quantity W τ corresponds to the generating functional of the correlation function (connected component) of the traces of the EMT. The Wess-Zumino action 1 was shown to be the source term [6] of the Euler anomaly. Above R stands for the non-dilaton curvature background. More precisely, using arguments of conformal anomaly matching it was shown that the difference of the UV and IR dilaton effective action, with with g * (UV,IR) ≡ g(∞, 0), contains a term proportional to S WZ times the sought after quantity ∆a ≡ a(µ UV ) − a(µ IR ) [6]. Hence determining ∆a reduces to finding ∂ ln µ W τ . Note that the second equality in (9) follows from (C.1) in the limit m → 0 and using dg/β = d ln µ.

Dilaton dependent conformal factor
In this work a theory is considered which can be reinterpreted as a free field theory in a conformally flat backgroundg which carries the information on the RG flow parameters. Above δ ρλ denotes the flat Euclidian metric with positive signature. The adaption to Minkowski space is straightforward as it results in the appearance of various factors of i only. By using (C.5) from the appendix 2 3 and (9) we may write a more explicit formula for the difference of the Euler anomaly ∆a where where | S WZ denotes the following projection The coefficientsã andc depend on the dynamics of the theory. The Wess-Zumino action S WZ is defined in (8) and we follow the rule that the tilde denotes geometric quantities, e.g.Ẽ 4 and H 2 , evaluated in the background metricg µν . 4 The quantity ∆a is determined onceã andc 1 The Wess-Zumino action, above, is analogous to the Wess-Zumino term [14] of pions in connection with the the axial anomaly. 2 Note that metric (10) is not a physical or geometric metric as it does not transform like (1) under Weylrescaling s(τ ) → s(τ + α) unless s(x) = x. The latter is the case in [6],ĝ ρλ = e −2τ (x) δ ρλ , and constitutes one of the differences with respect to our approach. 3 The subscript τ refers to the VEV of the trace of the EMT with respect to the partition function (7). 4 Conformal flatness ofg implies thatW 2 = 0. are known. In the next section we will discuss a very simple toy model that illustrates these ideas and will serve as a stepping stone for the N = 1-computation.

Weyl anomaly of free scalar field in conformally flat background
We consider a scalar field theory on a flat space, focusing solely on the kinetic term thereby ignoring other contributions. The usefulness of which will, hopefully, become clear in section 3.1. Taking Z(µ) → Z(µe τ (x) ) amounts to passing to S τ (5). The factor Z(µe τ (x) ) can be absorbed into the metric by a local Weyl-rescaling by choosing α = s in (1) with The theory then becomes a field theory on a conformally flat space with metricg µν (15) Above D (s) ρ = ∂ ρ − (∂ ρ s) denotes the Weyl-covariant derivative 5 analogous to the covariant derivative in gauge theories. The coefficients of the trace anomaly (2) for a theory with metric g ρλ and one free scalar field are given by (cf. [11,15] or the explicit computation in appendix or equivalently Θ τ = a free (0) (Ẽ 4 − 2˜ R ). As stated earlier, the coefficient c ′ is of no importance for this work and is therefore discarded. The Euler density in terms of s is given by Using the explicit form s = − 1 2 ln Z(µe τ ) the Euler term becomes where here and below we use the abbreviationγ ≡ d d log µ γ and the following expressions have been used. The quantity ∆a is obtained by integrating over d ln µ and projecting on S WZ . In doing so γ andγ can be treated as being space-independent, since expanding γ(µe τ ) = γ(µ)+O(τ (x)) leads to terms which are not contained in S WZ . Furthermore it is then clear that the first line in (19) can be discarded since it is a total derivative and therefore inequivalent to 5 Adding the Weyl-covariant derivatives is equivalent to the replacement → − 1 6R which is the usual conformally coupled scalar in a curved space of metricg ρλ . the S WZ (8) bulk-term. In order to project the second line of (19) on S WZ (8) it is convenient (following [6], [7] under which all four-derivative invariants vanish, except for Using (12) and performing the integral over d ln µ we get where and γ IR,UV ≡ γ(g * IR,UV ) are the values of the anomalous dimensions at the respective fixed points. For further reference the final result (23) is stated with explicit coefficients A 1 and A 2 This result constitutes an important intermediate result for the derivation of ∆a| N =1 .

N = 1 supersymmetric gauge theory
The theory considered in this section is a N = 1 supersymmetric gauge theory with flavour symmetry SU (N f ) × SU (N f ) and gauge group SU (N c ). The action can be written in terms of the usual vector superfield V and matter superfields (Φ f ,Φ f ) as, e.g. [16], where W 2 is the supersymmetric gauge field kinetic term, g is referred to as the holomorphic coupling constant parametrisation and d 6 z and d 8 z include integration over the fermionic superspace variables. The main tool in deriving ∆a| N =1 is the use of the Konishi anomaly [17][18][19]. The latter is illustrated in appendix A as a method to derive the NSVZ beta function. In section 3.1 the Konishi anomaly is used to write the Wilsonian action such that the RG flow can be absorbed into the metric. This procedure makes it amenable to the free field theory computation in the dilaton background discussed in section 2.2.

∆a| N =1 from Dilaton effective Action and Konishi anomaly
We consider the N = 1 supersymmetric gauge theory with Wilsonian effective action given in (A.1). Choosing a rescaling factor, with γ * = −b 0 /N f (A. 6), on the matter fields the Konishi turns the action into the following form where µ ′ > µ is an arbitrary scale which can be thought off as a UV cut-off Λ UV . Crucially, the RG flow is absorbed into the precoefficientẐ(µ) in front of the matter term. Eq. (28) is the analogue of the action (14) for the scalar field to the degree that the running of the theory is parametrised by a coefficient in front of the matter kinetic term. Again following the procedure in (5) a dilaton is introduced througĥ To keep the procedure manifestly supersymmetric, following [20] the dilaton is promoted to a (chiral) superfield T such that Above ω is the axion and the bar stands for projection on to the lowest component of the multiplet. It will be seen thatẐ in (28) can be absorbed into the background geometry by a local Weyl-rescaling. To preserve local SUSY invariance the Weyl transformations are promoted to super-Weyl transformations. Under the latter, the trW 2 -term is invariant whereas the matter term transforms as follows (c.f [21]) with the superfield A = α + iβ + . . . being the super-Weyl parameter corresponding to α in (1). Note that such a formalism is automatically local Weyl-invariant and that there is no need to introduce the Weyl-covariant derivatives as in (15). Furthermore, the transformation (32) amounts to a Weyl-rescaling of the vielbein 7 7 Under a super-Weyl transformation, the supersymmetric generalization of vielbein transforms as E a µ → e − A 2 e − A † 2 E a µ , which corresponds to the standard Weyl transformation e a µ → e −α e a µ after projecting on the lowest component of E a µ . In the interest of clarity we would like to add that e a µ = δ a µ on flat space.
The action S W (µe T ) (28) can then be written in a manifestly locally supersymmetric form; cf. section 6.3 in [22]. Eq. (34) results iñ Notice that the UV scale µ ′ is arbitrary and that therefore a physical quantity like Θ τ should not depend on it. Since the geometric termsẼ 4 andH 2 are independent of µ ′ , 8 the form of (13) implies thatã andc are µ ′ -independent and therefore constants. This means thatã andc assumed the values at the (free) UV fixed point and the geometric quantities are to be evaluated in the background metricg ρλ carrying the dynamic information. This allows to recycle, in large parts, the computation in section 2.2 as outlined below. A free theory in a curved background is in particular conformal and therefore free of thẽ R 2 -term (i.e.c = 0). Since the dilaton couples to the matter part only, the trace anomaly is exhausted by the free field theory computation of the matter-fields in the curved background with metric (35). Equivalence to the example in the previous section is achieved through the formal replacement Z →Ẑ (following from (35)) which implies γ → δγ ≡ γ − γ * and the change in the number of degrees of freedom ν. More precisely the matter superfield consists of a complex scalar and a Weyl fermion which contribute [11] ν ≡ 2 C -scalar in units of a real scalar field. This number has to be multiplied by the number of colours 2N f (two matter-field per flavour) and N c (the SU (N c ) Casimir of the adjoint representation). Hence ∆a is given by 2N f N c ν∆a| (25) γ UV,IR →δγ UV,IR . Now, (γ UV , γ IR ) = (0, γ * ) implies (δγ UV , δγ IR ) = (−γ * , 0) and therefore We note that (37) is indeed the same as the non-perturbative result quoted in (Eq.4.18) in [1] when taking into account the explicit form of γ * (A.6). The formula above is valid in the conformal window 3/2N c < N f < 3N c where the UV theory is asymptotically free and the IR theory acquires a non-trivial fixed point. Within these boundaries the anomalous dimension γ * takes on the values −1 to 0 and the quantity ∆a is therefore manifestly positive in accordance with the a-theorem. The latter has been proven for N = 1 supersymmetric theories by using R-symmetries and is known as a-maximization [23]. The adaption to gauge groups other than SU (N c ), provided they are asymptotically free, amounts to replacing the SU (N c )-Casimir N c by the corresponding Casimir of the group.

Discussion and Conclusion
In this short paper we rederived the difference of the Euler term in N = 1 supersymmetric gauge theories (37) valid in the conformal window. By an appropriate rescaling of the matter superfield and choosing the Weyl-parameter α (1) to equal the logarithm of the matter prefactor (35), the computation was shown to be equivalent to one of the (free) UV theory in a curved background carrying the information on the flow. This allowed for ∆a| N =1 to be computed from the free field theory example, in section 2.2, with a simple formal replacement for γ IR and γ UV . It is noted that the structure of ∆a| N =1 is completely given by the Wess-Zumino term of the dilaton effective action. The aspect of matching the computation with a free theory bears some resemblance with the original AFGJ-derivation [1] in that independence on an RG-scale is exploited in evaluating certain quantities in the UV where they correspond to free field theory computations. An extension to the non-supersymmetric case is not straightforward because it relies on the one-loop exactness of the rescaling anomaly in supersymmetric gauge theories. From sections A and 3.1 it is seen that an exact expression of ∆a in non-supersymmetric theories is related to finding an exact beta function. The Konishi anomaly is a rescaling anomaly which in N = 1 supersymmetric theories is, by holomorphicity, bound to the axial anomaly. The latter is generally one-loop exact by topological protection of the axial charge.
In non-supersymmetric theories there is no holomorphicity and the Konishi anomaly is an unknown function which could be determined order by order in perturbation theory. An extension of some of the ideas in this paper to non-supersymmetric theories, in direction other than the Konishi anomaly, is presented elsewhere [24]. We end the paper with remarks of the speculative and qualitative kind. Reformulating a gauge theory as a free theory in a curved background is reminiscent of the anti-de Sitter space/conformal field theory duality which has given rise to a lot of work and inspiration over the past two decades. The extension to theories with more than one relevant coupling is not immediate. One might wonder whether bi-gravity, whose renormalisation group flow has been studied in [25], might be a possible avenue for a theory with two relevant couplings. A practical requirement is that the UV theory is to asymptotically free in order to retain computability.
where b 0 ≡ 3N c − N f and µ ′ > µ is an arbitrary scale which can be identified with the UV cut-off Λ UV . Rescaling the matter fields by is accompanied by the Konishi anomaly [17][18][19], and leads to the effective action [26] S W (µ) = d 6 z 1 where the running has been removed from the matter term and all the running is absorbed in front of the gauge field term which in this case defines the running gauge coupling to be The µ ′ -independence of g(µ) implies an RGE which solves to the holomorphic NSVZ beta function [27][28][29] β Above we have used the following notation In the range 3/2N c < N f < 3N c the theory is in the so-called conformal window. Theories of the latter kind are asymptotically free in the UV and flow to a non-trivial (γ IR = γ * = 0) IR fixed point. The lower bound 3/2N c follows from the unitarity bound on the scaling dimension of the composite squark field ∆qq = 2 + γ * ≥ 1 and the upper bound of 3N c is derives form the requirement of asymptotic freedom.

B. Free theory trace anomaly in dilaton background field
In this appendix the trace anomaly of a free scalar field theory (16)  This expression requires regularisation since it is UV-divergent as t → 0. Noting that the mass dimension of the t-variable is two, a UV cutoff Λ UV is introduced as follows Using W = W (µ/Λ UV ) and (C.5) one gets 9 where b 4 is a coefficient of the asymptotic Heat Kernel expansion Tr(e −t∆ (0) ) = where the geometric quantitiesR andẼ 4 are defined with respect to the metricg ρλ (s) given above. The result quoted in (17) follows by comparing the equation above to (2).

C. Renormalisation group equations for W
In this appendix we summarise the RG equation obeyed by W and how they relate to the trace anomaly. For future reference and completeness an explicit scale symmetry breaking term in form of a matter mass term is added. The quantum vacuum transition amplitude W obeys an RG equation The adaption of these equation to W τ involves replacing µ → µe τ everywhere. Note, if τ is made space-dependent then g(µe τ ) and m(µe τ ) and the partial derivatives are to be replaced by functional derivatives ∂ ∂g → d 4 x δ δg(x) and ∂ ∂m → d 4 x δ δm(x) respectively.
A definition of the trace of the EMT is given by where g(x) denotes the determinant of the metric. Combining (C.4) with (C.2) the following equations are obtained where the subscripts "anom" and "expl" refer to anomalous and explicit scale breaking respectively.